THE LIBRARY

OF

THE UNIVERSITY
OF CALIFORNIA

PRESENTED BY

PROF. CHARLES A. KOFOID AND
MRS. PRUDENCE W. KOFOID

u>

A

COURSE OF LECTURES

ON

NATURAL PHILOSOPHY,

&c.

VOLUME I TEXT.

COURSE OF LECTURES

ON

NATURAL PHILOSOPHY

AND THE

MECHANICAL ARTS.

BY THOMAS YOUNG, M.D.

A NEW EDITION, WITH REFERENCES AND NOTES,

BY THE

REV. P. KELLAND, M.A., F.R.S., LOND. & EDINB.,

J, ETC. IN THI

fcp ftumerous (Engrabings on

IN TWO VOLUMES.
VOLUME I. - TEXT.

. 4! >. '

r

LONDON:
PRINTED FOR TAYLOR AND WALTON,

UPPER GOWER STREET.
1845.

Printed by J. & H. COX, BROTHERS (LATE COX & SONS),
74 & 75, Great Queen Street, Lincoln's-Inn Fields.

PREFACE BY THE EDITOR.

THE Lectures which are now a second time presented to the public,
are so well known, and so justly celebrated, amongst those who are
most capable of judging of their merits, that very little need be offered
by way of preface to this volume. Whether we regard the depth of
Dr. Young's learning, the extent of his research, the accuracy of his
statements, or the beauty and originality of his theoretical views, in
whatever way we contemplate these Lectures, our admiration is equally
excited. They embody a complete system of Mechanical Philosophy,
drawn from original sources, and illustrated by a hand capable of
reducing them to the most perfect subjection. Unlike other popular
writers, who, for the most part, either take the sciences at second
hand, or content themselves simply with extracting the discoveries
and adopting the hypotheses of more distinguished philosophers, Dr.
Young travelled over the whole literature of science, and whilst we
are astonished at the rich store of materials which he has collected, we
find nothing more prominent than the impress of his own acute and
powerful mind. It is particularly conspicuous in his treatise on
motion and force, which, with their applications to the useful arts,
forms the first part of these Lectures. In comparing this treatise
with others of similar pretension, we are forcibly impressed with the
fact, that whilst their authors have been driven to popularize from in-
ability to grapple with mathematical researches, Dr. Young has been
enabled to do so from his thorough mastery of those researches. It
combines correctness with simplicity. The popular reader may trust
to it as always based on right principles, and calculated to pave his
way to a more extensive and intimate research ; the mathematical
reader will find in it the clearest statement of arguments which
have already been presented to him in another form. The remaining
parts of these Lectures are equally valuable on account of the origi-
nality of the views which they unfold, and of the unity and simplicity

a 2

M363122

iv PREFACE BY THE EDITOR.

which they give to physical science. Here will be found, for the first
time, correct notions on capillary action. Here Dr. Young developed
the beautiful principle of interference, " that fine discovery," to use the
words of M. Arago, " which will render his name imperishable."

What Dr. Young has done cannot, however, be better explained
than in 'his own Preface, of which it is sufficient to remark, that the
Author has in no instance over-estimated the importance of his labours.

It only remains to add a few words relative to the present edition.
In some of the subjects treated of, considerable advances have been
made since the Lectures were first published. To render the work as
complete as possible, the Editor has supplied brief expositions of what-
ever additional discoveries have been made, which are printed along
with the Lecture on that branch to which they belong, and distin-
guished by being inclosed within brackets. They serve, for the most
part, to complete the subject according to the plan of the Author. In
the case of Electricity, and its kindred branches, so much addition
has been made within the last half-century, that it would greatly
exceed the necessary limits to treat of those sciences satisfactorily.
All that has been attempted is to offer a very brief sketch of the
nature of the extension of those sciences, without entering at all into
details.

The authors to whom Dr. Young directly refers in the Lectures are
given at the foot of the page, the name and date of the work being
added, and in many cases the page which is referred to. Accompany-
ing these references will be found others to authors who have treated
on the same subject. At the end of each Lecture is given a table of
additional authorities, a portion of which have been extracted from Dr.
Young's own catalogue. Indeed these tables embrace every important
work which the catalogue contains, and, except in Meteorology and
Astronomy, reference to all the most valuable memoirs found in the
different scientific transactions. In the excepted cases, the lists were
too extensive, and too little suited to the character of this work to be
given entire, whilst abridgement would have answered no useful purpose.
With respect to the additions which the Editor has made to this branch,
they will be found to be very extensive ; and it is believed the whole
forms a tolerably complete body of scientific literature. There must
necessarily be expected some important omissions, but it is hoped they
are not numerous. For the guidance of those who shall consult these
catalogues, it is necessary to point out the fact, that condensation has

PREFACE BY THE EDITOR. v

been an object, so that a repetition of reference to the same work has
been, as much as possible, avoided. Thus most of the works which are
mentioned under one branch of the mechanical sciences, embrace many
others under which they are not quoted. For example, the autho-
rities on central forces (Lect. IV.) do not include Laplace, Lagrange,
and others, because these authors have been already given in Lect. II.,
as treating on a branch of the same subject. This want of repetition
may be considered a defect, but it must be remembered, that the cata-
logue has already extended to several thousand articles. On the
whole, it is expected that this edition will supply a work greatly
needed, in which correct exposition is combined with extensive
research.

Great pains have been taken to render the Table of Contents and
Index as complete and accurate as possible. In framing these, the
Editor has received the valuable assistance of Mr. Stewart, which he
begs thankfully to acknowledge.

AUTHOR'S PREFACE.

HAVING undertaken to prepare a course of Lectures on natural philo-
sophy, to be delivered in the theatre of the Royal Institution, I thought
that the plan of the institution required something more than a mere com-
pilation from the elementary works at present existing ; and that it was
my duty to collect from original authors, to examine with attention, and
to digest into one system, every thing relating to the principles of the me-
chanical sciences, that could tend to the improvement of the arts subser-
vient to the conveniences of life. I found also, in delivering the lectures,
that it was most eligible to commit to writing, as nearly as possible, the
whole that was required to be said on each subject ; and that, even when
;an experiment was to be performed, it was best to describe that experiment.!
1 uninterruptedly, and to repeat the explanation during its exhibition.'
Hence it became necessary that the written lectures should be as clearly
and copiously expressed, and in a language as much adapted to the com-
prehension of a mixed audience, as the nature of the investigations would
allow ; and that each experiment, which was to be performed, should also
be minutely described in them. If, therefore, there was any novelty either
in the matter or the arrangement of the lectures, as they were delivered
for two successive years, it is obvious that they must have possessed an
equal claim to the attention of a reader, had they been published as a
book ; and upon resigning the situation of Professor of Natural Philosophy,
I immediately began to prepare them for publication.

I had in some measure pledged myself^ in the printed syllabus of the
lectures, to make a catalogue of the best works already published on the
several subjects ; with references to such passages as appeared to be most
important : it was therefore necessary > as well for this purpose, as in order
to procure all possible information that could tend to the improvement
of the work, to look over a select library of books entirely with this
view, making notes of the principal subjects discussed in them, and exa-
mining carefully such parts as appeared to deserve more than ordinary
attention. Hence arose a catalogue of references ; respecting which it is
sufficient to say, that the labour of arranging about twenty thousand,
articles in a systematic form, was by no means less considerable than that
of collecting them. The transactions of scientific societies, and the best
and latest periodical publications, which have so much multiplied the
number of the sources of information, constituted no small part W the
collection, which was thus to be reduced into one body of science.

viii AUTHOR'S PREFACE.

With the addition of the materials acquired in making this com-
pilation, and of the results of many original investigations, to which
they had given rise, it hecame almost indispensable to copy the whole of
the lectures once more, and to exchange some of them for others, which
were wholly new ; at the same time all possible pains were taken to dis-
cover and to correct every obscurity of expression or of argument.

Drawings were also to be made, for representing to the reader the appa-
ratus and experiments exhibited at the time of delivering the lectures, for
showing the construction of a variety of machines and instruments con-
nected with the different subjects to be explained, and for illustrating them
in many other ways. These figures have been extended to more than
forty plates, very closely engraved, and the execution of the engravings has
been minutely superintended. But the text of the lectures has been made
so independent of the figures, that the reader is never interrupted in the
middle of a chain of reasoning, but is referred, at the end of a paragraph,
to a plate, which has always a sufficient explanation on the opposite page.

The bulk of this work is not so great as to require, for its entire perusal,
any unreasonable portion of time or of labour. There may, however, be
some persons who would be satisfied with attending to those parts in which
it differs most from former publications, without having leisure or inclina-
tion to study the whole. To such it may be desirable to have those sub-
jects pointed out which appear to the author to be the most deserving of
their notice.

I The fundamental doctrines of motion have, in the first place, been more

II immediately referred to axioms simply mathematical than has hitherto been
1 usual ; and the application of these doctrines to practical purposes has

perhaps in some instances been facilitated. The passive strength of
materials of all kinds has been very fully investigated, and many new
conclusions have been formed respecting it, which are of immediate
' importance to the architect and to the engineer, and which appear to con-
tradict the results of some very elaborate calculations.

The theory of waves has been much simplified, and somewhat extended,
and their motions have been illustrated by experiments of a peculiar nature.
A similar method of reasoning has been applied to the circulation of the
blood, to the propagation of sound, either in fluids or in solids, and to the
vibrations of musical chords ; the general principle of a velocity corre-
sponding to half the height of a certain modulus being shown to be appli-
cable to all these cases, and a connexion has been established between the
sound to be obtained from a given solid, and its strength in resisting a
flexure of any kind ; or, in the case of ice and water, between the sound
in a solid and the compressibility in a fluid state.

The doctrine of sound, and of sounding bodies in general, has also
received some new illustrations, and the theory of music and of musical
intervals has been particularly discussed.

With respect to the mathematical part of optics, the curvature of the
images formed by lenses and mirrors, has been correctly investigated, and
the inaccuracy of some former estimations has been demonstrated. .

In the department of physical optics, the phenomena of halos and

AUTHOR'S PREFACE. ix

parhelia have been explained upon principles not entirely new, but long
forgotten ; the functions of the eye have been minutely examined, and the |
mode of its accommodation to the perception of objects at different dis- \
tances ascertained ; the various phenomena of coloured light have been
copiously described, and accurately represented by coloured plates ; and
some new cases of the production of colours have been pointed out, and
have been referred to the general law of double lights, by which a great
variety of the experiments of former opticians have also been explained ;
and this law has been applied to the establishment of a theory of the nature
of light which satisfactorily removes almost every difficulty that has
hitherto attended the subject.

The theory of the tides has been reduced into an extremely simple
form, which appears to agree better with all the phenomena than the more
intricate calculations which they have commonly been supposed to require.
With respect to the cohesion and capillary action of liquids, I have had
the good fortune to anticipate Mr. Laplace in his late researches, and I
have endeavoured to show that my assumptions are more universally
applicable to the facts, than those which that justly celebrated mathema- I
tician has employed. I have also attempted to throw some new light on
the general properties of matter in other forms ; and on the doctrine of
heat which is materially concerned in them ; and to deduce some useful
conclusions from a comparison of various experiments on the elasticity of
steam, on evaporation, and on the indications of hygrometers. I have
enumerated, in a compendious and systematical form, the principal facts
which have been discovered with respect to galvanic electricity; and I
have fortunately been able to profit by Mr. Davy's most important expe-
riments, which have lately been communicated to the Royal Society, and
which have already given to this branch of science, a much greater per-
fection, and a far greater extent, than it before possessed. The historical
part of the work can scarcely be called new, but several of the circum-
stances which are related, have escaped the notice of former writers on the
history of the sciences.

Besides these improvements, if I may be allowed to give them that
name, there are others, perhaps of less importance, which may still be
interesting to those who are particularly engaged in those departments of
science, or of mechanical practice, to which they relate. Among these
may be ranked, in the division of mechanics, properly so called, a simple
demonstration of the law of the force by which a body revolves in an
ellipsis ; another of the properties of cycloidal pendulums ; an examina-
tion of the mechanism of animal motions ; a comparison of the measures
and weights of different countries ; and a convenient estimate of the effect
of human labour : with respect to architecture, a simple method of
drawing the outline of a column : an investigation of the best forms for
arches ; a determination of the curve which affords the greatest space for
turning ; considerations on the structure of the joints employed in car-
pentry, and on the firmness of wedges ; and an easy mode of forming a
kirb roof : for the purposes of machinery of different kinds, an arrange-
ment of bars for obtaining rectilinear motion ; an inquiry into the most

x AUTHOR'S PREFACE.

eligible proportions of wheels and pinions ; remarks on the friction of
wheel work, and of balances ; a mode of finding the form of a tooth for
impelling a pallet without friction ; a chronometer for measuring minute
portions of time ; a clock scapement ; a calculation of the effect of tem-
perature on steel springs ; an easy determination of the best line of
draught for a carriage ; an investigation of the resistance to be overcome
by a wheel or roller ; and an estimation of the ultimate pressure pro-
duced by a blow.

In the hydraulic and optical part, may be enumerated an overflowing
lamp ; a simplification of the rules for finding the velocity of running
water ; remarks on the application of force to hydraulic machines ; a
mode of letting out air from water pipes ; an analysis of the human
voice ; and some arrangements for solar microscopes, and for other optical
instruments of a similar nature.

In the astronomical and physical division of the work, will be found a
general rule for determining the correction on account of aberration ;
a comparison of observations on the figure of the earth ; a table of the
order of electrical excitation ; a chart of the variation of the compass, and
of the trade winds : formulae for finding the heat of summer and winter ;
remarks on the theory of the winds ; and a comparative table of all the
mechanical properties of a variety of natural bodies.

A few of these subjects have been more fully discussed in the miscel-
laneous papers, which have already been published, in the Philosophical
Transactions and elsewhere, and which are now reprinted with corrections
and additions ; others are summarily investigated in the mathematical
elements, which form a part of the second volume, or in the remarks
which are inserted, in their proper places, in the catalogue of references.

The arrangement of the whole work is probably different in many
respects from any other that has yet been adopted ; the extent of the
subjects, which have been admitted, rendered it necessary to preserve a very
strict attention to a methodical and uniform system ; and it is presumed
that this arrangement will be considered as in itself of some value, espe-
cially in a work calculated to serve as a key, by means of which, access
may be obtained to all the widely scattered treasures of science ; and which
will enable those, who are desirous of extending their researches in any
particular department, to obtain expeditiously all the information that
books can afford them.

It will not be thought surprising that the execution of this plan, allow-
ing for some professional engagements of a different kind, and for a
variety of accidental interruptions, should have occupied more than three
years, from the resignation of the professorship to the publication of the
work. Some part of it is in its nature incapable of permanent perfection,
since the catalogue must require to be continually extended by the
enumeration of new publications; and it might perhaps be desirable that
an appendix should be added to it, at least every ten years ; but the
lectures themselves may be expected to remain tolerably commensurate
to the state of the sciences for a much longer period ; since, in investiga-
tions so intimately connected with mathematical principles, the essential

AUTHOR'S PREFACE, xi

improvements will always bear a very small proportion to the number of
innovations. I do not, however, mean to assert, that the catalogue is by
any means complete, even with regard to older works, but I believe that
the references which it contains, are at least sufficient to lead those who
may consult the passages quoted, to the works of every author of eminence
that has treated of the respective subjects. Nor do I profess to have
excluded all references that are of little importance ; but I trust that the
number which I have admitted will be found inconsiderable ; and it would
have been very difficult to have rejected any of them, without some
chance of omitting others of greater value.

Whatever the deficiencies of this work may be, I think it right to ob-
serve, that my present pursuits will not allow me to look forwards to any
period, at which I shall be able to remove them, or even to attend to the
correction of the press, or the revision of the engravings, in case of the
necessity of a second edition.

I have already begun to collect materials for a work, in a form nearly
similar, relating to every department of medical knowledge : this work
will not, however, be speedily ready for publication ; it will be compara-
tively more concise than these lectures, in proportion to what has been said
and written respecting physic, but, I hope, much more complete, with
regard to all that is known with certainty, and can be applied with
utility.

WELBECK-STREET, 30th March, 1807.

CONTENTS.

*** The matter within brackets [ ] has been supplied by the Editor.

PART THE FIRST. — MECHANICS.

LECTURE I.
INTRODUCTION, page 1.

Objects of the Royal Institution; Dissemination of elementary knowledge, I.
Education of females ; Theory of practical mechanics, and of manufactures, 2.
Simplicity of useful theory, 3. Difficulty of making improvements ; Repository
of the Institution ; Library ; Journals ; Nature of the lectures, 4. Merits of
English philosophers, 5. Delivery of the lectures ; General view, 6. Division
of the lectures ; Synthetical method, 7... 11. Causation, 11. Induction; Erroneous
inductions; Newtonian rules of philosophizing, 12. Their insufficiency, 13.

LECTURE II.
ON MOTION, 13.

Definition of motion, 13. Absolute and relative motion ; All motion relative,
14. Quiescent space ; Direction of motion ; Laws of motion, 15. Time, 16,
17. Composition of motion; Space in motion; Result of two motions, 18.
Resolution of motion ; General result of a number of motions, 19.

LECTURE III.

ON ACCELERATING FORCES, 21.
Definition offeree ; Action of force, 21. Acceleration and retardation ; Velocity

22. Uniform force; Gravitation; Laws of falling bodies; Atwood's machine,

23. Space described ; Law of Galileo; General law of velocities, 24. Ascent;
Velocity due to a height, 25.

LECTURE IV.
ON DEFLECTIVE FORCES, 26.

Centrifugal force ; Sling ; Motion of a hoop, 26. Whirling table ; Laws of
central forces, 27; Keplerian laws, 28. Ellipsis; Projectiles, 29. Resolution
of oblique motion ; Horizontal range ; Best elevation, 30. Parabolic path ; Prac-
tice of gunnery ; Experiments of Robins, 31, 32.

b

xiv CONTENTS.

LECTURE V.
ON CONFINED MOTION, 32.

Motion limited by suspension, or by a smooth surface ; Effect of friction and of
rotatory motion, 32. Inclined plane; Descent in the chords of a circle;
Velocity of descent, 33. Ascending force ; Energy ; Cycloid ; Cycloidal pendulums,
34. Laws of pendulums ; Swiftest descent, 35. Circular pendulums ; Pendu-
lums with resistance; Revolving pendulums, 36. Composition of vibrations;
Regulator for steam engines; Circular road; Principle of the least action,
37, 38.

LECTURE VI.

ON THE MOTIONS OF SIMPLE MASSES, 38.

Definition of a moveable body, without regard to its extension ; Inertia [gravity] ;
Centre of inertia, 39. Its properties ; Reciprocal forces ; Quantity of motion, 40.
Momentum; Centre of inertia of a system; Motion of the centre of inertia, 41.
Action and reaction, 42. Newton's illustrations ; Magnitude of reciprocal forces,
43. Fall of a feather and of a piece of gold ; Lucretius ; Relation between forces
and distances ; Displacement of the earth by the effect of a machine, 44.

LECTURE VII.

ON PRESSURE AND EQUILIBRIUM, 45.

Pressure, a force counteracted ; Pressure and momentum incommensurable, 45.
Laws of pressure included in those of motion; Opposition of pressures, 46.
Equilibrium of mechanical powers ; Centre of gravity ; Stability of equilibrium,
47. Stability independent of equilibrium, 48. Situation and motions of the
centre of gravity of animals, 49. Levers of two kinds ; Fundamental property of
the lever, 50. Series of levers ; Bent levers ; Oblique levers ; Wheel and axis,
51. Wheels and pinions ; Double axis ; Pullies, 52. Blocks; Smeaton'spullies;
Oblique ropes, 53. Inclined plane ; Wedges, 54. Props, or shores ; Screws ;
Nuts ; Hunter's screw, 55. Determination of mechanical power from virtual
velocities, 56.

LECTURE VIIL
ON COLLISION, 57.

Motions of various bodies acting reciprocally ; Elastic bodies, 57. Nature of
repulsion ; Experiment on an ivory ball ; Apparatus for experiments on collision,
58. Inelastic bodies ; Energy, 59. Measure of force ; Relation of labour to
energy; Preservation of energy; or of ascending force, 60. Effect of a blow;
Rotation, 61. Billiards ; Reflection, 62.

LECTURE IX.

ON THE MOTIONS OF CONNECTED BODIES, 63.

!•> fTf,'", I

Rotatory power; Consideration of the square of the velocity, 63. Smeaton's
apparatus; Centre of gyration ; Centre of percussion and of oscillation, 64. Free
rotation ; Motion of a stick broken by a blow, 65. Preponderance, 66. Greatest

CONTENTS. xv

effect of machines ; Experiments, 67. Cautions with regard to the construction
of machines, 68. Comparison of animal with inanimate force; Regulation of
force ; Small momentum of machines, 69. Impossibility of a perpetual
motion, 70.

LECTURE X.
ON DRAWING, WRITING, AND MEASURING, 71.

Subjects preliminary to the consideration of practical mechanics ; Instrumental
geometry; Statics; Passive strength ; Friction; Drawing; Outline, 71. Pen;
Pencil ; Chalks ; Crayons, 72. Indian ink ; Water colours ; Body colours ;
Miniatures ; Distemper ; Fresco ; Oil, 73. Encaustic paintings ; Enamel ;
Mosaic work ; Writing, 74-. Materials for writing ; Pens, 75. Inks ; Use of
coloured inks for denoting numbers ; Polygraph ; Telegraph, 76. Geo-
metrical instruments ; Rulers ; Compasses, 77. Flexible rulers ; Squares ; Tri-
angular compasses ; Parallel rulers ; Marquois's scales, 78. Pantograph ;
Proportional compasses; Sector, 79. Measurement of angles; Theodolites;
Quadrants ; Dividing engine, 80. Vernier ; Levelling ; Sines of angles, 81 .
Gunter's Scale ; Nicholson's circle ; Dendrometer ; Arithmetical machines ;
Standard measures, 82. Quotation from Laplace ; New measures ; Decimal
divisions ; Length of the pendulum, and of the meridian of the earth, 83. Mea-
sures of time, 84. Objections ; Comparison of measures, 85. Instruments for
measuring ; Micrometrical scales ; Log lines, 86.

LECTURE XI.

ON MODELLING, PERSPECTIVE, ENGRAVING, AND PRINTING, 87.

Copying a statue ; Modelling; Casting, 87. Perspective; Mechanical perspec-
tive ; Geometrical perspective, 88. Orthographical projection, 89. Projections of a
sphere, 90. Invention of engraving ; Wood cuts ; Mode of engraving ; Ruling, 91 .
Me/zotinto; Etching, 92. Aquatinta; Musical characters ; Printing; Copying
letters ; Printing from stones, 93. Letterpress ; Stereotype printing, 94-.

LECTURE XII.
ON STATICS, 95.

Weighing ; English and French weights, 95, 96. Balances, 96. False balances ;
Weighing machines ; Steelyards, 97 ; Bent lever balances ; Spring steelyard ;
Dynamometer ; Animal actions ; Strength of muscles, 98. Instances of strength,
99. Progressive motion ; Running, 100. Pulling ; Sources of motion ; Work
of a labouring man, 101. Temporary exertions ; Horses, 102. Wind; Water;
Steam; Gunpowder; Measurement of small forces, 103,104-.

LECTURE XIII.

ON PASSIVE STRENGTH AND FRICTION, 104.

Immediate effects of force on a solid ; Extension and compression ; Rigidity,
105. Measure of elasticity, 106. Detrusion; Lateral adhesion; Flexure, 107.
Cause of irregularities; stiffness; Stiffness of beams; Hollow beams; Torsion,
108. Alteration ; Ductility, 109. Temper of metals ; Toughness ; Brittleness ;

b 2

xvi CONTENTS.

Fracture; Strength; Resilience, 110. Effect of velocity; Limit of strength
or resilience, 111. Qualities of natural bodies; Fracture by simple com-
pression; Strength of lateral adhesion, 112. Transverse force; Fracture
by flexure, 113. Comparative strength and resilience, 113, 114. Uses of
resistances of different kinds ; Coach springs ; Comparison of direct and trans-
verse strength, 114. Beam cut out of a tree; Hollow masts; Strongest forms of
beams, 115. Machine for measuring strength; Strength of different substances,

116. Inconvenience of bulk ; Friction; Lateral adhesion ; Uniformity of friction,

117. Usual magnitude of friction, 118. Best direction for draught; Stability
of a wedge or nail, 119. Resistance to penetration, 120.

LECTURE XIV.
ON ARCHITECTURE AND CARPENTRY, 121.

Architecture; Form of a column, 121. Eddystone lighthouse; Wall, 122.
Joints; Mortar; Arch, 123. Oblique pressure of earth, 124. Bridge; Flat arch,
125. Horizontal thrust; Piers; Blackfriars bridge; Dome, 126. St. Paul's
cathedral; Pantheon; Orders of architecture ; Gothic architecture, 127. Carpen-
try; Joints, 128. Scarfing ; Joggles ; Tenons ; Mortises ; Straps, 129. Inconve-
nience of transverse strains ; Roofs ; Kirb roof; Height of a roof, 130. Wooden
bridges; Centres of bridges ; Furniture; Parker's gates, 131.

LECTURE XV.
ON MACHINERY, 132.

Application of force, 132. Levers; Connected rods; Hooke's joint; Cranks,
133. Winches; Rectification of circular motion; Wheel work, 134. Teeth of
wheels, 135. Kinds of wheels, 136. Eccentric wheels; Sun and planet wheels;
Construction of wheels ; Weights and springs ; Fly wheels, 137. Air vessels, 138.

LECTURE XVI.
ON THE UNION OF FLEXIBLE FIBRES, 138.

Chain; Union by means of adhesion; Friction of a rope on a cylinder; Twist-
ing; Spinning; Rope-making, 139. Materials of ropes; Hemp, 140; Flax;
Cotton, 141. Silk; Wool; Weaving, 142. Crape; Cloth; Felts; Hats, 143.
Paper, 144.

LECTURE XVII.
ON TIMEKEEPERS, 144.

Clepsydrae, 144. Clocks ; Fly clocks, 145. Balances ; Chronometer with a
revolving pendulum, 146. Measurement of minute intervals of time; Pendulum ;
Balance spring; Principal requisites of a timekeeper; Sustaining force, 147.
Equalization of the force ; Intermediate spring or wheel; Scapement ; Crank, 148.
Crutch scapement ; Common watch scapement, 149. Dead beat scapement and
horizontal watch ; Friction of scapements ; Duplex scapement ; Common scape-
ment; Scapements of Harrison, Mudge, 150. Scapements of Haley, Camming,
Nicholson, Arnold, and Earnshaw; Isochronism of vibrations, 151. Properties of

CONTENTS. xvii

springs, 152. Expansion of pendulums ; Compensations for clocks, 153. Com-
pensations for watches ; Resistance of the air, 15*. Striking part ; Supports of
clocks ; Mutual influence of two clocks, 155.

LECTURE XVIII.
ON RAISING AND REMOVING WEIGHTS, 156.

Counteraction of gravitation; Levers, 156. Perrault's lever; Axis with a
winch ; Water whimsey ; Gin, 157. Capstan ; Double capstan ; Wheelvvork ;
String of buckets, 158. Pullies ; Inclined plane; Duke of Bridgwater's canal, 159.
Screws ; Cranes ; Walking wheels, 160. White's crane ; Weighing cranes ; Lewis;
Counterpoise for a chain ; Removing weights, 161. Porters ; Distribution of
weight: Simple dray, 162. Effect of agitation ; Oily substances; Rollers, 163.
Friction wheels ; Perrault's ropes ; Wheels of carriages, 164- . Magnitude of
wheels, 165. Line of draught; Conical wheels; Effect of springs, 166. Attach-
ment of horses ; Wheel ways, 167. String of baskets or carts, 168.

LECTURE XIX.

ON MODES OF CHANGING THE FORMS OF BODIES, 169.

Compression ; Presses ; Effect of momentum ; Printing press, 169. Sugar mill ;
Oil mills ; Hammering; Hydrostatic press; Extension; Laminating machine, 170.
Glazier's vice; Wire drawing; Pottery; Glassblowing ; Percussion; Forges;
Goldbeating ; Coining, 171. Stamping; Penetration, 172. Pile driving engine ;
Sling; Bow and arrow, 173. Whip; Division; Cutting instruments; Slitting
mill; Lathes, 174. Boring; Agricultural instruments; Mining; Sawing, 175.
Stonecutting; Grinding; Polishing, 176. Trituration ; Powder mills, 177. Agi-
tation; Threshing machines ; Corn mills, 178. Kneading; Levigating; Demoli-
tion ; Bolt drawer; Burning, 179. Blasting, 180.

LECTURE XX.
ON THE HISTORY OF MECHANICS, 180.

Origin of the Grecian learning in Egypt ; Thales ; Ionian school ; Italian school;
Pythagoras, 181. Democritus ; Invention of the arch ; Archytas and Eudoxus,
182. Aristotle ; Foundation of Alexandria ; Epicurus, 183. Archimedes ; Siege
of Syracuse, 184, 185. Athenaeus ; Ctesibius, 185. Vitruvius ; Middle ages;
British manufactures, 186. Anglonorman and Gothic architecture, 187. Roger
Bacon ; Clocks ; Engraving and Printing, 188. Leonardo da Vinci ; Bacon Lord
Verulam ; Galileo; Napier, 189. Laws of collision; Hooke ; Barrow; Newton,
190. Followers of Newton, 191. Modern mathematicians and mechanics ; Time-
keepers ; Journals ; Royal Institution, 192. Future prospects ; Use of a catalogue
of references, 193. Table of the chronology of mathematicians and mechanics, ta
face p. 194.

xviii CONTENTS.

PART THE SECOND.— HYDRODYNAMICS.

LECTURE XXI.
ON HYDROSTATICS, 195.

Hydrodynamics more dependent on experiment than mechanics ; Division of
the subject into Hydraulics, Acustics, and Optics, 195. Hydrostatics ; Definition
of a fluid and a liquid, 196. Surface of a gravitating fluid horizontal, 197. Sur-
face of a revolving fluid ; Pressure of a fluid ; Magnitude of hydrostatic pressure,
198. Hydrostatic paradox, 199. Blowing with the mouth and lungs ; Pressure
on the bank of a river ; Pressure on a concave surface, 200. Pressure of different
fluids ; Equilibrium of fluids with solids; Floating bodies, 201. Stability and os-
cillations of floating bodies ; Buoyancy, 202. Bodies falling in fluids ; Hooke's
hemisphere ; Flexible vessels, 203.

LECTURE XXII.
ON PNEUMATIC EQUILIBRIUM, 204.

Properties of the air, and of gases ; Mercurial column, 204. Steams and vapours ;
Weight of the air ; Experiments with the air pump, 205. Constitution of the at-
mosphere ; Measurement of heights ; Ascent of a balloon, 206. Pressure of the
atmosphere ; Magdeburg hemispheres ; Nature of suction, 207. Barometers, 208.
Compressibility of liquids, 209.

LECTURE XXIII.
ON THE THEORY OF HYDRAULICS, 210.

General principle of ascending force, 210. Bernoulli's inferences ; Velocity
of a jet of a fluid, 211. Ajutages of different kinds ; Contraction of a jet, 212.
Effect of a short pipe ; Diverging pipe ; Experiments of Bernoulli, Venturi, and
Matthew Young; Discharge through large apertures, 213. Vessels emptying
themselves ; Locks, 214. Siphons ; Discharge through a vertical pipe, 215. Ex-
planation; Limit of velocity; Whirlpool; Intermitting springs, 216. Ascending
jets; Oscillations of fluids, 217. Waves, 218. Reflection of waves ; Height of
waves ; Experimental exhibition of waves, 219. Divergence of waves ; Combina-
tions of waves ; Applications ; Elastic pipes ; Circulation of the blood, 220.

LECTURE XXIV.

ON THE FRICTION OF FLUIDS, 222.

Experiments of Du Buat ; Motions of rivers ; Friction and resistance, 222.
Examples of the velocity of rivers, 223. Velocity at different depths ; Weres, 224.
Changes and flexures of rivers ; Lateral friction ; Venturi's experiments, 225. Ball
supported by a jet ; Discharge of long pipes, 226. Bent pipes ; Dilatations of
pipes ; Effect of temperature, 227.

CONTENTS. xix

LECTURE XXV.

ON HYDRAULIC PRESSURE, 228.

Pressure of fluids in motion ; Counterpressure, 228. Magnitude of the pressure
and impulse of fluids ; Laws of hydraulic pressure, 229. Particular case of water
wheels ; Oblique impulse ; Distribution of pressure, 230. Elevation and depres-
sion produced by the motion of a floating body ; Form of a ship ; Body moving
below the surface, 231. Convex surfaces ; Hydraulic pressure of the air, 232.
Concave surfaces ; Great effect of an increase of velocity ; Reflection of a ball or
stone, 233.

LECTURE XXVI.

ON HYDROSTATIC INSTRUMENTS, AND HYDRAULIC ARCHITECTURE, 235.

Statics and architecture of fluids; Hydrostatic balance, 235. Hydrometer;
Glass globules ; Specific gravities of particular substances ; Mixtures, 236. Spirit
level ; Hydrostatic lamps ; Embankments, 237. Dikes ; Rivers ; Reservoirs, 238.
Flood gates ; Strength of sluices and flood gates, 239. Friction ; Canals ; Piers ;
Harbours, 240.

LECTURE XXVII.

ON THE REGULATION or HYDRAULIC FORCES, 241.

Machinery of fluids ; Water pipes ; Siphons, 241. Stopcocks and valves, 242.
Pitot's tube ; Hydrometric fly ; Captain Hamilton's hydraulic register ; Motions
of the air, 243. Weight and impulse of fluids ; Raising weights by the descent of
water; Effect of velocity ; Overshot wheel, 244. Undershot wheel ; Mechanical
power of a stream, 245. Breast wheel ; Second wheel ; Oblique wheels and wind-
mills, 246, 247. Smoke jack ; Kite, 247. Parent's mill ; Seamanship ; Sidewind ;
Form and arrangement of a vessel, 248. Stability of a ship, 249.

LECTURE XXVIII.

ON HYDRAULIC MACHINES, 250.

Machines for raising water ; Noria ; Bucket wheel ; Throwing wheel ; Rope
pump, 250. Venturi's drain ; Spiral pipes ; Screw of Archimedes ; Water screw,
251. Wirtz's spiral pump, 252. Centrifugal pump ; Pumps; Plunger pump, 253.
Forcing pump ; Mixed pump ; Pistons ; Bramah's press ; Sucking pump ; Bag
pump, 254. Lifting pump ; Sucking and forcing pump ; Air vessel ; Fire engine,
255. Roller pumps and slider pumps ; Arrangement of pipes ; Bead pump ; Cel-
lular pump; Chain pump, 256. Cranks; Wheels and rollers; Chinese walking
wheels; Inverted pump ; Hydraulic air vessels, 257. Fountain of Hero ; Atmos-
pheric machines, 258. Hydraulic ram, 259.

LECTURE XXIX.
ON PNEUMATIC MACHINES, 259.

Counteraction and application of pneumatic forces ; Torricellian vacuum ; Air
pump; Double barrel, 260. Smeaton's pump ; Experiments; Gages, 261. Pear
gage ; Condensers ; Diving bells ; Bellows, 262. Gasometers ; Shower bellows ;

xx CONTENTS.

Velocity of a blast, 263. Ventilation ; Corn fan, 264. Chimnies ; Furnaces ;
Balloons; Steam engines; Savery's engine, 265. Newcomen's arid Beighton's
engine ; Watt's improvements, 266. Power of Boulton and Watt's machines ;
Later alterations, 267. Gunpowder; Calculations of Bernoulli and of Count
Rumford ; Properties of a gun ; Bullets ; Shot, 268. Air gun ; Improvements
on steam engines ; Stray Park engine ; Cornish boiler, 269. D valve, 270. Ap-
plication of steam engine to navigation ; G. Dodd ; Explanations ; Marine engines,
271. High pressure engine ; Young's formula for the elasticity of steam; For-
mula of the Franklin Institute ; Trevithick, 272. Description of the locomotive
engine, 272, 273, 274-, 275.

LECTURE XXX.
ON THE HISTORY OF HYDRAULICS AND PNEUMATICS, 275.

Discoveries of Archimedes ; Ctesibius; Hero; Vitruvius, 276. Canals; Gun-
powder; Galileo; Torricelli; Castelli, 277. Mariotte; Guglielmini; Guericke;
Hooke ; Marquis of Worcester, 278. Huygens ; Pardies : Renaud ; James and
John Bernoulli ; Newton, 279. Poleni ; Bouguer ; D. Bernoulli, 280. John
Bernoulli; Maclaurin ; Robins, 281. Dalembert; Kaestner; Euler; Smeaton ;
Borda; Watt, 282. Specification of Mr. Watt's patent, 282, 283. Bossut; Juan;
Prony; Chapman; Romme; Hutton; Rumford, 284. DuBuat; Black; Mont-
golfier, 285. Chronological table, 286.

LECTURE XXXI.

ON THE PROPAGATION OF SOUND, 287.

Importance of acustics; Division of the subject; Definition of sound; Pro-
pagation of sound, 287. Velocity of sound ; Delineation of a sound, 288. Com-
pressibility of hard bodies ; Transmission of sound by different mediums, 289.
Correction on account of heat, 290. Transmission in gases of different kinds ; In
liquids; In solids, 291. Divergence of sound, 292. Reflection of sound ; Illus-
tration by waves of water; Speaking trumpet, 293. Whispering gallery; Invisible
girl ; Partial interception of sound ; Decay of sound, 294.

LECTURE XXXII.

ON THE SOURCES AND EFFECTS OF SOUND, 295.

Origin of a simple sound; Of a continued sound; Musical sounds derived
from vibrations, 295. Open pipes; Stopped pipes; Harmonic sounds, 296.
Effect of temperature ; Longitudinal sounds of solids ; Lateral vibrations ; Flexi-
ble cords and membranes, 297. Harmonic sounds of cords, 298. Loaded
wire ; Revolutions of cords ; Vibrations of elastic rods, 299. Vibrations of
plates, rings, and vessels, 300. Mixed vibrations of solids and fluids ; Sympa-
thetic sounds; Hearing, 301. Description of the ear, 302. Delicacy of the
ear, 30a

CONTENTS. xxi

LECTURE XXXIII.
ON HARMONICS, 304.

Theory of harmonics; Combinations of sounds, 304. Beats, 305. Grave
harmonics ; Concords ; Melody ; Rhythm ; Simple compositions, 306. Diatonic
scale ; Half notes or semitones, 307. Minor mode ; Discords ; Rules of accom-
paniment, 308. Temperament ; Distinction of the notes, 309.

LECTURE XXXIV.
ON MUSICAL INSTRUMENTS, 310.

Division of musical instruments, 310. Harp; Lyre; Harpsichord; Spinet;
Pianoforte; Dulcimer; Clavichord; Guitar, 31 L Violins of different kinds;
Vielle ; Trumpet Marigni ; Aeolian harp ; Human voice, 312. Drum ; Stacada,
313. Bell; Harmonica; Vox humana pipe; Simple wind instruments; Mixed
wind instruments, 314. History of Music ; Lyre; Hermes; Terpander; Pytha-
goras; Simonides; Tibia; Aristotle, 315. Ctesibius; Pope Gregory; Guido;
Bacon ; Galileo ; Mersenne ; Kircher ; Meibomius ; Wallis ; Newton, 316^
Brook Taylor ; Sauveur ; Lagrange ; Euler ; Bernoulli ; Dalembert ; Sounds of
rods ; Grave harmonics of Romieu and Tartini ; Sounds of pipes, 317. Chladni;
Laplace, 318. Chronological table, 319.

LECTURE XXXV.
ON THE THEORY OF OPTICS, 320.

Importance of optics; Division of the subject; Definition of light; Ray of
light, 320. Motion of light; Homogeneous mediums; Reflection; Refraction,
321. Polished surfaces, 322. Return of a ray; Refractive density: Index of
refractive power, 323. Intermediate refraction ; Total reflection ; Dioptrics and
catoptrics; Focus, 324. Plane speculum; Principal focus; Convergence by
reflection; Concave and convex mirrors, 325. Prism; Multiplying glass ; Lens;
Effects of lenses ; Focus of a lens, 326. Joint focus ; Image ; Optical centre ;
Curvature of the image, 327.

LECTURE XXXVI.

ON OPTICAL INSTRUMENTS, 328.

Divergence of light, 328. Photometers ; Measurement of refractive densities;
Instruments strictly optical ; Images formed by lenses and mirrors, 329. Mag-
nifiers; Simple microscopes; Globules; Illumination of an image; Burning
glasses, 330. Materials of lenses and mirrors ; Images visible in every direction ;
Camera obscura, 331. Solar microscope, 332. Lucernal microscope; Phantas-
magoria, 333. Astronomical telescope ; Double microscope ; Galilean telescope ;
Common day telescope; Dr. Herschel's telescope, 334;' Newtonian reflector;
Gregorian telescope; Cassegrain's telescope; Smith's microscope; Curvature of
images in telescopes, 335. Magnifying powers of telescopes ; Field glass ; Dou-
ble magnifier, 336. Aberration from colour ; Achromatic glasses ; Achromatic
eye piece ; Micrometers, 337. Divided speculum, 338.

xxii CONTENTS.

LECTURE XXXVII.
ON PHYSICAL OPTICS, 340.

Sources of light ; Combustion ; Slow decomposition ; Electricity ; Friction,
340. Solar phosphori ; Emission of light; Velocity of light, 341. Apparent
aberration ; Oblique reflection ; Diffraction ; Dispersion, 342. Colour ; Division
of the spectrum ; Light of different kinds, 343. Mixed lights ; Imitation of white
light ; Primitive colours, 344. Mixture of colours by rapid motion ; Combina-
tions ; Atmospherical refraction, 345. Horizontal refraction ; Rainbows, 346.
Halos and parhelia, 347. Refraction of ice ; Complicated halos ; Double refrac-
tion ; Iceland spar, 348. Second refraction ; Transparent plates, 349.

LECTURE XXXVIII.
ON VISION, 350.

Description of the eye, 350. Image on the retina ; Advantages of the arrange-
ment; Inversion of the image, 351. Instinct; Sensibility of the retina, 352.
Focus of the eye; Accommodation; Change in the crystalline lens, 353. Uses
of the iris ; Optometer ; Myopic sight, 354. Presbyopic sight ; Single vision ;
Judgment of distance, 355. Apparent magnitudes of the sun and moon ; Aerial
perspective ; Painting ; Panorama, 356. Duration of sensations ; Ocular spectra,
357.

LECTURE XXXIX.

ON THE NATURE OF LIGHT AND COLOURS, 359.

Theories respecting the nature of light ; Simple propagation, 359. Transparent
mediums, 360. Uniformity of velocity ; Reflection and refraction, 361. Partial
reflection ; Total reflection ; Sources of light, 362. Aberration ; Double refrac-
tion ; Dispersion, 363. Colours of thin plates ; Alternate union and extinction
of colours ; Light admitted by two holes, 364. Supposed dimensions of undula-
tions ; Correction ; Stripes in a shadow, 365. Light passing through a narrow
aperture ; Colours of striated surfaces ; Curved stripes of colours, 366. Fringes
near a shadow; Colours of thin plates, 367. Colours of natural bodies, 368.
Colours of mixed plates ; Supernumerary rainbows ; Colours of concave mirrors,
369. Agreement of the Huygenian theory with the phenomena ; Interference of
light, 370. Phenomena of polarized light; Double refraction, 371, 372. Re-
ferences, 372.

LECTURE XL.
ON THE HISTORY OF OPTICS, 374.

Knowledge of the ancients; Empedocles; Aristotle, 374. Archimedes;
Euclid; Ptolemy; Alhazen; Vitellio; R. Bacon; Janson, 375. Galileo,
Kepler ; Scheiner ; Rheita ; Maurolycus ; De Dominis ; Snellius ; Descartes ;
Fermat ; Leibnitz ; Barrow, 376. Boyle : Hooke ; Newton ; Grimaldi, 377. Bar-
tholin; Huygens; Roemer,378. Bradley; Bouguer; Porterfield; Jurin; Smith;
Dollond ; Hall, 379. Euler; Lambert, 380. Mathemetical opticians ; Mazeas ;
Dutour; Comparetti ; Priestley; Delaval, 381. R. Darwin; Atmospherical
refraction; Wollaston ; Ritter; Herschel; Laplace; Attempts of the author,
382. Chronological table, 385.

CONTENTS.

PART THE THIRD.— PHYSICS.

LECTURE XLI.

ON THE FIXED STARS, 387.

Division of the subjects of physics ; Astronomy, 387. Empty space, 388.
Fixed stars ; Light of the stars ; Figure ; Twinkling ; Number ; Magnitudes, 389.
Distances of the stars, 390. Clusters or nebulae ; Arrangement of the stars
in general ; Milky way, 391. Proper motions of the stars; Dr. Herschel's division
of stars and nebulae, 392. Changes of the stars, 393. Constellations ; Repre-
sentations of the stars ; Allocations, 394, 395.

LECTURE XLII.

ON THE SOLAR SYSTEM, 397.

The sun a star ; Progressive motion of the sun, 397. Orbit of the sun ; Ro-
tation ; Spots, 398. Solar heat ; Sun's attraction ; Solar atmosphere, 399.
Planets ; Ecliptics, 400. Change of position of the ecliptic ; Nodes ; Keplerian
laws, 401. Rotation of the planets; Precession of the equinoxes ; Nutation of
the earth's axis ; Proportional distances of the planets, 402. Mercury ; Venus ; The
earth ; Mars, 403. Juno ; Pallas ; Ceres ; Vesta ; Jupiter ; Saturn, 404. Georgian
planet; Unknown planets; Satellites; Moon, 405. Satellites of Jupiter, 406.
Ring of Saturn; Comets, 407. Number and orbits of the comets, 408.

LECTURE XLIII.

ON THE LAWS OF GRAVITATION, 409.

Newton's great discovery, 409. Attraction of spherical bodies ; Extent of the
force of gravity, 410. Sun's change of place ; Orbits of the planets ; Keplerian
laws ; Universality of gravitation, 411. Motions of the apsides and nodes ;
Changes of the ecliptic; Forms of the planets; Precession; Nutation, 412.
Lunar motions ; Disturbing force of the sun, 413. Acceleration of the moon's
motion ; Moon's rotation ; Orbits of comets ; Predictions of Halley and
Clairaut, 414, 415.

LECTURE XLIV.

.
ON THE APPEARANCES OF THE CELESTIAL BODIES, 415.

Apparent motions to be described after the real ones, 415. Motions of the stars
and sun ; Motions of the earth ; Apparent revolution of the sun ; Sun's apparent
diameter, 416. Length of summer and winter ; Day and night ; Sun's apparent
path; Centrifugal force, 417. Places of the stars ; Twilight; Relative positions
and phases of the planets, 418. Phases of the moon ; Lunar eclipses ; Eclipses
of the sun, 419. Series of eclipses, 420. Harvest moon; Eclipses of Jupiter's

xxiv CONTENTS.

satellites; Comets; Light of the heavenly bodies, 421. Planetary worlds ; Fon-
tenelle; Mercury; Venus, 422. Moon; Mars, 423. Newly discovered planets;
Jupiter ; Saturn ; Georgian planet, 424, 425.

LECTURE XLV.

ON PRACTICAL ASTRONOMY, 425.

Real motions neglected ; Situation of a point in the heavens, 425. Meridian ;
Astronomical instruments; Time; Sidereal day; Solar day; Equation of time,
426. Dialling ; Chronology ; Calendar, 427. Improvement suggested ; Repub-
lican calendar ; Metonic cycle, 428. Golden number ; Epact ; Moon's age ;
Julian period ; Astronomical time ; Quadrants ; Transit instruments, 429.
Hadley's quadrant ; Declinations ; Refraction and parallax ; Latitudes, 430.
Longitudes ; Lunar observations ; Distance of the sun, 431. Transits; Densities
of the sun and planets ; Artificial globe, 432. Planispheres ; Orreries, 433.

LECTURE XLVI.

ON GEOGRAPHY, 435.

Particular account of the earth ; Curvature of its surface ; Direction of the
plumb line; Ellipticity, 435. Measurements of degrees ; Zones, 436. Climates;
Sea and land ; Continents, 437. Rivers ; Elevations ; Mountains, 438. Dif-
ferent orders of mountains, 439. Internal parts of the earth ; Density of the
earth, 440.

LECTURE XL VII.

ON THE TIDES, 441.

Tides noticed by the ancients, 441. Daily changes ; Monthly changes ;
Yearly changes ; Connexion with the moon ; Effect of gravitation on a fluid
sphere, 442. Primitive lunar tides ; Comparison with a pendulum ; Direct and
inverted tides, 443. Tides of a lake; Resistance; Tides of the Atlantic, 444.
Particular modifications, 445. Tides of the channels and of rivers; Inferior and
superior tides, 446. Laws of elevation and of depression ; Mode of observing
the tides ; Solar tides ; Combination of tides ; Retardation of spring and neap
tides, 447. Increased height in converging channels, 448. Combinations in
particular ports ; Currents, 449. Tides of the atmosphere, 450.

LECTURE XLVIII.

ON THE HISTORY OF ASTRONOMY, 451.

Earliest astronomy; Signs of the zodiac, 451. Babylonian observations ; Chal-
deans; Hermes; Egyptians; Chinese; Indians, 452. Greeks; Thales; Pytha-
goras, 453. Meto ; Alexandrian school ; Eratosthenes ; Hipparchus, 454.
Ptolemy, 455. Arabians ; Persians ; Copernicus, 456. Tycho Brahe ; Kepler,
457. Napier ; Huygens ; Cassini ; Gravitation, 458. Newton's discoveries ;
Extract from Pemberton, 459. British astronomers ; Observatory at Greenwich,
460. Determinations of the longitude; Late discoveries, 461, 462. Chrono-
logical table, 463.

.

CONTENTS. xxv

LECTURE XLIX.

ON THE ESSENTIAL PROPERTIES OF MATTER, 464.
Importance of minute objects; Definition of matter, 464. Place of the investiga-
tion ; Essential and accidental properties of matter ; Extension ; Divisibility,
465. Actual division of matter, 466. Impenetrability, 467. Permeability;
Orders of substances ; Repulsion ; Apparent contact, 468. Laws of repulsion,
469. Dalton's hypothesis ; Repulsion of liquids and solids ; Reciprocality of
repulsion ; Inertia, 470. Gravitation ; Cause of gravitation ; Mathematical con-
ceptions, 471. Newton's opinion ; Constitution of a medium capable of pro-
ducing gravitation, 472. Difficulties, 473.

hits

LECTURE L.
ON COHESION, 473.

Accidental properties of matter; Laws of cohesion; Modification of cohesion
by heat, 474. Liquidity; Superficial cohesion, 475. Bubbles; Form of the surface
of a fluid, 476. Magnitude of the force of cohesion ; Ascent between two plates;
Capillary tubes, 477. Horizontal surface ; Detached portion of a liquid ; Lyco-
podium ; Attractions and repulsions of floating bodies, 478. Apparent cohesion
of plates ; Drop between plates ; Oil spreading on water ; Sponge ; Long column
supported by cohesion, 479. Cohesion of solids; More perfect union ; Solidity;
Cause of solidity, 480. Elasticity, 481. Stiffness ; Strength ; Softness ; Ductility;
Primary cause of cohesion, 482, 483.

. .

LECTURE LI.

.ill
ON THE SOURCES AND EFFECTS OF HEAT, 484.

Division of the subject of heat; Definition of heat and cold; Excitement of
heat; Condensation, 484. Friction; Count Rumford's experiments, 485.
Effect of velocity; Pictet's experiments; Heat from combustion, 486. Com-
munication of heat; Conducting powers; Fluids, 487. Radiation of heat; Mr.
Leslie's discoveries ; Differences of solar and culinary heat, 488. Invisible heat;
Equilibrium of radiant heat ; Apparent reflection of cold, 489. Refrangibility of
heat; Blackening rays, 490. Effects of heat; Temporary effects ; Expansion of
gases; Condensation; Expansion of fluids, 491. Diminution of cohesive powers;
Boiling; Slow evaporation ; Contraction, 492. Freezing; Expansion of solids ;
Liquefaction, 493. Cracks from heat ; Permanent effects of heat ; Glass drops ;
Tempering of metals, 494, 495.

: plli

LECTURE LII.

ON THE MEASURES AND THE NATURE OF HEAT, 496.
Measures of expansion ; Pyrometer ; Scale of heat ; Mixtures ; Sun's rays,
496. Expansion of solids and fluids; Thermometers; Wedgwood's thermo-
meter, 497. Different scales ; Temporary change of a thermometer ; Air ther-
mometers, 498. Capacities for heat ; Natural zero, 499. Theory of capacities ;
Chemical effects, 500. Latent heat; Mr. Davy's experiments ; Intimate nature of
heat ; Theory of caloric, 501 . Confutation ; Heat a quality ; Newton's opinion ;

xxvi CONTENTS.

Vibrations ; Mechanical effects of vibrations, 502. Chemical effects ; Comparison
with sound, 503. General inferences ; Additional remarks ; Thermomultiplier ;
Rock salt, 504. Polarization of light and heat ; Discoveries of Melloni and
Professor Forbes, 505. Theory of Heat ; References, 506.

LECTURE LIIL
ON ELECTRICITY IN EQUILIBRIUM, 507.

Utility of electrical hypotheses ; Division of the subject, 507. Supposed elec-
tric fluid; Its attractions and repulsions, 508. Conductors and nonconductors;
Positive and negative electricity ; Local electricity, 509. Distribution of electri-
city; Electricity of a sphere; Connected spheres, 510. Difference of hydrostatic
and electrical pressure; Attractions and repulsions, 511. Induced electricity;
Neutral point ; Effects of attraction and repulsion ; Currents of air ; Bodies elec-
trified in different degrees, 512. Charge; Discharge; Shock; Coated jar; Bat-
tery; Comparison of conducting powers, 513, 514.

LECTURE LIV.
ON ELECTRICITY IN MOTION, 516.

Effects and causes of electrical motions, and electrical apparatus ; Velocity ;
Spark, 516. Perforation of a jar ; Direction of the motion ; Opinions respecting
positive and negative electricity; Effects of electricity; Accumulation; Simple
current, 517. Electric light ; Heat, 518. Mechanical effects ; Chemical effects ;
Sensible effects, 519. Excitation of electricity; Electrics; Vapours; Tourmalin,
520. Galvanic electricity ; Chemical changes ; Galvanic combinations ; General
laws, 521. Particular facts ; Pile of Volta, 522. Troughs; Animal electricity;
Mr. Davy's discoveries, 523. Electrical nature of chemical attractions, 524.
Theory of the pile ; Efficacy of decomposable substances ; Electrical machines ;
Teylerian machine, 525. Electrophorus ; Cendenser ; Multiplier, 526. Doublers ;
Electrical balance; Quadrant electrometer ; Gold leaf electrometer, 527. Lane's
electrometer ; General observations, 528.

LECTURE LV.
ON MAGNETISM, 531.

Resemblance of magnetism and electricity ; Theory, 531 . Conducting powers ;
Magnetical substances, 532. Aurora borealis ; North and South poles ; Attrac-
tions and repulsions ; Polarity, 533. Arrangement of filings ; Directive force ;
Terrestrial magnetism ; Compass ; Dipping needle, 534. Illustration ; Temporary
magnetism ; Natural magnet ; Magnetic poles of the earth ; Diurnal changes, 535.
Variation of the declination ; Line of no declination ; Dip, 53G. Artificial mag-
nets ; Double touch, 537. Magnetic paste ; Division of a magnet ; Striking and
ringing a magnet; Hammering brass ; Solution in an acid, 538. Resemblance of
polarity to crystallization ; Additional remarks ; Discovery of Professor Oersted ;
Electro-magnetism, 539. Construction of the galvanometer ; Gumming; Nobili;
Action of the voltaic current, 540. Electro-magnetic telegraph ; Faraday ; Mag-
neto-electric machine, 541 ; Arago ; References, 542.

CONTENTS. xxvii

LECTURE LVI.
ON CLIMATES AND WINDS, 544.

Meteorology ; Division of the subject ; Climates ; Meteorological thermome-
ters, 544-. Immediate effects of the sun ; Prerost's calculations ; Variations of
temperature, 545. Slow changes ; Heat of the sea ; Effect of freezing and thaw-
ing ; Heat of the atmosphere, 54*6. Summer and winter ; Temperatures of differ-
ent places ; Local variations, 547. Winds ; Periodical winds ; Trade winds ;
Hadley ; Halley's theory, 548. Greater heat of the northern hemispheres ; West-
erly winds ; Local modifications ; Monsoons, 549. Land and sea breezes ; Hur-
ricanes ; Variations of the barometer, 550.

LECTURE LVII.

ON AQUEOUS AND IGNEOUS METEORS, 551.

Evaporation and its effects ; Theory of Deluc and Dalton ; Quantity of water
evaporating, 551. Precipitation; Moisture; Mediterranean, 552. Currents at
the Straights ; Attraction of moisture ; B. Prevost ; Hygrometers, 553. Natural
hygrometer; Water contained in air, 554. Visible vapour ; Dew ; Mists, 555. Rain ;
Indications of the barometer ; Effects of mountains, 556. Periodical rains ; Thun-
der and lightning ; Atmospherical electricity, 557. Thunder storms ; Conduc-
tors, 558. Sudden condensations; Waterspouts, 559. Aurora borealis ; Earth-
quakes and volcanos ; Volcanic countries, 560. Earthquakes of Calabria, 561.
Eruptions of Vesuvius, 562. Geological changes ; Reality of various changes ;
Effects of rivers and of the sea, 563. Shooting stars ; falling stones, 564, 565.

LECTURE LVIII.

ON VEGETATION, 565.

Sketch of natural history ; Minerals, 565. Vegetables ; Animals ; Distinctions
of animals and vegetables, 566. Description of a vegetable ; Germination, 567.
Parts of plants ; Vessels, 568. Motion of the sap ; Mr. Knight's experiments, 569.
Grafting; Diseases of plants, 570. Exposure to the air; Linnean system, 571.
System of Jussieu,572.

LECTURE LIX.

ON ANIMAL LIFE, 573.

Classification of animals, according to Linne, 573. Mammalia; Birds, 574.
Amphibia; Fishes; Insects, 575. Vermes, 576. Senses; Nutrition, 577. Ner-
vous system ; Nature of the nerves, 578. Diseases ; Natural cures, 579.

LECTURE LX.

ON THE HISTORY OF TERRESTRIAL PHYSICS, 580.

General retrospect ; Knowledge of the ancients ; Chinese ; Numa, 580. Thales ;
Anaximander ; Anaximenes ; Pythagoras ; Anaxagoras ; Democritus ; Heraclitus ;
Plato, 581. Aristotle; Epicurus, 582. R.Bacon; Discovery of the compass ;

xxviii CONTENTS.

Gesner; Aldrovandus ; Gilbert of Colchester; Variation of the compass; R.
Bacon, 583. Opinions of heat ; Drebel ; Harvey ; Circulation of the blood ; Baro-
meter ; Bauhins, 584. Ray ; Willughby ; Philosophical societies ; Variation charts ;
Electricity, 585. Linnean system ; Discoveries respecting heat ; Theory of mag-
netism and electricity, 586. Boscovich ; Hygrometry ; Galvanism, 587. Pile of
Volta ; Mr. Davy's experiments ; Dalton ; Rumford ; Leslie, 588. Herschel ;
Capillary tubes ; Laplace ; Advantages to be expected from modern institutions,
589, 590. Chronological table, to face p. 590.

ERRATA.

P. 16, line 31, for " but is " read " but it is."

P. 160, lines 25, 26, for " immediately " read " immediately."

P. 255, line 29, for " adjutage " read " ajutage."

P. 292, line 4, for " wagon " read " waggon."

P. 390, note, line 3, after " distances" insert " from another star, of the middle

point."
P. 396, Catalogues, insert " Groombridge's Catalogue of Circumpolar Stars, 4to,

Lond. 1838."

P. 404, line 24, for " asmosphere " read " atmosphere."
P. 505, line 41, for " Franenhofer's " read " Frauenhofer's."
P. 582, Une 33, for " indentical" read "identical."

ON

NATURAL PHILOSOPHY

AND

THE MECHANICAL ARTS.

LECTURE I.

INTRODUCTION.

IT is to be presumed, that most of those who honour the theatre of the
Royal Institution with their attendance, are already acquainted with the
nature of the objects which its founders and promoters have been endea-
vouring to attain : yet it appears to be by no means superfluous that I
should define with accuracy my own views of the utility that is likely to
be derived from it, and of the most effectual means of accomplishing its
purposes ; in order that we may be able to distinguish, without difficulty,
the most eligible track for our common progress through the regions of
science ; and that those who are desirous of accompanying me in the jour-
ney may know precisely what route we are to follow, and what depart-
ments will more particularly arrest our attention.

Societies, which are merely literary and philosophical, have in general
principally proposed to themselves to enlighten the understanding by the
discovery of unknown phenomena, and to exercise the reasoning powers by
opening new fields for speculation, Other associations have been more
particularly intended for the encouragement of the arts, of manufactures,
and of commerce. The primary and peculiar object of the Royal Insti-
tution of Great Britain is professedly of an humbler, but not of a less
interesting nature. It is to apply to domestic convenience the improve-
ments which have been made in science, and to introduce into general
practice such mechanical inventions as are of decided utility. But while
it is chiefly engaged in this pursuit, it extends its views, in some measure,
to the promotion of the same ends which belong to the particular pro-
vinces of other literary societies ; and it is the more impossible that such
objects should be wholly excluded, as it is upon the advancement of these
that the specific objects of the Institution must ultimately depend. Hence
the dissemination of the knowledge of natural philosophy and chemistry
becomes a very essential part of the design of the Royal Institution ; and

2 LECTURE I.

this department must, in the natural order of arrangement, be anterior to
the application of the sciences to practical uses. To exclude all know-
ledge but that which has already been applied to immediate utility, would
be to reduce our faculties to a state of servitude, and to frustrate the very

\ purposes which we are labouring to accomplish. No , discovery, however
remote in its nature from the subjects of daily observation, can with rea-

I son be declared wholly inapplicable to the benefit of mankind.

It has therefore always appeared to me, to be not only the best begin-
ning, but also an object of high and permanent importance in the plan of
the Institution, to direct the public attention to the cultivation of the
elementary doctrines of natural philosophy, as well speculative as prac-
tical. Those wrho possess the genuine spirit of scientific investigation, and
who have tasted the pure satisfaction arising from an advancement in
intellectual acquirements, are contented to proceed in their researches,
without inquiring at every step what they gain by their newly discovered,
, lights, and to what practical purposes they are applicable : they receive a
sufficient gratification from the enlargement of their views of the consti-
. tution of the universe, and experience, in the immediate pursuit of know-
ledge, that pleasure which others wish to obtain more circuitously by
its means. And it is one of the principal advantages of a liberal educa-
tion, that it creates a susceptibility of an enjoyment so elegant and so
Irational.

A considerable portion of my audience, to whose information it will be
my particular ambition to accommodate my lectures, consists of that sex
which, by the custom of civilized society, is in some measure exempted
from the more laborious duties that occupy the time and attention of the
other sex. The many leisure hours which are at the command of females
in the superior orders of society may surely be appropriated, with greater
satisfaction, to the improvement of the mind and to the acquisition of
knowledge, than to such amusements as are only designed for facilitating
the insipid consumption of superfluous time. The hours thus spent will
unquestionably become, by means of a little habit, as much more agreeable
at the moment, as they must be more capable of affording self-approbation
upon reflection. And besides, like the seasoning which reconciled the
Spartans to their uninviting diet, they will even heighten the relish for
those pursuits which they interrupt : for mental exercise is as necessary to
mental enjoyment as corporal labour to corporal health and vigour. In this
point of view the Royal Institution may in some degree supply the place of
a subordinate university, to those whose sex or situation in life has denied
them the advantage of an academical education in the national seminaries
of learning.

But notwithstanding the necessity of introducing very copiously specu-
lations of a more general nature, we must not lose sight of the original
objects of the Royal Institution ; and we must, therefore, direct our atten-
tion more particularly to the theory of practical mechanics and of manu-
factures. In these departments we shall find some deficiencies which may
without much difficulty be supplied from scientific principles ; and by an
ample collection and display of models, illustrative of machines and of

INTRODUCTION. 3

inventions of all kinds, we may proceed in the most direct manner to con-
tribute to the dissemination of that kind of knowledge which is most parti-
cularly our object. So that we must be more practical than academies of,
sciences, and more theoretical than societies for the improvement of arts ; |
while we endeavour at the same time to give stability to our proceedings by
an annual recurrence to the elementary knowledge which is subservient to
the purposes of both ; and, as far as we are able, to apply to practice the
newest lights which may from time to time be thrown on particular
branches of mechanical science. It is thus that we may most effectually
perform what the idolized sophists of antiquity but verbally professed, to
bring down philosophy from the heavens, and to make her an inhabitant of
the earth.

To those who are engaged in the practical cultivation of various arts
subservient to the conveniences of life, these lectures may be of some
utility, by furnishing them with well established principles, applicable to
a variety of cases which may occasionally occur to them, where a little
deviation from the ordinary routine of their profession may be necessary.
Unfortunately, the hands that execute are too often inadequately sup-
ported by the head that directs ; and much labour is lost for want of a
little previous application to the fundamental doctrines of the mechanical
sciences. Nor is any exorbitant portion of time or industry necessary for
this purpose ; for it happens singularly enough, that almost all practical
applications of science depend on principles easilyjilearnt ; and, except in
astronomy only, it has seldom been found that very abstruse investigations
have been of great importance to society. Our most refined analytical
calculations are by far too imperfect to apply to all possible cases of me-
chanical actions that can be proposed ; and those problems which most
frequently occur, may in general be solved by methods sufficiently
obvious ; although, from a want of proper order and perspicuity in the
treatment of first principles, it has often happened that the most ele-
mentary propositions have been considered as requiring great study and
application.

We may also be able to render an important service to society, and to
confer a still more essential benefit on individuals, by repressing the pre-
mature zeal of unskilful inventors. We need only read over the monthly
accounts of patents, intended for securing the pecuniary advantages of
useful discoveries, in order to be convinced what expense of time and for-
tune is continually lavished on the feeblest attempts to innovate and
improve. If we can be succcessful in convincing such inconsiderate
; enthusiasts of their real ignorance, or if we can shew them, that even their
own fairy ground has been pre-occupied, we may save them from impending ;
ruin, and may relieve the public from the distraction of having its atten-
tion perpetually excited by unworthy objects. The ridicule attendant on
the name of a projector has been in general but too well deserved ; for few,
very few, who have aspired at improvement, have ever had the patience
to submit their inventions to such experimental tests as common sense
would suggest to an impartial observer. We may venture to affirm that
out of every hundred of fancied improvements in arts or in machines,

B 2

4 LECTURE I.

ninety at least, if not ninety-nine, are either old or useless ; the object of
our researches is, to enable ourselves to distinguish and to adopt the hun-
dredth. But while we prune the luxuriant shoots of youthful invention,
we must remember to perform our task with leniency, and to show that we
wish only to give additional vigour to the healthful branches, and not
to extirpate the parent plant.

The Repository of the Royal Institution, as soon as it can be properly
furnished, will be considered as a supplementary room for apparatus, in
which the most interesting models, exhibited and described in the lectures,
will be placed for more frequent inspection, and where a few other articles
may perhaps deserve admission, which will not require so particular an
explanation. To those who have profited by the lectures, or who are
already too far advanced to stand in need of them, our rooms for reading
and for literary conversation may be a source of mutual instruction. Our
library in time must contain all those works of importance which are too
expensive for the private collections of the generality of individuals ; which
are necessary to complete the knowledge of particular sciences, and to
which references will occasionally be given in the lectures on those sciences.
Our journals, free from commercial shackles, will present the public,
from time to time, with concise accounts of the most interesting novelties
in science and in the useful arts ; and they will furnish a perpetual incite-
ment to their editors to appropriate, as much as possible, to their own
improvement, whatever is valuable in the publications of their cotempo-
raries. When all the advantages which may reasonably be expected from
this institution shall be fully understood and impartially considered, it is
to be hoped that few persons of liberal minds will be indifferent to its
success, or unwilling to contribute to it and to participate in it.

To that regulation, which forbids the introduction of any discussions
connected with the learned professions, I shall always most willingly submit,
and most punctually attend. It requires the study of a considerable portion
jof a man's life to qualify him to be of use to mankind in any of them ; and
fl nothing can be more pernicious to individuals or to society, than the
jj attempting to proceed practically upon an imperfect conception of a few
first principles only. In physic, the wisest can do but little, and the igno-
rant can only do worse than nothing : and anxiously as we are disposed
to seek whatever relief the learned and experienced may be able to afford
us, so cautiously ought we to avoid the mischievous interference of the
half-studied empiric : in politics and in religion, we need but to look back
on the history of kingdoms and republics, in order to be aware of the
mischiefs which ensue, when " fools rush in where angels fear to tread."

Deeply impressed with the importance of mathematical investigations,
both for the advancement of science and for the improvement of the mind, 1
thought it in the first place an indispensable duty to present the Royal Insti-
tution, in my Syllabus, with a connected system of natural philosophy,
on a plan seldom, if ever, before executed in the most copious treatises,
v ^Proceeding from the simplest axioms of abstract mathematics, the Syllabus
contains a strict demonstration of every proposition which I have found it
-^> ! necessary to employ throughout the whole extent of natural philosophy.

INTRODUCTION.

I In the astronomical part only, some obs^rj£atjons occur, i
mathematical evidence ; here, however, it was as impracticable as ~itl
would have been useless to attempt to enter into investigations, which in
many instances have been extended far beyond the limits even of Newton's
researches. But for the sake of those who are not disposed to undertake the
labour of following, with mathematical accuracy, all the steps of the
demonstrations on which the doctrines of the mechanical sciences are
founded, I shall endeavour to avoid, in the whole of this course of lectures,
every intricacy which might be perplexing to a beginner, and every
argument which is fitter for the closet than for a public theatre. Here I
propose to support the same propositions by experimental proofs : not that
I consider such proofs as the most conclusive, or as more interesting to a
truly philosophic mind than a deduction from general principles ; but because
there is a satisfaction in discovering the coincidence of theories, with
visible effects, and because objects of sense are of advantage in assisting
the imagination to comprehend, and the memory to retain, what in a more
abstracted form might fail to excite sufficient attention.

This combination of experimental with analogical arguments constitutes
the principal merit of modern philosophy. And here let the citizen of the
world excuse the partiality of an Englishman, if I pride myself, and con-
gratulate my audience, on the decided superiority of our own country, in
the first establishment, and in the subsequent cultivation, of the true phi-
losophy of the operations of nature. I grant that we have at times been
culpably negligent of the labours of others ; that we have of late suffered
our neighbours to excel us in abstract mathematics, and perhaps, in some
instances, in patient and persevering observation of naked phenomena. We
have not at this moment a taagrauge or a Laplace/: what we have I do not
think it necessary to enumerate : but there is a certain combination of
theoretical reasoning with experimental inquiry, in which Great Britain,
from the time of the reformation of philosophy, has never been inferior to
any nation existing. I need only refer to the Transactions of the Royal
Society, for abundant instances of the mode of investigation to which I
allude ; and I will venture to affirm, that their late publications are equal
in importance to any that have preceded. It was in England that a Bacon
| first taught the world the true method of the study of nature, and rescued
science from that barbarism' in which the followers of Aristotle, by a too
servile imitation of their master, had involved it ; and with which, even of
late, a mad spirit of innovation, under the name of the critical jjhilqsophy,
has, in a considerable part of Europe, again been threatening it. It was in
this country that Newton advanced, with one gigantic stride, from the re-
gions of twilight into the noon day of science. A Boyle and a Hooke, who
would otherwise have been deservedly the boast of their century, served but
as obscure forerunners of Newton's glories. After these, a c'ro'wd of eminent
men succeeded, each of great individual merit ; but, absorbed in the prose-
cution of the Newtonian discoveries, they chose rather to be useful by their
humble industry than to wander in search of the brilliancy of novelty. It
is difficult to judge of our coj,emporaries ; but we appear at present to be
in possession of more than one philosopher, whose names posterity will be

(5 LECTURE I.

eager to rank in the same class with the few that have been enumerated.
But it is not our present business to enter into the history of science ;
respecting what is supposed to be wholly unknown we can have little
curiosity : a short sketch of the progress of each branch of natural philoso-
phy will be more properly introduced after we have finished our investiga-
tion of the principal doctrines belonging to it.

With regard to the mode of delivering these lectures, I shall in general
intreat my audience to pardon the formality of a written discourse, in
favour of the advantage of a superior degree of order and perspicuity. It
would unquestionably be desirable that every syllable advanced should be
rendered perfectly easy and comprehensible even to the most uninformed ;
that the most inattentive might find sufficient variety and entertainment
in what is submitted to them to excite their curiosity, and that in all cases
the pleasing, and sometimes even the surprising-, should be united with the
instructive and the important. But whenever there appears to be a real
impossibility of reconciling these various objects, I shall esteem it better to
seek for substantial utility than temporary amusement ; for if we fail of
being useful for want of being sufficiently popular, we remain at least
respectable ; but if we are unsuccessful in our attempts to amuse, we
immediately appear trifling and contemptible. It shall, however, at all times
be my endeavour to avoid each extreme ; and I trust that I shall then only
be condemned when I am found abstruse from ostentation, or uninteresting
from supineness. The most difficult thing for a teacher is, to recollect how
much it cost himself to learn, and to accommodate his instruction to the
apprehension of the uninformed : by bearing in mind this observation, I
hope to be able to render my lectures more and more intelligible and
familiar ; not by passing over difficulties, but by endeavouring to facilitate
the task of overcoming them ; and if at any time I appear to have failed in
this attempt, I shall think myself honoured by any subsequent inquiries
that my audience may be disposed to make.

We have to extend our views over the whole circle of natural and arti-
ficial knowledge, to consider in detail the principles and application of the
philosophy of nature and of art. We are to discuss a great number of
subjects, to each of which a separate title and rank among the sciences has
sometimes been assigned ; and it is necessary, in order to obtain a distinct
conception of the foundation and relation of each subdivision, to pay par-
ticular attention to the order in which the sciences are to be treated, and to
the connexion which subsists between them, as well as to the degree of
importance which each of them claims, with regard either to theory or to
practice. To insist on the propriety of a distinct and logical order is unne-
cessary ; for however superfluous we may deem the scholastic forms of
rhetoric, it is confessedly advantageous to the judgment as well as to the
meniQry, to unite those things which are naturally connected, and to sepa-
rate those which are essentially distinct. When a traveller is desirous of
becoming acquainted with a city or country before unknown to him, he
naturally begins by taking, from some elevated situation, a distant view of
the distribution of its parts ; and in the same manner, before we enter on
the particular consideration of the subjects of our researches, it may be of

INTRODUCTION. 7

use to form to ourselves a general idea of the sciences and arts which are to
be placed among them.

Upon the advantages of mathematical and philosophical investigation in
general it is unnecessary to enlarge, because no liberal mind can require
any arguments to be convinced how much the judgment is strengthened, !
and the invention assisted, by habits of reasoning with caution and accu-
racy. The public opinion is rather, on the contrary, in danger, at least in
some parts of the world, of being too exclusively biassed in favour of
natural philosophy ; and has sometimes been inclined to a devotion too
much limited to science, without a sufficient attention to such literature as
an elegant mind always desires to see united with it. As to the practical
importance of philosophical theories of thqf arts, it may have been overrated
by some, but no person is authorised to amrm that it has been too highly
estimated, unless he has made himself master of every thing that theory is
capable of doing ; such a one, although he may in some cases be obliged to
confess the insufficiency of our calculations, will never have reason to com-
plain of their fallacy.

The division of the whole course of lectures into three parts was origi-
nally suggested by the periodical succession in which the appointed hours
recur : but it appears to be more convenient than any other for the regular
classification of the subjects. The general doctrines of motion, and their
application to all purposes variable at pleasure, supply the materials of the
first two parts ; of which the one treats of the motions of solid bodies, and
the other of those of fluids, including the theory of light. The third part
relates to the particular history of the phenomena of nature, and of the
affections of bodies actually existing in the universe, independently of the
art of man ; comprehending astronomy, geography, and the doctrine of the
properties of matter, and of the most general and powerful agents that
influence it.

The synthetical order of proceeding, from simple and general principles,
to their more intricate combinations in particular cases, is by far the most
compendious for conveying information with regard to sciences that are at
all referable to certain fundamental laws. For these laws being once
established, each fact, as soon as it is known, assumes its place in the
system, and is retained in the memory by its relation to the rest as a con-
necting link. In the analytical mode, on the contrary, which is absolutely
necessary for the first investigation of truth, we are obliged to begin by
collecting a number of insulated circumstances, which lead us back by
degrees to the knowledge of original principles, but which, until we arrive
at those principles, are merely a burden to the memory. For the pheno-
mena of nature resemble the scattered leaves of the Sibylline prophecies ;
a word only, or a single syllable, is written on each leaf, which, when sepa-
rately considered, conveys no instruction to the mind ; but when, by the
labour of patient investigation, every fragment is replaced in its appropriate
connexion, the whole begins at once to speak a perspicuous and a harmoni-
ous language.

Proceeding, therefore, in the synthetical order, we set out from the '
abstract doctrines of mathematics, relating to quantity, space, and number,

8 LECTURE I.

which we pass over, as supposed to be previously understood, or as suffi-
ciently explained in the mathematical elements, and go on to their imme-
diate application to mechanics and hydrodynamics, or to such cases of the
motions of solids and fluids as are dependent on arbitrary assumptions, that
is, where we do not confine our inquiries to any particular cases of existing
phenomena. By means of principles which are deducible in a satisfactory
manner from mathematical axioms, with the assistance only of the general
logic of induction^ we shall be able to draw such conclusions as are capable
^of giving us very important information respecting the operations of na-
I ture and of art, and to lay down such laws, as, to an uninformed person,
I it would appear to be beyond the powers of reason to determine without the
/^assistance of experiment. The affections of falling bodies and of projectiles,
the phenomena of bodies revolving round a centre, the motions of pendu-
lums, the properties of the centre of gravity, the equilibrium of forces in
machines of different kinds, the laws of preponderance, and the effects of
collision ; all these are wholly referable to axiomatical evidence, and are
frequently applicable to important uses in practice. Upon these founda-
tions we shall proceed ta the general principles of machinery, and the
application of forces of different kinds : we shall inquire what are the
principal sources of motion that we can subject to our command, and what
advantages are peculiar to each of them : and then, according to the
purposes for which they are employed, we shall separately examine the
principal machines and manufactures in which those forces are applied to
the service of mankind.

Such instruments and machines as are more or less immediately subser-
vient to mathematical purposes will be the first in order, including all the
mechanism of literature, the arts of writing, engraving and printing, in
their various branches, and the comparison of measures with each other
and with different standards ; the principles of perspective will also form a
useful appendage to the description of geometrical instruments. The deter-
mination of weights, and of the magnitude of moving forces of various
kinds, constituting the science of statics, will be the next subject, and will
be followed by the consideration of the retarding force of friction, and of
the passive strength of the various materials that are employed in building
and in machinery.

All these subjects are in part preparatory to the immediate examination
of the mechanical arts and manufactures, which are so numerous and com-
plicated as not to admit of regular arrangement without some difficulty :
they may howeve^ be divided into such as are principally employed for
resisting, for modifying, or for counteracting, any motion or force ; thus
architecture and carpentry are chiefly intended to resist the force of gravi-
tation : these comprehend the employments of the mason, the bricklayer,
the joiner, the cabinet maker, and the locksmith. In these departments it
is often of the utmost importance to the mechanic to recur, especially in
! works of magnitude, to philosophical principles ; and in many other cases,
fo where there is no need of much calculation, we may still be of service, by
collecting such inventions of ingenious artists as are convenient and elegant,
and which, although simple in their principles, are not obvious in their

INTRODUCTION. 9

arrangements ; and in the same manner we may be able, in taking- a gene-
ral view of other arts and manufactures, to explain their principles, where
theory is concerned, and to exhibit practical precedents, where the nature
of the subject requires no refined investigation.

The modification of motion and force includes its communication and
alteration, by joints of various kinds, by wheel work, and by cordage, and
its equalisation by means of timekeepers. The subject of wheelwork gives
considerable scope for mathematical research, and requires the more notice,
as it has often been inaccurately treated : the consideration of cordage leads
us to that of union by twisting and by intermixture of fibres ; including
the important arts of carding, combing, spinning, ropemaking, weaving,
fulling, felting, and papermaking ; which constitute the employment of
many millions of manufacturers of all ages and sexes, in every part of the
world, and by which the animal and vegetable productions of a large por-
tion of the surface of the globe are made to contribute, as well to the power
and riches of the individuals who supply them, as to the health and comfort
of the public that consumes them. The admirable art of the watch and
clock maker is a peculiarly interesting department of practical mechanics ;
it affords employment for mathematical investigation, for experimental
inquiry, and for ingenious invention ; and the perfection which it has
derived from a combination of these means, does honour as well to the
nations who have encouraged it as to the individuals who -have been en-
gaged in it.

To counteract the powers of gravitation and of friction, is the object of
such machines as are used for raising and removing weights : cranes, fric-
tion wheels, and carriages of all kinds, are referable to this head, and some
of them have been the subjects of much speculation and experiment.
Lastly, to overcome and to modify the corpuscular forces of cohesion and
repulsion, and to change the external forms of bodies, is the object of ma-
chinery intended for compression, extension, penetration, attrition, tritura-
tion, agitation, and demolition. For these purposes we employ presses,
forges, rolling, stamping, coining, and milling machines ; the processes of
digging, ploughing, and many other agricultural arts ; boring, mining,
grinding, polishing, and turning ; mills of various kinds, threshing mills,
corn mills, oil mills, and powder mills ; besides the chemical agents con-
cerned in blasting rocks, and in the operations of artillery. All these arts
are comprehended in the department of mechanics, which constitutes the
first division of this course. Not that we shall be able to enter at large faito
the detail of each ; but having formed a general outline, we may fill up its
particular parts with more or less minuteness, as we may find more or less
matter of importance to insert in each ; and those who wish to pursue the
subjects further, will every where be able to derive great assistance from
the authors whose works will be mentioned.

The doctrines of hydrodynamics relate to the motions and affections of
fluids, in which we no longer consider each distinct particle that is capable
of separate motion, but where we attend to the effect of an infinite number
of particles, constituting a liquid or aeriform aggregate. The general
theory of such motions will be premised, under the heads hydrostatics, or

10 LECTURE I.

the affections of liquids at rest ; pneumatostatics, or the properties of elastic
fluids at rest ; and hydraulics, or the theory of fluids in motion. The
practical application of this theory to hydraulic and pneumatic machines
is of very considerable importance, and is as interesting to the philosopher
as it is necessary to the engineer. The employment of the force of water
and wind to the best advantage, the draining of lands and mines, the supply
of water for domestic convenience, the manoeuvres of seamanship, the con-
struction of the steam engine, are all dependent upon hydrodynamical
principles, and are often considered as comprehended in the science of
hydraulics. Harmonics and optics, the remaining parts of this division, are
more insulated : the doctrine of sound, the theory of music, and the con-
struction of musical instruments, are as pleasing to the intellect in theory,
as they are gratifying to the senses in practice ; but the science of optics is
not less interesting, and at the same time far more useful ; the instruments
which it furnishes are of almost indispensable necessity to the navigator, to
the naturalist, to the physiologist, and even to the man of business or plea-
sure. It is perhaps in this science that many persons of the greatest
genius have been the most happily employed. The reasons for which it is
classed as a division of hydrodynamics will be explained hereafter.

The contemplation of the particular phenomena of nature, as they are
displayed in the universe at large, contributes perhaps less to the perfection
of any of the arts which are immediately subservient to profit or conve-
nience, than the study of mechanics and hydrodynamics. But the dignity
and magnificence of some of these phenomena, and the beauty and variety
of others, render them highly interesting to the philosophical mind ; at the
same time that some of them are of the utmost importance in their appli-
cation to the purposes of life. In all these respects the science of astro-
nomy holds the first rank ; its uses in assisting navigation, and in regulating
chronology, are beyond all calculation. Geography and hydrography, or
the particular histories of the earth and sea, are immediately connected
with astronomy. The discussion of the properties of matter in general,
and of the alterations of temperature to which all bodies are liable, has not
hitherto received a distinct appellation as a science ; but both these subjects
require a separate consideration, and afford a vast scope for speculation
and for observation. Electricity and magnetism are partly referable to the
affections of matter, and partly to the agency of substances which appear
to agree with common matter in some properties and to differ from it in
others. The phenomena produced by these agents are often such as excite
a high degree of curiosity to inquire into their causes, although the inquiry
too often terminates only in astonishment ; but we have reason to expect
considerable advancement in these sciences from the singular discoveries of
modern chemists. The utility of the philosophy of electricity is sufficiently
exemplified in the general introduction of conductors for securing us
against lightning, to say nothing of the occasional enployment of electricity
in medicine ; and since the important discovery of the compass, we have
only to lament that the changeable nature of magnetic effects so much
limits the utility of that instrument for nautical and geographical purposes.
Of meteorology and of geology our knowledge is hitherto very imperfect.

INTRODUCTION. 11

Notwithstanding many diffuse treatises which relate to them we cannot
boast of having reduced them to any determinate laws ; and yet there are
some meteorological facts which well deserve our attention. Natural history
is the last of the sciences that it will be necessary for us to notice. Some
may think it superfluous to attempt to give so superficial a sketch of this
most extensive subject as our plan will allow ; but it is still possible to
select some general observations respecting the methods of classification, as
well as the philosophy of natural history, which, although very concise,
may yet be in some measure instructive. This third division of the course
would properly include, together with the general properties of matter and
the particular actions of its particles, the whole science of chemistry ; but
the variety and importance of chemical researches demand a separate and
minute discussion ; and the novelty and beauty of many of the experi-
ments with which the labours of our cotemporaries have presented us, and
which will be exhibited in the theatre of the Royal Institution by the pro-
fessor of Chemistry, are sufficient to make this department of natural phi-
losophy the most entertaining of all the sciences.

Such is the whole outline of our plan, and such are the practical uses to
which the arts and sciences comprehended in it are principally applicable.
Before we proceed to the examination of its several parts, we must pause to
consider the mode of reasoning which is the most generally to be adopted. It
depends on the axiom which has always been essentially concerned in
every improvement of natural philosophy, but which has been more and
more employed, ever since the revival of letters, under the name induction,
and which has been sufficiently discussed by modern metaphysicians.
That like causes produce like effects, or that in similar circumstances
similar consequences ensue, is the most general and most important law of
nature ; it is the foundation of all analogical reasoning, and is collected
from constant experience by an indispensable and unavoidable propensity
of the human mind.

It does not appear that we can have any other accurate conception of causa-
tion, or of the connexion of a cause with its effect, than a strong impression
of the observation, from uniform experience, that the one has constantly
followed the other. We do not know the intimate nature of the connexion
by which gravity causes a stone to fall, or how the string of a bow urges
the arrow forwards ; nor is there any original absurdity in supposing it
possible that the stone might have remained suspended in the air, or that
the bowstring might have passed through the arrow as light passes through
glass. But it is obvious that we cannot help concluding the stone's weight
to be the cause of its fall, and that every heavy body will fall unless sup-
ported ; and the pressure of the string to be the cause of the arrow's mo-
tion ; and that if we shoot, the arrow will fly ; if we hesitated to make
these conclusions, we should often pay dear for our scepticism. This ex-
planation is sufficient to show the identity of the two expressions, that like
causes produce like effects, and that in similar circumstances similar con-
sequences ensue. And such is the ground of argument from experience,
the simplest principle of reasoning after pure mathematical truths, which
appear to be so far prior to experience, as their contradiction always im-
plies an absurdity repugnant to the imagination.

12 LECTURE I.

In the application of induction the greatest caution and circumspection
are necessary ; for it is obvious that, before we can infer with certainty
the complete similarity of two events, we must be perfectly well assured
that we are acquainted with every circumstance which can have any rela-
tion to their causes. The error of some of the ancient schools consisted
principally in the want of sufficient precaution in this respect ; for although
Bacon is, with great justice, considered as the author of the most correct
method of induction, yet, according to his own statement, it was chiefly the
guarded and gradual application of the mode of argument that he laboured
to introduce. He remarks that the Aristotelians, from a hasty observa-
tion of a few concurring facts, proceeded immediately to deduce universal
principles of science and fundamental laws of nature, and then derived
from these, by their syllogisms, all the particular cases which ought to
have been made intermediate steps in the inquiry. Of such an error we
may easily find a familiar instance. We observe that, in general, heavy
bodies fall to the ground unless they are supported ; it was therefore con-
cluded that all heavy bodies tend downwards ; and since flame was most
frequently seen to rise upwards, it was inferred that flame was naturally
and absolutely light. Had sufficient precaution been employed in observ-
ing the effects of fluids on falling and on floating bodies, in examining the
relations of flame to the circumambient atmosphere, and in ascertaining
the specific gravity of the air at different temperatures, it would readily
have been discovered that the greater weight of the colder air was the
cause of the ascent of the flame, — flame being less heavy than air, but yet
having no positive tendency to ascend. And, accordingly, the Epicureans,
whose arguments, as far as they related to matter and motion, were often
more accurate than those of their cotemporaries, had corrected this error ;
for we find in the second book of Lucretius a very just explanation of the
phenomenon.

" See with what force yon river's crystal stream
Resists the weight of many a massy beam.
To sink the wood the more we vainly toil,
The higher it rebounds, with swift recoil.
Yet that the beam would of itself ascend
No man will rashly venture to contend.
Thus too the flame has weight, though highly rare,
Nor mounts but when compelled by heavier air."

It may be proper to notice here those axioms which are denominated by
Newton * rules of philosophizing ; although it must be confessed that they
render us very little immediate assistance in our investigations. The
first is, that " no more causes are to be admitted as existing in nature than
are true and sufficient for explaining the phenomena to be considered :"
the second, " therefore effects of the same kind are to be attributed, as far
as is possible, to the same causes :" thirdly, " those qualities of bodies which
cannot be increased nor diminished, and which are found in all bodies
within the reach of our experiments, are to be considered as general

* PHncipia ; Introduction to Book III.

INTRODUCTION. 13

qualities of all bodies existing :" fourthly, " in experimental philosophy,
propositions collected by induction from phenomena, are to be esteemed
either accurately or very nearly true, notwithstanding any contrary hypo-
thesis, until other phenomena occur by which they may either be corrected
or confuted."

As an illustration of the remark, that these axioms, though strictly true,
are of little real utility in assisting our investigations, I shall give an in-
stance from the subject of electricity. Supposing that we wish to determine
whether or no the electric fluid has weight ; we are to inquire whether or no
gravitation is one of those properties which are described in the third rule,
and whether that rule will authorise us to apply it to the electric fluid, as
one of those qualities of bodies which cannot be increased nor diminished,
which are found in all bodies within the reach of our experiments, and
which are, therefore, to be considered as general qualities of all bodies
existing. Now it appears to be, in the first place, uncertain whether or no
the increase and diminution of gravity, from a change of distance, is strictly
compatible with the terms of the definition ; and, in the second place, we
are equally at a loss to decide, whether or no the electric fluid can with
propriety be called a body ; for it appears in some respects to be wholly
different from tangible matter, while it has other qualities in common with
it. Such are the difficulties of laying down general laws on so comprehen-
sive a scale, that we shall find it more secure to be contented to proceed
gradually by closer inductions in particular cases. We shall, however,
seldom be much embarrassed in the choice of a mode of argumentation. The
laws of motion, which will be the first immediate subjects of discussion,
have indeed sometimes been referred to experimental evidence ; but we
shall be able to deduce them all in a satisfactory manner, by means of our
general axiom, from reasonings purely mathematical, which, wherever they
are applicable, are unquestionably preferable to the imperfect evidence of
the senses, employed in experimental investigations.*

LECTURE II.

ON MOTION.

THE whole science of mechanics depends on the laws of motion, either
actually existing or suppressed by the opposition of the forces which tend
to produce it. The nature of motion requires, therefore, to be particularly
examined at the entrance of the science of mechanical philosophy ; and
although the subject is so abstract as to demand some effort of the attention,
being seldom capable of receiving much immediate illustration from the
objects of sense, yet we shall find it indispensable to our progress in the

* Consult Stewart's Philosophy of the Human Mind, 2 vols. 1818-21, v. 2. Brown
on Cause and Effect. Whewell's Philosophy of the Inductive Sciences, 2 vols.
1840.

14 LECTURE II.

investigation of many particular problems of importance, to obtain, in the
first place, a clear conception of the properties and affections of motions of
all kinds.

One of the ancient philosophers, on being asked for a definition of motion,
is said to have walked across the room, and to have answered, you see it,

j but what it is I cannot tell you. It does not, however, appear absolutely
^necessary to appeal to the senses for the idea of motion ; for a definition is
the resolution of a complex idea into the more simple elements which com-
pose it ; and, in the present instance, these elements are, the existence of
two points at a certain distance, and, after a certain interval of time,
the existence of the same points at a different distance ; the difference of the
distances being supposed to be ascertained according to that postulate of
geometry (which has in general been tacitly understood, but which I
have expressly inserted in the geometrical part of my syllabus), requiring
that the length of a line be capable of being identified, whether by the effect
of any object on the senses, or merely in imagination.

Motion, therefore, is the change of rectilinear distance between two points.*
Allowing the accuracy of this definition, it appears that two points are
necessary to constitute motion ; that in all cases when we are inquiring
whether or no any body or point is in motion, we must recur to some

, 'other point which we can compare with it, and that if a single atom existed
alone in the universe, it could neither be said to be in motion nor at rest.
This may seem in some measure paradoxical, but it is the necessary con-
sequence of our definition, and the paradox is only owing to the difficulty
of imagining the existence of a single atom, unsurrounded by innumerable
points of a space which we represent to ourselves as immoveable.

It has been for want of a precise definition of the term motion, that
many authors have fallen into confusion with respect to absolute and rela-
tive motion. For the definition of motion, as the change of rectilinear
distance between two points, appears to be the definition of what is com-
monly called relative motion ; but, on a strict examination, we shall find,
that what we usually call absolute motion is merely relative to some space

, which we imagine to be without motion, but which is so in imagination
only. The space which -we call quiescent is in general the earth's surface ;
yet we well know, from astronomical considerations, that every point of
the earth's surface is perpetually in motion, and that in very various
directions : nor are any material objects accessible to our senses which we
can consider as absolutely motionless, or even as motionless with regard to
each other ; since the continual variation of temperature to which all bodies
are liable, and the minute agitations arising from the motions of other
bodies with which they are connected, will always tend to produce some
imperceptible change of their distances.

When, therefore, we assert that a body is absolutely at rest, we only
mean to compare it with some large space in which it is contained : for
that there exists a body absolutely at rest, in as strict a sense as an abso-
lutely straight line may be conceived to exist, we cannot positively affirm ;
and if such a quiescent body did exist, we have no criterion by which it
* See Descartes Princip. Philos. Part ii. § 25.

ON MOTION. 16

could be distinguished. Supposing a ship to move at the rate of three miles
in an hour, and a person on board to walk or to be drawn towards the
stern at the same rate, he would be relatively in motion with respect to the
ship, yet we might very properly consider him as absolutely at rest : but
he would, on a more extended view, be at rest only in relation to the
earth's surface ; for he would still be revolving round the axis of the earth,
and with the earth round the sun, and with the sun and the whole solar
system, he would be slowly moving among the starry worlds which surround
them. Now with respect to any effects within the ship, all the subsequent
relations are of no consequence, and the change of his rectilinear distance
from the various parts of the ship is all that needs to be considered in deter-
mining those effects. In the same manner, if the ship appear, by compari-
son with the water only, to be moving through it with the velocity of
three miles an hour, and the water be moving at the same time in a con-
trary direction at the same rate in consequence of a tide or current, the
ship will be at rest with respect to the shore ; but the mutual actions of the
ship and the water will be the same as if the water were actually at rest
and the ship in motion.

It is not sufficient to observe the increase or decrease of distance of a
moving point from another single point only ; we must compare its succes-
sive situations with many other points surrounding it ; and for this
purpose these points must be at rest among themselves, in order to be
considered as belonging to a quiescent space or surface ; which may be
denned as a space or surface of which all the points remain always at equal
distances from each other without any external influence. In this sense
we must call the deck of the ship a quiescent surface, whether the ship be
at anchor or under sail ; but we must not consider a surface revolving
round a centre as a quiescent surface, for it will appear hereafter that no
such motion can exist without the influence of a centripetal force ; which
renders it improper for determining the affections of a moving body.

When a point is in motion with respect to a quiescent space, it is often
simply denominated a moving point, and the right line joining any two
of its places immediately contiguous to each other is called its direction.
If it remains continually in one right line drawn in the quiescent space,
that line is always the line of its direction ; if it describes several right lines,
each line is the line of its direction as long as it continues in it ; but if its
path becomes curved, we can no longer consider it as perfectly coinciding at
any time with a right line, and we must recur to the letter of the defini-
tion, by supposing a right line to be drawn through two successive points
in which it is found, and then if these points be conceived to approach each
other without limit we shall have the line of its direction. Now such a line
is called in geometry a tangent : for it meets the curve but does not cut it,
provided that the curvature be continued. (Plate I. ,Fig. 1 — 3.)

Having formed an accurate ^idea) of the nature of motion, and of the im-
port of the terms employed in speaking of its properties, we may proceed
to consider the mechanical laws to which it is subjected, and which are de-
rivable from the essence of tKe~aennitions that have been premised. The
first is, that a moving point never quits the line of its direction without a

16 LECTURE II.

II
disturbing causej for a right line being the same with respect to all sides,

no reason can be imagined why the point should incline to one side more '
than another ; and the general law of induction requires that the moving
point should preserve the same relations towards the points similarly
situated on every side of the line. This argument appears to be sufficiently
satisfactory to give us ground for asserting that the law of motion here laid
down may be considered as independent of experimental proof. It was once
proposed as a prize question by the Academy of Sciences at Berlin, to determine
jj whether the laws of motion were necessary or accidental ; that is, whether
they were to be considered as mathematical or as physical truths. Mauper-
tuis,* then president of the Academy, wrote an elaborate dissertation, in
which he endeavoured to deduce them from a complicated principle of the
production of every effect in the manner which requires the least possible
action, a principle which he supposes to be most consistent with the wise
economy of nature. But this principle has itself been shown to be capable
/v of accommodation to any other imaginable laws of motion, and the intricacy
of the theory tends only to envelope the subject in unnecessary obscurity ;
I the laws of motion appear to be easily demonstrable from the simplest
mathematical truths, granting only the homogeneity or similarity of matter
with respect to motion, and allowing the general axiom that like causes pro-
duce like effects. If, however, any person thinks differently, he is at liberty
to call these laws experimental axioms collected from a comparison of
various phenomena ; for we cannot easily reduce them to direct experi-
ments, since we can never remove from our experiments the action of all
disturbing causes; for either gravitation, or the contact of surrounding
^bodies, will interfere with all the motions which we can examine.

Having established the rectilinear direction of undisturbed motion, we
come to consider its uniformity. Here the idea of time enters into our sub-
ject. To define time in general is neither easy nor necessary ; but we must
have some measure of equal times. Our abstract idea of time depends on
the memory of past sensations ; but is obvious that the results of an intel-
lectual measure of the duration of time would be liable to the greatest
uncertainty. We may observe that on a journey the perpetual succession
of various objects will often make a week appear, upon retrospection, as
long as a month spent in a continuation of such employments as are uni-
form without being laborious ; the multitude of new impressions not only
serving to increase the apparent magnitude of the interval, by filling up its
vacuities, but tending also to dimmish the vivacity of the ideas which they
have superseded, and to give them the character of the fainter recollections
of an earlier date. We are therefore obliged to estimate the lapse of time
by the changes in external objects ; of these changes the simplest and most
convenient is the apparent motion of the sun, or rather of the stars, derived

* Hist, et Mem. de 1'Acad. de Berl. 1746, jp. 267] Collected Works of Maup.,
4 vols. Lyons, 1756, vol. iv; p. 31. Compare Leibnitz's Leipsic Acts, 1682.
D'Arcy on Maupertuis's Minimum of Action. Hist, et Mem. de Paris, 1749, p. 531.
H. 179, 1752, p. 503. Euler on the General Principles of Motion and Rest. Hist,
et Mem. de 1'Acad. de Berl. 1751, pp. 169, 199. Bertrand on the least Action, ib.
1753, p. 310. Malvezio on the Principle of Maupertuis, Com. Bon. VI. Opuscula,
p. 315. Euler, Dissertatio de Principio Min. Act. Berl. 1753.

ON MOTION. 17

from the actual rotation of the earth on its axis, which is not, indeed, an
undisturbed rectilinear motion, but which is equally applicable to every
practical purpose. Hence we obtain, by astronomical observations, the
well-known measures of the duration of time implied by the terms day,
hour, minute, and second.

Now the equality of times being thus estimated from any one motion, all
other bodies moving without disturbance will describe equal successive parts
of their lines of direction in equal times. And this is the second law of
motion, which, with the former law, constitutes Newton's first axiom
or law of motion :* " that every body perseveres in its state of rest or
uniform rectilinear motion, except so far as it is compelled by some force to
change it." It appears that this second law is strictly deducible from the
axioms and definitions which have been premised, and principally from the
consideration of the relative nature of motion, and the total deficiency of a
, criterion of absolute motion. For, since the velocity of a body moving
without resistance or disturbance is only a relation to another body, if the
second body has no mechanical connexion with the first, its state with
respect to motion can have no effect on the velocity of the first body, how-
ever great its comparative magnitude may be : and if a body is at rest, there
is nothing to determine it to begin to move either to the right hand or to
the left ; if it is at rest with respect to any other bodies it will remain in
the same condition, whatever the relative motions of those bodies may be
when compared with the surrounding objects ; and these relations can only
be preserved by its continuance in uniform rectilinear motion. This law is
also confirmed by its perfect agreement with all experimental observations,
although it is too simple to admit of an immediate proof. For we can
never place any body in such circumstances as to be totally exempt from
the operation of all accelerating or retarding causes ; and the deductions
from such experiments as we can make, would require in general, for the
accurate determination of the necessary corrections, a previous knowledge
of the law which we wish to demonstrate.

When, indeed, we consider the motion of a projectile, we have only to
allow for the disturbing force of gravitation, which so modifies the effect,
that the body deviates from a right line, but remains in the same vertical
plane ; whence we may infer that, in the absence of the force of gravita-
tion, the body would continue to move in every other plane in which its
motion began, as well as in the vertical plane ; since in that case all these
planes would be indifferent to it ; it must, therefore, remain in their com-
mon intersection, which could only be a straight line ; so that by thus
combining arguments with observation, we may obtain a confirmation of
the law of the rectilinear direction of undisturbed motion, partly founded on
direct experiment. Its uniformity is, however, still less subjected to im-
mediate examination : yet, from a consideration of ttye nature of friction
and resistance, combined with the laws of gravitation, we may ultimately
show the perfect coincidence of the theory with experiment. The ten-
dency of matter to persevere in this manner in the state of rest or of
uniform rectilinear motion is called its inertia.

* Principia, lib. i.
c

18 LECTURE II.

In all these cases it is of importance to attend to the composition
of motion, or the joint effect of more than one motion existing at the
same time. The existence of two or more motions at the same time,
in the same body, is not at first comprehended without some difficulty.
It is in fact only a combination or separation of relations that is con-
sidered; in the same manner as by combining the relation of son to
father, and brother to brother, we obtain the relation of nephew to uncle,
so, by combining the motion of a man walking in a ship, with the motion of
the ship, we determine the relative velocity of the man with respect to the
earth's surface. It is, however, necessary, for ascertaining these relations,
to consider the affections of a space or surface in motion, and to examine
how it may move in the most simple manner with respect to another space.

If any number of points move in parallel lines, describing equal spaces
in equal times, they are at rest with respect to each other ; for it may
easily be demonstrated that the rectilinear distance of each, from each of
the rest, remains unchanged ; and if all the points of a plane move in this
manner on another plane, either plane may be said to be in rectilinear
motion with respect to the other. This is easily exemplified by causing
one plane to move on another, so that two or more of its points shall
always remain in a given right line in the second plane : as when a [car-
penter's] square is made to slide along the straight edge of a board, the
surface of the square is in rectilinear motion with respect to the board.
(Plate I. Fig. 4.)

If, besides this general motion of the plane, any point be supposed to
have a particular motion in it, the point will have two motions with re-
spect to the other plane : the one in common with its plane, and the other
peculiar to itself ; and the joint effect of these motions, with respect to the
second plane, is called the result of the two motions. Thus, when a car-
riage moves on a perfectly [straight and] level road, all its points describe
parallel lines, and it is in rectilinear motion with respect to the road : its
wheels partake of this motion, but have also a rotatory motion of their
own ; and the result of the two motions of each point of the wheels is the
cycloid, or trochoid, that it describes in a quiescent vertical plane. (Plate
I. Fig. 5.)

When an arm is made to slide upon a bar, and a thread, fixed to
the bar, is made to pass over a pulley at the end of the arm next the
bar, to a slider which is moveable along the arm, the slider moves on
the arm with the same velocity as the arm on the bar ; but if the thread,
instead of being fixed to the slider, be passed again over a pulley which is
attached to it, and then brought back to be fixed to the arm, the motion of
the slider will be only half that of the arm ; and this will be true in what-
ever position the arm be fixed. Here we have two motions in the slider,
one in common with the arm, and the other peculiar to itself, which may
be either equal or unequal to the first ; and by tracing a line on a fixed
plane, with a point attached to the slider, we may easily examine the joint
result of both the motions. (Plate I. Fig. 6.)

The joint result of any two motions is the diagonal of the parallelogram
of which the sides would be described, in the same time, by the separate mo-

ON MOTION. .19

tions, that is, if we have two lines, representing the directions and velocities
of the separate motions, and from the remoter extremity of each draw a line
parallel to the other, the intersection of these lines will be the place of the
moving body at the end of the given time. This is the necessary consequence
of the co-existence of two motions in the sense that has been denned ; it is
also capable of a complete illustration by means of the apparatus that has
been described. (Plate I. Fig. 7.)

Any given motion may be considered as the result of any two or more
motions capable of composing it in this manner. Thus the line described
by the tracing point of our apparatus will be precisely the same, whether
it be simply drawn along in the given direction, or made to move on the
arm with a velocity equal to that of the arm, or when the arm is in a
different position, with only half that velocity. (Plate I. Fig. 8.)

This principle constitutes the important doctrine of the resolution of
motion. There is some difficulty in imagining a slower motion to contain, \
as it were, within itself, two more rapid motions opposing each other : but ;
in fact we have only to suppose ourselves adding or subtracting mathe-
matical quantities, and we must relinquish the prejudice, derived from our
own .feelings, which associates the idea of effort with that of motion.
When we conceive a state of rest as the result of equal and contrary
motions, we use the same mode of representation as when we say that a
cipher is the sum of two equal quantities with opposite signs ; for instance,
plus ten and minus ten make nothing.

[ The law of motion here established differs but little in its enunciation I
1 from the original words of Aristotle, in his mechanical problems.* He '
says, that if a moving body has two motions, bearing a constant propor-
tion to each other, it must necessarily describe the diameter of a parallelo-
gram, of which the sides are in the ratio of the two motions. It is obvious
that this proposition includes the consideration, not only of uniform
motions, but also of motions which are similarly accelerated or retarded :
and we should scarcely have expected that, from the time at which the
subject began to be so clearly understood, two thousand years would
have elapsed before this law began to be applied to the determination of
the velocity of bodies actuated by deflecting forces, which Newton has so
simply and elegantly deduced from it.

In the laws of motion, which are the chief foundation of the Principia,
their great author introduces at once the consideration of forces ; and the
first corollary stands thus : " a body describes the diagonal of a parallelo-
gram by two forces acting conjointly, in the same time in which it would
describe its sides by the same forces acting separately." It appears, how-
ever, to be more natural and perspicuous to defer the consideration of force
until the simpler doctrine of motion has been separately examined.

We may easily proceed to the composition of any number of different
motions by combining them successively in pairs. Hence any equable
motions, represented by the sides of a polygon, that is, of a figure consist-
ing of any number of straight sides, being supposed to take place in the
same moveable body in directions parallel to those sides, and in the order
* Mech. Prob. c. 24. See also Galileo, Dial. 4, Prop. 2.
c2

20 LECTURE II.

of going round the figure, destroy each other, and the body remains at
rest. We may understand the truth of this proposition by imagining each
motion to take place in succession for an equal small interval of time ;
then the point would describe a small polygon similar to the original one, and
would be found, at the end of every such interval, in its original situation.
When the motions to be combined are numerous and diversified, it is
often convenient to resolve each motion into three parts, reduced to the
directions of three given lines perpendicular to each other. It is easy to
find in this manner, by addition and subtraction only, the general result of
any number of motions. We may describe the flight of a bird ascending
in an oblique direction, by estimating its progress northwards or south-
wards, eastwards or westwards, and at the same time upwards, and we may
thus determine its place as accurately as by ascertaining the immediate
bearing and angular elevation of its path, and its velocity in the direction
of its motion.

LECT. II.— ADDITIONAL AUTHORITIES.

Wjallis's Ph. Tr. iii. 864. Mechanica, 4to, Lon. 1670. Opera, 3 vols. fol. Oxf. 1713,
i. 571. Varignon, Nouvelle Mecanique, 2 vols. 4to, Paris, 1725 (Posthumous). Hist, et
Mem. de 1'Acad. de Paris, 1714, p. 280, H. 87 ; 1733, x. 301. Roberval, ibid. vi.
1,68. Joh. Bernoulli, Opera, 4 vols. 4to, Lausanne, 1 742, iii. 1. Hermann, Phoro-
nomia, 4to, Amst. 1716. Courtivron's Researches, Hist, et Mem. de Paris, 1748,
p. 304 ; 1749, p. 15. Kraftii Mechanik, 2 vols. 4to, Soroe, 1762-4 ; also, 4to, Bat.
1772, and Dresd. 1787. See also LECT. XIX.

Elementary Treatises on Mechanics. — Qflohaulti, Physica Clarkii, 2 vols. Lond.
1799. Ferguson's Mechanics, 1799. Bossut, Traite de Mec. Paris, 1800. Eytelwein
ETaridbuch der Mechanik, Berl. 1801. Carnot, Principes Fondamentaux, Par.
1803. Bezout, Cours deMath. Gregory's Mechanics, 2 vols. plates, 1806. LePriol,
Introduction, Strasburg, 1806. Foucoeur, Mec. 1800. Gamier, Lefons, 1811. Emer-
son's Mech. Venturoli, Element! di Meccanica, 2 vols. Milan, 1817 ; translation by
Creswell, Camb. 1822. Vega Vorlesungen iiber die Mathematik, 4 vols. Wien.
1818-19. Farrar, Mech. Camb. U.S. 1825. Boucharlat, Mec. 1827. Leslie's Ele-
ments of Natural Philosophy, Edin. 1829. Biot, Notions Elementaires de Statique,
1829. Hachette (translation of Young} Resume Complet de Mecanique, 32, Par.
1829. Dandelin, Cours de Statique, Liege, 1830. Prichard's Theory of Statical
Couples, Camb. 1831. Renwick's Mechanics, New York, 1832. Poinsot, Elem. de
Statique, Monge, TraitS Elem. de Statique, 1834. Together with treatises by the
following authors, most of which have passed through several editions, and are well
known : —

Bridge, Wood, Whewell — Mechanical Euclid, Statics, Dynamics, fyc. Earn-
shaw's Statics, Dynamics. Walker, Young (J. R.), Lardner — (Library of Useful
Knowledge}. Lardner and Kater (Cabinet Cyclopaedia), Eland's Mechanical Pro-
blems. Walton's Do. Moseley's Illustrations of Mechanics, and Mechanics applied
to the Arts.

Treatises which embrace a wider range — Laplace, Traite de Mecanique Celeste,
5 vols. 4to, Paris, 1799-1825. Bowditch's Translation of Laplace's Celestial Mecha-
nics, with a Commentary, 3 vols. 4to, Boston, U.S. 1829-34./\Lagrange, Mecanique
Analytique, 2 vols. 4to, Paris, 1815. Prony, Legons de Mecanique analytique, 2 vols.
4to, Paris, 1815. Poisson, Traite de Mecanique, 2 vols. Paris, 1833. Harte's Trans-
lation of Poisson's Mechanics, 2 vols. Lond. 1843. Pontecoulant, Theorie Analytique
du Systeme du Monde, 3 vols. Paris, 1829-34. Somerville's (Mrs.) Mechanism of the
Heavens, Lond. 1831. Pratt, The Mathematical Principles of Mechanical Philoso-
phy, Camb. 1836.

LECTURE III.

ON ACCELERATING FORCES.

WE have hitherto only considered motion as already existing, without
any regard to its origin or alteration ; we have seen that all undisturbed
motions are equable and rectilinear ; and that two motions represented by
the sides of a parallelogram, cause a body to describe its diagonal by their
joint effect. We are now to examine the causes which produce or destroy
motion. Any cause of a change of the motion of a body with respect to a
quiescent space, is called a force ; that is, any cause which produces
motion in a body at rest, or which increases, diminishes, or modifies it in a
body which was before in motion. Thus the power of gravitation, which
causes a stone to fall to the ground, is called a force ; but when the stone,
after descending down a hill, rolls along a horizontal plane, it is no longer
impelled by any force, and its relative motion continues unaltered, until it is
gradually destroyed by the retarding force of friction. Its perseverance
in the state of motion or rest in consequence of the inertia of matter, has
sometimes been expressed by the term vis inertiae, or force of inertia ; but it
appears to be somewhat inaccurate to apply the term force to a property
which is never the cause of a change of motion in the body to which it
belongs.

It is a necessary condition, in the definition of force, that it be the cause
of a change of motior<with respect to) a quiescent space. For if the change
were only in the relative motion of two points, it might happen without
the operation of any force : thus, if a body be moving without disturbance,
its motion with respect to another body, not in the line of its direction,
will be perpetually changed : and this change, considered alone, would
[appear to] indicate the existence of a repulsive force ; and, on the other
hand, two bodies may be subjected to the action of an attractive force,
while their distance remains unaltered, in consequence of the centrifugal
effect of a rotatory motion. (Plate I. Fig. 9.)

The exertion of an animal, the unbending of a bow, and the communi-
cation of motion by impulse, are familiar instances of the actions of forces.
We must not imagine that the idea of force is naturally connected with
that of labour or difficulty ; this association is only derived from habit,
since our voluntary actions are in general attended with a certain effort,
which leaves an impression almost inseparable from that of the force that
it calls into action.

It is natural to inquire in what immediate, manner>any force acts, so as
to produce motion ; for instance, by what means the earth causes a stone
to gravitate towards it. In some cases, indeed, we are disposed to imagine
that we understand better [tolerably well] the nature of the action of a
force, as, when a body in motion strikes another, we conceive that the
impenetrability of matter is a sufficient cause for the communication of
motion, since the first body cannot continue its course without displacing

22 LECTURE III.

the second ; and it has been supposed that if we could discover any similar
impulse that might be the cause of gravitation, we should have a perfect
idea of its operation. But the fact is, that even in cases of apparent
jj impulse, the bodies impelling each other are not actually in contact j| and
if any analogy between gravitation and impulse be ever established, it will
not be by referring them both to the impenetrability of matter, but to the
intervention of some common agent, perhaps imponderable. It was
observed by Newton,* that a considerable force was necessary to bring two
pieces of glass into a degree of contact, which still was not quite perfect ;
and Profesor Robison t has estimated this force at a thousand pounds for
every square inch. These extremely minute intervals have been ascer-
tained by observations on the colours of the thin plate of air included
between the glasses ; and when an image of these colours is exhibited by
means of the solar microscope, it is very easily shown that the glasses are
separated from each other, by the operation of this repulsive force,
as soon as the pressure of the screws which confine them is diminished ;
the rings of colours dependent on their distance contracting their dimen-
sions accordingly. Hence it is obvious, that whenever two pieces of glass
strike each other, without exerting a pressure equal to a thousand pounds
on a square inch, they may affect each other's motion without actually
coming into contact. Some persons might perhaps be disposed to attribute
this repulsion to the elasticity of particles of air adhering to the glass, but
I have found that the experiment succeeds equally well in the vacuum of
the air pump. We must, therefore, be contented to acknowledge our total
ignorance of the intimate nature of forces of every kind ; and we are

/ first to examine the effect of forces, considering only their magnitude and

L direction, without any regard to their origin.

It was truly asserted by Descartes,^ that the state of motion is equally
natural with that of rest. When a body is once in motion, it requires no
foreign power to sustain its velocity. If, therefore, a moving body is sub-
jected to the influence of any force, which acts upon it in the line of its
direction, its motion will be either accelerated or retarded, accordingly as
the direction of the force coincides with that of the motion, or is opposed
to it. A stone, for instance, beginning to fall, or projected downwards,
by no means retains the same velocity throughout its descent, but acquires
more and more motion every instant. We well know that the greater
the height from which a body falls, the more danger there is of its destroy-
ing whatever opposes its progress. In the same manner, when a ball is
thrown upwards, it gradually loses its motion by the operation of gravita-
tion, which is now a retarding force, and at last begins again to descend.

It may here be proper to inquire what is the precise meaning of the
term velocity ; we appear indeed to understand sufficiently the common use
of the word, but it is not easy to give a correct definition of it. The velocity
of a body may be said to be the quantity or degree of its motion, indepen-
dently of any consideration of its mass or magnitude ; and it might
always be measured by the space described in a certain portion of time, for

* Optics, Book II. See also Huygens, Ph. Tr. No. 86.
f Robison's Mechanical Philosophy, Brewster's Ed., i. 250.
J Principia Philos. Part ii. § 26.

ON ACCELERATING FORCES. 23

instance a second, if there were no other motions than undisturbed or
uniform motions ; but the velocity may vary very considerably within the
second, and we must therefore have some other measure of it than the
space actually described in any finite interval of time. If, however, the
times be supposed infinitely short, the elements of space described may be
considered as the 'true measures of velocities. These elements, although
smaller than any assignable quantity, may yet be accurately compared with
each other ; and the reason that they afford a true criterion of the velo-
city is this, that the change produced in the velocity during so short an
interval of time, must be absolutely inconsiderable, in comparison with the
whole velocity, and the element of space becomes a true measure of the
temporary velocity, in the same manner as any larger portion of space may
be the measure of a uniform velocity.

When the increase or diminution of the velocity of a moving body is
uniform, its cause is called a uniform force ; the spaces which would be
described in any given time with the actual velocity uniformly continued,
being always equally increased or diminished by the action of such a force.
For example, if the velocities at the beginning of any two separate
seconds be such that the body would describe one foot and ten feet in the
respective seconds, if undisturbed, and the spaces actually described
become two feet and eleven feet, each being increased one foot, the accele-
rating force must be denominated uniform.

The power of gravitation, acting at or near the earth's surface, may,
without sensible error, be considered as such a force. Thus, if a body
begins to fall from a state of rest, it describes about 16 feet, or more cor-
rectly 16-jir in the first second ; if it begins a second with a velocity of 32
feet, it describes 32 and 16, or 48 feet in this second. The decrease of the
force of gravitation, in proportion to the squares of the distances from the
earth's centre, is barely perceptible, at any heights within our reach, by
the nicest tests that we can employ.

The velocity produced by any uniformly accelerating force, is propor-
tional to the magnitude of the force and the time of its operation con-
jointly.* When the forces are the same, a little consideration will convince
us that, since every equal portion of time adds equally to the velocity, the
whole velocity produced or destroyed must be in proportion to the whole
time ; and when the forces differ, the velocities differ in the same ratio ;
for the forces are only measured by the velocities which they generate.
Thus a double force, in a double time, produces a quadruple velocity.
That a force producing a double velocity is properly called a double force,
may be shown from the laws of the composition of motion ; for when the
equal sides of a parallelogram representing two separate forces or motions,
approach to each other, and at last coincide in direction, the diagonal of
the parallelogram, representing their joint effect, becomes equal to the sum
of the sides. (Plate I. Fig. 10.) t
, The machine invented by Mr. AtwoodJ (Plate I. Fig. 11) furnishes us

f*S^

* Galileo, Dialogues on Motion, Dial. III. Def.

f Halley, Ph. Tr.xvi. 0 (1686).

t On the Rectilinear Motion, and the Rotation of Bodies, Camb. 1804, p. 291.

24 LECTURE III.

with a very convenient mode of making experiments on accelerating forces.
The velocity produced by the undiminished force of gravity, is much too
great to be conveniently submitted to experimental examination ; but by
means of this apparatus we can diminish it in any degree -that is
required. Two boxes, which are attached to a thread passing over a
pulley, may be filled with different weights, which counterbalance each
other and constitute, together with the pulley, an inert mass, which is put
into motion by a small weight added to one of them. The time of descent
is measured by a second or half second pendulum, the space described
being ascertained by the place of a moveable stage, against which the bot-
tom of the descending box strikes ; and when we wish to determine im-
mediately the velocity acquired at any point, by measuring the space
uniformly described in a given time, the accelerating force is removed by
means of a ring which intercepts the preponderating weight, and the box
proceeds with a uniform velocity, except so far as the friction of the
machine retards it. By changing the proportion of the preponderating
weight to the whole weight of the boxes, it is obvious that we may change
the velocity of the descent, and thus exhibit the effects of forces of different
magnitudes. The most convenient mode of letting the weights go, with-
out danger of disturbance from their vibrations, is to hold the lowest
weight only, and to allow it to ascend at the instant of a beat of the pen-
dulum.

That the velocity generated is proportional to the time of the action of
the force, or that the force of gravitation, thus modified, is properly called
a uniform accelerating force, may be shown by placing the moveable ring
so as to intercept the same bar successively at two different points : thus the
space uniformly described in a second, by the box alone, is twice as great,
when the force is withdrawn after a descent of ten half seconds, as it is after
a descent of five. And if we chose to vary the weight of the bar, we might
show, in a similar manner, that the velocity generated in a given time is
proportional to the force employed.

We are next to determine the magnitude of the whole space described in
a given time with a velocity thus uniformly increasing. The law discovered
by Galileo,* that the space described is as the square of the time of descent,
and that it is also equal to half the space which would be described in the
same time with the final velocity,f is one of the most useful and interesting
propositions in the whole science of mechanics. Its truth is easily shown
from mathematical considerations, by comparing the time with the base,
and the velocity with the perpendicular of a triangle gradually increasing,
of which the area will represent the space described ; and we may observe,
by means of Atwood's machine, that a quadruple space is always described
in a double time, whatever may be the magnitude of the force. Of course,
if the forces vary, the spaces are as the forces and as the squares of the
times conjointly. (Plate I. Fig. 12.)

It may also be demonstrated, that if a body falls through one foot in a
second by means of a certain force, it will require a quadruple force to
make it fall through the same space in half a second ; and in general, where
* Dial. III. Prop. 2. f Ibid. Prop. 1.

ON ACCELERATING FORCES. 25

the spaces are equal, the forces are as the squares of the velocities. Wher-
ever the space and the force remain the same, whether the force be uniform
or not, the squares of any two velocities with which different bodies enter
the space, will receive equal additions while they pass through it.

When a force acts in a direction contrary to that of the moving body, we
may readily determine the retardation that it produces, by comparing the
motion with that of a body accelerated by the same force. For the degrees
by which an ascending body loses its motion, are the same as those by
which it is again accelerated at the same points when it has acquired its
greatest height and again descends. We may thus calculate to what height
a body will rise when projected upwards with a given velocity, and retarded
by the force of gravitation. Since the force of gravitation produces or de-
stroys a velocity of 32 feet in every second, a velocity of 320 feet, for
instance, will be destroyed in 10 seconds ; and according to what has been
premised, a body will fall in 10 seconds through a hundred times 16 feet,
or 1600 feet, which is therefore the height to which a velocity of 320 feet
in a second will carry a body moving without resistance in a vertical
direction. We may also obtain the same result by squaring one eighth of
the velocity : thus one eighth of 320 is 40, of which the square is 1600,
the height corresponding to the given velocity ; and this velocity is some-
times called the velocity due to the height.

LECT. III.— ADDITIONAL AUTHORITIES.

Galileo, Discorsi e Dimostrazioni Matematiche intorno a due Nuove Scienze,!
Leyd. 1638. Riccioli, Almagestum Novum, fol. 1641, ii. c. 21. Mersenni Cogi-
tata Physico-Mathematica, fol. Paris, 1644. Toricellius de Motu Gravium, 4to, Flor.
1644. Hooke on Falling Bodies, Birch, i. 195. Borellus de Motionibus a Gravi-
tate dependentibus, 4to, Reg. Jul. 1670. Halley on Gravity, Ph. Tr. 1686, xvi. p. 3.
Mariotte on the Fall of Heavy Bodies, Hist, et Mem. de FAcad. i. 249. Varignon,
ibid. ii. 96. See also x. 231, 242, and an. 1709, pp. 69, 267, H. 97 ; an. 1719,
p. 195, H. 77; an. 1720, p. 107, H. 97. Camus, an. 1726, p. 159, H. 73. Riccati
on the Effects of Attraction, Comm. Bon. ii. III. 143, 6 O. 138. Euler, Me-
chanics, 1736, Hist, et Mem. Berlin, 1748, p. 184. Nov. Comm. Petrop. ii. 144.
Cotes de Descensu Gravium, 4to, Camb. 1770.

On the Laws of Motion. — Hooke's Posthumous Works, p. 355. Euler, Mec. i. 8.
D'Alembert, Encyc. au mot Force. Laplace, Mecanique Celeste, liv. i. c. 2, § 5.
Robison, Mech. Ph. i. 121. Playfair's Outlines, 2 vols. Edin. 1816, vol.i. Home
and Stewart's Lit. and Phil. Essays, i. Whewell, Edin. Journ. of Science, No. 15.
Trans, of the Cambridge Philos. Soc. V. Hist, of the Inductive Sciences, ii. Me-
chanics, Camb. 1828, 3rd ed. Poisson, Mecanique, i. 278. Powell, Nature and
Evidence of the Primary Laws of Motion, Oxf. 1837.

26
LECTURE IV.

ON DEFLECTIVE FORCES.

IT has been shown that the velocity generated by an accelerating force,
is proportional to the time of its action, and the space described to the
square of the time. We are next to consider the more complicated cases of
the action of such forces. When they are directed to a certain point out of
the line of the motion which they affect, they become central forces, of
which we have an example in the force of gravitation, considered as it
governs the planetary motions ; and when this point becomes extremely
distant in comparison with the length of the body's path, so that the force
acts very nearly in parallel lines, the body comes under the denomination
of a projectile, as for instance a cannon ball projected horizontally or
obliquely.

An accelerating force, therefore, tending to a point out of the line of
direction of a moving body,* deflects it from that line, and is then usually
called a centripetal force. And the natural tendency of the body to perse-
vere in its rectilinear motion, unless opposed by such a force, is sometimes
called a centrifugal force. How far the term force is properly applicable
to the perseverance of a body in its rectilinear motion, may perhaps be
liable to dispute. If we allow the propriety of the appellation, we must
. extend the definition of the term force to any change of the relative motion
of two points, and we must also allow the inertia of a body to be justly deno-
*; minated a force. The fact, however, is certain, that all bodies revolving
round a centre, have a tendency to recede from the centre in the direction
of the tangent, and when this force is counterbalanced an equal centrifugal
force must be exerted.

The effects of a centrifugal force may be observed in the familiar instance
of a stone placed in a sling, which may be made to revolve in a vertical
direction, and even at the upper part of its orbit, may adhere, as it were,
notwithstanding its weight, to the sling which is above it, in consequence of
the excess of the centrifugal force above the force of gravitation.

It is also a centrifugal force that is the foundation of the amusement of a
boy driving a hoop. A hoop at rest, placed on its edge, would very quickly
fall to the ground ; but when it is moving forwards, a slight inclination
towards either side causes the parts to acquire a motion towards that side,
those which are uppermost being most affected by it ; and this lateral
motion, assisted sometimes by the curvature of the surface of the hoop,
causes its path to deviate from a rectilinear direction ; so that instead of
moving straight forwards it turns to that side towards which it began to
incline ; and in this position its tendency to fall still further is counter-
acted by the centrifugal force, and it generally makes several complete
revolutions before it falls. The motion of a bowl, with its bias, is of a
similar nature ; the centrifugal force counteracting the tendency to curvi-
* Galileo, Dial. II. p. 147.

ON DEFLECTIVE FORCES. 27

linear motion, so as to diminish it very considerably, until the velocity is
so much reduced as to suffer it to describe a path evidently curved, and
becoming more and more so as the motion is slower.

When a body is retained in a circular orbit by a force directed to its
centre, its velocity is every where equal to that which it would acquire in
falling, by means of the same force, if uniform, through half the radius,
that is, through one fourth of the diameter.* This proposition affords a
very convenient method of comparing the effects of central forces with
those of simple accelerating forces, and deserves to be retained in memory.
We may in some measure demonstrate its truth by means of the whirling
table : an apparatus which is arranged on purpose for exhibiting the pro-
perties of central forces, although it is more calculated for showing their
comparative than their absolute magnitude ; for accordingly as we place
the string on the pullies, the two horizontal arms may be made to revolve
either with equal velocities, or one twice as fast as the other. The sliding
stages, which may be placed at different distances from the centres, and
which are made to move along the arms with as little friction as possible,
are in a certain proportion to the weights, which are to be raised by means
of threads passing over pullies in the centres, as soon as the centrifugal
forces of the stages with their weights are sufficiently great; and the
experiment is to be so arranged, that when the velocity having been gradu-
ally increased, produces a sufficient centrifugal force, both stages may raise
their weights, and fly off at the same instant. But, for the present purpose,
one of the stages only is required, and the time of revolution may be mea-
sured by a half second pendulum. We may make the force, or the weight
to be raised, equal to the weight of the revolving body, and we shall find
that this body will fly off when its velocity becomes equal to that which
would be acquired by any heavy body in falling through a height equal to
half the distance from the centre, and as much greater as is sufficient for
overcoming the friction of the machine. (Plate I. Fig. 13.)

From this proposition we may easily calculate the velocity with which a
sling of a given length must revolve, in order to retain a stone in its place
in all positions ; supposing the motion to be in a vertical plane, it is obvious
that the stone will have a tendency to fall when it is at the uppermost point
of the orbit, unless the centrifugal force be at least equal to the force of
gravity. Thus, if the length of the sling be two feet, we must find the
velocity acquired by a heavy body in falling through a height of one foot,
which will be eight feet in a second, since eight times the square root of 1
is eight ; and this must be its velocity at the highest point ; with this
velocity it would perform each revolution in about a second and a half, but
its motion in other parts of its orbit will be greatly accelerated by the
gravitation of the stone.

It may also be demonstrated, that when bodies revolve in equal circles,

their centrifugal forces are proportional to the squares of their velocities.t

, Thus, in the whirling table, the two stages being equally loaded, one of

them, which is made to revolve with twice the velocity of the other, will

lift four equal weights at the same instant that the other raises a single one.

* Newton, Principia, I. Prop. 4, Cor. 9. f Ibid. I. Prop. 4, Cor. 1.

28 LECTURE IV.

But when two bodies revolve with equal velocities at different distances,
the forces are inversely as the distances ; consequently the forces are, in all
cases, directly as the squares of the velocities, and inversely as the dis-
tances.

If two bodies revolve in equal times at different distances, the forces by
which they are retained in their orbits are simply as the distances. If one
of the stages of the whirling table be placed at twice the distance of the
other, it will raise twice as great a weight when the revolutions are per-
formed in the same time.

In general, the forces are as the distances directly, and as the squares of
the times of revolution inversely.* Thus the same weight revolving in a
double time, at the same distance, will have its effect reduced to one fourth,
but at a double distance the effect will again be increased to half of its
original magnitude.

From these principles we may deduce the law which was discovered by
Keplerf in the motions of the planetary bodies, but which was first demon-
strated by NewtonJ from mechanical considerations. Where the forces
vary inversely as the squares of the distances, as in the case of gravitation,
the squares of the times of revolution are proportional to the cubes of the
distances. Thus if the distance of one body be four times as great as that
of another, the cube of 4 being 64, which is the square of 8, the jtime of its
revolution will be 8 times as great as that of the first body. It would be
easy to show the truth of this proposition experimentally by means of the
whirling table, but the proof would be less striking than those of the
simpler laws which have already been laid down.

Hitherto we have supposed the orbit of a revolving body to be a perfect
circle ; but it often happens in nature, as for instance in all the planetary
motions, that the orbit deviates more or less from a circular form ; and in
such cases we may apply another very important law which was also
discovered by Kepler ;§ that the right line joining a revolving body and its
centre of attraction, always describes equal areas in equal times, and the
velocity of the body is therefore always inversely as the perpendicular
drawn from the centre to the tangent. (Plate I. Fig. 14.)

The demonstration of this law invented by Newton, || was one of the most
elegant applications of the geometry of infinites or indivisibles ; a branch of
mathematics of which Archimedes laid the foundations, which Cavalleri**
and Wallistt greatly advanced, and which Newton^ brought near to per-
fection. Its truth may be in some measure shown by an experiment on the
revolution of a ball suspended by a long thread, and drawn towards a point
immediately under the point of suspension by another thread, which may

* Principia, I. Prop. 4, Cor. 2. f Harmonice Mundi, lib. v. cap. 3, § 8.

J Principia, I. Prop. 4, Cor. 6; and Prop. 15.

§ On the Motion of Mars, 1609, p. 194. || Principia, I. 1.

** Exercitationes Geometricse, Bonon. 1647.

ft Arithmetica Innnitorum, Op. fol. Ox. 1699, v. i. p. 365.

JJ Fluxions, Trans, by Colson, 4to, 1736, Ph. Trans. No. 432. Consult also
Taylor, Methodus Incrementorum, 4to, 1715. Maclaurin's Fluxions, 2 vols. 4 to,
Lond. 1742. Euler, Calculus Dif. et int. 4 vols. 4to, Pet. 1792. Lacroix, Traitc
du Calcul Dif. 3 vols. 4to, Paris. Lagrange, Calcul des Fonctions.

ON DEFLECTIVE FORCES. 29

either be held in the hand, or have a weight attached to it. The ball being
made to revolve, its motion becomes evidently more rapia when it is drawn
by the horizontal thread nearer to the fixed point, and slower when it is
suffered to fly off to a greater distance. (Plate II. Fig. 15.)

It was also discovered by Kepler* that each of the planets revolves in an
ellipsis, of which the sun occupies one of the foci. It is well known that an
ellipsis is an oval figure, which may be described by fixing the ends of a
thread to two points, and moving a tracing point so that it may always be
at the point of the angle formed by the thread ; and that the two fixed
points are called its foci. The inference respecting the force by which a
body may be made to revolve in an ellipsis, was first made by Newton ;t
that is, that the force directed to its focus must be inversely as the square
of the distance. We have no other experimental proof of this theorem than
astronomical observations, which are indeed perfectly decisive, but do not
i require to be here anticipated. (Plate II. Fig. 16.)

There is another general proposition which is sometimes of use in the
comparison of rectilinear and curvilinear motions. Two bodies being at-
tracted in the same manner towards a given centre, that is, with equal
forces at equal distances, if their velocities be once equal at equal distances,
they will remain always equal at equal distances whatever be their direc-
tions. J For instance, if one cannon ball be shot obliquely upwards, and
another perpendicularly upwards with the same velocity, the one will
describe a curve, and the other a straight line, but their velocities will
always remain equal, not at the same instants of time, but at equal distances
from the earth's centre, or after having ascended through equal vertical
heights, although in different directions. This proposition has usually
been made a step in the demonstration of the law of the force by which a
body is made to revolve in an ellipsis, but there is a much simpler method
of demonstrating that law, by means of some properties of the curvature of
the ellipsis.

In treating of the motion of projectiles, the force of gravitation may,
without sensible error, be considered as an equable force, acting in parallel
lines perpendicular to the horizon. In reality, if we ascend a mile from
the earth's surface, the actual weight of a body is diminished about a two
thousandth part, or three grains and a half for every pound ; and we may
discover this inequality by means of the vibrations of pendulums, which
become a little slower when they are placed on the summits of very high
mountains. On the other hand, a body not specifically heavier than water
gains more in apparent weight on account of the diminished density of the
atmosphere at great elevations, than it loses by the increase of its distance
from the earth. But both these differences may, in all common calcula-
tions, be wholly disregarded. The direction of gravity is always exactly
perpendicular to the horizon, that is, to the surface of the earth, which is
somewhat curved, on account of the earth's spheroidical figure ; but any
small portion of this surface may be practically considered as a plane, and
the vertical lines perpendicular to it as parallel to each other.

* On the Motion of Mars, c. 58. f Principia, I. 11.

J Prop. 40.

30 LECTURE IV.

The oblique motion of a projectile may be the most easily understood by
resolving its velocity into two parts — the one in a horizontal, the other in a
vertical direction. It appears from the doctrine of the composition of
motion, that the horizontal velocity will not be affected by the force of
gravitation acting in a direction perpendicular to it, and that it will, there-
fore, continue uniform ; and that the vertical motion will also be the same
as if the body had no horizontal motion. Thus, if we let fall from the
head of the mast of a ship a weight which partakes of its progressive
motion, the weight will descend by the side of the mast in the same manner
and with the same relative velocity as if neither the ship nor the weight
had any horizontal motion.

We may, therefore, always determine the greatest height to which a pro-
jectile will rise, by finding the height from which a body must fall in order
to gain a velocity equal to its vertical velocity, or its velocity of ascent ;
that is, by squaring one eighth of the number of feet that it would rise in the
first second if it were not retarded. For example, suppose a musket to be so
elevated that the muzzle is higher than the but-end by half of the length,
that is, at an angle of 30° ; and let the ball be discharged with a velocity
of 1000 feet in a second ; then its vertical velocity will be half as great, or
500 feet in a second ; now the square of one eighth of 500 is 3906, conse-
quently the height to which the ball would rise, if unresisted by the air, is
3906 feet, or three quarters of a mile. But, in fact, a musket ball actually
shot upwards, with a velocity of 1670 feet in a second, which would rise
six or seven miles in a vacuum, is so retarded by the air, that it does not
attain the height of a single mile.

We may easily find the time of the body's ascent from its initial velocity ;
for the time of ascent is directly proportional to the velocity, and may be
found in seconds by dividing the vertical velocity in feet by 32 ; or if we
divide by 16 only we shall have the time of ascent and descent ; and then
the horizontal range may be found, by calculating the distance described
in this time with the uniform horizontal velocity. Thus, in the example
that we have assumed, dividing 500 by 16 we have 31 seconds for the
whole time of the range ; but the hypotenuse of our triangle being 1000,
and the perpendicular 500, the base will be 886 feet ; consequently the hori-
zontal range is 31 times 886, that is, nearly 28,000 feet, or above 5 miles.
But the resistance of the air will reduce this distance also to less than one
mile.

It may be demonstrated that the horizontal range of a body, projected
with a given velocity is always proportional to the sine of twice the angle
of elevation : that is, to the £sine of the angle of] elevation of the muzzle of
the piece in a situation twice as remote from a horizontal position as its
actual situation. Hence it follows, that the greatest horizontal range will
be when the elevation is half a right angle ;* supposing the body to move
in a vacuum. But the resistance of the air increases with the length of
the path, and the same cause also makes the angle of descent much greater
than the angle of ascent, as we may observe in the track of a bomb. For
both these reasons, the best elevation is somewhat less than 45°, and some-
* Galileo, Dial. IV. Prop. 7, cor.

ON DEFLECTIVE FORCES. SI

times, when the velocity is very great, as little as 30°. But it usually
happens, in the operations of natural causes, that near the point where any
quantity is greatest or least, its variation is slower than elsewhere ; a small
difference, therefore, in the angle of elevation, is of little consequence to the
extent of the range, provided that it continue between the limits of 45° and
35° ; and for the same reason, the angular adjustment requires less accuracy
in this position than in any other, which, besides the economy of powder,
makes it the best elevation for practice. (Plate II. Fig. 17, 18.)

The path of a projectile, supposed to move without resistance, is always
a parabola. This interesting proposition was first discovered by Galileo :*
it follows very readily from the doctrine of the composition of motion, com-
bined with the laws which that philosopher established concerning the fall
of heavy bodies. If from any points of a given right line, as many lines be
drawn, parallel to each other, and proportional to the squares of the corre-
sponding segments of the first line, the curve in which all their extremities
are found, is a parabola. Now supposing the first line to be placed in the
direction of the initial motion of a projectile, and parallel vertical lines to be
drawn through any points of it, proportional to the squares of the segments
which they cut off, these lines will represent the effect of gravitation, during
the times in which the same segments would have been described, by the
motion of projection alone ; consequently the projectile will always be
found at the extremity of the vertical line corresponding to the time
elapsed, and will therefore describe a parabola. (Plate II. Fig. 17, 19.)

It is easy to show by experiment, that the path of a projectile is a para-
bola : if we only let a ball descend from a certain point, along a grobve, so
as to acquire a known velocity, we may trace on a board the parabola which
it will afterwards describe during its free descent ; and by placing rings at
different parts of the curve, we may observe that it will pass through them
all without striking them. (Plate II. Fig. 19.)

In practical cases, on a large scale, where the velocity of a projectile is
considerable, the resistance of the atmosphere is so great as to render the
Galilean propositions of little or no use ; and a complete determination of
the path, including all the circumstances which may influence it, is attended
with difficulties almost insuperable. It appears from Robins's experi-
ments, that the resistance of the air to an iron ball of 4| inches in diameter,
moving at the rate of 800 feet in a second, is equal to four times its weight,
and that where the velocity is much greater the resistance increases far
more rapidly.t But what must very much diminish the probability of our
deriving any great practical advantage from the theory of gunnery, is an
observation, made also by Mr. Robins, that a ball sometimes deviates three
or four hundred yards laterally, without any apparent reason ; J so that we
cannot be absolutely certain to come within this distance of our mark in any
direction. The circumstance is probably owing to an accidental rotatory mo-
tion communicated to the ball in its passage through tKe piece, causing after-
wards a greater friction from the air on one side than on the other ; and it may
'in some measure be remedied by employing a rifle barrel, which determines

* Dialogues on Motion, Dial. IV.
f Mathematical Tracts, 2 vols. 1761, i. 131. J Ibid. p. 150.

32 LECTURE V.

the rotation of the ball in such a manner that its axis coincides at first with
the path of the ball, so that the same face of the ball is turned in succession
every way. For the ordinary purposes of gunnery, an estimation governed
by experience is found to be the best guide ; at the same time there is no
doubt but that some assistance may be obtained from theory and from ex-
periment. Those who are desirous of pursuing the subject may find much
information relating to it, collected by Professor Robison, in the article
'Projectile' of the Encyclopaedia Britannica.

LECT. IV.— ADDITIONAL AUTHORITIES.

Central Forces. — Hooke, Birch's History of the Royal Society, ann. 1664-6, ii.
69, 90. Huygens de vi centrifuga, Op. post ; de causa gravitatis, 1690. Keill, Ph.
Tr. 1708, p. 174; 1714, p. 91. Demoivre, Ph. Tr. 1717, p. 622. Maclaurin,
Geom. Organ. 4to, 1720. Louville, Hist, et Mem. de 1'Acad. de Paris, 1720. Mau-
pertuis, ibid. 1732, p. 343, H. 112. Montigny, 1741, p. 280, H. 143. Bosco-
vich, Com. Bon. II. iii. 262. Waring, Ph. Tr. 1788, p. 67. Manchester Memoirs,
iv. 369 ; v. 101. Trembley, Hist, et Mem. de 1'Acad. de Berlin, 1797, p. 36.
Brinkley, Trans, of the Royal Irish Academy, viii. 215. Lagrange, Miscellanea
Taurinensia, ii. II. and iv. IV. Airy on Gravitation, 1834.

On Projectiles and Gunnery. — Frisius, Cosmographia, iv. Antwerp, 1584. Digges
on the Art of Great Artillery, 4to, 1624. Halley, Ph. Tr. 1686, p. 3 ; 1695, p. 68.
Bernoulli, Comm. Physico-Math. Paris, 1710. Keill, Ph. Trans. 1715, p. 91.
T. Simpson, ibid. 1748, p. 137. Robins, ibid. 1743, xlii. 437; and Mathematical
Tracts, 2 vols. 1761. Borda, Hist, et Mem.deFAcad. de Paris, 1769, p. 247, H. 116.
Glenie, History of Gunnery, Edin. 1776. Brown, The True Principles of Gunnery,
4to, 1777 (partly a translation from Euler). Hutton, Ph.Tr. 1778, p. 50 ; Tracts, 4to,
1786, v. 3. Pringle, A Discourse on the Theory of Gunnery, 4to, 1778. Thompson
(Count Rumford), Ph. Tr. 1781, p. 229 ; 1797, p. 222. Inman, An Introduction to
Naval Gunnery, Portsea, 1828.

LECTURE V.

ON CONFINED MOTION.

WE have hitherto considered the principal cases of motion, either un-
disturbed, or simply subjected to the action of an accelerating, retarding, or
deflective force. We now proceed to examine the effects of an additional
modification, which is introduced when the motion is limited to a given
line or surface of any kind ; the body either being supposed to slide on the
surface of a solid actually extended, or being confined to an imaginary sur-
face by its attachment to a thread, or still more narrowly restricted by
means of two threads, which allow it to move only in a given line. Sus-
pension is the most convenient mode of making experiments on confined
motion ; but it is not always easy to cause the body to remain in the sur-
face that is required ; and to confine it in this manner to a perfectly plane
surface is impossible. When we suffer a body to slide along any surface,
there is a loss of force from friction, from the production of rotatory
motion, or from both these causes combined. The effect of friction is

ON CONFINED MOTION. 33

obvious and well known ; and we may be convinced of the retardation at-
tendant on the production of rotatory motion, by allowing two cylinders,
of equal dimensions, to roll down an inclined plane : the one being covered
with sheet lead, the other having an equal weight of lead in its axis, and
being covered with paper, and both having similar projecting surfaces at
the ends, which come into contact with the plane, we may easily observe that
in the first cylinder, much more of the force is consumed in producing
rotatory motion, than in the second, and that it therefore descends much
more slowly. (Plate II. Fig. 20.)

When a body is placed on an inclined plane, the force urging it to
descend, in the direction of the plane, is to the whole force of gravity as
the height of the plane is to its length. This is demonstrable from the prin-
ciples of the composition of motion, and may also be shown experimentally
with great accuracy, when we consider the doctrine of the equilibrium of
forces. But the interference of friction will only allow us to observe, with
respect to the velocities produced, that they nearly approach to those which
the calculation indicates. Thus, if a plane be inclined one inch in 32, a
ball will descend on it in two seconds, instead of 64 feet, somewhat less
than two feet.

It may be deduced from the laws of accelerating forces, that when
bodies descend on any inclined planes of equal heights, but of different in-
clinations, the times of descent are as the lengths of the planes, and the
final velocities are equal. Thus a body will acquire a velocity of 32 feet in a
second, after having descended 16 feet either in a vertical line, or in an oblique
direction ; but the time of descent will be as much greater than a second as
the oblique length of the path is greater than 16 feet. This may be shown
by experiment, as nearly as the obstacles already mentioned will permit,
the times being measured by a pendulum or by a stop watch. (Plate II.
Fig. 21.)

There is an elegant proposition, of a similar nature, which is still more
capable of experimental confirmation ; that is, that the times of falling
through all chords drawn to the lowest point of a circle are equal. If two
or more bodies are placed at different points of a circle, and suffered to
descend at the same instant along as many planes which meet in the lowest
point of the circle, they will arrive there at the same time. (Plate II.
Fig. 22.)

The velocity of a body descending along a given surface, is the same as
that of a body falling freely through an equal height, not only when the
surface is a plane, but also when it is a continued curve, in which the
body is retained by its attachment to a thread, or is supported by any
regular surface, supposed to be free from friction.* We may easily show,
by an experiment on a suspended ball, that its velocity is the same when
it descends from the same height, whatever may be the- form of its path, by
observing the height to which it rises on the opposite side of the lowest
point. We may alter the form of the path in which it descends, by placing
pins at different points, so as to interfere with the thread that supports the
ball, and to form in succession temporary centres of motion ; and we shall
* Principia, i. 40.

D

34 LECTURE V.

find, in all cases, that the body ascends to a height equal to that from
which it descended, with a small deduction on account of friction. (Plate
II. Fig. 23.)

Hence is derived the idea conveyed hy the term living or ascending force ;
for since the height to which a body will rise perpendicularly, is as the
square of its velocity, it will preserve a tendency to rise to a height which
is as the square of its velocity whatever may be the path into which it is
directed, provided that it meet with no abrupt angle, or that it rebound at
each angle in a new direction without losing any velocity. The same idea
is somewhat more concisely expressed by the term energy, which indicates
the tendency of a body to ascend or to penetrate to a certain distance, in
opposition to a retarding force.

The most important cases of the motion of bodies, confined to given sur-
faces, are those which relate to the properties of pendulums. Of these
the simplest is the motion of a body in a cycloidal path. The cycloid
is a curve which has many peculiarities ; we have already seen that it is
described by marking the path of a given point in the circumference of a
circle which rolls on a right line. [p. 19.] Galileo was the first that con-
sidered it with attention, but he failed in his attempts to investigate its
properties.* It is singular enough, that the principal cause of his want of
success was an inaccurate experiment : in order to obtain some previous
information respecting the area included by it, he cut a board into a
cycloidal form, and weighed it, and he inferred from the experiment that
the area bore some irrational proportion to that of the describing circle,
while in fact it is exactly triple. In the same manner it has happened in
later times, that Newton, in his closet, determined the figure of the earth
more accurately than Cassini from actual measurement.t It was Huygens ^
that first demonstrated the properties of the cycloidal pendulum, which are
of still more importance in the solution of various mechanical problems,
than for the immediate purposes of timekeepers, to which that eminent
philosopher intended to apply them. (Plate I. Fig. 5.)

If a body be suspended by a thread playing between two cycloidal
cheeks, it will describe another equal cycloid by the evolution of the
thread, and the time of vibration will be the same, in whatever part of the
curve it may begin to descend. § Hence the vibrations of a body moving
in a cycloid are denominated isochronous, or of equal duration. This
equality may be shown by letting go two pendulous balls at the same in-

* On the authority of Toricelli, Op. 1644. Consult Wallis, Op. 3 vols. fol. Oxf.
1699, i. 543, and Ph. Tr. xix. Ill, 561. The cycloid was known to Cusanus 1454,
and to Bovillus 1500, a century before it was considered by Galileo. See Leibnitz,
Op. iii. 95, and British Magazine for 1800.

f Cassini, from his father's and M. Picard's measurements, proved that the earth
must be a spheroid, whose axis is greater than its equatorial diameter. Newton de-
duced the contrary from theory; and it is so in fact. See Mem. de 1'Acad. 1713,
1718. Newton's Principia, and Ph. Tr. 1725, pp. 33, 201, 239, 277, 344. Against
Mairan, Mem. de 1'Acad. 1720.

J Horologium Oscillatorium, fol. Paris, 1673.

§ Ibid. Compare Part I. with Prop. 25, Part II. In Birch's History of the
Royal Soc. is found an investigation of the same property by Lord Brouncker,
registered Jan. 22, 1662. The president was ordered to send a copy of it to Huy-
gens.

ON CONFINED MOTION. 35

slant, at different points of the curve, and observing that they meet at the
lowest point. (Plate II. Fig. 24.)

The absolute time of the descent or ascent of a pendulum, in a cycloid,
is to the time in which any heavy body would fall through one half of the
length of the thread, as half the circumference of a circle to its diameter.*
It is, therefore, nearly equal to the time required for the descent of a
body through -| of the length of the thread ; and if we suffer the pendulum
to descend, at the same moment that a body falls from a point elevated
one fourth of the length of the thread above the point of suspension, this
body will meet the pendulum at the lowest point of its vibration. (Plate
II. Fig. 24.)

Hence it may readily be inferred, that since the times of falling through
any spaces are as the square roots of those spaces, the times of vibration
of different pendulums are as the square roots of their lengths. Thus, the
times of vibration of pendulums of 1 foot and 4 foot in length, will be as
1 to 2 : the time of vibration of a pendulum 39-rW inches in length is one
second ; the length of a pendulum vibrating in two seconds must be four
times as great.

The velocity with which a pendulous body moves, at each point of a
cycloidal curve, may be represented, by supposing another pendulum to
revolve uniformly in a circle, setting out from the lowest point, at the same
time that the first pendulum begins to move, and completing its revolution
in the time of two vibrations ; then the height, acquired by the pendulum
revolving equably, will always be equal to the space described by the
pendulum vibrating in the cycloid. (Plate II. Fig. 24.)

It may also be shown, that if the pendulum vibrate through the whole
curve, it will everywhere move with the same velocity as the point of the
circle which is supposed to have originally described the cycloid, pro-
vided that the circle roll onwards with an equable motion.

All these properties depend on this circumstance, that the relative force,
urging the body to descend along the curve, is always proportional to the
distance from the lowest point ; and it happens in many other instances of
the action of various forces, that a similar law prevails : in all such cases,
the vibrations are isochronous, and the space described corresponds to the
versed sine of a circular are increasing uniformly, that is to the height of
any point of a wheel revolving uniformly on its axis, or rolling uniformly
on a horizontal plane.

The cycloid is the curve in which a body may descend in the shortest
possible time, from a given point to another obliquely below it.t It may
easily be shown that a body descends more rapidly in a cycloid than in the
right line joining the two points. This property is of little practical
utility ; the proposition was formerly considered as somewhat difficult to
be demonstrated, but of late, from the invention of new modes of calcula-

, * Huyg. Horol. Oscil. Part II. Prop. 25.

t Jo. Bernoulli, Acta Erudit. Lips. 1696, p. 269. Ja. Bernoulli, ibid. 1697, p.
211, and Opera, ii. 768. Euler, Acta Petrop. 1733, &c. &c. Lagrange, Miscella-
nea Taurinensia, vols. i. and ii. Consult Woodhouse's Isoperimetrical Problems,
Camb. 1810 ; or the article Variations in the Encyc. Brit.

D 2

3G LECTURE V.

tion, theorems of a similar nature have been much extended with great
facility. The experiment naturally suggests a familiar proverb, which
cautions us against being led away too precipitately by an appearance of
brevity and facility. (Plate II. Fig. 25.)

It has been found that the inconveniences, resulting from the compli-
cated apparatus necessary to introduce a cycloidal motion for the pen-
dulums of clocks, are more than equivalent to the advantage of perfect
isochronism in theory. For since in small cycloidal arcs the curvature is
nearly constant, the time of vibration of a simple circular pendulum must
be ultimately the same as that of a cycloidal pendulum of the same length ;
but in larger arcs, the time must be somewhat greater, because the circular
arc falls without the cycloidal, and is less inclined to the horizon at
equal distances from the lowest point. This may be shown by a compa-
rison of two equal pendulums, vibrating in arcs of different extent : it
may also be observed, by an experiment with two simple pendulums of
different lengths, that their times of vibration, like those of cycloidal
pendulums, are proportional to the square roots of their lengths ; a half
second pendulum being only one fourth as long as a pendulum vibrating
seconds.

We have been obliged to suppose the weight, as well as the inertia, of a
pendulum, to be referred to one point, since we are not at present prepared
to examine the effect of the slight difference between the situations and the
velocities of the different parts of the substances, employed in our experi-
ments. The nature of rotatory motion requires to be more fully under-
stood, before we can attend to the determination of the centres of oscillation
of bodies of various figures, that is, of the points in which their whole weight
may be supposed to be concentrated, with regard to its effect on the times
of their vibrations.

It is remarkable that the isochronism of pendulums, which is a property
so important in its application, may still be preserved, notwithstanding
the interference of a constant retarding force, such as the force of friction
is in many cases found to be. It has been shown by Newton,* that each
complete vibration of a cycloidal pendulum, retarded by a resistance of
this nature, will be shorter than the preceding one by a certain constant
space, but that it will be performed in the same time.

There is a great analogy between the vibrations of pendulums, and the
revolution of balls suspended from a fixed point. If a body, suspended
by a thread, revolve freely in a horizontal circle, the time of revolution
will be the same, whenever the height of the point of suspension above the
plane of revolution is the same, whatever be the length of the thread.
Thus, if a number of balls are fixed to threads, or rather wires, connected
to the same point of an axis, which is made to revolve by means of the
whirling table, they will so arrange themselves as to remain very nearly in
the same horizontal plane. (Plate II. Fig. 26.)

The time of each revolution of the balls is equal to the time occupied by
a double vibration of a pendulum, of which the length is equal to the
height of the point of suspension above the plane in which they revolve ;
* Principia, Book II. sec. 6.

ON CONFINED MOTION. 37

consequently all the revolutions will be nearly isochronous, while the
threads or wires deviate hut little from a vertical situation.* In fact, we
may imagine such a revolution to be composed of two vibrations of a
simple pendulum, existing at the same time, in directions at right angles
to each other ; for while a pendulum is vibrating from north to south, it
is liable to the impression of any force, capable of causing a vibration from
east to west ; and the joint result of both vibrations will be a uniform
revolution in a circle, if the vibrations are equal and properly combined ;
but if they are unequal, the joint vibration will be ultimately an ellipsis,
the joint force being directed to its centre, and always proportional to
the distance from that centre. (Plate II. Fig. 27.)

The near approach of these revolutions to isochronism has sometimes
been applied to the measurement of time, but more frequently, and more
successfully to the regulation of the motions of machines. Thus in Mr.
Watt's steam engines, two balls are fixed at the ends of rods in continual
revolution, and as soon as the motion becomes a little too rapid, the balls
rise considerably, and turn a cock which diminishes the quantity of steam
admitted. (Plate II. Fig. 28.)

The same laws are applicable to many other cases of rotatory motion ;
for instance, if we wish to determine the height, at which a ball, revolving
with a given velocity, will be retained in a spherical bowl, or the incli-
nation of a circular road, capable of counteracting the centrifugal force of
a horse, running round [in] it; (for the horse, like the ball of the
revolving pendulum, has a centrifugal tendency, which is greater as his
velocity is greater;) this centrifugal force, combined with the force of
gravity, composes a result, which, in the case of the pendulum, is com-
pletely counteracted by the force of the thread or wire, and must there-
fore be in the direction of the thread, and which obliges the horse to
place his legs in a similar direction, proceeding from an imaginary
point of suspension above ; since he would otherwise be liable to fall out-
wards, if his velocity were sufficiently great. But in order to withstand
the pressure of the horse's legs, the road must be in a direction perpendi-
cular to them ; otherwise its materials will naturally be forced outwards,
until they produce an elevation sufficient to give the road the required
form. Thus, if the diameter of the ring were 40 feet, and the horse
moved at the rate of 12 miles an hour, he would perform about 500 revo-
lutions in an hour, and half a revolution in three seconds and a half.
Now the length of a pendulum vibrating in &J seconds, must be 39 inches
multiplied by the square of 3^, or a little more [less] than 80 [40] feet :
the road must, therefore, be perpendicular to the direction of a line drawn
to it from a point 80 [40] feet above the centre of the ring ; and its
external circumference must be higher than its internal circumference by
one fourth [half] of its breadth. It would, however, be improper to have
a road of this form in a manege, since the horse must be taught to perform
all his evolutions on a perfect plane.

There is a general principle of curvilinear motion, which is in itself of

* Euler on a Rotatory Pendulum, Acta Petr. 1780, pp. 133, 164.

88 LECTURE VI.

little importance or practical utility, but which so far deserves to be
noticed, as it has been magnified by some philosophers into a fundamental
law of nature. Among all the curves that a body can describe, in moving
from one point to another, it always selects that, in which, if its velocity
be supposed to be everywhere multiplied by the distance that it describes,
the sum of the infinitely small products will be a minimum, that is, less
than in any other path that the body could take. For example, if a body
move freely, and therefore with a uniform velocity, in any regular curved
surface, it will pass from one part of the surface to another by the shortest
possible path. This has been called the principle of the least possible
action ; it is, however, merely a mathematical inference from the simpler
laws of motion, and if those laws were even different from what they are,
the principle would be true in another form, and in another sense of the
word action.*

LECT. V.— ADDITIONAL AUTHORITIES.

Confined Motion, Pendulum, Sfc. — Becherus de Nova Temporis Dimetiendi Ra-
tione, 4to, Lond. 1680. Brook Taylor, Ph. Tr. xxviii. 11. Graham and Camp-
bell's Experiments to determine the Difference in the Length of Isochronal Pendu-
lums at different Places, Ph. Trans. 1733, p. 302. Courtivron on a Circular Pendu-
lum, Hist, et Mem. de 1'Acad. de Paris, 1744, p. 384, H. 30. Lagrange on Iso-
chronous Curves, Mem. de 1'Acad. de Berlin, 1765, p. 361; 1770, p. 97. D'Alem-
bert, ibid. 1765, p. 381. Landen on Circular Pend. Ph. Tr. 1771, p. 308 ; 1775,
p. 287. Maseres, ibid. 1777, p. 215. Legendre, on do. Hist, et Mem. de 1'Acad.
de Paris, 1786, pp. 30, 637. Biot, on Tautoch. Curves, Bulletin dela Soc. Philo-
matique, No. 73. Carlini sulla Lunghezza del Pendolo, Cesaris Effemeridi,
1827, Milan. Bessel Untersuchungen tiber das Secunden Pendul, 4to, Berl.1828.
Piola sulla Teoria del Pen. Ces. EfFem. 1831-2.

Confined Motion with, Resistance. — Krafft on the Inclined Plane, Com. Petr. xii.
261 ; xiii. 100. Euler, ibid. xiii. 197. Kastner, ibid. Leips. Mag. ii. 1. Euler
on a Rotatory Pendulum with Res. A. Petr. 1780, IV. ii. 164. Airy, Transactions
of the Cambridge Philosophical Society, III. 111. Plana sur le Mouvement d'un
Pendule dans un milieu resistant, 4to, Turin, 1835. Challis. Trans, of the Camb.
Phil. Soc. vii. 333.

Properties of the Cycloid.— Pascal, Histoire de la Roulette. Carlo Dati, Let-
tera della vera Storia della Cicloide, 4to, Firenze, 1663. Groningius, Historia Cy-
cloidis. Lalouere, Geometria Promota, 4to, Tolosae, 1660. Young, An Essay
on Cycloidal Curves, 4to, 1800. Peacock's Examples to the Diff. Calc. I. Gre-
gory's Do. 134.

LECTURE VI.

ON THE MOTIONS OF SIMPLE MASSES.

HITHERTO we have considered the motions of one or more single points
or atoms only, without any regard to the bulk or mass of a moveable body :
but it now becomes necessary to attend also to the difference of the masses

* See p. 16.* Consult also Ampere sur 1' Application du Calcul des Variations
aux Prop, de Mec. 4to, Par. 1809.

ON THE MOTIONS OF SIMPLE MASSES. 39

of bodies in motion. This may however be done, without considering the
actual magnitude or extent of the body. We may easily conceive different
masses or bulks to be concentrated in a mathematical point ; and it is most
convenient to define a moveable body, as a moveable point or particle com-
posed of other elementary particles, differing only in number, and thus
constituting the proportionally different mass or bulk of the body.

Although in our experiments on motion we are obliged to have recourse
to material bodies, and although such bodies differ considerably from this
definition of a single moveable body, yet they serve sufficiently well to
represent such bodies, especially when they are small and regularly formed ;
and we are here considering the doctrine of motion rather in a mathe-
matical than in a physical sense ; so that we are able to neglect all such
properties of matter as are not immediately necessary to our purpose. In-
deed though the general properties of matter are usually placed at the
entrance of elementary works on mechanics, it has yet been found
necessary to omit the consideration of their effects, in examining the laws
and affections of motion. The forces of cohesion and repulsion, for exam-
ple, act, in general, in a very complicated manner, in almost all cases of
the communication of motion ; but to consider these operations minutely
in treating of collision, would be to involve the subject in very great
and very unnecessary difficulties ; and the complete investigation of these
properties of matter would require the employment of various branches of
mechanical and hydrodynamical science. We may therefore take a much
simpler course, by deferring entirely all theoretical consideration of actual
matter ; but in the mean time we must have, for our experimental illustra-
tions, some measure of the mass or bulk as here defined. We might employ
spherical bodies, composed only of homogeneous substances, that is, of sub-
stances of the same kind, and we might estimate the mass by the compara-
tive magnitude, imagining all the particles of each sphere to be united in
its centre. But it is more convenient to anticipate, from the gravitation of
matter, a measure of the mass derived from the weight : since it can be
proved that the weight of a body is proportional to its absolute quantity of
matter, supposing all matter to be alike in its affections relative to motion.
So that instead of numbering the particles of each body, the same purpose
is answered by determining their comparative weight.

Inertia, or a tendency to persevere in a state of rest, or of uniform recti-
linear motion, is a property attached to all matter, and may be considered
as proportional to the mass or weight of a body. When the motions of a
system of bodies are considered, their inertia may in some respects be
referred to a single point, which is called the centre of inertia. [See the
next paragraph.] The centre of inertia of two bodies is that point, in the
right line joining them, which divides it into two such portions, that the
one is to the other as the mass of the remoter body tp that of the adjacent
body. For instance, if one body weighs a pound, and another two pounds,
and their distance is a yard, then the centre of inertia is at the distance of
two feet from the smaller body, and one foot from the larger : and the dis-
tance of each is to the whole distance, as the weight of the other to the
whole weight. Also the products obtained by multiplying each weight by

40 LECTURE VI.

its distance are equal : thus two multiplied by one is equal to one multi-
plied by two. (Plate II. Fig. 29.)

This point is most commonly called the centre of gravity ; it has also
sometimes been denominated the centre of position. Since it has many
properties independent of the consideration of gravity, it ought not to derive
its name from gravitation, [but as custom has familiarized the term, we
deem it better to retain it.]

The centre of inertia [gravity] of any two bodies initially at rest, remains
at rest, notwithstanding any reciprocal action of the bodies ; that is, not-
withstanding any action which affects the single particles of both equally,
in increasing or diminishing their distance. For it may be shown, from
the principles of the composition of motion, that any force, acting in this
manner, will cause each of the two bodies to describe a space proportional
to the magnitude of the other body : thus a body of one pound will move
through a space twice as great as a body of two pounds weight, and the
remaining parts of the original distance will still be divided in the same
proportion by the original centre of inertia [gravity], which therefore still
remains the centre of inertia [gravity], and is at rest. And it follows also,
that if the centre of inertia [gravity] is at first in motion, its motion will
not be affected by any reciprocal action of the bodies.

This important property is very capable of experimental illustration ;
first observing, that all known forces are reciprocal, and among the rest the
action of a spring ; we place two unequal bodies so as to be separated when
a spring is set at liberty, and we find that they describe, in any given
interval of time, distances which are inversely as their weights ; and
that consequently the place of the centre of inertia [gravity] remains un-
altered. They may either be made to float on water, or may be suspended
by long threads ; the spring may be detached by burning a thread that
confines it, and it may be observed whether or no they strike at the same
instant two obstacles, placed at such distances as the theory requires ; or if
they are suspended as pendulums, the arcs which they describe may be
measured, the velocities being always nearly proportional to these arcs, and
accurately so to their chords. (Plate II. Fig. 30.)

The same might also be shown of attractive as well as of repulsive forces.
For instance, if we placed ourselves in a small boat, and pulled a rope tied
to a much larger one, we should draw ourselves towards the large boat with
a motion as much more rapid than that of the large boat, as its weight is
greater than that of our own boat ; and the two boats would meet in their
common centre of inertia [gravity], supposing the resistance of the water
inconsiderable.

Having established this property of the centre of inertia [gravity] as a
law of motion, we may derive from it the true estimate of the quantity of
motion in different bodies, in a much more satisfactory manner than it has
usually been explained. For since the same reciprocal action produces, in
a body weighing two pounds, only half the velocity that it produces in
a body weighing one pound, the cause being the same, the effects must be
considered as equal, and the quantity of motion must always be measured
by the joint ratio of mass to mass, and velocity to velocity ; that is, by the

ON THE MOTIONS OF SIMPLE MASSES. 41

ratio of the products, obtained by multiplying the weight of each body
by the number expressing its velocity ; and these products are called
the momenta of the bodies. We appear to have deduced this measure of
motion from the most unexceptionable arguments, and we shall have occa-
sion to apply the momentum thus estimated as a true measure of force ; at
the same time that we allow the practical importance of considering, in
many cases, the efficacy of forces, according to another criterion, when we
multiply the mass by the square of the velocity, in order to determine the
energy : yet the true quantity of motion, or momentum, of any body, is
always to be understood as the product of its mass into its velocity. Thus
a body weighing one pound, moving with the velocity of a hundred feet in
a second, has the same momentum and the same quantity of motion as a
body of ten pounds, moving at the rate of ten feet in a second.

We may also demonstrate experimentally, by means of Mr. Atwood's
machine [Plate I. Fig. 11], that the same momentum is generated, in a
given time, by the same preponderating force, whatever may be the quan-
tity of matter moved. Thus, if the preponderating weight be one sixteenth
of the whole weight of the boxes, it will fall one foot in a second instead of
16, and a velocity of two feet will be acquired by the whole mass, instead
of a velocity of 32 feet, which the preponderating weight alone would have
acquired. And when we compare the centrifugal forces of bodies revolving
in the same time at different distances from the centre of motion, we find
that a greater quantity of matter compensates for a smaller force ; so that
two balls connected by a wire, with liberty to slide either way, will retain
each other in their respective situations when their common centre of
inertia [gravity] coincides with the centre of motion ; the centrifugal force
of each particle of the one being as much greater than that of an equal
particle of the other, as its weight or the number of the particles is smaller.

But it is not enough to determine the centre of inertia [gravity] of two
bodies only, considered as single points ; since in general a much greater
number of points is concerned : we must therefore define the sense in which
the term is in this case to be applied. We proceed by considering the first
and second of three or more bodies, as a single body equal to both of them,
and placed in their common centre of inertia [gravity], ; determining the
centre of inertia [gravity] of this imaginary body and the third body, and
continuing a similar process for all the bodies of the system. And it
matters not with which of the bodies we begin the operation, for it may be
demonstrated that the point thus found will be the same by whatever steps
it be determined. When we come to consider the properties of the same
point as the centre of gravity [weight] we shall be able to produce an ex-
perimental proof of this assertion, since it will be found that there is only
one point in any system of bodies which possesses these properties. (Plate
III. Fig. 31.)

We may always represent the motion of the centre of inertia [gravity]
of a system of moving bodies, by supposing their masses to be united into
one body, and this body to receive at once a momentum equal to that of
each body of the system, in a direction parallel to its motion. This may'
often be the most conveniently done by referring all the motions of this

42 LECTURE VI.

imaginary body to three given directions, and collecting all the results into
three sums, which will represent the motion of the centre of inertia
[gravity] of the system.

We have already presupposed this proposition, when we have employed
material bodies of finite magnitude, that is, systems of material atoms, to
represent imaginary bodies of the same weight condensed into their centres ;
and it now appears that the velocity and direction of the motions of such
bodies as we have employed, agree precisely with those of our imaginary
material points. We cannot attempt to confirm this law by experiment,
because the deductions from the sensible consequences of an experiment
would require nearly the same processes as the mathematical demonstration.

It is obvious that the result of any number of uniform and rectilinear
motions thus collected, must also be a uniform and rectilinear motion.
The centre of inertia [gravity] of a system of bodies moving without dis-
turbance, is, therefore, either at rest, or moving equably in a right line.

The mass, or weight, of each of any number of bodies, being multiplied
by its distance from a given plane, the products, collected into one sum,
will be equal to the whole weight of the system, multiplied by the distance
of the common centre of inertia [gravity] from the same plane. And the
proposition will be equally true, if, instead of the shortest distances, we
substitute the distances from the same plane, measured obliquely, in any
directions always parallel to each other. This property is peculiarly appli-
cable to the consideration of the centre of gravity [weight], and affords also
the readiest means of determining its place in bodies of complicated forms.
(Plate III. Fig. 32.)

We have already seen that the place of the centre of inertia [gravity] of
two bodies is not affected by any reciprocal action between them ; and the
same is true of the actions of a system of three or more bodies. We might
easily apply our experiment on the reciprocal action of two bodies to a
greater number, but we should throw no further light on the subject, and
the mode of obtaining the conclusion would be somewhat complicated.

All the forces in nature, with which we are acquainted, act reciprocally
between different masses of matter, so that any two bodies repelling or at-
tracting each other, are made to recede or approach with equal momenta.
This circumstance is generally expressed by the third law of motion, that
action and reaction are equal. There would be something peculiar, and
almost inconceivable, in a force which could affect unequally the similar
particles of matter ; or in the particles themselves, if they could be pos-
sessed of such different degrees of mobility as to be equally moveable with
respect to one force, and unequally with respect to another. For instance,
a magnet and a piece of iron, each weighing a pound, will remain in equi-
librium when their weights are opposed to each other by means of a
balance ; they will be separated with equal velocities, if impelled by the
unbending of a spring placed between them, and it is difficult to conceive
that they should approach each other with unequal velocities in consequence
of magnetic attraction, or of any other natural force. The reciprocality of
force is therefore a necessary law in the mathematical consideration of
mechanics, and it is also perfectly warranted by experience. The contrary

ON THE MOTIONS OF SIMPLE MASSES. 43

supposition is so highly improbable, that the principle may almost as
justly be termed a necessary axiom, as a phenomenon collected from
observation.

Sir Isaac Newton * observes, in his third law of motion, that " reaction
is always contrary and equal to action, or, that the mutual actions of two
bodies are always equal, and directed contrary ways." He proceeds, " if
any body draws or presses another, it is itself as much drawn or pressed.
If any one presses a stone with his finger, his finger is also pressed by the
stone. If a horse is drawing a weight tied to a rope, the horse is also
equally drawn backwards towards the weight : for the rope, being dis-
tended throughout, will, in the same endeavour to contract, urge the horse
towards the weight and the weight towards the horse, and will impede the
progress of the one as much as it promotes the advance of the other." Now
although Newton has always applied this law in the most unexceptionable
manner, yet it must be confessed that the illustrations here quoted are
clothed in such language as to have too much the appearance of paradox.
When we say that a thing presses another, we commonly mean, that the
thing pressing has a tendency to move forwards into the place of the thing
pressed, but the stone would not sensibly advance into the place of the
finger, if it were removed ; and in the same manner we understand that a
thing pulling another has a tendency to recede further from the thing
pulled, and to draw this after it ; but it is obvious that the weight which
the horse is drawing would not return towards its first situation, with the
horse in its train, although the 'exertion of the horse should entirely cease ;
in these senses, therefore, we cannot say that the stone presses, or that the
\veight pulls, and we have no reason to offend the just prejudices of a be-
ginner, by introducing paradoxical expressions without necessity. Yet it is
true in both cases, that if all friction and all connexion with the surround-
ing bodies could be instantaneously destroyed, the point of the finger and
the stone would recede from each other, and the horse and the weight would
approach each other with equal quantities of motion. And this is what we
mean by the reciprocality of forces, or the equality of action and reaction.

The quantity of action of two attractive or repulsive bodies on each
other is partly dependent on their magnitude. When the bodies are of the
same kind, their mutual action is in the compound ratio of their bulks ;
that is, in the ratio of the products of the numbers expressing their bulks.
For instance, if two bodies, each containing a cubic inch of matter, attract
or repel each other with a force of a grain, and there be two other bodies,
the one containing two inches, the other ten, of the same matter, then the
mutual attraction or repulsion of these will be expressed by twenty grains ;
for each of the 10 inches is attracted by each of the two with a force of a
grain. And the mutual action of 3 and 10 will be 30, of 4 and 10, 40 ; so
that when one of the bodies remains the same, the, attraction will be
simply as the bulk of the other. Hence the quantity of matter, in every
body surrounding us, is considered as proportional to its weight ; for it is
inferred from experiment that all material bodies are equally subject to the
power of gravitation towards the earth, and are, in respect to this force, of
* Principia, Lib. I.

44 LECTURE VI.

the same kind. For the apparent difference in the velocity with which
different substances fall through the atmosphere, is only owing to the
resistance of the air, as is sometimes shown by an experiment on a feather
and a piece of gold falling in the vacuum of an air pump ; but the true
cause was known long before the invention of this machine, and it is dis-
tinctly explained in the second book of Lucretius :
" In water or in air when weights descend,

The heavier weights more swiftly downwards tend.

The limpid waves, the gales that gently play,

Yield to the weightier mass a readier way,

But if the weights in empty space should fall,

One common swiftness we should find in all."

We are therefore to suppose, that the different weights of equal bulks of
different substances depend merely on the greater or less number of parti-
cles contained in a given space, independently of any other characters that
may constitute the specific differences of those substances.

In some cases it is necessary to consider the sum of the masses of two
bodies, in order to estimate their mutual action ; that is, when we wish to
know the whole relative motion of two bodies with respect to each other ;
for here we must add together their single motions with respect to the
centre of inertia [gravity], which are inversely in the same ratio. This
consideration is sometimes of use in determining the action of the sun on
the several planets.

If two bodies act on each other with forces proportional to any power of
their distance, for instance to the square or the cube of the distance, the
forces will also be proportional to the same power of either of their dis-
tances from their common centre of inertia [gravity]. Thus, in the
planetary motions, when one body performs a revolution by means of the
attractive force of another, this other cannot remain absolutely at rest ; but
because it is more convenient to determine the effect of the attraction as
directed to a fixed point, we consider the force as residing in the common
centre of inertia [gravity] of the two bodies, which remains at rest, as far as
the mutual actions of those bodies only are concerned, and it may be shown,
that the force diminishes as the square of the distance of the bodies, either
from this point or from each other, increases. The reciprocal forces of two
bodies may therefore be considered as tending to or from their common
centre of inertia [gravity] as a fixed point ; but it often happens that the
difference of magnitude being very great, the motion of one of the bodies
may be disregarded. Thus we usually neglect the motion of- the sun, in
treating of the planetary motions produced by his attraction, although, by
means of very nice observations, this motion becomes sensible. But it is
utterly beyond the power of our senses to discover the reciprocal motion of
the earth produced by any terrestrial cause, even by the most copious erup-
tion of a volcano, although, speaking mathematically, we cannot deny that
whenever a cannon ball is fired upwards, the whole globe must suffer a
minute depression in its course. The boast of Archimedes was therefore
accompanied by an unnecessary condition : " give me," said he, " but a
firm support, and I will move the earth ;" but, granting him his support,

ON PRESSURE AND EQUILIBRIUM. 45

he could only have displaced the earth insensibly by the properties of his
machines ; and without any such support, when he threw rocks upon the
ships of Marcellus, he actually caused the walls of Syracuse and the island
of Sicily to move northwards, with as much momentum as carried his pro-
jectiles southwards against the Roman armaments.

LECT. VI.— ADDITIONAL AUTHORITIES.

Centre of Gravity. — Wallis de Centre Gravitatis Hyperbolae, Ph. Tr. 1672, p.
5074. Roberval on the Centres of Gravity of Solids, Hist, et Mem. de Par. vi. 270,
282. Lahireon the Motion of the Centre of Inertia, ibid. ix. 175. Laura Bassion
ditto, Com. Bon. iv. O. 74. Varignon on the Centre of Gravity of Spheres,
Hist, et Mem. de Paris, x. 508. Clairaut on Finding the Centre of Gravity, ibid.
1731, p. 159. Bossut on the Centres of Gravity of Cycloidal Surfaces and Solids,
Mem. Presentes, Paris, iii. 603. Gr. Fontana on the Axis of Equilib. and the
Centre of Gr. Atti dell' Academia di Siena, 4to, vi. 177. L'Huillier's Theorem
respecting the Centre of Gravity, Nov. Act. Petrop. 1786, 4to, H. 39. Kramp on
the Centre of Gravity of Sph. Triangles, Hindenburg's Archiv. ii. 296.

LECTURE VII.

ON PRESSURE AND EQUILIBRIUM.

WE have now examined the principal cases in which a simple force is
employed in the production of motion ; it is of equal consequence to attend
to the opposition of forces, where they prevent each other's action. A
force counteracted by another force, so that no motion is produced, becomes
a pressure : thus we continually exert a pressure, by means of our wreight,
upon the ground on which we stand, the seat on which we sit, and the bed
on which we sleep ; but at the instant when we are falling or leaping, we
neither exert nor experience a pressure on any part.

It was very truly asserted by the ancients, that pressure and motion are
absolutely incommensurable as effects ; for according to the definition of
pressure, the force appears to be what is called in logic a potential cause,
which is not in a state of activity : and since an interval of time must elapse
after the removal of the opposite force, before the first force can have
caused any actual motion, this effect of a finite time cannot with justice be
conceived to bear any proportion to the pressure, which is as it were a
nascent effect only. It is true that a large weight pressing on a spring,
may keep it bent, in exactly the same place into which a smaller weight,
falling on it with a certain velocity, would inflect 'it : but, to retain a
spring in a certain position, and to bend it into that position, are effects
absolutely incommensurable ; the one being a measure of the constant
repulsive force of the spring, bent to a certain point, the other of the sum of
the effects of the same spring in various degrees of flexure, for a certain

46 LECTURE VII.

time. Hence the smallest possible momentum is said to be more than
equivalent to the greatest possible pressure : a very light weight, falling
from a very minute distance, will force back a very strong spring, although
often through an imperceptible space only. But the impulse of a -stream of
infinitely small particles, like those of which a fluid is supposed to consist,
striking an obstacle in a constant succession, may be counteracted by a
certain pressure, without producing any finite motion.

Nothing, however, forbids us to compare two pressures, by considering
the initial motions which they would produce, if the opposition were
removed ; nor is there any difficulty in extending the laws of the composi-
tion of motion to the composition of pressure. For since we measure
forces by the motions which they produce, it is obvious that the composi-
tion of forces is included in the doctrine of the composition of motions ;
and when we combine three forces according to the laws of motion, there
can be no question but that the resulting motion is truly determined in all
cases, whatever may be its magnitude ; nor can any reason be given why
it should be otherwise, when this motion is evanescent, and the force
becomes a pressure. The case is similar to that of a fraction, which may
still retain a real value, when both its numerator and denominator become
less than any assignable quantity. Some authors on mechanics, and
indeed the most eminent, Bernoulli,* Dalembert,1* and Laplace,^ have
deduced the laws of pressure more immediately from the principle of the
equality of the effects of equal causes ; and the demonstration may be
found, in an improved form, in the article Dynamics of the Supplement
of the Encyclopaedia Britannica ; but its steps are still tedious and intri-
cate.

We are, therefore, to consider the momentum or quantity of motion
which would be produced by any force in action, as the measure of the
pressure occasioned by it when opposed ; and to understand by equal or pro-
portionate pressures, such as are produced by forces which would generate
equal or proportionate momenta in a given time. And it may be inferred
that two contrary pressures will balance each other, when the momenta
which the forces would separately produce in contrary directions, are
equal ; and that any one pressure will counterbalance two others, when it
would produce a momentum equal and contrary to the momentum which
would be derived from the joint result of the other forces. For, supposing
each [either] of two forces opposed to each other to act for an instant, and to
remain inactive for the next equal instant while the other force is exerted,
it is obvious that these effects will neutralise each other, so that the body
on which they are supposed to operate will retain its situation ; but such
an action is precisely half of the continued action of each force ; conse-
quently, since the halves completely counteract each other, the wholes will
do the same. And a similar mode of reasoning may be extended to any
number of forces opposed to each other.

* Com. Petrop. I. 126.

f On the Principles of Mechanics, Hist, et Mem. de 1'Acad. 1769, p. 278, and
Opuscula, I. and VI.

t Mecanique Celeste. See also Celestial Mechanics of Laplace (by Young),
p. 87.

ON PRESSURE AND EQUILIBRIUM. 47

It follows from the laws of the composition of motion, that the result of
two pressures, expressed by the sides of a parallelogram, will be repra-
sented by its diagonal,* and that, if a body remain at rest by means of
three pressures, they must be related to each other in magnitude as the
sides of a triangle parallel to their directions. This may be very com-
pletely shown by experiment. We attach three weights to as many
threads, united in one point, and passing over three pullies ; then by
drawing any triangle, of which the sides are in the directions of the
threads, or in parallel directions, we may always express the magnitude of
each weight by the length of the side of the triangle corresponding to its
thread. (Plate III. Fig. 33.)

The most important of the problems relating to equilibrium are such as
concern the machines which are usually called mechanical powers. We are
not, however, to enter at present into all the properties and uses of these
machines ; we have at first only to examine them in a state of rest, since the
determination of their motion requires additional considerations, and their
application to practice belongs to another subdivision of our subject.

There is a general law of mechanical equilibrium, which includes the
principal properties of most of these machines. If two or more bodies,
connected together, be suspended from a given point, they will be at rest
when their centre of inertia [gravity ] is in the vertical line passing through
the point of suspension. The truth of this proposition may easily be
illustrated by the actual suspension of any body, or system of bodies,
from or upon a fixed point ; the whole remaining in equilibrium, when
the centre of inertia [gravity] is either vertically below the point of sus-
pension, or above the point of support, or when the fixed point coincides
with the centre of inertia [gravity]. And whatever may be the form of a
compound body, it may be considered as a system of bodies connected
together, the situation of the common centre of the inertia [gravity] deter-
mining the quiescent position of the body. (Plate III. Fig. 34 . . 38.)

Hence the centre of inertia is called the centre of gravity ; and it may
be practically found, by determining the intersection of two lines which
become vertical in any two positions in which the body is at rest. Thus,
if we suspend a board of an irregular form from any two points succes-
sively, and mark the situation of the vertical line in each position, we may
find by the intersection the place of the centre of gravity : and it will
appear that this intersection will be the same whatever positions we
employ. (Plate III; Fig. 39.)

The consideration of the degree of stability of equilibrium is of material
importance in many mechanical operations. Like other variable quanti-
ties, the stability may be positive, negative, or evanescent. The equili-
brium is positively more or less stable, when the centre of gravity would
be obliged to ascend more or less rapidly if it quitted the vertical line :
the equilibrium is tottering, and the stability is negative, when the centre
of gravity would descend if it were displaced ; but when the centre of

* Seepage 19 and last page. For demonstrations of this property consult also
Poisson, Traite de Mecanique, i. 43. Duchayla, extracted in Pratt's Mec. p. 7,
note ; and WhewelTs Mechanics.

48 LECTURE VII.

gravity coincides with the centre of motion, or when its path would be a
horizontal right line, the equilibrium has been called insensible, but may
more properly be termed neutral, and the body will rest in any position,
without tending either to fall or to return to its original situation. It is
obvious that the centre of gravity cannot move without descending, when
it is vertically over the fixed point, nor without ascending, when it is
immediately below it ; so that in the one case the equilibrium is tottering,
and in the other stable. Hence we may understand the reason of fixing
the moveable handles of a vessel of any kind at its upper part, in order that
the centre of suspension may be always above the centre of gravity. If
they be fixed too low, the vessel will be liable to overset, unless there be
sufficient friction to retain it in its proper situation. (Plate III. Fig. 40.)

An oval surface, placed on a horizontal plane, is capable of a stable
equilibrium, when it rests on its side, or on the extremity of its lesser axis,
and of a tottering equilibrium, when it stands on the extremity of its
greater axis. But the equilibrium of a circle or a sphere is always neutral,
for, when disturbed, it neither recovers its first position, nor deviates
further from it. ' A flat body, resting on a sphere, will have its equilibrium
tottering or stable, accordingly as its centre of gravity is more or less than
the semidiameter of the sphere above the point of contact. (Plate III. Fig.
41, 42.)

The stability of a body supported on a flat basis of a given extent, is of a
different kind, and is independent of equilibrium. For here, if the centre
of gravity move either way, it must begin its motion in an inclined direc-
tion, instead of describing a curve which is initially horizontal. The
stability of such a body becomes less and less as it is more and more
inclined, till, when the centre of gravity is vertically over the margin of
the basis, there is a tottering equilibrium ; and if the inclination be still
further continued, the body will faU. (Plate III. Fig. 43.)

The broader the basis and the lower the centre of gravity, the steeper
must the path of that centre be, and consequently the greater the stability.
Thus the disposition of the weight in a carriage may considerably affect its
stability by altering the place of the centre of gravity. A waggon loaded
with iron is much less easily overturned than when it is loaded with an
equal weight of hay ; supposing the inequality of the road or any acci-
dental obstacle, to elevate one side of the waggon, it will always recover its
position, provided that the centre of gravity remain within the vertical line
passing through the point of contact of the lower wheel and the ground ;
and it is obvious that the higher the centre of gravity is situated the sooner
it passes this line. If the velocity of the motion were very great, the wheel
which is elevated might be lifted off the ground by the momentum, and the
centre of gravity might thus be carried beyond the vertical line, by means
of an obstacle which would not have overset the waggon, if it had been
moving slowly. (Plate III. Fig. 44.)

If a person be sitting or standing in a carriage, the part of the carriage
on which he sits or stands may be considered as representing the place of
his weight, provided that his situation be always perpendicular ; but if the
motion be rapid he will not be able to remain constantly in a posture per-

ON PRESSURE AND EQUILIBRIUM. 49

fectly erect, and the centre of gravity of the carriage with its passengers,
will be somewhat more elevated than it would be on this supposition.

The direction of the initial motion of the centre of gravity readily ex-
plains the suspension of a weight or a bucket of water, on a rod resting on
the end of a table, when another rod is employed to keep the bucket at
such a distance from the end of the first, that the centre of gravity may be
under the table ; for although the bucket seems suspended by its handle,
yet if the handle began to descend, the centre of gravity would be obliged
to rise ; consequently the whole will retain its position, and remain at rest.
(Plate III. Fig. 45.)

The apparent ascent of a loaded cylinder on an inclined plane, and the
motion of a roller composed of two united cones with a common axis,*
resting on the edge of a triangle which is inclined to the horizon, may be
easily understood from the same consideration. (Plate III. Fig. 46.)

We may also observe in the equilibrium of animals many circumstances
illustrative of the properties of the centre of gravity. When a person stands
on one foot and leans forwards, in the attitude which is usually exhibited
in the statues of Mercury, the other foot is elevated behind, in order to
bring back the centre of gravity so as to be vertically over some part of the
foot on which he stands. But on account of the convex and irregular form
of the foot, the basis that it affords is really very narrow ; hence, wheji we
attempt to stand on one foot, we find it often necessary to use a muscular
exertion, in order to bring the point of support to that side towards which
we are beginning to fall ; and when the basis is still more contracted, the
body never remains at rest, but, by a succession of actions of this kind,
sometimes too minute to be visible, it is kept in a state of perpetual vibra-
tion, without ever attaining such a position as would give it any degree of
positive stability ; and thus it may be conceived to be supported even on a
single point, recovering its position from time to time by means of a slight
degree of rotatory motion, which is produced by its flexure and by the
changes of the position of the extremities : hence, by habit, the arts of
rope-dancers and balancers are acquired. Sometimes, however, the position
of the balancer is not so difficult to be preserved as it appears, the curva-
ture of the wire in contact with the foot tending materially to assist him.

When we attempt to rise from a seat, we generally draw our feet inwards,
in order to bring the point of support into, or near, the vertical line passing
through the centre of gravity, and to create a tottering equilibrium, which
is favourable for the beginning of motion. And before we rise, we bend
the upper part of the body forwards, in order to procure a momentum,
capable of carrying the centre of gravity beyond the vertical line passing
through the point of support.

When a horse is walking, the centre of gravity is sometimes supported
only by two feet of the same side, yet for a time so short that its declension
towards the other side is easily recovered, after the legs on that side have

* Krafft on the apparent Ascent of a Double Cone, Nov. Com. Petrop. vi. 389.
Kastner on a Cylinder appearing to roll upwards. Deutsche Schriften Soc. Gott. 1 13.
On the motion of a double cone, see also Kostonov. Nov. Act. Petr. 1789, vii. 229.
Brunings Hind. Arch. ii. 321.

50 LECTURE VII.

resumed their activity. Some authors have thought it impossible that a
quadruped should stand for an instant with both feet of the same side
raised from the earth ; but when a horse is walking fast, it may very often
be observed that the print of the hind foot is considerably more advanced
than that of the fore foot, which has been raised to make way for it.

From the general law of the equilibrium of the centre of gravity, we
may deduce the properties of levers of all kinds. It follows, from the defi-
nition of this point, that if two bodies be attached to a straight rod of in-
considerable weight, they may be sustained in equilibrium by a fixed
point or fulcrum, which divides their distance into portions which are in-
versely as their weights. And it is obvious that if any other equivalent
forces be substituted for weights, acting at the same distance from the
fulcrum, and with the same inclination to the rod or lever, the conditions
of equilibrium will be precisely the same. Also, if either of the forces be
transferred to an equal distance on the other side of the fulcrum, and act
there in a contrary direction, the equilibrium will still remain. Hence we
have two principal kinds of levers ; the first, in which the fixed point or
fulcrum is between the points at which the forces or weights are applied ;
the second, where the forces are applied in contrary directions, on the
same side of the fulcrum. (Plate III. Fig. 47.)

The demonstrations of the fundamental property of the lever have been
very various. Archimedes himself has given us two.* Huygens,t Newton,^
Maclaurin,§ Dr. Hamilton, || and Mr. Vince,1T have elucidated the same
subject by different methods of considering it. The demonstration of
Archimedes, as improved by Mr. Vince, is ingenious and elegant, but it is
neither so general and natural as one of Dr. Hamilton's, nor so simple and
convincing as Maclaurin's, which it may be worth our while to notice. Sup-
posing two equal weights, of an ounce each, to be fixed at the ends of the
equal arms of a lever of the first kind ; in this case it is obvious that there
will be an equilibrium, since there is no reason why either weight should
preponderate. It is also evident that the fulcrum supports the whole
weight of two ounces, neglecting that of the lever ; consequently we may
substitute for the fulcrum a force equivalent to two ounces, drawing the
lever upwards ; and instead of one of the weights, we may place the end
of the lever under a firm obstacle, and the equilibrium will still remain,
the lever being now of the second kind. Here, therefore, the weight re-
maining at the other end of the lever counterbalances a force of two
ounces, acting at half the distance from the new fulcrum ; and we may
substitute for this force a weight of two ounces, acting at an equal distance
on the other side of that fulcrum, 'supposing the lever to be sufficiently
lengthened, and there will still be an equilibrium. In this case the fulcrum
will sustain a weight of three ounces, and we may substitute for it a force
of three ounces acting upwards, and proceed as before. In a similar

* Archimedes de ^Equiponderantibus, and de Planorum

t Demonstratio ^Squilibrii Bilancis, Hist, et Mem. Paris, 1693.

J Principia, Laws of Motion, cor. 2. § View of Newton's Philosophy.

(I The Properties of the Mechanic Powers Demonstrated, Ph. Tr. 1763, liii. 103.

«|| Ph. Tr. 1794, Ixxxiv. 33. Philosophical Essays, 12mo, Lond. 1767.

ON PRESSURE AND EQUILIBRIUM. . 51

manner the demonstration may be extended to any commensurable propor-
tion of the arms, that is, any proportion that can be expressed by numbers ;
and it is easy to show that the same law must be true of all ratios what-
ever, even if they happen to be incommensurable, such as the side of a
square compared to its diagonal, which cannot be accurately expressed by
any numbers whatever ; the forces remaining always in equilibrium when
they are to each other inversely as the distances at which they are applied.

It is sometimes more convenient to have a series of levers acting on each
other with a moderate increase of power in each, than to have a single
lever equivalent in its effect. We may also bend either arm of a lever in
any manner that we please, without altering its power, provided that the
direction of the force be perpendicular to the line drawn to the fulcrum ;
or if the force be applied obliquely, it may always be imagined to act at
the end of a lever equal in length to the perpendicular let fall from the ful-
crum on the direction of the force. Thus, if two levers are connected by a rope
or bar, when the direction of one of them nearly coincides with that of the
rope, a force applied transversely to the lever acts with a great mechanical
advantage against the rope ; but as the inclination increases, the advantage
gradually diminishes, and changes, at last, to an equal advantage on the
side of the rope and the other lever to which it is attached. When, there-
fore, a great force is required in the beginning of the motion, and after-
wards a much smaller force with a greater velocity, this apparatus may be
extremely convenient : thus, in opening a steam valve, the pressure of the
steam is at first to be overcome, and after this, little or no additional force
is required ; and Mr. Watt has very ingeniously applied this arrangement
of levers to the purpose in his steam engines. In the same manner, it is
necessary that the platten of a printing press, or the part which presses the
paper on the types, should descend from a considerable height, but it is
only at the instant of taking off the impression that a great force is re-
quired ; and both these ends are obtained by similar means in a press
lately invented by Lord Stanhope. (Plate III. Fig. 48, 49.)

The wheel and axis bear a very strong resemblance to the lever. If two
threads, or perfectly flexible and inextensible lines, be wound in contrary
directions round two cylinders, drums, or rollers, moveable together on
the same axis, there will be an equilibrium when the weights attached to
the threads, or the forces operating on them, are inversely as the radii of
the cylinders, or as the diameters of which they are the halves. It may
easily be understood that the weights have the same power in turning
round the cylinders, as if they were immediately attached to the arms of a
lever equal in length to their semidiameter, and that the conditions of
equilibrium will be the same. The demonstration may also be more im-
mediately deduced from the position of the centre of gravity immediately
below the axis of the cylinders, which requires, the weights to be inversely
as the radii. With respect to stability, the equilibrium is neutral, and the
cylinders will remain at rest in any situation. A single cylinder is also
often combined with a lever or winch, and in this case the radius of the
cylinder is to be compared with the length of the lever or winch. (Plate
III. Fig. 50.)

E2

52 LECTURE VII.

Systems of wheels and pinions of various kinds resemble, in their mecha-
nical properties, either a series of levers, or the combination of cylinders
which constitutes the wheel and axis ; but the form of the teeth may pro-
duce a difference in their action, which will be mentioned when the prac-
tical construction of wheelwork is discussed.

Sometimes the axis connected with a winch is composed of two cylinders,
one end of the rope being uncoiled from the smaller, while the other end
winds round the larger ; the weight being supported by a pulley running
in its angle. Here the conditions of equilibrium are easily determined
from the place of the centre of gravity, and the effect of the machine is the
same as if the weight were attached to a rope coiled round a simple
cylinder, of a diameter equal to half the difference of the diameters of the
double axis. The machine is, however, much stronger than such a
cylinder would be, and does not require so great a curvature in the ropes
employed. (Plate IV. Fig. 51.)

The laws of the equilibrium of pullies have been referred, by some
writers on mechanics, to those of the lever ; but the comparison is both
unnecessary and imperfect ; in the simple case of two equal weights at-
tached to a thread passing over a single pulley, which is the only one that
allows us to recur to the properties of the lever, the conditions of equili-
brium are axiomatically evident, without any further reasoning ; and in
more complicated cases the calculations proceed on perfectly different
grounds. We are, therefore, to consider a pulley as a cylinder, moving
on an axis, merely in order to change the direction of a thread, without
friction ; for whatever is demonstrable of pullies or their combinations,
would be equally true of as many perfectly smooth grooves, which do not
bear the most distant analogy to the lever.

Now when the direction of a thread is altered, by passing over any per-
fectly smooth surface, it communicates the whole force acting on it ; for
the resistance of a surface, without friction, can only be in a direction
perpendicular to itself and to the thread, and the operation of any force
remains undisturbed by a resistance which is always in a direction per-
pendicular to it.

A fixed pulley, therefore, has no effect in gaining a mechanical ad-
vantage ; but by means of a moveable pulley it is obvious that a weight
may be supported by two forces, each equivalent to half the weight,
applied in a vertical direction to the extremities of the thread ; and these
forces may be derived from two weights, if the thread be made to pass over
two fixed pullies in a proper position ; and if one of the ends be attached
to a fixed point, and the other remain connected to its weight, the equi-
librium will continue unimpaired, each portion of the thread still support-
ing one half of the original weight ; so that, by means of a single moveable
pulley, one body may retain in equilibrium another of double its weight.
(Plate IV. Fig. 52, 53.)

The modes of arranging pullies are very various, but the advantage
which they procure may always be estimated, from the consideration that
every part of the same thread must be equally stretched ; and where there
is only one thread, the weight will be divided equally among all the por-

ON PRESSURE AND EQUILIBRIUM. 53

tions which help to support the moveable block, each of them bearing a
weight equivalent to the force which is applied at the end of the thread.
In the common ship's blocks, the pullies or shieves are equal in magni-
tude, and placed side by side ; here their number cannot conveniently
exceed two or three, without causing an obliquity in the block, when the
force is applied to the rope. Mr. Smeaton,* for this reason, invented a
system of pullies, arranged in two rows in each block, one larger, and the
other smaller : the force being applied in the middle, the rope passes on the
larger pullies till it arrives at the last, then returns through the whole of
the smaller series to the opposite side, and comes back again on the larger,
to be finally attached in the middle. (Plate IV. Fig. 54... 56.)

If the diameters of all the pullies in both blocks be taken in the ratio of
the number of portions of the thread intervening between them and the
fixed extremity, their angular velocity will be equal, each of them turning
on its axis in the same time. They may therefore be fixed to a single axis
in each block ; and in this case the axis being longer, there will be less
accidental friction from its want of steadiness, and even the necessary fric-
tion may, perhaps, be somewhat diminished. (Plate IV. Fig. 57.)

If one end of a thread supporting a moveable pulley be fixed, and the
other attached to another moveable pulley, and the threads of this pulley
be similarly arranged, the weight will be counterpoised by a ppwer which
is found by halving it as many times as there are moveable pullies ; for it
is obvious that each of these pullies doubles the effect of the power.
(Plate IV. Fig. 58.)

There are also other arrangements, by which the effect of pullies may be
increased or diversified : for instance, where one end of each rope is attached
to the weight to be moved ; or where two of the pullies are connected by a
rope passing over a third ; but these methods are of little practical utility.
(Plate IV. Fig. 59, 60.)

We have hitherto supposed the ropes passing over the pullies to be either
perfectly or very nearly parallel to each other ; but when their directions
are oblique the forces applied to them require to be modified accordingly.
Thus, if two threads be attached to a weight, and passed over two pullies
fixed at a distance from each other, so that two equal weights may be
attached to their extremities, the depression of the first weight below either
pulley will be to its distance from the pulley, in the same proportion as
half of the weight to either of the other weights ; and if, instead of having
a weight attached to it, one end of the thread be fixed to a firm obstacle,
the effect will be precisely the same. A machine of this kind is sometimes
called a swig, perhaps by corruption from swing. (Plate IV. Fig. 61.)

If all the weights are unequal, we must draw a triangle of which the
three sides are in the same proportions as the weights ; and we may deter-
mine the directions of the threads by placing such a triangle, with the side
representing the middle weight in a vertical position.

A force may also be applied obliquely to a wheel and axis. Supposing a
'rope to be coiled obliquely round the axis, it will require, in order to pre-
serve the equilibrium, a force as much greater than would be sufficient, if
* Ph.Tr. 1752, xlvii. 404.

54 LECTURE VII.

it were simply applied in the direction of the motion, as the length of any
part of the rope uncoiled is greater than the perpendicular distance of its
extremity from the axis. So that when the rope becomes very oblique, a
great force is required, in order to counteract a much smaller one- acting
perpendicularly. This remark may be in some measure illustrated by
considering the method used by joiners and stone cutters for keeping a saw
straight : two ropes or braces are twisted together by means of a pin or
lever passing between them, and serve each other in place of an axis, round
which they are coiled obliquely, so that they act with great force, when
they are sufficiently tight and not too much twisted. (Plate IV. Fig. 62.)

It appears from the laws which have already been laid down, respecting
the motions of bodies on inclined surfaces, that a weight acting vertically
will hold in equilibrium another weight resting on an inclined plane, with-
out friction, when the first is to the second as the height of the plane to its
oblique length. The pressure on the plane is in this case to the weight
resting on it, as the horizontal length of the plane is to its oblique length.
This pressure may be measured experimentally, by substituting for the
resistance of the plane that of a thread perpendicular to it. (Plate IV.
Fig. 63.)

The same principles are applicable to the equilibrium of the wedge. A
wedge is a solid which has three plane faces inclined to each other, and
two triangular ends ; and we suppose the faces perfectly polished, so as to
be free from friction, and that no force can act on them otherwise than in
a perpendicular direction. Now in order that three forces, acting on the
faces or sides of a wedge, may hold each other in equilibrium, each of them
must be in proportion to the length of the side on which it acts : they must
also be applied at such parts that their directions may meet in one point ;
for otherwise they will not be completely opposed to each other, and a
rotatory motion will be produced. (Plate IV. Fig. 64.)

If each face of the wedge were conceived to be capable of receiving a
pressure, not only in a perpendicular direction, but in any other direction
at pleasure, as some authors have supposed, the instrument would lose its
essential character as a wedge ; but in such cases the proportion of the
forces required for the state of equilibrium may always be determined by
drawing a triangle with its sides parallel to their directions.*

It happens, however, not uncommonly, that the force actually operating
on the wedge is derived from another force acting in a direction more or
less oblique, as when a heavy body rests on one of the faces of the wedge
which is inclined to the horizon, the body being retained in its situation by
an obstacle or a thread which confines it to a vertical line, and the sliding
away of the wedge being prevented by a horizontal force. A wedge so
situated, and supposed to be capable of sliding without friction on a hori-
zontal surface, is sometimes called a moveable inclined plane, and it will
support the weight resting on it, if the horizontal force be to the weight as
the height of the plane is to its horizontal length. If the thread or the
obstacle helping to support the weight be placed in any other direction, the

* See Whewell's Mechanics.

ON PRESSURE AND EQUILIBRIUM. 55

magnitude of the forces must be determined from the general law of the
composition of three pressures. (Plate V. Fig. 65.)

If a prop or bar, leaning against a smooth vertical surface or wall, be
employed to support or to raise a weight, by means of a force which draAvs
its base along a smooth horizontal surface, the horizontal force must be to
the weight as the distance of the bottom of the prop from the wall to its
perpendicular height. And from similar principles, the conditions of the
equilibrium of arches, domes, and roofs may be determined. (Plate V.
Fig. 66, 67.)

The action of a screw depends on the same principles as that of an
inclined plane ;* for by rolling a thin and flexible wedge, for instance a
triangular piece of card, round a cylinder, we form a screw. We may
consider the force tending to turn the screw round its axis, as applied hori-
zontally to the base of the wedge, and the weight which is to be raised as
acting vertically on its inclined surface : the circumference of the cylinder
will represent the horizontal length of the wedge, and the distance between
the threads, measured in the direction of the axis, will be its height, pro-
vided that the threads be single ; consequently, the forces required for the
equilibrium are to each other as the height of one spire to the circumference
of the screw. But besides these forces, it is necessary that some obstacle
be present, which may prevent the body on which the screw acts from
following it in its motion round its axis ; otherwise there can be no equi-
librium. (Plate V. Fig. 68.)

The cylinder, which is the foundation of a screw, may be either convex
or concave, making a cylindrical or a tubular screw, and these, when fitted
together, are sometimes called a screw and a nut. The nut acts on the
screw with the same mechanical power as a single point would do, since it
only divides the pressure among the different parts of the spire. In general
the screw is applied in combination with a lever, in order to procure an
advantage in overcoming the friction, which is always considerable in the
simple screw and nut, and which would resist a force applied immediately
at the circumference, without any diminution of its power. Sometimes the
spires of a screw are made to act on the teeth of a wheel, when a very slow
motion of the wheel, or a very rapid motion of the screw, is required for
the purposes of the machine. (Plate V. Fig. 69, 70.)

The power of screws may be increased, in a great proportion, by means
of an arrangement invented by Mr. Hunter ;t which is somewhat similar,
in its operation, to the double axis already described. A cylindrical screw
is bored, and made at the same time a tubular screw, with a little difference
in the distances of the threads ; so that when it is turned within a fixed nut
it rises or sinks a little more or less than the internal screw which perfo-
rates it would rise or sink by the action of its own threads, and a weight
attached to this internal screw ascends, in each revolution, only through a
space equal to the difference of the height of the two coils. Here the ma-

** Leupold. Theat. Machin. t. 6, 7. Com. Bon. iii. 131, 304. Kastner on the
Screw. Commentationes Soc. Gott. 4to, 1795, xiii. M. i. 47, 1797, xiv. M. 3.
Ibid, de Theoria Cochlese. Diss. VI. 38. Nicholson's Jour. i. 158.

f Essay on a New Method of applying the Screw, Ph. Tr. 1781, Ixxi. 58.

56 LECTURE VII.

chine is analogous to a very thin wedge, of which the thickness is only
equal to the difference of the distances of the threads, and which of course
acts with a great mechanical advantage. It might in some cases be more
convenient to make two cylindrical screws, of different kinds, at, different
parts of the same axis, rather than to perforate it. The friction of such
machines is, however, a great impediment to their operation. (Plate V.
Fig. 71.)

In all the kinds of equilibrium that we have considered, and in all other
cases that can be imagined, it will be found that the forces, or rather
weights, opposed to each other, are so arranged that if they were put in
motion, their momenta in the direction of gravity would, in the first
instance, be equal and contrary, the velocity being as much greater as the
magnitude of the weight is smaller.* Thus, if an ounce weight, placed on
a lever, at the distance of four feet from the fulcrum, counterpoise a weight
of four ounces at the distance of one foot, the velocity with which the
ounce would descend, if the lever were moved, would be four times as
great as that with which the weight of four ounces would descend. A
single moveable pulley ascends with half the velocity of the end of the rope
which is drawn upwards, and acts with a force twice as great ; a block of
three shieves enables a weight to sustain another six times as great ; but
the velocity with which this weight ascends, is only one sixth of that with
which the smaller weight must descend. When a weight rests on an in-
clined plane, of wrhich the height is one half of the length, it may be
retained by the action of a weight of half its magnitude, drawing it up
the plane by means of a thread passing over a pulley. Here if the weight
ascended or descended along the oblique surface, its velocity, reduced to a
vertical direction, would be half as great as that of the smaller weight
which balances it.

Some authors have considered this law as affording a fundamental de-
monstration of the conditions of equilibrium in all possible cases.t For
since, wherever two weights are in equilibrium, if one of them descended,
the other must ascend with an equal quantity of motion, it appears absurd
to suppose that the force of gravitation could produce these two equal and
contrary effects at the same time. But it is more satisfactory to trace, in
every case, the steps by which the immediate actions of the different
weights are enabled to oppose each other ; and the general law may then
be inferred, by induction, from the agreement of the particular results, in
confirmation of the general reasoning which tends to establish its truth.

LECT. VII.— ADDITIONAL AUTHORITIES.

Mechanical Powers. — Roberval's Paradox, Leupold, Theatrum St. 4 t. 17. Lud-
lam's Essays, 1770.

Equilibrium. — Varignon on Composition of Forces, Hist. etMem. de Paris, 1714,
280, H. 87. Riccati, Comm. Bon. ii. II. 305 ; III. 215 ; v. II. 186. Foncenex,
Miscel. Taurin. ii. II. 299. Euler, Hist, et Mem. de 1'Acad. de Berlin, 1762, p. 265.

* Varro de Motu, Geneva, 1584, Th. 1.

f Lagrange, Mecanique Analytique, 4to, 1788, and 2 vols. 4to, 1811.

ON COLLISION. 57

Acta Petrop. iii. II. 106. Belidor, Ingenieur Fran?ais. Fuss, Nova Acta Pe-
trop. 1788, vi. 197. Nicholson's Journal, iv. 443.

Virtual Velocities. — Galileo, Dial. 1592. De Caus, les Raisons des Forces Mou-
vantes, Antwerp. Bp. Wilkins's Mathematical Magic, 1648. J. Bernoulli, in Va-
rignon's Mec. 1717. D'Alembert, Hist, et Mem. 1769, p. 278. Lagrange on a
Property of the Centre of Gravity, Ac. Berl. 1783, p. 290. Do. on Virtual Velo-
cities, Journal Poly technique, ii. V. 115. Fossombroni sul Principio delle Velocita
Virtuali, 4to, Flor. 1796. Essay on Virtual Velocities, Journal de Physique, xlviii.
210. Fourrier and Prony on Do. Journal Poly technique, ii. V. 20, 191. Buquoy,
Analytische Bestimmung des Gesetzes der Virtuellen Geschwindigkeiten, Leips.
1812. Do. Weitere Entwickerung, do. 1814. Do. Exposition d'un Nouveau
Principe General de Dynamique, dont le Principe des v. v. n'est q'un cas particulier,
4to, Paris, 1815. Pagani, Mem. de 1'Acad. de Bruxelles, 1825, iii. Gauss in
Crelle's Journal, Band 4. Mobius Lehrbuch der Statik, Leipz. 1837.

LECTURE VIII.

ON COLLISION.

HAVING inquired into the laws and properties of the motions and rest of
single bodies under the operation of one or more forces, and into the equi-
librium of these forces in different circumstances, we are next to examine
some simple cases of the motions of various moveable bodies acting recipro-
cally on each other. In all problems of this kind, it is of importance to
recollect the general principle already laid down respecting the centre of
inertia [gravity] that its place is not affected by any reciprocal or mutual
action of the bodies constituting the system.

Whenever two bodies act on each other so as to change the direction of
their relative motions, by means of any forces which preserve their
activity undiminished at equal distances on every side, the relative veloci-
ties with which the bodies approach to or recede from each other, will
always be equal at equal distances. For example, the velocity of a comet,
when it passes near the earth in its descent towards the sun, is the same
as its velocity of ascent in its return, although at different distances its
velocity has undergone considerable changes. In this case, the force acts
continually, and attracts the bodies towards each other ; but the force
concerned in collision, when a body strikes or impels another, acts only
during the time of more or less intimate contact, and tends to separate the
bodies from each other. When this force exerts itself as powerfully in
causing the bodies to separate as in destroying the velocity with which they
meet each other, the bodies are called perfectly elastic : when the bodies
meet each other without a re-action of this kind, they are called more or
less inelastic. Ivory, metals, and elastic gum, are highly, and almost
perfectly elastic : clay, wax mixed with a little oil, and other soft bodies,
are almost inelastic : and the effects of inelastic bodies may be imitated by
elastic ones, if we cause them to unite or adhere after an impulse, so as
to destroy the effect of the repulsive force which tends to separate them.

58 LECTURE VIII.

When two bodies approach to each other, their form is in some degree
changed, and the more as the velocity is greater. In general, the repulsive
force exerted is exactly proportional to the degree in which a body is com-
pressed ; and when a body strikes another, this force continues to be
increased until the relative motion has been destroyed, and the bodies are
for an instant at rest with respect to each other ; the repulsive action then
proceeds with an intensity which is gradually diminished, and if the
bodies are perfectly elastic they re-assume their primitive form, aud separate
with a velocity equal to that with which they before approached each other.
Strictly speaking, the repulsion commences a little before the moment of
actual contact, but only at a distance which in common cases is imper-
ceptible. The change of form of an elastic substance, during collision, is
easily shown by throwing a ball of ivory on a slab of marble or a piece of
smooth iron, coloured with black lead or printing ink ; or by suffering it
to fall from various heights : the degree of compression will then be indi-
cated by the magnitude of the black spot which appears on the ball. It
may be shown, from the laws of pendulums, that, on the supposition that
the force is proportional to the degree of compression, its greatest exertion
is to the weight of a striking body, as the height from which the body
must have fallen, in order to acquire its velocity, to half the depth of the
impression.

For making experiments on the phenomena of collision, it is most con-
venient to suspend the bodies employed by threads, in the manner of
pendulums ; their velocities may then be easily measured by observing the
chords of the arcs through which they descend or ascend, since the veloci-
ties acquired in descending through circular arcs are always proportional
to their chords ; and for this purpose, the apparatus is provided with a
graduated arc, which is commonly divided into equal parts, although it
would be a little more correct to place the divisions at the ends of arcs, of
which the chords are expressed by the corresponding numbers. (Plate V.
Fig. 72.)

The simplest case of the collision of elastic bodies is when two equal
balls descend through equal arcs, so as to meet each other with equal
velocities. They recede from each other after collision with the same
velocities, and rise to the points from which they before descended, with a
small deduction for the resistance of the surrounding bodies.

When a ball at rest is struck by another equal ball, it receives a velocity
equal to that of the ball which strikes it, and this ball remains at rest.
And if two equal balls meet or overtake each other with any unequal
velocities, their motions will be exchanged, each rising to a height equal
to that from which the other descended.

The effect of collision takes place so rapidly, that if several equal balls
be disposed in a right line in apparent contact with each other, and another
ball strike the first of them, they will all receive in succession the whole
velocity of the moving ball before they begin to act on the succeeding ones ;
they will then transmit the whole velocity to the succeeding balls, and
remain entirely at rest, so that the last ball only will fly off.

In the same manner, if two or more equal balls, in apparent contact, be

ON COLLISION. 59

in motion, and strike against any number of others placed in a line, the
first of the moving balls will first drive off the most remote, and then the
second the last but one, of the row of balls which were at rest : so that
the same number of balls will fly off together on one side, as descended to
strike the row of balls on the other side ; the others remaining at rest.

If the line of balls, instead of being loosely in contact, had been firmly
united, they would have been impelled with a smaller velocity, and the
ball striking them would have been reflected. For when a smaller elastic
body strikes a larger, it rebounds with a velocity less than its first velocity,
and the larger body proceeds also with a less velocity than that of the
body striking it. But if a larger body strikes a smaller, it still proceeds
with a smaller velocity, and the smaller body advances with a greater.

The momentum communicated by a smaller elastic body to a larger one
is greater than its own, and when the first body is of a magnitude compa-
ratively inconsiderable, it rebounds with a velocity nearly as great as the
velocity of its impulse, and the second body acquires a momentum nearly
twice as great as that of the first. When a larger body strikes a smaller
one, it communicates to it only as much momentum as it loses.

In the communication of motion between inelastic bodies, the want of a
repulsive force, capable of separating them with an equal relative velocity,
is probably owing to a permanent change of form ; such bodies receiving and
retaining a depression at the point of contact. When the velocity is too
small to produce this change of form, the bodies, however inelastic, may
usually be observed to rebound a little.

Bodies which are perfectly inelastic, remain in contact after collision ;
they must therefore proceed with the same velocity as the centre of inertia
[gravity] had before collision. Thus, if two equal balls meet, with equal
velocities, they remain at rest ; if one is at rest, and the other strikes it,
they proceed with half the velocity of the ball which was first in motion.
If they are of unequal dimensions, the joint velocity is as much smaller
than that of the striking ball, as the weight of this ball is smaller than the
sum of the weights of both balls. And in a similar manner the effects of
any given velocities in either ball may be determined.

It follows immediately from the properties of the centre of inertia [gra-
vity] that in all cases of collision, whether of elastic or inelastic bodies,
the sum of the momenta of all the bodies of the system, that is of their
masses or weights multiplied by the numbers expressing their velocities, is
the same, when reduced to the same direction, after their mutual collision,
as it was before their collision. When the bodies are perfectly elastic, it
may also be shown that the sum of their energies or ascending forces, in
their respective directions, remains also unaltered.

The term energy may be applied, with great propriety, to the product of
the mass or weight of a body, into the square of the number expressing its
velocity. Thus, if a weight of one ounce moves with the velocity of a foot
in ^a second, we may call its energy 1 ; if a second body of two ounces
have a velocity of three feet in a second, its energy will be twice the square
of three, or 18. This product has been denominated the living or ascend-
ing force [the vis viva], since the height of the body's vertical ascent is in

60 LECTURE VIII.

proportion to it ; and some have considered it as the true measure of the
quantity of motion ; but although this opinion has been very universally
rejected, yet the force thus estimated well deserves a distinct denomina-
tion. After the considerations and demonstrations which have been pre-
mised on the subject of forces, there can be no reasonable doubt with
respect to the true measure of motion ; nor can there be much hesitation in
allowing at once, that since the same force, continued for a double time, is
known to produce a double velocity, a double force must also produce a
double velocity in the same time. Notwithstanding the simplicity of this
view of the subject, Leibnitz,* Smeaton,t and many others have chosen to
estimate the force of a moving body by the product of its mass into the
square of its velocity ; and though we cannot admit that this estimation
of force is just, yet it may be allowed that many of the sensible effects of
motion, and even the advantage of any mechanical power, however it may
be employed, are usually proportional to this product, or to the weight of
the moving body, multiplied by the height from which it must have fallen,
in order to acquire the given velocity. Thus a bullet, moving with a
double velocity, will penetrate to a quadruple depth in clay or tallow : a
ball of equal size, but of one fourth of the weight, moving with a double
velocity, will penetrate to an equal depth : and, with a smaller quantity of
motion, will make an equal excavation in a shorter time. This appears at
first sight somewhat paradoxical : but, on the other hand, we are to con-
sider the resistance of the clay or tallow as a uniformly retarding force,
and it will be obvious that the motion, which it can destroy in a short
time, must be less than that which requires a longer time for its destruc-
tion. Thus also when the resistance, opposed by any body to a force tend-
ing to break it, is to be overcome, the space through which it may be bent
before it breaks being given, as well as the force exerted at every point of
that space, the power of any body to break it is proportional to the energy
of its motion, or to its weight multiplied by the square of its velocity.

In almost all cases of the forces employed in practical mechanics, the labour
expended in producing any motion, is proportional, not to the momentum,
but to the energy which is obtained ; since these forces are seldom to be
considered as uniformly accelerating forces, but generally act at some dis-
advantage when the velocity is already considerable. For instance, if it
be necessary to obtain a certain velocity, by means of the descent of a
heavy body from a height to which we carry it by a flight of steps, we
must ascend, if we wish to double the velocity, a quadruple number of
steps, and this will cost us nearly four times as much labour. In the same
manner, if we press with a given force on the shorter end of a lever, in
order to move a weight at a greater distance on the other side of the ful-
crum, a certain portion of the force is expended in the pressure which is
supported by the fulcrum, and we by no means produce the same mo-

* Acta Erudit. Lips. 1686.

t Ph. Tr. 1776, p. 450, and 1782, p. 337. See Desaguliers's Exp. Ph. ii. 92 ;
and Ph. Tr. 1723, xxxii. 269, 285. Eames on the Force of Moving Bodies, Ph. Tr.
1726, xxxiv. 188. Clarke in Ph. Tr. 1728, xxxv. 381. Zendrini, Sulla Inutilita
della Questione Intorno alia Misura delle Forze Vivi, 8vo, Venezia, 1804.

ON COLLISION. 61

mentum as would have been obtained by the immediate action of an equal
force on the body to be moved.

An elastic ball of 2 ounces weight, moving with a velocity of 3 feet in a
second, possesses an energy, as we have already seen, which may be ex-
pressed by 18. If it strike a ball of 1 ounce which is at rest, its velocity
will be reduced to 1 foot in a second, and the smaller ball will receive a
velocity of 4 feet : the energy of the first ball will then be expressed by 2,
and that of the second by 16, making together 18, as before. The mo-
mentum of the larger ball after collision is 2, that of the smaller 4, and the
sum of these is equal to the original momentum of the first ball.

Supposing the magnitude of an elastic body which is at rest to be
infinite, it will receive twice the momentum of a small body that strikes
it ; but its velocity, and consequently its energy, will be inconsiderable,
since the energy is expressed by the product of the momentum into the
velocity. And if the larger body be of a finite magnitude, but still much
greater than the smaller, its energy will be very small ; that of the smaller,
which rebounds with a velocity not much less than its original velocity,
being but little diminished. It is for this reason that a man, having a
heavy anvil placed on his chest, can bear, without much inconvenience, the
blow of a large hammer striking on the anvil, while a much slighter blow
of the hammer, acting immediately on his body would have fractured his
ribs, and destroyed his life. The anvil receives a momentum nearly twice
as great as that of the hammer ; but its tendency to overcome the strength
of the bones and to crush the man, is only proportional to its energy, which
is nearly as much less than that of the hammer, as four times the weight of
the hammer is less than the weight of the anvil. Thus, if the weight of
the hammer were 5 pounds, and that of the anvil 100, the energy of the
anvil would be less than [only] one fifth as great as that of the hammer,
besides some further diminution, on account of the want of perfect elas-
ticity, and from the effect of the larger surface of the anvil in dividing the
pressure occasioned by the blow, so as to enable a greater portion of the
chest to cooperate in resisting it.

When a body strikes another in a direction which does not pass through
its centre of gravity, the effect produced involves the consideration of
rotatory motion, since, in this case, the body is made to revolve on an axis.
But this can never happen when the body is spherical, and its surface
perfectly polished ; since every impulse must then be perpendicular to the
surface, and must consequently be directed to the centre of the body. If
the motion of a ball which strikes another is not directed to its centre, the
surface of contact must be oblique with respect to its motion, and the
second ball will only receive an impulse in a direction perpendicular to
this surface, while the first receives, from its reaction, an equal impulse in
a contrary direction, which is combined with its primitive mption. The
magnitude of this impulse may be determined by resolving the motion of
the first ball into two parts, the one parallel to the surface of contact, and
the other perpendicular ; the first part remaining always unaltered, the
second being modified by the collision. If, for example, the balls were
equal, this second part of the motion would be destroyed, and the remain-

62 LECTURE VIII.

ing motion would be in the direction of the surface of contact, and perpen-
dicular to that of the ball impelled.

Hence it follows, that if we wish to impel a billiard ball * in a given
direction, by the stroke of another ball, we have only to imagine a third
ball to be placed in contact with the first, immediately behind it in the
line of the required motion, and to aim at the centre of this imaginary ball ;
the first ball will then be impelled in the required direction, and the second
will also continue to move in a direction perpendicular to it.

By a similar resolution of the motion of an elastic ball, we may deter-
mine its path, when it is reflected from a fixed obstacle. That part of the
motion, which is in a direction parallel to the surface of the obstacle, re-
mains undiminished : the motion perpendicular to it is changed for an
equal motion in a contrary direction, and the joint result of these consti-
tutes a motion, in a direction which is equally inclined to the surface with
the first motion, but on the opposite side of the perpendicular. Of this we
have also a familiar instance in the motions of billiard balls ; for we may
observe, that a ball rebounds from the cushion in an angle equal to that in
which it arrives at it ; and if we wish that our ball, after reflection, should
strike another placed in a given situation, we may suppose a third ball to
be situated at an equal distance, on the other side of the cushion, and aim
at this imaginary ball : our ball will then strike the second ball, after re-
flection, with a direct impulse. We here suppose the reflection to take
place when the centre of the ball arrives at the cushion, while in fact the
surface only comes into contact with it ; if we wish to be more accurate,
we may place the imaginary ball at an equal distance beyond the centre of
a ball lying in contact with the nearest part of the cushion, instead of
measuring the distance from the cushion itself. (Plate V. Fig. 73.)

When the number of bodies, which meet each other, is greater, and their
magnitudes and motions are diversified, the calculation of the effects of
collision becomes very intricate, and the problem is scarcely applicable to
any practical purpose. Those who are desirous of pursuing the investiga-
tion as a mathematical amusement, will find all the assistance that they
require in the profound and elegant works of Maclaurin.

LECT. VIII.— ADDITIONAL AUTHORITIES.

Galileo, Op. i. 957, ii. 479. Wallis, Wren, Huygens, in Ph. Tr. 1668-69-71. In
the last, Wallis gives a correct view of momentum. Mariotte, Traite de la Percussion
des Corps, 12mo, Par. 1673. Borellus de vi Percussionis, 4to, Lugd. 1686. Saulmon
— Mairan— Molieres, Hist.et Mem. de Paris, 1721, pp. 23 ; 1722, p. 23, 38,40 ; 1726.
Gravesande, Essai d'uneNouvelleTheoriedu Choc des Corps fondee sur 1' Experience,
12mo, La Haye, 1722. Maclaurin's Fluxions, 2 vols. 4to, 1742. Milner, Ph. Tr.
1788, p. 344. Euler, Comm. Petr. v. 159 ; ix. 50. N. Comm. Petr. xv. 414 ; xvii.
315. Mem. de Berl. 1745, p. 21. Theoria Motus Corporum Solid. &c. &c.

* BiUiards, Encyclopedic Methodique, pi. 4 ; Art. Pausmerie, pi. 4, 5 ; Art.
Amusemens de Mecanique. Coriolis, Theorie Mathematique du Jeu de Billiard, 8vO.

63

LECTURE IX

ON THE MOTIONS OF CONNECTED BODIES.

THE motions of single bodies, acting in any manner on each other, which
we have been considering, as far as they belong to the effects of collision,
are of less importance to practical mechanics, than the affections of such
bodies as are united, so as either to revolve round a common centre, or to
participate in each other's motions by any kind of machinery.

It is only within half a century, that the phenomena and effects of rota-
tory motion have been sufficiently investigated. Newton committed a
mistake, which is now universally acknowledged, in his computation of the
precession of the equinoxes, for want of attending sufficiently to the subject ;
and it is of importance in the calculation of many of the effects of me-
chanical arrangements, that it should be treated in an accurate manner.

The effect of a moving body in producing motion in any other bodies, so
connected as to be capable of turning freely round a given centre, is jointly
proportional to its distance from that centre, and to its momentum in the
direction of the motion to be produced. Thus a body, of one pound weight,
moving with a velocity of one foot in a second, will have three times as
great an effect on a system of bodies, to which its whole force is communi-
cated, at the distance of one yard from the centre of their motion, as if it
acted only at the distance of a foot, on the same system of bodies : a double
weight, or a double velocity, would also produce a double effect. For,
supposing two unequal bodies to be connected by an inflexible line, and to
move with equal velocities in a direction perpendicular to that of the line,
it is demonstrable, from the principles of the composition of motion, that
they may be wholly stopped by an obstacle applied to the centre of gravity,
consequently their effects in turning the line round this point are equal ;
here the momenta are proportional to the weights, but the products obtained
by multiplying them by the distances from the centre, at which they act,
are equal : these products therefore represent the rotatory power of the
respective bodies. Hence in a connected system of bodies, revolving round
a given point, with equal angular velocities, the effect produced by the
rotatory motion of each body, as well as the force which is employed in
producing it, is expressed by the product of the mass multiplied by the
square of the velocity, since the velocity is in this case proportional to the
distance from the centre ; and this product is the same that I have denomi-
nated the energy of a moving body.

These propositions are of great use in all inquiries respecting the opera-
tions of machines ; and it is of importance to bear in mind, that although
the equilibrium of a system of bodies is determined by the equality of the
products of their weights into their effective distances on each side of the
centre, yet that the estimation of the mechanical power of each body, when
once in motion, requires the mass to be multiplied by the square of the
distance, or of the velocity. For this reason, together with some others,

64 LECTURE IX.

which have been already mentioned, some have considered the square of the
velocity as affording the true measure of force ; but the properties of
motion, concerned in the determination of rotatory power, are in reality
no more than necessary consequences of the simpler laws on which the
whole theory of mechanics is founded.

The effects of rotatory motion may be very conveniently examined, by
means of an apparatus similar to that which was employed for the same
purpose by Mr. Smeaton.* A vertical axis is turned by a thread passing
over a pulley, and supporting a scale with weights ; the thread may be
applied at different parts of the axis, having different diameters, and the
axis supports two arms, on which two leaden weights are fixed, at distances
which may be varied at pleasure. The same force will then produce, in
the same time, but half the velocity, in the same situation of the weights,
when the thread is applied to a part of the axis of half the diameter : and
if the weights are removed to a double distance from the axis, a quadruple
force will be required, in order to produce an equal angular velocity in a
given time. (Plate V. Fig. 74.)

When a number of connected bodies, or a single body of considerable
magnitude, is made to revolve round a centre, it is sometimes necessary to
inquire into what point their masses might be supposed to be concentrated
so as to preserve the same rotatory power with the same angular velocity.
This point is called the centre of gyration. In a circle, or any portion of a
circle, turning round its centre, the square of the distance of this point from
the centre, is half the square of the semidiameter ; and the whole effect of
the momentum of the circle upon an obstacle at its circumference, is exactly
half as great as that of an equal quantity of matter, striking the obstacle
with the velocity of the circumference.

There is another point, of which the determination is of considerable
utility in many mechanical problems : this is the centre of percussion ; or
the point at which an obstacle must be applied, in order to receive the whole
effect of a stroke of a body which is revolving round a given centre, with-
out producing any pressure or strain on the centre or axis of motion. In
a straight line, or a slender rod fixed at one extremity, the distance of this
point from the centre of motion is two thirds of the whole length.t

The same point is also the centre of oscillation, the distance of which
determines the time of oscillation or vibration of the body, suspended as
a pendulum upon the given centre of motion.^ It may easily be shown
that a rod a yard long, and of equable thickness, suspended at one ex-
tremity, vibrates in the same time as a ball suspended by a thread of which
the length is two feet. But if the rod were suspended on a centre at some
point within its extremities, the time of its vibration would be prolonged,
so as to become equal to that of a simple pendulum of much greater length.

* Ph. Tr. 1776, Ixvi. 450, and plate. See an examination of this paper in
Atwood, p. 382.

t Lahire, Hist, et Mem. Paris, ix. 175. Parent, ibid. 1700, H. 149. Bernoulli,
ibid. 1703, pp. 78, 272, H. 114 ; 1704, p. 136, H. 89. Clairaut, ibid. 1735, p. 281,
H. 92.

J Huygens, Hist, et Mem. de 1'Acad. x. 446, 462, and Hor. Osc. 121. John Ber-
noulli de Natura Centri Oscil. 1714. Taylor, Ph. Tr. 1713.

ON THE MOTIONS OF CONNECTED BODIES. 65

This may be illustrated by two balls fixed at the end of a rod, with a centre
of suspension moveable to any part of the rod, for as the centre approaches
the middle of the rod, the vibrations are rendered extremely slow. (Plate V.
Fig. 75.)

The rotatory motion of bodies not fixed on an axis might be considered
in this place, but the subject involves in its whole extent some intricacy of
calculation, and, except in astronomy, the investigation is scarcely ap-
plicable to any problems which occur in practice. We may, however,
examine a few of the simplest cases. If two bodies be supposed to be con-
nected by an inflexible line, and to be moving with equal velocities in
parallel directions ; if an immoveable obstacle be applied, so as to form a
fulcrum, at the common centre of gravity, they will, as we have already
seen, be wholly stopped ; but if the fulcrum be applied to any other part of
the line, one of the bodies will move forwards, and the other backwards, with
a velocity which may easily be determined by calculating their rotatory
power with respect to the fulcrum. If the fulcrum be applied at a point
of the line continued beyond the bodies, the one will lose and the other
gain velocity ; since the quantity of rotatory power will always remain
unaltered : that point only which is denominated the centre of oscillation
retaining its original velocity. Now the same inequality in the motion of
the bodies, and consequently the same angular velocity of rotation will be
produced, if the connected bodies be initially at rest, and the fulcrum be
applied to them with the same relative velocity. For example, if a straight
rod or wire receive an impulse at one end in a transverse direction, the
centre of oscillation, which is at the distance of two thirds of the length
from the end struck, will at the first instant remain at rest, consequently
the centre will move with one fourth of the velocity of the impulse, and
this must be the velocity of the progressive motion of the rod, since the
centre of gravity of any body which is at liberty moves always with an
equable velocity in a right line, while the whole rod will also revolve
equably round its centre, except such retardations as may arise from
foreign causes. In a similar manner the computation may be extended to
bodies of a more complicated form. Thus it has been calculated at what
point of each planet an impulse must have operated, in order to communi-
cate to it at one blow its rotation and its progressive motion in its orbit.*

Those who have asserted that the motion of the centre of gravity of a
body can only be produced by an impulse which is either wholly or partly
directed towards it, have obviously been mistaken. The centre of oscilla-
tion is the only point which remains at rest with regard to the first effect
of the stroke, and the centre of gravity, which never coincides with the
centre of oscillation, moves in the direction of the impulse, while the parts
beyond the centre of oscillation begin to move in a contrary direction.
Hence it is that a thin stick may be broken by a blow on the middle, with-
out injuring the glasses on which it is supported : fo'r the ends of the stick,
instead of being depressed by the stroke, would rise with half the velocity
of the body which strikes them, if the two portions were separated without

* John Bernoulli, Op. vol. 4, 284. Consult Whewell, Dynamics, 1823, c. 8.

F

66 LECTURE IX.

the loss of any force. But unless some art has been previously employed
in producing a partial separation, it will frequently be found that the stick
has strength enough to break the glasses before it gives way.

When an insulated body revolves round an axis in any direction, the
state of revolution cannot be permanent, unless the axis be so situated that
the centrifugal forces on each side of it balance each other.* It is obvious
that this must happen in a homogeneous sphere, whatever may be the
situation of the axis ; and it has been demonstrated, that when the body
is of an irregular form, there are at least three different axes, situated at
right angles to each other, round which the body may revolve in an equi-
librium either stable or tottering. It may also be shown that if a body,
revolving round any axis, receive at the same time an impulse whicli
would cause it to revolve round a second axis in another direction, the two
revolutions will be combined, and will form a single revolution round a
third axis in an intermediate position, which will remain at rest until it be
displaced by some new force, provided that it be one of the axes of permanent
revolution : so that no body can revolve round a moveable axis without a
continual disturbing force. And when an irregular body begins to move on
an axis incapable of equilibrium, its revolution will be gradually altered,
so as to approach continually to a revolution round one of the natural
axes ; but it will never pass beyond the state of equilibrium, as in many
other cases of deviation from such a state ; since the momentum pro-
duced by the excess of centrifugal force in one part of the revolution is
destroyed in another. For a similar reason, if a stick be thrown, in a
horizontal position, with a rotatory motion, it will fall in the same position
much more certainly than if it were thrown without any rotation ; for
any small disturbing force, which might be sufficient to turn it into a verti-
cal position during the course of its path, will only produce, when com-
bined with the rotatory motion, a slight change of the direction of the rota-
tion, which will confine the deviation of the stick from a horizontal posi-
tion within narrow limits.

The subject of preponderance, or of the action of wreights or forces coun-
teracted by other forces and incumbered with foreign matter to be put in
motion, requires for its discussion a previous knowledge of the simple
operation of forces, of the conditions of equilibrium, and of the estimation
of rotatory power. The consideration of the effects of preponderance
enables us to determine, in some circumstances, the best possible propor-
tions of the powers of machines for producing the required effects in the
most advantageous manner. For, in order that motion may be produced, it
is not sufficient that there be an equilibrium, in procuring which a part only
of the power is expended, but there must be an excess of force above that
which would be necessary for the equilibrium; and it is often of con-
sequence to know what portion of the power must be employed in each
wray, in order that the greatest effect may be produced in a given time.
We are sometimes told, that what we gain in power we lose in time. t In

* Segner, de Motu Turbinum, Halle, 1755, first pointed out the three natural
axes of rotation of all bodies. Their existence was demonstrated by Eujer in 1760.
See Hist, et Mem. de 1'Acad. 1758, p. 154; 1760, p. 176.

ON THE MOTIONS OF CONNECTED BODIES. 67

one sense indeed the remark is true ; thus one man can do no more by a
powerful machine in ten hours, than ten men can do by a weaker machine
in one hour ; but in other senses the assertion is often erroneous ; for by
increasing the mechanical advantage to a given degree we may in some
cases considerably increase the performance of a machine without adding
to the force.

According to the nature of the force employed, and to the construction
of the machine, a different calculation may be required for finding the best
proportions of the forces to be employed ; but a few simple instances will
serve to show the nature of the determination. Thus, in order that a
smaller weight may raise a greater to a given vertical height, in the
shortest time possible, by means of an inclined plane, the length of the plane
must be to its height as twice the greater weight to the smaller,* so that the
acting force may be twice as great as that which is simply required for
the equilibrium. This may be shown experimentally, by causing three
equal weights, supported on wheels, to ascend at the same time as many
inclined planes of the same height but of different lengths, by means of
the descent of three other equal weights, connected with the former three
by threads passing over pullies. The length of one of the planes is twice
its height, that of another considerably more, and that of a third less : if
the weights begin to rise at the same time, the first will arrive at the top
before either of the others. (Plate V. Fig. 76.)

If a given weight, or any equivalent force, be employed to raise another
equal weight by means of levers, wheels, pullies, or any similar powers,
the greatest effect will be produced if the acting weight be capable of sus-
taining in equilibrium a weight about twice and a half as great as itself.
This proposition may be very satisfactorily illustrated by an experiment.
Three double pullies being placed, independently of each other, on an axis,
round which they move freely, the diameters of the two cylindrical por-
tions which compose the first being in the ratio of 3 to 2, those of the
second as 5 to 2, and those of the third as 4 to 1, six equal weights are
attached to them in pairs, so that three may be raised by the descent of the
other three, on the principle of the wheel and axis. If then we hold the
lower weights by means of threads or otherwise, and let them go, so that
they may begin to rise at the same instant, it will appear evidently that
the middle pulley raises its wreight the fastest ; and consequently, that in
this case, the ratio of 5 to 2 is more advantageous than either a much less
or a much greater ratio. If the weight to be raised were very great in pro-
portion to the descending weight, the arrangement ought to be such that this
weight might retain in equilibrium a weight about twice as great as that
which is actually to be raised. If the descending weight were a hundred
times as great as the ascending weight, the greatest velocity would be
obtained in this case, by making the descending weight capable of holding
in equilibrium a weight one ninth as great as itself. (Plate VI. Fig. 77.)

The proportion required for the greatest effect is somewhat different,

when the heights through which both the weights are to move are limited,

as they usually must be in practical cases. Here, if we suppose the opera-

* Whewell's Dyn. c. 4, § 4.

F2

68 LECTURE IX.

tion to be continually repeated, the effect will be greatest in a given time,
when the ascending weight is between two thirds and one half of the exact
counterpoise to the descending weight: If, however, the force were accu-
mulated during the action of the machine, there would be no limit to the
advantage of a slow motion. Thus, if we have a stream of water filling a
single reservoir, which is to raise a weight by means of its descent, the
proportion here assigned will be the best for performing the most work in
a given time ; but if we chose to double our machine, so that one reservoir
should be filled during the descent of another, it would be proper to pro-
portion the weights in such a manner that the whole time required for
filling one of the reservoirs should be occupied in the descent and the re-
ascent of the other.

In all these cases, if great accuracy were required, it would be necessary
in the calculation to add to the mass to be moved the quantity of moveable
matter in the machine, reduced to a mean distance from the fulcrum or
centre, according to its rotatory power, in the same manner as the centre of
gyration is determined. But there is seldom occasion for such a degree of
precision. The magnitude of the pressure which is exerted on the fulcrum
during the motion of the connected bodies may always be determined, by
comparing the actual velocity of the centre of gravity with that of a body
descending without resistance.

These propositions and experiments must be allowed to require an atten-
tive consideration from those who are engaged in practical mechanics ; and
it is natural to suppose that the proportions laid down may be adopted with
safety, and employed with success, and that we may sometimes derive
important advantages from their application. But on more mature consi-
deration, we shall find some practical reasons for caution in admitting them
without material alterations.*

If a machine were constructed for raising a solid weight, and so arranged
as to perform its office in the shortest possible time with a given expense of
power, the weight would still possess, when it arrived at the place of its
destination, a considerable and still increasing velocity : in order that it
might retain its situation, it would be necessary that this velocity should
be destroyed ; if it were suddenly destroyed, the machinery would undergo
a strain which might be very injurious to it ; and if the velocity were
gradually diminished, the time would no longer be the same as is supposed
in the calculation. In the second place, the forces generally employed are
by no means uniformly accelerating forces, like that of gravitation, to which
the propositions which we have been considering are adapted : they are not
only less active when a certain velocity has once been attained, but they are
often capable of a temporary increase or diminution of intensity at pleasure.
We have seen the inconvenience of producing a great final velocity, on ac-
count of its endangering the structure of the machine : if therefore our per-
manent force be calculated according to the common rule, so as to be able
to maintain the equilibrium, and overcome the friction, the momentum or
inertia of the weights, when once set in motion, will be able to sustain that

* Consult S, Gravesande's Nat. Ph. i. c. 21. Euler, Ac. Berl. 1748. Blake,
Ph. Tr. 1759.

ON THE MOTIONS OF CONNECTED BODIES. 69

motion equably ; and it will not be difficult to give them a sufficient mo-
mentum, by a greater exertion of the moving force for a short space of
time, at the beginning : and this is in fact the true mode of operation of
many machines where animal strength is employed. Other forces, for
instance those of wind and water, regulate themselves in some measure, at
least with respect to the relative velocity of the sails and the wind, or the
floatboards and the water ; for we may easily increase the resistance until
the most advantageous effect is produced. Many authors, considering the
pressure of a stream of water as analogous to the impulse of a number of
unconnected particles striking the floatboards and then ceasing to produce
any further effect, have inferred that the force obtained by such an impulse
must be as the square of the relative velocity, and that the effect of an
undershot wheel must be the most advantageous when its velocity is one
third of that of the stream : but it will hereafter appear, that this estima-
tion of hydraulic force is by no means accurate. If we compare the
greatest velocity with which a man or a horse can run or walk without
fatigue, to the velocity of the stream, and the actual velocity of that part of
the machine to which the force is applied, to the velocity of the floatboards
of a water wheel, the strength which can be exerted may be represented,
according to the experiments of some authors, by the impulse of the stream
as supposed to be proportional to the square of the relative velocity ; con-
sequently the same velocity would be most advantageous in both cases, and
the man or horse ought, according to these experiments, to move, when his
force is applied to a machine, with one third of the velocity with which he
could walk or run when at liberty. This, for a man, would be about a
mile and a half an hour ; for a horse, two or three miles : but in general
both men and horses appear to work most advantageously with a velocity
somewhat greater than this.

Where a uniformly accelerating force, like that of gravitation, is em-
ployed in machines, it might often be of advantage to regulate its opera-
tion, so that it might act nearly in the same manner as the forces that we
have been considering ; at first with greater intensity, and afterwards with
sufficient power to sustain the equilibrium and overcome the friction only.
This might be done by means of a spiral barrel, like the fusee of a watch ;
and a similar modification has sometimes been applied by causing the
ascending weight, when it arrives near the place of its destination, to act
on a counterpoise, which resists it with a force continually increasing, by
the operation of a barrel of the same kind, so as to prevent the effect of
the shock which too rapid a motion would occasion.

On the whole, we may conclude, that on account of the limited velocity
which is usually admissible in the operation of machines, a very small
portion of the moving force is expended in producing momentum ; the
velocity of 3 miles an hour would be generated in a heavy body, descend-
ing by its own weight, in one seventh of a second, and a very short time is
generally sufficient for obtaining as rapid a motion as the machine or the
nature of the force will allow ; and when this has been effected, the whole
force is employed in maintaining the equilibrium and overcoming the
resistance : so that the common opinion, which has probably been formed

70 LECTURE IX.

without entering minutely into the consideration of the subject, and which
appears, when first we examine its foundation with accuracy, to lead to
material errors, is in great measure justified by a more profound investi-
gation.

To seek for a source of motion in the construction of a machine, betrays
a gross ignorance of the principles on which all machines operate. The
only interest that we can take in the projects which have been tried for
procuring a perpetual motion, must arise from the opportunity that they
afford us to observe the weakness of human reason ; to see a man spending
whole years in the pursuit of an object which a week's application to
sober philosophy might have convinced him was unattainable. The most
satisfactory confutation of the notion of the possibility of a perpetual
motion, is derived from the consideration of the properties of the centre of
gravity : we have only to examine whether it will begin to descend or to
ascend, when the machine moves, or whether it will remain at rest. If it
be so placed, that it must either remain at rest or ascend, it is clear, from
the laws of equilibrium, that no motion derived from gravitation can take
place : if it may descend, it must either continue to descend for ever with
a finite velocity, which is impossible, or it must first descend and then
ascend with a vibratory motion, and then the case will be reducible to
that, of a pendulum, where it is obvious that no new motion is generated,
and that the friction and resistance of the air must soon destroy the
original motion. One of the most common fallacies, by which the super-
ficial projectors of machines for obtaining a perpetual motion have been
deluded, has arisen from imagining that any number of weights ascending
by a certain path on one side of the centre of motion, and descending in
the other at a greater distance, must cause a constant preponderance on
the side of the descent : for this purpose the weights have either been fixed
on hinges which allow them to fall over at a certain point so as to become
more distant from the centre, or made to slide or roll along grooves or
planes which lead them to a more remote part of the wheel, from whence
they return as they ascend : but it will appear, on the inspection of such a
machine, that although some of the weights are more distant from the
centre than others, yet there is always a proportionally smaller number of
them on that side on which they have the greatest power ; so that these
circumstances precisely counterbalance each other. (Plate VI. Fig. 78.)

LECT. IX.— ADDITIONAL AUTHORITIES.

Lagrange, Hist, et Mem. de Berlin, 1773, p. 85. Landen, New Theory of Ro-
tatory Motion, Ph. Tr. 1777, p. 266 ; 1785, p. 311. Vince, Ph. Tr. 1780, p. 546.
Robison, Encyc. Brit. Art. Rotation. Fra^ais sur le Rotat. d'un Corps, 4to, Par.
1813. Raeb, De Motu Gyratorio, Trajecti adRhenum, 1834.

Rotation with Progression, D. Bernoulli, Comm. Petr. xiii. 94. Euler, xiii. 220,
and Acta Petr. ii. II. 162 ; 1781, v. II. 131 ; 1782, vi. I. 117, II. 107 ; 1783, I.
119 ; 1787, v. 149. Fuss, ibid. 176, and 1788, vi. 172. Prony sur le Mouvement
d'un Corps sollicite par des Puissances quelconques, 4to, 1800.

71

LECTURE X.

ON DRAWING, WRITING, AND MEASURING.

HAVING investigated all the general principles and laws of motion, and
of mechanical power, we may now proceed to the consideration of parti-
cular departments of practical mechanics. But before we can satisfactorily
compare the various forces which we are to employ or to oppose, we must
have some mode of determining their magnitude ; and we must begin by
examining the spaces which are measures of their action : a knowledge of
the instruments employed for delineation, and of the rules of perspective
projection, is also necessarily required as a previous step in the study of
practical mechanics. We have therefore to consider, as preliminary
subjects, first, the arts which may be expressed by the terms instrumental
geometry, or the geometry of mechanics : secondly, statics, or the mode
of ascertaining the magnitude of weights and of other active forces ; and
thirdly, the examination of the passive strength of materials of various
kinds, and of the negative force of friction.

The art of drawing can scarcely be distinguished by any correct defini-
tion from painting. In its simplest state, when we merely imitate an
original laid before us, it is called copying ; and in writing, we only copy
the letters of the alphabet. If we proceed in a mathematical manner in
the operation of drawing, we require a number of geometrical instruments,
which are still more necessary for the first construction of diagrams or
figures. In modelling and sculpture, a solid is simply imitated ; but when a
solid is represented on a plane, the principles of perspective are employed in
determining the position of the lines which are to form the picture. The
productions of the arts of drawing and writing are multiplied and per-
petuated by means of engraving and printing ; inventions which have been
the sources of inestimable advantage in the instruction and civilisation of
mankind.

In drawing, we may employ the pen, the pencil, chalks, crayons, inks,
water colours, or body colours ; we may paint in miniature, in distemper,
in fresco, in oils, in varnish, in wax, or in enamel ; and we may imitate
the effects of painting by mosaic work or by tapestry.

The first step in copying a drawing or in painting, is to procure a correct
outline : a master of the art can do this with sufficient accuracy, by such
an estimate of the proportions of the figures as the eye alone enables him
to form ; especially if he be assisted by lines which divide the original into
a number of squares, and enable him to transfer their contents to the corre-
sponding squares of the copy, which may in this manner be reduced or
enlarged, when it is required. But a copy may sometimes be more expe-
ditiously made by tracing immediately from the original, when the mate-
rials employed are sufficiently transparent to admit the outlines to be seen
through them ; or, where the original is of no value, by pricking a number

72 LECTURE X.

of points through it, so as to mark the copy, either at once, or by means
of charcoal powder ruhbed through the holes, which is called stenciling :
and for this purpose, an intermediate copy may be formed on semi-trans-
parent paper. Another method is to put a thin paper, rubbed with the
powder of black lead or of red chalk, between the original and the paper
intended for the copy, and to pass a blunt point over all the lines to be
traced, which produces correspondent lines on the paper ; this is called
calking. Where the work is large, it may be covered with a thin gauze,
and its outlines traced on the gauze with chalk, which is then to be placed
on the blank surface, and the chalk shaken off it in the way that a car-
penter marks a board with his line/"

The pen was formerly much used for making rough sketches, and it is
still sometimes employed for the same purpose, as weh1 as for assisting the
effect of the pencil. The appearances of uniform lights and shades must
necessarily be imitated in drawings with the pen, as well as engravings, by
a mixture of the whiteness of the paper with the blackness or colour of
the ink, the eye being too remote to distinguish minutely the separate lines
by which the effect is produced, although they do not entirely escape its
observation. In this respect, drawings in pencils and chalks have an
advantage over engravings ; these substances, after being laid on in lines,
are spread, by means of rubbers or stumps, of paper, leather, or linen, so
as to produce a greater uniformity of tint. Some, indeed, are of opinion
that engravings derive a great brilliancy from the hatches that are
employed in shading them, and that minute inequalities of colour make
every tint more pleasing. In drawings with chalk, however, the advan-
tage of rubbers is unquestionable. The lines of a drawing may be made
to have an appearance of greater freedom than those of an engraving ;
they should be parallel, and when they are crossed [the different sets
should be] moderately oblique to each other ; their direction should be
governed by that of the outline. Engravings in mezzotinto exhibit no
lines : but they are deficient in spirit and precision : the effect of aqua tinta
approaches much nearer to that of drawing, and it has a similar advan-
tage in the mode of producing its lights and shades. (Plate VI. Fig. 79.)

It is well known, that the best pencils are made of English black lead,
or plumbago. Of black chalks, the Italian is harder and more generally
useful than the French : red chalk has the disadvantage of not being
easily removed, either by bread or by Indian rubber, without leaving a
brownish mark. All these chalks are of the nature of a soft schistus or
slate : they may be made to adhere firmly to the paper by dipping the
drawings in milk freed from cream, or even in water only, which dissolves
the size or gum of the paper. Sometimes a grey paper is used, which
serves for a middle tint, and lessens the labour, the lights and shades only
being added in white and black chalks.

Crayons consist of colours mixed up with gum water, or other adhesive
substances, and usually also with some chalk, plaster, or pipe clay, so as
to be of a proper consistence for working in the manner of chalks. The

* Imison's Elements, ii. 240, 327.

ON DRAWING, WRITING, AND MEASURING. 73

principal inconvenience attending them is their want of adhesion to the
paper : the paper must therefore not be too smooth.*

For drawings washed in light and shade only, the material's employed
are Indian ink, the black liquor of the cuttle fish, or bistre which is ex-
tracted from soot : both these last produce a browner and richer tint than
the Indian ink.t In using these washes, as well as water colours, there is
a great diversity in the methods of different artists : some work with a dry
pencil, others with a full one : some begin all their coloured drawings in
black only, others use colours from the beginning. When a full pencil is
used, care must be taken that no part of the same tint dry sooner or later
than the rest. When body colours are employed, there is less difficulty in
producing a uniformity of tint than with water colours, each coat of the
colour being laid on in sufficient quantity to cover all that is below it without
mixing : hence it becomes easier to make any alterations that may be re-
quired. For water colours of all descriptions a certain quantity of gum is
used, and sometimes a size made of isinglass with a little sugar candy. Body
colours contain less gum than other water colours. :£ Besides paper, wood,
silk and cotton velvet are sometimes used for drawings in water colours.

In miniatures, the most delicate tints are laid on in points with simple
water colours; but for the draperies body colours are sometimes used.
They are commonly executed on ivory.

For painting in distemper the colours are mixed with a size made by
boiling shreds of untanned leather or of parchment, for several hours : this
method is chiefly employed for colouring walls or paper, but sometimes for
painting on cloth. For delicate purposes, the size may be made with
isinglass.

When a wall or ceiling is painted in fresco, the rough coat of the plaster
is covered with a coat of fine sand and lime as far as it can be painted before
it is dry, the colours being partly imbibed by this coat, and thus becoming
durable. When they have been once laid on, no alteration can be made,
without taking off the last coat of plaster, and each part must be completed
at once ; it is therefore always necessary to have a finished drawing for a
copy ; this is usually executed on paper, and is called a cartoon. The
colours can be only of earths or metallic oxids ; they are prepared as for
painting in distemper. The only paintings of the ancients, which have
been preserved, were executed in fresco.

The art of painting in oil was first discovered by Van Eyck of Bruges, §
towards the end of the 14th century : it has now become almost the only
manner in which paintings of magnitude are performed. The colours are
mixed with linseed or nut oil, and sometimes with oil of poppy seed, together
with a small portion of oil of turpentine to assist in drying them, and with

* Russel on Painting in Crayons, 4to. Encyclop. Meth. Arts et Metiers vi. Art.
Pastel. Contes Crayons, Ann. de Chimie, xx. 370. Lomet, ibid. xxx. 284. Nich.
Jour. iii. 216.

f Gill on Indian Ink, Ph. Mag. xvii. 210.

% t Handmaid to the Arts, 1758, Field's Chromatography. Mrs. Callcott's Es-
says towards a History of Painting, 1836.

§ On the authority of Vasari, c. 21 ; but it is probably incorrect. Consult
James's Flemish and Dutch Schools of Painting, or Haydon's Lectures, 1844, p.
265. Cennini, translated by Mrs. Merriefield, 1844 ; Tambroni's Preface, p. 49.

74 LECTURE X.

the occasional addition of other oily and resinous substances. The work
may be executed on wood, cloth, silk, paper, marble, or metals : these sub-
stances being first washed with size, and then primed with an oil colour,
which is usually white, but sometimes dark. Some painters have, however,
preferred a ground of distemper. The glare of the oil colours or of the var-
nish, which is added in order to give them brilliancy, is considered as an
inconvenience attending oil paintings ; and some of the colours are too
liable to fade or to blacken by the effect of time.

The encaustic paintings of the ancients were imperfect approximations
to the art of painting in oil. Wax or resins were employed for retaining
the colours in their places ; and they were applied by means of a moderate
heat.* An effect nearly similar is produced by dissolving the resins in
spirits of wine, as is done in painting in varnish. A much greater degree
of heat is required for paintings in enamel : for this purpose the colours are
mixed with a glass of easy fusion, and, when finely powdered, they are
usually applied with oil of turpentine, or sometimes oil of lavender, to a
ground of metal or porcelain ; they are afterwards fixed and vitrified by
exposure to the heat of a furnace.

Mosaic work is performed by putting together small pieces of stone or
baked clay of various colours, so as to imitate the effects of painting ;t in
tapestry and in embroidery, the same is done by weaving, or working in
threads of different kinds.

The art of writing is of great antiquity, but it is probably in all coun-
tries, and certainly in some, of a later date than that of drawing represent-
ations of nature. The Mexicans, at the first arrival of the Spaniards in
South America, are said to have employed drawings as a mode of conveying
intelligence : some of them simply resembling the objects to which they
related, others intended as hieroglyphics ; that is, like the ancient Egyptian
characters, of a nature intermediate between drawing and writing.;}; The
Chinese have always used arbitrary marks to represent whole words or the
names of external objects, not resembling the objects to which they relate,
nor composed of letters appropriated to constituent parts of the sound,
although they are said to be combined from a few hundred radical charac-
ters expressive of the most simple ideas. The art of writing with alpha-
betical letters must have been sufficiently understood in the age of Moses,
to serve the purpose of the promulgation of laws and of religion ; it is
generally supposed to have been invented by the Phenicians. Among the
Greeks it was in a very imperfect state until the time of the siege of Troy,
or about 3000 years ago. The Chinese- write from above downwards,
beginning on the right side ; the other eastern nations have always written
from right to left. The most ancient Greek inscriptions are turned alter-
nately backwards and forwards, the letters being reversed in the lines
which begin on the right side ; but the Greeks soon confined themselves to

* Pliny,!. 35, c. 11. Vitruvius, Architectura, 1. 7, c. 9, de Minii Temperatura.
Colebrooke, Ph. Tr. 1759, p. 40. Caylus on Encaustic Painting, Lond. Fabbroni
on Do. Ph. M. i. 23, 141. Gilbert's Annalen, v. 357.

t Ph. Mag. ix. 289.

t See the plate of Aztec Chronology from Carreri, in Encyc. Metr. vol. xix, pi.
28. Robertson's Hist, of America, ii. 284, 480. Humboldt, Voyage de Cordilleras.

ON DRAWING, WRITING, AND MEASURING. 75

that mode, which has been since adopted by all European nations, and
which appears to be in itself the most natural, at least for writing with a
pen, and with the right hand.*

The earliest methods of writing were probably such as rather deserve the
name of engraving ; the letters being cut in stone, in wood, on sheets of
lead, on bark, or on leaves. For temporary purposes, they were formed on
tablets of wax, with a point called a stile, and this practice was long con-
tinued for epistolary correspondence, and was not wholly out of use in the
fourteenth century. The stile was made of metal or of bone ; its upper
extremity was flattened for the purpose of erasing what had been written.
The Egyptian papyrus is said by Varro to have been first used for writing
at the time of the foundation of Alexandria ; the leaves of palms, the inner
bark of trees, or sometimes linen cloth, having been before employed. The
exportation of the papyrus was forbidden by Ptolemy, and in consequence
of this prohibition, skins of parchment or of vellum were first applied
to the purpose of writing at Pergamus, for the library of king Eumenes,
whence they were called membrana pergamena. To make the best paper,
the widest and finest leaves of the papyrus were matted together, united by
a vegetable glue, and pressed till they became sufficiently smooth ; the
coarser kinds were not used for writing, but for commercial purposes. In
China, paper is sometimes made of a thin and almost transparent mem-
brane taken from the bark of a tree. Paper of cotton was introduced into
Europe from the east in the middle ages : it has been since superseded by
that which is made of linen rags, and which is also an eastern invention ;
but for coarse and strong paper, old ropes of hemp are also used ; and
sometimes many other vegetable substances have been employed. The
strength and consistence of paper is owing to the lateral adhesion derived
from the intermixture of the fibres, assisted by the glutinous size, which is
also of use in obviating the bibulous quality of the paper, by filling up its
pores.t

Ivory, and prepared ass's skin, are sometimes employed for writing with
a black lead pencil ; for slates, a pencil of a softer kind of slate is used.
The ancient mathematicians usually constructed their diagrams on sand
for the instruction of their pupils.

Pens of goose quills, swan's quills, or crow quills, were known as early
as the seventh century : in Europe they have generally superseded the
reeds which were employed for writing by the ancients : but in India,
reeds, canes, and bamboos, are still in use. In China a hair pencil is
used instead of a pen.

* As Dr. Young distinguished himself by his researches on hieroglyphical writing,
we subjoin the following references to his works : — Museum Criticum, 8vo, Camb.
vol. ii. pp. 125, 329. Hieroglyphics, fol. Lond. by the Egyptian and Royal Societies
of Literature. Supp. to Ency. Brit. vol. iv. 38. Discoveries in Hier. Lit. 8vo. Lond.
1823. A sketch of the discoveries will be found in the Quarterly Review, vol. xliii.
p. 112; or in Ency. Metr. Art. Hieroglyphics. See also J. F. Champollion,
L'Egypte sous les Pharaons, 2 vols. 8vo. Par. 1814. Lettre a M. Dacier relative a
*!' Alphabet des Hieroglyphics, 8vo, Par. 1822. Precis du Systeme Hierog. 8vo.
Par. 1824 and 1828. Lettres relatives au Musee Royal Egyptien de Turin, Par.
1824 and 1826.

f Rombold on Paper, Berl. 1744. Lalande, L'Art de faire le Papier, fol. Par.
1761.

76 LECTURE X.

The inks of the ancients are said to have been made of a carbonaceous
substance, and the modern Indian ink owes its blackness to similar materials.
Common writing ink consists of a gallate of iron, suspended by means of
a little gum ; the sulfuric acid, which remains mixed with it, is probably
of no consequence to its blackness. It has been observed, that an abun-
dance of the gallic acid produces a much blacker colour than is obtained
where this acid is used in a smaller proportion. Mr. Ribaucourt's method
of making ink,* is to boil eight ounces of galls, and four of logwood, in
twelve pounds of water, until the quantity is reduced to one half ; and, hav-
ing strained the decoction, to add to it four ounces of sulfate of iron, one of
sulfate of copper, three of gum arabic, and one of sugar candy. But for
ordinary purposes, it is sufficient to infuse three ounces of galls for a day or
two in a pint of water, and to add to it an ounce of gum arabic, half an
ounce of green sulfate of iron, or copperas, and a drachm of sulfate of cop-
per, or blue vitriol, or even a much smaller quantity of gum and of copperas,
if a very fluid ink is required. The sulfate of copper produces a durable
stain, but it does not immediately add to the blackness of the ink : its
principal use is to counteract the tendency of the ink to become mouldy.
Sometimes a mercurial salt is employed for the same purpose, and a little
cotton, if the inkstand is too open, is also useful in preserving the ink ;
but the addition of spirits is often insufficient, and is liable to make the
ink run.

It has been proposed to use inks of different colours for indicating
different numbers ; so that by ten kinds of ink applied in different ways,
any numbers at pleasure might be expressed. Thus, in making an index
of the words of an author, each page might be readily covered with lines
of different colours drawn in different directions, so that each word, when
cut out, might indicate the page to which it belongs.

An ingenious instrument has lately been constructed, by means of
which copies may be multiplied with great facility ; it is called the poly-
graph, and consists of two or more pens, so connected by frames and
springs, as to move always in parallel directions, each having an inkstand
and a sheet of paper for itself.t In this manner five copies may be made
at once with tolerable facility, and the method may perhaps hereafter be
extended to a much greater number.

A mode of writing, perfectly different from any of those which have
been mentioned, is performed by means of the telegraph, which is justly
considered as the invention of the ingenious Dr. Hooke.:}; The ancients
had attempted something similar, by the exhibition of torches on elevated
situations ; but Dr. Hooke observes, that the addition of the telescope is
absolutely necessary for the practical success of the process; and the
directions which he gives for its performance differ very little from the
plan which has since been generally adopted, first in France, and after-
wards, with some variations, in this country. Dr. Hooke proposed the

* Repertory of Arts, ix. 125.

f Cotteneuve, Mem. de 1'Acad. Paris, 1763, H. 147.

J Ph. Tr. 1684. Philosoph. Exp. and Obs. bv Hooke, edited by Derham,
p. 142.

ON DRAWING, WRITING, AND MEASURING. 77

employment of alphabetical and other arbitrary characters ; at present it
is usual to have six boards,* each turning 011 its axis so as to appear or
disappear at pleasure : these admit of sixty-four combinations, which are
sufficient, besides indicating the letters of the alphabet, for every other
purpose that can be required. (Plate VI. Fig. 80, 81.)

Pens for drawing lines and figures differ sometimes from those which
are used for writing ; they are made of two plates of steel inclined to each
other, and adjusted by a screw ; or sometimes of a plate of tin folded up,
so as to include a receptacle for the ink ; or of a glass tube drawn to a
very fine point, and still remaining perforated. In all these pens, as well
as in common pens, the ink is retained by its cohesion, and by the
capillary attraction of the pen ; and it attaches itself to the paper by the
operation of similar powers.

It is by no means easy to comply strictly with that postulate of geometry
which requires us to draw a straight line from one point to another. The
edge of a ruler is made straight by the instrument called a plane, which is
worked with a considerable velocity, and therefore naturally tends to
move in a right line, besides that it is guided by the flatness of its lower sur-
face. We judge of the straightness of a line, by means of the well known
property of light, which moves only in right lines, so that if we look along
the edge of a ruler, we easily discover its irregularities ; and this may be
done with still greater accuracy, if we look through a small hole made
with a pin in a card. Rulers of silver, brass, or ivory, have a material
advantage over those of wood, as they are not liable to be spoilt by warp-
ing. A pen filled with ink cannot be applied close to the edge of a ruler
without inconvenience ; it is therefore best, for diagrams which require
great accuracy, to draw the lines first with a steel point, or a very hard
black lead pencil, and to finish them with ink if necessary. The paper
should also be fixed on a drawing board ; and plates of lead or copper may
be employed, instead of paper, for very delicate purposes. The carpenter's
chalk line is a useful instrument for supplying the place of a very long
ruler ; it becomes straight when it is stretched, because a right line is the
shortest distance between any two points.

For drawing a circle of a given radius we use compasses, with one point
generally of metal, the other of various descriptions, t Compasses are
sometimes made with a spring instead of a joint, and opened or shut by a
screw : sometimes a graduated arc is fixed in one leg, and passes through

* This species of telegraph was invented in 1695, by Lord G. Murray ; it was
adopted by the Admiralty until the end of the late war, when it was discontinued,
and the semaphore, consisting of two arms projecting from an upright
post, and working about pivots, was substituted in its place. In this
instrument each arm has seven different positions, which afford by
their combinations forty-nine different arrangements. Consult Edge-
worth, Trans. Roy. Irish Ac. vi. 95, 319. Nicholson's Journal, ii. 319.
Chappe, Breguet and Betancourt, Bulletin de la Soc. Ph. n. 16.
Mem. de 1'Institut III. H. 22. Ph. Mag. i. 312. Nocturnal Tele-
graph, Rep. of Arts. x. 28. Boaz's Patent Tel. ibid. xvi. 223. Ph.
Mag. xii. 84. Ency. Brit. Art. Tel. Pasley, Description of the Uni-
versal Telegraph, 1823. Chappe, Histoire dela Telegraphe, 2 vols. Paris, 1824.

f Duval's New Compasses, Mem. Paris, 1717, H. 83. Leup. Th. Art. t. 20,
a. b.

78 LECTURE X.

the other ; and when great accuracy is required, hair compasses may be
employed, having a joint with a spring in one of the legs, which is bent a
little by means of a fine screw. Beam compasses* are useful for drawing
circles of larger radii : they have also the advantage of being steadier than
the common compasses, and of admitting readily the application of a gra-
duated scale, so as to indicate the measure of the radius of the circle which
is described. Sometimes, for drawing portions of very large circles, two
wheels, differing a little in diameter, are fixed on a common axis, and thus
made to revolve round a point, which is more or less distant, accordingly
as the wheels are set at a greater or less distance on the axis, the surface
of the wheels tracing the circles on the paper ; or two rulers joined toge-
ther, so as to form an angle, are made to slide against two points, or
edges, projecting from a third ruler ; so that the angular point remains
always in the arc of a circle. The same effect may be produced, somewhat
more commodiously, by means of a thin piece of elastic wood, which
is made to assume any required curvature by the action of screws
applied to different parts of its concavity : it would, however, be more
simple and accurate to employ only one screw, in the middle of the arc,
and to make the flexible ruler, or bow, every where of such a thickness as
to assume a circular form in its utmost state of flexure : it would then
retain the circular form, without a sensible error, in every other position.
(Plate VI. Fig. 82... 85.)

For drawing a line perpendicular to another, we often employ a square ;
and if we use a rectangular drawing board, there is an additional conve-
nience in making the square to slide on its margin. Rulers also, of various
descriptions, are commonly made rectangular, in order to answer occasion-
ally the same purpose.

Triangular compasses are sometimes used for laying down a triangle
equal to a given triangle ;t and by repeating the operation, any figure
which can be divided into triangles, may be copied without the intersection
of arcs ; but the same end is more commonly obtained by pricking off the
figure with a steel point. (Plate VI. Fig. 86.)

Various properties of parallel lines are employed in constructing parallel
rulers : a parallelogram with jointed angles is the most commonly used ;
two equal rulers being united by equal cross bars placed in an oblique
position, and turning on pins fixed in the rulers : the instrument is much
improved by adding a third ruler, similarly united to the second, for then
the obliquity of one of the two motions may be made to correct that of the
other. A simple cylinder, or a round ruler, answers the purpose in a rough
manner, and two small rollers, fixed on the same axis, are also sometimes
attached to a flat ruler, and cause it to move so as to be always in parallel
positions. A very useful instrument for drawing parallel lines, at any
given distances, is now generally known by the name of Marquois's scales,
although it is by no means of late invention ;£ by sliding a triangle along
a graduated ruler, we read off the divisions on an amplified scale with great

* Shuckburgh, Ph. T. 1798.
f Leupold, Th. Art. t. 28. J Ibid. t. 21, a.

ON DRAWING, WRITING, AND MEASURING. 79

accuracy ; but where the distances of the lines are great, the obliquity of
this motion is a considerable inconvenience. The ruler or square of the
drawing board affords us lines parallel to each other, in a certain position ;
and if it be made with a joint, or as the workmen call it, bevilled, it may
be employed for the same purpose in all other directions. The systems of
lines, on which music is written, are drawn at one stroke by a pen with
five orifices, usually made of brass. It was long since proposed to rule a
whole page at once, with a more complicated pen of the same kind, and
the greatest part of the paper on which music is written in this country,
is actually ruled by such a machine, for which a patent has been taken
out. (Plate VI. Fig. 87, 88.)

The pantograph is used for copying figures, and at the same time
reducing or enlarging them ; it consists of four rulers, two of them
united by a joint at the extremities, and receiving at the middle the other
two, which are but half as long, and are also united together so as to form
with the others a jointed parallelogram, of which two of the sides are
produced beyond the angles ; if holes be made in these, and in one of
the shorter rulers, so situated as to be in the same right line in any
position of the instrument, they will remain in a right line in any other
position, and they will always divide this line in the same proportion : so
that if one of the holes be placed on a fixed axis or pin, a tracing point
inserted in another, and a pencil in the third, any figure delineated by
the pencil will be similar to that which is described by the tracing
point. And instead of holes in the rulers, they may be furnished with
sliding sockets, to receive the axis, the point, and the pencil. (Plate VI.
Fig. 89.)*

Proportional compasses are also of great use in reducing lines and
figures to a different scale.t This instrument consists of two legs, pointed
at each end, and turning on a centre which slides in a groove common to
both legs, and is furnished with an index. The divisions of the scale are
so laid down that the centre may divide the length of the legs from
point to point in a given proportion ; hence by the properties of similar
triangles, when the legs are opened to any extent, the intervals between
each pair of points must be to each other in the same ratio as the por-
tions of the legs. Sometimes a screw is added, for the sake of adjusting
the centre with greater accuracy ; and it is usual to lay down scales for
dividing the circumference of a circle into a given number of parts, and
for some other purposes ; but the instrument might be much improved
by inserting, in the common scale, fractional or decimal divisions between
the whole numbers, so that the legs might be divided, for example, in
the ratio of 2 to 3, 3 to 4, or 4 to 5, or of 10 to 11, 12 or 13, at pleasure.
(Plate VI. Fig. 90.)

The use of the sector depends also on the properties of similar triangles.

* Leup. Th. Art. t. 26. Langlois's Pantograph : Machines Approuves par 1'Ac.
7 vols. 4to, 1735-1777, vii. 207. Sike's Pantograph, Mem. Par. 1778, invented by
Sclieiner, who describes it in his Pantographice. An improved instrument for the
same purpose is described by Prof. Wallace, in the Trans, of the Roy. Soc. of Edin.
vol. xiii. and termed by him the Eidograph.

f Leon, da Vinci MSS. Leup. Th. Ar. 121.

80 LECTURE X.

The scale of equal parts, which is laid down on each leg, beginning from
the centre, serves to determine the length of the legs of two equilateral
triangles, in any required proportion to each other, according to the
division which we mark, and the transverse distances from the corre-
sponding points are necessarily in the same proportion. Thus, if we have
any line in a figure which we wish to call three feet, or three inches, we may
take the interval with a pair of common compasses, and open the sector to
such an angle, that it may extend from the third division of one leg to
that of the other ; then all the other divisions of the scale will furnish us
with the lengths corresponding to any distances that we may wish to lay
down. The other scales usually engraved on the sector are principally
intended for trigonometrical calculations on similar principles. (Plate VII.
Fig. 91.)

The magnitude of angles admits an easy determination and description,
by the comparison of the respective arcs with a circle, or with a right angle.
We may divide an angle geometrically, by continual bisection, into parts
as small as may be required, and by numbering these parts we may
define any angle, with an error smaller than any assignable quantity.
Bisections of this kind are sometimes actually employed in the construc-
tion of instruments ; for instance, in one of the arcs of the mural quad-
rant of the observatory at Greenwich, the right angle is divided into 96
parts, by the continual bisection of one sixth of the circle. There are
also some practical methods of dividing angles into three or more equal
parts, which are sufficiently accurate for many purposes, . although it is
well known that in theory the perfect trisection of an angle is beyond the
reach of plane geometry. This trisection is necessary in the common
division of the circle into 360 degrees, a number which was probably
chosen because it admits a great variety of divisors, and because it nearly
represents the diurnal and annual motion of the sun among the stars.
The circle being divided into 6 parts, the chord of each of which is equal
to the radius, these parts are divided into 60 degrees, each degree into 60
minutes, and each minute into 60 seconds : further than this we cannot
easily carry the accuracy of our determination, although in calculations
we sometimes descend as far as tenths or even hundredths of a second.
The decimal division of a right angle, which has been lately adopted in
France, appears to have very little advantage for the purposes of calcula-
tion, beyond the common method, and its execution in practice must be
much more difficult.

Whole circles, or theodolites, divided into degrees and their parts, quad-
rants and sextants, are usually employed in measuring angles ; and protrac-
tors, semicircles, and lines of chords, in laying them off. The most convenient
of quadrants for general use is Hadley's reflecting instrument,* which is in
fact an octant or a sextant, but in which, for reasons depending on optical
principles, each degree of the arc is reckoned for two.

For the graduation of all instruments of this kind, of moderate dimen-
sions, Mr. Ramsden's dividing engine is of great utility ;t the instrument

. * Ph. Tr. 1731, p. 147. f Description, 4to, 1787. Rozier, i. 147.

ON DRAWING, WRITING AND MEASURING. 81

being fixed on the revolving plate of the engine, its arc is made to advance
under the cutting tool by very minute steps, regulated by the turns of a
screw, of which each revolution is divided into a considerable number of
equal parts. The largest and finest instruments are, however, still usually
divided by hand ; that is, by means of compasses. Some artists have first
divided a straight plate, and then made a hoop of it, which has served as
a standard for further processes. An arc of 7° 10', of which the chord is
one-eighth of the radius, may be employed as a test of the accuracy of the
work. A micrometer screw is often used in large instruments as a substi-
tute for the minutest divisions ; * a moveable part of the index being
brought to coincide with the nearest point marked in the arc, by turning
the screw through a part of its revolution, which is measured by means of
a graduated circle. But a simpler method of reading off divisions with
accuracy in common instruments, is the application of a vernier, an appa-
ratus so called from its inventor. The space occupied by eleven divisions
of the scale being divided into ten parts on the index, the coincidence of
any of the divisions of the index with those of the scale, shows, by its
distance from the end, the number of tenths that are to be added to that of
the entire divisions. (Plate VII. Fig. 92.)

There are several ways of measuring the angular elevation of an object
above the horizon ; at sea, the apparent horizon, formed by the surface
of the water, affords the most convenient determination ; but since the
spectator is somewhat elevated above the convex surface of the sea, the
apparent horizon is necessarily lower than the true horizon, and a correc-
tion is therefore required according to the height. In the open sea this
correction may be determined by measuring the whole angle above and
below the apparent horizon [respectively], and taking one fourth of the
difference for the dip or depression. On shore,- a plumb line is the simplest
instrument for determining the situation of the horizon, and its accidental
vibrations may be prevented by suspending the weight in water or in oil.
For small instruments, a spirit level, of which the operation depends on
hydrostatical principles, is capable of greater delicacy than a plumb line.
It readily indicates, when well made, an error of a single second ; but it
requires some attention to avoid inequalities of temperature, which would
tend to disturb its figure. Well rectified ether is found, on account of its
perfect fluidity, to be the best liquid for a spirit level. An artificial hori-
zon is a reflecting surface employed for obtaining an image as much below
the horizon as the object is above it, and for measuring the angular dis-
tance of this image from the object: sometimes a plane speculum of glass
or metal is used for this purpose, being previously adjusted by a spirit
level ; and sometimes the surface of mercury, treacle, or tar, protected from
the wind by a vessel with holes in it, or by a glass cover, either detached, or
simply floating on the mercury, when this liquid is employed.

It is in many cases simpler and more convenient to estimate angles, not
by the arcs subtending them, but by their sines, or the perpendiculars
falling from one leg on the other. Thus, it is usual among miners, to say

* Hooke's Lectures, Lambert iiber die Branderschen micrometer, 12 Aug. 1769.
Hornblower, in Nich. Jour. vi. 247.

82 LECTURE X.

that the ground rises or falls one foot, or one yard, in ten, when the sine of
the angle of its inclination to the horizon is one tenth of the radius. Angles
of different magnitudes are indeed proportional to the arcs, and not to the
sines, so that in this sense the sine is not a true measure of the comparative
magnitude of the angle ; but in making calculations, we are more frequently
obliged to employ the sine or cosine of an angle than the angle or arc
itself. It is, however, easy to pass from one of these elements to the others,
by means either of trigonometrical tables, or of the scales engraved on the
sector.

The sines, tangents, and secants laid down on the sector, may be em-
ployed according to the properties of similar triangles, in the computation
of proportions. The same purpose is answered by Gunter's scale, by the
sliding rule, and by the logarithmic circles of Clairaut and of Nicholson,*
which are employed mechanically in the same manner as a table of loga-
rithms is used arithmetically, the proportion of any two numbers to each
other being determined by the distance of the corresponding divisions on
the scale ; so that if we wish to double or to halve a number, we have only
to find the distance from 1 to 2, and to lay it off from the given number
either way. (Plate VII. Fig. 93, 94.)

The measurement of angles is at once applied to the estimation of dis-
tances in the dendrometer or engymeter ;t a part of the instrument forms a
base of known dimensions, and the angle at each extremity of this base
being measured with great accuracy, the distance of the object may be
inferred from an easy calculation, or from a table. The most complete
instruments of this kind have twTo speculums for measuring the difference
of the angles at once, in the manner of Hadley's quadrant. Telescopic
scales or micrometers are also sometimes used for measuring angles sub-
tended by distant objects, of which the magnitude is known or may be
estimated, for example, by the height of a rank of soldiers, and inferring
at once the distance at which they stand.

Arithmetical and even algebraical machines, of a much more complicated
nature, have been invented and constructed with great labour and ingenuity ;
but they are rather to be considered as mathematical toys, than as instru-
ments capable of any useful application.^

An angle, when once measured, can be verbally and numerically de-
scribed, by reference to the whole circle as a unit : but for the identification
of the measure of a right line, we have no natural unit of this kind, and it
is therefore necessary to establish some arbitrary standard with which any
given lengths and surfaces may be compared. It might be of advantage in
the communication between different countries to fix one single standard to
be employed throughout the world, but this does not appear to be practi-

* Hist, et Mem. de Paris, 1727, H. 142. Nich. Journal, v. 40. Ph. Tr. 1753,
p. 96 ; 1787, p. 246.

f Pitt's Dendrometer, Repertory of Arts, ii. 238. Fallen's Engymeter, Zach.
Monatliche Correspondenz, vi. 46.

J Napier's Reckoning Rods, Leup. Th. Ar. t. 13. Robertson on GunCer's
Scale, Ph. Tr. 1753, p. 96. Nicholson's Logistic Circle and Scales, Ph. Tr. 1787,
p. 246 ; Herschel's Description of Babbage's Calculating Machine, Transactions of
the Cambridge Philosophical Society, iv. 425.

ON DRAWING, WRITING, AND MEASURING. 83

cally possible, even if it were determined what the standard ought to be.
" The observation of the isochronism of the small vibrations of a pendulum,
and the ease and certainty with which the length of a pendulum vibrating
seconds may be ascertained, have suggested," says Mr. Laplace,* in his
account of the system of the world, " the idea of employing this length
as a universal measure. We cannot reflect on the prodigious number of
measures in use, not only among different nations, but even in the same
country, their capricious and inconvenient divisions, the difficulty of deter-
mining and comparing them, the embarrassment and the frauds which they
occasion in commerce, without regarding, as one of the greatest benefits
that the improvements of the sciences, and the ordinances of civil govern-
ments can render to humanity, the adoption of a system of measures of
which the divisions being uniform, may be easily employed in calculations,
and which may be derived, in a manner the least arbitrary, from a funda-
mental magnitude indicated by nature itself. A nation that would intro-
duce such a system of measures, would unite to the advantage of reaping
the first fruits of the improvement, the pleasure of seeing its example
followed by other countries, of which it would thus become the benefactor :
for the slow but irresistible empire of reason must at length prevail over
national jealousies, and over all other obstacles that are opposed to a mea-
sure of which the convenience is universally felt. Such were the motives
that determined the constituent assembly to intrust the Academy of Sciences
with this important charge. The new system of weights and measures is
the result of the labours of the Committee, seconded by the zeal and infor-
mation of several members of the national representation.'!*

" The identity of the calculation of decimal fractions and of whole
numbers, leaves no doubt with respect to the advantage of the division of
measures of all kinds into decimal parts : it is sufficient, in order to be
convinced of this, to compare the difficulty of compound multiplication and
division, with the facility of the same operations where whole numbers only
are concerned, a facility that becomes still greater by means of logarithms,
of which the use may also be rendered extremely popular by simple and
cheap instruments. The decimal division was therefore adopted without
hesitation ; and in order to preserve the uniformity of the whole system, it
was resolved to deduce every thing from the same linear measure and its
decimal divisions. The question was then reduced to the choice of this
universal measure, to which the name of metre was to be given.

" The length of the pendulum, and that of a meridian of the earth, are
the two principal standards that nature affords us for fixing the unit of
linear measures. Both of these being independent of moral revolutions,
they cannot experience a sensible alteration without very great changes in
the physical constitution of the earth. The first method, which is of easy
execution, has the inconvenience of making the measuf e of length depend
on two elements, heterogeneous with respect to itself and to each other,
gravitation, and time ; besides that the division of time into small portions

* Systeme du Monde, liv. i. c. 12.

f Report on the choice of a unit of measure, by Borda, Lagrange, Laplace,
Monge, and Condorcet, Mem. de 1'Acad. Paris, 1788. H. 7-17.

G2

84 LECTURE X.

is wholly arbitrary. It was resolved, therefore, to employ the second
method, which," says Mr. Laplace, " appears to he of very high antiquity ;
it is so natural to man to refer measures of distance to the dimensions of
the globe which he inhabits, in order that, in transporting himself from
place to place, he may know, by the denomination of the space passed
through alone, the relation of this space to the entire circumference of the
earth. This method has also the advantage of making nautical measures
correspond at once with celestial ones. The navigator has often occasion
to compare with each other the distance that he has passed over, and the
arc of the heavens corresponding to that distance ; it is therefore of conse-
quence that these measures should be readily obtained from each other, by
altering only the place of the units. But, for this purpose, the funda-
mental unit of linear measures must be an aliquot part of the terrestrial
meridian, which must correspond to one of the divisions of the circum-
ference of a circle. Thus the choice of the metre was reduced to that of
the unit of angular measure, and the right angle, as constituting the limit
of the inclination of two lines to each other, was considered as entitled to
the preference.

" The arc, which was measured in 1740, from Dunkirk to the Pyren-
nees, might have served for finding the magnitude of the quadrant of the
meridian ; but a new and more accurate measurement of a larger arc was
more likely to excite an interest in favour of the new measures. Delambre
and Mechain were therefore intrusted with the direction of the operations
for measuring an arc from Dunkirk to Barcelona,* and after making a
proper correction for the ellipticity of the earth, according to the measure-
ment of the arc in Peru, the quadrant was determined to be equal to
5,130,740 of the iron toise used at the equator, its temperature being 61^° of
Fahrenheit : the ten-millionth part of this quadrant was taken for the unit
or metre. A standard was deposited in the custody of the legislative body,
adjusted at the temperature of melting ice. In order to be able always to
identify this length, without recurring to an actual measurement of the arc,
it was of importance to compare it very accurately with that of the pen-
dulum vibrating seconds, and this has been done with great care by
Borda, at the observatory of Paris. The unit of measures of land is the
are, or 100 square metres : a cubic metre of wood is called a stere, and a
cubic decimetre, or a cube of which the side is one tenth of a metre, is a
litre, or measure of fluids.

" Uniformity appeared to require that the day should be divided into ten
hours, the hour into a hundred minutes, and the minute into a hundred
seconds. This division, useful as it will be to astronomers, is of less
advantage in civil life, where arithmetical operations are seldom performed
on the parts of time ; and the difficulty of adapting it to clocks and
watches, together with our commercial relations with foreign countries,

* Delambre, Base du Systeme Metrique, 3 vols. 4to, Paris. A fourth volume, the
work of MM. Biot and Arago, was added in 1821. They extended the survey tojthe
island of Formentera. Consult also Reports to the National Institute. Rozier's
Journal, xliii. 169. Jour, de Phys. xliv. (1), 81. Bulletin de la Soc. Phil. n. 28.
Nich. Jour. iii. 316. Ph. Mag. i. 269 ; and the article, Figure of the Earth, by
Airy, in the Encyclopaedia Metropolitana.

ON DRAWING, WRITING, AND MEASURING. 35

have suspended its introduction for the present. We may, however,
expect that it will ultimately be brought into general use."

Such is Mr. Laplace's account of the new system of measures, the result
of the joint labours of many of the ablest mathematicians on the continent.
There is not at present any great probability that it will ever be employed
in this country. It is of little consequence from what the original unit has
been derived, unless we can with ease and accuracy recur to its origin :
and whether a standard has been first adjusted according to the circum-
ference of the globe, or to the foot of an individual hero, the facility of
comparing other measures with it is the same. It is confessed that the
pendulum affords the readiest method of recovering the standard when
lost ; and if it was necessary for the Committee of the French Academy
to determine a unit absolutely new, it would perhaps have been more
eligible to fix on one which was independent of any ulterior comparison,
than to seek for an ideal perfection in attempting to copy from a more
magnificent original ; to say nothing of the uncertainty with regard to the
ellipticity of the earth, and the probable irregularity of its form in various
respects. On the other hand, it must be allowed, that the correct deter-
mination of the length of the pendulum has sometimes been found more
difficult than Mr. Laplace's statement would lead us to suppose it, and we
cannot depend on any measurement of it as totally exempt from an error
of the ten thousandth part of the whole.

The metre, as definitively established by the government of France, is
equal to 39-nyVV English inches, measured, as it has been usual in this
country, on a standard scale of brass, at the temperature of 62° of Fah-
renheit; while the French, on the contrary, reduce the length of their
measures to that which they would acquire at the freezing point. Hence
ten thousand inches are nearly 254 metres, a thousand feet 305 metres.
The length of the pendulum vibrating seconds in London, was found by
George Graham, from a mean of several experiments, all agreeing very
nearly together, to be 39-rVo- inches. This is also nearly a mean between
the length which may be deduced, with proper corrections, from Borda's
experiments at Paris,* and Mr. Whitehurst's experiments made in Lon-
don, t with the apparatus invented by Mr. Hatton,;}; where the length
ascertained is the difference between the lengths of two pendulums vibrat-
ing in different times. Mr. Whitehurst's measures, however, require some
corrections, which Mr. Nicholson has pointed out. The fall of a heavy
body in the first second appears, from this determination of the length of
the pendulum, to be sixteen feet one inch and a tenth.

Of the old French measure, 15 inches made nearly 16 English, and 76,
very exactly 81 ; the toise was 76T%3A inches. In Germany the Rhinland
foot is generally used ; 100 of these feet make 103 English.

A wine gallon contains 231 cubic inches ; an ale gallon is the content of
10 yards of a cylindrical inch pipe.

^ See Base du Systeme Metrique, vol. iii.

f Whitehurst's Attempt to obtain Measures of Length from the Measurement of

Time, 4to, Lond. 1787. Do. on Pendulums, 1792.

I Hatton's Machine for finding a Standard. Trans, of the Soc. of Arts, I. 238.

86 LECTURE X.

A variety of instruments are used for the immediate comparison of the
standard measure or its parts, with other lengths or distances. Such are
scales, simple and diagonal,* verniers, micrometer screws, beam compasses,
rods, lines, chains, and measuring wheels. The greatest accuracy has
generally been supposed to be obtained, in large distances, by means of
rods, made of glass or of platina, in order to be less susceptible of such
changes as are produced by variations of temperature ; General Roy,t how-
ever, found that a steel chain was as little liable to error, as any mode
that he could employ ; and those who have continued the extensive survey
which he began, even prefer it to every other. J For the comparison of
standards, and for determining small distances with great precision, beam
compasses, or scales with sliding indices, furnished with microscopes and
cross wires, have been constructed by the artists of this country: in
France a lever has sometimes been introduced, its longer arm having an
ample range of motion, corresponding to a very minute difference in the
length of the substance which acts on the shorter arm. But for common
purposes the diagonal scale is sufficiently accurate, and may be applied
without the error of the thousandth of an inch : in cases where a very
delicate vernier or a micrometer screw is applied, a magnifier is usually
required. Mr. Coventry has, however, succeeded in making simple scales
which are accurate enough to measure the ten thousandth of an inch.
He draws parallel lines on glass, at this distance, which are in some parts
sufficiently regular, although they can only be seen by the help of a power-
ful microscope : but those which are at the distance of the five thousandth
of an inch are much more correct and distinct. For dividing rectilinear
scales of all kinds, Mr. Ramsden § constructed a machine which acts by
the turns of a screw : others have employed an apparatus resembling Mar-
quois's parallel rulers. (Plate VII. Fig. 95... 97.)

The motion of a ship at sea is measured by a log line, or a rope divided
by knots into equal parts, and attached to a log, which is retained nearly
at rest by the resistance of the water. Attempts have also been made to
cause a little waterwheel to turn by the motion of the ship, and to measure
both the rate and the distance run ; and an instrument has been invented
for doing the same upon hydraulical principles ; raising the water of a gage
to different heights, by means of the pressure occasioned by the relative
motion of the ship and the water, and discharging at the same time a
small stream into a reservoir, with a velocity proportional to that of the
ship.

LECT. X.— ADDITIONAL AUTHORITIES.

Drawing and Painting. — Leonardo da Vinci, Trattato della Pittura, 4to, Rome,
1817; Translation by Rigaud. Junii de Pictura Veterum, fol. Rotterd. 1694. Du-
fresnoy, Art of Painting. De Piles, Do. 1706. Bardwell, Do. 4to, Lond. 1706.

* Hooke on Diagonal Divisions, Animad. on Hevelius. Wallis on Do. Phil.
Tr. 1674, ix. 243.

f Roy's Account of the Measurement of a Base on Hounslow Heath, Ph. Tr.
1785, Ixxv. 385.

J Ramsden's Steel Chain, Ph.Tr. 1785, p. 394.

§ Ramsden's Description of an Inst. for dividing Lines, 4to, 1779.

I

MODELLING, PERSPECTIVE, ENGRAVING, PRINTING. 87

Lahire, Hist, et Mem. ix. 405, 431, 464. Reynolds's Discourses, Burnet's ed.
4to, 1842. Cooper on the Painting of the Ancients, Manch. Mem. iii. 510. Ra-
phael Mengs Obras, 4to, Madrid, 1780. In Italian, by D'Arezza. Wincklemann,
Histoire del' Art chez les Anciens (Trad, de VAllemande), 3 vols. 4to, Paris, 1790-
1803. Burnet' s Hints on Composition, 4to, 1827. Lanzi, Storia Pittorica, Trans,
by Roscoe, 1828. Rosini, Storia della Pittura Italiana, 4 vols. 8vo. already pub-
lished, with plates fol.

Writing. — Nouveau Traite de Diplomatique, 6 vols. 4to, Paris, 1750-65. Butt-
ner on the Alphabets of all Nations, Nov. Com. Gott. 1776, p. 106. Astle's Origin
and Progress of Writing, 4to, Lond. 1803.

Measuring Instruments, 8?c.— Bion on Math. Insts. 1723, and 4to, Paris, 1752.
Adams's Essays, Lond. 1797. Smeaton on the Graduation of Insts. Ph. Tr. 1786,
p. 1. Ludlam do. 4to, 1786. Gounella do. Pistoia, 1816.

Modes of obtaining a Standard, fyc. — Condamine on an Invariable Measure,
Hist, et Mem. de Paris, 1747, p. 489, H. 83. Remarks on Experiments with
Pendulums. Nich. Journ. iii. 29.

Comparison of Measures. — Comparison of French and English Measures, Ph.
Tr. 1742, p. 185. Of English Standards, Ph. Tr. 1743, p. 541. Gray on the
Measures of Scotland, Ed. Ess. i. 200. Shuckburgh on a Standard of Weights and
Measures, Ph. Tr. 1798, p. 133. Kater, Ph. Tr. 1818-19-21. Hall and Foster,
Ph. Tr. 1823. Sabine, Ph. Tr. 1828-29. Baily, Report on the New Standard Scale
of the Astronomical Society, Trans. Royal Astr. Soc. 1836, ix. See also Lect. XII.

LECTURE XI.

ON MODELLING, PERSPECTIVE, ENGRAVING, AND PRINTING.

WE have examined the principal instruments and materials employed
for drawing and for measuring ; we are now to consider, first, the methods
of copying solids, and of projecting their images 011 a plane surface ; and
secondly, the arts of perpetuating the works of the pen and of the pencil
by engraving and printing.

When it is required to make a copy of a solid of an irregular form, as for
example of a statue, we must determine the situation of a sufficient num-
ber of points to guide us in our work with accuracy, by means of an
instrument capable of being fixed in any required situation ; so that the
extremity of a sliding bar or pin may be in contact with each point in the
original, and then removed to a similar part of another frame, on which the
copy is placed, a perforation being made, by degrees, in the block, so as to
suffer the pin to arrive at its proper place, at which it stops. (Plate VII.
Fig. 98.)

The model of a statue is generally first made of clay, and a cast of this
taken immediately in plaster of Paris, since the clay would crack and
change its form in drying. This mode of copying by means of plaster is
exceedingly useful in various departments of the mechanical arts : the
original is well oiled and placed in a proper vessel ; a mixture of prepared
plaster and water, of the consistence of cream, is then poured on it ; this
in a short time hardens, and is divided into several parts, in such a manner
as not to injure the original figure in its removal. These pieces, being
again united, form a mould for the ultimate cast. Sometimes a small

88 LECTURE XI.

figure is first modelled in a mixture of wax, turpentine, and oil ; and a
mould being formed on this, the ultimate cast is made either of plaster, or
of a composition of wax with white lead and a little oil, which serves as
an imitation of marble.

We have, however, much less frequent occasion to make an exact copy
of a solid of any kind, than to represent its appearance by means of per-
spective delineation. Supposing ourselves provided with proper materials
for drawing, we may easily imitate, with the assistance of a correct eye,
and a hand well exercised, the figures and relative positions of objects
actually before us, by delineating them in the same form as they would
appear to be projected on a transparent surface placed before the eye.
Considering the simplicity of this process, it is almost surprising that the
doctrine of perspective should have been supposed to require a very serious
study, and that material errors should have been committed with respect
to it, by men whose general merit in other departments of painting is by
no means contemptible. But it must be confessed, that when, instead of
imitating objects immediately before us, the pencil is employed in embody-
ing imaginary forms, calculated either for beauty or for utility, a great
degree of care and attention may be necessary in order to produce a true
representation of objects, which are either absent, or have no existence :
and here memory and fancy only will scarcely ever be sufficient, without a
recurrence to mathematical principles. To architects therefore, and to
mechanics in general, a knowledge of perspective is almost indispensable,
whenever they wish to convey by a drawing an accurate idea of their
projected works.

If any assistance be required for the delineation of an object actually
before us, it may easily be obtained in a mechanical manner, by means of
a frame with cross threads or wires interposed between the eye and the
object. The eye is applied to an aperture, which must be fixed, in order
to preserve the proportions of the picture ; and which must be small, in
order that the threads and the more distant objects may be viewed at the
same time with sufficient distinctness. The paper being furnished with
corresponding lines, we may observe in what division of the frame any
conspicuous point of the object appears, and may then represent its image
by a point similarly situated among the lines drawn on our paper ; and
having obtained, in this manner, a sufficient number of points, we may
complete the figures by the addition of proper outlines. Sometimes, for the
delineation of large objects requiring close inspection, it has been found
useful to employ two similar frames, the one a little smaller than the
other, and placed at a certain distance from it ; so that every part of the
object, when seen through the corresponding divisions of both frames,
appears in the same manner as if the eye were situated at a very remote
point. It was in this manner that the elegant anatomical figures of Albinus
were executed. (Plate VII. Fig. 99.)

But if it be required to lay down, in the plane of a picture, the projection
of an object of which the actual dimensions and situation are given, we
may obtain the requisite measures from the properties of similar triangles,
and the consideration of the rectilinear motion of light. We may consider

MODELLING, PERSPECTIVE, ENGRAVING, PRINTING. .89

our picture as a reduced copy of a projection formed on an imaginary plane,
which, as well as the picture, is generally supposed to he in a vertical situa-
tion, and which stands on the horizontal plane, at the point where the
ohjects to be represented hegin. In order to find the position of the image
of a given right line, we must determine the point in which a line parallel
to it passing through the place of the eye cuts the plane of the picture ; this
is called the vanishing point of the given line and of all other lines parallel
to it, since the image of any such line, continued without limit, will he a
right line directed to this point, hut never passing it. When the lines to he
represented are parallel to the picture, the distance of their vanishing point
becomes infinite, and their images are also parallel to the lines and to each
other. The centre of the picture, or that point which is nearest to the eye,
is the vanishing point of all lines perpendicular to the picture ; through
this point it is usual to draw a horizontal and a vertical line : we may then
lay off downwards on the vertical line the distance of the eye from the
picture, in order to find the point of distance, which serves to determine
the position of any oblique lines on a horizontal plane : for if we draw a
ground plan of any object, considering the picture as a horizontal surface,
we may find the vanishing point of each of its lines, by drawing a line
parallel to it through the point of distance until it meets the horizontal
vanishing line. (Plate VII. Fig. 100, 101.)

In order to find the position of the image of a given point of a line, we
must divide the whole image in such a manner that its parts may be to each
other in the same proportion as the distance of the given point and of the
eye, from the plane of projection. This may be readily done, when a
ground plan has been first made, by drawing a line "from any point in the
plan to the point of distance, which will cut the whole image of the line ia
the point required. (Plate VII. Fig. 102.)

When it is required to determine a point in a line parallel to the picture,
we may suppose a line to be drawn through it perpendicular to the picture,
and, by finding the image of this line, we may intersect the former image
in the point required. It is thus that the height of any number of columns
or figures, at different distances, may be readily determined. (Plate VIII.
Fig. 103.)

The projection of curvilinear figures is most conveniently effected by
drawing across them parallel lines, which form small squares or rectangles,
throwing these divisions into perspective, and tracing a curve through the
corresponding points. There are also methods of determining mathemati-
cally, or of drawing mechanically the ellipsis, which results from the
projection of a circle, in a given position, but they are considerably intri-
cate, and a steady hand is seldom in want of them. (Plate VIII. Fig. 104.)

This system of perspective must necessarily be employed when we wish
to represent objects which appear to us under angles of considerable mag-
nitude, and to give them as much as possible the appearance of an imitation
o£ nature. But for almost all purposes of science, and of mechanical
practice* the most convenient representation is the orthographical projection,
where the distance of the eye from the plane is supposed to be increased
without limit, and the rays of light passing to the eye to be parallel to each

90 LECTURE XL

other. In order to represent any object in this manner, we must assume
one line for the direction of the centre of the picture, to which the images
of all lines perpendicular to the plane of projection must he parallel, and
another for that of the point of distance, hy means of which we may
measure the first lines, #s if that point were actually within reach ; and in
this manner we may determine the place of any number of points of the
object to be delineated. (Plate VIII. Fig. 105.)

If we wish to apply the mechanical method of drawing by the assistance
of a frame to this mode of representation, instead of a fixed aperture for a
sight, or a second frame of smaller dimensions, we must employ a second
frame of the same magnitude with the first, in the manner which has
already been described. Professor Camper* has censured Albinus for not
adopting this method in his figures : but subjects so large as those which
he has represented would have had less of the appearance of nature, if they
had been projected orthographically, nor would such projections have been
materially more instructive.

It frequently happens, that in geographical and astronomical drawings
we have occasion to represent, on a plane, the whole or a part of a spherical
surface. Here, if we employ the orthographical projection, the distortion
will be such that the parts near the apparent circumference will be so much
contracted as to render it impossible to exhibit them with distinctness. It
is, therefore, more convenient, in this case, to employ the stereographical
projection, where the eye is supposed to be at a moderate distance from the
object. The place of the eye may be assumed either within or without the
sphere at pleasure ; and according to the magnitude of the portion which
we wish to represent, the point, from which the sphere may be viewed with
the least distortion, may be determined by calculation. But in these cases
all circles obliquely situated on the sphere must be represented by ellipses :
there is, however, one point in which the eye may be placed, which has the
peculiar and important advantage, that the image of every circle, greater or
lesser, still remains a circle. This point is in the surface itself, at the
extremity of the diameter perpendicular to the plane of projection ; and
this is the point usually employed in the stereographical projection of the
sphere, which serves for the geometrical construction of problems in spheri-
cal trigonometry. The projection of the whole surface of the sphere would
occupy an infinite space, but within the limits of the hemisphere, the
utmost distortion of the linear measure is only in the proportion of 2 to 1,
each degree at the circumference of the figure occupying a space twice as
great as at the centre. The angles, which the circles form in crossing each
other, are also correctly represented. (Plate VIII. Fig. 106.)

For projecting figures on curved or irregular surfaces, the readiest method
is to trace cross lines on them, with the assistance of such a frame as has
been described for drawing in perspective, representing the appearance of
uniform squares or rectangles, and to delineate in each of these the corre-
sponding parts of the object, or of the drawing which serves as a copy. .

The arts of writing and drawing, in all their varieties, are extended in

* Cogan's Translation of Camper, on the connection between Anatomy and the
Arts of Painting, Sculpture, &c. 4to, Lend. 1794.

MODELLING, PERSPECTIVE, ENGRAVING, PRINTING. 91

their performance, and perpetuated in their duration, by means of en-
graving and printing. If there is any one circumstance to which we can
peculiarly attribute the more rapid progress of general civilisation in mo-
dern than in ancient times, it is the facility of multiplying copies of literary
productions of all kinds, by the assistance of these arts. The distinguishing
character of printing consists in the employment of moveable types : the art
of engraving is more simple, and in some of its forms more ancient. The
Romans were in the habit of using seals and stamps, for marking letters
and words oh wax and on pottery; it was usual in the middle ages to
employ perforated plates of metal as patterns for guiding a brush, by means
of which the capital letters were inserted in some manuscripts, and the
Chinese are said to have been long in possession of the art of printing books
from wooden blocks.* It was in this form that printing was first intro-
duced into Europe, in the beginning of the fifteenth century. There seems
to have been formerly a method of engraving on wood with greater ease
and accuracy than is now practised ; the hatches may be observed in old
wooden cuts to cross each other more frequently and with greater freedom,
than in modern works, although some have conjectured, with considerable
appearance of probability, that these old engravings were in reality etched
in relief on metal. The art of engraving on wood is, however, at present
in a high degree of perfection in this country, and blocks are still frequently
used for mathematical diagrams and other simple figures : for although
they are somewhat more expensive than copper plates, they wear much
longer, and they have the advantage of being printed off at the same time
with the letter press, and of being included in the same page with the text
to which they belong, since the ink is applied to the projecting parts only,
both of these cuts and of the common printing types.f

The method of engraving on plates of pewter or of copper, and of taking
impressions, by means of the portion of ink retained in the furrows cut by
the graver, was also introduced in the fifteenth century. For dry engraving,
the drawing, if it is not executed in black lead, is generally prepared by
passing a pencil over its principal features, and the outline is transferred to
the plate, which has a thin coat of white wax laid on it, by placing the
drawing on it, and rubbing it with a burnisher ; sometimes a drawing in
Indian ink, especially if freed from a part of its gum, may be transferred
in this manner without the application of a pencil. When written charac-
ters are to be engaved, the plate is laid on a cushion, so as to be readily
turned under the graver, which is a great convenience in forming curved
lines.

In laying on equable shades of considerable extent, much labour is saved
by the use of a ruling machine, which enables us to draw lines, at any re-
quired distance, very accurately parallel, and either straight, or following
each other's gentle undulations, in order to avoid Jhe appearance of stiffness.

, * Du Halde, Description de 1'Empire de la Chine, 4to, 1736. Zani, Material!
per Servire alia Storia dell' Incisione in Rame ein Legno, Parma, 1802.

f An account of the re-discovery of the mode of decarbonizing steel so as to ren-
der it capable of being engraved on, will be found in the Tr. of the Soc. of Arts,
vol. xli.

92 LECTURE XI.

This machine, like the dividing engine, is sometimes adjusted by the revolu-
tions of a screw, and sometimes by the oblique motion of a triangular
slider. Besides the cutting graver, which is of a prismatic form, terminated
by an oblique surface, other instruments are occasionally employed ; the
dry needle makes a very fine line, and leaves the metal that it has displaced
to be rubbed off by another tool. Sometimes a number of detached exca-
vations are formed by a pointed instrument, and the projections are after-
wards removed ; this is called stippling. A burnisher and some charcoal
are required for erasing the strokes of the graver, when it is necessary, and
for polishing the surface. It is seldom, however, that a plate is begun and
completed by dry engraving only.

For engraving in mezzotinto, the plate is roughened, by scraping it in
every direction with a tool made for the purpose, so that an impression
from it, in this state, would be wholly dark ; the lights are then inserted,
by removing the inequalities of the surface, in particular parts, by means
of a smooth scraper and a burnisher. As the plate wears in printing, some
of these parts are liable to have the grain a little raised again, so that the
lights are less clear in the later impressions than in the proofs. It is well
known, that in common engravings the proofs are usually the darkest
throughout.

The most expeditious and most generally useful mode of working on
copper, is the process of etching. The plate, being covered with a proper
varnish, is usually blackened with smoke, and the drawing is placed on it,
with the interposition of a paper rubbed over with red chalk, which, when
the drawing is traced with a wooden point, adheres to the varnish, in the
form of the outline : or if it is required that the ultimate impression be
turned the same way as the drawing, an intermediate outline must be
procured in the same manner on a separate paper, and then transferred to
the plate. All the outlines thus marked are traced with needles, which
make as many furrows in the varnish, and leave the copper bare : the
shades are inserted with the assistance of the ruling machine, wherever
parallel lines can be employed. The plate thus prepared, and furnished
with an elevated border of a proper consistence, is subjected to the action
of the diluted nitric acid, until all the parts are sufficiently corroded, care
being taken in the mean time to sweep off the air bubbles as they collect,
and to stop out, or cover with a new varnish, the lighter parts, which are
soonest completed. When the varnish is removed, the finishing touches
are added with the graver : and if the plate requires further corrosion, the
varnish may sometimes be replaced, without filling up the lines, by apply-
ing it on a ball or cushion, taking care to avoid any oblique motion. It is
said that the acid sometimes operates so as to undermine the metal on each
side, and to render the furrows wider as they become deeper, and that for
this reason in etchings, as well as in mezzotintos, the latter impressions
are sometimes darker than the proofs ; but this is by no means universally
true. It is well known to chemists, that glass may be corroded in a similar
manner by means of the fluoric acid.

An etching may also be expeditiously executed by using a varnish
mixed with mutton fat, and drawing upon a paper laid on the plate ; the

MODELLING, PERSPECTIVE, ENGRAVING, PRINTING. 93

varnish then adheres to the hack of the paper, under the lines which are
drawn, and is immediately removed when the paper is taken off, without
the use of needles. Sometimes the outlines only are etched, and the plate
is finished in mezzotinto.

In the mode of engraving called aqua tinta, the outline having heen first
etched, the shades are also produced by corrosion, the parts being prepared
by various methods, so as to be partially protected from the action of the
acid. Sometimes a little resin, very finely powdered, is sifted on the plate,
which is then sufficiently warmed to make the particles adhere to it ; some-
times it is varnished with a spirituous solution of resin, which cracks
throughout in drying : and if a strong line be any where required, it may
be traced with a mixture of whiting with some adhesive substance, before
the varnish is laid on ; this will cause it to break up at that part ; or the
varnish may be partially removed, by rubbing it with spirits or with an
essential oil. The lighter parts may be covered, during the corrosion, with
a second varnish, which defends them from the acid. This mode of en-
graving succeeds very well in imitating the effect of drawings, but the
plates are soon worn out. In order to judge of the state of the work, an
impression of any part of the plate may be taken off, by pouring on it a
little plaster of Paris mixed with water.

Musical characters are usually stamped with punches ; in this country,
on plates of pewter, but in France generally on copper. Mr. Rochon *
has invented a machine for stamping letters on copper, instead of printing,
but the method does not appear to have been practically employed.

In whatever way the plate may have been engraved, when an impression
is to be taken from it, it is covered with printing ink of the finest kind, by
means of stuffed balls, and then wiped, chiefly with the hand, so that the
ink is wholly removed from the polished surface : it is then placed, with
the moistened paper, on a board, between flannels, and strongly pressed
in passing between two wooden rollers. By frequent use the plate loses its
sharpness, and sometimes requires to be retouched ; hence arises the
greater value of first impressions ; but by proper precautions in cleaning
the plate, its delicacy may be preserved for a long time.

An impression, while it is moist, may be reversed, by passing it through
the press with another paper. And by writing with a peculiar ink, even
common letters may be thus copied on thin paper, and the impression will
be legible on the opposite side. Mr. Montbret proposes to put some sugar
candy into the ink, and to take a copy on unsized paper by means of a
hot iron.f

A simple and elegant method of multiplying drawings has been lately
introduced by Mr, Andre. The drawings are made with an unctuous com-
position, in the form of a crayon or of an ink, on a soft stone of a calca-
rious nature, somewhat like a stone marie. When the drawing is finished,
the stone is moistened, and imbibes so much Svater that the [unctuous]
printing ink will not adhere to it, except at the parts where the crayon or
tne ink has been applied [for neither will water adhere to grease nor grease
to water] ; and in this manner an impression is procured, which has much
* Nich. Jour. 4to, Ui. 61. f Ibid. 8vo, i. 147.

94 LECTURE XI.

of the freedom and spirit of an original drawing. When the ink is used,
a little acid is afterwards applied to the stone, in order to corrode its inter-
mediate parts ; and the bold style of the impression much resembles that of
the old wooden cuts.

The art of printing with separate types was invented soon after the
introduction of wooden blocks into Europe.* The improvement was great
and important. The year 1443, or 1444, is considered as the date of the
oldest printed book ; but the precise time and place of the invention remain
somewhat doubtful: the art, however, advanced towards perfection by
very rapid steps. The letters are first cut, in a reversed form, on steel
punches ; with these a matrix of copper is stamped, and the matrix forms
the lower part of the mould in which the types are cast ; the metal is a
composition of lead and antimony, which is easily fusible. Thus the
printed sheet is the fourth form of the letter, reckoning from the original
engraving on the punch : in the stereotype printing, lately invented, or
rather improved and revived, it is the sixth. In this method, when a form
for the side of a sheet has been composed, made up, corrected, and locked
up by wedges in the chase or iron frame which confines it, a mould of the
whole is formed in fine plaster, and as many repetitions of it may be cast
very thin, in type metal, as will serve to print for the use of a century,
without the expense of keeping a large quantity of types made up, or of
providing paper for a numerous impression at once.

The modes of arranging the types in boxes or cases, of composing the
separate lines on the stick, and making them up by degrees into pages and
forms, of correcting the press, of applying the ink, and taking off the
impression, are entirely calculated for the simplicity and convenience of
the manual operations concerned, and require little or no detailed expla-
nation.

LECT. XL—ADDITIONAL AUTHORITIES.

Sculpture, Painting, Sfc. — Behnes's Machine for Sculpture, Tr. of the Soc. of
Arts, XXXVII. Jesuit's Perspective, 4to. Brook Taylor's Linear Perspective
1715 and 1811. Monge Geometric Descriptive, 4to, Paris. Edwards's Perspec-
tive, 4to, 1803. Creswell's Perspective, Camb. 1812. Courtonne, Deidier, Laine,
Ozanam, Faucaud, Lavit, Traites de Perspective. Laurent, Theorie de la Peinture,
1827. Montabert, Dessein Lineaire enseigne aux Ouvriers, 1831. Bardwell,
1834. Rider, 1836. Hall, Practical Geometry, &c. 1841. A brief Elementary
Treatise on Projections is given in the Appendix to Maddy's Astronomy, Camb.
1826.

Engraving. — Evelyn's Art of Engraving, 1662. Papillon, Traite Historique de
laGravure en Bois, 1766. Lowry's Ruling Machine, Nich. Jour. ii. 523. Accum
on Etching on Glass, ibid. iv. 1. Bartsch Peintre Graveur, 21 vols. Vienna,
1808. Ottley's Hist, of Engraving, 2 vols. 4to, 1816. Hullmandel's Manuel of
Lithography, 1820. Englemann, Manuel du Dessinateur Lithographique, Par. 1824.
Bregault, do. 1827.

The inventor of Lithography was Alois Senefelder, of Munich. Andre was asso-
ciated with Senefelder, but has no claim to the invention. A patent for fifteen years
was granted to Senefelder in 1799. The art has now arrived at a high state of per-
fection.

* Consult Hansard's Typographia, 1825.

95

LECTURE XII,

ON STATICS.

THE examination of the magnitude of the various forces employed in
practical mechanics, constitutes the doctrine of statics. The term statics,
in a strict sense, implies the determination of weights only; hut it may
without impropriety be extended to the estimation of forces of all kinds,
especially active forces, that can be compared with weights, in the same
manner as the term hydrostatics comprehends every thing that relates to
the equilibrium of fluids. The measurement of the passive strength of the
materials employed, the changes produced in them by the forces which they
resist, and the laws of the negative force of friction, are also subjects imme-
diately introductory to the particular constructions and uses of machinery,
and nearly connected with the department of statics.

The art of weighing is peculiarly important, as it furnishes us with the
only practical mode of determining the quantity of matter in a given body.
We might indeed cause two bodies to meet each other with known veloci-
ties, and from the effects of their collision we might determine their com-
parative momenta, and the proportion of their masses ; but it is obvious
that this process would be exceedingly troublesome, and incapable of great
accuracy ; we therefore recur to the well known law of gravitation, that
the weight of every body is proportional to the quantity of matter that it
contains, and we judge of its mass from its weight. If all bodies were of
equal density, we might determine their masses from their external dimen-
sions ; but we seldom find even a single body which is of uniform density
throughout ; and even if we had such a body, it would in general be much
easier to weigh it correctly than to measure it.

The weight of a body is commonly ascertained, by comparing it imme-
diately with other weights of known dimensions ; but sometimes the
flexure of a spring is employed for the comparison. Standard weights
have generally been deduced from a certain measure of a known substance,
and in particular of water. According to the most accurate experiments,
when the barometer is at 30 inches, and Fahrenheit's thermometer at 62°,
12 wine gallons of distilled water weigh exactly 100 pounds avoirdupois,
each containing 7000 grains troy ; and a cubic inch weighs 252 £ grains.
A hogshead of water, wine measure, weighs, therefore, 525 pounds, and a
tun 2100 pounds, which is nearly equal to a ton weight. Mr. Barlow *
supposes that the tun measure of water contained originally 32 cubic feet,
and weighed 2000 pounds, which was also called a ton weight, the gallon
being somewhat smaller than it is at present, and the cubic foot weighing
exactly 1000 ounces, or 62£ pounds. A quarter' of wheat weighed about
a quarter of a ton, and a bushel as much as a cubic foot of water. A
chaldron of coals was also considered as equivalent to a ton, although it

* On the analogy between English weights and measures of capacity, Ph. Tr.
1740, p. 457.

96 LECTURE XII.

now weighs nearly half as much more. But at the mean temperature of this
climate, or 52°, a cubic foot of distilled water weighs only 998 ounces.
The avoirdupois ounce appears to agree very nearly with the ancient Roman
ounce. Of the old French weight, 100 pounds made 108 English pounds
avoirdupois. The gramme of the new weights is a cubic centimetre of
pure water at its greatest density, that is about the temperature of 39° of
Fahrenheit ; it is equal to 15£ English grains : hence the chiliogramme is
2£ pounds, and five myriogrammes are nearly a hundred weight. Five
grammes of silver, including one tenth of alloy, make a franc, which is one
eightieth better than the old franc or livre, and is intrinsically worth nearly
ninepence three farthings English.

The instruments usually employed for the comparison of weights are
either balances or steelyards. In the common balance, the weights of the
substances compared are equal ; in a compound weighing machine, we use
weights which are smaller, in a certain proportion, than those which they
represent: in the steelyard, a single weight acquires different values at
different parts of the arm, and in the bent lever balance the position of
the arms determines the magnitude of the counterpoise. The spring steel-
yard measures the weight, by the degree of flexure that it produces in a
spring.

The beam of a common balance must have its arms precisely equal.
The scales, being freely suspended from fixed points in the beam, act on
them always in the direction of gravity ; and the effect is the same as
if the whole weight were concentrated in those points. The beam sup-
ports the scales, and is itself supported by means of fine edges of hard
steel, working on steel, agate, or garnet, in order that the motion may be
free, and the distances of the points precisely defined. The best beams are
made of two hollow cones of brass, united at their bases ; they are lifted
off their supports when the balance is not used, in order to avoid accidental
injuries ; the scales also are supported, so as not to hang from the beam,
until they have received their weights. According to the position of the
fulcrum, with respect to the points of suspension of the scales, the equili-
brium of the balance may be either stable, neutral, or tottering ; or if the
beam be too flexible, it may pass from one of these states to the other by
the effect of the weights. The stable equilibrium is the most usual and the
best, because it gives us an opportunity of determining the degree of in-
equality of the weights, by the position in which the centre of gravity
rests, or by the middle point of the vibrations of the beam, which are
sometimes measured by an index pointing to a graduated arc. If, how-
ever, the fulcrum be too much elevated above the centre of gravity, the equi-
librium may be too stable, and may require too great an inequality in order
to produce a sensible preponderance. If, on the contrary, by the elevation
of the points of suspension of the scales, the equilibrium be rendered tot-
tering the lower scale will not rise, even if it be somewhat less loaded than
the upper ; and steelyards of this construction have sometimes been em-
ployed, in order to impose on the purchaser by the appearance of an ample
weight. It is necessary, where great accuracy is desired, to bring the
equilibrium very near the state of neutrality, and to make the vibrations

ON STATICS. ^97

of the beam slow and extensive, whether the scales have weights in them
or not : for this purpose a small weight is sometimes inclosed within the
beam, which is raised or depressed at pleasure by a screw, so as to bring
the centre of gravity of the whole moveable apparatus as near to the ful-
crum as may be required for the occasion. Mr. Ramsden's balance, made
for the Royal Society, is capable of weighing ten pounds, and turns with
one ten millionth part of the weight.* (Plate VIII. Fig. 107... 109.)

The arms of a balance have sometimes been made unequal for fraudulent
purposes, the weight being placed nearer to the fulcrum than the substance
to be weighed. It is obvious that the fraud may be detected, by changing
the places of the contents of the two scales. In such a case, if a counter-
poise to the same weight be determined in each situation, the sum of both
will be greater than twice the weight ; and the purchaser would be sure of
having even more than his due, by requesting the seller to weigh half in
the one scale and half in the other. For example, if one arm of the beam
were only three fourths as long as the other, the counterpoise to a weight
of twelve ounces would be nine ounces in one scale, and sixteen in the
other, making together twenty five instead of twenty four ounces, (Plate
VIII. Fig. 110.)

Supposing the beams of a balance to be accidentally unequal, either in
length or in weight, we may still weigh in it with accuracy, by making a
perfect counterpoise of any kind to a weight, and then removing the
weight and putting in its place as much of the substance to be weighed as
is sufficient to restore the equilibrium.

The weights may also be reduced, or increased, in proportion to the
length of the arms, if they differ from each other, care being taken to put
the weights always into the same scale. This is actually performed in
weighing machines, where a composition of levers is employed, in order to
enable us to determine the weight of large masses by means of weights of
moderate dimensions. (Plate IX. Fig. 111.)

When the effective lengths of one or both arms of the beam are capable
of being varied by changing the points of suspension according to the
divisions of a scale, the instrument is called a steelyard. Where one
weight only is used, it is not necessary that the two arms should exactly
balance each other, since the divisions may be so placed as to make the
necessary adjustment ; but it is sometimes convenient to have two or three
weights of different magnitudes, and for this purpose the instrument
should be in equilibrium without any weight. In such cases, great accu-
racy may be obtained by applying a small weight at the end, in the form
of a micrometer screw. (Plate IX. Fig. 112.)

The arms of a balance, though constant in length, may vary in effect
without limit, if they can sufficiently alter their inclination to the horizon ;
for no weight, however great, acting on the arm of a bent lever, can make
it perfectly vertical, since, in this position, the weight may be over-
powered by the minutest counterpoise acting on the other arm. The centre
of1 gravity being, in the common balance, very nearly in a right line
between the weights, in order that it may be immediately below the
* Rozier's Journal, xxxiii. 144.
H

98 LECTURE XII.

fulcrum, the arm must have a very considerable angular motion for a
slight inequality of the weights ; but in the bent lever balance, the centre
of gravity is at such a distance from the fulcrum, that a moderate motion
of the arms may bring it into the vertical line. This motion is measured
by an index on a graduated arc, which gives the instrument a considerable
range ; and where expedition is particularly desired, it may often be used
with advantage ; but if the weights to be determined are large, the scale
becomes very much contracted, and the instrument requires to be levelled
with great accuracy. A counterpoise acting on a spiral or conical barrel
has also been applied to a similar purpose ; it is capable of a scale some-
what more extended than a bent lever balance, but it is less simple, and
scarcely more accurate. (Plate IX. Fig. 113.)

A spring, which is usually of a spiral form, being made to support a
hook by the intervention of a graduated bar, the divisions of this bar,
which are drawn out beyond the fixed point, indicate the weight sup-
ported by the hook. This instrument is called a spring steelyard. Mr.
Hanin's * spring steelyard has a long index, which revolves on a centre,
and shows at once the weight according to the standards of different coun-
tries. The divisions of the scales in moderate flexures of the spring are
nearly equal : hence it may be inferred, that the space through which a
spring is bent, and consequently its curvature or change of curvature, is
simply proportional to the force which acts on it, and that the vibrations of
a weight supported by a spring, must, like those of a cycloidal pendulum,
be performed in equal times, whatever may be their magnitude. The
strength of all springs is somewhat diminished by heat, and for each degree
of Fahrenheit that the temperature is raised, we must deduct about one
part in five thousand from the apparent weight indicated by the spring
steelyard. (Plate IX. Fig. 114.)

The spring steelyard affords us the most convenient method of measur-
ing the immediate intensity of the forces exerted by animals of different
kinds, in the labour which they perform. When it is adapted for this
purpose, it is sometimes called the dynamometer. We may also estimate
the force of an animal which is employed in drawing a distant boat or car-
riage, by the inclination of the rope or chain to the horizon, compared
with the weight of that portion of it which the animal supports, that is, of
the part which extends to the point where the curve becomes horizontal.*!*

All animal actions, or, at least, all the external actions of animals, are
ultimately dependent on the contractions and relaxations of the fleshy
parts, which are called muscles. The operation of the particular muscles
belongs properly to the science of physiology ; but their mechanism may
in general be understood from the properties of the lever and of the
^ centre of gravity. The bones are the levers, the joints the fulcrums, and
the force is applied by the muscles, which are usually attached to the bones
by the intervention of tendinous cords. When a muscle contracts in the
direction of its fibres, it becomes at the same time thicker, and its total bulk

* Hist. etMem. de Paris, 1765, H. 135.

•f Consult Morin, Description des Appareils Chronometriques, et des Ap. Dyna-
mometriques. Metz, 1838.

ON STATICS. 99

is little if at all diminished : when it relaxes itself, it is merely passive, for
the fibres, being extremely flexible, can have little or no efl&ct in separating
the parts to which they are attached ; this separation is generally performed
by the action of other muscles which are called the antagonists of the first,
but sometimes by elastic ligaments, or by other means. The bone forms a
lever of the second kind, where the two forces opposing each other are on
the same side of the fulcrum. In general the insertion of a muscle is much
nearer to the fulcrum than the point of action, and the obliquity of its
direction gives it a still greater mechanical disadvantage with regard to
rotatory power ; but it is more convenient in the animal economy to pro-
duce a great contractile force than a great extent in the original motion.
For instance, when the arm is raised by the exertion of the deltoid muscle
of the shoulder, a very strong contraction takes place in the muscle, but
the action is only continued through a short space ; had the contractile
power been weaker and more extensive, the shoulder must have been made
higher, in order to give it sufficient purchase, and the projection would
have been inconvenient.

Borelli* has calculated that the immediate force of the biceps, or double-
headed muscle which bends the arm, is equivalent to about 300 pounds, and
that of the muscles which raise the lower jaw, above 500 in man, but in
beasts of prey far greater. It is obvious that in muscles of the same kind
the strength must be as the number of fibres, or as the extent of the surface
which would be formed by cutting the muscle across ; and it is not im-
probable that the contractile force of the muscles of a healthy man is
equivalent to about 500 pounds for each square inch of their section. The
weakest man can lift with his hands about 125 pounds, a strong man 400.
Topham, a carpenter, mentioned by Desaguliers, could lift 800 pounds.
He rolled up a strong pewter dish with his fingers ; he lifted with his teeth
and knees a table six feet long, with a half hundred weight at the end.
He bent a poker, three inches in circumference, to a right angle, by striking
it upon his left fore arm ; another he bent and unbent about his neck ; and
snapped a hempen rope two inches in circumference. A few years ago
there was a person at Oxford who could hold his arm extended for half a
minute, with half a hundred weight hanging on his little finger. A young
gentleman, who has distinguished himself as a pedestrian by going 90 miles
in 19 hours, has also lifted two hundred weights, one in each hand, and
made them meet over his head.

Sometimes feats of strength apparently extraordinary have been ex-
hibited by men who have not really been possessed of any material supe-
riority. Desagulierst relates, that one of them used to withstand the force
of two horses drawing at a girdle passed round his middle, while his feet
acted on a firm obstacle. By falling suddenly backwards, in an oblique
position, he broke a rope which was fixed a little before his feet. He
supported one or two men by forming his body into an arch ; and by a
harness fitted to his hips, he sustained a cannon weighing two or three

* De Motu Animalium, 4to, Lugd. Batav. 1710, p. 30 et seq.
f Course of Experimental Philosophy, 2 vols. 8vo, Lond. 1763, i. 266, &c.

H2

100 LECTURE XII.

thousand pounds. In all these cases the muscles principally employed are
the extensors of ^he legs and thighs, hut the passive strength of the hones is
more concerned than the active force of the muscles. In the instance,
mentioned by Lahire,* of a young man who raised an ass from the ground
by cords tied to the hair of his head, the sensibility of the nerves of the
skin must have been diminished by habit, so as to allow the hair to be thus
forcibly extended without immoderate pain.

The application of animal force is usually performed by means of a
progressive motion. The muscles employed in this process are in general,
if not always, the strongest of the body, both by nature and by habit ; so
that when force alone is required, it is most advantageously obtained from
their exertions. In walking, the centre of gravity is moved forwards with
a velocity nearly uniform. If the legs were perfectly inflexible, the centre
of gravity would describe, in succession, portions of circles, of which each
leg would alternately be the radius : but if the velocity were great enough
to create a centrifugal force more than equivalent to the force of gravity,
the pressure would be removed from each leg after the first instant of its
touching the ground ; the path would become parabolic instead of circular,
and the walking would be converted into running : for the difference be-
tween walking and running is this, that in running, one foot is removed
from the ground before the other touches it ; while in walking, the hind-
most foot is only raised after the foremost has touched the ground. Now
supposing the length of the inflexible leg three feet, the centrifugal force
would become equal to the weight, with a velocity which would be acquired
by a heavy body in falling through a foot and a half, that is, near 10 feet
in a second, or 7 miles an hour ; and this is the utmost velocity with
which it would be mechanically possible to walk with inflexible legs. But
the flexibility of the legs makes the progressive motion much more uni-
form, by softening the angles of the path which the centre of gravity
describes, and rendering it either more or less curved at pleasure ; so that
it becomes mechanically if not physically possible, to walk with a velocity
somewhat greater than 7 miles an hour, and to run or dance with as small
a velocity as we please, since we may make the path of the centre of gravity
somewhat less, or much more curved, than a circle described on the point
of the foot as a centre. (Plate IX. Fig. 115, 116.)

The flexions and extensions of the legs are also almost the only means
by which an impulse is given to the body ; if the legs were perfectly
inflexible, it would be extremely difficult, although not absolutely impos-
sible, to obtain a progressive motion. The centre of gravity is principally
impelled forwards in the beginning of the ascending part of the curve
which it describes in walking, by the action of the leg which is left behind,
but in running or hopping, by that of the only foot which touches the
ground at any one time. When we thrust against any obstacle, or draw a
rope in a horizontal or in a descending direction, the body is inclined
forwards, and in some cases its action is limited by the effect of the weight
of the body reduced to the direction of the line of draught : but we much

* Hist, et Mem. de 1'Acad. 1699, p. 153, H. 96.

ON STATICS. 101

more usually draw or pull in an ascending direction, so that our whole
muscular force may be exerted without any limit of this kind.*

It happens, however, very frequently, that we have occasion for motions
of such a nature as to be more conveniently performed by the hands and
arms than by the action of walking or running ; and where delicacy is
required rather than strength, the form of the hand and fingers gives the
human species a great superiority over all other animals, although by no
means, as some authors have supposed, an advantage equivalent to that of
the higher perfection of the intellectual powers. It is true, as we may
observe in the manufactories of this country, that machinery has been
invented by which a power of any kind may be converted to purposes
seemingly the most intricate and refined ; and after all that has been done
by a Watt and an Arkwright, it is difficult to determine a positive limit to
the ingenuity of mechanical invention.

It is necessary to consider, in examining the different sources of motion,
not only the immediate magnitude of the forces which they produce, but
also the velocity with which they are capable of acting, and the time for
which that action can be continued. The daily work of a labouring man,
of middle age, and in good health, will serve as a convenient unit for the
comparison of moving powers of all kinds. It may be most easily remem-
bered in this form : a man can raise a weight of 10 pounds to the height of
10 feet in a second, and can continue this labour for 10 hours a day. The
actual velocity of the man's motion must vary according to the mode in
which his force is applied ; but we suppose that velocity to be such as to
give the greatest effect under the circumstances of the machine. This is a
moderate estimate of the work of a labourer, without any deduction for
friction. Desagulierst states the performance of a man working at a
winch, with the assistance of a fly, as considerably greater, but he does not
allege any correct experiments in support of his estimate. Professor
Robison, however, mentions a hydraulic machine in which the effect was
actually more than one tenth greater, without making any allowance for
friction ; so that it is probable, considering the loss both from friction and
from the momentum with which the water must have been disengaged,
that the immediate performance was at least one third more than this
unit : the machine was worked by a light man carrying a weight, and
walking backwards and forwards on a lever. According to Mr. Bucha-
nan's J experiments, an action like that of ringing bells produced an effect
about one third greater than turning a winch, and the action of rowing, an
effect four ninths greater ; but it does not appear that these experiments
were continued for a whole day ; and the greatest number of observations
make the daily performance of workmen considerably less. It is indeed
seldom that the muscles employed in progressive motion are so much
exerted as in the arrangement described by Professor Robison. A Chinese,
in the operation called sculling, is said to beat a European at his oar.
•For a short time a much greater effect than this may be produced by a

* See Mairan on the position of the legs in walking, Hist, et Mem. de Par. 1721,
t Desaguliers, vol. i. pp. 254, 255. J Repertory of Arts, xv. 319.

102 LECTURE XII.

great exertion :* thus a man weighing above 160 pounds can ascend by
means of steps at the rate of more than three feet in a second, for a quarter,
or perhaps half a minute ; and this is an effort five times as great as that
which can be continued for a day. Usually, however, where the hands are
chiefly employed, whether in turning a winch, or in pumping, it is only
possible to exert a double, or at most a triple action, for a minute or two :
thus, although a machine may only enable a man to raise a hogshead of
water in a minute to the height of ten feet for a whole day, yet it is easy
to work it so rapidly for a single minute as to raise double the quantity, or
to raise a single hogshead to a height of twenty feet. The whole exertion
of force must be a little greater than that which is thus estimated, because
a certain degree of superfluous momentum must be generated in removing
weights from one situation to another : but this loss is usually incon-
siderable.

The action of carrying a load horizontally requires an exertion of a
different kind, and admits of no direct comparison with the application of
a constant force to overcome the gravitation of a weight, or any other
immediate resistance. The work of a labourer thus employed is however
confined within moderate limits. A strong porter can carry 200 pounds at
the rate of three miles an hour ; and, for a short distance, even 300 pounds :
a chairman carries 150 pounds, and walks four miles an hour : and in
Turkey it is said that there are porters, who, by stooping forwards, and
placing the weight very low on their backs, are enabled to carry from 700
to 900 pounds. The subjects of Mr. Coulomb's t experiments appear to
have been either weaker or more inactive than the generality of porters in
this country: he calculates that the most advantageous load for a man of
common strength is about a hundred weight ; or, if he is to return without
a burden, 135 pounds.

The daily work of a horse is equal to that of five or six men : its imme-
diate force is something greater, but it cannot support the labour of more
than 8 hours a day, when drawing with a force of 200 pounds, or of 6
hours when with a force of 240, walking two miles and a half an hour. It
is generally supposed that in drawing up a steep ascent a horse is only
equivalent to 3 or 4 men, and the employment of horses in walking wheels,
where the action is similar to that of ascending a hill, has for this reason been
condemned. For men, on the contrary, an ascent of any kind appears to
afford a favourable mode of exertion. But, perhaps, the weight of the
carriage, and of the horse itself, has not always been sufficiently considered
in the comparison. The strength of a mule is equal to that of three or four
men. The expense of keeping a horse is in general about twice or three
times as great as the hire of a day labourer ; so that the force of horses
may be reckoned about half as expensive as that of men. The horse
Childers is said, although, perhaps, without sufficient authority, to have
run an English mile in a single minute ; his velocity must in this case
have been 88 feet in a second, which would have been sufficient to carry

* See Amontons, Hist, et Mem. de 1'Acad. 1703.

t On the Daily Labour of Men, Nich. Jour. iii. 416.

I
ON STATICS. i03

him on an inclined plane without friction, or in a very long sling, to the
perpendicular height of 120 feet.*

A large windmill, on which Mr. Coulombt made many experiments, was
capable, on an average, of working eight hours a day ; its whole perform-
ance was equivalent to our estimate of the daily labour of 34 men ; 25
square feet of the sails doing the work of one labourer. The expense of
the machinery, with its repairs, would probably amount to less than half
the expense of a number of horses capable of exerting the same force.
Where a stream of water can be procured, its force is generally more con-
venient, because more regular, than that of the wind.

A steam engine of the best construction, with a thirty inch cylinder has
the force of forty horses ; and, since it acts without intermission, will per-
form the work of 120 horses, or of 600 men, each square inch of the
piston being nearly equivalent to a labourer. According to Mr. Boulton,
the consumption of a bushel, or 84 pounds of coals, will raise 48,000 cubic
feet of water 10 feet high, which is equivalent to the daily labour of 8^
men, or perhaps more : the value of this quantity of coals is seldom more
than that of the work of a single labourer for a day ; but the expense of
the machinery generally renders a steam engine somewhat more than half
as expensive as the number of horses for which it is substituted. Accord-
ing to other accounts, a 24 inch cylinder, being equivalent to about 72
horses, requires only a chaldron of coals in a day, each bushel doing the
work of ten men.

The force of gunpowder is employed with advantage where a very
powerful action is required for a short space, as in dividing rocks, or in
generating a great velocity in a projectile. As a source of momentum or
energy only, this power is by no means economical, the daily labour of a
man being equivalent to the effect of about 40 pounds of powder ; but the
advantage of artillery consists in having the force communicated by means
of an elastic fluid extremely rare, which is capable of generating a very
great velocity in the ball only, without any waste of power in producing a
useless momentum in any other substance.

The comparative force of different kinds of gunpowder is determined by
an eprouvette or powder proof; the effect is measured by the angular
motion of a little wheel, a projecting part of which is impelled by the
explosion of a small quantity of the powder, while the friction of a spring
or a weight creates a resistance which may be varied if it be required. The
absolute force of a given quantity of powder may be ascertained either by
suspending a cannon as a pendulum, and measuring its angular recoil ; or
by shooting into a large block, and finding the velocity which is imparted
to it by the ball.^

For measuring very small attractive or repulsive forces, with great accu-

* Messrs. Boulton and Watt caused experiments to be made with the strong
horses used in the breweries in London, and from the result of their trials, they
assigned 33,0001bs. raised one foot per minute, as the value of a horse's power. This
is the unit of engine power now universally adopted. — Lardner on the Steam Engine,
1840, p. 288.

f Hist, et Mem. 1781, p. 65. Theorie des Machines Simples, 4to, 1821.

j See the latter part of Lect. IV.

104

LECTURE XIII.

racy, the most convenient test is furnished by the effects of twisting. An
arm or beam is suspended horizontally by a long wire, and the force
required to cause the beam to make one or more revolutions being ascer-
tained, we may divide the circle described by its extremities into as many
parts as we think proper, and the force required to bring the beam into
any position will always be proportional, without a sensible error, to the
magnitude of the part of the circle intercepted between the given position,
and that in which the arm would naturally rest. When the force is of
such a nature as to be capable of producing a vibration, the body on
which it acts being suspended by the thread of a silkworm or of a spider,
we may compare its magnitude with that of gravitation, by observing the
time required for each vibration, and determining the operation of the force
according to the laws of pendulums. It is in this manner that the forces
concerned in the effects of electricity and of magnetism have been measured
by Mr. Coulomb.

LECT. XII.— ADDITIONAL AUTHORITIES.

Balances. — Lahire, Hist, et Mem. de 1'Acad. ii. 9 ; ix. 42. Roberval's New
Balances, ibid. x. 343. Emerson's Mechanics. Troughton's Balance, Nich.
Jour. iii. 233.

Steelyard.— Hooke's Steelyard, Birch's History, iv. 242. Roemer's Danish
Steelyard. Machines Approuvees, i. 79. Pictet on Paul's Steelyard, Ph. Mag.
iii. 408.

Weights and Measures. — Whitehurst, An Attempt to obtain an invariable Stand-
ard of Length, &c. 4to, 1787. Adams (John Quincey), Report on Weights and
Measures. Washington, U.S. 1821. Hassler on Do. Wash. 1832. Report of the
Franklin Institute on Do. 1834. Pasley on the expediency of simplifying Weights,
&c. Lond. 1834. Clark on Weights and Measures, Westminster Review, No. 31.
Parliamentary Reports, fol. 1758, 1759, 1814, 1819, 1820, 1821.

Animal Mechanics, &fc. — Perrault on Animal Mechanics, Hist, et Mem. de Paris,
i. 181. Parent on Do. ibid, 1702, H. 95. Amontons on Moving Powers, ibid. 1703.
D. Bernoulli on the Muscles and Nerves, Com. Petr. i. 297. Ray, The Wisdom of
God manifested in the Works of Creation. Derham, Physico -Theology. 1712.
Paley's Natural Theology. Cuvier, Regne Animale. Bell, Animal Mechanics, Lib.
Useful Knowledge. Do. Bridgewater Treatise on the Hand.

Inanimate Force.— Smeaton on the Effect of Wind and Water, Ph. Tr. 1759,
p. 100. Reprinted, 8vo, Lond.

LECTURE XIII.

ON PASSIVE STRENGTH AND FRICTION.

THE passive strength of the materials employed in the mechanical ails
depends on the cohesive and repulsive forces of their particles, and on the
rigidity of their structure. The consideration of the intimate nature c>f
these forces belongs to the discussion of the physical properties of matter ;
but the estimation of their magnitude, and of their relative value in various
circumstances, is of undeniable importance to practical mechanics, and

ON PASSIVE STRENGTH AND FRICTION. 105

requires to be examined as a continuation of the subject of statics. The
retarding force of friction is very nearly allied to some kinds of passive
strength, and may be in great measure explained from similar conside-
rations.

The principal effects of any force acting on a solid body may be reduced
to seven denominations ; extension, compression, detrusion, flexure, tor-
sion, alteration, and fracture. When a weight is suspended below a fixed
point, the suspending substance is extended or stretched, and retains its
form by its cohesion assisted by its rigidity : when the weight is supported
by a block or pillar placed below it, the block is compressed, and resists
primarily by a repulsive force, but secondarily also by its rigidity. The
effect here called detrusion is produced when a transverse force is applied
close to a fixed point, in the same manner as the blades of a pair of
scissors act on the pin, and the force which resists this operation is prin-
cipally the rigidity or lateral adhesion of the strata of the substance, but
it could scarcely be effectual without some degree of cohesive and repul-
sive force. When three or more forces are applied to different parts of
any substance they produce flexure, that is, they bend it, some of its
parts being extended and others compressed. In torsion or twisting, the
central particles remain in their natural state, while those which are
in opposite parts of the circumference are detruded or displaced, in op-
posite directions. The operation of forces applied in any of these ways
may produce a permanent alteration or change of figure in substances
sufficiently soft, and perhaps, in a certain degree, in all substances : this
change is sometimes called by workmen settling or taking a set. But the
limit of all these effects is fracture, which is the consequence of the appli-
cation of any force capable of overcoming the strength of the substance,
and to which the generality of writers on mechanics have hitherto confined
their attention.

The forces by which the form of any substance is changed may also be
divided into two kinds, simple pressure and impulse ; but it is only with
regard to fracture that it will be necessary to take the force of impulse
into consideration.

Extension and compression follow so nearly the same laws, that they
may be best understood by comparison with each other. The cohesive and
repulsive forces which resist these effects, depend almost as much on the
solidity or rigidity of the substances, as on the attractions and repulsions
which are their immediate causes : for a substance perfectly liquid,
although its particles are in full possession of their attractive and repulsive
powers, may be extended or compressed by the smallest force that can be
applied to it. It is not indeed certain that the actual distances of the par-
ticles of all bodies are increased when they are extended, or diminished
when they are compressed : for these changes are generally accompanied
by contrary changes in other parts of the same substance, although pro-
bably in a smaller degree. We may easily observe that if we compress a
piece of elastic gum in any direction, it extends itself in other directions ;
and if we extend it in length, its breadth arid thickness are diminished.

If the rigidity of a body were infinite, and all lateral motions of its

106 LECTURE XIII.

particles were prevented, the direct cohesion alone would be the measure
of the force required to produce extension, and the direct repulsion, of the
force required to produce compression ; in this respect indeed, the actual
rigidity of some substances may be considered as infinite, wherever the
extension or compression is moderate, and no permanent alteration of
form is produced ; and within these limits these substances may be called
perfectly elastic. If the cohesion and repulsion were infinite, and the
rigidity limited, the only effect of force would be to produce alteration of
form : and such bodies would be perfectly inelastic, but they would be
harder or softer according to the degree of rigidity.

It is found by experiment, that the measure of the extension and com-
pression of uniform elastic bodies is simply proportional to the force which
occasions it ; at least when the forces are comparatively small. Thus if a
weight of 100 pounds lengthened a rod of steel one hundredth of an inch,
a weight of 200 would lengthen it very nearly two hundredths, and a
weight of 300 pounds three hundredths.* The same weights acting in a
contrary direction would also shorten it one, two, or three hundredths
respectively. The former part of this law was discovered by Dr. Hooke,
and the effects appear to be perfectly analogous to those which are more
easily observable in elastic fluids.

According to this analogy, we may express the elasticity of any sub-
stance by the weight of a certain column of the same substance, which may
be denominated the modulus of its elasticity, and of which the weight is
such, that any addition to it would increase it in the same proportion as
the weight added would shorten, by its pressure, a portion of the sub-
stance of equal diameter. Thus if a rod of any kind, 100 inches long,
were compressed 1 inch by a weight of 1000 pounds, the weight of the
modulus of its elasticity would be 100 thousand pounds, or more accu-
rately 99,000, which is to 100,000 in the same proportion as 99 to 100. In
the same manner, we must suppose that the subtraction of any weight
from that of the modulus will also diminish it, in the same ratio that the
equivalent force would extend any portion of the substance. The height
of the modulus is the same for the same substance, whatever its breadth
and thickness may be : for atmospheric air, it is about 5 miles, and for
steel nearly 1500. This supposition is sufficiently confirmed by experi-
ments to be considered at least as a good approximation : it follows that
the weight of the modulus must always exceed the utmost cohesive
strength of the substance, and that the compression produced by such a
weight must reduce its dimensions to one half : and I have found that a
force capable of compressing a piece of elastic gum to half its length will
usually extend it to many times that length, and then break or tear it ;
and also that a force capable of extending it to twice its length will only
compress it to two thirds. In this substance, and others of a similar nature,
the resistance appears to be much diminished by the facility by which a
contrary change is produced in a different direction ; so that the cohesion
and repulsion thus estimated appears to be very weak, unless when the
rigidity is increased by a great degree of cold. It would be easy to ascer-
* See S'Gravesande's Elem. Physices, lib. i.

ON PASSIVE STRENGTH AND FRICTION. 107

tain the specific gravity of such a substance in different states of tension
and compression, and some light might he thrown by the comparison, on
the nature and operation of the forces which are concerned. It has indeed
been asserted that the specific gravity of elastic gum is even diminished by
tension, so that the actual distances of the particles cannot, in this case, be
supposed to be materially increased.

It is difficult to compare the lateral adhesion, or the force which resists
the detrusion of the parts of a solid, with any form of direct cohesion. This
force constitutes the rigidity or hardness of a solid body, and is wholly
absent from liquids, although their immediate cohesion appears to be nearly
equal to that of solids. Some experiments have been made on the fracture
of bodies by means of detrusion, but it does not appear that the force
necessary to produce a temporary derangement of this kind has ever been
examined : it may be inferred, however, from the properties of twisted
substances, that the force varies in the simple ratio of the distance of the
particles from their natural position, and it must also be simply propor-
tional to the magnitude of the surface to which it is applied.

The most usual, as well as the most important effect produced by the
application of force, is flexure. When a force acts on a straight column in
the direction of its axis, it can only compress or extend it equally through
its whole substance ; but if the direction of the force be only parallel to
the axis, and applied to some point more or less remote from it, the com-
pression or extension will obviously be partial : it may be shown that in
a rectangular column, when the compressing force is applied to a point
more distant from the axis than one sixth of the depth, the remoter surface
will no longer be compressed but extended ; and it may be demonstrated
that the distance of the neutral point from the axis is inversely as that of
the point to which the force is applied. From the effect of this partial
compression, the column must necessarily become curved : and the curva-
ture of the axis at any point will always be proportional to its dis-
tance from the line of direction of the force, not only while the column
remains nearly straight, but also when it is bent in any degree that the
nature of the substance will allow. If the column was originally bent,
any force, however small, applied to the extremities of the axis will
increase the curvature according to the same law, but if the column was
originally straight, it cannot be kept in a state of flexure by any lon-
gitudinal force acting precisely on the axis, unless it be greater than a
certain determinate force which varies according to the dimensions of the
column. It is not however true, as some authors have asserted, that every
column pressed by such a force must necessarily be bent ; its state when it
is straight and submitted to the operation of such a force will resemble a
tottering equilibrium, in which a body may remain at rest until some
external cause disturbs it. The figure of a cplumn naturally straight,
but bent a little by a longitudinal force, will coincide with that of the
harmonic curve, in which the curvature is as the distance from the basis.
(Plate IX. Fig. 117... 121.)

Considerable irregularities may be observed in all the experiments
which have been made on the flexure of columns and rafters exposed to

108 LECTURE XIII.

longitudinal forces ; and there is no doubt but that some of them were
occasioned by the difficulty of applying the force precisely at the extremi-
ties of the axis, and others by the accidental inequalities of the substances,
of which the fibres must often have been in such directions as to constitute
originally rather bent than straight columns.

When a rod, not very flexible, is fixed at one end in a horizontal posi-
tion, the curvature produced by its own weight is every where as the
square of the distance from the other end : and if a rod be simply sup-
ported at each end, its curvature at any point will be proportional to the
product of the two parts into which that point divides it. But when the
weights are supposed to be applied to any given points of the rod only, the
curvature always decreases uniformly between these points and the points
of support. (Plate IX. Fig. 122, 123.)

The stiffness of any substance is measured by the force required to cause
it to recede through a given small space in the direction of the force. It is
only necessary to consider this property with regard to forces applied
transversely. In such cases the stiffness is directly as the breadth and the
cube of the depth of the beam, and inversely as the cube of its length.*
Thus if we have a beam which is twice as long as another, we must make
it, in order to obtain an equal stiffness, either twice as deep or eight times
as broad. The property of stiffness is fully as useful in many works of
art as the ultimate strength with which a body resists fracture : thus for a
shelf, a lintel, or a chimney piece, a great degree of flexure would be
almost as inconvenient as a rupture of the substance.

When a beam is supported at both ends, its stiffness is twice as great as
that of a beam of half the length firmly fixed at one end ; and if both ends
are firmly fixed the stiffness is again quadrupled. For if the whole beam
were inverted and supported by a fulcrum in the middle, each half would
resemble a separate beam fixed at one end, and the fulcrum would bear
the sum of two equal weights placed at the extremities, disregarding that
of the beam ; and consequently the same flexure will be produced by
placing a double weight on the middle of the beam in an inverted position.
If both ends were firmly fixed, the curvature would be every where as the
distance from the middle of each half, the whole being in the same state
as four separate beams fixed at their extremities : each of these beams
would be eight times as stiff as beams of twice the length, and the whole
beam, in this state, would be eight times as stiff as if the ends were simply
supported. It is, however, difficult to fix the ends of a beam so firmly as
to increase its resistance in this proportion, unless it be continued both
ways considerably beyond the supports.

It is evident that a tube or hollow beam of any kind, must be much
stiffer than the same quantity of matter in a solid form : the stiffness is
indeed increased nearly in proportion to the square of the diameter, since
the cohesion and repulsion are equally exerted with a smaller curvature,
and act also on a longer lever.

Torsion, or twisting, consists in the lateral displacement or detrusion of
the opposite parts of a solid, in opposite directions, the central particles
* Robison's Mechanical Philosophy, art. Strength of Materials, § 386.

ON PASSIVE STRENGTH AND FRICTION. 109

only remaining in their natural state. We might consider a wire as com-
posed of a great number of minute threads, extending through its length,
and closely connected together ; if we twisted such a wire, the external
threads would be extended, and, in order to preserve the equilibrium, the
internal ones would be contracted ; and it may be shown that the whole
wire would be shortened one fourth as much as the external fibres would
be extended if the length remained undiminished ; and that the force
would vary as the cube of the angle through which the wire is twisted.
But the force of torsion, as it is determined by experiment, varies simply
as the angle of torsion ; it cannot, therefore, be explained by the action of
longitudinal fibres only ; but it appears rather to depend principally, if
not entirely, on the rigidity or lateral adhesion which resists the detrusion
of the particles. If a wire be twice as thick as another of the same length,
it will require sixteen times as much force to twist it once round ; the
stiffness varying as the fourth power of the diameter, that is, as the square
of its square. But if the length vary, it is obvious that the resistance to
the force of torsion will be inversely as the length.

A permanent alteration of form is most perceptible in such substances as
are most destitute of rigidity, and approach most to the nature of fluids.
It limits the strength of materials, with regard to practical purposes,
almost as much as fracture, since in general the force which is capable of
producing this effect, is sufficient, with a small addition, to increase it till
fracture takes place. A smaller force than that which has first produced
an alteration of form, is seldom capable either of increasing, or of removing
it, a circumstance which gives such materials as are susceptible of an alter-
ation of this kind, a great advantage for many purposes of convenience and
of art. The more capable a body is of a permanent alteration of form, the
more ductile it is said to be ; pure gold and silver, lead, annealed iron and
copper, wax when warm, glass when red hot, and clay when moist, possess
considerable ductility. Wood admits of little permanent change of form,
except in a green state, although it sometimes settles a little, when it has
been exposed to pressure. Even stone will become permanently bent in
the course of years, as we may observe in old marble chimney pieces.
But the most ductile of all solid substances appears to be a spider's web.
Mr. Bennet twisted a thread of this kind many thousand times, and
shortened it more than a fourth of its length, yet it showed no disposition
to untwist.*

A ductile substance acquires the same cohesive and repulsive powers
with regard to its new form, as it possessed in its original state ; and when
the alteration of form has once commenced, those powers are neither in-
creased nor diminished by continuing the operation ; the degree of flexure
or torsion required for producing a further alteration, appears also to be
little varied : thus if the spider's web could at first be twisted only one half
round, so as to retain the power of returning to its original state, without
a,ny permanent alteration of form, it would never acquire the power of
returning more than half a revolution, however it might be twisted. From

* Experiments on a New Suspension of the Magnetic Needle, Ph. Tr. 1792,
Ixxxii. 82.

110 LECTURE XIII.

a want of attention to this consideration, a late respectable author has
called in question, without sufficient reason, the accuracy of Mr. Bennet's
experiments.

A variation of ductility in any substance, does not appear to depend on
any change in the magnitude of the ultimate powers of cohesion and repul-
sion. Steel, whether perfectly hard, or of the softest temper, resists flexure
with equal force, when the deviations from the natural state are small : but
at a certain point the steel, if soft, begins to undergo an alteration of form :
at another point it breaks if much hardened ; but when the hardness is
moderate, it is capable of a much greater curvature without either perma-
nent alteration or fracture ; and this quality, which is valuable for the
purposes of springs, is called toughness, and is opposed to rigidity and
brittleness on the one side, and to ductility on the other. There may,
however, be an apparent difference in the stiffness of some substances in
different states, arising from the greater facility with which their dimen-
sions are extended in one direction while they are contracted in another :
thus elastic gum appears to possess a much greater degree of stiffness when
its hardness is increased by cold than when it is at a more elevated tempe-
rature ; but the change produced in this case by heat is not an increase of
that ductility which facilitates a permanent alteration of form, but rather
of the toughness which allows a temporary change of figure, continuing
only while the force is applied. The effect of forging and of wiredrawing
tends to lessen the ductility of metals, and to render them tough, and even
rigid : so that in hammering copper and brass, and in drawing wire, it is
necessary to anneal the metals more than once by fire, in order to restore
their ductility, which is lessened by the operation. The corrosion of the
surface of a metal by an acid is also said to render it brittle ; but it is not
impossible that this apparent brittleness may be occasioned by some irre-
gularity in the action of the acid.

The last effect of force on solid materials is their fracture, which, as well
as the former changes, may be produced either by impulse or by pressure
alone. The action which resists pressure is called strength, and that which
resists impulse may properly be termed resilience. The strength of every
body is in the joint ratio of its immediate cohesion and repulsion, or elas-
ticity, and of its toughness, or the degree in which it may be extended,
compressed, or otherwise deranged, without a separation of its parts. The
resilience is jointly proportional to its strength and its toughness, and is
measured by the product of the mass and the square of the velocity of a
body capable of breaking it, or of the mass and the height from which it
must fall in order to acquire that velocity ; while the strength is merely
measured by the greatest pressure that it can support in a state of rest.

The simplest way in which a body can be broken is by tearing it asunder.
The cohesive force continues to be increased as long as the tenacity of the
substance allows the particles to be separated from each other without a
permanent alteration of form ; when this has been produced, the same
force, if its action is continued, is generally capable of causing a total solu-
tion of continuity ; and sometimes a separation takes place without any
previous alteration of this kind that can be observed.

ON PASSIVE STRENGTH AND FRICTION. Ill

It follows from the nature of resilience, that a hody of a pound weight,
falling from the height of a yard, will produce the same effect in breaking
any substance, as a body of three pounds falling from the height of a foot ;
so that here, as well as in the estimation of mechanical power, it is the
energy and not the momentum, that is to be considered as the measure of
the effect. If we know the strength of any substance, and the degree in
which it is capable of extension, we may easily determine its resilience
from a consideration of the laws of pendulums. For the same weight
which would break it by pressure, will acquire a sufficient impulse for
breaking it, if it fall from a height equal to half the space through which
the substance may be extended, supposing the direction of the stroke to be
horizontal, so that its effect may not be increased by the force of gravity.
Thus if the pressure of a weight of 100 pounds broke a given substance
after extending it through the space of an inch, the same weight would
break it by striking it with the velocity that would be acquired by the fall
of a heavy body from the height of half an inch, and a weight of one
pound would break it by falling from a height of 50 inches.

It is obvious that the cohesive strength, as well as the resilience, of any
substance must be simply proportional to the magnitude of its transverse
section, that is, of the surface of fracture. Some experiments appear to show
that it increases in a greater proportion than this surface, others that it
increases in a smaller proportion ; but it is probable that in both cases
some accidental irregularities must have interfered, and that a wire two
inches in diameter is exactly four times as strong as a wire one inch in
diameter. The length has no effect either in increasing or in diminishing
the cohesive strength ; but the resilience is proportional to the length, since
a similar extension of a longer fibre produces a greater elongation.

There is however a limit beyond which the velocity of a body striking
another cannot be increased without overcoming its resilience and breaking
it, however small the bulk of the first body may be, and this limit depends
on the inertia of the parts of the second body, which must not be dis-
regarded when they are impelled with a considerable velocity. For it is
demonstrable that there is a certain velocity, dependent on the nature of a
substance, with which the effect of any impulse or pressure is transmitted
through it ; a certain portion of time, which is shorter accordingly as the
body is more elastic, being required for the propagation of the force through
any part of it ; and if the actual velocity of any impulse be in a greater
proportion to this velocity than the extension or compression, of which the
substance is capable, is to its whole length, it is obvious that a separation
must be produced, since no parts can be extended or compressed which are
not yet affected by the impulse, and the length of the portion affected at
any instant is not sufficient to allow the required extension or compression.
Thus if the velocity with which an impression is transmitted by a certain
kind of wood be 15,000 feet in a second, and it f>e susceptible of compres-
sion to the extent of ^-y of its length, the greatest velocity that it can resist
will be 75, feet in a second, which is equal to that of a body falling from a
height of about 90 feet. And by a similar comparison we may determine
the velocity which will be sufficient to penetrate or to break off a substance

112 LECTURE XIII.

in any other manner ; if we calculate the velocity required to convey the
impulse from one part of the substance to the other, and ascertain the
degree in which it can have its dimensions altered without fracture.

It is easy to understand, from this statement, the different qualities of
natural bodies with respect to hardness, softness, toughness, and brittleness.
A column of chalk, capable of supporting only a pound, will perhaps be
compressed by it only a thousandth part of its length ; a column of elastic
gum, capable of suspending a pound, may be extended to more than twice
its length, the elastic gum will therefore resist the energy of an impulse
incomparably greater than the chalk. A diamond, so hard as to resist an
enormous pressure, may be broken by a moderate blow, with a small
hammer. A weight of 1000 pounds, moving with a velocity of one foot in
a second, and acting on a small surface of a board, may possess sufficient
energy to break or to penetrate it ; with a velocity of 100 feet in second, a
weight of T^ of a pound will possess the same energy, and produce the same
effect, if it act on a similar surface ; but if the wood be so constituted as
to be wholly incapable of resisting a velocity of 100 feet in a second, it may
be penetrated by a weight of -^-^ of a pound as well as by one tenth, and
by a moderately soft body as well as by a harder one. The whole board,
however, if at liberty, would receive a much greater momentum from the
impulse of the large weight, than from that of the small one, its action
being continued for a much longer time. And it is for this reason that a
ball shot by a pistol will perforate a sheet of paper standing upright on a
table, without overturning it.

The strength, or rather hardness, of a substance exposed to the action of
a force that tends to compress it, must not be confounded with its resistance
to a force applied longitudinally and tending to produce flexure. A slender
rod of wood, when it yields to a longitudinal pressure, commonly bends
before it breaks, and gives way at last to the force by a transverse fracture ;
but a column of stone or brick, and even a thick pillar of wood, is crushed
without bending, and generally by a smaller force than that which would
produce or continue a flexure. In this case the parts slide away laterally,
and in a rectangular pillar ; if the texture of the substance is uniform, and
not fibrous, the surfaces of fracture will make nearly a right angle with
each other, supposing the resistance arising from the lateral adhesion in the
direction of any surface or section, to be simply proportional to that sec-
tion ; but if this force, like that of friction, is increased by a pressure
which tends to bring the parts into closer contact, the angle left after frac-
ture must be more acute. (Plate X. Fig. 124, 125.)

The power of the force of lateral adhesion in resisting fracture, is con-
sidered by Mr. Coulomb as nearly equal to that of the direct cohesion of
the same substance, or a little greater ; while Professor Robison* makes it
twice as great. If, however, this force be supposed to be simply equal to
the direct cohesion, it may be inferred that the strength of a square bar in
resisting compression is twice as great as its cohesive strength, allowing
that the fracture takes place in the surface of least resistance. It is, how-
ever, seldom that the strength with which a body resists compression, is in
* Strength of Materials, arts. 372, 373.

ON PASSIVE STRENGTH AND FRICTION. 113

so great a proportion as this to its cohesive strength ; and where the sub-
stance is in any degree composed of fibres, they must naturally produce
great irregularities by their flexure. The strength in resisting compression,
must, according to this statement, be simply proportional to the magnitude
of the section of the substance, although some experiments on freestone
appear to indicate, that when the section is increased, the strength is in-
creased in a greater proportion ; and there is no reason to suppose that it
can be influenced either way by the length. A cylindrical or prismatic
form is therefore the best that can be given to materials of a given bulk, in
order to enable them to resist a force which tends to crush them, except
that the additional pressure of their own weight on the lower parts, re-
quires that those parts should be a little stronger than the upper parts. It
appears, also, that something is gained by making the outline a little con-
vex externally ; for it may be demonstrated, that for a column or upright
beam to be cut out of a slab of equable thickness, supposing the strength
to be independent of pressure, the strongest form is a circle. (Plate X.
Fig. 126, 127.)

When a body is broken by a transverse force applied very near to a
fixed point, its lateral adhesion is overpowered by the effect which we have
called detrusion, and its strength in this case is, therefore, generally some-
what greater than its direct cohesive strength. But when the part to which
the force is immediately applied is at a distance from the fixed point
greater than about one sixth of the depth, the fracture is no longer the
immediate consequence of detrusion, but of flexure.

Flexure is the most usual manner in which fracture is produced ; the
superficial parts on the convex side are most extended, and usually give
wray first, except in soft fibrous substances, such as moist or green wood,
which is more easily crushed than torn ; and in this case the concave side
fails first, and becomes crippled, and the piece still remains suspended by
the cohesion of the fibres. After the convex surface has been cracked, the
whole substance is usually separated, but not always ; for example, a
triangular beam, with one of the edges uppermost, may be charged with
such a weight that the upper edge may be divided and the lower part may
remain intire.

When a column or rafter is broken by the operation of a longitudinal
pressure, the stiffness of the column being once overcome, a small addition
of force is usually sufficient to produce fracture, unless the pressure has
been applied to a part more or less distant from the axis ; for in this case
a moderate force may produce a moderate flexure, and a much greater
force may be required to break the column. But in general, the stiffness
of columns is of more consequence than their strength in resisting trans-
verse fracture.

The strength of beams of the same kind, and fixed in the same manner,
in resisting a transverse force, is simply as their fcreadth, as the square of
their depth, and inversely as their length.* Thus, if a beam be twice as
biDad as another, it will also be twice as strong, but if it be twice as deep,
it will be four times as strong ; for the increase of depth not only doubles
* Robison's Mech. Phil. i. § 374, &c.

114 LECTURE XIII.

the number of the resisting particles, but also gives each of them a double
power, by increasing the length of the levers on which they act. The
increase of the length of a beam must also obviously weaken it, by giving
a mechanical advantage to the power which tends to break it ; and some
experiments appear to show that the strength is diminished in a proportion
somewhat greater than that in which the length is increased.

The strength of a beam supported at both ends, like its stiffness, is twice
as great as that of a single beam of half the length, which is fixed at one
end ; and the strength of the whole beam is again doubled if both the ends
are firmly fixed.

The resilience of a prismatic beam, resisting a transverse impulse, follows
a law very different from that which determines its strength, for it is
simply proportional to the bulk or weight of the beam, whether it be
shorter or longer, narrower or wider, shallower or deeper, solid or hollow.
Thus a beam ten feet long will support but half as great a pressure, with-
out breaking, as a beam of the same breadth and depth, wThich is only five
feet in length ; but it will bear the impulse of a double weight striking
against it with a given velocity, and will require that a given body should
fall from a double height in order to break it.

It is therefore of great consequence in the determination of the form and
quantity of the materials to be employed for any mechanical purpose, that
we should consider the nature as well as the magnitude of the forces which
are to be resisted. Stiffness, strength, or resilience, may be separately or
jointly required in various degrees. For a ceiling, stiffness would be prin-
cipally desirable ; for a door, strength ; for the floor of a ball room, resi-
lience ; for a coach spring, resilience and flexibility, that is, resilience with-
out stiffness. An observatory should be as stiff as possible, a ship as strong
as possible, a cable as resilient as possible.

It is a common remark, that a floor which shakes is the strongest ; and,
improbable as it appears at first sight, it may perhaps be founded in truth ;
for if the absolute strength of a stiff and a shaking floor were equal, the
shaking floor would bear the effects of motion with the least injury. It is
possible that a stiff floor, which would support a numerous assembly,
might give way at a ball ; while a more resilient one, which would be
suited for dancing, might be destroyed by a crowded concert.

A coach spring, divided into plates, has the same power of resisting,
without being broken, the momentum of the carriage, arising from sudden
elevations and depressions, as it would possess if it formed one entire
mass, while its greater flexibility allows it to regulate these motions in a
much more gradual and gentle manner. A single piece of timber may
perhaps, sometimes, have too much of the flexibility of a coach spring, its
strata sliding, in some degree, on each other ; in such a case its stiffness
and strength may be increased by binding it very firmly with hoops.

The transverse strength of a perfectly elastic substance, fixed at one end,
is to its direct cohesive strength as the depth of the substance to six times
its length. This proportion is equally applicable to such substances as re-
sist compression more strongly than extension ; for their immediate repul-
sive force is probably not greater than their cohesive force, when their

ON PASSIVE STRENGTH AND FRICTION. 115

dimensions are equally changed, so that the middle of the beam is always
in its natural state ; and when the curvature is sufficient to overcome the
cohesive force, the whole beam must give way. When, however, the sub-
stance is less capable of resisting compression than extension, the concave
surface gives way first, and the strength depends immediately on the repul-
sive strength of the substance. This is perhaps the reason that, in experi-
ments on beams of oak, the transverse strength has seldom been found in a
greater ratio to the whole cohesive strength than that of the depth to nine
times the length.

It may be inferred from the consideration of the nature of the different
kinds of resistance which have been explained, that if we have a cylindrical
tree a foot in diameter, which is to be formed into a prismatic beam by
flattening its sides, we shall gain the greatest stiffness by making the
breadth or thickness 6 inches, and the depth 10£, the greatest strength by
making the breadth 7 inches and the depth 9|, and the greatest resilience
by making the beam square. The stiffness and the strength of the beam
may be much increased by cutting the tree into four pieces, turning their
edges outwards, and uniting them so as to make a hollow beam : but it
will require great strength of union to make the whole act as one piece,
and the resilience of the beam will be rather diminished than increased by
the operation.

The adoption of the hollow masts and beams which an ingenious me-
chanic has lately introduced, requires, therefore, some caution. For where
an impulse is to be resisted, such a mast is no stronger than a solid mast of
the same weight, and much weaker than a solid mast of the same diameter.
The force of the wind is, however, rather to be considered as constituting
a pressure than a finite impulse, except when a sudden squall carries a
loose sail before it with considerable velocity. A similar caution may also
be extended to some other attempts to make improvements in naval archi-
tecture : it is a common opinion, and perhaps a well-founded one, that
flexibility is of great advantage to a ship's sailing ; if therefore we sacrifice
too much resilience to strength, and too much of both to stiffness, we may
perhaps create greater evils than those which we wish to avoid.

We have hitherto supposed the beams of which the strength has been
compared, to be prismatic, that is, of equal breadth and thickness through-
out, which is not only the simplest form in theory but the most generally
useful in practice. If, however, we have the power of giving any form
that we please to materials of a certain weight, which may often be done
where several smaller pieces are to be cut out of a larger one, or a larger
one to be composed of several smaller ones, or where the materials are
either ductile or fusible, it is frequently possible to determine a more ad-
vantageous form than that of an equable beam or column. For since the
extension which the parts of the substance admit without giving way, is the
limit of their strength, if the depth of a beam be everywhere equal, and the
curvature unequal, the fracture will first take place where the curvature
is greatest, and the superfluous strength of the other parts will be lost ; so
that, in order to have the greatest strength that a given quantity of mate-
rials is capable of affording in a beam of given length, the form must be

i2

116 LECTURE XIII.

such that the strength may be everywhere equal, the tension of the surface
being equal throughout ; and the depth must be as much smaller as the cur-
vature is greater. It is also necessary to consider whether the substance is
likely to be crushed, and whether it is liable to be broken by detrusion
rather than by flexure. Sometimes the depth of the beam may be limited,
and sometimes its breadth ; or it may be required that the breadth and
depth may be always equal or proportional to each other, and the force
may be either applied at one end of the beam or it may be equally divided
throughout its length ; it may also principally depend on the weight of the
substance itself ; and the strongest form will be different according to the
different conditions of its application. In the most common cases, the
outline must be either triangular or parabolic, as if the point of the triangle
were rounded off; but the curves required are sometimes of much more
difficult investigation. (Plate X. Fig. 128... 147.)

The strength of bodies is sometimes employed in resisting torsion, as in
the case of the axles of wheels and pinions, rudders of ships, and screws of
all kinds : but there is seldom occasion to determine their absolute strength
in resisting a force thus applied ; if they are sufficiently stiff, their parts
are not often separated by any violent efforts.

In order to investigate the strength of the various substances employed
for the purposes of the mechanical arts, it is most convenient to use a
machine furnished with proper supports, and gripes, or vices, for holding
the materials, and with steelyards for ascertaining the magnitude of the
force applied, while the extension or compression is produced by a screw
or a winch, with the intervention of a wire, a chain, or a cord : provision
ought also to be made for varying the direction of the force, when the
flexure of the materials renders such a change necessary. (Plate XI. Fig.
148.)

According to the experiments of various authors, the cohesive strength of
a square inch of razor steel is about 150 thousand pounds, of soft steel 120,
of wrought iron 80, of cast iron 50, of good rope 20, of oak, beech, and
willow wood, in the direction of their fibres, 12, of fir 8, and of lead about
3 thousand pounds : the cohesive strength of a square inch of brick 300,
and of freestone 200. Teak wood, the tectona grandis, is said to be still
stronger than oak.

The weight of the modulus of the elasticity of a square inch of steel, or
that weight which would be capable of compressing it to half its dimen-
sions, is about 3 million pounds ; hence it follows, that when a square inch
of steel is torn asunder by a weight of 150,000 pounds, its length is first in-
creased to one twentieth more than its natural dimensions.

The strength of different materials, in resisting compression, is liable to
great variation. In steel, and in willow wood, the cohesive and repulsive
strength appear to be nearly equal. Oak will suspend much more than
fir ; but fir will support twice as much as oak ; probably on account of
the curvature of the fibres of oak. Freestone has been found to support
about 2000 pounds for each square inch, oak in some practical cases more
than 4000.

The strongest wood of each tree is neither at the centre nor at the cir-

ON PASSIVE STRENGTH AND FRICTION. 117

cumference, but in the middle between both ; and in Europe it is generally
thicker and firmer on the south-east side of the tree. Although iron is
much stronger than wood, yet it is more liable to accidental imperfections ;
and when it fails, it gives no warning of its approaching fracture. The
equable quality of steel may be ascertained by corrosion in an acid ; but
there is no easy mode of detecting internal flaws in a bar of iron, and we
can only rely on the honesty of the workman for its soundness. Wood,
when it is crippled, complains, or emits a sound, and after this, although
it is much weakened, it may still retain strength enough to be of service.
Stone sometimes throws off small splinters when it is beginning to give
way ; it is said to be capable of supporting by much the greatest weight
when it is placed in that position, with respect to the horizon, in which it
has been found in the quarry.

It is obvious that when the bulk of the substance employed becomes very
considerable, its weight may bear so great a proportion to its strength as to
add materially to the load to be supported. In most cases the weight in-
creases more rapidly than the strength, and causes a practical limitation of
the magnitude of our machines and edifices. We see also a similar limit
in nature : a tree never grows to the height of 100 yards ; an animal is
never strong enough to overset a mountain. It has been observed that
whales are often larger than any land animals, because their weight is
more supported by the pressure of the medium in which they swim.

The force of friction which resists the sliding of different bodies on each
other, seems to be intimately connected with that lateral adhesion or ri-
gidity which is opposed to the internal displacement of the parts of a
single body, by the effect which we have denominated detrusion ; and
when the friction is considered as resisting pressure rather than motion, it
approaches still more nearly to the same force. It is probably derived in
great measure from the strength of the protuberant particles, which must
be broken, bent, or compressed by the motion of the bodies on each other :
but it is not always that the existence of such particles can be asserted,
much less can they be made perceptible to the senses, and we can only ex-
amine the effects which they may be supposed to produce, by immediate
experiments on the forces required to counteract them. Such experiments
have been made on a very extensive scale by Musschenbroek* and
Coulomb,f and many of their results have been confirmed by Mr. Vince,J
in a simple and elegant manner.

With a few exceptions, the friction of all solid bodies is either perfectly,
or very nearly, a uniformly retarding force, neither increasing nor di-
minishing when the relative velocity of the bodies concerned is changed.
The friction of some rough substances is a little increased with the velocity,
but, as they become more polished, this variation disappears. When, how-
ever, the motion is wholly extinct, and the bodies remain in contact with
each other, their adhesion is usually greater thkn the friction, and by a
continuation of the contact, it may become twice or even thrice as great,

* Introductio ad Philosophiam Naturalem, 2 vols. 4to, Leyd. 1762, i. 145.

f Mem. des Savans Etrangers, x. 161.

£ On the Motion of Bodies affected by Friction, Ph. Tr. 1785, kxv. 165.

118 LECTURE XIII.

especially where the surfaces are large and the substances but moderately
hard.

The truth of the assertion, that friction is a uniformly retarding force,
may be shown very conveniently by means of Atwood's machine for ex-
periments on accelerated motion. By suffering the axis of the pulley to
rest on the surface of any fixed substance, we may subject it to a friction
of which the magnitude may be varied by different methods ; and we shall
find that the motions of the boxes still indicate the action of a uniformly
accelerating force, the spaces described being always proportional to the
squares of the times of descent ; it follows therefore, that since the ope-
ration of gravity is uniform, that of friction which is deducted from it at
each instant, must also be uniform, in order that the remaining acceleration
may follow the same law.

The uniformity of the force of friction may also be shown by the descent
of a flat substance on an inclined plane : if the body be caused to begin its
descent with a certain velocity, it will be retarded when the resistance is
greater than the relative force of gravity : in this case the retardation will
continue until it is wholly stopped, the resistance not diminishing with the
velocity. If, on the contrary, the relative weight overpowers the resistance
at first, the motion will be continually accelerated, the resistance not being
increased by the increase of the velocity. But since every experiment of
this kind must be performed in the presence of the air, the resistance of
this fluid, which follows another law, will in the end prevent the ac-
celeration.

It may in general be asserted, with some exceptions, that the force of
friction is simply proportional to the weight or pressure that brings the
substances concerned into contact, independently of the magnitude of their
surfaces : but Mr. Coulomb has observed that in many cases there is,
besides this force, another resistance, amounting to several pounds for each
square foot of the surface, which is independent of the pressure ; and by
calculating these forces separately, we may probably always ascertain the
whole resistance with sufficient accuracy. This constant portion is usually
much smaller than that which varies with the weight, and in all common
cases it may be safely neglected, and the friction of stone on stone may be
called equal to one half of the pressure, that of wood on wood one third,
and that of metal on metal one fourth ; and this may serve as an estimate
sufficiently accurate for calculating the effects of machines ; although, if
their parts were perfectly adjusted to each other, and all the surfaces well
polished, the friction would not in general exceed one eighth of the
pressure, whatever might be the nature of the materials. The application
of unctuous substances lessens the friction in the first instance ; but unless
they are frequently renewed, they sometimes tend rather to increase it.

The simplest mode of ascertaining the magnitude of the friction of two
bodies, is to incline their common surface to the horizon until the one
begins to slide on the other : this point determines the magnitude of their
adhesion ; but in order to find that of their friction when they are in mo-
tion, they must be first separated, and then allowed to move on each other,
while the whole apparatus is gently agitated. The friction will then be to

ON PASSIVE STRENGTH AND FRICTION. 119

the pressure, as the height of the inclined plane to its horizontal length,
when the inclination is barely such as to allow the continuance of any
motion which is imparted to the substance placed on the plane.

It follows from the doctrine of the resolution of force, that when any
body is to be drawn along a horizontal surface, which produces a resistance
proportionate to the pressure, a part of the force may be advantageously
employed in diminishing the pressure produced by the weight of the body ;
hence, in order for the most advantageous application of the force, its di-
rection must be inclined to the horizon, and it may be demonstrated that
the inclination must be the same with that of a plane on which the relative
weight of the body is precisely equal to the friction. Thus, if we can de-
termine the inclination of a road which is barely sufficient for a carriage to
descend on it by its own weight, the same inclination will be the best pos-
sible for the application of any force by which the carriage is to be drawn
along a horizontal road of the same materials.

It is obvious that an inclined plane on which a weight rests by means of
an adhesion proportionate to the pressure, can never be forced backwards
by any increase of that pressure, since the resistance increases in the same
proportion, and continues always sufficient to prevent the relative motion
of the weight and the inclined plane. Two such planes, put together,
would constitute a wedge, which would be equally incapable of giving
way to a pressure applied to its opposite surfaces, each of them possessing
similar properties with respect to friction. Thus, if the friction or adhe-
sion were exactly one eighth of the pressure, the height of the inclined
plane would be one eighth of its length, and the back of the wedge one
fourth. Such a wedge would therefore possess a perfect stability with
respect to any forces acting on its inclined surfaces. But the effects of
agitation, and the minute tremors produced by percussion, have a great
tendency to diminish the force of adhesion, by interrupting the intimacy of
contact : and where a pin, a nail, or a screw is required to retain its
situation with firmness, the inclination of the surfaces must be smaller
than the angle of such a wedge as is barely capable of affording a sufficient
resistance in theory.

It appears, therefore, that the force of lateral adhesion, acting between
two bodies in contact, is of great importance in all mechanical arts ; the
firmness of architecture and of carpentry depends in great measure on it.
This kind of resistance being equally powerful, when the force is applied
in the direction of the surface, to whatever part of the surface it may tend,
it follows that any body which is subjected to friction on all sides, will
retain its situation with the same force that was used in overcoming the
friction in order to bring it into that situation, or rather with a greater
force, since the lateral adhesion is generally a little greater than the fric-
tion : so that a cylindrical wire cannot be withdrawn from a perforation in
a board, by any direct force less than that which'was employed in intro-
ducing it ; and this kind of stability, together with that of a wedge or nail
resisting a lateral pressure, constitutes the security of the lighter structures
of carpentry, while those of architecture receive a great part of their

120 LECTURE XIII.

firmness from the accumulation of weight, which makes the resistance of
their lower parts to any lateral motion almost insuperable.

When a hard body penetrates another, or when a substance is ground
away by the attrition of another, the force which opposes the motion, is
to be considered, like the force of friction, as a uniformly retarding force.
There is no reason for imagining the stiffness of a bar, whether longer or
shorter, to depend on the velocity of the body that bends it, and the space
through which it may be bent, without breaking, is also limited only by
the toughness of the materials. In the same manner, when the internal
parts of a solid are broken and displaced by the penetration of another, or
its external parts abraded by its attrition, the resistance is the same, what-
ever the velocity may be, and the space described by the body before its
velocity is destroyed, is always proportional to the square of that velocity,
or to the energy which results from a combination of the proportions of
the velocity and the momentum.

LECT. XIII.— ADDITIONAL AUTHORITIES.

Passive strength. — Buffinger on the Strength of Beams, Comm. Petr. iv. 164.
Muschenbroek, Systeme de Physique, par Lafond, Par. 1760. Buffon on the
Strength of Timber, Hist, et Mem. de Paris, 1738, p. 169, H. 54; 1740, p. 453 ;
1741, p. 292. Duhamel on do. ibid. 1742, p. 335 ; 1768, p. 534, H. 29. Jurin
on the Elastic Force of Springs, Ph. Tr. 1744, p. 46. Emerson's Fluxions, 343,
Mechanics, 4to, 1758. Euler, Novi Com. Petr. 1757. Acta Petr. 1758. Belidor,
Architecture Hydraulique, I. ii. 92. Jo. Bernoulli on the Extension of Threads,
&c. Hist, et Mem. de Berlin, 1766, pp. 78, 108. Coulomb on the Force of Tor-
sion, Hist, et Mem. de Paris, 1784, p. 229. Gauthey on the Strength of Stones,
Rozier's Journal, iv. 402. Dupin sur la Flexibilite, la Force, et 1'Elasticite des
Bois, Journal de 1'Ecole Poly technique, x. 137. Rennie, Ph. Tr. 1818. Barlow
on the Strength of Timber, 1824 ; Iron, 1835. Do. do. Second Report, 1835.
Tredgold on the Strength of Iron, Lond. 1824. Hodgkinson's Memoirs of the Lit.
and Phil. Soc. of Manchester, vols. iv. and v.

Friction. — Amontons on the Resistance of Mach. Hist, et Mem. 1699, p. 206,
H. 104 ; 1700, p. 47 ; 1703, H. 105 ; 1704, pp. 173, 206. Parent, do. ibid. 1700,
H. 149 ; 1704. Sauveur on the Friction of Ropes coiled round a Cylinder, ibid.
1703, p. 305. Varignon, do. ibid. 1717, p. 195, H. 68. Euler on Friction, Hist,
et Mem. de Berlin. 1748, pp. 122, 133. Novi Com. Petrop. vi. 233; xx. 304, 327.
Bernoulli, ibid. xiv. i. 249. Hedin, Dissertatio Physico-Mechanica de Frictione, 4to,
Upsal, 1770. Ximenes, Teoria e Pratica delle Resistenze de' Solidi ne' loro Attriti,
2 vols. 4to, Pisa, 1782. Library of Useful Knowledge, Mechanics, Third Treatise.
Morin, Nouvelles Experiences sur le Frottement, 3 vols. 4to, Paris, 1843. The
newest and best authority.

121

LECTURE XIV.

ON ARCHITECTURE AND CARPENTRY.

THE subjects which we have lately examined, are to be considered as
preliminary to the particular departments of practical mechanics. The
first division of these is to consist of such as are employed in resisting
forces of various kinds, but they may almost all be referred, without in-
convenience, to the general heads of architecture and carpentry, of which
the principal business is to resist the force of gravitation. Architecture,
in its most extensive sense, may be understood as comprehending carpen-
try, but the term is more usually applied to the employment of those ma-
terials, which are only required to resist the effects of a force tending
principally to produce compression, while the materials used by carpenters
are frequently subjected to the operation of a force which tends to extend
or to bend them : the works of architects being commonly executed in stone
or brick, and those of carpenters in wood, besides the occasional use of
iron and other metals, in both cases.

The simplest problem in mechanical architecture appears to be, to de-
termine the most eligible form for a column. The length and weight being
supposed to be given, it is of importance to investigate the form which
affords the greatest possible strength ; but it is somewhat difficult to ascer-
tain the precise nature and direction of all the forces which are to be
resisted. If we consider the column as a beam fixed in the ground, and
impelled by a transverse force, it ought to be much tapered, and reduced
almost to a point at its extremity ; but it is seldom that any force of this
kind can be powerful enough to do more than overcome the weight alone
of the column, and it is only necessary to regard the load which presses
vertically on it ; and whether we consider the force as tending to bend or
to crush it, the forms commonly employed will appear to be sufficiently
eligible. Lagrange seems to have been misled by some intricacies of ma-
thematical investigation,* too remote from physical accuracy, when he
calculated that a cylinder was the strongest form for resisting flexure ;
that form approaches in reality much more nearly to an oblong spheroid,
of which the outline is elliptical. The consideration of the flexure of a
column is, however, of little practical importance in architecture, for
upon a rough estimate of the properties of the materials usually employed,
it may be computed that a column of stone must be about forty times as
high as it is thick, in order to be capable of being bent by any weight
which will not crush it ; although a bar of wood or of iron may be bent
by a longitudinal force, if its length exceed about twelve times its thick-
ness. The force may therefore be considered as 'tending only to crush the
column ; and since the inferior parts must support the weight of the
superior parts in addition to the load which presses on the whole column,
their thickness ought be somewhat increased ; and it appears from a con-
* Melanges de Turin, v. ii. 123.

122 LECTURE XIV.

sideration of the direction in which the fracture is most easily effected,
that the outline ought to be made a little convex externally, and more
curved above than below, which is the usual, although not the universal
practice; an elliptic arc is perhaps the most eligible outline, or a curve
formed by bending a ruler fixed at the summit of the column ; sometimes
the form is made to differ little from a cone, but such a figure is very
inelegant. The diminution of the thickness amounts in general to about
one sixth or one seventh of the whole, and sometimes to one fourth.
(Plate XI. Fig. 149.)

For a light house, where a great force of wind and water was to be
resisted, Smeaton chose a curve with its concavity turned outwards.* If
we calculated what would be the best form for a wooden pillar, intended
to remain always immersed in the water to a certain depth, we should find
that a cone or pyramid would possess the greatest possible strength for
supporting the motion of the water ; and a cone more acute than this
would be equally capable of resisting the force of the wind, supposing it
to be less active than that of the water ; the part below the water might,
therefore, be widened so as to become a portion of a more obtuse cone, the
upper part remaining more slender ; and the greatest agitation of the sea
being near its surface, the basis of the pillar might be a little contracted,
so as to have the outline of the lower part a little convex outwards, if the
depth of the water were considerable. But in the case of a building of stone,
the strength often depends as much on the weight of the materials as
on their cohesive power : and the lateral adhesion, which is materially
influenced by the weight, constitutes a very important part of the strength.
For resisting a force which tends to overset the building, the form in
which the weight gives the greatest strength is that of a conoid, or a solid
of which the outline is a parabola, concave towards the axis : and for pro-
curing, by means of the weight, a lateral adhesion which is everywhere
proportional to the force, the form must be cylindrical. So that in a
building circumstanced as we have supposed the pillar to be, there ap-
pears to be no reason for making either portion of the outline taken sepa-
rately convex towards the axis, although the angular junction of the two
portions of cones might very properly be rounded off; and the upper
parts might be a little enlarged if it were desirable to reduce the thickness
of the walls. But the Eddystone light house is completely above the level of
the sea, although in stormy weather every part of it is exposed to the action
of the waves, the water being sometimes thrown up to a much greater height
than that of the light house : so that it may be considered as exposed to
the force of a fluid more and more powerful as it is nearer to the founda-
tion ; and in this point of view its form differs but little from that which
the most accurate theory would point out ; but it is probably a little weaker
about the middle of its height, or somewhat lower, than in any other part.
(Plate XI. Fig. 150.)

A wall must be reduced in thickness as it rises, for the same reason as a
column is diminished ; and if the wall is a part of a house, it must Le
reduced in a still greater degree, since the load, which is to be supported by
* On the Eddystone Lighthouse, fol. Lond. 1791, PI. ix.

ON ARCHITECTURE AND CARPENTRY. 123

it at different parts of its height, is usually much varied by the weight of
the floors and of the contents of the apartments. But sometimes the
obliquity of the surface of the wall may become inconvenient, by promoting
the growth of moss and weeds. In building a wall, the first precaution
that is required, is to dig deep enough to ascertain the nature of the ground ;
the next, to lay a sufficiently extensive and firm foundation ; and it has
been very properly recommended that where a well is wanted, it should be
dug before the foundations of the house are laid, in order to examine the
qualities of the different strata which are to support them. The disposition
of the stones or bricks, is not a matter of indifference ; the strength is obvi-
ously greatest when all the surfaces are either horizontal or vertical ; for if
they are oblique, they must have a tendency to slide away laterally, and the
wall must be very liable to crack : hence the reticulated walls, sometimes
employed by the ancients, of which all the joints were oblique, possessed
but little durability. If the materials are thrown together without order,
they press on the parts in contact with them ; but occasionally, as in the
case of piers or quays, this circumstance may be of some advantage in
opposing external pressure ; or at least the effect of such a pressure may
remove the inconvenience which would otherwise arise from the irregularity
of the structure.

In some cases it is necessary to unite the stones of a building mechani-
cally, either by cramps of iron, fixed by means of melted lead, or by other
methods, similar to those which are more usually employed in carpentry.
Mr. Smeaton was obliged to fix the stones of his light house to the rock and
to each other, by dovetail joints, and to connect each horizontal tier with
the tier below it, by pins of wood passing through the stones, with wedges
driven in at each end, to make them expand, and tie the stones fast
together. But, in general, it is sufficient to employ mortar, made of lime
or terras and sand, of which the utility depends principally on the firmness
and cohesive strength that it acquires in consequence of its chemical pro-
perties. Sometimes the whole structure is composed of a mass which is at
first soft, but hardens as it dries ; in this manner mud walls are built ; and
the materials called pise are of a similar nature. (Plate XI. Fig. 151.)

The wall or column, when raised, must in general help to support a
single lintel or beam, an arch, a dome, or a roof of carpentry. The strength
of the lintel depends more on the nature of the substance than on any art
employed in forming it, excepting the precaution to give it as much depth
as is convenient, especially towards the middle, if the depth be anywhere
unequal ; but the construction of an arch affords considerable scope for the
exertion of mechanical science.

The simplest theory of the arch, supporting itself in equilibrium, is that
of Dr. Hooke,* the greatest of all philosophical mechanics. The arch,
when it has only its own weight to bear, may be considered as the inversion
of a chain suspended at each end ; for the chain Kangs in such a form that
the weight of each link is held in equilibrium by the result of the two forces
acting at its extremities ; and these forces or tensions are produced, the one

* Hooke, De Potentia Restitutiva, 1678, p. 31. See Waller's Life of Hpoke,
prefixed to the edition of his posthumous Works, Lond. 1705, p. 21.

124 LECTURE XIV.

by the weight of the portion of the chain below the link, the other by the
same weight increased by that of the link ; both of them acting originally
in a vertical direction. Now supposing the chain inverted, so as to consti-
tute an arch of the same form and weight, the relative situations of all the
lines, indicating the directions of the forces, will remain the same, the
forces acting only in contrary directions, so that they are compounded in a
similar manner, and balance each other on the same conditions, but with
this difference, that the equilibrium of the chain is stable, and that of the
arch tottering. This property of the equilibrium renders an accurate
experimental proof of the proposition somewhat difficult ; but it may be
shown that a slight degree of friction is sufficient for retaining in equili-
brium an arch formed by the inversion of a chain of beads. The figure is
called a catenaria, when the links are supposed to be infinitely small, and
the curvature is greatest at the middle of the chain.* It is not at all
necessary to the experiment that the links of the chain be equal ; the same
method may be applied to the determination of the form requisite for an
equilibrium, whatever may be the length or weight of the constituent parts
of the arch ; and when the arch is to be loaded unequally in different parts,
we may introduce this circumstance into the experiment, by suspending
proportional weights from different parts of the chain. Thus we may
employ wires or other chains to represent the pressure, and adjusting them
by degrees, till their extremities hang in a given line, we may find the form
which will best support the weight of the materials, the upper surface or
extrados of the arch being represented by the same line in an inverted
position, while the original chain shows the form of the intrados, or of the
curve required for the arch stones themselves. In common cases, the form
thus determined will differ little from a circular arc, of the extent of about
one third of a whole circle, rising from the abutments with an inclination
of 30° to the vertical line, and it never acquires a direction much more
nearly perpendicular to the horizon. It usually becomes more curved at
some distance below the summit, and then again less curved. (Plate XI.
Fig. 152... 154.)

But the supposition of an arch resisting a weight which acts only in a
vertical direction, is by no means perfectly applicable to cases which
generally occur in practice. The pressure of loose stones and earth,
moistened as they frequently are by rain, is exerted very nearly in the same
manner as the pressure of fluids, which act equally in all directions : and
even if they were united into a mass, they would constitute a kind of
wedge, and would thus produce a pressure of a similar nature, notwith-
standing the precaution recommended by some authors, of making the
surfaces of the arch stones vertical and horizontal only. This precaution
is, however, in all respects unnecessary, because the effect which it is
intended to obviate, is productive of no inconvenience, except that of

* For its properties see D. Gregory, Ph. Tr. xix. 637, and xxi. 419. Clairaut
on Catenariae, Miscellanea, Berolin, 1743, vii. 270. Krafft, Novi Com. Petrop: v.
145. Cantezzani, Com. Bon. vi. O. 265. Legendre, Mem. de Paris, 1786, p. 20.
Fuss. N. A. Pet. 1794, xii. 145. The elementary works of Poisson, Traite de Me-
canique, and Whewell's and Earnshaw's Mechanics.

ON ARCHITECTURE AND CARPENTRY. 125

exercising the skill of the architect. The effect of such a pressure only
requires a greater curvature near the abutments, reducing the form nearly
to that of an ellipsis, and allowing the arch to rise at first in a vertical
direction.

A bridge must also be so calculated as to support itself without being in
danger of falling by the defect of the lateral adhesion of its parts, and in
order that it may in this respect be of equal strength throughout, its depth
at each point must be proportional to the wreight of the parts beyond it.
This property belongs to the curve denominated logarithmic, the length
corresponding to the logarithm of the depth. If the strength were af-
forded by the arch stones only, this condition might be fulfilled by giving
them the requisite thickness, independently of the general form of the arch :
but the whole of the materials employed in the construction of the bridge,
must be considered as adding to the strength, and the magnitude of the
adhesion as depending in great measure on the general outline.

We must examine in the next place what is the most advantageous form
for supporting any weight which may occasionally be placed on the bridge,
in particular at its weakest part, which is usually the middle. Supposing
the depth at the summit of the arch and at the abutments to be given, it
may be reduced considerably in the intermediate parts, without impairing
the strength, and the outline may be composed of parabolic arcs, having
their convexity turned towards each other. This remark also would be
only applicable to the arch stones, if they afforded the whole strength of
the bridge, but it must be extended in some measure to the whole of the
materials forming it.

If therefore we combine together the curve best calculated for resisting
the pressure of a fluid, which is nearly elliptical, the logarithmic, and the
parabolic curves, allowing to each its due proportion of influence, we may
estimate, from the comparison, which is the fittest form for an arch in-
tended to support a road. And in general, whether the road be horizontal
or a little inclined, we may infer that an ellipsis, not differing much from
a circle, is the best calculated to comply as much as possible with all the
conditions. (Plate XI. Fig. 155.)

The tier of bricks cut obliquely, which is usually placed over a window
or a door, is a real arch, but so flat as to allow the apparent outline to be
horizontal. Mr. Coulomb observes, that the greatest strength is obtained
by causing all the joints to tend to a single point : * but little dependence
can be placed on so flat an arch, since it produces a lateral thrust which
may easily overpower the resistance of the wall. For the horizontal force
required to support each end of any arch, is equal to the weight of a
quantity of the materials which are supported by its summit, supposed to
be continued, of their actual depth, to the length of a semidiameter of the
circle of which the summit of the arch is a portion. This simple calcu-
lation will enable an architect to avoid such accidents, as have too often
happened to bridges for want of sufficient firmness in the abutments. The
equilibrium of a bridge, so far as it depends only on the form of the arch,
is naturally tottering, and the smallest force which is capable of deranging
* Theorie des Machines Simples, 4to, 1821, p. 355 (reprint}.

126 LECTURE XIV.

it may completely destroy the structure ; but when the stones or blocks
composing it have flat surfaces in contact with each other, it is necessary
that the line expressing the direction of the pressure be so much disturbed,
as to exceed at some part the limits of these surfaces, before the blocks can
be displaced. When this curve, indicating the general pressure which
results from the effect of a disturbing force, combined with the original
thrust, becomes more remote from the centre of the blocks than one sixth
of their depth, the joints will begin to open on the convex side, but the
arch may still stand, while the curve remains within the limits of the
blocks.

It is desirable that the piers of bridges should be so firm, as to be
able not only to support the weight of half of each adjoining arch, but also
to sustain, in case of the failure of one of those arches, the horizontal
thrust of the other ; and the same condition is obviously necessary for the
stability of walls of any kind which support an arched or vaulted roof,
wherever there is no opportunity of assisting the strength by ties or chains
of any kind. There are two ways in which such a pier or wall may give
way : it may either be overset, or caused to slide away horizontally ; but
since the friction or adhesion which resists the horizontal motion is usuallv
greater than one third of the pressure, it seldom happens that the whole
thrust of the arch is so oblique as not to produce a sufficient vertical pres-
sure for securing the stability in this respect ; and it is only necessary to
make the pier heavy enough to resist the force which tends to overset it.
It is not, however, the weight of the pier only, but that of the half of the
arch which rests on it, that resists any effort to overset it, and in order that
the pier may stand, the sum of these weights, acting on the end of a lever
equal to half the thickness of the pier, must be more than equivalent to the
horizontal thrust, acting on the whole height of the pier. The pier may
also be simply considered as forming a continuation of the arch, and the
stability will be preserved as long as the curve, indicating the direction of
the pressure, remains within its substance.

The arches of Black Friars bridge are of an oval form, composed of cir-
cular arcs, and differing but little from ellipses ; the arch stones are so
large that the pressure in any direction might be very greatly increased
without causing the general result to exceed the limits of their magnitude,
or even to approach very near to their surfaces. (Plate XII. Fig. 156.)

The construction of a dome is less difficult than that of an arch, since
the tendency of each part to fall is counteracted, not only by the pressure
of the parts above and below, but also by the resistance of those which are
situated on each side. A dome may therefore be erected without any
temporary support like the centre which is required for the construction
of an arch, and it may at last be left open at the summit, without standing
in need of a keystone, since the pressure of the lower parts is sufficiently
resisted, by the collateral parts of the same horizontal tier, to prevent the
possibility of their falling in, or of their forcing out the upper parts. The
weight of the dome may however force out its lower parts, if it rises irt a
direction too nearly vertical; and supposing its form spherical, and its
thickness equable, it will require to be confined by a hoop or chain as soon

ON ARCHITECTURE AND CARPENTRY. 127

as the span becomes eleven fourteenths of the whole diameter. But if the
thickness of the dome be diminished as it rises, it will not require to be
bound so high : thus, if the increase of thickness in descending begin at
about 30 degrees from the summit, and be continued until, at about 60
degrees, the dome becomes a little more than twice as thick as at first, the
equilibrium will be so far secure ; and at this distance it would be proper
to employ either a chain or some external pressure, to preserve the sta-
bility, since the weight itself would require to be increased without limit,
if it were the only source of pressure on the lower parts. (Plate XII.
Fig. 157.)

The dome of St. Paul's cathedral is elliptical, and is built of wood, and
confined by strong chains, consisting of iron bars ; that of the Pantheon at
Rome is nearly circular, and its lower parts are so much thicker than its
upper parts, as to afford sufficient resistance to their pressure : they are
supported by walls of great thickness, and furnished with many projections
which answer the purpose of abutments and buttresses. (Plate XII. Fig.
158, 159.)

A knowledge of the parts and proportions usually assigned to columns
and to buildings in general, and of their technical names and divisions,
belongs rather to the subject of ornamental than to that of useful architecture ;
and the consideration of symmetry and elegance is in great measure foreign
to that of the mechanical properties of bodies, which it is our present
business to investigate. The five orders of ancient architecture are found
to differ considerably in their proportions, in the different remains of
Greek and Roman edifices ; but there always remain some characteristic
distinctions : the Tuscan is known by its strength and simplicity, without
any peculiar ornament ; the Doric by its triglyphs, or triangular grooves,
above each column, imagined to represent the ends of beams ; the Ionic by
the large volutes, and the Corinthian by the foliage, respectively envelop-
ing their capitals ; and the Composite usually by the combination of both
these characters ; each order being lighter than the preceding, and being
sometimes employed with it in the upper parts of the same building. In
general, the length of the Tuscan column, with its capital, is equal to about
seven diameters of the base, that of the Doric eight, of the Ionic nine,
and of the Corinthian and Composite ten diameters. (Plate XII. Fig.
160... 164.)

The Gothic architects appear to have been superior to the Greeks in the
mechanical arrangement of the parts of their edifices, so as to produce the
most advantageous effect in preserving the general equilibrium. They
made every essential member of their buildings a constituent part of their
system of ornament, and even those embellishments, which, by a super-
ficial observer, might be deemed useless or prejudicial, are frequently cal-
culated, either by their strength or by their weight, to serve some bene-
ficial purposes. The pointed arch is not in all cases well calculated for
equilibrium, but when it has a pillar resting on its summit, it is exceedingly
strong. The most celebrated of modern architects have sometimes been
less successful than those of the middle ages ; and for want of paying suf-
ficient attention to mechanical principles, have committed such errors in

128 LECTURE XIV.

their attempts to procure an equilibrium, as have been followed by the
most mischievous consequences. Examples of this might be pointed out in
the bridges of our own country and the churches of others ; but if we are
masters of the true theory of pressure, we shall be able to avoid similar
errors, without examining the particular circumstances which have oc-
casioned these accidents. (Plate XII. Fig. 165.)

The principles of equilibrium, which are employed in architecture, are
equally applicable to many cases in carpentry ; and where the work is
principally calculated to withstand a thrust, there is little difference in
the operation of the forces concerned ; but where a tie is introduced, that
is, a piece which resists principally by its cohesive strength, the parts often
require to be arranged in a different manner. The general principle, that
three forces, in order to retain each other in equilibrium, must be propor-
tional to the sides of a triangle corresponding to their directions, is suf-
ficient for determining the distribution of pressure in almost all cases that
can occur. The conclusions which have been drawn from this principle,
and from other similar considerations, respecting the strength of materials,
will also be of great use in directing us how to determine the best forms for
beams, rafters, and timbers of all kinds, and how to arrange and connect
them in the best manner with each other.

The employment of the cohesive strength of materials in carpentry in-
troduces a difficulty which scarcely exists in architecture. Two blocks,
placed on each other, resist the force of a weight compressing them, as ef-
fectually as if they formed but one piece : but they have no sensible cohe-
sion to enable them to withstand a force tending to separate them, and if
they are required to co-operate by their cohesive strength, some mode of
uniting them must be found. For this purpose, it is generally necessary to
sacrifice a considerable portion of the strength of the materials employed.
The most usual mode is to place the ends of the pieces side by side, first
reducing their dimensions, where a regular outline is required ; and to
procure a firm adhesion between them by means of external pressure, or to
employ the natural adhesion of some parts which are made to project be-
yond the rest in each piece, and receive in their interstices the correspond-
ing projections of the other piece.

Where the adhesion is produced by external pressure only, it is of ad-
vantage to subdivide the joints into a considerable number of parts, as is
usually done in the masts of ships, and to make the junction of any two
pieces, following each other in the same line, as distant as possible from
any other junction ; for in this manner, the loss of strength may be di-
minished almost without limit, provided that the distance between the
joints be great enough to afford a firm adhesion to each part. The junction
may also be formed by an oblique line ; but the obliquity must be so great
that any lateral pressure may increase the stability of the wedge, the length
being in a greater proportion to the depth than the pressure to the adhesion
that it occasions ; and the pieces must be pressed together very forcibly by
means of hoops or bolts. (Plate XIII. Fig. 166... 168.)

Where the natural adhesion of some projecting parts in each piece is em-
ployed, the projections must be sufficiently long to secure their strength,

ON ARCHITECTURE AND CARPENTRY. 129

and they must be as little prominent as possible, partly because the con-
tiguous piece must be excavated for their reception, and partly because
their strength is diminished when they project more than one sixth of their
length. A beam united to another in this manner is said to be scarfed.
(Plate XIII. Fig. 169.)

In order to preserve the strength of a compound beam, intended to re-
sist a transverse action in a particular direction, it is necessary to avoid, as
much as possible, reducing the depth of the beam in that direction, and to
secure the union with the greatest care on the convex side of the beam,
which is stretched by the operation of the force. Where no inconvenience
can result from the projection of a piece on one side, it is easy to preserve
the strength unimpaired, by splicing or fishing it on the convex side ; and
if the depth of the piece added be only half as great as that of the original
beam, the strength will be somewkat increased by the operation, supposing
the two ends to meet each other without any connection. Such pieces re-
quire, however, to be firmly united, either by pins passing through them,
or by blocks or joggles let in to a certain depth, in order to prevent their
sliding on each other ; and this mode of union is stronger than scarfing
them, because it does not diminish the depth. (Plate XIII. Fig. 170, 171.)

Where the pieces to be connected together are in different directions, the
end of one of them is usually reduced in its size, and becomes a tenon,
while a mortise is cut in the other for its reception, and the joint is also
often secured still more firmly by a strap of iron. If a joist be let into a
beam, at its upper edge, and made very tight by wedges, the strength of
the beam will not be materially diminished ; but the vicissitudes of mois-
ture and dryness may very much impair the firmness of the union, and
the end of the joist may fail in dry weather to afford sufficient resistance
to the flexure of the beam : so that in some cases it might be more ad-
visable to cut the mortise near the middle of the depth of the beam. If
two pieces meet obliquely, and one of them exerts a thrust against the
other, the simplest mode of opposing this thrust is to bind them toge-
ther by a strap of iron fixed to the second piece ; this strap renders it
impossible for the first to advance without having its extremity crushed ;
it is also common to make a mortise in the second piece, a part of which
serves as an abutment for the first ; and for this purpose the piece must be
continued far enough beyond the abutment to give the projection sufficient
force of adhesion, a condition which is the more easily fulfilled when the
action of the strap produces a pressure on it. The assistance of a strap
is still more indispensable where the pieces are perpendicular to each other,
and the force tends to draw one of them away from the other ; in this case
the mortise may be made a little wider at the remoter part, and the end of
the tenon may be made to fit it by driving in wedges, in the same manner
as Mr. Smeaton united his blocks of stone ; but, a large mortise would
weaken the beam too much, and a strong strap or hoop is usually required
^additional security. Such a strap ought always to be as straight as
possible, so as to act only in the direction of the force to be resisted : it
has been too customary to accommodate the strap to the form of the beams, or
to make it deviate in other ways from a right line : but wherever a strap

130 LECTURE XIV.

is bent in any direction, to a distance from a right line equal only to its
depth in that direction, its strength is so reduced, as not to exceed one
seventh of what it would have been if it had remained straight. (Plate XIII.
Fig. 172.. .174.)

It is equally necessary in all other cases which occur in carpentry, to
avoid as much as possible a transverse strain, the disadvantage of which is
obvious from the great inferiority of the strength of any substance, resisting
a transverse force, to its primitive cohesive or repulsive strength. For
similar reasons it is proper to avoid employing a very open angle at a point
where a load is supported, the great obliquity of the two pieces forming the
angle requiring them to exert a great force in order to oppose a much
smaller one. Allowance must also be made for the contraction of the
timber, and care must be taken that it do not so alter the arrangement of
the parts, as to bring a disproportionate strain on a point not calculated to
support it. If the two pieces forming an obtuse angle consisted, either
wholly or partly, of wood cut across the grain, and the piece joining their
extremities were cut in the usual manner, the oblique pieces would contract
considerably more as they became drier, and the angle would become more
obtuse, so that the strain, produced by a given weight, would be greater
than in the original state of the triangle. Sometimes the work is liable to
be deranged by the operation of a lateral force, which may have appeared
too trifling to produce any considerable effect, but which may still destroy
the greater part of the strength, by causing the resistances to deviate from
the plane of the forces which they are intended to oppose.

The framing of a roof is one of the most common and most important
subjects for the employment of the theory of carpentry. If the rafters
were simply to abut on the walls, they would force them outwards ; a tie
beam is therefore necessary, to counteract the thrust. In order to enable
the tie beam to support a weight, a king post is suspended from the rafters ;
and frequently braces are again erected from the bottom of the king post,
to support the middle of the rafters. Sometimes a flat or less inclined
portion is placed in the middle, forming a kirb or mansard roof, somewhat
resembling an arch ; this form has the advantage, when it is properly propor-
tioned, of lessening the transverse strain on the rafters by making them
shorter ; but this purpose is answered equally well by the addition of the
braces which have been already mentioned. A kirb roof affords, however, a
greater space within, than a plain roof of the same height, and produces also
somewhat less strain on the tie beam or on the abutments : the tie beam may
be suspended from it by a king post and two queen posts, descending perpen-
dicularly from the joints ; and the place of the king post may be supplied
by a cross beam uniting the heads of the queen posts and keeping them at
a proper distance ; this beam may also be suspended by a shorter king-
post from the summit. Such a roof appears to be more advantageous
than it has been commonly supposed. (Plate XIII. Fig. 175. ..177.)

The angle of inclination of a roof to the horizon usually varies in
different climates : in Italy the height is generally less than one fourtli of
the breadth ; in England it was formerly three fourths, but it now com-
monly approaches much more to the Italian proportion. In northern

ON ARCHITECTURE AND CARPENTRY. 131

climates, a steep roof is required on account of falls of snow, which
greatly increase the lateral thrust of the rafters ; for the horizontal force
exerted by a roof is always proportional to the length of a line perpendi-
cular to the rafter, descending from its extremity till it meets another
similar line drawn from the opposite rafter ; and this perpendicular is
obviously much increased when the roof becomes very flat. But for bear-
ing the transverse strain which tends to break the rafters themselves, a low
roof is stronger than a high one, supposing the number of braces and
queen posts equal on both ; for if we have to support a given weight by a
beam or rafter, whether it be placed in the middle, or equally divided
throughout the length, we neither gain nor lose force by lengthening
the beam and raising it higher, while the horizontal span continues the
same, since the obliquity lessens the effect of the weight precisely in the
same ratio that the length of the beam diminishes its strength ; but by
lengthening the beam we also add to the weight which is to be supported,
and we thus diminish the strength of the roof. It must be observed, in
calculating the strength of a rafter, that the slight flexure produced by
the transverse strain, has a material effect in diminishing its strength in
resisting a longitudinal force ; and this diminution must be determined
according to the principles that have been laid down respecting the equili-
brium of elastic substances.

Wooden bridges, and the temporary centres on which arches of stone
are supported during their construction, depend nearly on the same prin-
ciples as roofs : the external parts usually support a thrust, and the
internal act as ties ; but the abutments are generally capable of withstand-
ing a horizontal thrust without inconvenience, so that by their assistance
the strain on the ties is considerably diminished. Great strength may
also be obtained, where it is practicable to support each part of the centre
by two beams, in the direction of chords, bearing immediately on the abut-
ments. (Plate XIV. Fig. 178, 179.)

The various articles of household furniture belong to subordinate
branches of carpentry, but their form is in general more accommodated to
convenience and elegance than to strength and durability. Yet, even in
making a chair, there is room for error and for improvement ; the same
principles that direct us in framing a roof are capable of application here ;
but if they were implicitly followed, they would lead us to the employ-
ment of bars crossing each other in an inelegant manner. Doors, gates,
locks, and hinges, are either parts of the carpenter's employment, or
appendages to his works ; and it is possible that, by attentive considera-
tion, improvements might be made in all of them. Mr. Parker has de-
voted much time and labour to the subject of gates, with their hinges and
fastenings, and has presented to the Royal Institution a very useful col-
lection of models, which show the result of his investigations.*

* Parker on Gates, Lond. 1801, Rep. of Arts, ii. II. 50.

K2

132 LECTURE XV.

LECT. XIV.— ADDITIONAL AUTHORITIES.

Architecture. — Perrault's Vitruvius, fol. Par. 1673. Newton's do. 2 vols. fol.
1772. Hall's Essay, 4to, 1813. Rickman's Gothic Architecture, 1825. Willis
on the Architecture of the Middle Ages, Camb. 1835. Britton's Dictionary of Archi-
tecture, 1830-8. Hope's Essay, 2 vols. 1835. Pugin's various works.

Strongest forms of Columns and Walls. — Euler on the Strength of Columns,
Hist, et Mem. de Berlin, 1757, p. 252. Acta Petr. ii. I. 121, 146, 163. Belidor,
Architecture Hydraulique, ii. I. 420. Coulomb, Mem. des Savans Etrangers, vii.
Theorie des Mach. Simples, 1821. Prony sur la Poussee des Terres, 4to, Par.
1802. Prony sur les Murs de Revetement, 4to, 1802.

Practical Architecture. — Rondelet, L'Art de Batir, 3 vols. 4to. Par. 1804. Bor-
gnis, Traite Elementaire de Construction appliquee & 1'Architecture Civile, 4to, Par.
1823. Chambers's Civil Architecture, by Gwilt, 2 vols. 1825. Bullet, Nouvelle
Architecture Pratique, par Jay, 2 vols. 1825. Navier sur 1'Application de la Me-
canique a 1'Etablissement des Constructions, &c. 1833. Hosking on Architecture
and Building, from Encyc. Brit. 4to, 1835. Nicholson's Principles and Practice of
Architecture, 3 vols. 1836.

Carpentry in general. — Fuss on the Strains of Framed Carpentry, Acta Petr.
1778, ii. I. 194. Encyclopedic Methodique, Arts et Metiers, art. Charpentier.
Robison's Mech. Phil. Tredgold's Principles of Carpentry, 1820. Nicholson's
Mechanic's Companion, 1824. Carpenter's Guide, 4to, 1828.

Arches, Domes, and Bridges. — Lahire on Arches, &c. Hist, et Mem. de Paris,
1702, p. 94, H. 119; 1712, p. 69, H. 74. Couplet on the Thrust of Arches, do.
1729, p. 79 ; H. 75 ; 1730, p. 117, H. 107. Labelye on Westminster Bridge, 1739.
Euler on the Strength of a Model, Nov. Com. Petr. xx. 271. Belidor, Arch.
Hydr. ii. II. 415. Gauthey, Construction des Fonts. Peronnet sur les Fonts de
Neuilly, d'Orleans, &c. fol. 1782-8. Berard, Theorie de 1'Equilibre des Voutes,
4to, Par. 1810. Wiebekings' Wasserbaukunst, 1812. Ware, Tracts on Vaults and
Bridges, 1822. Barres, Nouveau Systeme des Fonts a Grandes Portees, 4to, Paris,
1827. Navier, Memoires sur les Fonts Suspendus, 4to, Par. Belidor, Science des
Ingenieurs, 4to, Paris, 1830. (Navier's Ed.)

The student is particularly referred to Robison's Mechanical Philosophy, vol. i.
p. 369 to the end, for details on the subjects discussed in this Lecture.

LECTURE XV.

ON MACHINERY.

HAVING taken a general view of those branches of practical mechanics
in which forces are to be resisted, we are next to consider the modifications
of forces and of motions ; and in the first place, the modes of applying
forces, of changing their direction and intensity, and of communicating
them to different parts of our machines by the intervention of rods, joints,
cranks, wheelwork, ropes, or other flexible substances ; in the second place,
the structure of these substances, and the methods by which the union of
flexible fibres in general may be effected ; and in the third place, the regu-
lation and equalisation of motion, by means of clocks and watches.

The modes of applying mechanical forces are almost as various as .the
machines that are constructed and the purposes for which they are em-
ployed : but in general, the strength of men is applied by means of levers
or winches, or by walking wheels which slide beneath them as they attempt

ON MACHINERY. 133*

to ascend ; and that of other animals, by a horizontal arm projecting from
a vertical axis to which they are harnessed, and sometimes also by causing
them to walk on or in a moveable wheel. Many of these arrangements
may however be very conveniently considered as belonging to the particular
objects for which each machine is constructed, especially to the modes of
raising weights by cranes, and of grinding substances by mills.

When motion is simply communicated to a substance placed before the
moving body, such materials must be employed as are capable of exerting
a repulsive force or a thrust ; and these are generally of the same kind as
are sometimes concerned in the operations of architecture, but more com-
monly in those of carpentry, particularly metal and wood. But when the
body to be moved is behind the moving power, and is pulled along by it,
chains or ropes are sometimes more convenient. In the union of wood for
moveable machinery, it is generally advisable to avoid employing pins or
bolts of metal ; for these, by their superior weight and hardness, sometimes
injure the wood in contact with them, and become loose.

When the direction of the motion communicated is also to be changed,
levers or cranks may be employed, united by joints or hinges of various
kinds. Sometimes a long series of connected rods is suspended by other
rods or chains, so as to convey the effect of the force to a considerable dis-
tance ; in this case the motion is generally alternate, when, for example,
pumps are worked by means of a waterwheel at a distance from the shaft
in which the pumps are placed. In this arrangement, there is no necessary
loss on account of the alternation of the motion of the rods ; for if they are
suspended at equal distances from a number of fixed points, they will move
backwards and forwards in the manner of a single pendulum ; but the
magnitude of the friction is the principal inconvenience produced by the
weight of the series. Where a lever is employed for changing the direction
of a great force, its strength may be increased by the addition of a frame
projecting in the direction of its depth; and if the lever is bent, a cross
piece uniting its arms is still more requisite. (Plate XIV. Fig. 180... 182.)

For the communication of a rotatory motion, Dr. Hooke's universal joint*
is sometimes of use, especially when the inclination is not required to
be materially changed ; but if the obliquity is great, the rotation is not
communicated equably to the new axis at all points of its revolution. This
joint is formed by a cross, making the diameters of two semicircles, one of
which is fixed at the end of each axis. (Plate XIV. Fig. 183.)

The best mode of connecting a rotatory motion with an alternate one
is, in all common cases, to employ a crank, acting on one end of a long rod
which has a joint at the other. If the rotatory motion of the crank be
equable, the progressive motion of the rod will be gradually accelerated and
retarded, and for a considerable part of the revoluion the force exerted will
be nearly uniform : but if we attempted to communicate at once to the rod
its whole velocity in each direction, as has sometimes been done by inclined
planes, or by wheelwork, the motion would become extremely irregular,

* Hooke, Animadversions on Hevelius' Machina Coelestis, p. 73, 4to, Lond. 1674 ;
and a Description of Helioscopes and other Instruments, 4to, Lond, 1676, p. 14.

134 LECTURE XV.

and the machinery would be destroyed by the strain. (Plate XIV.
Fig. 184.)

On the other hand it must be observed that the force applied to a ma-
chine may, in general, be divided into two portions ; the one employed in
opposing another force, so as to produce equilibrium only, the other in
generating momentum. With respect to the first portion, a single crank
has the inconvenience of changing continually the mechanical advantage of
the machine ; with respect to the second, its motion in the second quarter
of its revolution is accelerated, instead of being retarded, by the inertia
which this portion of the force is intended to overcome : and from a com-
bination of both these causes, the motion must necessarily be rendered very
irregular. They may, however, be completely removed, by employing always
cranks in pairs, one of them being fixed so as to make a right angle with
the other, which is also the best position for two winches to be turned by two
labourers ; since the point of the circle, in which a man can exert his
greatest strength, is nearly at the distance of a right angle, or a little more,
from the point at which his force is smallest.

An alternate motion may be communicated to a rod, so that the force
may be either uniformly exerted, or varied according to any given law, by
means of an inclined surface formed into a proper curve, and acting on a
friction wheel fixed to the rod ; and a single plane surface, placed ob-
liquely, would answer sufficiently well for this purpose. But in such
cases, as well as when a crank is used, it is necessary to employ other
means for supporting the rod in its proper situation ; this may either be
done by additional friction wheels, or in a more elegant manner, by such an
arrangement of jointed rods as will cause the extremity of one of them to
move in a curve which does not sensibly differ from a right line. If we fix
two pins in a beam, so as to connect to it two equal rods, of which the
extremities are joined by a third, and the end of this third rod which is
nearest to the centre of the beam be connected to a second beam of a proper
length, the opposite end of the rod will initially describe a right line ; and
for this purpose the length of the second beam must be to the distance of
the nearest pin from the centre as that distance is to the distance of the
pins from each other. The same effect may also be produced by means of
a frame, made of two pieces, each a yard long, united by joints to each
other, and to two other pieces of a foot each ; one of the first pieces being
fixed, if the shorter piece opposite to it be produced to the length of four
feet, its extremity will move at first in a right line. The proportions of
the rods may also be made more convenient than these, and others may be
added to them, if it be required, which may make a line move so as to
remain always in parallel directions. (Plate XIV. Fig. 185... 188.)

But of all the modes of communicating motion, the most extensively
useful is the employment of wheelwork, which is capable of varying its
direction and its velocity without any limit.

Wheels are sometimes turned by simple contact with each other ; some-
times by the intervention of cords, straps, or chains, passing over theni ;
and in these cases the minute protuberances of the surfaces, or whatever
else may be the cause of friction, prevents their sliding on each other.

ON MACHINERY. 135

Where a broad strap runs on a wheel, it is usually confined to its situation,
not by causing the margin of the wheel to project, but, on the contrary,
by making the middle prominent : the reason of this may be understood
by examining the manner in which a tight strap running on a cone would
tend to run towards its thickest part. Sometimes, also, pins are fixed in
the wheels, and admitted into perforations in the straps ; a mode only
practicable where the motion is slow and steady. A smooth motion may
also be obtained, with considerable force, by forming the surfaces of the
wheels into brushes of hair. (Plate XV. Fig. 189.)

More commonly, however, the circumferences of the contiguous wheels
are formed into teeth, impelling each other, as with the extremities of so
many levers, either exactly or nearly in the common direction of the cir-
cumferences ; and sometimes an endless screw is substituted for one of the
wheels. In forming the teeth of wheels, it is of consequence to deter-
mine the curvature which will procure an equable communication of
motion, with the least possible friction. For the equable communication
of motion, two methods have been recommended ; one, that the lower part
of the face of each tooth should be a straight line in the direction of the
radius, and the upper a portion of an epicycloid, that is, of a curve de-
scribed by a point of a circle rolling on the wheel, of which the diameter
must be half that of the opposite wheel ; and in this case it is demonstrable
that the plane surface of each tooth will act on the curved surface of the
opposite tooth so as to produce an equable angular motion in both wheels :
the other method is, to form all the surfaces into portions of the involutes
of circles, or the curves described by a point of a thread which has been
wound round the wheel, while it is uncoiled ; and this method appears to
answer the purpose in an easier and simpler manner than the former.*
It may be experimentally demonstrated that an equable motion is pro-
duced by the action of these curves on each other : if we cut two boards
into forms terminated by them, divide the surfaces by lines into equal or
proportional angular portions, and fix them on any two centres, we shall
find that as they revolve, whatever parts of the surfaces may be in
contact, the corresponding lines will always meet each other. (Plate XV.
Fig. 190... 192.)

Both of these methods may be derived from the general principle that
the teeth of the one wheel must be of such a form that their outline may
be described by the revolution of a curve upon a given circle, while the
outline of the teeth of the other wheel is described by the same curve
revolving within a second circle. It has been supposed by some of the
best authors that the epicycloidal tooth has also the advantage of com-
pletely avoiding friction ; this is however by no means true, and it is
even impracticable to invent any form for the teeth of a wheel, which will
enable them to act on other teeth without friction. In order to diminish
it as much as possible, the teeth must be as small and as numerous as is
consistent with strength and durability ; for the effect of friction always
increases with the distance of the point of contact from the line joining

* For a demonstration of these propositions, see Airy on the Teeth of Wheels,
Trans, of the Camb. Phil. Soc. ii. 279.

13G LECTURE XV.

the centres of the wheels. In calculating the quantity of the friction,
the velocity with which the parts slide over each other has generally been
taken for its measure : this is a slight inaccuracy of conception, for, as
we have already seen, the actual resistance is not at all increased by in-
creasing the relative velocity ; but the effect of that resistance in retarding
the motion of the wheels, may be shown from the general laws of
mechanics, to be proportional to the relative velocity thus ascertained.
When it is possible to make one wheel act on teeth fixed in the concave
surface of another, the friction may be thus diminished in the proportion
of the difference of the diameters to their sum. If the face of the teeth,
where they are in contact, is too much inclined to the radius, their mutual
friction is not much affected, but a great pressure on their axes is pro-
duced ; and this occasions a strain on the machinery, as well as an increase
of the friction on the axes.

If it is desired to produce a great angular velocity with the smallest pos-
sible quantity of wheelwork, the diameter of each wheel must be between
three and four times as great as that of the pinion on which it acts. Where
the pinion impels the wheel, it is sometimes made with three or four teeth
only, but it is much better in general to have at least six or eight ; and
considering the additional labour of increasing the number of wheels, it
may be advisable to allot more teeth to each of them than the number re-
sulting from the calculation ; so that we may allow 30 or 40 teeth to a
wheel acting on a pinion of 6 or 8. In works which do not require a great
degree of strength, the wheels have sometimes a much greater number of
teeth than this ; and on the other hand, an endless screw or a spiral acts as
a pinion of one tooth, since it propels the wheel through the breadth of one
tooth only in each revolution. For a pinion of six teeth, it would be
better to have a wheel of 35 or 37 than 36 ; for each tooth of the wheel
would thus act in turn upon each tooth of the pinion, and the work would
be more equally worn than if the same teeth continued to meet in each
revolution. The teeth of the pinion should also be somewhat stronger than
those of the wheel, in order to support the more frequent recurrence of
friction. It has been proposed for the coarser kinds of wheelwork, to di-
vide the distance between the middle points of two adjoining teeth into
30 parts, and to allot 16 to the tooth of the pinion, and 13 to that of the
wheel, allowing 1 for freedom of motion.

The wheel and pinion may either be situated in the same plane, both
being commonly of the kind denominated spur wheels, or their planes may
form an angle : in this case one of them may be a crown or contrate wheel,
or both of them may be bevilled, the teeth being cut obliquely. According
to the relative magnitude of the wheels, the angle of the bevil must be dif-
ferent, so that the velocities of the wheels may be in the same proportion
at both ends of their oblique faces : for this purpose the faces of all the
teeth must be directed to the point where the axes would meet. (Plate XV.
Fig. 193, 194.)

In cases where a motion not quite equable is required, as it sometimes
happens in the construction of clocks, but more frequently in orreries, the
wheels may either be divided a little unequally, or the axis may be placed

. ON MACHINERY. 137

a little out of the centre ; and these eccentric wheels may either act on
other eccentric wheels, or, if they are made as contrate wheels, upon a
lengthened pinion. (Plate XV. Fig. 195, 196.)

An arrangement is sometimes made for separating wheels which are in-
tended to turn each other, and for replacing them at pleasure ; the wheels
are said to he thrown by these operations out of gear and into gear again.

When a wheel revolves round another, and is so fixed as to remain
nearly in a parallel direction, and to cause the central wheel to turn round
its axis, the apparatus is called a sun and planet wheel. In this case, the
circumference of the central wheel moves as fast as that of the revolving-
wheel, each point of which describes a circle equal in diameter to the dis-
tance of the centres of the two wheels : consequently, when the wheels are
equal, the central wheel makes two revolutions, every time that the ex-
terior wheel travels round it. If the central wheel be fixed, and the ex-
terior wheel be caused to turn on its own centre during its revolution, by
the effect of the contact of the teeth, it will make in every revolution one
turn more with respect to the surrounding objects, than it would make, if
its centre were at rest, during one turn of the wheel which is fixed : and
this circumstance must be recollected when such wheels are employed in
planetariums.

Wheels are usually made of wood, of iron, either cast or wrought, of
steel, or of brass. The teeth of wheels of metal are generally cut by means
of a machine ; the wheel is fixed on an axis, which also carries a plate fur-
nished with a variety of circles, divided into different numbers of equal
parts, marked by small excavations ; these are brought in succession
under the point of a spring which holds the axis firm, while the intervals
between the teeth are expeditiously cut out by a revolving saw of steel.
The teeth are afterwards finished by a file ; and a machine has also been
invented for holding and working the file. (Plate XV. Fig. 197.)

It is frequently necessary in machinery to protract the time of ap-
plication of a given force, or to reserve a part of it for future use. This is
generally effected by suffering a weight to descend, which has been previ-
ously raised, or a spring to unbend itself from a state of forcible flexure, as
is exemplified in the weights and springs of clocks and watches. The com-
mon kitchen jack is also employed for protracting and equalising the ope-
ration of a weight : in the patent jack the same effect is produced by an
alternate motion, the axis being impelled backwards and forwards, as in
clocks and watches, by means of an escapement, and the place of a balance
spring being supplied by the twisting and untwisting of a cord.

In these machines, as well as in many others of greater magnitude, the
fly wheel is a very important part, its velocity being increased by the ope-
ration of any part of the force which happens to be superfluous, and its ro-
tatory power serving to continue the motion when the force is diminished
or withdrawn. Thus, when a man turns a winch, he can exert twice as
much force in some positions as in others, and a fly enables him in some
crises to do nearly one third more work. In the pile engine also, without
the help of the fly, the horses would fall for want of a counterpoise, as soon
as the weight is disengaged. Such a fly ought to be heavy, and its motion

138 LECTURE XVI.

must not be too rapid, otherwise the resistance of the air will destroy too
much of the motion ; but in the kitchen jack, as well as in the striking
part of a clock, where the superfluous force is purposely destroyed, the fly
is made light, and strikes the air with a broad surface. An effect similar
to that of a fly and a spring is sometimes produced in hydraulic machines
by the introduction of an air vessel, the air contained in which is com-
pressed more or less according to the intensity of the force, and exerts a
more uniform pressure in expelling the fluid which is forced irregularly
into it.

LECT. XV.— ADDITIONAL AUTHORITIES.

Lahire's Mech. Par. 1695. Maudey's Mechanical Powers, 1709. Leupold, Thea-
trum Machinarum, 9 vols. fol. Leipz. 1724. . . . Euler on the Theory of Machines,
Com. Petr. x. 67. Nov. Com. Petr. iii. 254 ; viii. 230. Hist, et Mem. de Berlin,
1747, 1752. Camus, Cours de Mathematiques, Par. 1766. Berthelot, Mecanique
appliquee aux Arts, 2 vols. 4to, 1773. Jacobsons Technologisches Worterbuch,
von Rosenthal, Berl. 1787. Person, Recueil de Mecanique, 4to, Paris, 1802.
Banks on the Power of Machines, Kendal, 1803. Guenyveau, Essai sur la Science
des Machines, Lyons, 1809. Lippi, Principj Pratici di Meccanica, Napoli, 1811.
Lauz et Betancourt, Essai sur la Composition des Machines, 4to, Par. 1819. Bprgnis,
Traite Complet de Mechanique appliquee aux Arts, 7 vols. 4to. Paris, 1818-20.
Dictionnaire de do. 4to, 1823. Hachette, Traite Elementaire des Machines, 1828.
Robison, art. Machinery. Coriolis, Calcul de 1'Effet des Machines, 1829. Navier,
Resume des Lemons donnees & 1'Ecole des Ponts et Chaussees, 1833. Prony, Me-
moire sur un Moyen de convertir les Mouvemens, &c. 4to, 1837. Whewell's Me-
chanics of Engineering, Camb. Willis's Principles of Machinery, Camb. 1841.
Poncelet, Introduction a la Mecanique Industrielle, Metz et Paris, 1841. Moseley,
The Mechanical Principles of Engineering, 1843.

WheelworJc. — Hooke's Perfection of Wheelwork. Cutlerian Lectures, No. 2,
Animadversions on Hevelius, 4to, 1674, p. 70. Lahire on the Teeth of Wheels,
Hist, et Mem. de Paris, ix. 90, 283, 292. Camus on do. ibid. 1733, p. 117, H.
81 ; and Cours de Mathematiques, 4 vols. translated, 1806. Euler on do. Nov. Com.
Petr. v. 299 ; ii.207. Ferguson's Lectures, by Brewster, 2 vols. 1806. Buchanan,
Essay on the Teeth of Wheels, 1808. Trans, of the Soc. of Civil Engineers, ii. 89.

LECTURE XVI.

ON THE UNION OF FLEXIBLE FIBRES.

THE strength of cordage, and of other substances which are employed in
the communication of motion where flexibility is required, as well as the
utility of other flexible materials which serve for furniture or for clothing,
depends principally upon the lateral adhesion produced by twisting, or by
the intermixture of fibres. The union of flexible fibres, therefore, being
frequently subservient to the communication of motion, and the machinery
usually employed for producing it, belonging immediately to the subject
of the modification of motion, we may with propriety consider at'present,

ON THE UNION OF FLEXIBLE FIBRES. 139

as far as our plan will allow us, those important branches of the mechanical
arts, of which the object is to effect a union of this kind.

When a chain is made of wire, each link is separately bent, and remains
united with the neighbouring links in virtue of its rigidity : but the fibres
of vegetable and of animal substances must be united by other means. For
this purpose we have recourse to the force of friction, or rather of lateral
adhesion, and the fibres are so disposed, that besides the mutual pressure
which their own elasticity causes them to exert, any additional force
applied in the direction of the length of the aggregate, tends to bring the
parts into closer contact, and to augment the adhesion, in the same manner
as we have already seen that a wedge and a screw may be retained in their
situations. The simple art of tying a knot, and the more complicated pro-
cesses of spinning, ropemaking, weaving, and felting, derive their utility
from this principle. : .,.

When a line is coiled round a cylinder, for instance, in letting down a
weight by means of a rope which slides on a post, or on such a grooved
cylinder as is sometimes employed to enable a person to lower himself from
a window in cases of fire, the pressure on the whole circumference is to the
weight, as twice the circumference to the diameter ; supposing, for ex-
ample, that the friction of rope on metal were one tenth of the pressure,
then a single coil of rope round a cylinder of metal would support about
two thirds of the weight ; or if the weights acting on the different ends are
different, the adhesion may be a little greater or less than in this proportion,
according to the manner in which the rope is applied. If such a rope made
two or three coils, it would be impossible to apply a force sufficient to cause
it to slide in the grooves.

From considering the effect of a force which is counteracted by other
forces acting obliquely, we may understand both the effect of twisting, in
binding the parts of a rope together, and its inconvenience, in causing the
strength of the fibres to act with a mechanical disadvantage. The greater
the obliquity of the fibres, the greater will be their adhesion, but the greater
also will be their immediate tension, in consequence of the action of a given
force in the direction of the rope : so that after employing as much ob-
liquity and as much tension, as is sufficient to connect the fibres firmly in
all cases of relaxation and of flexure, and to prevent in some measure the
penetration of moisture, all that is superfluously added tends to overpower
the primitive cohesion of the fibres in the direction of their length.*

The mechanism of simple spinning is easily understood ; care is taken,
where the hand is employed, to intermix the fibres sufficiently, and to
engage their extremities as much as possible in the centre ; for it is obvious
that if any fibre were wholly external to the rest, it could not be retained
in the yarn ; in general, however, the materials are previously in such a
state of intermixture as to render this precaution unnecessary. Where
we have a number of single continuous fibres, as- in reeled silk, they are
sufficiently connected by twisting, and we have no need of spinning. In
bc4h cases such machinery has been invented for performing the necessary
operations, as is both honourable and lucrative to the British nation.
* See Hooke's Experiments on Cordage Birch., ii. 393.

140 LECTURE XVI.

A single thread or yarn, consisting of fibres twisted together, has a ten-
dency to untwist itself ; the external parts are the most strained in the
operation and at first shorten the thread, until the internal parts have no
longer room for spreading out laterally, as they must necessarily do when
their length is diminished ; the elasticity of all the parts, therefore, resists,
and tends to restore the thread to its natural state. But if two such
threads are retained in contact at a given point of the circumference of
each, this point is rendered stationary by the opposition of the equal forces
acting in contrary directions, and becomes the centre, round which both
threads are carried by the remaining forces, so that they continue to twist
round each other till the new combination causes a tension capable of
counterbalancing the remaining tension of the original threads. Three,
four, or more threads may be united nearly in the same manner : a strand
consists of a considerable number of yarns thus twisted together, generally
from sixteen to twenty five, a hawser of three strands, a shroud of four,
and a cable of three hawsers or shrouds. Shroud laid cordage has the dis-
advantage of being hollow in the centre, or of requiring a greater change of
form in the strands to fill up the vacuity, and in undergoing this change,
the cordage stretches, and is unequally strained. The relative position and
the comparative tension of all the fibres in these complicated combinations
are not very easily determined by calculation ; but it is found by expe-
rience to be most advantageous to the strength of the ropes to twist the
strands, when they are to be compounded, in such a direction as to untwist
the yarns of which they are formed ; that is, to increase the twist of the
strands themselves : and probably the greatest strength is obtained when
the ultimate obliquity of the constituent fibres is the least, and the most
equable. This advantage is obtained in a considerable degree by Mr. Hud-
dart's* method of adjusting the length of the strand to its position in the
rope, and his registered cordage appears to derive a decided superiority
from this arrangement of the strands. A very strong rope may also be
made by twisting five or six strands round a seventh as an axis ; the central
strand, or heart, is found after much use to be chafed to oakum ; it should
be more twisted than the rest, in order to allow it to extend a little ; such
ropes are, however, unfit for running rigging, or for any use in which they
are liable to be frequently bent.

Ropes are most commonly made of hemp, but various other vegetables
are occasionally employed ; the Chinese even use woody fibres, and the
barks of trees furnish cordage to other nations ; we have indeed in this
country an example of the use of the bark of the lime tree, which is
employed for garden matting. The finest hemp is imported from Riga and
St. Petersburg. The male and female flowers of hemp are on different
plants ; the male plants are soonest ripe, and require to be first pulled.
They are prepared for dressing by being exposed to the air, and the fibrous
part is separated from the dry pulp by beating and hackling. In spinning
the yarn, the hemp is fastened round the waist ; the wheel is turned by an
assistant, and the spinner, walking backwards, draws out the fibres with

* Huddart's Patent registered Cordage, Rep. of Arts, xii. 80. Remarks on do.
4to. 1800.

ON THE UNION OF FLEXIBLE FIBRES. 141

his hands. When one length of the walk has been spun, it is immediately
reeled, to prevent its untwisting. The machines employed in continuing
the process of ropemaking are of simple construction, hut hoth skill and
attention are required in applying them so as to produce an equable
texture in every part of the rope. The tendency of two strands to twist, in
consequence of the tension arising from the original twist of the yarns, is
not sufficient to procure an equilibrium, because of the friction and rigidity
to be overcome ; hence it is necessary to employ force in order to assist this
tendency, and the strands or ropes afterwards retain spontaneously the
form which has thus been given them : the largest ropes even require
external force in order to make them twist at all.

The constituent ropes of a common cable, when separate, are stronger
than the cable in the proportion of about 4 to 3 ; and a rope worked up
from yarns 180 yards in length to 135 yards, has been found to be stronger
than when reduced to 120 yards, in the ratio of 6 to 5. The difference
is owing partly to the obliquity of the fibres, and partly to the unequal
tension produced by twisting. Mr. Huddart's ropes of 100 yarns lose
but about one eighth of the whole strength of the yarns ; and his experi-
ments appear to show that similar ropes made in the common manner
retain only one half of their original strength. The tarring of ropes,
although sometimes necessary for their preservation from decay, is found
to lessen their strength, probably because it produces partial adhesions
between some of the fibres, which cause them to be disproportionally
strained. A rope is also said to be weaker when wet than when dry,
perhaps because the water enables the fibres to slide more readily on
each other, or because the presence of water is in general favourable to
separation of any kind. A good hempen rope will support, without
danger, one fifth as many tons as the square of its circumference contains
inches.*

Flax is weaker than hemp, but not less extensively useful. Its growth
considerably exhausts the strength of the soil which produces it ; its
cultivation is encouraged in this country by a bounty from government,
and a large quantity is also imported from the north of Europe. The
plant, while green, is laid in water for ten days, and undergoes a chemical
change, which softens the pulpy part, without injuring the strength of the
fibres, and renders it more easy, when it has been dried and exposed to
the air for a fortnight, to separate the two substances in the process of
dressing it. This is performed by beating it with the edge of a flat piece
of wood, the stroke being oblique, and nearly in the direction of the fibres,
and afterwards combing it, in order to reduce the fibres into regular order,
and to prepare them for spinning. The refuse, consisting of the shorter
fibres, is tow.

Cotton is a fine fibrous substance, that envelopes the seeds of a plant.
The best is brought from the isle of Bourbon ; but by far the greatest
quantity from the West Indies, although the Turkish dominions as well as
the East Indies furnish us with a considerable supply. It is usually
white, but there is a yellow kind, which is used for nankeens. It is
* See Duhamel, Traite de la Corderie Perfectionne"e, 4to, Paris.

142 LECTURE XVI.

separated from the seeds by means of rollers, between which it passes, and
leaves the seeds behind. It is then beaten, on a flake, or a stool covered
with a texture of cord. Next, it is carded, either by hand, the fibres
being drawn into regular order by cards, that is, by brushes of fine pointed
wire; or, more commonly, by machinery, the cards being disposed in
cylinders which revolve nearly in contact with each other. The drawing
or roving machine then draws it into long flakes, a state preparatory to
its being spun by Sir Richard Arkwright's machines or jennies, which
form at once forty threads by the labour of one person.

The silkworm is bred in the greatest abundance in Italy and in Asia ;
it has lately been introduced very successfully into the British possessions
in the East Indies. The principal food of the caterpillar is the white
mulberry tree, which is too delicate to thrive well in northern climates : in
Italy the trees are planted in beds, like willows, and the foliage is cut as
it is wanted. The room in which the worms are fed, is kept at the tem-
perature of 80 degrees of Fahrenheit. The eggs of a former year are
hatched either by animal heat, or by that of the sun ; at the age of six
weeks, the caterpillars begin to spin, first a light external texture, which is
carded and spun for coarse silk, and afterwards a compact oval pod or
cocoon, of one continued thread. The threads of several cocoons are
reeled off at the same time : for this purpose they are generally put into
warm water, which kills the chrysalis : but when it is preserved, it soon
turns to a moth, which lives but a few -days, taking no food, and dies after
producing eggs for the next season.

The silk is either yellow or white, but the white is an accidental variety
only. By repeated washings, the yellow silk is bleached, and that which
is originally white, acquires a more perfect whiteness. Soap is also used
for removing a gummy substance that accompanies the silk of the cocoons.

Wool is distinguished into two principal varieties, long and short wool.
The longest is from Lincolnshire ; it is combed, by means of instruments
furnished with a double row of long and sharp teeth of iron or steel ; it is
repeatedly drawn from one comb to the other, heat being used in the pro-
cess, and also a little oil. The fleeces of long wool are generally heavier
than those of short wool, but less valuable on account of their coarseness ;
they are used for worsteds, and for cloths in which the separate threads
remain visible, as stuffs, shalloons, serges, and tammies. Short wool, on
the contrary, is carded, and is used for cloths in which the individual
threads are concealed by the projecting fibres.

The principal use of thread and yarn, when spun, is for the purpose of
weaving. The same force of lateral adhesion that retains the twisted
fibres of each thread in their situations, is here also employed in giving
firmness to the cloth ; and this adhesion is generally increased by the
action of any external force, tending to strain the whole texture.

The first step in weaving is to form a warp, which consists of threads
placed side by side, continued through the length of the piece, and suffi-
cient in number to constitute its breadth. This being wound on a beam
or roller, in the loom, the threads are drawn through a harness, consisting
of loops formed by twine fixed to bars or frames, which elevates and

ON THE UNION OF FLEXIBLE FIBRES. 143

depresses the threads in succession by means of treadles, moved by the
feet, in an order which is different, according to the different nature of the
intended work ; the cross thread or woof, being thrown between them at
each alternation, by means of a shuttle, and forced into its place by a
batten or comb made of wire or reeds, while the piece, in proportion as it
is completed, is rolled upon a second beam opposite to the first.

Crape is composed of threads which are so strongly twisted, as to have
a disposition to curl, and in weaving it, moisture is sometimes employed,
in order to obviate this tendency during the process. Woollen cloth, when
woven, is rendered stronger and more compact by means of the fulling
mill, in which it is beaten by heavy hammers of wood, at the same time
that fullers' earth, or alcaline substances of animal origin, are applied in
order to cleanse it. In this operation both its length and breadth are
diminished, and it is reduced to a texture approaching to that of felt.
The reason of the contraction is probably this, that all the fibres are bent
by the operation of the hammer, but not all equally, and those which have
been the most bent are prevented by their adhesion to the neighbouring
fibres from returning to their original length. After fulling, the cloth is
roughened by means of teasels, which are cultivated for the purpose ; and
the most projecting fibres are cut away by the operation of shearing.

The lateral adhesion of fibres of various kinds gives strength also to
felted substances, assisted, as some assert, by minute barbs, with which
the fibres of furs are said to be furnished. The whole strength is, how-
ever, much inferior to that of cloth ; partly because the fibres are in
general much shorter, and partly because their arrangement is less accu-
rately adjusted.*

The materials commonly used for felting are the furs of rabbits and
beavers, mixed with each other, and with sheep's wool, in various pro-
portions, according to the quality required. A very fine fur has lately
been discovered on the skin of a species of seal, mixed with its hair, and it
has been employed not only for felting, but also for spinning and weaving
into a cloth resembling the shawls of the East Indies. The fur of the
rabbit is also mixed with a coarser hair, which is separated from it, by
being first pulled off from the skins, with a sharper knife. The materials
to be felted are intimately mixed by the operation of bowing, which de-
pends on the vibrations of an elastic string; the rapid alternations of
its motion being peculiarly well adapted to remove all irregular knots and
adhesions among the fibres, and to dispose them in a very light and
uniform arrangement. This texture, when pressed under cloths and
leather, readily unites into a mass of some firmness ; this mass is dipped
into a liquor containing a little sulfuric acid, and when intended for a hat,
is moulded into a large conical figure, which is reduced in its dimensions
by working it with the hands, and is formed into a flat surface with
several concentric folds, which are still more compacted in order to make
the brim and the circular part of the crown, and forced on a block which
selves as a mould for the cylindrical part. The black dye is composed of
* On Hatmaking, see Nich. Jour. 4to, i. 67 ; iii. 22, 73.

144 LECTURE XVII.

logwood, sulfate of iron, and a little acetate of copper, or verdigris ; and
the stiffening is a thin glue.

The texture of paper is scarcely different from that of felt, except that
its fibres are less visible to the naked eye. To make white paper, linen rags
are ground with warm water in a mill, into a paste of the consistence of
cream : a portion of the paste is taken up in a wire sieve, which is passed
obliquely through it, and this, being a little shaken, subsides into a sheet,
which is turned out on a piece of flannel ; a number of sheets being thus
formed, they are then pressed, first writh the interposition of flannel, and
afterwards alone, while they are still moist. For thick paper, two or more
sheets are laid on each other before the first pressing. To fill up the pores
of the paper, and to increase its strength, a size is employed, which is
generally made by boiling shreds of parchment or untanned leather.
Sometimes the size is added after printing on the paper, but this is only
done in works of inferior elegance, and in this country not at all.

Such are the principal cases of the union of flexible fibres, for the
different purposes of strength or of convenience. Their importance is such
that they might be esteemed worthy of a more detailed consideration ; but
we are not likely to make any material improvements in these departments
of mechanical art by the application of theoretical refinements.*

LECTURE XVII.

ON TIMEKEEPERS.

THE measurement of time by clocks and watches is a very important
and interesting department of practical mechanics. The subject is inti-
mately connected with the consideration of astronomical instruments, but
it is not essentially dependent on astronomical principles.

Time is measured by motion ; but in order that motion may be a true
measure of time, it must be equable. Now a motion perfectly free and
undisturbed, and consequently uniform, is rendered unattainable by
the resistances inseparable from the actual constitution of material sub-
stances. It becomes therefore necessary to inquire for some mode of
approximating to such a motion. Astronomical determinations of time,
which are the most accurate, can only be made under particular circum-
stances, and even then they assist us but little in dividing time into small
portions.

The first timekeepers somewhat resembled the hour glasses which are still
occasionally employed ; they measured the escape of a certain quantity, not
of sand but of water, through a small aperture. In these clepsydrae, "it
* For additional authorities, see Lect. XIX.

ON TIMEKEEPERS. 145

appears from Vitruvius's account that wheelwork was employed,* and the
hour was shown on a graduated scale ; the graduations were also probably
so adjusted as to correct the error arising from the inequality of the velocity
occasioned by the variation of the height of the water in the reservoir. This
inconvenience was however sometimes wholly avoided by means of a con-
stant stream, which kept the vessel full, or still more elegantly, by the
siphon of Hero, which was a bent tube supported by a float, so that its
lower orifice, at which the water was discharged, was always at a certain
distance below the surface. Dr. Hooke proposed to keep the reservoir full,
by means of a semicylindrical counterpoise, t so that the time might be
determined either from the measure or weight of the quantity of water
discharged, or from the position of the counterpoise. Various other modes
might also be devised for making cheap and simple timekeepers on similar
principles, dependent on the motion of various liquids or elastic fluids ; but
great accuracy could scarcely be expected from them. A candle sometimes
serves as a coarse measure of time ; and by burning a thread which passes
through it, it may easily be made to answer the purpose of an alarm.

Clocks and watches are machines in which wheelwork is employed for
the measurement of time, being driven by a weight or by a spring, and
regulated by a pendulum or a balance. Watches differ from clocks, in
being portable, and this condition excludes the pendulum and the weight
from their construction.

It is conjectured that the Saracens had clocks which were moved by
weights, as early as the eleventh century. J Trithemius mentions an orrery,
moved by a weight, and keeping time, which was sent, in 1232, by the
Sultan of Egypt, as a present to the Emperor Frederick II. Wallingf ord ?
in 1326, had made a clock which was regulated by a fly.§ The use of such
a fly in equalising motion depends on the resistance of the air, which
increases rapidly when the velocity is increased, and therefore prevents any
great inequality in the motion as long as the moving power varies but
little ; and if the action of the weight were transmitted with perfect regu-
larity by the wheels, and the specific gravity of the air remained unaltered
by pressure or by temperature, a fly clock might be a perfect machine, the
weight being always exactly counterbalanced by the resistance of the air,
attending a certain velocity of the fly ; and it might even be possible to
regulate the inequalities of the action of the weight, by causing the fly to
open and shut or to turn on an axis, by means of a spring, according to the
magnitude of the resistance. The unequal density of the air would how
ever still remain uncompensated ; and in this respect a liquid would be a
better medium than an elastic fluid. For experiments which are but of
short duration and which require great precision, a chronometer regulated
by a simple fly is still a useful instrument. Mr. Whitehurst's || apparatus
for measuring the time occupied in the descent of heavy bodies, is governed

* See Derham, The Artificial Clockmaker, 1696, p. 85.
• f Lampas, 4to, 1677, p. 42.

J Beckmann, History of Inventions, 4 vols. translated by Johnstone, vol. i.
§ Epitome Conrardi Gesneri, p. 604.
|| Ph. Tr. 1794, p. 2.

146 LECTURE XVII.

by a fly ; the index is stopped by the machinery, and points out the time
elapsed without an error of the hundredth part of a second.

The alternate motion of a balance, thrown backwards and forwards by
the successive actions of a wheel impelling its pallets, is also capable of
producing a degree of uniformity in the motion of the wheel ; for the force
operating on the pallet is consumed in destroying a velocity in one direc-
tion, and in generating a velocity in the contrary direction ; and the space
in which it acts being nearly the same in all cases, the velocity generated
will also be nearly the same at all times, as long as the force remains the
same. The addition of a balance to a clock was made soon after the year
1400, for Arnault* who died in 1465, describes a planisphere constructed
by his master De Fondeur, which had a balance with a scapement like that
of a common watch, but without a spring. Such a balance vibrates much
more slowly than a balance provided with a spring ; if the balance spring
of a common watch be removed, the hands will pass over the space of about
twenty eight minutes in an hour.

It is said that before the pendulum was used, a balance wheel was some-
times suspended in a horizontal position by a thread passing through its
axis, which coiled round it and caused it to rise and fall as it oscillated
backwards and forwards. This mode of regulation differed but little in
principle from the modern pendulums, but it was more complicated and
less accurate. Huygens, in somewhat later times, constructed a clock with
a revolving weight, which rose higher, and increased the resistance, when-
ever an augmentation of the force increased the velocity ; and he caused
the thread which supported the weight, to bend round a curve of such a
form as to preserve the equality of the revolutions.

A chronometer may be constructed on this principle for measuring small
portions of time which appears to be capable of greater accuracy than
Mr. Whitehurst's apparatus, and by means of which an interval of a
thousandth part of a second may possibly be rendered sensible. If two
revolving pendulums be connected with a vertical axis, in such a manner as
to move two weights backwards and forwards accordingly as they fly off
to a greater or smaller distance, the weights sliding, during their revolution,
on a fixed surface, a small increase of velocity will considerably increase
the distance of the weights from the axis, and consequently the effect of
their friction, so that the machine will be immediately retarded, and
its motion may thus be made extremely regular. It may be turned by
a string coiled round the upper part, and this string may serve as a
support to a barrel, sliding on a square part of the axis, which will conse-
quently descend as it revolves. Its surface, being smooth, may be covered
either with paper or with wax, and a pencil or a point of metal may be
pressed against it by a fine spring, so as to describe always a spiral line on
the barrel, except when the spring is forced a little on one side by touching
it slightly, either with the hand, or by means of any body of which the
motion is to be examined, whether it be a falling weight, a vibrating cord
or rod, or any other moving substance. In this manner, supposing a bar-

* Venturi, Essai sur les Ouvrages de L. da Vinci, p. 28, quoting MS. No. 7295, in
the National Library of Paris.

ON TIMEKEEPERS. 147

rel a foot in circumference to revolve in two seconds, each hundredth of an
inch would correspond to the six hundredth part of a second ; and the scale
might be still further enlarged if it were necessary. (Plate XV. Fig. 198.)

By means of this instrument we may measure, without difficulty, the
frequency of the vibrations of sounding bodies, by connecting them with a
point, which will describe an undulated path on the roller. These vi-
brations may also serve in a very simple manner for the measurement of
the minutest intervals of time ; for if a body, of which the vibrations are
of a certain degree of frequency, be caused to vibrate during the revolution
of an axis, and to mark its vibrations on a roller, the traces will serve as a
correct index of the time occupied by any part of a revolution, and the
motion of any other body may be very accurately compared with the
number of alternations marked in the same time, by the vibrating body.
For many purposes, the machine, if heavy enough, might be turned by a
handle only, care being taken to keep the balls in a proper position, and it
would be convenient to have the descent of the barrel regulated by the
action of a screw, and capable of being suspended at pleasure.

But for the general purposes of timekeepers, all other inventions have
been almost universally superseded by the pendulum and the balance
spring, or pendulum spring. About the year 1000, Ibn Junis, and
the other Arabian astronomers were in the habit of measuring time,
during their observations, by the vibrations of pendulums; but they
never connected them with machinery. The equality of the times
occupied by these vibrations, whether larger or smaller, was known to
Galileo* in 1600, and some time before 1633, he proposed that they
should be applied to the regulation of clocks. But Sanctorius, in his
commentary on Avicenna, describes an instrument to which he had him-
self applied the pendulum in 1612. Huygens made the same application
only in 1658, which is the date of his work on the subject. In the same
year, Hooke applied a spring to the balance of a watch ; t and soon after,
he conceived the idea of improving timekeepers sufficiently for ascertaining
the longitude at sea, J but he was interrupted in the pursuit of his plan.
Hooke was also probably the first that employed for a clock a heavy
weight vibrating in a small arc ; an arrangement from which the peculiar
advantages of a pendulum are principally derived.

The objects which require the greatest attention in the construction of
timekeepers, are these ; to preserve the moving power or sustaining force
as equable as possible, to apply this force to the pendulum or balance in
the most eligible manner, and to employ a pendulum or balance of which
the vibrations are in their nature as nearly isochronous as possible. In
clocks, the sustaining force, being generally derived from a weight, is al-
ready sufficiently equable, provided that care be taken that the line by
which it is suspended may be of equal thickness throughout, and may act

* Mem. dell' Acad. del Cimento. The date is there stated as 1583.

t Cassini laid claim to this invention, in behalf of Huygens, but Hooke proves
that he had not only conceived it but sent it to Huygens fifteen years before, who
wrote a letter against it as impracticable. Philosoph. Exp. &c. by Hooke, p. 388.
The main merit of the application of the pendulum to clocks probably belongs to
Huygens. + Ibid. p. 4.

L2

148 LECTURE XVII.

on a perfect cylinder. But in some clocks, and in all watches, the moving
power is a spring. One of the first clock springs is said to have been an
old sword blade ; a clock with such a spring was lately preserved at Brus-
sels : the spring which is at present used, is a thin elastic plate of steel,
coiled into a spiral form. Every spring exerts the more force as it is more
bent ; in order to correct this inequality, the chain or cord by which it acts
on the work is wound on a spiral fusee ; so that in proportion as the force
is lessened, it is applied to a larger cylinder or a longer lever. The gene-
ral outline of the fusee must be nearly such that its thickness at any part
may diminish in the same proportion as it becomes more distant from the
point at which the force would cease altogether, the curve being that which
is denominated a hyperbola ; but the workmen have in general no other
rule than an habitual estimation.* (Plate XV. Fig. 199.)

Notwithstanding all possible precautions in the immediate application of
the weight or spring, the irregular action of the teeth of the wheels, the in-
creasing tenacity of the oil usually employed, and other accidental dis-
turbances, make it still desirable to procure a further equalisation of the
force ; which is sometimes obtained in clocks by raising the loaded arm of
a lever to a given height whence it may descend ; and in watches, by
bending a spring into a given position from which it may return, so as to
limit with great precision the propelling force employed in each vibration.
The necessity of applying oil is sometimes in great measure removed by
jewelling the holes in which the axes or verges run; a perforation being
made in a plate of ruby, and a diamond applied upon this, in contact with
the end of the axis ; the hardness and high polish of these stones tending
very considerably to diminish the friction.

There are also different methods of continuing the action of the force
while the clock or watch is wound up : a spring is interposed between the
fusee and the wheel impelled by it, a little inferior in force to the original
weight or spring, so as to remain always bent, until, when the pressure of
the main spring is removed, it begins to act upon a fixed point on one side,
and upon the wheel of the fusee on the other, so that it propels the work for
a short time with a force nearly equal to that of the main spring. Some-
times also the spring is wound up by causing a small wheel to revolve
round the centre of the fusee, having its teeth engaged on one side in those
of a wheel which makes a part of the fusee, and on the other side with the
internal teeth of a hoop connected with the work, so that the same pressure
which winds up the spring tends also to turn the hoop round, and to con-
tinue the motion. (Plate XVI. Fig. 200.)

The scapement, by which the sustaining force is communicated to the
pendulum or balance, demands a greater exertion of skill and accuracy
than any other part of a timekeeper. Sometimes the alternate motion of
the pendulum has been produced by the action of a crank, but this con-
struction subjects it too much to the irregularities of the wheel work, and
is liable to several other objections. A crank cannot properly be called a
scapement, for according to the etymology of the term, the pendulum must

* Lahire on the Figure of Fusees, Hist, et Mem. de Paris, ix. 102. Varignon
on do. ibid. 1702, p. 192, H. 122.

ON TIMEKEEPERS. 149

escape for a time from the action of the wheelwork, and in general, the
more independent its motion is rendered the better is the effect of the
machine. The simplest forms in common use are the crutch scapement
for a clock, and the pallets with a vertical wheel for a watch ; the dead
beat scapement, and the cylinder with a horizontal wheel, are improve-
ments on these ; and the detached scapement is a still further refinement.

The crutch scapement, called by the French the anchor scapement, is
an arch in the plane of the scape wheel, and parallel to that in which the
pendulum vibrates, supporting at each extremity a pallet, of which the
face is a plane, and which is impelled in its turn by the teeth of the scape
wheel. The faces are so inclined that the pallets are alternately forced, by
the action of the teeth, to retire from the centre of the wheel : and great
care is taken in making the teeth exactly at equal distances, so that they
may fall regularly on the pallet, immediately after the disengagement of
the teeth on the other side from the opposite pallet. (Plate XVI. Fig. 201.)

In the common watch, the axis of the balance is parallel to the plane of
the scape wheel, which is a contrate or crown wheel, and the flat pallets
are fixed on the axis of the balance at the opposite parts of the circum-
ference of the scape wheel. (Plate XVI. Fig. 202.)

In both these cases the impulse given to one pallet carries the opposite
pallet with some force against the approaching tooth, and drives the wheel
a little backwards with a visible recoil. Here the sustaining power, being
applied principally at the extremities of the vibrations, disturbs their iso-
chronism or the equality of the times in which they are performed, by
partially increasing the force. We may recollect that, in order that all
vibrations, of whatever magnitude may be performed in equal times, the
force must be exactly proportional to the distance from a given point,
consequently if an additional force be applied near the extremities of the
vibration only, the longer vibrations will occupy less time than the shorter ;
and we may observe that, by adding to the force of the spring of a common
watch, with the key, we may accelerate its motion, at the same time that
the angular magnitude of the vibration is increased. The motion of the
balance also, being slowest at the extremities of its vibration where the
sustaining force is applied, is more affected by the inequalities of this force
than if it were subjected to its action through an equal space in the middle
of the vibration. Yet a good clock on this construction may keep time
without an error of the ten thousandth part of the whole, and a watch
within a two thousandth. In the common watch scapement there is little
friction, for the force acts almost perpendicularly on the pallet ; it appears
to have been the oldest scapement, and was employed before the applica-
tion of springs to balances : it requires a considerable extent of motion in
the balance, and cannot therefore well be applied to clocks with such pen-
dulums as vibrate in small arcs. The crutch scapement, on the contrary,
cannot be applied immediately to a vibration in a very large arc ; but by
the, interposition of a lever with a roller, or of a part of a wheel with a
pinion, it may be adapted to the balance of a watch ; and some watches
thus constructed by Emery, Letherland and others, appear to have suc-
ceeded very well.

150 LECTURE XVII.

To avoid the inconveniences of the recoiling scapements, Mr. Graham
invented or introduced the dead beat for the clock, and the cylinder for
the watch.* In both of these, the tooth of the scape wheel rests, during
the greater part of the vibration, on a cylindrical surface, and acts on the
inclined plane for a short time only in the middle of each vibration ; so
that a change of the sustaining power scarcely produces a sensible derange-
ment of the isochronism ; for which ever way we turn the key of a hori-
zontal watch, as long as it continues to go, the frequency of its vibrations
is scarcely affected. A good horizontal watch will keep time within about
a ten thousandth part, especially if a little oil be frequently applied to it,
or if the cylinder be made of a ruby : and the timekeeper in the obser-
vatory at Greenwich with a dead beat scapement, made by Graham, varies
from true time only two parts in a million. (Plate XVI. Fig. 203, 204.)

Still, however, the friction of the teeth of the scape wheel on the cylin-
der or pallet, and the tenacity of the oil, where it is employed, may in-
terfere in a slight degree with the time of vibration, especially by the
irregularities to which they are liable. Since friction is always increased
by an increase of pressure, the effect of any addition to the sustaining
force must tend in some degree to retard the vibrations, even if the friction
be somewhat less increased than the force propelling the balance. In order
to obviate this retardation, the surfaces on which the teeth rest, have some-
times been so formed as to create a slight recoil ; but this construction does
not appear to have been very successful in practice. The friction may,
however, be considerably diminished by the duplex scapement, apparently
so called from the double series of teeth employed. The teeth of the more
prominent series are detained on a cylinder so small as to be unfit for re-
ceiving an impulse from them, the balance is therefore impelled by the other
series of teeth, acting on a pallet at a greater distance from its axis. The
French have sometimes employed a construction somewhat similar, which
they call the comma scapement, the teeth first resting on a small arch of
repose, and then impelling the curved surface of a pallet extending to a
considerable distance beyond it. In both these cases the single pallet,
which is impelled by a tooth of a simple form, requires less labour in the
execution than a number of larger teeth, each of which is to be finished
with great accuracy : but watches on these constructions, especially those
with the comma scapement, are too liable to be stopped by any sudden
motion, although the duplex scapement begins to be frequently employed
for pocket timekeepers. (Plate XVI. Fig. 205.)

Mr. Harrison avoided all friction on the pallet, by connecting it with
the pendulum by means of a slender spring, so flexible as to follow the
motion of the scape wheel to a sufficient extent without sliding on its
teeth. But the construction which is most usually employed where the
greatest accuracy is required, is the detached scapement: in which the
teeth of the scape wheel always rest on a detent, excepting a short interval
when it is unlocked in order to impel the pallets. Mr. Mudget employed a
detached scapement actuated by a subsidiary spring, of which the force is

* See Nich. Journal, 4to, ii. 49.

f Mudge on a Scapement, 1763. On a Timekeeper, 4to, 1799.

ON TIMEKEEPERS. 151

scarcely liable to any variation ; the detent being unlocked by the motion
of the balance. Mr. Haley* has refined still further on this construction,
by causing the subsidiary spring to unlock the wheel in its return, so that
the balance is relieved from this action, which may sometimes produce a
slight irregularity. These constructions are, however, much too delicate
for common pocket watches. In a clock, Mr. Gumming has employed a
detached scapement, in which a lever is raised to a certain height by each
tooth of the scape wheel, and acts immediately on the pendulum in its
descent in the middle of the vibration. The scape wheel is unlocked by
the pendulum during its ascent, and a variation of the pressure may, there-
fore, produce a very slight inequality in the motion of the pendulum. Mr.
Nicholson has attempted to remove this cause of error, by a construction
in which the scape wheel only assists the pendulum in raising the lever ;
but it depends on the degree of force applied, to determine what part of
the weight the scape wheel shall sustain ; this scapement cannot, therefore,
by any means be considered as detached. It is, however, easy to remove
the defect of Mr. Gumming' s scapement, if it can be called a defect, by a
method similar to that which Mr. Haley has applied to watches ; each
tooth of the wheel being unlocked by the descent of the lever on the op-
posite side, at the moment that it ceases to act on the pendulum, and
remaining inactive until the pendulum meets it. (Plate XVI. Fig. 206, 207.)

The detents of the scapements of Mudge and Gumming are parts of the
pallet, but in the timekeepers now commonly made by Arnold, Earnshaw,
and others, the tooth is detained by a pallet or pin projecting from a lever,
the point of which is forced back by the balance, at the moment that the
pallet presents itself to another of the teeth. Mr. Arnold employs an
epicycloidal tooth, acting on a single point of the pallet ;t Mr. Earnshaw
makes a flat surface of the tooth first act on the point of the pallet, and
then the point of the tooth on a flat surface of the pallet.^ In other
respects there is little difference in these scapements ; and both the artists
have been judged worthy of a public reward for their success. (Plate
XVI. Fig. 208, 209.)

The last of the three principal objects, which require the attention of the
watchmaker, is to employ a pendulum or balance of which the vibrations
are in their nature perfectly isochronous. For this purpose the weight of
the pendulum ought to move in a cycloidal arc, but the difficulty of pro-
ducing such a motion in practice is much greater than the advantage derived
from it, and a circular vibration, confined to a small arc, is sufficiently
isochronous for all practical purposes. The error of such a vibration is
nearly proportional to the square of the arc described by the pendulum,
and amounts to a second and a half in a day of 24 hours, for a single degree
on each side the point of rest ; so that a pendulum keeping true time in
an arc of three degrees, would gain 13i seconds if the arc were very much

* Haley's Patent Timekeeper, Repertory of Arts, vi. 145.

• f Explanation of Mr. Arnold's Timekeeper. Questions proposed by the Board
of Long, relative to the same, 4to, 1804-5.

J Explanation of Mr. Eamshaw's Timekeeper. Questions proposed by the
Board of Long, relative to the same, 4 to, 1804-5.

152 LECTURE XVII.

contracted or made cycloidal, and would lose 10£ seconds by having the
vibration extended to an arc of four degrees. In order to avoid the friction
which would be occasioned by the motion of the pendulum on an axis, it is
usually suspended by a flexible spring which is wholly free from friction.
The elasticity of this spring adds a minute force to the power of gravitation
which acts on the pendulum, and this force must be considered, when the
length of a simple pendulum is compared with the frequency of its vibra-
tions. It does not, however, interfere with the equality of the vibrations
among each other ; for in all springs, Dr. Hooke's general law,* that the
force increases as the degree of flexure, is found for moderate oscillations
to be perfectly accurate ; such a force, therefore, accelerates the larger and
the smaller vibrations precisely in the same degree. But in balances, it is
desirable to have the velocity and the extent of the vibration as great as
possible, in order that the motion may be the less influenced by the ine-
qualities of the sustaining power ; and in large excursions, Dr. Hooke's
law is not so precisely true ; there must also necessarily be some inaccuracy
from the loss of a certain portion of the force in generating the momentum
of the spring itself, which, when the form is spiral, introduces great intricacy
into the calculation of the properties of the vibration. Yet it has been
found by experiment that a certain kngth may be determined for almost
every spring, which will afford vibrations either perfectly or veiy nearly
isochronous. In order that the weight or inertia of the spring may inter-
fere the less with the regularity of its motion, it is sometimes tapered and
made thinner at the extremity: it is now also usual in the best watches to
employ a spring coiled into a cylindrical form, like that of the spring of a
bell, of which the motion appears to be somewhat more regular than that
of a flat spiral. This was indeed the original construction, but was pro-
bably laid aside on account of the space which it required. The balance
springs are made of the finest steel, and the best are manufactured in this
country, although the French are said to have the art of making their main
springs of a better temper than ours. Sometimes the balance spring- is
made of an alloy of gold and copper ; these springs are very elastic, but
they are too liable to break. Mr. Earnshaw observes that the strength of
a spring always diminishes a little as it wears ; and endeavours to derive
a compensation for this diminution of strength, by employing a spring of
such a form, that the vibrations in small arcs may be a little more frequent
than in larger ones, in order that when the presence of dust and the
tenacity of the oil contract the extent of the vibrations, this contraction
may tend to produce an acceleration which compensates for the diminished
force of the spring. But it is perhaps more eligible to make every com-
pensation, as far as possible, independent of circumstances foreign to the
cause of the error. The strength of the spring is found to be less impaired
by use when it is hardened than when the steel is softer. It sometimes
happens, that from a sudden motion, or from some other accidental circum-
stance, the balance of a timekeeper may be thrown beyond the point at
which the pallets are impelled by the scape wheels, and the whole motioli

* Hooke, De Potentia Restitutiva, 4to, Lond. 1678. This law was published by
him about the year 1660, in the form of an anagram.

ON TIMEKEEPERS. 153

may from this cause be interrupted. To prevent this accident, a small bar
or pin is usually fixed on the balance spring, which is carried outwards
when the vibration begins to be extended too far, and stops the further
progress of the balance by intercepting a pin which projects from it. This
arrangement is called banking the balance.

We have already seen that the squares of the times of vibration of two
pendulums are proportional to their lengths ; so that if we add to a pen-
dulum one hundredth part of its length, we increase the time of its vibration
very nearly one two hundredth. But since all bodies are expanded by heat,
the variable temperature of the atmosphere must necessarily produce
changes of this kind in the motions of pendulums, and it may be observed
that a clock goes somewhat more slowly in summer than in winter. The
same expansion has a similar effect in the motion of a balance, and the
increase of temperature produces also a diminution of the elastic force of
the spring itself. There is, however, a great difference in the expansibilities
of various substances ; dry deal is one of the least expansible, and is there-
fore often used for the rods of pendulums. Brass expands one part in a
hundred thousand for every degree of Fahrenheit, or a little more or less
than this, accordingly as it contains more or less zinc. Glass and platina
are less than half as expansible as brass, iron about two thirds, and
mercury three times as much. A pendulum of brass would therefore
make one vibration in ten thousand less at 70° than at 50°, and would lose
8^ seconds in a day : a balance regulated by a spring would lose much
more ; for I have observed that vibrations governed by the elasticity of
steel have lost in frequency as much as one ten thousandth part for a
single degree of Fahrenheit ; and Berthoud informs us, that where a clock,
probably with a pendulum of steel, loses 20 seconds by heat, a watert loses
eight minutes.

Mr. Graham appears to have been the first that attempted to compen-
sate for the effects of temperature by the different expansibilities of various
substances. He employed for a pendulum, a tube partly filled with mer-
cury ; when the tube expanded by the effect of heat, the mercury expanded
much more ; so that its surface rose a little more than the end of the pen-
dulum was depressed, and the centre of oscillation remained stationary.*
This mode of compensation is still practised with success; but the
gridiron pendulum is more commonly used ; it was the invention, of
Harrison, t who combined seven bars, of iron or steel, and of brass, in such
a manner, that the bars of brass raised the weight as much as the bars of
iron depressed it. At present five bars only are usually employed, two of
them being of a mixture of zinc and silver, and three of steel. Mr. Ellicott$
suspended a pendulum at the extremity of a lever, which was supported*
by a pillar of brass much nearer to the fulcrum ; as the pendulum ex-
panded, the end of the lever was raised in the same degree, and the weight

* A Contrivance to avoid the Irregularities of a Clock's Motion, Ph. Tf. 1726.
xxxiv. 40.

t In 1726. An account is to be found in the Minutes of the Roy. Soc. for 1749,
and in Ph. Tr. xlvii. 521. See also Harrison's work, with preface by Maskelyne,
4to, Lond. 1767.

J Ph. Tr. 1752, xlvii. 479.

154 LECTURE XVII.

remained at its original distance from the point of suspension, which was
determined hy a fixed plate, transmitting the slender spring, as usual, be-
tween two opposite edges. The same effect is produced more simply by
suspending the pendulum from the summit of a bar nearly parallel to it,
and of the same substance with itself, resting on a fixed support, and either
of the same length with the pendulum, or a little longer, accordingly as
the distance of the fixed plate from the point of support of the bar, is
determined by materials which may be considered as nearly of an inva-
riable length, or as liable to a certain degree of expansion. (Plate XVI.
Fig. 210.)

All these methods of compensation are peculiar to clocks ; for watches,
it is usual to unite together two metals which differ in expansibility, so as
to form a compound plate ; one side of the . plate is commonly of steel,
the other of brass, and it is obvious that any increase of temperature, by
causing the brass to expand more than the steel, must bend the whole plate.
Such a plate is variously applied ; the most accurate method, which is
employed by Arnold and other modern artists, is to make it a part of the
balance itself, fixing a weight on its extremity, which is brought nearer to
the centre, by the increase of curvature of the plate, whenever the ex-
pansion of the arms of the balance tends to remove it further off. The
best way of making the plate appears to be to turn a ring of steel, and to
immerse it in melted brass, and then to turn away what is superfluous of
the brass. The magnitude of the weight, and the length of the plate, may
easily be so regulated as to compensate not only for the expansion pro-
duced by heat, but also for the diminution of the elasticity of the spring.
Sometimes also a plate has been applied in such a way as to shorten the
spring when the temperature is increased, by an operation similar to that
which serves to regulate a common watch, the clip that determines the
effective length of the spring, being moved backwards and forwards ; and
a similar effect has also been produced by dividing this clip into two parts,
one of which is fixed to a compound plate, and is made to approach the
other, so as to confine the spring more narrowly and thus diminish its
length, upon an increase of temperature. (Plate XVI. Fig. 211.)

The flexure of a compound plate has also been applied in a simple and
elegant manner by Mr. Nicholson to the pendulum of a clock, by causing
it to support the upper extremity of the pendulum. The plate is placed
horizontally, the brass being uppermost, and carries the pendulum in the
middle, while the ends rest on two fixed points, of which the distance may
be adjusted with great accuracy, so that when the temperature is in-
creased, the curvature of the plate may raise the rod of the pendulum,
enough to keep the weight or bob at a constant distance below the fixed
point, which determines its upper extremity. (Plate XVI. Fig. 212.)

The resistance, opposed to the motion of a pendulum by the air, affects
in some degree its velocity, and the variation of the density of the atmo-
sphere must therefore also produce some irregularities in timekeepers :
they are, however, too small to be sensible. Derham* found that fhe
resistance of the air accelerated the motion of a half second pendulum
* Ph. Tr. 1704, xxiv. 1785,

ON TIMEKEEPERS. 155

about four vibrations in an hour, by diminishing the arc in which it
vibrated : and when the vibrations were restored to their original magni-
tude, the resistance of the air produced a retardation of eight vibrations in
the same time. But a heavy pendulum, vibrating in a small arc, is very
little affected by this resistance.

Besides these more essential parts of the watchmaker's art, there are
several subordinate considerations which require his attention ; the striking
part in particular occupies, in clocks, and in repeating watches, no incon-
siderable portion of the bulk of the machine. But the apparatus employed
on these occasions requires neither refinement of invention nor delicacy of
execution. In old clocks, the number of hours struck is usually deter-
mined by the revolution of a certain portion of a wheel, which supports an
arm, and allows the hammer to strike, until at a proper time it falls into
a notch. In watches, and in more modern clocks, the same effect is pro-
duced by means of a spiral of 12 teeth, revolving once in 12 hours.

It is of considerable importance to the accurate performance of a good
clock, that it should be firmly fixed to a solid support. Any unsteadiness
in the support causes the point of suspension to follow the motion of the
pendulum, and enlarges the diameter of the circle of which the pendulum
describes an arc ; it must, therefore, tend in general to retard the motion of
the clock. Sometimes, however, an unsteady support may be of such a
nature as to accelerate the motion ; and an observation of this kind, made
by Berthoud, has suggested to Bernoulli a theory of compound vibrations,
which may perhaps be true in some cases, but is by no means universally
applicable to every case. On account of some circumstances of this kind,
it happens that when two clocks are placed near each other, and rest in
some degree on the same support, they have often a remarkable effect on
each other's vibrations, so as to continue going for several days, without
varying a single second, even when they would have differed considerably
if otherwise situated : and it sometimes happens that the clock which goes
the more slowly of the two will set the other in motion, and then stop
itself ; a circumstance which has been explained from the greater frequency
of the vibrations of a circular pendulum when confined to a smaller arc, the
tendency of the pendulums to vibrate in the same time causing the shorter
to describe an arc continually larger and larger, and the longer to contract
its vibrations, until at last its motion entirely ceases.* This sympathy has
some resemblance to the alternate vibrations of two scales hanging on
the same beam, one of which may often be observed to stop its vibrations
when the other begins to move, and to resume its motion when its com-
panion is at rest ; but it is still more analogous to the mutual influence of
two strings, or even two organ pipes, which, though not separately tuned
to a perfect unison, still influence each other's vibrations in such a manner
as to produce exactly the same note when they sound together.

* Ellicott, Ph. Tr. 1739, p. 126, describes the interference of two pendulums—
tbe one set the other in motion — the one stopped the other, &c.

156 LECTURE XVIII.

LECT. XVII.— ADDITIONAL AUTHORITIES.

Cumming's Elements of Clock and Watch Work, 4 to, Lond. 1766. Lepaute,
Traite d'Horlogerie, 4to, Par. 1767. Berthoud's Works, viz. Essai sur 1'Horlo-
gerie, 2 vols. 4to, Paris, 1763. Traite des Horloges Marines, 4to, 1773. Surl'In-
vention, &c. des Machines proposees en France pour la Determination des Longi-
tudes par la Mesure du Temps, 4to, 1773. Supplement, 1787. Trait6 des Mon-
tres a Longitudes, 4to, 1792. Suite, 1797. Supplement, 1807. Robison, Mech.
Phil. Reid, Treatise on Clock and Watch Work, Edin. 1826. Prony, Note sur un
Nouveau Moyen de regler la Duree des Oscillations des Pendules, 4to, Paris.
Jurgensen, MSmoires sur 1'Horlogerie Exacte, 4to, Paris, 1832.

LECTURE XVIII.

ON RAISING AND REMOVING WEIGHTS.

THE methodical arrangement of our subject leads us, after having con-
sidered the modifications of force, to those machines which are intended for
counteracting it, or for producing motion in opposition to an existing force.
The simplest of the forces to be counteracted is gravitation, and it is one
of the most common employments of mechanical powers to raise a weight
from a lower to a higher situation. This operation is also intimately
connected with the modes of overcoming the corpuscular force of friction or
adhesion, which constitutes the principal difficulty in removing bodies
horizontally from place to place ; for if we had only to produce motion in
an unresisting mass of matter, a loaded waggon might in time be drawn
along by a silk worm's thread. The raising and removing of weights,
therefore, together with the modes of avoiding friction in general, constitute
the first part of the subject of the counteraction of forces, and the remain-
ing part relates to the machinery intended for overcoming the other cor-
puscular powers of bodies by such operations as are calculated to change
their external forms.

Machines for raising weights, which involve only the mechanics of solid
bodies, are principally levers, capstans, wheels, pullies, inclined planes,
screws, and their various combinations in the form of cranes.

A lever is a very simple instrument, but of most extensive utility in
raising weights to a small height. We may recollect that levers are distin-
guished into two principal kinds, accordingly as the power and weight
are on different sides or on the same side of the fulcrum ; the forces
counteracting each other being in the one case in the same direction, in the
other, in opposite directions. Thus, when a man lifts a stone by means of
a lever of the first kind, resting on a fulcrum between himself and the
stone, he presses down the end of the lever, and the utmost force that fre
can apply is equal to the whole weight of his body ; but when he thrusts
the lever under the stone, so that its extremity bears on the ground, it
becomes a lever of the second kind, and in order to raise the stone, he must

ON RAISING AND REMOVING WEIGHTS. 157

now draw the end of the lever upwards. In this direction, a strong man
can exert a force equivalent to twice his weight ; consequently the second
kind of lever possesses here a temporary advantage over the first ; although,
if the operation were continued, the workman would he more fatigued by
raising even the same weight hy this method, than if he could conveniently
apply his weight to a lever of the first kind : and for this purpose, cross
bars have sometimes been added to levers, in order to enable several work-
men to stand on them with advantage at once. A bent lever operates
precisely with the same power as a straight one, provided that the forces be
applied in a similar manner with respect to its arms : and in all cases, the
forces capable of balancing each other are inversely as the distances of the
points of action from the fulcrum. Some addition of force is necessary for
overcoming the equilibrium and producing motion, but the velocity of the
motion being seldom of much consequence, a small preponderance is usually
sufficient.

The principal inconvenience of the lever is the short extent of its action :
this may, however, be obviated by means of the invention of Perrault,
in which two pins are fixed in the lever, at a short distance from each
other, sliding in two pairs of vertical grooves provided with ratchets, so that
when the long arm of the lever is pulled by means of a rope, the nearer pin
serves as a fulcrum, and the more distant one is elevated at the same time
with the weight, and is detained in its place by the click ; but when the
rope is slackened, the weight sinks a little, and raises the pin which first
served as a fulcrum, to a higher place in its groove. The same effects may
also be produced by catches or clicks resting upon ratchets on the opposite
sides of a single upright bar, which passes through a perforation in the
lever. There must, however, be a considerable loss of force from the con-
tinual intermission of the motion. (Plate XVII. Fig. 213.)

An axis with a winch, that is, a lever bent at the end, is known from
the common machine for raising a bucket out of a well. A vertical or
upright axis with two or more levers inserted into it, becomes a capstan.
In these cases, if we wish to estimate the force with accuracy, we must add
to the radius of the axis half the thickness of the rope, when we compare
it with the arm of the lever.

Sometimes the weight of a reservoir or bucket of water is employed for
raising another bucket, filled with coals or other materials, by means of a
rope or chain coiled round a cylinder or drum, or two drums of different
sizes. This machine is called a water whimsey : when the bucket of
water has reached the bottom, a valve is opened by striking against a pin,
and lets out the water. In a machine of this kind employed in the Duke
of Bridgwater's coal works, the water descends thirty yards and raises a
smaller quantity of coals from a depth of sixty. In such cases, supposing
the action to be single, and the stream of water to be unemployed during
the descent of the reservoir, a considerable preponderance may be advan-
tageously employed in giving velocity to the weights, provided that the
machinery be not liable to injury from their impulse.

An erect axis or drum, turned by the force of horses walking in a circle,
is used for raising coals and other weights, and is called a gin, probably by

158 LECTURE XVIII.

corruption from engine : the buckets being attached to the opposite ends of
a rope which passes round the drum, and which is drawn by means of its
adhesion to the drum. One of the buckets descends empty while the other
is drawn up full, and when the motions of the buckets are to be changed,
the horses are turned, or the wheels are made to impel the axis in a con-
trary direction when any other moving power is employed.

When a ship's anchor is weighed, the cable itself would be too large to
be bent round the capstan ; it is therefore connected with it by means of an
endless rope, called a messenger. As the messenger is coiled round the
lower part of the capstan, it quits the upper part ; so that its place becomes
lower and lower, till at last it has no longer room on the capstan ; it is
therefore necessary to force it up from time to time : this is called surging
the messenger ; it is commonly done by beating it, and to facilitate the
operation, the capstan is made somewhat conical. It has been proposed to
employ lifters in different parts of the circumference, which are raised
once in each revolution, by passing over an inclined plane, with the inter-
position of friction wheels ; a patent has been taken out for the invention,
and it has already been introduced in the navy. Some experienced judges,
however, are of opinion, that it would be better and more simple to employ
a capstan so much tapered that the tension of the rope itself, guided only
by a pulley, might always be sufficient to bring the messenger into its
place. *

The capstan, which consists of two cylinders of different sizes, on the
same axis, with a rope passing from the smaller one over a pulley which is
connected with the weight, and returning to be wound up by the larger one,
is very powerful in its operation ; but it requires a great length of rope for
a small extent of motion. (Plate IV. Fig. 61.)

Wheelwork is employed in a variety of ways for raising weights : its
powers are in all cases derived from the same principles as the actions of
levers, each wheel and pinion being considered as composed of a series of
bent levers of which the axis is the common fulcrum, and which act in suc-
cession on the teeth of the next wheel. The simplest combination of wheel-
work used for this purpose constitutes a jack ; a bar which is furnished
with teeth on one side, being raised by the last pinion. Such instruments
were not unknown even to the ancients ; the barulcust described by Hero
was a machine of this nature.

A series of buckets connected by ropes and passing over a wheel, is often
employed for raising water to a small height, and sometimes even for solid
substances in the state of powder, in particular for raising flour in a corn
mill ; and in this case the flour must be brought within reach of the buckets
by means of a revolving spiral, which pushes it gradually forwards.
When a weight of any kind is raised in buckets distributed through the
circumference of a wheel, the force required for retaining the weight in
equilibrium, is as much less than the weight, as the diameter of a circle is.
less than half the circumference, the remainder of the weight being sup-
ported by the axis of the wheel.

* See Hamilton's Rep. of Arts, ii. II. 126.

f Brugmans, Commentat. Gott. 1784, vii. M. 75.

ON RAISING AND REMOVING WEIGHTS. 159

Pullies, and their combinations in blocks, are universally employed on
board of ships. They are very convenient where only a moderate increase
of power is required ; but in order to procure a very great advantage, the
number of separate pullies or sheaves must be very much multiplied ; a
great length of rope must also be employed ; and it is said that in a pair of
blocks with five pullies in each, two thirds of the force are lost by the fric-
tion and the rigidity of the ropes. The inconvenience resulting from a
large number of pullies, may, however, as we have already seen, be con-
siderably lessened when they are arranged in Mr. Smeaton's manner,* the
acting rope being introduced in the middle, so as to cause no obliquity in
the block. Tackles, or combinations of pullies for raising weights, are
most conveniently supported on shore by means of shears, which consist of
three rods or poles, resting on the ground, and meeting each other in the
point of suspension. For raising stones in building, two poles are em-
ployed, with a rope fixed to their summit which keeps them in a proper
position ; their lower ends are usually connected by a third pole which
serves as an axis. (Plate IV. Fig. 56. Plate XVII. Fig. 214.)

Sometimes a pulley is drawn horizontally along a frame, setting out
from the point where the rope is fixed, so that while the bucket is raised,
it is also transferred diagonally to the opposite end of the scaffolding.
This apparatus is used in some of the Cornish stream works, in which the
earth of a whole valley is raised in order to be washed for the separation of
tin ore. (Plate XVII. Fig. 215.)

A fixed inclined plane is often of use in assisting the elevation of great
weights by means of other machinery. It is supposed that in all the
edifices of remote antiquity, where great masses of stone were employed,
as in the pyramids of Egypt and the druidical temples of this country,
these vast blocks were elevated on inclined planes of earth, or of scaffold-
ing, with the assistance also of levers and rollers. Inclined planes are
frequently used for drawing boats out of one canal into another ; and
sometimes the local circumstances are such that this may be done with
great convenience, merely by allowing a loaded boat to descend and to turn
the axis which raises an empty one. An example of this may be seen, on
a large scale, in the Duke of Bridgwater's canal.1* This canal is extended,
above ground, for forty miles on one level : an underground navigation,
twelve miles long, joins it at Worsley, leading to the coal mines under
"Walkden moor. At a height of 35£ yards above this is another subter-
raneous portion, nearly six miles in length. The connection between these
levels is formed by an inclined plane ; the boats are let down loaded, and
proceed three miles along the tunnel into the open canal. The inclined
plane is fixed in a stratum of stone, which fortunately has the most eligible
inclination of 1 in 4, and is 33 yards in thickness, affording the most ad-
vantageous means of fixing every part of the machinery with perfect
security. The whole length of the plane is 151 yards, besides a lock of
18 yards at the upper end. (Plate XVII. Fig. 216.)

^Inclined planes are also universally employed for facilitating the ascent

* Ph. Tr. 1751, p. 494.

t Consult Tr. of the Soc. of Arts, xviii. 288 ; Nich. Jour. iv. 486.

160 LECTURE XVIII.

of heights by men or by animals ; they may either be uniform, as roads,
or the general inclination of the surface may be superseded by the for-
mation of separate steps or stairs. The inclination of the surface may be
governed by the proportion of the strength of the animal to its weight,
the force required to support any weight on a plane being to the whole
weight as the height of the plane to its length ; and if the plane be a little
less inclined than the exact equilibrium would require, the animal will be
able to acquire a sufficient velocity at first to carry it easily up the ascent
with a motion nearly equable. The strength of a labourer may be advan-
tageously employed in ascending a given height by a flight of steps, and
placing himself on a stage which may raise a weight by its descent ; but
it appears that the force of other animals is less calculated for exertions
of this kind.

The screw is not often immediately applied to the elevation of weights ;
although sometimes a number of screws have been used for raising by slow
degrees a large and unmanageable weight, for instance, that of an obelise :
and a perpetual screw is frequently employed in giving motion to wheel-
work. Such machines possess a considerable mechanical advantage, but
they are subject to much friction, and are deficient in durability. Mr.
Hunter's double screw might be applied with advantage, if the extent of
the motion required were extremely small ; but this limitation confines its
utility within very narrow bounds.

A crane is a machine for raising weights by means of a rope or chain
descending from an arm which is capable of horizontal motion, and passing
over a pulley to be wound up on an axis. The axis is turned, either im-
mediateiy, or with the interposition of wheelwork, by a winch, by the
horizontal bars of a windlass, or by a walking wheel, and sometimes by
the force of wind, of water, or of steam. A walking wheel is an advan-
tageous mode of employing the strength of a labourer, but the bulk of the
machine is sometimes inconvenient and detrimental ; when, however, the
man walks upon the wheel, and not within it, this objection is in great
measure obviated. A walking wheel requires to be provided with some
method of preventing the dangerous consequences of the rapid descent of
the weight, in case of an accidental fall of the labourer : for this purpose,
a catch is usually employed, to prevent any retrograde motion ; a bar has
also sometimes been suspended from the axis of the wheel, on which the
man may support himself with his hands, and other similar precautions
have been adopted. Sometimes the plane of a walking wheel is but little
inclined to the horizon, and the man walks on its flat surface. In
either case the labour of horses, asses, or oxen, may be substituted for
that of men : but for cranes this substitution would be very disadvan-
tageous, since much force would be lost in stopping frequently so bulky a
machine as would be required. The employment of a turnspit dog is an
humble example of the same operation, and even goats appear to have
been sometimes made to climb in a similar manner. In a walking wheel
used for raising water at Carisbrook Castle, in the Isle of Wight, the woik
was performed by the same individual ass for the whole of forty-five years
preceding 1771. Walking wheels have also been invented, on which horses

ON RAISING AND REMOVING WEIGHTS. 161

were to act externally with their fore feet or hind feet only ; but they
have seldom, if ever, been applied to practical purposes. In general it is
advisable that walking wheels for quadrupeds should present to them a
path as little elevated as possible ; and it might probably be of advantage
to harness them either to a fixed point or to a spring or weight, which
would enable them to exert a considerable force even in a horizontal direc-
tion ; but, probably, after all, they might be more advantageously employed
in a circular mill- walk. (Plate XVII. Fig. 217.)

Mr. White's crane* affords a good specimen of an oblique walking
wheel ; the force may be varied accordingly as the labourer stands at a
point more or less distant from the centre ; and in order to avoid accidents,
a break is always acting on the axis of the wheel by its friction, except
when it is removed by the pressure of the man's hand on a lever upon
which he leans as he walks. The force is also varied in some cranes by
changing the pinion which acts on the principal wheel, and an expanding
drum has been contrived for the same purpose, consisting of a number of
bars moveable in spiral grooves, so as to form a greater or smaller cylinder
at pleasure. In order to place the weight in any situation that may be
required, the pulley may be made to slide horizontally on the gib or arm.
(Plate XVII. Fig. 218.)

A model of a crane was exhibited some years ago to the Royal Society,
in which a large wheel fixed to a short axis was made to roll round on
a plane, while the lower end of its axis was connected by a joint with
another axis in a vertical position : then the wheel, having to describe a
circumference somewhat larger than its own, was turned slowly, and there-
fore powerfully, round its axis, and the motion was communicated to the
fixed axis. The machine, however, appears to be more curious than
useful.

Sometimes a steelyard has been combined with a crane, for weighing
goods at the same time that they are raised by it. A small crane, fixed in
a carriage, is convenient for loading and unloading goods. In France, the
carts used on the wharfs are generally so long as to reach the ground be-
hind when depressed, and to furnish an inclined plane, along which the
goods are raised by a lever and axis, or a kind of capstan, fixed in front.

For taking hold of stones which are to be raised by means of a rope, a
hole is sometimes formed in them, wider within than at its opening, and in
this a lewis is inserted, consisting of two inverted wedges, separated by a
plug, to which they are fastened by a pin. (Plate XVII. Fig. 219.)

When a rope or chain which is to raise a weight, is so long as to require
a counterpoise, the effect of this may be varied according to the length of
the rope which is unbent, by hanging it on a second rope or chain, which
acts on a spiral fusee, slowly turned by a wheel and pinion.

The use of cranes is so extensive and so indispensable, that their forms
have been often multiplied on account of local circumstances, or even from
caprice ; but the constructions which have been described appear to be of the
most general utility, and from them it will be easy to judge of others.

When weights of any kind are simply to be removed from one situation
* Trans, of the Soc. of Arts, x. 230,
M

162 LECTURE XVIII.

to another, the most natural and obvious method, if they are portable, is to
carry them. There is, however, some scope for theory even in this common
operation, and we have seen that calculations have been made in order to
determine the most advantageous burden for a porter to carry, but the
experience of a few trials would in general be a better guide. Some carry
weights on their heads, others on their shoulders, others low down on their
backs ; and according to the situation of the burden, they bend forwards
or backwards, so that the common centre of gravity of the weight and
the body comes immediately or very nearly over some part of the ground
between their feet. The difficulty of carrying a weight at the extremity
of a long rod is easily understood from the properties of the lever, and
the same principles will enable us to determine the distribution of a load
between two porters, in whatever way they may carry it. Supposing the
weight to be placed on a porter's horse or hand barrow, and at equal dis-
tances from both extremities, each of the men will support an equal portion
of it ; but if it be nearer to the one than to the other, the load will be dis-
tributed in the same proportion as the poles are divided by the centre of
the burden. For instance, if the weight were 300 pounds, and it were
one foot distant from the one, and two from the other, the first would have
to carry 200 pounds, and the second 100. If the porters ascend a hill, or
a flight of steps, the distribution of the load will remain the same, provided
that the centre of the weight lie in the plane of the poles. But if the weight
consists of a large body placed on that plane, the centre of gravity being
above it, the effect of an inclination to the horizon may materially change
the distribution of the load, since the pressure will always be determined
by the distance of the ends of the poles from the line passing perpendicu-
larly through the centre of gravity ; so that if the elevation were sufficient,
the whole burden might rest on the lower porter. And in the same manner,
if the weight were suspended below the poles, the inclination would cause
a greater proportion of the load to be borne by the upper porter. The force
is, however, only thus distributed as long as the arms of the porters con-
tinue parallel to each other ; but the inequality would naturally be lessened
by a change of the directions in which they would act ; it would only be
necessary that those directions should meet in some part of the vertical line
passing through the centre of gravity ; the magnitude of each force would
then be determined by the length of the side of a triangle corresponding to
its direction, and the load might be either equally or unequally divided,
according to the positions of the arms. (Plate XVII. Fig. 220, 221.)

A man can carry in general a weight four or five times as great as that
which he can raise continually in a vertical direction with the same velo-
city : so that we may consider the resistance to be overcome as a kind of
friction which amounts to about a fourth or a fifth of the weight. If we
attempted to draw a weight along a horizontal surface, the resistance of
the surface would often not only impede the motion, but also injure the
texture of the substance to be moved. This injury may, however, be
avoided by the interposition of a simple frame or dray, and the dray may
be armed with a substance subject to little friction, as with iron : the fric-
tion may also be somewhat further diminished by making the outline of

ON RAISING AND REMOVING WEIGHTS. 163

the dray a little convex below, so that a slight agitation may be continually
produced during its motion. Sometimes the simple expedient of placing a
load on two poles of elastic wood, the thickest ends of which are supported
by the horse, and the thinner drag on the ground, is of use both in dimi-
nishing the friction by confining it to a smaller and smoother surface, and
in equalising the motion by the flexibility of the poles.

It often happens that agitation of any kind enables us to lessen consi-
derably the friction between two bodies, especially when they are elastic.
If we wish, for instance, to draw a ring along an iron rod, by a thread
which is nearly perpendicular to it, we may exert all our strength in vain
if we apply it by slow degrees, since the increase of force continues to in-
crease the adhesion. But if we pull the ring suddenly, and then slacken
the thread, it rebounds from the rod by its elasticity, and in this manner it
slides readily along by a continuance of alternations. In such a case, how-
ever, it would be more natural, if the thread were sufficiently heavy, to
give it a serpentine motion which would draw the ring in a more oblique
direction. It is said that when a screw is fixed very firmly in a piece of
iron, it may be extricated much more easily while the iron is filed in some
neighbouring part. The agitation thus produced probably operates in a
manner somewhat similar to that of the rod.

Friction may in general be considerably diminished by the interposition
of oily substances, where the surfaces are of such a nature as to admit of
their application. Thus common oil, tallow, or tar, are usually interposed
between metals which work on each other. It is necessary to attend to
the chemical properties of the oil, and to take care that it be not of such
a nature as to corrode the metals employed, especially where the work
requires great accuracy. Tallow is liable to lose its lubricating quality
unless it be frequently renewed. Between surfaces of wood, soap is some-
times applied, but more commonly black lead which becomes highly
polished. The advantages of canals and of navigation in general, are prin-
cipally derived from the facility with which the particles of fluids make
way for the motion of bodies floating on them.

The interposition of rollers or of balls bears some resemblance to the
application of fluids. Supposing the surfaces to be flat and parallel, a
roller moves between them without any friction : but it has still to over-
come the resistance occasioned by the depression which it produces in the
substance on which it moves, and which is greater or less according to
the softness and want of elasticity of the substance, If the substance were
perfectly elastic, the temporary depression would produce no resistance,
because the tendency to rise behind the roller would be exactly equivalent
to the force opposing its progress before ; and the actual resistance only
arises from a greater or smaller want of elasticity in the materials con-
cerned. The continued change of place of the rollers is often a material
objection to their employment ; their action may in some cases be pro-
longed by fixing wheels on their extremities, as well as by some other
arrangements ; but these methods are too complicated to afford much
practical utility. Rollers may also be placed between two cylinders, the
one convex and the other concave, and the friction may in this manner

M2

164 LECTURE XVIII.

be wholly removed, whatever may be the magnitude of the rollers.
(Plate XVII. Fig. 222, 223.)

The effect of friction in any machine being always diminished, in pro-
portion as the velocity of the parts sliding on each other is diminished, it
is obvious that by reducing the dimensions of the axis of a wheel as much
as possible, we also reduce the friction. When the pressure on the axis is
derived principally from the weight of the wheel itself, the friction may
be lessened by placing the wheel in a horizontal position and making the
axis vertical ; for in this manner the weight may be supported on an axis
ending in a very small surface, and the effect of the friction on this surface
will be about one third less than if it acted at the circumference. The
velocity of the parts sliding on each other may be still more reduced by
placing each extremity of the axis on another wheel, or between two
wheels, on which the axis rolls as they turn round, so that the friction is
transferred to the axis of these wheels of which the motion is very slow.
But when a great weight is to be supported, it is necessary that the friction
wheels be very strong and very accurately formed ; for if their surface
were irregular they might stand still, and their use would be destroyed.
(Plate XVIII. Fig. 224.)

Perrault* attempted to avoid all friction by supporting the axis of a
wheel in the coil of a rope, which allowed it to turn while the whole wheel
ascended and descended ; but the stiffness of a rope occasions in general
even a greater resistance than the friction for which it is substituted.

The wheels of carriages owe a great part of their utility to the diminu-
tion of friction, which is as much less in a carriage than in a dray, as the
diameter of the axle is less than that of the wheel, even supposing the dray
to slide on a greased surface of iron. The wheels also assist us in drawing
the carriage over an obstacle, for the path which the axis of the wheel
describes is always smoother and less abrupt than the surface of a rough
road on which the wheel rolls. It is obvious that both these advantages
are more completely attained by large wheels than by smaller ones ; the
dimensions of the axis not being increased in the same proportion with
those of the wheel, and the path of the axis, to which that of the centre
of gravity is similar, consisting of portions of larger circles, and conse-
quently being less curved ; and if the wheels are elastic and rebound from
an obstacle, the difference is still increased. It is, however, barely possible,
that the curvature of the obstacle to be overcome may be intermediate
between those of a larger and of a smaller wheel ; and in this case the
higher wheel will touch a remoter part of the obstacle, so that the path
of the axis will form an abrupt angle, while the smaller wheel follows
the curve, and produces a more equable motion ; this, however, is a case
of rare occurrence, and an advantage of little importance. (Plate XVIII.
Fig. 225, 226.)

The greater part of the resistance to the motion of a carriage very
frequently arises from the continual displacement of a portion of the
materials of the road, which do not react on the wheels with perfect elasfi-

* Machines approuve*es par 1' Academic, i. 13. Leupold, Th. Mach. t. xiv. xv.
Desaguliers Ph. Tr. xxxvi. 222.

ON RAISING AND REMOVING WEIGHTS. 165

city, but undergo a permanent change of form proportional to the loss of
force. Hence, in a soft sand, although the axles of the wheels may move
in a direction perfectly horizontal, the draught becomes extremely heavy.
The more the wheel sinks, the greater is the resistance, and if we suppose
the degree of elasticity of the materials and their immediate resistance at
different depths to be known, we may calculate the effect of their reaction
in retarding the motion of the carriage. Thus, if the materials were
perfectly inelastic, acting only on the preceding half of the immersed por-
tion of the wheel, and their immediate pressure or resistance were simply
proportional to the depth, like that of fluids or of elastic substances, the
horizontal resistance would be to the weight nearly as the depth of the
part immersed to two thirds of its length ; but if the pressure increased
as the square of the depth, which is a more probable supposition, the re-
sistance would be to the weight as the depth to about four fifths of the
length ; the pressure may even vary still more rapidly, and we may con-
sider the proportion of the resistance to the weight as no greater than that
of the depth of the part immersed to its length, or of half this length to
the diameter of the wheel ; and if the materials are in any degree elastic,
the resistance will be lessened accordingly. But on any of these sup-
positions, it may be shown that the resistance may be reduced to one half,
either by making a wheel a little less than three times as high, or about
eight times as broad as the given wheel. This consideration is of parti-
cular consequence in soft and boggy soils, as well as in sandy countries ;
thus, in moving timber in a moist situation, it becomes extremely advan-
tageous to employ very high wheels, and they have the additional con-
venience that the timber may be suspended from the axles by chains,
without the labour of raising it so high as would be necessary for placing
it upon a carriage of any kind. (Plate XVIII. Fig. 227.)

But the magnitude of wheels is practically limited, by the strength or
the weight of the materials of which they are made, by the danger of
overturning when the centre of gravity is raised too high, and in the case
of the first pair of wheels of a four wheeled carriage, by the inconvenience
that would arise, in turning a corner, with a wheel which might interfere
with the body of the carriage. It is also of advantage that the draught of
a horse should be in a direction somewhat ascending, partly on account of
the shape of the horse's shoulder, and partly because the principal force
that he exerts is in the direction of a line passing through the point of
contact of his hind feet with the ground. But a reason equally strong
for having the draught in this direction is, that a part of the force may
always be advantageously employed in lessening the pressure on the
ground ; and to answer this purpose the most effectually, the inclination
of the traces or shafts ought to be the same with that of a road on which
the carriage would begin or continue to descend by its own weight only.*
In order to apply the force in this manner to both pairs of wheels, where
there are four, the line of draught ought to be directed to a point half way

* Couplet, Reflexions sur le Tirage des Charrettes, Hist, et Mem. de Paris,
1733, p. 49, H. 82. Deparcieux sur le Tirage des Chevaux, ib. 1760, p. 263,
H. 151.

166 LECTURE XVIII.

between them, or rather to a point immediately under the centre of gravity
of the carriage ; and such a line would always pass above the axis of the
fore wheels. If the line of draught pass immediately through this axis,
the pressure on the hind wheels will remain unaltered ; and if the traces
or shafts be fixed still lower, the pressure on the hind wheels will even be
somewhat increased by the draught. It is evident, therefore, that this
advantage cannot be obtained if the fore wheels are very high ; we may
also understand that in some cases the common opinion of the eligibility
of placing a load over the fore wheels rather than the hind wheels, may
have some foundation in truth. When several horses are employed, the
draught of all but the last must be nearly horizontal ; in this case the
flexure of the chain brings it into a position somewhat more favourable
for the action of the horses ; but the same cause makes the direction of its
attachment to the waggon unfavourable; further than this there is no
absolute loss of force, but it appears to be advisable to cause the shaft
horse to draw in a direction as much elevated as possible ; and on the
whole it is probable that horses drawing singly have a material advantage,
when they do not require additional attendance from the drivers.

The practice of making broad wheels conical has obviously the disadvan-
tageous effect of producing a friction at each edge of the wheel, when the
carriage is moving in a straight line ; for such a wheel, if it moved alone,
would always describe a circle round the vertex of the cone to which it
belongs. When the wheels are narrow, a slight inclination of the spokes
appears to be of use in keeping them more steady on the axles than if they
were exactly vertical ; and when, by an inclination of the body of the
carriage, a greater proportion of the load is thrown on the lower wheel, its
spokes, being then in a vertical position, are able to exert all their strength
with advantage. The axles being a little conical, in order that they may
not become loose, or may easily be tightened as they wear, it is necessary
that they should be bent down so that their lower surfaces may be hori-
zontal, otherwise the wheels would press too much on the linch pin. For
this reason, the distance between the wheels should be a little greater above
than below, and their surfaces of course slightly conical. (Plate XVIII.
Fig. 228.)

It has been proposed to fix the wheels to their respective axles, to con-
tinue the axles as far as the middle of the carriage only, and to cause
them to turn on friction wheels or rollers ; a plan which may succeed if
the apparatus is not too complicated for use ; but in fact the immediate
friction on the axles is not great enough to render this refinement neces-
sary. If both opposite wheels were fixed to a single axis, one of them
would be dragged backwards and the other forwards, whenever the motion
deviated from a straight line ; and a similar effect actually takes place in
those carriages which are supported on a single roller.

The effect of the suspension of a carriage on springs is to equalise its
motion, by causing every change to be more gradually communicated to it
by means of the flexibility of the springs, and by consuming a certaili
portion of every sudden impulse in generating a degree of rotatory motion.
This rotatory motion depends on the oblique position of the straps sus-

ON RAISING AND REMOVING WEIGHTS. 167

pending the carriage, which prevents its swinging in a parallel direction ;
sucli a vibration as would take place if the straps were parallel would be
too extensive unless they were very short, and then the motion would be
somewhat rougher. The obliquity of the straps tends also in some mea-
sure to retain the carriage in a horizontal position ; for if they were
parallel, both being vertical, the lower one would have to support the
greater portion of the weight, at least according to the common mode of
fixing them to the bottom of the carriage ; the spring, therefore, being
flexible, it would be still further depressed. But when the straps are
oblique, the upper one assumes always the more vertical position, and conse-
quently bears more of the load ; for when a body of any kind is supported
by two oblique forces, their horizontal thrusts must be equal, otherwise
the body would move laterally ; and in order that the horizontal portions
of the forces may be equal, the more inclined to the horizon must be the
greater : the upper spring will, therefore, be a little depressed, and the
carriage will remain more nearly horizontal than if the springs were
parallel. The reason for dividing the springs into separate plates has
already been explained : the beam of the carriage, that unites the wheels,
supplies the strength necessary for forming the communication between
the axles : if the body of the carriage itself were to perform this office, the
springs would require to be so strong that they could have little or no effect
in equalising the motion, and we should have a waggon instead of a coach.
The ease with which a carriage moves depends not only on the elasticity
of the springs but also on the small degree of stability of the equilibrium,
of which we may judge in some measure, by tracing the path which the
centre of gravity must describe when the carriage swings. (Plate XVIII.
Fig. 229.)

The modes of attaching horses and oxen to carriages are different in
different countries, nor is it easy to determine the most eligible method.
When horses are harnessed to draw side by side, they are usually attached
to the opposite ends of a bar or lever ; and if their strength is very unequal
the bar is sometimes unequally divided by the fulcrum, the weaker horse
being made to act on the longer bar, and being thus enabled to counteract
the greater force of his companion. But even without this inequality a
compensation takes place, for the centre on which the bar moves is always
considerably behind the points of attachment of the horses ; and when one
of them falls back a little, the effective arm of the lever becomes more per-
pendicular to the direction of his force, and gives him a greater power,
while the opposite arm becomes more oblique, and causes the other horse
to act at a disadvantage ; so that there is a kind of stability in the equili-
brium. If the fulcrum were further forwards than the extremity of the
bar, the two horses could never draw together with convenience. (Plate
XVIII. Fig. 230.)

In mining countries and in collieries, it is usual, for facilitating the mo-
tion of the carriages employed in moving the ore or the coals, to lay wheel-
ways of wood or iron along the road on which they are to pass ; and this
practice has of late been extended in some cases as a substitute for the
construction of navigable canals. Where there is a turning, the carnages

168 LECTURE XVIII.

are usually received on a frame supported by a pivot, which allows them
to be turned with great ease. In particular situations, these waggons are
loaded by little carts, rolling without direction down inclined planes, and
emptying themselves ; they are also provided with similar contrivances for
being readily unloaded, when they arrive at the place of their destination.
The carriages used for drawing loaded boats over inclined planes, where
they have to ascend and again to descend, are made to preserve their level
by having at one end four wheels instead of two, on the same transverse
line ; the outer ones as much higher than the pair at the other end, as the
inner ones are lower; and the wheelway being so laid that either the
largest or the smallest act on it, accordingly as the corresponding part of
the plane is lower or higher than the opposite end. It is possible that
roads paved with iron may hereafter be employed for the purpose of expe-
ditious travelling, since there is scarcely any resistance to be overcome,
except that of the air, and such roads would allow the velocity to be
increased almost without limit.

For removing earth from one situation to another, a series of baskets has
sometimes been hung on two endless ropes, moving on pullies of such a
form as to suffer the bars supporting the baskets to pass freely over them ;
the baskets being moved by means of a winch acting on the rope by a
wheel like one of the pullies. Sometimes also a series of little carts has
been connected by ropes, and drawn in a circle or oval up and down an
inclined plane. These methods may be adopted in making roads where a
hill is to be levelled, and the materials are to be employed in filling up the
valley below ; but in such cases two carts, connected by a cylinder or
windlass, are generally sufficient ; and they may be arranged in the same
manner as the carriages for removing boats on an inclined plane.

LECT. XVIII.— ADDITIONAL AUTHORITIES.

Machine employed for clearing the Port of Toulon. Belidor, Architecture Hy-
draulique, ii. II. pi. 20. Ferguson on a Crane, Ph.' Tr. 1763, liv. 24. Redely-
kheid, Machine & creuser les Pores, fol. La Hague, 1774. Suspended Scaffolding,
Encyclopedic JMethodique, pi. iv. Peintre en Batimens. Hall's Crane, Trans, of
the Soc. of Arts, vol. xii.

On Wheel Carriages.— On the Benefit of High Wheels, Ph. Tr. 1685, xv. 856.
Lahire on the Magnitude of Wheels, Hist, et Mem. de Paris, ix. 116. Parent, do.
1712, p. 96. Reaumur, do. 1724. p. 300. Dupin de Chenonceau, do. 1753, H.
301. Emerson's Mech. p. 194. Boulard and Margueron on Broad Wheels, Ro-
zier's Jour. xix. 424. Jacob on Wheel Carriages, &c. 2 vols. 4to, 1773-4. Anstice
on do. 1790. Bailey, Plates of Machines approved by the Society of Arts, 2 vols.
fol. 1782. Rizzetti, Riforma de' Cam di quattro Ruote, Trevigi, 1785. Edgeworth,
Tr. R. Ir. Aca. 1788, ii. 73. Lamber, Hindenburg's Archiv, ii. 51. Grobert sur
les Voitures a deux Roues, 1797. A. Young, Annals of Agriculture, xviii. Fuss,
Versuch einer Theorie des Widerstandes zevey-und-vier-radiger Fuhrwerke. Co-
penhag. 1798. Ph. Mag. xiii. 115. Anderson's Institutes of Physics, quoted by
Cavallo, Nat. Ph. Cumming on Conical Wheels, 4to, 1804. Board of Agriculture,
ii. 351. Repertory of Arts, xiii. 256. Imison's Elements of Science and Art,
2 vols, 1803, i. 129. Ferguson's Lect. by Brewster, ii. 296.

169

LECTURE XIX.

ON MODES OF CHANGING THE FORMS OF BODIES.

THE corpuscular forces by which bodies retain their peculiar forms of
aggregation, require in many cases to be counteracted or modified by
mechanical processes : thus we have frequent occasion to compress bodies
into a smaller space, to augment their dimensions in a particular direc-
tion, to divide their substance, either partially or totally, in given lines or
surfaces, or to destroy their general form by reducing them into more
minute portions ; and we may consider these subjects as principally refer-
able to the effects of compression, extension, penetration, division, attri-
tion, digging, boring, agitation, trituration and demolition. The two first
of these articles depend on such a change as we have examined in consider-
ing the strength of materials, under the name of alteration, the remainder
on fracture.

The instruments peculiarly intended for compression are in general of
the description of presses ; and the most common act by means of a screw.
The friction on the screw interferes considerably with the power of the
machine ; but it is of use in keeping the press fixed in a situation into
which it has been brought by force. The screw is always turned by a
lever ; for without this assistance, however powerful it might be, the fric-
tion would render it almost useless. When great force is required, the
screw is made as close as is consistent with the strength of its spires.
Mr. Hunter's double screw may also be used with advantage, where only
a small extent of motion is required. The screw of a printing press or of a
stamping press, is, on the contrary, open, and it is caused to descend with
considerable momentum, the handle being loaded with a weight. Wher-
ever a force is so employed as to produce an impulse which acts on any
body, the momentum which is the result of the action of the force for a
certain time, is usually much more powerful than the simple pressure ;
the degree of its efficacy depends, however, on the degree of compressibi-
lity of the substance. Thus, if a heavy body fall from a certain height so
as to acquire a momentum in consequence of the force of gravity, it will
ultimately exert on the substance upon which it falls a force about as
much greater than its weight, as the space through which the surface of
the substance struck is depressed, by means of the impulse, is less than
twice the height from which the body has fallen ; and unless either the
substance is very compressible, or the height very small, this force must be
incomparably greater than the pressure of the weight only.

For a printing press, a single heavy roller is 'sometimes made to pass
over the paper when it has been laid on the types ; and since the whole
action of such a roller is confined to a small part at any one time, it is said
to exert sufficient force and to perform its work more equably than a
common press ; but its operation must be comparatively slow. A common

170 LECTURE XIX.

mangle for linen acts nearly in a similar manner. In calendering mills,
the force of a spring is employed for exerting a pressure on the block with
which the materials are glazed.

The copper plate printing press, and the machine for copying letters, are
composed of two rollers parallel to each other, pressing on the substance
which is interposed, and which is brought into its situation partly by the
friction of the surface of the roller and partly by external force.

The rollers by which sugar canes are pressed, are in general situated
vertically, the middle one of three being turned by horses, by mules, or by
water, and the canes being made to return round it so as to pass through
both interstices in succession. It appears to be of some advantage in presses
of this kind that all the rollers should be turned independently of their
action on the materials interposed, since the friction of two rollers may
tend to draw the materials into the space between them, with more regu-
larity and greater force, than the action of a single roller would do. For
this reason, it may be advisable to retain the toothed wheels turning the
rollers, even when their axes are not firmly fixed but held together by
an elastic hoop. (Plate XVIII. Fig. 231.)

In oil mills, a still greater momentum is applied to the purpose of com-
pression than in the printing press : hammers, or long wooden beams placed
vertically, are raised by a water wheel, and suffered to fall on wedges
which act very forcibly on the materials contained in bags on each side.

Compression is also sometimes performed by the operation of hammer-
ing : thus, cast brass is generally hammered before it is used, in order to
increase its strength ; the hammer renders it so much stiffer, that if it is
necessary to preserve its ductility, it must be frequently annealed by
exposure to heat. Anvils and vices are necessary appendages to the
hammer ; their use depends principally on their firmness, which is chiefly
derived from weight in the one case, and from strength in the other ; and
pincers may be considered as portable vices.

For the purpose of producing a continued pressure on such substances
as have a tendency to contract their dimensions under the operation of a
press, a spring has been interposed between the press and the materials,
which is capable of pursuing them with a certain degree of force : the
utility of such an arrangement must, however, be extremely limited.
Mr. Bramah has applied a well known law of hydrostatics to the construc-
tion of a very useful press, which is simple, powerful, and portable.

Extension is seldom performed by forces that tend immediately to in-
crease the dimensions of the substance only : it is generally procured by
reducing the magnitude of the substance in another direction, sometimes
by means of pressure, but more effectually by percussion. The rollers of
the press employed for laminating metals are turned by machinery, and
are capable of being moved backwards and forwards in order to repeat the
operation on the same substance ; their distance is adjusted by screws
which are turned at once by pinions fixed on the same axis, in order that
they may be always parallel. In this manner lead, copper, and silver, ate
rolled into plates, and a thin plate of silver being soldered to a thicker one
of copper, the compound plate is submitted again to the action of the

ON MODES OF CHANGING THE FORMS OF BODIES. 171

' press, and made so thin as to be afforded at a moderate expense. The
glazier's vice is a machine of the same nature for forming window lead :
the softness of the lead enables it to assume the required shape, in conse-
quence of the pressure of the rollers or wheels ; and the circumference of
these wheels is indented, in order to draw the lead along by the correspond-
ing elevations. (Plate XVIII. Fig. 232.)

In drawing wire, the force is originally applied in the direction of the
extension, but it produces a much stronger lateral compression, by means of
the conical apertures through which the wire is successively drawn. For
holding the large wire, pincers are at first used, which embrace it strongly
while they pull, and open when they advance to a new position, the inter-
ruption being perhaps of use, by enabling the pincers to acquire a certain
momentum before they begin to extend the wire ; but afterwards, when the
wire is finer, it is simply drawn through the aperture from one wheel or
drum to another. During the operation, it requires frequent annealing,
which causes a scale to form on its surface ; and this must be removed by
rolling it in a barrel with proper materials ; for the application of an acid
is said to injure the temper of the metal. Copper is sometimes drawn into
wire so large as to serve for the bolts used in shipbuilding, especially for
sheathing ships' bottoms. Silver wire, thinly covered with gold, is ren-
dered extremely fine, and then flattened, in order to be fit for making gold
thread : the thickness of the gold is inconceivably small, much less than
the millionth part of an inch, and sometimes only a ten millionth.

In order to form the handles of vessels of earthenware, the clay is forced
through a hole of a proper shape in an iron box. The operation of the
potter's wheel consists in great measure of compression and extension, per-
formed by the hands ; the vessels are finished, when they are partly dry,
in a lathe, or by other instruments ; some kinds of earthenware are formed
in a mould only.

When a thread or a plate of glass is extended in a semifluid state, it has
a tendency to preserve an equable thickness throughout : this is derived
from the effect of the air in cooling it, the thinnest parts becoming imme-
diately a little colder than the rest, and consequently harder, so that they
retain their thickness, until the neighbouring parts are brought into a
similar state.

Extension is performed by means of percussion, in forges and in the
common operation of the smith's hammer. In forges, the hammers are
raised by machinery, and thrown forcibly against a spring, so as to recoil
with great velocity. With the help of this spring, the hammer sometimes
makes 500 strokes in a minute, its force being many times greater than
the weight of the hammer. Such forges are used in making malleable
iron, in forming copper plates, and in manufacturing steel. (Plate XVIII.
Fig. 233.)

Gold is beaten between the intestines of animals/ on a marble anvil ; for
this purpose it is alloyed with copper or silver. It is reduced to the thick-
iies"s of little more than the three hundred thousandth of an inch. Silver
leaf is about the hundred and sixty thousandth : it is made of silver without
alloy.

The operation of coining depends also principally on an extension of the

172 LECTURE XIX.

metal into the recesses of the die ; it is performed by a strong pressure, f
united with a considerable impulse, communicated by a screw like that of
a printing press ; and sometimes the impression is formed by the repeated
blows of a hammer only.

Thin plates of silvered copper are moulded into any figure that may be
required, by being placed between two corresponding stamps, of which the
one is fixed, and the other attached to the bottom of a heavy hammer. The
hammer is raised and suffered to fall in a right line, by means of pincers,
which open when they have acquired a certain height. Sometimes the
contact, produced by the forcible impulse of a die, is sufficiently intimate to
cause a thin plate of silver to cohere permanently with a surface of iron ;
and this mode of uniting metals is actually employed in some manu-
factures.

The operations of perforating, cutting, turning, boring, digging, sawing,
grinding, and polishing, resemble each other, in great measure, with respect
to the minute actions of the particles of bodies which they have to overcome.
Penetration is generally performed in the first instance by the effect which
we have called detrusion, where the magnitude of the penetrating substance
is considerable : but when a fine point or edge is employed, it probably first
tears the surface where it is most depressed, and then acts like a wedge on
the portions of the substance left on each side, with a force so much the
greater as the edge is thinner. The resistance opposed by a solid, or even
by a soft substance, to the motion of a body tending to penetrate it, appears
to resemble in some measure the force of friction, which is nearly uniform,
whether the motion be slow or rapid, destroying a certain quantity of
momentum in a certain time, whatever the whole velocity may be, or what-
ever may be the space described. Hence arises the advantage of giving a
great velocity to a body which is to penetrate another, the distance to which
a body penetrates being as the square of its velocity, or as its energy ; and a
certain degree of energy being required in order to make it even penetrate at
all. It is true that when we exchange a slow motion for a more rapid one,
by the immediate action of any mechanical power, we can only obtain the
same energy from the same power, for we must diminish the mass in the
same proportion as the square of the velocity is increased ; but a very small
part of the force which is consumed in the operation of a machine of any
kind, is employed in generating momentum ; by much the greatest part is
spent in overcoming resistances which vary but little with the velocity ; a
small portion only of the resistance increasing in proportion to the square
of the velocity ; so that by applying a triple force, we may obtain more
than a double velocity, and more than a quadruple effect : and besides it
has already been observed that when the velocity begins to exceed a certain
limit, the effect is increased in a much greater proportion than that of its
square. The same work is also performed with less pressure, and less strain
on the machinery, where a great velocity is employed. It is on account of
the efficacy of velocity in facilitating penetration, that soft substances, mov-
ing very swiftly, will readily perforate much harder ones ; and for the
saine reason a gunshot wound, and even the loss of a limb, takes place with
so little disturbance of the neighbouring parts, that it is sometimes scarcely
felt. The advantage of an impulse, however inconsiderable, above a pres-

ON MODES OF CHANGING THE FORMS OF BODIES. 173

sure, however great, may be easily understood from the ease with which a
moderate blow of a hammer causes a nail to penetrate a substance, into
which the whole force of the arm could not have thrust it.

In the engine for driving the piles, or upright beams, used for the founda-
tions of buildings in water, or in soft ground, the weight is raised slowly to
a considerable height, in order that, in falling, it may acquire sufficient
energy to propel the pile with efficacy. The same force, if applied by very
powerful machinery immediately to the pile, would perhaps produce an
equal effect in driving it, but it would be absolutely impossible in practice
to construct machinery strong enough for the purpose, and if it were pos-
sible, there would be an immense loss of force from the friction. For ex-
ample, supposing a weight of 500 pounds, falling from a height of 50 feet,
to drive the pile 2 inches at each stroke ; then, if the resistance be con-
sidered as nearly uniform, its magnitude must be about 150 thousand
pounds, and the same moving power, with a mechanical advantage of 300
to 1, would perform the work in the same time. But for this purpose some
parts of the machinery must be able to support a strain equivalent to the
draught of 600 horses. In the pile driving engine, the forceps, or tongs,
sometimes called the monkey, or follower, is opened as soon as the weight
arrives at its greatest height ; and at the same time a lever detaches the
drum, employed for raising the weight, from the axis or windlass, at which
the horses are drawing ; the follower then descends after the weight, un-
coiling the rope from the drum, and the force of the horses is employed in
turning a fly-wheel, until the connexion with the weight is again restored.
(Plate XVIII. Fig. 284.)

When we throw a stone, or a missile weapon of any kind, with the hand,
the stone can acquire no greater velocity than the hand itself, accompanied
by the neighbouring part of the arm ; so that the whole velocity must be
produced in a mass of matter comparatively very large. A sling enables
us to throw a stone or a ball much further ; for here the stone may be
moved with a velocity far greater than the hand that impels it, although
the action of the force on the stone is indirect, and the resistance of the air
considerable. An elastic bow, furnished with a strong and light string,
enables us to apply to an arrow or to a ball the whole force of our arms,
unencumbered with any considerable portion of matter, that requires to be
moved with the arrow ; hence a very great velocity may be obtained in
this manner. An air gun possesses the same advantage in a still greater
degree, and the force of fired gunpowder excels perhaps all others from its
concentrating an immense force in the form of an inconceivably light elastic
fluid ; of course a ball impelled by this force, becomes a most effectual
instrument in penetrating the most refractory substances. We may easily
calculate the velocity of an arrow, by comparing its motion with that of a
pendulum, if we know the proportion of its weight to the force that bends
the bow ; including in the weight a small addition for the inertia of the
bow and bowstring ; the height to which the arrow will rise, being about as
much greater than the space through which the bowstring acts on it, as the
greatest force applied in drawing the bow is greater than twice the weight
to be moved.

174 LECTURE XIX.

The action of a whip, either on the air, or on a solid body, depends on
the increase of velocity occasioned by the successive transmission of the
motion from a thicker to a thinner portion of its flexible substance, so that
at last, the energy of the lash, and of its knots, gives it a sufficient capa-
bility of exciting sound or of inflicting pain.

The instruments generally employed for the division of solid bodies, are
wedges, chisels, knives, and scissors ; they sometimes act by pressure only,
but they are more powerful when impulse is added to it. Hatchets, planes,
saws, and files, always act with some rapidity. Cutting instruments are in
general very thin wedges, but the edge itself is usually much more obtuse ;
Mr. Nicholson* has estimated the angle, formed ultimately by the surfaces
constituting the finest edge, at about 56 degrees. Knives are sometimes
fixed on wheels, so as to revolve in a direction oblique to their edges, as in
some machines for cutting chaff, where the straw is also drawn forwards,
through a space variable at pleasure, during each revolution of the knife.
An instrument of a similar nature has also been invented for the purpose of
cutting weeds under water.

For the edges of all cutting instruments, steel is principally employed.
After being hardened, by plunging it when red hot into cold water, it is
tempered, by laying it on a heated iron, or more accurately, by Mr. Stod-
art'st method, of immersing it in a metallic composition in the state of
fusion. When its surface has acquired a yellow tinge, it is fit for edge tools,
and the degree of heat proper for watch springs is indicated by a blue
colour. The backs of knives are often made of iron, which is less brittle
than steel : these substances are generally welded together, by hammering
them when red hot ; but sometimes, in large instruments, a back of iron is
only rivetted on.

The iron employed for making nails and other small articles, is first
rolled into flat bars, and then cut into narrow rods, by causing it to pass
between the cylinders of the slitting mill, the surfaces of which are formed
into rectangular grooves, and which are placed close to each other, so that
the prominent parts of the one are opposed to the depressions of the other,
and the bars are divided by the pressure of the opposite forces acting trans-
versely at the same points, so as to separate them by the effect which we
have already considered under the name detrusion. The same machinery
also generally works a pair of large shears for cutting bars of any kind.
(Plate XVIII. Fig. 235.)

The lathe is an elegant instrument, in which a considerable relative
velocity is produced between the tool and the substance to be cut, by the
revolution of this substance on an axis, while the tool is supported by a
rest. Ornamental lathes admit of a great variety of mechanical contriv-
ance, but they are of little practical use, except for amusement. Picture
frames are, however, sometimes turned in oval lathes ; and in the manufac-
ture of buttons, machines of a similar nature are occasionally employed.
The effect of every lathe of a complicated construction depends on a certain
degree of motion of which its axis is capable : if this motion be governed by

* Nich. Jour. 8vo, i. 47, 210.

f Nich. Jour. 4to, iv. 127. See also i. 380, 468, 575 ; ii. 64, 102.

ON MODES OF CHANGING THE FORMS OF BODIES. 175

a screw, a screw of any diameter may be turned by its assistance ; if by a
frame producing an elliptic curve, any number of ovals, having the same
centre, may be described at once ; and if a moveable point connected with
the work be pressed by a strong spring against a pattern of any kind, placed
at one end of the axis, a copy of the same form may be made at the other
end of the axis.

The process of boring is a combination of penetration and division, and
sometimes of attrition. Awls, gimlets, screws, augers, and centrebits, are
various forms of borers. The drill has the advantage of a rapid motion,
communicated by the drill bow, which turns it round by means of a little
wheel or pulley. In boring cannon, the tool is at rest, while the cannon
revolves, and by this arrangement the bore of the cannon is formed with
much more accuracy than according to the old method of putting the borer
in motion ; perhaps because the inertia of so large a mass of matter as con-
stitutes the cannon, assists in denning the axis of revolution with more
accuracy. The borer is pressed against the cannon by a weight hung
on the arm of a bent spring, and during the operation the outside is also
turned into its intended shape by the application of proper instruments.
Cylinders for steam engines are cast hollow, and afterwards bored ; but in
this case the borer revolves, and the cylinder remains at rest.

Ploughs, spades, pickaxes, mattocks, harrows, and other agricultural
instruments, resemble in their operation the chisel and the wedge : the
numerous diversities in their form and the complications of their structure,
are determined more by the various modifications of their action, required
for particular purposes, than by any material difference in the mode of
application of the principles on which they depend. (Plate XVIII.
Fig. 236.)

The process of mining is a combination of boring and digging. Shafts
are sunk, levels are driven, and drains are carried off, by the help of picks
or pickaxes, wedges, and hammers, the rocks being also sometimes loosened
by blasting with gunpowder. In searching for coal, a shaft is sunk
through the uppermost soft strata, and the rock is then bored by striking it
continually with an iron borer terminating in an edge of steel, which is in
the mean time turned partly round ; and at proper intervals a scoop is let
down to draw up the loose fragments. In this manner a perforation
is sometimes made for more than a hundred fathoms, the borer being
lengthened by pieces screwed on to it ; it is then partly supported by a
counterpoise, and is worked by machinery ; if it happens to break, the
piece is raised by a rod furnished with a hollow cone, like an extinguisher,
which is driven down on it. Sometimes the borer is furnished with knives,
which are made to act on any part at pleasure, and to scrape off a portion
of the surrounding substance, which is collected in a proper receptacle.

For sawing wood on a large scale, sawing mills are very advantageously
employed, being usually driven by water. Several saws are generally
fixed in a frame, parallel to each other ; they are worked up and down by
a cfank, and at every alternation, a wheel is drawn round a little by a
catch, or click, and moves forwards the frame which supports the timber.
When the machine is employed for cutting the fellies which form the cir-

176 LECTURE XIX.

cumference of wheels, the frame supporting the timber is made to turn
round a centre. A circular saw is used in the construction of blocks and
pullies ; and in order to make the motion more secure from the effect of
accidental irregularities, the wheels are made to turn each other by contact
only, without teeth. The machinery for making blocks, in the 'Royal
dock yard at Portsmouth, has been lately much improved and enlarged ;
it is worked by a steam engine, the action of which is applied to a great
variety of purposes. The advantage of a saw which revolves continually
appears to be very considerable, since a .much greater velocity may be
given to it than can be obtained when the motion is alternate. Such a saw
has also sometimes been applied to cutting off piles under water.

In mills for sawing marble into slabs, the saws are drawn backwards
and forwards horizontally : they are made of soft iron, without teeth ; and
sand being applied to them, with water, during the operation, the sand
is partly imbedded in the iron, and grinds away the marble.

Granite is worked by driving a number of thin wedges very gradually
into it, at various parts of the section desired ; and sometimes wedges of
wood are employed, which being moistened by water, their expansion
separates the parts from each other.* It is also said that many stones
may be divided by drawing lines on them with oil, and then exposing
them to heat. Perhaps some processes of this kind might be performed
with advantage under water ; it is well known that glass may be cut in a
rough manner under water, without much difficulty, by a common pair
of scissors.

For reducing the magnitude of a substance in a particular part, instru-
ments of attrition are used ; rasps, files, grindstones, and hones ; and of
all these the immediate actions appear to resemble those of chisels and
saws. The hatches of files are cut with a hard chisel while the steel is
soft, and the files are afterwards hardened. In using the grindstone, water
is applied, in order to avoid the inconvenience produced by too much heat ;
and sometimes tallow is substituted for water with equal advantage : but oil
is not found to answer the same purpose ; and it has been conjectured that
the cold continually occasioned by the melting of the tallow at the point
of friction, serves as a substitute for the cooling effect of the evaporation
of the water. For grinding and polishing steel, the grindstones are made
to revolve, either vertically or horizontally, with a velocity so great as to
describe sometimes as much as 60 feet in a second. The steel is also in
some cases drawn backwards and forwards horizontally on a circular sur-
face, and in order that the action may be equally divided throughout the
surface, it is allowed to revolve on an axis by means of the friction ; its
motion being confined to one direction by the action of a catch.

Various substances, chiefly of mineral origin, are also used, on account
of their hardness, as intermediate materials for grinding and polishing
others. These are diamond dust, corundum, emery, tripoli, putty, glass,
sand, flint, red oxid of iron, or crocus martis, and prepared chalk ; they
are sometimes applied in loose powder, and sometimes fixed on leather,

* See Herschel's Discourse on the Study of Natural Philosophy, p. 48.

ON MODES OF CHANGING THE FORMS OF BODIES. 177

wood or paper. Cuttle fish bone, and seal skin, are furnished by the
animal kingdom, and Dutch rushes by the vegetable ; these are employed
chiefly in polishing wood or ivory.

Marble is made smooth by rubbing one piece on another, with the in-
terposition of sand ; the polishing blocks are sometimes caused to revolve
by machinery in a trough in which the marble is placed under water, and
are drawn at the same time gradually to and from the centre ; or the slab
itself, with the frame on which it rests, is drawn slowly backwards and
forwards, while the blocks are working on it. Granite is polished with
iron rubbers, by means of sand, emery, and putty ; it is necessary to take
care during the operation that the water, which trickles down from the
rubbers, and carries with it some of the iron, may not collect below the
columns, and stain them ; but this inconvenience may be wholly avoided
by employing rubbers of glass.

Optical lenses are fixed on blocks by means of a cement, and ground
with emery, by a tool of proper convexity or concavity ; if they are small,
a large number is fixed on the blocks at the same time. The tool is some-
times first turned round its axis by machinery, and when the lenses are
to be finished, a compound motion is given to it by means of a crank ; and
in order to make it more smooth, the wheels turn each other by brushes
instead of cogs. The point of the lens where its two surfaces are parallel,
is determined by looking through it at a minute object, while it is fixed on
a wheel with a tubular axis, and shifting it, until the object no longer ap-
pears to move ; a circle is then described, as it revolves, in order to mark
its outline.

Machines for trituration, by means of which the larger masses of matter
are crushed, broken, or ground, into smaller parts, are in general compre-
hended under the denomination of mills. After the pestle and mortar,
the simplest machine of this kind appears to be the stamping mill ; the
stampers resemble the hammers of the mill employed in the extraction of
oils from seeds, and the machine is used for reducing to powder the ores of
metals, and sometimes also barks, and linseed ; the surface of the stampers
being armed Avith iron or steel. But barks and seeds are more usually
ground by the repeated pressure of two wheels of stone, rolling on an axis
which is forced in a horizontal direction round a fixed point. A noble-
man of distinguished rank and talents has lately employed for a mortar
mill, a wheel of cast iron, formed of two portions of cones, joined at their
bases : after thirty revolutions, the mortar being sufficiently ground, a bell
rings, and the horse stops.

The materials for making gunpowder are also ground by a wheel re-
volving in a trough : in order to corn them, they are moistened, and put
into boxes with a number of holes in their bottoms, and these boxes being
placed side by side, in a circular frame, suspended by cords, the frame is
agitated by a crank revolving horizontally, and the paste shaken through
the holes : the corns are polished by causing them to revolve rapidly within
a barrel.

A revolving barrel is used for forming and polishing small round bodies
of different kinds, and it is often employed in agriculture as a churn for

178 LECTURE XIX.

making butter. The purpose of agitation is perhaps more effectually
answered by an alternate motion, which has sometimes been produced in a
barrel churn, by means of a cord attached to a heavy pendulum.

Threshing machines are of two kinds ; the one consists of a number of
flails, beating the corn nearly in the same manner as they are used by
labourers: in the other, which is more commonly employed in this
country, the corn is drawn along by two revolving rollers, and caused to
pass between a cylinder and its concave cover, while a number of blocks,
projecting from the surface of the cylinder, beat or rub out the grains
very effectually from the ears ; the corn falls out at the lower part, and is
winnowed by a fan which the machine turns at the same time. In this
manner it is said that a horse will thresh about 100 bushels of corn in a
day. It is commonly reckoned the work of a labourer to thresh about six
bushels in a day. (Plate XVIII. Fig. 237.)

Some kinds of grain are occasionally ground in mills of iron or steel,
which consist of a solid cylinder or cone turning within a hollow one, both
the surfaces being cut obliquely into teeth. But the common mill for
grinding corn is composed of two circular stones of silicious grit, placed
horizontally ; the upper one revolves with considerable velocity, and is
supported by an axis passing through the lower one, at a distance variable
at pleasure. When the diameter is five feet, the stone usually makes about
90 revolutions in a minute ; if the velocity were greater, the flour would
be too much heated. The corn is shaken out of a funnel or hopper, by
means of projections from the revolving axis which strike against the ori-
fice ; it passes through the middle of the upper millstone, and is readily
admitted between the stones ; the lower stone is slightly convex, and the
upper one somewhat more concave, so that the corn passes over more than
half the radius of the stone before it begins to be ground : after being re-
duced to powder, it is discharged at the circumference, its escape being
favoured by the convexity of the lower stone, as well as by the centrifugal
force. The surface of the stones is cut into grooves, in order to make them
act more readily and effectually on the corn. The resistance, in grinding
wheat, has been estimated at about a thirty-fifth of the weight of the mill-
stone. The stones have sometimes been placed vertically, and the axis
supported on friction wheels : but the common position appears to be more
eligible for mills on a large scale. It is said that a man and a boy can
grind by a hand-mill a bushel of wheat in an hour ; in a watermill, the
grinding and dressing of a bushel of wheat is equivalent to the effect of
20,160 pounds of water falling through a height of 10 feet, which is about
as much as the work of a labourer for a little more than half an hour. In
a windmill, when the velocity is increased by the irregular action of the
wind, the corn is sometimes forced rapidly through the mill, without being
sufficiently ground. There is an elegant method of preventing this, by
means of the centrifugal force of two balls, which fly out as soon as the
velocity is augmented, and as they rise in the arc of a circle, allow the end
of a lever to rise with them, while the opposite end of a lever descends with
the upper millstone, and brings it a little nearer to the lower one. The
bran or husk is separated from the flour by sifting it in the bolting mill,

ON MODES OF CHANGING THE FORMS OF BODIES. 179

which consists of a cylindrical sieve, placed in an inclined position and
turned by machinery. (Plate XVIII. Fig. 238.)

When the flour is made into bread, the dough requires to be kneaded :
for this purpose a machine is sometimes used in which four or more bars,
parallel to the axis of motion, are turned round by means of a walking
wheel. The dough is placed in a circular trough, in which the bars
revolve not quite in the middle, so as to approach in each revolution to one
of its sides, and thus the dough is perpetually compelled to change its
form.

A machine of nearly the same construction is employed for levigating
flints, after they have first been made red hot, and plunged into cold water,
in order to render them friable. They are mixed, when it is necessary,
with other large stones, and the water, in which the process is performed,
carries off the powder, and deposits its coarser parts in a short time, while
the finer remain much longer suspended, and are thus separated from the
rest.

When a mechanical structure is to be demolished, or a natural substance
to be broken into smaller parts, we have often occasion to employ the col-
lected force of men, the powers of machinery, or the expansive force of
chemical agents. Battering rams, or wooden beams suspended by ropes
and armed with iron, which were used by the warriors of antiquity in be-
sieging a town, are now generally superseded by the introduction of
artillery, although they may perhaps still afford, in some cases, a more
economical and equally powerful mode of operation. The same mo-
mentum, and the same energy, may be given to a battering ram at a less
expense than to a cannon ball ; but it is probable that the efficacy of a
cannon ball is chiefly owing to the augmentation of its velocity beyond
that limit, which is the utmost that the substance to be destroyed can sus-
tain without giving way, independently of the mass of the body which
strikes it.

For demolishing smaller aggregates, pincers, hammers, and crows, are
generally sufficient ; to these sometimes more complicated instruments are
added. Thus, for example, several machines have been invented for draw-
ing out ship's bolts. A hook which grapples like the common instrument
for drawing teeth, has been applied for holding them firmly, and sometimes
a screw, turned by means of wheelwork, has been used for gaining a force
sufficient to overcome their adhesion. In all such cases, however, the effect
of percussion has a considerable advantage ; and even if other means are
employed, it is of use to begin with lessening the firmness of the adhesion
by the blows of a hammer ; and in this manner a screw may be extracted,
which is so firmly attached by its rust as to be immoveable by other
means.

The expansive force of heat is frequently of great service in dividing
rocks, or in destroying old buildings. This is sometimes done simply by
the application of fire, as in the mine of Rammelsberg in the Hartz, where
the stratum containing the ore is of such a nature, partly, perhaps, on ac-
count of the combustible matter which enters into its composition, that, by
the effect of a large quantity of fuel which is burnt in the vast excavation,

N2 "

180 LECTURE XX.

of which it forms the side, it is rendered so friable as to be worked with
ease. More commonly, however, the force of gunpowder is employed, and
rocks are generally blasted with great convenience by an explosion of this
powerful agent. A hole being bored to the depth of three or four feet, the
powder is placed at the bottom, and a wire being introduced, small stones
and sand are rammed round it, and the wire is withdrawn, leaving a com-
munication for firing the powder by means of a train of sufficient length to
insure the safety of the workman. It is said that the explosion is more
efficacious when the powder does not fill the whole of the cavity ; this,
however, appears to require confirmation. The chemical powers which are
the ultimate causes of the operation of gunpowder, belong to a department
of philosophy which it is not our business to investigate : but the elasticity
of the gases and vapours which are extricated, as modified by the heat
which accompanies their production, will be considered and explained in
the subsequent divisions of this Course of Lectures.

LECT. XIX.— ADDITIONAL AUTHORITIES.

The subjects embraced in this Lecture and Lecture XVI. are of so miscellaneous
a nature, that a detailed list of authorities would be very tedious. We refer for ge-
neral information to the Encyclopedic Methodique Arts et Metiers. Machines ap-
prouvees par 1' Academic Royale des Sciences, 4to and fol. 1735-89. Bailey's Plates
of Machines, approved by the Society of Arts, 2 vols. fol. 1782. Repertory of Arts,
1794 Nicholson's Journal, 1797 Philosophical Magazine, 1798 An-
nals of Philosophy, 1800 Mechanics' Magazine. Newton's Patents. Engi-
neers' and Architects' Journal. The Encyclopaedias Britannica and Metropolitan,
&c. &c.

Treatises. — Borgnis, Theorie de la Mecanique Industrielle, 4to, Par. 1821. Du-
pin, Introduction d'un Nouveau Cours de Geometric et de Mecanique appliquees
aux Arts, 1824. Second Discours, 1825. Geometric et Mecanique appliquees aux
Arts, 3 vols. 1825-8. Christian, Traite de Mecanique Industrielle, 3 vols. 4to,
1822-5. Hachette, Traite Elementaire des Machines, 4to, 1828. Barlow on Ma-
nufactures and Machinery, 4to, 1836. Ure's Dictionary of Arts, Manufactures,
and Mines, 1839. Supplement, 1844.

LECTURE XX.

ON THE HISTORY OF MECHANICS.

THE order which we have pursued in considering the various depart-
ments of mechanical science, has been in great measure synthetical, dic-
tated by the plan of proceeding logically from the most simple principles to
their more complicated combinations, so as to build at every step on foun-
dations which had been firmly laid before : and this method is unquestion-
ably the best adapted for the expeditious progress of a student in sciences
with which he is unacquainted. But having once acquired a certain
degree of knowledge, he is anxious to be informed by what steps that
knowledge was originally obtained, and to what individuals mankind is in-
debted for each improvement that has been successively made. Hence,

ON THE HISTORY OF MECHANICS. 181

although we cannot attempt to enter into a complete history of mechanics,
it may still be satisfactory to take a short retrospect of a few of the most
remarkable eras in mechanical philosophy, and in those parts of mathe-
matics on which it immediately depends.

It is universally allowed that the Greeks derived the elements of mathe-
matical, mechanical, and astronomical learning from Egypt and from the
East.* Diogenes Laertius, who appears to be very desirous of claiming for
his countrymen the merit of originality, does not deny that Thales and
Pythagoras acquired much of their knowledge in their travels. Thales of
Miletus is the first that can be supposed to have introduced these studies
into Greece. Moeris, who was probably a king of Egypt, and Theuth or
Thoth, a native of the same country, are mentioned as having laid the foun-
dations of geometry ; but the science could scarcely have extended, in
those ages, further than was barely necessary for the measurement of land :
since Thales, or even a later philosopher, is said to have first discovered
that two lines drawn from the extremities of the diameter of a circle, and
meeting in any other part of its circumference form with each other a right
angle. Thales was one of the seven whom antiquity distinguished by the
appellation of wise men ; he flourished about 600 years before the Christian
era, and he was the father of the Ionian school, the members of which, in
subsequent times, devoted themselves more particularly to the study of
moral than of natural philosophy.

The Italian school, on the contrary, which was founded by Pythagoras,
appears to have been more inclined to the study of nature and of its laws ;
although none of the departments of human knowledge were excluded from
the pursuits of either of these principal divisions of the Grecian sages, until
Socrates introduced into the Ionian school a taste for metaphysical specu-
lations, which excluded almost all disposition to reason coolly and clearly
on natural causes and effects. To Pythagoras philosophy is indebted for
the name which it bears ; his predecessors had been in the habit of calling
themselves wise, he chose to be denominated a lover of wisdom only. He
had studied under Pherecydes, and Pherecydes under Pittacus : but with
respect to mathematical and mechanical researches, it does not appear that
either of his teachers had made any improvements. On his return from
his travels in Egypt and the East, in the time of the last Tarquin, about
500 years before Christ, he found his native country Samos under the do-
minion of the tyrant Polycrates, and went as a voluntary exile to seek a
tranquil retreat in a corner of Italy. At Croto, says Ovid, he studied and
taught the laws of nature.

" From human view what erst had lain concealed
His piercing mind to open light revealed ;
To patient toil his ardent soul constrained,
Of Nature's richest stores possession gained :
And thence, with glowing heart and liberal hand,
He dealt her treasures o'er the listening land.
• The wondering crowd the laws of nature hears,

And each great truth in silent awe reveres."

* See Kelland's Lectures on Demonstrative Mathematics, Edinb. 1843. Lect. I.

182 LECTURE XX.

However erroneous the opinion may be, that Pythagoras was acquainted
with the laws of gravitation, it is certain that he made considerable im-
provements both in mathematics and in mechanics, and in particular that
he discovered the well known relation between the hypotenuse and the
sides of a right angled triangle, and demonstrated that the square of the
hypotenuse is always equal to the sum of the squares of the sides. This
theorem is more essential to the perfection of geometry than any other pro-
position that can be named ; and if we may judge by the story of his
having sacrificed a hecatomb to the Muses on occasion of the discovery, he
seems to have had a foresight of the magnificence of the edifice that was in
subsequent times to-be built on this foundation.

Democritus of Abdera lived about a century after Pythagoras, whose
works he studied and whose principles he adopted. He appears to have
been possessed of very extensive knowledge and profound learning ; but
little remains of his works excepting their titles. Some have attributed to
him the invention of the method of arranging stones so as to form an arch.
Seneca thinks that so simple an invention must have been practised in
earlier ages : but Mr. King has endeavoured to show that its general intro-
duction in building was of much later date. Architecture and other
mechanical arts had however been considerably advanced some time before
this period, if it is true that Ctesiphon or Chersiphron, who built the
temple of Ephesus, was contemporary with Croesus and Thales. It is un-
certain at what time bridges of stone were first built ; and it is doubtful
whether the art of building bridges of wood was very well understood in
those ages : for according to Herodotus, it was commonly believed that
Thales avoided the necessity of procuring a passage over the Halys for the
army of Croesus, by encamping them on its banks, and cutting a channel
for the river in their rear, although the historian himself is of opinion, that
they passed over bridges which already existed. Curtius speaks of a bridge
of stone over the Euphrates at Babylon, which appears to have been built
long before the time of Alexander, whose expedition he relates ; and it is
scarcely probable that a stone bridge could have withstood the impulse of
so rapid a river, if it had been supported by columns only, without arches,
since they must have left too small a space for the passage of the water. If
however, we may believe Herodotus, whom Mr. King has quoted, this was
in reality a kind of drawbridge. According to this author, it was built by
Nitocris, the immediate successor of Semiramis : the stones were united by
iron and lead, and beams were laid across them which were removed at
night, in order to prevent the mutual depredations of the inhabitants of dif-
ferent parts of the city. We are informed by Pliny that Ctesiphon lowered
his large blocks of stone by placing them on heaps of sand bags, and
letting out the sand by degrees ; it does not appear how he raised them, but
the inclined plane seems to afford the simplest and most obvious method.

Archytas of Tarentum and Eudoxus of Cnidus were also Pythagoreans.
They were the first that attempted to make the mathematical sciences
familiar by popular illustrations ; and Archytas is said by some to have
invented the pulley and the screw. They lived nearly 150 years after
Pythagoras, and geometry had made in the mean time very rapid advances,

ON THE HISTORY OF MECHANICS. 183

for the properties of the conic sections were well known to these philoso-
phers. " The first persons," says Plutarch, " that cultivated the method
of organic geometry, were of the school of Eudoxus and Archytas. These
philosophers introduced elegance and variety into science, by illustrations
derived from sensible objects, and made use of mechanical contrivances for
expediting and familiarising the solutions of problems, which, if more
mathematically treated, are complicated and difficult : each of them in-
vented a method of determining in this manner the magnitude of two
mean proportionals between two given lines, by the assistance of certain
curves and sections. Plato by no means approved of their mode of pro-
ceeding, and reprehended them severely, as giving up and perverting the
most essential advantages of geometry, and causing the science to revert
from pure and incorporeal forms to the qualities of sensible bodies, sub-
jected to narrow and servile restraints. It was for this reason that practi-
cal mechanics were separated from geometry, and were long neglected by
philosophers, being considered as a department only of the art of war."

Aristotle, who was almost the last of the Ionian school, flourished a
little less than half a century after Archytas ; he was perhaps the author
of no original discoveries relating to the principles of mechanics, but we
find, in his treatise on this science, the law of the composition of motion
very distinctly laid down ; he makes, however, some mistakes respecting
the properties of levers. His general merit in elegant literature, as well as
in natural history and natural philosophy, is too well known to require
encomium.

The foundation of Alexandria commences a period memorable for
science in general, but more particularly for mathematics and astronomy.
Dinocrates was the architect whom Alexander employed in laying out and
in building this celebrated city. Among those who studied in this school,
the sciences are indebted to none more than to Euclid, who lived about 300
years before our era. It is uncertain how much of his Elements may
have been derived from his own investigations ; but the masterly manner
in which this well known work is arranged, and the precision and accuracy
which reign in every part of it, demand almost as great a share of praise
as is due to original discovery.

Epicurus was a contemporary of Euclid, and is considered as the last of
the Pythagorean or Italian philosophers. The penetration that he dis-
covered in assigning the true causes to many mechanical phenomena, his
explanations of which are copied by Lucretius, is sufficient to induce us
to look forwards with impatience to the publication of such of his works,
as have lately been discovered among the manuscripts of Herculaneum.
Apollonius of Perga lived about half a century later ; the elegance and
extent of his investigations of the most abstruse properties of the conic
sections left but little to be added to them by more modern geometricians.
The architect Philo appears to have been more'ancient than Apollonius ;
but he is not the Philo whose essay on warlike engines is published in the
collection of the Ancient mathematicians ; since this author was a pupil
of Ctesibius.

For the demonstration of the fundamental properties of the lever and of

184 LECTURE XX.

the centre of gravity ; for the discovery of the laws of hydrostatics, and
of the modes of determining the specific gravities of bodies ; for the con-
struction of the first cranes and of the first planetarium ; and for those
improvements of the methods of mathematical investigation which have
been the basis of every modern refinement in analytical calculation ; for
all these additions to our knowledge and our powers, we are indebted to
Archimedes. On a character so conspicuous, we can with pleasure dwell
long enough to attend to some particulars of his history, which are related
by Plutarch in his account of the siege of Syracuse ; omitting, however,
such details as are evidently fabulous. "Archimedes," says Plutarch,
" armed with his own inventions only, made light of the splendour of the
Roman preparations, and of the glory of the name of Marcellus. And
these were inventions that he even considered as of subordinate value, as
geometrical playthings, which had been the amusements of his leisure hours.
It was king Hiero that first induced him to transfer a portion of his science
from intellectual to material objects, and to condescend in some degree to the
comprehension of the multitude, by giving a sensible form to those truths
which in their abstract state are discoverable only to the reasoning faculty.
Archimedes, who was a friend and a relation of Hiero, had asserted that
any weight whatever might be moved by any given power : and depending
on the validity of his arguments, had given scope to his imagination, and
boasted that if he had another earth to which he could step over, he would
draw the whole of the present globe out of its place. Hiero, surprised at
the boldness of his assertion, requested him to give some substantial proof
of its truth, by moving a great weight with a small power ; upon this
Archimedes procured a ship, which was with great labour drawn up on the
shore, and having completely manned and freighted her, he seated himself
at a distance, and by lightly touching the first movement of a machine, he
drew her along as smoothly and as safely as if she had been sailing in the
deepest water. Hiero, full of astonishment, and admiring the powers of
mechanical art, prevailed on Archimedes to construct such engines both of
defence and of offence, as might be of use to him in case of a siege : for
these, however, Hiero, who lived a life of peace and prosperity, was not so
unfortunate as to have occasion ; but they now became highly valuable to
the Syracusans, and they were of the more advantage, as their inventor
was present to direct their use. And in fact the whole people of Syracuse
constituted but a part of Archimedes's corporeal machinery, and he was
the soul that moved and governed the whole. All other arms were deserted,
and they employed his engines alone, both for their own defence, and for the
annoyance of the enemy. In short the Romans soon became so terrified,
that if they saw a stick or a rope upon the walls, they cried out that it was
some machine of Archimedes, and immediately fled ; so that Marcellus
at last determined to desist from attempting to take the place by assault,
and resolved to blockade it only.

" Archimedes, however, had such depth of intellect, and such sublimity
of mind, that notwithstanding he had obtained by these inventions, the
credit and glory of an intelligence rather divine than human, he thought it
unworthy of him to leave any written treatise on the subject, considering

ON THE HISTORY OF MECHANICS. 185

practical mechanics and every art that is concerned in satisfying the wants
of life, as ignoble and sordid ; and resting all his hopes of fame on those
works, in which the magnificent and the elegant are exhibited uncontami-
nated by the imperfections of the material world : works that are com-
parable to nothing else that the mind of man has produced ; in which the
subject only contends with the mode of treating it, the magnitude and
beauty of the one being rivalled by the accuracy and vigour of the other.
It is impossible that propositions more difficult and important should be
deduced from simpler and purer elements. Some attribute this excellence
to his natural genius, others to his indefatigable application, which has
given to every thing that he has attempted the appearance of having been
performed with ease. For we might ourselves search in vain for a demon-
stration of his propositions ; but so smooth and direct is the way by which
he leads us, that when we have once passed it, we fancy that we could
readily have found it without assistance. We may, therefore, easily give
credit to what is said of him, that being as it were fascinated by this
domestic syren that bore him company, he often neglected his food and
his clothing ; that when sometimes brought by compulsion to the baths, he
used to draw his figures in the ashes of the fire places, and to make his
calculations upon the cosmetics that were employed by the attendants ;
deriving, like a true votary of the muses, every pleasure from an intellec-
tual origin. Among all his beautiful discoveries, he is said to have chosen
that of the proportion of the sphere and cylinder for his sepulchral honours ;
requesting of his friends that they would place on his tomb a cylinder con-
taining a sphere, and inscribe on it the ratio which he had first determined.

" By artifice, and through the thoughtlessness and security of a day of
festivity, the Romans at length obtained possession of Syracuse, and in the
pillage, although orders had been issued that the life of Archimedes should
be spared, he was killed by a private soldier. His death is variously
related, but all accounts agree that Marcellus was deeply concerned for his
loss, that he held his assassin in abhorrence, and conferred distinguished
favours on his surviving relations." This event is supposed to have hap-
pened about 212 years before the birth of Christ ; and the cultivation of
mechanical philosophy, which had been continued for four hundred years
with increasing success, was almost wholly interrupted for eighteen cen-
turies.

There lived, however, in the mean time, some mathematicians and
mechanics of considerable merit. A work on warlike machines, addressed
to Marcellus by Athenaeus, is still extant, and may be found in the splendid
collection of writers on military mechanics entitled Mathematici Veteres.
Ctesibius of Alexandria was about a century later than Archimedes ; he
enriched hydraulics with several valuable machines ; although he contri-
buted little to the advancement of theoretical investigation. Hero was of
the same school, and his pursuits were similar ; some of his treatises on
hydraulics, pneumatics, and mechanics, are published in the collection of
Ancient mathematicians, and some others are still extant in manuscript.
We are informed by Pappus, that Hero and Philo had referred the proper-
ties of the lever, the wheel and axis, the pulley, the wedge, and the screw,

186 LECTURE XX.

to the same fundamental principle ; so that the theory of the mechanical
powers began at that time to be extremely well understood. The treatises
of Hero on pneumatics and on automatons contain many very ingenious
inventions, but they are rather calculated for amusement than for utility ;
among them is a cupping instrument, which operates nearly in the manner
of an air pump. A work of Bito, on warlike machinery, addressed to
king Attalus, is included in the same collection.

Vitruvius was an author of great general knowledge : he lived under
one of the earliest of the Caesars, and the greatest part of our information
respecting the mechanics of antiquity has been derived from his works.
Apollodorus was employed by Trajan, in building a bridge over the
Danube, in the year 102 ; he has left a treatise on besieging a town, which
is to be found among the Ancient mathematicians. Diophantus, Pappus,
and Proclus, were mathematicians of eminence : Diophantus confined him-
self in great measure to arithmetic and pure geometry ; but the last book
of Pappus's collections is devoted to mechanics, and Proclus wrote a
treatise on motion, which is still extant.* The rudiments of algebraical
notation and calculation may be found in the works of Diophantus ; but
the Arabians appear to have first practised the method of denoting quanti-
ties in general by literal characters ; they made, however, no considerable
advances, and mathematics in general remained nearly stationary until the
time of the revival of letters.

During the long interval, in which learning and science were involved
in the darkness of the middle ages, the arts subservient to the convenience
of life were also in great measure neglected. It is evident from many
remains of antiquity, that various manufactures had attained in Greece
and at Rome, a high degree of perfection ; but the irruptions of the bar-
barians were as effectual in suppressing the refinements of civilisation, as
in checking the pursuit of literary acquirements : our own country was
not the earliest in recovering the arts which had been lost, but it has
always received with open arms those who have excelled in them ;
and the improvements which have been made, within a few centuries, in
the British manufactures, have obtained for them a celebrity unrivalled by
those of any other nation. The ancient Britons are supposed to have
made, in common with the other Celtic nations, coarse cloths and felts of
wool, and perhaps some articles of linen ; their chariots of war, wrhich are
mentioned by Caesar, could not have been executed without some skill in
the arts of the carpenter and the smith. The Romans introduced a certain
degree of civilisation into England, but it appears to have been in a great
measure forgotten soon after they left the country. In the seventh cen-
tury, several architects and workmen were brought from the continent by
Wilfrid and Biscop ; they restored the practice of building with stone,
which had been generally superseded by wood, and laid the foundation for
other improvements. In the time of king Alfred, the English goldsmiths
began to excel, and before the conquest, the woollen manufactures had
acquired a considerable degree of perfection. The paper now in use w»as
introduced about the year 1100 ; it was probably imported from the con-
* De Motu Disputatio, Basileae, 1531.

ON THE HISTORY OF MECHANICS. 187

tinent, since the linen manufacture was little advanced in England till 150
years later ; but embroidery was much practised, although in the 12th
century silks were principally woven in Sicily. The manufactory of cloth
was considerably improved, in the 14th century, by the establishment of
Kempe and other Flemish weavers in England : and many of the arts
were benefited, about the same time, by the invention of the method of
drawing wire, which was first introduced at Nuremberg. In the succeed-
ing century, the increasing number of hands employed in various manu-
factures, suggested to some mind of superior penetration the great principle
of the division of labour, by which each individual is enabled to acquire
so high a degree of perfection in a very limited branch of each manufac-
ture, that the whole work is performed much more perfectly, as well as
more expeditiously, than if it had been begun and completed by any one
person, even of greater abilities and experience. The invention of the
modern spinning wheel is attributed to Jiirgen of Brunswick, and the year
1530 is assigned as its date : England soon profited by the improvement ;
many manufacturers took refuge in this country from the Duke of Alva's
persecutions in Flanders, and before the end of the century a new modifi-
cation of the art of weaving was introduced by Lee of Cambridge, who
invented the stocking loom, imitating the texture of the knit stockings,
which were first manufactured in Spain about the year 1550. Mills for
drawing wire and for slitting iron were also first erected in the sixteenth
century ; Birmingham and Sheffield were even at that time, according to
Camden, celebrated for their manufactures; and the machinery which
has been since introduced at different periods in those places, affords a
facility and expedition which astonish every unexperienced spectator. The
names of Watt and of Boulton have acquired a just celebrity from their
refined improvements ; but many other mechanics of inferior rank have
exhibited a degree of ingenuity which would have done honour to the
most distinguished talents. The manufactures of Manchester are also of
considerable antiquity ; but they are very greatly indebted to the inven-
tions of Ark wright and his followers, which have also been introduced in
many other parts of the united kingdom. The importance of these
improvements may be estimated from the quantity of cotton which is
annually imported into Great Britain ; in 1787, it amounted to 23 million
pounds, and gave employment to 420 thousand manufacturers; in 1791,
it was increased to 32 millions : about one half is consumed in white goods,
one fourth in fustians, and the remainder in hosiery, mixtures, and candle
wicks. But the woollen manufactory affords a subsistence to above a
million persons, who receive annually for their work about nine millions
sterling, and employ as much wool as is worth about three.*

In architecture, the Anglonorman style prevailed in this country from
the conquest to the beginning of the thirteenth century ; the arch was fre-
quently employed, and its form was semicircular. 'The Gothic architecture,
distinguished by its pointed arches, which is said to have originated from
thG Saracens, was first introduced into England about the year 1170, and

* See Baines's History of the Cotton Manufacture, 1835 ; or art. Cotton Manu-
facture, Encyc. Brit.

188 LECTURE XX.

was more and more generally adopted for about three centuries. Of the
architects of this school, two of the most celebrated were William of Sens,
and Walter of Coventry : the most elegant specimen of its performances
is, perhaps, King's College Chapel at Cambridge, which was founded by
Henry the Sixth, and begun in the year 1441. The Cathedral of Lincoln
appears to have been one of the earliest Gothic edifices : Westminster Abbey
was finished about 1285, the Minster of York was begun a few years after-
wards ; and it is difficult to determine which of these three buildings most
deserves the attention of the antiquary and the architect, or whether the
Cathedral at Canterbury may not be equal to either of them.

In the midst of an age of darkness, an insulated individual arrests our
attention by merits of no ordinary kind. Roger Bacon was born at
Ilchester, in the year 1214 ; it is well known that his experiments had led
him to a discovery of the properties of gunpowder, although he humanely
concealed the nature of its composition from the public, and described it
only in an enigma : the extent of his optical knowledge has been variously
estimated, but it was unquestionably much greater than that of the ancient
philosophers. He appears, however, to have had some companions in his
mechanical pursuits ; he declares that he had seen chariots which could
move with incredible rapidity, without the help of animals ; he describes a
diving bell ; and he says that he had been informed, on good authority, that
machines had been made, by the assistance of which men might fly through
the air. Cimabue, who first began to revive the long neglected art of
painting, was contemporary with Bacon. The use of oil in painting is
commonly supposed to have been introduced by Van Eyck, but there are
traces, in the records of this country, of its employment as early as the
year 1239.*

The clepsydrae or water timekeepers of the ancients appear to have been
gradually transformed, in the middle ages, into the clocks of the Saracens
and of the Arabians : and these were introduced into Europe in the
thirteenth century. About the year 1290, turret clocks were erected at
Westminster and at Canterbury. The first clock, of which we know the
construction, is that which was made by Wallingford in 1326, and which
was regulated by a fly ; and the second that of Defondeur, or Fusorius,
with a simple balance, made about 1400. But it appears that some portable
watches had been constructed in the beginning of the fourteenth century ;
and about the year 1460, several clock makers are said to have come to
England from Flanders.

The art of engraving on metal, and of printing with the rolling press, is
supposed to have been invented in the year 1423. Some attribute the art
of printing with types, to Laurentius Coster of Haerlem,t who, as they
say, in 1430, employed for the purpose separate blocks of wood, tied
together with thread. Gensfleisch, one of his workmen, went to Mentz,
and was there assisted by Gutenberg, who invented types of metal. But
the best authors appear to disbelieve this story ; and Gutenberg, in partner-

* See Lect. XI.

t Ellis, Ph. Tr. xxiii. 1416. See also Ph. Tr. xxiii. 1507, Boxhoin, de Origine
Artis Typographicae.

ON THE HISTORY OF MECHANICS. 189

ship with Fust and Schaeffer, is the first that is universally allowed to
have practised the art.* It was introduced into this country by William
Caxton.

Leonardo da Vinci, the most accomplished man of his age, was born
about the year 1443, and excelled not only in painting and poetry, but
also in architecture, mathematics, and mechanics. The state of practical
mechanics in this and the subsequent centuries may be estimated from
Ramelli's collection of machines, which contains several curious and
useful inventions ; some of them long since forgotten, and even lately
proposed again as new.

The works of Bacon, Lord Verulam, although not immediately tending
to the advancement of mathematics or of mechanics, are universally allowed
to have conduced very materially to the improvement of every branch of
science, by the introduction of a correct and conclusive method of philo-
sophical argument and inquiry. Guido Ubaldi published, in 1577, a
treatise on mechanics, not wholly exempt from inaccuracies, and in the
following year a valuable commentary on the works of Archimedes : some
of the properties of projectiles were about the same time rather imagined
than demonstrated by Tartalea : Benedetti soon after began to reason
correctly respecting the principles of mechanics ; but it was reserved for
Galileo to lay the foundations of the discoveries, which have succeeded
each other with increasing rapidity for more than two centuries. He
investigated, in the year 1589, the laws of accelerating forces, and showed
the nature of the curve which is described by a projectile ; he inferred from
observation the isochronism of the vibrations of a pendulum, and the
principle was soon after applied by Sanctorius to the regulation of time-
keepers. Stevinus, a Dutchman, was the first that clearly stated the
important law by which the equilibrium of any three forces is determined ;
and the properties of the centre of gravity were successively investigated by
Lucas Valerius, Lafaille, and Guldinus, who made some additions to
the elegant propositions of Archimedes which relate to it.t

The application of the more intricate parts of the mathematics, to prac-
tical purposes of all kinds, has become incomparably easier and more
convenient since the invention of logarithms. This important improvement
was made by Baron Napier ; his tables were published in 1614 : J and they
were reduced to a still more useful form by the labours of Briggs§ and
of Gunter.|| Descartes, about the same time, was making considerable

* Fischer sur les Monumens Typographiques de Gutenberg, 4to, Mentz, 1802.

f The authors here mentioned occupy a prominent position in the History of Me-
chanics. We therefore add a list of their principal works. Lord Bacon's Works,
a new edition by Basil Montagu, 14 vols. 1825-31. Guido Ubaldi Mechanicorum
liber, fol. Pesaro. Tartalea Nuova Scienza, 4to, Venice, 1537. Quesiti et Inventi
Diversi, 1544. Benedettus Diversarum Speculationum liber, fol. Taurini, 1585.
Galileo Opera, 4 vols. Padova, 1744. See Lect. IV. Stevinus, Beghinselen der
Waagconst, 1586. (Euvres Mathematiques, 2 vols. fol.. Ley de. 1634. Lucas Va-
lerius, DeCentro Gravitate Solidorum, 4to, Romse, 1604. Lafaille, Theoremata de
Centre Gravitatis, 4to, Antwerp, 1632. Guldinus de Centro Gravitatis, fol.
Vienna;, 1635.

J Mirifici Logarithmorum Canonis Descriptio, 4to, Edinb. 1614.

§ Arithmetica Logarithmica, fol. Lond. 1624.

|| Works, 4to, 1 680. The tables of logarithms in common use are, Taylor's,
Collet's, Hutton's, and Babbage's.

190 LECTURE XX.

additions to the science of algebra, and the mathematics were soon after
enriched by Cavalleri's invention of the method of indivisibles. This
method was founded on the principles introduced by Archimedes, it was
further improved by Wallis, and it led to the still more valuable invention
of the fluxional analysis.

The laws of collision were investigated nearly at the same time in England
by Wren and Wallis, and in France by Huygens. After the discoveries
of Archimedes and of Galileo, those of Huygens hold the third place, in the
order of time, among the greatest benefits that have been conferred on
science. His theory of cycloidal pendulums and his doctrine of central
forces were the immediate foundations of Newton's improvements.

Hooke was as great in mechanical practice and in ingenious contrivance,
as Huygens was in more philosophical theory ; he was the first that applied
•.• the balance spring to watches, and he improved the mode of employing
pendulums in clocks ; the quadrant, the telescope, and the microscope,
were materially indebted to him ; he had the earliest suspicions of the true
nature of the cause that retains the planets in their orbits ; and the multi-
tude of his inventions is far too great to be enumerated in a brief history of
the progress of science.

The composition of motion, and several other mechanical and optical
subjects, are elegantly treated in the lectures published by the learned
Doctor Barrow.* He was professor of mathematics at Cambridge, and
voluntarily resigned his chair to make way for his successor, the pride of
his country, and the ornament of mankind. Sir Isaac Newton t was born
at Woolsthorpe in Lincolnshire, on Christmas day in 1642, the year of
Galileo's death. At the age of 12 he was sent to school at Grantham, and
at 18 to Cambridge. He made some important improvements in algebraical
analysis, and laid the foundation of his admirable method of fluxions,
before he was 24 years old ; but his modesty prevented him from imme-
diately publishing any work on these subjects. His first optical experi-
ments were also made in the year 1666, and they were communicated to the
Royal Society, then in its infancy, on his admission as a member in 1672.
The theory of gravitation, and the mechanics of the universe, are developed
in his Mathematical Principles of Natural Philosophy, first published in
1687. The following year he was chosen representative of the university
of Cambridge in parliament, and in 1696 he was placed, upon the recom-
mendation of the Earl of Halifax, in a lucrative situation in the Mint.
From 1703 until his death in 1727, he continued president of the Royal
Society, and enjoyed, to the age of 80, an uninterrupted state of good
health. He was knighted by Queen Anne, in 1705, and died possessed of a
considerable fortune. " He had the singular happiness," says Mr. Fonte-
nelle, "of obtaining, during his life, all the credit and consideration to which
his sublime researches and his fortunate discoveries entitled him. All men
of science, in a country which produces so many, placed Newton, by a kind
of acclamation, at their head ; they acknowledged him for their chief and

* Lectiones Mathematics xxiii. Lond. 1685.

f See Brewster's Life of Newton. A new edition, containing many important
facts hitherto unknown, is anxiously expected. Consult also Tumor's Collections
for the History of Grantham, 4to, Lond. 1806.

ON THE HISTORY OF MECHANICS. 191

J their master ; no opponent, nor even a cool admirer, dared to appear. His
philosophy was adopted throughout England, and it is supported in the
* Royal Society, and in all the excellent productions of the members of that
Society, with as much confidence, as if it had heeii consecrated by the
respect of a long course of ages." A remarkable instance of the extent and
refinement of Newton's mathematical acquirements may be found in a paper
of a celebrated modern mathematician, on the subject of atmospherical
refraction ; Mr. Kramp* observes, with a mixture of surprise and doubt,
that Newton appears to have been acquainted with those methods of alge-
braical calculation which he had himself pursued ; at the same time he
says that this is almost incredible, since " he must have discovered certain
improvements in the higher analysis which were unknown even to Euler,
and to every other mathematician before Laplace."

Although Newton was unquestionably the first inventor of the method of
fluxions, yet Leibnitz, whether he had received any hints of Newton's
ideas, as there is some reason to suspect, or whether his investigations were
wholly independent of those of Newton, was the first that published any
work on the subject ; and he extended its application to many important
problems, earlier, perhaps, than any English mathematician. James and
John Bernoulli also pursued the same studies with considerable success,
and the general laws of mechanics were very elegantly investigated, and
successfully applied by these three contemporary philosophers on the con-
tinent, while Machin, Cotes, Halley, and Demoivre, were applying them-
selves to similar pursuits in this country. Perrault, Lahire, Amontons,
and Parent, members of the Parisian academy of sciences, were the authors
of many useful investigations relating to practical mechanics ; but few of
them were made public till after the year 1700 ; some of their inventions
made their appearance much later, in the valuable collection of machines
approved by the academy, and some of them have been inserted in the
useful work published by Leupold, at Leipzig, under the title of a Theatrum
Machinarum. Throughout the last century, the transactions of various
societies, established for the promotion of science, became every year more
numerous, and the publication of the literary journals of Leipzig and of
Paris formed a mode of communication which was extremely serviceable
in facilitating the dissemination of all new discoveries.

The philosophy of Newton assumed also a more popular and attractive
form in the writings of Clarke,t Pemberton,^ Maclaurin, § and Musschen-
broek, || and the lectures of S'Gravesande and Desaguliers; at the same
time that its more refined investigations were pursued with success in this
country by Maclaurin and Simpson, and on the continent by Hermann,
Daniel Bernoulli, Leonard Euler, and Clairaut. Maclaurin, Bernoulli,
and Euler, had the honour of sharing with each other the prize, proposed
by the academy of sciences at Paris, for the best essay on the intricate
subject of the tides; but a premature death prevented Maclaurin from

* Hindenburgs Archiv. ii. 380, 499.

• f Demonstration of some Sections of Newton's Prin. 1730.
t View of Sir I. Newton's Ph. 4to, 1728.

§ Account of Sir I. Newton's Philosophical Discoveries, 4to, Lond. 1748.
|| Introductio ad Phil. Nat. 2 vols. Leyd. 1762.

192 LECTURE XX.

long pursuing the career which he began so successfully. Bernoulli and
Euler continued for many years to vie with each other for the elegance
and extent of their researches : Euler appears to have been the more pro-
found mathematician, and Bernoulli the more accurate philosopher.

The latter half of the eighteenth century was in many respects extremely
auspicious to the progress of the sciences ; the names of Dalembert, Landen,
Waring, Frisi, * Robison, Lagrange, and Laplace, deserve to be enumerated
in the first class of mathematicians and theoretical mechanics ; those of
Smeaton, Wedgwood, and Watt are no less distinguished for their success
in improving the practice of the useful arts and manufactures. The union
of all these objects, into one system of knowledge, was effected, on a mag-
nificent scale, in the Encyclopedic, a work which does as much honour to
the labour and genius of some of its authors, as it reflects disgrace on the
principles and politics of others. The Society for the encouragement of
arts, manufactures, and commerce, was established in London about the
same time that the Encyclopedic began to appear at Paris, and its pre-
miums and publications have, without doubt, excited a degree of attention to
the subjects of practical mechanics, and agricultural, as well as commercial
improvements, which must have been beneficial both to individuals and to
the public. The academy of Paris began to print, in 1762, a collection of
the descriptions of arts and trades of all kinds, on a still more extended
scale than had been attempted in the Encyclopedic ; the work was carried
to a very considerable length, but it by no means comprehends all the
articles which were intended to compose it.

The construction of watches has been so much improved by the artists
both of this country and of France, that they have been rendered capable
of affording very essential service to navigation, especially since the astro-
nomical methods of determining a ship's place have been brought to such
a degree of perfection, as greatly to facilitate the frequent correction of
the accidental errors of the timekeeper. The first artist that constructed
watches, sufficiently accurate for the determination of the longitude, was
William Harrison, who was indebted to himself alone for his education
and his inventions ; in 1765 he received for his labours, from the Board of
Longitude, the promised reward of ten thousand pounds.

There has scarcely been a period, in any age of the world, in which the
sciences and literature in general, have been so rapidly promoted and so
universally disseminated, as within the last forty years. This advance-
ment has partly been the cause, and partly the effect, of the great multi-
plication of scientic journals, cyclopaedias, and encyclopaedias, which have
been annually increasing since the beginning of the Journal de Physique in
1773 ; supported by the interest which they have derived, in great measure,
from the new and amusing discoveries and improvements which have been
made in chemistry and natural history : some of the most copious of these
works have had a sale unprecedented even for books of more moderate
extent.

The charter of the Royal Institution is dated in 1799 ; its foundation
will not perhaps make an era in the history of the refinements of science ;

* Pauli Frisii Opera, 3 vols. 4to, Mediolani, 1782-5.

ON THE HISTORY OF MECHANICS. 193

but if it be hereafter found to have given notoriety to what is useful, and
popularity to what is elegant, the purposes of those who established it will
not have been frustrated.

After all that has been effected by the united labours and talents of the
philosophers who have been mentioned, and of many more, who, though
less fortunate, have yet been highly meritorious, there is still ample oppor-
tunity for the employment of genius and industry in following their steps.
To suppose that little or nothing remains to be done, betrays a want either
of knowledge, or of courage. The experimental researches of some of the
greatest philosophers have been very imperfectly conducted, and the most
interesting results may be expected from repeating and diversifying them.
Whatever advances our neighbours may have made beyond us, in intricate
calculations and combinations, we are still able to vie with them, and shall
probably long remain so, in the accuracy of our instruments, and in the
art of using them with precaution and with success.

When, however, we contemplate the astonishing magnitude to which a
collection of books in any department of science may even at present be
extended, and the miscellaneous nature of the works in which many of the
most valuable disquisitions have been communicated to the public, together
with the natural disposition to indolence, which a high degree of civilisa-
tion too frequently encourages, there is the greatest reason to apprehend,
that from the continual multiplication of new essays, which are merely
repetitions of others that have been forgotten, the sciences will shortly be
overwhelmed by their own unwieldy bulk, that the pile will begin to totter
under its own weight, and that all the additional matter that we heap on
it, will only tend to add to the extent of the basis, without increasing the
elevation and dignity of the fabric. Having been impressed, from con-
tinued experience, with the truth of this observation, I have employed no
small portion of time and labour, in order to obtain an effectual remedy
for the evil ; and I trust that, in future, every one who is desirous of en-
larging the sphere of our knowledge, with respect to any branch of science,
connected with the subject of these Lectures, will find it easy, by consult-
ing the authors who will be quoted in my catalogue of references, to collect
that previous knowledge of all that has been already done with the same
view, which, in justice to himself, he ought to acquire before he enters on
the pursuit, or at any rate, in justice to the public, before he calls on the
world at large to participate in his improvements and discoveries.

LECT. XX.— ADDITIONAL AUTHORITIES.

History of Mechanics. — P. Vergilius, De Inventpribus rerum, Basle, 1521.
Sprat's History of the Royal Society, 4to, Lond. 1667.x Histoire des Ouvrages des
Savans. Journal des Savans, Sep. 1688. Harris, Lexicon Technicum, 3 vols. fol.
1704, &c. Pancirollus, History of Memorable Things, 3 vols. 12mo, 1715. Reg-
nault, Origine Ancienne de la Physique Nouvelle, 3 vols. Amst. 1735. Goguet,
Origine des Lois, des Arts, et des Sciences, 3 vols. 4to, 1755. Mattaire, Mar-

(chand, Bowyer, Ames, Lemoine, and Lucombe, on the History of Printing. Birch's
History of the Royal Society, 4 vols. 4to, 1756. Rollin's History of the Arts and
Sciences of the Ancients, 3 vols. 1768. Priestley's Chart of Biography. Diction-

194 LECTURE XX.

naire des Origines des Inventions Utiles, 6 vols. 12mo. Par. 1777. Brugmans on
the Mechanics of the Ancients, Comm. Gott. 1784, vii. M. 75. Mongez on Ancient
Coining. Roz. Journal de Physique, xl. 426. Dutens on the Origin of Discoveries,
4to. Delambre, Rapport Historique sur les Progres des Sciences Mathematiques
depuis 1789. Beckmann's Hist, of Inventions (translated by Johnstone), 4 vols.
1797. Poppe, Geschichte der Uhrmackerkunst, 1801. Montucla and Lalande,
Histoire des Mathematiques, 4 vols. 4to, Paris, 1802. Bossut's History of Mathe-
matics, translated by Bonnycastle, Lond. 1803. Libes, Histoire des Progres de la
Physique, 4 vols. 1810. Hutton's Mathematical and Philosophical Dictionary,
2 vols. 4to, 1815. Powell's History of the Physical and Mathematical Sciences
(Cab. Cyc.), 1834. Whewell's History of the Inductive Sciences, 3 vols. Lond.
1837.

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[To face page 1Q40

PART II.

LECTURE XXI.

ON HYDROSTATICS.

THE mechanical properties and affections of fluids, and the laws, and
phenomena of their motions, are to be the subjects of the second division
of this Course of Lectures. Although these properties are in reality
derived from the same fundamental principles as the doctrines of pure
mechanics, they are yet in great measure incapable of being referred, in a
demonstrative and accurate manner, to the operation of simple and general
causes. We are therefore frequently under the necessity of calling in the
assistance of experimental determinations ; and for this reason, as well as
others, the science of hydrodynamics may with propriety hold a middle
rank, between mathematical mechanics and descriptive physics. In treat-
ing of the mechanics of solid bodies, we are able to begin with axioms or
self-evident truths, almost inseparable from the constitution of the human
mind ; to deduce from them the general laws of motion, and to apply these
laws, with little chance of error, to every combination of circumstances in
which we have occasion to examine their consequences ; and it requires
only a certain degree of attention and of mathematical knowledge, to be
perfectly convinced of the justice of all our conclusions, without any
reference to experimental proof. But here our abstract reasonings begin to
fail ; and whether from the imperfection of our modes of considering the
mechanical actions of the particles of fluids on each other, or from the
deficiencies of our analytical calculations, or, as there is more reason to
suppose, from a combination of both these causes, all attempts to reduce
the affections of fluids to a perfect mechanical theory have been hitherto
unsuccessful. At the same time it will appear, that by a proper mixture
of calculation with experiment, we may obtain sufficient foundations for
all such determinations as are likely to be of any practical utility.

The whole of the subjects, which will be classed under the denomination
Hydrodynamics, may be divided into three general heads ; Hydraulics,
Aoustics, and Optics ; terms which are sufficiently understood, as relating
to the common properties of fluids, to sound, and to light ; but which do
not allow of a very strict definition, without a still further division. The

o2

196 LECTURE XXI.

first subdivision which we shall consider, will relate to the laws of the
equilibrium of fluids, or of the opposition of forces acting on them without |
producing actual motion, comprehending hydrostatics, or the doctrine of '
the equilibrium of liquids, either within themselves or with moveable bodies ;
and pneumatostatics, or the equilibrium of elastic fluids. The actual
motions of fluids will be considered in the second subdivision : and the
third will relate to the instruments and machines in which the principles
of hydrostatics, hydraulics, and pneumatics, are applied to the purposes of
the arts or of domestic convenience. The science of hydraulics must be
allowed to be of as great importance to civil life, and especially to a mari-
time nation, as any department of practical mechanics. Let us only reflect
for a moment to what the metropolis of England would be reduced, if
deprived of pipes for the conveyance of water, of pumps, and of fire
engines ; and how much the commerce of the whole kingdom has been
facilitated by the formation of navigable canals, and we shall soon be con-
vinced of the obligations that we owe to the art of modifying the motion
of water, and to the principles of hydraulics on which that art depends.

The facts concerned in acustics and harmonics, or the doctrine of sound
and the science of music, are not exclusively dependent on the characteristic
properties of fluids. In these departments, although we can by no means
explain with precision the manner in which every appearance is produced,
we shall still find a variety of very beautiful phenomena, which have
indeed been too generally neglected, and supposed to be of the most
abstruse and unintelligible nature ; but which, when carefully examined,
will appear to be much more within the reach of calculation, than the
simplest doctrines of hydraulics. We may also apply some of these
phenomena to a very complete explanation of an extensive class of facts
in optics, which, in whatever other way they are considered, are inextri-
cably obscure. Whether this explanation may or may not be admitted as
satisfactory, it deserves at least a fair examination; it would, therefore,
be impossible to assign to the science of optics an earlier place in the order
of the system, even if we agree with those who imagine that all the pheno-
mena of light depend on causes wholly deducible from the mechanics of
solid bodies.

We must commence the subject of hydrostatics, or the doctrine of the
equilibrium of liquids, with a definition of the essential characteristics of a
fluid substance. The most eligible definition appears to be, that a fluid is
a collection of material particles, which may be considered as infinitely
small, and as moving freely on each other in every direction, without
friction. Some have defined a fluid as a substance which communicates
pressure equally in all directions ; but this appears to be a description of a
property derivable from the former assumption, which is certainly more
simple ; and although it may be somewhat difficult to deduce it mathe-
matically, in a manner strictly demonstrative, yet we may obtain from
mathematical considerations a sufficient conviction of its truth, without
assuming it as a fundamental or axiomatic character.* A fluid which has

* See Miller's Hydrostatics, Camb. 1831. Challis's Report on Hydrostatics and
Hydrodynamics, Brit. Assoc. 1833, p. 134.

ON HYDROSTATICS. 197

no immediate tendency to expand when at liberty, is commonly considered
as a liquid : thus water, oil, and mercury, are liquids ; air and steam are
fluids, but not liquids.

We shall for the present consider a liquid as without either compressi-
bility or expansibility : and we must neglect some other physical properties
essential to liquids, such as cohesion and capillary attraction ; although in
reality the partigles of liquids are found, by very nice experiments, to be
subject to the same laws of elasticity which we have already examined with
regard to solids, and are possessed also of cohesive powers, which essentially
distinguish them from elastic fluids, and which resist any force tending
directly to separate the particles from each other, while they admit any
lateral motion with perfect facility. In treating of hydrostatics, therefore,
we suppose the fluids concerned to be of uniform density throughout ; and
as far as elastic fluids agree with this description, they are subject to the
same laws with liquids ; on the other hand, all fluids, as far as they are
compressible, possess properties similar to those which will hereafter be ex-
amined, when we investigate the subject of pneumatic equilibrium.

The first law of hydrostatics which arrests our attention, is this, that the
surface of every homogeneous gravitating fluid when at rest, is horizontal.
If any part of the surface were inclined to the horizon, the superficial par-
ticles would necessarily tend towards its lowest part, in the same manner
as if they moved without friction on the inclined surface of a solid. And
if any two portions of the surface of the fluid are separated, as in two
branches of a tube or pipe, however they may be situated, the fluid cannot
remain at rest unless the surfaces be in the same level plane : for if we
imagine such a tube, containing water, to be made of ice, and to be
immersed in a large reservoir of water, and then thawed, the water will
make a part of the general contents of the reservoir, and consequently will
remain at rest, if its surfaces are level with that of the reservoir : and it is
obvious that the tube has acquired no new power of supporting it from
being thawed ; consequently, the water would have remained in equili-
brium at the same height in the original state of the solid tube. The
experimental proof of this proposition is easy and obvious, and the property
affords one of the most usual modes of determining a horizontal surface.
But when we compare the heights of fluids occupying tubes of different
magnitudes, it is necessary, if the tubes are small, to apply a slight cor-
rection on account of the actions of the tubes on the fluids which they con-
tain, which are more apparent as their diameters are smaller. The same
cause produces also a curvature in each separate surface, which is always
visible at the point of contact with the tube or vessel. (Plate XIX. Fig.
239.)

If several separate fluids of different kinds be contained in the same
vessel, they will never remain at rest unless all the surfaces intervening
between them be horizontal ; and this is in fact the state of the surface of
common liquids, which is exposed to the pressure of the atmosphere.

The power of gravitation, strictly speaking, does not act precisely in
parallel lines, so that the surface of lakes, instead of being perfectly plane,
becomes, like that of the earth, a little convex. It is obvious that the sur-

198 LECTURE XXI.

face of a fluid must always be perpendicular to the direction of the joint t
results of all the forces which act on it ; and since the earth turns round
on its axis, the centrifugal force resulting from its motion is combined with
that of gravity, in determining the position of the general surface of the
ocean.

A similar combination of a centrifugal force with gravitation may be
observed when a bucket is suspended by a rope, and caused to turn round
on its axis by twisting the rope : the direction of the joint forces is such
that the surface, in order to be perpendicular to it, must assume a parabolic
form. When also any number of different fluids are made to revolve in
the same manner, or when they are inclosed in a glass globe and turned by
means of the whirling table, the surfaces which separate them, acquire
always the forms of parabolic conoids, when the axis remains in a vertical
position ; but if the axis be in any other position, the situation of the sur-
face will be of more difficult determination. (Plate XIX. Fig. 240.)

In all these cases the equilibrium is stable ; for if any part of the fluid
be raised above the surface, it will immediately tend to return to its level.
But if a heavier fluid were contained in a bent tube or siphon, with its
legs or branches opening downwards, and immersed in a lighter fluid, the
equilibrium would be tottering, since, if it were once disturbed, it would
never be restored. (Plate XIX. Fig. 241.)

From these principles, we may infer that the pressure of a fluid on every
particle of the vessel containing it, or of any other surface, real or im-
aginary, in contact with it, is equal to the weight of a column of the fluid
of which the base is equal to that particle, and the height to its depth below
the surface of the fluid. Thus if we have a vessel of water one foot deep,
each square foot of the bottom will sustain the pressure of a cubic foot of
water, or nearly 1000 ounces : if we have a vessel of mercury an inch in
depth, each square foot will sustain a pressure of one twelth part of a cubic
foot of mercury, or 1130 ounces ; the atmosphere presses on each square
foot of the earth's surface with a force of about 34,000 ounces, which is
equivalent to the pressure of a column of mercury 30 inches high. The
pressure of the water on a small portion of the lowest part of the side of the
vessel containing it, is also equal to the weight supported by an equal por-
tion of the bottom ; but we cannot estimate the force sustained by any
large portion of the side, without considering the different depths below the
surface at which its different parts are situated.

It is obvious that if we conceive a fluid to be divided by an imaginary
surface of any kind, the particles contiguous to it are urged on either side
by equal forces, the fluid below resisting them, and pressing them upwards
with as much force as the fluid above presses them downwards, their own
weight being comparatively inconsiderable, for without this equality of
pressures they could not possibly remain at rest. And if we employ a
vessel of such a form as to occupy the place of any superior portion of the
fluid, the pressure against that part of the vessel which is thus substituted
will be the same that before supported the weight of the fluid removed ;
and in order that all may remain in equilibrium, the vessel must itself
exert an equal pressure on the fluid below it ; so that the pressure on the

ON HYDROSTATICS. 199

bottom will be the same as if the vessel had remained in its original state,
'and were filled to the same height with the fluid. (Plate XIX. Fig. 242.)

In order to understand this the more readily, we may suppose the portion
of the fluid, instead of being removed, to have been congealed into a solid mass
of equal density ; it is obvious that this congelation of the fluid would not
have altered the quantity of its pressure ; it would, therefore, have re-
mained in equilibrium with the water below ; the mass might also be
united with the sides of the vessel, so as to form a part of it, without in-
creasing or diminishing any of the pressures concerned : and we should
thus obtain a vessel similar to that which was the subject of our investi-
gation, the pressure on the bottom being always the same as if the mass,
supposed to be congealed, had remained fluid. Thus, the pressure on the
base of a conical or pyramidical vessel, full of water, is three times as
great as the weight of the water, since its content is one third of that of a
column of the same height, and standing on the same base. (Plate XIX.
Fig. 243.)

In this manner the smallest given quantity of any fluid contained in a
pipe may be made to produce a pressure equivalent to any given weight,
however large, which rests on the cover of a close vessel communicating
with the pipe, and this may be done either by diminishing the diameter of
the pipe, and increasing its height, while the weight is supported by a sur-
face of a certain extent, or by increasing the magnitude of this surface,
without adding to the height of the pipe ; for in either case the ultimate
force of the fluid, in supporting the weight, will be equal to the weight of
a column of the same height, standing on the whole surface which is sub-
jected to its action. And if the effect of the column be increased by any
additional pressure, independent of its weight, that pressure may be
represented by supposing the height of the column to be augmented ; and
the effect of the additional pressure will also be increased in proportion to
the magnitude of the surface which supports the weight. It is on this
principle that the pressure of water has been applied by Mr. Bramah to
the construction of a very convenient press.* (Plate XIX. Fig. 244.)

Although this property of fluids is the cause of some results which would
scarcely be expected by a person not accustomed to reflect on the subject,
and has, therefore, not improperly, been called the hydrostatic paradox,
yet it depends wholly on the general and acknowledged principles of
mechanical forces ; nor can we agree with those authors, who have asserted
that a very small quantity of a fluid may, " without acting at any mechani-
cal advantage " whatever, be made to balance a weight of any assignable
magnitude : for the immediate operation of the force very much resembles,
in the most common cases, the effect of a wedge, or of a moveable inclined
plane ; thus, a wedge remains in equilibrium, when the forces acting on
each side are in proportion to its length, like the hydrostatic pressure on a
vessel of a similar form. The conditions of the equilibrium of fluids may
also be determined, in all cases, from the general law of the descent of the
centre of gravity to the lowest point. Thus, it is easy to show that even
when two branches of a tube are of unequal diameter, a fluid must stand at
* He obtained a patent for this press in 1796.

200 LECTURE XXI.

the same height in both of them, in order to remain in equilibrium : for if
any portion be supposed to stand, in either leg, above the surface of the
fluid in the other leg, it is obvious that its centre of gravity may be lowered,
by removing so much of it as will raise the fluid in the opposite leg to its
own level, the situation of the fluid below remaining unaltered ; conse-
quently the centre of gravity of the whole fluid can never acquire its lowest
situation, unless both the surfaces are in the same level.

The air, and all other elastic fluids, are equally subject with liquids to
this general law. Thus, a much greater force is required, in order to
produce a blast of a given intensity with a large pair of bellows, than with
a smaller pair ; and for the same reason, it is much easier to a glassblower,
when he uses a blowpipe, to employ the muscles of his mouth and lips,
than those of his chest, although these are much more powerful. If we
estimate the section of the chest at a foot square, it will require a force of
seventy pounds to raise a column of mercury an inch high, by means of the
muscles of respiration, but the section of the mouth is scarcely more than
eight or nine square inches, and a pressure of the same intensity may here
be produced by a force of about four pounds. The glassblower obtains,
besides, the advantage of being able to continue to breathe during the
operation, the communication of the chest with the nostrils remaining
open, while the root of the tongue is pressed against the palate.

It is obvious that the pressure on each square inch of the side of a vessel,
or on each square foot of the bank of a river, continually increases in
descending towards the bottom. If we wish to know the sum of the
pressures on all the parts of the side or bank, we must take some mean
depth by which we can estimate it ; and this must be the depth of the
point which would be the centre of gravity of the surface, if it were
possessed of weight. Thus, if we had a hollow cube filled with water, the
centre of gravity of each side being in its middle point, the pressure on
each of the upright sides would be half as great as the pressure on the
bottom, that is, it would be equal to half the weight of the water contained
in the cube.

If, however, we wished to support the side of the cube externally by a
force applied at a single point, that point must be at the distance of one
third of the height only from the bottom. For the pressure at each point
may be represented by aline equal in length to its depth below the surface,
and a series of such lines may be supposed to constitute a triangle, of
which the centre of gravity will indicate the place of the centre of pressure
of the surface ; and the height of the centre of gravity will always be one
third of that of the triangle. It is easily inferred, from this representation,
that the whole pressure on the side of a vessel, or on a bank, of a given
length, is proportional to the square of the depth below the water to which
it extends. (Plate XIX. Fig. 245.)

The magnitude of the whole pressure on a concave or convex surface
may also be determined by the position of its centre of gravity ; but such
a determination is of no practical utility, since the portions of the forcbs
which act in different directions must always destroy each other. Thus
the perpendicular pressure on the whole internal surface of a sphere filled

ON HYDROSTATICS. 201

with a fluid, is three times as great as the weight of the fluid ; but the
force tending to burst the sphere, in the circumference of any vertical
circle, is only three fourths of that weight.

If two fluids are of different specific gravities, that is, if equal bulks of
them have different weights, their opposite pressures will counterbalance
each other, when their heights above the common surface are inversely as
their specific gravities ; for it is obvious that the greater density of the one
will precisely compensate for its deficiency in height. Thus, a column of
mercury, standing at the height of 30 inches in a tube, will support the
pressure of a column of water, in another branch of the tube, exactly 34
feet high : since the weight of 30 cubic inches of mercury is equal to that
of 408 cubic inches of water. (Plate XIX. Fig. 246.)

We have hitherto considered the properties of fluids in contact with
solids which are immoveable, and of invariable form ; but it often happens
that they act on substances which are moveable ; and they are sometimes
contained in vessels of which the form is susceptible of variation ; in these
cases, other considerations are necessary for the determination of the equi-
librium of fluids and solids with each other ; and in the first place the
properties of floating bodies require to be investigated.

When a solid body floats in a fluid, it displaces a quantity of the fluid
equal to itself in weight ; and every solid which is incapable of doing this,
must sink. For in order that the solid may remain at rest, the pressure of
the fluid below it, reduced to a vertical direction, must be precisely equal to
its weight ; but before the body was immersed, the same pressure was
exerted on the portion of the fluid which is now displaced, and was exactly
counterbalanced by its weight ; consequently that weight was equal to the
weight of the floating body.

Since the force which supports the weight of a floating body, is the pres-
sure of the fluid immediately below it, if this pressure be removed or
diminished, the body may remain at rest below the surface of the fluid,
even when it is specifically lighter. Thus a piece of very smooth wood wrill
remain, for some time, in contact with the flat bottom of a vessel of water,
until the water insinuates itself beneath it ; and it will contimie at the
bottom of a vessel of mercury, without any tendency to rise, since the
mercury has no disposition to penetrate, like water, into any minute inter-
stices which may be capable of admitting it. And, for a similar reason, if
the pressure of the incumbent fluid be removed from the upper surface of a
solid substance wholly immersed in it, the solid may remain suspended,
although heavier than an equal bulk of the fluid. Thus, if a tube or vessel
of any kind, open above and below, have a bottom of metal, ground so as to
come into perfect contact with it, without being fixed, the bottom will
appear to adhere to the vessel when it is immersed to a sufficient depth in
water, the vessel remaining empty.

In order that a floating body may remain in equilibrium, it is also neces-
sary that its centre of gravity be in the same vertical line with the centre of
gravity of the fluid displaced ; otherwise the weight of the solid will not be
completely counteracted by the pressure of the fluid. The nature of the
equilibrium, with respect to stability, is determined by the position of the

202 LECTURE XXI.

metacentve, or centre of pressure, which may be considered as a fixed point
of suspension or support, for the solid body. It is obvious that when the
lower surface of the body is spherical or cylindrical, the metacentre must
coincide with the centre of the figure, since the height of this point, as well
as the form of the portion of the fluid displaced, must remain invariable in
all circumstances, and the nature of the equilibrium will depend on the
distance of the centre of gravity above or below the centre of the sphere or
cylinder. And the place of the metacentre may always be determined
from the form and extent of the surface of the displaced portion of the
fluid, compared with its bulk and with the situation of its centre of gravity.
For example, if a rectangular beam be floating on its flat surface, the
height of the metacentre above the centre of gravity will be to the breadth
of the beam, as the breadth to twelve times the depth of the part immersed.
Hence, if the beam be square, it will float securely when either the part
immersed or the part above the surface is less than TVo- of the whole ; but
when it is less unequally divided by the surface of the fluid, it will overset.
If, however, the breadth be so increased as to be nearly one fourth greater
than the depth, it will possess a certain degree of stability whatever its
density may be. (Plate XIX. Fig. 247.)

When the equilibrium of a floating body is stable, it may oscillate back-
wards and forwards in the neighbourhood of the quiescent position : and
the oscillations will be the more rapid in proportion as the stability is
greater in comparison with the bulk of the body. Such oscillations may
also be combined with others which take place in a transverse direction : a
ship, for example, may roll on an axis in the direction of her length, and
may pitch, at the same time, upon a second axis in the direction of the
beams. Besides these rotatory vibrations, a floating body which is suffered
to fall into a fluid, will commonly rise and sink several times by its own
weight ; and in all these cases, the vibrations of any one kind, when they
are small, are performed nearly in equal times : but various and intricate
combinations may sometimes arise, from the difference of the times in
which the vibrations of different kinds are performed.

When a solid body is wholly immersed in a fluid, and is retained in its
situation by an external force, it loses as much of its weight as is equiva-
lent to an equal bulk of the fluid. For, conceiving the fluid which is
displaced by the body, to have been converted into a solid by congelation,
it is obvious that it would retain its situation, and the difference of the
pressures of the fluid on its various parts would be exactly sufficient to
support its weight. But these pressures will be the same if a body of any
other kind be substituted for the congealed fluid ; their buoyant effect may,
therefore, be always estimated by the weight of a portion of the fluid equal
in bulk to the solid. Thus, when a little figure, containing a bubble of air,
is immersed in a jar of water, which is so covered by a bladder that it may
be compressed by the hand, the bulk of the figure with its bubble is
diminished by the pressure, it is, therefore, less supported by the water, and
it begins to sink ; and when the hand is removed it immediately rises
again. (Plate XIX. Fig. 248.)

While a body is actually rising or sinking in a fluid, with an accelerated

ON HYDROSTATICS. 203

motion, the force of gravity being partly employed in generating momentum
either in the fluid or in the solid, the whole pressure on the bottom of the
vessel is necessarily somewhat lessened. Hence the apparent weight of a
jar of water will suffer a slight diminution, while a bullet is descending in
it, or while bubbles of air are rising in it, but the difference can seldom be
great enough to be rendered easily discoverable to the senses.

It sometimes happens that a solid body is partly supported by a fluid,
and partly by another solid ; of this we have an example in one of
Dr. Hooke's* ingenious inventions for keeping a vessel always full. A
half cylinder, or a hemisphere, being partly supported on an axis, which is
in the plane of the surface of the fluid, its weight is so adjusted as to be
equal to that of a portion of the fluid of half its magnitude : when the
vessel is full it is half immersed, and exerts no pressure on the axis : it
descends as the fluid is exhausted, and its tendency to turn round its axis
can only be counteracted by the pressure of the fluid on its flat side, as long
as the surface of the remaining portion of the fluid retains its original level.
(Plate XIX. Fig. 249.)

When a fluid is contained in a vessel of a flexible nature, the sides of the
vessel will always become curved, in consequence of the pressure, and the
more in proportion as the pressure is greater ; the form of the curved sur-
face will also be such that the common centre of gravity of the fluid and
the vessel may descend to the lowest point that the circumstances of the
case allow ; this form is generally of too intricate a nature to be determined
by calculation : no mathematician has hitherto been able to investigate, for
example, the curvature which a square or rectangular bag of leather will
assume when filled with water or with corn. "When, indeed, one dimension
only of a vessel is considered, for instance, when the bottom of a cistern is
supposed to be flexible, and to be fixed at two opposite sides, while the ends
are simply in contact with upright walls, without allowing the water to
run out, the nature of the curve may be determined with tolerable facility,
whether the weight of the bottom itself be considered or not. If the weight
be exactly equal to that of the water, the form of a semicircle will agree
with the conditions of equilibrium, as Bernoulli has demonstrated, sup-
posing the fixed points at the distance of its diameter ; but if the weight of
the bottom be neglected, the curvature will be everywhere proportional to
the distance below the surface, the form being the same as that of an elastic
rod, bent by two forces in the direction of the surface. The same principles,
with a slight difference in the calculations, will serve to determine the forms
adapted to the equilibrium of arches intended for supporting the weight of
superincumbent fluids, or of such soft materials as approach nearly in their
operation to more perfect fluids. (Plate XIX. Fig. 250.)

LECT. XXI.— ADDITIONAL AUTHORITIES.

Works on Hydrodynamics, 8fc. not referred to in the Lectures. — Switzer's
Hydrostatics, 2 vols. 4to, Loud. 1729. Wolfius, Elementa Matheseos, 5 vols. 4to,
Geneva, 1732-41. D'Alembert, TraitS de 1'Equilibre et du Mouvement des Fluides,

* Birch's History of the Royal Society, ii. 155.

204 LECTURE XXII.

4to, Paris 1744. Cotes's Hydrostatical and Pneumatical Lectures, 1747. Euler
on Hydrostatics, &c. Hist, et Mem. de Berlin, 1755, p. 217, &c. Nov. Com.
Petr. xiii. xiv. xv. Lecchi, Idrostatica ed Idraulica, Milan, 1765. Kastner, An-
fangsgriinde der Hydrodynamik, Gott. 1769. Bossut, Traite d'Hydrodynamique,
2vols. 1777. Lambert on the Constitution of Fluids. Hist, et Mem. de Berlin,
1784, p. 299. Bernard, Hydraulique, 4to, Paris, 1787. Langsdorfs, Theorie der
Hydrodynamischen Grundlehren, Frankf. 1787. Hydraulik, 4to, Altenb. 1794.
Parkinson's Hydrostatics, 4to, 1789. Burja, Grundlehren der Hydrostatik, 1790.
Eytelwein's Handbuch der Mechanik und Hydraulik, Berlin, 1801 ; translated by
Nicholson. Mollet Hydraulique Physique, Paris, 1810. Raccolta di Autori Ita-
liani che Trattono del Moto dell' Acque, 19 vols. 4to, Bologna, 1821-4. Gauss,
Principia Generalia Theorise, fig. Fluid, in Statu yEquilib. 4to, Gott. 1830.

Elementary Treatises will be found in many of the works on mechanics already
mentioned, besides which are the following : — Francoeur, Paris. Vince, Camb. 1812.
Bland, Camb. 1824. Moseley, Camb. ; Miller, Camb. 1831. Webster, Camb.
Moreau, 4to, Brest, 1830 ; together with the treatises in Brewster's Cyclopaedia,
&c. &c.

LECTURE XXII.

ON PNEUMATIC EQUILIBRIUM.

THE laws of the pressure and equilibrium of liquids, which are the
peculiar subjects of hydrostatics, are also applicable in general to fluids of
all kinds, as far as they are compatible with the compressibility of those
fluids, or with their tendency to expand.

Elastic fluids are distinguished from liquids by the absence of all cohesive
force, or by their immediate tendency to expand when they are at liberty.
Such are atmospheric air, steam, and gases of various kinds ; and the consi-
deration of these fluids, in the state of rest, constitutes the doctrine of pneu-
matostatics, or of the equilibrium of elastic fluids.

That the air is a material substance, capable of resisting pressure, is easily
shown by inverting an empty jar in water, and by the operation of trans-
ferring airs and gases from vessel to vessel, in the pneumatic apparatus
used by chemists. The tendency of the air to expand is shown by the
experiment in which a flaccid bladder becomes distended, and shrivelled
fruit recovers its full size, as soon as the external pressure is removed from
it, by the operation of the air pump : and the magnitude of this expansive
force is more distinctly seen, when a portion of air is inclosed in a glass
vessel, together with some mercury, in which the mouth of a tube is
immersed, while the other end is open, and without the vessel ; so that
when the whole apparatus is inclosed in a very long jar, and the air of the
jar is exhausted, the column of mercury becomes the measure of the expan-
sive force of the air. (Plate XIX. Fig. 251.)

If the diameter of the tube, in an apparatus of this kind, were very
small in comparison with the bulk of the air confined, the column of mer-
cury would be raised, in the ordinary circumstances of the atmosphere,
to the height of nearly 30 inches. But supposing the magnitude of the

ON PNEUMATIC EQUILIBRIUM. 205

tube such, that the portion of air must expand to twice its natural bulk,
•before the mercury acquired a height sufficient to counterpoise it, this
height would be 15 inches only. For it appears to be a general law of
all elastic fluids, that their pressure on any given surface is diminished
exactly in the same proportion as their bulk is increased. If, therefore, the
column of mercury in the vacuum of the air pump were 60 inches high, the
air would be reduced to half its natural bulk ; and for the same reason, the
pressure of a column of 30 inches of mercury in the open air will reduce
any portion of air to half its bulk, since the natural pressure of the atmo-
sphere, which is equal to that of about 30 inches of mercury, is doubled by
the addition of an equal pressure. In the same manner the density of the
air in a diving bell is doubled at the depth of 34 feet below the surface of
the water, and tripled at the depth of 68 feet. This law was discovered by
Dr. Hooke ;* he found, however, that when a very great pressure had been
applied, so that the density became many times greater than in the natural
state, the elasticity appeared to be somewhat less increased than the density;
but this exception to the general law has not been confirmed by later and
more accurate experiments, t

Not only the common air of the atmosphere and other permanently elastic
gases, but also steams and vapours of all kinds, appear to be equally subject
to this universal law : they must, however, be examined at temperatures
sufficient to preserve them in a state of elasticity ; for example, if we wished
to determine the force of steam twice as dense as that which is usually pro-
duced, we should be obliged to employ a heat 30 or 40 degrees above that of
boiling water, we should then find that steam of such a density as to support,
when confined in a dry vessel, the pressure of a column of 30 inches of
mercury, would be reduced to half its bulk by the pressure of a column
of 60 inches. But if we increased the pressure much beyond this, the
steam would be converted into water, and the experiment would be at an
end.

That the air which surrounds us is subjected to the power of gravitation,
and possesses weight, may be shown by weighing a vessel which has been
exhausted by means of the air pump, and then allowing the air to enter, and
weighing it a second time. In this manner we may ascertain the specific
gravity of the air, even if the exhaustion is only partial, provided that we
know the proportion of the air left in the vessel to that which it originally
contained. The pressure derived from the weight of the air is also the cause
of the ascent of hydrogen gas, or of another portion of air which is rarefied
by heat, and carries with it the smoke of a fire ; and the effect is made
more conspicuous, when either the hydrogen gas, or the heated air, is con-
fined in a balloon. The diminution of the apparent weight of a body by
means of the pressure of the surrounding air, is also shown by the destruc-
tion of the equilibrium between two bodies of different densities, upon
their removal from the open air into the vacuum of an air pump. For
this purpose a light hollow bulb of glass may be exactly counterpoised in
tty3 air by a much smaller weight of brass, with an index, which shows,

* Birch's History of the Royal Society, 1678, iii. 384, 387.

t Rickmann on the Compression of the Air by Ice, Nov. Com. Petr. ii. 162.

206 LECTURE XXII.

on a graduated scale, the degree in which the large ball is made to prepon -
derate in the receiver of the air pump, by the rarefaction of the air, less-1
ening the buoyant power which helps to support its weight. (Plate XIX.
Fig. 252.)

From this combination of weight and elasticity in the atmosphere, it fol-
lows, that its upper parts must be much more rare than those which are
nearer to the earth, since the density is everywhere proportional to the
whole of the superincumbent weight. The weight of a column of air one
foot in height is one twenty eight thousandth of the whole pressure ; conse-
quently that pressure is increased one twenty eight thousandth by the addi-
tion of the weight of one foot, and the next foot will be denser in the same
proportion, since the density is always proportionate to the pressure ; the
pressure thus increased will therefore still be equal to twenty eight thousand
times the weight of the next foot. The same reasoning may be continued
without limit, and it may be shown, that while we suppose the height to
vary by any uniform steps, as by distances of a foot or a mile, the pressures
and densities will increase in continual proportion ; thus, at the height of
about 3000 fathoms, the density will be about half as great as at the earth's
surface ; at the height of 6000, one fourth ; at 9000, one eighth as great.
Hence it is inferred that the height in fathoms may be readily found from
the logarithm of the number expressing the density of the air ; for the
logarithm of the number 2, multiplied by 10,000, is 3010, the logarithm of
4, 6020, and that of 8, 9031 ; the corresponding logarithms always in-
creasing in continual proportion, when the numbers are taken larger and
larger by equal steps. (Plate XIX. Fig. 253.)

Hence we obtain an easy method of determining the heights of mountains
with tolerable accuracy : for if a bottle of air were closely stopped on the
summit of a mountain, and being brought in this state into the plain
below, its mouth were inserted into a vessel of water or of mercury, a
certain portion of the liquid would enter the bottle ; this being weighed, if
it were found to be one half of the quantity that the whole bottle would
contain, it might be concluded that the air on the mountain possessed only
half of the natural density, and that its height was 3000 fathoms. It ap-
pears also, from this statement, that the height of a column of equal density
with any part of the atmosphere, equivalent to the pressure to which that
part is subjected, is every where equal to about 28,000 feet.

Many corrections are, however, necessary for ascertaining the heights of
mountains with all the precision that the nature of this kind of measure-
ment admits ; and they involve several determinations, which require a
previous knowledge of the effects of heat, and of the nature of the ascent of
vapours, which cannot be examined with propriety at present.

We may easily ascertain, on the same principles, the height to which a
balloon will ascend, if we are acquainted with its bulk and with its weight :
thus, supposing its weight 500 pounds, and its bulk such as to enable it to
raise 300 pounds more, its specific gravity must be five eighths as great as
that of the air, and it will continue to rise, until it reach the height ,at
which the air is of the same density : but the logarithm of eight fifths,
multiplied by 10,000, is 2040 ; and this is the number of fathoms contained

ON PNEUMATIC EQUILIBRIUM. 207

in the height, which will, therefore, be a little more than two miles and a
quarter. It may be found, by pursuing the calculation, that at the dis-
tance of the earth's semidiameter, or nearly 4000 miles above its surface,
the air, if it existed, would become so rare, that a cubic inch would
occupy a space equal to the sphere of Saturn's orbit : and on the other
hand, if there were a mine about 42 miles deep, the air would become as
dense as quicksilver at the bottom of it.

It appears, therefore, that all bodies existing on or near the earth's
surface may be considered as subjected to the pressure of a column of air
28,000 feet high, supposing its density everywhere equal to that which it
possesses at the earth's surface, and which is usually such, that 100 wine
gallons weigh a pound avoirdupois, creating a pressure equal to that of 30
inches of mercury, or 34 feet of water, and which amounts to 14| pounds
for each square inch. This pressure acts in all directions on every
substance which is exposed to it : but being counterbalanced by the
natural elasticity of these substances, it produces in common no apparent
effects ; when, however, by means of the air pump, or otherwise, the pres-
sure of the air is removed from one side of a body while it continues to
act on the other, its operation becomes extremely evident. Thus, when two
hollow hemispheres, in contact with each other, are exhausted of air, they
are made to cohere with great force ; they are named Magdeburg hemis-
pheres, because Otto von Guerike, of Magdeburg,* constructed two such
hemispheres, of sufficient magnitude to withstand the draught of the em-
peror's six coach horses, pulling with all their force to separate them. By
a similar pressure, a thin square bottle may be crushed when it is suf-
ficiently exhausted, and a bladder may be torn with a loud noise : and
the hand being placed on the mouth of a vessel which is connected with the
air pump, it is fixed to it very forcibly, when the exhaustion is performed,
by the pressure of the air on the back of the hand ; the fluids also, which
circulate in the bloodvessels of the hand, are forced towards its lower sur-
face, and the effect which is called suction is produced in a very striking
manner. It is on the same principle that cupping glasses are employed, a
partial exhaustion being procured by means of the flame of tow, which
heats the air, and expels a great part of it : so that the remainder, when it
cools, is considerably rarefied.

It was Galileo that first explained the nature of suction from the effects
of the pressure of the atmosphere ;t and his pupil Torricelli J confirmed his
doctrines by employing a column of mercury, of sufficient height to over-
come the whole pressure of the atmosphere, and to produce a vacuum in
the upper part of the tube or vessel containing it. In the operation of

* Schotti, Mechanica Hydraulico-Pneumatica, 4to, 1657. Ottonis Guericke,
Experimenta Nova Magdeburgica, fol. Amst. 1670.

f We may doubt whether this is not saying too much. Galileo proved that the
air has weight, and not, as was then believed, intrinsic levity. He actually weighed a
portion in a flask, but his determination of the specific gravity of air is not, as we
might conjecture, very accurate. Opere, iii. 47. The gravity of air was, however,
knoton to the ancients. See Aristotle, De Coelo, lib. iv.

J On this subject, see Pascal, Traite de rEquilibre des Liqueurs, Par. 1669, p. 177.
Cartesii Opera, ii. 243, 246 ; andMontucla Histoire des Mathematiques, ii. 203.

208 LECTURE XXII.

sucking up a fluid through a pipe, with the mouth or otherwise, the pres-
sure of the air is but partially removed from the upper surface of the fluid,
and it becomes capable of ascending to a height which is determined by the
difference of the densities of the air within and without the cavity
concerned : thus, an exhaustion of one fourth of the air of the cavity would
enable us to raise water to the height of 8£ feet, and mercury to 7i inches,
above the level of the reservoir from which it rises. We can draw up a
much higher column of mercury by sucking with the muscles of the mouth
only, than by inspiring with the chest, and the difference is much more
marked than the difference in the forces with which we can blow : for in
sucking, the cavity of the mouth is very much contracted by the pressure
of the external air, and the same force, exerted on a smaller surface, is
capable of counteracting a much greater hydrostatic or pneumatic pres-
sure.

When a tube of glass about three feet long, closed at one end and open
at the other, is filled with mercury, and then immersed in a bason of the
same fluid, the pressure of the atmosphere is wholly removed from the
upper surface of the mercury in the tube, while it continues to act on the
mercury in the bason, and by its means on the lower surface of the column
in the tube. If such a tube be placed under the receiver of an air pump,
the mercury will subside in the tube, accordingly as the pressure of the
atmosphere is diminished ; and if the exhaustion be rendered very perfect,
it will descend very nearly to the level of the open bason or reservoir.
When the air is readmitted, the mercury usually rises, on the level of the
sea, to the height of about 30 inches ; but the air being lighter at some
times than at others, the height varies between the limits of 27 and 31
inches. This well known instrument, from its use in measuring the weight
of the air, is called a barometer. In the same manner a column of water
from 30 to 35 feet in height may be sustained in the pipe of a pump ; but
if the pipe were longer than this, a vacuum would be produced in the
upper part of it, and the pump would be incapable of acting.

In order to observe the height of the mercury in the barometer with
greater convenience and accuracy, the scale has sometimes been amplified
by various methods ; either by bending the upper part of the tube into an
oblique position, as in the diagonal barometer, or by making the lower part
horizontal, and of much smaller diameter than the upper, or by making the
whole tube straight and narrow, and slightly conical, or by placing a float
on the surface of the mercury in the reservoir, and causing an axis which
carries an index, to revolve by its motion. But a good simple barometer,
about one third of an inch in diameter, furnished with a vernier, is perhaps
fully as accurate as any of these more complicated instruments. In order
to exclude the air the more completely from the tube, the mercury must at
least be shaken in it for a considerable time, the tube being held in an
inverted position ; and where great accuracy is required, the mercury must
be boiled in the tube. The reservoir most commonly employed is a flat
wooden box, with a bottom of leather ; the cover, which is unscrewejl at
pleasure, being cemented to the tube. Sometimes a screw is made to act
on the leather, by means of which the surface of the mercury is always

ON PNEUMATIC EQUILIBRIUM. 209

brought to a certain level, indicated by a float, whatever portion of it may
be contained in the tube ; but the necessity of this adjustment may be
easily avoided, by allowing the mercury to play freely between two hori-
zontal surfaces of wood, of moderate extent, and at the distance of one
seventh of an inch : the height may then be always measured from the
upper surface, without sensible error. But if the surfaces were closer
than this, the mercury would stand too high in the tube. (Plate XIX.
Fig. 254.)

The same method which is employed for determining the relation be-
tween the heights and densities of elastic fluids, may be extended to all
bodies which are in any degree compressible, and of which the elasticity is
subjected to laws similar to those which are discoverable in the air and
in other gases: and it is not improbable that these laws are generally
applicable to all bodies in nature, as far as their texture will allow them to
submit to the operation of pressure, without wholly losing their form.
Water, for example, has been observed by Canton* to be compressed one
twenty two thousandth of its bulk by a force equal to that of the pressure
of the atmosphere ; consequently this force may be represented by that of
a column of water 750 thousand feet in height ; the density of the water
at the bottom of a lake, or of the sea, will be increased by the pressure of
the superincumbent fluid ; and supposing the law of compression to
resemble that of the air, it may be inferred that at the depth of 100 miles,
its density would be doubled ; and that at 200 it would be quadrupled.
The same measures would also be applicable to the elasticity of mercury.
But there is reason to suppose that they are in both cases a little too
small.

LECT. XXII.— ADDITIONAL AUTHORITIES.

Pascal, Nouvelles Experiences touchant la Vuide, 4to, 1647. Tables of the Com-
pression of Air under Water, Ph. Tr. vi. 2192, 2239. Sinclair, Ars Magna Gravi-
tatis et Levitatis, 4to, Rotterd. 1669. Mariotte, sur la Nature del' Air, 1676. Mari-
otte and Homberg on the Weight of Air. Hist, et Mem. de Paris, ii. 41. Homberg,
ii. 105. Wallis, Ph. Tr. 1685, p. 1002. Halley, ibid. 1686, p. 106. Derham, ibid.
1698, p. 2. Desaguliers, ibid. No. 386. Senguerd de Aeris Natura, 4to, Lond. 1699.
Cassini, Hist, et Mem. de Paris, 1705, p. 61. Lahire, ibid. 110, H. 10. Amontons,
ibid. 119, H. 10. Varignon, ibid. 1716, p. 107, H. 40. Forssell, Dissertatio Physica
de Barometro, 4to, Upsal, 1747. Scheuchzer, Ph. Tr. xxxv. 537, 577. De Luc, sur
les Modifications de T Atmosphere, 1 772. Shuckburgh, Observations made to ascertain
the Heights of Mountains by the Barometer, Ph. Tr. 1777, p. 513 ; 1778, p. 681.
Roy, ibid. 1777, p. 513. Playfair, Ed. Tr. i. 87. Dalton, Manch. Mem. v. Assier
Perricat, Nouveau Traite des Barometres, 1802. Lindenau, Tables Barometriques,
4to, Gotha, 1809. Biot. do. 1811. Ramond, sur la Formule Barometrique de la
Mecanique Celeste, 4to, 1811. Winckler, Tables Barometriques, 4to, Halle,
1820 and 1826. Carlini's do. Milan, 1823. Duvillard's do. Paris, 1826. Olt-
mann's do. Stuttg. 1830. Galbraith's do. Edinb. 1833.

* Phil. Trans. 1762, p. 640 ; 1764, p. 261. See also Perkins, Ph. Tr. 1826, p.
561 ; and (Ersted's Report of the British Association for 1833 ; Trans, of Sections,
p. 353.

210

LECTURE XXIII.

ON THE THEORY OF HYDRAULICS.

HAVING considered the principal cases of the equilibrium of fluids, both
liquid and aeriform, we proceed to examine the theory of their motions.
Notwithstanding the difficulties attending the mathematical theory of
hydraulics, so much has already been done, by the assistance of practical
investigations, that we may in general, by comparing the results of former
experiments with our calculations, predict the effect of any proposed
arrangement, without an error of more than one fifth, or perhaps one tenth
of the whole : and this is a degree of accuracy fully sufficient for practice,
and which indeed could scarcely have been expected from the state of the
science at the beginning of the last century. Many of these improvements
have been derived from an examination of the nature and magnitude of
the friction of fluids, which, although at first sight it might be supposed to
be very inconsiderable, is found to be of so much importance in the appli-
cation of the theory of hydraulics to practical cases, and to affect the
modes of calculation so materially, that it will require to be discussed, here-
after, in a separate lecture.

There is a general principle of mechanical action, which was first
distinctly stated by Huygens,* and which has been made by Daniel
Bernoulli t the basis of his most elegant calculations in hydrodynamics.
Supposing that no force is lost in the communication of motion between
different bodies, considered as belonging to any system, they always acquire
such velocities in descending through any space, that the centre of gravity
of the system is capable of ascending to a height equal to that from which
it descended, notwithstanding any mutual actions between the bodies. The
truth of this principle may easily be inferred from the laws of collision,
compared with the properties of accelerating and retarding forces. Thus,
if an elastic ball, weighing 10 ounces, and descending from a height of 1
foot, be caused to act in any manner on a similar ball of one ounce, so as
to lose the whole of its motion, the smaller ball will acquire a velocity
capable of carrying it to the height of 10 feet. It is true that some other
suppositions must be made, in applying this law to the determination of
the motions of fluids, and that in many cases it becomes necessary to sup-
pose that a certain portion of ascending force or energy is lost in conse-
quence of the internal motions of the particles of the fluid. But still, with
proper restrictions and corrections, the principle affords us a ready method
of obtaining solutions of problems, which, without some such assistance,
it would be almost impossible to investigate. The principal hypothesis
which is assumed by Bernoulli, without either demonstration, or even the
appearance of perfect accuracy, is this, that all the particles of a flui4 in

* Horologium Oscillatorium, Pars 4, Hypoth. 1.
f Hydrodynamica, 4to, Strasb. 1738.

ON THE THEORY OF HYDRAULICS. 211

motion, contained in any one transverse section of the vessels or pipes
'through which it runs, must always move with equal velocities ; thus, if
water be descending through a vessel of any form, either regular or irre-
gular, he supposes the particles at the same height to move with the same
velocity ; so that the velocity of every particle in every part of a cylindrical
vessel 10 inches in diameter, through which a fluid is moving, must be one
hundredth part as great as in passing through a circular orifice, an inch in
diameter, made in its bottom. It is evident that this cannot possibly be
true of the portions of the fluid nearest the bottom of the vessel, since the
particles most distant from the orifice must be nearly at rest, while those
which are immediately over the orifice are in rapid motion ; but still the
calculations founded on the hypothesis agree tolerably well with experi-
ments. In this case the actual descent, in any instant, may be estimated
by the removal of the quantity discharged, from the surface of the fluid to
the orifice, since the intermediate space remains always occupied. The
ascending force thus obtained is to be distributed throughout the fluid,
according to the respective velocities of its different portions ; and it may
easily be shown, that when the orifice is small, the part which belongs to
the fluid in the vessel is wholly 'inconsiderable in comparison with the
ascending force required for the escape of the small portion which is flow-
ing through the orifice, and the whole ascending force may, therefore, be
supposed to be employed in the motion of this portion ; so that it will
acquire the velocity of a body falling from the whole height of the surface
of the reservoir, or the velocity due to that height. It appears, also, that
very nearly the same velocity is acquired by almost the first particles that
escape from the orifice, so that no sensible time elapses before the jet flows
with its utmost velocity.

This velocity may be found, as we have already seen, by multiplying
the square root of the height of the reservoir, expressed in feet, by 8, or
more correctly, by 8^ ; thus, if the height be 4 feet, the velocity will be
sixteen feet in a second ; if the height be 9 feet, the velocity will be 24,
the squares of 2 and 3 being 4 and 9 ; and if the height were 14 feet, the
velocity would be 80 feet in a second, and a circular orifice an inch in
diameter would discharge exactly an ale gallon in a second. In the same
manner, the pressure of the atmosphere being equal to that which would be
produced by a column of air of uniform density 28,000 feet high, the air
would rush into a vacuum with a velocity of more than 1800 feet in a
second.

The velocity is also equal, whatever may be the direction of the stream ;
for since the pressure of fluids acts equally in all directions at equal depths,
the cause being the same, the effect must also be the same. And if the
motion be occasioned by a pressure derived from a force of any other kind,
the effect may be found by calculating the height of a column of the fluid,
which would be capable of producing an equal pressure. When also the
force arises from the difference of two pressures, the velocity may be deter-
mined in a similar manner. Thus, the pressure of a column of water 1 foot
in height, would force the air through a small orifice with a velocity of 230
feet in a second, corresponding to the height of 830 feet ; a column of mer-

212 LECTURE XXIII.

cury 1 inch high, would produce the same effect as a reservoir of water
more than thirteen times as high, and the force of the air confined in a
closed bottle under the receiver of the air pump, will cause a jet to rise to
the same height as a column of mercury which measures the difference of
the elasticities of the air in the bottle and in the receiver.

But these calculations are only confirmed by experiment in cases when
the ajutage through which the fluid runs is particularly constructed ; that
is, when it is formed by a short tube, of which the sides are so curved that
the particles of the fluid may glide along them for some distance, and es-
cape in a direction parallel to the axis of the stream. A short cylindrical
pipe is found to answer this purpose in some measure ; but the end may be
more completely obtained by a tube nearly conical, but with its sides a
little convex inwards, so as to imitate the shape which a stream or vein of
water spontaneously assumes when it runs through an orifice in a thin
plate : for in such cases the stream contracts itself, after it has passed the
orifice, for the distance of about half its diameter, so that at this point its
thickness is only four fifths as great as at its passage ; and the quantity
discharged is only five eighths as great as that which the whole orifice
would furnish, according to the preceding calculation : instead, therefore,
of multiplying the square root of the height by 8, we may employ the
multiplier 5 for determining the actual discharge. But the velocity, where
the stream is most contracted, is only one thirtieth less than that which is
due to the whole height ; and when the jet is discharged in a direction
nearly perpendicular, it rises almost as high as the surface of the fluid in
the reservoir.

This contraction of the stream, and the consequent diminution of the
discharge, is unquestionably owing to the interference of the particles of the
fluid coming from the parts on each side of the orifice, with those which
are moving directly towards it ; and the effect is more perceptible when
the orifice is made by a pipe projecting within the reservoir, so that some
of the particles approaching it must acquire in their path a motion contrary
to that of the stream. It would be possible to obtain an approximate cal-
culation of the magnitude of this contraction, from the equilibrium which
must subsist between the centrifugal forces of the particles, as they pass
out of the orifice, describing various curves, according to their various
situations, and the pressure required for the contraction of the internal
parts of the stream, which obliges the particles to move more rapidly as
they proceed, and which must be proportional to the height required for
producing this acceleration. (Plate XX. Fig. 255.)

When a short cylindrical tube is added to the orifice, it is probable that
the motion of the fluid within the tube is still in some measure similar :
but the vessel must now be supposed to be prolonged, and to have a new
orifice at the end of the tube, at which the particles cannot arrive by any
lateral motions, and which will, therefore, not be liable to a second con-
traction : the discharge may, therefore, be estimated nearly according to
the true measure of this orifice ; the original pressure of the fluid continuing
to act until the stream escapes.

The effect of a short pipe in increasing the discharge, ceases when the

ON THE THEORY OF HYDRAULICS. 213

water separates from its sides, so that it is no longer filled by the stream :
'since there is then nothing to distinguish its motion from that of a stream
passing through a simple orifice : but the increase is not owing merely to
the cohesion of the water to the sides of the pipe ; for the effect, as I have
found by experiment, is nearly the same in the motion of air as in that of
water. The contraction caused by the motion of the water at the entrance
of the short pipe, may be considered simply as a contraction in the pipe
itself, and the subsequent part of the pipe either as cylindrical or as nearly
conical : for in this case it follows, from the general law on which Ber-
noulli's calculations are founded, that as long as the fluid remains in one
mass, the discharge will be nearly the same, as if the mouth of the pipe
were the only orifice, supposing that no force is lost : and the exceptions
which Bernoulli has made to the general application of the principle in
such cases, although partly supported by experiments, have been extended
somewhat further, both by himself and by other authors, than those ex-
periments have warranted. In the case of a diverging conical pipe, or of a
pipe with a conical termination, the discharge is found to be considerably
greater than that which a cylindrical pipe would produce, but not quite so
great as would be produced on the supposition that no force is lost. (Plate
XX. Fig. 256.)

This analogy between the effects of a cylindrical and conical pipe is
strongly supported by the experiments of Venturi,* compared with those
of Bernoulli. Bernoulli found that when a small tube was inserted into
any part of a conical pipe, in which the water was flowing towards the
wider end, not only none of the water escaped through the tube, but the
water of a vessel, placed at a considerable distance below, was drawn up by
it ;t Venturi observed the same, when the tube was inserted into the side
of a cylindrical pipe, near to its origin ; and in both cases air was absorbed,
as well as water, so that cohesion could not be in any manner concerned.^
But the pressure of the atmosphere is generally necessary for all effects of
this kind, and both VenturiJ and Dr. Matthew Young§ have observed,
that a short pipe has no effect, in increasing the discharge through an
orifice, in the vacuum of an air pump : but even if the difference were
sometimes found to exist in the absence of atmospherical pressure, it might
be produced by an accidental cohesion, like that which sometimes causes a
column of mercury to remain suspended in similar circumstances. (Plate
XX. Fig. 257.)

The effect of ajutages of different kinds, on the quantity of water dis-
charged through an orifice of a given magnitude, may be most conveniently
exhibited by placing them side by side at the same height in a reservoir,
and suffering the water to begin to flow at the same moment through any
two of them ; the quantities discharged in a given time will then obviously
indicate the respective velocities. If a very long pipe were employed, some
time would be required before the velocity became uniform ; but in such

* Surla Communication Laterale du Mouvement dans les Fluides, Par. 1797.

t Hydrodyn p. 47. See D'Alembert, Trait6 des Fluides, Art. 149.

J Exp. 2 and 7.

§ Transactions of the Royal Irish Academy, ii. 8} ; vii. 53.

214 LECTURE XXIII.

cases the retardation arising from friction is so considerable as to cause a
still greater deviation from the quantity which would be discharged by a'
shorter pipe in the same time.

When the aperture through which a fluid is discharged, instead of being
every way of inconsiderable magnitude, is continued throughout the height
of the vessel, and is every where of equal breadth, the velocity must
be materially different at different parts of its height ; but we may find the
quantity of the discharge, by supposing the whole velocity equal to two
thirds of the velocity at the lowest point. And we may find the quantity
discharged by an orifice not continued to the surface, but still of consider-
able height, by subtracting from the whole discharge of an orifice so
continued, that which would have been produced by such a portion of it,
as must be shut up in order to form the orifice actually existing. But in
this case, the result will seldom differ materially from that which is found
by considering the pressure on the whole orifice, as derived from the height
of the fluid above its centre.

When a cylindrical vessel empties itself by a minute orifice, the velocity
of the surface, which is always in the same proportion to the velocity of
the fluid in the orifice, is, therefore, uniformly retarded and follows in its
descent the same law as a heavy body projected upwards, in its ascent ;
consequently the space actually described, in the whole time of descent, is
equal to half of that which would have been described, if the initial motion
had been uniformly continued ; and in the time that such a vessel occupies
in emptying itself, twice the quantity of the fluid would be discharged if it
were kept full by a new supply. This may be easily shown, by filling two
cylindrical vessels, having equal orifices in their bottoms, and while the one
is left to empty itself, pouring into the other the contents of two other equal
vessels in succession, so as to keep it constantly full ; for it will be seen
that both operations will terminate at the same instant.

A similar law may be applied to the filling of a lock from a reservoir of
constant height ; for in all such cases, twice as long a time is required for
the effect, as would be necessary if the initial velocity were continued. The
immersion of the orifice in a large reservoir has been found to make no
difference in the magnitude of the discharge, so that the pressure may
always be estimated by the difference of the levels of the two surfaces.
Thus, when a number of reservoirs communicate with each other by ori-
fices of any dimensions, the velocity of the fluid flowing through each
orifice being inversely as the magnitude of the orifice, and being produced
by the difference of the heights of the fluid in the contiguous reservoirs, this
difference must be every where as the square of the corresponding velocity.
But if the reservoirs were small, and the orifices opposite and near to each
other, a much smaller difference in the heights of the surfaces would be
sufficient for producing the required velocity. The same circumstances
must be considered, in determining the velocity of a fluid forced through a
vessel divided by several partitions, with an orifice in each ; if the orifices
are small in proportion to their distance from each other, and if they &re
turned in different directions, each orifice will require an additional pres-
sure, equivalent to the whole velocity produced in it : but if the partitions

ON THE THEORY OF HYDRAULICS. 215

occupy a small part only of the vessel, and are placed near to each other,
' the retardation will be much less considerable. Cases of this kind occur
very frequently in the passage of water through the pipes and valves of
pumps, and it is, therefore, of consequence to avoid all unnecessary expan-
sions, as well as contractions, in pipes and in canals, since there is always
a useless expense of force in restoring the velocity which is lost in the
wider parts.

When a siphon or bent tube is filled with a fluid, and its extremities are
immersed in fluids of the same kind contained in different vessels, if both
their surfaces are on the same level, the whole remains at rest ; but if
otherwise, the longer column in the siphon preponderates, and the pressure
of the atmosphere forces up the fluid from the higher vessel, until the equi-
librium is restored ; provided, however, that this pressure be sufficiently
powerful : for if the height of the tube were more than 34 feet for water,
or than 30 inches for mercury, the pressure of the atmosphere would be
incapable of forcing up the fluid to its highest part, and this part remaining
empty, the fluid could no longer continue to run. (Plate XX. Fig. 258.)

If the lower vessel be allowed to empty itself, the siphon will continue
running as long as it is supplied from the upper, with a velocity nearly
corresponding to the height of that portion of the fluid in the longer leg,
which is not counterbalanced by the fluid in the shorter : that is, to the
height of the surface of the upper vessel above that of the lower one, or
above the end of the siphon, when it is no longer immersed ; for the height
of the pipe is in all cases to be considered as constituting a part of that
height which produces the pressure. Thus the discharge of a pipe, descend-
ing from the side or bottom of a vessel, is nearly the same as from a similar
horizontal pipe, inserted into a reservoir of the whole height of the descend-
ing pipe and of the fluid above it ; and this is true even when the depth of
the vessel is inconsiderable in comparison with the length of the pipe, if its
capacity is sufficient to keep the pipe running full. It appears at first sight
extremely paradoxical, that the whole water discharged, each particle of
which is subjected to the action of gravitation in a pipe 16 feet long, for
half a second only, should acquire the velocity of 32 feet in a second, which
would require, in common circumstances, the action of the same force of
gravitation for a whole second, and this fact may be considered as favour-
able to the opinion of those who wish to estimate the magnitude of a force,
rather by the space through which it is continued, than by the time during
which it acts ; but if we attend to the nature of hydrostatical pressure, we
shall find that the effect of the column on the atmosphere is such as to pro-
duce, or to develope, a portion of accelerating force which is actually
greater than the weight of the particles immediately concerned. If a doubt
could be entertained of the truth of this theory, it might be easily removed
by recurring to the general law of ascending force, since it follows from
that law, that each particle, which descends in any manner through the
space of 16 feet, must acquire, either for itself or for some other particles,
af power of ascending to the same height ; and on the other hand, the event
of the experiment confirms the general law. For if we fix a shallow funnel
on a vertical pipe, and pour water into it, so as to keep it constantly full,

216 LECTURE XXIIT.

while the pipe discharges itself into a reservoir out of which the water runs
through a second pipe, placed horizontally, of exactly the same dimensions
with the first, the height at which the water in the reservoir becomes
stationary, will be very nearly equal to the height of the funnel above its
surface, so that the same height produces the same velocity in both cases.
(Plate XX. Fig. 259.)

We may understand the action of the forces immediately concerned in
this experiment, by attending to the mutual effects of the water and of the
atmosphere. The water entering the orifice must immediately acquire a
velocity equal to that of the whole water in the pipe, otherwise there would
be a vacuum in the upper part of the pipe, which the pressure of the at-
mosphere will not permit ; and this pressure, considered as a hydrostatic
force, is equal to that which would be derived in any other way from a
column of the same height with the pipe, since the weight of the water in
the pipe is wholly employed in diminishing the counterpressure of the
atmosphere below, not only in the beginning, when it is at rest, but also
while it is in motion ; for that motion being uniform throughout its descent,
the power of gravitation is expended in producing pressure only : so that
the pressure of the atmosphere on the water in the funnel becomes com-
pletely analogous to the pressure of a reservoir of water, of the same height
with the pipe. The circumstance which causes the appearance of paradox
in this experiment, exists also in the simplest case of the discharge of
water ; for it may be shown that the portion of accelerating force actually
employed in generating the velocity with which a stream is discharged
through a small orifice, is twice as great as the pressure of the fluid on a
part of the vessel equal in extent to the orifice ; and in the same manner the
quantity of force exerted by the atmosphere on the water in the funnel, as
well as that with which the descending fluid impels the air below, is equal
to twice the weight of the quantity existing at any time in the pipe.

There is, however, a limit, which the mean velocity in such a pipe can
never exceed, and which is derived from the magnitude of the pressure of
the atmosphere. For the water cannot enter the pipe with a greater
velocity than that with which it would enter an exhausted pipe, and which
is produced by the whole pressure of the atmosphere ; and this pressure
being equivalent to that of a column of water 34 feet high, the velocity de-
rived from it is about 47 feet in a second : so that if the vertical pipe were
more than 34 feet long, there would be a vacuum in a part of it near the
funnel.

Wherever a pipe of considerable length descends from a funnel, if the
supply of the fluid be scanty, and especially if it approach the orifice ob-
liquely, the pressure of the amosphere, and the centrifugal force of the
particles which must necessarily revolve round the orifice, will unite in
producing a vacuity in the centre ; and when this happens, the discharge
is considerably diminished.

In order that a siphon may run, it is obvious that it must first be filled ;
and when it is once filled, it will continue to run till the reservoir ia
exhausted, as far as the level of its upper orifice. And from this cir-
cumstance, the phenomena of some intermitting springs have been ex-

ON THE THEORY OF HYDRAULICS. 217

plained,* which only begin to run when the reservoirs from which they
originate have been filled by continued rains, and then go on to exhaust
them, even though the weather may be dry. From a combination of
several such siphons and reservoirs, a great number of alternations may
sometimes be produced. (Plate XX. Fig. 260.)

Since the velocity of a stream or jet issuing in any direction, out of a
simple orifice or a converging one, is nearly equal to that of a heavy body
falling from the height of the reservoir, it will rise, if directed upwards,
very nearly to the same height, excepting a slight difference occasioned by
the resistance of the air, and by the force which is lost in producing the ve-
locity with which the particles must escape laterally, before they begin to
descend. The truth of this conclusion is easily confirmed by experiment.
(Plate XX. Fig. 261.)

If a jet issue in an oblique or in a horizontal direction, its form will be
parabolic, since every particle tends, as a separate projectile, to describe the
same parabola in its range : and it may be demonstrated, that if it be
emitted horizontally from any part of the side of a vessel, standing on a
horizontal plane, and a circle be described, having the whole height of the
fluid for its diameter, the jet will reach the plane, at a distance from the
vessel twice as great as the distance of that point of the circle, through
which it would have passed, if it had continued to move horizontally.
And if the jet rise in any angle from the bottom of the vessel, the utmost
height of its ascent will be equal to that of the point in which it would meet
the same semicircle, if it continued to move in a right line, and the hori-
zontal range will be equal to four times the distance, intercepted between
the same point and the side of the vessel. This law is equally true with
regard to simple projectiles : but the experiment is most conveniently ex-
hibited in the motion of a jet. (Plate XX. Fig. 262.)

We have hitherto considered the motions of fluids as continued princi-
pally in the same direction ; but they are frequently subjected to alternations
of motion, which bear a considerable analogy to the vibrations of pendu-
lums ; thus, if a long tube be immersed in a fluid, in a vertical direction,
and the surface of the fluid within the tube be elevated a very little, by
some external cause, the whole contents of the fluid will be urged down-
wards by a force which decreases in proportion to the elevation of the
surface above the general level of the vessel, and when both surfaces have
acquired the same level, the motion \vill be continued by the inertia of the
particles of the fluid, until it be destroyed by the difference of pressures,
which now tends to retard it ; and this alternation will continue until the
motion be destroyed by friction and by other resistances. It is also obvious,
that since any two vibrations in which the forces are proportional to the
spaces to be described, are performed in equal times, these alternations
will require exactly the same time for their completion, as the vibrations
of a pendulum of which the length is equal to that of the whole tube ; for
the relative force in the tube is to the whole force of gravity as the elevation

* Regnault, Philosophic Conversations (English edition), ii. 125. Dechales, De
FontibusNaturalibus, Tr. 7, Prop. 15. Desaguliers, Ph. Tr. No. 384. Atwell, Ph.
Tr. xxxvii. 301.

218 LECTURE XXIII.

or depression is to the whole length of the tube. Hence it follows, that if
two such tubes were united below, so as to form a single bent tube, the'
vibrations might take place in the whole compound tube, in the same
manner, and in the same time, as in each of the separate tubes ; nor would
the effects be materially altered if any part of the middle of the tube were
in a horizontal or in an oblique direction, provided that the whole length
remained unaltered. In such a tube, also, all vibrations, even if of con-
siderable extent, would be performed in the same time, and would long
remain nearly of the same magnitude ; but in a single tube, open below,
the vibrations would continually become less extensive, and their duration
would also be altered as well as their extent; besides the unavoidable
resistances, which would in both cases interfere with the regularity of the
effects.

But it does not appear that the laws of the vibrations of fluids in pipes
will at all serve to elucidate the phenomena of waves. Sir Isaac Newton *
has supposed that each wave may be compared with the fluid oscillating in
a bent pipe ; but the analogy is by far too distant to allow us to found any
demonstration on it. The motions of waves have been investigated in a
new and improved manner by Mr. Lagrange ;t and I have given a concise
demonstration of a theorem similar to his, but perhaps still more general
and explicit. It appears from these determinations, that supposing the
fluids concerned to be infinitely elastic, that is, absolutely incompressible,
and free from friction of all kinds, any small impulse communicated to a
fluid, would be transmitted every way along its surface with a velocity
equal to that which a heavy body would acquire in falling through half
the depth of the fluid ; and I have reason to believe, from observation and
experiment, that where the elevation or depression of the surface is con-
siderably extensive in proportion to the depth, the velocity approaches
nearly to that which is thus determined, being frequently deficient one
eighth or one tenth only of the whole ; in other cases, where a number of
small waves follow each other at intervals considerably less than the depth,
I have endeavoured to calculate the retardation which must be occasioned
by the imperfect elasticity or compressibility of the fluid ; but it seems
probable that the motion of small waves is still much slower than this
calculation appears to indicate.

Whatever corrections these determinations of the velocity of waves may
be found to require, the laws of their propagation may still be safely
inferred from the investigation. Thus, it may be shown, supposing the
waves to flow in a narrow canal of equable depth, that, whatever the
initial figure of the waves may be, every part of the surface of the fluid
will assume in succession the same form, except that the original elevations
and depressions, extending their influence in both directions, will produce
effects only half as great on each side, and those effects will then be con-
tinued until they are destroyed by resistances of various kinds. It may
also be inferred that the surface of a fluid thus agitated by any series of
impressions, will receive the effects of another series, in the same manner

* Principia, Lib. II. Prop. 46.

f Mecanique Analytique, 2de Partie, § xi.

ON THE THEORY OF HYDRAULICS. 219

as a horizontal surface, and that the undulations, thus crossing each other,
'will proceed without any interruption, the motion of each particle being
always the sum or difference of the motions belonging to the separate
series.

Supposing two equal and similar series of waves to meet each other in
such a canal, in opposite directions, the point in which their similar parts
meet must be free from all horizontal motion, so that any fixed obstacle in
an upright position would have the same effect on the motions of the fluid
on either side as the opposition of a similar series ; and this effect con-
stitutes the reflection of a series of waves, which is easily observed, when
they strike against a steep wall or bank ; and when this reflection is
sufficiently regular, it is easy to show that the combination of the direct
with the reflected motions must constitute a vibration of such a nature,
that the whole surface is divided into portions which appear to vibrate
alternately upwards and downwards, without any progressive motion,
while the points which separate the portions remain always in their natural
level. (Plate XX. Fig. 263.)

But those series of waves which are usually observable in any broad
surface, and which constitute a number of concentric circles, are usually
reflected in such a manner as to appear to diverge after reflection from a
centre beyond the surface which reflects them, and to be subject to all those
laws, which are more commonly noticed in the phenomena of reflected light ;
but as these laws are of more practical importance in their application to
optics, than to hydraulics, it is unnecessary at present to examine their
consequences in detail. It may, however, be easily understood, that a new
series of waves, proceeding from a centre at the same distance behind the
reflecting surface as the centre of the original series is before it, would pro-
duce precisely the same effect as a fixed obstacle ; consequently the law of
reflection at equal angles is a very simple inference from this mode of rea-
soning. (Plate XX. Fig. 264.)

When a series of waves proceeds in an equable canal, it is obvious that
the centre of gravity of the whole fluid neither rises nor falls; from this
analogy, as well as from the general application of the law of ascending
force, it is probable that in all cases of the propagation of waves, the place
of the centre of gravity remains unaltered ; so that when a circular wave
spreads further and further from its centre, its height is not diminished in
the same ratio as its diameter is increased, but the square of its height
only varies in this proportion ; that is, a wave which is a yard in diameter,
and an inch high, will retain a height of half an inch, when its diameter is
increased to four yards.

Many of the phenomena of waves may be very conveniently exhibited,
by means of a wide and shallow vessel, with a bottom of glass, surrounded
by sides inclined to the horizon, in order to avoid the confusion which would
arise from the continual reflections produced by perpendicular surfaces.
The waves may be excited by the vibrations of an elastic rod or wire,
loaded with a weight, by means of which its motions may be made more or
less rapid at pleasure ; and the form and progress of the waves may be
easily observed, by placing a light under the vessel, so that their shadows

220 LECTURE XXIII.

may fall on a white surface, extending in an inclined position above. In
this manner the minutest inflections of the surface of the water may be
made perfectly conspicuous. (Plate XX. Fig. 265.)

By means of this apparatus, we may examine the manner in which a
wave diverges, when a portion of it has been intercepted on either side or on
both sides. Thus, if a wave is admitted, by an aperture which is very
narrow in proportion to its own breadth, into the surface of a part of the
water which is at rest, it diverges from the aperture as from a new centre ;
but when the aperture is considerably wider than the wave, the wave con-
fines its motion in great measure to its original direction, with some small
divergence, while it is joined on each side by fainter circular portions,
spreading from the angles only. (Plate XX. Fig. 266.)

When two equal series of circular wraves, proceeding from centres near
each other, begin their motions at the same time, they must so cross each
other in some parts of their progress, that the elevations of the one series
tend to fill up the depressions of the other ; and this effect may be actually
observed, by throwing two stones of equal size into a pond at the same
instant ; for we may easily distinguish, in favourable circumstances, the
series of points in which this effect takes place, forming continued curves,
in which the water remains smooth, while it is strongly agitated in the
intermediate parts. These curves are of the kind denominated hyperbolas,
each point of the curve being so situated with respect to its foci, as to be
nearer to one than the other by a certain constant distance. (Plate XX.

Fig. 267.)

The subject of waves is of less immediate importance for any practical
application than some other parts of hydraulics ; but besides that it is
intimately connected with the phenomena of the tides, it affords an elegant
employment for speculative investigation, and furnishes us with a sensible
and undeniable evidence of the truth of some facts, which are capable of
being applied to the explanation of some of the most interesting phenomena
of acustics and optics.

It may be shown, by steps nearly similar to those by which the velocity
of the motions of waves is investigated, that a fluid which is contained in an
elastic pipe, and which receives an impulse at any part of the pipe, will
transmit its effects with the same velocity as a wave would have in a
reservoir, of that depth which measures the elasticity of the pipe, that is,
with half the velocity which a body would acquire in falling from the
height at which a portion of the fluid connected with the contents of the
pipe, would stand in a vertical tube. It is in this manner that the blood is
transmitted, by means of the impulse given to it by the heart through the
blood vessels ; the pulse moves on with great rapidity, the elastic force of
the vessels being considerably assisted by the temporary actions of the mus-
cular coats of the arteries, which cause a contraction more rapid than the
dilatation ; while the whole mass of the arterial blood continues at the same
time to advance with a much smaller velocity ; like the slow stream of a
river, on the surface of which undulations are continually propelled with
motions independent of its own.

ON THE THEORY OF HYDRAULICS. 221

LECT. XXIII.— ADDITIONAL AUTHORITIES.

Theory of Hydraulics. — See Lect. XXI. Baliani, De Motu Gravium, 4to,
Geneva, 1646. Sturmius de Clepsydrarum Phenomenis et Eflfectibus, 4to, 1674.
Mariotte, Traite du Mouvement des Eaux, Paris, 1686. Varignon, Hist, et Mem.
de Paris, ii. 162, 1703, p. 238, H. 125. Picard, ibid. vii. 323. Lahire, ibid. x.
162, 264. Saulmon, ibid. 1712, p. 279, H.77; 1714, p. 381, H. 102; 1715, p. 61,
H. 61 ; 1716, p. 244, H. 68. Polenus, De Motu Aquae Mixto, 4to, Patavii, 1717.
Da Castellis per quse Derivantur Fluviorum Aquae, Pat. 1718. Desaguliers on the
Running of Water in Pipes, Ph. Tr. 1726, p. 77. Eames on the Estimation of
Force in Hydraulic Experiments, ibid. 1727, p. 343. D. Bernoulli on the Motion,
Action, and lateral Pressure of Fluids. Com. Petr. ii. Ill, 304 ; iv. 194. Pitot,
Hist, et Mem. 1730, p. 336, H. 110. Guglielmini, Com. Bon. i. 545. Couplet,
Hist, et Mem. 1732, p. 113, H. 107. Dufay, ibid. 1736, p. 191, H. 118. Clare
on the Motion of Fluids, 1737. Jo. Bernoulli on the Motion of Water in Pipes,
Com. Petr. ix. 3, 19; x. Opuscula, 4. Krafft, ibid. x. 207. Ja. Bernoulli,
Opera, vol. iv. Petit Vandin, Mem. des Savans Etrangers, i. 261. Euler on the
Motion of Water in Pipes. Hist. etMem. de Berlin, 1752, p. 111. On the Re-
action of Water in Pipes. Nov. Com. Petr. vi. 312. Borda on the Discharge of
Fluids. Hist, et Mem. 1766, p. 579, H. 143. Kastner, Nov. Com. Gott. 1769,
i. 45. Matteuci, Com. Bon. vi. 286 ; Michelotti, Sperienze Idrauliche, 2 vols. 4to.
Turin, 1771. D'Alembert, Opuscules, vi. Lagrange, Hist, et Mem. de Berlin,
1781, p. 151. Ximenes, Nuove Sperienze Idrauliche. Riccati, Memorie di Ma-
tematica e Fisica della Societa Italiana, 4to, Verona, iii. 238. Lorgna, ibid. iv. 369.
On Weres, ibid. v. 313. On Castelli's Principle, ibid. vi. 218. Bonati, ibid. v. 501.
Stratico, ibid. v. 525. Girard on the Pressure of running Water, Journal de
Physique, xlii. 429. Banks, Manchester Mem. v. 398. Young on the Discharge of
a vertical Pipe, Journal of the Royal Institution, vol. i. Eytelwein, Me'm. de Berlin,
1814, 1815. Prony, Journal de 1'Ecole Poly technique, vol. iii. Bidone, Experiences
sur la Forme et la Direction des Veines et des Courans d'Eau. Mem. de Turin,
1822, 1824, p. 281 ; 1830, p. 229 ; 1838, p. 1 ; et Mem. della Soc. Ital. vol. xxi.
D'Aubuisson, Traite du Mouvement de 1'Eau dans les Tuyaux de Conduite, Paris,
1827. Traite d'Hydraulique a TUsage des Ingenieurs, Paris, 1834. Annales de
Chimie, 1830, p. 225. Corancez, Theorie du Mouvement de 1'Eau dans les Vases,
Paris, 1830. Mallet, Notices Historiques, 1830. Navier, Mem. de 1'Acad. 1830,
vol. ix. Hachette, Experiences sur le Mouvement des Fluides, Paris, 1830.
Poncelet et Lesbros, Experiences Hydrauliques sur les Lois de 1'Ecoulement de
1'Eau, 4to, Paris, 1832. Savart, Comptes Rendus, 1833. Rennie, Ph. Tr. 1831,
and Report of the British Association, 1833, p. 153.

Waves. — Laplace, Mem. de 1'Acad. 1776. D'Alembert, Encyclopedic, art.
Onde. Lagrange, Mem. de 1'Acad. de Berlin, 1781, 1786, p. 192. Flaguergues,
Journal des Savans, Oct. 1789. Bremontier, Recherches sur le Mouvement des
Ondes, Paris, 1809. Poisson, Mem. de 1'Institut, 1816, vol. viii. Cauchy, Mem.
des Savans Etrangers, vol. i. Bidone, Mem. de Turin, 1826, p. 195. Weber,
Wellenlehre auf Experimente Gegrundet, Leipz. 1825. Challis, Trans, of the Camb.
Phil. Soc. vols. iii. and v. Earnshaw, ibid. vi. 203. Green, ibid. vol. vi. Russell,
Trans, of the Roy. Soc. of Edinb. v. xiv. p. 47. Kelland, ibid. vol. xiv. p. 497 ;
xv. 101. Report of the British Assoc. 1837, p. 417. Airy on Tides and Waves,
Encyc. Metrop.

222

LECTURE XXIV.

ON THE FRICTION OF FLUIDS.

WE have hitherto considered the motions of fluids independently of the
resistance which they undergo from the vessels containing them and from
the surfaces in contact with them, as well as from the interference of the
neighbouring particles with each other ; there is, however, a variety of
cases of very common occurrence, in which these frictions most materially
affect the results of our calculations ; so that before this subject was labo-
riously and judiciously investigated by the Chevalier du Buat,* it was
almost impossible to apply any part of our theoretical knowledge of hy-
draulics to practical purposes.

The effect of friction is particularly exemplified by the motions of rivers,
in which almost the whole force of gravity is employed in overcoming it.
When the inclination and the dimensions of a river continue uniform, the
velocity is also every where equal ; for otherwise the depth would become
unequal : here, therefore, the force of gravitation must be an exact counter-
poise to the resistance which is to be overcome, in order that the water may
flow with its actual velocity : this velocity having been originally derived
from the effect of a greater inclination near the origin of the river. When
the river is thus proceeding, with an equable motion, it is said to be in train ;
and it is obvious that no increase of its length will produce any alteration
in its velocity. There is, therefore, a very material difference between the
course of a river, and the descent' of a body, with an accelerated motion,
along an inclined surface. For when a solid body is placed on an inclined
plane, the force of friction is either great enough to overpower its relative
weight, and to retain it at rest, or else the friction is constantly less than
the gravitation, and the motion is always accelerated. But the resistance
to the motions of fluids arises principally from different causes ; not from
the tenacity of the fluids, which, where it exists, is a force nearly uniform
like that of friction, but principally from the irregular motions and mutual
collisions of their particles; and in this case, according to the laws of
mechanics, it must vary nearly in proportion to the square of the velocity.
For when a body is moving in a line of a certain curvature, the centrifugal
force is always as the square of the velocity ; and the particles of water in
contact with the sides and bottom of a river or pipe, must be deflected, in
consequence of the minute irregularities of the surfaces on which they slide,
into nearly the same curvilinear paths, whatever their velocity may be, so
that the resistance, which is in great measure occasioned by this centrifugal
force, must also vary as the square of the velocity. Thus also the curva-
ture assumed by the outline of a stream of water issuing from a simple
orifice which constitutes the contraction already described, is very nearly

* Principes d'Hydraulique, 1786, and Svols. 1816.

ON THE FRICTION OF FLUIDS. 223

the same, whatever the velocity may be : nor does the friction increase with
the pressure, as is demonstrated by an experiment of Professor Robison
on the oscillations of a fluid through a bent tube, terminated by two bulbs,
which were performed in the same time, whether the tube was in a hori-
zontal or in a vertical position. Mr. Coulomb has also proved the same
fact by experiments on the vibrations of bodies immersed in fluids, and
suspended by twisted wires ; he finds that precisely at the surface, the
friction is somewhat greater than at any depth below it : he also considers
a certain part of the friction as simply proportional to the velocity, and
a very small portion only, in common fluids, as perfectly independent
of it.*

It is obvious that wherever the friction varies as the square of the velo-
city, or even when it increases in any degree with the velocity, there must
always be a limit, which the velocity can never exceed by means of any
constant force, and this limit must be the velocity at which the resistance
would become equal to the force. It is for this reason that a light body,
descending through the air, soon acquires a velocity nearly uniform ; and
if it be caused, by any external force, to move for a time more rapidly, it
will again be speedily retarded, until its velocity be restored very nearly to
its original state. In the same manner the weight of the water in a river,
which has once acquired a stationary velocity, is wholly employed in over-
coming the friction produced by the bottom and the banks.

From considering the effect of the magnitude of the surface exposed to
the friction of the water, in comparison with the whole quantity contained
in the river, together with the degree in which the river is inclined to the
horizon, we may determine, by following the methods adopted by Mr.
Buat, the velocity of any river of which we know the dimensions and the
inclination. Supposing the whole quantity of water to be spread on a hori-
zontal surface, equal in extent to the bottom and sides of the river, the
height at which it would stand, is called the hydraulic mean depth ; and
it may be shown that the square of the velocity must be jointly propor-
tional to the hydraulic mean depth, and to the fall in a given length. If
we measure the inclination by the fall in 2800 yards, the square of the
velocity in a second will be nearly equal to the product of this fall multi-
plied by the hydraulic mean depth. For example, in the Ganges, and in
some other great rivers, the mean depth being about 30 feet, and the fall 4
inches in a mile, the fall in 2800 yards will be about 6£ inches, which,
multiplied by 360 inches, gives 2340 inches for the square of the mean ve-
locity, and 48 £ inches, or about four feet, for the mean velocity in a second,
that is, not quite three miles an hour, which is the usual velocity of rivers
moderately rapid. If, however, great precision were required in the deter-
mination, some further corrections would be necessary, on account of the
deviation of the resistance from the exact proportion of the squares of the
velocities ; since the friction, as we have already seen, does not increase
quite so fast as this.
Jit is obvious that the friction of a fluid, moving on the surface of a solid

* Hist, et Mem. de Paris, 1784, p, 229. Mem. de 1'Institut, vol. Hi. Phil.
Mag. vii. 183.

224 LECTURE XXIV.

alone, would not produce any material retardation of its motion, if the par-
ticles of the fluid themselves were capable of moving on each other without
the least resistance ; for in this case a small portion of the fluid, in imme-
diate contact with the solid, might remain at rest, and the remaining mass
of the fluid might slide over this portion without any retardation. It ap-
pears, however, that the water in contact with the bottom of a river moves
with a very considerable velocity, and the water next above this only a
little faster, so that the velocity increases almost uniformly as we ascend
towards the surface. It follows, therefore, that the resistance must be much
greater where the particles of water slide on each other, than where they
glide along the surface of a solid. This internal friction operates gradually
throughout the water ; the surface being retarded by the particles immedi-
ately below it, those particles by the next inferior stratum, and each stra-
tum being actuated, besides its own relative weight, by the friction of the
water above, tending to draw it forwards, and by that of the water below
tending still more to retard it ; the retardation being communicated from
below upwards, in such a manner as to be every where equivalent to the
relative weight of the water above the part considered. It appears from
observation, that when we have determined the mean velocity in English
inches, we may find the superficial velocity, very nearly, by adding to it
its square root, and the velocity at the bottom, by subtracting from it the
same number : thus the square root of 48f being nearly 7, the superficial
velocity of the Ganges will be about 55 inches, or 4 feet 7 inches in a
second, and the velocity at the bottom 41 f. There are, however, frequent
irregularities in the proportions of the velocities at different depths, and it
has sometimes been observed, perhaps on account of the resistance of the
air, that the velocity is a little less immediately at the surface, than a few
inches below it.

For similar reasons, the velocity of a river is also generally greater in
the middle than at the sides ; and the motion of the particles in the middle
must be retarded, not only by those which are below them, but also by
those on each side, while these, on the contrary, are dragged on by the
water in the middle : the middle parts tend, therefore, to draw the sides
towards them, which they cannot do, without lowering the surface of the
fluid next to the banks in such a degree as to make the difference of level
an equivalent to this tendency to approach the middle. This appears to be
the reason that the surface of a large river may generally be observed to
be slightly convex, or a little elevated in the middle.

The course of a river is sometimes interrupted by a were or a fall, natu-
ral or artificial ; in such cases the velocity may be calculated in the same
manner as when a fluid is discharged from a reservoir through an aperture
of considerable height : supposing the whole section of the were to be such
an aperture, in a vessel so much higher, that the velocity of a fluid issuing
from it at the upper part of the aperture would be precisely equal to the
actual velocity of the river. The extent of the swell caused by a were, or
by any partial elevation thrown across the bed of a river, may also 'be
found by first determining the height at which the surface must stand
immediately above the were, and then calculating the inclination of the

ON THE FRICTION OF FLUIDS. 225

surface which will be required for producing the actual velocity in the
• river thus made deeper ; which of course will determine the situation of
the surface where the water approaches the were ; and this surface, which
is more nearly horizontal than the general surface of the river, will be so
joined to it as to have a curvature nearly uniform throughout.

It appears from calculations of the effects of various changes in the
dimensions of rivers, as well as from immediate observation, that a con-
siderable diminution of the breadth of a river at a particular place, will
often produce but a small elevation of its surface. The velocity, however,
may sometimes be considerably increased by such a change, and where the
bottom is of a loose nature, its particles may be carried away by means of
the increased velocity, and the bed of the river may be deepened.

Where a river bends in a considerable degree, it is generally remarked
that the velocity of the water is greater near the concave than the convex
side of the flexure, that is, at the greatest distance from the centre of its
curvature. This effect is probably occasioned by the centrifugal force,
which accumulates the water on that side ; so that the banks are under-
mined, and the channel is deepened by its friction. Some authors have
been led to expect that the velocity would be greater nearest to the convex
bank, because the inclination of the surface must be a little greater there ;
but the effect of the accelerating force in any short distance is inconsiderable,
and it is more than compensated by the want of depth. It may easily be
understood, that all angles and flexures must diminish the general velocity
of the river's motion, and the more as they are more abrupt.

It has sometimes been imagined, that because the pressure of fluids is
propagated equally in all directions, their motions ought also to diverge in
a similar manner ; but this opinion is by no means well founded, even
with respect to those particles which receive their motions in an unlimited
reservoir from the impulse of a stream which enters it. An experi-
ment, which sets this fact in a clear point of view, was made long ago
by Hauksbee. * He produced a very rapid current of air, by means of a
vessel, into which three or four times as much air as it naturally con-
tained had been condensed by means of a syringe, and causing the current
to pass through a small box, in which the bason of a barometer was
placed, the mercury was depressed more than two inches, in consequence
of the rarefaction which the current produced in the air of the box. (Plate
XXI. Fig. 268.)

Professor Venturi has also made several experiments of a similar nature
on the motion of water : he observes that not only the water in contact with
a stream is drawn along by it, but that the air in the neighbourhood of a jet
is also made to partake of its motion. When the mouth of a pipe through
which a stream of water is discharged, is introduced into a vessel a little
below the surface of the water which it contains, and is allowed to escape
by ascending an inclined surface placed opposite to the pipe and leading
over the side of the vessel, the stream not only ascends this surface without
leaving any portion of itself behind, but carries also with it the whole of

* Hauksbee, Physico- Mechanical Experiments, 4to, Lond. 1709, p. 89. See
Leslie's examination of the experiment, art. Barometer. Supp. Encyc. Brit. p. 129.

226 LECTURE XXIV.

the water of the vessel, until its surface becomes level with the lowest part
of the stream. (Plate XXI. Fig. 269.)

The effect of a jet of water, in drawing towards it a current of air, is in
some measure illustrated by an experiment which is often exhibited among
the amusements of hydraulics. A ball of cork, or even an egg, being
placed in the middle of a jet, which throws up a pretty large stream to a
moderate height, the ball, instead of falling or being thrown off, as it might
naturally have been expected to do, remains either nearly stationary or
playing up and down, as long as the experiment is continued. Besides the
current of air which Venturi has noticed, and which tends to support the
ball in a stable equilibrium, the adhesion of the water, combined with its
centrifugal force in turning round the ball, assists in drawing it back,
when it has declined a little on either side, so that the stream has been
principally in contact with the other side. A similar effect may be observed
in the motions of the air only, as I have shown by some experiments of
which an account is published in the Philosophical Transactions.* Thus,
if we bend a long plate of metal into the form of the letter S, and sus-
pend it in the middle by a thread, so that it may move freely on its centre,
and if we then blow on its convex surface with a tube directed obliquely
towards the extremity, instead of retreating before the blast, it will on the
contrary appear to be attracted ; the pressure of the atmosphere being
diminished by the centrifugal force of the current, which glides along the
convex surface, because it finds a readier passage in the neighbourhood of
the solid, towards which it is urged by the impulse of the particles of the
air approaching it on one side, and by the defect of pressure on the other
side, occasioned by the removal of a certain portion of the air which it
carries with it. (Plate XXI. Fig. 270, 271.)

From considerations similar to those by which the velocity of a river is
determined, we may calculate the quantity of water discharged from a pipe
of any given dimensions, and in any position. The same expressions will
serve for estimating the magnitude of the friction in both cases ; the pipe
being considered as a small river of which the mean depth is one fourth of
its diameter : but a part only of the force of gravity is now expended in
overcoming the friction, the rest being employed in producing the momen-
tum of the water. We may obtain a sufficiently accurate determination of
the velocity, by supposing the height of the reservoir above the orifice of
the pipe to be diminished in the same proportion as the diameter of the
pipe would be increased by adding to it one fiftieth part of the length, and
finding the whole velocity corresponding to four fifths of this height. Thus,
if the diameter of the pipe were one inch, and its length 100 inches, we
must suppose the effective height to be reduced to one third by the friction,
and the discharge must be calculated from a height four fifths as great as
this, which may be considered as a reduction derived from the interference
of the particles entering the pipe, with each other's motions. If the diameter
of the pipe had been two inches, the height must only have been supposed
to be reduced to one half by the friction ; such a pipe would, therefore,

* Outline of Experiments and Inquiries respecting Sound and Light, Ph. Tr.
1800, p. 106.

ON THE FRICTION OF FLUIDS. 227

discharge about five times as much water as the former, although of only
twice the diameter ; and this circumstance requires the attention of all those
who are concerned in regulating the distribution of water by pipes- for
domestic use, or for any other purpose.

In such cases it becomes also frequently necessary to attend to the angle
in which a small pipe is inserted into a larger ; whenever a pipe is bent,
there is a loss of force according to the degree of flexure and to the velocity
of the water, which may be calculated, if it be required ; but if a pipe be
fixed into another through which the water is moving very rapidly in a
direction contrary to that of the stream, its discharge will not only be much
smaller than if the directions more nearly coincided, but sometimes such a
pipe will discharge nothing at all ; on the contrary, like the air in Hauks-
bee's experiment, the water which it contains may be dragged after the
stream in the larger pipe.

The bad effect of unnecessary dilatations, as well as contractions, in aque-
ducts and in pipes, may be understood from what has been already said of
the loss of force attendant on every change of velocity ; a circumstance of
a similar nature sometimes happens in the animal economy. When an
artery is dilated so as to form an aneurism, it has been observed that the
artery is usually distended above the cavity ; and this effect is easily un-
derstood from the actual increase of resistance which the aneurism pro-
duces, united perhaps with the previous debility of the artery.

Mr. Gerstner* has found by some very accurate observations on the
motion of water in very small pipes, that the resistance is considerably
affected by the temperature at which the experiment is performed ; but in
the cases of rivers, and of such pipes as are commonly used in practice, no
variations of temperature to which they can be liable, will produce any
sensible effects. His experiments indicate a resistance, where the tubes are
very small, which follows a law so different from that which is observed in
more common cases, that it appears to be owing to some other cause : this
cause is perhaps the capillary attraction of the open end of the tube, and it
is the more probable that the resistance depends on some such circumstance,
as there is reason to think that the irregularity may be in great measure
removed by placing the tube wholly under water.

LECT. XXIV.— ADDITIONAL AUTHORITIES.

Castelli, della Mensura dell' Acque correnti, 4to, Rome, 1628. Toricellius^ de
Motu gravium naturaliter accelerate, 1643. Varennius, by Jurin and Shaw, 1765.
Guglielmini, Aquarum fluentium Mensura, 2 vols. 4to, Bonon, 1690-91. Epistolse
duse Hydrostatics, 4to, Bonon, 1692. Della Naturadi Fiumi, 2 vols. Milan, 1821.
Polenus, see Lect. XXIII. Jurin, De Motu Aquse fluentis, 4to, Venetii, 1724.
Frisius, Del Mododi regolare i Fiumi, 4to, Lucca, 1762 ; Paris, 1774 ; Lond. 1818.
Lorgna, Ricerca intorno alia Distribuzione della Velocita nella Sectione de'
Fiumi, 4to, Verona, 1771. Stattleri Physica, 8 vols. Augsb. 1772. Euler on the
Motion of Rivers. Hist, et Mem. de Berlin, 1760, p. 101. Ximenes on the
Velocity of Rivers. Atti dell' Academia di Siena, iii. 16 ; vi. 31 ; vii. 1. Les-

* * On the Discharge of Water at different Temperatures. Abhandlungen der
Bohmischen Gesellschafft der Wissenchaften, 4to, Prag. 1798. Gilbert's Journal,
v. 160.

Q2

228 LECTURE XXV.

pinasse and Frisi on do. Rozier's Journal, ix. 145, 398 , xi. 58. Fabre sur les
Torrens and les Rivieres, 4to, Paris, 1797. Silberschlag, Theorie des Fleuves. Ro-
bison's Mechanical Philosophy, art. Rivers. Eytelwein's Experiments with the
Hydraulic Quadrant. Sammlung zurBaukunst 1799. Girard, Essai sur le Mouve-
ment des Eaux courantes, 1804. Recherches sur les Eaux Publiques, &c. Prony,
Mem. des Savans Etrangers, 1815. Tadini, Del Movimento delle Acque correnti,
4to, Milan, 1816. Hachette, De 1'Ecoulement des Fluides Aeriformes, Annales de
Chimie. 1827, and Paris, 1830. Genieys, Essai sur les Moyens de conduire, d'ele-
ver, et dedistribuer les Eaux, 4to, 1839.

LECTURE XXV.

ON HYDRAULIC PRESSURE.

THE mutual effects of fluids and moveable solids on each other depend
principally on the laws of hydraulic pressure, and of the resistance of fluids,
which have been considered by Bernoulli as constituting a separate depart-
ment of hydrodynamics, under the name of hydraulicostatics, and which
are of the utmost practical importance, since the application of the powers
of wind or water to the working of mills, and to the navigation of ships, are
wholly dependent on them. The impulse of a fluid differs very materially
from that of a solid, for in the motions of solids, the least possible finite
momentum must overpower the strongest possible pressure; but since the
particles of fluids are supposed to be infinitely small, the momentum of a
fluid stream may always be balanced by a certain determinate pressure,
without producing motion in the solid opposed to it ; so that this division of
the subject of hydraulics has nothing analogous to it in simple mechanics.
It is true that when a certain quantity of a fluid is made to concentrate its
action almost instantaneously, its effect is nearly similar to that of a solid,
for here the essential distinction derived from the successive action of the
particles no longer exists. Thus, when a stream of fluid filling a pipe acts
suddenly on an obstacle at the end of it, it requires to be resisted by a force
far greater than that which originally caused its motion, unless the action
of the force be continued through a considerable space ; and for this reason
the strength of the pipe ought to be so calculated as to be able to resist
this action ; its intensity may, however, be easily diminished by means of
an air vessel communicating with the pipe, which will allow the motion
to be changed in a less abrupt manner. But in the principal cases which
we are about to consider, the action of the fluid on the solid is supposed
to be confined to such of its particles as are nearly in contact with the
surface.

When a part of the weight of any fluid is expended in producing a motion
in any direction, an equal force is deducted from its pressure on the vessel
in that direction : for the gravitation employed in generating velocity,
cannot at the same time be causing pressure ; and when the motion produced
is in any other direction than a vertical one, its obliquity must be imme-

ON HYDRAULIC PRESSURE. 229

diately derived from the reaction of the vessel, or of some fixed obstacle ; for
"it is obvious that a vertical force, like that of gravity, cannot of itself pro-
duce an oblique or a horizontal motion.

If a small stream descends from the bottom of a vessel, the weight expended
in producing its motion is equal to that of a column of the fluid standing
on a base equal to the contracted orifice, and of twice the height of the
vessel. Thus, if the vessel be 16 feet high, the velocity of the stream will
be 32 feet in a second, and a column 32 feet in length will pass through the
orifice in each second, with the whole velocity derivable from its weight
acting for the same time ; so much, therefore, of the pressure of the fluid in
the reservoir must be expended in producing this motion, and must of course
be deducted from the whole force with which the fluid acts on the bottom of
the reservoir ; in the same manner as when two unequal weights are con-
nected by means of a thread passing over a pulley, and one of them begins
to descend, the pressure on the pulley is diminished by a quantity, which is
as much less than the sum of the weights, as the velocity of their common
centre of gravity is less than the velocity of a body falling freely. If the
stream issue from the vessel in any other direction, the effect of the dimi-
nution of the pressure in that direction will be nearly the same as if the
vessel were subjected to an equal pressure of any other kind in a contrary
direction ; and if the vessel be moveable, it will receive a progressive or
rotatory motion in that direction. Thus, when a vessel or pipe is fixed on a
centre, and a stream of water is discharged from it by a lateral orifice, the
vessel turns round at first with an accelerated motion, but on account of the
force consumed in producing the rotatory motion, in successive portions of
the water, the velocity soon becomes nearly stationary. (Plate XXI. Fig.
272.)

From similar reasoning it appears, that the effect of a detached jet on a
plane surface perpendicular to it must be equivalent to the weight of a
portion of the same stream equal in length to twice the height which is
capable of producing the velocity. And this result is confirmed by expe-
riments : but it is necessary that the diameter of the plane be at least four
times as great as that of the jet, in order that the full effect may be produced.
When also a stream acts on an obstacle in a channel sufficiently closed on
all sides to prevent the escape of any considerable portion of water, its
effect is nearly the same as that of a jet playing on a large surface. But if
the plane opposed to the jet, be only equal to it in diameter, or if it be
placed in an unlimited stream, the whole velocity of the fluid column will
not be destroyed, it will only be divided and diverted from its course, its
parts continuing to move on, in oblique directions ; in such cases the pres-
sure is usually found to be simply equivalent to the weight of a column equal
in height to the reservoir, the surface being subjected to a pressure nearly
similar to that which acts on a part of the bottom of a vessel, while a stream
is descending through a large aperture in another part of it. (Plate XXI.

Fig. 273.)

* It is obvious that in all these cases, the pressure varies as the square of

the velocity, since the height required to produce any velocity is proper-

230 LECTURE XXV.

tional to its square. This inference was first made in a more simple man- <
ner, from comparing the impulse of a fluid on a solid with that of a number
of separate particles striking the surface of the body, each of which would
produce an effect proportional to its velocity, while the whole number of
particles acting in a given time, would also vary in the same ratio. If the
solid were in motion, and the fluid either in motion or at rest, it is obvious
that the relative velocity of the solid and the fluid with regard to each other,
would be the only cause of their mutual effects, and that the hydraulic
pressure or resistance must be dependent on this velocity alone, except so
far as the limited dimensions of the reservoir containing the fluid, might pro-
duce a difference in the internal motions of its particles in different cases.
Thus, where the fluid is so confined that the whole of the stream acts on a
succession of planes, each portion into which it is divided may be considered
as an inelastic solid, striking on the surface exposed to it with a certain
velocity ; and in this case the force must be considered as simply propor-
tional to the relative velocity, and not to its square. For want of this con-
sideration, the effects of water wheels have frequently been very erroneously
stated.

When a jet strikes a plane surface obliquely, its force, in impelling the
body forwards, in its own direction, is found to be very nearly proportional
to the height to which the jet would rise, if it were similarly inclined to the
horizon. But when a plane is situated thus obliquely with respect to a
wide stream, the force impelling it in the direction of the stream is some-
what less diminished by the obliquity, at least if we make allowance for its
intercepting a smaller portion of the stream : thus, if the anterior part of a
solid be terminated by a wedge more or less acute, the resistance, according
to the simplest theory of the resolution of forces, might be found by
describing a circle on half the base of the wedge as a diameter, which
would cut off* a part from the oblique side of the wedge that would be the
measure of the resistance, the whole side representing the resistance to the
same solid without the wedge : but the resistance is always somewhat more
than this, and the portion to be added may be found, very nearly, by
adding to the fraction thus found one ten millionth of the cube of the
number of degrees contained in the external angle of the wedge. (Plate
XXI. Fig. 274.)

The pressure of a fluid striking perpendicularly on a plane surface, has
been found to be very different at different parts of the surface ; being
greatest at the centre, and least towards the edges ; so that if an aperture
be made in the centre of a circular plane, covering the mouth of a bent
tube, the fluid within it will rise half as high again as if the whole mouth
were open. It is also observable, that two bodies, equal and similar in the
form of the part meeting the fluid, undergo very different degrees of
resistance according to the forms of their posterior terminations, and that a
thin circular plate is much more retarded than a long cylinder of the same
diameter. These circumstances are utterly inexplicable upon the vague
approximation of supposing the resistance produced by the immediate im-
pulse of separate particles of the fluid on the solid ; but they are no longer

ON HYDRAULIC PRESSURE. 231

surprising, when we consider the true mode of action of continuous fluids,
'since all the motion which is produced by the fluid in the solid or by the
solid in the fluid is communicated much more by means of pressure than
by immediate impulse. The minute operations of this pressure are too
intricate to be accurately developed, but we may observe in general, that
when a body moves along the surface of a resisting medium at rest, or
when an obstacle at rest is opposed to a fluid in equable motion, the pres-
sure is increased before the moving substance, and diminished behind it ;
so that the surface is elevated at the one part and depressed at the other,
and the more as the velocity is greater. Now it is obvious that the
pressure must be greatest where the elevation is greatest, and hence a
perforation at the centre of the surface indicates a greater pressure than at
the circumference. Behind the body this pressure becomes negative, and
has sometimes been called nonpressure ; hence it happens that a tube,
opening in the centre of the posterior surface, exhibits the fluid within it
depressed below the level of the general surface of the water. Thus, if we
suppose the velocity of a body, terminated by perpendicular surfaces, to be
8 feet in a second, it will require the pressure of about a foot, to produce
such a velocity, and we may, therefore, expect an elevation of about a foot
before the body, and an equal depression behind it ; consequently an
equivalent difference must be found in the pressure of the water at any
equal depths on the anterior and posterior surfaces of the body. The water
elevated before the body escapes continually towards each side, and the
deficiency behind is also filled up in some measure by the particles rushing
in and following the body : but there is in both cases, a certain quantity of
water which moves forwards, and constitutes what is called the dead water :
before, where it is usually most observable, it forms an irregular triangle,
of which the sides are convex inwards. If the posterior part of the body
be formed like a wedge, the water on each side will be advancing to fill up
the vacuity, even while it remains in contact with the sides, and the nega-
tive pressure will be considerably diminished. For this reason the bottoms
of ships are made to terminate behind in a shape somewhat resembling a
wedge ; and the same economy may be observed in the forms of fishes,
calculated by nature for following their prey with the greatest possible
rapidity. In general, fishes, as well as ships, are of a more obtuse form
before than behind, but it is not certain that there would be any material
difference in the resistance in a contrary direction, although some experi-
ments seem to favour such an opinion. Perhaps if the natural form of the
dead water moving before an obtuse body, were ascertained, it might serve
to indicate a solid calculated to move through the water with the least
resistance ; for the water must naturally assume such a form for its own
motions, and the friction of fluids on solids being less than that of fluids
moving within themselves, the resistance would be diminished by substi-
tuting a solid of the same form for a fluid.* (Plate XXI. Fig. 275.)

Supposing a body to move through a fluid at a considerable depth below
•

* Consult Russel, Trans, of the Roy. Soc. of Edin. vol. xiv. p. 47.

232 LECTURE XXV.

its surface, there will still be an elevation before and a depression behind it,
the less in height and the greater in extent, as the depth at which the body*
is situated is greater. Such an elevation appears to be in some measure
analogous to the effect of a low were thrown across a river, which raises its
surface, and produces a swell.

If two or more bodies differently formed, the resistances to the motions
of which had been ascertained, were caused to move through a fluid in
contact with each other, it is obvious that the paths described by the
particles of the fluid in gliding by them, must be very materially altered
by their junction ; and it seems natural to expect that the joint disturbance
produced in the motions of the fluid, when the surfaces are so united as to
form a convex outline, would be somewhat less than if each surface were
considered separately. Accordingly, it is found that no calculation, de-
duced from experiments on the resistance opposed to oblique plane surfaces,
will determine with accuracy the resistance to a curved surface. It appears
from experiment that the resistance to the motion of a sphere is usually
about two fifths of the resistance to a flat circular substance of an equal
diameter. The resistance to the motion of a concave surface is greater
than to a plane, and it is easily understood, that since the direction in
which the particles of the fluid recede from the solid, must be materially
influenced by the form of the solid exposed to their action, their motion in
this case must be partly retrograde, when they glide along towards the
edges of the concave surface, and a greater portion of force must have
been employed, than when they escape with a smaller deviation from their
original direction. (Plate XXI. Fig. 276.)

For some reason which is not well understood, the hydraulic pressure of
the air appears to be somewhat greater in proportion to its density, than
that of water. It has been found that the perpendicular impulse of the
air on a plane surface, is more than equivalent to the weight of a column
of air of a height corresponding to the velocity, and the excess is said by
some to amount to one third, by others to two thirds of that weight. The
resistance appears also to be a little greater for a large surface, than for a
number of smaller ones which are together of equal extent.

The resistance or impulse of the air on each square foot of a surface
directly opposed to it, may in general be found, with tolerable accuracy,
in pounds, by dividing the square of the velocity in a second, expressed
in feet, by 500. Thus, if the velocity were 100 feet in a second, the pressure
on each square foot would be 20 pounds ; if 1000 feet, 2000 pounds. For
a sphere of a foot in diameter, we may divide the square of the velocity
by 1600. We may also find, in a similar manner, the utmost velocity that
a given body can acquire or retain in falling through the air ; for the
velocity at which the resistance is equal to the weight must be its limit.
Thus, if a sphere one foot in diameter weighed 100 pounds, the square of
its utmost velocity would be 160,000, and the velocity itself 400 feet in a
second ; if a stone of such dimensions entered the atmosphere with a greater
velocity, its motion would very soon be reduced to this limit ; and a lighter
or a smaller body would move still more slowly. The weight of Mr.

ON HYDRAULIC PRESSURE. 233

Garnerin's parachute,* with its whole load, was about a quarter of a
pound for each square foot, the square of its greatest velocity must, there-
fore, have been about 125, and the velocity 11 feet in a second, which is no
greater than that with which a person would ascend, in leaping from a
height of two feet, without stooping. Mr. Garnerin found the velocity
even less than this, and it is not improbable that the concave form of the
parachute might considerably increase the resistance. Thus, Mr. Edge-
worth found that a plate 9 inches long, when bent into an arc of which
the chord was 7£, had the resistance increased more than one seventh, t
The diminution of the resistance of the air by the obliquity of the surface is
still less than that of the resistance of water : thus, the resistance on the
oblique surfaces of a wedge is not quite so much less than the resistance
on its base, as its breadth is less than the length of those surfaces.

When the velocity of a body moving through an elastic fluid is very
great, the resistance is increased in a much greater proportion than the
square of the velocity : thus the retardation of a cannon ball moving with
a velocity of 1000 feet in a second, or a little more, becomes suddenly
much greater than the calculation indicates. The reason of this change
appears to be, that the condensation of the air before the ball is necessarily
confined to a smaller portion which is very intensely compressed, because
the effect of the impulse can only spread through the air with a certain
velocity which is not much greater than that of the ball ; and this smaller
portion of air must necessarily be much more condensed than a larger
portion would have been. Thus, when a cannon ball moves slowly, its
effect at any instant is in some degree divided throughout all that part of
the atmosphere which the sound of the report has reached ; and if the ball
follows the sound very speedily, it is obvious that the portion of the air
before the ball which partakes of the effect, must be very small. The
sound is observed to be propagated with a velocity of about 1130 feet in a
second, and a cannon ball may be discharged with a velocity of 2000 ; but
one half of this is very speedily lost, so as to be wholly useless with regard
to the effect of the ball. If, therefore, we wish to increase the range of a
cannon ball, we must increase its weight ; for the resistance increases only
in proportion to the surface of the ball, while the weight is determined by
its solid content.

It is not easy to explain, in a manner perfectly satisfactory, the reflection
of a cannon ball, or of a stone, which strikes the surface of the sea, or of
a piece of water, in an oblique direction. We may, however, assign some
causes which appear to be materially concerned in this effect. In the first
place the surface of the water, acting at first for some time on the lower
part of the ball, produces, by its friction, a degree of rotatory motion, by
means of which the ball, as it proceeds, acts upon the mass of water which
is heaped up before it, and is obliged by a similar friction to roll upwards,
so that it mounts again to a much greater height than it could possibly

.* Nich. Jour. i. 523, 8vo. iii. 57. Gilbert's Jour. xvi. 156, 164, 257. See the
article Aeronautics, Supp. to Encyc. Brit,
f Ph. Tr. 1783, kxiii. 136.

234 LECTURE XXV.

have attained by the mere hydrostatic pressure of the water at a depth so
inconsiderable. But a more powerful cause than this appears to be the*
continual succession of new surfaces which are to be depressed, and which
may be supposed to react on the ball, so as to produce the same effect as a
more intense pressure would have done, if it had continued stationary ;
and the mutual action of the water and the ball may be compared to the
impulse of an oblique stream, moving with the velocity of the ball, which
would impel it much more powerfully than the simple hydrostatic pressure
at a much greater depth. It happens in this case, as in many others, that
the effects which appear to be the most familiar to us, do not by any means
admit the clearest and simplest explanation.

LECT. XXV.— ADDITIONAL AUTHORITIES.

L'Hopital on the Solid of least Resistance, Hist, et Mem. de Paris, 1699, p. 107,
H. 95. Craig on do. Ph. Tr. 1701, p. 746. Varignon on Motions in a resisting
Medium, Hist, et Mem. 1707, p. 382, H. 139; 1708, pp.212, 250, 302, 419, H.
123 ; 1709, 1710, 1711, p. 248, H. 87. Desaguliers on the Resistance of the Air,
from Exp. in St. Paul's Cathedral, Ph. Tr. 1719, No. 362. Pitot on the Oblique
Impulse of Fluids, Hist, et Mem. 1727, p. 49, H. 137. D. Bernoulli on Pres-
sure and Resistance, Com. Petr. iii. 214, iv. 136, v. 106, viii. 99, 113. Euler on
Friction and Resistance, Nov. Com. Petr. vi. 338, viii. 197. Bouguer on the
Solid of least Resistance, Hist. etMem. 1733, p. 85, H. 86 ; 1767, p. 504, H. 110.
On Impulse of Fluids, ibid. 1746, p. 237, H. 289. Manoeuvre des Vaisseaux, 4to,
1757. Krafft on the Impulse of a Vein of Water, Com. Petr. viii. 253 ; xi. 233.
D'Alembert, Essai sur la Resistance des Fluides, 4to, 1752. Silvabelle on the
Solid of least Resistance, Mem. des Savans Etrangers, iii. 639. Borda on the Re-
sistance of Fluids, Hist, et Mem. de Paris, 1763, p. 358, H, 118 ; 1767, p. 495, H.
145 ; 1769, p. 247. Lambert, Hist, et Mem. de Berlin, 1765, p. 102. Don Jorge
Juan, Examen Maritime, 2 vols. 4to, Madrid, 1771. Nouvelles Experiences sur la
Resistance des Fluides, par MM. D'Alembert, De Condorcet, et Bossut, 1777.
Bossut's Experiments, Hist, et Mem. 1778, p. 353, H. 38. Mann's Experiments
on the Resistance in shallow Canals, &c. Ph. Tr. 1779, pp. 555, 629. Euler on
the Impulse of a Vein of Fluid, in his Comment on Robins, 1783. Lagrange on do.
Mem. de Turin, 1784-5. Michelotti on do. Melanges de Turin, 1788, App. 121.
Legendre's Example of the Solid of least Resistance, Hist. etMem. 1786, p. 21,
App. 121. Lorgna,Mem. della Soc. Ital. 4to, 418. Vince on the Resistance of Fluids,
Ph. Tr. 1795, p. 24; 1798, p. 1. Gerstner's Theory of the Impulse of Water,
Abhandlungen derBomischen Gesellschaft, 1795. Experiments of the Society for the
Advancement of Naval Architecture, 4to, Lond. Charnock's Hist, of Marine Ar-
chitecture, 3 vols. 4to, 1800. Morosi on the Impulse of a Vein of Fluid, Mem.
dell' Institute Lombardo-Veneto, 1812, pp. 119, 305. Brunacci, Mem. della Soc.
Ital. 1816-17. Macneill's Canal Navigation, 4to, 1833. Beaufoy's Nautical and
Hydraulic Experiments, 4to, 1834.

235

LECTURE XXVI.

ON HYDROSTATIC INSTRUMENTS AND HYDRAULIC
ARCHITECTURE.

WE have now examined the fundamental laws of the principal depart-
ments of hydrodynamics, which may be considered as constituting the
theory of the science : we are next to proceed to the application of this
theory to a variety of practical purposes. Following the same general
order as we have observed in mechanics, our first division will be analogous
to the subject of statics, and will relate to hydrostatic instruments ; the
second to architecture, containing some particulars respecting canals and
embankments ; the third to machinery, comprehending the modification
and application of the force of fluids considered as inelastic ; the fourth
and the fifth to the methods of raising and removing weights, in which the
principal hydraulic and pneumatic machines will be respectively explained,
and, as a part of this subject, the application of pneumatic force will also
be examined.

The principles of hydrostatics are very frequently applied to the deter-
mination of the specific gravities of the various productions of nature or
of art. The diminution of the apparent weight of a solid body upon
immersion into a fluid, affords an easy method of comparing its density
with that of the fluid. For the weight of the solid being previously
determined, if we examine how much that weight is diminished by plung-
ing the body in pure water, we shall have the weight of an equal bulk of
water: and thence we may immediately obtain the proportion of the
specific gravity of the body to that of water, which is the usual
standard of comparison. And if we weigh a solid of given magnitude,
for instance, a ball of glass, first in water, and then in any other fluid, the
quantities of weight lost in each case will be in the same proportion as
the specific gravities of the two fluids. A balance adapted for such exami-
nations is called a hydrostatic balance ; on one side it has a scale as usual,
and on the other a loop of fine wire or of horse hair, for holding the solid
to be weighed, which may be changed occasionally for a ball of glass,
suspended in a similar manner : sometimes also a dish is added for holding
any loose substances which will sink in water, proper counterpoises being
used as equivalents for the weight of the dish either in air or in water ;
and when a body lighter than water is examined, a weight of known
magnitude and density is employed for sinking it. (Plate XXI. Fig. 277.)

The specific gravities of any substances, and in particular of such as are
lighter than water, may also be very conveniently determined by means of
a common balance, employing a phial with a conical ground stopple, filling
it first with water, and then either with a given fluid, or with a portion of
the solid of which the weight has been ascertained, together with as much
water as is sufficient to exclude all the air.

236 LECTURE XXVI.

For the speedy examination of a variety of fluids, differing but little in
specific gravity from some known standard, an hydrometer may be very"
conveniently employed. This instrument is said to have been invented by
Archimedes : it consists of a hollow ball, with a weight below it, and a
slender stem above, so graduated as to express the specific gravity of the
fluid by the degree to which it sinks. Sometimes the instrument is sunk
to a certain mark, by means of weights placed in a dish at the end of the
stem ; or different weights are fixed to it below, while the graduations of
the scale are still observed ; and it may even be applied to finding the
specific gravities of solids, the solid being first placed in the dish at the end
of the stem, and then in a second dish which is suspended from the bulb
below the water. (Plate XXI. Fig. 278.)

Another mode of ascertaining the specific gravities of fluids differing but
little from each other in density, is to have a series of globules of glass, so
loaded as to correspond to the specific gravities indicated by as many
numbers, which are marked on them ; and throwing several of them
together into the fluid, to observe which of them remains nearly stationary
without either rising to the surface or sinking. This method, though not
expeditious, appears to be very secure from error : the globules are sold by
patent, adapted for the measurement of the strength of spirituous liquors.

In whatever manner we compare the specific gravities of bodies with
that of water, it is necessary, for very accurate experiments, either that the
water be employed at the temperature of the air when moderately warm,
or that a proper correction should be made for its change of bulk at dif-
ferent temperatures. Platina, the densest known substance, is 23 times as
heavy as distilled water, gold 19J, mercury 13|, lead 11£, silver 11, copper
9, iron and steel 71 , stony substances usually about 2£, rectified spirits £,
naphtha, the lightest liquid, ^ cork about •£, common air -g-j-^, steam s £0 0,
and pure hydrogen gas -r^^nr* From this comparison the weight of a cubic
foot of any of these substances may be easily determined ; since a cubic
foot of water weighs nearly 1000 ounces avoirdupois, or more nearly 998 ;
thus a cubic foot of gold would weigh about 195,000 ounces, and be worth
above 60,000 pounds sterling ; a cubic foot of iron weighs 7750 ounces, and
a cubic foot of common stone about 2500.

The method of measuring the bulk of solid bodies by immersing them in
a fluid, was applied, by its inventor Archimedes, to the detection of a fraud
in the composition of a mixed metal :* and at present the principal use of
hydrometers is for ascertaining, by the specific gravity of a compound of
alcohol and water, the proportional quantities of its ingredients. But in all
experiments of this kind, it is necessary to be aware, that a considerable
change of the joint bulk of two substances is often produced by their mix-
ture : and that in general their dimensions are considerably contracted.
Thus, 18 gallons of water, and 18 of alcohol, instead of 36 gallons, make
only 35, consequently the specific gravity of the compound is one 35th
greater than the mean of the specific gravities of the ingredients. And in
some cases the whole dimensions of a single substance may even be co'u-

* Vitruvius, Architect. 1. ix, c. 13.

ON HYDROSTATIC INSTRUMENTS, &c. 237

traded by the addition of another substance : thus iron, by the addition of
erne eighth of its bulk of platina, becomes contracted one fortieth of that
bulk.

The use of the spirit level depends on the tendency of all fluids to pre-
serve a horizontal surface, and the freedom with which the particles of
fluids move on each other, renders it an instrument capable of the greatest
delicacy. A tube which is very slightly curved, being nearly filled with
alcohol or ether, and then perfectly closed, the bubble will always rise to
the highest part of the tube, and will never be stationary at the point which
is marked as its proper place, unless the instrument be very accurately
horizontal, or in the same position in which the mark was adjusted. The
surface of the bubble, especially when it is small, cannot, in a strict sense,
be called perfectly horizontal, since its form approaches nearly to that of a
sphere ; but in order that the centre of gravity of the water may attain the
lowest possible situation, the bubble must necessarily occupy the highest
point of the tube. (Plate XXI. Fig. 279.)

The principles of hydrostatics have been employed in various ways for
supplying lamps with oil. It is found that a lamp will burn, without con-
suming any considerable portion of its wick, as long as it is amply supplied
with oil ; hence it becomes desirable that it should always be level with
the surface of the reservoir, and this may be effected sufficiently well by
placing the wick at the edge of a very large vessel, or at the end of a tube
projecting from such a vessel, or from a vessel closed above, and opening
only by an orifice below, which lets in the air as the oil escapes through it.
But all these methods are often attended with inconveniences of various
kinds, especially where the lamp is to be employed like a candle, and
placed on a table. A French artist has applied a little pump, which is
worked by means of a spring, for raising the oil from a vessel under the
lamp ; but this refinement is too complicated to be practically useful. Mr.
Keir's lamp * contains a divided cavity, one part of which is filled with oil,
and the other with a saline or saccharine fluid of greater density, so that
when the oil contained in the upper part of the tube is exhausted, its place
is partly supplied by a fresh portion, which is forced up in consequence of
the descent of the denser fluid in a much larger vessel. Still, however, the
surface must be lowered by degrees ; but by combining the invention with
Dr. Hooke's semicylindrical counterpoise,t a little modified, the height of
this fluid may be so regulated, that the surface of the oil may remain
almost invariable, until the reservoir is quite exhausted. For this purpose,
the centre of gravity of the counterpoise must be a little higher than the
line which bisects it ; and its specific gravity must be about three fourths
as great as that of the fluid ; and in this manner it may be made to raise
the surface of the heavier fluid, in proportion as a greater quantity of it
escapes, to supply the place of the oil ; and to keep it always at a sufficient
height above the surface which separates it from the oil, so that the wick
may be amply and almost uniformly supplied. (Plate XXI. Fig. 280.)

•The art of embankment is a branch of architecture entirely dependent on

* Nich. Jour. iii. 467. t Lampas, p. 188.

238 LECTURE XXVI.

hydrostatical and hydraulic principles. In Holland and in some parts of
Germany, this art is indispensable to the existence of large tracts of
country ; and even in this island it has heen of extensive utility, in gaining
and securing ground on the sea coast. The construction of canals, and
the management of rivers and harbours, are also dependent on the -same
principles ; and these important subjects have been discussed by various
writers, in many copious treatises, expressly devoted to hydraulic archi-
tecture.

When a bank or dike is to be constructed, it must be composed of ma-
terials capable of resisting, by their weight, the effort of the fluid to over-
turn them ; by their lateral adhesion, the force tending to thrust them aside
horizontally ; and by their density and tenacity, the penetration of the
water into their substance. If the water be in motion, they must also be
able to resist its friction, without being carried away by it, and they must
be arranged in such a form, as to be least liable to be undermined. For
many of these reasons, the surface of the bank exposed to the water
must be inclined to the horizon : the line expressing the general direction
of the pressure of the water ought to be confined entirely within its sub-
stance, so that no force thus applied may be able to overturn it as a whole ;
and this condition will always be fulfilled, when the sides of the bank make
an angle with each other not less than a righl^angle. The pressure acting
on a bank thus inclined will also tend to condense the materials, and to
increase their lateral adhesion, and the particles will become less liable to
crumble away by their weight, than if the surface were more nearly ver-
tical. For embankments opposed to the sea, a bank much inclined has also
the additional advantage of breaking the force of the waves very effectually.
An embankment of this kind is usually furnished with drains, formed by
wooden pipes or by brickwork, closed by falling doors or valves, which
allow the water to flow out at low water, but do not permit the tide to enter.
To prevent the penetration of the water, clay is often used, either mixed
with gravel or sunk in a deep trench cut on each side of the canal or re-
servoir. (Plate XXI. Fig. 281.)

The greater or less velocity of a river must determine what substances
are capable of withstanding its tendency to disturb them ; some are carried
away by a velocity of a few inches in a second, others remain at rest when
the velocity amounts to several feet. But in general, the velocity of a river
is sufficient to produce a gradual transfer of the particles of its bed, which
are shifted slowly downwards, towards the sea, being occasionally deposited
in those parts where the water has least motion, and serving at last to form
the new land, which is always advancing into the sea, on each side of the
mouth of a large river. It has been recommended as a good form for a
navigable river or canal, to make the breadth of the horizontal bottom one
fifth of that of the surface, and the depth three tenths. (Plate XXI.
Fig. 282.)

If a canal or a reservoir were confined by a perpendicular surface of
boards, and it were required to support it by a single prop, the prop should
be placed, as we have already seen, at the distance of one third of the whole
height from the bottom ; but it would be always more convenient in prac-

ON HYDROSTATIC INSTRUMENTS, &c. 239

tice to fix the side of the reservoir at the bottom, than to allow the whole
pressure to be supported by the prop, and it might also be strengthened by
means of ribs, thicker below than above, so as to produce an equal strength
throughout, wherever the prop might be placed ; but if the side were
formed of a single plank of uniform thickness, the strain would be most
equally divided by placing the prop very near the middle of its height.

The strength of the materials employed for flood gates and sluices requires
to be determined according to the principles which have been laid down,
in treating of the passive strength of substances used for purposes simply
mechanical ; but the calculations become in this case much more intricate.
Thus, if we have a circular plate or plank, of a uniform elastic substance,
constituting the bottom of a pipe or cistern, and simply supported at the
circumference, a very complicated calculation is required for determining
the proportion of its strength to that of a square plate of the same breadth,
supported only at two opposite ends, since at each point of the circular
piece, there are two curvatures which require to be considered. The square
plate will support a column of fluid twice as heavy as the weight which
would break it, if placed at its centre ; and if I have been correct in the
calculation, a circular plate will support a height of water nearly sixteen
sevenths as great as a square plate. But for ordinary purposes, it will be
sufficient to consider the strength as derived only from the resistance opposed
to the flexure in one direction, since the additional strength, obtained from the
lateral supports, may very properly be neglected, as only assisting in afford-
ing that additional security which is always necessary, to compensate for
any accidental defects of the materials. It has been asserted that the
strength of a square plate is doubled when it is supported on both sides ;
but this appears to be a mistake.

We may, therefore, be contented with determining the strain on the ma-
terials in that direction in which they afford the greatest resistance, either
from the shorter distance between the supports, or by the disposition of the
fibres ; and it will be always most eligible to combine these circumstances,
so that the fibres of the wood may be arranged in the direction of the short-
est dimensions of the sluice. If a sluice be supported above and below
only, the greatest strain will be at the distance of about three sevenths of
its height from the bottom ; and it is at this point that the greatest strength
is required. But if the boards forming the sluice be fixed across it, in hori-
zontal directions, their strength must be greatest at the bottom. (Plate
XXI. Fig. 283.)

In the construction of flood gates, the principles of carpentry must be ap-
plied in a manner nearly similar to that which serves for the determination
of the best forms of roofs. The flood gates, if they are double, without a
solid obstacle between them, must meet at an angle : and when this angle
is very open, the thrust against the walls or hinges must necessarily be very
great. If, however, the angle were too acute, the flood gates would require
to be lengthened, and in this case their strength would be far more dimi-
nj^hed than that of a roof similarly elevated, since the hydrostatic pressure
acts always with full force in a perpendicular direction. The thickness

240 LECTURE XXVI.

required for each flood gate may be determined in the same manner as the
thickness of a sluice.

Where a sluice board of considerable dimensions is to be occasionally
raised, it may be necessary to ascertain the force which will be required for
overcoming its friction ; this friction is nearly proportional to the whole
pressure of the water, and may be found, with sufficient accuracy, in pounds,
by multiplying the square of the depth of the sluice, in feet, by 10. Thus,
if the depth be 3 feet, the friction or adhesion will be about 90 pounds for
each foot of the breadth.

If the side of a canal gives way, it is sometimes of consequence to pre-
vent, as much as possible, the escape of the water. For this purpose it is
usual to have doors or valves in various parts of the canal, which, when the
water is at rest, lie nearly flat at the bottom ; but when it begins to run
over them, with a considerable velocity, they are raised by its force, and
put a stop to its motion.

The utility of the introduction of canals into a commercial country may
be estimated in some measure by the effect of the same labour, employed in
removing weights by land carriage and by water. Thus, a single horse can
scarcely draw more than a ton weight on the best road, but on a canal, the
same horse can draw a boat of 30 tons at the same rate.

The construction of piers and quays, and the management of harbours,
are also important departments of hydraulic architecture ; it often happens
that besides the application of the general principles of mechanics and
hydrostatics to these purposes, the peculiar circumstances of the case may
indicate to an ingenious artist a mode of performing the required work in
an effectual and economical manner. We may find a good example of such
an arrangement, in the account given, by Mr. Smeaton, of the method
which he adopted for the improvement of the port of Ramsgate,* and which
indeed resembles some that had been before employed in similar cases : by
forming a large excavation, which is furnished with flood gates, and is con-
stantly filled at high water, he has procured a number of artificial torrents,
which escape through the sluices, and become powerful agents for carrying
away the matter deposited by the sea, and tending to impede the navigation
of the harbour.

LECT. XXVL— ADDITIONAL AUTHORITIES. (See LECT. XXIV.)

Specific Gravities. — Marinus Ghetaldus, Promotus Archimedes, 4to, Romse,
1603. Boyle's Works, 1772. Tables of Specific Gravities, Ph. Tr. xv. 927;
xvii. 694 ; xxvii. 206, 511 ; xxxiii. 114 ; xlv. 416, the last by Davis. Brisson, Pe-
santeur Spec, des Corps, 4to, Paris, 1787. Ramsden on the Sp. Gr. of Fluids, 4to,
1792. Atkins on Sp. Gr. 4to, 1803.

Hydrometers.— Boyle's, Ph. Tr. 1675, p. 329. Moncorie's, Birch, i. 257. Horn-
berg's Areometer, Hist, et Mem. 1699, p. 46. Irwin, Ph. Tr. 1721, p. 223.
Fahrenheit, Areometri Descriptio, ibid. 1724, p. 140. Desaguliers on Clarke's
Hydrometer, ib. 1730, p. 277. Gesner de Hydroscopico, Zurich, 1754. On
Areometers, Hist, et Mem. 1768, p. 435 ; 1770, p. 526 ; Ph. Tr. 1778, p. 509 ;
1788, p. 582 ; 1793, p. 145. Roz. Jour, xxxiii. 241. Mem. della Soc. Ital. vii. 79.
Annales de Chimie, xxi. 3 (Guytoris), xxvi. 3, 132 ; xxviii. 3, 282 ; xxxi. 12^ ;

* Smeaton on Ramsgate Harbour, Lond. 1791.

ON THE REGULATION OF HYDRAULIC FORCES. 241

xxxiii. 3. Nicholson's Journal, i. 37 (Baume's), 110. Nicholson's Hyd. Manch.
Mem. ii. 370. Nat. Ph. ii. 16. Quin's, Tr. Soc. Arts, yiii. 198. Schmidt's,
Gren's Journal der Physik, vii. 186. Charles's Biot's Traite de Physique, i. 114;
Benoit Theorie Generale des Pese -liqueurs, 1821.

Hydraulic Architecture. — Belidor Sommaired'un Cours d' Architecture Hydrau-
lique, Paris, 1720. Architecture Hydraulique, 4 vols. 4to, 1737-53. Erskine, A
Dissertation on Rivers, &c. Loud, 1770. Prony, Nouvelle Architecture Hydraulique,
2 vols. 4to, Paris, 1790-6. Gilly, Grundriss zu den Vorlesungen iiber Wasserbau-
kunst, Berlin, 1795. Wiebekung, Wasserbaukunst, 4to, Darmst. 1798. Smeaton's
Reports. Coulomb sur les Moyens d'Executer sous 1'Eautoutes sortes de Travaux
Hydrauliques sans employer aucun Epuisement, 1819. Delaistre, Science des Inge-
nieurs, 2 vols. 4to, Paris, 1825. Crisp, A Treatise on Marine Architecture, 1826.
Aster's Constructions Hydrauliques, fol. Paris, 1828. Beaudemoulin, Recherches
sur la Fondation des Ouvrages Hyd. 4to, 1829.

LECTURE XXVII.

ON THE REGULATION OF HYDRAULIC FORCES.

THOSE modifications of the motions of fluids which are employed either
for conducting them from place to place, or for applying their powers to
the production of mechanical effects, may be considered as constituting a
separate division of practical hydraulics, which is analogous to the subject
of general machinery in practical mechanics.

A supply of water may be obtained from a reservoir situated above the
level at which it is wanted, whatever its distance may be, either by means
of open canals, or aqueducts, or of closed pipes. Where an uninterrupted
declivity cannot be obtained, it is necessary to employ pipes, which may be
bent upwards or downwards at pleasure, provided that no part of them be
more than thirty feet above the reservoir, and when the pipe is once filled,
the water will continue to flow from the lower orifice ; but it is best in all
such cases to avoid unnecessary angles ; for when the pipe rises and falls
again, a portion of the air, which is always contained in water, is frequently
collected in the angle, and very materially impedes the progress of the
water through the pipe. When the bent part is wholly below the orifices
of the pipe, this air may be discharged by various methods. The ancients
used small upright pipes called columnaria, rising from the convexity of
the principal pipe, to the level of the reservoir, and suffering the air to
escape without wasting any of the water. It may however frequently be
inconvenient or impossible to apply a pipe of this kind ; and the same pur-
pose may be answered, by fixing on the pipe a box containing a small valve,
which opens downwards, and is supported by a float, so as to remain shut
while the box is full of water, and to fall open when any air is collected in
it. (Plate XXI. Fig. 284.)

J[f the pipe were formed into a siphon, having its flexure above both
orifices, it would be necessary to bend it upwards at the extremities, in order
to keep it always full ; but in this case the accumulation of the air would

R

242 LECTURE XXVII.

be extremely inconvenient, since it would collect so much the more copiously
as the water in the upper part of the pipe would be more free from pres-
sure, and neither of the methods which have been mentioned would be of
any use in extricating it. It has been usual in such cases to force a quan-
tity of water violently through the pipe, in order to carry the air with it ;
but perhaps the same effect might be produced much more easily, by making
a small airtight valve in the upper part of the pipe, opening outwards, and
a stopcock immediately before it : the stopcock being suddenly turned as
often as might be necessary, the momentum of the water in the pipe would
probably carry it forwards with sufficient force to throw out the air ; or if it
were necessary, external pressure might be added, and the air might even in
this manner be discharged by the valve much more readily than without it.
But it might be still simpler to have a pretty large vessel of water screwed
on to the pipe, which would not be filled with air for a considerable time ;
and which, when full, might be taken off and replenished with water. (Plate
XXI. Fig. 285.)

The diameter of a pipe required for conveying a given quantity of water
to a given distance may be calculated from the experiments of Mr. Buat,
which have been already mentioned. Pipes are usually made of wood, of
lead, or of cast iron, but most commonly of lead ; and of late tinned copper
has been employed with considerable advantage. A pipe of lead will
bear the pressure of a column of water ] 00 feet high, if its thickness be
one hundredth of its diameter, or even less than this ; but when any
alternation of motion is produced, a much stronger pipe is required, and
it is usual to make leaden pipes of all kinds far thicker than in this pro-
portion.

The form and construction of stopcocks and valves are very various, ac-
cording to their various situations and uses. Stopcocks usually consist of
a cylindrical or conical part, perforated in a particular direction, and
capable of being turned in a socket formed in the pipe, so as to open or shut
the passage of the fluid, and sometimes to form a communication with either
of two or more vessels at pleasure. A valve is employed where the fluid
is to be allowed to pass in one direction only, and not to return. For
water, those valves are the best which interrupt the passage least ; and none
appears to fulfil this condition better than the common clack valve of
leather, which is generally either single, or divided into two parts ; but it
is sometimes composed of four parts, united so as to form a pyramid, nearly
resembling the double and triple valves which are formed by nature in the
hearts of animals. A board, or a round flat piece of metal, divided un-
equally by an axis on which it moves, makes also a very good simple valve.
Where a valve is intended to intercept the passage of steam, it must be of
metal ; such a valve is generally a flat plate, with its edge ground a little
conically, and guided in its motion by a wire or pin. For air, valves are
commonly made of oiled silk, supported by a perforated plate or grating.
(Plate XXI. Fig. 286, 287.)

Before we consider the application of the force of fluids in motion to prac-
tical purposes, we must attend to the methods of measuring the velocity of
their motions. This may be done either by a comparison with linear mea-

ON THE REGULATION OF HYDRAULIC FORCES. 243

sures, or by instruments founded on the laws of hydraulic pressure. One
of the best of such instruments is the tube invented by Pitot,* and improved
by Buat.t A funnel is presented to the stream, and the water in a vertical
tube connected with it is elevated above the level of the river, nearly to the
height corresponding to the velocity : but it is said that the result will be
less liable to error, if the funnel be covered by a plate with a small orifice in
its centre, the elevation being in this case always half as great again as the
height due to the velocity. Other instruments intended for the|same pur-
pose, require some previous experiments for determining the degree in
which they are affected by different velocities ; in this manner the hydro-
metrical fly is adjusted ;£ the impulse of the water on two inclined planes
turning an axis to which they are fixed, and by its means a series of wheels,
with an index, which expresses the space described during the time of
observation. Instruments similar to these have also sometimes been em-
ployed, for measuring the relative velocity with which a ship under way
passes through the water ; and an apparatus, resembling Pitot' s, has been
adapted to this purpose by Captain Hamilton, with the addition of a tube
inserted into it on a level with the surface of the water, which continually
discharges a small stream into a reservoir with a velocity regulated by the
pressure, and consequently equal or proportional to that of the ship itself.§
In this manner he obtains an accurate register of the whole distance
described, including the effect of all the variations of the velocity. If the
orifice be small, it will be necessary to attend to the temperature of the
water, since the discharge is considerably retarded by any considerable
degree of cold. But when the aperture which determines the magnitude
of the discharge is wholly under water, as Captain Hamilton has placed
it, this source of error is probably much diminished. (Plate XXII. Fig.
288, 289.)

The motions of the air may also be measured by instruments similar to
those which are employed for determining the velocity of streams of water.
The direction of the wind is sometimes indicated by a wind dial, consisting
simply of an index, connected by wheels with a common vane or weather-
cock. Its velocity may be found by means of wind gages of different kinds : ||
these are sometimes constructed by opposing a flat surface to the wind, the
pressure being measured by the flexure of a spring, or by the winding up
of a weight on a spiral barrel ; and sometimes by receiving the stream in
the mouth of a funnel, so as to raise a column of water, in a vertical
tube, to a height equivalent to the pressure, or to condense a quantity of
air inclosed in a cavity, to a degree which is indicated by the place of
a small portion of mercury, moving in a horizontal tube, which leads to
the cavity. A little windmill, like the hydrometrical fly, may also be

* Hist, et Mem. del'Acad. de Par. 1732, p. 263, H. 103.
f Principes d'Hydraulique, vol. ii. See also

Langsdorffs Hydraulik, PI. 25.

Brouckner's Machine, Hist, et Mem. de Paris, 1750, H. 169. Woltmann,
Theorie des Hydrometrischen Fliigels, Hamb. 1790.
"

.

§ "Papers on Naval Architecture, Repert. ii. I. 355.
|| Such as Lind's Wind-Gage, Ph. Tr. 1775, p. 353.

R2

244 LECTURE XXVII.

employed for measuring the velocity of the wind, with the assistance of
a watch.*

The principal methods of applying the force of fluids to useful purposes
are to employ their weight, their impulse, or their pressure. The weight of
water may be applied, by collecting it in a reservoir which alternately
ascends and descends, by causing it to act within a pipe on a moveable
piston, or by conducting it into the buckets of a revolving wheel; its
impulse may be directed either perpendicularly or obliquely against a
moveable surface ; and its pressure may be obtained, without any imme-
diate impulse, by causing a stream to flow horizontally out of a moveable
pipe which revolves round an axis. The force of the air can only be applied
by means of its impulse, and this may be employed either perpendicularly
or obliquely.

When water is collected in a single reservoir, which serves to work a
pump or to raise a weight, the mode of its operation may be determined
from mechanical considerations only ; and it is obvious that if we are de-
sirous of preserving the whole force of the water, we must employ a second
reservoir to be filled during the descent of the first, which may either
descend in its turn, or empty itself into the first when it has ascended
again to its original situation. The action of a column of water inclosed in
a pipe, is of a nature nearly similar to that of such a reservoir, excepting
that the apparatus is more liable to friction ; the arrangement of its parts is
nearly similar, although in an inverted position, to that which is more com-
monly employed for raising water by means of pumps. But both these
methods of employing the weight of water are in great measure confined to
those cases in which it is to be procured in a small quantity, and may be
allowed to descend through a considerable height, and when the circum-
stances do not allow us to employ machines which require a greater space.

We have seen that in order to determine the effect of any force employed
in machinery, we must consider not only its magnitude, but also the velo-
city with which it can be brought into action, and we must estimate the
ultimate value of the power, by the joint ratio, or the product, of the force
and the velocity. Thus, if we had a corn mill, for example, in which we
wished the millstone to revolve with a certain velocity and to overcome a
given resistance, and supposing that this effect could be obtained by means
of a certain train of wheels from a given source of motion ; if the velocity
of the motion at its source be reduced to one half, we must double the
diameter of one of the wheels by which the force is communicated, in order
to give the millstone the desired velocity, and thus we must introduce a
mechanical disadvantage, which can only be compensated by a double in-
tensity in the force at its origin.

If we apply this estimation of effect to the motion of an overshot wheel,

* Huygens, Mach. Approuvees, i. 71. Sir C. Wren's "Weather-wiser, Birch's
Hist, of the Roy. Soc. i. 341. Hookes in his Philos. Experiments, &c. edited by
Derham, p. 41. Whewell's, Trans, of the Camb. Ph. Soc. vol. vi. Osier's, Report
of the Br. Ass. vol. vii. Sections, p. 33, and Description of a Self-registering Ane-
mometer, &c. 4to, Birmingham, 1839.

ON THE REGULATION OF HYDRAULIC FORCES. 245

we shall find that the velocity of the wheel, and consequently its hreadth,
and the magnitude of its buckets, is perfectly indifferent with respect to the
value of its operation : for supposing the stream to enter the buckets with
the uniform velocity of the wheel, the quantity of water in the wheel at any
one time, and consequently the pressure, must be inversely as the velocity,
so that the product of the force into the velocity will be the same, however
they may separately vary. If, however, the velocity were to become very
considerable, it would be necessary to sacrifice a material part of the fall,
in order that the water might acquire this velocity before its arrival at the
wheel ; but a fall of one foot, or even less, is sufficient for producing any
velocity that would be practically convenient : and it is obvious, on the
other hand, that a certain velocity may be procured from a wheel moving
rapidly, with less machinery than from another which moves more slowly.
In general the velocity of the surface of the wheel is between two and six
feet in a second ;* and whether it be greater or smaller, the force actually
applied will always be equal in effect to the weight of a portion of the
stream employed, equal in length to the height of the wheel. In order to
avoid the resistance which might be occasioned by the stagnant water below
the wheel, it is a good practice to turn the stream backwards upon its
nearer half, so that the water, when discharged, may run off in the general
direction of its motion. (Plate XXII. Fig. 290.)

If we suffer the stream of water to acquire the utmost velocity that the
whole fall can produce, and to strike horizontally against the floatboards of
an undershot wheel, or if we wish to employ the force of a river running in
a direction nearly horizontal, the wheel must move, in order to produce the
greatest effect, with half the velocity of the stream.t For the whole quan-
tity of water impelling the floatboards is nearly the same, whatever may be
the velocity, especially if the wheel is properly inclosed in a narrow chan-
nel, and hence it is easy to calculate that the greatest possible effect will be
produced when the relative velocity of the stream, striking the floatboards,
is equal to the velocity of the wheel itself. The pressure on the floatboards
is equal to that of a stream containing the same quantity of water, and
striking a fixed obstacle with half the velocity, that is, such a stream as
escapes from the wheel, which must be twice as deep or twice as wide as
the original stream, since its motion is only one half as rapid ; and a column
of such a stream, of twice the height due to its velocity, that is, of half the
height of the fall, being, as we have already seen, the measure of the
hydraulic pressure, this force will be precisely half as great as that of a
similar column, acting on an overshot wheel, which moves with the same
velocity.;}: But the stream thus retarded will not retain the other half of
its mechanical power ; since its greatest effect will be in the same propor-
tion to that of an equal stream acting on an overshot wheel with one fourth
of the fall of the former : and the remaining fourth of the power is lost in

* Smeaton, Ph. Tr. 1759, li. 134, deduces from experiments a little more than
three feet in a second, and observes, that high wheels (24 feet, or the like) may de-
viate more from this velocity than low, without materially affecting their work.

f Do. ibid. p. 122, gives the best proportion as 2 : 5. Compare Robison, Mech.
Phil. ii. 625. J Ibid. p. 130.

246 LECTURE XXVII.

producing the change of form of the water, and in overcoming its friction.
In whatever way we apply the force of water, we shall find that the me-
chanical power which it possesses must be measured by the product of
the quantity multiplied by the height from which it descends :* for exam-
ple, a hogshead of water capable of descending from a height of 10 feet,
possesses the same power as 10 hogsheads descending from a height of one
foot ; and a cistern filled to the height of 10 feet above its orifice possesses
100 times as much power as the same cistern filled to the height of one
foot only.

When, therefore, the fall is sufficiently great, an overshot wheel is far
preferable to an undershot wheel, and where the fall is too small for an
overshot wheel, it is most advisable to employ a breast wheel, which par-
takes of its properties, its floatboards consisting of two portions meeting at
an angle so as to approach to the nature of buckets, and the water being
also in some measure confined within them by the assistance of a sweep or
arched channel which follows the curve of the wheel, without coming too
nearly into contact with it so as to produce unnecessary friction. When
the circumstances do not admit even of a breast wheel, we must be con-
tented with an undershot wheel ; it is recommended, for such a wheel, that
the floatboards be so placed as to be perpendicular to the surface of the
water at the time that they rise out of it ; that only one half of each should
ever be below the surface, and that from three to five should be immersed
at once, according to the magnitude of the wheel. Sometimes, however, it
has been thought eligible to employ a much smaller number ; thus the
water wheel which propels Mr. Symington's steam-boatt has only six
floatboards in its whole circumference. (Plate XXII. Fig. 291, 292.)

Since the water escaping from an undershot wheel still retains a part of
its velocity, it is obvious that this may be employed for turning a second
wheel, if it be desirable to preserve as much as possible of the force. In this
case, by causing the first wheel to move with two thirds of the velocity of
the stream, the whole effect of both will be one third greater than that of a
single wheel placed in the same stream ; but it must be considered that
the expense of the machinery will also be materially increased.

Considerable errors have frequently been made by mathematicians and
practical mechanics in the estimation of the force of the wind or the water
on oblique surfaces ; they have generally arisen from inattention to the
distinction between pressure and mechanical power. It may be demon-
strated that the greatest possible pressure of the wind or water, on a given
oblique surface at rest, tending to turn it in a direction perpendicular to
that of the wind, is obtained when the surface forms an angle of about 55°
with the wind ; but that the mechanical power of such a pressure, which
is to be estimated from a combination of its intensity with the velocity of
the surface, may be increased without limit by increasing the angle of
inclination, and consequently the velocity. The utmost effect that could be
thus obtained would be equal to that of the same wind or stream acting on
the floatboards of an undershot wheel : but since in all practical cases the

* Smeaton, Ph. Tr. li. 116, 131 ; and Ixvi. 450.
f See Journal of the Royal Institution, vol. i.

ON THE REGULATION OF HYDRAULIC FORCES. 247

velocity is limited, the effect will be somewhat smaller than this : for
example, if the mean velocity of the sails or floatboards be supposed equal
to that of the wind, the mechanical power will be more than four fifths as
great as that of an undershot wheel, that is, in the case of a windmill,
more than four fifths of the utmost effect that can be obtained from the
wind. In such a case Maclaurin has shown that the sails ought to make
an angle of 74° with the direction of the wind : * but in practice it is
found most advantageous to make the angle somewhat greater than this,
the velocity of the extremities of the sails being usually, according to
Mr. Smeaton, more than twice t as great as that of the wind. It appears,
therefore, that the oblique sails of the common windmill are in their nature
almost as well calculated to make the best use of any hydraulic force as
an undershot wheel ; and since they act without intermission throughout
their whole revolution, they have a decided advantage over such machines
as require the sails or fans to be exposed to a more limited stream of the
wind, during one half only of their motion, which is necessary in the
horizontal windmill, where a screen is employed for covering them while
they are moving in a direction contrary to that of the wind : and such
machines, according to Smeaton, % are found to perform little more than
one tenth of the work of those which are more usually employed. The
sails of a common windmill are frequently made to change their situation
according to the direction of the wind, by means of a small wheel, with
sails of the same kind, which turns round whenever the wind strikes on
either side of it, and drives a pinion turning the whole machinery ; the
sails are sometimes made to furl or unfurl themselves, according to the
velocity of the wind, by means of a revolving pendulum, which rises to a
greater or less height, in order to prevent the injury which the flour would
suffer from too great a rapidity in the motion, or any other accidents which
might happen in a mill of a different nature. The inclination of the axis
of a windmill to the horizon is principally intended to allow room for the
action of the wind at the lower part, where it would be weakened if the
sails came too nearly in contact with the building, as they must do if they
were perfectly upright. When it is necessary to stop the motion of a
windmill, a break is applied to the surface of a large* wheel, so that its
friction operates with a considerable mechanical advantage. Water wheels
with oblique floatboards are sometimes used with good effect in China and
in the south of France : for tide wheels, such floatboards have the advan-
tage that they may be easily made to turn on a hinge with the stream, so
as to impel the wheel in the same direction whether the tide be flowing or
ebbing. (Plate XXII. Fig. 293.)

A smoke jack is a windmill in miniature ; a kite affords a very familiar
example of the effect of the oblique impulse of the air, of which the action
first causes a pressure perpendicular to the surface of the kite, and this
force, combined with the resistance of the string, produces a vertical result
capable of counteracting the weight of the kite. (Plate XXII. Fig. 294.)

* Maclaurin's Account of Sir I. Newton's Philos. Discoveries, art. 29.
f Nearer three times. See Smeaton, Ph. Tr. 1759, li. 163.
t Ibid. p. 172.

248 LECTURE XXVII.

The counterpressure of the water, occasioned by the escape of a stream
from a moveable reservoir, was applied by Parent* to the purpose of turn-
ing a millstone, and various other authors have described machines of a
similar nature : they may be constructed with little or no wheel work, and it
does not appear to be necessary that much of the force of the water sliould be
lost in their operation ; but they have never been practically employed
with success, nor have they perhaps ever had a fair trial.

The art of seamanship depends almost entirely on the management of
the forces and resistances of air and water, and if the laws of hydraulic
pressure, with respect to oblique and curved surfaces, were more completely
ascertained, we might calculate not only what the motions of a ship would
be under any imaginable circumstances, but we might also determine pre-
cisely what would be the best possible form of a ship, and what the best
arrangement of her rigging.

When a ship is sailing immediately before the wind, little or no art is
required in setting her sails, and her velocity is only limited by that of the
wind and by the resistance of the water : but for sailing with a side wind,
it becomes necessary that the immediate force of the wind should be con-
siderably modified.

If we had a circular vessel or tub, with a single mast, and a sail perfectly
flat, and if the sail were placed in a direction deviating but little from that
of the wind, the tub would begin to move in a direction nearly at right
angles to that of the wind, since the impulse of the wind acts almost
entirely in a direction perpendicular to that of the sail : but the slightest
inequality of the dimensions of the sail, or of the force of the wind, would
immediately disturb the position of the vessel ; and in order to avoid this
inconvenience, it would be necessary to have a moveable body projecting
into the water, so as to create a resistance by means of which the vessel
might be steered, and the sail confined to its proper place : and this might
be done more effectually by changing the form of the vessel from round to
oval ; it would then also have the advantage of moving much more easily
through the water in the direction of its length than a circular vessel of
equal size, and of creating still more resistance in a transverse direction, so
that when urged by an oblique force, it would move in some measure
obliquely, but always much more nearly in the direction of its length than
of its breadth. The angular deviation from the track of the ship is called
its lee way, and if we know the direction of the sails, and the actual pro-
portions of the resistances opposed to the ship's motion in different direc-
tions, we may calculate from these resistances the magnitude of the angular
deviation or lee way : but hitherto such calculations have generally indi-
cated a lee way three or four times as great as that which has been
observed. The use of the keel is not only to assist in confining the motion
of the ship to its proper direction, but also to diminish the disposition to
vibrate from side to side, which would interfere with the effect of the sails,
and produce many other inconveniences. When the principal force of the
wind is applied to the anterior part of the ship, her head would be naturally

* Hist, et Mem. de Paris, 1704. See Euler, Hist, et Mem. de Berlin, 1750,
1751, 1752. Waring, American Transactions, iii. 185.

ON THE REGULATION OF HYDRAULIC FORCES. 249

turned from the wind if the rudder were not made to project from the stern
in a contrary direction, and to present the surface of an inclined plane to
the water which glides along the keel, so as to preserve the ship, by means
of the pressure which it receives, in any direction that may be required for
her manoeuvres. Commonly, however, although the sails may be so
arranged that the principal force of the wind appears to be on the fore part
of the ship, the curvature of the sails, or some other cause, throws the pres-
sure further backwards, and the action of the rudder is necessary to prevent
the ship's head turning towards the wind. (Plate XXII. Fig. 295.)

When a ship is steering in this manner on a side wind, the effect of the
wind has a natural tendency to overset her, and if she is too crank, that is,
deficient in stability, she cannot sail well, otherwise than directly before the
wind. The place of the centre of gravity, compared with that of the meta-
centre, or imaginary centre of pressure, determines the degree of stability,
and the most general way of increasing it is to lessen the weight of the
upper part and of the rigging of the vessel, to diminish her height, or to
increase her breadth, and to stow the ballast as low as possible in the hold.
Too little attention has frequently been paid to this subject, as well as to
many other departments of naval architecture ; and although mere theore-
tical investigations have hitherto been but of little service to the actual
practice of seamanship, yet it cannot be doubted that an attention to what
has already been discovered of the laws of hydrodynamics, as well as to the
principles of mechanics in general, must be of great advantage to the navi-
gator, in enabling him to derive from his own experience all the benefits
which a correct mode of reasoning is capable of procuring him.

LECT. XXVII.— ADDITIONAL AUTHORITIES.

Force of Water. — Segner, Exercitationes Hydraulics, 4to, Gott. 1747. J. A.
Euler, Enodatio Qusestionis de Molis, Gott. 1754. Lambert on Mills, Hist, et
Mem. de Berlin, 1755. On Water-wheels, ibid. 1755, pp. 49, 70, 82. Mallet on
do. Ph. Tr. 1767, pp. 57, 372. Borda on do. Hist. etMem. de Paris, 1767, p. 270,
H. 149. Bossut on do. ibid. 1769, pp. 288, 477, H. 121. Fabre, Essai sur la
Maniere la plus avantageuse de construire les Machines Hydrauliques, 4to, 1783.
Buchanan on Water-wheels, Ph. Mag. x 278 ; xi. 79. Essay on Millwork, 2 vols.
1823. L'Huillier sur rArtd'Employerl'Eaucomme Moteur des Roues, Paris, 1823.

Force of Wind. — Hooke on the Sails of Mills and Ships, Philosophical Collections,
No. 3, p. 61. Lahire on Windmills, Hist, et Mem. de Paris, ix. 96. Euler on
do. Nov. Com. Petr. vi. 41. Hist, et Mem. de Berlin, 1756, p. 165. Bourrier's
Horizontal Mill, Hist, et Mem. de Paris, 1762, H. 190. Maiziere's Windmill, ibid.
1767, H. 185. Coulomb on Windmills and the form of their Sails, ibid. 1781,
p. 65, H. 41. Repertory of Arts, iv. 12 ; vii. 6, ii. I. II. 13, Phil. Mag. iv. 174. See
also Leupold's Theatrum Hydraulicum, Bailey's Machines, Machines Approuvees,
Emerson's Mechanics, and Encyclep. Method, art. Meunier.

Seamanship. — John Bernoulli's Theory of the Manoeuvres of Ships, Hist, et
Mem. de Paris, 1714, H. 107. Pitot on do. ibid. 1731, H. 81. Bouguer, ibid.
1754, p. 342, H. 91 ; 1755, p. 355, 481, H. 83, 135. Clairaut, ibid. 1760, p. 171,
H. 141. Bouguer, de la Manoauvre des "Vaisseaux, 4to, 1757. Euler, Scientia
Navalis, 2 vols. 4to, Petrop. 1749. Theorie de la Man. des Vais., Pet. 1773.
Romme, Art de la Marine, 4 to, Paris, 1787. Hutchison's Seamanship, 4to, 1794.
Chapman on Canal Navigation, 4to, 1797. Bezout, Traite de Navigation, Paris.
1814.

Naval Architecture. — Meibomius de Triremium Fabrica, 4to, Amst. 1671. Du-

250 LECTURE XXVIII.

hamel, Architecture Navale, 4to, Paris, 1758. Gordon's Principles of Naval Archi-
tecture, Lond. 1784. Chapman, Traite de la Construction des Vaisseaux, trans-
lated by Inman. Also, Essays, in Papers on Naval Architecture. Atwood on the
Stability of Ships, Ph. Tr. 1796, p. 46; 1798, p. 201. Euler on the Construc-
tion of Vessels, by Sir G. Shee. Trans. Roy. Ir. Acad. vi. 15. Watson's Elements
of Naval Architecture, fol. 1805.

LECTURE XXVIII.

ON HYDRAULIC MACHINES.

WE shall apply the denomination of hydraulic machines to such only,
as are intended for counteracting the gravity of water, that is, for raising it
from a lower situation to a higher.. The simplest of these are buckets,
bucket wheels, and friction ropes ; moveable pipes are the next in order ;
and pumps of various kinds constitute the most extensive and the most
important part of the subject. Besides these and some other similar
machines, hydraulic air vessels and artificial fountains will also require to
be examined.

A series of earthen pitchers, connected by ropes, and turned by trundles
or pinions, over which they pass, has long been used in Spain under the
name of noria : in this country buckets of wood are sometimes employed
in a similar manner. A bucket wheel is the reverse of an overshot water-
wheel, and the water may be raised by buckets nearly similar to those
which are calculated for receiving it in its descent : sometimes the buckets
are hung on pins, so as to remain full during the whole ascent ; but these
wheels are liable to be frequently out of repair. Sometimes the reverse of
an undershot wheel or rather of a breast wheel, is employed as a throwing
wheel, either in a vertical or in an inclined position. Such wheels are
frequently used for draining fens, and are turned by windmills ; the float-
boards are not placed in the direction which would be best for an undershot
wheel, but on the same principle, so as to be perpendicular to the surface
when they rise out of it, in order that the water may the more easily flow
offthem.* (Plate XXII. Fig. 296.. .298.)

Instead of a series of buckets connected by ropes or chains, a similar effect
is sometimes produced by a simple rope, or a bundle of ropes, passing over
a wheel above, and a pulley below, moving with a velocity of about 8 or 10
feet in a second, and drawing a certain quantity of water up by its fric-
tion. It is probable that the water commonly ascends with about half the
velocity of the rope, and on this supposition we might calculate its depth on
the rope by comparing its relative motion with that of a little river : but
the rules which serve for calculating the velocity of rivers, do not perfectly
agree in this case with the results of direct experiments ; for the friction

* Vitruvius, Architecture, 1. 10, c. 9, translated by Newton, 2 vols. fol. London.

ON HYDRAULIC MACHINES. 251

required for elevating the quantity raised by such a machine, appears
from calculation to correspond to a velocity about twice as great as the
actual relative velocity. While the water is principally supported by
the friction of the rope, its own cohesion is amply sufficient to prevent
its wholly falling, or being scattered, by any accidental inequality of the
motion. (Plate XXII. Fig. 299.)

The lateral friction of water has been applied in a very simple manner
by Venturi* to the draining of land by means of a stream which runs
through it, allowing the stream to acquire sufficient velocity to carry it
over an inclined surface, and to drag with it a certain portion of water
from the lowest part of this surface : but the quantity of water raised
in this manner must be very inconsiderable, and the loss of force by fric-
tion very great.

A system of spiral pipes may be placed in the plane of a wheel, receiving
the water at its circumference, and raising it by degrees, as the wheel turns,
towards the axis, where it is discharged ; the motion of the wheel being
usually derived from the same stream which supplies the pipes : but the
height to which the water is raised by this machine is very small in
proportion to its bulk. A single pipe wound spirally round a cylinder
which revolves on an axis in an oblique situation, has been denominated
the screw of Archimedes,t and is called in Germany the water snail. Its
operation, like that of the flat spiral, may be easily conceived by imagining
a flexible pipe to be laid on an inclined plane, and its lower part to be
gradually elevated, so that the fluid in the angle or bend of the pipe may
be forced to rise ; or by supposing a tube, formed into a hoop, to be rolled
up the same plane, the fluid being forced by the elevation of the tube
behind it to run as it were up hill. This instrument is sometimes made by
fixing a spiral partition round a cylinder, and covering it with an external
coating, either of wood or of metal ; it should be so placed with respect to
the surface of the water as to fill in each turn one half of a convolution ;
for when the orifice remains always immersed, its effect is much dimi-
nished. It is generally inclined to the horizon in an angle of between 45
and 60 degrees : hence it is obvious that its utility is limited to those cases
in which the water is only to be raised to a moderate height. The spiral
is seldom single, but usually consists of three or four separate coils, forming
a screw which rises slowly round the cylinder. (Plate XXII. Fig. 300,
301.)

An instrument of a similar nature is called by the Germans a water
screw ; it consists of a cylinder with its spiral projections detached from
the external cylinder or coating, within which it revolves. This machine
might not improperly be considered as a pump, but its operation is pre-
cisely similar to that of the screw of Archimedes. It is evident that some
loss must here be occasioned by the want of perfect contact between the

* Prop. 9.

t Vitruvius, 1. 10, c. 11. Pitot, Hist, et Mem. de Paris, 1736, p. 173, H. 110.
TSuler, Nov. Com. Petr. v. 259. Hennert, Dissertation sur la vis d'Archimede,
Berl. 1767. Pattu, Journal des Mines, 1815, xxxviii. 321. Gregory's Mechanics,
ii. 343.

252 LECTURE XXVIII.

screw and its cover ; in general, at least one third of the water runs back,
and the machine cannot be placed at a greater elevation than 30° ; it is
also very easily clogged by accidental impurities of the water : yet it has
been found to raise more water than the screw of Archimedes, when the
lower ends of both are immersed to a considerable depth ; so that if the
height of the surface of the water to be raised were liable to any great vari-
ations, the water screw might be preferable to the screw of Archimedes.
(Plate XXII. Fig. 302.)

When a spiral pipe, consisting of many convolutions, arranged either in
a single plane, or in a cylindrical or conical surface, and revolving round a
horizontal axis, is connected at one end by a watertight joint with an as-
cending pipe, while the other end receives during each revolution nearly
equal quantities of air and water, the machine is called a spiral pump. It
was invented about 1746, by Andrew Wirtz, a pewterer at Zurich, and it
is said to have been used with great success at Florence and in Russia : it
has also been employed in this country by Lord Stanhope, and I have
made trial of it for raising water to a height of forty feet.'* The end of the
pipe is furnished with a spoon, containing as much water as will fill half a
coil, which enters the pipe a little before the spoon has arrived at its
highest situation, the other half remaining full of air, which communicates
the pressure of the column of water to the preceding portion, and in this
manner the effect of nearly all the water in the wheel is "united, and be-
comes equivalent to that of the column of wrater, or of water mixed with
air, in the ascending pipe. The air nearest the joint is compressed into a
space much smaller than that which it occupied at its entrance, so that
where the height is considerable, it becomes advisable to admit a larger
portion of air than would naturally fill half the coil, and this lessens the
quantity of water raised, but it lessens also the force required to turn the
machine. The joint ought to be conical, in order that it may be tightened
when it becomes loose, and the pressure ought to be removed from it as
much as possible. The loss of power, supposing the machine well con-
structed, arises only from the friction of the water on the pipe, and the
friction of the wheel on its axis ; and where a large quantity of water is to
be raised to a moderate height, both of these resistances may be rendered
inconsiderable. But when the height is very great, the length of the spiral
must be much increased, so that the weight of the pipe becomes extremely
cumbersome, and causes a great friction on the axis, as well as a strain on
the machinery : thus, for a height of 40 feet, I found that the wheel
required above 100 feet of a pipe which was three quarters of an inch in
diameter ; and more than one half of the pipe being always full of water,
we have to overcome the friction of about 80 feet of such a pipe, which will
require 24 times as much excess of pressure to produce a given velocity, as
if there were no friction. The centrifugal force of the water in the wheel
would also materially impede its ascent if the velocity were considerable,
since it would be always possible to turn it so rapidly as to throw the
whole water back into the spoon. The machine which I had erected being

* Sulzer's Sammlungen Vermischeln Schriften, 1754. Ziegler, Gesellschaft zu
Zurich, vol. iii. Nicander, Schwed. Abhand. 1783.

ON HYDRAULIC MACHINES. 253

out of repair, I thought it more eligible to substitute for it a common
'forcing pump, than to attempt to make any further improvement in it,
under circumstances so unfavourable. But if the wheel with its pipes were
entirely made of wood, it might in many cases succeed better : or the pipes
might be made of tinned copper, or even of earthenware, which might be
cheaper and lighter than lead. (Plate XXII. Fig. 303.)

The centrifugal force, which is an impediment to the operation of
Wirtz's machines, has sometimes been employed, together with the pressure
of the atmosphere, as an immediate agent in raising water, by means of the
rotatory pump. This machine consists of a vertical pipe, caused to revolve
round its axis, and connected above with a horizontal pipe, which is open
at one or at both ends, the whole being furnished with proper valves to pre-
vent the escape of the water when the machine is at rest. As soon as the
rotation becomes sufficiently rapid, the centrifugal force of the water in the
horizontal pipe causes it to be discharged at the end, its place being sup-
plied by means of the pressure of the atmosphere on the reservoir below,
which forces the water to ascend through the vertical pipe. It has also
been proposed to turn a machine of this kind by the counterpressure of
another portion of water, in the manner of Parent's mill, where there is
fall enough to carry it off.* This machine may be so arranged that,
according to theory, little of the force applied may be lost ; but it has
failed of producing in practice a very advantageous effect. (Plate XXIII.
Fig. 304.)

A pump is a machine so well known, and so generally used, that the de-
nomination has not uncommonly been extended to hydraulic machines of
all kinds ; but the term, in its strictest sense, is to be understood of those
machines in which the water is raised by the motion of one solid within
another, and this motion is usually alternate, but sometimes continued so as
to constitute a rotation. In all the pumps most commonly used, a cavity
is enlarged and contracted by turns, the water being admitted into it through
one valve, and discharged through another.

One of the simplest pumps for raising a large quantity of water to a small
height, is made by fitting two upright beams or plungers, of equal thickness
throughout, into cavities nearly of the same size, allowing them only room
to move without friction, and connecting the plungers by a horizontal beam
moving on a pivot. The water being admitted, during the ascent of each
plunger, by a large valve in the bottom of the cavity, it is forced, when the
plunger descends, to escape through a second valve in the side of the cavity,
and to ascend by a wide pipe to the level of the beam. The plungers ought
not to be in any degree tapered, because of the great force which would be
unnecessarily consumed, in continually throwing out the water, with great
velocity, as they descend, from the interstice formed by their elevation.
This pump may be worked by a labourer, walking backwards and forwards,
either on the beam or on a board suspended below it. By means of an ap-
paratus of this kind, described by Professor Robison,tan active man, loaded
with a weight of thirty pounds, has been able to raise 580 pounds of water

* West in Tilloch's Ph. Mag. vol.xi.

f Mechanical Philosophy, art. Pump, ii. 671.

254 LECTURE XXVIII.

every minute, to a height of 11^ feet, for ten hours a day, without fatigue ;
this is the greatest effect produced by a labourer that has ever been correctly
stated by any author ; it is equivalent to somewhat more than 1 1 pounds
raised through 10 feet in a second, instead of 10 pounds, which is a fair
estimate of the usual force of a man, without any deduction for friction.
(Plate XXIII. Fig. 305.)

It is obvious that if the plungers were so well fitted to the cavity as to
prevent the escape of any water between them, the ascending pipe might
convey the water to any required height ; the machine would then become
a forcing pump, and the plungers might be shortened at pleasure, so as to
assume the form of a piston sliding within a barrel. The piston might also
be situated above the level of the reservoir, and in this case the water would
be forced up after it by the pressure of the atmosphere to the height of
about 30 feet, but not much further : and even this height would be some-
what too great for practice, because the water might sometimes follow the
piston in its ascent too slowly. Such a pump, partaking of the nature of a
forcing and a sucking pump, is sometimes called a mixed pump. In
Delahire's pump, the same piston is made to serve a double purpose, the rod
working in a collar of leathers, and the water being admitted and expelled
in a similar manner, above and below the piston, by means of a double ap-
paratus of valves and pipes.* (Plate XXIII. Fig. 306.)

For forcing pumps of all kinds, the common piston, with a collar of
loose and elastic leather, is preferable to those of a more complicated struc-
ture : the pressure of the water on the inside of the leather makes it suffi-
ciently tight, and the friction is inconsiderable. In some pumps the leather
is omitted, for the sake of simplicity, the loss of water being compensated
by the greater durability of the pump ; and this loss will be the smaller
in proportion as the motion of the piston is more rapid. (Plate XXIII.
Fig. 307.)

Mr. Bramah has very ingeniously applied a forcing pump, by means of
the well known properties of hydrostatic pressure, to the construction of a
convenient and powerful press. The water is forced, by a small pump, into
a barrel in which it acts on a much larger piston ; consequently this piston
is urged by a force as much greater than that which acts on the first pump
rod, as its surface is greater than that of the small one. (Plate XXIII.
Fig. 308.)

In the common sucking pump, the valve through which the water
escapes is placed within the piston itself, so that the same barrel serves for
the ascent of the water, which rises in one continued line while the piston
is raised, and rests on the fixed valve while it is depressed. The velocity of
the stroke ought never to be less than 4 inches in a second, nor greater than
two or three feet ; the stroke should also be as long as possible, in order to
avoid unnecessary loss of water during the descent of the valves. The di-
ameter of the pipe through which the water rises to the barrel, ought not to
be less than two thirds of the diameter of the barrel itself. (Plate XXIII.
Fig. 309.)

A bag of leather has also been employed for connecting the piston of a

* Hist, et Mem. de 1'Acad. 1716, p. 322.

ON HYDRAULIC MACHINES. 255

pump with the barrel, and in this manner nearly avoiding all friction : but
it is probable that the want of durability would be a great objection to such
a machine. (Plate XXIII. Fig. 310.)

Where the height, through which the water is to be raised, is consider-
able, some inconvenience might arise from the length of the barrel through
which the piston rod of a sucking pump would have to descend, in order
that the piston might remain within the limits of atmospheric pressure.
This may be avoided by placing the inoveable valve below the fixed valve,
and introducing the piston at the bottom of the barrel. Such a machine is
called a lifting pump : in common with other forcing pumps, it has the
disadvantage of thrusting the piston before the rod, and thus tending to
bend the rod, and produce an unequal friction on the piston, while, in the
sucking pump, the principal force always tends to straighten the rod.
(Plate XXIII. Fig. 311.)

The rod of a sucking pump may also be made to work in a collar of
leather, and the water may be forced through a valve into an ascending
pipe. By applying an air vessel to this, or to any other forcing pump, its
motion may be equalised, and its performance improved ; for if the orifice
of the air vessel be sufficiently large, the water may be forced into it,
during the stroke of the pump, with any velocity that may be required,
and with little resistance from friction, while the loss of force, from the
frequent accelerations and retardations of the whole body of water, in a
long pipe, must always be considerable. The condensed air, reacting on
the water, expels it more gradually, and in a continual stream, so that the
air vessel has an effect analogous to that of a fly wheel in mechanics.
(Plate XXIII. Fig. 312.)

If, instead of forcing the water to a certain height through a pipe, we
cause it to form a detached jet, we convert the forcing pump into a fire
engine : and in general two barrels, acting alternately, are connected, for
this purpose, with the same air vessel ; so that the discharge is thus
rendered very nearly uniform. The form of the adjutage, or orifice of the
pipe, is by no means indifferent to the effect of the machine, since the
height of the jet may be much increased by making it moderately con-
tracted, and a little conical rather than cylindrical. When the air vessel
is half filled with water, the height of such a jet will be about 30 feet, when
two thirds filled, about 60, the height being always nearly proportional to
the degree of condensation of the air, or to the excess of its density above
that of the surrounding atmosphere. Sometimes a double forcing pump, or
fire engine, is formed by the alternate rotatory motion of a flat piston
within a cylindrical barrel ; the axis of its motion coinciding with that of
the barrel, and the barrel being divided by a partition into two cavities,
which are filled and emptied in the same way as the separate barrels of the
common fire engine. The mechanical advantage of this machine is nearly
the same as that of the more usual constructions, but it appears to be some-
what more simple than a common engine of equal force. The partition
may be extended throughout the diameter of the cylinder, the opposite pairs
of cavities being made to communicate with each other, and thus both sides
of the piston may be employed at once. (Plate XXIII. Fig. 313.)

256 LECTURE XXVIII.

A piston placed in a similar manner has sometimes been made to revolve
continually, and to force the water through a pipe by means of a slider or A
spring, which intercepts its passage in any other direction. Machines of this
kind have been invented and rein vented, by Ramelli,* Cavalleri,f Amontons,J
Prince Rupert,t Dr. Hooke, Mr. Bramah,§ and Mr. Gwynn. Mr. Gwynn's
engine, which has been employed in many cases with considerable success,
consists of a piston or roller nearly elliptical, well fitted to the cylinder
within which it revolves, with a valve pressed lightly against it by a spring,
which causes a considerable part of the water contained in the cylinder to
be forced in each revolution into the pipe : the whole machine is made of
brass ; the spring requires very little force, for the pressure of the water
on the valve keeps it always close to the roller, and the friction arising
from this cause is even an objection to the machine. The stream, although
never wholly intermitted, is, however, by no means uniform in its velocity.
(Plate XXIII. Fig. 314... 317.)

The pipes, through which water is raised by pumps of any kind, ought
to be as short and as straight as possible ; thus, if we had to raise water to
a height of 20 feet, and to carry it to a horizontal distance of 100 by
means of a forcing pump, it would be more advantageous to raise it first
vertically into a cistern 20 feet above the reservoir, and then to let it run
along horizontally, or find its level in a bent pipe, than to connect the
pump immediately with a single pipe carried to the place of its destination.
And for the same reason a sucking pump should be placed as nearly over
the well as possible, in order to avoid a loss of force in working it. If
very small pipes are used, they will much increase the resistance, by the
friction which they occasion.

Water has been sometimes raised by stuffed cushions, or by oval blocks
of wood, connected with an endless rope, and caused by means of two
wheels or drums, to rise in succession in the same barrel, carrying the
water in a continual stream before them ; but the magnitude of the friction
of the cushions appears to be an objection to this method. From the re-
semblance of the apparatus to a string of beads, it has been called a bead
pump, or a paternoster work. When flat boards are united by chains,
and employed instead of these cushions, the machine may be denominated
a cellular pump ; and in this case the barrel is usually square, and placed
in an inclined position, but there is a considerable loss from the facility
with which the water runs back. The chain pump generally used in the
navy is a pump of this kind, with an upright barrel, through which
leathers, strung on a chain, are drawn in constant succession ; these pumps
are only employed, when a large quantity of water is to be raised, and
they must be worked with considerable velocity in order to produce any

* Artificiose Machine, fol. Paris, 1588.

t Exercit. Geomet. p. 541. Birch; i. 285.

J Machines et Inventions Approuvees par 1'Academie, 7 vols. 4to, 1735, v. i.

to which work we refer for the description of numerous hydraulic machines by Per-
rault, Cusset, Joly, Francini, Cordamoy, Gay, L' Heureux, Joue, Martenot, Mar-
chand, Auger, Ublemann, Laesson, Denisart, Ledemoust, Boulogne, Saulm, Gallon,
Deparcieux, Gensanne, Dupuy, Amy, &c.

§ Repertory of Arts, ii. 73.

ON HYDRAULIC MACHINES. 257

effect at all. Mr. Cole has improved the construction of the chain pump,
so as materially to increase the quantity of water raised by it.* (Plate
XXIII. Fig. 318.)

It is frequently necessary to procure alternate motion in pumps by
means of wheel-work, and for this purpose the application of a crank is the
most usual and perhaps the best method. Provided that the bar by which
it acts be sufficiently long, very little will be lost by the obliquity of its
situation, and it is easy, by means of rollers, or of a compound frame, to
confine the head of the pump rod to a rectilinear motion. When any
other mode is employed, it must be remembered that the motion of the
pump rod ought always to be slower at the beginning of each alternation,
since a considerable part of the force is consumed in setting the water in
motion, especially where the pipe is long, and the velocity considerable.
But it may happen that, from the nature of hydraulic pressure under
other circumstances, the resistance may be nearly equal throughout the
stroke : for example, when the motion of the piston is slow in comparison
of that of the water in the pipe, or when the force employed in producing
velocity is inconsiderable, in comparison with that which is required for
counteracting the pressure. In such cases it may sometimes be eligible to
employ inclined surfaces of such forms as are best adapted to communicate
the most advantageous velocity to the pump rod by their pressure on a
roller, which may be confined to its proper direction by the same means as
when a crank is used. (Plate XIV. Fig. 184... 187.)

The Chinese work their cellular pumps, or bead pumps, by walking on
bars which project from the axis of the wheel or drum that drives them,
and whatever objection may be made to the choice of the machine, the
mode of communicating motion to it must be allowed to be advantageous.

Pumps have sometimes been worked by means of the weight of water
acting within a barrel, which resembles a second pump placed in an in-
verted position. The only objection to the machine appears to be the
magnitude of the friction, and even this inconvenience may perhaps be
inconsiderable. The invention is by no means modern,t but it is best
known in Germany under the name of HolTs machine,^ and it has been
introduced into this country by Mr. Westgarth§ and Mr. Trevithick.||
A chain pump, or a series of buckets, may also be applied, in a manner
nearly similar, to the working of machinery of any kind. (Plate XXIII.
Fig. 319.)

The mediation of a portion of air is employed for raising water, not
only in the spiral pump, but also in the air vessels of Schernnitz.«[ A
column of water, descending through a pipe into a closed reservoir full of
air, obliges the air to act, by means of a pipe leading from the upper part

* London Magazine for 1768, p. 499.

t It is figured in Fludd's Naturae Simia, Oppenheim, 1618, p. 467.
J Hist. etMem. 1760, H. 160.

§ Bailey's Machines, ii. 52. Smeaton, Transactions of the Society of Arts,
vol. v.
•|| Nich. Jour. 8vo, i. 161.

^ Wolfe's Description of Hero's Fountain at Schemnitz, Ph. Tr. 1762, p. 547.
Poda's do. Prag. 1771. Nich. Jour. iv. 8, 117.

s

258 LECTURE XXVIII.

of the reservoir or air vessel, on the water in a second reservoir, at any
distance either below or above it, and forces this water to ascend through-
a third pipe to any height less than that of the first column. The air
vessel is then emptied, and the second reservoir filled, and the whole opera-
tion is repeated. The air must, however, acquire a density equivalent to
the pressure, before it can begin to act ; so that if the height of the
columns were 34 feet, it must be reduced to half its dimensions before any
water would be raised ; and thus half of the force would be lost ; in the
same manner, if the height were 68 feet, two thirds of the force would be
lost. But where the height is small, the force lost in this manner is not
greater than that which is usually spent in overcoming friction and other
imperfections of the machinery employed ; for the quantity of water,
actually raised by any machine, is not often greater than half the power
which is consumed. The force of the tide, or of a river rising and falling
with the tide, might easily be applied by a machine of this kind, to the
purposes of irrigation. (Plate XXIII. Fig. 320, 321.)

The fountain of Hero precisely resembles in its operation the hydraulic
vessels of Schemnitz, which were probably suggested to their inventor by
the construction of this fountain.* The first reservoir of the fountain is
lower than the orifice of the jet ; a pipe descends from it to the air vessel,
which is at some distance below, and the pressure of the air is communi-
cated, by an ascending tube to a third cavity, containing the water which
supplies the jet. Many other hydraulic and pneumatic instruments, in-
tended for amusement only, and some of them of much more complicated
structure, are also described in the works of Hero. (Plate XXIII. Fig.
322.)

The spontaneous vicissitudes of the pressure of the air, occasioned by
changes in the weight and temperature of the atmosphere, have been ap-
plied, by means of a series of reservoirs furnished with proper valves, to
the purpose of raising water by degrees to a moderate height. But it
seldom happens that such changes are capable of producing an elevation in
the water of each reservoir of more than a few inches, or at most a foot or
two, in a day ; and the whole quantity raised must, therefore, be very
inconsiderable.

The momentum of a stream of water, flowing through a long pipe, has
also been employed for raising a small quantity of water to a considerable
height.

The passage of the pipe being stopped by a valve, which is raised by the
stream, as soon as its motion becomes sufficiently rapid, the whole column
of fluid must necessarily concentrate its action almost instantaneously on
the valve ; and in this manner it loses, as we have before observed, the cha-
racteristic property of hydraulic pressure, and acts as if it were a single
solid ; so that, supposing the pipe to be perfectly elastic and inextensible,
the impulse must overcome any pressure, however great, that might be
opposed to it, and if the valve open into a pipe leading to an air vessel, a
certain quantity of the water will be forced in, so as to condense the air,

* See Heronis Spiritalium Liber, Lat. & F. Commandino, 4to, Par. 1583; or
Veter. Math. Op. fol. 1693.

ON PNEUMATIC MACHINES. 259

more or less rapidly, to the degree that may be required for raising a por-
•tion of the water contained in it to any given height. Mr. Whitehurst
appears to have been the first that employed this method ;* it was after-
wards improved by Mr. Boulton ;t and the same machine has lately at-
tracted much attention in France under the denomination of the hydraulic
ram of Mr. Montgolfier.J (Plate XXIII. Fig. 323.)

LECT. XXVIII.— ADDITIONAL AUTHORITIES.

Strada, Wasserkunsten, fol. Frankfort, 1617 ; Cologne, 1623. De Caus, Inven-
tions Hydrauliques, translated into English by Leak, fol. 1659. Morland, Elevation
des Eaux, 1685. Papin's Engine for Raising Water, Ph. Tr. 1686, p. 283. Recueil,
Cassel, 1695. In the vols. of the Hist. et,Mem. de 1' Acad. de Paris are the follow-
ing : — Lafaye's Hydraulic Machine, 1717, p. 67, H. 70. Mey and Meyer's do.
1726, H. 71. Lebrun's do. 1731, H. 91 ; his Piston, 1735, H. 102. Drussen's
Puinp, ibid. Renon's Mach. ibid. Bertier's Mach. ibid. Pitot's Theory of Pumps,
1735, p. 327, H. 72 ; 1739, p. 393 ; 1740, p. 511. Camus on the Best Application
of Buckets, 1739, p. 157, H. 49 ; on the Best Proportion of Pumps, 1739, p. 287,
H. 49. Gensanne on Pumps, 1741, H. 163. Geffrier's Hyd. Mach. 1743, H. 168.
Thillay's Fire Engine, 1746, H. 120. Bonnet's, 1749, H. 182. Jacquet's Piston,
1752, H. 148. D'Arcy, 1754, p. 699, H. 138. Veltman, 1756, H. 129. Varan,
1760, H. 162. Limbourg, 1761, H. 154. Loritt's Endless Chain, 1761, H. 161.
Deparcieux, 1762, p. 1, H. 182. Nollet's Pumps, 1766, H. 150, Borda on Pumps,
1768, p. 418, H. 122. Quentin, 1769, H. 130. Bertier, 1770, H. 117. Recueil
d' Ouvrages Curieux de Math, et de Mec. ; ou Description du Cabinet de M. Grol-
lier de Serviere par son Petit Fils, Lyons, 1719. Briickmann and Weber's Elemen-
tar Maschine, Cassel, 1720. Beighton on the Water Works at London Bridge, Ph.
Tr. 1731, p. 5. Churchman's Engine, Ph. Tr. 1734, p. 402. Weidler, Tractatus de
Mach. Hydraul. Leipz. 1728. Besson's Theatre, Lyons, 1579. Bocker's Theatrum,
1661. Tielen en Von der Host's do. Policy's do. Amst. 1737. Van Zyl's do.
Amst. 1761. Euler on Pumps, Hist, et Mem. de Berlin, 1752, pp. 149, 185.
Landriani on the Rope Pump, Geneva, 1782. Perronet, Description des Projetsdes
Fonts de Neuilly, 1783. Baaden, Theorie der Pumpen, 4to, Bayr, 1797. Close's
Method of Raising Water, Nich. Jour. iv. 293, 493, 8vo. i. 145. Application of the
Siphon, iv. 547 ; v. 22, 8vo. i. 27. Person, Recueil de Mechanique, 4to, Paris, 1802.
Dietot's Danaide, Thomson's Annals of Philosophy, ii. 412. Ewbank's Descriptive
and Historical Account of Hydraulic Machines, New York, 1842, very copious,
interesting, and curious.

LECTURE XXIX.

ON PNEUMATIC MACHINES.

PNEUMATIC machines are such as are principally dependent, in their
operation, upon the properties of elastic fluids ; they may be calculated
either for diminishing or increasing their density and pressure, as air
pumps and condensers ; or for directing and applying their force, as bel-
lows, ventilators, steam engines, and guns.

. * Ph.Tr. 1775, Ixv. 277.

f Repertory of Arts, 1798, vol. ix.

J Journal de Physique, xlvi. 143. Brunacci, Trattato dello Ariete Hydraulico,
4to, Milan, 1813.

s2

2GO LECTURE XXIX.

The density and pressure of the air may be diminished, or the air may be
perfectly or very nearly withdrawn from a given space, either by means of •
a column of mercury, or by the air pump. The ancients sometimes ex-
hausted a vessel imperfectly by the repeated action of the mouth, and
preserved the rarefaction by the assistance of a stopcock. The Torricellian
vacuum, obtained by inverting a receiver filled with mercury, and fur-
nished with a descending tube at least 30 inches long, is the most perfect
that can be procured ; but there is generally a portion of air adhering to
the vessels, and mixed with the mercury, which may often be considerably
diminished by agitation, but can only be completely expelled by boiling
the mercury for some time in the vessel and its tube, previously to their
inversion. (Plate XXIV. Fig. 324.)

The construction of an air pump greatly resembles that of a common
sucking pump for raising water ; but the difference in the operation to be
performed requires a difference in several particular arrangements. The
objects are, to rarefy or exhaust the air as completely, as expeditiously, and
as easily, as possible. In order that the exhaustion may be complete, it is
necessary that no air remain in the barrel when the valve is opened, and
that the process be very long continued. For, supposing all the parts of an
air pump to be perfectly well fitted, and the exhaustion to be carried on for
any length of time, the limit of its perfection will be a rarefaction expressed
by the proportion of the air remaining in the barrel, when the piston is down,
to the whole air that the barrel is capable of containing ; for such will be
the rarity of the air in the barrel when the piston is raised. It becomes,
therefore, of consequence to lessen the quantity of this residual air as much
as possible : and at the same time to take care that the valve may be capable
of being accurately closed and easily opened, or that a stopcock may be
occasionally substituted for it, which may be opened and shut by external
force, when the elasticity of the air remaining is too small to lift the valve.
In pumping water from a well, we raise an equal quantity at each stroke,
but in the air pump, we withdraw at most only equal bulks of the air dif-
ferently rarefied, so that the quantity extracted is continually diminished
as the operation proceeds. Thus, if one tenth of the air were exhausted by
the first stroke, only nine tenths as much, that is, one tenth of the remain-
der, would be drawn out by the second ; hence, in order that the process
may be expeditious, it is of importance to have the barrel as large as pos-
sible in proportion to the receiver. In cases where the presence of aqueous
vapour would be of no consequence, the exhaustion might be made very
rapidly by filling the whole apparatus with water, which was the me-
thod first employed by Otto von Guericke, the inventor of the modern
air pump.

In order to lessen the labour of the operation, two barrels may be em-
ployed, and so connected as to work alternately ; in this manner the
pressure of the atmosphere, acting on both pistons at once, opposes no
resistance to their motion in either direction. In Smeaton's pump* a

* Ph. Tr. 1751-2, xlvii. 415. See also the Dutch translation of Dr. Priestley's
Observations and Experiments on different kinds of air, vol. ii. 1781. Cavallo, Ph.
Tr.1783, p. 435.

ON PNEUMATIC MACHINES. 261

single barrel has nearly the same advantage, the rod of the piston working
'in a collar of leathers with oil, and the air heing excluded from the upper
part of the barrel by a valve, through which the air passes when the piston
is raised near to the top ; so that in the descent of the piston there is
a vacuum above it, and the air below opens the valve much earlier, and
passes more completely through it, than in the common air pump ; and
the piston is only exposed to the whole pressure of the atmosphere
during the discharge of the air through the upper valve. (Plate XXIV.
Fig. 325.)

That the air is really removed by the operation of the air pump, may be
demonstrated by various experiments, which show the absence of its
resistance, of its buoyant effect, and of its pressure ; such are the descent of
a guinea and a feather at the same time, the equal duration of the motion
of two fly wheels, with their plates placed in different directions, the prepon-
derance of the larger of two bodies which balance each other in the open
air, the descent of mercury or of water in a barometrical tube, the playing
of a fountain urged by the expansion of a portion of confined air, and the
ebullition of ether, or of water moderately warm. (Plate XXIV. Fig.
326, 327.)

The degree of perfection of the vacuum formed by the air pump, or the
rarity of the air remaining in the receiver, is measured by gages of different
kinds. The simplest gage is a short tube filled with mercury, and inverted
in a bason of the same fluid ; in this the mercury begins to descend when
the elasticity of the air becomes diminished in the proportion of the height
of the gage to that of the barometer ; but on account of the capillary at-
traction of the particles of mercury for each other, there is a depression
within the tube, differing in quantity according to its magnitude, which
renders it difficult to observe the exact situation of the surface when the
height of the column is very small, although, if that height were correctly
ascertained, the allowance to be made for the depression might easily be
calculated. It is, however, more usual to employ the long barometer gage,
in which the pressure is removed from the upper surface of the column of
mercury in proportion as the exhaustion proceeds, and the height to which
it is raised by the pressure of the external atmosphere, is compared with
that of a common barometer, the difference always indicating the density
of the air left in the receiver. Sometimes also a bent tube is employed in-
stead of the short gage, the difference of the height in its two branches indi-
cating the pressure ; and this instrument has the advantage of requiring no
correction on account of capillary attraction, since the depressions of the
two columns exactly counterbalance each other. But in all these cases the
mercury must be well boiled in the tubes ; and in the bent tube, or siphon
gage, the operation is somewhat difficult.

The pressure indicated by a gage of any kind depends on the elasticity
of the whole of the fluid remaining in the receiver ; but this fluid is not
always atmospheric air alone. In all common temperatures, water, and
iriany other liquids, have the property of emitting a vapour which possesses
a very sensible degree of elasticity ; so that if either water, or any moist
substance, be present under the receiver, it will be impossible to procure a

262 LECTURE XXIX.

total absence of pressure, the short mercurial gage commonly standing at
the height of at least half an inch, in the best pumps. Hence, the vacuum
may be made more perfect when the receiver is ground to the plate of the
pump, with the interposition of an unctuous substance, than when it is
placed on wet leather, as it has sometimes been usual to do. The quantity
of atmospherical or incondensable air actually existing in the receiver,
whether mixed with vapour or alone, is measured by means of Smeaton's
pear gage,* which is left open under the receiver during the exhaustion,
and having its orifice then plunged, by means of a wire passing through a
collar of leather, into a bason of mercury, receives, upon the readmission
of the air, as much of the mercury as is sufficient to fill it, leaving only in
a tube rising from the neck of the gage, the small quantity of air which
had before filled the whole cavity, so that from the space occupied by this
air, compared, by means of previous measurements, with the capacity of the
gage, the degree of exhaustion of the pump with respect to air may be es-
timated. It is said that in an air pump of Cuthbertson's construction, such
a rarefaction has been procured that the air sustained but one hundredth
part of an inch of mercury,t that is, it was expanded to nearly 3000 times
its original bulk. The pear gage often indicates a much more complete
exhaustion, but this measurement relates only to the quantity of dry air
present.* (Plate XXIV. Fig. 328.)

A condenser is the reverse of an air pump ; and sometimes the same
machine is made to serve for both purposes ; but the condenser requires
more strength than the air pump, and less delicacy. The gage for measur-
ing the degree of condensation is a small portion of air contained in a gra-
duated cylindrical tube, the space that it occupies being indicated by a drop
of mercury which confines it. (Plate XXIV. Fig. 329.)

Diving bells were formerly supplied with air by means of barrels let
down continually from the surface of the water, and taken into the bell by
the divers ; but it is now more usual to force down a constant stream by
means of a pump resembling a condenser in its construction and operation ;
the heated air is suffered to escape by a stopcock at the upper part of
the bell. When proper care is taken to lower the machine gradually,
the diver can support the pressure of an atmosphere of twice or thrice
the natural density. It would be advisable that every diver should be
provided with a float of cork, or with a hollow ball of metal, which
might be sufficient to raise him slowly to the surface, in case of any
accident happening to the bell ; for want of a precaution of this kind,
several lives have been lost from confusion in the signals. § (Plate XXIV.
Fig. 330.)

Bellows are commonly made of boards connected by leather, so as to
allow of alternately increasing and diminishing the magnitude of their
cavities, the air being supplied from without by a valve. The blast must

* Ph. Tr. 1752, p. 420.

t Cuthbertson, Description of an improved Air Pump, 1783, §38.

* See Nairne's Account of some Experiments made with an Air Pump, Ph. Tr.
1777, p. 622. Roz. Journ. xi. 159 ; xxv. 261.

§ See Halley's Art of Living under Water, Ph. Tr. 1716, p. 492; 1721, p. 177.
Healy on Diving Bells, Ph. Mag. xv. 9.

ON PNEUMATIC MACHINES. 263

be intermitted while the cavity is replenished ; and in order to avoid this
inconvenience, a second cavity is sometimes added, and loaded with a weight,
which preserves the continuity of the stream. If great uniformity be
required in the blast, it will be necessary to take care that the cavity be so
formed as to be equally diminished while the weight descends through equal
spaces ; but notwithstanding this precaution, there must always be an
additional velocity while the new supply of air is entering from the first
cavity. Sometimes the construction of the bellows resembles that of a
forcing pump ; and then, if the barrel is single, a second barrel, loaded with
a weight, must be provided, in order to equalise the blast : or a vessel
inverted in water, and either loaded or fixed, may supply the place of the
second barrel. The first cavity may also be formed of a similar inverted
vessel, suspended to a beam, so as to be moved up and down in the water,
and such a machine is much used, in large founderies, under the name of
hydraulic bellows. The quantity of water employed may be much dimi-
nished, and the operation expedited, by introducing, in the centre of the
inverted vessel, a fixed solid, or an internal inverted vessel, capable of
nearly filling up the cavity of the moveable vessel when it is in its lowest
position, so that the water only occupies a part of the interstice between the
vessels. (Plate XXIV. Fig. 331.)

The gasometer differs little from the hydraulic bellows, except that it is
provided with stopcocks instead of valves, and the moveable cylinder is
supported by a counterpoise, which, in the best kind, acts on a spiral fusee,
calculated to correct the difference of pressure arising from the greater or
less immersion of the cylinder. (Plate XXIV. Fig. 332.)

A shower of water, or even an irregular stream, being conveyed through
a descending pipe, plunged into the water of a reservoir, a large quantity of
air is carried down with the water, and rises to the upper part of an
inverted vessel which surrounds the pipe, whence it may be conveyed
through another pipe, in a rapid stream, for any required purpose ; and
the water escapes at the bottom of the air vessel into the general reser-
voir, from the surface of which it runs off. The quantity of air sup-
plied by these shower bellows is, however, small. (Plate XXIV. Fig.
333.)

The velocity of the blast produced by any pressure, forcing the air
through a pipe of moderate dimensions, may readily be determined from
the height of a column of air equivalent to the pressure. Thus if the
hydraulic bellows were worked with a constant pressure of four feet of
water, the velocity would correspond to a height of about 3300 feet,
and the air would move through a space of about 460 feet in a second. But
in this calculation no allowance is made for any of the causes which
diminish in all cases the discharge of fluids, and the velocity actually
observed is only five eighths as great as that which corresponds to the
height ; that is, in the example here given, 285 feet in a second, when the
air escapes through a small orifice ; but when it moves in a pipe, about
three fourths, or 345 feet. If the pipe were of considerable length, there
would also be a diminution of velocity on account of friction. In
some bellows actually employed, a pressure equivalent to nine feet of

264 LECTURE XXIX.

water is applied, and in this case the velocity must be about 500 feet
in a second.

Bellows may be used for the ventilation of a mine, either by forcing
air into it, or by drawing it out through a pipe connected with the valve.
The wind may also be received by the mouth of a tube a little conical, and
may be made to cause a current where it is conveyed ; such an instrument
is sometimes called a windsail, or a horse head. It has been proposed to
draw the air up through a pipe by the lateral friction of a current of air
received by such a funnel, but the effect would probably be too small to be
of much practical utility.

A corn fan is turned by the hand or by machinery ; its simplest opera-
tion is to cause a portion of air to revolve with it, and to create a wind in
the direction of its circumference. But when a small fan is made to revolve
with great rapidity, as in Papin's Hessian bellows, the centrifugal force
causes the air admitted at the centre to rush towards the circumference,
and to pass with great velocity through a pipe inserted there. The com-
mon ventilator placed in windows, which revolves in the same manner as
a smoke jack, in consequence of the impulse of a current of air, serves
only to retard a little the entrance of that current, to disperse it in some
measure in different directions, and to prevent any sudden increase of the
intensity of the draught ; but it has little or no power of acting on the air,
so as to prevent the decrease of the velocity of the current. (Plate XXIV.
Fig. 384.)

The operation of heat affords us also a very effectual mode of ventilation.
Its action upon air at common temperatures occasions an expansion of
about T^ly for every degree that Fahrenheit's thermometer is raised ; the air
becomes in the same proportion lighter, and the fluid below it is conse-
quently relieved from a part of its weight : the pressure of the surrounding
atmosphere, .therefore, preponderates, and the lighter column is forced
upwards. When the shaft of a mine communicates with the external air
at two different heights, there is generally a sufficient ventilation from the
difference of the temperatures of the air in the shaft, and of the surrounding
atmosphere: for the temperature of the earth is nearly invariable, it
therefore causes the air in the shaft to be warmer in winter than the
external air, and colder in summer ; so that there is a current upwards in
winter, and downwards in summer ; and in the more temperate seasons,
the alternations take place in the course of the day and night. For a
similar reason there is often a current down a common chimney in sum-
mer ; but when the fire is burning, the whole air of the chimney is heated,
and ascends the more rapidly as the height is greater. It would be easy,
from the principles of hydraulics, if the length of the chimney, and the
mean temperature of the air in it were given, to calculate the velocity of
the draught : thus if the height of the chimney were 50 feet, and the air
contained in it 10 degrees hotter than the external air, the expansion would
be one fiftieth, and the pressure of the whole column being diminished one
fiftieth, the difference would be equivalent to a column of one foot ifc
height, and such a column would represent the pressure causing the
draught, which might, therefore, be expected to have a velocity of 0 feet

ON PNEUMATIC MACHINES. 265

in a second. If the room were perfectly closed, the air contained in it
would by degrees become so much lighter than the external air, as would
be equivalent to one foot of the height of the column causing the pres-
sure, and the current would then stop ; if fresh air were gradually ad-
mitted by a small orifice, the current would again go on, but the air in
the room would always remain somewhat rarer than the external at-
mosphere, unless a fresh supply were admitted through ample openings.

The object of a chimney is not so much to ventilate the room, as to pro-
vide a sufficiently rapid supply of air for maintaining the process of com-
bustion, and to carry off the products of that process : hence, it is desirable
to allow as little air as possible to enter the chimney without passing
through the fire ; and this is the best general mode of avoiding smoky
chimnies. For wind furnaces, the flue should be as equable as possible,
throughout its height, or widened rather than contracted in its ascent, and
free from any considerable angles.

The ascent of a balloon is an effect of the same kind as that of air in a
chimney, and arises sometimes from the same* cause, when the air within it
is expanded by heat ; but more commonly from the greater rarity of hydro-
gen gas, with which the balloon is filled, and which, when pure, is only one
thirteenth as heavy as atmospherical air, but as it is commonly used, about
one fifth or one sixth.

The steam engine is perhaps the most magnificent effort of mechanical
power ; it has undergone successive changes, and it appears to have been
brought very near to perfection by the improvements of Mr. Watt. The
pressure of steam was first applied by the Marquis of Worcester,* and
afterwards by Savery,f to act immediately on the surface of water contained
in a close vessel, and this water was forced, by the elasticity of the steam, to
ascend through a pipe. But a great degree of heat was required for raising
water to any considerable height by this machine ; for in order that steam
may be made capable of supporting, in addition to the atmospherical pres-
sure, a column of 34 feet of water, its temperature must be raised to 248°
of Fahrenheit, and for a column of 68 feet, to 271° ; such a pressure, also,
acting on the internal surface of the vessels, made it necessary that they
should be extremely strong ; and the height to which water could be drawn
up from below, when the steam was condensed, was limited to 33 or 34 feet.
A still greater objection was, however, the great quantity of steam neces-
sarily wasted, on account of its coming into contact with the cold water
and the receiver, the surfaces of which required to be heated to its own
temperature, before the water could be expelled ; hence a tenth or a twen-
tieth part only of the steam produced could be effective ; and there would
probably have been a still greater loss, but for the difficulty with which
heat is conducted downwards in fluids. These inconveniences were in

* See p. 278. There is reason to believe that Hooke, in 1678, was master of the
principle ; for he gives in a cypher the outline of " a very extraordinary invention in
mechanics, above the chimeras of perpetual motion, for several uses." The cypher is
expressed by Pondere premit aer vacuum quod ab igne relictum est. Waller's Life
of Hooke, p. 21.

f Ph. Tr. 1699, p. 228, with a plate of the engine. Improvements on it by De
Moura, Ph. Tr. 1752, p. 436.

266 LECTURE XXIX.

great measure avoided in Newcomen's engine,* where the steam was gra-
dually introduced into a cylinder, and suddenly condensed by a jet of
water, so that the piston was forced down with great violence by the pres-
sure of the atmosphere, which produced the effective stroke : this effect
was, however, partly employed in raising a counterpoise, which descended
upon the readmission of the steam, and worked a forcing pump in its re-
turn, when water was to be raised. The condensation, although rapid, was,
however, neither instantaneous, nor complete, for the water injected into
the cylinder had its temperature considerably raised by the heat emitted
by the steam during its condensation ; it could only reduce the remaining
steam to its own temperature, and at this temperature it might still retain
a certain degree of elasticity ; thus, at the temperature of 180° steam is
found to be capable of sustaining about half the pressure of the atmosphere,
so that the depression of the piston must have been considerably retarded
by the remaining elasticity of the steam, when the water was much heated.
The water of the jet was let off when the piston was lowest, and was after-
wards pumped up to serve the boiler, as it had the advantage of being
already hot. This engine, with Beighton's apparatus for turning the cocks,
was until lately in general use, and it is still very frequently employed. In
this, as well as in other steam engines, the boiler is furnished with a safety
valve, which is raised when the force of the steam becomes a little greater
than that of the atmospheric pressure ; and it is supplied with water by
means of another valve, which is opened when the surface of the water within
it falls too low, by the depression of a block of stone which is partly supported
by the water. (Plate XXIV. Fig. 335, 336.)

The cylinder of Beighton's machine is necessarily much cooled by the
admission of the jet, and by exposure to the air. Mr. Watt has avoided
this inconvenience by performing the condensation in a separate vessel, into
which a small jet is flowing without intermission ; and by introducing the
steam alternately above and below the piston, the external air is wholly ex-
cluded ; the piston rod working in a collar of leathers, so that the machine
has a double action, somewhat resembling that of Lahire's double pump ;
and the stroke being equally effectual in each direction, the same cylinder,
by means of an increased quantity of steam, performs twice as much work
as in the common engine. We might also employ, if we thought proper, a
lower temperature than that at which water usually boils, and work in this
manner with a smaller quantity of steam ; but there would be some diffi-
culty in completely preventing the insinuation of the common air. On the
other hand, we may raise the fire so as to furnish steam at 220° or more,
and thus obtain a power somewhat greater than that of the atmospheric pres-
sure ; and this is found to be the most advantageous mode of working the
engine ; but the excess of the force above the atmospheric pressure cannot
be greater than that which is equivalent to the column of water descending
to supply the boiler, since the water could not be regularly admitted in
opposition to such a pressure. The steam might also be allowed to expand
itself within the cylinder for some time after its admission, and in this
manner it appears from calculation that much more force might be obtained
* His patent is dated 1705..

ON PNEUMATIC MACHINES. 267

from it than if it were condensed in the usual manner as soon as its ad-
mission ceases ; but the force of steam thus expanding is much diminished
by the cold which always accompanies such an expansion, and this method
would be liable to several other practical inconveniences.

The peculiarities of Mr. Watt's construction require also some other ad-
ditional arrangements ; thus, it is necessary to have a pump, to raise not
only the water out of the condenser, but also the air, which is always ex-
tricated from the water during the process of boiling. If the water em-
ployed has been obtained from deep wells or mines, it contains more air
than*iisual, and ought to be exposed for some time in an open reservoir be-
fore it is used ; for it appears that the quantity of air, which can be con^-
tained in water, is nearly in proportion to the pressure to which it is sub-
jected. The admission of the steam into the cylinder is regulated by the
action of a double revolving pendulum. The piston is preserved in a situ-
ation very nearly vertical by means of a moveable parallelogram, fixed on
the beam, which corrects its curvilinear motion by a contrary curvature.
In the old engines, a chain working on an arch was sufficient, because
there was no thrust upwards. When a rotatory motion is required, it may
be obtained either by means of a crank, or of a sun and planet wheel,* with
the assistance of a fly wheel ; this machinery is generally applied to the
opposite end of the beam ; but it is sometimes immediately connected with
the piston, and the beam is not employed. The cylinder is usually inclosed
within a case, and the interval is filled with steam, which serves to confine
the heat very effectually. (Plate XXIV. Fig. 337.)

The steam engines of Messrs. Boulton and Watt are said to save three
fourths of the fuel formerly used ; and it appears that only one fourth of
the whole force of the steam is wasted. Such a machine, with a thirty
inch cylinder, performs the work of 120 horses, working 8 hours each in
the day.

When the water producing the condensation is to be raised from a great
depth, a considerable force is sometimes lost in pumping it up. Hence
Mr. Trevithick t has attempted, as Mr. Watt had indeed long before pro-
posed, to avoid entirely the necessity of condensation, by employing steam
at a very high temperature, and allowing it to escape, when its elasticity is
so reduced by expansion, as only to equal that of the atmosphere : the air
pump is also unnecessary in this construction, and for a small machine, it
may perhaps succeed tolerably well. But there must always be a very
considerable loss of steam, and although the expense of fuel may not be
increased quite in the same proportion as the elasticity of the steam, yet
the difference is probably inconsiderable. A great number of less essential
alterations have also been made in Mr. Watt's arrangements by various
engineers, but they have generally been calculated either for obtaining some
subordinate purpose of convenience, or for imposing on the public by a
fallacious appearance of novelty. (Plate XXIV. Fig. 338.)

The force of steam, or of heated vapour, is probably also the immediate
•

* After the expiry of Wasbrough's patent for the crank, the sun and planet wheel
was discontinued in Watt's engines, and is now never used.

t Repertory of Arts, vol. iv.

268 LECTURE XXIX.

agent in the astonishing effects produced by the explosion of gunpowder.
The initial elasticity of the fluid by which a cannon ball is impelled, ap-
pears, from Bernoulli's calculation, to be at least equal to ten thousand
times the pressure of the atmosphere, and upon the most moderate compu-
tation, from Count Rumford's experiments, to be more than three times as
great as this. The quantity of moisture, or of water of crystallization, con-
tained in the powder, is certainly too small to furnish steam enough for so
great an effect. We have no reason to suppose that the elasticity of a given
quantity of any aeriform fluid or vapour is increased more than about one
five hundredth for each degree of Fahrenheit that its temperature ib ele-
vated ; and if we suppose the heat to be raised to more than 5000 degrees,
the force of each grain of water converted into steam will only be increased
tenfold ; so that if the elasticity were 40 thousand times as great, the den-
sity must be 4 thousand times as great as that of ordinary steam, and the
whole space must be filled with an aqueous vapour almost twice as dense
as water itself. It is, therefore, probable that some other parts of the
materials assume, together with the water, the state of vapour, and possess
in this form a much greater elasticity than that of the steam : for the quan-
tity of fluids permanently elastic, which are extricated, must be allowed to
be wholly inadequate to the effect.

The force of fired gunpowder is found to be very nearly proportional to
the quantity employed ; consequently, if we neglect the consideration of the
resistance of the atmosphere, the square of the velocity of the ball, the height
to which it will rise, and its greatest horizontal range, must be directly as
the quantity of powder, and inversely as the weight of the ball. Count
Rumford,* however, found that the same quantity of powder exerted some-
what more force on a large ball than on a smaller one.

The essential properties of a gun are to confine the elastic fluid as com-
pletely as possible, and to direct the motion of the bullet in a rectilinear
path ; and hence arises the necessity of an accurate bore. The advantage
of a rifle barrel is principally derived from the more perfect contact of the
bullet with its cavity ; it is also supposed to produce a rotation round an
axis in the direction of its motion, which renders it less liable to deviations
from its path on account of irregularities in the resistance of the air. The
usual charge of powder is one fifth or one sixth of the weight of the ball,
and for battering, one third. When a 24 pounder is fired with two thirds
of its weight of powder, it may be thrown almost four miles, the resistance
of the air reducing the distance to about one fifth of that which it would
describe in a vacuum.

Bullets of all kinds are usually cast in separate moulds: shot are
granulated by allowing the lead, melted with a little arsenic, to pass
through perforations in the bottom of a vessel, and to drop in a shower
into water. The patent shot fall in this process through a height of 120
feet : the roundest are separated by rolling them down an inclined plane
slightly grooved, those which are of an irregular form falling off at the sides.

Condensed air may also be employed for propelling a bullet by means

* New Experiments upon Gunpowder, by Benjamin Thompson, Ph. Tr. 1781,
p. 229. Consult Dalton, Manchester Memoirs, vol. v.

ON PNEUMATIC MACHINES. 269

of an air gun, an instrument of considerable antiquity, but of little utility.
It- is obvious that no human force can so far increase the density of
air as to make its elasticity at all comparable to that of the fluid evolved
by fired gunpowder, and even if it were reduced to such a state, its effects
would still be far inferior to those of gunpowder : for the utmost velocity,
with which it could expand itself, would not exceed 1300 feet in a second,
and it would, therefore, be incapable of imparting to a ball a velocity even
as great as this, while the vapour of gunpowder impels a heavy ball with
a velocity of more than 2000 feet in a second. When, however, it is
considered that by far the greatest part of such a velocity as this is use-
lessly employed, and that the mechanical power which is practically
obtained from gunpowder is much more expensive than an equivalent
exertion of any of the ordinary sources of motion, it must be allowed
that the force of condensed air may possibly be applied in some cases with
advantage, as a substitute for that of gunpowder. (Plate XXIV. Fig. 339.)

[The improvements which have been effected in the construction of con-
densing steam engines since the time of the publication of these Lectures,
are neither few nor unimportant. As, however, most of them are con-
nected with details, rather than with principles, it will not be necessary to
give a very specific account of them. They consist of alterations in the
construction of furnaces and the regulation of the fire ; better forms of the
boiler and its appendages ; simpler modes of effecting a communication
between the boiler, the cylinder, and the condenser, by a new form of the
valves, and an improved way of opening them ; and, lastly, more accurate
methods of fitting the different portions together, so as to lose less heat and
to waste less steam. Many of these improvements are the results of prac-
tical experience. About 1811, the proprietors of some of the Cornish mines
established a system of inspection of their engines, the efficacy of which is
fully evinced by the work of the registrar and inspector, Captain Lean.
He mentions the following instance of it,* " relative to Stray Park engine,
a single engine, on Boulton and Watt's construction, of sixty-inch cylinder.
When this engine was first put on the report in 1811, its duty was below
1C millions : during eight months, ending with April 1813, it had consumed
17,633 bushels of coal, performing the average duty of 21 '5 millions, and
worked at the rate of 5 strokes per minute : during eight months, ending
with April 1814, it had consumed only 12,671 bushels of coal, performed
the average duty of 30'5 millions, and worked at the rate of 5'7 strokes per
minute." And from the same work it appears, that the average duty had,
up to 1834, increased from 26'5 to 90 millions ; the duty being the number
of pounds which are raised one foot high by a bushel of coals. The eco-
nomy of the Cornish boiler and its appendages is due, in a great measure,
to the extent of surface which is presented to the flame. This is effected
by a number of flues, external and internal, the latter somewhat analogous
to those of the locomotive boiler, which will be described presently. The
firg is laid on in large masses, and allowed to consume slowly, whilst the]

* Historical Statements of the Improvements on the Duty of Engines in Corn-
wall. By T. Lean & Brother, 1836. Introd. p. 11.

270

LECTURE XXIX.

([space which the evolved gases, &c. have to travel before they quit the
neighbourhood of the water, enables them both to be thoroughly consumed,
and to part with all their heat to advantage. The communications between
the cylinder and boiler, and cylinder and condenser, are commonly made by
means of a sliding valve, which, from its shape, is known by the- name of
the D valve. It is seen in figures (1) and (2),

Fig. 1.

Fig. 2

and consists of nothing more than a slide G of this shape,

placed in the steam chest, the opening be being sufficiently

wide to allow a free communication between the passage

which leads to the eduction pipe T, and one of the passages

to the cylinder, whilst it closes the latter from the steam chest.

Thus, in figure 1, the communication is between the bottom

of the cylinder and the condenser, whilst steam is entering to the top of the

cylinder. In figure 2 it is the reverse.

The apparatus by which the valve is slid up and down is seen at
figure (3),

ON PNEUMATIC MACHINES. 271

[the rod EF, which moves the valve, in figs. 1 and 2, being united to it at O.
•I* is called an eccentric, and consists of a hollow circle working on a solid
one, the centre of motion of which is not the centre of the circle. As the
centres of both circles always coincide, and that of the solid circle revolves
about the centre of motion, the rod CM will be moved forwards and back-
wards, and consequently 0 will move upwards and downwards, and effect
the different communications and interruptions by the aid of the D valve.

The application of the steam engine to navigation, which took place about
the time of the appearance of Dr. Young's work, was a result so obvious and
necessary, that it required the development of no new principles, and but
little refinement in the application of those already recognised, to bring it
about. The size and weight of the machinery no doubt offered a considerable
obstacle at first, inasmuch as the force obtained might bear too small a pro-
portion to the mass to be moved and the resistance to be overcome, to render
it economical. And this in fact appears to have been the case with the
earliest application of steam power to navigation, that of Mr. Symington.
But the perseverance of Fulton, Henry Bell, and others, obviated all these
difficulties, whilst successive improvements both in the arrangements and
the construction of the different parts of the machinery, have rendered
the expenditure of fuel very much less than it was a quarter of a cen-
tury ago.

For a considerable time steam vessels plied only on rivers, not daring to
venture into the open sea, and nautical men, for the most part, entertained
the opinion that they were unfitted to brave it. George Dodd, an enter-
prising but unfortunate man, decided this point. He came down to Glas-
gow and fitted up a little vessel of 75 tons burthen, with a steam engine of
14 horse power, in which he started with a crew of five seamen, two engine
men, and a boy, for London. Although the voyage was stormy, it was
safely performed in 122 hours (exclusive of stoppages). Dodd was emi-
nent as engineer ; he projected Waterloo Bridge and the Thames Tunnel,*
purposing to carry it across from Gravesend to Tilbury, at the estimated
cost of under £16,000 ! t Yet, with talent, energy, and courage, he almost
literally died in the streets a beggar. His active mind led him into dis-
astrous schemes — failure impoverished him and drove him to intempe-
rance, which ended in destitution and premature death.

The adaptation of the steam engine to the propelling of vessels is now
so universally known, that a very brief description of the mode of effecting
it will suffice. Across the deck of the vessel is carried a shaft, to the ex-
tremities of which paddle wheels are fixed, the action of which every one
is familiar with. On this shaft two cranks are constructed at right angles
to each other, on which the connecting rods of the two engines respectively
work. By this contrivance a tolerable uniformity of action is produced,
without the aid of a fly wheel, the one engine being in its position of
greatest effect when the other is in the contrary position. The principal
feature in the construction of marine engines, as compared with land ones,
consists in the reversal of the beam, to prevent the inconvenience of its]

* Stuart's Anecdotes, p. 534. Probably his father was the projector,
t Nicholson's Journal, ii. 239, 473.

272 LECTURE XXIX.

[protruding above deck. To connect it with the piston rod, cross pieces are
attached to the top of the latter, which extend beyond the cylinder, aiid
are, on each side of it, united by parallel motions to connecting rods 4which
communicate with the beam.

In this country the steam is applied to marine engines at a temperature
not greatly exceeding the boiling point ; but in America the case is other-
wise, the steam being often applied at such a temperature as to produce
double the pressure it does under ordinary circumstances.

Where lightness is an object, the condensing apparatus is altogether
done away with, steam of a high temperature being employed, which after
it has done its work, is allowed to escape into the air. An engine on this
principle is designated a high pressure engine. The pressure of steam in-
creases very rapidly with its temperature, because its density increases at
the same time. The law which connects the two is given empirically by
Dr. Young (vol. ii. p. 398), as

d—(l + -0029/)7,

d being the depth of mercury in atmospheres of 30 in. each, which would
press as much as steam at a temperature f of Fahrenheit above 212°.
Thus, at 212°, d = 1 atmosphere = weight of 30 in. of mercury = about
15lbs. per square inch; at 250°, /= 38, d= (1-1102)7 = a little more
than 2, or the pressure is more than doubled. Many analogous formulae
have been proposed at different times. That of the Franklin Institute is

For the purpose of inland transport, the condensing engine is inappli-
cable, on account of the weight of the condensing apparatus. As early as
1802, Mr. Trevi thick constructed a high-pressure engine, in which the
boiler and apparatus formed one machine ; but it was soon found that the
roughness of common roads prevented the use of such an engine, and
finally destroyed it. Mr. Trevithick consequently turned his attention
to railroads ; but a difficulty arose, which gave much unnecessary trouble,
from the fact of its being almost imaginary. The adhesion of the wheels
was not supposed to be sufficient to prevent their slipping. To obviate
tthis, various devices were put in requisition,* such as rack work, pro-
jecting moveable feet, &c. Experience finally taught that the friction of
the driving wheels .is more than sufficient, in ordinary cases, to prevent
slipping, provided a considerable portion of the weight be made to press
on them. This being established, the construction of the working engine
presents no insuperable difficulties. The following is a brief description
of one of the most approved forms of the locomotive.

The first thing to be attained is the supply of a large quantity of
steam at a high temperature, and from a small apparatus. To effect this,
as large a surface as possible must be exposed to the action of the fire,
and the fuel itself must be kept in a vigorous state of combustion by a
great draught. These objects are attained by perforating the boiler, which
is cylindrical, from end to end, by upwards of 100 hollow tubes of about
two inches diameter. Through these the flame and heated air find their
way from the grate to the chimney ; thus imparting heat to a vast surface]
* See Gordon's Treatise on Elemental Locomotion, 1832.

ON PNEUMATIC MACHINES.

273

[of water with which they come in contiguity. To accelerate the draught,
the steam ejected from the cylinder, which has still considerable force, is
emitttyl up the chimney, thus producing a rapid current in that direction.
A section of the engine is given in figure (4.)*

* From Tredgold's work on the Steam Engine, edited by Woolliouse, 2 vols. 4to,

274 LECTURE XXIX.

[It will be seen, that by far the larger part of the machine is the boiling
apparatus; the working machinery occupying only the comparative!}
small space below. That portion of the boiler which contains water is
shaded in the figure ; and the tubes are seen at E, by which the flame
penetrates the whole body of the boiler from the fire-box C to the
smoke-box F. From every part of the surface of the water, steam is rapidly
and constantly emitted; but it has no way of escape from the boiler,
except tjy ascending the steam dome T, in which the mouth of the steam
pipe df Is situated. After entering the steam pipe, it has to traverse the
whole length of the boiler d'SS before it reaches the cylinder. The .object
of this arrangement is to separate from the steam a quantity of water,
which, being raised by the violence of ebullition, would otherwise be
carried along to the cylinder. The same arrangement facilitates the
regulation of the steam, by bringing it into the immediate neighbourhood
of the engine driver, who is enabled to increase or diminish the supply which
is furnished to the cylinder, by means of a winch hf acting on a valve e'.
As the steam pipe is everywhere inclosed in steam, there is no loss of tem-
perature on this account, except a very trifling amount due to the time
which elapses between the production of the steam, and its application to do
its work. Two safety valves are placed in the upper part of the boiler ; one
at 0, loaded with a constant weight, and out of the reach of the conductor ;
the other at N.

In the steam chest at U is the D valve, admitting steam to the front or
back of the cylinders W, which are horizontal, and alternately suffering it
to escape by the waste port up the blast pipe />, to increase the draught of
the chimney, as already mentioned. The construction of the working
machinery is of the most simple kind. An axle, bent so as to form two
cranks, at right angles to each other, is attached to the two driving wheels.
These are larger than the other wheels, of which there are usually two
pair, provided with flanges or rims on the inside of their circumference, for
the purpose of retaining the machine on the rail. Thus the axle and
driving wheels of this engine are analogous to the shaft and paddle wheels
of the marine engine. There is no beam, but the piston rods Y being con-
fined by guide bars, which allow them to play backwards and forwards
through the space of about 18 inches, are attached immediately at their
extremities to the connecting rods which act on the cranks. These being
at right angles to each other, the force is equalised as in the marine engine.
The valve machinery consists, as usual, of an eccentric and levers, but in
the locomotive, each cylinder is provided with two sets of eccentrics, the
one being the reverse of the other, that is, tending to move the valve back
when the other would move it forwards. A lever is in the direction of the
driver, by means of which one set may be thrown out of gear, whilst
the other is thrown in. The rod of the eccentric which is not in gear is
seen at/". It terminates in a Y, so that when raised it will readily catch
the working levers (at M, fig. 3). By this means the action of the engine
can be instantaneously reversed.

It will be seen from this description that the locomotive is by far the
most simple form of the steam engine. As, however, it is applied to per-
form work in which great speed is necessary ; so much so as to require that]

ON THE HISTORY OF HYDRAULICS, &c. 275

[the piston rods move backwards and forwards three or four times in a
'second, and therefore, that each of the cylinders he filled and emptied six
or eight times in the same interval ; it is evident that the utmost accuracy
of workmanship is requisite, not merely to prevent useless expenditure of
fuel, but even to keep the machine in action at this speed at all.]

For further information on the Steam Engine, consult Partington's History and
Description of the Steam Engine, 1822. Stuart's Descriptive History of do. 1824.
Historical and Descriptive Anecdotes of Steam Engines and their Inventors, 2 vols.
12mo, 1829. Farey on the Steam Engine, 4to, 1827. Gilbert, Progressive Improve-
ment^ on the Efficiency of Steam Engines in Cornwall, Ph. Tr. 1830. Coriolis,
Journal de 1'Ecole Polytechnique, vol. 13. Brewster's Journal, Nos. 17 and 19.
Birkbeck and Aldcock on the Steam Engine. Renwick on do. New York, 1830.
Tredgold on do. a new edition by Woolhouse, 2 vols. 4to, 1838, with various subse-
quent appendices. De Pambour on the Theory of do. Lardner on do. 7th edition,
1840. Russel on do. from Encyc. Brit. The last four treatises contain all that
could be desired on this subject.

On its Applications, see Jonathan Hulls' Description of a New Machine, 1737.
Buchanan on Steam Navigation, Glasgow, 1816. Dodd on do. 1816. Wood's
Practical Treatise on Railways. Marestier, Mem. sur les Bateaux a Vapeur, 1824.
Cleland's Hist. Account of the Steam Engine, and its Application to propelling
Vessels, 1825. Seguin, Mem. sur la Navigation a Vapeur, 1828. Brees's Railway
Practice, 1838.

LECT. XXIX.— ADDITIONAL AUTHORITIES.

Air Pumps, Condensers, 8fc. — Boyle on the Spring and Weight of the Air, 4to,
Oxf. 1663, and Opera, passim. Varignon on an Air Pump, Hist, et Mem. de Paris,
x. 285. Leupolds Beschreibung der Luftpumpe, 4to, Leipz. 1710-12. Nollet, Hist,
et Mem. 1740, pp. 385, 567 ; 1741, p. 338, H. 145. Lowitz iiber die Eigenschaf-
ten derLuft, 1754. Coulomb on Condensing with an Air Pump, Roz. Journ. xvii.
301. Ingenhousz Vermischte Schriften, p. 197. Hindenburg de Antlia Baaderiana,
4to, Leipz. 1787. De Antlia Nova, 4to, Leipz. 1789. Goth. Mag. v. II. 81.
Prince's Air Pump, Trans, of the American Academy, vi. 235. Van Marum's
Simple Air Pump, Gilb. Jour. i. 379. Mackenzie's Air Pump, Nich. Jour. ii. 28.

LECTURE XXX.

ON THE HISTORY OF HYDRAULICS AND PNEUMATICS.

NOTWITHSTANDING a few observations and experiments made by
Aristotle and his predecessors, the properties of fluids had scarcely been
the subjects of much accurate investigation before the time of Archimedes.
The progress which the science of hydrostatics in particular made under
this eminent mathematician, does the highest honour to his genius and
penetration. His treatise on floating bodies, although the theorems which
it contains are not so general as they have been rendered since the late
improvements in the methods of calculation, still affords us instances of
very ingenious determinations of the equilibrium of floating bodies of
different forms, grounded on the true principles of the opposition of the
general directions of the weight of the body and of the pressure of the
fluid ; and in this manner he has shown in what cases the equilibrium of

T2

276 LECTURE XXX.

conical and conoidal solids will be stable, and in what cases unstable.
Archimedes was the inventor of the mode of measuring the bulk of a solid
by immersing it in a fluid : to us, indeed, there appears to have beef\ little
difficulty in the discovery, but the ancients thought otherwise. Vitruvius
observes that this invention indicates a degree of ingenuity almost in-
credible. The philosopher himself is said to have valued it so highly,
that when it first occurred to him, in a public bath, he hastened home in
an ecstasy without recollecting to clothe himself, in order to apply it to
the determination of the specific gravity of Hiero's crown and to the
detection of the fraud of the maker, who had returned the crown 'equal
in weight to the gold that was given him, but had adulterated it with
silver, and imagined that on account of the complicated form of the work,
which rendered it almost impossible to determine its bulk by calculation,
he must infallibly escape conviction. The hydrometer, which has some-
times been attributed to Hypatia, a learned Greek lady of Constantinople,
is mentioned by Fannius,* an early writer on weights and measures, and
is ascribed by him to Archimedes.

The forcing pump, or rather the fire engine, was the invention of
Ctesibius of Alexandria, the greatest mechanic of antiquity after Archi-
medes. He is also said to have invented the clepsydra, for the hydraulic
measurement of time, and Philot informs us that he constructed an air
gun, for propelling a stone, or rather a ball, by means of air previously
condensed by a syringe. The ball was not immediately exposed to the
action of the air, but was impelled by the longer end of a lever, while the
air acted on the shorter. Ctesibius is said to have been the son of a barber,
and to have had his attention turned to mechanics and pneumatics, by
being employed to fit a shutter, with a counterpoise sliding in a wooden
pipe, for his father's shop window.^

Hero was a cotemporary, and a scholar of Ctesibius ; he describes, in
his treatise on pneumatics, a number of very ingenious inventions, a few
of which are calculated for utility, but the greater part for amusement
only ; they are principally siphons variously concealed and combined,
fountains, and water organs, besides the syringe and the fire engine. The
description of this engine agrees precisely with the construction which is
at this day the most usual ; it consists of two barrels, discharging the
water alternately into an air vessel ; and it appears from Vitruvius, that
this was the original form in which Ctesibius invented the pump. Hero
supposes the possibility of a vacuum in the intervals of the particles of
bodies, observing that without it no body could be compressible ; but he
imagines that a vacuum cannot exist throughout a perceptible space, and
thence derives the principle of suction. The air contained in a given
cavity may be rarefied, he says, by sucking out a part of it, and he
describes a cupping instrument, which approaches very nearly to the
nature of an imperfect air pump. (Plate XXIV. Fig. 324.)

After the time of Ctesibius and Hero, the science of hydraulics made

* Rhemnius Fannius Palsemon de Ponderibus et Mensuris.

f Duten's Inquiry into the Origin of the Discoveries attributed to the Moderns,
Lond. 1769, p. 186.

I Vitruvius, ix. 9. A figure of tbe clepsydra is given in Perrault's translation.

ON THE HISTORY OF HYDRAULICS, &c. 277

little further progress until the revival of letters. The Romans had water
mills in the time of Julius Caesar, which are described by Vitravius ; and
it appears that their aqueducts were well built, and their waterpipes well
arranged. Pipes of lead were, however, less frequent than at present,
from an apprehension of the poisonous quality of the metal, which was
not wholly without foundation.* Some say that the ancients had no
chimnies, but whatever may be the authorities, the opinion is extremely
improbable.

It was in the middle ages that navigable canals began to be considerably
multiplied, first in China, and afterwards in other parts of the world.
The canal from the Trent to the Witham, which is the oldest in England,
is said to have been dug in 1134. The date of the earliest windmills has
been referred to the year 1299. The invention of gunpowder possesses
perhaps an equal claim with the art of printing, to the honour of being
considered as constituting the most marked feature that distinguishes the
character of ancient from that of modern times ; its introduction must
necessarily have tended to produce material alterations, and perhaps im-
provements, in the habits of nations and of individuals. It is said to have
been known long since to the Chinese, and our countryman, Roger Bacon,
was evidently acquainted with its properties ; but it was not actually em-
ployed in Europe or in its neighbourhood till about the year 1330 ; and the
earliest artillery appears to have been that which was used by the Moors,
at the siege of Algesiras, in 1334. King Edward had four pieces of cannon
at the memorable battle of Cressy, in 1346.

About the year 1600, Galileo made the important discovery of the effects
of the weight and pressure of the atmosphere,t in the operation of suction,
and in various other phenomena. Before his time, it was generally sup-
posed that water was raised by a sucking pump, on account of the im-
possibility of the existence of a vacuum : if, however, a vacuum had been
impossible in nature, the water would have followed the piston to all
heights, however great, but Galileo found that the height of its ascent was
limited to about 34 feet, and concluded that the weight of a column of
this height was the measure of the magnitude of the atmospherical pres-
sure. His pupil Torricelli afterwards confirmed the explanation, by
showing that a column of mercury was only supported when its weight
was equal to that of a column of water standing on the same base ; hence
the vacuum obtained by means of mercury is often called the Torricellian
vacuum. Torricelli corrected also, in 1644, the mistake of Castelli respect-
ing the quantities of water discharged by equal orifices, at different
distances below the surface of the water in the reservoir. Castelli's ex-
periments, made about 1640, were the first of the kind, and some of them
really tended to the improvement of the science of hydraulics, but others
appeared to show that a double height of the head of water produced a
double discharge. Torricelli's more accurate observations proved that a

* It is an important circumstance in reference to the action of water on lead, that
it^is more injurious in proportion to the purity of the water. That which contains
less than gg^th of salts in solution, cannot be safely conducted in lead pipes without
certain precautions. Christison, Trans. Roy. Soc. Edin. xv. 265.

f See note, p. 207. t

278 LECTURE XXX.

quadruple height was required in order to produce a double velocity ; and
his assertions were afterwards fully confirmed by Mariotte and by
Guglielmini.* j

A little before the year 1654, Otto von Guericke, of Magdeburg, first
constructed a machine similar to the air pump, by inserting the barrel of a
fire engine into a cask of water, so that when the water was drawn out by
the operation of the piston, the cavity of the cask remained nearly void
of all material substance. But finding that the air rushed in between or
through the staves of the cask, he inclosed a smaller cask in a larger one,
and made the vacuum in the internal one more complete, while the inter-
vening space remained filled with water ; yet still he found that the water
was forced into the inner cask through the pores of the wood. He then
procured a sphere of copper, about two feet in diameter, and was exhausting
it in the same way, wThen the pressure of the air crushed it, with a loud
noise. This machine was more properly a water pump than an air pump,
but the inventor soon after improved his apparatus, and made all the expe-
riments which are to this day the most usually exhibited with the air
pump, such as the apparent cohesion of two exhausted hemispheres, the
playing of a jet by means of the expansion of a quantity of air inclosed in
a jar, the determination of the air's weight, and others of a similar
nature. He also observed that for very accurate experiments, the valve
of the pump might be raised at each stroke by external force ; and he
particularly noticed the perpetual production of air from the water that
he generally employed, which caused an imperfection in the vacuum. An
account of his experiments was first published in different works, by
Caspar Schott,t and afterwards by himself, in his book entitled Experimenta
nova Magdeburgica, printed in 1672 at Amsterdam.

In the year 1658, Hooke finished an air pump for Boyle, in whose labo-
ratory he was an assistant : it was more convenient than Guericke' s, but
the vacuum was not so perfect ; yet Boyle's numerous and judicious expe-
riments gave to the exhausted receiver of the air pump the name of the
Boylean vacuum, by which it was long known in the greatest part of
Europe. Hooke' s air pump had two barrels, and with some improvements
by Hauksbee,^ it remained in common use until the introduction of Smea-
ton's pump, which, however, has not wholly superseded it. The theory of
pneumatics was also considerably indebted to Hooke's important experi-
ments on the elasticity of the air, which were afterwards confirmed and ex-
tended by Mariotte and Amontons, in France, by Hales in this country,
and by Richmann at Petersburg.

About the same time the first steam engine was constructed by the cele-
brated Marquis of Worcester. Hints of the possibility of such a machine
had been given a hundred years before, by Matthesius,§ in a collection of
sermons entitled Sarepta, and at a subsequent period by Brunau;|| but the

* See authorities in Lect. XXIV.

f Magia Universalis, 4vols.4to, Wurtzb. 1657. Mechanica Hydraulico-pneu-
matica, 4to, 1657. Technica Curiosa, 4to, Norimbergee, 1664.

J Hauksbee, Physico-Mechanical Experiments, 4to, Lond. 1709, p. 1.

§ Kepler in Bergmannische's Journal, 1791, ii. 263.

|| Hints towards a Steam Engine, in 1627, Nich. Jour. vii. 311.

ON THE HISTORY OF HYDRAULICS, &c. 279

Marquis of Worcester professes to have carried the project into full effect,
as We are informed by his account of what he called a fire water work,
which fts one of his Century of Inventions, first published in 1663,* and
which is thus described : " I have taken a piece of a whole cannon, whereof
the end was burst, and filled it three quarters full of water, stopping and
screwing up the broken end, as also the touch hole ; and making a constant
fire under it, within 24 hours it burst, and made a great crack : so that
having a way to make my vessels so that they are strengthened by the
force within them, and the one to fill after the other, I have seen the water
run l&e a constant fountain stream forty foot high. One vessel of water,
rarefied by fire, driveth up forty of cold water : and a man that tends the
work is but to turn two cocks, that one vessel of water being consumed,
another begins to force and refill with cold water, and so successively, the
fire being tended and kept constant, which the self same person may like-
wise abundantly perform in the interim between the necessity of turning
the said cocks." The machine was, however, not at that time prac-
tically introduced, and it was soon forgotten ; Savery's engines were
constructed in a manner precisely similar, some time before 1700 ; and
it is uncertain whether he adopted the Marquis of Worcester's ideas,
or reinvented a similar machine. About 1710, the piston and cylinder
were invented by Newcomen, and with Beighton's apparatus for turning
the cocks by its own motion, the engine remained nearly stationary for
many years.

As early as the year 1667, the pressure of fluids in motion, and the re-
sistance opposed by fluids at rest to the motion of solid bodies, were expe-
rimentally examined by Huygens, and some other members of the Parisian
Academy. Pardies, whose works were published in 1673, attempted to
determine, although upon some inaccurate suppositions, the effects of the
wind on a ship's sails under different circumstances. His principles were
adopted by Renaud, who published a work on the subject in 1689.t Their
imperfections were, however, soon after pointed out by Huygens, and by
James Bernoulli; and, in 1714, John Bernoulli published an extensive
treatise on the manoeuvres of ships, which at last compelled Renaud to
submit to so many united authorities.

It must be confessed that the labours of Newton added fewer improve-
ments to the doctrines of hydraulics and pneumatics than to many other
departments of science ; yet some praise is undeniably due both to his com-
putations and to his experiments relating to these subjects. No person
before Newton had theoretically investigated the velocity with which fluids
are discharged, and although his first attempt was unsuccessful, and the
method which he substituted for it in his second edition is by no means free
from objections, yet either of the determinations may be considered in some
cases as a convenient approximation ; and the observation of the contraction
of a stream passing through a simple orifice, which was then new, serves to
reconcile them in some measure with each other. His modes of considering
the* resistance of fluids are far from being perfectly just, yet they have led
to results which, with proper corrections, are tolerably accurate ; and his
* Invention, 68. f Manoeuvres des Vaisseaux.

280 LECTURE XXX.

determination of the oscillations of fluids in bent tubes, was a good begin-
ning of the investigation of their alternate motions in general.

The accurate experiments of Poleni were published in 1718. He has the
merit of having first distinctly observed that the quantity of water dis-
charged by a short pipe is greater than by a simple orifice of the same
diameter ; although there is some reason to suppose that Newton was before
acquainted with the circumstance.

In 1 727, Mr. Bouguer received a prize from the Academy of Paris for his
essay on the masts of ships, which is said to be ingenious, but by no means
practically useful. He was, probably, tempted by this encouragement to
continue his application to similar studies ; and, about twenty years after-
wards, he published his valuable essay on the construction and manoeuvres
of ships, which appears to have superseded all that had been done before
respecting the subjects of his investigation.

The first researches of Daniel Bernoulli concerning the properties and
motions of fluids, bear also the date of 1727. This justly celebrated man
was as happy in his application of mathematics to natural philosophy, as he
was ready and skilful in his calculations. The greatest part of his hydraulic
theorems are founded on the principle first assumed by Huygens, and called
by Leibnitz the law of living or ascending force, which is confessedly only
true where there is no loss of velocity, from the imperfection of the elasticity
of the bodies concerned ; for it is only with this limitation that the motions
of any system of bodies are always necessarily such as to be capable of
carrying the common centre of gravity to the height from which it has de-
scended while the bodies have been acquiring their motions. This law of
ascending force is of considerable utility in facilitating the solution of a great
variety of problems. It is certain that mechanical power is always to be
estimated by the product of the mass of a body into the height to which it
is capable of ascending ; and whatever objections may have been made to the
employment of this product as the measure of the force of a body in motion,
which is indeed an expression inconsistent with a correct definition of the
term force, yet it must be confessed, on the other hand, that some of the best
English mathematicians have fallen into material errors for want of paying
sufficient attention to the general principle. Bernoulli estimates very justly
in this manner the mechanical power of a variety of natural and artificial
agents, and among the rest he examines that of gunpowder ; but, from an
accidental combination of errors, he states the force of a pound of gun-
powder as equivalent to the daily labour of 100 men, while, in fact, the
effect which is actually obtained from two tons of powder is no greater than
that which is here attributed to a pound. His calculations of the motions
of fluids, in some very intricate cases, are very ingenious and satisfactory,
and they are in general sufficiently confirmed by well imagined experiments.
He examines the force of the wind acting on the sails of a windmill, but by
another mistake in calculation, which Maclaurin has detected,* of two
angles which answer the conditions of the determination, he has taken the
wrong one, and assigned that position of the sail as the most effectual, which
produces absolutely no effect at all.

* Fluxions, 2 vols. 4to, Edin. 1742, art. 914.

ON THE HISTORY OF HYDRAULICS, &c. 281

It may be objected to Bernoulli's calculations, that some of the circum-
stances which are necessarily neglected in them, produce a very material
effect in the results of all experiments ; but it must be allowed that the
corrections required on account of this unavoidable omission, may easily
be deduced from simple experiments, and then applied to the most compli-
cated cases. It is, however, a more material objection, that the fundamental
law of the preservation of ascending force can only be adopted with certain
limitations ; thus, when a small stream passes through a large reservoir, Ber-
noulli is obliged to suppose the whole of its force consumed by the resistance
which it meets. The immediate mode, in which the accelerating forces must
be supposed to act, remains also wholly undetermined ; and it was princi-
pally for this reason, that John Bernoulli attempted to substitute, for his
son's calculations, a method of deducing the motions of fluids more imme-
diately from the gravitation of their different parts. The peculiarity of
John Bernoulli's mode of investigation consists in his imagining the weight
of each individual particle to be transferred to the surface of the fluid,
causing there a pressure in the direction of gravity; and he examines
the manner in which this force must operate, in order to produce every
acceleration which is required for the motion of fluids in vessels of all
imaginable forms.

Maclaurm, in his treatise of fluxions, investigated several of the proper-
ties of fluids in his usual concise and elegant manner. His remarks on the
positions of the sails of windmills and of ships are peciiliarly interesting :
he added much to what had been done respecting the effects of the wind,
and showed the possibility of arranging the sails of a ship in such a manner
as to make her advance with a greater velocity than that of the wind itself.
At that time, however, the science of hydraulics had been too little assisted
by experiments to be capable of affording determinations of all questions
which are of very frequent occurrence in practice. An application was
made to Maclaurin, and at the same time to Desaguliers,* a man of con-
siderable eminence in the mechanical sciences, respecting the quantity of
water that might be brought, by a train of pipes of certain dimensions, to
the city of Edinburgh. The project was executed with a confidence founded
on their opinions, but the quantity actually obtained was only about one
sixth of Desaguliers's calculation, and one eleventh of Maclaurin' s. At a
still later period, the French Academicians were consulted respecting a
great undertaking of a similar nature ; and their report was such as to
dissuade the projectors from making the attempt, which was consequently
at the point of being abandoned, till a celebrated practical architect insisted,
from a rough estimation, deduced from his general experience, that more
than double the quantity assigned by the Academicians might be obtained ;
and the event justified his assertion.

The experiments and calculations of Robins, respecting the resistance of
the air and the operation of gunpowder, deserve to be mentioned with
commendation on account of their practical utility ; but he appears to have
.been less successful in his theoretical investigations than Daniel Bernoulli
had been a few years before.

* Robison's Mech. Phil. See Desaguliers's Course of Exp. Ph. vol. ii. p. 126.

282 LECTURE XXX.

Dalembert attempted, in his treatise on the motions of fluids, which was
published in 1744, to substitute for the suppositions of John Bernoulli, a
more general law, relating to all changes produced in the motions of a
system of bodies by their mutual actions on each other ; but his calculations
are more intricate and less easily understood, than some others which are
capable of an application equally extensive. The late Professor Kaestner
of Gottingen has defended Bernoulli against Dalembert's objections with
some success, and has in many instances facilitated and extended Ber-
noulli's theory; but there is often a singular mixture of acuteness and
prolixity in this author's works.* By the side of an intricate and difficult
fluxional calculation, he inserts a long string of logarithms for performing
a simple multiplication ; and in a work which comprehends the whole
range of the mathematical sciences, he does not venture to determine the
square root of 10 without quoting an authority.

About the same time, the profound Leonard Euler applied himself, with
some success, to the examination of the motions of fluids, particularly as
they are connected with the subjects of seamanship and naval architecture ;
but the investigations of Euler are in general more remarkable for mathe-
matical address than for philosophical accuracy and practical application ;
although his calculation of the resistance of the air to the motions of pro-
jectiles may be employed with considerable advantage by the gunner.

The beginning of the modern experimental improvements in hydraulics
may perhaps be dated from the investigations of Smeaton respecting the
effects of wind and water, which were published in the Philosophical
Transactions for 1759. His observations are of material importance
as far as they are capable of immediate application to practice, but he has
done little to illustrate their connexion with the general principles of me-
chanics. It was Mr. Borda that first derived from a just theory, about 10
years after, the same results, respecting the effects of undershot water
wheels, as Smeaton had obtained from his experiments. Before this time,
the best essay on the subject of water wheels was that of Elvius, published
in 1742 ; his calculations are accurate and extensive ; but they are founded,
in great measure, on the imperfect suppositions respecting the impulse of a
stream of water, which were then generally adopted.

Our countryman Mr. Watt obtained, in 1769, a patent for his improve-
ments of the steam engine, which includes almost every essential change
that has been made since the time of Beighton. On a subject so important,
it cannot be superfluous to insert the words of the inventor, whose admirable
application of the sciences to practical purposes, most justly entitles him to
a rank among philosophical mechanics, not inferior to that of Ctesibius and
Dr. Hooke,

" My method of lessening the consumption of steam, and consequently
fuel, in fire engines," says Mr. Watt, in his specification of his patent,
"consists of the following principles. First, that vessel in which the
powers of steam are to be employed to work the engine, which is called
the cylinder in common fire engines, and which I call the steam vessel,
must, during the whole time the engine is at work, be kept as hot as the
* Dissertationes Math, et Phys. 4to, Altenb. 1776.

ON THE HISTORY OF HYDRAULICS, &c. 283

steam that enters it ; first, by inclosing it in a case of wood, or any other
•materials that transmit heat slowly ; secondly, by surrounding it with
steam or other heated bodies ; and thirdly, by suffering neither water, nor
any other substance colder than the steam, to enter or touch it during that
time. Secondly, in engines that are to be worked wholly or partially by
condensation of steam, the steam is to be condensed in vessels distinct from
the steam vessels, or cylinders, although occasionally communicating with
them ; these vessels I call condensers ; and, whilst the engines are working,
these condensers ought at least to be kept as cold as the air in the neigh-
bourhood of the engines, by application of water, or other cold bodies.
Thirdly, whatever air or other elastic vapour is not condensed by the cold
of the condenser, and may impede the working of the engine, is to be drawn
out of the steam vessels, or condensers, by means of pumps, wrought by the
engines themselves, or otherwise. Fourthly, I intend, in many cases, to
employ the expansive force of steam to press on the pistons, or whatever
may be used instead of them, in the same manner as the pressure of the
atmosphere is now employed in common fire engines : in cases where cold
water cannot be had in plenty, the engines may be wrought by this force
of steam only, by discharging the steam into the open air after it has done
its office. Fifthly, where motions round an axis are required, I make the
steam vessels in form of hollow rings or circular channels, with proper
inlets and outlets for the steam, mounted on horizontal axles, like the
wheels of a water mill ; within them are placed a number of valves, that
suffer any body to go round the channel in one direction only ; in these
steam vessels are placed weights, so fitted to them as entirely to fill up a
part or portion of their channels, yet capable of moving freely in them by
the means herein after mentioned or specified. When the steam is admitted
in these engines between the weights and the valves, it acts equally on both,
so as to raise the weight to one side of the wheel, and, by the reaction of
the valves, successively, to give a circular motion to the wheel, the valves
opening in the direction in which the weights are pressed, but not in the
contrary ; as the steam vessel moves round, it is supplied with steam from
the boiler, and that which has performed its office may either be discharged
by means of condensers, or into the open air. Sixthly, I intend, in some
cases, to apply a degree of cold, not capable of reducing the steam to water,
but of contracting it considerably, so that the engines may be worked by
the alternate expansion and contraction of the steam. Lastly, instead of
using water to render the piston or other parts of the engines air and steam
tight, I employ oils, wax, resinous bodies, fat of animals, quicksilver, and
other metals, in their fluid state."

It is probable that the rotatory engines described by Mr. Watt, although
they appear to produce some advantage in theory, will never be generally
introduced, on account of the difficulty of constructing steam vessels so
large, and of so complicated a form, as would be necessary, in order to give
full effect to the machine. The term of this patent was prolonged by act
o£ parliament until the year 1799 ; but although the legal privilege of the
original manufacturers is expired, yet the superiority of their workmanship
still gives their engines a decided preference.

284 LECTURE XXX.

Much of the labour of the later writers on hydraulics has been em-
ployed on the determination of the resistance of fluids to bodies of different •
forms which move through them ; a subject which derives great importance
from its immediate application to the manoeuvres of ships. The most
extensive experiments on these subjects were made by Bossut and some
other members of the Academy of Sciences. About the same time Don
George Juan, a gentleman who had enjoyed the best possible opportunity
for actual observation and practical study in serving with Ulloa, published
at Madrid his Examen Maritimo, which appears to be the most ingenious
and useful treatise on the theory and practice of seamanship that ha« yet
appeared. But unfortunately his deductions, however refined and diversi-
fied, are principally founded on a mistaken theory respecting the effects of
hydraulic pressure ; since he tacitly assumes, in his fundamental pro-
position on the subject, that a double force, acting in a given small space,
will produce a double velocity ; while it is well known that in such cir-
cumstances a quadruple force would be required. Hence he derives some
conclusions which indicate that the resistance must vary very materially
at different depths below the surface of the water, and alleges in support
of the assertion a few imperfect experiments of Mariotte and of his own,
in which some accidental circumstances not noticed may easily have caused
great irregularities. Mr. Prony, in his Architecture Hydraulique, appears
to have followed Juan ; and Professor Robison very justly observes, in
speaking of this work, that if the pressure of the water alters the magni-
tude of the resistance at different depths, that of the atmosphere ought by
no means to be omitted in the calculation. But if a more correct mathe-
matician and mechanic would take the pains to model Juan's book anew,
to correct his errors, and to adapt his modes of calculation to the laws of
resistance previously deduced from accurate experiments rather than from
theory, there is no doubt but that the work thus modified might essentially
improve the science of seamanship. He alleges indeed that the results of his
calculations are in almost every instance rigidly conformable to observa-
tion and experience, but it is probable that where such a coincidence really
exists, it must be owing to some combination of errors compensating each
other ; and it is indeed very possible that his calculations, with all their
errors, may approach nearer to the truth than the imperfect approximations
which had been before employed. Juan has generally made use of the
English weights and measures, on account of their convenience in compu-
tations respecting the descent of falling bodies and the impulse of water.

The works of Chapman and of Romme, upon various departments of
seamanship, possess also considerable merit. These authors appear to
have avoided the errors of Juan, but without entering so minutely into the
detail of nautical operations as he has done.

The accurate experiments of Dr. Hutton and of Count Rumford on the
force of fired gunpowder and the resistance of the air, deserve to be men-
tioned as affording valuable materials to the speculative investigator, and
useful information to the practical gunner. Robins had very erroneously
supposed that the whole of the effects of gunpowder might be derived
from the expansive force of fluids permanently elastic ; but Vandelli soon

ON THE HISTORY OF HYDRAULICS, &c. 285

after maintained a contrary opinion in the commentaries of Bologna,* and
Count Rumford has very satisfactorily shown the insufficiency of the
agents considered by Robins, although he has been unsuccessful in at-
tempting to deduce the whole force from the elasticity of aqueous vapour
alone.

The theory of practical hydraulics, as affected by friction, may be con-
sidered as having been begun and completed by the highly meritorious
labours of the Chevalier du Buat. He had some assistance in expressing
the results of his experiments by means of general rules or formulae, and
these, although they agree sufficiently well with the experiments, have not
always been reduced to the simplest and most convenient forms ; nor have
they been much improved either by Langsdorf or Eytelwein in Germany,
or by Robison in this country, who have gone over nearly the same ground
with each other, and have shown the way in which the results of Buat's
investigations may be applied to a variety ofoases, which occur in hydrau-
lic architecture.

One of the latest inventions which require to be mentioned in speaking
of the history of pneumatics, is that of the aerostatic globe or air balloon.
The suggestions of Lohmeier,t of Albertus, and of Wilkins,^ respecting
the various modes of passing through the air, had long remained disre-
garded as idle speculations ; and Rosnier, who, in the 17th century,
descended obliquely over some houses, by means of wings, was wholly
unable to employ them in ascending. § Dr. Black had exhibited in his
lectures a bladder filled with hydrogen gas, and floating in the air by
means of its smaller specific gravity, many years before Montgolfier con-
ceived the idea of applying a similar machine to the elevation of human
beings into the aerial regions. It was in 1783 that this project was first
executed, and persons of a warm imagination were disposed to believe that
the discovery would be of great importance to the convenience of mankind. ||
But if we coolly consider the magnitude of the force with which the wind
unavoidably impels a surface so large as that of a balloon, we shall be con-
vinced of the absolute impossibility of counteracting it, in such a manner,
as to direct the balloon in any course, materially different from that of the
wind which happens to blow. With this limitation, the invention may
still in some cases be capable of utility, wherever we are only desirous
of ascending to a great height, without regarding the place in which we
are to descend : or where we wish to attain only a height so moderate that
the machine may be kept by ropes in the situation which is desired. In
France the balloon has lately been employed with considerable success as
a meteorological observatory; Mr. Biot and Mr. Gay Lussac having
ascended to a height of above four miles, for the laudable purpose of ascer-
taining some facts relating to the constitution of the atmosphere, and to
the magnetic properties of the earth.

* iii. 92 ; iv. 106. f De Artificio Navigandi per Aerem, 1676.

J Mathematical Magic, 1680. § Hooke, Ph. Coll. No. 1, p. 15.
'|| Montgolfier, Discours sur TAerostate, Paris, 1784. P. de Rozier, Premiere
Experience de la Montgolfiere, 4to, Paris, 1784.

28G

LECTURE XXX.

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287

LECTURE XXXI,

ON THE PROPAGATION OF SOUND.

THE theory of sound, which constitutes the science of acustics, is on
many accounts deserving of particular attention, for it not only involves
many interesting properties of the motions of elastic substances, but it also
affords us considerable assistance in our physiological inquiries respecting
the nature and operation of the senses. The subject has usually been con-
sidered as exceedingly abstruse and intricate, but the difficulty has in some
measure originated from the errors which were committed in the first
inquiries respecting it ; and many of the phenomena belonging to it are so
remarkable and so amusing, as amply to repay the labour of examining
them by the entertainment that they afford. We shall consider first the
nature and propagation of sound in general, secondly, the origin of par-
ticular sounds, and the effects of single sounds ; thirdly, the consequences
of the combinations of sounds variously related, constituting the doctrine
of harmonics, and fourthly, the construction of musical instruments, and
the history of the science of acustics.

Sound is a motion capable of affecting the ear with the sensation peculiar
to the organ. It is not simply a vibration or undulation of the air, as it is
sometimes called ; for there are many sounds in which the air is not con-
cerned, as when a tuning fork or any other sounding body is held by the
teeth : nor is sound always a vibration or alternation of any kind ; for
every noise is a sound, and a noise, as distinguished from a continued
sound, consists of a single impulse in one direction only, sometimes without
any alternation ; while a continued sound is a succession of such impulses,
which, in the organ of hearing at least, cannot but be alternate. If these
successive impulses form a connected series, following each other too
rapidly to be separately distinguished, they constitute a continued sound,
like that of the voice in speaking ; and if they are equal among themselves
in duration, they produce a musical or equable sound, as that of a vi-
brating cord or string, or of the voice in singing. Thus, a quill striking
against a piece of wood causes a noise, but, striking against the teeth of a
wheel or of a comb, a continued sound ; and if the teeth of the wheel are
at equal distances, and the velocity of the motion is constant, a musical
note.

Sounds of all kinds are most usually conveyed through the medium of
the air ; and the necessity of the presence of this or of some other material
substance for its transmission is easily shown by means of the air pump ;*
for the sound of a bell struck in an exhausted receiver is scarcely per-
ceptible. The experiment is most conveniently performed in a moveable
receiver or transferrer, which may be shaken at pleasure, the frame which

* Hauksbee, Ph. Tr. 1705, xxiv. 1902, and xxvi. 367. Biot, Mem. d'Arcueil,
ii. 97. See Tr. R. S. E. v. 34. Saussure, Voyage dans les Alpes, vii. 377.

288 LECTURE XXXI.

suspends the bell being supported by some very soft substance, such as
cork or wool. As the air is gradually admitted, the sound becomes stronger
and stronger, although it is still much weakened by the interposition of
the glass : not that glass is in itself a bad conductor of sound, but the
change of the medium of communication from air to glass, and again from
glass to air, occasions a great diminution of its intensity. It is perhaps on
account of the apparent facility with which sound is transmitted by air,
that the doctrine of acustics has been usually considered as immediately
dependent on pneumatics, although it belongs as much to the theory of the
mechanics of solid bodies as to that of hydrodynamics. •

A certain time is always required for the transmission of an impulse
through a material substance, even through such substances as appear to
be the hardest and the least compressible. It is demonstrable that all
minute impulses are conveyed through any homogeneous elastic medium,
whether solid or fluid, with » uniform velocity, which is always equal to
that which a heavy body would acquire by falling through half the height
of the modulus of elasticity, that is, in the case of the air, half the height
of the atmosphere, supposed to be of equal density ; so that the velocity of
sound passing through an atmosphere of a uniform elastic fluid must be
the same with that of a wave moving on its surface. In order to form a
distinct idea of the manner in which sound is propagated through an
elastic substance, we must first consider the motion of a single particle,
which, in the case of a noise, is pushed forwards, and then either remains
stationary, or returns back to its original situation ; but in the case of a
musical sound, is continually moved backwards and forwards, with a
velocity always varying, and varying by different degrees, according
to the nature or quality of the tone ; for instance, differently in the notes
of a bell and of a trumpet. We may first suppose for the sake of sim-
plicity, a single series of particles to be placed only in the same line with
the direction of the motion. It is obvious that if these particles were ab-
solutely incompressible, or infinitely elastic, and were also retained in
contact with each other by an infinite force of cohesion or of compression,
the whole series must move precisely at the same time, as well as in the
same manner. But in a substance which is both compressible and
extensible or expansible, the motion must occupy a certain time in being
propagated to the successive particles on either side, by means of the
impulse of the first particle on those which are before it, and by means of
the diminution of its pressure on those which are behind ; so that when the
sound consists of a series of alternations, the motion of some of the par-
ticles will be always in a less advanced state than that of others nearer to
its source, while at a greater distance forwards, the particles will be in
the opposite stage of the undulation, and still further on, they will again
be moving in the same manner with the first particle, in consequence of
the effect of a former vibration.

The situation of a particle at any time may be represented by supposing
it to mark its path on a surface sliding uniformly along in a transverse dJ-
rection. Thus, if we fix a small pencil in a vibrating rod, and draw a
sheet of paper along, against the point of the pencil, an undulated line will

ON THE PROPAGATION OF SOUND. 289

be marked on the paper, and will correctly represent the progress of the
vibration. Whatever the nature of the sound transmitted through any
medium may be, it may be shown that the path thus described will also
indicate the situation of the different particles at any one time. The
simplest case of the motion of the particles is that in which they observe
the same law with the vibration of a pendulum, which is always found op-
posite to a point supposed to move uniformly in a circle : in this case the
path described will be the figure denominated a harmonic curve ; and it may
be demonstrated that the force impelling any particle backwards or for-
warfls, will always be represented by the distance of the particle before or
behind its natural place ; the greatest condensation and the greatest direct
velocity, as well as the greatest rarefaction and retrograde velocity, happen-
ing at the instant when it passes through its natural place.

We are ready to imagine that very hard bodies transmit motion instan-
taneously, because we have no easy means of measuring the interval of
time that elapses between the action of pushing the end of a rod, and the
protrusion of an obstacle at the other end, or between the instant of pulling
a bell rope, and that of the ringing of the bell. But it is demonstrable that
in order to transmit an impulse in a time infinitely small, the hardness of
the substance must be infinitely great, and it must be absolutely incom-
pressible and inextensible by any force, which is a property not discover-
able in any natural bodies : the hardest steel and the most brittle glass
being very susceptible both of extension and compression.

The least elastic substance that has been examined, is perhaps carbonic
acid gas,* or fixed air, which is considerably denser than atmospheric air
exposed to an equal degree of pressure. The height of the atmosphere,
supposed to be homogeneous, is in ordinary circumstances, and at the sea
side, about 28,000 feet, and in falling through half this height a heavy
body would acquire a velocity of 946 feet in a second. But from a com-
parison of the accurate experiments of Derham,t made in the day time,
with those of the French Academicians,^: made chiefly at night, it appears
that the true velocity of sound is about 1130 feet in a second, which agrees
very nearly with some observations made with great care by Professor
Pictet. This difference between calculation and experiment has long
occupied the attention of natural philosophers, but the difficulty appears
to have been in great measure removed by the happy suggestion of
Laplace,§ who has attributed the effect to the elevation of temperature,
which is always found to accompany the action of condensation, and to
the depression produced by rarefaction. It is true that a greater change
of temperature would be required than Mr. Dalton's experiments on the
compression of air appear to indicate ; but those experiments do not per-

* It is sulphurous acid, in which the velocity is 229 -2 ft. Rees, Dissertatio de
Celeritate Soni, 4to. Trajecti ad Rhenum, 1819. Journal de Physique, 1821, p. 40.

t Ph.Tr. 1708, p. 2, concludes that the velocity is 1142ft. per second.

J Hist. etMem. del'Acad. 1738-9. Here the effect of the wind was first taken
into account: vel. = 1106ft. at 43° of temp. The actual velocity at the freezing
t&np. is about 1090 ft. per second. The increase of velocity is 1 '136 ft. for every
degree of temperature, on Fahrenheit's scale.

§ See Poisson, Journal de 1'Ecole Poly technique, cah. xiv. Biot, Journal de Phy-
sique, Iv. 173. Mem. d'Arcueil, ii. 94.

290 LECTURE XXXI.

fectly agree among themselves ; and the observation which has been made
in France, that a heat, sufficient to set tow on fire, may be produced/ by
the operation of a condensing syringe, seems to show that Mr. Dalton's
results are somewhat below the truth. In this manner the theory may
be completely reconciled with experiments ;* we may estimate the modulus
of the air's effective elasticity, which is the measure of its immediate
force, from the velocity which is thus observed, and its height will appear
to be 39,800 feet, instead of 27,800, which is the supposed height of the
atmosphere. This velocity remains unchanged by any alternation of pres-
sure indicated by the barometer, but it may be affected by a change of
temperature. For when an elastic fluid is compressed, its elasticity is
increased in the same ratio as its density ; and the height of a homo-
geneous atmosphere equivalent to the pressure, remains the same, conse-
quently the velocity calculated from that height remains unaltered ; but
the identity of the acceleration, from the effect of heat which has been
mentioned, can only be inferred from observation : this identity may, how-
ever, be satisfactorily shown, by means of experiments on the sounds of
organ pipes, which are intimately connected with the velocity of the
transmission of sound through the air, and which are found to remain
precisely the same, however the air may be rarefied or condensed. The
Academicians del Cimento inclosed an organ pipe, with bellows worked by
a spring, in the receiver of an air pump and of a condenser, and they
found that, as long as the sound was audible, its pitch remained unchanged.
Papint screwed a whistle on the orifice which admits the air into the
receiver of the air pump, and I have fixed an organ pipe in the same
manner ; and the result agreed with the experiment of the academicians.
But if the density of the air is changed, while its elasticity remains unal-
tered, which happens when it is expanded by heat, or condensed by cold,
the height of the column, and consequently the velocity, will also be
altered ; so that for each degree of Fahrenheit's thermometer the velocity
will vary about one part in a thousand. Bianconi J has actually observed
this difference of velocity according to the different heights of the thermo-
meter, and it may be shown less directly by means of the sounds of pipes ;
but it has not been accurately determined whether or no the correction on
account of the effect of compression in causing heat, remains unaltered,
although Bianconi's experiments agree very well with the supposition that
no material change takes place in this respect. The velocity of sound
must also be in some measure influenced by the quantity of moisture con-
tained in the atmosphere : it must be a little diminished by cold fogs,
which add to the density, without augmenting the elasticity, and increased
by warm vapours, which tend to make the air lighter ; and these two
opposite states are probably often produced in succession in wind instru-
ments blown by the mouth, the air within them being at first cold and
damp, and afterwards warm and moist.

In pure hydrogen gas, the velocity of sound ought, from calculation, to
be more than three times as great as in common air, but the difference does

* Clement and Desormes, Journal de Physique, 1819, p. 34.
t Birch, iv. 379. J Comm. Bonon. ii. I. 365.

ON THE PROPAGATION OF SOUND. 291

not appear to have been so great in any experiment hitherto made on the
soiftids of pipes in gases of different kinds. For such experiments, the
comparative specific gravity of the gas may be most conveniently ascer-
tained by Mr. Leslie's method of observing the time employed in emptying a
vessel through a small orifice, by means of the pressure of an equal column
of water ; according to the simple theory, the velocities of the gas thus
discharged ought to be in the same proportion as the respective velocities
with which sounds would be transmitted by them ; and if any variation
from this proportion were discovered, it must be attributed to the different
degrees of heat produced by condensation in the different fluids. Steam,
at the temperature of boiling water, is only one third as heavy as common
air ; consequently the velocity of sound in steam must be nearly three
fourths greater than in air.

It does not* appear that any direct experiments have been made on the
velocity with which an impulse is transmitted through a liquid, although
it is well known that liquids are capable of conveying sound without diffi-
culty ; Professor Robison informs us, for example, that he heard the sound
of a bell transmitted by water at the distance of 1200 feet. It is, however,
easy to calculate the velocity with which sound must be propagated in any
liquid of which the compressibility has been measured. Mr. Canton has
ascertained that the elasticity of water is about 22,000 times as great as
that of air ; t it is, therefore, measured by the height of a column which is
in the same proportion to 34 feet, that is 750 thousand feet, and the velocity
corresponding to half this height is 4900 feet in a second. In mercury,
also, it appears from Mr. Canton's experiments, J that the velocity must be
nearly the same as in water, in spirit of wine a little smaller. These
experiments were made by filling the bulb of a thermometer with water,
and observing the effects of placing it in an exhausted receiver, and in con-
densed air ; taking care to avoid changes of temperature, and other sources
of error : the fluid rose in the tube when the pressure was removed, and
subsided when it was increased. A slight correction is, however, required,
on account of the expansion and contraction of the glass, which must have
tended to make the elasticity of the fluids appear somewhat greater than it
really was.

It is also well known that solid bodies in general are good conductors of
sound : thus any agitation communicated to one end of a beam is readily
conveyed to the ear applied to the other end of it. The motion of a troop

* Since the above was written, experiments have been made on the velocity of
sound in water, by M. Beudant, at Marseilles, and MM. Colladon and Sturm, (a) in
the Lake of Geneva. The care with which the latter series of experiments were con-
ducted, and the distance to which the sounds were transmitted, amounting to about
four leagues, entitles them to confidence. The sounds were made by bells rung
under the water on one side of the lake, which were heard on the other side by the
intervention of a tube, closed at one end and open at the other ; the closed end being
immersed in the water, so that a column of air transmitted the sound to the ear
above the water. By a great number of experiments, it appears that the velocity is
4708 feet per second, in water of the temperature 46'6° of Fahrenheit.

*t 21,740, according to Canton, Ph. Tr. 1762, lii. 640; 1764, liv. 261.

J Ibid.

(a) Annales de Chimie, vol. xxxvi. Comptes Rendus, xiii. 439.
u2

292 LECTURE XXXI.

of cavalry is said to he perceived at a greater distance by listening with the
head in contact with the ground, than by attending to the sound conveyecl
through the air ; and we may frequently observe that some parts of the
furniture of a house are a little agitated by the approach of a wagon, before
we hear the noise which it immediately occasions. The velocity with
which impulses are transmitted by solids, is in general considerably greater
than that with which they are conveyed by the air : Mr. Wunsch* has
ascertained this by direct observations on a series of deal rods closely
united together, which appeared to transmit a sound instantaneously, while
a sensible interval was required for its passing through the air : I have also
found that the blow of a hammer on a wall, at the upper part of a high
house, is heard as if double by a person standing near it on the ground, the
first sound descending through the wall, the second through the air. It
appears from experiments on the flexure of solid bodies of all kinds, that
their elasticity, compared with their density, is much greater than that of
the air : thus, the height of the modulus of elasticity of fir wood, is found,
by means of such experiments, to be about 9,500,000 feet, whence the
velocity of an impulse conveyed through it must be 17,400 feet, or more
than three miles, in a second. It is obvious, therefore, that in all common
experiments such a transmission must appear perfectly instantaneous.
There are various methods of ascertaining this velocity from the sounds
produced under different circumstances by the substances to be examined,
and Professor Chladnit has in this manner compared the properties of a
variety of natural and artificial productions.

We have hitherto considered the propagation of sound in a single right
line, or in parallel lines only ; but it usually happens, at least when a sound
is transmitted through a fluid, that the impulse spreads in every direction,
so as to occupy at any one time nearly the whole of a spherical surface.
But it is impossible that the whole of this surface should be affected in a
similar manner by any sound, originating from a vibration confined to a
certain direction, since the particles behind the sounding body must be
moving towards the centre, whenever the particles before it are retreating
from the centre ; so that in one half of the surface the motions may be
called retrograde or negative, while in the other they are direct or positive,
consequently at the sides, where these portions join, the motions can be nei-
ther positive nor negative, and the particles must remain at rest ; the mo-
tions must also become gradually less and less sensible as they approach to
the limit between the two hemispheres. And this statement may be con-
firmed by an experiment on the vibration of a body of which the motion is
limited to a certain direction, the sound being scarcely audible when the ear
is in a direction precisely perpendicular to that of the vibration.

The sound thus diverging must always be spread through a part of a
spherical surface, because its velocity must be equal in every direction, so
that the impulse will always move forwards in a straight line, passing
through the centre of the sphere, or the vibrating body. But when a hemi-

* Berlin Memoirs, 1788, p. 87.

f Traite d'Acoustique, Paris, 1809, p. 319. See Herschel's remark on these
results, Encyc. Met. art. Sound, p. 773.

ON THE PROPAGATION OF SOUND. 293

spherical pulse arrives at the surface of a plane solid obstacle, it is reflected
precisely in the same manner as we have already seen that a wave of water
is reflected, and assumes the form of a pulse proceeding from a centre at an
equal distance on the opposite side of the surface. This reflection, when it
returns back perpendicularly, constitutes what is commonly called an
echo ; but in order that the echo may be heard distinctly, it is necessary
that the reflecting object be at a distance moderately great, otherwise the
returning sound will be confused with the original one ; and it must either
have a smooth surface, or consist of a number of surfaces arranged in a
suitable form ; thus there is an echo not only from a distant wall or rock,
but frequently from the trees in a wood, and sometimes, as it is said, even
from a cloud.

If a sound or a wave be reflected from a curved surface, the new direction
which it will assume may be determined, either from the condition that the
velocity with which the impulse is transmitted must remain unaltered, or
from the law of reflection, which requires that the direction of the reflected
pulse or wave be such as to form an angle with the surface, equal to that
which the incident pulse before formed with it. Thus if a sound or wave
proceed from one focus of an ellipsis, and be reflected at its circumference,
it will be directed from every part of the circumference towards the other
focus, since the distance which every portion of the pulse has to pass over
in the same time, in following this path, is the same, the sum of the lines
drawn from the foci to any part of the curve being the same ; and it may
also be demonstrated that these lines form always equal angles with the
curve on each side. The truth of this proposition may be easily shown by
means of the apparatus already described for exhibiting the motions of the
waves of water ; we may also confirm it by a simple experiment on a dish
of tea : the curvature of a circle differs so little from that of an ellipsis of
small eccentricity, that if we let a drop fall into the cup near its centre, the
little wave which is excited will be made to converge to a point at an equal
distance on the other side of the centre. (Plate XXV. Fig. 340, 341.)

If an ellipsis be prolonged without limit, it will become a parabola : hence
a parabola is the proper form of the section of a tube calculated for collect-
ing a sound which proceeds from a great distance into a single point, or
for carrying a sound nearly in parallel directions to a very distant place.
It appears, therefore, that a parabolic conoid is the best form for a hearing
trumpet, and for a speaking trumpet ; but for both purposes the parabola
ought to be much elongated, and to consist of a portion of the conoid re-
mote from the vertex ; for it is requisite, in order to avoid confusion, that
the sound should enter the ear in directions confined within certain limits :
the voice proceeds also from the mouth without any very considerable di-
vergence, so that the parts of the curve behind the focus would in both cases
be wholly useless. A trumpet of such a shape does not very materially
differ from a part of a cone ; and conical instruments are found to answer
sufficiently well for practice ; it appears, however, unnecessary to suppose,
as Mr. Lambert has done, that they differ essentially in principle from
parabolic trumpets.* It is not yet perfectly decided whether or no a speak-
* On Acoustic Instruments, Hist, et Mem. de Berlin, 1763, p. 87.

294 LECTURE XXXI.

ing trumpet has any immediate effect in strengthening the voice, inde-
pendently of the reflection of sound. (Plate XXV. Fig. 342.)

An umbrella, held in a proper position over the head, may serve to collect
the force of a distant sound by reflection, in the manner of a .hearing
trumpet ; but its substance is too slight to reflect any sound very perfectly,
unless the sound fall on it in a very oblique direction. The whispering
gallery at St. Paul's produces an effect nearly similar, by a continued repe-
tition of reflections. Mr. Charles's paradoxical exhibition of the Invisible
Girl * has also been said to depend on the reflection of sound ; but the de-
ception is really performed by conveying the sound through pipes, artVully
concealed, and opening opposite to the mouth of the trumpet, from which
it seems to proceed.

When a portion of a pulse of sound is separated by any means from the
rest of the spherical or hemispherical surface to which it belongs, and pro-
ceeds through a wide space, without being supported on either side, there is
a certain degree of divergence, by means of which it sometimes becomes
audible in every part of the medium transmitting it : but the sound thus
diverging is comparatively very faint ; and more so indeed than the effect
of a wave of water, admitted under similar circumstances, into a wide re-
servoir, which we have already examined. Hence, in order that a speaking
trumpet may produce its full effect, it must be directed in a right line to-
wards the hearer : and the sound collected into the focus of a concave
mirror is far more powerful than at a little distance from it, which could
not happen if, as some have erroneously supposed, sound in all cases tended
to spread equally in all directions. The sounds that enter a room, in which
there is an open window, are generally heard by a mixture of this faint
divergence with the reflection from various parts of the window and of the
room, and with the effect of the impulse transmitted through the walls.
This diverging portion, however faint, probably assists in preserving the
rectilinear motion of the principal sound, and gradually gains a little ad-
ditional strength at the expense of this portion.

The decay of sound is the natural consequence of its distribution through-
out a larger and larger quantity of matter, as it proceeds to diverge every
way from its centre. The actual velocity of the particles of the medium
transmitting it, appears to diminish simply in the same proportion as the
distance from the centre increases ; consequently their energy, which is to
be considered as the measure of the strength of sound, must vary as the
square of the distance ; so that, at the distance of ten feet from the sounding
body, the velocity of the particles of the medium becomes one tenth as
great as at the distance of one foot, and their energy, or the strength of the
sound, only one hundredth as great.

LECT. XXXI.— ADDITIONAL AUTHORITIES.

Sound in general. — Mersenne, Harmonie Universelle, fol. Paris, 1636. Lahire,
Hist, et Mem. de Paris, 1716, p. 262, H. 66. Hales, Doctrina Sonorum, 4to, Lorfd.
1778. Dr. T. Young on Sound and Light, Ph. Tr. 1800, p. 106. Huddlestone's

* See Nich. Jour. 1802, p. 56 ; 1807, p. 69.

ON THE SOUECES AND EFFECTS OF SOUND. 295

Observations on Sound, Nich. Jour. 8vo, i. 329. Armi, Ristretto di Fatti Acustici,
4to, Rom. 1821, Append. 1822.

Propagation of Sound.— Walker on the Velocity of Sound, Ph. Tr. 1698, xx.
433. Mairan, Hist, et Mem. de Paris, 1737, H. 1. Cassini, ibid. 1738, p. 128,
H. 1, 1739. La Condamine, ibid. 1745, p. 448, and Introd. Hist. &c. 1751, p. 98.
Euler, Hist, et Mem. de Berlin, 1765, p. 335. Winckler, Tentamina circa Soni
Celeritatem, 4to, Leipz. 1763. Blagden, Ph. Tr. 1784, p, 201. Miiller, Gotting.
Gelehrte Anzeigen, 1791. Espinosa and Bauza, Annales de Chimie, vii. 93. Ben-
zenburg, Gilbert's Annalen, new series, v. 383. Arago, &c. Connoissance des
Temps, 1825, p. 361. Goldingham, Ph. Tr. 1823, p. 96. Moll, &c. ibid. 1824,
p. 424. Gregory, Trans. Camb. Ph. Soc. 1824, ii. 120. Myrbach and Stampfer,
Jahrbuch des Instit. zu Wien, vol. vii.

Propagation in Gases.— Perolle, Melanges de Turin, 1786, iii. Corr. 1 ; 1790,
v. Corr. 195. Dulong, Annales de Chimie, vol. xli.

LECTURE XXXII.

ON THE SOURCES AND EFFECTS OF SOUND.

THE examination of the origin of sound might naturally be deemed an-
terior to the inquiry respecting its propagation ; but it will appear, that the
properties of many of the most usual sources of sound depend immediately
on the velocity with which an impulse of any kind is transmitted through an
elastic medium ; it was therefore necessary to consider this velocity, before
the production of sound in general could be discussed.

The origin of a simple sound, without any alternation, requires very
little investigation : it appears that the only condition necessary for its
production is a sufficient degree of velocity in the motion or impulse which
occasions it. A very moderate velocity must be sufficient for producing an
impression on the ear ; there is reason to believe that, when the sound is
continued, it may remain audible with a velocity of no more than one hun-
dredth of an inch in a second, and perhaps even with a velocity much
smaller than this ; but at its origin, it is probable that the velocity of the
motion, constituting a sound, must always be considerably greater.

A continued sound may be produced by a repetition of separate im-
pulses independent of each other, as when a wheel strikes in rapid succes-
sion the teeth of a pinion, so as to force out a portion of air from between
them ; when a pipe, through which air is passing, is alternately opened
and shut, either wholly or partially, by the revolution of a stopcock or
valve ; or when a number of parallel surfaces are placed at equal distances
in a line nearly perpendicular to them, and a noise of any kind is reflected
from each of them in succession ; a circumstance which may often be ob-
served when we are walking near an iron railing, an acute sound being
heard, which is composed of such reflections from the surfaces of the
palisades.

Musical sounds are, however, most frequently produced by the alternate
motions of substances naturally capable of isochronous vibrations, and these

296 LECTURE XXXII.

substances may be either fluids or solids, or instruments composed of a com-
bination of fluids with solids. The resonance of a room or passage is on? of
the simplest sources of a musical sound ; the walls being parallel, the impulse
is reflected backwards and forwards continually at equal intervals of time,
so as to agree with the definition, and to produce the effect, of a musical
sound. When we blow obliquely and uniformly into a cylindrical pipe
closed at one end, it is probable that the impulse or condensation must
travel to the bottom and back before the resistance is increased ; the cur-
rent of our breath will then be diverted from the mouth of the pipe for an
equal time, which will be required for the diminution of the resistance by
the discharge of the condensed air, so that the whole time of a vibration
will be equal to the time occupied by an impulse of any kind in passing
through four times the length of the pipe. An open pipe may be considered
nearly as if it consisted of two such pipes, united at their closed ends, the
portions of air contained by them being agitated by contrary motions, so as
always to afford each other a resistance similar to that which the bottom of
the stopped pipe would have furnished. It is probable that when an
open pipe is once filled with air a little condensed, the oblique current
is diverted, until the effect of the discharge, beginning at the remoter end,
has returned to the inflated orifice, and allowed the current to re-enter the
pipe. Where the diameter of the pipe is different at different parts of its
length, the investigation of the sound becomes much more intricate ; but it
has been pursued by Daniel Bernoulli* with considerable success, although
upon suppositions not strictly consistent with the actual state of the motions
concerned.

In the same manner as an open pipe is divided by an imaginary basis,
so as to produce the same sound with a stopped pipe of half the length, a
pipe of any kind is capable of being subdivided into any number of such
pipes, supposed to meet each other's corresponding ends only ; and, in
general, the more violently the pipe is inflated, the greater is the number
of parts into which it subdivides itself, the frequency of the vibrations being
always proportional to that number. Thus, an open pipe may be divided
not only into two, but also into four, six, eight, or more portions, producing
the same sounds as a pipe of one half, one third, one fourth, or any other
aliquot part of the length ; but a stopped pipe cannot be divided into any
even number of similar parts, its secondary sounds being only those of a
pipe of which the proportion is determined by the odd numbers, its length
being, for example, one third, one fi/th, or one seventh of the original
length. These secondary notes are sometimes called harmonics ; they are
not only produced in succession from the same pipe, but they are also often
faintly heard together, while the fundamental note of the pipe continues to
sound. When the pipe has a large cavity connected with it, or consists
principally of such a cavity, with a small opening, its vibrations are usually
much less frequent, and it is generally incapable of producing a regular
series of harmonics.

It is obvious from this statement of the analogy between the velocity of

* Hist, et Mem. de 1'Acad. 1762, p. 431, H. 170. See Euler, Nov. Com. Petr.
xvi. 281. Hauy, Traite de Physique, i. 316. Biot, do. ii. 111.

ON THE SOURCES AND EFFECTS OF SOUND. 297

sound and the vibrations of the air in pipes, that they must he affected in
• a swiilar manner by all alterations of temperature. Thus the frequency of
the vibrations of a pipe must be increased nearly in the ratio of 33 to 34 by
an elevation of 30 degrees of Fahrenheit's thermometer ; and if this change
be accompanied by a transition from dampness to simple moisture, the sound
will be still more altered.

Dr. Chladni has discovered that solids of all kinds, when of a proper form,
are capable of longitudinal vibrations, exactly resembling in their nature
those of the air in an organ pipe, having also their secondary or harmonic
noter* related to them in a similar manner. These vibrations are always far
more frequent than those of a column of air of equal length, the velocity
with which an impulse is transmitted by a solid of any kind being usually
from 5 to 16 times as great as the velocity of sound in air, so that the
longitudinal sounds are always extremely acute when they are produced by
substances of moderate length. These sounds afford, perhaps, the most ac-
curate mode of determining the velocity of the transmission of an impulse
through any elastic substance, and of obtaining from that velocity the exact
measure of its elasticity ; they may be easily exhibited by holding a long
bar or wire of iron or brass in the middle, and striking it at one end with a
small hammer in the direction of its length.

The vibrations by which solid bodies most usually produce sound are,
however, not longitudinal, but lateral, and they are governed either by a
tension derived from the operation of a weight, or of some other external
force, or by the natural elasticity of the substance. The vibrations of ex-
tended substances resemble most in their properties those of elastic fluids,
and they occur the most frequently in practice, although the vibrations
produced by the elasticity of the substance may be considered as the most
natural.

Vibrations derived from tension are either those of cords or musical
strings, or those of membranes : but the vibrations of membranes afford
little variety, and have not hitherto been very accurately investigated, the
drum being almost the only instrument in which they are concerned ; they
do not however appear to differ materially in their properties from the
vibrations of strings. A musical string or cord is supposed to be perfectly
flexible, and of uniform thickness, to be stretched between two fixed points
by a force incomparably greater than its own weight, and to vibrate in a
single plane through a minute space on each side of its natural position.
Its motions may then be traced through all their stages, by comparing the
cord to a portion of an elastic medium of the same length, contained
between two bodies capable of reflecting any impulse at each end ; for
example, to a portion of air situated between two walls, or inclosed in a
pipe stopped at both ends ; for the vibration of such a medium will be
performed in the time occupied by any impulse in travelling through twice
its length ; and the vibration of the cord will be performed in the same
time, supposing the height or depth of the medium equal to the length of
a, portion of the cord, of which the weight is equivalent to the force
applied to stretch it, and which may be called with propriety the modulus

298 LECTURE XXXII.

of the tension. If the cord be at first bent into a figure of any kind, and
then set at liberty, the place of any part of it at every subsequent time will
be such, that it will always be in a right line between two points moving
along the figure each way with the appropriate velocity ; but in order to
pursue this determination, we must repeat the figure of the cord on each
side of the fixed points in an inverted position, changing the ends as well
as the sides. Hence it appears that, at the end of a single vibration, the
whole cord will assume a similar figure on the opposite side of its natural
place, but in an inverted position, and after a complete or double vibration,
it will return precisely to the form which it had in the beginning. » The
truth of this result is easily shown by inflecting any long cord near one of
its ends, having first drawn a line under its natural position, and it will
then be evident that the cord returns in each vibration nearly to the point
of inflection, and passes at that end, but to a much shorter distance on the
opposite side of the line, while at the other end its excursions are greatest
on the opposite side of the line. The result of the calculation of the fre-
quency of vibration agrees also perfectly with experiment, nor is the
coincidence materially affected by the inflexibility or elasticity of the
string, by the resistance of the air, nor by the slight increase of the
tension occasioned by the elongation of the string when it is inflected :
thus, if the weight or force causing the tension of a string were equal
or equivalent to the weight of 200 feet of the same string, that is, if the
modulus of tension were 200 feet long, the velocity corresponding to half
this height would be 80 feet in a second ; and every impulse would be con-
veyed with this velocity from one end of the string to the other, so that if
the string were 1 foot long, if would vibrate 40 times in a second, if 6
inches, 80 times, and if it were 40 feet long, only once in a second. Hence,
it is obvious that the time of vibration of any cord is simply proportional
to the length ; and this may be shown either by means of such vibrations
as are slow enough to be reckoned, or by a comparison with the sounds of
pipes, or with other musical sounds. But if the tension of a cord of
given length were changed, it would require to be quadrupled in order to
double the frequency of vibration ; and if the tension and length remained
unaltered, and the weight of the cord were caused to vary, it would also
be necessary to make the weight four times as great in order to reduce the
frequency of vibration to one half.

It appears from the mode of tracing the progress of a vibration, which
has already been laid down, that every cord vibrates in the same manner
as if it were a part of a longer cord, composed of any number of such
cords, continually repeated in positions alternately inverted ; consequently
if a long cord be initially divided into any number of such equal portions,
its parts will continue to vibrate in the same manner as if they were sepa-
rate cords ; the points of division only remaining always at rest. Such
subordinate sounds are called harmonics : they are often produced in violins
by lightly touching one of the points of division with the finger, when the
bow is applied, and in all such cases it may be shown, by putting sma.ll
feathers or pieces of paper on the string, that the remaining points of

ON THE SOURCES ANI> EFFECTS OF SOUND. 299

division are also quiescent, while the intervening portions are in motion.*
1 (Pfete-XXV. Fig. 343.)

These harmonic sounds are also generally heard together with the funda-
mental sound of the cord, and it is, therefore, necessary, in such cases, to
consider the subordinate vibrations as combined with a general one. It is
not, however, universally true that the fundamental sound must always be
accompanied by all the harmonics of which the cord is susceptible ; for I
have found that by inflecting the cord exactly at any point in which the
cord might be divided into a number of equal parts, and then suffering it
to Vibrate, we lose the effect of the corresponding harmonic. There is
some difficulty in explaining the reason of the distinct production of these
sounds, in cases where the theory appears to indicate a single and simple
vibration only ; but it appears to be most probable that they usually
become audible in consequence of some imperceptible irregularity in the
form or weight of the cord, which is just sufficient to derange the perfect
coincidence of the actual motions with those which the theory indicates?
without producing a discordance capable of offending the ear. That a
cord irregularly loaded may have the relations of its harmonics disturbed,
may easily be understood by considering the effect of a small weight placed
at one of the points of division, which will obviously retard the principal
vibration, without materially affecting that of the portions terminated by
it. An abrupt and irregular agitation appears also in many cases to make
the secondary notes more audible, while a gradual and delicate impulse,
like that of the wind on the strings of an Aeolian harp, produces a sound
almost entirely free from subordinate harmonics.

It usually happens that the vibration of a cord deviates from the plane
of its first direction, and becomes a rotation or revolution, which may be
considered as composed of various vibrations in different planes, and which
is often exceedingly complicated. These vibrations may be combined in
the first instance in a manner similar to that which has been already ex-
plained respecting the vibrations of pendulums ; and if the motion of the
cord be supposed to follow the same law as that of a pendulum, the result
of two entire vibrations thus united, may be either a vibration in an inter-
mediate direction, or a revolution in a circle or in an ellipsis. But besides
these compound vibrations of the whole cord, it is also frequently agitated
by subordinate vibrations, which constitute harmonic notes of different
kinds, so that the whole effect becomes very intricate ; as we may observe
by a microscopic inspection of any luminous point on the surface of the
cord, for instance the reflection of a candle in the coil of a fine wire wound
round it. The velocity of the motion is such that the path of the luminous
point is marked by a line of light, in the same manner as when a burning
coal is whirled round ; and the figures, thus described, are not only different
at different parts of the same chord, but they often pass through an amusing
variety of forms during the progress of the vibration ; they also vary con-
siderably according to the mode in which that vibration is excited. (Plate
XXV. Fig. 344, 345.)

The vibrations immediately dependent on elasticity are those of rods,
* Wallis, Op. II. 466. Sauveur, Hist, et Mem. de 1'Acad. 1701.

300 LECTURE XXXII.

plates, rings, and vessels. These admit of much greater variety, and are
of more difficult investigation than the vibrations of cords. A rfc4 nifty
be either wholly loose, or fixed at one end only, or at both ; and it may
either be loosely fixed, in situation only, or firmly fixed, in direction as
well as in situation ; and these conditions may be variously combined with
each other ; the rod may also have a variety of secondary vibrations besides
the principal or fundamental sound. All these cases have been examined
by various mathematicians : the subject was begun by Daniel Bernoulli,*
and much extended by Euler,t some of whose conclusions have been cor-
rected by Riccati ; J and Chladni § has compared them all with experinent.
The sounds produced by the same rod, either under different circum-
stances, or as harmonics which may be heard at the same time, are scarcely
ever related to each other in any simple proportion, except that when a rod
is loosely fixed at both ends, the frequency of the vibrations of the
subordinate notes is expressed by the series of the squares of the
natural numbers, as 1, 4, 9, and 1G. But the times occupied by any
similar vibrations of rods, similarly circumstanced, are always directly as
the squares of their lengths, and inversely as their depths. When the rod
is wholly at liberty, two at least of its points must be at rest, and these are
at the distance of about one fifth of its length from either end : in the next
sound of the same rod, the middle point is at rest, with two others near the
ends. There is by no means the same regularity in the progress of the
vibrations of rods of different kinds as in those of cords ; it can only
happen in particular cases that the rod will return after a complete
vibration to its original state, and these cases are probably such as seldom
occur in nature.

The vibrations of plates differ from those of rods in the same manner as
the vibrations of membranes differ from those of cords, the vibrations
which cause the plate to bend in different directions being combined with
each other, and sometimes occasioning singular modifications. These vi-
brations may be traced through wonderful varieties by Professor Chladni' s
method of strewing dry sand on the plates, which, when they are caused to
vibrate by the operation of a bow, is collected into such lines as indicate
those parts, which remain either perfectly or very nearly at rest during the
vibrations. Dr. Hooke|| had employed a similar method, for showing the
nature of the vibrations of a bell, and it has sometimes been usual, in mili-
tary mining, to strew sand on a drum, and to judge, by the form in which
it arranges itself, of the quarter from which the tremors produced by
countermining proceed. (Plate XXV. Fig. 346... 348.)

The vibrations of rings and of vessels are nearly connected with those of
plates, but they are modified in a manner which has not yet been suf-

* Comm. Petr. iii. 62. Nov. Comm. Petr. xv. xvi. 257.

t Comm. Petr. vii. 99. Nov. Comm. Petr. x. 243 ; xvii. 381 ; 1780, iv. 11.99.
Acta Petr. iii. I. 103.

J Mem. della Soc. Jtal. i. 444.

§ Entdeckungen iiber die Theorie des Klanges, Leipz. 1787. Acta Ac. Electr
Mogunt. Erford, 1796. Neue Schriften der Berl. Gesell. 1799. Traite d'Acous-
tique, 1809, PI. 3.. .7. Neue Beytrage zur Akustik, 1817.

1| Birch's Hist, of the Roy. Soc. ii. 475.

ON TH SOURCES AND EFFECTS OF SOUND. 301

ficiently in\ssatigated. A glass, or a bell, divides in general into four
porlioTS? vibrating separately, and sometimes into six or eight ; they may
readily be distinguished by means of the agitations excited by them in a
fluid contained in the glass. It is almost unnecessary to observe, that the
fluid thus applied, by adding to the mass of matter to be moved, makes the
vibration slower, and the sound more grave.

In some cases the vibrations of fluids and solids are jointly concerned in
the production of sound : thus, in most of the pipes of an organ denomi-
nated reed pipes, the length of a tongue of metal is so adjusted as to be
capable of vibrating in the same time with the air contained in the pipe.
Sometimes, however, the air only serves to excite the motion of the solid,
as in some other organ pipes, which are usually much shorter than would
be required for producing the proper note alone, and probably in the
glottis, or organ of the voice of animals. On the other hand, the alternate
opening and shutting of the lips, in blowing the trumpet or French horn,
can scarcely be called a vibration, and the pitch of the sound is here de-
termined by the properties of the air in the pipe only. The vibrations of a
solid may be excited by an undulation propagated through a fluid ; thus,
when a loud sound strikes against a cord, capable of vibrating, either ac-
curately, or very nearly, with the same frequency, it causes a sympathetic
tone, resembling that from which it originated ; and the cord may pro-
duce such a sound either by vibrating as a whole, or by dividing itself into
any number of equal parts. Thus, if the damper be raised from any of the
strings of a harpsichord, it may be made to vibrate, by striking or singing
• any note, of which the sound corresponds either to that of the whole string,
or to that of any of its aliquot- parts. Sometimes also two cords that are
very nearly alike, appear, when sounding together, to produce precisely
the same note, which differs a little from each of those which the cords
would produce separately ; and a similar circumstance has been observed
with respect to two organ pipes placed near each other. In these cases the
vibrating substances must affect each other through the medium of the air ;
nearly in the same manner as two clocks, which rest on the same support,
have been found to modify each other's motions, so as to exhibit a perfect
coincidence in all of them.

It is uncertain whether any fibres in the ear are thus sympathetically agi-
tated in the process of hearing, but if there are any such vibrating fibres,
their motions must necessarily be of short duration, otherwise there would
be a perpetual ringing in our ears, and we should never be able to judge
accurately of the termination of a sound. Besides, a sympathetic vibration
may be excited not only by a sound producing vibrations of equal fre-
quency, but also by a sound, of which every alternate, or every third or
fourth vibration, coincides with its motions : it would, therefore, be
difficult to distinguish such sounds from each other, if hearing depended
simply on the excitation of sympathetic vibrations. It is true that we
generally distinguish, in listening to a loud and deep sound, precisely such
notes as would be thus produced ; but it is only when the sounding body
is capable of affording them from the nature of its vibrations ; for we may
listen for them in vain in the sound of a bell or of a humming top. There

302 LECTURE XXXII.

is, however, no doubt that the muscles, with which the difi^ent parts of
the ear are furnished, are concerned in accommodating the tensionN/f soihe '
of them to the better transmission of sound ; and it is equally certain that
their operation is not absolutely necessary in the process.

The external ear serves in some measure for collecting the undulations
of sounds transmitted through the air, and reflecting them into the auditory
passage, at the bottom of which they strike against the membrane of the
tympanum or drum, which, being larger and more moveable than some of
the subsequent parts, is capable of transmitting a stronger impulse than
they would immediately receive. In the same manner we may oftew feel
the tremors produced in a sheet of thick paper, held in the hand, by the
agitation of the air, derived from a loud sound, which would not otherwise
have affected the organ of touch. The impulse received by the membrane
of the tympanum is conveyed by the hammer and anvil, two small bones,
which together constitute a kind of bent lever, through a third minute flat-
tened bone, to a fourth called the stirrup, which serves merely as a handle
to its basis, a plate covering the orifice of a cavity called the vestibule, and
communicating the impulse to the mucous fluid which fills this cavity.
The fluid of the vestibule, thus agitated, acts immediately on the termi-
nations of the nerves, which form a loose membranous tissue, almost float-
ing in it, while another portion of them is distributed on the surface of
three semicircular tubes or canals, opening at both ends into the cavity,
and a third portion supplies the cochlea, a detached channel, which appears
to be arranged with singular art as a micrometer of sound. It resembles
the spiral convolutions of a snail shell, and if uncoiled, would constitute
two long conical tubes connected at their summits, the base of one opening
into the vestibule, that of the other being covered by a membrane only,
which separates the fluid from the air contained in the general cavity of
the ear, or the tympanum. It is evident from the properties of fluids
moving in conical pipes, that the velocity of any impulse affecting the fluid
at the base of the cone must be extremely increased at its vertex, while
the flexibility of the membrane at the base of the second channel allows
this motion to be effected without difficulty. It has also been supposed
that a series of fibres are arranged along the cochlea, which are susceptible
of sympathetic vibrations of different frequency according to the nature of
the sound which acts on them ; and, with some limitations, the opinion
does not appear to be wholly improbable. We must, however, reason with
great caution respecting the functions of every part of the ear, since its
structure varies so much in different animals, that we cannot pronounce
with certainty respecting the indispensable necessity of any one arrange-
ment for the perfection of the sense. And even in the case of the human
ear, many of these parts may be spared without great inconvenience ; thus,
we hear very perfectly, by means of impressions communicated to the
teeth, and through them to the large bones of tjie head ; and even when the
membrane of the tympanum, and all the small bones of the ear have been
destroyed by disease, the undulations of the air still continue to affect tho
organ in the usual manner.* (Plate XXV. Fig. 349... 351.)
* Douglas, De Aure Humana, 4to, Bonon. 1704.

ON THll SOURCES AND EFFECTS OF SOUND. 303

~* ^Such is tjj^d^licacy of the organs of hearing in their perfect state, that
we reaHtfy distinguish not only the frequency of the vibrations of a sound,
whether constant or variable, and its loudness or softness, but also the
quality of tone, depending on the law which governs each separate vibra-
tion, and which constitutes the difference between instruments of different
kinds, or different instruments of the same kind, or even the same instru-
ment differently employed. Thus, we can distinguish very accurately the
voices of our friends, even when they whisper, and those modifications of
the same voice which constitute the various vowels and semivowels, and
whicil with the initial and final noises denominated consonants, compose
the words of a language. We judge also, without an error of many degrees,
of the exact direction in which the sound approaches us ; but respecting
the manner in which the ear is enabled to make this discrimination, we
cannot reason upon any satisfactory grounds.

LECT. XXXII.— ADDITIONAL AUTHORITIES.

Vibrations of— 1. Fluids. — Euler on the Motion of Air in Pipes, Nov. Com.
Petr. xvi. 281. Chladni, Ph. Mag. iv. 275. Delarive on the Sounds from Hydrogen
Gas, Jour, de Physique, Iv. 165 ; Nich. Jour. 8vo, iv. 23. Higgins on do. Nich.
Jour. 8vo, i. 129. Biot, Mem. d'Arcueil, ii. 99. Leslie, Trans. Camb. Ph. Soc.
i. 267.

2. Cords. — Sauveur on the Sounds of Cords, Hist, et Mem. de Paris, 1713,
p. 324, H. 68. Jo. Bernoulli, Com. Petr. iii. 13. D. Bernoulli, Com. Petr. iii.
62; Hist, et Mem. de Berlin, 1753, pp. 147, 173. Bernoulli on the Vibrations of
Unequal and Compound Cords, Hist, et Mem. de Berlin, 1765, p. 281 ; Nov. Com.
Petr. xvi. 257. Euler on the Vibrations of Cords, Hist, et Mem. de Berlin, 1748,
p. 69, &c. &c. D'Alembert, ibid. 1747, pp. 214, 220 ; 1750, p. 355 ; 1763, p. 235.
Voigton the Nodes of Cords, Ph. Mag. iv. 347. Pellisov, Poggendorf's Annalen,
xix. 237.

3. Surfaces. — Biot, Mem. de 1'Institut, iv. 21. Jo. Bernoulli on the Vibrations
of Rectangular Plates, Nov. Act. Petr. 1787, v. 197. Voigt, Ph. Mag. iii. 389.
Comparisons with Chladni's Experiments. Savart, Annales de Chimie, vol. xii.
&c. &c. Kastner's Archiv, B. 8. Faraday, Ph. Tr. 1831, p. 237. Wheatstone,
ibid. 1833, p. 593. Tomlinson, Records of General Science, vol. ii.

Vibrations in general. — On Numbering the Vibrations of Sound, Com. Bon. i.
180. Poisson sur la Theorie du Son, Journal de 1'Ecole Poly technique, torn. xiv. ;
sur le Mouvement dans les Tuyaux Cylindriques, Mem. de 1'Acad. 1818-19, Me-
canique, ii. 693. Savart, Annales de Chimie, xliv. 337, xlvii. 69. Cagniard Latour,
Annales de Chimie, Ivi. 280. Blein sur la Theorie des Vibrations, 4to, 1827, 8vo,
1831. Trevelyan on the Vibrations of Heated Metals, Ph. Mag. 1832, vi. 141,
1833. Faraday, Jour, of the Royal Inst. vol. iv. Forbes, Ph. Mag. vol. iv. Trans-
actions of the Royal Soc. of Edin. vol. xii. Eisenlohr Lehrbuch der Physik, Mann-
heim, 1836. Dove's Repertorium der Physik, Band. vi. 1842. On Reflection of
Waves, Annales de Chimie, Ixxi. 20. Fechner, Repertorium der Physik, Band i.

Interference of Vibrations. — Dr. Young pointed out the fact, that a tuning fork
held vertically at a short distance from the ear, and turned on its axis, emits a louder
or softer sound, according to its position — the vibrations of the two prongs tending
alternately to strengthen and to diminish each other's effect. Mr. Hopkins (Trans.
Camb. Ph. Soc. v. 231) exhibited a similar interference to the eye.

Ear and Hearing. — Perrault on the Organ of Hearing, Hist, et Mem. de Paris, i.
158. Duverney, ibid. i. 256. Treatise on do. Lond. 1737. Valsalvade Aure, 4to,
Bologna, 1704. Mairan on the Effect of Sound on the Ear, Hist, et Mem. de Paris,
1?37, p. 49, H. 97. Nollet on the Hearing of Fishes, Hist, et Mem. de Paris, 1743.
Klein on do. Ph. Tr. 1748, p. 233. Arderon on do. ibid. 1748, p. 149. Camper
on do. Mem. des Savans Etrangers, vi. 177. Hunter on do. Ph. Tr. 1782, p. 379.
Geoffroy on the Hearing of Reptiles, Mem. des Savans Etrangers, ii. 164. Haller,

304 LECTURE XXXIII.

Physiol. Elliott on Vision and Hearing, 1780. Vicq d'Azyr ofc^e Hearing of
Birds, Hist, et Mem. de Paris, 1778, p. 381, H. 5. Galvani on do. Corn^Dn. vi.
O. 420. Scarpa de Auditu et Olfactu, fol. Patav. 1789. Comparetti de Aure
Interna, 4to, Paris, 1789. Home on the Membrana Tympani, Ph. Tr. 1800, p. 1.
Cooper on do. ibid. 1800, p. 151 ; 1801, p. 435. Gough on the Method of judging
of the Position of Sonorous Bodies, Manch. Mem. v. 622. Darwin's Zoonomia, ii.
487. Saunders's Anatomy of the Ear, 1806. Ramdohr, Magazin der Gess. Nat.
Freunde, Berlin, 1811, p. 389. Cuvier's Report on a Paper of Flourens, Annales
de Chimie, xxxix. 104. Muncke, Kastner's Archiv, vii. 1. Wollaston on Sounds
inaudible to certain Ears, Ph. Tr. 1820. Weber de Aure, Lips. 1820. Wheat-
stone, Journal of Science, 1827, vol. ii. Breschet, Recherches sur 1'Organe de
1'Ouie, 1836. Cyclopaedia of Anat. and Phys. art. Organ of Hearing, by Jones.
Lincke, Handbuch der Ohren heilkunde, Leipz. 1837. *

LECTURE XXXIII.

ON HARMONICS.

THE philosophical theory of harmonics, or of the combinations of
sounds, was considered by the ancients as affording one of the most refined
employments of mathematical speculation ; nor has it been neglected in
modern times, but it has been in general either treated in a very abstruse
and confused manner, or connected entirety with the practice of music, and
habitually associated with ideas of mere amusement. We shall, however,
find the difficulties by no means insuperable, and the subject will appear to
be worthy of attention, not only on its own account, but also for the sake
of its analogy with many other departments of science.

It appears both from theory and from experience, that the transmission
of one sound does not at all impede the passage of another through the
same medium. The ear too is capable of distinguishing, without difficulty,
not only two sounds at once, but also a much greater number. The mo-
tions produced by one series of undulations being wholly indifferent with
respect to the effect of another series, and each particle of the medium being
necessarily agitated by both sounds, its ultimate motion must always be
the result of the motions which would have been produced in it by the
separate sounds, combined according to the general laws of the composition
of motion, which are the foundation of the principal doctrines of mechanics.
When the two sounds, thus propagated together, coincide very nearly in
direction, the motions belonging to each sound may be resolved into two
parts, the one in the common or intermediate direction, and the other
transverse to it ; the latter portions will obviously be very small ; they will
sometimes destroy each other, and may always be neglected in determining
the effect of the combination, since the ear is incapable of distinguishing a
difference in the directions of sounds which amounts to a very few degrees
only. Thus, when two equal undulations, of equal frequency, coincide in
this manner, and when the particular motions are directed the same way
at the same time, the velocities in each direction are added together, and

ON HARMONICS. 805

jthe joint effecj><£ doubled, or perhaps quadrupled, since it appears that the
'sfeno^r-dfsound ought to be estimated from the squares of the velocities
of the particles : but when the particular motions of the two sounds coun-
teract each other, both their effects are wholly destroyed. These combina-
tions resemble the effects of the waves of water in similar circumstances,
which we have already examined, and they may be illustrated by drawing
two curved lines representing the motions which constitute the sounds, in
the same manner as we have already supposed them to be described, by a
vibrating particle, on a surface moving uniformly in a transverse direction ;
these figures being placed side by side, the joint effect may be represented
by a third curve drawn in such a direction as to be always in the middle
between the corresponding points of the first two. A similar result, but
still more strongly marked, may be obtained mechanically, by cutting two
boards or plates of any kind into the form of the curves, and then dividing
one of them into a number of thin pieces or sliders by lines perpendicular
to the general direction of the curve, or to the termination of the plate
which is parallel to it : the bottom of these sliders being then placed on the
other curve, their general outline will represent the effect of the combina-
tion. We may assume for this purpose the form of the harmonic curve,
which represents the motions of a body vibrating like a pendulum, and
which probably agrees very nearly with the purest and simplest sounds.
(Plate XXV. Fig. 352.)

If the two undulations differ a little from each other in frequency, they
alternately tend to destroy each other, and to acquire a double, or perhaps
a quadruple force, and the sound gradually increases and diminishes in
continued succession at equal .-intervals. This intension and remission is
called a beat, and furnishes us with a very accurate mode of determining
the proportional frequency of the vibrations, when the absolute frequency
of one of them is known, or the absolute frequency of both when their pro-
portion is known ; since the beats are usually slow enough to lqe reckoned,
although the vibrations themselves can never be distinguished. Thus, if one
sound consisted of 100 vibrations in a second, and produced with another
acuter sound a single beat ^n?" every second, it is obvious that the second
sound must consist of 101 vibrations in a second. Again, if we have two
portions of a similar cord equally stretched, or two simple pipes, of which
the lengths are in the proportion of 15 to 16, they will produce a beat in 15
vibrations of the longer ;* and if we count the number of beats in 15 seconds,
we shall find the number of vibrations in a single second. The easiest way
of procuring two such strings or pipes, in practice, is to tune them by a
third, so that they may be respectively -t and -| Of its length ; the vibrations
of the third pipe in a second will also be equal to the number of beats of the
first two in 12 seconds. (Plate XXV. Fig. 353.)

When the beats of two sounds are too frequent to be heard as distinct

* For the times of performing a vibration are as the lengths of the cords or pipes,
and therefore 15 of the latter correspond to 16 of the former. Now an interval
between two beats is that interval which occurs between one relative state of the two
cords or pipes and the return to the same state. Hence this interval is that due to
16 vibrations of the shorter, or 15 of the longer.

300 LECTURE XXXIII.

augmentations of their force, they have the same effect as another impulse?
which recur in regular succession, and produce a musical note^-w^zh has

been denominated a grave harmonic. Thus, two sounds in the proportion
of 4 to 5, produce, when they are both very low or grave, an audible suc-
cession of beats ; but when they are higher or more acute, a grave harmonic,
which may be separately distinguished as a third sound by an attentive ear.
Those combinations of sounds which produce beats distinctly audible, have
in general a harsh and coarse effect, and are called discords ; but those of
which the vibrations are so related, as to have a common period after a few
alternations, and which may be observed to produce a third sound, consti-
tute concords, which are in themselves the more perfect as the common
periods are shorter. (Plate XXV. Fig. 353.)

The natural association of the secondary sounds, which generally ac-
company almost all musical notes, serves, in some measure, as a foundation
for the science of harmonics, the same sounds as are thus frequently con-
nected in nature, being found to be agreeable when united by art. But it
appears to depend still more immediately on a love of order, and a predi-
lection for a regular recurrence of sensations, primitively implanted in the
human mind. Hence, when two sounds are heard together, those propor-
tions are the most satisfactory to the ear which exhibit a recurrence of a
more or less perfect coincidence at the shortest intervals, expressed by the
smallest numbers of the separate vibrations ; for though we cannot im-
mediately estimate the magnitude of the vibrations, yet the general effect of
a regular or irregular succession necessarily produces the impression of
sweetness or harshness. The same sounds as form the best accompaniment
for each other, are also in general the most agreeable for melodies, consist-
ing of a succession of single notes ; their intervals are, however, too large
to be sufficient for the purposes of music, and they require to be mixed with
other sounds which are related to them in a manner nearly similar.

The same constitution of the human mind which fits it for the perception
of harmony, appears also to be the cause of the love of rhythm, or of a re-
gular succession of any impressions whatever, at equal intervals of time.
Even the attachment to the persons and places to which we are accustomed,
and to habits of every kind, bears a considerable resemblance to the same
principle. The most barbarous nations have a pleasure in dancing ; and
in this case, a great part of the amusement, as far as sentiment and grace
are not concerned, is derived from the recurrence of sensations and actions
at regular periods of time. Hence not only the elementary parts of music,
or the single notes, are more pleasing than any irregular noise, but the
whole of a composition is governed by a rhythm, or a recurrence of periods
of greater or less extent, generally distinguished by bars, which are also the
constituent parts of larger periods, and are themselves subdivided into
smaller. An interruption of the rhythm is indeed occasionally introduced,
but merely for the sake of contrast ; nearly in the same manner as, in all
modern pieces of music, discords are occasionally mixed with concords, in
order to obtain an agreeable variety of expression.

In a simple composition, all the intervals are referred to a single funda-
mental or key note. Thus, any air which can be played on a trumpet or

J ON HARMONICS. 307

dti a bud£*5irtJrn, must consist of the harmonics of a single sound only : and
when teiaccompaniment is performed by a French horn, the length of the
instrument is first adjusted to the principal note, and all the sounds which
it is to produce are selected from this natural series. But the notes consti-
tuting the most natural scale are not, without exception, comprehended
among the harmonics ; they are, however, all immediately dependent on a
similar relation. A sound of which the vibrations are of equal frequency
with those of another, is called a unison ; if two vibrations occur for every
one of the fundamental note, the sound is called its superior octave, being
the eighth of those which are commonly considered as filling up the scale ;
and on account of its great resemblance to the fundamental note, it is de-
scribed by the same letter of the alphabet, or by the same syllable ; so that
all audible sounds are considered as repetitions of a series contained within
the interval of an octave. One third part of the string or pipe gives the
fifth above the octave ; one fourth the double octave, and one fifth of the
string its third. Thus we obtain the common accord or chord, or the har-
monic triad, consisting of the fundamental note, with its third and fifth,
which produces the most perfect harmony, and which also contains the
constituent parts of the most simple and natural melodies. But we are
still in want of intermediate steps for the scale ; these are supplied by
completing first, the triad of the fifth, which gives us the second, and the
seventh, of which 9 and 15 vibrations correspond respectively to 8 of the
fundamental, and which may also be found in the ascending series of
natural harmonics; and in the second place, by adding the fourth and
sixth in such proportions as with the octave to make up another perfect
triad ; the respective notes consisting of 4 and 5 vibrations, while the fun-
damental note makes 3, and being no where found among the natural har-
monics. The complete scale is, therefore, formed by these harmonic triads
contiguous to and connected with each other ; the middle one being the
triad of the key note, the superior one that of its fifth, which is sometimes
called the dominant or governing note, and the inferior one that of the
fourth, or subdominant. This scale is derived from principles so simple,
that it may properly be considered as a natural arrangement, and it
appears to be found with little variation in barbarous as well as in civil-
ized countries. (Plate XXV. Fig. 354.)

A long piece would, however, be too monotonous, unless the funda-
mental note were sometimes changed ; we may, therefore, take at pleasure
one of the auxiliary triads for the principal harmony, and we may continue
the modulation or progression, until every note of the scale becomes in suc-
cession a key note. But, in order to fill up the intervals of these several
scales in just proportion, it becomes necessary to add several new notes to
the first series ; for instance, if we take the seventh for a key note, we shall
want five new sounds within the octave, making twelve in the whole,
which is the number usually employed in music. The interval between
any two adjoining sounds of the twelve is called a semitone or half note,
two semitones making a tone or note ; these terms are, however, sometimes
employed with various subordinate distinctions and limitations. The five
additional sounds have no separate names, but they are denominated from

x2

V

308 LECTURE XXXIII.

the neighbouring notes on either side, with the addition of fiTe<^tni js

or flat, accordingly as they are a semitone higher or lower than the notes

of which they bear the names.

For still further variety, we sometimes change the place of the middle
note of each triad, placing the minor third, or the interval expressed by the
ratio of 5 to 6, below the major, which is in the ratio of 4 to 5 ; and the
scale thus formed is called the scale of the minor mode, in contradistinction
to the major, the three principal thirds being depressed a semitone. Some-
times, however, the alteration is made in the third of the key note only,
especially in ascending, in order to retain the seventh of the major scale,
which leads so naturally into the -octave, as to be sometimes called the
characteristic semitone of the key ; and it is for this reason, that the triad,
in which it is found, is called the accord of the dominant, which, in all
regular compositions, immediately precedes the termination in the key
note.

The major and minor triads, with the discord of the flat seventh, may be
considered as constituting the foundation of all essential harmonies. The
flat seventh is principally used with the major triad, in transitions from
the fundamental key into its fourth, to which that seventh naturally
belongs as a concord ; so that it serves to introduce the new key, by
strongly marking the particular note in which it differs from the old one ;
and in such cases the flat seventh always descends into, or is followed by,
the third of the new key, and the third of the first triad ascends into the
new key note. Other discords are also sometimes introduced, but they are
in general either partial continuations of a preceding, or anticipations of a
following accord. Two different parts of a harmony are never allowed, in
regular and serious compositions, to accompany each other in successive
octaves or fifths, since such a succession is found to produce a disagreeable
monotony of effect, except when a series of octaves is continued for some
time, so as to be considered as a repetition of the same part.

These are almost the only principles upon which the art of accompani-
ment, as well as the general theory of practical music, is founded. Many
prolix treatises have been written on the subject, but they only contain
particular illustrations of the application of these principles, together with
a few refinements upon them. The art of composition, however, depends
much more on a good taste, formed by habitual attention to the best
models, and aided, perhaps, by some little natural predisposition, than
upon all the precepts of science, which teach us only how to avoid what is
faulty, without instructing us in the mode of attaining what is beautiful or
sublime.

It is impossible to assign any such proportions for the twelve sounds
thus employed, that they may be perfectly appropriate to all the capacities
in which they are used ; their number is, therefore, sometimes considerably
increased ; and in some instruments they may be varied without limit, at
the performer's pleasure, as in the voice, in instruments with finger boards,
and in some wind instruments ; but in many cases this is impracticable,
nor could any imaginable alteration make all the intervals perfect, unless
every note were varied, whenever we returned to it by steps different from

/ ON HARMONICS. 309

,U*ose by whi^inve had left it. The simplest mode of arranging the twelve
' s0un<|^j*s*'to divide the octave into twelve equal intervals, all the notes
being in the same proportion to those which immediately follow them :
this is called the equal temperament, because the imperfection is equal in
all keys. In this system of temperament, the fifths, which consist of
seven semitones, are a little too flat, that is, the interval is a little too
small ; the minor thirds, consisting of three semitones, are also too flat,
and the major thirds too sharp. But it has generally been esteemed best
to preserve some keys more free from error ; partly for variety, and partly
because some are more frequently used than others : this cannot, however,
be done without making some of the scales more imperfect than in the equal
temperament. A good practical mode of performing it, is to make six
perfect fifths, in descending from the key note of the natural scale, and
six ascending fifths equally imperfect among themselves. We thus retain
a slight imperfection in the scales most commonly used, and make the keys
which are most remote from them considerably less perfect. Another
method, which is perhaps somewhat more easily executed, is to make the
fifth and third of the natural scale perfectly correct, to interpose between
their octaves, the second and sixth, so as to make three fifths equally tem-
pered, and to descend from the key note by seven perfect fifths, which will
complete the scale. Any of these modes of temperament may be actually
executed, either by the estimation of a good ear, or, still more accurately,
by counting the frequency of the beats which the notes make with each
other.*

For denoting precisely the absolute as well as the relative frequency of
the sounds of the different octaves, we employ the first seven letters of the
alphabet ; A being the key note of the minor mode, in the scale of natural
notes, and C of the major. The peculiar characters used in music are
generally disposed on five or more lines, with their intervening spaces,
each implying a separate step in the scale, setting out from any line at
pleasure, which is marked with an ill formed G, a C, or an F : a sharp or
a flat implying that all the notes written on the line, or in the space,
to which it belongs, are to be raised or depressed a semitone, and a natural
restoring the note to its original value. The actual frequency of the vibra-
tion of any note, according to the pitch most usually employed, may be
found, if we recollect to call a noise, recurring every second, the first C,
then the C denoted by the mark of the tenor cliff will be the ninth, con-
sisting of 256 vibrations in a second. The fifth, consisting of sixteen
vibrations, will be nearly the lowest audible note, and the fourteenth the
highest note used in music, but the sixteenth, consisting of above 30,000
vibrations in a second, may perhaps be an audible sound. The frequency
of the vibrations of the other notes may easily be calculated from the
known relations which they bear to the note thus determined. (Plate
XXV. Fig. 355.)

* Consult Marpurg's Anfangsgriinde der Theoretischen Musik, 4to, Leipz. 1757.
Versuch iiber die Temperatur, Bresl. 1776. Cavallo, Ph. Tr. 1788, p. 238. Robi-
son's Mech. Phil.

3JO LECTURE XXXIV.

LECT. XXXIII.— ADDITIONAL AUTHORITIES.

Zarlino, Institution! Harmoniche, fol. Venice, 1558. Salinas, do. fol. Salamanca,
1577. Tigrini, II Compendio della Musica, 4to, Venice, 1588. Cartesii Musicse
Compendium, Utr. 1650. Menzoli, Musica Speculativa, 4to, Bologna, 1670.
Salmon on Music, Lond. 1672. Dechales, Cursus Mathematicus, 3 vols. fol. Lyons,
1674. Holder on the Natural Grounds and Principles of Harmony, Lond. 1694.
Wallis, Ph. Tr. 1698, pp. 80, 249. Henfling's Musical System, Miscel. Berol. i. 265.
Malcolm on Music, Edin. 1721. Rameau, Traite de 1' Harmonic, 4 to, Paris, 1722.
Systeme de Musique, 4to, 1726. Euler, Tentamen Novae Theorise Musicse, 4to,
Petrop. 1729 ; also Hist, et Mem. de Berlin, 1764, pp. 165, 175 ; Novi Com. Petr.
xviii. 330. Montvallon, Hist, et Mem. de Paris, 1742, H. 117. Smith's Harmonies,
Camb. 1749. Serre, Principes d'Harmonie. Esteves on Temperament, Mem. des
Savans Etrangers, ii. 113. Romieu on do. Hist, et Mem. 1758, p. 483. Avison on
Musical Expression, 12mo, 1752. Antoniotto on Music, 2 vols. fol. 1760. Doni,
Opere, 3 vols. fol. 1763. BaiUiere de Laisement, Theorie de la Musique, 4to, Paris,
1764. Jamard, do. 1768. Holden, 4to, Lond. 1770. Kirnberger, Kunst der
Reinen Satzes, 4to, Berlin, 1771. Sulzer's Theorie der Schonen Kiinste, 4 vols.
Leipz. 1772. Lambert on Temperament, Hist, et Mem. de Berlin, 1774, p. 55.
Bemetzreider, Traite de Musique, Paris, 1776 ; Essai sur 1'Harmonie, 1781. Van-
dermonde, Systeme d'Harmonie, 1778. Choron, Abrege des Principes de Composi-
tion, 2 vols. fol. Paris. Steele's Prosodia'.Rationalis, 4to, Lond. 1779. Pizzali, La
Scienza de' Suoni e dell' Armonia, 4to, Venice, 1782. Young on Compound Sounds,
Nich. Jour. 8vo, ii. 264 ; iii. 145 ; iv. 72, 101. Weber, Theorie der Tonsetzkunst,
4 vols. Mainz. Shield's Introduction to Harmony, 4to, 1800. Kollman's New
Theory of Harmony, 4to, 1806. Busby's Treatises, v. y. Catel, Traite d'Har-
monie, 1808. Raymond, Bases Physico-math. de 1'Art Musical, Paris, 1813.
Morel, La Musique expliquee, 1816. Macdonald's Treatise on the Harmonic
System, fol. Lond. 1822. Nathan's Essay, 1823. Baldwin's Science of Music,
12mo, 1829. Blein, Principes de Melodic, 1832. Prony, Introduction Elementaire
aux Intervalles Musicaux, 4to, 1832. Beethoven, Etudes d'Harmonie, 2 vols. Paris,

1833. Albrechtsberger, Methode Elementaire d'Harmonie, translated into English,

1834. Woolhouse's Essay on Musical Intervals, 12mo, 1835. Busset, La Musique
expliquee, 1836. Fetis, La Musique mise a la Portee de tout le Monde, 1836.
Graham's Essay on Musical Composition, Edin. 1838.

LECTURE XXXIV.

ON MUSICAL INSTRUMENTS.

THE application of the theory of harmonics to practice depends on the
construction of musical instruments of different kinds : of these we shall
only be able to take a cursory view, and we shall afterwards attend to the
historical order of the most remarkable steps, by which both the theory and
practice of music have been advanced to a high degree of refinement.

Musical instruments may be most conveniently arranged, accordingly as
they are principally calculated for exciting sound by the vibrations of
cords, of membranes, of elastic plates, or of the air ; or by the joint effects
of the air and a solid body vibrating together. The essential varieties o,f
stringed instruments are found in the harp, the harpsichord, the pianoforte,
the clavichord, the guitar, the violin, the vielle or monochord, and the

ON MUSICAL INSTRUMENTS. 311

n all these, the immediate force of the sound of the strings
is increased by means of a sounding board, which appears to be agitated by
their motion, and to act more powerfully on the air than the strings could
do alone.

In the harp, the sound is produced by inflecting the string with the finger,
and suffering it to return to its place. The lyre of the ancients differed
from the harp only in its form and compass, except that the performer
sometimes used a plectrum, which was a small instrument, made of ivory,
or some other substance, for striking the strings. Each note in the harp
has £ separate string ; and in the Welsh harp there are two strings to each
note of the principal scale, with an intermediate row for the semitones ; but
in the pedal harp, the half notes are formed by pressing pins against the
strings, so as to shorten their effective length. Instead of this method, an
attempt has lately been made to produce the semitones by changing the
tension of the strings, which is said to have succeeded tolerably well, al-
though it appears at first sight somewhat unpromising.

In the harpsichord, and in the spinet, which is a small harpsichord, the
quill acts like the finger in the harp, or the plectrum in the lyre, and it is
fixed to the jack by a joint with a spring, allowing it without difficulty to
repass the string, which is here of metal. Sometimes leather is used instead
of quills ; and this serves to make the tone more mellow, but less powerful.
Besides two strings in unison, for each note, the harpsichord has generally
a third which is an octave above them. Different modifications of the tone
are sometimes produced by striking the wire in different parts, by bringing
soft leather loosely into contact with its fixed extremity, and by some other
means. When the finger is removed from the key, a damper of cloth falls
on the string, and destroys its motion. In all instruments of this kind, the
perfection of the tone depends much on the construction and situation of
the sounding board : it is usually made of thin deal wood, strengthened at
different parts by thicker pieces fixed below it.

In the pianoforte, the sound is produced by a blow of a hammer, raised
by a lever, which is as much detached from it as possible. The dulcimer,
or hackbrett of the Germans, is also made to sound by the percussion of
hammers, but they are simply held in the hand of the performer.

The clavichord, the clavier of the Germans, differs from other keyed
instruments in the manner in which the length of the string is determined ;
it is attached at one end to a bridge, and at the other to a pin or screw as
usual ; but the effective length is terminated on one side by the bridge, and
on the other by a flat wire projecting from the end of the key, which strikes
the string, and at the same time serves as a temporary bridge as long as the
sound continues : the remaining portion of the string is prevented from
sounding by being in contact with a strip of cloth, which also stops the
whole vibration as soon as the hammer falls. The instrument is capable of
great delicacy and neatness of expression, but it is deficient in force. The
guitar is generally played with the fingers, like a harp ; but each string is
made to serve for several notes, by means of frets, or cross wires, fixed to
the finger board, on which it is pressed down by the other hand. But in
the pianoforte guitar, hammers are interposed between the fingers and the

312 LECTURE XXXIV.

strings, acting like those of the pianoforte. The mandolinfess^d lute
species of the guitar : and the arch lute was a very powerful instrument of
the same kind, formerly much used in full pieces.

In the violin, and in other instruments resembling it, all the strings are
capable of having their length altered at pleasure, by being pressed down
on the finger board. The sound is produced by the friction of the bow,
rubbed with resin : the string is carried forwards by its adhesion to the
bow, and when its resistance has overcome this adhesion, it begins to return
in opposition to the friction ; for the friction of bodies in motion is gene-
rally less than their adhesion when they are at rest with respect to £ach
other, besides that the contact of the string with the bow is usually in great
measure interrupted by subordinate vibrations, which may be distinguished,
by the assistance of a microscope, in the manner already described ; but
when the string changes once more the direction of its motion, it adheres
again to the bow, and is accelerated by it as before. The original instru-
ment appears to have been the viola or tenor, its diminutive the violino, its
intensitive, expressing a greater bulk, the violone or double bass, and the
diminutive of this, the violoncello, or common bass. The viola di gamba
had one or more long strings separate from the finger board, serving as an
occasional accompaniment.

The vielle, or monochord, commonly called the hurdy gurdy, has frets
which are raised by the action of the fingers on a row of keys ; and instead
of a bow, the string is made to vibrate by the motion of a wooden wheel :
there is a second string serving as a drone, producing always the same
sound ; this is furnished with a bridge loosely fixed, which strikes continu-
ally against the sounding board, and produces a peculiar nasal effect. The
trumpet marine, or trumpet Marigni, was a string of the same kind, which
was lightly touched at proper points, so as to produce harmonic notes only;
it was impelled by a bow. The aeolian harp, when agitated by the wind,
affords a very smooth and delicate tone, frequently changing from one to
another of the harmonics of the string, accordingly as the force of the wind
varies, and as it acts more or less unequally on different parts of the string.
(Plate XXV. Fig. 356.)

The human voice depends principally on the vibrations of the mem-
branes of the glottis, excited by a current of air, which they alternately
intercept and suffer to pass ; the sounds being also modified in their sub-
sequent progress through the mouth. Perhaps the interception of the air
by these membranes is only partial ; or it may be more or less completely
intercepted in sounds of different kinds : the operation of the organs con-
cerned is not indeed perfectly understood, but from a knowledge of their
structure, we may judge in some measure of the manner in which they are
employed.

The trachea, or windpipe, conveys the air from the chest, which serves
for bellows : hence, it enters the larynx, which is principally composed of
five elastic cartilages. The lowest of these is the cricoid cartilage, a strong
ring, which forms the basis of the rest : to this are fixed, before, the thy<
reoid cartilage, and behind, the two arytaenoid cartilages, composing
together the cavity of the glottis, over which the epiglottis inclines back-

/ON MUSICAL INSTRUMENTS. 313

as if^g^ends from its origin at the upper part of the thyreoid cartilage,
fne glottis are extended its ligaments, contiguous to each other
before, where they are inserted into the thyreoid cartilage, but capable of
diverging considerably behind whenever the arytaenoid cartilages separate.
These ligaments, as they vary their tension, in consequence of the motions
of the arytaenoid cartilages, are susceptible of vibrations of various fre-
quency, and as they vibrate, produce a continuous sound. Properly speak-
ing, there are two ligaments on each side ; but it is not fully understood
how they operate ; probably one pair only performs the vibrations, and the
other* assists, by means of the little cavity interposed, in enabling the air to
act readily on them, and in communicating the vibrations again to the air.
(Plate XXVI. Fig. 357, 358.)

The vowels and semivowels are continuous sounds, chiefly formed by
this apparatus in the glottis, and modified either in their origin or in their
progress by the various arrangements of the different parts of the mouth.
Of simple vowels sixteen or eighteen may be enumerated in different lan-
guages : in the French nasal vowels the sound is in part transmitted
through the nostrils, by means of the depression of the soft palate : the
perfect semivowels differ from the vowels only in the greater resistance
which the air undergoes in its passage through the mouth ; there are also
nasal and seminasal semivowels. The perfect consonants may be either
explosive, susurrant, or mute; the explosive consonants begin or end
with a sound formed in the larynx, the others are either whispers, or mere
noises, without any vocal sound. By attending to the various positions
of the organ, and by making experiments on the effects of pipes of dif-
ferent forms, it is possible to construct a machine which shall imitate very
accurately many of the sounds of the human voice ; and this has indeed
been actually performed by Kratzenstein * and by Kempelen.t (Plate
XXVI. Fig. 359.)

Although the vibrating ligaments of the glottis may be anatomically
denominated membranes, yet their tension is probably confined to the
direction of their length, and their action is, therefore, the same with that
of a simple string or cord. But in the case of a tambourine and a drum,
the membrane is stretched in every direction, and the force of tension con-
sequently acts in a different manner. The principal character of such in-
struments is their loudness, derived from the magnitude of the surface which
strikes the air, and the short duration of the sound, on account of the great
resistance necessarily produced by the air's reaction.

Musical instruments which produce sounds, by means of vibrations de-
pending on the elasticity of solid bodies, are less frequently employed than
others ; they have a peculiar character of tone, which is by no means un-
pleasant, but which renders them less fit to be mixed with other instru-
ments, since their secondary harmonics are in different proportions. Such
is the stacada, a series of cylinders of glass, or of metal, struck either im-

t * Journal de Physique, xxi. 358. Acta Petr. 1780, iv. II. H. 16.

f Ueber den Mechanismus der Menschlichen Sprache, Vienna, 1791. On this
subject consult Willis, Trans. Camb. Phil. Soc. iii. 231. Purkinje on the Physio-
logy of Speech, Cracow, 1836.

314 LECTURE XXXIV. \ ^

mediately with hammers, or by means of keys ; the tuning>«^the
the cymbal, and the bell. Bells and other similar instruments are usually
made of a mixture of copper and tin, with a little brass or zinc, which is
more highly elastic than either of the component parts taken separately.
The harmonica consists of a series of vessels of glass, either placed side by
side, or fixed on a common axis, and made to sound by the friction of the
fingers, and sometimes by that of rubbers of cork. The vibrations of an
elastic plate, agitated by a current of air, which it continually admits and
excludes, constitute the sound of the vox humana and regal organ pipes,
resembling the human voice as much in their effects as in the mechanism
on which they depend. (Plate XXVI. Fig. 360... 362.)

Of simple wind instruments, in which the quality of the sound is deter-
mined by the vibrations of the air, the principal are the syrinx, the flute,
the flageolet, the diapason organ pipe, whether open, stopped, or with a
chimney, the humming top, and the cavity of the mouth in whistling, or
in playing on the Jew's harp. The pipes of the syrinx are adjusted to their
respective notes by cutting them, or filling them up, until they are reduced
to a proper length ; and the effective length of the flute and flageolet is
altered at pleasure by opening or shutting the holes made at proper dis-
tances in them ; the opening a hole at any part having the same effect as
if the pipe were cut off a little beyond it, and the elevation of the tone being
somewhat greater as the hole is larger. The instruments differ little except
in the mechanism by which the breath is directed in such a manner as to
excite a sound ; and the flageolet, when furnished with bellows, becomes a
bagpipe. The tongue of the Jew's harp is an elastic plate, but the sound,
which it immediately produces, serves only as a drone ; its vibration, how-
ever, appears to act like the motion of the bow of a violin in exciting
another sound : this sound, although faint, is still sufficiently musical, and
appears to be determined by the magnitude of the cavity of the mouth,
nearly in the same manner as that of the humming top, or as the sound of
the same cavity produced in whistling, by a current of air which is forced
through it. (Plate XXVI. Fig. 363... 367.)

In mixed wind instruments, the vibrations or alternations of solid bodies
are made to cooperate with the vibrations of a given portion of air. Thus,
in the trumpet, and in horns of various kinds, the force of inflation, and
perhaps the degree of tension of the lips, determines the number of parts
into which the tube is divided, and the harmonic which is produced. In
the serpent, the lips cooperate with a tube, of which the effective length may
be varied by opening or shutting holes, and the instrument which has been
called an organized trumpet appears to act in a similar manner ; the trom-
bone has a tube which slides in and out at pleasure, and changes the actual
length of the whole instrument. The hautboy and clarinet have mouth-
pieces of different forms, made of reeds or canes ; and the reed pipes of an
organ, of various constructions, are furnished with an elastic plate of metal,
which vibrates in unison with the column of air that they contain. An
organ generally consists of a number of different series of such pipes, se,
arranged, that by means of registers the air proceeding from the bellows
may be admitted to supply each series, or excluded from it, at pleasure, and

1

/ ON MUSICAL INSTRUMENTS. 315

is ope£*!a, when the proper key is touched, which causes all the pipes
* belon^in^to the note, in those series of which the registers are open, to
sound at once. These pipes are not only such as are in unison, but fre-
quently also one or more octaves above and below the principal note, and
sometimes also twelfths and seventeenths, imitating the series of natural
harmonics. But these subordinate sounds ought to be comparatively faint,
otherwise their irregular interference would often occasion an intolerable
discord, instead of the grand and sublime effect which this instrument is
capable of producing, when it is judiciously constructed and skilfully
employed. (Plate XXVI. Fig. 368.)

The practice of music appears to be of earlier origin than either its
theory, or any attention to the nature and general phenomena of sound.
The first lyre, with three strings, is said to have been invented in Egypt
by Hermes, under Osiris, between the years 1800 and 1500 before Christ ;*
but a tradition so remote, concerning a personage so enveloped in fable,
can scarcely be considered as constituting historical evidence : we cannot,
therefore, expect to ascertain with any certainty the proportions of these
strings to each other ; some suppose that they were successive notes of the
natural scale, others that they contained the most perfect concords ;
perhaps in reality each performer adjusted them in the manner which best
suited his own fancy. The trumpet is said to have been employed about the
same time ; its natural harmonics might easily have furnished notes for the
extension of the scale of the lyre, but it does not appear that the ancients
ever adopted this method of regulating the scale. The lyre with seven
strings is attributed to Terpander,t about 700 years before our era, and
two centuries afterwards, either Pythagoras, or Simonides, completed the
octave, which consisted of intervals differing very little from the modern
scale, the key note being nearly in the middle of the series. { In subse-
quent times the number of the strings was much increased ;§ the modula-
tions, and the relations of the intervals, became very intricate, and were
greatly diversified in a variety of modes or scales, which must have afforded
an inexhaustible supply of original and striking melodies, but which could
scarcely admit so many pleasing combinations as our more modern
systems. Although it is certain that the ancients had frequent accom-
paniments in perfect harmony with the principal part, yet they had no
regular art of counterpoint, or of performing different melodies together ;
nor does it appear that they ever employed discords. The tibia of the
ancients resembled a hautboy or clarinet, for it had a reed mouth piece,
about three inches long ; the same performer generally played on two of
these instruments at once. There were, however, several varieties of the
tibia ; and it is not improbable that some of them may have had the simple
mouth piece of the flageolet.

The first philosophical observer of the phenomena of sound, after Pytha-
goras, appears to have been Aristotle ; he notices a great variety of curious
facts in harmonics among his mechanical problems ; and he entertained a

' * Rollin's History of the Arts and Sciences (trans.), 4 vols. 8vo, Lond. 1737.
f Ibid. i. 156. J Jamblichus, Vita Pythag.

§ See Aristophanes, Nubes.

316 . LECTURE XXXIV.

very correct idea of the true nature of the motions of the\^r constituti^
sound. He knew that a pipe or a cord of a double length p"r^duT;ed > a
sound of which the vibrations occupied a double time ; and that the pro-
perties of concords depended on the proportions of the times occupied by
the vibrations of the separate sounds. It is not indeed improbable that at
least as much as this was known to Pythagoras, since he established cor-
rectly the numerical ratios between various sounds ; but so little justice has
been done to his discoveries by the imperfect accounts of them which have
been preserved, that we cannot expect to be able to ascertain his opinions
on any subject with accuracy. t

The invention of the organ, by Ctesibius of Alexandria, about 2000
years ago, forms a remarkable epoch in harmonics. The larger instruments
of this kind were furnished with hydraulic bellows, the smaller with
bellows of leather only ; and they had keys which were depressed, like
those of the modern organs, by the fingers of the performer, and which
opened valves communicating with the pipes.

The modern system of music is one of the few sciences, if such it can
be called, which owe their improvement to the middle ages. The old
ecclesiastical music was probably founded in great measure on that of the
Greeks ; its peculiar character consisted in the adoption of any note of the
scale at pleasure for a key note, without altering materially the other
intervals ; and in this manner they obtained a variety much resembling
that of the modes or kinds of music in use among the ancients. Pope
Gregory, about the year 600, distinguished the notes by literal characters ;
the rules of counterpoint were formed by degrees from the experience
of the ecclesiastical musicians ; and early in the eleventh century, Guido
of Arezzo, otherwise called Aretin the monk, introduced, together with
some improvements in the theory and practice of music, a new method of
naming the notes by syllables.

Some curious experiments on sound may be found in the works of
Bacon, but they added very little to the true theory of acustics, and some
of them are not perfectly accurate. Galileo* rediscovered what was well
known to Aristotle, respecting the nature of sound ; for the words of Ari-
stotle had been so much misunderstood and misinterpreted, that he could
have profited but little by them. His cotemporaries Mersennet and
Kircher^ made a variety of very ingenious experiments and observations,
on sound and on sounding bodies, many of them unknown to authors of
later date. The theory of the ancient music was very accurately investi-
gated, in the middle of the 17th century, by Meibomius : § our countryman
Wallis, also, besides employing much learning and penetration in the illus-
tration of the ancient music, observed some insulated facts in harmonics
which were new and interesting. ||

Sir Isaac Newton's propositions^ respecting the velocity of the pro-
pagation of sound were the beginning of all the more accurate inves-

* Op. iii. 58. t Harmonicorum Liber, Par. 1635.

+ Musurgia, 2vols. fol. Rom. 1650. Phonurgia, fol. 1673. ,

§ Musicse Antiq. Scrip. Meibomii, 2 vols. 4to, Amst. 1652.

II Opera, vol. iii. and Cl. Ptolemsei Op. a Wallis, 4 to, Oxf. 1682.

11 Principia, lib. ii. Prop. 46, &c.

J

ON MUSICAL INSTRUMENTS. 317

tiga\ions j<^atingto acustics. It must not be denied that these propositions
contain some very inconclusive reasoning respecting the nature of the
motions constituting sound, hy which the determination of a particular
case is erroneously extended into a general solution of the problem. The
velocity is, however, truly calculated, because it is in fact independent of
the particular nature of the vibration, and all that is wanting to generalise
the proposition is the remark, that if the velocity of sound is the same in
all cases, it must be such as the calculation indicates. An error nearly
similar was committed by Brook Taylor,* who in the year 1714 investi-
gated the time occupied by the vibration of a string or cord upon a
particular supposition, which he considered as a necessary condition, but
which in fact confined the inquiry to a limited case. It happens, however,
that the same determination of the frequency of vibration is equally true
in all possible cases. Sauveur obtained, about the same time, a similar con-
clusion from reasoning still less accurate : his merits with respect to the
theory of acustics in general are, however, by no means contemptible.
Lagranget and Euler^ have corrected and much extended the investi-
gations of Newton, and of Taylor, and Bernoulli§ and Dalembert|| have
also materially contributed to the complete examination and discussion of
the subject.

About the year 1750, Daniel Bernoulli succeeded in obtaining a solution
of a problem still more difficult than those which relate to the motions of
cords : he determined the frequency of the vibrations of an elastic rod
fixed at one end, as well as the relations of its subordinate sounds. The
solution is not indeed absolutely general, but it may perhaps be adapted to
all possible cases, by considering the effect of a combination of various
sounds produced at the same time. Euler has also great merit in extend-
ing and facilitating the mathematical part of this investigation, although he
has committed several mistakes respecting the mechanical application of it,
some of which he has himself corrected, and others have been noticed by
Riccati and Chladni.

The grave harmonics produced by the combination of two acute sounds
were noticed about the same time by Romieu and by Tartini, but first by
Romieu : ^[ their existence is not only remarkable in itself, but particularly
as it leads to some interesting consequences respecting the nature of sound
and hearing in general.

Bernoulli has also investigated, in a very ingenious manner, the sounds
produced by the air in pipes of various forms, although confessedly on
suppositions deviating in some measure from the truth : the results of his

• * De Motu Nervi Tensi, Ph. Tr. 1713, xxviii. 26. Methodus Incrementorum,
Lond. 1715. t Mel. de Turin, i. ii. & UL

J Hist, et Mem. de Berlin, 1748, 1753, 1759, p. 185, &c. ; 1765, p. 355. Nov.
Com. Petr. ix. xvii. xix. ActaPetr. 1779, p. 2; 1780, p. 2; 1781, p. 1. Mel. de
Turin, vol. iii. § See Lect. XXXI.

|| Hist, et Mem. de Berlin, 1747, 1750, 1753, 1763. Opuscula, i. & iv.

IT Mem. de 1'Acad. de Montpellier, 1751. See Tartini, Trattato diMusica, Pad.
1754; and Mercadier de Belesta, Systeme de Musique, Paris, 1776; or Matthew
Young's Enquiry into the principal Phenomena of Musical Strings, Dublin, 1784,
p. 2, sect. vi. The existence of the grave harmonic was first noticed by Sorge,
Anweisung zur Stimmung der Orgelwerke, &c. Hamburg, 1744.

318 LECTURE XXXIV. V

computations have, however, been amply confirmed by the expt^Jmen^pl 4
Lambert* on the sounds of flutes. '"

Dr. Chladni's method of examining the sounds of plates has afforded a
very interesting addition to our knowledge of the nature of vibrations ; his
discovery of the longitudinal sounds of solids is of considerable importance,
and he is said to be engaged in an extensive work on the subject of acustics
in general.t Some remarks which I have made in the Philosophical Trans-
actions may perhaps also be considered as tending to illustrate the vi-
brations of cords. The latest improvement which deserves to be mentioned,
with respect to the theory of sound, is Laplace's explanation of the increase
of its velocity on account of the effect of heat, which appears to afford a
satisfactory explanation of a difficulty so much the more important, as it
tended to lessen our confidence in every part of a theory, which differed so
widely from the most accurate and best established observations.

1.

LECT. XXXIV.— ADDITIONAL AUTHORITIES.

Musical Instruments. — Sauveur on the Composition of Organ Pipes, Hist, et
Me*m. de Paris, 1702, p. 308, H. 90. Carre, ibid. 1702, H. 136. Weber, Poggen-
dorf's Annalen, xvi. xvii. 193. Savart, Mem. surla Construction des Instrumens a
Cordes, &c. Paris, 1819.

Human Voice.— Dodart, Hist, et Mem. 1700, p. 244, H. 17 ; 1706, pp. 136, 388 ;
1707, p. 66, H. 18. Ferrein, ibid. 1741, p. 409, H.51. Vicq d'Azyr, ibid. 1779,
p. 178, H. 5. Liscovius, Theorie der Stimme, Leipz. 1814. Savart, Annales de
Chimie, xxx. 64, &c. Biot, Precis Elementaire de Physique, 1824. Fechner's
German Trans, of do. Chladni, Gilbert's Ann. Ixxvi. 187. Mayer, Meckel's
Archiv, 1826. Willis on the Mechanism of the Human Larynx, Tr. Camb. Ph. Soc.
iv. 323. Bennati, Recherches sur la Mechanisme de la Voix Humaine, Paris, 1832.
Sir C. Bell, Ph. Tr. 1832. Muncke in Gehler's Physik Worterbuch, viii. 373.
Rush, The Philosophy of the Human Voice, Philadelphia, 1833. Malgaigne, Archiv.
Gen. de Med. 25. Lauth, Mem. de 1'Acad. Royale de Me"d. 1835. Lehfeldt, Dis.
de Vocis Formatione, Berol. 1835. Bishop, Ph. Mag. 1836. Mayer, Outlines of
Physiology, 1837. Miiller's Handbuch der Physiol. ii. 179, English translation,
1838, p. 1002.

Voice of Birds. — Duvernay, Hist, et Mem. ii. 4. Herissant, ibid. 1753, p. 279,
H. 107. Parsons, Ph. Tr. 1766, p. 204. Barrington, ibid. 1773, p. 249. Dau-
benton, Hist, et Mem. 1781, p. 369, H. 12. Cuvier, Bulletin de la Societe Philo-
mat. No. 15. Le9ons d'Anatomie Comparee, torn. iv. lee. 28. Latham, Trans, of
the Linnaean Soc. Savart, Annales de Chimie, xxx. 64, and Froriep's Not. 331.

History. — Dodart on Ancient and Modern Music, Hist, et Mem. 1706, p. 388.
Pepusch on the Genera and Species of Music among the Ancients, Ph. Tr. 1746,
p. 266. Styles on do. ibid. 1760, p. 695. Hawkins's History of Music, 5 vols. 4to,
1776. Burney's Hist, of Music, 4 vols. 4to, 1789. Forkel, Allgemeine Litteratur
der Musik, Leipz. 1792. Jones, Asiatic Researches, iii. 55. Busby's Hist, of
Music, 2 vols. 1819.

* Observations sur les Flutes, Mem. de Berlin, 1775.
f The work is Traite d'Acoustique, Paris, 1809.

ON MUSICAL INSTRUMENTS.

319

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LECTURE XXXV.

ON THE THEORY OF OPTICS.

THE science of optics is one of the most elegant, and the most important
branches of natural and mechanical philosophy. It presents us with
experiments attractive by their beauty and variety, with investigations
affording an ample scope for mathematical refinements, and with instru-
ments of extensive utility both in the pursuit of other sciences, and in
the common employments of life ; nor is there any department of the
study of nature in which an unprejudiced observer is more convincingly
impressed with the characteristic marks of the perfect works of a supremely
intelligent Artist.

We shall first consider the essential properties which we discover in
light, and which are the basis of our calculations, together with the con-
clusions immediately deducible from those- properties ; and next, the ap-
plication of these laws to practical purposes, in the construction of optical
instruments. We shall afterwards proceed to examine the more compli-
cated phenomena, which are derived from the same laws, and which are
observed as well in natural as in artificial circumstances, constituting the
subdivision of physical optics. The description of the eye, and the ex-
planation of the sense of vision, by means of which all these effects are
connected with the human mind, is properly a continuation of the subject
of physical optics : the intimate nature of light will be the next subject
of investigation, and a historical sketch of the progress of the science of
optics will conclude the second part of this course of lectures.

In order to avoid all hypothesis in the beginning, it will be necessary to
define light from its sensible qualities. The sensation of light is sometimes
produced by external pressure on the eye ; we must exclude this sensation
from the definition of light, and must therefore call light an influence
capable of entering the eye, and of affecting it with a sense of vision. A
body, from which this influence appears to originate, is called a luminous
body. We do not include in this definition of the term light the invisible
influences which occasion heat only, or blacken the salts of silver, although
they both appear to differ from light in no other respects than as one kind
of light differs from another ; and they might probably have served the
purpose of light, if our organs had been differently constituted.

A ray of light is considered as an infinitely narrow portion of a stream
of light, and a pencil as a small detached stream, composed of a collection
of such rays accompanying each other. As a mathematical line is some-
times conceived to be described by the motion of a mathematical point, so
a ray of light may be imagined to be described by the motion of a point
of light. We cannot exhibit to the senses a single mathematical lino,
except as the boundary of two surfaces ; in the same manner, we cannot

ON THE THEORY OF OPTICS. 321

?-vliibit a single- ray of light, except as the confine between light and dark-
nesspor as the lateral limit of a pencil of light.

When light passes through a space free from all material substances, it
moves, with great velocity, in a direction perfectly rectilinear ; when also
it passes through a material substance perfectly uniform in its structure,
it probably always moves in a similar manner. But in many cases its
motions are much interrupted. Those substances through which light
passes the most freely, and in straight lines, are called homogeneous trans-
parent mediums. Perhaps no medium is, strictly speaking, absolutely
transparent ; for even in the air, a considerable portion of light is inter-
cepted.* It has been estimated that of the horizontal sunbeams, passing
through about 200 miles of air, one two thousandth part only reaches us ;
and that no sensible light can penetrate more than 700 feet deep into the
sea ; a length of seven feet of water having been found to intercept one
half of the light which enters it.

It is possible that mediums, not in other respects identical, may be
homogeneous with respect to the transmission of light ; for example, a
glass may be filled with a fluid of such a density, that the light may pass
uninterruptedly through their common surface ; but it generally happens,
that whenever the nature of the medium is changed, the path of light
deviates from a straight line ; thus, the apparent places of the sun and
stars are changed by the effect of the atmosphere, because the light, by
which we judge of their situations, is deflected, in its passage out of the
empty space beyond the atmosphere, first into the rarer and then into the
denser air. In the same manner, when we view a distant object over a
fire or a chimney, it appears to dance and quiver, because the rays of light,
by which it is seen, are perpetually thrown into new situations, by the
different changes of the density of the air in consequence of the action
of heat.

When rays of light arrive at a surface which is the boundary of two
mediums not homogeneous, they continue their progress without deviating
from those planes in which their former paths lay, and which are perpen-
dicular to the surface of the mediums ; but they no longer retain the same
direction, a part of them, and sometimes nearly the whole, is reflected back
from the surface, while the remaining part is transmitted and refracted,
or bent. The name refraction is derived from the distortion which it
occasions in the appearance of an object viewed in part only by refracted
light : thus an oar, partially immersed in water, appears to be bent, on
account of the refraction of the light by which its lower part is seen, in its
passage out of the water into the air.

There is no instance of an abrupt change of the density of a medium,
without a partial reflection of the light, passing either into the denser or
into the rarer medium ; and the more obliquely the light falls on the
surface, the greater, in general, is the reflected portion. No body is so

* Consult Bouguer, Traite d'Optique sur la Gradation de la Lumiere, 4to, Par.
17£0. Herschel's Description of an Actinometer, Ed. Jour, of Sci. iii. 107.
Pouillet, Mem. sur la Chaleur Solaire, Comptes Rendus, July 9, 1838. Forbes on
the Extinction of the Solar Rays in passing through the Atmosphere, Ph. Tr. 1842,
p. 225.

y

322 LECTURE XXXV.

black as to reflect no light at all, and to be perfectly invisible in a strou^,
light ; although at the surface separating two very rare bodies, ffs two
kinds of gas, the reflection is too faint to be perceptible ; but in this case
the separation is seldom perfectly abrupt.

The angles of incidence and reflection are the angles made by a ray of
light, before and after its reflection, with a line perpendicular to the re-
flecting surface ; and these angles are always equal to each other ; conse-
quently the inclination of the rays to the surface remains also the same.
The quantity of light reflected, when other circumstances are equal,
appears to be always greatest when the difference of the optical or refrac-
tive density of the two substances is greatest. Thus the reflection from
the common surface of glass and water is much weaker than from a surface
of glass exposed to the air. Metals in general reflect a great proportion
of the light falling on them, and even the reflection from the common
surface of glass and mercury appears to be but little weaker than the
reflection from the surface of mercury immediately exposed to the air, so
that the optical density of the metals must be exceedingly great.

It appears also that a portion of the light falling on a reflecting surface
is always transmitted, at least to a certain depth, notwithstanding the
apparent opacity of any large masses of the substance. Thus, if we cover
a small hole of a window shutter with the thinnest leaf gold, we shall find
that it transmits a greenish light, which must have passed the reflecting
surface, but which, if the gold had been but one ten thousandth of an inch
in thickness, would have been wholly intercepted, and probably almost in
the same manner as by passing through 700 feet of water. In transparent
substances, however, the greater part of the light penetrates to all distances
with little interruption, and all rays of the same kind, thus transmitted by
the same surface, form with the perpendicular an angle of refraction which
is ultimately in a certain constant proportion to the angle of incidence ;
that is, for instance, one half, three fourths, or two thirds, according to the
nature of the surface. Thus, if the refractive properties of the substance
were such, that an incident ray, making an angle of one degree with the
perpendicular, would be so refracted as to make an angle of only half a
degree with the same line, another ray, incident at an angle of two degrees,
would be refracted, without sensible error, into an angle of one degree. But
when the angles are larger, they vary from this ratio, their sines only pre-
serving the proportion with accuracy : for example, if the angle of inci-
dence at the supposed surface were increased to 90°, the angle of refraction
would be 30° only, instead of 45°. Rays of the same kind are in general
distinguished by the same colour, although some rays which differ from
each other in refrangibility, have scarcely a discernible difference of colour ;
and it is possible, on the other hand, to find a surface at which the ratio of
the angles is the same for rays of all kinds. (Plate XXVI. Fig. 369, 370.)
In order to obtain the effects of regular reflection and transmission, we
must have perfectly smooth and polished substances ; for all rough bodies,
and sometimes even such as to the touch seem tolerably smooth, have their
surfaces divided into innumerable eminences and depressions, constituting,
in reality, as many separate surfaces, disposed in all imaginable directions,

, ON THE THEORY OF OPTICS. 323

co that from the equality of the angles of incidence and reflection with
resp'&At to each of these surfaces, the light must be scattered every way,
and no regularity can be observed in its direction. It is true that by con-
tinuing the mechanical operation of polishing, we only render these sur-
faces more minute and more numerous ; but when they are so much
reduced in magnitude as not to be elevated or depressed more than about
the millionth part of an inch, they appear to become, for some physical
reason, incapable of acting separately, and only to conspire in the general
effect.

In* all cases of refraction, as well as of reflection, if the ray of light
return directly backwards in the same line to the surface, it would proceed,
after a second refraction or reflection, in the direction precisely opposite
to that in which it first was incident, so that the same lines would mark its
path in both cases. Thus, if we stand before a looking glass, with one eye
shut, and cover its place on the glass with a finger, the same finger will
hide the other eye as soon as it is shut and the first is opened in its place ;
and a similar effect might be observed if the glass were under water, or
behind any other refracting substance. (Plate XXVI. Fig. 371.)

The medium, in which the rays of light are caused to approach nearest
to the line perpendicular to its surface, is said to have the greatest refrac-
tive density. In general there is a considerable analogy between this
refractive density and the specific gravity of the substance : thus water is
more refractive than air, and glass than water. But inflammable bodies
are usually more refractive than bodies of the same specific gravity, which
are not inflammable ; and it is well known that from the high refractive
power of the diamond, in proportion to its actual density, Sir Isaac New-
ton most ingeniously conjectured that it was combustible, as more modern
experiments have actually shown it to be. It is still more singular that
he also imagined, from the same analogy, that water consists of a combi-
nation of oily or inflammable particles, with others earthy or not inflam-
mable. In the order of refractive density, beginning from the lowest, or a
vacuum, we have airs and gases of different rarities, water, which is the
least refractive of all liquids, and which is still less refractive when frozen
into ice : alcohol, oils, glass, and lastly the diamond ; but probably some
metallic substances are much more refractive than even the diamond.

The refractive powers of different substances, are usually estimated by a
comparison of the refractions produced at their surfaces in contact with
the air, which, in all common experiments, has the same sensible effect as
a vacuum or an empty space ; the ratio of the angles of refraction and inci-
dence, when small, and that of their sines, in all cases, being expressed by
the ratio of 1 to a certain number, which is called the index of the refrac-
tive density of the medium. Thus, when a ray of light passes out of air
into water, the sines of the angles are in the ratio of 3 to 4, or of 1 to £,
which is, therefore, the index of the refractive density of water. In the
same manner, for crown glass, the ratio is that of 2 to 3, and the index
l^ ; but for flint glass it is somewhat greater, the ratio being nearly that
of 5 to 8.

It may easily be shown that a refractive substance, limited by parallel

Y2

324 LECTURE XXXV.

surfaces, must transmit a ray of light, after a second refraction at its pos-.
terior surface, in a direction parallel to that in which it first passed through
the air. It is also found by experiment that such a substance, interposed:
between any two mediums of different kinds, produces no alteration in the
whole angular deviation of a ray passing from one of them into the 'other.
Hence it may be inferred, that the index of refraction at the common sur-
face of any two mediums is the quotient of their respective indices. For
instance, a plate of crown glass being interposed between water on one side
and air on the other, it produces no change in the direction of a ray of light
entering the water ; and the index of refraction at the common surfe^e of
glass and water is -£. (Plate XXVI. Fig. 372, 373.)

There is one remarkable consequence of the general law by which the
angles of incidence and refraction are related, that when the angle of inci-
dence exceeds a certain magnitude, the refraction may become impossible ;
and in this case the ray of light is wholly reflected, in an angle equal to
the angle of incidence. Thus, if the law of refraction required the sine of
the angle of refraction to be twice as great as that of incidence, this con-
dition could not take place if the angle of incidence were greater than 30°,
so that when a ray passing within a dense medium falls very obliquely on
its surface, it must be wholly reflected ; and the greater the density of the
medium, the more frequently will the light be totally reflected. This re-
flection is more perfect than any other ; the diamond owes much of its
brilliancy to it : the great refractive density of this substance not only
giving a lustre to its anterior surface, but also facilitating the total reflec-
tion of such rays as fall obliquely on its posterior surface. If we hold a
prism near a window, in a proper position, we may observe that its lower
surface appears to be divided into two parts, the one much brighter than
the other ; the common partial reflection taking place in one, and the total
reflection in the other. The two surfaces are separated by a coloured arch :
it is coloured, because the total reflection commences at different angles for
the rays of different colours ; and it is curved, because the points, at which
the light passing to the eye forms a given angle with the surface, do not lie
in a straight line ; and if we throw a light on a wall by a reflection of this
kind, we may easily observe, as we turn the prism, the point at which the
brightness of the image is very conspicuously increased. (Plate XXVI.
Fig. 374.)

Such are the principal properties which we discover in light. Before we
consider their immediate application to optical instruments, we must ex-
amine the general theory of refraction and reflection at surfaces of dif-
ferent kinds, or the doctrines of dioptrics and catoptrics.

The rays,, which constitute a pencil of light, are sometimes parallel to
each other, sometimes divergent from a point, and sometimes convergent to
a point. The intersection of the directions of any two or more rays of light
is called their focus ; and the focus is either actual or virtual, accordingly
as they either meet in it, or only tend to or from it. Thus, a small luminous
object may represent an actual focus of diverging rays, since the light
spreads from it in all directions; and the small surface into which the
image of such an object, or of the sun, is collected by a lens or mirror,

ON THE THEORY OF OPTICS. 325

lua'j ^represent the actual focus of converging rays. It was to such an
image of the sun that the term focus, meaning a fire place, was first applied.
But if the rays tending to this focus be intercepted and made to diverge, the
point will then be their virtual focus, since they will never actually arrive
at it, being made to diverge as if they proceeded from a new point, which
will also be a virtual focus. When the divergence or convergence of rays
of light is altered by refraction or reflection at any surface, the foci of the
incident and refracted or reflected rays are called conjugate to each other :
the ^ew focus is also called the image of the former focus. Thus, in the
case already mentioned, where the convergence of the rays to one focus is
converted into divergence from another, the two virtual foci are conjugate
to each other ; and the original focus of the lens or mirror is conjugate to
the place of the sun, or of the luminous object. If the object had been put
in the place of its image, the image would then have occupied that of the
ohject ; a property which follows from the direct return of every ray of
light through the path by which it has arrived, and which may easily be
illustrated by experimental confirmation. (Plate XXVII. Fig. 375.)

Whenever light is reflected by a plane surface, the conjugate foci are at
equal distances from it, and in the same perpendicular. Thus, every point
of an image in a looking glass is perpendicularly opposite to the correspond-
ing point of the object, and is at the same distance behind the looking glass
as the point of the object is before it. (Plate XXVII. Fig. 376.)

The focus into which parallel rays are collected, or from which they are
made to diverge, is called the principal focus of a surface or substance.
The sun is so distant, that the rays proceeding from any point of his sur-
face, affect our senses as if they were perfectly parallel, and the principal
focal distance of a surface or substance may often be practically determined
by measuring the distance of the image of the sun or of any other remote
object, which is formed by it.

In order that the rays of light, proceeding from or towards any one
point, may be made to converge by reflection towards another, the form
of the surface must be elliptical, parabolic, or hyperbolic ; there are also
curves of still more intricate forms, which possess the same property with
respect to refraction. A small portion, however, of any of these curves,
differs very little from a circle ; and a spherical surface is almost universally
substituted in practice for all of them, except that the mirrors of large
reflecting telescopes are sometimes made parabolical.

The principal focus of a spherical reflecting surface, whether convex or
concave, is half way between the surface and its centre. If a luminous
point be placed in the centre of a concave mirror, the rays will all return
to the same point ; if the point be beyond the centre, the image will be
between the centre and the principal focus, its distance from that focus
being always inversely as that of the radiant point. Such a focus is never
absolutely perfect, for the rays are never collected from, the whole surface
of the mirror into the same point, except when both the point and its image
are in the centre : but, provided that the surface be only a small portion
of that of the whole sphere, the aberration will be too small to be easily

32G LECTURE XXXV.

observed : and the same is true of the foci produced by refracting sur£*ces.
(Plate XXVII. Fig. 377, 378.)

When a ray of light passes through two surfaces forming an angle with
each other, including a denser medium, as in the case of a prism of 'glass,
it is always deflected from the angle in which the two surfaces meet. A
greater number of surfaces, placed in different directions, constitute what
is sometimes called a multiplying glass, each of them bending the rays of
light into a different direction. (Plate XXVII. Fig. 379, 380.)

A lens is a detached portion of a transparent substance, of whic^ the
opposite sides are regular polished surfaces, of such forms as may be
described by lines revolving round a common axis. These lines may be
portions of circles, of ellipses, hyperbolas, or of any other curves, or they
may be right lines. But in general, one of the sides is a portion of a
spherical surface, and the other either a portion of a spherical surface or
a plane ; whence we have double convex, double concave, planoconvex,
planoconcave, and meniscus lenses. The figures of all these are suffi-
ciently described by their names, except that the term meniscus, which
properly implies a little moon or crescent, is applied in general to all lenses
which are convex on the one side and concave on the other, although they
may be thicker at the edges than in the middle. Sometimes, however, a
lens of this kind is distinguished by the term concavoconvex. A lens is
generally supposed, in simple calculations, to be infinitely thin, and to be
denser than the surrounding medium. (Plate XXVII. Fig. 381.)

The general effect of a lens may be understood, from conceiving its sur-
face to coincide at any given point with that of a prism ; for if the angle
of the prism be external, as it must be when the lens is convex, the rays
will be inflected towards the axis ; but if the base of the prism be external,
and the lens concave, the rays will be deflected from the axis : so that a
convex lens either causes all rays to converge, or lessens their divergence,
and a concave lens either causes them to diverge, or lessens their conver-
gence. (Plate XXVII. Fig. 382.)

The principal focus of a double convex or double concave lens, of crown
glass, is at the distance of the common radius of its surfaces ; and the focal
length of a planoconvex lens is equal to the diameter of the convex surface.
If the radii of the surfaces are unequal, their effect will be the same as if
they were each equal to the harmonic mean between them, which is found
by dividing the product by half the sum ; or, in the meniscus, by half the
difference. Thus, if one of the radii were two inches, and the other six, the
effect would be the same as that of a lens of three inches radius ; and if it
were a meniscus, the same as that of a lens of six inches. (Plate XXVII.
Fig. 383, 384.)

The focal length of a lens of flint glass, of water, or of any other sub-
stance, may be found, by dividing that of an equal lens of crown glass by
twice the excess of the index of refraction above unity. Thus, the index
for water being 1£, we must divide the radius by f, or increase it one half,
for the principal focal distance of a double convex or double concave lens
of water.

I

ON THE THEORY OF OPTICS. 327

\yiien a radiant point is at twice the distance of the principal focus
' from fc convex lens, the image is at an equal distance on the other side ;
when the radiant point is nearer than this, the image is more remote, the
distance of the image from the principal focus nearest to it being always
inversely as the distance of the object from the principal focus on the
opposite side. (Plate XXVII. Fig. 385.)

The joint focus of two lenses, in contact with each other, is also found by
multiplying together their separate focal lengths, and dividing the product
by their sum or difference, accordingly as they agree or differ with respect
to convexity and concavity.

We have hitherto considered the place of the focus only in relation to a
single point, placed in the axis of the lens or mirror ; but it is equally
necessary to attend to other points, out of the principal axis ; for in order
to form a picture, the rays from a great number of such points must be
collected into as many distinct points of the image. Some of the rays
proceeding from every radiant point must be considerably bent, in order to
be collected into a common focus ; others remain nearly straight ; and if
we can discover which of the rays are ultimately either in the same line
with their original direction, or in a direction parallel to it, we may
determine the line in which the image of the point in question is to be
found. For this purpose we employ the property of the optical centre,
which is a point so situated, that all rays which pass through it, or tend
towards it, while they are within the lens, must ultimately acquire a direc-
tion parallel to their original direction. In some cases, the optical centre
may be without the lens, but no practical inconvenience results from
supposing it to be always situated within the lens, especially when its
thickness is inconsiderable ; so that all rays which pass through the middle
point of the lens must proceed, without sensible error, in the same straight
line, and the image of any radiant point must consequently be found some-
where in this line : but in the case of a mirror, the centre of its figure is
also the optical centre. Now when any radiant point is removed a little
from the axis of a lens or mirror, the distance of its image is in general a
little diminished, but the difference is too small to be observable in common
cases. We may, therefore, suppose it to be at the same distance as if the
point remained in the axis, or even to be in a plane crossing the axis
perpendicularly at that distance, so as to form part of a flat image, of
which the magnitude is determined by straight lines drawn from the ex-
tremities of the object through the centre of the lens. This is, however, an
approximation which is only admitted for the greater convenience of com-
putation and representation, the image being almost always in reality
considerably curved. (Plate XXVII. Fig. 386.)

LECT. XXXV.— ADDITIONAL AUTHORITIES.

, Optics in general. — Euclidis Optica, 4to, Paris, 1557. Faulhaber, Descriptio
Inst. Geom. et Opt. 4to, Frankf. 1610. Kepler, Dioptrice, 4to, Augsb. 1611.
Aquilo, Op. Autw, 1613. Schneider de Luce, 1616. Mathise Buchholdii Lucis
Contemp. 4 to, 1630. Descartes, Dioptrique, 1637. Bwllialdus de Natura Lucis,

328 LECTURE XXXVI.

fcl. Par. 1638. Zucchius, 2 vols. 4to, Lugd. 1652-6. Thomasius, 4to, l/)53.
Lichtner, 4to, 1653, and 4to, 1654. Balthasar, 4to, 1656. Mancini, Bolog.*1660.
De la Chambre, La Lumiere, Paris, 1662. Vossius de Nat. Lucis, 4to, Amst. 1662.
Kohlhausen, Lips. 1663. Grandorgeeus, 4to, Cad. 1664. Fabri, Synopsis Optica,
Lyon, 1667. Saggi del Acad. di Cimento, 1667. Cherubin, Dioptrique, fol. Paris,
1671. Kirchmaier de Luce, Misc. Acad. Nat. Cur. 1677, App. p. 219. ' Moly-
neux, Dioptrica Nova, 4to, Lond. 1694. Hartsoeker, Essai de Diop. 4to, Par.
1694. Gregorii (D.) Catop. et Diop. Oxf. 1695. Huygens, Op. Post. Lugd. 1704.
Craig< Optica Analytica, 1708. Rizetti de Luminis Affectionibus, Ven. 1727.
Smith's Optics, 4to, Camb. 1738. (German by Kastner, Altenb. 1775.) Euler,
Dioptrica, 3 vols. 4to, Petrop. Martin's Optics, 1750. Courtivron, Traite, Paris,
1752. LacaiUe, 1756. Deincken, Alt. 1757. Bischoff, Ulm, 1760 & 1772. Al-
garotti (trans.}, The Philosophy of Sir I. Newton explained in Six Dial, on Light,
12mo, Glasg. 1765. Thomin's Traite d' Op. Paris, 1769. Harris's Optics, 4to,
1775. Scherfer, 4 vols. 4to, Vindob. 1775. Biirja, Berlin, 1793. Karstens,
Lehr. derMathem. Theil viii. Rampinellius, Optique, Brix. 1760. Emerson's Op-
tics, 1768. Ferguson's, 1770. Stack's, Dub. 1811. Settele, Elem. di Ottica, 2 vols.
Rom. 1818-19. Nobili, Milan, 1820. Maseres, Scriptores Optici, 4to, Lond.
1823. Bourgeois, Manuel d'Optique Experimental, 2 vols. 12mo. 1823. Brew-
ster's Optics, Edin. Encyc., Cab. Cyc., and Encyc. Brit. Herschel's, Encyc. Me-
tropolit. (complete and good), Transl. in French by Verhulst, with Supplement by
Quetelet, 3 vols. 1826. Amondieu, Lehr. der Optik, Leipz. 1827. Rottger, Halle,
1828. Prechtl. Wien, 1829. Higgins, Lond. 1829. Coddington on Reflection
and Refraction, Camb. 1829. Littrow, Dioptrik, Wien, 1830. Wood's Optics, Camb.
v. y. Lloyd on Light, 1831. Powell's Optics, Oxf. 1833. Schmidt, Gott. 1834.
Johnson's Optical Investigations, Oxf. 1835. Phelps's Optics, Camb. 1835. Grif-
fin's, Camb. 1840. Bartlett's, New York, 1841.

LECTURE XXXVI.

ON OPTICAL INSTRUMENTS.

AMONG the great variety of instruments depending on optical principles,
it is most consistent with our plan to attend first to those which may be
denominated optical measures, which are calculated either for the determi-
nation of the quantity or intensity of light itself, or for the examination of
the properties of various material substances with respect to light. Reflect-
ing quadrants and circles, which are often used in astronomical and
nautical observations, although they derive their utility in some measure
from optical laws, may most properly be considered as belonging to the
subject of practical astronomy.

It is a problem of frequent occurrence in economical investigations, to
compare the intensity of the light afforded by any two luminous objects.
For this purpose it is necessary to assume as a principle, that the same
quantity of light, diverging in all directions from a luminous body, remains
undiminished at all distances from the centre of divergence. Thus, we
must suppose that the quantity of light falling on every body is the same
as would have fallen on the place occupied by its shadow : and if there
were any doubt of the truth of the supposition, it might be confirmed by
some simple experiments. It follows that since the shadow of a square

ON OPTICAL INSTRUMENTS. 329

inch of any surface, occupies, at twice the distance of the surface from the
luminous point, the space of four square inches, the intensity of the light
diminishes as the square of the distance increases. We can judge with
tolerable accuracy of the equality of two lights by the estimation of the
eye, but we cannot form any idea of the proportions of lights of different
intensities : if, however, we remove two sources of light to such distances
from an object, that they may illuminate it in equal degrees, we may con-
clude that their original intensities are inversely as the squares of their
distances. Count Rumford's * photometer performs this very conveniently,
by casting two shadows of a given object near each other, on the same
surface, the lights being removed to such distances that the shadows
appear equally dark. (Plate XXVII. Fig. 387, 388.)

For determining the refractive density of solids, it has been usual to form
them into a prism, and to measure the angular deviations which they
produce ; and for fluids, to inclose them either in a hollow prism, or between
two meniscus lenses, and to measure the angular deviations produced by
the prisms, and the focal distances of the lenses. But in most cases, Drv
Wollaston's apparatus t is far preferable to both these methods: it is
arranged for ascertaining the angle at which light, moving within a certain
dense transparent substance, begins to be totally reflected from the common
surface of that substance and the solid or fluid which is to be examined. Thus,
if we first measure the angle at which light begins to be totally reflected
from the posterior surface of a prism of glass, in contact with air, we may
readily determine its refractive power ; and then, having caused a drop of
a fluid to adhere to that surface, or fixed a solid to it by a small portion of
some fluid denser than itself, we may observe, as we turn the prism round
its axis, at what angle the drop or spot begins to disappear, and may thence
calculate the refractive density of the substance ; and even without actual
measurement of the angle, we may readily compare the disappearance of
the drop or spot with that of others placed near it, of which the properties
are known. Dr. Wollaston has, however, rendered the process still easier
and more simple, by employing a rectangular prism of glass, with sights
fixed to a jointed frame, of such a construction as to enable him to read off,
by a vernier, without any calculation, the index of the refractive power of
any substance less dense than glass. (Plate XXVII. Fig. 389.)

All instruments strictly optical are employed for forming an image of an
external object : the simplest are mirrors and lenses, which form a single
image only, either actual or virtual, and sometimes depict it on a surface
calculated for receiving and exhibiting it. Other instruments repeat the
image once or more under several forms, in general enlarging it con-
tinually ; and these are either microscopes or telescopes, which present us
with great diversity in their arrangements, and in the appurtenances sub-
servient to their uses.

It is a general rule, that when an image of an actual object is formed by

any lens or speculum, if the rays converge to an actual focus, the image is

inverted ; but erect, if they diverge from a virtual focus, and the object

and image subtend equal angles at the centre of the lens or speculum.

* Ph. Tr. 1794, Ixxxiv. 67, f Ibid. 1802, p. 365.

330 LECTURE XXXVI.

Hence, a convex lens and a concave mirror form an inverted image,
smaller than the object, whenever the object is at a greater distance than,
twice the principal focal length ; but larger, when the object is within this
distance ; and when it is within the principal focal distance, the magnified
image is virtual and erect, and may be seen by looking into the concave
mirror, or by looking through the lens towards the object. But a concave
lens and a convex mirror always form a virtual image of a real object,
which is erect and smaller than the object. (Plate XXVII. Fig. 390...
394.)

When the object is precisely in the principal focus of a convex lens'br a
concave mirror, the virtual image becomes infinitely distant ; so that from
whatever point in the neighbourhood of the lens it may be viewed, it must
subtend the same angle, which is always equal to that which the object
subtends at the centre of the lens : and since this angle may easily be
much greater than that under which the object can be conveniently viewed
by the naked eye, such a lens or mirror is often used as a simple micro-
scope ; and its magnifying power may be ascertained from a comparison
of the angles which the object and image subtend. Thus, if a person
cannot see a minute object with the naked eye at a distance less than eight
inches, a lens of half an inch focal length will represent it to him in an
angle 16 times as great : but if he can see it without the lens at the dis-
tance of four inches, the lens will magnify it to his eye but eight times.
Supposing, however, the eye to be applied close to the lens, the object may
be viewed a little within the focal distance, and its apparent angular mag-
nitude may be increased 17 times instead of 16, and 9 times instead of 8.
(Plate XXVII. Fig. 395, 396.)

Since the magnifying power of a lens is the greater, the smaller its
focus, it is usual to employ the minutest lenses that can be ground, and
sometimes a small globule is formed by fusion in a lamp. Even a drop of
water, placed in the perforation of a plate, makes a tolerable magnifier ;
and it has been proposed to substitute for water a transparent varnish,
which is less liable to evaporate.

Supposing the whole light that proceeds from a distant object, and falls
on a lens or speculum, to be collected in the image, its intensity must be
increased in the ratio of the surface of the lens or speculum to that of the
image. The image is greater in proportion as the object is greater ; conse-
quently the degree of condensation produced by any lens is the greater as
the object is smaller, thus if the diameter of a lens were an inch, and the
image of the sun formed by it were also an inch in diameter, the density of
the light would be unaltered ; but the image of a star would be infinitely
brighter than the direct light of the star falling on the lens. The illumina-
tion of any image formed by a lens or mirror, supposing no light to be
lost, is always the same as would be produced by the direct light of the
surface of the lens or mirror, if it were equally luminous with the surface
of the object which emits the light. It may also be shown, that when two
lenses are of similar forms, their focal lengths being proportional to their
diameters, they must produce the same degree of illumination in the image :
but as far as the heat excited may be supposed to be a measure of the

ON OPTICAL INSTRUMENTS. 331

quantity of light, this conclusion is not confirmed by experiment : it is
probable, however, that the greater heat, produced by a larger lens, is only
derived from the greater extent of surface exposed at once to the solar
rays.

Lenses are most commonly made of glass, but sometimes of rock crystal,
or of other transparent substances. It is difficult to find glass, especially
flint glass, for large lenses, sufficiently free from veins : it has been pro-
posed to suffer the melted glass to cool without agitation, and to cut the
lens out of any of its strata taken in a horizontal direction ; but this method
appears to be liable to several practical objections. Mirrors are made either
of glass, coated with an amalgam of mercury and tin, or of metal, as of
platina, of silver, or of an alloy of copper and tin, to which a little arsenic
and silver are sometimes added. Mirrors of metal are more perfect than
those of glass, because they are free from the inconvenience of a double re-
flection ; but they are more expensive, and are liable to tarnish. Where a
large mirror is required, with a wreak reflection only, we may employ a
single surface of glass, the back of the piece being covered with a black
coating of some substance differing little from glass in its refractive density,
by means of -which the second reflection is avoided. Dr. Wollaston has
very ingeniously applied the effect of the reflection of two plane surfaces,
inclined to each other, to the construction of an instrument for drawing,
which he calls a camera lucida.* He usually employs the internal re-
flection of a prism of glass, of which the four surfaces are ground so as to
form proper angles with each other. The image formed by the first sur-
face is inverted, and the second reflection restores it to its original posi-
tion, but places it in a direction which is at right angles with the direction
of the object ; so that when we look down through the prism on a sheet of
paper, we see a perfect picture of the objects immediately before us, while
at the same time, the aperture through which we look, is only partly occu-
pied by the edge of the prism, the remaining part being left open, or simply
covered with a lens, for the admission of the direct rays of light by which
we may see, at the same time, the paper and the pencil to be employed for
making a drawing or a copy of any object placed before us.

When the image formed by a lens or mirror is received on a smooth but
unpolished surface, which is capable of irregular reflection, it is visible in
every direction. Such an image is exhibited in the camera obscura, the
solar microscope, and the magic lantern, or lucernal microscope.

The general effect of the camera obscura f is the same as may often be
observed in a dark room, where there is a small hole in the window shutter :
the great masses of light and shade, before the windows, being represented
in an inverted position, in the parts of the room diametrically opposite to
them, which are illuminated in different degrees, according to the quantity

* Nich. Jour. 8vo, xvii. 1. Compare Wren, Ph. Tr. 1669, iv. 898. Peacock,
ibid. Ixxv. 366. Ramsden and Jones, Phil. Jour, xxviii. Brewster's Account of
New Phil. Insts. An account of the modifications which Prof. Amici has effected in
\his instrument is given in the Edin. Jour, of Sci. v. 157. Chevalier, Notice sur
1'Usage des Chambres Obscures et des Chambres Claires, Par. 1829. Ludicke,
Gilb. Ann. xlii. 338.

f Invented by Baptista Porta, Magia Naturalis, p. 12, Lug. Bat. 1650.

332 LECTURE XXXVI.

of light which can reach them in straight lines from the external objects.
A lens, of a focal length somewhat smaller than the distance of the surface
on which the picture is projected, renders the images much more distinct ;
but some of them are unavoidably imperfect and ill denned, unless the ob-
jects happen to be situated at the same distance from the aperture ; for the
focus of the lens can never be adjusted at once to nearer and more remote
objects ; nor would the picture be rendered more natural by such an
adjustment, for it would present to the eye at one view, with equal distinct-
ness, objects which never can be seen at once without some degree of con-
fusion. Sometimes the picture is intercepted, by a speculum placed ob-
liquely, and is thrown upwards on the surface of a plate of ground glass,
upon which its outline may be traced with a black lead pencil, and an im-
pression may be taken from it on moist paper, which will represent the
natural situation of the objects without inversion. Another arrangement
is, to place the lens horizontally, with a speculum above it, which throws
the image through the lens, upon a flat surface placed below, on which the
objects may be delineated in their natural position, but not without some
impediment from the interception of the light by the hand and the instru-
ment employed. Such a surface, however, ought not to be perfectly flat,
in order to afford the most distinct image, although by means of a meniscus
lens, with a cover admitting the light only through a small aperture near
its centre, on the principle of Dr. Wollaston's periscopic spectacles,* an
image nearly flat might be obtained ; but in this case too much of the light
would be excluded. It has been usual to consider the image of a very dis-
tant object, formed by a convex lens, as a portion of a spherical surface of
which every part is equally distant from the centre of the lens ; but this
estimate is extremely erroneous, for the effect of the obliquity of the
different pencils of rays materially increases the curvature of the image.
In fact no pencil of rays, falling obliquely on a spherical surface, can be
collected any where to a perfect focus : the image of a circle would become
most distinct at one distance, and that of its diameter at another ; but for
both these images, the surface ought to be much more curved than that
which has been usually considered, and the mean of the curvatures re-
quired for them, which must be the best form for the ground or bottom of
a camera obscura, is equal to that of a sphere of which the radius is three
eighths of the focal distance, when a double convex lens of crown glass is
employed. (Plate XXVIII. Fig. 397. . .399.)

In the solar microscope, an image is formed on a wall or screen, by
means of a lens of small focal length, near to which the object is placed, so
that the image is very much magnified. For this purpose the room must
be darkened, and the object strongly illuminated by the sun's light, which
is condensed by means of a large lens, and sometimes by two or more lenses
placed at a distance from each other ; but care must be taken to avoid
burning the object by bringing it exactly into the focus ; and, on the other
hand, if it be much beyond the focus, the light will be thrown upon a small
part of the image only ; the best arrangement appears to be, to bring the*
focus of the condensing lenses very near to the small lens ; and in order to
* Nicli. Jour. vii. 143, 241.

ON OPTICAL INSTRUMENTS. 333

adjust the instrument in the most convenient manner, the distances of all
the lenses ought to he moveahle at pleasure : the want of this precaution is
a material defect in the usual construction of the instrument. The specu-
lum which first receives the light must be capable of motion in all angular
directions, in order to allow us to accommodate its position to the change-
able place of the sun ; and the adjustment has sometimes been performed
by means of a heliostate, an instrument calculated for turning the speculum
by clockwork, into such a position as always to reflect the sun's light in
the required direction. An easier method would be to employ two specu-
lum^ the one moveable round an axis parallel to that of the earth, and re-
flecting the sun's light into the direction of its axis, the other fixed, and
changing this direction into any other that might be required. When an
opaque object is to be examined, the light may be thrown on it either by a
plane mirror placed obliquely, or by a perforated concave mirror ; and if
the object is small, the concave mirror appears to be the more eligible.
(Plate XXVIII. Fig. 400.)

By night, a lamp with a large lens before it, may supply the place of
the sun's light, and the instrument will become a lucernal microscope,
which, when painted glass sliders are employed as objects for the amuse-
ment of children, is called a magic lantern : and this, exhibited on a
larger scale, and projecting an image on a semitransparent screen of taffetas,
instead of a wall, has of late been the source of much entertainment under
the name of the phantasmagoria, a term which implies the raising of
spectres. In order to favour the deception, the sliders are made perfectly
opaque, except where the figures are introduced, the glass being covered,
in the light parts, with a more or less transparent tint, according to the
effect required. Several pieces of glass may also be occasionally placed
behind each other, and may be made capable of such motions as will nearly
imitate the natural motions of the objects which they represent. The
figures may also be drawn with water colours on thin paper, and after-
wards varnished. By removing the lantern to different distances, and
altering at the same time more or less the position of the lens, the image
may be made to increase or diminish, and to become more or less distinct
at pleasure, so that to a person unaccustomed to the effects of optical in-
struments, the figures may appear actually to advance and retire. In
reality, however, these figures become much brighter as they are rendered
smaller, while in nature the imperfect transparency of the air causes them
to appear fainter when they are remote than when they are near : this
imperfection might be easily remedied by the interposition of some semi-
opaque substance, which might gradually be caused to admit more light as
the figure became larger, or by uncovering a larger or a smaller portion of
the lamp, or of its lens. Sometimes, by throwing a strong light upon an
actual opaque object, or on a living person, its image is formed on the
curtain, retaining its natural motions : but in this case the object must be
considerably distant, otherwise the images of its nearer and remoter parts
will never be sufficiently distinct at once, the refraction being either too
great for the remoter, or too small for the nearer parts : and there must
also be a second lens, placed at a sufficient distance from the first to allow

834 LECTURE XXXVI.

an inverted image to be formed between them, and to throw a second
picture of this image on the screen, in its natural erect position, unless the
object be of such a nature that it can be inverted without inconvenience.
This effect was very well exhibited at Paris by Robertson ; he also com-
bined with his pictures the shadows of living objects, which imitate toler-
ably well the appearance of such objects in a dark night, or by moonshine :
and while the room was in complete darkness, concealed screens were pro-
bably let down in various parts of it, on which some of the images were
projected ; for they were sometimes actually situated over the heads of the
audience. (Plate XXVIII. Fig. 401.) r-

In almost all telescopes and compound microscopes, the image formed
by one lens or mirror stands in the place of a new object for another.
The operation of such instruments may be illustrated by placing a screen
of fine gauze at the place of the image, which receives enough light to
make the image visible in all directions, and yet transmits enough to form
the subsequent image. The simplest of such instruments is the astrono-
mical telescope. Here the object glass first forms an actual inverted image,
nearly in the principal focus of the eye glass, through which this image is
viewed as by a simple microscope, and therefore still remains apparently
inverted. In order to find the angular magnifying power, we must divide
the focal length of the object glass by that of the eye glass : this quotient
is consequently the greater as the focal length of the object glass is greater,
and as that of the eye glass is smaller : but the power of the instrument
cannot be increased at pleasure by lessening the focal length of the eye
glass, because the object glass would not furnish light enough to render the
view distinct, if the magnifying power were too great. (Plate XXVIII.
Fig. 402.)

The double or compound microscope resembles in its construction the
astronomical telescope, except that the distance of the lenses much exceeds
their joint focal length; and the angular magnitude is greater than when
the same object is viewed through the eye glass alone, in proportion as the
first image is further from the object glass than the object itself. (Plate
XXVIII. Fig. 403.)

In the Galilean telescope or opera glass, a concave eye glass is placed so
near the object glass that the first image would be formed beyond it, and
near its principal focus ; and the second image, formed by the eye glass,
which is the virtual image viewed by the eye, being on the opposite side of
the centre, is inverted with respect to the first image, and erect with respect
to the object. In this case also the magnifying power is indicated by the
quotient of the numbers expressing the focal lengths of the glasses. (Plate
XXVIII. Fig. 404.)

The inverted image of the astronomical telescope may be made erect by
means of an additional eye glass. In the common day telescope of Rheita,
two such eye glasses are employed, of nearly equal focus, which have the
advantage of procuring a greater extent in the field of view ; they are
usually so placed as to have little or no effect on the magnifying power.
(Plate XXVIII. Fig. 405.)

Dr. Herschel's reflecting telescopes resemble, in their effects, the simple

ON OPTICAL INSTRUMENTS. 335

astronomical telescope ; a concave speculum or mirror being substituted
for the object glass, and the eye glass being so placed as to magnify the
image formed by the speculum. But since the speculum, if it received the
principal rays perpendicularly, would send them back in the same direc-
tion, it is necessary, in this construction, to have them reflected somewhat
obliquely, the speculum being a little inclined to the axis of the telescope,
in order that the light may have free access to it. An arrangement of this
kind was proposed long ago by Maire,* but it has been very little employed
before Dr. Herschel's time. This excellent philosopher and mechanic has
carried the perfection of his telescopes to a degree far exceeding all that
could have been expected from the labours of former opticians. His in-
struments allow him to extend the linear dimensions of his objects several
thousand times : but he commonly finds it more eligible to employ only
powers of 5 or 600, which afford a much stronger illumination. (Plate
XXVIII. Fig. 406.)

The Newtonian reflector has a plane speculum placed in its axis, at the
inclination of half a right angle, which intercepts the rays about to form
the image, and throws them into the focus of an eye glass fixed in the side
of the tube. The plane speculum which he employed was the posterior
surface of a rectangular prism of glass, which produces a total reflection :
but Dr. Herschel has found that the sources of error are diminished by
wholly omitting this speculum. (Plate XXVIII. Fig. 407.)

In the Gregorian telescope, the object speculum is perforated, and the
image formed by it is received into the focus of a smaller concave speculum,
which returns it to be viewed through the aperture by the eye glasses. It
has been objected to this form of the reflecting telescope, which is the first
that was invented, that the best part of the speculum is sacrificed by the
perforation. But Dr. Herschel has found that the image formed by the
external part of a speculum is in general more perfect than that which is
formed by the central part. (Plate XXVIII. Fig. 408.)

For the smaller concave speculum of Gregory, Mr. Cassegraint substi-
tuted a convex one, placing it within the focal distance of the large specu-
lum, so as to form the first actual image nearly in the same place as the
second image of the Gregorian telescope ; but this image is inverted. The
instrument has some advantage in theory, with respect to the perfection of
the focus ; but it is little used. (Plate XXVIII. Fig. 409.)

Dr. Smith's reflecting microscope resembles Cassegrain's telescope, but
the rays of light are first admitted through a perforation in the small
speculum, that part of them which tends to fall immediately on the eye
being intercepted by a screen. The convexity of the one mirror is nearly
equal to the concavity of the other ; and the instrument, although seldom
employed, is said to succeed extremely well. (Plate XXVIII. Fig. 410.)

The image of a very distant object, formed by a speculum of any kind,
is considerably less curved than that which is depicted by a lens of equal

* Mach. Approuv. vi. 61. Herschel on his Forty-foot Telescope, Ph. Tr. 1795,
p. 347. See also Herschel, ibid. 1782, p. 173; 1786, p. 499; 1800, p. 49; 1803,
p. 214.

t Journal des Savans, 1672. See Newton, Ph. Tr. 1672, p. 4056.

336 LECTURE XXXVI.

focal length. There is a similar imperfection in the nature of the focus of
oblique pencils, but it is confined within narrower limits, the remotest part
of the image in which any radiating lines would be most distinctly repre-
sented, being a flat surface, and the nearest, in which circles would become
most distinct, being a part of a sphere touching the speculum : so that the
radius of the mean curvature is equal to the focal distance. (Plate
XXVIII. Fig. 411.)

The magnifying power of a refracting telescope may often be measured
by comparing the diameter of the object glass with that of the narrowest
space into which the beam of light is contracted beyond the eye glass, pro-
vided that none of the light has been intercepted in its passage through the
telescope : for the object will be viewed through the telescope in an angle
as much greater than that which it naturally subtends, as the diameter of
the object glass is greater than that of this contracted pencil, which may be
considered as an image of the object glass. But in the Galilean telescope
this method cannot be employed, since no such image is formed. The field
of view in a simple telescope, or the angular magnitude of that part of an
object which can be seen through it at once, is nearly equal to the magni-
tude of the eye glass as seen from the object glass.

If a lens be added to any refracting telescope at the place of the first
image, it will have no effect either on the place or on the magnitude of any
subsequent image, but it will enlarge the field of view, by throwing more
pencils of light on the original eye glass. If, however, the image fell
exactly on such a lens, it would be liable to be impaired by any accidental
impurities of its substance or on its surface, every opaque particle inter-
cepting the whole of the light belonging to one of its points, which would
not happen if the image were at a small distance from the lens. A field
glass is, therefore, usually placed, both in telescopes, and in the common
compound microscope, a little nearer to the object glass than the place of
the first image. The best places for the various lenses, in an eye piece,
are partly determined from similar considerations, but they require also in
general to be adjusted by experiment, for several circumstances are con-
cerned in the performance of a telescope, which are almost too intricate for
practical calculation, although some assistance may certainly be obtained
from theory with regard to the most important of them. The curvature of
the image produced by any lens has already been mentioned : it may be
in some measure remedied by Mr. Ramsden's method of placing a plano-
convex lens a little beyond the image, with its flat side turned towards it.
Mr. Ramsden* also employs an -eye piece constructed on this principle in-
stead of a simple microscope, under the name of a double magnifier. The
aberration of the different parts of any single pencil of rays, from the cor-
responding point of the image, requires also to be considered in the con-
struction of telescopes : its magnitude is such, in the case of a double convex
lens of crown glass, that those parts of a pencil of parallel rays which fall
on it near the circumference meet each other in a point, which is within
the true focus, by a distance a little more than half as great again as the
thickness of the lens. In an image formed by a concave speculum of equal
* Ph. Tr. 1783, Ixxiii. 94.

ON OPTICAL INSTRUMENTS. 337

focal length, this aberration would be only -^ as great ; it may, however,
1 be almost entirely corrected, in refracting telescopes, by employing proper
proportions in the dimensions of the various lenses. (Plate XXVIII. Fig.
412, 413.)

A still more important aberration, from which reflecting telescopes are
also wholly free, is that which arises from the different refrangibilities of
the rays of light of different colours, which form an infinite number of
images, neither agreeing perfectly in situation nor in magnitude, so that
the objects are rendered indistinct by an appearance of colours at their
edgesrf this imperfection, however, Mr. Dollond has in great measure
obviated, by his achromatic object glasses : * the construction of which de-
pends on the important discovery, that some kinds of glass separate the
rays of different colours from each other much more than others, while the
whole deviation produced in the pencil of light is the same. Mr. Dollond
combined, therefore, a concave lens of flint glass with a convex lens of
crown glass, and sometimes with two such lenses ; the concave lens of flint
glass being sufficiently powerful to correct the whole dispersion of coloured
light produced by the crown glass, but not enough to destroy the effect of
its refraction, which was still sufficient to collect the rays of light into a
distant focus. For this purpose, it is necessary that the focal lengths of the
two lenses should be in the same proportion as the dispersive powers of the
respective substances, when the mean deviations of the pencils are equal ;
that is, in the case of the kinds of glass commonly used, nearly in the ratio
of 7 to 10. Sometimes also the chromatic aberration, that is, the error
arising from the different refrangibilities of the different rays, is partially
corrected in an eye piece, by placing a field glass in such a manner as con-
siderably to contract the dimensions of the image formed by the least
refrangible rays, which is nearest to the eye glass, and to cause it to subtend
an equal angle with the image formed by the most refrangible rays,
this image being little affected by the glass. (Plate XXVIII. Fig. 414,
415.)

The apparent magnitude of an object viewed through a telescope, may
be measured, with great accuracy, by a scale or by wires, introduced at the
place of the last image, reducing afterwards the angle thus ascertained
according to the magnifying power. Care must, however, be taken to
avoid as much as possible the distortion which usually accompanies any
curvature of the image ; and the wires, one of which is sometimes made
moveable by means of a micrometer screw, must be sufficiently illuminated
to be distinctly visible. Sometimes a scale is introduced, which, from the
apparent magnitude of a known object, such as that of a man of ordinary
height, or of a portion of a wall built with bricks of the usual size, enables
us at once to read off its actual distance, which is expressed on the scale in
hundreds of yards. The angular magnitude of an object seen through a
telescope may also be found, by viewing at the same time, with the other
eye, either a scale, or any other object of known dimensions, placed at a
given distance : the lucid disc micrometer of Dr. Herschel t is employed

* Ph. Tr. 1758, 1. 733, and 1765, Iv. 54.
f Ibid. 1782, Ixxii. 163; 1785, p. 46.

338 LECTURE XXXVI.

in this manner for judging of the magnitude of the celestial bodies. The
divided object glass micrometer affords another mode of measurement.:*
the object glass being divided into two semicircular portions, one of which
slides on the other ; each portion acts as a separate lens, and two images
of every part of the object being formed, the angular distance of any two
points is determined by bringing their images together, and measuring the
displacement of the moveable portion of the object glass which is required
for procuring the coincidence. Sometimes also a similar purpose is
answered by inserting a divided glass in the eye piece, which acts nearly
on the same principle, and which seems to be somewhat less liable to flrror.
In a reflecting telescope of Cassegrain's construction, Mr. Ramsden t has
also produced the same effect by dividing the convex speculum, and
causing a part of it to turn round an axis. All these arrangements parti-
cularly deserve the attention of those who are employed in practical astro-
nomy and in geography, since the advancement of these sciences much
depends on the accuracy of the telescopic and microscopic measures, which
are performed by means of optical instruments. (Plate XXVIII. Fig.
416, 417.)

LECT. XXXVI.— ADDITIONAL AUTHORITIES.

Photometry. — Marie, Nouvelle Decouverte en Lumiere, 1701. Mairan, Hist,
et Mem. 1721. Celsius, Nouvelle Idee sur la Mesure de la Lumiere, ibid. 1735,
H. 5. Euler, Hist, et Mem. de Berlin, 1750, p. 280. Karsten, Photometric,
Greifswald, 1777. Fontana, Mem. della Soc. Ital. i. 111. Fossombroni, Sull'
Intensita del Lume, fol. Arezzo, 1781. Langsdorf, Grundlehren der Photometric,
2 vols. Erlangen, 1803-5. Leslie's Photometer, Nich. Jour. iii. 461. Ritchie's,
Ph. Mag. v. 139. Potter's, ibid, new series, iii. 284. Xavierde Maistre's, Bibliot.
Univ. 1832, p. 323. Osann's, Pogg. Ann. xxxiii. 405. Steinheil's, ibid, xxxiv. 644.

Measurement of Refractive Powers. — J. A. Porta, De Refractione, 4to, Neap.
1583. Lahire on the Refraction of Ice, Hist, et Mem. ix. 328, x. 172 ; of Oil and
Water, ix. 382, 577. Lowthorp on the Refraction of Air, Ph. Tr. 1699, p. 339.
Cassini on do. Hist, et Mem. 1700, p. 78, H. 112. Hauksbee on the Refraction of
Fluids, Ph. Tr. 1710, p. 204. Euler on do. Hist, et Mem. de Berlin, 1756, p.
235; of Glass, 1766, p. 202. J. A. Euler, ibid. 1762, pp. 279, 302, 318, 328.
Cadet and Brisson, Hist, et Mem. 1777, p. 541. Biot and Arago, Memoire sur
les Affinites des Corps pour la Lumiere, Mem. de 1'Institut, 1806, ii. 301. Frau-
enhofer, Bestimmung des Brechungs und Farbenszerstreuungs-vermbgens Ver-
schiedener Glasarten Schum. Astron. Abhandl. 1815. Hartmann, in Schum. Astr.
Nachr. vii. 265. Malus, in Gilb. Ann. xxxi. 225. Marx, in Schweigger's Jour. v.
385, Ixi. 46. Arago, Annales de Chimie, vol. i. &c. Dulong, Mem. sur les Pouvoirs
Refringens des Fluides Elastiques, Mem. de 1' Acad. 1825.

Construction of Mirrors.— Mudge on the Best Composition of Metals, Ph. Tr.
1777, p. 296. Potter on Improvements in Casting and Working, Ph. Mag. 1831,
iv. 13, vi. 228. Lord Oxmantown, ibid. ix. 213.

Theory of Lenses. — Kastner, Com. Gott. i. 185, ii. 183. Lagrange surlaTheorie
des Lunettes, Hist, et Mem. de Berlin, 1778, p. 162. Bohnenberger, Zeitschrift fur
Astron. i. 277, 385. Von Munchow, ibid. ii. 448. Gauss, ibid. iv. 345. Mobius,
inCrelle's Jour. v. 113 ; Schleiermacher, in Poggendorf's, xiv. 1. Schulten, Supp.
a la Theorie des Verres Simples, Vedensk Aph. 1821, p. 265. Herschel, Ph. Tr.
1821, p. 222. Hamilton on a System of Rays, Tr. Roy. Ir. Ac. 1824, &c. Barlow,
Ph. Tr. 1827, p. 231. Santini, Teoria degli Stromenti Ottici, Padua, 1828. Lub-
bock, Ph. Mag. vii. 161.

Reflecting Goniometer.— Wollaston, Ph. Tr. 1809, p. 253. Malus, Mem. d'Ar-

* Savery and Dollond, Ph. Tr. xlviii. 165, 178, 551.

t Ramsden, Description of two new Micrometers, Ph. Tr. 1779, Ixix. 419.

ON OPTICAL INSTRUMENTS. 339

cueil, iii. 122. Studer, Gilb. Ann. Ixvi. 8. Von Reise, Vorschlage zu einem Neuen
Goniometer, Bonn, 1829.
'Kaleidoscope. — Art. in Ed. Encyc. by the inventor, Brewster.

Sextant. — Hooke, Animadversions on Hevelius, 4to, 1674, Birch, ii. 394. Had-
ley's Inst. Ph. Tr. 1731, p. 147; 1732, p. 32. Dollond's Alterations on do. ibid.
1772, p. 99. Atwood's Theory of do. ibid. 1781, p. 395. Encke on do. Astron.
Jahrbuch, 1830, p. 285. Adie on Metallic Reflectors for do. Proceedings of Roy.
Soc. Ed. 1845.

Microscope. Treatises. — Hooke's Micrographia, fol. 1665. Grindelius, Micros.
Nov. Norimb. 1687. Joblot, Description des Plusieurs Micros. Paris, 1708.
Observations Micros. 1754. Wideburg, De Micros. Solari, Erlang. 1755. Due de
Chaulnes, Descrip. d'un Micros, fol. Paris, 1768. Brander, Beschreibung zweier
Mic.,Augsb. 1769. Baker on the Mic. 1769. Martin's Optical Essays, 1770.
Descrip. of a Graphical Perspective and Micrometer, 1771. De la Barre, Mem. sur
les Mic. 1777. Gleichen, Vom Sonnenmicros. 4to, Nurimb. 1781. Tiedemann,
Stuttgart, 1785. Adam's Essays on the Mic. 4to, 1798. Villars, Mem. sur le
Mic. Paris, 1806. Amici, Mem. di Micros. Modena, 1818. Goring and Pritchard's
Micros. Illustrations, 1830. In Journals.— Fabri's, Ph. Tr. 1668, p. 842. Leu-
wenhoek's, ibid. 1673, p. 6037. Huygens's, Hist, et Mem. x. 427. Wilson's, Ph.
Tr. 1702, p. 1241. Adams's, ibid. 1710, p. 24. Baker's, ibid. 1736, p. 442.
Lieberkuhn's Solar do. ibid. 1740, p. 516, and Hist, et Mem. de Berlin, 1745, p. 14.
Euler's, Nov. Com. Petr. iii. 363. Aepinus's, ibid. ix. 316. Zeiher's, ibid. x. 299.
Selva's, Hist, et Mem. de Paris, 1769, H. 129. Brewster's, Ph. Mag. iii. 74, viii. 316,
and New Philosophical Insts. Rossi's, Baumgartners Zeitsch. v. 95. Ehrenberg's,
Pogg. Ann. xxiv. 188. Wollaston's Mic. Doublet, Ph. Tr. 1829, p. 9. Codding-
ton's, Tr. Camb. Ph. Soc. iii. 421. Lister on the Mic., Ph. Mag. 1831, v. 169.
Chevalier and Goring, ibid. p. 224, &c.

Telescope. — Mersenne, Universse Geometrise Synopsis, 4to, Paris, 1644. Hooke,
Auzout, and Campani, Ph. Tr. 1665-6, i. Huygens, ibid. 1684, p. 668. Hist, et
Mem. x. 351, &c. Hadley's Telescope (has a rectangular prism instead of the plane
mirror of Newton's), Ph. Tr. 1723, p. 382. Euler, Hist, et Mem. de Berlin, 1747
— 1767 (various memoirs). Kratzenstein and Euler on the Iconantidiptic Teles.
Acta Petr. iii. I. 192, 201. Hertel, Anweisung Teles, zu Verfertigen, Halle, 1747.
Clairaut on the Improvement of Teles. Hist, et Mem. 1756, p. 380, H. 112 ; 1757,
p. 524, H. 153 ; 1762, p. 578, H. 160. Scherfer on Dioptrical Tel. by Hardy,
1768. Rochon's Achromatic Tel. Hist, et Mem. 1773, p. 299 ; Reflecting do. Ph.
Mag. ii. 19, 170. Lagrange on the Theory of do. Mis. Taur. iii. II. 152 ; Hist.
et Mem. de Berlin, 1778, p. 162. Navarre's Tel. Hist, et Mem. 1778, H.56.
Fuss on Tel. 4to, Leipz. 1778. Oriani, Mem. de la Soc. Ital. iii. 664. Biirja, Hist,
et Mem. de Berlin, 1797, ii. 8, 1798, p. 3. Blair (Fluid Lenses) Tr. Roy. Soc. Ed.
iii. 3. Repertory of Arts, vii. 15. Kater, Comparison of Cassegrain's with Gre-
gory's, Ph. Tr. 1813, p. 206. Kitchiner, Practical Observations on Telescopes. &c.
1818. Guilio, Mem. di Torino, xvi. 128. Brewster's, Phil. Mag. vii. 323. Lord
Oxmantown, ibid. ix. 25, new series, ii. 136.

Micrometers. — Huygens, Systema Saturnium, Hag. Com. 1659. Auzout and
Hooke, Ph. Tr. 1665-6, i. 123. Hooke on Gascoigne's Screw Mic. ibid. ii. 541.
Lefevre's Mic., Mach. Appr. ii. 103. Kirckius's, Mis. Berl. i. 202. Cassini's, Hist,
et Mem. 1724, p. 347. Fouchy's, Mach. Appr. vi. 45. Aepinus's, Hist, et Mem.
de Berlin, 1756, p. 365. Wilcke's, Schwed. Abhand. 1772, p. 56. Boscovich's,
Ph. Tr. 1777, p. 789. Kohler's, Bode's Jahrbuch, 1785, p. 155. Smeaton's, Ph. Tr.
1787, p. 318. Rochon's (Rock Crystal), Nov. Act. Petr. 1788, H. 37. Jour, de
Phy. liii. 169. Cavallo's, Ph. Tr. 1791, p. 283. Wollaston's, ibid. 1813, p. 119 ;
also 1820, p. 126. Dollond's, ibid. 1821, p. 101. Brewster's, Ph. Mag. i. 104; iv.
164. Treatise on New Philosophical Instruments, ed. 1813, pp. 48, 173. Amici's,
Mem. della Soc. Ital. xvii. II. 344. Frauenhofer's, Schumacher's Astr. Nach. ii. 361,
364. Hansen on do. Gotha, 1827. Bessel on do. Schumacher, viii. 397. Stein-
heil's, ibid. v. 359.

z2

840

LECTURE XXXVIL

ON PHYSICAL OPTICS.

HAVING examined the general theory of optics, and the construction of
optical instruments, we are now to consider those properties and affections
of light, which rather belong to its natural history than to its mechanical
effects ; to trace its relations to the particular phenomena of nature ; to
investigate the manner in which it is connected with our sensations, and
to inquire on what intimate mode of action the various effects of light
depend. All these subjects may be properly comprehended under the
denomination of physical optics, but we shall find it convenient to reserve
each of the two last for a separate examination. The sources of light, the
velocity of its motion, its interception and extinction, its dispersion into
different colours ; the manner in which it is affected by the variable den-
sity of the atmosphere, the meteorological appearances in which it is
concerned, and the singular properties of particular substances with regard
to it, will be the first subjects of our investigation.

The sources from which light is commonly derived, are either the sun or
stars, or such terrestrial bodies as are undergoing those changes which
con^itute combustion. The process of combustion implies a change in
which a considerable emission of light and heat is produced ; but it is not
capable of a very correct definition : in general it requires an absorption,
or at least a transfer, of a portion of oxygen : but there appear to be some
exceptions to the universality of this distinction ; and it has been observed
that both heat and light are often produced where no transfer of oxygen
takes place, and sometimes by the effect of a mixture which cannot be
called combustion.

Light is also afforded, without any sensible heat, by a number of vege-
table and animal substances, which appear to be undergoing a slow decom-
position not wholly unlike combustion. Thus decayed wood, and animal
substances slightly salted, often afford spontaneously a faint light, without
any elevation of temperature ; and it is not improbable that the light of the
ignis fatuus may proceed from a vapour of a similar nature.

The effects, which are commonly attributed to the motions of the elec-
trical fluid, are often attended by the production of light ; and violent or
rapid friction frequently seems to be the immediate cause of its appearance.
But it is difficult to ascertain whether friction may not be partly concerned
in the luminous phenomena attributed to electricity, or electricity in the
apparent effects of friction. Light is sometimes produced by friction with a
much lower degree of heat than is required for combustion, and even when
it is accompanied by combustion, the heat produced by the union of these
causes may be very moderate : thus it is usual in some coal mines, ,to
obtain a train of light by the continual collision of flint and steel, effected
by the machine called a fire wheel, in order to avoid setting fire to the

ON PHYSICAL OPTICS. 341

inflammable gas emitted by the coal, which would be made to explode if
it came near the flame of a candle.

There is a remarkable property, which some substances possess in an
eminent degree, and of which few, except metals and water, are entirely
destitute.* These substances are denominated solar phosphori; besides
the light which they reflect and refract, they appear to retain a certain
portion, and to emit it again by degrees till it is exhausted, or till its emis-
sion is interrupted by cold. The Bolognan phosphorus was one of the first
of these substances that attracted notice ; it is a sulfate of barytes, found
in the state of a stone ; it is prepared by exposure to heat, and is after-
wards made up into cakes : these, when first placed in a beam of the sun's
light, and viewed afterwards in a dark room, have nearly the appearance
of a burning coal, or a red hot iron. Burnt oyster shells, t and muriate
of lime have also the same property, and some specimens of the diamond
possess it in a considerable degree. From the different results of experi-
ments apparently accurate, made by different persons, there is reason to
conclude that some of these phosphori emit only the same kind of light as
they have received, while others exhibit the same appearances, to whatever
kind of light they may have been exposed. Sometimes it has even been
found that light of a particular colour has been most efficacious in exciting
in a diamond the appearance of another kind of light, which it was natu-
rally most disposed to exhibit. The application of heat to solar phosphori
in general expedites the extrication of the light which they have borrowed,
and hastens its exhaustion ; it also produces, in many substances, which
are not remarkable for their power of imbibing light, a temporary scintil-
lation or flashing, at a heat much below ignition : the most remarkable
of these are fluor spar in powder, and some other crystallized substances.
It appears that luminous bodies in general emit light equally in every
direction, not from each point of any of their surfaces, as some have
supposed, but from the whole surface taken together, so that the surface,
when viewed obliquely, appears neither more nor less bright than when
viewed directly.^

However light of any kind may have at first originated, there is reason
to believe that the velocity with which it passes through a given medium
is always the same. It has been ascertained by the astronomical observa-
tions of Roemer and of Bradley, that each ray of light, emitted by the
sun, arrives at the earth in eight minutes and one eighth, when the earth
is at its mean distance of about 95 millions of miles. Roemer § deduced
this velocity from observations on the eclipses of the satellites of Jupiter,
and Bradley 1 1 confirmed it by his discovery of the cause of the apparent
aberration of the fixed stars.

* See Cellio, La Pietra Bolognese Preparata, Rom. 1680. Beccari de Phospho-
ris, 4to, Bolog. 1744. T. Wedgwood, Ph. Tr. 1792, p. 28.

f See Bartholinus de Luce Animalium, 1669. Boyle on the Light of Fish, &c.
Ph. Tr. ii. 581, 605, 1672 ; vii. 5107. Works, iii. 304. Canton, Ph. Tr. 1768,
p. 337 ; 1769, p. 446. Hulme, ibid. 1800, p. 161 ; 1801, pp. 403, 426.
• t Hauksbee, on the Production of Light from Phosphorus in vacua, Ph. Tr.
xxiv. p. 1865.

§ Hist, et Mem. x. 399, Ph. Tr. 1677, xii. 893.

|| Ph. Tr. 1728, xxxv. 637.

342 LECTURE XXXVII.

This aberration is produced by the effect of the revolution of the earth
in its orbit, combined with that of the progressive motion of light. Since
light proceeds always in right lines, when its motion is perfectly undis-
turbed, if a fine tube were placed so as to receive a ray of light, passing
exactly through its axis when at rest, and then, remaining in the same
direction, were moved transversely with great velocity, it is evident that
the side of the tube would strike against the ray of light in its passage,
and that in order to retain it in the axis, the tube must be inclined, in the
same manner as if the light, instead of coming in its actual direction, had
also a transverse motion in a contrary direction to that of the tube. The
axis of a telescope, or even of the eye, may be considered as resembling
such a tube, the passage of the light through the refracting substances not
altering the necessary inclination of the axis. In various parts of the
earth's orbit, the aberration of any one star must be different in quan-
tity and in direction ; it never exceeds 20 seconds each way, and must
therefore, in common observations, be wholly insensible. (Plate XXIX.
Fig. 418.)

The quantity of light, which is reflected by a substance of any kind,
depends not only on the nature of the substance, but also on the obliquity
of its incidence : and it sometimes happens, that a surface, which reflects
a smaller portion of direct light than another, reflects a greater portion
when the light falls very obliquely on its surface. Bouguer found that
the surface of water reflected only one fifty fifth part of the light fall-
ing perpendicularly on it, that of glass one fortieth, and that of quick-
silver more than two thirds : but when the obliquity was as great as possi-
ble, the water reflected nearly three fourths of 'the incident light, and the
glass about two thirds only.

Of the light which passes by a dense substance of any kind, the greatest
part pursues its course undisturbed, but there is always a certain divergence,
which has been called by Grimaldi diffraction, and by Newton inflection.
This effect is usually attended by the production of colours, and will
therefore require to be more particularly considered hereafter.

The separation of colours by refraction is one of the most striking of all
optical phenomena. It was discovered by Newton* that white light is a
compound of rays of different kinds, mixed in a certain proportion, that
these rays differ in colour and in refrangibility, that they constitute a
series, which proceeds by gradual changes from red to violet, and that
those substances which appear coloured when placed in white light, derive
their colours only from the property of reflecting some kind of rays most
abundantly, and of transmitting or extinguishing the rest. Dr. Herschelt
has added to this series rays of heat less refrangible than the red, and
Hitter J and Dr. Wollaston § have discovered, beyond the violet, other still
more refrangible rays, which blacken the salts of silver.

* Ph. Tr. 167J, vi. 3075 ; vii. 4059, 4087, 5004, 5012, 5084 ; viii. 6086, 6108,
&c. &c. Opuscula, ii. 181.

f On Heat and Light, Ph. Tr. 1800.

J Gilbert's Ann. vii. 527 ; xii. 409.

§ Ph. Tr. 1802, p. 365. See also Scheele on Air and Fire (trans.}, Loud. 1780,
§ 66.

ON PHYSICAL OPTICS. 343

, It has generally been supposed, since the time of Newton, that when the
rays of light are separated as completely as possible by means of refrac-
tion, they exhibit seven varieties of colour, related to each other with
respect to the extent that they occupy, in ratios nearly analogous to those
of the ascending scale of the minor mode in music. The observations
were, however, imperfect, and the analogy was wholly imaginary. Dr.
Wollaston * has determined the division of the coloured image or spectrum,
in a much more accurate manner than had been done before : by looking
through a prism, at a narrow line of light, he produces a more effectual
separation of the colours, than can be obtained by the common method of
throwing the sun's image on a wall. The spectrum formed in this manner
consists of fo.ur colours only, red, green, blue, and violet, which occupy
spaces in the proportion of 16, 23, 36, and 25, respectively, making toge-
ther 100 for the whole length ; the red being nearly one sixth, the green
and the violet each about one fourth, and the blue more than one third of
the length. The colours differ scarcely at all in quality within their
respective limits, but they vary in brightness ; the greatest intensity of
light being in that part of the green which is nearest to the red. A narrow
,line of yellow is generally visible at the limit of the red and green, but its
I breadth scarcely exceeds that of the aperture by which the light is ad-
, mitted, and Dr. Wollaston attributes it to the mixture of the red with the
green light. There are also several dark lines t crossing the spectrum
within the blue portion and its neighbourhood, in which the continuity of
the light seems to be interrupted. This distribution of the spectrum Dr.
Wollaston has found to be the same, whatever refracting substance may
have been employed for its formation ; and he attributes the difference
which has sometimes been observed in the proportions, to accidental varia-
tions of the obliquity of the rays. The angular extent of the spectrum
formed by a prism of crown glass is one 27th of the deviation of the red
rays ; by a prism of flint glass, one 19th. (Plate XXIX. Fig. 419.)

In light produced by the combustion of terrestrial substances, the spec-
trum is sometimes still more interrupted ; thus, the bluish light of the

* Ph. Tr. 1802, p. 365.

f This fact did not excite the attention which it merited at the time of its dis-
covery. Several years afterwards, M. Fraunhofer, of Munich, by viewing the
spectrum formed from a narrow line of solar light, when in its purest state, at
the angle of minimum deviation, discovered that it was crossed by a very great
number of dark lines, not separating different colours, but mixed up with them,
without any order. In solar light they are nearly 600 in number, and with
the same kind of light always retain the same places, but are very different for dif-
ferent kinds of light ; and even that of the sun, after it has been transmitted through
nitrous acid gas, exhibits very different lines from what it did previously. By far the
readiest mode of viewing such lines, is to cause sun-light to pass through a bottle of
this gas before it falls on the prism. Since these lines always retain their places in
the spectrum, they afford the most accurate method of determining the refractive
and dispersive powers of bodies, to which purpose Fraunhofer himself applied them.
See Fraunhofer, Bestimmung des Brechungs und F-arbenszerstreuungs-vermogens
verschiedener Glasarten. Miincher Akad. Abhand. 1821,' xxii. Brewster's Obser-
yations on the Lines produced by the Earth's Atmosphere and by the Action of
Nitrous Acid Gas, Tr. Roy. Soc. Ed. xii. 519. Ed. Jour, of Sci. No. XV. 7. Mil-
ler, ibid. ii. 381. Rudberg, Pogg. Ann. xxxv. 523. Wheatstone (Electrical Light),
Ph. Mag. vii. 299.

344 LECTURE XXXVII.

lower part of a flame of a candle is separated by refraction into five parcels
cf various colours; the light of burning spirits, which appears perfectly
blue, is chiefly composed of green and violet rays ; and the light of a
candle into which salt is thrown abounds with a pure yellow, inclining to
green, but not separable by refraction. The electrical spark furnishes also
a light which is differently divided in different circumstances. (Plate
XXIX. Fig. 420.)

If the breadth of the aperture viewed through a prism is somewhat
increased, the space occupied by each variety of light in the spectrum is
augmented in the same proportion, and each portion encroaches on the
neighbouring colours, and is mixed with them : so that the red is suc-
ceeded by orange, yellow, and yellowish green, and the blue, is mixed on
the one side with the green, and on the other with the violet ; and it is in
this state that the prismatic spectrum is commonly exhibited. (Plate
XXIX. Fig. 421.)

When the beam of light is so much enlarged as to exceed the angular
magnitude of the spectrum, it retains its whiteness in the centre, and is ter-
minated by two different series of colours at the different ends. These series
are still divided by well marked lines : on the one hand the red remains
unmixed ; the space belonging to the green and blue becomes a greenish
yellow, nearly uniform throughout, and here the appearance of colour ends,
the place of the violet being scarcely distinguishable from the neighbouring
white light : on the other hand, the space belonging to the red, green, and
blue of the simple spectrum, appears of a bluish green, becoming more and
more blue till it meets the violet, which retains its place without alteration.
This second series is also the same that accompanies the limit of total
reflection at the posterior surface of a prism. (Plate XXIX. Fig. 422.)

Sir Isaac Newton observed that the effect of white light on the sense of
sight might be imitated by a mixture of colours taken from different parts
of the spectrum, notwithstanding the omission of some of the rays naturally
belonging to white light. Thus, if we intercept one half of each of the four
principal portions into which the spectrum is divided, the remaining halves
will still preserve, when mixed together, the appearance of whiteness ; so
that it is probable, that the different parts of those portions of the spectrum,
which appear of one colour, have precisely the same effect on the eye. It
is certain that the perfect sensations of yellow and of blue are produced
respectively, by mixtures of red and green and of green and violet light, and
there is reason to suspect that those sensations are always compounded of
( the separate sensations combined ; at least, this supposition simplifies the
theory of colours : it may, therefore, be adopted with advantage, until it be
found inconsistent with any of the phenomena ; and we may consider white
light as composed of a mixture of red, green, and violet only, in the pro-
portion of about two parts red, four green, and one violet, with respect to
the quantity or intensity of the sensations produced.*

* So WUnsch, Versuche iiber die Farben, Leipz. 1792. Mayer, in an essay De
Affinitate Colorum, pub. 1722, refers all colours to red, yellow, and blue : and thk
is the more common hypothesis. See Guyot, Recreations, Par. 1769. Goethe, Far-
benlehre, 1810. Brewster, Tr. Roy. Soc. Ed. xii. 123. Nollet, Lesons de Phy-
sique, v. 388, considers the three colours to be orange, green, and indigo.

. ON PHYSICAL OPTICS, 345

If we mix together, in proper proportions, any substances exhibiting
tliese colours in their greatest purity, and place the mixture in a light
sufficiently strong, we obtain the appearance of perfect whiteness ; but in
a fainter light the mixture is grey, or of that hue which arises from a com-
bination of white and black ; black bodies being such as reflect white light
but in a very scanty proportion. For the same reason, green and red sub-
stances mixed together usually make rather a brown than a yellow colour,
and many yellow colours, when laid on very thickly, or mixed with black,
become brown. The sensations of various kinds of light may also be com-
bined in a still more satisfactory manner, by painting the surface of a circle
with different colours, in any way that may be desired, and causing it to
revolve witi^such rapidity, that the whole may assume the appearance of a
single tint, or of a combination of tints, resulting from the mixture of the
colours. (Plate XXIX. Fig. 423... 426.)

From three simple sensations, with their combinations, we obtain seven
primitive distinctions of colours ; but the different proportions in which
they may be combined, afford a variety of tints beyond all calculation.
The three simple sensations being red, green, and violet, the three binary
combinations are yellow, consisting of red and green ; crimson, of red and
violet ; and blue, of green and violet ; and the seventh in order is white
light, composed by all the three united. But the blue thus produced, by
combining the whole of the green and violet rays, is not the blue of the
spectrum, for four parts of green and one of violet make a blue, differing
very little from green ; while the blue of the spectrum appears to contain
as much violet as green : and it is for this reason that red and blue usually
make a purple, deriving its hue from the predominance of the violet.

It would be possible to exhibit at once to the eye the combinations of any
three colours in all imaginable varieties. Two of them might be laid down
on a revolving surface, in the form of triangles, placed in opposite direc-
tions, and the third on projections perpendicular to the surface, which,
while the eye remained at rest in any one point, obliquely situated, would
exhibit more or less of their painted sides, as they passed through their
different angular positions ; and the only further alteration, that could be
produced in any of the tints, would be derived from the different degrees of
light only. The same effect may also be exhibited by mixing the colours
in different proportions, by means of the pencil, beginning from three
equidistant points as the centres of the respective colours. (Plate XXIX.
Fig. 427.)

The ordinary atmospherical refraction cannot be determined in the
usual manner from the knowledge of its density, and of the angular direc-
tion of the incident or refracted light, since the constitution of the atmo-
sphere is such, that its density varies every where with its height, and the
curvature of the earth's surface causes the inclination of the strata through
which the ray passes to be perpetually changed ; the difference of tempera-
ture at different elevations increases also the difficulty of an exact calcula-
tion, and it is only very lately that Mr. Laplace,** by a comparison of
I astronomical with meteorological observations, has given a satisfactory

* Mec. Cel. iv. 268.

346 LECTURE XXXVII.

solution of the problem in all its extent. But for practical uses, the refrac-
tion may be determined with sufficient accuracy by an approximation
which is easily remembered ; the deviation being at all altitudes one sixth
part as great as the refracted ray would undergo at the horizontal surface
of a medium six times as dense as the air. When a celestial object appears
exactly in the horizon, it is actually more than half a degree below it,
since the refraction amounts to 33 minutes, when the barometer stands at
29-^V inches, and Fahrenheit's thermometer at 50°.

The accidental variations of the temperature of the air at different parts,
produce, however, great irregularities in its refraction, especially near the
horizon. The most remarkable of these is occasioned by the rarefaction of
the air in the neighbourhood of the surface of water, of a budding, or of
the earth itself, in consequence of which a distant object appears to be
depressed instead of being elevated, and is sometimes seen at once both
depressed and elevated, so as to appear double, one of the images being
generally in an inverted position, as if the surface possessed a reflective
power; and there seems indeed to be a considerable analogy between this
kind of refraction and the total reflection which happens within a denser
medium. These effects are known by the appellations looming, mirage,
• and Fata Morgana ; they may be very completely imitated, as Dr. Wollas-
ton as shown,* by looking at a distant object along a red hot poker, or
through a saline or saccharine solution with water and spirit of wine
floating on it. The effect of refraction on the apparent places of terrestrial
objects must be frequently disturbed by circumstances of this kind ; but
its magnitude is usually about one tenth of the angular distance of the
object, considered as a part of the earth's circumference. (Plate XXIX.
Fig. 428, 429.)

The atmospherical phenomena of rainbows and halos present us with
examples of the spontaneous separation of colours by refraction. The
rainbow is universally attributed to the refraction and reflection of the
sun's rays in the minute drops of falling rain or dew, and the halos,
usually appearing in frosty atmospheres, are in all probability produced by
the refraction of small triangular or hexagonal crystals of snow. It is
only necessary, for the formation of a rainbow, that the sun should shine on a
dense cloud or a shower of rain, in a proper situation, or even on a number
of minute drops of water, scattered by a brush or by a syringe, so that the
light may reach the eye after having undergone a certain angular deviation,
by means of various refractions and reflections ; and the drops so situated
must necessarily be found somewhere in a conical surface, of which the
eye is the vertex, and must present the appearance of an arch. The light,
which is reflected by the external surface of a sphere, is scattered almost
equally in all directions, setting aside the difference arising from the
greater efficacy of oblique reflection ; but when it first enters the drop, and
is there reflected by its posterior surface, its deviation never exceeds a
certain angle, which depends on the degree of refrangibility, and is, there-
fore, different for light of different colours ; and the density of the light being
the greatest at the angle of greatest deviation, the appearance of a lumi-
* Ph. Tr. 1800, p. 239. See also ibid. 1803, p. 1.

ON PHYSICAL OPTICS. 347

nous arch is produced by the rays of each colour at its appropriate dis-
tance. The rays which never enter the drops produce no other effect,
than to cause a brightness, or haziness round the sun, where the reflection is
the most oblique ; those which are once reflected within the drop exhibit
the common internal or primary rainbow, at the distance of about 41
degrees from the point opposite to the sun ; those which are twice reflected,
the external or secondary rainbow, of 52° ; and if the effect of the light,
three times reflected, were sufficiently powerful, it would appear at the
distance of about 42 degrees from the sun. The colours of both rainbows
encroach considerably on each other ; for each point of the sun may be
considered as affording a distinct arch of each colour, and the whole disc
as produciflg^an arch about half a degree in breadth for each kind of light ;
so that the arrangement nearly resembles that of the common mixed spec-
trum. There is, however, another cause of a further mixture of the
colours ; the arch of any single colour, which belongs to any point of the
sun, is accurately defined on one side only, while on the other it becomes
gradually fainter, the breadth of the first minute containing about five times
as much light as a minute at the distance of a quarter of a degree ; the
abrupt termination is on the side of the red, that is, without the inner bow,
and within the outer, so that, for this reason, the order of colours partakes,
in some degree, of the nature of the red termination of a broad beam of light
seen through a prism ; but it is more or less affected by this cause, on
account of some circumstances, which will be explained when we examine
the supernumerary rainbows, which sometimes accompany the bows more
commonly observed. A lunar rainbow is much more rarely seen than a
solar one, but its colours differ little, except in intensity, from those of the
common rainbow. (Plate XXIX. Fig. 430.)

In the highest northern latitudes, where the air is commonly loaded with
frozen particles, the sun and moon usually appear surrounded by halos or
coloured circles, at the distances of about 22 and 46 degrees from their
centres; this appearance is also frequently observed in other climates,
especially in the colder months, and in the light clouds which float in the
highest regions of the air. The halos are usually attended by a horizontal
white circle, with brighter spots, or parhelia, near their intersections with
this circle, and with portions of inverted arches of various curvatures ; the
horizontal circle has also sometimes anthelia, or bright spots nearly opposite
to the sun. These phenomena have usually been attributed to the effect
of spherical particles of hail, each having a central opaque portion of a
certain magnitude, mixed with oblong particles, of a determinate form,
and floating with a certain constant obliquity to the horizon. But all
these arbitrary suppositions, which were imagined by Huygens,* are in
themselves extremely complicated and improbable, and are wholly unau-
thorised by observation. A much simpler, and more natural, as well as more
accurate explanation, which was suggested at an earlier period by Mariotte,t
had long been wholly forgotten, until the same idea occurred to me,^

* Ph. Tr. 1670, v. 1065. Op. Rel. vol.ii.
f Trait6 des Couleurs, Paris, 1686. OEuv. i. 272.
x J Jour, of the Roy. Inst. ii. 4.

348 LECTURE XXXVII.

without any previous knowledge of what Mariotte had done. The natural «
tendency of water to crystallize, in freezing, at an angle of GO degrees, is
sufficiently established to allow us to assume this as the constant angle of
the elementary crystals of snow, which are probably either triangular or
hexagonal prisms : the deviation produced by such a prism differs very
little from the observed angle at which the first circle is usually seen ; and
all the principal phenomena, which attend this circle, may be explained,
by supposing the axis of the crystals to assume a vertical or a horizontal
position, in consequence of the operation of gravity : thus the parhelia,
which are sometimes a little more distant from the sun than the halo, are
attributed by Mariotte to the refraction of the prisms which are situated
vertically, and produce a greater deviation, on account of tb?iobliquity of
the rays of light with respect to their axes. The horizontal circle may be
deduced from the reflection, or even the repeated refractions of the vertical
facets ; the anthelia from two refractions with an intermediate reflection,
and the inverted arch from the increase of the deviation, in the light
passing obliquely, through prisms lying in a horizontal position. The
external circle may be attributed either to two successive refractions
through different prisms, or with greater probability, as Mr. Cavendish
has suggested to me, to the effect of the rectangular terminations of the
single crystals. The appearance of colours, in halos, is nearly the same
as in rainbows, but less distinct ; the red being nearest to the luminary,
and the whole halo being externally very ill defined. (Plate XXIX. Fig.
431, 432.)

From the observed magnitude of these halos, I had concluded that the
refractive power of ice must be materially less than that of water, although
some authors had asserted that it was greater ; and Dr. Wollaston after-
wards fully confirmed this conclusion by means of the very accurate
instrument which has already been described : his measurement agreeing
precisely with the mean of the best observations on these halos ; so that
ice must be considered as the least refractive of any known substances not
aeriform.

Sometimes the figures of halos and parhelia are so extremely compli-
cated, as to defy all attempts to account for the formation of their different
parts : but if we examine the representations which have been given, by
various authors, of the multiplicity of capricious forms frequently assumed
by the flakes of snow, we shall see no reason to think them inadequate to
the production of all these appearances. (Plate XXIX. Fig. 433, 434.)

The most singular of all the phenomena of refraction is perhaps the
property of some natural substances, which have a double effect on the
light transmitted through them, as if two mediums of different densities
freely pervaded each other, the one only acting on some of the rays of
light, the other on the remaining portion. These substances are usually
crystallized stones, and their refractions have sometimes no further pecu-
liarity ; but the rhomboidal crystals of calcarious spar, commonly called
Iceland crystals, possess the remarkable property of separating such pencils
of light, as fall perpendicularly on them, into two parts, one of them only
being transmitted in the usual manner, the other being deflected towards

ON PHYSICAL OPTICS. 349

the greater angle of the crystal.* It appears from the experiments of
Hi5ygens,t confirmed and extended by Dr. Wollaston, £ that the medium,
which causes the unusual refraction, has a different refractive power,
according to the direction in which the light passes through it, and that
if an oblate or flattened spheroid be described within a crystal, its axis
being in the middle of one of the obtuse solid angles, and its principal
diameters in the proportion of 9 to 10, the refractive power, with respect
to light passing in any direction, will always be inversely as the diameter
of the spheroid which is parallel to it ; and where it is greatest, will be
equaLgto that of the medium which produces the usual refraction, of which
the index is -§• . A ray of light, falling perpendicularly on any surface of
the spar, its point of incidence being considered as the centre of the spher-
oid, will meef the surface of the spheroid at the point where it is parallel
to that of the spar ; and a ray incident on the same surface in any other
direction, will preserve a relation to the perpendicular ray, which is nearly
the same as in ordinary refraction. (Plate XXIX. Fig. 435.)

It is also remarkable, that the two portions of light, thus separated, will
not be further subdivided by a transmission through a second piece, pro-
vided that this piece be in a position parallel to that of the first ; but if it
be placed in a transverse direction, each of the two pencils will be divided
into two others ; a circumstance which appears to be the most unintel-
ligible of any that has been discovered respecting the phenomena of double
refraction.

The appearances of colours, which are produced by transparent plates
of different thicknesses, and of those which are seen in light variously
diffracted or inflected, will be more conveniently examined, when we in-
vestigate the intimate nature of light, since the general explanation of these
colours, which will then be given, will enable us to follow them through
all their varieties, with much more ease than could be done at present,
without the help of some theory respecting their origin.

LECT. XXXVII.— ADDITIONAL AUTHORITIES.

Colour and dispersion. — Castelli, Optica Colorum, 1740. Euler, Hist, et Mem.
de Berlin, 1753, p. 294. Acta Petr. i. I. 174. Nov. Com. Petr. xii. 166. Dol-
lond, Ph. Tr. 1759, p. 733. Beguelin, Mem. sur les Prismes Achromatiques, Hist,
et Mem. de Berlin, 1762, p. 66. Lambert's Farben-pyramide, 4to, Berl, 1772.
Rochon, Recueil de Mecanique, p. 279. Comparetti de Luce et Coloribus, 4to,
Pad. 1787. Gruber iiber die Strahlenbrechung, 4to, Dresd. 1787. Seebeck,
Schweig. Jour. 1810, p. 1. Mollweide, Demonstratio Prop, quse th. Col. Hewtoni
Fundamenti loco est, Lips. 1811. Hoppe, Versuch einer ganz neuen Theorie der
Entstehung Sammtlicher Farben, Breslau, 1824. Deal, Nouvelle Essai sur la Lum.
et les Couleurs, 1827. Talbot, Ph. Mag. iii. 45 ; iv. 112, &c. Brewster, Ph. Tr.
1836, &c. Helwag, Newton's Farbenlehre, Liibeck, 1835. Rudberg, Pogg. Ann-
ix. 483. An account of Amici's prismatic telescope will be found in Quetelet's Sup-

* Bartholin on Iceland Crystals (quibus mira et insolita refractio detegitur), Co-
penhagen, 1669. Ph. Tr. v. 2039.
' t Traite de la Lumiere par C. H. D. Z. A Leyde, 1690.

I Ph.Tr. 1802, p. 381. On this subject see Beccaria, Ph. Tr. 1762, p. 486 i
Brewster, Edin. Ph. Jour. i. 289 ; ii. 167, &c. &c. See also Lect. XXXIX.

350 LECTURE XXXVIII.

plement to Herschel, p. 432. Becquerel on the Constitution of the Solar Spectrum,
Scientific Memoirs, iii. 537.

Atmospheric Refraction. Ordinary. — Cassini, Novissimae Motuum Solis Epne-
merides a Malvasia supputatae, 1G61. Hist, et Mem. i. 103, 1700, p. 39, H. 112 ;
1714, p. 33, H.61; 1742, p. 203, H. 72; 1743, p. 249, H. 140. Halley, Ph.
Tr. 1721, p. 169, with Newton's table. Lacaille, Hist, et Mem. 1755, p. 547. H.
111. Lambert, Route de la Lumiere par les Airs, A la Haye, 1758. Lagrange,
Hist.etMem.de Berl. 1772, p. 259. Maskelyne, Ph. Tr. 1777, p. 722. Her-
schel, ibid. 1785, p. 88. Oriani, Ephem. Mediol. 1788. Hennert, Hind. Arch.ii.
1, 129. Kramp, Analyse des Refractions Astronom. et Terr. 4to, Strasb. 1799.
Humboldt's Voy. i. 134. Bessel, Fundamenta Astronomiae, fol. Regiom. 1818,
pp.28, 43. Konigsb. Beobacht. vii. 38; viii. 22. Svanberg, Nov. Act. Upsal,
ix. 89. Plana, Recherches Analytiques, 4to, Turin, 1823. Ivory, Ph. Tr. ,1823,
p. 409. T. Young, ibid. 1824, p. 159. Forster, Ph. Mag. 1824, p. 192.

Extraordinary. — Mariotte on the Rainbow, Hist, et Mem. i. 189. Halley on
do. Ph. Tr. 1698, p. 193 ; 1700, p. 714. Weidler de Parheliis, 4tq, Wittemb.
1738. Biisch, Tractatus duo Optici, Hamb. 1788. Huddart on Horizontal Re-
fractions, Ph. Tr. 1797, p. 29. Latham, ib. 1798, p. 357. Monge on the Mirage
in Egypt, Ann. de Ch. xxix. 207. Vince on Horizontal Refraction, Ph. Tr. 1799,
p. 13. Biot, Mem. del'Inst. i. 266. Brandes, Beobachtungen uber die Strahlen-
brechung, Oldenb. 1807. Frauenhofer, Theorie der Hofe, &c. Schumacher's
Ast. Abh. iii. 33. Arago, Bullet. Univ. 1825.

LECTURE XXXVIII.

ON VISION.

THE medium of communication, by which we become acquainted with
all the objects that we have been lately considering, is the eye ; an organ
that exhibits to an attentive observer, an arrangement of various sub-
stances, so correctly and delicately adapted to the purposes of the sense of
vision, that we cannot help admiring, at every step, the wisdom by which
each part is adjusted to the rest, and made to conspire in effects, so remote
from what the mere external appearance promises, that we have only been
able to understand, by means of a laborious investigation, the nature and
operations of this wonderful structure, while its whole mechanism still
remains far beyond all rivalship of human art.

The eye is an irregular spheroid, not very widely differing from a
sphere ; it is principally composed of transparent substances, of various
refractive densities, calculated to collect the rays of light, which diverge
from each point of an object, to a focus on its posterior surface, which is
capable of transmitting to the mind the impression of the colour and
intensity of the light, together with a distinction of the situation of the
focal point, as determined by the angular place of the object. (Plate
XXX. Fig. 436.)

The first refraction happens at the surface of the cornea, or that trans-
parent coat which projects forwards from the ball of the eye : but the
cornea, being very nearly of equable thickness, has little effect by its own
refractive power, and serves only to give a proper form to the aqueous

ON VISION. 351

humour, which fills its concavity, and distends it. This humour is par-
tially divided by the uvea or iris, which is of different colours in different
persons, having a perforation in its centre, called the pupil. Immediately
behind the uvea, and closely connected to its base, are the ciliary processes,
the summits of which hang like a short fringe, before the crystalline lens,
a substance much more refractive than the aqueous humour, and increas-
ing in density towards its centre. The remaining cavity is filled by an
aqueous fluid, lodged in a cellular texture of extremely fine membrane,
and called the vitreous humour. The retina lines the whole posterior
partaof this cavity ; it is semitransparent, and is supported by the choroid
or chorioid coat, a very opaque black or brown membrane, continued from
the uvea and ciliary processes ; but immediately where the retina is con-
nected witUPbhe optic nerve, the choroid is necessarily perforated ; and at
this part a small portion of the retina is nearly insensible. The whole is
surrounded by an opaque continuation of the cornea, called the sclerotica.^

The rays of light which have entered the cornea and passed through the
pupil, being rendered still more convergent by the crystalline lens, are
collected into foci on the retina, and form there an image, which, according
to the common laws of refraction, is inverted, since the central rays of each
pencil cross each other a little behind the pupil ; and the image may easily
be seen in a dead eye, by laying bare the posterior surface of the retina.
(Plate XXX. Fig. 437.)

By means of this arrangement of the various refracting substances,
many peculiar advantages are procured. The surface of the cornea only,
if it had been more convex, could not have collected the lateral rays of a
direct pencil to a perfect focus, without a different curvature near its
edges ; and then the oblique pencils would have been subjected to greater
aberration, nor could they have been made to converge to any focus on the
retina. A second refraction performs both these offices much more com-
pletely, and has also the advantage of admitting a greater quantity of
light. If also the surfaces of the crystalline lens thus interposed, had been
abrupt, there would have been a reflection at each, and an apparent
haziness would have interfered with the distinct view of every luminous
object; but this inconvenience is avoided by the gradual increase of
density in approaching the centre, which also makes the crystalline equiva-
lent to a much more refractive substance of equal magnitude ; while, at
the same time, the smaller density of the lateral parts prevents the usual
aberration of spherical surfaces, occasioned by the too great refraction of
the lateral rays of direct pencils, and causes also the focus of each oblique
pencil to fall either accurately or very nearly on the concave surface of
the retina, throughout its extent.

Opticians have often puzzled themselves, without the least necessity, in
order to account for our seeing objects in their natural erect position, while
the image on the retina is in reality inverted : but surely the situation of a
focal point at the upper part of the eye could be no reason for supposing the
object corresponding to it to be actually elevated. We call that the lower
end of an object which* is next to the ground ; and the image of the trunk
of a tree ,being in contact with the image of the ground on the retina, we

352 LECTURE XXXVIII.

may naturally suppose the trunk itself to be in contact with the actual
ground : the image of the branches being more remote from that of the
ground, we necessarily infer that the branches are higher and the trunk
lower : and it is much simpler that we should compare the image of the
floor with the image of our feet, with which it is in contact, than with the
actual situation of our forehead, to which the image of the floor on the
retina is only accidentally near, and with which indeed it would perhaps
be impossible to compare it, as far as we judge by the immediate sensa-
tions only.*

We might indeed call in experience to our assistance, and habitually
correct the errors of one sense by a comparison with the perceptions of
another. But it appears that some philosophers have been too hasty in
supposing that the use of all our senses is derived from experience alone,
and in disbelieving the existence of instinct independent of it. Without
any other authority than that of their own imaginations, they have denied
the observation recorded by Galen, on the instincts of a kid, which is suffi-
ciently credible to counterbalance much more than bare assertion. The
instant after its birth, accompanied by the loss of its mother, the little
animal ran to some green vegetables, and having first smelt them, chewed
and swallowed them. The kid could have been taught by no experience to
be tempted by the sight, to act with the proper muscles of locomotion, to go
near and smell, and to be induced by the smell to masticate, and by the
taste to swallow and digest its food, had it not been provided with some
fundamental instinct, by the same intelligence which so calculated the
adjustments of the eye, that the lens should be able to produce a perfect
image of every object, and that the retina should be of that precise form,
which is exactly suited to the reception of the image to be depicted on it.

The whole surface of the retina appears to be usually occupied by such
an image, but it is not all of equal sensibility ; a certain portion only, near
the axis, is capable of conveying distinct impressions of minute objects.
But the perfection of this limited distinctness is a far greater advantage to
us, than a more extensive field of moderately accurate vision would have
been ; for by means of the external muscles, we can easily so change the
position of the eye, that the image of any object before us may be made
to fall on the most sensible part of the retina. We may readily observe
the want of sensation at the entrance of the optic nerve, by placing two
candles so that the distance of each from the eye may be about four times
their distance from each other: then if we direct our right eye to the
left hand candle, the right hand candle will be lost in a confused mass of
faint light, its image on the retina falling on the point at which its sensi-
bility is deficient, t

* Consult Berkeley on Vision, Dub. 1709. Lecat, Traite des Sens, 1767. Wal-
ter, Berlin Mem. 1788, p. 3. Wells, Essay on Single Vision, 1791. Wollaston,
Ph. Tr. 1824, p. 222. Berthold, Ueber das Aufrecht-erscheinen der Gesichtsob-
jecte, Gott. 1830. Bartels,"Beitrage zur Phys. des Gesichtsinnes, Berl. 1834.
Volkmann, doTLeipz. 1836.

t A better way of doing this is to make two blots on a sheet of paper, about four
inches apart, and to look attentively with the right eye on that which lies to the left
hand ; the eye being placed right over it. When the eye is raised to the height of

ON VISION. 363.

When the attention is not directed to any particular object of sight, the
refractive powers of the eye are adapted to the formation of an image of
objects at a certain distance only, which is different in different individuals,
and also generally increases with increasing age. * Thus, if we open our
eyelids suddenly, without particular preparation, we find that distant
objects only appear as distinct as we are able to make them ; but by an
exertion of the will, the eye may be accommodated to the distinct percep-
tion of nearer objects, yet not of objects within certain limits. Between
the ages of 40 and 50, the refractive powers of the eye usually begin to
diminish, but it sometimes happens that where they are already too great,
the defect continues unaltered to an advanced age. It appears also that
after 50 or 60, the power of changing the focus of the eye is always much
impaired, amjtfstmietimes wholly lost.

The mode, in which the accommodation of the eye to different distances
is effected, has long been a subject of investigation and dispute among
opticians and physiologists, but I apprehend that at present there is little
further room for doubting that the change is produced by an increase of
the convexity of the crystalline lens, arising from an internal cause. The
arguments in favour of this conclusion are of two kinds ; some of them
are negative, derived from the impossibility of imagining any other mode
of performing the accommodation, without exceeding the limits of the
actual dimensions of the eye, and from the examination of the eye in its
different states by several tests, capable of detecting any other changes if
they had existed : for example, by the application of water to the cornea,
which completely removes the effect of its convexity, without impairing
the power of altering the focus, and by holding the whole eye, when
turned inwards, in such a manner as to render any material alteration of
its length utterly impossible. Other arguments are deduced from positive
evidence of the change of form of the crystalline, furnished by the parti-
cular effects of refraction and aberration which are observable in the
different states of the eye ; effects which furnish a direct proof that the
figure of the lens must vary ; its surfaces, which are nearly spherical in
the quiescent form of the lens, assuming a different determinable curvature
when it is called into exertion. The objections which have been made to
this conclusion are founded only on the appearance of a slight alteration
of focal length in an eye from which the crystalline had been extracted ;
but the fact is neither sufficiently ascertained, nor was the apparent change
at all considerable : and even if it were proved that an eye without the
lens is capable of a certain small alteration, it would by no means follow
that it could undergo a change five times or ten times as great, t

about 11 inches, the second spot disappears as though it had passed under a curtain :
on continuing to lift the head, the spot will reappear when the eye is .about 15 inches
from the paper. This was pointed out by Mariotte, Ph. Tr. 1G68, p. 668 ; 1670,
p. 1023. On the vanishing of images at points not coincident with the entrance of
the optic nerve, consult Brewster's Jour, of Sci. iii. 289.

* On the effects of attention in vision see Purkinje, Beobachtungen zur Physio-
logic der Sinne, vol. i. Prag. 1823 ; vol. ii. Berlin, 1825. Heermann, Ueber die
BHdung der Gesichtsvorstellungen, Hanover, 1835.

t Consult Pemberton, De Facultate Oc. ad Diversas Dist. se Acoommodandi,
Lug. Bat. 1719. Camper, De Oculo Humane, Lug. Bat. 1742. Albinus, Lug.

2 A

354 LECTURE XXXVIII.

The iris serves, by its variable magnitude, to exclude more or less of the
light falling on the cornea, when its intensity would otherwise be too great ;
hence the pupil is usually smallest by day, and its increased magnitude at
night sometimes gives the eye a greater apparent lustre. The iris also inter-
cepts such rays as would fall on parts incapable of refracting them regu-
larly ; and by its contraction when a nearer object is viewed, it lessens the
confusion which would arise, in such eyes as cannot accommodate them-
selves sufficiently, from the magnitude of the imperfect focal points on the
retina. Such a contraction almost always accompanies the diminution of
the focal length, even in a perfect eye, and it may easily be rendered visible
by walking gradually up to a looking glass, and observing the magnitude
of the pupil as we approach nearer and nearer to our image. It would be
difficult to assign a reason for this change of the state of theTpupil within
the limits of perfect vision, unless we allowed the irregularity of the form
assumed by the marginal parts of the crystalline lens. The iris is also
peculiarly useful in excluding such parts of lateral pencils of light as
fall very obliquely on the cornea, and are too much refracted, while a
smaller pencil only, which enters the eye more directly, is admitted into
the pupil.

The refractive powers and properties of the eye may be very conveniently
ascertained by means of an instrument to which I have given the name
optometer, a term first employed in a sense nearly similar by Dr. Porter-
field.* If two or more separate parcels of the rays of the same pencil be
admitted at distant parts of the pupil, they will only be reunited on the
retina when the focus is perfect, so that if we look through two small per-
forations, or slits, at a minute object, to the distance of which the eye is
not accommodated, it will appear as if double ; and when the object is a
line directed nearly towards the eye, each point of it will appear double,
except that which is at the distance of perfect vision, and an image of two
lines will be seen, crossing each other in this point ; so that the measure-
ment of the focal length of the eye is immediately performed by inspection
of the optometer only. The scale may be extended by the addition of a
lens, which enables us to produce the effect of a longer line, while the
instrument still remains portable.

When the eye is possessed of too great a refractive power for the distinct
perception of distant objects, the pupil is generally large, so that the confu-
sion of the image is somewhat lessened by partially closing the eyelids ;
and from this habit an eye so formed is called myopic. In such cases, by
the help of a concave lens, the divergence of the rays of light may be
increased, and a virtual image may be formed, at a distance so much
smaller than that of the object as to afford perfect vision. For a long

Bat. 1746. Le Roy, Mem. sur la M6e. par lequel 1'CEil s'Accommode, Hist, et
Mem. 1755, p. 594. Gibers, De Oculi Mutationibus Intends, 4to, Gott. 1780.
Young, Ph. Tr. 1793, p. 169 ; 1801, p. 23. De Corp. Hum. Viribus Conserva-
tricibus, Gott. 1780. Hunter, Ph. Tr. 1794, p. 21. Home, ibid. 1800, p. 146.
Brewster in Ed. Jour, of Science, i. 77. Treviranus, zur Anat. der Sinneswerk-
zeuge, 1828. Kolrausch on Treviranus' Hypoth. 1837. Luchtman, De Mutatione
Oculi, Tr. ad Rhenum, 1832. Simonoff, Jour, de Physiol. iv. 260.
* Edinb. Med. Essays, iv. 185.

ON VISION. 355

sighted or presbyopic eye, on the contrary, a convex lens is required, in
order to obtain a virtual image at a greater distance than the olyect ; and
it often happens that the rays must be made not only to diverge less than
before, but even to converge towards a focus behind such an eye, in order
to make its vision distinct. Presbyopic persons have in general a small
pupil, and, therefore, seldom acquire the habit of covering any part of it
with their eyelids.

When the images of the same object fall on certain corresponding points
of the retina in each eye,* they appear to the sense only as one ; but if
they fall on parts not corresponding, the object appears double ;f and in
general, all objects at the same distance, in any one position of the eyes,
appear ajlke either double or single. The optical axes, or the directions
of the rays falling on the points of most perfect vision, naturally meet at
a great distance ; that is, they are nearly parallel to each other, and in
looking at a nearer object we make them converge towards it, wherever it
may be situated, by means of the external muscles of the eye ; while in
perfect eyes the refractive powers are altered, at the same time, by an
involuntary sympathy, so as to form a distinct image of an object at the
given distance. This correspondence of the situation of the axes with the
focal length is in most cases unalterable ; but some have perhaps a power
of deranging it in a slight degree, and in others the adjustment is imper-
fect : but the eyes seem to be in most persons inseparably connected toge-
ther with respect to the changes that their refractive powers undergo,
although it sometimes happens that those powers are originally very dif-
ferent in the opposite eyes.

These motions enable us to judge pretty accurately, within certain
limits, of the distance of an object ; and beyond these limits, the degree
of distinctness or confusion of the image still continues to assist the judg-
ment. We estimate distances much less accurately with one eye than
with both, since we are deprived of the assistance usually afforded by the
relative situation of the optical axes ; thus we seldom succeed at once in
attempting to pass a finger or a hooked rod sideways through a ring, with
one eye shut. Our idea of distance is also usually regulated by a know-
ledge of the real magnitude of an object, while we observe its angular
magnitude ; and on the other hand a knowledge of the real or imaginary
distance of the object often directs our judgment of its actual magnitude.
The quantity of light intercepted by the air interposed, and the intensity
of the blue tint which it occasions, are also elements of our involuntary
calculation : hence, in a mist, the obscurity increases the apparent distance,
and consequently the supposed magnitude, of an unknown object. We
naturally observe, in estimating a distance, the number and extent of the

* On corresponding points of the two retinae, see Newton, Op. Qu. 15. Wol-
laston, Ph. Tr. 1824. On single vision, see Le Clerc, Paris, 1679. Wells, Lond.
1791. Herholt, Kopenhag. 1814. Wollaston, Ph. Tr. 1824, p. 222. Twining,
Ed. Jour. ix. 143.

f The most simple mode of witnessing this is to place a small wafer on a pane of
a window, and to look attentively through that pane at a well-defined object without
so as to fix the direction of the axes of the eyes. The spot will be distinctly
doubled.

356 LECTURE XXXVIII.

intervening objects ; so that a distant church in a woody and hilly country
appears more remote than if it were situated in a plain ; and for a similarr
reason, the apparent distance of an object seen at sea, is smaller than its
true distance. The city of London is unquestionably larger than Paris ;
but the difference appears at first sight much greater than it really is ; and
the smoke, produced by the coal fires of London, is probably the principal
cause of the deception.

The sun, moon, and stars, are much less luminous when they are near
the horizon, than when they are more elevated, on account of the greater
quantity of their light that is intercepted, in its longer passage through
the atmosphere : we also observe a much greater variety of nearer objects
almost in the same direction : we cannot, therefore, help imagining them
to be more distant, when they rise or set, than at other times ; and since
they subtend the same angle, they appear to be actually larger. For similar
reasons the apparent figure of the starry heavens, even when free from
clouds, is that of a flattened vault, its summit appearing to be much nearer
to us than its horizontal parts, and any of the constellations seems to be
considerably larger when it is near the horizon than when in the zenith. *
(Plate XXX. Fig. 438.)

The faculty of judging of the actual distance of objects is an impedi-
ment to the deception, which it is partly the business of a painter to pro-
duce. Some of the effects of objects at different distances may, however,
be imitated in painting on a plane surface. Thus, supposing the eye to be
accommodated to a given distance, objects at all other distances may be
represented with a certain indistinctness of outline, which would accom-
pany the images of the objects themselves on the retina : and this indis-
tinctness is so generally necessary, that its absence has the disagreeable
effect called hardness. The apparent magnitude of the subjects of our
design, and the relative situations of the intervening objects, may be so
imitated by the rules of geometrical perspective as to agree perfectly with
nature, and we may still further improve the representation of distance by
attending to the art of aerial perspective, which consists in a due observa-
tion of the loss of light, and the bluish tinge, occasioned by the interposi-
tion of a greater or less depth of air between us and the different parts of
the scenery.

We cannot indeed so arrange the picture, that either the focal length
of the eye, or the position of the optical axes, may be such as would be
required by the actual objects : but we may place the picture at such a
distance that neither of these criterions can have much power in detecting
the fallacy ; or, by the interposition of a large lens, we may produce nearly
the same effects in the rays of light, as if they proceeded from a picture at
any required distance. In the panorama, which has lately been exhibited
in many parts of Europe, the effects of natural scenery are very closely
imitated : the deception is favoured by the absence of all other visible
objects, and by the faintness of the light, which assists in concealing the
defects of the representation, and for which the eye is usually prepared, by

* Hooke on the Horizontal Moon, Birch, iii. 503, 507.

ON VISION. 357

being long detained in the dark winding passages, which lead to the place
<5f exhibition.

The impressions of light on the retina appear to be always in a certain
degree permanent, and the more so as the light is stronger ; but it is uncer-
tain whether the retina possesses this property merely as a solar phosphorus,
or in consequence of its peculiar organization. The duration of the im-
pression is generally from one hundredth of a second to half a second, or
more ; hence a luminous object revolving in a circle makes a lucid ring ;
and a shooting star leaves a train of light behind it, which is not always
real. If the object is painfully bright, it generally produces a permanent
spot, which continues to pass through various changes of colour for some
time, wiiVut much regularity, and gradually vanishes : this may, how-
ever, be considered as a morbid effect.

When the eye has been fixed on a small object of a bright colour,
and is then turned away to a white surface, a faint spot, resembling in
form and magnitude the object first viewed, appears on the surface, of a
colour opposite to the first, that is, of such a colour as would be produced
by withdrawing it from white light ; thus a red object produces a bluish
green spot ; and a bluish green object a red spot. The reason of this
appearance is probably that the portion of the retina, or of the sensorium,
that is affected, has lost a part of its sensibility to the light of that colour,
with which it has been impressed, and is more strongly affected by the
other constituent parts of the white light. A similar effect is also often
produced, when a white, or grey object is viewed on a coloured ground,
even without altering the position of the eye : the whole retina being
affected by sympathy nearly in the same manner as a part of it was
affected in the former case. These appearances are most conveniently
exhibited by means of the shadows of objects placed in coloured light : the
shadow appearing of a colour opposite to that of the stronger light, even
when it is in reality illuminated by a fainter light of the same colour. It
seems that the eye cannot perfectly distinguish the intensity of a colour,
either when the light is extremely faint, as that of many of the fixed stars,
which Dr. Herschel has found to be strongly coloured, or when the light '
is excessively vivid ; and that when a considerable part of the field of vision
is occupied by coloured light, it appears to the eye either white, or less
coloured than it is in reality : so that when a room is illuminated either
by the yellow light of a candle, or by the red light of a fire, a sheet of
writing paper still appears to retain its whiteness ; and if from the light of
the candle we take away some of the abundant yellow light, and leave or
substitute a portion actually white, the effect is nearly the same as if we
took away the yellow light from white, and substituted the indigo which
would be left : and we observe accordingly, that in comparison with the
light of a candle, the common daylight appears of a purplish hue. (Plate
XXX. Fig. 439.. .441.)

LECT. XXXVIII.-ADDITIONAL AUTHORITIES.

Vision.— Fabricius ab Aquapendente, fol. Yen. 1600. Scheineri, Oculus 4to
Rom. 1652. Cherubin, Vision parfaite, 1678. Briggs, Ph. Tr. 1683, p. 17l!

358 LECTURE XXXVIII.

Laurentius, Mis. Ac. Nat. Cur. 1684, App. 157. Trabers, Nervus Opticus, fol. Vien.
1690. Bernoulli, Com. Petr. i. 314. Scarella, Com. Bon. v. I. 110; ii. 446 ; vi.
O. 344. Bonati, Mem. della Soc. Ital. ii. 676. Gauteron, Mem. de Montpellier, i.
23. Wiinsch, Visus Phoen. qusedam, 4to, Lips. 1776. Adams on Vision, 1792.
DuTour, Mem. de Tlnstitut, iii. 514; iv. 499; v. 677; vi. 241. Horn on the
Seat of Vision, 1813. Muhlibach, Inquisitio de Visus Sensu. Vindob. 1816. Sir

C. Bell on the Motions of the Eye, Ed. Ph. Jour. xii. 371 ; Ph. Tr. 1823, pp. 166,
289. Brewster on do. Ed. Jour, of Sc. ii. 1 ; 3rd Series, ii. 168 ; v. 259. Smith,
ibid. v. 52 and 3rd Series, i. 249. Lehot, Nouvelle Theorie de la Vision, 1825.
Miiller, Vergleichende Physiologic des Gescihtsinnes, Leipz. 1826. Plagge, Hec-
ker's Annallen, 1830, p. 404. Hanow, Danz. Nat. Ges. Neue Sam. i. 1. Quete-
let, Pog. Ann. xxxi. 494. Mbser on Vision, &c, Scientific Memoirs, iii. 422.
Mackenzie, Physiology of Vision, 1841.

Structure of the Eye Vasali, De Hum. Corp. fabrica, Bas. 1543. Leeuwen-

hoek on the Crystalline Lens, Ph. Tr. 1684, p. 780. On the Eyes of Insects, ibid.
1698, p. 169. On the Eyes of Whales, &c. ibid. 1704, 1723. B.rigg'4 Ophthal-
mographia, 1686. Zahn, Oculus artificialis, fol. Nuremb. 1702. Petit on the
Chambers of the Eye. Hist, et Mem. de Paris, 1723, p. 38, H. 19 ; 1728, pp.
206, 289, H. 17. On the Capsule of the Crystalline, ibid. 1730, p. 435, H. 33 ; viii.
612. On the Crystalline in different Animals, ibid. 1730, p. 4, H. 33. Eye of the
Turkey, ibid. 1735, p. 123. Of the Owl, ibid. 1736, p. 121. Of the Frog and
the Tortoise, ibid. 1737, p. 142. Appel, De Oculi Humani Fabrica, Lug. Bat.
1740. Haller, Disquisitiones Anatom. 6 vols. 4to, Gott. 1746. Zinn, Descriptio
Oculi Humani, 4to, Gott. 1753. Von Grimm, De Visu, Gott. 1758. Albinus,
Mussch. Introd. ii. 744. Haseler, Ueber das Menschliche auge. Hamb. 1771.
Horrebow, De Oculo Humano, Hafn. 1792. Monro, Treatises on the Brain, the
Eye, and the Ear, 4to, Edin. 1797. Rudolph, De Oculi Partibus, 4to, Greifsw. 1801.
S. T. Sommering Abbildungen des Menschlichen Auges, fol. Frank. 1801. Che-
nevix, Ph. Tr. 1803, p. 195. Schreger, Anatomie des Auges. Leipz. 1810. Blu-
menthal, De Externis Oculi Integumentis, 4to, Berol. 1812. Bock, Beschreibung
des f iinfter Nervenpaares Meissen, 1817. Hegar, De Oculi Partibus, Gott. 1818.

D. W. Sommering Com. Gott. 1818. Home, Ph. Tr. 1822, p. 76. Brewster,
Ed. Ph. Jour. i. 42 ; Ph. Tr. 1833, p. 323. Knox, ibid. ix. 358 ; x. 323, 338.

Achromatism of the Eye. — D'Alembert, Opusc. de Math. viii. 324. Maskelyne,
Ph. Tr. Ixxix. 256. Tortual, Meckel's Archiv. 1830, p. 129. Powell, Report of
Br. Ass. 1833, p. 374. Frauenhofer, Gilb. Ann. Ivi. 304. Brewster, Phil. Mag.
ix. 358. Powell in Reply, ibid. vi. 247.

Duration of Impressions. — Segner, De Raritate Luminis. Gott. 1740. D'Arcy,
Hist, et Mem. ix. 614. Roget, Ph. Tr. 1825, p. 131. ^Plateau, Dissertation sur
quelques Proprietes des Impressions produites par la Lumiere sur 1'Organe de la Vue,
Liege, 1829. Annales de Chimie, liii. 304. Stamfer, Die Stroboskopischen
Scheiben. Wien, 1833. Homer on the Daedalium, Ph. Mag. iv. 36. Wheatstone
on the Velocity of Electrical Light, Ph. Tr. 1835, p. 583. Description of the
Kaleidophon, Quart. Jour, of Science, xi. 344. See also Faraday, Jour, of Roy.
Inst. i. 205. Dandelin, Mem. de Bruxelles, ii. 169. Talbot, Ph. Mag. iv. 113.
Addams, ibid. v. 373. Dove, Pogg. Ann. xxxv. 379.

Miscellaneous. — Buffon on Accidental Colours, Hist, et Mem. 1743, p. 147.
Darwin on Ocular Spectra, Ph.Tr. 1786, p. 313. Brewster on the Optical Illusion
of the Conversion of Cameos into Intaglios, Ed. Jour, of Sci. iv. 99. Ph. Mag.
Wollaston on the Direction of the Eyes in a Portrait, Ph. Tr. 1824, p. 247. New-
ton (Sir I.) on Ocular Spectra, Ed. Jour, of Sc. iv. 75. Brewster on do. Ph. Mag.
iv. 353. On the Influence of successive Impulses on the Retina, ibid. iv. 241.
Plateau Sur le Phenomene des Couleurs accidentelles, Ann. de Ch. liii. 386. Essai
d'une Theorie generate comprenant les Couleurs accidentalles, &c. ibid. Iviii. 337.
Chevreuil sur 1'Influence que deux Couleurs peuvent avoir 1'une sur 1'autre, Mem.
de I'lnstit. xi. 448. Dalton on some Facts relating to the Vision of Colours,
Manch. Mem. v. 28 ; Dalton could not distinguish blue from pink by daylight,
but by candlelight the pink appeared red. Tortual, Ueber die Escheinung des
Schattens, Berl. 1830.

359

XECTURE XXXIX,

ON THE NATURE OF LIGHT AND COLOURS.

THE nature of light is a subject of no material importance to the con-
cerns of life or to the practice of the arts, but it is in many other respects
extremely interesting, especially as it tends to assist our views both of the
natflre of our sensations, and of the constitution of the universe at large.
The examination of the production of colours, in a variety of circum-
stances, it f:?timately connected with the theory of their essential properties,
and their causes ; and we shall find that many of these phenomena will
afford us considerable assistance in forming our opinon respecting the
nature and origin of light in general.

It is allowed on all sides, that light either consists in the emission of very
minute particles from luminous substances, which are actually projected,
and continue to move with the velocity commonly attributed to light, or
in the excitation of an undulatory motion, analogous to that which con-
stitutes sound, in a highly light and elastic medium pervading the universe ;
but the judgments of philosophers of all ages have been much divided with
respect to the preference of one or the other of these opinions. There are
also some circumstances which induce those, who entertain the first hypo-
thesis, either to believe, with Newton,* that the emanation of the par-
ticles of light is always attended by the undulations of an etherial medium,
accompanying it in its passage, or to suppose, with Boscovich,t that the
minute particles of light themselves receive, at the time of their emission,
certain rotatory and vibratory motions, which they retain as long as their
projectile motion continues. These additional suppositions, however neces-
sary they may have been thought for explaining some particular pheno-
mena, have never been very generally understood or admitted, although no
attempt has been made to accommodate the theory in any other manner to
those phenomena.

We shall proceed to examine in detail the manner in which the two
principal hypotheses respecting light may be applied to its various proper-
ties and affections ; and in the first place to the simple propagation of light
in right lines through a vacuum, or a very rare homogeneous medium. In
this circumstance there is nothing inconsistent with either hypothesis ; but
it undergoes some modifications, which require to be noticed, when a por-
tion of light is admitted through an aperture, and spreads itself in a slight
degree in every direction. In this case it is maintained by Newton that
the margin of the aperture possesses an attractive force, which is jcapable
of inflecting the rays : but there is some improbability in supposing that
bodies of different forms and of various refractive powers should possess
an equal force of inflection, as they appear to do in the production of these

* * Ph. Tr. vii. 5087.

f Dissertatio de Lumine, Part II. 1748 ; and Theoria Philosopbia Naturalis, 4to»,
'Venice, 1763, p. 230.

SCO LECTURE XXXIX.

effects ; and there is reason to conclude from experiments, that such a »
force, if it existed, must extend to a very considerable distance from tfie
surfaces concerned, at least a quarter of an inch, and perhaps much more,
which is a condition not easily reconciled with other phenomena. In the
Huygehian system of undulation, this divergence or diffraction is illus-
trated by a comparison with the motions of waves of water and of sound,
both of which diverge when they are admitted into a wide space through
an aperture, so much indeed that it has usually been considered as an ob-
jection to this opinion, that the rays of light do not diverge in the degree
that would be expected if they were analogous to the waves of water. 'But
as it has been remarked by Newton,* that the pulses of sound diverge less
than the waves of water, so it may fairly be inferred, that '^ae- ^till more
highly elastic medium, the undulations, constituting light, must diverge
much less considerably than either. (Plate XX. Fig. 26G.)

With respect, however, to the transmission of light through perfectly
transparent mediums of considerable density, the system of emanation
labours under some difficulties. It is not to be supposed that the particles
of light can perforate with freedom the ultimate atoms of matter, which
compose a substance of any kind ; they must, therefore, be admitted in all
directions through the pores or interstices of those atoms ; for if we allow
such suppositions as Boscovich's, that matter itself is penetrable, that is,
immaterial, it is almost useless to argue the question further. It is cer-
tain that some substances retain all their properties when they are reduced
to the thickness of the ten millionth of an inch at most, and we cannot there-
fore suppose the distances of the atoms of matter in general to be so great
as the hundred millionth of an inch. Now if ten feet t)f the most trans-
parent water transmits, without interruption, one half of the light that enters
it, each section or stratum of the thickness of one of these pores of matter
must intercept only about one twenty thousand millionth, and so much must
the space or area occupied by the particles be smaller than the interstices
between them, and the diameter of each atom must be less than the hun-
dred and forty thousandth part of its distance from the neighbouring par-
ticles ; so that the whole space occupied by the substance must be as little
filled as the whole of England would be filled by a hundred men, placed at
the distance of about thirty miles from each other. This astonishing
degree of porosity is not indeed absolutely inadmissible, and there are
many reasons for believing the statement to agree in some measure with
the actual constitution of material substances ; but the Huygenian hypo-
thesis does not require the disproportion to be by any means so great, since
the general direction and even the intensity of an undulation would be
very little affected by the interposition of the atoms of matter, while these
atoms may at the same time be supposed to assist in the transmission of
the impulse, by propagating it through their own substance. Euler indeed
imagined that the undulations of light might be transmitted through the
gross substance of material bodies alone, precisely in the same manner as
sound is propagated ; but this supposition is for many reasons inadmis-'
sible.

* Op. Qu. 28.

ON THE NATURE OF LIGHT AfrD COLOURS. 3G1

A very striking circumstance, respecting the propagation of light, is the
uniformity of its velocity in the same medium. According to the projec-
tile hypothesis, the force employed in the free emission of light must be
about a million million times as great as the force of gravity at the earth's
surface ; and it must either act with equal intensity on all the particles of
light, or must impel some of them through a greater space than others,
if its action be less powerful, since the velocity is the same in all cases ;
for example, if the projectile force is weaker with respect to red light than
with respect to violet light, it must continue its action on the red rays to a
greater distance than on the violet rays. There is no instance in nature
besides of a simple projectile moving with a velocity uniform in all cases,
whateve^iiir1 ^be its cause, and it is extremely difficult to imagine that so
immense a force of repulsion can reside in all substances capable of
becoming luminous, so that the light of decaying wood, or of two pebbles
rubbed together, may be projected precisely with the same velocity as the
light emitted by iron burning in oxygen gas, or by the reservoir of liquid
fire on the surface of the sun. Another cause would also naturally inter-
fere with the uniformity of the velocity of light, if it consisted merely in
the motion of projected corpuscles of matter ; Mr. Laplace has calculated,*
that if any of the stars were 250 times as great in diameter as the sun,
its attraction would be so strong as to destroy the whole momentum of the
corpuscles of light proceeding from it, and to render the star invisible at a
great distance ; and although there is no reason to imagine that any of the
stars are actually of this magnitude, yet some of them are probably many
times greater than our sun, and therefore large enough to produce such a
retardation in the motion of their light as would materially alter its effects.
It is almost unnecessary to observe that the uniformity of the velocity of
light, in those spaces which are free from all material substances, is a
necessary consequence of the Huygenian hypothesis, since the undulations
of every homogeneous elastic medium are always propagated, like those
of sound, with the same velocity, as long as the medium remains un-
altered.

On either supposition, there is no difficulty in explaining the equality of
the angles of incidence and reflection ; for these angles are equal as well
in the collision of common elastic bodies with others incomparably larger,
as in the reflections of the waves of water and of the undulations of sound.
And it is equally easy to demonstrate, that the sines of the angles of inci-
dence and refraction must be always in the same proportion at the same
surface, whether it be supposed to possess an attractive force, capable of
acting on the particles of light, or to be the limit of a medium through
which the undulations are propagated with a diminished velocity. There
are, however, some casfe of the production of colours, which lead iis to
suppose that the velocity of light must be smaller in a denser than in a
rarer medium ; and supposing this fact to be fully established, the exist-
ence of such an attractive force could no longer be allowed, nor could the
System of emanation be maintained by any one.f

* Zachs Geographische Ephemeriden, iv. 1.

f Arago put this remark to the test, Annales de Chimie, Ixxi. 49.

362 LECTURE XXXIX.

The partial reflection from all refracting surfaces is supposed by Newton
to arise from certain periodical retardations of the particles of light,
caused by undulations, propagated in all cases through an ethereal me-
dium. The mechanism of these supposed undulations is so complicated,
and attended by so many difficulties, that the few who have examined
them have been in general entirely dissatisfied with them ; and the internal
vibrations of the particles of light themselves, which Boscovich has
imagined, appear scarcely to require a serious discussion. It may, there-
fore, safely be asserted, that in the projectile hypothesis this separation of
the rays of light of the same kind by a partial reflection at every refract-
ing surface, remains wholly unexplained. In the undulatory system, on
the contrary, this separation follows as a necessary consequence. It is
simplest to consider the ethereal medium which pervades any transparent
substance, together with the material atoms of the substance, as constituting
together a compound medium denser than the pure ether, but not more
elastic ;* and by comparing the contiguous particles of the rarer and the
denser medium with common elastic bodies of different dimensions, we
may easily determine not only in what manner, but almost in what degree,
this reflection must take place in different circumstances. Thus, if one of
two equal bodies strikes the other, it communicates to it its whole motion
without any reflection ; but a smaller body striking a larger one is re-
flected, with the more force as the difference of their magnitude is greater ;
and a larger body, striking a smaller one, still proceeds with a diminished
velocity ; the remaining motion constituting, in the case of an undulation
falling on a rarer medium, a part of a new series of motions which neces-
sarily returns backwards with the appropriate velocity ; and we may
observe a circumstance nearly similar to this last in a portion of mercury
spread out on a horizontal table ; if a wave be excited at any part, it will
be reflected from the termination of the mercury almost in the same
manner as from a solid obstacle.

The total reflection of light, falling, with a certain obliquity, on the
surface of a rarer medium, becomes, on both suppositions, a particular case
of refraction. In the undulatory system, it is convenient to suppose the
two mediums to be separated by a short space in which their densities
approach by degrees to each other, in order that the undulation may be
turned gradually round, so as to be reflected in an equal angle ; but this
supposition is not absolutely necessary, and the same effects may be ex-
pected at the surface of two mediums separated by an abrupt termination.

The chemical process of combustion may easily be imagined either to
disengage the particles of light from their various combinations, or to agi-
tate the elastic medium by the intestine motions attending it : but the
operation of friction upon substances incapable jof undergoing chemical
changes, as well as the motions of the electric fluid through imperfect
conductors, afford instances of the production of light in which there

* Some modern writers have adopted the contrary hypothesis, that the ethereal
medium which pervades a substance is of the same density as it is in void space,
but that its elasticity is different. See Neumann, Memoirs of the Academy of
Berlin, vol. xxii. for 1835, and Annalen der Physik, xxv. 418.

ON THE NATURE OF LIGHT AND COLOURS. 363

seems to be no easy way of supposing a decomposition of any kind. The
phenomena of solar phosphor! appear to resemble greatly the sympathetic
sounds of musical instruments, which are agitated by other sounds con-
veyed to them through the air : it is difficult to understand in what state
the corpuscles of light could be retained by these substances so as to be
reemitted after a short space of time ; and if it is true that diamonds are
often found, which exhibit a red light after having received a violet light
only, it seems impossible to explain this property, on the supposition of the
retention and subsequent emission of the same corpuscles.

Tbe phenomena of the aberration of light agree perfectly well with the
system of emanation ; and if the ethereal medium, supposed to pervade
the earth ^nd its atmosphere, were carried along before it, and partook
materially in its motions, these phenomena could not easily be reconciled
with the theory of undulation. But there is no kind of necessity for such
a supposition : it will not be denied by the advocates of the Newtonian
opinion that all material bodies are sufficiently porous to leave a medium
pervading them almost absolutely at rest ; and if this be granted, the
effects of aberration will appear to be precisely the same in either hypo-
thesis.

The unusual refraction of the Iceland spar has been most accurately
and satisfactorily explained by Huygens, on the simple supposition that
this crystal possesses the property of transmitting an impulse more rapidly
in one direction than in another ; whence he infers that the undulations
constituting light must assume a spheroidical instead of a spherical form,
and lays down such laws for the direction of its motion, as are incompar-
ably more consistent with experiment than any attempts which have been
made to accommodate the phenomena to other principles. It is true that
nothing has yet been done to assist us in understanding the effects of a
subsequent refraction by a second crystal,* unless any person can be satis-
fied with the name of polarity assigned by Newton to a property which he
attributes to the particles of light, and which he supposes to direct them in
the species of refraction which they are to undergo : but on any hypothesis,
until we discover the reason why a part of the light is at first refracted in
the usual manner, and another part in the unusual manner, we have no
right to expect that we should understand how these dispositions are con-
tinued or modified, when the process is repeated.

In order to explain, in the system of emanation, the dispersion of the
rays of different colours by means of refraction, it is necessary to suppose
that all refractive mediums have an elective attraction, acting more
powerfully on the violet rays, in proportion to their mass, than on the red.
But an elective attraction of this kind is a property foreign to mechanical
philosophy, and when we use the term in chemistry, we only confess our
incapacity to assign a mechanical cause for the effect, and refer to an ana-
logy with other facts, of which the intimate nature is perfectly unknown
to us. It is not indeed very easy to give a demonstrative theory of the
dispersion of coloured light upon the supposition of undulatory motion ;
but we may derive a very satisfactory illustration from the well known
* See additional remarks at the end of this Lecture.

364 LECTURE XXXIX.

effects of waves of different breadths. The simple calculation of the velo- •
city of waves, propagated in a liquid perfectly elastic, or incompressible,
and free from friction, assigns to them all precisely the same velocity, what-
ever their breadth may be : the compressibility of the fluids actually exist-
ing introduces, however, a necessity for a correction according to the
breadth of the wave, and it is very easy to observe, in a river or a pond of
considerable depth, that the wider waves proceed much more rapidly than
the narrower. We may, therefore, consider the pure ethereal medium as
analogous to an infinitely elastic fluid, in which undulations of all kinds
move with equal velocity, and material transparent substances, o\\ the
contrary, as resembling those fluids, in which we see the large waves ad-
vance beyond the smaller ; and by supposing the red li^t t&» consist of
larger or wider undulations and the violet of smaller, we may sufficiently
elucidate the greater refrangibility of the red than of the violet light.*

It is not, however, merely on the ground of this analogy that we may be
induced to suppose the undulations constituting red light to be larger than
those of violet light : a very extensive class of phenomena leads us still more
directly to the same conclusion ; they consist chiefly of the production of
colours by means of transparent plates, and by diffraction or inflection,
none of which have been explained upon the supposition of emanation, in a
manner sufficiently minute or comprehensive to satisfy the most candid
even of the advocates for the projectile system ; while on the other hand
all of them may be at once understood, from the effect of the interference
of double lights, in a manner nearly similar to that which constitutes in
sound the sensation of a beat, when two strings forming an imperfect
unison, are heard to vibrate together.

Supposing the light of any given colour to consist of undulations of a
given breadth, or of a given frequency, it follows that these undulations
must be liable to those effects which we have already examined in the case
of the waves of water and the pulses of sound. It has been shown that
two equal series of waves, proceeding from centres near each other, may be
seen to destroy each other's effects at certain points, and at other points to
redouble them ; and the beating of two sounds has been explained from a
similar interference. We are now to apply the same principles to the
alternate union and extinction of colours. (Plate XX. Fig. 267.)

In order that the effects of two portions of light may be thus combined,
it is necessary that they be derived from the same origin, and that they
arrive at the same point by different paths, in directions not much devi-
ating from each other. This deviation may be produced in one or both of
the portions by diffraction, by reflection, by refraction, or by any of these
effects combined ; but the simplest case appears to be, when a beam of
homogeneous light falls on a screen in which there are two very small holes
or slits, which may be considered as centres of divergence, from whence the

* See Cauchy, Memoire sur la Dispersion de la Lumiere, Prague, 1835. Powell,
Ph. Mag. vi. 16, 107, 189, 262. Ph. TV. 1835, p. 249, &c. ; and Essay on the Un-
dulatory Theory, as applied to the Dispersion of Light. Challis, Ph. Mag. viii.
Kelland, Trans. Camb. Ph. Soc. vi. 153. Difference of colour was referred to dif-
ference of velocity by Melvil, Ph. Tr. 1753, p. 262, and Essays, ii. 12.

ON THE NATURE OF LIGHT AND COLOURS. 365

light is diffracted in every direction. In this case, when the two newly
formed beams are received on a surface placed so as to intercept them, their
light is divided by dark stripes into portions nearly equal, but becoming-
wider as the surface is more remote from the apertures, so as to subtend
very nearly equal angles from the apertures at all distances, and wider also
in the same proportion as the apertures are closer to each other. The
middle of the two portions is always light, and the bright stripes on each
side are at such distances, that the light coming to them from one of the
apertures, must have passed through a longer space than that which comes
from^the other, by an interval which is equal to the breadth of one, two,
three, or more of the supposed undulations, while the intervening dark
spaces correspond to a difference of half a supposed undulation, of one and
a half, of two ctnd a half, or more.

From a comparison of various experiments, it appears that the breadth
of the undulations constituting the extreme red light must be supposed to
be, in air, about one 36 thousandth of an inch, and those of the extreme
violet about one 60 thousandth ; the mean of the whole spectrum, with
respect to the intensity of light, being about one 45 thousandth. From
these dimensions it follows, calculating upon the known velocity of light,
that almost 500 millions of millions of the slowest of such undulations must
enter the eye in a single second. The combination of two portions of white
or mixed light, when viewed at a great distance, exhibits a few white and
black stripes, corresponding to this interval : although, upon closer inspec-
tion, the distinct effects of an infinite number of stripes of different
breadths appear to be compounded together, so as to produce a beautiful
diversity of tints, passing by degrees into each other. The central white-
ness is first changed to a yellowish, and then to a tawny colour, succeeded
by crimson, and by violet and blue, which together appear, when seen at a
distance, as a dark stripe ; after this a green light appears, and the dark
space beyond it has a crimson hue ; the subsequent lights are all more or
less green, the dark spaces purple and reddish ; and the red light appears
so far to predominate in all these effects, that the red or purple stripes
occupy nearly the same place in the mixed fringes as if their light were
received separately.

The comparison of the results of this theory with experiments fully esta-
blishes their general coincidence ; it indicates, however, a slight correction
in some of the measures, on account of some unknown cause, perhaps con-
nected with the intimate nature of diffraction, which uniformly occasions
the portions of light proceeding in a direction very nearly rectilinear, to be
divided into stripes or fringes a little wider than the external stripes, formed
by the light which is more bent. (Plate XXX. Fig. 442, 443.)

When the parallel slits are enlarged, and leave only the intervening
substance to cast its shadow, the divergence from its opposite margins still
continues to produce the same fringes as before, but they are not easily
visible, except within the extent of its shadow, being overpowered in other
parts by a stronger light ; but if the light thus diffracted be allowed to fall
on the eye, either within the shadow or in ite neighbourhood, the stripes

366 LECTURE XXXIX.

will still appear ; and in this manner the colours of small fibres are pro-
bably formed. Hence if a collection of equal fibres, for example a lock- of
wool, be held before the eye when we look at a luminous object, the series
of stripes belonging to each fibre combine their effects, in such a manner,
as to be converted into circular fringes or coronae. This is probably the
origin of the coloured circles or coronae sometimes seen round the sun
and moon, two or three of them appearing together, nearly at equal dis-
tances from each other and from the luminary, the internal ones being,
however, like the stripes, a little dilated. It is only necessary that the air
should be loaded with globules of moisture, nearly of equal size among
themselves, not much exceeding one two thousandth of an inch in diameter,
in order that a series of such coronae, at the distance of two or three degrees
from each other, may be exhibited. (Plate XXX. Fig. 44-fT)

If, on the other hand, we remove the portion of the screen which sepa-
rates the parallel slits from each other, their external margins will still
continue to exhibit Jhe effects of diffracted light in the shadow on each
side ; and the experiment will assume the form of those which were made
by Newton on the light passing between the edges of two knives, brought
very nearly into contact ; although some of these experiments appear to
show the influence of a portion of light reflected by a remoter part of the
polished edge of the knives, which indeed must unavoidably constitute a
part of the light concerned in the appearance of fringes, wherever their
whole breadth exceeds that of the aperture, or of the shadow of the fibre.

The edges of two knives, placed very near each other, may represent the
opposite margins of a minute furrow, cut in the surface of a polished sub-
stance of any kind, which, when viewed with different degrees of obliquity,
present a series of colours nearly resembling those which are exhibited
within the shadows of the knives : in this case, however, the paths of the
two portions of light before their incidence are also to be considered, and
the whole difference of these paths will be found to determine the appear-
ance of colour in the usual manner : thus when the surface is so situated,
that the image of the luminous point would be seen in it by regular reflec-
tion, the difference will vanish, and the light will remain perfectly white,
but in other cases various colours will appear, according to the degree of
obliquity. These colours may easily be seen, in an irregular form, by
looking at any metal, coarsely polished, in the sunshine ; but they be-
come more distinct and conspicuous, when a number of fine lines of
equal strength are drawn parallel to each other, so as to conspire in their
effects.*

It sometimes happens that an object, of which a shadow is formed in a
beam of light, admitted through a small aperture, is not terminated by
parallel sides ; thus the two portions of light, which are diffracted from
two sides of an object, at right angles with each other, frequently form
a short series of curved fringes within the shadow, situated on each side
of the diagonal, which were first observed by Grimaldi,t and which are

* Young's Introduction to Medical Literature, 1813, p. 559.

•f* Physico-Mathesis de I^imine, Coloribus et Iride, Bonon. 1665.

ON THE NATURE OF LIGHT AND COLOURS. 367

completely explicable from the general principle, of the interference of the
two portions encroaching perpendicularly on the shadow. (Plate XXX.
Fig. 445.)

But the most obvious of all the appearances of this kind is that of the
fringes which are usually seen beyond the termination of any shadow,
formed in a beam of light, admitted through a small aperture : in white
light three of these fringes are usually visible, and sometimes four ; but
in light of one colour only, their number is greater ; and they are always
much narrower as they are remoter from the shadow. Their origin is
easity deduced from the interference of the direct light with a portion of
light reflected from the margin of the object which produces them, the
obliquity of its incidence causing a reflection so copious as to exhibit a
visible effect, however narrow that margin may be ; the fringes are, how-
ever, rendered more obvious as the quantity of this reflected light is
greater. Upon this theory it follows that the distance of the first dark
fringe from the shadow should be half as great as that of the fourth, the
difference of the lengths of the different paths of the light being as the
squares of those distances ; and the experiment precisely confirms this calcu-
lation, with the same slight correction only as is required in all other cases ;
the distances of the first fringes being always a little increased. It may
also be observed, that the extent of the shadow itself is always augmented,
and nearly in an equal degree with that of the fringes : the reason of this
circumstance appears to be the gradual loss of light at the edges of every
separate beam, which is so strongly analogous to the phenomena visible in
waves of water. The same cause may also perhaps have some effect in
producing the general modification or correction of the place of the first
fringes, although it appears to be scarcely sufficient for explaining the
whole of 'it. (Plate XXX. Fig. 446.)

A still more common and convenient method of exhibiting the effects of
the mutual interference of light, is afforded us by the colours of the thin
plates of transparent substances. The lights are here derived from the
successive partial reflections produced by the upper and under surface of
the plate, or when the plate is viewed by transmitted light, from the direct
beam which is simply refracted, and that portion of it which is twice [or
more times] reflected within the plate. The appearance in the latter case
is much less striking than in the former, because the light thus affected is
only a small portion of the whole beam, with which it is mixed ; while in
the former the two reflected portions are nearly of equal intensity, and may
be separated from all other light tending to overpower them. In both
cases, when the plate is gradually reduced in thickness to an extremely
thin edge, the order of colours may be precisely the same as in the stripes
and coronae already described ; their distance only varying when the
surfaces of the plate, instead of being plane, are concave, as it frequently
happens in such experiments. The scale of an oxid, which is often
formed by the effect of heat on the surface of a metal, in particular of
h'on, affords us an example of such a series formed in reflected light : this
scale is at first inconceivably thin, and destroys none of the light reflected,
it soon, however, begins to be of a dull yellow, which changes to red, and

368 LECTURE XXXIX.

then to crimson and blue, after which the effect is destroyed by the opacity
which the oxid acquires. Usually, however, the series of colours produce0d
in reflected light follows an order somewhat different : the scale of oxid
is denser than the air, and the iron below than the oxid ; but where the
mediums above and below the plate are either both rarer or both denser
than itself, the different natures of the reflections at its different surfaces
appear to produce a modification in the state of the undulations, and the
infinitely thin edge of the plate becomes black instead of white, one of the
portions of light at once destroying the other, instead of cooperating with
it. Thus when a film of soapy water is stretched over a wine glass, and
placed in a vertical position, its upper edge becomes extremely thin, and
appears nearly black, while the parts below are divided by horizontal lines
into a series of coloured bands ; and when two glasses, one of which is
slightly convex, are pressed together with some force, the plate of air
between them exhibits the appearance of coloured rings, beginning from
a black spot at the centre, and becoming narrower and narrower, as the
curved figure of the glass causes the thickness of the plate of air to increase
more and more rapidly. The black is succeeded by a violet, so faint as to
be scarcely perceptible ; next to this is an orange yellow, and then crim-
son and blue. When water or any other fluid, is substituted for the air
between the glasses, the rings appear where the thickness is as much less
than that of the plate of air, as the refractive density of the fluid is
greater ; a circumstance which necessarily follows from the proportion of
the velocities with which light must, upon the Huygenian hypothesis, be
supposed to move in different mediums. It is also a consequence equally
necessary in this theory, and equally inconsistent with all others, that
when the direction of the light is oblique, the effect of a thicker plate must
be the same as that of a thinner plate, when the light falls perpendicularly
upon it ; the difference of the paths described by the different portions of
light precisely corresponding with the observed phenomena. (Plate XXX.
Fig. 447... 449.)

Sir Isaac Newton supposes the colours of natural bodies in general to be
similar to these colours of thin plates, and to be governed by the magni-
tude of their particles. If this opinion were universally true, we might
always separate the colours of natural bodies by refraction into a number of
different portions, with dark spaces intervening ; for every part of a thin
plate which exhibits the appearance of colour, affords such a divided
spectrum, when viewed through a prism. There are accordingly many
natural colours in which such a separation may be observed ; one of the
most remarkable of them is that of blue glass, probably coloured with
cobalt, which becomes divided into seven distinct portions. It seems,
however, impossible to suppose the production of natural colours perfectly
identical with those of thin plates, on account of the known minuteness of
the particles of colouring bodies, unless the refractive density of these par-
ticles be at least 20 or 30 times as great as that of glass or water ; which is
indeed not at all improbable with respect to the ultimate atoms of bodies,
but difficult to believe with respect to any of their arrangements consti-
tuting the diversities of material substances.

ON THE NATURE OF LIGHT AND COLOURS. 369

•% The colours of mixed plates constitute a distinct variety of the colours of
thin plates, which has not been commonly observed. They appear when
the interstice between two glasses nearly in contact, is filled with a great
number of minute portions of two different substances, as water and air,
oil and air, or oil and water ; the light which passes through one of the
mediums, moving with a greater velocity, anticipates the light passing
through the other; and their effects on the eye being confounded and
combined, their interference produces an appearance of colours nearly
similar to those of the colours of simple thin plates, seen by transmission ;
but a£ much greater thicknesses, depending on the difference of the refrac-
tive densities of the substances employed. The effect is observed by hold-
ing the glasses between the eye and the termination of a bright object, and
it is most conspicuous in the portion which is seen on the dark part beyond
the object, being produced by the light scattered irregularly from the sur-
faces of the fluid. Here, however, the effects are inverted, the colours
resembling those of the common thin plates seen by reflection ; and the
same considerations on the nature of the reflections are applicable to both
cases. (Plate XXX. Fig. 450.)

The production of the supernumerary rainbows, which are sometimes
seen within the primary and without the secondary bow, appears to be
intimately connected with that of the colours of thin plates. We have
already seen that the light producing the ordinary rainbow is double, its
intensity being only greatest at its termination, where the common bow
appears, while the whole light is extended much more widely. The two
portions concerned in its production must divide this light into fringes ;
but unless almost all the drops of a shower happen to be of the same mag-
nitude, the effects of these fringes must be confounded and destroyed ; in
general, however, they must at least cooperate more or less in producing
one dark fringe, which must cut off the common rainbow much more
abruptly than it would otherwise have been terminated, and consequently
assist the distinctness of its colours. The magnitude of the drops of rain,
required for producing such of these rainbows as are usually observed, is
between the 50th and the 100th of an inch ; they become gradually nar-
rower as they are more remote from the common rainbows, nearly in the
same proportions as the external fringes of a shadow, or the rings seen in
a concave plate.* (Plate XXX. Fig. 451.)

The last species of the colours of double lights, which it will be neces-
sary to notice, constitutes those which have been denominated, from
Newton's experiments, the colours of thick plates, but which may be
called, with more propriety, the colours of concave mirrors. The anterior
surface of a mirror of glass, or any other transparent surface placed before
a speculum of metal, dissipates irregularly in every direction two portions
of light, one before and the other after its reflection. When the light falls
obliquely on the mirror, being admitted through an aperture near the
centre of its curvature, it is easy to show, from the laws of reflection, that
the two portions, thus dissipated, will conspire in their effects, throughout

* Young's Exp. and Obs. relative to Physical Optics, Ph. Tr. 1804, p. 1. Potter,
Math. Considerations on the Rainbow, Tr. Camb. Ph. Soc. vi. 141.

2 B

370 LECTURE XXXIX.

the circumference of a circle, passing through the aperture ; this circle will
consequently be white, and it will be surrounded with circles of colours
very nearly at equal distances, resembling the stripes produced by diffrac-
tion. The analogy between these colours and those of thin plates is by no
means so close as Newton supposed it ; since the effect of a plate of any con-
siderable thickness must be absolutely lost in white light, after ten or
twelve alternations of colours at most, while these effects would require
the whole process to remain unaltered, or rather to be renewed, after
many thousands or millions of changes. (Plate XXX. Fig. 452.)

It is presumed, that the accuracy, with which the general law of the
interference of light has been shown to be applicable to so great a variety
of facts, in circumstances the most dissimilar, will be allowed to establish
its validity in the most satisfactory manner. The full confirmation or
decided rejection of the theory, by which this law was first suggested, can
be expected from time and experience alone ; if it be confuted, our
prospects will again be confined within their ancient limits, but if it be
fully established, we may expect an ample extension of our views of the
operations of nature, by means of our acquaintance with a medium, so
powerful and so universal, as that to which the propagation of light must
be attributed.

[The principle of interference which Dr. Young advanced in this lecture
and elsewhere, has done much towards establishing the undulatory theory
as a true physical theory. This principle explains in the most satisfac-
tory way, not only the colours of thin plates, the fringes which accompany
shadows, and the like, but more refined and complicated phenomena, such
as those produced by placing gratings of different forms before the object
glass of a telescope. The simplest form in which the operation of inter-
fering light is witnessed, and consequently the most direct mode of com-
paring theory witlpi experiment, is to suffer a small pencil of light to fall
on a prism of a very large angle (say 179°). The two sides of this prism
constitute two prisms of an angle of \° each, and serve to bend the same
pencil so as to render it virtually two. By receiving the light from these
two pencils on any eye piece, it is evident that, in different parts of the
field of view, the one will mix with the other in different states of distance
from the original focus. In the centre both will have travelled the same
distance, and there will be a white bar formed by their mixture. On each
side of this, at a certain distance, the one will have travelled further than
the other by half the length of a wave. Here the motions of the one will
be the reverse of those of the other — the one, for instance, tending to raise
a particle of the undulating medium, whilst the other tends to depress it,
and by the same amount. The result is, that no motion at all ensues, and
we are presented with a dark bar : and so on. Moreover, as the lengths
of the waves are different for different colours, the next bright bar will not
be quite white, the space requisite to allow the one pencil to be in advance
of the other l?y a whole undulation (which is equivalent to not being in
advance of it at all), being less for the violet rays than for the red. We
find, consequently, a coloured fringe ; and as we recede from the centre,
the bars become more and more coloured, until the dark of the one alto-]

ON THE NATURE OF LIGHT AND COLOURS. 371

[gether obliterates the light of the other colour. Nothing can be more
satisfactory than the explanation which the theory affords of such pheno-
mena, and, whilst we do not assert that it has as yet brought every
observed fact within its pale, yet it does not appear that the arguments
which were raised against it have any power to shake it. Regarding it
as true, we shall adopt its language in giving a very brief sketch of the
phenomena of polarized light.

It has been stated that when a pencil of light falls on a surface of Ice-
land spar, it is divided into two. Huygens, by the hypothesis that one
series of waves diverges into a spheroid, whilst the other diverges into a
sphere, gave a most satisfactory explanation of the course of the two rays ;
and his conclusions were confirmed by the accurate measurements of Wol-
laston.* Dr. Young t perceived that this difference of divergence must
arise from a difference of elasticity within the crystal. Combining this
with the idea of Newton, that a ray of light possesses sides, the hypothesis
of a transverse vibration is a natural result. Dr. Young advanced this
hypothesis about 1817, and from that period the progress of the theory
has been rapid and satisfactory. The hypothesis consists in supposing
that the particles of light do not, like those whose motions constitute sound,
oscillate in the direction of the wave, but transversely to it, so as more to
resemble those of the particles of water which move up and down whilst
the wave advances horizontally. The explanation of double refraction is
now quite simple. A ray of light falls on the surface of a crystal, the
elasticity of which is different in different directions. The motions, con-
sequently, are not all transmitted with the same velocity, and as the index
of refraction depends on the velocity, one set of vibrations will, on emer-
gence, be totally separated from another. Moreover, the light on emerging
is quite different from common light. In each ray it consists only of vibra-
tions in one direction. Suppose, therefore, one of these rays to fall on a
second crystal placed in a similar position with the first, it will not now be
divided into two, but will emerge just as it entered. Light which consists of
vibrations in one direction only is termed polarized light. It was discovered
by Malus that light reflected from the same face of unsilvered glass is more
or less polarized ; and Brewster ascertained that it is perfectly so, when the
tangent of the angle of incidence is equal to the refractive index, and also
that the transmitted ray is partially polarized. Moreover, Seebeck and Biot
discovered a property of the tourmaline, that when it is cut into slices,
whose surfaces are parallel to the axis of the crystal, it absorbs one of the
two rays, and consequently transmits a polarized ray only. Thus we are
presented with various ways of effecting the polarization of light. The
simplest to understand is that by the tourmaline, and to it we shall conse-
quently refer. On looking through a plate of tourmaline, the effect to the
eye is similar to that produced by a bit of coloured glass. If a second
plate of tourmaline be placed on the first, so that their axes are parallel to
each other, the same is true. But if the axis of the one be perpendicular
to that of the other, the one horizontal, the other vertical, the compound
plate becomes opaque. The first suffering only horizontal vibrations to]
* Ph. Tr. 1802, p. 381. f Quarterly Review, 1809, ii.344.

2B2

372 LECTURE XXXIX.

[pass through it, the second only vertical ones. Another remarkable pro- t
perty of crystals was discovered by M. Arago, that of depolarizing light.
A plate of Iceland spar cut perpendicular to the axes, and placed between
two tourmalines, exhibits a beautiful series of concentric rings broken
by a dark or bright rectangular cross. This complex phenomenon admits
of the readiest explanation. Suppose the axes of the tourmalines at right
angles to each other. The light which has passed through the first con-
sists of horizontal vibrations only. These fall on the plate of Iceland
spar, which being symmetrical relative to its axis, those vibrations which
fall perpendicularly on it pass through without suffering any modification.
They are subsequently stopped by the second tourmaline, and hence a
dark horizontal band. For a nearly similar reason there is a dark vertical
band. The direction of motion of the particles in these cases is either
coincident with or at right angles to a plane which passes through the ray
and the axis of the crystal. But in other places, the direction of motion
is oblique to such a plane, and the ray is doubly refracted, so that on
emergence it consists of two, which on being united no longer form a
polarized ray as before. The second tourmaline, consequently, is incapa-
ble of wholly absorbing this ray, and thus we are presented with brightness.
Moreover, the distance from the centre at which the maxima of brightness
occur depends on the length of the wave ; these maxima are therefore
recurring, and form rings, which, since the waves of different colours are
different in length, must be coloured.

Such is an outline of the explanation afforded by the principles of this
lecture to the phenomena of double refraction and polarization. To enter
into detail exceeds our limits. We must refer the reader to Airy's
Mathematical Tracts, or Lloyd's Lectures on Light, the latter being a
popular treatise. In Gehler's Physikalisches Worterbuch, art. Undula-
tions (1842), is a tolerably complete analytical investigation of the subject.]

LECT. XXXIX.— ADDITIONAL AUTHORITIES.

Diffraction. — Hooke, Ph. Tr. 1672. Newton's Op. lib. 3. Maraldi, Hist, et
M6m. 1723, p. 111. Dutour, Mein. des Sav. Etr. v. 636. Stratico, Saggi di Pa-
dova, ii. 185. Jordan on the Inflections of Light, Lond. 1799. Arago, Ann. de
Chimie, i. 199, 332; Sur la Scintillation des Etoiles, xxvi. 431. Rapport sur
quelques Mem. xi. 5. Fresnel, Mem. de 1'Acad. v. Annales de Ch. xi. 246, 337.
Frauenhofer, Neue Modification des Lichtes, Miinchen, 1818. Gilbert's Ann. Ixxiv.
337. Mayer, Phaenom. ab Inflexione Luminis pendent. Com. Gott. 1820, p. 49.
T. Young on Frauenhofer's Experiments, Ed. Jour, of Sc. New Series, i. 112.
Airy, Camb. Tr. vol. v.

Coloured Rings.— Boyle, Experiments touching Colours, 1663. Hooke, Microg.
and Birch's Hist. iii. 29, 53. Newton, Op. lib. 2. Jordan on the Colours of thin
transparent Bodies, 1800. T. Young on the Colours of thin Plates, Jour, of the
Roy. Inst. i. 241. Introduction to Medical Literature, p. 556. W. Herschel, Ph.
Tr. 1807, pp. 180, 189, 338; 1810, p. 365. Knox, ibid. 1815, p. 161. Arago,
Mem. d'Arcueil, iii. 223. Brewster, Ed. Tr. 1815, xii. 191. Airy on a Modifica-
tion of Newton's Rings, Camb. Tr. iv. 219, 409.

Colours by Reflection. — Brewster on the Optical Phenomena of Mother-of- Pearl,
Ph. Tr. 1814, p. 397. Colours of grooved Surfaces, ibid. 1829, p. 301.

Miscellaneous.— Babinet, Mem. d'Op. Compte Rendue, 1837, p. 638.

ON POLARIZED LIGHT.
Apparatus.— Biot on Tourmaline, Ann. de Chimie, 1815. Seebeck, in Biot's

ON THE NATURE OF LIGHT AND COLOURS. 373

Traite de Ph. vol. iv. Marx (two tourmalines) Schweigger's Jahrb. xix. 167.
A'iry on a new Analyser, Camb. Tr. Nicol's Polarizing Prism, Edinb. Ph. Jour.
xx. 83. Hachette, Descrip. de 1'App. de M. No'rrenberg. Bullet, dela Soc. Philom.
1833. Dove, Pogg. Ann. xxxv. 596. Scientific Memoirs, i. 86.

Polarization by Reflection. — Malus sur une Propriete de la Lumiere Reflechie,
Mem. d'Arcueil, ii. 143, 254. Mayer, Com. Gott. 1813, p. 1. Brewster on the
Laws of Polarization by Reflection, Ph. Tr. 1815, p. 125 ; 1830, pp. 69, 145. See-
beck on do. 4to, Berlin, 1830; and Pogg. Ann. xx. 27; xxi. 22, 290; xxii. 126 ;
and xxxviii. 276.

Polarization by Refraction.— Brewster, Ph.Tr. 1816, p. 46; 1830, pp. 133, 145.

Circular Polarization by Rrflection. — Fresnel, Ann. de Chimie, xxix. 175.

Elliptic Polarization. — Fresnel, Mem. sur la Loi des Modifications que la Re-
flexion imprime & la Lumiere polarisee, Ann. de Chimie, xlvi. 225. Brewster, El-
liptic Pol. exhibited in the Action of Metals upon Light, Ph. Tr. 1830, p. 28.
Neumann, Theorie, Pogg. Ann. xxvi. 89.

Depolarization. — Arago, Mem. sur une Modification, qu'eprouvent les Rayons
Lumineux dans leur Passage a travers certains Corps Diaphanes, Mem. de 1'Jn-
stitut, xii. 93 (1811). Brewster, New Phil. Inst. 1813. On the Affections of Light
transmitted through crystallized Bodies, Ph. Tr. 1814, p. 187, Laws of Pol. in
Crystals, 1818, p. 199. Biot, Mem. de 1'Institut, 1812, i. 1, II. i. ; 1816, p. 275 ;
1818, p. 135. Mem. d'Arcueil, iii. 132. Lloyd on the Phenomena by Light
passing along the axes of biaxal Crystals, Ph. Mag. xi. 112, 207. Potter, ibid. ;
together with various memoirs on the action of different crystals, such as Brewster
on Agate, Ph. Tr. 1813, p. 101 ; on Calcspar, &c. 1814, p. 203; 1815, p. 270.
Ed. Tr. viii. 270. Apophyllite, ibid. ix. 317. Glauberite, ibid. xi. 273. Analcime,
1822. Amethyst, ix. 139. Muriate of Soda, &c. viii. 157. Topaz, Camb. Tr.
1822, ii. 1. Lithion Mica, Ed. Jour. ii. 205. Oxhaverite, No. 13, p. 115.
Diamond, ibid. iii. 98 ; Ph. Mag. vii. 245. Haytorit, Ed. Jour. vi. 301. Fresnel
on Rock Crystal, Ann. de Ch. xxviii. 147. Herschel on do. Camb. Tr. i. 43.
Herschel on Borax, Quetel. Corres. vii. 77. Bicarbonate of Potash, art. Light,
§ 1082, Apophyllite, Camb. Tr. i. 241. Airy on Quartz, Camb. Tr. iv. 79, 199.
Miller, Crystals of Oblique Prismatic System, Camb. Tr. v. 3.

Compressed and heated Glass, 8?c.— Brewster, Ph. Tr. 1814, p. 436 ; 1815, pp.
1, 60 ; 1816, pp. 46, 311 ; Ed. Tr. viii. 353. Effect of Comp. on Crystals, ibid,
viii. 281. Effect of Heat on do. Mitscherlich, Pogg. Ann. viii. 519; x. 137;
xli. 213. Brewster, Ph. Mag. i. 417. Rudberg, Pogg. Ann. xxvi. 291. Neumann,
ibid. xxxv. 81. Seebeck, Schweig. Jour. vii. 284. Effect of Vibration on Glass,
Biot, Ann. de Ch. xiii. 151. Double Refraction of do. Fresnel, Annales de Ch.
xx. 376. Brewster, Ph. Tr. 1830, p. 87. Guerard, Compte Rendue, xix. 474.

Fluids.— Biot, Traite de Phy. iv. 536 ; Ann. de Chimie, Iii. 58, 72.

Undulatory Theory. — Besides many memoirs already mentioned, see the follow-
ing Treatises : — Young, Supp. to Encyc. Brit. art. Chromatics. Fresnel, Supp.
a la Traduction Fransoise de la 5me ed. du Traite de Chimie de Thomson, par
Riffault, Paris, 1822. Herschel's art. Light, in the Encyc. Metrop. and the French
Translation of it by Quetelet and Verhulst. Airy's Tract on the Undulatory
Theory, in his Tracts, 2nd edition, Camb. 1831. Schwerd, Die Beugungserschei-
nungen aus den FundamentaJgesetzen der Undulations Theorie analytisch ent-
wickelt, 4to, Munich, 1835. Powell, The Undulatory Theory applied to Disper-
sion, &c. 184. Lloyd's Lectures, Dublin, 1836-41.

Memoirs. — Laplace sur le Mouvement de la Lumiere dans les Corps Diaphanes,
Mem. de 1'Inst. 1809, p. 300. Malus, Theorie de la Double Refraction, 4to, 1810.
Fresnel, Annales de Chimie, 1815. Expl. de Refraction, ibid. xv. 379. Note sur
le Calcul des Teintes que la Polarisation developpe dans les Lames crystallisees,
ibid. xvii. pp. 102, 167, 312. Des Anneaux colores, ibid. xxii. 129. Arago, ibid. i.
199. Fresnel sur la Double Refraction, Mem. de 1'Inst. 1827, vii. 45. Navier, sur
le Mouvement des Corps elastiques, ibid. vii. 375. Poisson, do. viii. and x.
Cauchy on do. ibid. ix. 114. Theorie de la Lumiere, ibid. x. 293. Exercises de
Math. v. 19, &c. Ampere, Ann. de Ch. xxxix. 113. On the Laws of Refraction,
Mem. del'Inst. xiv. 235. Neumann, Theorie der Doppleten Strahlenbrechung, Pogg.
Ann. xxv. 418. On Crystalline Reflection, 4to, Berl. 1837, and Berlin Mem. xxii.
•1. Challis, Ph. Mag. xi. 161. Hamilton, Theory of Systems of Rays, Ir. Tr. xv.
69; xvi. 1, 94. M'Cullagh on Double Refraction, ibid. xvi. Geometrical Pro-
positions applied to the Wave Theory of Light, ibid, xviii. On Crystalline Reflec-

374 LECTURE XL.

tion, ibid, xviii. Kelland on the Transmission of Light in crystallized Media, Camb.
Tr. vi. 323. On Reflection, Ed. Tr. xiv. 393 ; xv. 37, 511. On the Aggregate
Effect of Interference, Camb. Tr. vii. Ed. Tr. xv. 315. Green on Reflection,
&c. Camb. Tr. vii.

LECTURE XL.

<i

ON THE HISTORY OF OPTICS.

THE science of optics is not one of those which had been cultivated with
the greatest diligence and success by the philosophers of antiquity ; almost
every refinement relating to it has originated in the course of about two
centuries ; and some of its greatest improvements have been made within
these fifty years. The reflection of the rays of light is indeed an occur-,
rence too frequent and too obvious to have escaped the notice even of the
earliest observers : a river or a fountain was the first mirror ; its effect was
easily imitated by speculums of metal : and as soon as any philosophical
attention was paid to the phenomenon, it was easy to collect the equality
of the angles of incidence and reflection ; but although it was well known
that an oar, partially immersed in water, no longer appeared straight, it
was long before any attempts were made to ascertain the relation between
the angles of incidence and refraction. The Greeks were, however,
acquainted with the properties of the burning glass, which was sold as a
curiosity in the toy shops ; for it is well known, that one of the per-
sonages introduced by Aristophanes,* proposes to destroy the papers
[writing in wax] of his adversary by the assistance of this instrument.
The magnifying powers of lenses were, however, but little understood,
although it is scarcely credible that they could have escaped the notice of
a person in possession of a burning glass ; it appears from Seneca that the
Romans at least were informed of the effects of spherical refracting sub-
stances, and it is not improbable that some use was occasionally made of
them in the arts.

Empedocles is perhaps the first person on record that wrote systemati-
cally on light. He maintained that it consisted of particles projected from
luminous bodies, and that vision was performed both by the effect of these
particles on the eye, and by means of a visual influence emitted by the eye
itself. Both of these doctrines were combated by Aristotle,t who thought
it absurd to suppose that a visual influence should be emitted by the eye,
and that it should not enable us to see in the dark ; and who considered it
as more probable that light consisted in an impulse, propagated through a
continuous medium, than in an emanation of distinct particles. Light, he

• * Nubes.

t -De Sensu, ii. &c. But compare Meteor, i. 6 ; iii. 4, 5 ; and see authorities in
Kelland 's Lectures, p. 6.

ON THE HISTORY OF OPTICS. 375

says, is the action of a transparent substance ; and if there were absolutely
no medium between the eye and any visible object, it would be absolutely
impossible that we should see it.

It is said that Archimedes made a compound burning mirror, of suffi-
cient power to set on fire the Roman ships ; in this form the story is
scarcely probable, although the possibility of burning an object at a great
distance by a collection of plane mirrors has been sufficiently shown by
the experiments of Buffon.* It is, however, not unlikely that Archimedes
was acquainted with the properties of reflecting surfaces,t and that he
confirmed his theories by some experimental investigations. The work on
catoptrics, attributed to Euclid, contains the determination of the effects of
reflecting surfaces of different forms ; but it is not supposed to be genuine.
The existence and the magnitude of the atmospheric refraction were well
known to Ptolemy, and a treatise of this astronomer on the subject is still
extant in manuscript.

The mathematical theory of optics, or the science of dioptrics and
catoptrics, made some advances in the middle ages from the labours of
Alhazen and Vitellio.;j: Alhazen was mistaken in some of his proposi-
tions respecting refraction ; Vitellio, a native of Poland, gave a more
correct theory of this subject, and constructed a table of refractive densi-
ties, showing the supposed proportions of the angles of incidence and
refraction in the respective mediums. §

The invention of the magic lantern is attributed to Roger Bacon, and
the lens was soon afterwards commonly applied to the assistance of de-
fective sight. It has been much disputed whether or no Bacon was ac-
quainted with telescopes ; the prevalent opinion is, that the passages,
which have been alleged to prove it, are insufficient for the purpose ; but
there is reason to suspect, from the testimony of Recorde,|| who wrote in
1551, not only that Bacon had actually invented a telescope, but that
Recorde himself knew something of its construction. Digges also, in a
work published in 1571, IF has a passage of a similar nature, and from
Bacon's own words it has been conjectured that an instrument resem-
bling a telescope was even of much higher antiquity. But the first person
who is certainly known to have made a telescope, is Janson, a Dutchman,
whose son, by accident placing a concave and a convex spectacle glass at
a little distance from each other, observed the increased apparent mag_
nitude of an object seen through them ; the father upon this fixed two
such glasses in a tube a few inches long, and sold the instrument in this
form.** He also made some telescopes of greater powers, and one of his

* Ph. Tr. 1748, p. 504.

f See Kircher's Ars Magna Lucis et Umbrae, 4to, 1646. Parsdhs, Ph. tr.
1754, p. 621.

+ Opticse Thesaurus per Risnerum, fol. Basle, 1572.

§ Kepler, in his Paralipomena ad Vitellionem, 4to, Frankf. 1604, laboured un-
successfully to discover the true law of refraction.

|| See Ph. Mag. xviii. 245.

^f Pantometria.

** Borellus, De Vero Telescopii Tnventore, 4to, Hagse, 1655.

376 LECTURE XL.

family discovered a satellite of Jupiter with them.* Galileo t had heard
of the instrument, but had not been informed of the particulars of its con-
struction ; he reinvented it in 1609, and the following year;}; rediscovered
also the satellite which Janson had seen a little before.

It was, however, Kepler § that first reduced the theory of the telescope
to its true principles ; he laid down the common rules for finding the focal
lengths of simple lenses of glass ; he showed how to determine the magni-
fying power of the telescope, and pointed out the construction of the simple
astronomical telescope, which is more convenient for accurate observations
than the Galilean telescope, since the micrometer may be more easily applied
to it ; a third glass, for recovering the erect position of the object, was after-
wards added by Scheiner, and a fourth, for increasing the field of«view, by
Rheita. Kepler made also some good experiments on the nature of coloured
bodies, and showed the inverted situation of the image formed on the retina
of the eye. Maurolycus || of Messina had demonstrated, in 1575, that the
pencils of light are brought to focal points on the retina ; Kepler's obser-
vations were thirty or forty years later.

The next great step in optics was made by De Dominis,1F who in 1611
first explained the cause of the interior or primary rainbow, and this was
soon followed by a still more important discovery respecting the nature of
refraction, first made by Snellius, who ascertained, about 1621, that the
sines of the angles of incidence and refraction are always in the same pro-
portion to each other at the same surface ; he died, however, in 1626,
without having made his discovery public. Descartes is generally supposed
to have seen Snellius's papers, although he published the law of refraction**
without acknowledging to whom he was indebted for it. Descartes also
explained the formation of the secondary rainbow, ft and truly determined
the angular magnitude of both the bows from mathematical principles ; he
did not, however, give a sufficient reason for the production of colours in
either case. Descartes imagined light to consist in motion, or rather pres-
sure, transmitted instantaneously through a medium infinitely elastic, and
colours he attributed to a rotatory motion of the particles of this medium.^
He supposed that light passed more rapidly through a denser medium than
through a rarer; other philosophers about the same time maintained a
contrary opinion, without deciding with respect to any general theory of
light : thus Fermat and Leibnitz deduced, on this supposition, the path of
refracted light from the natural tendency of every body to attain its end

* Borellus, De Vero Telescopii Inventore, 4to, Hagse, 1655, p. 40. ThatBo-
rellus had no just grounds for this statement is shown by Moll, Journal of the Roy.
Inst. Nos. 2 and 3 ; and Drinkwater, Ph. Mag. 1832, i. 14. The credit of the dis-
covery of Jupiter's satellites is certainly due to Galileo.

t Opere, ii. 4.

£ Ibid. p. 17, and Nuncius Sidereus, Venet. 1610.

§ Dioptrice, 4to, Augsb. 1611.

II Theoremata deLumine, 4to, Lugd. 1613.

1[ De Radiis visis in Iride, 4to, Venet. 1611. But see Descartes, Meteorum, cap.
viii. p. 196.

* Specim. Dioptrices, chap. ii. § 7. See Huygens, Dioptrica, p. 2.

ft Spec. Meteorum, chap. viii. ++ De Lumine, chap. i.

ON THE HISTORY OF OPTICS. 377

by the shortest possible way ;* and Barrowf derived the same law, in a
more geometrical manner, from a similar hypothesis respecting the velocity
of light, by considering a pencil of light as a collection of collateral rays
influencing each other's motions. We are indebted to this learned mathe-
matician for the first accurate investigation of the properties of refracting
and reflecting surfaces, and for the most general determination of the situ-
ations of focal points.

The industrious Mr. Boyle J had noticed with attention the phosphores-
cence of diamonds, the colours produced by the effect of scratches on the
surfaces of polished metals, and the diversified tints which a bubble or a
film of soapy water usually assumes. His assistant, Dr. Hooke, investi-
gated th«se and other similar appearances with still greater accuracy, and
proposed, in his Micrographia, which was published in 1665, a theory of
light considerably resembling that of Descartes : he supposes that light is
an impulse propagated through a medium highly, but not infinitely,
elastic ; § that refraction is produced by the readier transmission of light
through the denser medium, and that difference of colour consists in the
different law of the particular impulse constituting coloured light, so that
red and blue differ from each other in the same manner as the sound of a
violin and of a flute. He explained the colours of thin plates from the
interference of two such pulses partially reflected from the upper and under
surface; || but the hypothesis which he assumed respecting the nature of
colours, renders this explanation wholly inadequate, nor were the pheno-
mena at that time sufficiently investigated for a complete solution of the
difficulties attending them.

It was still believed that every refraction actually produces colour, instead
of separating the colours already existing in white light ; but in the year
1666, Newton first made the important discovery of the actual existence of
colours of all kinds in white light, which he showed to be no other than a
compound of all possible colours, mixed in certain proportions with each
other, and capable of being separated by refraction of any kind.

About the same time that Newton was making his earliest experiments
on refraction, Grimaldi's treatise on light appeared ;^[ it contained many
interesting experiments and ingenious remarks on the effects of diffraction,
which is the name that he gave to the spreading of light in every direction,
upon its admission into a dark chamber, and on the colours which usually
accompany these effects. He had even observed that in some instances the
light of one pencil tended to extinguish that of another, but he had not
inquired in what cases and according to what laws such an interference
must be expected.

The discoveries of Newton were not received without some controversy
either at home or abroad ; the essential points of his theory were, however,
soon established, but Dr. Hooke very warmly opposed the hypothesis which

* See Maupertius, Hist, et Mem. de Paris^ 1744, p. 417, H. 53.

f Lectiones Opticae, 4 to, Cantab. 1674.

J Works, iii. 304. § Microg. p. 56.

|| Microg. p. 65.

^ Physico-Mathesis de Lumine, Bonon. 1665.

378 LECTURE XL.

Newton had suggested respecting the nature and propagation of light.* On
this subject Newton professed himself by no means tenacious ; he was not,
however, convinced by Dr. Hooke, and disliked the dispute so much, that
he deferred the publication of his treatise on optics till after Hooke's death
in 1703. Veiy soon after his first communication to the Royal Society, in
1672, he had sent them a description of his reflecting telescope, t which was
perhaps the first that had been constructed with success, although Gregory^
had invented his instrument some years before, and a plan of a similar
kind had been suggested by Eskinard § as early as 1615. The principal
parts of the treatise on optics had been communicated at different times to
the Royal Society ; besides the experiments on refraction and the theory of
the rainbow, they consist of an elegant analysis of the colours of thin trans-
parent substances, in which the phenomena are reduced to their simplest
forms, and of a collection of miscellaneous experiments on the colours pro-
duced in cases of inflection or diffraction.

With respect to the nature of light, the theory which Newton adopted
was materially different from the opinions of most of his predecessors. He
considered indeed the operation of an ethereal medium as absolutely neces-
sary to the production of the most remarkable effects of light, but he denied
that the motions of such a medium actually constituted light ; he asserted,
on the contrary, that the essence of light consisted in the projection of mi-
nute particles of matter from the luminous body, and maintained that this
projection was only accompanied by the vibration of a medium as an acci-
dental circumstance, which was also renewed at the surface of every re-
fractive or reflective substance.

In the mean time Bartholin had called the attention of naturalists and
opticians to the singular properties of the Iceland crystal, and had hastily
examined the laws of its unusual refraction. On this subject Huygens had
been much more successful : his analysis of the phenomena of the double
refraction is a happy combination of accurate experiment with elegant
theory ; it was published in 1690, making a part of his treatise on light,
the fundamental doctrines of which he had communicated to the Academy
of Paris in 1678. They scarcely differ in their essential parts from those
of our countryman Dr. Hooke, but the subject of colours Huygens has left
wholly untouched. Roemer had then lately made the discovery of the
immense velocity with which light passes through the celestial regions, by
observing the apparent irregularities of the eclipses of Jupiter's satellites ;
and Huygens readily admitted this property into his system ; although
Hooke, 1 1 by a singular caprice, professed himself more ready to believe
that the propagation of light might be absolutely instantaneous, than that
its motion could be successive, and yet so inconceivably rapid. The merits

* Birch, iii. 10, 52. Ph. Tr. viii. 5084, 6086.

t Ph. Tr. 1672, pp. 4004, 4032; 1673, p. 6087.

i Optica Promote, 1663.

§ Eskinard's Century of Optical Problems.

|| Lectures of Light, in Waller's Life and Works of Hooke, p. 77. From a
passage in the Micrographia, p. 56, it is evident not only that Hoake was ready to
admit the fact of the finite velocity of light, when proved, but that he anticipated
both the manner of proof and the result.

ON THE HISTORY OF OPTICS. 379

of JHuygens in the mathematical theory of optics were no less considerable
than in the investigation of the nature of light ; his determinations of the
aberrations of lenses were the first refinement on the construction of tele-
scopes, but with respect to the theory of halos and parhelia he was less
successful than Mariotte had been some years before.

In the year 1720, Dr. Bradley had the good fortune to discover both the
existence and the cause of the aberration of the fixed stars. He had for
some time observed an irregularity in the places of the stars, which he was
wholly unable to explain, and the idea of attributing it to a combination of
the effect of the earth's motion in its orbit, with the progressive motion of
light, occurred to him first as he happened to observe the apparent di-
rection of* the wind on board of a boat which was moving in a transverse
direction. He also determined with accuracy the magnitude of the at-
mospherical refraction,* which had been theoretically investigated by
Newton and by Taylor,t but never before practically ascertained with
sufficient precision. The formula, which Bradley appears to have deduced
from observation only, agrees precisely with an approximation which was
obtained by Simpson J from calculation ; but it cannot be considered as
rigidly accurate.

The optics of Bouguer were first published in 1729, and an improved
edition appeared thirty years afterwards ; the merits of this author in the
examination of the properties of a variety of substances, with respect to
the transmission and reflection of light in different circumstances, and in
the comparison of lights of different kinds, require to be mentioned with
the highest commendation. § Dr. Porterfield's|| investigations of the
functions of the eye tended greatly to illustrate the economy of this admi-
rable organ, and some valuable remarks of Dr. Jurin on the same subject
were soon after published in Dr. Smith's elaborate treatise on optics, which
contains all that had been done at that time with respect to the mathema-
tical part of the science.

The invention of achromatic telescopes is with justice universally attri-
buted to our countryman Mr. Dollond,^[ but there is reason to believe that
he was not absolutely the first author of the improvement. Mr. Hall, a
gentleman of Worcestershire, is said to have discovered, about the year
1 729, Sir Isaac Newton's mistake, in supposing that the rays of different
colours must of necessity be equally separated by all surfaces which pro-
duce an equal mean refraction ; and by combining the different dispersive
properties of different kinds of glass, he constructed, in 1733, several com-
pound object glasses, which were calculated not only for avoiding all ap-
pearance of colour, but also for correcting the imperfect refractions of the
spherical surfaces of the separate lenses. He did not, however, make known
the particulars of his investigations, and his invention was soon wholly
forgotten. It was in consequence of a discussion ** with Euler, Klingen-

* See Ph.Tr. 1787, p. 156. f Methodus Incrementorum, p. 108.

J Mathematical Dissertations, 4to, 1743, p. 46.

• § Hist, et Mem. 1726, H. 11 ; 1757, p. 1. Optique, 4to, Paris, 1760.
|| On the Eye and Vision, 2 vols. Edin. 1759.

^ Ph. Tr. 1. 735. Compare Newton's Optics, book i. part ii. Prop. 3, Exp. 8.
** Hist, et Mem. 1756-7.

380 LECTURE XL.

stierna, and some other mathematicians, that Mr. Dollond was led to make
experiments on the refraction of different kinds of glass ; these gentlemen
had not questioned the general truth of Newton's opinion respecting the
dispersion of the different colours, but Euler had asserted that the eye itself
produced a refraction free from the appearance of colour, and Klingen-
stierna* had shown the possibility of producing a deviation by refraction,
without a separation of colour, according to the laws of refraction laid
down by Newton himself. When Dollond had once discovered the mate-
rial difference which exists between the dispersive properties of flint glass
and of crown glass, it was easy to produce the combination required ; but
this ingenious artist was not satisfied with the advantage of freedom from
colours only ; he adjusted the forms and apertures of his lensts in the
most skilful manner to the correction of aberrations of various kinds, and
he was also particularly fortunate in being able to obtain, about the time
of his discovery, a glass of a quality superior to any that has been since
manufactured.

This opinion of Euler respecting the eye was, however, by no means
well founded, for the eye acts very differently on rays of different colours,
as we may easily observe by viewing a minute object in different parts of
a beam of light, transmitted through a prism. It must be allowed that
this great mathematician was less fortunate in his optical theories than in
many other departments of science ; his mathematical investigations of the
effects of lenses are much more intricate and prolix than the subject
actually requires, and with respect to the nature and propagation of light,
he adopted several paradoxical opinions. Assuming the theory of Huygens,
with the additional hypothesis respecting the nature of colours, which had
been suggested by Newton, and maintained by Pardies and Malebranche,
that is, that the difference of colours, like that of tones in music, depends
on the different frequency of the vibrations constituting light ; he imagined
that opaque bodies are not seen by reflected light, but that their particles
are agitated by the impulse of the light which falls on them, and that the
vibrations of these particles render the bodies again visible in every direc-
tion ; he also conceived that the undulations of light are simply propagated
through the solid substances of transparent mediums, in the same manner
as sound travels through the air. But on these suppositions, all bodies
would have the properties of solar phosphori, and the refraction of the
rarest of natural bodies would be incomparably greater than that of the
densest is actually found to be : and on the whole, although the character
of Euler has been so highly and so deservedly respected as to attach a cer-
tain degree of authority to all his opinions, so that in this instance the
name of Huygens has been almost superseded by that of Euler, yet in fact
he has added no argumentative evidence whatever to the theory, but, by
inaccurate and injudicious reasoning, has done a real injury to the cause
which he endeavoured to support.

The researches of Lambert t may be considered as a continuation of

* Schwed. Abhand. 1754, xvi. 300; 1760, xxii. 79. Ph. Tr. 1760, p. 944 ; and
Tentameu de Corrig. Aberrat. Luminis, 4to, Petrop. 1762.
f Photometria, Augsb. 1760.

ON THE HISTORY OF OPTICS. 381

those of Bouguer ; they present us with many interesting observations on
the natural history of light, and the properties of various bodies with regard
to it. Mr. Lambert first ascertained that a luminous surface emits its light
very nearly with equal intensity in all directions, so that any part of it
appears almost equally brilliant to an eye placed in any direction, while
the light thrown by each square inch or square foot of the surface in any
direction differs according to the obliquity of that direction. The mathe-
matical theory of optics is considerably indebted to the labours of
Clairaut,* Dalembert,t and Boscovich ; ^ Jeaurat, § Beguelin,|| Redern,*[
anc1, Kliigel** have also continued the investigation ; their calculations
may be of considerable utility to the practical optician, but it requires the
ingenuity of a Dollond or a Ramsden to apply the whole of the results to
any useful purposes.

The experiments of Mazeas ft on the colours of thin plates are mere re-
petitions of those of Newton under disadvantageous circumstances ; Mr.
DutourJ^ has, however, considerably diversified and extended these expe-
riments, as well as those on the colours which are produced in diffracted
light, yet without obtaining any general results of importance. Compa-
retti's§§ experiments on inflection have every appearance of accuracy, but
they are much too intricate to be easily compared with each other, or with
those of former observers.

The late Dr. Priestley |||| rendered an essential service to the science of
optics, considered as a subject for the amusement of the general reader, by
an elegant and well written account of the principal experiments and
theories, which had been published before the year 1770. But this work
is very deficient in mathematical accuracy, and the author was not suffi-
ciently master of the science to distinguish the good from the indif-
ferent.

Mr. Delaval's^T^T experiments on colours appear to show very satisfac-
torily, that all the colouring substances, in common use, owe their tints to
rays, which are separated from white light, during its passage through
them, and not, as Newton supposed, to the reflection of a particular colour
from the first surface. It has been observed that Kepler and Zucchius ***
had long ago made experiments nearly similar to those of Mr. Delaval.

* Hist, et Mem. de Paris, 1756, vii.380, H. 112; 1757, p. 524, H. 153; 1762,
p. 578, H. 160.

t Ibid. 1764, p. 75, H. 175; 1765, p. 53, H. 119; 1767, p. 43, H. 153.
Opuscules, vol. i. Hist, et Mem. de Berlin, 1769, p. 254.

t Com. Bon. v. II. 265.

§ Hist, et Mem. de Paris, 1770, p. 461, H. 103.

|| On the Improvement of Telescopes, Hist, et Mem. de Berlin, 1762, pp. 66,
343 ; 1764, p. 7 ; 1769, p. 1 ; 1784, H. 40.

H On Object-Glasses, ibid. 1759, p. 89 ; 1760, p. 3; 1761, p. 3.

** On do. Comm. Gott. 1795, xiii. 2, 28. Gilbert's Annalen, xxiv. 265, 276.
Analytische Dioptrik, 4to, Leipz. 1778.

ft Hist, et M6m. de Berl. 1752, p. 262. Mem. des Savans Etrangers, ii.26.

U Ibid. vols. iv. and v. Rozier's Journal, i. 368; ii. 11, 249; v. 120, 230;
vi. 135, 330, 341,412.

§§ De Luce Inflexa et Coloribus, 4to, Pad. 1787.

HI) The History and present State, &c. of Vision, Light, and Colours, 4to, Lond.
1772.

HH Manch. Mem. ii. 131. *** Optica Philosophia, 2 vols. 4to, Lugd. 1652-6.

382 LECTURE XL.

Dr. Robert Darwin's* investigation of the effects of strong lights on the eye
appears to comprehend almost all possible varieties of these ocular spectra,
but it does not lead to any fundamental analogy, capable of explaining the
most intricate of them.

The phenomena of the unusual atmospheric refraction, which frequently
produces double or triple images of objects seen near a heated surface, have
been successively illustrated by Mr. Huddart,t Mr. Vinc«,^ and Dr. Wol-
laston, § so that at present there appears to be little doubt remaining with
respect to their origin. Dr. Wollaston's instrument for the measurement
of refractive densities, very much facilitates the examination of the optical
properties of substances of various kinds : he has applied it very success-
fully to the confirmation of Huygens's theory of double refraction^; he has
corrected the common opinion respecting the division of the prismatic
spectrum ; he discovered, without being acquainted with the observations
of Ritter, the dark rays which blacken the salts of silver ; and he has re-
marked a singular property in some natural as well as artificial crystals,
which appear of one colour when viewed in the direction of the axis, and
of another when in a transverse direction.

To Dr. Herschel the sciences of optics and astronomy are equally in-
debted. He has carried the construction of the reflecting telescope to a
degree of perfection, far exceeding all that had been before attempted, and
the well known improvements which astronomy has derived from his ob-
servations are numerous and important. In the course of his researches
for the attainment of his more immediate objects, he has also had the good
fortune to discover the separation of the rays of heat from those of light by
means of refraction ; a fact which has been sufficiently established by the
experiments of several other persons.

The investigations of Mr. Laplace, relating to atmospherical refraction,
may be considered as the latest application of refined mathematics to the
purposes of optics and of astronomy. I have myself attempted to attain a
degree of certainty, in attributing the changes of the refractive powers of
the eye to a variation in the form of the crystalline lens ; I have discovered
a general law of the mutual action of two portions of light interfering with
each other, to which no exception has yet been shown ; and by reviving a
theory of light similar to that of Hooke and Huygens, with an improve-
ment originally suggested by Newton, respecting the nature of colours, 1
have endeavoured to obtain a satisfactory explanation of many circum-
stances, which appear, upon a minute examination, to be in every other
hypothesis difficulties absolutely insuperable. It cannot be expected that
all objections to such a system will at once be silenced, but if a full and
candid discussion only of the facts which I have advanced, should be ex-
cited, I trust that the science of optics will be essentially benefited, even
if the theory should be ultimately confuted.

* Ph. Tr. 1786, p. 313. > Ph. Tr. 1797, p. 29.

t Ph. Tr. 1799, p. 13. § Ibid. 1800, p. 239 ; 1803, p. 1.

ON THE HISTORY OF OPTICS. 383

For the history of optics consult Priestley's Hist. 4to, 1772. Pringle on the
Invention of the Telescope, 4to, Lond. 1778, Ph. Mag. xviii. 245 ; xix. 66, 176,
232, 344 ; xx. 14. Venturi, Comm. sopra la Storia e la Teoria dell' Ottica, Bolog.
1814. Meister, Nov. Comm. Gott. v. V. 141 ; VI. 189. Arago, Ann. de Ch. xiv.
434. Lloyd, Report of Brit. Assoc. 1835. Powell, British Annual. 1837.

WORKS ON NATURAL PHILOSOPHY IN GENERAL, NOT QUOTED IN THE PRECEDING

LECTURES.

Sennerti, Philosophia Naturalis, 4to, Wittenb. 1618. Herigone, Cursus Math.

5 vcis. Paris, 1634-7, Etten, Mathematicall Recreations, by Oughtred, 1652.
Jungii Doxoscopise Physicse Minores, 4to, Hamb. 1663. Power's Experimental
Philosophy, 4to, 1664. Duchess of Newcastle's Do. fol. 1666. Senguerdi, Philo-
sophia N«eturalis, 4to, Leyd. 1685. Paul Hoste, Hydrostatique, &c., the part on
Naval Tactics translated by Capt. J. D. Boswall, 4to, Ed. 1834. Hoffmanni Lexi-
con Universale, 4 vols. fol. Leyd. 1698. Muys, Elementa Physices, 4to, Amst. 1711.
Scheuchzer's Naturwissenchaft, 2 vols. 8vo, Tur. 1711. Nieuwentyt's Religious
Philosopher, 3 vols. 8vo, 1719. Verdries Conspectus Philos. Nat. Giess. 1720.
Wolff's Niitzliche Versuche, 3 vols. Halle, 1721-43. Vernunftige Gedanken, 3
vols. Halle, 1723-5. Keill's Natural Philosophy, 1726. Pemberton's New-
tonian Philosophy, 4to, 1728. Crivelli Fisica, 2 vols. 4to, Ven. 1731-2. Mo-
liere's Lemons de Physique. Teichmeyeri Philos. Nat. 4to, Jena, 1733. Ham-
bergi Elementa Physices, Jena, 1735. Helsham's Lectures on Natural Philosophy,
1739. Bulfingeri Elementa Physices, Leipz. 1742. Nollet, Lefons de Phy-
sique, 6 vols. 12mo, Paris, 1743. Segner's Einleitung, 1746, 1770. Ruther-
forth's Natural Philosophy, 2 vols. 4to, 1748. Kraftii Prselectiones, 3 vols.
Tubing. 1750. Kriiger's Naturlehre, Halle, 1750. Saverien, Dictionnaire de
Math, et de Physique, 2 vols. 4to, Paris, 1753. Winkler's Natural Phil, (trans.),
1757. Martin's, 1781. Jones's, 1762. Guyton de Morveau, Essais de Physique,
12mo, Dijon, 1762. Hennert, Cursus Math. 6 vols. Traj. ad Rhen. 1768-75. Eu-
ler's Letters to a German Princess (trans.), 2 vols. 1795, 1802. Karsten's Lehr-
begriff, Greifsw. 1764. Anfangsgriinde der Naturlehre, Halle, 1790, &c. Row-
ning's Natural Philosophy, 2 vols. 1765. Sigaud de la Fond, Physique, Paris,
1767, 1771. Silberschlags, Ausgesuchte Versuche, Berl. 1768. Hamberger's
Naturlehre, Jena, 1774. Bookman's Naturlehre, Carlsr. 1775. Senebier, Art.
d'Observer, 2 vols. Geneve, 1775. Ferguson's Lectures, 1776. Goldsmith's
Exp. Ph. 2 vols. 1776. Sauri, Cours de Phy. 4 vols. 12mo, 1777. Gabler's
Naturlehre, 4 vols. Munich, 1778. Richter's Lehrbuch, 1779. De Luc, Lettres
Physiques, 4 vols. La Haye, 1779. Turner's Introduction to Arts and Sciences.
Marivetz, Physique du Monde, 5 vols. Paris, 1780-7. Nollet, Le9ons de Phy.

6 vols. 12mo, 1783-6. Bruckhausen's Physik, von Bergmann. Schurer, Elemens de
Physique, Strasb. 1786. Van Swinden, Positiones Physicse, 2 vols. Harderwick,
1786. Nicholson's Nat. Phil. 2 vols. Lond. 1787. Serrati, Fisica, Flor. 1787.
Kratzenstein's Physik, Copenh. 1787. Gren's Naturlehre, Halle, 1788. Ingen-
housz, Nouvelles Experiences, Par. 1789. Hobert's Naturlehre, Berlin, 1789.
Ciscar Maquinas y Maniobras, fol. Madrid, 1791. Kliigel's Naturlehre, Berlin,
1792. Button's Dissertations on Nat. Phil. 4to, Ed. 1792. Geissler's Beschrei-
bung der Neuesten Instrumenten, 3 vols. Zittau, 1792-7. Hube, Naturlehre,

2 vols. Leipz. 1793. Erxleben's Naturlehre von Lichtenberg, Gott. 1794. An-
derson's Institutes of Physics, Glasgow, 1795. Gregory's Economy of Nature,

3 vols. 1796. Barruel, Physique en Tableaux, 4to, Paris, An. 7. Enfield's
Nat. Phil. 4to, 1799. Adams's Do. 4 vols. 1799. Walker's Do. 4to, 1799.
Brisson, Dictionnaire de Physique, 6 vols. An. 8. Traite de Phy. 1803. Biisch,
Mathematik zum Niitzen, 2 vols. Hamb. 1800. Berard, Melanges, Par. An. 9.
Jacotot, Cours de Physique, 2 vols. Paris, An. 9. Libes, Traite de Physique,
3 vols. 8vo, 1801. Cavallo's Nat. Phil. 4 vols. 1800. Imison's Elements,
2 vols. 8vo, 1803. Kett's Elements of Knowledge, 2 vols. 1803. Button's
Recreations, 4 vols. 1803. Pujoulx, Le9ons de Physique, Par. 1805. Sage,
Institutions de Phy. 3 vols. 1811-12. Barlow's Math. Diction. 1814. Pre-

384 LECTURE XL.

vost, Deux Traites de Phys. Geneve, 1818. Mollet, Cours de Phys. 2 vols. Lyon,
1822. Babinet, Resume Complet de Physique, 2 vols. 32mo, Paris, 1825. Des-
pretz, Traite de Phys. 1827. Natural Philosophy (Lib. of Useful Knowledge),
1829. Fischer, Traite de Physique, trad, de 1'Allemande, par Biot, Par. 1830.
Peclet, Traite de Phys. 2 vols. 1830. Dupre, Traite de Phy. 2 vols. Rennes, 1831.
Beudant, Traite de Phy. Paris, 1832. Regnaud, Cours de Math, et de Phy. 2 vols.
1832. George, Cours de Phy. appliquee aux Arts, Nancy, 1832. Kastner,
Grundziige der Physik, 2 vols. Nuremberg, 1832-3. Quetelet, Positions de Phy-
sique, 3 vols. 12mo, Brux. 1834. Pinault, Traite de Phy. 2 vols. 1836. Lame,
Cours de Phy. 3 vols. Paris, 1836. Somerville (Mrs.), The Connection of the
Physical Sciences, 12mo, Lond. 1838. Pouillet, Traite de Physique, 1842. PeschePs
Physics (trans.), 1845.

ON THE HISTORY OF OPTICS.

385

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PART III

LECTURE XLI.

ON THE FIXED STARS.

THE departments of natural philosophy, which are to be the subjects of
the third and last division of these lectures, are included in the description
implied by the term physics, or the history of the particular phenomena of
nature ; and the account which will be given of these phenomena, will be
accompanied by as much of mechanical theory and analogical reasoning,
as can be applied to them with sufficient certainty, and without too great
intricacy of calculation.

The science of astronomy might, without any great impropriety, have
been considered as a part of mechanics ; but there are circumstances
intimately connected with it, for the complete investigation of which, a
knowledge of the motions of fluids in general, and also of optics, is abso-
lutely necessary. It could not, therefore, hold any other place in a strict
order of arrangement, than that which is here allotted to it ; and, since it
will not be in our power to enter completely into a mathematical examina-
tion of all the motions of the heavenly bodies, although we shall be able to
pursue the detail of the most remarkable appearances which they exhibit,
we may for this reason more properly consider such a view of astronomy
as belonging to descriptive than to theoretical philosophy. This method of
treating the subject is sometimes denominated plain astronomy, in contra-
distinction to the mechanical theory of the science, which is called physical
astronomy ; but it is obvious that in the sense which we are at present
annexing to the word physics, that which is commonly called plain
astronomy must be termed physical or descriptive, and what is usually
called physical, must be denominated mathematical astronomy. We shall,
therefore, confine ourselves in great measure to descriptive astronomy,
and shall take only a general view of the laws of gravitation, as an illus-
tration of the phenomena previously described. After having considered
the magnificent objects of astronomy, which are scattered throughout the
universe, we descend to geography, or the particular history of the terra-
queous globe, and to the tides, produced by the influence of the celestial
bodies on the ocean ; and then, quitting the affections of the larger features

2c2

388 LECTURE XLI.

of the matter that constitutes the earth, we come naturally to the properties
and powers of its individual particles, and to the phenomena of heat,
electricity, and magnetism, which are either qualities of matter, or depen-
dent on substances differing in some respects from common matter ; and
in the next place, to the combination of all these substances and actions in
meteorology, and in the phenomena of vegetable and animal life, a general
view of which will complete our discussions on the subject of physics.
The science of chemistry, or the doctrine of the qualities of particular
kinds of matter, might be said to belong to the investigation of the proper-
ties of matter in general ; but this science is of too great extent1' and
importance to occupy a subordinate place in a system of natural philoso-
phy, and must, therefore, be considered as requiring a separate i^urse of
study.

In our astronomical inquiries, we shall first examine the phenomena of
the heavens and earth in their simplest form, not as they immediately
appear to our observation, but as they are shown by unexceptionable
proofs to be naturally arranged. The stars and sun, the planets and their
satellites, and lastly the comets, will be severally described ; the causes of
the motions of these bodies will be superficially indicated ; their sensible
effects with respect to the inhabitants of the earth will be shown, and the
practical modes of determining their .situations and orbits will be ex-
plained.

When we begin to consider, on a large scale, the affections of matter
and of space, we are impressed, at the first sight, with the inconceivable dis-
proportion between the magnitude of space and of sensible matter ; and we
are naturally led to inquire if the apparently void expanse of the universe
is wholly without all matter or all substance. The atmospheres of the planets
cannot indeed be said absolutely to terminate at any given point, but they
must become rare beyond all imagination at a very moderate distance.
The substance which produces the sensation of light must, however, be
every where found, at least without any sensible interval ; for if an eye
were placed in any point of the regions of unbounded space, wherever
human investigation or fancy can penetrate them, some luminous object
would at each instant be visible to it, and, in general, objects without
number might be seen in every direction. Light, therefore, must be every
where present, whether we suppose it to consist of separate projected cor-
puscles, or to be an affection of a highly elastic ether, pervading the uni-
verse in a state so rare, that although it constitutes a continuous medium,
it suffers all bodies to move through it without sensible resistance, and is
admitted even into their pores with perfect freedom ; and if we follow
Newton's opinion of the nature of light, we must suppose both such an
ethereal medium, nearly at rest, and the particles of light also, moving
swiftly through it, to exist together in all places ; to say nothing of the
possibility of the coexistence of a thousand other unseen and unknown
substances, essences, and influences, in the same individual place, which
may for ever set at defiance the pride of a presumptuous philosophy,
that would aspire to comprehend, within its own contracted sphere, the
whole extent of the mighty work of the creation.

ON THE FIXED STARS. 389

, The expanse of the universe is strewed, at immense distances, with de-
tached portions of a substance, which we suppose to be matter, constituting
stars, or suns, planets, and comets ; bodies which certainly agree with each
other in the power of emitting or reflecting light, and which, in all proba-
bility, have many other properties in common. Such of these as emit
then- own light, are called fixed stars ; and this appears to be the only
criterion that we can apply to a star : for the word fixed is only to be
understood in a comparative sense.

The stars must necessarily shine by their own light ; for if we grant that
they consist of gravitating matter, it must be allowed that no star could be
near enough to another to be seen by reflected light, without a very sensible
change of the places of both in consequence of their mutual gravitation, nor
would Jfbe possible, on account of their immense distance from us, to dis-
tinguish two such bodies from each other. It follows also, on the same
supposition of the universality of the force of gravity, that the form of the
stars must be nearly spherical.

The light of the stars appears to the naked eye to be generally white ;
being too faint to excite the idea of a particular colour ; but when it is
concentrated by Dr. Herschel's large speculums, it becomes, in various
stars, of various hues ; and indeed to the naked eye some of the stars
appear a little redder and others a little bluer. The cause of the twinkling
of the stars is not fully ascertained, but it is referred, with some proba-
bility, to changes which are perpetually taking place in the atmosphere,
and which affect its refractive density. It is said that in some climates,
where the air is remarkably serene, the stars have scarcely any appearance
of twinkling.*

Above two thousand stars are visible to the naked eye ; and when a tele-
scope is employed, their number appears to increase without any other
limit then the imperfection of the instrument. Dr. Herschel has observed
in the milky way above ten thousand stars in the space of a square degree.
Lucretius and Dr. Halley t have argued that their number must be abso-
lutely infinite, in order that all of them may remain at rest by the oppo-
sition of attractions acting in every possible direction ; but we are by no
means certain that they do remain in perfect equilibrium.

Of the actual magnitude of the stars we can give no exact account ; but
they are divided into seven or more orders, according to the degrees of their
apparent brightness. There is, however, reason to suppose, from the quan-
tity of light emitted by the brightest stars, that some of them are much
larger than the sun. Those stars which are below the sixth magnitude are
scarcely visible without the help of telescopes. The distances of all the stars
from us and from one another are so great, as not to be capable of being
immediately compared with their diameters ; for no star subtends an angle
large enough to be ascertained by direct observation. The more perfect the
instruments that we employ, the smaller are the apparent diameters of the
fixed stars. Dr. Herschel found that one of the stars of the first magnitude,
when viewed in his best telescopes, appeared to be about one third of a

* See Garcia, Hist, et Mem. 1743, H. 28 ; and Michell, Ph. Tr. 1767, p. 234.
f Ph. Tr. 1720, xxxi. 22.

390 LECTURE XLI.

second in diameter. But there is always a limit to the perfection of the
focus of the telescope and of the eye, and, however accurate both may be,
the image of every radiant point will occupy on the retina a space of a
certain magnitude, not depending on that of the object : so that it will per-
haps be for ever impossible to measure any angle, which is only a very
small fraction of a second. (Plate XXXI. Fig. 453, 454.)

There is, however, reason to suppose, that the angle subtended by the
nearest stars is in reality more than a hundred times less than the angle
measured by Dr. Herschel, for it may be conjectured that our distance
from the nearest stars is about a hundred million million miles ; taking
about one third of a second for the annual parallax of the earth, that is,
for the change of the apparent places of some of the fixed stars in conse-
quence of the earth's annual motion.* This seems to be nearly the' utmost
amount of an annual parallax that could wholly have escaped observation ;
for Dr. Herschelf supposes that, by means of double stars, a parallax of
one tenth of a second only might become sensible, and even this has never
yet been discovered ; on the other hand, if the parallax were really much
smaller than this, it would be necessary to suppose the actual magnitude or
splendour of the brightest stars to be incomparably greater than that of the
sun ; for at the distance of a hundred million million miles, our sun would
appear, according to Lambert's calculations, but about one fourth as bright
as Saturn, or like a star of the second or third magnitude only. Perhaps,
indeed, the stars may differ as much from each other in magnitude as the
planetary bodies, but it is somewhat more natural to imagine them more
nearly equal, until we have some reason for supposing any material inequal-
ity in their dimensions. At any rate there is little doubt, that the diversity
of their apparent magnitudes is principally owing to their different dis-
tances ; perhaps none of them are much nearer to each other than the
nearest to us ; and there may still be a very great variety in their actual
dimensions. There can be only twelve points on the surface of a sphere as
far from each other as from the centre J ; in a sphere of twice the radius,
there may be about 50 points at the same distance ; in a sphere of three
times the radius, more than 100 : and it has been observed that these
numbers do not greatly differ from the actual numbers of the stars of the

* The accuracy of modern instruments establishes the existence of a sensible paral-
lax to one star at the least. By means of an excellent heliometer, Bessel has obtained
a series of distances of the two stars which constitute the double star 61 Cygni,
from which he concludes that this star has a sensible parallax of about one-third of a
second. Other astronomers have attacked the subject with vigour, and amongst the
rest, Mr. Henderson has made out a highly probable parallax to a Centauri. A dis-
cussion of this subject will be found in Fockens's Commentatio Ast. de annua stel.
paral. Lugd. 1835 ; and in Mr. Main's Report on the present State of our Know-
ledge of the Parallax of the fixed Stars, Trans, of the Astron. Soc. vol. xii.

See also Clairaut, Hist, et Mem. 1739, p. 358. Schubert, Bode's Jahrbuch, 1796.
Piazzi, Mem. della Soc. Ital. 1805, xii. 1809. Calendrelli, Opusc. Astr. 1806.
Brinkley, Ir. Tr. 1815, p. 25. Ph. Tr. 1821. Ast. Soc. vol. i. Pond. ibid. 1817.
J. Herschel, Ph. Tr. 1826, p. 266 ; 1827. Struve, Introd. to Duplicium Mensura,
&c. fol. Dorpat, 1827. Bessel, Astronomische Nachrichten, vol. xvi. Taylor,
Madras Obs. vol. ii. Airy, Ast. Soc. vol. x. Henderson, ibid. vol. xi.

t Ph. Tr. 1782, Ixxii. 82.

I Halley, Ph. Tr. 1720, p. 22. Kastner, Dissert. Math.

ON THE FIXED STARS. 391

first, second, and third magnitudes ; although it is true that they are not
by, any means placed at equal angular distances from each other. But
from a comparison of the light of different stars, we may infer, that if their
real magnitudes are nearly equal, their distances must increase much
faster than in this arithmetical progression ; that is, that the stars of the
second magnitude are more than twice as remote as those of the first, and
those of the third more than three times as remote. Mr. Michell found
the light of Sirius between 400 and 1000 times as great as that of a star of
the sixth magnitude ;* consequently, supposing these stars actually equal,
their distances must differ in the ratio of 1 to 20 or 30 ; since light always
diminishes in proportion to the square of the distance of the luminous ob-
ject. The light of stars of different magnitudes, situated near each other,
may ^Blf compared by viewing them through two apertures of different
sizes, cut in cards, one held before each eye, the apertures being reduced to
such magnitudes, that the stars may appear equally bright ; and the com-
parison may be extended to the light of the sun, by finding a star and a
planet of equal brightness, and calculating what proportion of the sun's
light must be reflected by the planet, upon the most probable supposition
respecting the disposition of its surface to reflect more or less of the light
which falls on it.

The stars are in general dispersed without any regular order, but we may
observe in many parts of the heavens that a number of them are so much
nearer together than to the rest, as to form a cluster or nebula. The an-
cients had noticed some of the most conspicuous nebulae, but Huygens t
first directed the attention of modern astronomers to the large one situated
in the constellation Orion. Herschel^ has now given us catalogues of 2500
nebulae : many of them can be resolved by very high magnifying powers
into separate stars ; but others appear to consist of a luminous matter,
spread uniformly in the neighbourhood of the several stars to which they
seem to belong. (Plate XXXI. Fig. 455... 463.)

It has been conjectured that all stars are disposed in nebulae, and that
those which appear to us to be more widely separated, are individual stars
of that particular nebula in which we are placed, and of which the mar-
ginal parts may be observed, in the form of a lucid zone, which is called
the milky way, being too distant to allow the single stars to be perceived
by the naked eye. This opinion was first suggested by Professor Kant,
the author of the system of metaphysics called the critical philosophy.
The idea was adopted by Lambert,§ who considers the largest stars as con-
stituting a distinct nebula placed among a multitude of others, which toge-
ther produce the appearance of a continued zone ; and Dr. Herschel has
investigated very particularly the figure of a single nebula, which would be
capable of being projected into the form of the milky way.|| We must not,

* An Enquiry into the probable Parallax of the fixed Stars, Ph. Tr. 1767, Ivii.
234.

f Systema Saturnium, p. 8. Ph. Tr. 1716, p. 390.

J Ph. Tr. 1786, Ixxvi. 457 ; Ixxix. 212 ; 1802, p. 477. Catalogue of Nebulae in
Connaissance de Tems for 1783 and 1784. See also J. Herschel, Ph. Tr. 1833, &c.

§ Photometria, § 1139, 1140. || Ph. Tr. 1784, Ixxiv. 437.

392 LECTURE XLI.

however, suppose that each of Dr. HerschePs 2500 nebulae can be at all
comparable in magnitude to this supposed nebula, since many of them are
almost as much resolved by the telescope into single stars as the milky way
itself ; which would be utterly impossible, if the stars which they con-
tain were equally numerous with those of the nebula to which the milky
way belongs. Supposing all the stars of this nebula to be as remote £rom
each other as the nearest of them are from the sun, it may be calculated
that the most distant are about 500 times as far from us as the nearest, and
that light, which is probably 15 or 20 years in travelling to us from Sirius,
would be nearly twenty thousand in passing through the whole diameter of
the milky way. A nebula of the same size as this, appearing like a diffused
light of a degree in diameter, must be at such a distance, that its light would
require a million years to reach us. (Plate XXXI. Fig. 464.)

The stars are not, properly speaking, absolutely fixed with respect to
each other, for several of them have particular motions, which have been
discovered by a comparison of accurate observations, made at very distant
times. Arcturus, for instance, has a progressive motion, amounting to
more than two seconds annually. * Dr. Maskelyne found, that out of 36
stars, of which he ascertained the places with great precision, 35 had a
proper motion. Mr. Michellt and Dr. Herschel :£ have conjectured, that
some of the stars revolve round others which are apparently situated very
near them ; and perhaps even all the stars may in reality change their
places more or less, although their relative situations, and the directions of
their paths may often render their motions imperceptible to us.

Respecting all these arrangements of stars into different systems, Dr.
Herschel § has lately entered into a very extensive field of observation and
speculation, and has divided them into a number of classes, to each of
which he has assigned a distinct character. Some he supposes, like our
sun, to be insulated stars, beyond the reach of any sensible action of the
gravitation of others ; and around these alone he conceives that planets
and comets revolve. Double stars, in general, he imagines to be much
nearer to each other, so as to be materially affected by their mutual gravi-
tation, and only to preserve their distance by means of the centrifugal
force derived from a revolution round their common centre of inertia ; an
opinion which, he thinks, is strongly supported by his own observations of
some changes in the positions of double stars. Others again he supposes
to be united in triple, quadruple, and still more compound systems. A
fourth class consists of nebulae like the milky way, the clusters of stars
being rounded, and appearing brightest in the middle. Groups of stars
Dr. Herschel distinguishes from these by a want of apparent condensation

* Halley, Ph. Tr. No. 355. Cassini, Mem. de 1'Acad. 1738, p. 231. Monnier,
ibid. 1767, p. 417, proves that the latitude of Arc. varies at the rate of two seconds
annually; and that the longitude decreases at the rate of 60 seconds in 100 years.
See also Mem. de 1'Acad. 1769, p. 21. La Caille, Fundamenta Astron. pp.169,
187. Hornsby, Ph. Tr. 1773, Ixiii. 93, is of opinion that his deductions prove that
the obliquity of the ecliptic has become less.

t Ph. Tr. 1767, Ivii. 234 ; 1784, p. 35.

I Ibid. 1783, Ixxiii. 247.

§ On the Construction of the Heavens, Ph. Tr. 1785, Lxxv. 213.

ON THE FIXED STARS. 393

about a centre of attraction ; and clusters by a still greater central com-
pression. A seventh class includes such nebulae as have not yet been
resolved into stars, some of which Dr. Herschel supposes to be so remote,
that the light emitted by them must actually have been two millions of
years in travelling to our system. The nebulae of another description
resemble stars surrounded by a bur, or a faint disc of light ; a diffused
milky nebulosity, apparently produced by some cause distinct from the
immediate light of any stars, is the next in order : and Dr. Herschel has
distinguished other more contracted nebulous appearances, in different
states of condensation, into the classes of nebulous stars and planetary
neDulae, with and without bright central points. Many of these distinc-
tions are perhaps too refined to be verified by common observers ; but the
discovery of the existence of double and triple stars, revolving round a
common centre, will, if it be confirmed, add one more to the catalogue of
Dr. Herschel's important improvements.*

It is, however, fully ascertained, that some of the stars have periodical
changes of brightness, which are supposed to arise either from the tem-
porary interposition of opaque bodies revolving round them, or still more
probably, from a rotatory motion of their own, which brings at certain
periodical times a less luminous part of the surface into our view. Thus,
the star Algol, which is usually of the second magnitude, becomes, at
intervals of 2 days and 21 hours each, of the fourth only, and occupies 7
hours in the gradual diminution and recovery of its light, f A less pro-
bable conjecture respecting this change of brightness was advanced by
Maupertuis,^ who imagined that the disc of the star might be greatly
flattened by a rapid rotation, and its edge occasionally presented to us, in
consequence of the disturbances produced by the attraction of planets re-
volving round the luminary. Other irregular variations may possibly be
occasioned by the appearance and disappearance of spots, occurring, like
the spots of the sun, without any determinate order or assignable cause ;
and many stars have in the course of ages wholly disappeared, and some-
times have been again recovered ; others have made their appearance for a
short time, where no star had before been seen. Such a temporary star was
observed by Hipparchus, 120 years before our era, and the circumstance
suggested to him the propriety of making an accurate catalogue of all the
stars, with their respective situations, which is still extant, having been
preserved by Ptolemy, who added 4 stars to the 1022 that it contained. In
1572, Cornelius Gemma discovered a new star in Cassiopeia, § which was
so bright as to be seen in the day time, and gradually disappeared in six-
teen months. Kepler, in 1604, observed a new star in Serpentarius, more

* Catalogue of Double Stars, Ph. Tr. 1782, p. 112 ; 1785, p. 40; 1811, 1814,
1817. On their changes, ibid. 1803, p. 339 ; 1804, p. 353. Also Mem. of
the Astronomical Society, 1822. Bessel, Konigsberg Obs. Pat. 10. Astronomische
Nachrichten, No. 88. Struve, Catalogus Novus Stellarum Duplicium, &c. fol.
Dorpat, 1827. J. Herschel, Ph. Tr. 1824, 1830. Ast. Soc. 1821, &c. &c. South,
Ph. Tr. 1824-6.

t Goodricke, Ph. Tr. 1783, Ixxiii. 474, and Ixxiv. 287. See also Ixxvi. 48; and
,Lalande, Hist, et Mem. 1788, p. 240.

J Ph. Tr. 1732, p. 240.

§ See Ph. Tr. 1715, xxix. 354.

394 LECTURE XLI.

brilliant than any other star or planet, and changing perpetually into all
the colours of the rainbow, except when it was near the horizon ; it re-
mained visible for about a year. Many other new stars have also been
observed at different times.*

For describing the particular fixed stars according to their relative situ-
ations, it is necessary to consider them as they are visible to the inhabitants
of the earth. They have been divided, for the sake of convenience, into
parcels, making up imaginary forms, denominated constellations. This
division is of very remote antiquity, and though it may be useless, and some-
times even inconvenient, for the purposes of minute observation, yet( for
a general recollection of the great features of the heavens, these arbitrary
names and associations cannot but greatly assist the memory. It is also
usual to describe particular stars by their situation with respec^'to the
imaginary figure to which they belong, or, more commonly, at present, by
the letters of the Greek alphabet, which were first applied by Bayert in
1603, and in addition to these, by the Roman letters, and by the numbers
of particular catalogues.

There are two principal modes of representing the stars ; the one by
delineating them on a globe, where each star occupies the spot in which
it would appear to an eye placed in the centre of the globe, and where the
situations are consequently reversed, when we look on them from without,
in the same manner as a word appears reversed when seen from the back
of the paper ; the other mode is by charts, which are generally so arranged
as to represent the stars in positions similar to their natural ones, or as
they would appear on the internal concave surface of the globe. Some-
times also the stars have been delineated as they would be projected on
imaginary surfaces, without any reference to a globe ; for instance, on the
surfaces of transparent cones or cylinders. The art of constructing all
such projections belongs to the subject of perspective.

In describing the particular stars, it will be most convenient to begin
with such as never set in our climates, and we may then refer the situa-
tions of others to their positions with respect to these.

The great bear is the most conspicuous of the constellations which never
set ; it consists of seven stars, placed like the four wheels of a waggon, and
its three horses, except that the horses are fixed to one of the wheels. The
two hind wheels are the pointers, which direct us to the pole star, in the
extremity of the tail of the little bear ; and further on, to the constellation
Cassiopeia, which is situated in the milky way, where it is nearest to the
pole, and which consists of several stars, nearly in the form of the letter
W. The two northernmost wheels of the great bear, or wain, point at the
bright star Capella, the goat, in Auriga. Descending along the milky
way from Cassiopeia, if we go towards Capella, we come to Algenib, in
Perseus ; and a little further from the pole we find Algol, or Medusa's
head ; but if we take the opposite direction, we arrive at Cygnus, the
swan ; and beyond it, a little out of the milky way, is the bright star

* See Ph. Tr. 1715, p. 354 ; 1780, p. 338 ; 1786, p. 189 ; 1792, p. 24 ; 1795,,
p. 166; 1796, p. 452.

f Baieri Uranometria, Augsb. 1603.

ON THE FIXED STARS. 395

Lyra. The dragon consists of a chain of stars partly surrounding the
liitle bear ; and between Cassiopeia and the swan is the constellation
Cepheus.

Near Algenib, and pointing directly towards it, are two stars of Andro-
-*«feda, and a third is a little beyond them. A line drawn through the
great bear and Capella passes to the Pleiades, and then, turning at a right
angle towards the milky way, reaches Aldebaran, or the bull's eye, and the
shoulders of Orion, who is known by his belt, consisting of three stars,
placed in the middle of a quadrangle. Aldebaran, the Pleiades, and Algol,
make the upper, and Menkar, or the whale's jaw, with Aries, the lower
poiits of a W. In Aries we observe two principal stars, one of them with
a smaller attendant.

A Utfe drawn from the pole, midway between the great bear and Capella,
passes to the twins and to Procyon ; and then, in order to reach Sirius, it
must bend across the milky way. Algol and the twins point at Regulus,
the lion's heart, which is situated at one end of an arch, with Denebola at
the other end.

The pole star and the middle horse of the wain direct us to Spica Vir-
ginis, considerably distant ; the pole and the first horse nearly to Arcturus,
in the waggoner, or Bootes. Much further southwards, and near the milky
way, is Antares, in the scorpion, forming, with Arcturus and Spica, a
triangle, within which are the two stars of Libra. The Northern crown
is nearly in a line between Lyra and Arcturus, and the heads of Hercules
and Serpentarius are between Lyra and Scorpio.

In the milky way, below the part nearest to Lyra, and on a line drawn
from Arcturus through the head of Hercules, is Aquila, making with Lyra
and Cygnus a conspicuous triangle. The last of the three principal stars
in Andromeda makes, with three of Pegasus, a square, of which one of the
sides points to Fomalhaut, situated at a considerable distance in the south-
ern fish, and in the neighbourhood of the whale, which has already been
mentioned.

By means of these allineations, all the principal stars that are ever
visible in Britain may be easily recognized. Of those which never rise
above our horizon, there are several of the first magnitude ; Canopus, in
the ship Argo, and Achernar, in the river Eridanus, are the most brilliant
of them ; the feet of the centaur, and the crosier are the next ; and ac-
cording to Humboldt's observations, perhaps some others may require to
be admitted into the same class. (Plate XXXVI. XXXVII.)

LECT. XLI.— ADDITIONAL AUTHORITIES.

Treatises on Astronomy. — Albumasar, Introd. ad Ast. 4to, Aug. 1489. Coper-
nici Astr. reformata, 4to, Amst. 1617. Tychonis de Brahe, Ast. Progymnasmatum,
4to, Prag. 1603. De Mundi Phaenomenis, 1610. Epistolse, 1610. Lansbergius,
4to, Middleb. 1619. Galilaei Dialogus de Systemate Mundi, 4to, 1635. Kepleri
Epitome Ast. Copernicanse, Franco?. 1635. Riccioli, Almagestum Novum, 2 vols.
fol. Bonon. 1657. Wardi Ast. Geomet. Lond. 1656. Duhamel, Ast. Phys. 4to,
. Paris, 1660. Mercator, Institutionum Ast. Lib. II. Lond. 1676. Petit, Traite" de
PUnivers Materiel, 3 vols. 12mo, 1729-30. Simpson's Essays, 4to, 1740. Cassini,
Elemens d'Ast. 4 vols. 4to, Paris, 1742. Wright's Theory of the Universe, 4to,

396 LECTURE XLI.

1742. Long's Ast. 2 vols. 4to, Camb. 1742-64. Lemonnier, Institutions Astr.
4to, Paris, 1746. Lacaille's Elements (trans.), Lond. 1750. Werdler's Instit.
Astr. 4to, Witt. 1754. Hill's Dictionary, 4to, 1754. D'Alembert, Sur le Systeme
du Monde, 3 vols. 4to, Paris, 1754-6. Stewart's (M.) Tracts, Ed. 1761. Harris's
Ast. Dial. 12mo, 1766. Condorcet, Systeme du Monde, 4to, Par. 1768. Hennert,
Elem. Ast. Traj. ad Rhenum, 1768. Kohl's Einleitung, 8vo, Greifsw. 1768. - Keil's
Introd. to Ast. 1769. Lambert, Systeme du Monde, Bouillon, 1770. DicquenW ve,
Ast. Paris, 1771. Segner, Astron. Vorlesungen, 4to, Halle, 1775-6. Hellmuth,
Sternwissenschaft, Brunsw. 1776. Lalande's Astr. 3 vols. 4to, Par. 1792. Bode,
Anleitung zur Kentniss des Gestirnten Himmels, Berl. 1792. Biirja, Lehrbuch der
Ast. 5 vols. Berl. 1794—1806. Vince's Complete System of Ast. 3 vols. 4to, Camb.
1797; Elements, Camb. 1816. Ewing's Ast. Ed. 1797. Riidiger, Handbuch der
Rechnenden Ast. 4 vols. Leipz. 1802-4. Cagnoli, Notizie Ast. 2 vols. 12mo, Mod.
1802. Gregory's (O.) Ast. Lond. 1803. Hassenfratz, Cours de Physique Celyte,
Paris, 1803. Mollet, Etude du Ciel, Lyon, 1803. Oriani, Opusc. Ast. Milan,
1806. Monteiro, Mem. sur 1'Ast. Pratique, 4to, Paris, 1808. Marechal s'*- le
Systeme de I'Univers, Paris, 1810. Biot, Traite d'Ast. Physique, 3 vols. fc>jO-ll.
Bohnenberger, Ast. Tubing. 1811. Schon, Grundriss der Gesammten Theoretischen
Ast. Niirnb. 1811. Brandes, Die Vornehmsten Lehren der Ast. Leipz. 1813.
Woodhouse,^ Treatise on Ast. 2 vols. Camb. 1821. Elements, Camb. 1812. De-
lambre, Abrege d'Ast. Paris, 1813. Astronomic Theorique et Pratique, 3 vols. 4to,
Paris, 1814. Brinkley's Astronomy, Dublin, 1819. Littrow, Theoretische und
Practische Astronomic, 3 vols, Wien, 1821-7. Ferguson's Astronomy, by Brewster,
2 vols. Edin. 1821. Schubert, Traite d' Astronomic, 3 vols. 4to, St. Petersb. 1822.
Pearson's Astronomy, 2 vols. 4to. Lond. 1824-9. Baily's Tables and Formulae, 1827.
Farrar's Astronomy, Camb. N. E. 1827. Hassler's System of the Universe, New
York, 1828. Jambon's Astronomic, 1828. Santini, Elementi di Ast. 2 vols. Pad.
1830. Francoeur, Astr. Pratique, 1830. Malkin's Astron. (Lib. of Useful Know-
ledge), 1830. Quetelet, Astr. Populaire, Brux. 1832. Marcoz, Astr. Solaire, Paris,
1832. Veley, Astr. Elementaire, Lausanne, 1833. Herschel's Astr. 12mo, 1833.
Whewell's Bridgwater Treatise, 1833. Bailly's Resume, 32mo, 1835. Moseley's
Lectures on Astr. 1839. Maddy's Astr. by Hymers, Camb. 1840. Nichol's Archi-
tecture of the Heavens. Solar System.

Collections.— Transactions of the Royal, Astronomical and other Societies. Bode's
Astron. Jahrbuch, Berlin, 1788 .... Sammlung Astr. Abhandlungen, 1793 ....
Effemeridi Astr. di Milano, 1774 .... by Cesaris and others ; Zach's Monatliche

Correspondenz, Gotha, 1800 Lindenau und Bohnenberger, Zeitschrift fiir

Astr. Tubingen, 1816.... Connaissance des Temps, Paris, 1679. Schumacher's
Astr. Nachrichten, 4to, Altona, 1823 Annuaire de 1'Acad. de Bruxelles, 18mo.

Catalogues of Stars, fyc. — Alfonsus, Tabulae Astr. 4to, Venice, 1492, 1503. Kep-
leri Tabulae Rudolphinae, fol. Ulm, 1627. Lansbergius Uranometria, 4to, Mid-
dleb. 1631 ; Tabulae, fol. 1632. Riccioli, Almagestum Novum, 2 vols. fol. Bon.

1651. Wing's Ephemeris, 1669 Halley, Catal. Stel. Austral. 4to, Lond. 1679 ;

Astr. Tables, 4to, 1752. Lahire, Ephemerides ad Ann. 1701, 4to, 1700. Tab.
Astr. 4to, 1727. FlamsteediiHistoriaCselestis, 3 vols. fol. Lond. 1725. Winston's
Lect. with Tables, 1728. Manfred! Ephem. 4to, Bon. 1739. Lemonnier, Histoire
Celeste, 4to, Paris, 1741 ; Observations, 4 vols. fol. 1757-73. Dopplemaieri Atlas

C&elestis, Nuremb. 1742. Zanotta's Ephemeris, 4to, Bonon. 1750 Hell's

Ephem. Vienna, 1757. Lacaille, Coelum Australe, 4to, Paris, 1763. Obs. faites
au Cap de Bonne Esperance. Hallenstein, Obs. Pekini factse, 2 vols. 4to,
Vindob, 1768. Ludlam's Cambridge Observations, 4to, Camb. 1769. Darguier,
Obs. faites a Toulouse, 2 vols. 4to, Avignon, 1777. Bugge, Obs. Havni, 4to,
Havnise, 1784. Wollaston, Astron. Catal. fol. 1789. Fasciculus Astr. 4to, 1800.
Herschel's (Caroline) Catalogue, fol. Lond. 1798. Bode, Uranographia, fol. 1801 ;
also Trois Cat. de 1'Ascension droite et de la Declinaison de 17240, 5505, et de 5877
Etoiles, 4to, Berlin, 1801-5. Histoire Celeste, Paris, 1801. Piazzi's Cat. fol.
Palermo, 1803-14. Cagnoli, Cat. de 501 Etoiles, 4to, Modene, 1807. Mayer's,
1826. Baily's Catalogue of 2881 Stars, 4to, 1827. Plana, Obs. a Turin, 4to, Turin,
1817, &c. Harding, Atlas Cselestis, fol. Gott. 1822. Caturegli, Ephem. 4to, Bon.
1823. David, Astr. Beobach. Prag. 1823. Brioschi, Comentari Astr. 4to, Napoli,

1824-6. Robinson, Obs. at Armagh, 4to, Lond. 1829 Argelander, Obs.

Astr. 2 vols. fol. Helsingf. 1830-1. Positiones 560 Stel. 4to. 1835. Johnson, Obs. •
at Helena, 4to, St. Helen. 1832. Cat. of 606 Stars of Southern Hemisphere, 4to,
Lond. 1835. Rumker, Catal. from Obs. at Paramatta, 4to, Hamb. 1832. Taylor,

ON THE SOLAR SYSTEM. 397

Obs. at Madras, 1832. Bianchi, Atti del Osserv. di Modena, fol. 1834. Richard-
sdn, Obs. at Paramatta, 4to, Lond. 1835. Cerquero, Obs. en San Fernando, fol.
S. F. 1835, Henderson, Decl. of 172 Fixed Stars observed at the Cape of Good
Hope, 4to, Edin. 1835.

•-To these we must add the volumes which are issued from the observatories of

'trafnwich (Airy), Cambridge (Challis), Edinburgh (Henderson), Dorpat (Struve),

Oxford (Johnson), Berlin (Encke), Konigsberg (Bessel), Altona (Schumacher),

Paris (Arago), Vienna (Littrow), Palermo (Cacciatore) ; the Nautical Almanac,

&c. &c.

LECTURE XLII.

ON THE SOLAR SYSTEM.

THE most conspicuous of all the celestial bodies, which we have been
examining, is the sun, that magnificent luminary which occupies the centre
of the system that comprehends our earth, together with a variety of other
primary and secondary planets, and a still greater number of comets.

The sun agrees with the fixed stars in the property of emitting light con-
tinually, and in retaining constantly its relative situation with very little
variation ; it is probable also that these bodies have many other properties
in common. The sun is, therefore, considered as a fixed star compara-
tively near us ; and the stars as suns at immense distances from us : and
we infer from the same analogy, that the stars are possessed of gravitation,
and of the other general properties of matter ; they are supposed to emit
heat as well as light ; and it has with reason been conjectured that they
serve to cherish the inhabitants of a multitude of planetary bodies revolving
round them.

The sun, like many other stars, has probably a progressive motion,
which is supposed,* from a comparison of the apparent motions of a great
number of the stars, to be directed towards the constellation Hercules.
It is beyond all question that many of the stars have motions peculiar to
themselves, and it is not certain that any of them are without such motions :
it is, therefore, in itself highly probable that the sun may have such a
motion. But Dr. Herschelt has confirmed this conjecture by arguments
almost demonstrative. He observes that the apparent proper motions of 44
stars out of 56 are very nearly in the direction which would be the result
of such a real motion of the solar system : and that the bright stars
Arcturus and Sirius, which are probably the nearest to us, have, as they
ought to have, the greatest apparent motions. Besides, the star Castor
appears, when viewed with a telescope, to consist of two stars, of nearly
equal magnitude; and though they have both a considerable apparent
motion, they have never been found to change their distance a single

* Mayer, De Motu Fixarum, Getting. 1760. Wilson, Thoughts on general Gra-
vitation, and Views thence arising as to the state of the Universe, 1777. Lalande,
Mem. del'Acad. 1776.

f Herschel, Ph. Tr. 1783, Ixxiii. 247.

398 LECTURE XLII.

second ; a circumstance which is easily understood if both their apparent
motions are supposed to arise from a real motion of the sun, but which
is much less probable on the supposition of two separate and independent
motions.

Besides this progressive motion, the sun is subjected to some small ch&iige
of place, dependent on the situations of the planetary bodies, which was
long inferred from theory only, but which has been actually demonstrated
by modern observations. Supposing all the planets to be in conjunction,
or nearly in the same direction from the sun, the common centre of inertia
of the system is at the distance of about a diameter of the sun fron}3 his
centre : and since the centre of inertia of the whole system must be undis-
turbed by any reciprocal actions or revolutions of the bodies comper: Jg it,
the sun must describe an irregular orbit round this centre, his greatest
distance from it being equal to his own diameter. We may form an idea
of the magnitude of this orbit by a comparison with the orbit of the
moon : a body revolving round the sun, in contact with his surface, must
be nearly twice as remote from his centre as the moon is from the earth,
and the sun's revolution round the common centre of gravity of the system
must therefore be, where it is most remote, at four times the distance of
the moon from the earth.

The sun revolves on his axis in 25 days 10 hours, with respect to the
fixed stars : this axis is directed towards a point about half way between
the pole star and Lyra, the plane of the rotation being inclined a little
more than 7° to that in which the earth revolves. The direction of this
motion is from west to east, terms which we can only define from our pre-
supposed knowledge of the stars, by saying that the motion is such, that a
point of the sun's surface at first opposite Aries, moves towards Taurus.
Nor have we any better mode of describing north and south, or right and
left : we can only say comparatively, that if we are placed with our heads
northwards, and looking towards the centre, our right hands will be east-
wards, and our left westwards. All the rotations of the different bodies
which compose the solar system, as far as they have been ascertained, are
in the same direction, and all their revolutions, excepting those of some of
the comets, of which the motions are retrograde, and those of some of the
satellites of the Georgian planet, which revolve in planes so distant from
those of the other planetary motions, that the directions of their revolutions
can scarcely be called either direct or retrograde.

The time and direction of the sun's rotation is ascertained by the change
of the situation of the spots,* which are usually visible on his disc, and
which some astronomers suppose to be elevations, but others, apparently
on better foundations, to be excavations or deficiencies in the luminous
matter covering the sun's surface. These spots are frequently observed to
appear and disappear, and they are in the mean time liable to great varia-
tions, but they are generally found about the same points of the sun's
surface. Lalande t imagines that they are parts of the solid body of the

* Discovered by Fabricius. See his treatise De Maculis in Sole observatis, Wit-
tenb. 1611.

f Hist, et Mem. 1776. Brugnatelli, Bibliot. Fisic. i. 55.

ON THE SOLAR SYSTEM. 399

sun, which, by some agitations of the luminous ocean, with which he
conceives the sun to be surrounded, are left nearly or entirely bare. Dr.
Wilson* and Dr. Herschel are disposed to consider this ocean as consisting
jjiher of a flame than of a liquid substance, and Dr. Herschel attributes
''iybr-Fspots to the emission of an aeriform fluid, not yet in combustion, which
displaces the general luminous atmosphere, and which is afterwards to
serve as fuel for supporting the process ; hence he supposes the appear-
ance of copious spots to be indicative of the approach of warm seasons
on the surface of the earth, and he has attempted to maintain this opinion
by (historical evidence. The exterior luminous atmosphere has an appear-
aacfc somewhat mottled ; some parts of it, appearing brighter than others,
hav;r ^nerally ^een called faculae ; but Dr. Herschel distinguishes them
by the names of ridges and nodules. The spots are usually surrounded
by margins less dark than themselves, which Dr. Herschel calls shallows,
and which he considers as parts of an inferior stratum consisting of
opaque clouds, capable of protecting the immediate surface of the sun
from the excessive heat produced by combustion in the superior stratum,
and perhaps of rendering it habitable to animated beings. (Plate XXXI.
Fig. 465. ..469.)

But if we inquire into the intensity of the heat which must necessarily
exist wherever this combustion is performed, we shall soon be convinced
that no clouds, however dense, could impede its rapid transmission to the
parts below. Besides, the diameter of the sun is 111 times as great as that
of the earth ; and at its surface, a heavy body would fall through no less
than 450 feet in a single second ; so that if every other circumstance per-
mitted human beings to reside on it, their own weight would present an
insuperable difficulty, since it would become nearly thirty times as great as
upon the surface of the earth, and a man of moderate size would weigh
above two tons. Some of the most celebrated astronomers have imagined,
from the comparative light of different parts of the sun's disc, or apparent
surface, that he is surrounded by a considerably dense and extensive atmo-
sphere, imperfectly transparent ; conceiving that, without such an atmo-
sphere, the marginal parts, which are seen most obliquely, must appear
considerably the brightest ; but this opinion is wholly erroneous, and the
inferences which have been drawn from it, respecting the sun's atmosphere,
are consequently without foundation.

We are, however, assured, by direct observation, of the existence of some
aerial substance in the neighbourhood of the sun, producing the appearance
called the zodiacal light, which is sometimes seen, nearly in the plane
of the sun's rotation on its axis, extending beyond the orbit of Mercury.
It is said to have been first distinctly described in Childrey's Britannia
Baconica, a work published in 1661 ; and it was afterwards more par-
ticularly observed by Cassini,t Mairan,* and others. In the torrid zone it

* Ph. Tr. 1774, p. 1 ; 1783, p. 144. See also ibid. vi. 2216, 2295, and 3020
Cassini, Mem. de 1'Acad. x. 581. Herschel, Ph. Tr. 1795, p. 46 ; 1801, pp. 265,
354. Mossotti, Cesaris Effemeridi, 1820-1. Nicollet, Connoissance des Temps

t Hist, et Mem. vii. 119 ; viii. 193.

£ Mairan, Traite de 1' Aurore Boreale, Suite des Mem. de 1'Acad. Par. 1731 and
1751, 4to, Paris, 1733.

400 LECTURE XLII.

is almost constantly visible ; and in these climates it may often be distin-
guished in the beginning of March, after the termination of twilight,
exhibiting the appearance of a narrow triangle, somewhat rounded off, of a
whiteness resembling the milky way, ascending from the sun as a base, like
the projection or section of a very flat spheroid, and extending to a disWi^e
of more than 50° from the sun. The whole orbit of Venus never subtends
so great an angle from the earth as 96°, consequently this substance must
occasionally involve both Mercury and Venus ; and if it were not extremely
rare, it would produce some disturbance in their motions ; while in fact it
does not appear to impede the progress even of the tails of the comets, which
are probably themselves of very inconsiderable density. It cannot ££•„ a
continuous fluid atmosphere, revolving with the same velocity as the;ran ;
for the gravitation of such an atmosphere would cause it to assume a form
more nearly spherical ; and the only probable manner in which it can be
supposed to retain its figure, is by means of a revolution much more rapid
than the sun's rotation. Some persons have attributed the appearance to
the refraction of the earth's atmosphere only ; but if it arose from any such
cause as this, its direction could scarcely be oblique with respect to the
horizon, and it is highly improbable that it should always happen to coin-
cide with the plane of the sun's rotation. (Plate XXXI. Fig. 470.)

The sun is accompanied in his progressive motion among the fixed stars
by ten [eleven] planetary bodies, of different magnitudes, revolving round
him, from west to east, in orbits approaching to circles, and visible to us by
means of the light which they receive from him. These are Mercury, Venus,
the Earth, Mars, Juno, Pallas, Ceres [Vesta], Jupiter, Saturn, and the
Georgian planet. It is unnecessary to adduce at present any arguments to
prove the actual existence or direction of any of these motions ; their com-
plete agreement with the visible phenomena of the heavens, and with the
laws of gravitation, will hereafter appear to afford sufficient evidence of the
accuracy of the received theory of the arrangement of the solar system. The
motion of the earth is the most unanswerably proved by the apparent aber-
ration of the fixed stars, derived from the different directions of this motion
at different times, and corresponding precisely with the known velocity of
light, deduced from observations of a very different kind. That the planets
receive their light from the sun, is undeniably shown by the appearance of
the discs of many of them, when viewed through a telescope, those parts of
their surfaces only being luminous, on which the sun shines at the time of
observation.

These planets are neither all in one plane, nor does any one of them
remain precisely in the same plane at all times ; but their deviations from
their respective planes are inconsiderable, and they are commonly repre-
sented by supposing each planet to revolve in a plane passing through the
sun, and the situation of this plane to be liable to slight variations. There
is, however, a certain imaginary plane, determinable from the situations,
the velocities, and the masses of the planets, which, like the centre of inertia,
never changes its position on account of any mutual actions of the bodies of
the system, and this plane of inertia is called the fixed ecliptic. Its
situation is nearly half way between the orbits of Jupiter and of Saturn ;

ON THE SOLAR SYSTEM. 401

' and it is inclined in a small angle only to the plane of the earth's orbit,
which is called the earth's ecliptic, or simply the ecliptic.

The ecliptic passes through the constellations denominated the signs of

the zodiac, between Aries, the Pleiades, the twins, and Regulus, to the north,

yuft^Aldebaran, Spica, and Antares, to the south. Its position has varied

^ Mowly in the course of many ages, so that its northmost point is now more

^ than one third of a degree more remote from the pole star than it was in

the time of Eratosthenes, who observed its place 230 years before the birth

ot Christ. It appears from Lagrange's calculations, that the limit of its

greatest possible variation is about 10 or 11 degrees. The ecliptic is sup-

po&ji to be divided into twelve angular parts, or signs, each containing

thh JW degrees : they are named Aries, Taurus, Gemini, Cancer, Leo, Virgo,

Libra, Scorpio, Sagittarius, Capricornus, Aquarius, Pisces. Those who

prefer the cadence of a Latin distich, in order to assist the memory, may

repeat them thus, —

Sunt Aries, Taurus, Gemini, Cancer, Leo, Virgo,
Libraque, Scorpius, Arcitenens, Caper, Amphora, Pisces.

The planes of the orbits of the other primary planets, excepting the three
[four] minute planets lately discovered, intersect the ecliptic in small
angles, and the lines of intersection are called lines of the nodes. The nodes
of all the planets move very slowly, but not quite uniformly, from east to
west, that is, with respect to the fixed stars. At present the inclinations of
all the orbits appear to be somewhat diminishing : that of the orbit of
Jupiter is less by 6 minutes than it was in the time of Ptolemy.

The orbit of each planet is very nearly an ellipsis, one of the foci of which
coincides with the sun, or rather with the common centre of inertia of the
sun and planet. The extremities of the greater axis, where the orbit is
furthest from the sun and nearest to it, are called the upper and the lower
apsis, or the aphelion and perihelion ; the mean distance being at either
end of the lesser axis ; and the distance of the centre of the ellipsis from the
sun is called the eccentricity. The slight deviations of the planets from
these elliptic paths are expressed by considering the apsides as moveable,
and this motion is direct, that is, from west towards east, in the case of
all the planets except Venus, of which the aphelion has a retrograde motion,
with respect to the fixed stars.

The elliptic motion of the planets was first discovered by Kepler ; and
he found that a right line, joining the sun and any planet, describes always
equal areas in equal times. The observations, on which Kepler* founded
these important laws, were made principally on the planet Mars. He
determined by calculation, upon the supposition which was then generally
adopted, of a motion in an eccentric circle, what must be nearly the situ-
ation of the planet, with respect to the sun, that is, its heliocentric place, and
observing its geocentric place, with respect to the earth, he was thus able to
construct a triangle representing the situation of the three bodies ; repeating
this operation in various parts of the orbit, he discovered its form ; and
Raving done this, the velocity of the motion in different parts of the orbit

* See Lect. IV. and Kepler, Astronomia Nova, fol. Pragse. 1609.

2D

402 LECTURE XLII.

was easily determined from the apparent change of place in a given time.
(Plate XXXII. Fig. 471.)

The same astronomer also ascertained, that the squares of the times of
revolution of the different planets are in proportion to the cuhes of thejr
mean distances from the sun. For example, if one planet were four ti^iev
as distant as another, it would revolve in a period eight times as long,
since the cube of 4 is equal to the square of 8 ; thus Mars is nearly
four times as remote from the sun as Mercury, and the Georgian planet,
four times as remote as Jupiter, and their periods are nearly eight times a"s
long respectively. i

It is probable that all the planets have a rotatory motion from w
east, either perfectly or very nearly equable.* This motion has
observed in Venus, the Earth, Mars, Jupiter, and Saturn : and from some
phenomena of the satellites of the Georgian planet, Mr. Laplace thinks that
it may also be assumed as nearly certain that this planet has also a rota-
tory motion. The figure of the planets is spheroidical ; they are more or
less flattened at the poles, as they revolve more or less rapidly on their
axes. These axes retain, with a very slight deviation, a situation always
parallel, in every part of the orbits.

But, in the course of time, the gradual change of the position of the axis
produces a sensible effect. In the case of the earth, this effect is denomi-
nated the precession of the equinoxes. The equinoctial points are the
intersections of the apparent ecliptic, or the path of the sun in the heavens,
with the plane of the equinoctial, which is perpendicular to the earth's
axis and which passes through the equator on the earth's surface ; these
points of intersection have a retrograde motion, from east to west, on the
ecliptic. This motion was discovered by Hipparchus, in the year 128
before Christ, from a comparison of his own observations with those of
Timocharis, made 155 years before ; and since the time of Hipparchus,
the equinoctial points have receded about 26^°. Hence it happens that
the constellations called the signs of the zodiac, are now at a considerable
distance from those divisions of the ecliptic which bear the same names.

The earth's axis has also a small periodical change of inclination, or a
nutation, performed in about 19 years, and amounting in the whole to 18
seconds only. Its existence was determined by Newton from theory,
although he failed in the attempt to ascertain its quantity with accuracy ;
it was first actually observed by Dr. Bradley,t about the year 1747. The
absolute direction of the axis in the heavens is also liable to some variation,
in the course of many ages, but this change has not always been sufficiently
distinguished from the change of the position of the ecliptic. The inclina-
tion of the equator to the ecliptic is now very nearly 23° 28*.

In order to retain in memory a general idea of the proportional distances
of the primary planets from the sun, we may call that of the earth 10 and
that of Saturn 100 ; the distance of Mercury will then be 4, to which we
must add 3 for Venus, making 7 ; twice 3 or 6 for the earth, making 10 ;
twice 6 or 12 for Mars, making 16 ; twice 12 or 24, making 28, for the three

* Herschelon the Rotation of the Planets, Ph. Tr. 1781, p. 115.
f Ph. Tr. 1748, p. 1.

ON THE SOLAR SYSTEM. 403

[four] small planets, Juno, Pallas, and Ceres [Vesta], twice 24 or 48, making
5$, for Jupiter ; twice 48 or 96 for Saturn, making 100 ; and twice 9G or 192,
making 1Q6, for the Georgian planet ; and these sums will represent the
d; £°nces, without any material exception, in the nearest integer numbers.

,/yTne planet Mercury is little more than one third as large as the earth in
Diameter. He performs his revolution in somewhat less than three months,

v at about two fifths of the distance of the earth. His orbit is more eccentric,
t>^d more inclined to the ecliptic, than those of any of the planets except
the three [four] small ones lately discovered ; the eccentricity being one
fift^of the mean distance, and the inclination 7°. Of his density and his
rocl&on we know nothing but from conjecture.*

"Ve1'.7.s is very nearly as large as the earth; Dr. Herschel thinks her
even a little larger. Her revolution occupies about 7 months, her distance
from the sun being about seven tenths of that of the earth, and her orbit
nearly circular, inclined in an angle of 3° 24' to the ecliptic. Mr. Schroeterf
attributes to her mountains much higher than those of the earth, he has
observed strong indications of an atmosphere surrounding her, and he
assigns for her rotation on her axis the period of 23 hours 21 minutes.
Her density has been estimated from the perturbations, occasioned by her
attraction, in the motions of the other planets, and it has been supposed
to be a little less than that of the earth.

The distance of the earth from the sun is about 95 million English miles ;
and this determination is generally supposed to be so far accurate, that
there is no probability of an error of more than a million or two, at most,
although some authors are still disposed to believe that the distance may be
even greater than a hundred millions. The period of its revolution, with
respect to the equinoctial points, which are the usual standard of compari-
son, since their situation determines the annual return of the seasons, is
365 days, 5 hours, 48 minutes, and 48 seconds; and this is called its
tropical revolution ; that of its absolute or sidereal revolution is 365 days,
6 hours, 9 minutes, and 8 seconds ; the difference, which is 20 minutes and
20 seconds, being the time occupied in passing over the space, through
which the equinoctial points have retreated in the course of the tropical
year. By a day, we always understand the time which elapses during the
rotation of the earth with respect to the sun ; a sidereal day is about four
minutes shorter.

At a distance from the sun exceeding that of the earth by one half, the
planet Mars revolves, in about a year and seven eighths. He is of half
the earth's linear dimensions : he has spots which change their form, and,
therefore, probably, an atmosphere. Dr. Herschel J found his rotation per-
formed in 39 minutes more than a day ; his equator inclined 28° 42' to
the plane of his orbit, and his figure so much flattened at the poles, that
his axis is -^th shorter than his equatorial diameter. From this form,

* Consult Lalande, Mem. de 1'Inst. v. 442.

,f Beobachtungen, 4to, Erfurth, 1793. Aphroditographische Fragmenten, 4 to,
Helm. 1796. Journal de Physique, xlviii. 459. Beytrage, 8vo, Berlin, 1788. Ph.
Tr. 1792, Ixxxii. 309 ; 1795, Ixxxv. 117.

J Herschel, On the Planet Mars, Ph. Tr. 1781, p. 115 ; 1784, p. 223.

2 D 2

404 LECTURE XLII.

compared with the time of his rotation, it may be inferred that his
density must be very unequal in different pails : Laplace supposes it
from calculation to be on the whole about three fourths as great as that of
the earth.

In the interval between Mars and Jupiter, and nearly at the
where, from a dependance on the regularity of the progression already men-
tioned, a number of astronomers had for some years been seeking for a
primary planet, the observations of Mr. Piazzi,* Dr. Olbers,t and M* .
Harding ;£ have placed three very small bodies, differing but little in their
mean distance and their periodical time. They have named them Cf-res,
Pallas, and Juno :§ none of them subtends an angle large enough ttV'je
measured by our best instruments ; and all the circumstances af-'their
motions are yet but imperfectly established. Juno, however, appears to be
somewhat less remote than the other two : all their orbits are considerably
inclined to the ecliptic, especially that of Pallas, which is also extremely
eccentric. Dr. Herschel does not admit, that they deserve the name of
planets, and chooses to call them asteroids.

Jupiter is the largest of all the planets, his diameter being 11 times as
great as that of the earth, and the force of gravitation at his surface being
triple the terrestrial gravitation. He revolves in about 12 years, at a little
more than five times the earth's distance from the sun. His rotation is
performed in less than ten hours, his equator being inclined about three
degrees to his ecliptic, which makes an angle of 1° 19' with ours. His belts
are supposed by many to be clouds in his asmosphere ; they seem to have
a rotation somewhat slower than that of the planet.

The diameter of Saturn is ten times as great as that of the earth, but, on
account of the smaller density of his substance, the force of gravity at his
surface scarcely exceeds its force at the surface of the earth. He revolves
in 29 years and a half, in an orbit inclined 2^° to the ecliptic, at the dis-
tance of 9 1 semidiameters of the earth's orbit : his rotation occupies only
10^ hours, and his equator is inclined about 30° to our ecliptic. The most
remarkable circumstance attending him is the appearance of a double ring,||
which is suspended over his equator, and revolves with a rapidity almost
as great as that of the planet. His figure appears also, according to Dr.
Herschel's observations, to be extremely singular ; deviating very consi-
derably from that of an elliptical spheroid, which is the form assumed by
all the other planets that appear flattened, and approaching in some degree
to a cylinder with its angles rounded off. Such a form can only be derived
from some very great irregularities in the density of the internal parts of
his substance.

* Discovered Ceres, 1st Jan. 1801. Zach's Mon. Corresp. iv. 53.

t Disc. Pallas, 28th March, 1802.

I Disc. Juno, 1st Sept. 1804.

§ A fourth, named Vesta, was discovered by Olbers, on 19th March, 1807.

|| Pound. Ph. Tr. 1732, p. 240. Laplace, Memoire sur la Theorie de 1'An-
neau de Saturne. Herschel, Ph. Tr. 1790, pp. 4, 427 ; 1792, p. 1 ; 1794, p. 48 ;
1805-6-8. Bessel makes the inclination of the ring to our ecliptic to be 28° 22',
Berlin, Ephem. 1814, 1822. He estimates the mass of the ring at T}g of that of
Saturn, Ast. Nach. Nos. 193-4-5.

ON THE SOLAR SYSTEM. 405

• The Georgian planet, discovered by Dr. Herschel * in 1780, sometimes

also called Herschel, and sometimes Uranus, revolves in 83| years, at a

distance from the sun equal to 19 times that of the earth. Its diameter is

at little more than 4 times that of the earth, and the weight of bodies at its

aw face a little less than here. Notwithstanding its dimensions are by no

/means comparatively small, it appears to us as a star of the sixth or seventh

„ / magnitude, and is seldom seen by the naked eye. Its orbit approaches

.very near to the ecliptic ; its disc is said to be somewhat flattened, and it is

supposed to revolve with considerable rapidity.

These ten [eleven] planetary bodies are the only ones hitherto discovered
^'i \ch have any title to be considered as primary planets, that is, as bodies
reviving round the sun, in orbits so nearly circular, as to remain always
within the reach of our observation. It has been conjectured that the
number of planets may in reality be much greater, that not only many
small and perhaps invisible bodies may be revolving in the intervals of the
planets with which we are acquainted, but that larger bodies also may
belong to our system, which never approach within such a distance as to be
seen by us. Some have even bestowed names, borrowed from the ancient
mythology, on these imaginary planets ; but the idea of such an appropria-
tion of terms is rather to be regarded as belonging to the regions of poetical
fiction than to those of solid philosophy.

The largest and the most remote of the primary planets have their
attendant satellites, or secondary planets, accompanying them in their
respective revolutions round the sun, and moving, at the same time, in
subordinate orbits, round the primary planets. The earth is attended by
the moon, Jupiter by four moons or satellites, Saturn by seven, besides his
ring, and the Georgian planet by six moons. All these satellites move in
the direct order of the signs, and in planes not very remote from the eclip-
tic, excepting those of the Georgian planet, which revolve in planes nearly
perpendicular to the ecliptic. Each of these planets thus becomes the cen-
tral luminary of a little system of its own, in which the motions and the
periods observe the same general laws as prevail in the solar system at large.
Of the 28 primary and secondary planets, we are indebted to Dr. Herschel
for the knowledge of 9 ; the Georgian planet, with its six satellites,t and
the two innermost moons of Saturn.

The motions of some of these satellites, in particular of those of Jupiter
and of the moon, are of considerable importance for the assistance they
afford us in determinations of time, and of the relative situations of places.
They are subjected to considerable irregularities, but the united labours ol
various astronomers have enabled us to calculate all their motions with the
greatest accuracy.

The moon performs a complete sidereal revolution in 27 days 7| hours,

* Account of a Comet, Ph. Tr. 1781, Ixxi. 492. Herschel, Ph. Tr. 1783, p. 1.
Bode, Von dem neu Entdeckten Plan. Berl. 1784. Lexell, do. 4to, Petersb. Wurm,
Gotha, 1791. Robison, Ed. Tr. i. 305.

• f An Account of the Discovery of the Satellites of the Georgian Planet, Ph. Tr.
1787, p. 125 ; 1788, p. 364 ; 1798, p. 47. Account of the Discovery of a Sixth and
Seventh Satellite of the Planet Saturn, Ph. Tr. 1790, p. 1 , 427.

406 LECTURE XLII.

and a synodical revolution, during which she returns to the same position
with respect to the earth and sun, in 29 days 12f hours ; a period which
constitutes a lunation, or a lunar month. Her orbit is inclined to the
ecliptic in an angle of a little more than five degrees, hut this inclination is

liable to great variations : the place of its nodes is also continually caji \
ing, their motion being sometimes retrograde, and sometimes direct, but on^ ^»
the whole the retrograde motion prevails. The form of the moon's orbit is
irregularly elliptic, and the velocity of its motion deviates considerably
from the Keplerian law of the description of equal areas in equal timer; ;
the apsides, or the extremities of the greater axis of the ellipsis, which ,>re
called the apogee and perigee, have on the whole a direct motion. Frory a
comparison of modern observations with the most ancient, the mean motion
of the moon is found to be somewhat accelerated.

The moon revolves on her own axis with a very equable motion, and
the period of her rotation is precisely equal to the mean period of her
revolution round the earth ; so that she always presents to us the same
portion of her surface, excepting the apparent librations produced by her
unequal velocities in her orbit, and by the position of her axis, which is
inclined 1° 43' to the ecliptic, and sometimes as much as 7° to her own
orbit. Her distance from the earth is about 240,000 miles ; her diameter
-^r of that of the earth, or 21GO miles ; and the weight of bodies at her
surface is supposed to be about one fifth of their weight at the surface of
the earth,

The surface of the moon presents to us, when viewed with a telescope,
a great diversity of light and shade, the principal features of which are
visible even to the naked eye. Many of these inequalities resemble very
strongly the effects of volcanos ; several astronomers have imagined that
they have seen volcanos actually burning in the unenlightened part of the
planet ; and Dr. Herschel's instruments have enabled him to obtain satis-
factory evidence of the truth of the conjecture.* The appearance of a
perforation, which Ulloa supposed that he observed near the margin of
the Moon's disc, in a solar eclipse, has been attributed by some to a volcano
actually burning. Dr. Halley and Mr. Weidlerf have also observed
flashes of light on the dark part of the moon, considerably resembling the
effect of lightning. The height of the lunar mountains has been com-
monly supposed to exceed very considerably that of the mountains of the
earth ; but Dr. Herschel J is of opinion that none of them are so much as
two miles high. The names, which have been given by astronomers to
various parts of the moon's surface, are of some utility in the observation
of the progress of an eclipse.

Of the satellites of Jupiter, § some are a little larger, and others smaller
than the moon : they all revolve in planes inclined between 2£° and 3£° to
the orbit of the planet, and they are therefore always seen nearly in the
same line. It is inferred, from some periodical changes of light which they

* An Account of the Volcanos in the Moon, Ph. Tr. 1787, Ixxvii. 229. See also
Ph. Tr. 1794, pp. 84, 429, 435.

t Ibid. 1739, p. 228. J Ibid. 1780, p. 507.

§ Marii Mundus Jovialis, ito, Nuremb. 1611. Herschel, l'h. Tr. 1797, p. 332.

ON THE SOLAR SYSTEM. 407

undergo, that, like our moon, they always present the same face to their
primary planet.

The ring of Saturn is inclined 31 degrees to our ecliptic ; of his seven

satellites, six are nearly in the same plane with the ring ; hut the plane

• of Jhe seventh or outermost satellite is hut half as much inclined to the

i^ ^'j/ecliptic. The ring has heen observed by Dr. Herschel to revolve in 10 1

hours, which is considerably less than the time that would be occupied by

.the revolution of a satellite at the same distance. The planes of the six

s^ellites of the Georgian planet are nearly perpendicular to the ecliptic ;

anil some of their revolutions are supposed to be rather retrograde than

**iet.*

\ -sides the bodies which revolve completely round the sun, within the
limits of our observation, there are others, of which we only conclude from
analogy, that they perform such revolutions. These are the comets ; they
generally appear attended by a nebulous light, either surrounding them as
a coma, or stretched out to a considerable length as a tail ; and they some-
times seem to consist of such light only. Their orbits are so eccentric,
that in their remoter situations the comets are no longer visible to us,
although at other times they approach much nearer to the sun than any
of the planets : for the comet of 1680, when in its perihelion, was at the
distance of only one sixth of the sun's diameter from his surface. Their
tails are often of great extent, appearing as a faint light, directed always
towards a point nearly opposite to the sun : it is quite uncertain of what
substance they consist ; and it is difficult to determine which of the con-
jectures respecting them can be considered as the least improbable ; it is
possible that, on account of the intense cold, to which the comets are sub-
jected in the greatest part of their revolutions, some substances, more light
than any thing we can imagine on the earth, may be retained by them in
a liquid, or even in a solid form, until they are disengaged by the effect
of the sun's heat : but we are still equally at a loss to explain the rapidity
of their ascent : for the buoyancy of the sun's atmosphere cannot possibly
be supposed to be adequate to the effect ; and on the whole there is, per-
haps, reason to believe that the appearances are derived from some cause,
bearing a considerable analogy to the fluid, supposed to be concerned in
the effects of electricity. It is probable that the density of the nucleus,
or the body of the comet itself, is comparatively small, and its attraction
for the tail consequently weak, so that it has little tendency to reduce
the tail, even if it consists of a material substance, to a spherical form :
for since some comets have no visible nucleus at all, there is no difficulty
in supposing the nucleus, when present, to be of very moderate density,
and perhaps to consist of the same kind of substance as constitutes the tail
or coma, in a state of somewhat greater condensation. If, therefore, it
should ever happen to a planet to fall exactly in the way of a comet, of
which there is but very little probability, it is to be supposed that the
inconvenience suffered by the inhabitants of the planet might be merely
.temporary and local : the chances are, however, much greater, that a comet

* J. Herschel on the Satellites of Uranus, Mem. of the Ast. Soc. vol. viii.

408 LECTURE XLII.

might interfere in such a manner with a planet, as to deflect it a little from i
its course, and retire again without coming actually into contact with it.

Nearly 500 comets are recorded to have been seen at different times, and
the orbits of about a hundred have been correctly ascertained : but we
have no opportunity of observing a sufficient portion of the orbit of Sny v
comet, to determine with accuracy the whole of its form as an ellipsis^ 7
since the part which is within the limits of our observation does not sensibly N
differ from the parabola, which would be the result of an ellipsis prolonged
without end.

Two comets at least, or perhaps three, have been recognized in t^ir
return. A comet appeared in 1770, which Prosperin* suspected to iipvT?
in an orbit materially different from a parabola : Mr. Lexellf determjUed
its period to be 5 years and 7 months, and its extreme distances i/o be
between the orbits of Jupiter and of Mercury ; but it does not appear that
any subsequent observations have confirmed his theory. It has, however,
been calculated, that supposing the theory correct, it must afterwards have
approached so near to Jupiter as to have the form of its orbit entirely
changed.

Dr. Halley J foretold the return of a comet about 1758, which had
appeared in 1531, in 1607, and in 1682, at intervals of about 75 years ;
and with Clairaut's§ further correction for the perturbations of Jupiter
and Saturn, the time agreed within about a month. The mean distance
of this comet from the sun must be less than that of the Georgian planet ;
so that by improving our telescopes still more highly, we may, perhaps,
hereafter be able to convert some of the comets into planets, so far as
their remaining always visible would entitle them to the appellation.
Dr. Halley also supposed the comet of 1680 to have been seen in 1106, in
531, and in the year 44 before Christ, having a period of 575 years ; and
it has been suspected that the comets of 1556 and 1264 were the same, the
interval being 292 years ; a conjecture which will either be confirmed or
confuted in the year 1848. Some persons have even doubted of the perfect
coincidence of the orbits of any comets, seen at different times, with each
other, and have been disposed to consider them as messengers forming a
communication between the neighbouring systems of the sidereal world,
and visiting a variety of stars in succession, so as to have their courses
altered continually, by the attraction towards many different centres ; but
considering the coincidence of the calculation of Halley and Clairaut with
the subsequent appearance of the comet of 1759, this opinion can scarcely
be admitted to be in any degree probable with respect to the comets in
general, however possible the supposition may be in some particular cases.
(Plate XXXII. Fig. 472... 475. Plate XXXIII. Fig. 476... 485.)

* On Com. 1770, 4to, Upsal. 1776.

t Lexell, Mem. de 1'Acad. Par. 1776, p. 638 ; and Disquisitio de Temp. per. Co-
metse An. 1770, Ph. Tr. 1779, p. 68.

t Ph. Tr. 1705, p. 1882 ; and Gregory's Elements of Ast. 1726. See History of
Halley's Comet, with an Account of its Return in 1835, as predicted by MM. Damoi-
seau and De Pontecoulant, translated from the French of M. De Pontecoulant, by
Gold, 1835. Moseley's Lect. on Ast. 1839.

§ Journal des Savans, 1759. See his Theoriedu Mouvement des Cornetes, Paris,

ON THE LAWS OF GRAVITATION. 409

i

LECT. XLIL— ADDITIONAL AUTHORITIES.

Descriptions. Sun. — Hausen, 4to, Leipz. 1726. Schroter, 4to, Erfurt. 1789.
Woodward, Washington, 1801.

JL*"°,o». — Hevelii Selenographia, fol. Dantz. 1667. Cassini, Carte de la Lune.

Mylivis, Uber die Atmosphare des Mondes, 4to, 1746. Mayer's Cosmographische

% J/achrichten. 1748, p. 379. Von dem Mondkiigeln, 4to, Numb. 1750. Schroter's

rSelenotopographische Fragmenten, 2 vols. 4to, Gott. 1791. Beer's Map of the

Moon.

V Orbits. — See Lect. 43 ; also Primary Planets. — Halley's Mode of determining the
Orwts, Ph. Tr. 1676, p. 683. Huygenii Cosmotheros, 4to, Hag. 1698. Varignon,
HisA et Mem. 1 700, p. 224, H. 78. Cacciatore, Sull' Origine della Sistema Solare,
Palermo, 1826.

'/'• Kepler's Prob.— Keill, Ph. Tr. 1713, p. 1 ; Machin, ibid. 1783, p. 205.
Stewart, Ed. Ess. ii. 105. Lagrange, Hist, et Mem. de Berlin, 1764, p. 204.
Sejour, Hist, et Mem. 1790, p. 401. Ivory, Ed. Tr. v. 203. Brinkley, Ir. Tr. vi.
349 ; ix. 83.

Secondary Planets. — Clairaut on the Moon's Orbit, Hist, et Mem. 1743, p. 17,
H. 123 ; 1748, p. 421. Theorie de laLune, 4to,Petersb. 1752. Stewart, The Dis-
tance of the Sun deduced by Theory, Ed. 1763. Mayer, Theoria Lunse, 4to, Lond.
1767. Euler, do. Laplace, Hist, et Mem. 1784, p. 1 ; 1785, Errata; 1786,
p. 235 ; 1788, p. 249 ; 1789, p. 1, 237.

Comets. — Bartholinus de Cometis, 4to, Copen. 1665. Lubinietz, Theatrum
Cometicum,fol. Amst. 1668. Hevelius, Cometographia, fol. Gedani, 1668. Cassini,
Sur la Comete de 1680, 4to, Paris, 1681. Bernoulli, Conamen Novi Systematis
Cometarum, 12mo, Amst. 1682. Whiston's Praelectiones, 1710. Lemonnier, La
Theorie des Cometes, Paris, 1743. Heinsius, Ueber den Comet, von 1743, 4to,
Petersb. 1744. Loys des Cheseaux, do. Lausanne, 1744. Martin's Theory of Comets,
4to, 1757. Lambert, Insigniores Orbitae Cometarum Proprietates, Augsb. 1761.
Wideburg, Ueber den Com. Jena, 1769. Lambert on the Apparent Orbit of
Comets, Hist, et Mem. de Berlin, 1771, p. 352. Oliver on Comets, Salem, 1772.
Laplace, Mem. des Sav. Etr. 1773, p. 503. Dionis de Sejour, Hist, et Mem. 1774,
H. 78; Essai, Paris, 1775. Condorcet, Dissertation, 4to, Utr. 1780. Pingre",
Cometographie, 2 vols. Paris, 1783. Piazzi, Della Cometa del 1811, 4to, Palermo,
1812. Englefield on Comets, 4to, 1793. Legendre, Sur les Orbites des Cometes,
4to, Paris, 1806 ; Supplement, 4to, 1820. Schroter, Ueber den Grossen Com. von
1811, Gott. 1815. Cacciatore, Delia Com. di 1819, Palermo, 1819. Lubbock, On
the Orbit of a Comet, Mem. of the Ast. Soc. 1829. Encke, Ueber die Nachste
Wiederker des Cometen von Ponsin Jahr. 1832, Altona, 1831. Airy on Encke's
Comet, Camb. 1832. Littrow on do. Wien, 1832. Stratford, Ephemeris of Halley's
Comet, 1835. Arago, Des Cometes, 18mo, Par. 1834. Virlet, do. 18mo, Avesnes,
1835. Mime on Comets, Ed.

LECTURE XLIII.

ON THE LAWS OF GRAVITATION.

IT was first systematically demonstrated by Sir Isaac Newton, that all
the motions of the heavenly bodies, which have been described, may be
deduced from the effects of the same force of gravitation which causes a
heavy body to fall to the earth ; he has shewn that in consequence of this
universal property of matter, all bodies attract each other with forces de-
creasing as the squares of the distances increase ; and of later years the
same theory has been still more accurately applied to the most complicated

410 LECTURE XLIII.

phenomena. We are at present to take a general view of the operation of
this law, in the same order in which the affections of the celestial bodies
have been enumerated. It will not be possible to investigate mathemati-
cally the effects of gravity in each 'particular motion, but we may in some
measure illustrate the subject, by considering in what manner astronoTners.
have proceeded in their explanations and calculations, and we may entV ,
sufficiently into the principles of the theory, to understand the possibility
of its applications.

The bodies which exist in nature are never single gravitating points ; <Cnd
in order to determine the effects of their attraction, we must suppos^ the
actions of an infinite number of such points to be combined. It was sltajvn
by Newton, that all the matter of a spherical body, or of a spherical sur-
face, may be considered, in estimating its attractive force on other matter,
as collected in the centre of the sphere. The steps of the demonstration
are these : a particle of matter, placed at the summit of a given cone or
pyramid, is attracted by a thin surface, composed also of attractive matter,
occupying the base of the cone, with equal force, whatever may be the
length of the cone, provided that its angular position remain unaltered ;
hence it is easily inferred that if a gravitating point be placed any where
within a hollow sphere, it will remain in equilibrium, in consequence of
the opposite and equal actions of the infinite number of minute surfaces,
terminating the opposite pyramids into which the sphere may be divided ;
it is also demonstrable, by the assistance of a fluxional calculation, that a
point, placed without the surface, is attracted by it, precisely in the same
manner, as if the whole matter which it contains were collected in the
centre ; consequently the same is true of a solid sphere, which may be
supposed to consist of an infinite number of such hollow spheres. If,
however, the point were placed within a solid sphere it would be urged
towards the centre, by a force which is simply proportional to its distance
from that centre. This proposition tends very much to facilitate all calcu-
lations of the attractions of the celestial bodies, since all of them are so
nearly spherical, that their action on any distant bodies is the same as if
the whole of the matter of which they consist were condensed into their
respective centres; but if the force of gravity varied according to any
other law than that which is found to prevail, this simplification would no
longer be admissible, even with respect to a sphere.

It can scarcely be doubted that the power of gravitation extends from
one fixed star to another, although its effects may in this case be much too
inconsiderable to be perceived by us. It may possibly influence the pro-
gressive motions of some of the stars ; and if, as Dr. Herschel supposes,
there are double and triple stars revolving round a common centre, they
must be retained in their orbits by the force of gravity. Dr. Herschel
also imagines that the motion of our sun is in some measure derived from
the same cause, being directed nearly towards a point in which two strata
of the milky way meet ; the attraction of the stars, other things being
equal, must, however, be proportional to their brightness, and that part of
the heavens, to which the sun is probably moving appears to afford less
light than almost any other part, nor does the hemisphere, of which it is

ON THE LAWS OF GRAVITATION. 411

the centre, abound so much in bright stars as the opposite hemisphere. If
Sirius is a million times as far from the sun as the earth, and if he should
descend towards the sun by means of their mutual gravitation only, he
would move, on a rough estimate, but about 40 feet in the first year, and
in 1000 years only 8000 miles. It has been conjectured that the mutual
?. -ravitatioii of the stars of a nebula is sometimes the cause of the peculiar
form of the aggregate, which somewhat resembles that of a drop of a liquid
,ield together by its cohesion ; but unless the form of the nebula was ori-
gii.ftlly spherical, it could scarcely have acquired that form from the opera-
tioiitof gravity, since the spherical form of a drop is owing as much to the
elasticity as to the attractive force of the particles of water, and it would
be necessary, in order to preserve the analogy, that the stars should also
be floating in an incompressible fluid.

The sun's change of place, dependent on the relative situation of the
planets, is so inconsiderable, that it escaped observation until its existence
had been deduced from theory. Not but that this change would be suffi-
ciently conspicuous if we had any means of detecting it, since it may
amount in the whole to a distance equal to twice the sun's diameter, or
seven times the distance of the moon from the earth ; and this change is
readily deducible from the general and unquestionable law of mechanics,
that the place of the centre of inertia of a system cannot be changed by any
reciprocal or mutual action of the bodies composing it, the action of gra-
vity being found to be perfectly reciprocal. But the earth accompanies the
sun in great measure in this aberration, and the other planets are also more
or less affected by similar motions ; so that the relative situations are much
less disturbed than if the sun described this irregular orbit by the ope-
ration of a cause foreign to the rest of the system.

The simple revolution of a body, in a given plane, indicates, at first
sight, the existence of an attractive force directed to some point within the
orbit ; and the Keplerian law of the equality of the areas described in equal
times, by a line drawn from each planet to the sun, agrees precisely with
what is demonstrable of the effects of central forces, and points at once
to the sun as the centre of attraction of the system. And since the orbits
of the planets are elliptical, and the sun is placed in one of the foci of
each, it may be mathematically proved that the force directed to the sun
must increase in proportion as the square of the distance decreases.

The times of the revolutions of the planets are also in perfect con-
formity with the laws of gravitation, that is, the squares of the times are
proportional to the cubes of the distances from the sun. It was easy to
infer, from what Huygens had already demonstrated of centrifugal forces,
that this Keplerian law must be true of bodies revolving in circles by the
force of gravitation ; but Newton first demonstrated the same proportion
with respect to elliptic orbits, and shewed that the time of revolution in an
ellipsis is equal to the time of revolution in a circle, of which the diameter
is equal to the greater axis of the ellipsis, or the semidiameter to the mean
distance of the planet.

The universality of the laws of gravitation, as applied to the different
planets, shews also that the matter, of which they are composed, is equally

412 LECTURE XLIII.

subjected to its power ; for if any of the planets contained a portion of an
inert substance, requiring a force to put it in motion, and yet not liable to
the force of gravitation, the motion of the planet would be materially dif-
ferent from that of any other planet similarly situated.

The deviations of each planet from the plane of its orbit, and the motions
of its nodes, or the points in which the orbit intersects the plane of th
ecliptic, as well as the motions of the aphelion, or the point where the orbit*
is remotest from the sun, have also been deduced from the attractions of tb j
other planetary bodies ; but the calculations of the exact quantities of tfefese
perturbations are extremely intricate. In general, each of the disturbing
forces causes the nodes to have a slight degree of retrograde motion ; but/m
account of the peculiar situation of the orbits of Jupiter and Saturn, it
happens that the retrograde motion of Jupiter's node, on the plane of the
orbit of Saturn, produces a direct motion on the ecliptic, so that the action
of Saturn tends to lessen the effect of the other planets in causing a retro-
grade motion of Jupiter's nodes on the ecliptic.

The secular diminution of the obliquity of the ecliptic, or that slow
variation of its position, which is only discovered by a comparison of very
distant observations, is occasioned by the change of position of the earth's
orbit, in consequence of the attractions of the other planets, especially of
Jupiter. It has been calculated that this change may amount, in the
course of many ages, to 10° or 11°, with respect to the fixed stars ; but the
obliquity of the ecliptic to the equator can never vary more than two or
three degrees, since the equator will follow, in some measure, the motion
of the ecliptic.

The mutual attraction of the particles of matter, composing the bulk of
each planet, would naturally dispose them, if they were either wholly or
partially fluid, to assume a spherical form : but their rotatory motion would
require, for the preservation of this form, an excess of attraction in the
equatorial parts, in order to balance the greater centrifugal force arising
from the greater velocity of their motion : but since the attractive force of
the sphere on the particles at an equal distance from its centre is every
where equal, the equatorial parts would necessarily recede from the axis,
until the greater number of particles, acting in the same column, compen-
sated for the greater effect of the centrifugal force. The form would thus
be changed from a sphere to an oblate or flattened spheroid ; and the sur-
face of a fluid, either wholly or partially covering a solid body, must as-
sume the same figure, in order that it may remain at rest. The surface of
the sea is therefore spheroidical, and that of the earth deviates so far only
from a spheroidical figure, as it is above or below the general level of the
sea. (Plate XXXIV. Fig. 486.)

The actions of the sun and moon, on the prominent parts about the
earth's equator, produce a slight change of the situation of its axis, in the
same manner as the attractions of the other planets occasion a deviation
from the plane of its orbit. Hence arises the precession of the equinoxes,
or the retrograde motion of the equinoctial points, amounting annually to
about 50 seconds. The nutation of the earth's orbit is a small periodical
change of the same kind, depending on the position of the moon's nodes ; in

ON THE LAWS OF GRAVITATION. 413

consequence of which, according to Dr. Bradley's original observations,
the pole of the equator describes in the heavens a little ellipsis, of which the
diameters are 16 and 20 seconds. The same cause is also concerned in
modifying the secular variation of the obliquity of the ecliptic : and on the
other hand, this variation has a considerable effect on the apparent preces-
'ion of the equinoxes. On account of the different quantity of the preces-
sion at different times, the actual length of the tropical year is subjected to
*» slight variation : it is now 4 or 5 seconds shorter than it was in the time
of * fttpparchus. The utmost change, that can happen from this cause,
amounts to 43 seconds.

The exact computation of the moon's motion is one of the most difficult,
as well as the most important problems in astronomy ; but it is easy to
understand, in general, how the difference in the quantity and direction of
the sun's actions on the moon and earth, may cause such a derangement of
the moon's gravitation towards the earth, that the inclination of the orbit
must be variable, that the nodes must have a retrograde, and the apsides a
direct motion ; and that the velocity of the moon must often be different
from that which she would have, according to the Keplerian law, in a Sim-
ple elliptic orbit.

For, the sun's attraction as far as it acts equally on the earth and the
moon, can have no effect in disturbing their relative position, being always
employed in modifying their common annual revolution ; but the difference
of the forces, occasioned by the difference of distances, always tends to
diminish the effect of their mutual attraction ; since the sun acts more
powerfully on the nearer than on the remoter of the two bodies. The dif-
ference of the directions, in which the sun acts on the earth and the moon,
produces also a force, which tends, in some degree, to bring them nearer to-
gether ; but this force is, on the whole, much smaller than the former ; and
the result of both these disturbing forces is always directed to some 'point
in the line which joins the earth and the sun, on the same side of the earth
with the moon. It is obvious that when the nodes are also in this line, the
disturbing force can have no effect, either on their position, or on the in-
clination of the orbit, since it acts wholly in the plane of that orbit ; but
when they are in any other situation, the disturbing force must cause a
deviation from the plane, towards the side on which the sun is situated, so
that the inclination of the orbit increases and decreases continually and
equally ; but whatever may be the position of the nodes, it will appear
that they must recede during the greater part of the moon's revolution, and
advance during the smaller. (Plate XXXIV. Fig. 487.)

When the disturbing force tends to separate the earth and moon, it de-
ducts from the gravitation of the moon towards the earth a portion which
increases with the distance, and therefore causes the remaining force to
decrease more rapidly than the square of the distance increases ; and the
reverse happens when the disturbing force tends to bring the earth and
moon nearer together ; but the former effect is considerably greater than
Jbhe latter. Now in the simple ellipsis, when the body descends from the
mean distance, the velocity continually prevails over the attractive force,
so as to turn away the direction of the orbit more and more from the

414 LECTURE XLIII.

revolving radius, until, at a certain point, which is called the lower apsis,
it becomes perpendicular to it : but if the central force increase in a greater
proportion than is necessary for the description of the ellipsis, the point
where the velocity prevails over it will be more remote than in the ellipsis ;
and this is expressed by saying that the apsis moves forwards. When, on v
the contrary, the force varies more slowly, the apsis has a retrograded
motion. Since, therefore, the force attracting the moon towards the earth, ,
increases, on the whole, a little more rapidly than the square of the distance'
decreases, the apsides must have, on the whole, a direct motion. An4 a
similar theory is applicable to the mutual perturbations of the prinfary
planets. (Plate XXXIV. Fig. 488.)

The secular acceleration of the moon's mean motion, which had long
presented a difficulty amounting almost to an exception, against the suffi-
ciency of the theory of gravitation, has at last been satisfactorily deduced
by Mr. Laplace from the effect of the gradual change of the eccentricity of
the earth's orbit, which is subject to a very slow periodical variation, and
which causes a difference in the magnitude of the sun's action on the lunar
revolution.

The perfect coincidence of the period of the moon's rotation, with that
of a mean revolution, has been supposed to be in some degree an effect of
the attraction exerted by the earth on a prominent part, of her surface ;
there are, however, many reasons to doubt of the sufficiency of the expla-
nation. If the periods had originally been very nearly equal, we might
imagine that the motion of the earth would have produced a libration or
oscillation in the position of the moon, retaining it always within certain
limits with respect to the earth ; no libration is, however, observed, that
can be derived from any inequality in the moon's rotation ; and it has
very properly been suggested that the same attraction towards the earth
ought to have made the moon's axis precisely perpendicular to the plane of
her orbit, instead of being a little inclined to it. At the same time the
appearance of a similar coincidence, in the periods of the rotation and re-
volution of many other satellites, makes it probable that some general
cause must have existed, which has produced the same effect in so many
different cases.

The orbits of the comets afford no very remarkable singularity in the
application of the laws of gravity, excepting the modifications which depend
on their near approach to the parabolic form, and the great disturbance
which their motions occasionally suffer, when they happen to pass through
the neighbourhood of any of the larger planets. The velocity of a comet
in its perihelion is such, that its square is twice as great as the square of
the velocity of a body revolving in a circle at the same distance. It was
determined by Halley and Clairaut, that the attractions of Jupiter and
Saturn would delay the return of the comet of 1759 about 618 days ; and
the prediction was accomplished within the probable limits that they had
assigned for the error of the calculation. The labours of Clairaut have
indeed in many respects improved the science of mathematical astronomy ;>
he was the first that obtained a complete determination of the effects of the
mutual actions of three gravitating bodies, disturbing each other's motions ;

APPEARANCES OF THE CELESTIAL BODIES. 415

and his investigations, which were founded on those of Newton, led the
way to still further improvements and refinements, which have been since
made in succession by Euler, Lagrange, and Laplace.

LECT. XLIIL— ADDITIONAL AUTHORITIES.

Analytical Investigations on the Theory of Gravitation. — Euler, Theoria Motuum
ilanet. 4to, Berl. 1744. D'Alembert, Hist, et Mem. 1745, p. 365: Recherches
sur vla Precession des Equinoxes, &c. 4to, Paris, 1749 : Recherches sur le Systeme
du ^onde, 3 vols. 4to, Paris, 1754-6. Bailly, Essai sur la Theorie des Sat. de Jup.
4to, f766. Silvabelle, Ph. Tr. 1754, p. 385. Walmsley on Perturbations, ibid.
1756, p. 700; 1761, p. 275. Laplace on the Secular Variations of the Planets,
Hist, et Mem. 1772, i. 343, H. 67 ; 1784, p. 1 ; 1787, p. 267 : on the Theory of
Jupiter and Saturn, ibid. 1785, p. 33; 1786, p. 201. Lagrange on the Secular
Variations of the Nodes and Inclinations, ibid. 1774, p. 97, H. 39 ; 1780, p. 285,
H. 38. Dionis de Sejour, Traite Analyt. des Mouvemens Apparens des Corps
Celestes, 2 vols. 4to, Paris, 1786-9. Fuss on the True Anomaly N. A. Petr. 1785,
iii. 302. Cousin, Ast. Physique, 4to, 1787. Schubert on the Obliquity of the
Ecliptic, ibid. 1792, x. 433. Gauss, Theoria Motus Corporum Cselestum, 4to,
Hamb. 1809. Plana, Memoirs on the Coeff. of the great Inequality of Jupiter and
Saturn, 4to, Turin, 1826-28-29-32. Theorie de la Mouvement de la Lune, 3 vols.
4to, Turin, 1832. Airy's Tracts, Camb. 1831. Cauchy, Sur la Mec. Cel. 4to,
Lilhog. Lubbock on the Theory of the Moon and Perturbations of the Planets,
1833-6. Hansen, Theoria Motus Lunse, 4to, Gotha.

The number of essays on this subject is so very great, and they are scattered so
widely over the surface of all the transactions of the learned societies of Europe, that
we can do no more than direct the reader to consult their pages. He will find many
valuable memoirs in the introduction to the different observations : in the Effemerides
of Cesaris, Hell, &c. ; in Schumacher's Astronomic Nachrichten ; in Crelle's and
other Journals. The standard works are Newton's Principia, and the treatises given
under Lect. II. at the foot of p. 20. The subject is treated popularly in Airy's Gra-
vitation, 12mo, 1834.

LECTURE XLIV.

ON THE APPEARANCES OF THE CELESTIAL BODIES.

WE are next to proceed to examine the sensible effects produced by those
motions which we have first considered in their simplest state, and after-
wards with regard to their causes and their laws. Many authors have
chosen rather to pursue a contrary method, and have attempted to imitate
the original and gradual development of the primitive motions from their
apparent effects. But no conception is sufficiently clear, and no memory
sufficiently strong, to comprehend and retain all these diversified appear-
ances with accuracy and facility, unless assisted by some previous idea of
the real changes which produce them, or by some temporary hypothesis
respecting them, which may have been of use in its day for the better
connection of the phenomena, although it does not at present deserve to
be employed for a similar purpose, in preference to simpler and better
theories, which happen to be historically of a later date.

416 LECTURE XLIV.

The proper motions of the fixed stars, as they are subjected to our
observation, undergo two modifications ; the one from the relative direc-
tion of the motion, by which it may be more or less concealed from our
view ; the other from the proper motion of the sun, and the planets attend-
ing him. This motion has indeed only been inferred from the apparent
motions of a great number of stars, which are either partly or totally
referrible to it, and which could scarcely have agreed so correctly as they
do, if they had arisen from the real and separate motion of each star.

Among the motions of the primary planets, that of the earth itself
requires a principal share of our attention. The apparent places or the
fixed stars are not sensibly affected by the earth's annual revolution : if
any of them had been considerably less remote than they are, it is probable
that this motion would have occasioned a sensible annual parallax, or a
change of their relative situation, according to the earth's place in its orbit
round the sun ; for if this orbit, viewed from any of the stars, subtended an
angle even of a single second, the place of that star might be observed to
vary a second at different times of the year. Dr. Hooke supposed at one
time that he had discovered such a parallax, but later observations have
not confirmed those of Dr. Hooke. The stars have, however, a small aberra-
tion, in consequence of the progressive motion of the earth in its orbit,
combined with the limited velocity of light ; and the standard of com-
parison being the earth's axis, its nutation must also in some degree affect
the apparent places of the stars. It was in endeavouring to ascertain the
annual parallax, that Dr. Bradley discovered both the aberration of light
and the nutation of the earth's axis.

The revolution of the earth, in its orbit round the sun, produces the
apparent motion of the sun among the stars, by which he describes his
annual path in the ecliptic, with an apparent angular velocity equal to the
angular velocity of the earth, which varies considerably at various times.
It required some investigation of the magnitudes and distances of the hea-
venly bodies, to be convinced that the sun and stars had not in reality the
motion which a superficial inspection of the heavens would naturally lead
a spectator to attribute to them ; but it is at present perfectly unnecessary
to enter into arguments to prove that the true cause of these apparent
motions is the real motion of the earth. The effect of the earth's annual
revolution is the change of place of the sun among the fixed stars : it is
obvious that the sun will always appear, when viewed from the earth, in
a place diametrically opposite to that in which the earth would appear, if
seen from the sun : consequently, since the earth and sun remain in the
same plane, the apparent path of the sun will mark the same circle among
the stars as the earth would appear to describe, if viewed from the sun,
that is, the ecliptic. If the light of the stars were much stronger, or that
of the sun much weaker, we might see him pass by the stars in each part
of the ecliptic, as we do the moon ; but we are now obliged to observe
what stars are in turn diametrically opposite to the sun, or at certain dis-
tances from him, and thus we obtain a correct knowledge of his path.

The sun's apparent diameter is larger by one thirtieth in January than
in June ; of course the earth is so much nearer to the sun in winter than

APPEARANCES OF THE CELESTIAL BODIES. 417

in summer ; and since the revolving radius of the earth's orbit describes
equal areas in equal times, the angular motion must increase as the square
of the distance diminishes, or about twice as fast as the distance itself
diminishes ; so that the whole variation of the apparent diurnal motion of
the sun is one fifteenth of his mean motion : hence, the sun passes through
the winter half of the ecliptic in a time 7 or 8 days shorter than the summer
half. According to the different situations of the earth, with respect to the
plane of the sun's equator, his rotation on his axis causes the paths of his
spots to assume different forms ; when the earth is in that plane, the paths
appear straight, but in all other situations, elliptical.

The rotation of the earth on its axis produces the still more obvious
vicissitudes of day and night ; and, in combination with its annual motion,
occasions the change of seasons. Since the axis remains always parallel
to itself, and is inclined to the plane of the ecliptic in an angle of about
66f °, the plane of the equator, which is perpendicular to the axis, must
pass twice in the year through the sun. . When this happens, the limit of
illumination, or the circle which separates the dark portion of the earth
from the enlightened part, will then pass through the poles ; and as the
earth turns on its axis, each point of its surface must remain for an equal
length of time in light and in darkness. Hence the points of the ecliptic,
in which the sun is situated at such times, are called the equinoctial points.
At all other times, one pole of the earth is in the light, and the other in the
shadow ; and all the points of the earth nearest to the illuminated pole have
their day longer than their night, while the parts on the opposite side of
the equator, which are consequently nearer to the unenlightened pole, have
their day shorter. The parts nearest to the poles have also one of their
days and one of their nights protracted to a period of several common
days, or even months, whenever they revolve entirely within the limit of
illumination. (Plate XXXIV. Fig. 489.)

The sun appears to describe every day a circle in the heavens, more or
less distant from the plane of the equator, according to the actual situation
of the earth's axis ; this distance being always the same as that of the
poles from the limit of illumination, and never exceeding 23£° ; so that
by determining the sun's path at the time of the equinoxes, or the apparent
place of the equinoctial in the heavens, for any given point on the earth's
surface, we may represent the sun's path at any other time by a smaller
circle parallel to it. Speaking however, more correctly, the sun's apparent
path is a spiral, formed by the continuation of these supposed circles into
each other.

The effect of the centrifugal force, derived from the earth's rotation, is
perceptible at the equator, in the retardation of the vibrations of pendu-
lums. The whole centrifugal force at the equator is found by computation,
to be -g-i-g. of the force of gravity ; but the diminution of the force of gravi-
tation appears, by experiments on pendulums, to be -^ ; this diminution
being the sum of the centrifugal force, and of the decrease of gravity on
account of the oblate figure of the earth, the equatorial parts being further
removed from its centre, and the force of gravity being less powerful there.
The changes of inclination in the earth's axis are observable in the places

2 E

418 LECTURE XLIV.

of the equinoctial points, and in the situation of the plane of the earth's '
equator with respect to the fixed stars ; and the secular diminution of the
obliquity of the ecliptic is discoverable by a comparison of distant obser-
vations on the sun's apparent motion, and on the places of the fixed stars
with respect to the ecliptic.

For the phenomena of twilight, we are principally indebted to the light
reflected by the atmosphere ; when the sun is at a certain distance only
below the horizon, he shines on some part of the air immediately visible
to us, which affords us a portion of reflected light. The distance at wlych
this may happen, has been variously estimated, and it is perhaps actually
different in different climates, being a little greater in countries near tlio
poles than in those which are nearer the equator ; there is also sometimes
a secondary twilight, when the parts of the atmosphere, which reflect a
faint light on the earth, are themselves indebted for this light to an earlier
reflection. Some have assigned 18° as the limit of twilight, and on this
supposition, allowing for refraction, the atmosphere must be capable of
reflecting sensible light at the height of about 40 miles. Mr. Lambert,* on
the contrary, makes the limit only about 6£°. The duration of twilight
is greater or less as the sun moves more or less obliquely with respect to
the horizon ; it is, therefore, shortest near the time of the equinoxes, since
the equinoctial intersects the horizon less obliquely than any lesser circle
parallel to it. (Plate XXXIV. Fig. 490, 491.)

The revolutions of the primary planets, combined with that of the
earth, necessarily produce the various relations, in which they are either
in opposition or conjunction, with respect to each other or to the sun, and
in which the apparent motion is direct or retrograde, or the planet is sta-
tionary, according to the directions and the comparative velocities of the
real motions. If the earth were at rest, the inferior planets would appear
to be stationary when they are at the greatest elongation or angular dis-
tance from the sun ; but, on account of the effect of the earth's motion,
Venus is stationary at an elongation of about 29°, while her greatest
elongation is between 45° and 48°. The greatest elongation of Mercury, in
each revolution, is from 28|° to IT^0, according to the position of his orbit,
which is very eccentric. All these appearances are precisely the same as
if the sun actually revolved round the earth, and the planets accompanied
him in his orbit, performing at the same time their several revolutions
round him ; and the path which would thus be described in the heavens,
and which is of a cycloidal nature, represents correctly the true positions
of the planets with respect to the earth. The apparent angular deviation
from the ecliptic, or the latitude of the planet, is also greater or less,
accordingly as the earth is nearer or remoter to the planet, as well as
according to the inclination of its orbit and its distance from the node.
(Plate XXXIV. Fig. 492... 494.)

The various appearances of the illuminated discs, especially of the in-
ferior planets, and the transits of these planets over the sunj depend on

* Photometria, § 987. See Lacaille on the Length of Twilight at the Cape, Hi?t.
et Mem. 1751, p. 544, H. 158. Bergmann, Schwed. Abhand. 1760, p. 237. Opusc.
v. 331; vi.l.

APPEARANCES OF THE CELESTIAL BODIES. 41!)

their positions in their orhits, and on the places of the nodes, with respect
to the earth. Jupiter, Saturn, and the Georgian planet, are so remote in
comparison of the earth's distance from the sun, that they appear always
fully illuminated. Venus is brightest at an elongation of ahout 40° from
the sun,* in that part of her orhit which is nearest to the earth ; she then
appears like the moon when 5 days old, one fourth of her disc being illu-
minated ; she casts a shadow, and may even be seen in the day time in our
climates, if she happens to be far enough north ; a circumstance which
occurs once in about 8 years. In order that there may be a transit of
Venus over the sun, she must be within the distance of l.J° of her node at
the time of conjunction, otherwise she will pass either to the north or to
the south of the sun, instead of being immediately interposed between
him and the earth.

The phases and eclipses of the moon are very obviously owing to the
same causes ;t that part of the moon only, on which the sun shines, being
strongly illuminated, although the remaining part is faintly visible, by
means of the light reflected on it from the earth ; it is, therefore, most
easily seen near the time of the new moon, when the greatest part of the
earth's surface turned towards the moon is illuminated. The parts of the
moon which are immediately opposed to the earth, appear to undergo a
libration, or change of situation, of two kinds, each amounting to about
7 degrees ; the one arising from the inequality of the moon's velocity in
her orbit at different times, the other from the inclination of the axis of
her rotation to her orbit ; besides these changes, the diurnal rotation of the
earth may produce, to a spectator situated on some parts of it, a third
kind of libration, or a change of almost two degrees in the appearance of
the moon at her rising and setting. (Plate XXXIV. Fig. 495.)

When the moon passes the conjunction, or becomes new, near to the
node, she eclipses the sun, and when she is full, or in opposition in similar
circumstances, she herself enters the earth's shadow. The earth's shadow
consists of two parts, the true shadow, within which none of the sun's
surface is visible, and the penumbra, which is deprived of a part only of
the sun's light ; the true shadow forms a cone terminating in a point at a
little more than 3£ times the mean distance of the moon ; the penumbra,
on the contrary, constitutes, together with the shadow, a portion of a cone
diverging from the earth without limit ; but the only effect of this imper-
fect shadow is, that it causes the beginning of a lunar eclipse to be incapa-
ble of very precise determination ; for the limit of the darkened part of
the moon, as it appears in the progress of the eclipse, is that of the true
shadow, very little enlarged by the penumbra. The true shadow, where
the moon crosses it, is about 80 minutes in diameter, as seen from the
earth, while the moon herself is only 30. This shadow is not, however,
wholly deprived of the sun's light ; for the atmospheric refraction inflects
the light parsing nearest to the earth, in an angle of 66 minutes, and causes
a great part of the shadow to be filled with light of a ruddy hue, by means

* Halley, Ph. Tr. 1716, p. 466. Kies, Hist, et Mem. de Berlin, 1750, 218.
t Kastner on the Phases of the Moon, Com. Gott. 1780, Hi. M. 1.

2 E2

420 LECTURE XLIV.

of which the moon remains still visible to us, the cone of total darkness
extending to somewhat less than two thirds of the moon's distance. But
it has sometimes happened, probably from the effect of clouds occupying
the greatest part of our atmosphere, that the moon has totally disappeared.
(Plate XXXIV. Fig. 496.)

When the sun is eclipsed, it depends on the situations of the earth and
moon in their orbits, whether the sun or moon subtends the greatest angle
as seen from the earth ; since at their mean distances their apparent dia-
meters are each about half a degree. If the sun's apparent diameter is the
greater, the eclipse, when the centres coincide, must be annular, the margin
of the sun's disc being still visible in the form of a ring : when the moon's
apparent diameter is greater than the sun's, the eclipse, if central, becomes
total ; but still a ring of pale light is seen round the disc,* which has been
attributed to the effect of the sun's atmosphere, since that of the moon is
probably too inconsiderable to produce the appearance ; a red streak t is
also sometimes observed at the margin, before the actual emersion of the
sun. The degree of darkness depends on the situation of the place of obser-
vation within the shadow, on account of the greater or less illumination of
the atmosphere within view : sometimes a considerable number of stars
may be seen during a total eclipse of the sun.

It is obvious that, since the earth is much larger than the moon, the
whole shadow of the moon will only pass over a part of the earth's surface :
and that no solar eclipse can be visible in the whole of the hemisphere
turned to the sun : while lunar eclipses, on the contrary, present the same
appearance wherever the moon is visible. In the same manner, to a spec-
tator on the moon, an eclipse of the earth, or a transit of the moon's shadow
over the earth's disc, would have nearly the same appearance wherever he
might be stationed ; but an eclipse of the sun by the earth would be total
to that part of the moon's surface only, which to us appears dark at the
same time. (Plate XXXIV. Fig. 497. . .499.)

The moon's nodes arrive very nearly at the same situation with respect
to the earth after 223 lunations, or revolutions of the moon, which are per-
formed in 18 years of 365 days each, 15 days, 7 hours, and 43f minutes ;
so that after a period of about 18 years, the series of eclipses recommences
nearly in the same order. This circumstance was observed by the ancients,
and is mentioned by Ptolemy and by Pliny. When the full moon happens
within 74° of the node, there must be a lunar eclipse and there may be an
eclipse at the distance of 13° from the node. An eclipse of the sun may
happen when the moon changes, or comes into conjunction with the sun,
at any distance within 17l° of the node. The mean number of eclipses
which occur in a year is about 4 ; and there are sometimes as many as 7 :
there must necessarily be two solar eclipses, but it is possible that there may
not be even one lunar. In speaking of the magnitude of the part of the
sun or moon eclipsed, it is usual to consider the whole diameter as divided

* Duillier, Ph. Tr. 1706, p. 2241. Halley, ibid. 1715, p. 245. Lahire, Hist, et
M6m. 1715, p. 161, H. 47. Ulloa, ibid. 1779, p. 105.
t Ph. Tr. 1706, p. 2240 ; 1748, p. 490.

APPEARANCES OF THE CELESTIAL BODIES. 421

into 12 parts, called digits, each of which contains 30 minutes : thus if one
fifth part of the diameter were dark, the extent of the eclipse would be
called 2 digits 12 minutes.

The moon travels through the heavens with a motion contrary to their
apparent diurnal revolution. Hence she rises and sets, on an average, about
three quarters of an hour later every day. The least possible difference
between the times of the moon's rising on two successive days, is, in Lon-
don, 17 minutes ; and this circumstance occurs once in about 19 years,
which is nearly the period of the moon's nodes with respect to the heavens :
the greatest possible difference is 1 hour 17 minutes. But it happens every
month that the difference becomes greater and less by turns, and when the
least difference is at the time of the full moon, it is usually called the har-
vest moon. In parts nearer to the poles, the moon often rises at the same
hour on two succeeding days.

The eclipses of the satellites of Jupiter exhibit appearances extremely in-
teresting for their utility in identifying the same instant of time in different
places.* On account of the small inclination of their orbits to the plane of
Jupiter's orbit, the first three never pass the shadow without being plunged
into it, and the fourth but seldom ; while those of Saturn are much less
frequently liable to be eclipsed, on account of their greater deviation from
the plane of his ecliptic. These satellites are also frequently hidden be-
hind the body of the planet, and this circumstance constitutes an occul-
tation : hence it happens that we can never see both the immersion of the
first satellite into the shadow of Jupiter, and its emersion from it ; but
both the immersion and emersion of the three outer satellites are sometimes
observable. The ring of Saturn exhibits a variety of forms according to
its angular position : it disappears to common observation when either its
edge or its dark side is presented to us : but to Dr. Herschel's telescopes it
never becomes invisible ; the light reflected from the planet being probably
sufficient for illuminating in some measure the side not exposed to the sun's
direct rays.

The comets are seen for a short time, and are again lost to our view ;
their tails are in general situated in the planes of their orbits, following
them in their descent towards the sun, and preceding them in their ascent,
with a slight degree of curvature in their forms ; they must also appear to
us more or less arched, and of different extent, according to their distances,
and to the angular position of the orbits with respect to the ecliptic.

The proportion of the light afforded by the different heavenly bodies has
been variously estimated by various authors ; but there is little reason to
call in question the accuracy of the experiments and calculations of Mr.
Bouguer. He states the intensity of the moon's light as only one three
hundred thousandth of that of the sun. These calculations have been ex-
tended by Euler and by Lambert ; Eulert considers the direct light of the
sun as equal to that of 6560 candles of a moderate size, supposed to be
placed at the distance of 1 foot from the object : that of the moon to the

• * Wargentin, A New Method of determining the Longitude from the Eel. of Jup.
Sat. Ph. Tr. 1766, p. 278.
t Hist, et M&n. de Berlin, 1750, p. 280.

422 LECTURE XLIV.

effect of 1 candle, at the distance of 7 2 feet ; the light of Venus to a candle
at 421 feet, and of Jupiter to a candle at 1620 feet ; so that the sun would
appear as bright only as Jupiter if he were removed to a distance 131 thou-
sand times as great as his present distance. (Plate XXXIV. Fig. 500.)

When we reflect on the magnificence of the great picture of the universe,
the outlines of which we have been considering, we are lost in the contem-
plation of the immensity of the prospect, and returning to the comparatively
diminutive proportions of our individual persons, and of all the objects with
which we are most immediately connected, we cannot help feeling our own
insignificance in the material world. The mind, notwithstanding, endea-
vours to raise itself above the restraints which nature has imposed on the
body, and to penetrate the abyss of space in search of congenial existences.
But in speculations of this kind, reason and argument must give way to
conjecture and imagination ; and thus, from natural philosophy, our ima-
ginations wander into the regions of poetry ; and it must be confessed that
the union of poetical embellishment with natural philosophy is seldom very
happy. A poet has few facts to communicate, and these he wishes to
expand and diversify ; he dwells on a favourite idea, and repeats it in a
thousand emblematical forms ; his object is to say a little, very elegantly,
in very circuitous, and somewhat obscure terms. But the information,
which the natural philosopher has to impart, is too copious to allow of pro-
lixity in its detail ; his subjects are too intricate to be compatible with
digressions after amusement, which, besides interrupting, are too likely to
enervate the mind ; and if he is ever fortunate enough to entertain, it
must be by gratifying the love of truth, and satisfying the thirst after
knowledge. We have, however, a favourable specimen of highly orna-
mented philosophy in Fontenelle's Plurality of Worlds ;* a work which
must be allowed to convey much information in a very interesting form,
although somewhat tinctured with a certain frivolity which is not always
agreeable, We need not attempt to accompany all the flights of Fonte-
nelle's imagination ; it will be sufficient for our purpose to pursue his
ideas in a simple enumeration of the most remarkable phenomena, that
would occur to a spectator placed on each of the planets.

Of Mercury we know little except the length of his year, which is
shorter than three of our months. Supposing all our heat to come from
the sun, it is probable that the mean heat on Mercury is above that of
boiling quicksilver ; and it is scarcely possible that there should be any
point about his poles where water would not boil. The sun's diameter
would appear, if viewed from Mercury, more than twice as great as to us
on the earth.

Venus must have a climate far more temperate than Mercury, yet much
too torrid for the existence of animals or vegetables, except in some cir-
cumpolar parts ; her magnitude and diurnal rotation differ but little from
those of the earth, and her year is only one third shorter : so that her sea-
sons, and her day and night, must greatly resemble ours. The earth, when
in opposition to the sun, must be about four times as bright as Venus ever
appears to us, and must, therefore, always cast a shadow ; it must be fre-
* 12mo, 1686 ; par Lalande, 1800.

APPEARANCES OF THE CELESTIAL BODIES. 423

quently, and perhaps generally, visible in the day ; and together with the
moon, must exhibit a very interesting object. The atmosphere of Venus is
supposed to be nearly like our own, or somewhat more rare.

The inhabitants of the moon, if the moon is inhabited, must be capable
of living with very little air, and less water : there is reason to think their
atmosphere less than a mile high, and it is never clouded : so that the sun
must shine without intermission for a whole fortnight on the same spot,
without having his heat moderated by the interposition of air, or by the
evaporation of moisture. The want of water in the moon is not, as some
have supposed, the necessary consequence of the want of an atmosphere ;
but it is inferred partly from the total absence of clouds, and partly from the
irregular appearance of the margin of the moon, as seen in a solar eclipse :
no part of it being terminated by a line sufficiently regular to allow us to
suppose it the surface of a fluid. The earth must always appear to occupy
nearly the same part of the sky, or rather to describe a small oval orbit round
a particular point, exposing a surface 13 times as great as that of the moon
appears to us. This large surface, suspended, with phases continually
changing, like those of the moon, must afford, especially when viewed with
a telescope, an excellent timepiece ; the continents and seas coming gradu-
ally and regularly into view, and affording a variety equally pleasing and
useful. To us such a timepiece would be of inestimable value, as it would
afford us an easy method of discovering the longitude of a place, by com-
paring its motion with the solar time: but in the moon, the relative
position of the earth and sun, or of the earth and stars only, would be
sufficient for determining the situation of any place in sight of the earth ;
if, however, there are no seas and no navigation, astronomical observations
of this kind would be of very little utility. The assistance of the earth's
phases in the measurement of time might, however, still be very useful for
many purposes, to the inhabitants of the nearer half of the moon ; and
probably the remoter part is much deserted, for in their long night of half
a month, they must be extremely in want of the light reflected from the
earth, unless the inhabitants have the faculty of sleeping through the whole
of their dark fortnight. The surface of the moon appears to be very rocky
and barren, and liable to frequent disturbances from volcanos. These
have been supposed to project some of their contents within the reach of the
earth's attraction, which they might easily do, if they could throw them out
with a velocity of about eight thousand feet in a second, which is only four
times as great as that of a cannon ball : and these stones, falling through
the atmosphere, might very possibly generate so much heat, by compressing
the air, as to cause the appearance of fiery meteors, and to fall in a state of
ignition. The appearance of the moon, as viewed through a good telescope,
is extremely well imitated by Mr. Russel's lunar globe, which is also capa-
ble of exhibiting, with great accuracy, the changes produced by its libra-
tions.

The climate of Mars is as much colder than ours, as that of Venus is
.warmer ; in other respects there is no very striking difference : the incli-
nation of his axis to his ecliptic being nearly the same as that of the earth's
axis, the changes of seasons must be nearly like our own. Dr. Herschel

424 LECTURE XLIV.

has observed a constant appearance of two bright spots or circles near the
poles of Mars, which he attributes to the ice and snow perpetually sur-
rounding them. It is not, however, probable that water could remain fluid
in any part of Mars, and even quicksilver and alcohol would, perhaps, be
frozen in his temperate climates. It is pretty certain that Mars has an
atmosphere, and his dark spots seem to be occasioned by clouds : this
atmosphere may, perhaps, also be the cause of the ruddy hue of his
light.

It appears to be doubtful, whether either of the three little planets newly
discovered can be sufficiently solid, to give a firm footing to any material
beings : we should probably weigh only a few pounds each if transported
there. According to Dr. Herschel's opinion, neither Ceres nor Pallas is
much larger than a good Scotch estate, although they must, sometimes,
appear to each other as planets of a most respectable size. The light
reflected from Ceres is of a more ruddy hue than that of Pallas : both of
these planets are attended by more or less of a nebulosity, proceeding, per-
haps, from copious atmospheres ; and in this respect, as well as in the great
inclination of their orbits, they appear to have some affinity to comets.
It is tolerably certain that neither of them is 200 miles in diameter ; and
Juno is also probably about the same size.

It is obvious that the most striking features of the heavens, when con-
templated from Jupiter, would be the diversified positions and combina-
tions of his satellites : their light must be faint, but yet of service ; and to
a traveller on the surface of this vast globe they must afford useful infor-
mation, as well with respect to time as to place. Our little earth must
probably be always invisible to a spectator situated on Jupiter, on account
of its apparent proximity to the sun, in the same manner as a planet at
half the distance of Mercury would be invisible to us. The year of Jupiter
must contain nearly ten thousand of his days, and that of Saturn almost
thirty thousand Saturnian days. Besides the vicissitudes of the seven
satellites revolving round Saturn, his ring must afford, in different parts of
his surface, very diversified appearances of magnificent luminous archer,
stretched across the heavens, especially in that hemisphere which is on the
same side of the ring with the sun.

From the Georgian planet the sun must be seen but as a little star, not
one hundred and fiftieth part as bright as he appears to us. The axis of
this planet being probably near to the plane of its ecliptic, it must be
directed twice in the year towards the sun, and the limit of illumination
must approach to the equator, so that almost every place on his surface
must sometimes remain, for a great number of diurnal revolutions, in light
and in darkness ; the most moderate climates having one night, in their
long year, equal in duration at least to several of our years : and it must be
confessed that this planet would afford but a comfortless habitation to those
accustomed to our summer sunshine, even if it were possible to colonise it.
(Plate XXXIV. Fig. 501.)

On the whole, we are tempted, from an almost irresistible analogy, to con-,
elude that the planets are all in some manner or other inhabited ; but at the
same time we can scarcely suppose that a single species of terrestrial animals

ON PRACTICAL ASTRONOMY. 425

or even vegetables could exist in any of them ; their minerals may, per-
haps, resemble ours, and if the stones which Mr. Howard has analyzed are
really lunar productions, we have proofs that the moon at least contains
some substances resembling those which compose the earth ; but the seas
and rivers of the other planets must consist of some fluid unknown to us,
since almost all our liquids would either be frozen, or converted into
vapour, in any of them.

LECT. XLIV.— ADDITIONAL AUTHORITIES.

Librations of the Moon. — Cassini, Hist, et Mem. 1721, p. 168, H. 53. Lalande,
ibid. 1764, p. 555, H. 112. Sejour, ibid. 1776.

Eclipses. — Hevelius, Ph. Tr. i. 369 ; v. 2023. Louville's Geometrical Mode of
calculating Eclipses, Hist, et Mem. 1724, p. 63, H. 74. Gersten's Meth. Ph. Tr.
1744, p. 22. Lalande on the Effect of Ellipticity, Hist, et Mem. 1756, p. 364,
H. 96 ; 1763, p. 413. Lambert, Table Ecliptique, 12mo, Berlin, 1765. Boscovich
de Solis et Lunae Defectibus, 4to, Lond. 1760. Jeaurat on the Projection of Eclipses,
Mem. des Sav. Etr. iv. p. 818. Goudin, Mem. sur les Eclipses du Soleil, 4to,
1803. Lubbock, Elementary Treatise on the Computation of Eclipses, 1835.

LECTURE XLV.

ON PRACTICAL ASTRONOMY.

IT is generally most convenient in practical astronomy to neglect the real,
and to consider only the apparent motions of the sun, the stars, and planets,
for the visible effects must be precisely the same, whether the sun or the
earth perform a revolution in the plane of the ecliptic, and whether the
earth actually move on its axis, or the whole of the celestial bodies move
round it in a day. We may, therefore, suppose the sun to move, as he
appears to do, from west to east in the ecliptic, so as to advance almost a
degree in 24 hours, and from east to west, together with all the stars and
planets, so as to perform a whole revolution in a day. Speaking more
correctly, the sun appears to describe, in every sidereal day, a spiral, which
differs a little from a circle, and is also about a degree shorter, so that about
four minutes more are required for the return of the sun to the same part
of the heavens, and the completion of a solar day.

In order to determine the place of any point in the heavens, it is usual to
compare its situation either with the plane of the earth's equator, or with the
ecliptic ; its angular distance from the equator being called its declination,
and from the ecliptic, its latitude ; these distances must be measured in
planes perpendicular to those of the equator or ecliptic, and the distances of
.these planes from their intersection, or from the equinoctial point in Aries,
are called respectively the right ascension and the longitude of the point to
be described. For the stars, the decimation and right ascension are most

426 LECTURE XLV.

usually laid down ; but with respect to the sun and the planets, performing
their revolutions in or near the ecliptic, it is most convenient to calculate
their latitude and longitude.

The plane passing through the earth's axis and the place of a spectator is
the plane of the meridian of that place ; and a plane touching the earth
in any point is its horizon. With respect to the appearances of the fixed
stars, this plane may he considered as passing through the earth's centre
in the same direction : and the difference is scarcely sensible with respect
to the sun and the primary planets ; but in observations of the moon's
place, these planes must be carefully distinguished. (Plate XXXV. Fig.
502.)

The instruments requisite for astronomical observations are principally
referrible to geometrical or to optical apparatus, or to the measurement of
time. Particular constructions and combinations are, however, necessary
for the accommodation of quadrants, graduated circles, telescopes, and
transit instruments, to the uses of observatories ; and astronomical observa-
tions are as necessary to the correct determination of time, as artificial
timekeepers are useful for other astronomical purposes.

The most accurate standard of time is the diurnal rotation of the earth
on its axis, as ascertained by its situation with respect to the fixed stars.
The time elapsing between two successive passages of any star over the
same meridian, which constitutes a sidereal day, may be measured with
great precision ; and the star may for this purpose be observed, with almost
equal accuracy, in any other situation, and sometimes with greater con-
venience. The length of the sidereal day may be considered as perfectly
constant, the inequalities arising from the aberration of light, and from
the nutation of the earth's axis, being too small to be easily discovered ;
but the correction for the aberration may be applied when it is neces-
sary. For observations of this kind, it is usual to have a clock adjusted
to sidereal time, which not only admits of a more ready comparison with
the transits or passages of any one star over the meridian, but, by the
difference of the times of any two transits, shows at once the difference
of right ascension of the stars or planets, expressed in time instead of
degrees.

The solar days are not only about four minutes longer than the sidereal
days, but they are also unequal among themselves ; and this inequality
arises from two causes ; the one, that even if the sun's motion in the
ecliptic were uniform, his diurnal changes of right ascension would be
different at different times, and the difference between his path in every
sidereal day, and a whole circle, depending on this change, would also
vary ; the other that the sun's motion in the ecliptic is actually variable,
consequently the diurnal change of right ascension is liable to a double
inequality. Hence it happens that the solar time agrees at four instants
in the year only with the mean time, derived from supposing the whole
365 days to be divided into as many equal parts ; the difference is called
the equation of time, and amounts sometimes to as much as 16 minutes.
The term equation is commonly applied in astronomy to any small quan-
tity, which is to be added to, or subtracted from, another quantity ; thus

ON PRACTICAL ASTRONOMY. 427

it is usual, in calculating the place of a planet, to find from the tables of
its motion, the mean place, in which it would be found if its orbit were
circular, and thence to derive the true place, by means of various correc-
tions called equations. In France the solar time is considered as the true
time, and is used for all civil purposes, so that the clocks are some-
times embarrassed with a complicated apparatus, calculated for imitating
the inequalities of the actual apparent motion of the sun. (Plate XXXV.
Fig. 503.)

The art of dialling consists principally in projecting, on a given surface,
such lines as will coincide with the shadow of an index or gnomon parallel
to the earth's axis, at intervals corresponding to the different hours of the
day : so that nothing more is necessary for the construction of a dial, than
to determine the intersections of the surface on which the dial is to be
constructed, writh planes passing through the edge of the gnomon, and
situated at equal angular distances from each other : thus, supposing the
plane of the dial perpendicular to the gnomon, and parallel to the equinoc-
tial, the hour lines of the dial will be at equal distances from each other ;
but in other cases their distances will be unequal, and must be determined
either by calculation or by construction. A point may also be used as a
gnomon, as well as a line ; but in this case the hour lines must cover a
larger portion of the surface, in order that the shadow of the point may
always fall among them. (Plate XXXV. Fig. 504... 506.)

The changes of the seasons depend on the return of the sun to the same
position with respect to the equator, or on the length of the tropical year,
so called from the tropics, which are circles supposed to be parallel to the
equator, and between which the sun travels continually backwards and
forwards, appearing to remain for some time, when he is near them, with
very little change of declination ; whence the time when the sun touches
either tropic is called the solstice. The vicissitudes of light and darkness
depending also on the solar day, it is necessary, for the regulation of the
civil calendar, to establish the proportion between the periods of the solar
day and the tropical year ; and since the tropical year exceeds the time of
365 days, by 5 hours, 48 minutes, and 48 seconds, it is usual to add to the
common year an intercalary day once in about four years. The ancient
Egyptians reckoned only 365 days in a year, and their nominal new year
arrived continually earlier and earlier, so that after 1507 of their years, it
would have happened successively on each of the 365 days, and returned
to the original place : the same mode of computation was also adopted by
the Greek astronomers. The Romans inserted intercalary days, at first
without much regularity, according to the direction of their augurs, until
the time of Julius Caesar ; who, observing that the year was almost 6 hours
longer than 365 days, added a day every fourth year to the calendar, by
reckoning twice the day in February called sexto calendas Martias, whence
this year of 366 days was denominated a bissextile year. The new moon
immediately following the winter solstice, in the 707th year of Rome, was
.made the first of January of the first year of Caesar ; the 25th of December
in his 45th year is considered as the date of the Nativity of Christ, and
Caesar's 46th year is reckoned the first of our era. The preceding year is

428 LECTURE XLV.

commonly called by astronomers the year 0, but by chronologists the year
1 before Christ. The correction introduced by Caesar was, however, too
great, the error being exactly 7 days in 900 years ; so that in 1582 it
amounted to about 12 days. This error was not wholly removed by Pope
Gregory, who reformed the calendar ; he omitted 10 days only of the usual
reckoning, in order to bring back the course of the moveable feasts to the
same state, in which they had been established by the Nicene council, in
the fourth century. He determined at the same time that the last year
of every century should be passed without an intercalary day, excepting
that of every fourth century, which was still to be bissextile. Thus every
year divisible by four, without a remainder, is in general a bissextile or
leap year, but the last year of every century must be a common year,
unless the number of the century be divisible by 4 ; the year 1800 being
a common year, and 2000 a bissextile. In this manner 27 Julian bissex-
tiles are omitted in 3600 years, while the true length of the year would
require the omission of 28 ; but so small a difference can be of no material
consequence. The Persians had introduced into their calendar, in the
llth century, an intercalation still more accurate ; they make 8 bissex-
tiles only every 33 years, reckoning four common years together instead
of three, at the end of this period, so that in 132 years they have 32 leap
years instead of 33 ; and the error is only a day in about five thousand
years. If any change in the Gregorian calendar were thought necessary,
it would be easy to make the last year of every fourth and fifth century
alternately a bissextile, and this correction would be quite as accurate as
it is possible for our calculations to render it. The adoption of the
Gregorian calendar in this country was for some time delayed by religious
prejudices ; one of the best-founded objections to it was, that 2 days of
the real error was still unconnected ; but better arguments at last over-
came these difficulties, and the new style was introduced on the 14 Sep-
tember 1754, which would have been called, according to the old style, the
third.

Any tolerable approximation of this kind, when once generally estab-
lished, appears to be more eligible than the mode which was lately adopted
in France, where the republican year began at the instant of the midnight
preceding the sun's arrival at the autumnal equinox. Mr. Lalande very judi-
ciously observes, that there are several years, in which the sun will pass the
equinox so near to midnight, that it is not at present in the power of calcu-
lation to determine on what day the republican year ought to begin ; and
perhaps these arguments have co-operated with others in facilitating the
restoration of the ancient calendar.

The revolutions of the sun and moon are not very obviously commen-
surable, the solar year containing 12 lunations and almost 11 days ; but
Meto discovered, more than 2000 years ago, that 19 solar years contain
exactly 235 lunations ; and this determination is so accurate, that it makes
the lunar month only about half a minute too long. Hence it happens,
that in every period of 19 years, the moon's age is the same on the same
day of the year. The number of the year, in the Metonic cycle, is called
the golden number, the calendar of Meto having been ordered, at the cele-

ON PRACTICAL ASTRONOMY. 429

bration of the Olympic games, to be engraved in letters of gold on a pillar
of marble. At present, if we add 1 to the number of the year, and divide
it by 19, the remainder will be the golden number ; thus, for 1806, the
golden number is 2. If we subtract 1 from the golden number, then mul-
tiply by 11, and divide by 30, the remainder will be the epact, which is the
moon's age on the first of January, without any material error ; thus, for
1806, the epact is 11, and the moon is actually 11 days old on the first of
January.

From a combination of chronological periods of various kinds, Scaliger
imagined the Julian period, as an epoch to which all past events might
with convenience be referred, beginning 4713 years before the birth of
Christ. Laplace proposes, as a universal epoch, the time when the earth's
apogee was at right angles with its nodes, in the year 1250, calling the
vernal equinox of that year the first day of the first year. But the fewer
changes of this kind that we make, the less confusion we introduce into our
chronology. The astronomical year begins at noon on the 31st of Decem-
ber, and the date of an observation expresses the days and hours actually
elapsed from that time. Thus, the first of January, 1806, at 10 in the
morning, would be called, in astronomical language, 1805 December 31
days 22 hours, or more properly 1806 January 0 day 22 hours.*

For ascertaining, by immediate measurement, the position of any of the
heavenly bodies, it is usual to determine its meridian altitude by quadrants,
and the time of its passing the meridian by transit instruments. The large
quadrants, generally used for this purpose in observatories, are fixed to
vertical walls, in order to give them greater stability, and are thence
called mural quadrants ; sometimes a smaller portion of an arc only is
adapted for observations near the zenith, under the name of a zenith sector.
A transit instrument is a telescope so fixed on an axis as to remain always
in the plane of the meridian : the axis being perpendicular to this plane,
and consequently in a horizontal position, and directed east and west.
Those who are in the constant habit of observing with attention, can esti-
mate, in this manner, the precise time of the passage of a celestial object
over the meridian ; without an error of the tenth of a second, so that its
angular right ascension may be thus determined within about a second of
the truth. A very convenient mode of adjusting a transit instrument is to
direct it to the north polar star, at the same time that the last of the three
horses in the wain is perpendicularly above or below it : this process, in
1751, gave precisely the true meridian ; but since that time, the pre-
cession of the equinoxes, which produces a slight change in the places
of the stars, has made it necessary to wait 1 minute 13£ seconds for
every ten years that have elapsed. Thus, in 1806, if we wait 6£
minutes, the pole star will then be precisely in the meridian, and will
serve for the correct adjustment of the instrument. (Plate XXXV. Fig.
507-.. 510.)

* On the Calendar, consult Sauveur, Hist, et Mem. 1732, H. 94. Lord Mac-
' clesfield, Ph. Tr. 1750, p. 417. Lalande, Hist, et Mem. 1789, p. 95. Halma, Sur
la Reduction des Annees des Anciens a la Forme des Notres, 4to, 1819. Sur les
Mois Macedoniens, 4to, 1820. Encyc. Brit. art. Calendar.

430 LECTURE XLV.

The quadrant in most common use, especially for nautical observations,
was first proposed by Newton,* but improved, or perhaps reinvented, by
Hadley.t Its operation depends on the effect of two mirrors which bring
both the objects, of which the angular distance is to be measured, at once
into the field of view ; and the inclination of the speculums by which this
is performed serves to determine the angle. The ray proceeding from one
of the objects is made to coincide, after two reflections, with the ray coming
immediately from the other, and since the inclination of the reflecting sur-
faces is then half the angular distance of the objects, this inclination is read
off on a scale in which every actual degree represents two degrees of
angular distance, and is marked accordingly. There is also a second fixed
speculum, placed at right angles to the moveable one, when in its remotest
situation, which then produces a deviation of two right angles in the ap-
parent place of one of the objects, and which enables us, by moving the
index, to measure any angle between 180° and 90°. This operation is
called the back observation ; it is however seldom employed, on account of
the difficulty of adjusting the speculum for it with accuracy. The reflect-
ing instrument originally invented by Hooke was arranged in a manner
somewhat different. (Plate XXXV. Fig. 511.)

From the meridian altitude of any point, it is easy, when the elevation of
the pole is known, to deduce its declination ; and its right ascension may
be found from the time of its passage over the meridian after that of the
equinoctial point, allowing 15 degrees for each sidereal hour. (Plate
XXXV. Fig. 512.)

In all astronomical observations it is necessary to make proper cor-
rections, according to the rules of optics, for the effects of atmospherical
refraction ; and also, in observations on the moon more especially, for those
of parallax, or the difference of the apparent place of the luminary with
respect to the earth's centre, and to the place of the spectator, which is
equal to the angle subtended at the centre of the luminary by the semidia-
meter of the earth passing through the place of observation ; since all cal-
culations of the geocentric places of the heavenly bodies are referred to the
earth's centre. This angle, which is to be added to the apparent altitude,
amounts sometimes, in the case of the moon, when near the horizon, to
more than a degree ; the refraction, which is in a contrary direction, and is
to be subtracted from the altitude, being at the horizon about 33 minutes.
(Plate XXXV. Fig. 513.)

The most important applications of practical astronomy are in the de-
termination of the latitudes and longitudes of places on the earth's surface.
The latitude, which is the angular distance of the place from the equator,
or the angle formed by the plane of its horizon with the earth's axis, is
easily ascertained by finding the meridian altitude of a body, of which the
declination is known ; since, by deducting or adding the declination, we
have at once the elevation of the equinoctial, or of the plane of the equator
above the horizon, and subtracting this from a right angle, we find the
elevation of the pole, or the latitude. (Plate XXXV. Fig. 512.)

It is also common to determine the latitude of a place by means of two
* Ph. Tr. 1742, p. 155. f See Lect. XXXVI.

ON PRACTICAL ASTRONOMY. 431

altitudes observed at different times in the same day, noticing accurately
the interval of time that elapses between the observations. This method
has a great advantage in cloudy weather, when it is not possible to ensure
an observation of a meridian altitude.

The longitude of a place, or the relative position of its meridian, is by no
means so readily determined. For this purpose it becomes necessary to
ascertain the time that elapses between the passages of a given point in the
heavens over its meridian and some other meridian which serves as a
standard of comparison. Thus, if the sun arrives three hours later at the
meridian of any place than at the meridian of London, that place must
necessarily be 45 degrees west of London, or in 45° west longitude : and if
we know, when it is noon at the given place, that it is three o'clock in the
afternoon at Greenwich, we may be certain that we are in some part of a
meridian 45° west of that of Greenwich. Had we perfect timekeepers, we
might easily adjust them to the time of our first meridian, and then, by
comparison with the usual determinations of time in any other place, to
which they might be carried, the longitude of this place might be found
with perfect accuracy. Such timekeepers as we have are indeed suffi-
ciently correct, to be of considerable utility, but it is necessary to compare
them frequently with astronomical observations of phenomena, which occur
at times capable of a correct calculation. Sometimes the transits of Mer-
cury and Venus, or the eclipses of the moon, are employed for this purpose,
but more usually the eclipses of the satellites of Jupiter ; these, however,
cannot be well observed without a more powerful telescope than can be
employed at sea ; and the theory of the moon's motion, has of late years
been so much improved, that her distance from the sun or from a fixed
star can be calculated, with sufficient accuracy, for determining the time in
London or at Paris without an error of one third of a minute ; so that
supposing the observation could be rendered perfectly correct, the longitude
might be thus ascertained within about one twelfth of a degree, or at most
five nautical miles.

The observed parallax of the sun and moon may be employed for the
determination of their distances from the earth ; but in the case of the sun,
the simple comparison of his calculated with his apparent altitude is in-
sufficient for ascertaining the magnitude of the parallax with accuracy.
Sometimes the parallax of Mars, which is considerably greater than the
sun's, has been directly measured ; but the most correct mode of ascer-
taining the actual dimensions of the solar system is, to observe a transit
of Venus over the sun's disc, at two places situated in opposite parts of
the earth's surface. For, since the diurnal motion of some parts of the
earth is directed the same way with the motion of Venus in her orbit,
and that of others the contrary way, the different effects of these motions
must furnish a mode of comparing the rotatory velocity of the earth,
with the progressive velocity of Venus, and consequently of inferring,
from the known velocity with which the earth's surface revolves, the
actual velocity of Venus, and her distance from the sun ; whence the
distances of all the other planets may be readily deduced. (Plate XXXV.
Fig. 514.)

432 LECTURE XLV.

Our countryman Horrox* was the first that particular!}' attended to the
phenomena of a transit of Venus over the sun's disc : Dr. Halley,t when
he observed a transit of Mercury at St. Helena, thought that he could
ascertain the times of immersion and emersion without an error of a single
second ; and hence he concluded, that by means of a transit of Venus, the
sun's distance might be determined within a five hundredth part. The
most advantageous places for the experiment being such as differ most in
longitude, and are most remote from each other, Captain Cook was sent
by the British government to the South Seas, in the years 1761 and 1769,
in order to observe the transits of Venus in the island of Otaheite.
These observations were compared with those which were made at Ward-
huys, in Danish Lapland ; the difference of the times occupied by the
transit at these places was found to be 23 minutes 10 seconds, and from
this comparison, corrected by a number of collateral observations, the sun's
mean parallax was found to be 8 seconds and two thirds, or perhaps 8f ;
for it does not appear that we are sure of having avoided even an error of
one fortieth part of the whole ; although Mr. Laplace's determination of
the sun's distance, from the lunar motions, agrees very well with that
which is usually considered as the result of the observations of the transit
of Venus. J

The comparative densities of the sun, and of such planets as have satel-
lites, may be calculated from the periods and distances of the bodies revolv-
ing round them ; the densities of the other planets have sometimes been
assigned from conjecture only, but of late years the mathematical theory
of the planetary perturbations has been rendered so perfect, that some
dependence may perhaps be placed on the density assigned to them from
calculations of this kind. It was formerly supposed that the densities of
the planets were regularly greater as they were nearer to the sun ; but it
is now certain that the Georgian planet is more dense than Saturn, and it
is probable that Venus is somewhat less dense than the earth. The mass
of the moon is deduced from a comparison of the effects of her attraction on
the earth and sea with those of the sun's attraction.

The artificial globe serves as a useful instrument for determining, in a
rough manner, without calculation, the affections of the heavenly bodies at
particular times ; their places being first ascertained from tables, or, in the
case of the sun, merely from a scale on the globe's horizon, or on its surface.
We have only to adjust the elevation of the pole of the globe in such a
manner, that its axis may form the same angle with its horizon as the axis
of the earth does with the real horizon of the place ; then finding a point
on its surface corresponding to the place of the sun or planet, we may
represent its apparent motion by the motion of this point, and the time
occupied by that motion will be shewn by the index of the globe ; thus we
may find the length of the day and night, and the time and place of rising

* See Hevelius, Mercurius in Sole visus Gedani An. 1661, cuiannexa est Venus
in Sole visa An. 1639, Liverpolise, a J. Horroxio, fol. Ged. 1662.

t De Parallax! ope Veneris Determinanda, Ph. Tr. 1716, p. 454.

I Euler on the Sun's Parallax, computed by Lexell, Ph. Tr. 1772, p. 69, makes
it 8"-55 ; Laplace, from the moon's motion, makes it 8"-6.

ON PRACTICAL ASTRONOMY. 433

and setting ; and by means of a graduated circle, perpendicular to the hori-
zon, we may measure the altitude of the sun or planet at any other time,
and also its azimuth, or the distance of this circle from the north or south
point of the horizon. If we have a ring of any kind parallel to the horizon,
and 33 minutes below it, we may consider this ring as the apparent hori-
zon, allowing for the effects of refraction ; if it be still 15 or 16 minutes
lower, it will represent the rising or setting of the extreme margin of the
sun or moon : we might also have a circle about a degree above either of
these, which might represent the sensible or apparent horizon, with regard
to the moon, including the correction for her parallax ; and a similar ring,
placed still lower, would show the duratkm of twilight, on any supposition
that might be formed respecting the depression of the sun required for pro-
ducing total darkness. By means of the celestial globe, the apparent
motions of the fixed stars may be represented in a manner nearly similar,
proper attention being paid to the situation of the sun in the ecliptic, as
determining the time corresponding.

Many of these operations may also be performed with equal convenience
with a planisphere, which is a stereographical projection of the globe on a
plane surface. Professor Bode's planisphere comprehends in one view all
the stars that are ever visible at Berlin : he has added to it a moveable cir-
cle, representing the horizon of that place, carrying with it the circles of
altitude and azimuth, delineated on a transparent paper, which is adjusted,
by graduations at the margin of the chart, to the day and hour for which
we wish to ascertain the apparent places of the heavenly bodies. Any
other chart of the stars, having the pole in its centre, may be applied to a
similar use, by cutting out a circle, or a part of a circle, to represent the
horizon of a place of which the latitude is given ; and if the stars are pro-
jected, as is usual, on two equal charts, they must have two equal arcs to
represent the respective parts of the horizon belonging to them. A simple
construction may also often be made to serve for solving many problems of
a similar nature. (Plate XXXV. Fig. 515, 516. Plate XXXVI. Fig. 517.
Plate XXXVII. Fig. 518.)

For representing the real as well as the apparent motions of the different
parts of the solar system, planetariums or orreries have sometimes been
employed, in which the comparative periods of the revolutions have been
expressed by various combinations of wheelwork. Of these instruments
Archimedes was the original inventor, and Huygens revived them, with
many improvements, in modern times. The construction of the large pla-
netarium, which has been made in the house of the Royal Institution, was
principally directed by Mr. Pearson. I suggested to him, that the instru-
ment might be placed in a vertical position, and that the eccentricities of
the planetary orbits might be shown by the revolution of short arms, retained
in their situation by weights, and their deviation from the plane of the
ecliptic by inclining the axes of these arms, in a proper angle, to the plane
of the instrument. The other parts of the arrangement, which have any
claim to novelty, were entirely of Mr. Pearson's invention, and he appears
to have rendered the instrument in many respects more accurate than any
other planetarium that has ever been constructed.*

* On this subject see the article Planetarium, by Pearson, in Rees's Cyclopaedia.

2 F

434 LECTURE XLV.

LECT. XLV.— ADDITIONAL AUTHORITIES.

Apparatus in general. — Hevelii Organographia Astr. fol. Ghent, 1673. Hooke's
Animadversions on Hevelius's Machina Cselestis, 4to, 1674. Maskelyne's Remarks,
Ph. Tr. 1764, p. 348. Due de Chaulnes, Hist, et Mem. 1765, p. 411, H. 65. Bird,
The Method of Dividing Astr. Insts. 4to, 1767. Magellan, Collection deTraites sur
des Instr. d'Astr. 2 vols. 4to, 1775, 1780. Descrip. des Nouveaux Instr. a Re-
flexion, 4to, London, 1779. Descrip. des Octants et Sextants Anglois, 4to, Paris,
1775. Ludlam on Bird's Method of Dividing, 4to, 1786. Piazzi, della Specola Ast. di
Palermo, fol. Pal. 1792. Troughton, Zach's Mon. Corres. ii. 207. Kluber, Die
Sternwarte zu Mannheim beschreiben, 4to, Mann. 1811. Quetelet, Sur 1'Obs. de
Bruxelles, Corresp. Mathem. vol. vii. Simms, A Description of Mathematical In-
struments, 1834.

Telescope Wires, Sfc. — See Lect. XXXVI. Herschel's Micrometers, Ph. Tr.
1782, p. 163; 1783, p. 4. Wollaston on a System of Wires, ibid. 1785, p. 346.
Rittenhouse on Spiders' Webs, Am. Tr. ii. 181. Ussher on Illuminating the Wires
of a Transit, Ir. Tr. 1788, p. 13.

Transit.— Derham's, Ph. Tr. 1704,1578. Roemer's, Mis. Berl. 1727, p. 276.
Gensanne's, Hist, et Mem. 1736, H. 120. Mach. App. vii. 55. Wollaston, Ph. Tr.
1793, p. 133. Ramsden's, ibid. 1795, p. 419.

Equatorial Instruments.— Short, Ph. Tr. 1749, p. 241. Nairne's, ibid. 1771,

?. 107. Haupoin's, Jour, de Physique, xlii. 286. Shuckburgh on the Equatorial
nst. Ph. Tr. 1793, p. 67. Struve, Beschreibung des zu Dorpat Refractors von
Frauenhofer, fol. Dorpat, 1825.

Mural Circle. — Bird, The Method of constructing Mural Quadrants, 4to, Lond.
1768. Cesaris, De Quadrante Murale, quern Spec. Med. construx. J. Ramsden,
4to, Mediol. 1794. Ramsden and Berge's Zenith Sector, Ph. Tr. 1803, p. 383.

Observations, with their corrections. — Refraction — See Lect. XXXVII.

Aberration, ^c.— Zach's Tables, 2 vols. 4to, Gotha, 1806-7, Marseille, 1812-13.
Baily, Tables for Precession, Aberration, and Nutation, 4to, Lond. 1827. Bessel,
Tabulae Regiomontanse, 1830.

Parallax. — Halley, De Parallaxi Solis ope Veneris determinanda, Ph. Tr. 1716,
p. 454. Boscovich, ibid. 1760, p. 865. On the Transit of 6th June, 1761, Ph. Tr.
1761, lii. 173, 582, 611.... ; 1763, pp. 300, 467 ; 1764, p. 152 ; 1765, p. 326;
1766, p. 244; 1771, p. 574 ; 1768, pp.107, 154, 355. Chappe d'Auterouche,
Mem. 4to, St. Petersb. 1762. Rb'hl, Von den Durchgangen der Venus, Greifsw.
1768. Lalande on the Solar ParaUax, Hist, et Mem. 1771, p. 776, H. 83. Cook,
Ph. Tr. 1771, p. 433. Euler, ibid. 1772, p. 69.

Solar Tables.— Cassini's, 4to, Paris, 1740. Lacaille's, Vienn. 1763. Delambre's,
Bureau des Longitudes, 4to, 1806. Zach's, 4to, Gotha, 1809. Carlini's (Milan Ef .),
1810-11. Burckhardt's, Conn, des Temps, 1816. South and Airy's, Ph. Tr.
1826-7. Bessel's, Astr. Nachr. 1828.

Lunar Tables. — Hell's, Vienn. 1763, Mayer's, 4to, Lond. 1787. Mendoza's,
4to, Lond. 1801. Burg's, Bureau des Long. 4to, Paris, 1806. Zach's Tables
abrege"s pour Paris, Florence, 1809. Burckhardt's, 4to, Paris, 1826. Damoiseau's,
fol. Paris, 1828.

Tables of Mercury. — Lindenau's, 4to, Gotha, 1813.

Venus.— Lindenau's, 4to, Gotha, 1810. Reboul's, 4to, Marseille,

1811.

Mars. — Lindenau's, 4to, Eisenb. 1811.

New Planets.— Zach's of Ceres, Ph. Mag. xii. 360 ; xv. 190. Carlini's,

Milan, 1818.

Jupiter. — Delambre's of Jup. and Sat. 4to, 1789. Delambre's New

Tables of his Satellites, 4to, 1817.

Saturn. — Bouvard's of Jup. and Sat. 4to, Paris, 1808.

Uranus. — Bouvard's, 1821. Herschel on his Satellites, Ph. Tr. 1815.

Weisse, Coordinate Mercurii, Veneris, Martis, Jovis, Saturni, et Urani calculate,
4to, Cracow, 1826.

435

LECTURE XLVI.

ON GEOGRAPHY.

FROM the consideration of the stars, the sun, and the planets in general,
we are now to descend to that of the earth, the particular planet which we
inhabit, and which we can examine more minutely than the other parts of
the solar system. Its external form, its divisions, whether astronomical or
natural, its most remarkable features, and its internal structure, will require
to be separately investigated.

The general curvature of the earth's surface is easily observable in the
disappearance of distant objects, and in particular, when the view is limited
by the sea, the surface of which, from the common property of a fluid,
becomes naturally smooth and horizontal : for it is well known that the
sails and rigging of a ship come into view long before her hull, and that
each part is the sooner seen as the eye is more elevated. On shore, the fre-
quent inequalities of the solid parts of the earth usually cause the prospect
to be bounded by some irregular prominence, as a hill, a tree, or a build-
ing, so that the general curvature is the less observable.

The surface of a lake or sea must be always perpendicular to the direc-
tion of a plumb line, which may be considered as the direction of the force
of gravity ; and by means either of a plumb line or of a spirit level, we may
ascertain the angular situation of any part of the earth's surface with
respect to a fixed star passing the meridian ; by going a little further north
or south, and repeating the observation on the star, we may find the differ-
ence of the inclination of the surfaces at both points ; of course, supposing
the earth a sphere, this difference in latitude will be the angle subtended at
its centre by the given portion of the surface, whence the whole circum-
ference may be determined ; and on these principles the earliest measure-
ments of the earth were conducted. The first of these, which can be con-
sidered as accurate, was executed by Picart* in France, towards the end of
the seventeenth century.

But the spherical form is only an approximation to the truth ; it was
calculated by Newton, and ascertained experimentally by the French Aca-
demicians, sent to the equator and to the polar circle, that, in order to
represent the earth, the sphere must be flattened at the poles, and promi<r
nent at the equator. We may therefore consider the earth as an oblate
elliptic spheroid ; the curvature being greater, and consequently every
degree shorter, at the equator, than nearer the poles. If the density of the
earth were uniform throughout, its ellipticity, or the difference of the
length of its diameters, would be -3-^5- of the whole ; on the other hand, if
it consisted of matter of inconsiderable density, attracted by an infinite force
in the centre, the ellipticity would be only ^-y ; and whatever may be
the internal structure of the earth, its form must be between these limits,
* Hist, et Mem. vii. I. 46.
2 F2

43(5 LECTURE XLVI.

since its internal parts must necessarily be denser than those parts which
are nearer the surface. If indeed the earth consisted of water or ice,
equally compressible with common water or ice, and following the same
laws of compression with elastic fluids, its density would be several thou-
sand times greater at the centre than at the surface ; and even steel would
be compressed into one fourth of its bulk, and stone into one eighth, if it
were continued to the earth's centre ; so that there can be no doubt but
that the central parts of the earth must be much more dense than the super-
ficial. Whatever this difference may be, it has been demonstrated by
Clairaut,* that the fractions expressing the ellipticity and the apparent
diminution of gravity at the equator must always make together -^fj.-, and
it has been found, by the most accurate observations on the lengths of
pendulums in different latitudes, that the force of gravity is less powerful
by -,-^ff at the equator than at the pole, whence the ellipticity is found
to be -g-J-3- of the equatorial diameter, the form being the same as would
be produced, if about three eighths of the whole force of gravity were
directed towards a central particle, the density of the rest of the earth being
uniform.

This method of determining the general form of the earth is much less
liable to error and irregularity, than the measurement of the lengths of
degrees in various parts, since the accidental variations of curvature pro-
duced by local differences of density, and even by superficial elevations,
may often produce considerable errors in the inferences which might be
deduced from these measurements. For example, a degree measured at
the Cape of Good Hope, in latitude 33° south, was found to be longer than
a degree in France, in latitude 46° north, and the measurements in Austria,
in North America, and in England, have all exhibited signs of similar
irregularities. There appears also to be some difference in the length of
degrees under the same latitude, and in different longitudes. "We may,
however, imagine a regular elliptic spheroid to coincide very nearly with
any small portion of the earth's surface, although its form must be some-
what different for different parts : thus, for the greater part of Europe, that
is, for England, France, Italy, and Austria, if the measurements have been
correct, this osculating spheroid must have an ellipticity of Tf^.

The earth is astronomically divided into zones, and into climates. The
torrid zone is limited by the tropics, at the distance of 23° 28' on each side
of the equator, containing all such places as have the sun sometimes
vertical, or immediately over them ; the frigid zones are within the polar
circles, at the same distance from the poles, including all places which
remain annually within the limit of light and darkness, for a whole diurnal
rotation of the earth, or longer ; the temperate zones, between these, have
an uninterrupted alternation of day and night, but are never subjected to
the sun's vertical rays. At the equator, therefore, the sun is vertical at the
equinoxes, his least meridian altitude is at the solstices, when it is 66° 32',
that is, more than with us at midsummer, and this happens once on the
north and once on the south side of the hemisphere. Between the equator

* Sur la Figure de la Terre, Paris, 1743. Airy's Tracts, Figure of the Earth,
art. 62 ; or his article, Figure of the Earth, in the Encyclopedia Metropolitana.

ON GEOGRAPHY. 437

'""and the tropics, he is vertical twice in the year, when his declination is equal
to the latitude of the place, and his least meridian altitudes, which are
unequal between themselves, are at the solstices. At the tropics, the
meridian sun is vertical once only in the year, and at the opposite solstice,
or the time of midwinter, his meridian altitude is 43° 4', as with us in
April, and the beginning of September. At the polar circles, the sun
describes on midsummer day a complete circle, touching the north or south
point of the horizon ; and in midwinter he shows only half his disc above
it for a few minutes in the opposite point ; that is, neglecting the elevation
produced by refraction, which, in these climates especially, is by no means
inconsiderable. At either pole, the corresponding pole of the heaven being
vertical, the sun must annually describe a spiral, of which each coil is
nearly horizontal, half of the spiral being above the horizon, and half
below ; the coils being much opener in the middle than near either end.

The climates, in the astronomical sense of the word, are determined by
the duration of the longest day in different parts of the earth's surface ;
but this division is of no practical utility, nor does it furnish any criterion
for judging of the climate in a meteorological sense.

The natural division of the surface of the globe is into sea and land :
about three fourths of the whole being occupied by water, although pro-
bably no where to a depth comparatively very considerable, at most of a
few miles on an average. The remaining fourth consists of land, elevated
more or less above the level of the sea, interspersed in some parts, with
smaller collections of water, at various heights, and, in a few instances,
somewhat lower than the general surface of the main ocean. Thus tho
Caspian sea is said to be about 300 feet lower than the ocean, and in the
interior part of Africa there is probably a lake equally depressed.

We cannot observe any general symmetry in this distribution^ of the
earth's surface, excepting that the two large continents, of Africa and South
America, have some slight resemblance in their forms, and that each of
them is terminated to the eastward by a collection of numerous islands.
The large capes projecting to the southward have also a similarity with
respect to their form and the islands near them : to the west the continents
are excavated into large bays, and the islands are to the east : thus Cape
Horn has the Falkland Islands, the Cape of Good Hope Madagascar, and
Cape Comorin Ceylon, to the east. (Plate XLII., XLIII.)

The great continent, composed of Europe, Asia, and Africa, constitutes
about a seventh of the whole surface of the earth, America about a six-
teenth, and Australasia or New South Wales about a fiftieth ; or, in
hundredth parts of the whole, Europe contains 2, Asia 7, Africa 6, America
6, and Australasia 2, the remaining 77 being sea ; although some authors
assign 72 parts only out of 100 to the sea, and 28 to the land. These pro-
portions may be ascertained with tolerable accuracy by weighing the paper
made for covering a globe, first entire, and then cut out according to the
terminations of the different countries : or, if still greater precision were
required, the greater part of the continents might be divided into known
portions of the whole spherical surface, and the remaining irregular por-
tions only weighed.

438 LECTURE XL VI.

The general inclinations and levels of the continents are discovered by
the course of their rivers. Of these the principal are, the River of Amazons,
the Senegal, the Nile, the River St. Laurence, the Hoangho, the River La-
plata, the Jenisei, the Mississippi, the Volga, the Oby, the Amur, the
Oronooko, the Ganges, the Euphrates, the Danube, the Don, the Indus, the
Dnieper, and the Dwina ; and this is said to be nearly the order of their
magnitudes. But if we class them according to the length of country
through which they run, the order will, according to Major Rennel's calcu-
lation, be somewhat different : taking the length of the Thames for unity,
he estimates that of the River of Amazons at 15|, the Kiang Kew, in China,
15$, the Hoangho 13$, the Nile 12$, the Lena 11$, the Amur 11, the Oby
10$, the Jenisei, 10, the Ganges, its companion the Burrampooter, the river
of Ava, and the Volga, each 9$, the Euphrates 8$, the Mississippi 8, the
Danube 7, the Indus 5$, and the Rhine 5£.

We may form a tolerably accurate idea of the levels of the ancient con-
tinent, by tracing a line across it in such a direction as to pass no river,
which will obviously indicate a tract of country higher than most of the
neighbouring parts. Beginning at Cape Finisterre, we soon arrive at the
Pyrenees, keeping to the south of the Garonne and the Loire. After taking
a long turn northwards to avoid the Rhine, we come to Swisserland, and
we may approach very near to the Mediterranean in the state of Genoa,
taking care not to cross the branches of the Po. We make a circuit
in Swisserland, and pass between the sources of the Danube and of the
branches of the Rhine in Swabia. Crossing Franconia, we leave Bohemia
to the north, in order to avoid the Elbe, and coming near to the borders of
Austria, follow those of Hungary, to the south of the Vistla. The Dnieper
then obliges us to go northwards through Lithuania, leaving the Don
wholly to the right ; and the Volga, to pass still further north, between
Petersburg and Moscow, a little above Bjelesero. We may then go east-
wards to the boundary of Asia, and thence northwards to Nova Zembla.
Hence we descend to the west of the Oby, and then to the east of the
branches of the Volga, and the other inland rivers flowing into the lake
Aral and the Caspian sea. Here we are situated on the widely extended
elevation of India, in the neighbourhood of the sources of the Indus : and,
lastly, in our way from hence towards Kamschatka, we leave the Jenisei
and Lena on the left, and the Ganges, the Kiang Kew, the Hoangho, and
the Amur to the right.

The direction of the most conspicuous mountains is, however, a little
different from this, the principal chain first constitutes the Pyrenees, and
divides Spain from France, then passes through Vivarais and Auvergne, to
join the Alps, and through the south of Germany to Dalmatia, Albania,
and Macedonia ; it is found again beyond the Euxine, under the names of
Taurus, Caucasus, and Imaus, and goes on to Tartary and to Kamschatka.
The peninsula of India is divided from north to south by the mountains
of Gate, extending from the extremity of Caucasus to Cape Comorin. In
Africa, Mount Atlas stretches from Fez to Egypt, and the mountains of
the moon run nearly in the same direction ; there is also a considerable
elevation between the Nile and the Red Sea. In the new world, the neigh-

ON GEOGRAPHY. 439

" bourhood of the western coast is in general the most elevated ; in North
America the Blue mountains, or Stony mountains, are the most consi-
derable ; and the mountains of Mexico join the Andes or Cordeliers, which
are continued along the whole of the west coast of South America.

There are several points in both hemispheres from which we may
observe rivers separating to run to different seas ; such are Swisserland,
Bjelosero, Tartary, Little Tibet, Nigritia or Guinea, and Quito. The
highest mountains are Chimboracao and some others of the Cordeliers in
Peru, or perhaps Descabesado in Chili, Mont Blanc, and the Peak of Tene-
riffe. Chimbora9ao is about 7000 yards, or nearly 4 miles, above the level
of the sea ; Mont Blanc 5000, or nearly 3 miles ; the Peak of Teneriffe
about 4000, or 2 miles and a quarter ; Ophir, in Sumatra, is said to be
5 or 6 hundred feet higher. It has, however, been asserted that some of
the snowy mountains, to the north of Bengal, are higher than any of those
of South America. The plains of Quito, in Peru, are so much elevated,
that the barometer stands at the height of 15 inches only, and the air is
reduced to half its usual density. But none of these heights is equal to
a thousandth part of the earth's semidiameter, and the greatest of them
might be represented on a six inch globe by a single additional thickness
of the paper with which it is covered. Mount Sinai in Japan, Mount
Caucasus, Etna, the Southern Pyrenees, St. George among the Azores,
Mount Adam in Ceylon, Atlas, Olympus, and Taurus are also high
mountains : and there are some very considerable elevations in the island
Owhyhee. Ben Nevis, in Scotland, is the loftiest of the British hills, but
its height is considerably less than a mile. (Plate XXXVIII. Fig. 519.)

The most elevated mountains, excepting the summits of volcanos, con-
sist of rocks, more or less mixed, without regular order, and commonly of
granite or porphyry. These are called primary mountains; they run
generally from east to west in the old world, and from north to south in
the new ; and many of them are observed to be of easier ascent on the east
than on the west side. The secondary mountains accompany them in the
same direction, they consist of strata, mostly calcarious and argillaceous,
that is, of the nature of limestone and clay, with a few animal and vege-
table remains, in an obscure form, together with salt, coals, and sulphur.
The tertiary mountains are still smaller ; and in these, animal and vegetable
remains are very abundant; they consist chiefly of limestone, marble,
alabaster, building stone, mill stone, and chalk, with beds of flint. Where
the secondary and tertiary mountains are intersected by vallies, the oppo-
site strata often correspond at equal heights, as if the vallies had been cut
or washed from between them, but sometimes the mountains have their
strata disposed as if they had been elevated by an internal force, and their
summits had afterwards crumbled away, the strata which are lowest in the
plains being highest in the mountains. The strata of these mountains are
often intermixed with veins of metal, running in all possible directions,
and occupying vacuities which appear to be of somewhat later date than
the original formation of the mountains. The volcanic mountains inter-
rupt those of every other description without any regularity, as if their
origin were totally independent of that of all the rest.

440 LECTURE XLVI.

The internal constitution of the earth is little known from actual
observation, for the depths to which we have penetrated are comparatively
very inconsiderable, the deepest mine scarcely descending half a mile per-
pendicularly. It appears that the strata are more commonly in a direction
nearly horizontal than in any other ; and their thickness is usually almost
equable for some little distance ; but they are not disposed in the order of
their specific gravity, and the opinion of their following each other in a
similar series, throughout the greater part of the globe, appears to rest on
very slight foundations.

From observations on the attraction of the mountain Shehallion, Dr.
Maskelyne* inferred the actual mean density of the earth to be to that of
water as 4^ to 1, judging from the probable density of the internal sub-
stance of the mountain, which he supposed to be a solid rock. Mr.
Cavendish f has concluded more directly, from experiments on a mass of
lead, that the mean density of the earth is to that of water as 5^ to 1.
Mr. Cavendish's experiments, which were performed with the apparatus
invented and procured by the late Mr. Michell, appear to have been con-
ducted with all possible accuracy, and must undoubtedly be preferred to
conclusions drawn from the attraction of a mountain, of which the internal
parts are perfectly unknown to us, except by conjectures founded on its
external appearance. Supposing both series of experiments and calcula-
tions free from error, it will only follow that the internal parts of Shehal-
lion are denser, and perhaps more metallic, than was before imagined.
The density assigned by Mr. Cavendish is not at all greater than might be
conjectured from observations on the vibrations of pendulums ; Newton
had long ago advanced it as a probable supposition that the mean density
of the earth might be about 5 or 6 times as great as that of water, and the
perfect agreement of the result of many modern experiments with this con-
jecture affords us a new proof, in addition to many others, of the accuracy
and penetration of that illustrious philosopher. $

LECT. XLVI.— ADDITIONAL AUTHORITIES.

Figure of the Earth.— Snellius, Eratosthenes Batavus de Terrse Ambitus Quanti-
tate, Lugd. Bat. 1617. Norwood, The Seaman's Practice, 4to, Lond. 1637. Ric-
cioli, Geog. et Hydrog. fol. Bon. 1661. Cassini, De la Figure de laTerre, 12mo,
Amst. 1723. Hist, et Mem. 1735, p. 255 ; 1736, p. 64, H. 80. Maupertuis, Ph.
Tr. 1733, 1736, p. 302. Hist, et Mem. 1737, p. 389, H. 90. La Figure de la
Terre determinee, Paris, 1738. Examen des Ouvrages faits pour cet Objet, Amst.
1741. Clairaut on the Figure of Planets of unequal density, Ph. Tr. 1738, p. 277.
Celsius, De Figura Telluris, 4to, Upsal, 1738. Bouguer, La Fig. de la Terre, 4to,
Paris, 1749. Justification de do. 4to, Paris, 1752. Lettre sur do. 1754. War-
gentin, Schwed. Abhand. 1749, p. 243 ; 1750, pp. 3, 83 ; Ph. Tr. 1777, p. 162. La

* Ph. Tr. 1775, p. 501. Button's Calculations, ibid. 1778, p. 689. See also
Zach, L' Attraction des Montagnes determinee par des Observations faites en 1810,
pres de Marseilles, 2 vols. Avignon, 1814.

t Ph.Tr. 1798, p. 469.

I Cavendish's Experiment has been repeated by Reich, Versuche iiber die Mitt-
lere Dichtigkeit der Erde, Freiburg, 1838, and by Baily, Memoirs of the Astrono-
mical Society, vol. xiv. who concludes that the mean density of the earth is 5-6747
times that of water.

ON THE TIDES. 441

Condamine, Journal du Voyage a 1'Equateur, 4to, Paris, 1751. Mesure des Trois
Premiers Degres, 4to, Paris, 1751. Lacaille, Hist, et Mem. 1751, p. 425, H. 158 ;
1755, p. 53. Frisii Disquisitio de Fig. et Mag. Tel. Milan, 1752. Boscovich, De
Expeditione, &c. 4to, Romse, 1755. Laplace, Mem. des Sav. Etr. 1773, p. 503 ;
Hist, et Mem. 1783, p. 17. Beccaria, Gradus Taurinensis, 4to, Aug. Taur. 1774.
Hassencamp, Geschichte, Rinteln. 1774. Gerlach, Gestalt der Erde, Vienna, 1782.
Roy, Meas. of a Base at Hounslow Heath, Ph. Tr. 1785, p. 385 ; on the Relative
Situations of Greenwich and Paris, 1787, p. 188 ; 1790, p. 11 1. Cassini, &c. on do.
4to, Paris, 1790. Herschel on do. Ph. Tr. 1826. Lorgna, Geographia, Verona,
1789. Delambre, see Lect. X. Survey by Williams, Mudge, and Dalby, Ph. Tr.
1795, p. 414; 1797, p. 432 ; 1800, p. 539 ; 1803, p. 383. Kastner's Mathema-
tische Geographic, Gott. 1795. Lambton, Asiatic Researches, vii. 312. Melander-
hiem and Svanberg, Zach's Mon. Corresp. i. 372 ; ii. 250, 257 ; vii. 561. Svanberg,
Exposition des Operations faites en Lapponie, Stockholm, 1805. Low, Dissertation,
Lugd. 1808. Ivory, Ph. Tr. 1809, 1831, p. 109 ; 1834, p. 491. Krayenhoff, Precis
des Operations faites en Hollande, 4to, La Haye, 1815. Cagnoli, Method of ascer-
taining the Fig. of the Earth by Occupations (trans.), Lond. 1819. Puissant, Traite
de Geodesic, 3 vols. 4to, 1819-27. Principes du Fig. du Terrain, &c. 4to. Arago,
Recueil d' Observations, 4to, 1821. Carlini, Relazione delle Operation! intrapese in
Italia, Milan, 1822. Sabine, Acct. of Experiments with Pendulums, 4to, 1825, and
Ph. Tr. 1828-29. Operations Geodesiques executees en Piemont et en Savoie,
2 vols. 4to, Milan, 1825-7. Brousseaud, Mem. sur la Mesure d'un Arc du Paral-
lele, 1825. Goldingham, Madras Obs. Papers, fol. 1827, Ph. Tr. 1822. Schmidt,
Lehrbuch der Mathematischen Geographic, Gott. 1829. Ivory, Ph. Tr. 1809, 1831,
p. 109; 1834, p. 491. Francoeur, Geodesic, 1835.

Figures of the Planets. — Maupertuis sur la Figure des Astres, Paris, 1732. La-
grange, Hist, et Me"m. de Berlin, 1773, p. 121 ; 1775, p. 273; 1792, p. 258.
Laplace, Hist, et Mem. de Paris, 1782, p. 113, H. 43. Legendre, ibid. 1784,
p. 370; 1789, p. 372.

Navigation. — See Lect. XXVII. Duillier, Navigation improved, fol. Lond.
1728. Maupertuis, Astronomie Nautique, Paris, 1743. Lemonnier, do. 1771.
Juan, Examen Maritimo, 2 vols. 4to, Madrid, 1771. Robertson's Navigation,
2 vols. Lond. 1786. Moore's, 1796. Lalande, Abrege' de Nav. 4to, 1793. Mac-
kay's Nav. 2 vols. 1793. Bowditch's, Lond. 1809. Bouguer, Traite de Nav. 4to,
1814. Norie's Nav. Lond. 1822. Kelly's Spherics and Naut. Ast. 1822. Riddle's
Nav. 1824. Inman's, Portsea, 1826.

LECTURE XLVII.

ON THE TIDES.

THE form and structure of the solid parts of the globe have afforded but
few remarkable features capable of arresting our attention, except the
general distribution of land and water, and the permanent differences of
elevation of different parts of the earth. But the sea exhibits a series of
phenomena far more interesting to the mathematical philosopher, because
they admit of a methodical investigation, and of a deduction from general
causes, the action of which may be traced in detail. For the height of the
surface of the sea at any given place is observed to be liable to periodical
variations, which are found to depend on the relative position of the moon,
combined in some measure with that of the sun. These variations are
* called tides ; they were too obvious to escape the observation even of the
ancients, who inhabited countries where they are least conspicuous : for

442 LECTURE XLVII.

Aristotle mentions the tides of the northern seas, and remarks that they"
vary with the moon, and are less conspicuous in small seas than in the
ocean : Caesar, Strabo, Pliny, Seneca, and Macrobius give also tolerably
accurate accounts of them.

There are in the tides three orders of phenomena which are separately
distinguishable ; the first kind occurs twice a day, the second twice a
month, and the third twice a year. Every day, about the time of the
moon's passing over the meridian, or a certain number of hours later, the
sea becomes elevated above its mean height, and at this time it is said to
be high water. The elevation subsides by degrees, and in about six hours
it is low water, the sea having attained its greatest depression ; after this
it rises again when the moon passes the meridian below the horizon, so
that the ebb and flood occur twice a day, but become daily later and later
by about 50 £ minutes, which is the excess of a lunar day above a solar one,
since 28£ lunar days are nearly equal to 29| solar ones.

The second phenomenon is, that the tides are sensibly increased at the
time of the new and full moon ; this increase and diminution constitute
the spring and neap tides; the augmentation becomes also still more
observable when the moon is in its perigee or nearest the earth. The
lowest as well as the highest water is at the time of the spring tides ; the
neap tides neither rise so high nor fall so low.

The third phenomenon of the tides is the augmentation which occurs at
the time of the equinoxes ; so that the greatest tides are when a new or
full moon happens near the equinox, while the moon is in its perigee. The
effects of these tides are often still more increased by the equinoctial winds,
which are sometimes so powerful as to produce a greater tide before or
after the equinox, than that which happens in the usual course, at the time
of the equinox itself.

These simple facts are amply sufficient to establish the dependence of the
tides on the moon ; they were first correctly explained by Newton as the
necessary consequences of the laws of gravitation, but the theory has been
still further improved by the labours of later mathematicians. The
whole of the investigations has been considered as the most difficult of all
astronomical problems ; some of the circumstances depend on causes
which must probably remain for ever unknown to us; and unless we
could every where measure the depth of the sea, it would be impossible to
apply a theory, even if absolutely perfect, to the solution of every difficulty
that might occur. A very injudicious attempt has been made to refer the
phenomena of the tides to causes totally different from these, and depending
on the annual melting of the polar ice ; the respectability of its author is
the only claim which it possesses even to be mentioned ; and a serious
confutation of so groundless an opinion would be perfectly superfluous.

A detached portion of a fluid would naturally assume, by its mutual
gravitation, a spherical form, but if it gravitate towards another body at
a distance, it will become an oblong spheroid of which the axis will
point to the attracting body ; for the difference of the attraction of this
body on its different parts will tend to separate them from each other in
the greatest part of the sphere, that is, at all places within the angular

ON THE TIDES. 443

distance of 79 £° from the line passing through the attracting body, either
in the nearer or in the remoter hemisphere ; but to urge them to-
wards the centre, although with a smaller force, in the remaining part.
Hence, in order that there may be an equilibrium, the depth of the fluid
must be greatest where its gravitation, thus composed, is least ; that is, in
the line directed towards the attracting body, and it may be shown that it
must assume the form of an oblong elliptic spheroid.

If the earth were wholly fluid, and the same part of its surface were
always turned towards the moon, the pole of the spheroid being imme-
diately under the moon, the lunar tide would remain stationary, the
greatest elevation being at the points nearest to the moon and furthest from
her, and the greatest depression in the circle equally distant from these
points ; the elevation being, however, on account of the smaller surface to
which it is confined, twice as great as the depression. The actual height
of this elevation would probably be about 40 inches, and the depression 20,
making together a tide of 5 feet. If also the waters were capable of
assuming instantly such a form as the equilibrium would require, the
summit of a spheroid equally elevated would still be directed towards the
moon, notwithstanding the earth's rotation. This may be called the
primitive tide of the ocean ; but on account of the perpetual change of
place which is required for the accommodation of the surface to a similar
position with respect to the moon, as the earth revolves, the form must be
materially different from that of such a spheroid of equilibrium. The
force employed in producing this accommodation may be estimated by
considering the actual surface of the sea as that of a wave moving on the
spheroid of equilibrium, and producing in the water a sufficient velocity to
preserve the actual form. We may deduce, from this mode of considering
the subject, a theory of the tides which appears to be more simple and satis-
factory than any which has yet been published ; and by comparing the
tides of narrower seas and lakes with the motions of pendulums suspended
on vibrating centres, we may extend the theory to all possible cases.

If the centre of a pendulum be made to vibrate, the vibrations of the
pendulum itself, when they have arrived at a state of permanence, will be
performed in the same time with those of the centre ; but the motion of the
pendulum will be either in the same direction with that of the centre, or in
a contrary direction, accordingly as the time of this forced vibration is
longer or shorter than that of the natural vibration of the pendulum ; and
in the same manner it may be shown that the tides either of an open ocean
or of a confined lake may be either direct or inverted with respect to the
primitive tide, which would be produced if the waters always assumed the
form of the spheroid of equilibrium according to the depth of the ocean,
and to the breadth as well as the depth of the lake. In the case of a direct
tide the time of the passage of the luminary over the meridian must coin-
cide with that of high water, and in the case of an inverted tide with that
of low water.

In order that the lunar tides of an open ocean may be direct, or synchro-

' nous, its depth must be greater than 13 miles, and for the solar tides than

14. The less the depth exceeded these limits, the greater the tides would

444 LECTURE XLVII.

be, and in all cases they would be greater than the primitive tides. But m
fact the height of the tides in the open ocean is always far short of that
which would be produced in this manner ; it is therefore improbable that
the tides are ever direct in the open ocean, and that the depth of the sea
is so great as 13 miles.

In order that the height of the inverted or remote lunar tides may be five
feet, or equal to that of the primitive tides, the depth of the open sea must
be 6 1 miles; and if the height is only two feet, which is perhaps not far
from the truth, the depth must be 3 miles and five sevenths.

The tides of a lake or narrow sea differ materially from those of the open
ocean, since the height of the water scarcely undergoes any variation in the
middle of the lake ; it must always be high water at the eastern extremity
when it is low water at the western ; and this must happen at the time
when the places of high and low water, with respect to the primitive
tides, are equally distant from the middle of the lake. (Plate XXXVIII.
Fig. 520.)

The tides may be direct in a lake 100 fathoms deep and less than 8
degrees wide ; but if it be much wider, they must be inverted. Supposing
the depth a mile, they will be direct when the breadth is less than 25° ; but
if a sea, like the Atlantic, were 50 or 60 degrees wide, it must be at least
four miles deep, in order that the time of high water might coincide with
that of the moon's southing.

Hitherto we have considered the motion of the water as free from all
resistance ; but where the tides are direct, they must be retarded by the
effect of a resistance of any kind ; and where they are inverted, they must
be accelerated ; a small resistance producing, in both cases, a considerable
difference in the time of high water.

Where a considerable tide is observed in the middle of a limited portion
of the sea, it must be derived from the effect of the elevation or depression
of the ocean in its neighbourhood ; and such derivative tides are probably
combined in almost all cases with the oscillations belonging to each parti-
cular branch of the sea. Mr. Laplace supposes that the tides, which are
observed in the most exposed European harbours, are produced almost
entirely by the transmission of the effect of the main ocean, in about a day
and a half ; but this opinion does not appear to be justified by observation ;
for the interval between the times of the high water belonging to the same tide,
in any two places between Brest and the Cape of Good Hope, has not been
observed to exceed about twelve hours at most ; nor can we trace a greater
difference by comparing the state of the tides at the more exposed situations
of St. Helena, the Cape Verd Islands, the Canaries, the Madeiras, and the
Azores, which constitute such a succession as might be expected to have
indicated the progress of the principal tide, if it had been such as Mr.
Laplace supposes. The only part of the ocean which we can consider as
completely open, lies to the south of the two great continents, chiefly
between the latitudes 30° and 70° south, and the original tide, which hap-
pens in this widely extended ocean, where its depth is sufficiently uniform,
must take place, according to the theory which has been advanced, at some '
time before the sixth lunar hour. It sends a wave into the Atlantic, which

ON THE TIDES. 445

is perhaps 12 or 13 hours in its passage to the coast of France, but cer-
tainly not more. This tide, which would happen at the sixth lunar hour
after the moon's transit, if there were no resistance, is probably so checked
by the resistance, that the water begins to subside about the fourth, and in
some seas even somewhat earlier, although in others it may follow more
nearly its natural course. There is scarcely a single instance wrhich
favours the supposition of the time of high water in the open sea being
within an hour of the moon's southing, as it must be if the depth were
very great ; so that neither the height of the tides nor the time of high
water will allow us to suppose the sea any where quite so deep as 4
miles.

The tide entering the Atlantic appears to advance northwards at the rate
of about 500 miles an hour, corresponding to a depth of about 3 miles, so
as to reach Sierra Leone at the 8th hour after the moon's southing ; this
part of Africa being not very remote from the meridian of the middle of
the south Atlantic ocean, and having little share in the primitive tides of
that ocean. The southern tide seems then to pass by Cape Blanco and
Cape Bojador, to arrive at Gibraltar at the 13th hour, and to unite its
effects with those of other tides at various parts of the coast of Europe.

We may therefore consider the Atlantic as a detached sea about 3500
miles broad and 3 miles deep ; and a sea of these dimensions is susceptible
of tides considerably larger than those of the ocean, but how much larger
we cannot determine without more accurate measures. These tides would
happen on the European coasts, if there were no resistance, a little less
than five hours after the moon's southing, and on the coast of America, a
little more that seven hours after ; but the resistance opposed to the motion
of the sea may easily accelerate the time of high water in both cases about
two hours, so that it may be a little before the third hour on the western
coasts of Europe and of Africa, and before the fifth on the most exposed
parts of the eastern coast of America ; and in the whole of the Atlantic,
this tide may be combined more or less both with the general southern
tide, and with the partial effects of local elevations or depressions of the
bottom of the sea, which may cause irregularities of various kinds. The
southern tide is, however, probably less considerable than has sometimes
been supposed, for, in the latitudes in which it must originate, the extent
of the elevation can only be half as great as at the equator ; and the Islands
of Kergulen's Land and South Georgia, in the latitudes of about 50° and
55°, have their tides delayed till the 10th and llth hours, apparently
because they receive them principally from distant parts of the ocean,
which are nearer to the equator.

On the western coasts of Europe, from Ireland to Cadiz, on those of
Africa, from Cape Coast to the Cape of Good Hope, and on the coast
of America, from California to the streights of Magellan, as well as in the
neighbouring islands, it is usually high water at some time between two
and four hours after the moon's southing ; on the eastern coast of South
, America between four and six, on that of North America between seven
and eleven ; and on the eastern coasts of Asia and New Holland between
four and eight. The Society islands are perhaps too near the middle of the

44G LECTURE XLVII.

Pacific ocean to partake of the effects of its primitive tide, and their tide,
being secondary, is probably for this reason a few hours later. At the
Alniirantes, near the eastern coast of Africa, the tide is at the sixth hour ;
but there seem to be some irregularities in the tides of the neighbouring
islands.

The progress of a tide may be very distinctly traced from its source in
the ocean into the narrow and shallow branches of the sea which constitute
our channels. Thus the tide is an hour or two later at the Scilly Islands
than in the Atlantic, at Plymouth three, at Cork, Bristol, and Weymouth
four, at Caen and Havre six, at Dublin and Brighthelmstone seven, at
Boulogne and Liverpool eight, at Dover near nine, at the Nore eleven, and
at London bridge twelve and a half. Another portion appears to proceed
round Ireland and Scotland into the North Sea ; it arrives from the Atlantic
at Londonderry in about three hours, at the Orkneys in six, at Aberdeen
in eleven, at Leith in fourteen, at Leostoffe in twenty, and at the Nore in
about twenty-four, so as to meet there the subsequent tide coming from
the south. From the time occupied by the tide in travelling from the
mouth of the English channel to Boulogne, at the rate of about 50 miles an
hour, we may calculate that the mean depth of the channel is about 28
fathoms, independently of the magnitude of the resistances of various
kinds to be overcome, which require us to suppose the depth from 30 to 40
fathoms. In the great river of Amazons, the effects of the tides are still
sensible at the streights of Pauxis, 500 miles from the sea, after an interval
of several days spent in their passage up ; for the slower progressive motion
of the water no more impedes the progress of a wave against the stream,
than the velocity of the wind prevents the transmission of sound in a con-
trary direction. (Plate XXXVIII. Fig. 521.)

Such are the general outlines of the lunar tides ; they are, however, liable
to a great variety of modifications, besides their combination with the tides
produced by the sun. When the moon is exactly over the equator, the
highest part of the remoter, or inferior, as well as of the nearer or superior
tides, passes also over the equator, and the effect of the tide in various lati-
tudes decreases gradually from the equator to the pole, where it vanishes ;
but when the moon has north or south declination, the two opposite
summits of the spheroid describe parallels of latitude, remaining always
diametrically opposite to each other. Hehce the two successive tides must
be unequal at every place except the equator, the greater tide happening
when the nearer elevation passes its meridian ; and the mean between both
is somewhat smaller than the equal tides which happen when the moon
passes the equator. This inequality is, however, much less considerable
than it would be if the sea assumed at once the form of the spheroid
of equilibrium ; and the most probable reasons for this circumstance, are,
first, that our tides are partly derived from the equatorial seas ; secondly,
that the effects of a preceding tide are in some measure continued so
as to influence the height of a succeeding one; and, thirdly, that the
tides of a narrow sea are less affected by its latitude than those of a wide
ocean. The height of the sea at low water is the same whatever the
moon's declination may be. There is also a slight difference in the tides,

ON THE TIDES. 447

according to the place of the moon's nodes, which allows her declination to
be greater or less, and this difference is most observable in high latitudes,
for instance, in Iceland ; since, in the neighbourhood of the poles, the tides
depend almost entirely on the declination.

In all these cases, the law of the elevation and depression of each tide may
be derived, like that of the vibrations of a pendulum and of a balance, from
the uniform motion of a point in a circle. Thus, if we conceive a circle to
be placed in a vertical plane, having its diameter equal to the whole mag-
nitude of the tide, and touching the surface of the sea at low water, the
point, in which the surface meets the circumference of the circle, will
advance with a uniform motion, so that if the circle be divided into 12
parts, the point will pass over each of these parts in a lunar hour. It
sometimes happens, however, in confined situations, that the rise and fall
of the water deviates considerably from this law, and the tide rises some-
what more rapidly than it falls ; and in rivers, for example in the Severn,
the tide frequently advances suddenly with a head of several feet in height.
These deviations probably depend on the magnitude of the actual displace-
ment of the water, which in such cases bears a considerable proportion to
the velocity of the tide, while in the open ocean a very minute progressive
motion is sufficient to produce the whole elevation. The actual progress of
the tides may be most conveniently observed, by means of a pipe descending
to some distance below the surface, so as to be beyond the reach of super-
ficial agitations, and having within it a float, carrying a wire, and indicating
the height of the water on a scale properly divided.

We have hitherto considered the tides so far only as they are occasioned
by the moon ; but in fact the tides, as they actually exist, depend also on
the action of the sun, which produces a series of effects precisely similar to
those of the moon, although much less conspicuous, on account of the
greater distance of the sun, the solar tide being only about two fifths of the
lunar. These tides take place independently of each other, nearly in the
same degree as if both were single ; and the combination resulting from
them is alternately increased and diminished, accordingly as they agree, or
disagree, with respect to the time of high water at a given place ; in the
same manner as if two series of waves, equal among themselves, of which
the breadths are as 29 to 30, be supposed to pass in the same direction over
the surface of a fluid, or if two sounds similarly related be heard at the same
time, a periodical increase and diminution of the joint effect will in either
case be produced. Hence are derived the spring and neap tides, the effects
of the sun and moon being united at the times of conjunction and oppo-
sition, or of the new and full moon, and opposed at the quadratures, or first
and last quarters. The high tides at the times of the equinoxes are pro-
duced by the joint operation of the sun and moon, when both of them are
so situated as to act more powerfully than elsewhere.

The lunar tide being much larger than the solar tide, it must always
determine the time of high and low water, which, in the spring and neap
tides, remains unaltered by the effect of the sun ; so that in the neap tides,
the actual time of low water is that of the solar high water ; but at the
intermediate times, the lunar high water is more or less accelerated or

448 LECTURE XLVII.

retarded. The progress of this alteration may easily be traced by means of
a simple construction. If we make a triangle of which two of the sides are
two feet and five feet in length, the external angle which they form being
equal to twice the distance of the luminaries, the third side will show pre-
cisely the magnitude of the compound tide, and the halves of the two
angles opposite to the first two sides the acceleration, or retardation, of the
times of high water belonging to the separate tides respectively. Hence it
appears that the greatest deviation of the joint tide from the lunar tide
amounts to 11° 48' in longitude, and the time corresponding, to 47 minutes,
supposing the proportion of the forces to remain always the same ; but in
fact the forces increase in proportion as the cubes of the distances of their
respective luminaries diminish, as well as from other causes ; and in order
to determine their joint effects, the lengths of the sides of the triangle must
be varied accordingly. In some ports, from a combination of circum-
stances in the channel, by which the tides reach them, or in the seas, in
which they originate, the influence of the sun and moon may acquire
a proportion somewhat different from that which naturally belongs to
them : thus at Brest, the influence of the moon appears to be three times as
great as that of the sun ; when it is usually only twice and a half as great.
(Plate XXXVIII. Fig. 522.)

The greatest and least tides do not happen immediately at the times of the
new and full moon, but at least two, and commonly three tides after, even
at those places which are most immediately exposed to the effects of the
general tide of the ocean. The theory which has been advanced will
afford us a very satisfactory reason for this circumstance ; the resistance
of fluids in general is as the square of the velocity, consequently it must
be much greater for the lunar than for the solar tide, in proportion to the
magnitude of the force, and the acceleration of the lunar tide produced
by this cause must be greater than that of the solar ; hence it may happen
that when the lunar tide occurs two or three hours after the transit of the
moon, the solar tide may be three or four hours after that of the sun,
so as to be about an hour later, at the times of conjunction and opposition,
and the tides will be highest when the moon passes the meridian about an
hour after the sun ; while at the precise time of the new and full moon, the
lunar tide will be retarded about a quarter of an hour by the effect of the
solar tide.

The particular forms of the channels, through which the tides arrive at
different places, produce in them a great variety of local modifications ;
of which the most usual is, that from the convergence of the shores of the
channels, the tides rise to a much greater height than in the open sea.
Thus at Brest the height of the tides is about 20 feet, at Bristol 30, at
Chepstow 40, at St. Maloes 50 ; and at Annapolis Royal, in the Bay of
Fundy, as much sometimes as 100 feet ; although perhaps in some of these
cases a partial oscillation of a limited portion of the sea may be an imme-
diate effect of the attraction of the luminary. In the Mediterranean the
tides are generally inconsiderable, but they are still perceptible ; at Naples
they sometimes amount to a foot, at Venice to more than two feet, and in
the Euripus, for a certain number of days in each lunation, they are very

ON THE TIDES. 449

distinctly observable, from the currents which they occasion. In the West
Indies, also, and in the gulf of Mexico, the tides are less marked than in
the neighbouring seas, perhaps on account of some combinations derived
from the variations of the depth of the ocean, and from the different
channels by which they are propagated.

In order to understand the more readily the effects of such combinations,
we may imagine a canal, as large as the river of Amazons, to communi-
cate at both its extremities with the ocean, so as to receive at each an equal
series of tides, passing towards the opposite extremity. If we suppose the
tides to enter at the same instant at both ends, they will meet in the middle,
'and continue their progress without interruption : precisely in the middle
the times of high and low water belonging to each series will always coin-
cide, and the effects will be doubled ; and the same will happen at the
points where a tide arrives from one extremity at the same instant that an
earlier or a later tide comes from the other ; but at the intermediate points
the effects will be diminished, and at some of them completely destroyed,
where the high water of one tide coincides with the low water of another.
The tides at the port of Batsha in Tonkin have been explained by Newton
from considerations of this nature. In this port there is only one tide in a
day ; it is high water at the sixth lunar hour, or at the moon's setting,
when the moon has north declination, and at her rising, when she has
south declination ; and when the moon has no declination there is no tide.
In order to explain this circumstance, we may represent the two unequal
tides which happen in succession every day, by combining with two equal
tides another tide, independent of them, and happening only once a day ;
then, if a point be so situated in the canal which we have been considering,
that the effects of the two equal semidiurnal tides may be destroyed, those
of the daily tides only will remain to be combined with each other ; and
their joint result will be a tide as much greater than either, as the diagonal
of a square is greater than its side ; the times of high and low water being-
intermediate between those which belong to the diurnal tides considered
separately. Thus, in the port of Batsha, the greater tide probably arrives
at the third lunar hour directly from the Pacific ocean, and at the ninth
from the gulf of Siam, having passed between Sumatra and Borneo ; so that
the actual time of high water is at the sixth lunar hour. The magnitude of
this compound tide is by no means inconsiderable ; it sometimes amounts
to as much as 13 feet. (Plate XXXVIII. Fig. 523, 524.)

Besides the variations in the height of the sea, which constitute the tides,
a current is observed in its most exposed parts, of which the general direc-
tion is from east to west. This current comes from the Pacific and Indian
oceans, round the Cape of Good Hope, along the coast of Africa, then
crosses to America, and is there divided and reflected southwards towards
the Brazils, and northwards into the Gulf stream which travels round the
gulf of Mexico, and proceeds north eastwards into the neighbourhood of
Newfoundland, and then probably eastwards and south eastwards once
more across the Atlantic. It is perhaps on account of these currents that
the Red Sea is found to be about 25 feet higher than the Mediterranean :
their direction may possibly have been somewhat changed in the course of

2G

450 LECTURE XLVII.

many ages, and with it the level of the Mediterranean also ; since the floor
of the cathedral at Ravenna is now several feet lower with respect to the sea
than it is supposed to have been formerly, and some steps have been found
in the rock of Malta, apparently intended for ascending it, which are at
present under water.

The atmosphere is also liable to elevations and depressions analogous to
those of the sea, and perhaps these changes may have some little effect on
the winds and on the weather ; but their influence must be very incon-
siderable, since the addition of two or three feet to the height of the atmo-
sphere at any part can scarcely be expected to be perceptible. The height of
an aerial tide must be very nearly the same with the observed height of the
principal tides of the sea ; and the variation of atmospherical pressure, which
is measured by the difference between the actual form and the spheroid of
equilibrium, must be equivalent to the weight of a column of about 10 feet
of air, or only -^ of an inch of mercury. A periodical variation five times
as great as this has indeed been observed near the equator, where the state
of 'the atmosphere is the least liable to accidental disturbances ; but this
change cannot in any degree be referred to the effect of the moon's action,
since it happens always about the same hour of the day or night. The
atmosphere is also affected by a general current from east to west, like
that of the sea, and there is reason, from astronomical observations, to
suppose that a similar circumstance happens in the atmosphere of Jupiter.
These currents, as well as the general current of the sea, have been attri-
buted, by some astronomers, to the immediate attraction of the sun and
moon, and of the satellites of Jupiter, which they have supposed to act in
the same manner as the attraction of the sun operates in retarding the
lunar motions ; but the fact is, that, according to Mr. Laplace, the disturb-
ing force of the sun produces this effect on the moon only in proportion as it
increases her distance from the earth ; consequently no such retardation
can possibly be produced by the force of gravitation in the rotation of the
sea or of the atmosphere, and the whole effect must be attributed to the
operation of meteorological causes, producing first the trade winds, and
secondly occasioning, by means of the friction of these winds, a similar
motion in the sea. In the case of the atmosphere of Jupiter, the effects of
heat can indeed scarcely be supposed to be very perceptible, and the rota-
tion of this planet being extremely rapid, it is not at all impossible that the
satellites may exert an action on the atmosphere somewhat analogous to
the retardation of the moon's motion by the disturbing force of the sun.

LECT. XLVII.— ADDITIONAL AUTHORITIES.

Borro, DelFlusso e Reflusso del Mare, Fiorenza, 1577. Moray on Observing the
Tides, Ph. Tr. 1665-6, i. 298. Colepresse's Obs. at Plymouth, ibid. 1668, iii. 632.
Davenport on the Tides at Tonqueen, ibid. 1684, p. 677. Halley on do. ibid. 1684,
p. 685. Newtoni Principia, and Halley's Remarks, ibid. 1697, p. 445. Prize
Essays on the Tides, by Cavalleri, Bernoulli, Maclaurin, and Euler, Hist, et Mem.
Prix iv. VI.. . IX. Le Seur's edition of Newton's Principia. Jones and Saumare*
on the Tides in the Thames, Ph. Tr. 1726, p. 68. Wright on an irregular Tide in the
Forth (the Leaky), ibid. 1750, p. 412. Toaldo, Tabula Barometri ^Estusque Maris,

ON THE HISTORY OF ASTRONOMY. 451

4to, Pavia, 1773. On the Tides in the Adriatic, Ph. Tr. 1777, p. 144. Lalande,
Traite du Flux et du Reflux de la Mer, 4to, Paris, 1781. Laplace, Mec. Celeste,
lib. v. Lubbock, Ph. Tr. 1831, p. 379 ; 1832, pp. 51, 595 ; 1833, p. 19 ; 1834,
p. 143; 1835, p. 275 ; 1836, pp. 57, 217 ; 1837, p. 97. Whewell's Cotidal Lines,
Ph. Tr. 1833, p. 147. Researches on the Tides, ibid. 1834, p. 15 ; 1835, p. 83 ;
1836, pp. 1, 131, 289; 1837, pp. 75, 227; 1838, p. 231 ; 1839, pp. 151, 163;
1840, pp. 161,255. Palmer's Tide Gauge, Ph. Tr. 1831, p. 209. Bunt's, ibid.
1838, p. 249. Daussy, Connaissance des Temps, 1834 (a low barometer is accom-
panied with high tides). Bunt, Eleventh Report of Brit. Ass.

LECTURE XLVIII.

ON THE HISTORY OF ASTRONOMY.

WE have now taken a general view of the most striking phenomena of
the universe at large, of the great features of the solar system, and of the
peculiarities of the planet which we inhabit, with respect both to its solid
and to its fluid parts. All these are departments of astronomy, and we shall
conclude our examination of the subject with a summary of the history of
the science, principally extracted and abridged from Laplace's Exposition
du syst&me du monde.

In all probability the astronomy of the earliest ages was confined to
observations of the obvious motions and eclipses of the sun and moon, the
rising, setting, and occupations of the principal stars, and the apparent
motions of the .planets. The progress of the sun was followed, by remarking
the stars as they were lost in the twilight, and perhaps also by the variation
of the length of the shadow of a detached object, observed at the time of
the day when it was shortest. In order to recognise the fixed stars, and
their different motions, the heavens were divided into constellations ; and
twelve of these occupied the zone denominated the zodiac, within the limits
of which the sun and planets were always found.

The entrance of the sun into the constellation aries, or the ram, denoted,
in the time of Hipparchus, the beginning of the spring ; and as the season
advanced, the sun continued his progress through the bull, the twins, and
the other signs in order ; some of which appear to have been denominated
from their relation to the agriculture and to the climates of the countries
in which they were imagined, and others from the celestial phenomena
attending the sun's passage through them ; the crab, for example, denot-
ing his retrograde motion after the time of the solstice, and the balance
the equality of day and night at the autumnal equinox. But the motion
of the equinoctial points having changed in some degree the course of
the seasons with regard to the stars, the signs of the ecliptic, by which
the places of the sun and planets are described, no longer coincide pre-
cisely with the constellations of the zodiac from which they derive their
names.

The most ancient observations of which we are in possession, that are

2 G 2

452 LECTURE XLVIII.

sufficiently accurate to be employed in astronomical calculations, are those
made at Babylon in the years 71 9 and 720 before the Christian era, of three
eclipses of the moon. Ptolemy,* who has transmitted them to us, employed
them for determining the period of the moon's mean motion, and, therefore,
had probably none more ancient on which he could depend. The Chal-
deans, t however, must have made a long series of observations before they
could discover their Saros or lunar period of 6585^ days, or about 18 years,
in which (as they had learnt at a very early time) the place of the moon,
her node and apogee return nearly to the same situation with respect to
the earth and sun ; and of course a series of nearly similar eclipses recurs.
The observations attributed to Hermes indicate a date seven hundred years
earlier than those of the Babylonians, but their authenticity appears to be
extremely doubtful.

The Egyptians J were very early acquainted with the length of the year,
as consisting nearly of 365 days and a quarter, and they derived from it
their Sothic period of 1460 years, containing 365 days each. The accurate
correspondence of the faces of their pyramids with the points of the compass
is considered as a proof of the precision of their observations :§ but their
greatest merit was the discovery that Mercury and Venus revolve round
the sun, and not round the earth, as it had probably been before believed :||
they did not, however, suppose the same of the superior planets. (Plate
XXXVIII. Fig. 525, 526.)

In Persia and in India, the origin of astronomy is lost in the darkness
which envelopes the early history of those countries. We find the annals
of no country so ancient and so well authenticated as those of China, which
are confirmed by an incontestable series of historical monuments. The
regulation of the calendar, and the prediction of eclipses, were regarded in
this country as important objects, for which a mathematical tribunal was
established at a very early period. But the scrupulous attachment of the
Chinese to their ancient customs, extending itself even to their astronomy,
has impeded its progress, and retained it in a state of infancy. The Indian
tables indicate a much higher degree of perfection in the early state of the
science than it had attained in China ; but we have every reason to believe
that they are not of very remote antiquity. " Here," says Mr. Laplace, IT
who must be allowed to be free from prejudices in favour of established
opinions, " I am sorry to be obliged to differ from an illustrious philoso-
pher, Mr. Bailly, who, after having distinguished his career by a variety of
labours useful to the sciences, and to mankind at large, fell a victim to the
most sanguinary tyranny that ever disgraced a civilised nation. The
Indian tables are referred to two principal epochs, which are placed the
one 3102 years before Christ, the other 1491. These are connected by the
mean motions, and not the true motions, of the sun, the moon, and the
planets ; so that one of the epochs must necessarily be fabulous. The cele-

.* Ptol. Almagest. 1. 4, c. 6.

t Suidas, Lexicon (Saros). Pliny, Hist. Nat. 1. 2, c. 13.

J Giraud, Journal des Savans, 1760.

§ Mem. del'Acad. 1710. c

|| Macrobius, Comm. in Somn. Scip. 1.1, c. 9.

f Exposition du Systeme du Monde, 2nd edit. p. 239.

ON THE HISTORY OF ASTRONOMY. 453

brated author, who has been mentioned, has sought to establish, in his
treatise on Indian astronomy, that the former of these epochs is founded on
observation. But if we calculate from our own improved tables, we shall
find that the general conjunction of the sun, moon, and planets, which the
Indian tables suppose, in reality never happened, although it may be
deduced, according to those tables, by ascending from the later series. The
equation of the sun's centre, depending on the eccentricity of the earth's
orbit, appears indeed to indicate a still higher antiquity ; but its magni-
tude, as deduced from eclipses, must have been affected by a contrary error
with respect to the moon's place : and the determination of the mean motion
of the moon seems to make it probable that these tables are even of a later
date than Ptolemy."

In astronomy, as well as in other sciences, the Greeks were the disciples
of the Egyptians ; they appear to have divided the stars into constellations
13 or 1400 years before Christ. Newton attributes this arrangement to
Chiron,* and he supposes that he made the middle of the constellations cor-
respond to the beginning of the respective signs. But until the time of the
foundation of the school of Alexandria, the Greeks treated astronomy as a
science purely speculative, and indulged themselves in the most frivolous
conjectures respecting it. It is singular that amidst the confusion of sys-
tems heaped up on each other, without affording the least information to
the mind, it should never have occurred to men of so great talents, that th
only way to become accurately acquainted with nature, is to institute
experimental inquiries throughout her works.

Thales of Miletus, who was born in the year 640 before Christ, having tra-
velled and studied in Egypt, founded, on his return, the Ionian school of
philosophy, in. which he taught the sphericity of the earth,f and the obli-
quity of the ecliptic with respect to the equator.^ He also explained
the true causes of eclipses,§ which he was even able to foretel,|| unques-
tionably by means of the information that he had obtained from the Egyp-
tian priests.

Pythagoras of Samos was born 590 years before Christ ; he probably
profited by the information which Thales had acquired, and travelled also
into Egypt for his further improvement.^ It is conjectured that he was
acquainted with the diurnal and annual motions of the earth,** but he did
not publicly profess the true system of the world. It was taught after his
death, by his disciple Philolaus, about the year 450, as well as by Nicetas,ft
and by others of the school. They considered all the planets as revolving
round the sun,^ and as inhabited globes ; and they understood that the

* Chronology, p. 25.

t Plutarch, de Placit. Philos. 1. 2, c. 9, 10.

t Diogenes Laertius, Life of Thales. Plutarch, Conviv. Sept. Sapient. Proclus,
Comm. in Euc. 1. 1.

§ Plutarch, de Placit. 1. 2, c. 21, 24, 28.

|| Herod, l.-l, &c. Pliny, Nat. Hist. 1. 2, c. 12. Riccioli, Almagest. Nov. i.
363. Costard, Ph. Tr. xlviii. 17. Baily, ibid. 1811.

fl Jamblichus, Vita Pythag. ** Aristotle, de Coelo, 1. 2, c. 13.

* ft Cicero, Qusest Acad. 1. 4, §39.

Jt Plin. Hist. Nat. 1. 2, c. 22. Macrob. in Somn. Scip. 1. 1. c. 19. Greeorii
Prtef. ad Ast. &c.

454 LECTURE XLVIII. /

comets were only eccentric planets.* Some time after this, the lunar
period of Meto was publicly made known at the Olympic games, and was
universally adopted as the basis of the calendar. (Plate XXXVIII. Fig.
527.)

The next occurrence which deserves to be noticed, with respect to astro-
nomy, is the foundation of the school of Alexandria, which waa the first
source of accurate and continued observations. Upon the death of Alex-
ander and the subsequent division of his empire, the province of Egypt
fell to the lot of Ptolemy Soter ; a prince whose love of science, and whose
munificence towards its professors, attracted to his capital a great number
of learned men from various parts of Greece. His son, Ptolemy Phila-
delphus, continued and increased the benefits conferred on them by his
father, and built the magnificent edifice which contained, together with
the celebrated library, collected by Demetrius Phalereus, an observatory,
furnished with the necessary books and instruments.t The first astro-
nomers, who were appointed to occupy this building, were Aristyllus and
Timocharis ; they flourished about 300 years before Christ, and observed
with accuracy the places of the principal stars of the zodiac. £ Aristarchus
of Samos was the next : he imagined a method of finding the sun's
distance, by observing the portion of the moon's disc that is enlightened,
when she is precisely in the quadrature, or 90° distant from the sun ; and
although he failed in his attempt to determine the sun's distance with
accuracy, yet he showed that it was much greater than could at that time
have been otherwise imagined ; and he asserted that the earth was but as
a point in comparison with the magnitude of the universe. § His esti-
mation of the distance of the sun is made by Archimedes the basis of a
calculation of the number of grains of sand that would be contained in the
whole heavenly sphere, intended as an illustration of the powers of
numerical reckoning, and of the utility of a decimal system of notation,
which was the foundation of the modern arithmetic. ||

Eratosthenes, the successor of Aristarchus,^" is known by his observation
of the obliquity of the ecliptic, and his measurement of a certain portion
of the earth's circumference ; the whole of which he determined to be
250,000 stadia ; but the length of his stadium is uncertain. Ptolemy,
calculating perhaps from the same measures, or from some others still
more ancient, calls it 180,000 ; which, if the stadium is determined from
the Nilometer at Cairo, and from the base of the pyramid, is within one
thousandth part of the truth, the length of the base of the pyramid being
equal to 400 Egyptian cubits, or to 729 feet 10 inches English.

Hipparchus of Bithynia flourished at Alexandria about the year 140
before Christ. Employing the observations of Timocharis, and comparing
them with his own, he discovered the precession of the equinoxes. He
also observed that the summer was 9 days longer than the winter, and that

* Aristotle, Meteor. 1. 1, c. 6. t Strabo, Geog. 1. 13.

I Ptolemy, Almagest. 1. 6, c. 3.

§ Arist. Sam. de Magnit. et Dist. Solis et Lunse, 4to, Pis. 1572. Wallis's Op.
vol. iii.

|| Archimedes, Arenarius, ed. Paris, 1615, p. 449.
^ Cleomedes, Cyc. Th. 1. 1, c. 10, in Arati Op. Oxf. 1672, fol.

ON THE HISTORY OF ASTRONOMY. 455

)

the solstices divided each of these seasons a little unequally. In order to
explain this, Hipparchus supposed the sun to move uniformly in an
eccentric circle, the distance of its centre from that of the earth being -^ of
the radius, and placed the apogee in the sixth degree of gemini. Probably
the annual equation of the moon, which has some influence on the time of
eclipses, was the cause of his making the eccentricity too great ; had he
assumed it but one fifth part less, the supposition would have represented
the sun's place with tolerable accuracy. Hipparchus appears to have been
the first that employed astronomical observations for determining the lati-
tudes and longitudes of places.

The interval of three centuries, which elapsed between Hipparchus and
Ptolemy, offers us little that is remarkable in the progress of astronomy,
except the reformation of the calendar by Julius Caesar, who was assisted
in making the arrangement by Sosigenes, an astronomer of the same
school that gave birth to all the preceding discoveries, as well as to the
improvements of Ptolemy. This great astronomer was born at Ptolemais
in Egypt, and flourished about the year 140 of our era. He continued the
vast project, begun by Hipparchus, of reforming the whole science which
he studied. He discovered the evection of the moon, or the change of her
velocity, occasioned by the position of the apogee with respect to the sun ;
he determined the quantity of this equation with great precision ; and in
order to represent it, he supposed the moon to perform a subordinate revo-
lution in an epicycle, or a smaller circle, of which the centre was carried
round in the line of the general orbit, which he considered as an eccentric cir-
cle. This mode of approximation is exceedingly ingenious ; it is said to have
been the invention of Apollonius of Perga, the mathematician, and although
it sometimes becomes complicated, yet it is very convenient for calculation ;
and it may be employed with advantage in the representation of the plane-
tary motions by machinery. Ptolemy adopted the most ancient opinion
with respect to the solar system, supposing all the heavenly bodies to
revolve round the earth ; the moon being nearest, then Mercury, Venus,
the Sun, Mars, Jupiter, and Saturn. This opinion had long been the most
general, although some astronomers had placed Mercury and Venus at
greater distances than the sun, and some attributed to the earth a diurnal
motion only ; but the doctrine of the Pythagoreans appears to have been
wholly exploded or forgotten. Ptolemy determined the quantity of the
precession of the equinoxes from a comparison of his own observations
with those of Hipparchus ; but he made it smaller than the truth ; and he
probably formed his table of the places of the stars by applying this
erroneous correction to the tables of Hipparchus, in order to accommodate
them to his own time. Both these errors may, however, be otherwise
explained, by supposing him to have followed Hipparchus in the length of
the tropical year, which being somewhat too great, caused an error in the
calculation of the sun's place, to which that of the stars was referred ; but
upon this supposition, he must also have been mistaken in three obser-
vations of the place of the equinoctial points. Ptolemy's principal work is
nis mathematical system of astronomy, which was afterwards called the
great syntax or body of astronomy, and is at present frequently quoted by

456 LECTURE XLVIII.

the Arabic name Almagest. He also wrote a treatise on optics, in which
the phenomena of atmospherical refraction are described, and which is
extant is manuscript in the National library at Paris.* (Plate XXXVIII.
Fig. 528.)

Ptolemy was the last as well as the greatest of the Alexandrian astrono-
mers, and the science made no further progress till the time of the Arabians.
The first of these was Almamoun, the son of the celebrated Aaron Reschid ;
he reigned at Bagdad in 814, and having conquered the Greek emperor,
Michael the Third, he made it a condition of peace, that a copy of the
works of each of the best Greek authors should be delivered to him ; and
among them were the works of Ptolemy, of which he procured an Arabic
translation. Almamoun also observed the obliquity of the ecliptic, and
measured the length of a degree in the plains of Mesopotamia.

Among the astronomers protected by this prince and his successors,
Albategni was the most eminent/)- He ascertained with great accuracy,
in 880, the eccentricity of the solar motion, and discovered the change of
the place of the sun's apogee, or of the earth's aphelion.

Ibn Junis made his observations at Cairo, about the year 1000 ; he was
a very assiduous astronomer, and determined the length of the year within
2 seconds of the truth. At this time the Arabians were in the habit of
employing, in their observations, the vibrations of a pendulum.

The Persians soon after applied themselves to astronomy ; and in the
eleventh century they invented the approximation of reckoning 8 bissex-
tiles in 33 years, which was afterwards proposed by Dominic Cassini as an
improvement of the Gregorian calendar. The most illustrious of this
nation was Ulugh Beigh, who observed in his capital Samarcand, about
the year 1437, with very elaborate instruments. In the mean time
Cocheouking had made in China some very accurate observations, which
are valuable for the precision with which they ascertain the obliquity of
the ecliptic : their date is about 1278.

It was not long after the time of Ulugh Beigh, that Copernicus laid the
foundation of the more accurate theories which modern improvements have
introduced into astronomy. Dissatisfied with the complicated hypotheses
of the Ptolemaean system, he examined the works of the ancients, in quest
of more probable opinions. He found from Cicero that'Nicetas and other
Pythagoreans had maintained, that the sun is placed in the centre of the
system, and that the earth moves round him in common with the other
planets. He applied this idea to the numerous observations which the
diligence of astronomers had accumulated, and he had the satisfaction to
find them all in perfect conformity with this theory. He quickly discarded
the Ptolemaean epicycles, imagined in order to explain the alternations of
the direct and retrograde motions of the planets; in these remarkable
phenomena, ^.Copernicus saw nothing but the consequences necessarily
produced by the combination of the motions of the earth and planets round

* Composition Mathematique, Gr. et Fr. 2 vols. 4to, Paris, 1813. See also
Table Chronologique des Regnes, &c. trad, de 1'Allemagne de M. Ideler par Halma,
4to, Paris, 1819. Hypotheses et Epoques des Planetes de Ptol. &c. ibid. 4to, 1820.'
Tables Manuelles Astron. Gr. et Fr. 4to, 1823.

t See Halley's Dissertation, in Ph. Tr. xvii. 913.

ON THE HISTORY OF ASTRONOMY. 457

the sun ; and from a minute examination of these circumstances he calcu-
lated the relative distances of the planets from the sun, which till then had
remained unknown. In this system, every thing had the marks of that
beautiful simplicity which pervades all the works of nature, and which,
when once understood, carries with itself sufficient evidence of its truth.
Copernicus was born at Thorn, in Polish Prussia, in the year 1475 ; he
studied in Italy ; he taught mathematics at Rome, and afterwards settled
on a canonicate at Frauenberg, where, in 36 years of retirement and medi-
tation, he completed his work on the celestial revolutions, which was
scarcely published when he died.*

About this time, William the Fourth, Landgrave of Hesse Cassel, not
only enriched astronomy by his own observations, but also exerted his
influence with Frederic, King of Denmark, to obtain his patronage for the
celebrated Tycho Brahe. Frederic agreed to give him the little island
Huen, at the entrance of the Baltic, where Tycho built his observatory of
Uraniburg, and, in a period of 21 years, made a prodigious collection of
accurate observations. After the death of his patron, his progress was
impeded, and he sought an establishment at Prague, under the emperor
Rudolph. Here he died soon after, at the age of 55. Struck with the
objections made to the system of Copernicus, principally such as were
deduced from a misinterpretation of the scriptures, he imagined a new
theory, which, although mechanically absurd, is still astronomically correct ;
for he supposed the earth to remain at rest in the centre, the stars to revolve
round it, together with the sun and all the planets, in a sidereal day, and
the sun to have, besides, an annual motion, carrying with him the planets
in their orbits. Here the apparent or relative motions are precisely the
same as in the Copernican system ; the argument that Tycho Brahe drew
from the scriptures in favour of his theory was, therefore, every way
injudicious ; for it is not to be imagined that any thing but relative motion
or rest could be intended in the scriptures, when the sun is said to move, or
to stand still. But in the Copernican system, there was an evident regu-
larity in the periods of all the planets, that of the earth being longer than
that of Venus, and shorter than that of Mars, which were the neighbouring
planets on each side ; and when Tycho imagined the sun to move round
the earth, this analogy wras entirely lost. Tycho Brahe was the discoverer
of the variation and of the annual equation of the rnoon, the one being an
irregularity in its velocity, dependent on its position with respect to the
sun, the other a change in the magnitude of all the perturbations produced
by the sun, dependent on his distance from the earth. (Plate XXXVIII.
Fig. 529.)

Kepler was the pupil and assistant of Tycho, whose observations were
the basis of his important discoveries : he succeeded him in his appoint-
ments at Prague, and enjoyed the title of Imperial Mathematician. Adopt-
ing the Copernican system, which was then becoming popular, he pro-
ceeded to examine the distances of the celestial bodies from each other at
various times ; and after many fruitless attempts to reconcile the places of
the planets with the supposition of revolutions in eccentric circles, at last

* De Revolutionibus Orbium Coelestium, fol. 1543.

458 LECTURE XLVIII.

\

discovered that their orbits are ellipses, and demonstrated, chiefly from his
observations on the planet Mars, that the revolving radius, or the line
drawn from the sun to the planet, always describes equal areas in equal
times. By comparing the periods and the mean distances of the different
planets with each other, he found, after 17 years calculation, that the
squares of the times of revolution are always proportional to the cubes of
the mean distances from the sun.*

Kepler died in 1630 : before his death he had the satisfaction of applying
his theory to the motions of the satellites of Jupiter, which, as well as the
phases of Venus, and the spots of the sun, had lately been discovered in
Italy by the telescopic observations of Galileo. This great man, celebrated
as well for his theory of projectiles, as for his zealous defence of the
Copernican system, was born at Pisa in 1564, and lived to the age of 78,
full of that enthusiasm which made him despise the threats of the Inquisi-
tion, and submit patiently to its persecutions. He died in 1642, the year
in which Newton was born.

The invention of logarithms, by Baron Napier, requires to be noticed
for its importance to practical astronomy, and the laborious observations of
Hevelius deserve also to be mentioned with commendation. The dis-
coveries of the form of the ring of Saturn, and of one of his satellites, by
Huygens, and of four more, together with the belts and rotation of Jupiter,
by Dominic Cassini, were among the early improvements derived from
the introduction of the telescope. But, without dwelling on any of these
subjects, we hasten to the establishment of the system of gravitation,
which has immortalised the name of Newton, and done unrivalled honour
to the country that gave him birth.

The mutual attraction of all matter seems to have been suspected by the
Epicureans, but Lucretius never speaks of it in such terms as are sufficient
to convey by any means a distinct idea of a reciprocal force. Gregory, in
the preface of his Astronomy, has endeavoured to prove that Pythagoras
must have been acquainted even with the law of the decrease of gravita-
tion ; and Lalande appears to assent to his arguments ; but they rest only
on the bare possibility that Pythagoras might have deduced an analogy
from the tension of cords, which we have no reason to suppose that he
even completely understood : and this merely because he fancifully
imagined, that there was a correspondence between the planets and the
strings of a lyre. But the nature of gravitation had long been in some
measure suspected ; Plutarch had asserted that the moon is retained by it
in her orbit, like a stone in a sling; and Bacon, Copernicus, Kepler,
Fermat, and Roberval were aware of its efficacy. Bacon, in his Novum
organum, calls the descent of heavy bodies the motion of " general con-
gregation," and attributes the tides to the attraction of the moon. Kepler
mentions also the perfect reciprocality of the action of gravitation, and
considers the lunar irregularities as produced by the attraction of the
sun. But our most ingenious countryman, Dr. Hooke, was still more
decided in attributing the revolutions of the planets to the combination of
a projectile motion with a centripetal force ; he expresses his sentiments

* See Lect. IV.

ON THE HISTORY OF ASTRONOMY. 459

on the subject very clearly in his Attempt to prove the motion of the earth,
published in 1674, and had his skill in mathematics been equal to his
practical sagacity, he would probably have completed, or at least have
published the discovery before his great cotemporary.

It must be confessed that Newton's good fortune was equal to his talents
and his application ; for had he lived earlier, he might probably have con-
fined his genius to speculations purely mathematical ; had he been later,
his discoveries in natural philosophy might have been anticipated by
others ; and yet Newton would perhaps have improved still more on their
labours than they have done on his. It was in 1676, when he was 34
years old, that he first demonstrated the necessary connexion of the
planetary revolutions in elliptic orbits, with an attractive force varying
inversely as the square of the distance. But he had collected the law of
the force, from the discoveries of Kepler respecting the periods of the dif-
ferent planets, some time before 1671, as he asserts to Dr. Halley, and, to
the best of his recollection, about 1668, although in his Principia he allows,
with the most laudable candour, to Wren, Hooke, and Halley, the merit
of having made the same discovery, without any connexion with each
other's investigations, or with his own. The manner, in which Newton
was led to attend particularly to the subject, is thus related by Pemberton,
in the preface to his View of Sir Isaac Newton's philosophy.

" The first thoughts," says Pemberton, * "which gave rise to his Prin-
cipia, he had, when he retired from Cambridge in 1666, on account of the
plague. As he sat alone in a garden, he fell into a speculation on the
power of gravity ; that as this power is not found sensibly diminished
at the remotest distance from the centre of the earth, to which we can rise,
neither at the tops of the loftiest buildings, nor even on the summits of
the highest mountains ; it appeared to him reasonable to conclude, that
this power must extend much further than was usually thought ; why not
as high as the moon ? said he to himself ; and if so, her motion must be
influenced by it ; perhaps she is retained in her orbit thereby. However,
though the power of gravity is not sensibly weakened in the little change
of distance, at which we can place ourselves from the centre of the earth ;
yet it is very possible that so high as the moon this power may differ
much in strength from what it is here. To make an estimate, what might
be the degree of this diminution, he considered with himself, that if the
moon be retained in her orbit by the force of gravity, no doubt the primary
planets are carried round the sun by the like power. And by comparing
the periods of the several planets with their distances from the sun, he
found, that if any power like gravity held them in their courses, its strength
must decrease in the duplicate proportion of the increase of distance. This
he concluded by supposing them to move in perfect circles concentrical to
the sun, from which the orbits of the greatest part of them do not much
differ. Supposing, therefore, the power of gravity, when extended to the
moon, to decrease in the same manner, he computed whether that force
would be sufficient to keep the moon in her orbit. In this computation
being absent from books, he took the common estimate in use among geo-
* View of Newton's Philosophy, 1728, Preface.

460 LECTURE XLVIII.

graphers and our seamen, before Norwood had measured the earth, that 60
English miles were contained in one degree of latitude on the surface of the
earth. But as this is a very faulty supposition, each degree containing
ahout 69^ of our miles, his computation did not answer expectation ; whence
he concluded that some other cause must at least join with the action of
the power of gravity on the moon. On this account he laid aside for that
time any further thoughts upon this matter. But some years after, a letter
which he received from Dr. Hooke, put him on inquiring what was the real
figure, in which a body let fall from any high place descends, taking the
motion of the earth round its axis into consideration. Such a body, having
the same motion, which by the revolution of the earth the place has from
whence it falls, is to be considered as projected forwards, and at the same
time drawn down to the centre of the earth. This gave occasion to his
resuming his former thoughts concerning the moon ; and Picart, in France,
having lately measured the earth, by using his measures, the moon appeared
to be kept in her orbit purely by the power of gravity ; and consequently,
that this power decreases as you recede from the centre of the earth, in the
manner our author had formerly conjectured. Upon this principle he
found the line described by a falling body to be an ellipsis, the centre of
the earth being one focus. And the primary planets moving in such orbits
round the sun, he had the satisfaction to see, that this inquiry, which he
had undertaken merely out of curiosity, could be applied to the greatest
purposes. Hereupon he composed near a dozen propositions relating to the
motion of the primary planets about the sun. Several years after this,
some discourse he had with Dr. Halley, who at Cambridge made him a
visit, engaged Sir Isaac Newton to resume again the consideration of this
subject ; and gave occasion to his writing the treatise which he published
under the title of Mathematical principles of natural philosophy. This
treatise, full of such variety of profound inventions, was composed by him,
from scarce any other materials than the few propositions before men-
tioned, in the space of one year and a half."

The astronomers of Great Britain have not been less diligent in the prac-
tical, than successful in the theoretical part of the science. The foundation
of the observatory at Greenwich was laid in 1675, some years before the
completion and publication of the discoveries of Newton. It is with the
erection of this edifice that the modern refinements in practical astronomy
may be said to have commenced ; its immediate object was to assist in the
perfection of the science of navigation, and the series of observations, which
have been made in it, has afforded an invaluable fund of materials to
astronomers of every country. A reward had been proposed, more than
half a century before, by Philip the Third, of Spain, for the discovery of a
mode of determining the longitude of a ship at sea ; and the states of Hol-
land had followed his example ; a large reward was also offered by the
French government in the minority of Louis the Fifteenth. In 1674,
Mr. St. Pierre, a Frenchman, had undertaken to determine the longitude
of a place from observations of the moon's altitude, and King Charles the,
Second had been induced to appoint a commission to examine his propo-
sals. Mr. Flamsteed was consulted by the commissioners, and was added to

ON THE HISTORY OF ASTRONOMY. 461

their number : he showed the disadvantages of the method proposed hy
Mr. St. Pierre, and the inaccuracy of the existing tables of the lunar
motions, as well as of the catalogues of the places of the stars, but expressed
his opinion, that, if the tables were improved, it would be possible to deter-
mine the longitudes of places with sufficient accuracy by lunar observations.
The king, being informed of Flamsteed's representations, is said to have
replied with earnestness, that he " must have the places of the stars anew
observed, examined, and corrected, for the use of his seamen ;" upon this
Flamsteed was appointed Astronomer Royal, with a salary of £100 a year,
and it was proposed to have an observatory built either in Hyde Park, or
at Chelsea college ; but, upon Sir Christopher Wren's recommendation,
the situation of Greenwich Park was preferred.

In the year 1714, the British Parliament offered £20,000 for a determi-
nation of the longitude of a ship at sea, without an error of 30 miles, and
a smaller sum for a less accurate method, appointing at the same time a
Board of Longitude for the examination of the methods which might be
proposed. Under this act several rewards were assigned, and in 1774, it
was superseded by another, which offers £5000 for the invention of any
timekeeper, or other method, capable of determining the longitude of a
place within one degree, and £10,000 if within 30 miles ; and a reward of
£5000 to the author of any lunar tables, which should be found within 15
seconds of the truth ; allowing the Board also the power of granting smaller
sums at their discretion. Timekeepers are at present very commonly em-
ployed in the British navy, and some of them have been capable of deter-
mining the longitude within half a degree, after having been two or three
months at sea. The lunar tables, which have been employed for the
Nautical Almanacs, are those of Professor Mayer, who adopted the methods
of calculation invented by Leonard Euler ; but the tables of Mr. Burg, of
Vienna, are still more accurate, and are said to be always within about ten
seconds of the truth.

The progress of astronomy, since the death of Newton, in 1727, has been
fully adequate to what its most sanguine votaries could have hoped. The
great discoveries of the aberration of the fixed stars, and of the nutation
of the earth's axis, were made by our countryman Bradley, with the
assistance of the instruments for which he was indebted to the delicate
workmanship of our artists. Among these the names of Bird, Short,
Sisson, Graham, Dollond, Harrison, and Ramsden have long been cele-
brated throughout Europe. The geographical operations, which have been
performed in every part of the globe, have been chiefly conducted by the
liberality of the French and English governments, although other countries
have not been deficient in taking their share of the labour. Observations
of the transit of Venus were made with great care in the south seas
by British navigators, whom the munificence of our present sovereign
enabled >to undertake so arduous a voyage for this express purpose ; and we
are indebted to the fund which was granted on the occasion, as well as to
the zeal of the Astronomer Royal, for the experiments on the attraction of
mountains, which were instituted after their return. In this country also,
Dr. Herschel, besides many other important additions to our astronomical

462 LECTURE XLVIII.

knowledge, has discovered a primary planet, and eight secondary ones,
unknown before. The astronomers of Sicily and Germany have, however,
the honour of the first discovery of the three humbler members of the solar
system which have been last introduced to our acquaintance, Ceres by
Piazzi, Pallas by Olbers, and Juno by Harding: and the mathemati-
cians of France have excelled all their predecessors in the elaborate and
refined application of the theory of gravitation, to the investigation of the
most minute and intricate details of the celestial motions.

For the latest improvement that has been made in astronomy we are
also indebted to the zeal and ingenuity of Dr. Olbers, who, in pursuit of
an opinion which he had formed, respecting the origin of the three small
planets from the separation of a larger one into fragments, has been in the
habit of examining monthly that part of the heavens in which he supposes
the event to have taken place, and through wrhich each of the bodies must
necessarily pass. He has had the good fortune to discover, in this manner,
a fourth planet [Vesta], which nearly resembles the three others in its
appearance, except that it seems to be considerably larger.

LECT. XLVIII.— ADDITIONAL AUTHORITIES.

Gassendus, Tychonis Brahei Vita, 4to, Hagse Comitum, 1655. Blegny, Le Mes-
sager Celeste, 12mo, Paris, 1681. Champollion, Resume de Chronologie, 32rno,
Paris, 1730. De L'Isle, Mem. pour servir £ Histoire de 1'Astr. &c. 4to, St. Petersb.
1738. Weidler, Historia Astronomise, 4to, Vitembergse, 1741. Heilbroner, Hist.
Math, ab Orbe Condito ad Seculum XV. 4to, Lipsise, 1742. Heathcote, Historia
Astronomies, Cantab. 1747. Costard on the Chinese Astr. Ph. Tr. 1747, p. 476.
Hist, of Astr. 4to, Lond. 1767. Esteve, Hist, de 1'Ast. 3 vols. 12mo, Paris, 1755.
Bernoulli, Lettres Astronomiques, Berlin, 1771. Bailly, Histoire de 1' Astronomic
Anc. et Mod. 3 vols. 4to, Paris, 1775-9. Do. Indienne, 4to, 1787 ; Abrege, 1805.
Blair's History of Geography, 12mo, 1784. Schaubach, Geschichte der Greichischen
Ast. Gott. 1802. Lalande, Bibliographic Astronomique, 4to, 1803. Small's
Account of the Discoveries of Kepler, 1803. Ideler, Historische Untersuchungen
iiberdie Ast. Beob. der Alten. Berlin, 1806. Voiron, Hist, de 1'Ast. depuis 1781
jusqu'a 1811, 4to, 1810. Cassini, Memoires pour servir al'Hist. des Sciences, 4to,
Paris, 1810. Gautier, Essai Historique sur le Probleme des trois Corps, 4to, Paris,
1817. Delambre, Histoire de 1' Astronomic Ancienne, 2 vols. 4to, Paris, 1817 ; du
Moyen Age, 4to, 1819 ; Moderne, 2 vols. 4to, 1821 ; de 18e Siecle, publiee par
Mathieu, 4to, 1826. Laplace, Precis de 1'Hist. de 1'Ast. 1821. Rigaud's Memoirs
of Bradley, 4to, Oxf. 1832 ; Suppl. 4to, Oxf. 1833. Rothman's History of Astr.
(Lib. of Useful Knowledge), 1832. Airy's Report on Astr. Brit. Assoc. 1832.
Baily's Account of Flamsteed, 4to, Lond. 1835.

ON THE HISTORY OF ASTRONOMY.

463

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464

LECTURE XLIX.

ON THE ESSENTIAL PROPERTIES OF MATTER.

THE objects, which have lately occupied our inquiries, are the most
sublime and magnificent that nature any where exhibits to us, and the
contemplation of them naturally excites, even in an uncultivated mind, an
admiration of their dignity and grandeur. But all magnitude is relative ;
and if we examine with more calm attention, we shall find still greater
scope for our investigation and curiosity, in the microscopic, than in the
telescopic world. Pliny has very justly observed, that nature no where
displays all her powers with greater activity, than in the minutest objects
perceptible to our senses ; and we may judge how wide a field of research
the corpuscular affections of matter afford, from the comparatively small
progress that has hitherto been made in cultivating it. For while the
motions of the vast bodies, which roll through the heavens, have been
completely subjected to the most rigorous calculations, we know nothing,
but from experience only, of the analogies by which the minute actions of
the particles of matter are regulated. It is probable, however, that they
all depend ultimately on the same mechanical principles. We have seen,
for example, that the widely extended elevations and depressions of the
ocean, which are raised by the attractive powers of the two great lumina-
ries, and cover at once a half of the globe, are governed and combined
according to the same laws which determine the motions of the smaller
waves excited by different causes in a canal, the rapid tremors of a medium
transmitting sound, or the inconceivably diminutive undulations which
are capable of accounting for the phenomena of light, and which must be
exerted in spaces as much smaller than those of sound, as a grain of sand
is smaller than a mountain. Thus the annihilation of the effects of the
semidiurnal changes of the tide, and the preservation of the diurnal change,
in the harbour of Batsha, may be explained precisely in the same manner
as the reflection of red light from a transparent substance, of such a thick-
ness, as to be capable of destroying a portion of violet light under the same
circumstances.

We are at present to descend from the affections of the large masses of
matter, which form the great features of the universe, to the particular
properties of the matter which constitutes them, as far as they are common
to all matter in general ; but those properties which are peculiar to certain
kinds of matter only, being the subjects of chemical science, are not to be
included in the discussion. If we are asked for a definition of matter, it
will be somewhat difficult to avoid all circuitous expressions. We may
make gravitation a test of matter, but then we must say, that whatever is
attracted by other matter, is also to be denominated matter, and this sup-
poses the subject of our definition already known ; besides that the property*
of attraction may also possibly belong to substances not simply material ;

ON THE ESSENTIAL PROPERTIES OF MATTER. ^465

for the electrical fluid, if such a fluid exists, is probably attracted by mat-
ter, and yet it seems to be different in most respects from any modification
of common matter. A similar difficulty would occur if we attempted to
define matter by its impenetrability or mutual repulsion, or if we consi-
dered every thing as material that is capable of affecting the senses. We
must, therefore, take it for granted that matter is known without a defi-
nition, and we may describe it as a substance occupying space, or as a
gravitating or ponderable substance.

It cannot be positively determined whether matter is originally of one
kind, owing its different appearances only to the form and arrangement of
its parts ; or whether there are various kinds of simple matter, essentially
distinct from each other ; but the probability appears to be in favour of
the former supposition. However this may be, the properties of matter
are by no means so simple in their nature, nor so easily reducible to gene-
ral laws, as the more mathematical doctrines of space and motion ; and
since our knowledge of them depends more on experience than on abstract
principles, they may properly be considered as belonging to particular
physics. We have found no inconvenience from the omission of the doc-
trine of matter as a part of the subject of mechanics ; although, in treating
of the strength of materials, as subservient to practical mechanics, it was
necessary to consider the effects of some of these properties as deduced from
experiment ; but it will appear that it was impossible to examine their
origin and mutual connexion, without supposing a previous knowledge of
many other departments of natural philosophy.

We may distinguish the general properties of matter into two principal
classes, those which appear to be inseparable from its constitution, and those
which are only accidental, or which are not always attached to matter of
all kinds. The essential properties are chiefly extension and divisibility,
density, repulsion, or impenetrability, inertia, and gravitation ; the acci-
dental properties are in great measure dependent on cohesion, as liquidity,
solidity, symmetry of arrangement, cohesive elasticity, stiffness, toughness,
strength, and resilience.

The extension of matter can scarcely be considered as a property sepa-
rate from its impenetrability, unless we conceive that it can occupy space,
without excluding other bodies from it. This opinion has indeed been
maintained by some philosophers, who have imagined that the minute
particles which they suppose to constitute light, may penetrate the ulti-
mate atoms of other matter without annihilating or displacing them ;
and if this hypothesis were admitted, it would be necessary to consider
each particle of matter as a sphere of repulsion, extended without being
impenetrable.

The divisibility of matter is great beyond the power of imagination, but
we have no reason for asserting that it is infinite ; for the demonstrations
which have sometimes been adduced in favour of this opinion, are obvi-
ously applicable to space only. The infinite divisibility of space seems to
be essential to the conception that we have of its nature ; and it may be
strictly demonstrated, that it is mathematically possible to draw an infinite
number of circles between any given circle and its tangent, none of which

2H

460 LECTURE XLIX.

shall touch either of them, except at the general point of contact ; and that
a ship, following always the same oblique course with respect to the meri-
dian, for example, sailing north eastwards, would continue perpetually to
approach the pole without ever completely reaching it. But when we
inquire into the truth of the old maxim of the schools, that all matter is
infinitely divisible, we are by no means able to decide so positively. New-
ton observes, that it is doubtful whether any human means may be suffi-
cient to separate the particles of matter beyond a certain limit ; and it is
not impossible that there may be some constitution of atoms, or single
corpuscles, on which their properties, as matter, depend, and which would
be destroyed if the units were further divided ; but it appears to be more
probable that there are no such atoms ; and even if there are, it is almost
certain that matter is never thus annihilated in the common course of
nature.*

It remains to be examined how far we have any experience of the actual
extent of the divisibility of matter ; and we shall find no appearance of any
thing like a limit to this property. The smallest spherical object, visible to
a good eye, is about -jnjVu- of an inch in diameter ; by the assistance of a
microscope, we may perhaps distinguish a body one hundredth part as
large, or ir^^rjra. of an inch in diameter. The thickness of gold leaf is
less than this, and the gilding of lace is still thinner, probably in some
cases not above one ten millionth of an inch ; so that -jV^ of a grain would
cover a square inch, and a portion barely large enough to be visible by a
microscope, might weigh only the 80 million millionth part of a grain.t
A grain of musk is said to be divisible into 320 quadrillions of parts, each
of which is capable of affecting the olfactory nerves. There are even living
beings, visible to the microscope, of which a million million would not
make up the bulk of a common grain of sand. But it is still more remark-
able, that, as far as we can discover, many of these animalcules are as
complicated in their structure as an elephant or a whale. It is true that
the physiology of the various classes of animals is somewhat more simple as
they deviate more from the form of quadrupeds, and from that of the
human species ; the solid particles of the blood do not by any means vary
in their magnitude in the same ratio with the bulk of the animal ; and
some of the lower classes appear to approximate very much to the nature
of the vegetable world. But there are single instances that seem wholly
to destroy this gradation : Lyonnet has discovered a far greater variety of
parts in the caterpillar of the willow butterfly, than we can observe in
many animals of the largest dimensions ; and among the microscopic
insects in particular, we see a prodigality of machinery, subservient to the
various purposes of the contracted life of the little animal, in the structure
of which nature appears to be ostentatious of her power of giving perfection
to her minutest works.

If Newton's opinion, respecting the origin of the colours of natural
bodies in general, were sufficiently established, it would afford us a limit

* Consult Wollaston, Ph. Tr. 1822.

t SeeHalley, Ph. Tr. 1693, p. 540. Nicholson, ibid. 1789, p. 286. Button's
transl. of Montucla's Mathematical Recreations, 4 vols. Lond. 1803, vol. iv. p. 80.

ON THE ESSENTIAL PROPERTIES OF MATTER. 467

to the divisibility of matter with respect to coloured substances ; for the
colours of thin transparent substances, which he considers as resembling those
of most other substances, are no longer observable, in any known medium,
when the thickness is less than about ^mAnnr of an inch. But we have
positive evidence that coloured substances may be reduced to dimensions
far below this limit ; besides the instance of the gilt wire, which has already
been mentioned, a particle of carmine may still retain its colour, when its
thickness is no more than one thirty millionth of an inch, or one sixtieth
part of the limit deduced from the supposition of Newton ; and it is there-
fore scarcely possible that the colours of such substances can precisely
resemble those of thin plates, although they may perhaps still be in some
measure analogousv to them.

Impenetrability is usually attributed to matter, from the common obser-
vation that two bodies cannot occupy the same place at once. And it is
thus that we distinguish matter from space ; for example, when we dip
an inverted jar into mercury, the air contained in the jar depresses the
surface of the mercury, and prevents its occupying the space within the
jar : but if the jar had been void of matter, like the space above the mer-
cury of a barometer, nothing would have prevented its being filled by the
mercury, as soon as either its weight or the pressure of the atmosphere,
urged it to enter the jar.

But it does not appear that our senses are fully competent to extend this
proposition to all substances, whether material or not. We cannot prove
experimentally that the influence of gravitation is incapable of pervading
even the ultimate particles of solid matter, for this power appears to suffer
no diminution nor modification, when a third body is interposed between
the two gravitating masses. In the same manner, a magnet operates as
rapidly on a needle, through a plate of glass or of gold, whatever its thick-
ness may be, as if a vacuum only intervened. It may, however, be
inquired if the gold or the glass has not certain passages or pores, through
which the influence may be transmitted : and it may be shown, in many
instances, that substances, apparently solid, have abundant orifices into
which other substances may enter ; thus mercury may easily be made to
pass through leather, or through wood, by the pressure of the atmosphere,
or by any other equal force : and, however great we may suppose the pro-
portion of the pores to the solid matter, it may be observed, that it requires
only a more or less minute division of the matter, to reduce the magnitude
of the interstices between the neighbouring particles within any given
dimensions. Thus platina contains, in a cubic inch, above 200 thousand
times as many gravitating atoms as pure hydrogen gas, yet both of these
mediums are free from sensible interstices, and appear to be equally con-
tinuous ; and there may possibly be other substances in nature that contain
in a given space 200 thousand times as many atoms as platina ; although
this supposition is not positively probable in all its extent ; for the earth is
the densest of any of the celestial bodies with which we are fully ac-
quainted, and the earth is only one fourth as dense as if it were composed
entirely of platina ; so that we have no reason to believe that there exists

2n2

4G8 LECTURE XLIX.

in the solar system any considerable quantity of a substance even so dense
as platina.

Besides this porosity, there is still room for the supposition, that even the
ultimate particles of matter may be permeable to the causes of attractions
of various kinds, especially if those causes are immaterial : nor is there
anything in the unprejudiced study of physical philosophy that can induce
us to doubt the existence of immaterial substances ; on the contrary we see
analogies that lead us almost directly to such an opinion. The electrical
fluid is supposed to be essentially different from common matter; the
general medium of light and heat, according to some, or the principle of
caloric, according to others, is equally distinct from it. We see forms of
matter, differing in subtility and mobility, under the names of solids,
liquids, and gases ; above these are the semimaterial existences which
produce the phenomena of electricity and magnetism, and either caloric or
a universal ether ; higher still perhaps are the causes of gravitation, and
the immediate agents in attractions of all kinds, which exhibit some phe-
nomena apparently still more remote from all that is compatible with
material bodies ; and of these different orders of beings, the more refined
and immaterial appear to pervade freely the grosser. It seems therefore
natural to believe that the analogy may be continued still further, until it
rises into existences absolutely immaterial and spiritual. We know not
but that thousands of spiritual worlds may exist unseen for ever by human
eyes ; nor have we any reason to suppose that even the presence of matter,
in a given spot, necessarily excludes these existences from it. Those who
maintain that nature always teems with life, wherever living beings can be
placed, may therefore speculate with freedom on the possibility of indepen-
dent worlds ; some existing in different parts of space, others pervading
each other, unseen and unknown, in the same space, and others again to
which space may not be a necessary mode of existence.

Whatever opinion we may entertain with respect to the ultimate impe-
netrability of matter in this sense, it is probable that the particles of matter
are absolutely impenetrable to each other. This impenetrability is not
however commonly called into effect in cases of apparent contact. If the
particles of matter constituting water, and steam, or any other gas, are of
the same nature, those of the gas cannot be in perfect contact ; and when
water is contracted by the effect of cold, or when two fluids have their
joint bulk diminished by mixture, as in the case of alcohol or sulfuric
acid, and water, the particles cannot have been in absolute contact before,
although they would have resisted with great force any attempt to com-
press them. Metals too, of all kinds, which have been melted, become
permanently more dense when they are hammered and laminated. A still
more striking and elegant illustration of the nature of repulsive force is
exhibited in the contact of two pieces of polished glass. The colours of
thin plates afford us, by comparison with the observations of Newton, the
most delicate micrometer that can be desired, for measuring any distances
less than the ten thousandth of an inch : it was remarked by Newton him-
self, that when two plates of glass are within about this distance of each

ON THE ESSENTIAL PROPERTIES OF MATTER. 469

other, or somewhat nearer, they support each other's weight in the same
manner as if they were in actual contact, and that some additional force is
required in order to make them approach still nearer ; nor does it appear
probable that the contact is ever perfect, otherwise they might be expected
to cohere in such a manner as to become one mass. Professor Robison*
has ascertained by experiment the force necessary to produce the greatest
possible degree of contact, and finds it equivalent to a pressure of about a
thousand pounds for every square inch of glass. It is therefore obvious
that in all common cases of the contact of two distinct bodies, it must be
this repulsive force that retains them in their situation. I have found that
glass, placed on a surface of metal, exhibits this force nearly in the same
degree as if placed on another piece of glass ; it is also independent of the
presence of air ; but under water it disappears.

The existence of a repulsive force, extending beyond the actual surface
of a material substance, being proved, it has been conjectured by some
that such a force, unconnected with any central atom, may be sufficient for
producing all the phenomena of matter. This representation may be
admitted without much difficulty, provided that it be allowed that the
force becomes infinite at or near the centre ; but it has been sometimes
supposed that it is every where less than infinite, and consequently that
matter is not absolutely impenetrable ; such a supposition appears however
to lead to the necessity of believing that the particles of matter must some-
times be annihilated, which is not a very probable opinion.

The magnitude of the repulsive force by which the particles of any single
body are enabled to resist compression, increases nearly in proportion to
the degree of compression, or to the decrease of the distances between the
particles. This is almost a necessary consequence of any primary law
that can be imagined, for the immediate actions of the particles : for
instance, if the repulsion increased either as the square or as the cube of
the distance diminished, the effect of a double change of dimensions would
always be nearly a double change of the repulsive force ; that is, if an
elastic substance were compressed one thousandth part of its bulk, it would
in either case resist twice as much as if it were only compressed one two
thousandth.

It is obvious that if the particles of matter are possessed of a repulsive
force decreasing in any regular proportion with the increase of distance,
they can never remain at rest without the operation of some external
pressure, but will always retain a tendency to expand. This is the case of
all elastic fluids, the density of which is found to vary exactly as the com-
pressing force, whence it may be demonstrated, that the primary repulsive
force of the particles must increase in the same proportion as the distance
decreases. It follows also that this force can only be exerted between such
particles as are either actually or very nearly in contact with each other;
since it requires no greater pressure, acting on a given surface, to retain a
gallon of air in the space of half a gallon, than to retain a pint in the space

* Robison's Mech. Phil. vol. i. Corpuscular Action, art. 241. See also Huy-
gens, Ph. Tr. No. 86. Hauksbce, ibid. 1709, p. 306.

470 LECTURE XLIX.

of half a pint ; which could not be, if the particles exercised a mutual
repulsion at all possible distances.

Mr. Dalton* has proposed a singular theory respecting the constitution
and mutual repulsion of elastic fluids ; he imagines that when any two
gases of different kinds are mixed, the particles of each gas repel only the
similar particles of the same gas, without exerting any action on those of
the other gas, except when the ultimate solid atoms chance to interfere.
The idea is ingenious and original, and may perhaps be of use in connect-
ing some facts together, or in leading to some other less improbable suppo-
sitions ; but it may easily be shown, that Mr. Dalton' s hypothesis cannot
possibly be true in all its extent, since it would follow from it, that two
portions of gases of different kinds, could not exist, for a sensible time, in
the same vessel, without being uniformly diffused throughout it, while the
fact is clearly otherwise ; for hydrogen gas remains, when left completely
at rest, a very considerable time above, and carbonic acid gas below a
portion of common air writh which it is in contact ; nor is there any cir-
cumstance attending the mixture of gases, which may not be explained
without adopting so paradoxical an opinion. Mr. Dalton thinks that,
from the laws of hydrostatics, no two gases, not chemically united, could
remain mixed, if their particles acted mutually on each other : but the
laws of hydrostatics do not apply to the mixture of single particles of fluids
of different kinds ; since they are only derived from the supposition of a
collection of particles of the same kind.

In liquids and in solids, this repulsive force appears at first sight to be
wanting ; but when we consider that the particles both of liquids and of
solids are actuated by the attractive force of cohesion, we shall see the
necessity of the presence of a repulsive force, in order to balance it ; it is,
therefore, probable .that the particles of aeriform fluids still retain their
original repulsive powers, when they are reduced to a state of liquidity or
of solidity, by being subjected to the action of a second force which causes
them to cohere.

The mutual repulsion of the particles of matter is a reciprocal force,
acting equally, in opposite directions, on each of the bodies concerned. It
scarcely requires either experiment or argument to show, that if two
bodies repel each other, neither of them will remain at rest, but both of
them will move, with equal quantities of motion. Thus, if a portion of
condensed air be made to act upon the bullet of an air gun, it will force the
gun backwards with as much momentum as it impels the bullet forwards.

Inertia is that property of matter, by which it retains its state of rest or
of uniform motion, with regard to a quiescent space, as long as no foreign
cause occurs to change that state. This property depends on the intimate
constitution of matter ; it is generally exhibited by means of the force of
repulsion, which enables a body in motion to displace another, in order to
continue its motion, or by means of some attractive force, which causes two
bodies to approach their common centre of inertia with equal momenta.

* Manchester Memoirs, vol. v. See also Graham, Edin. Tr. 1831 ; Thomson,
Phil. Mag. 3rd Ser. vol. iv. p. 321, by whom the hypothesis of Dalton is established.

ON THE ESSENTIAL PROPERTIES OF MATTER. 471

Another universal property of matter is reciprocal gravitation, of which
the force is directly in the joint proportion of the quantities of matter at-
tracting each other, and inversely as the square of their distance. In order
to prove that the gravitation towards a given substance, for instance, the
weight of a body, or its gravitation towards the earth, is precisely in pro-
portion to the mass or inertia of the moveable matter of which it consists,
Sir Isaac Newton made two equal pendulums, with hollow balls of equal
size : in order that the resistance of the air might be the same with respect
to both, he placed successively within the balls a variety of different sub-
stances, and found that the time of vibration remained always the same ;
whence he inferred that the attraction was proportional in all cases to the
quantity of matter possessing inertia. For if any of these substances had
contained particles capable of receiving and communicating motion, yet
without being liable to gravitation, they would have retarded the vibrations
of the pendulum, by adding to the quantity of matter to be moved, without
increasing the moving force. The law of gravitation, which indicates the
ratio of its increase with the diminution of the distance, is principally
deduced from astronomical observations and computations : it is the
simplest that can be conceived for any influence, that either spreads from a
centre, or converges towards a centre ; for it supposes the force acting on
the same substance to be always proportional to the angular space that it
occupies.

Newton appears to have considered these laws of gravitation, which he
first discovered, rather as derivative than as original properties of matter ;
and although it has often been asserted that we gain nothing by referring
them to pressure or to impulse, yet it is undoubtedly advancing a step in the
explanation of natural phenomena, to lessen the number of general prin-
ciples ; and if it were possible to refer either all attraction to a modification
of repulsion, or all repulsion to a modification of attraction, we should
make an improvement of the same kind as Newton made, when he reduced
all the diversified motions of the heavenly bodies to the universal laws of
gravitation only. We have, however, at present little prospect of such a sim-
plification.

It has been of late very customary to consider all the phenomena of
nature as derived from the motions of the corpuscles of matter, agitated by
forces varying according to certain intricate laws, which are supposed to
be primary qualities, and for which it is a kind of sacrilege to attempt to
assign any ulterior cause. This theory was chiefly introduced by Bosco-
vich,* and it has prevailed very widely among algebraical philosophers, who
have been in the habit of deducing all their quantities from each other by
mathematical relations, making, for example, the force a certain function
or power of the distance, and then imagining that its origin is sufficiently
explained ; and when a geometrician has translated this language into his

* De Viribus Vivis, 4to, 1745; De Lumine, 4to, 1748; De Lege Continuitatis,
4 to, 1754; De LegeVirium in Natura existentium, 4to, 1755; De Divisibilitate
Materise et Principiis Corporum, 4to, 1757 ; Theoria Philosophise Naturalis, 4to,
1763, p. 4, Venice. See also Benvenutus, Physicse Generalis Synopsis, 1754.

472 LECTURE XLIX.

own, and converted the formula into a curve, with as many flexures and
reflections as the labyrinth of Daedalus, he imagines that he has depicted
to the senses the whole procedure of nature. Such methods may often be
of temporary advantage, as long as we are contented to consider them as
approximations, or as classifications of phenomena only ; but the grand
scheme of the universe must surely, amidst all the stupendous diversity of
parts, preserve a more dignified simplicity of plan and of principles, than
is compatible with these complicated suppositions.

" To show," says Newton, in the preface to the second edition of his
Optics, " that I do not take gravity for an essential property of bodies, I
have added one question concerning its cause, choosing to propose it by way
of a question, because I am not yet satisfied about it for want of experi-
ments." In the query here mentioned, he proceeds from the supposition of
an elastic medium, pervading all space ; a supposition which he advances
with considerable confidence, and which he supports by very strong argu-
ments, deduced as well from the phenomena of light and heat, as from the
analogy of the electrical and magnetic influences. This medium he supposes
to be much rarer within the dense bodies of the sun, the stars, the planets,
and the comets, than in the empty celestial spaces between them, and to
grow more and more dense at greater distances from them, so that all these
bodies are naturally forced towards each other by the excess of pressure.

The effects of gravitation might be produced by a medium thus consti-
tuted, if its particles were repelled by all material substances with a force
decreasing, like other repulsive forces, simply as the distances increase ; its
density would then be every where such as to produce the appearance of an
attraction varying like that of gravitation. Such an ethereal medium
would therefore have the advantage of simplicity, in the original law of
its action, since the repulsive force which is known to belong to all matter,
would be sufficient, when thus modified, to account for the principal pheno-
mena of attraction.

It may be questioned whether a medium, capable of producing the effects
of gravitation in this manner, would also be equally susceptible of those
modifications which we have supposed to be necessary for the transmission
of light. In either case it must be supposed to pass through the apparent
substance of all material bodies with the most perfect freedom, and there
would, therefore, be no occasion to apprehend any difficulty from a retard-
ation of the celestial motions ; the ultimate impenetrable particles of matter
being perhaps scattered as thinly through its external form as the stars
are scattered in a nebula, which has still the distant appearance of a
uniform light and of a continuous surface : and there seems no reason
to doubt the possibility of the propagation of an undulation through the
Newtonian medium with the actual velocity of light. It must be remem-
bered that the difference of its pressure is not to be estimated from the
actual bulk of the earth or any other planet alone, but from the effect of the
sphere of repulsion of which that planet is the centre ; and we may then de-
duce the force of gravitation from a medium of no very enormous elasticity.

We shall hereafter find that a similar combination of a simple pressure

ON COHESION. 473

•**(
with a variable repulsion is also observable in the force of cohesion ; and

supposing two particles of matter, floating in such an elastic medium,
capable of producing gravitation, to approach each other, their mutual
attraction would at once be changed from gravitation to cohesion, upon the
exclusion of the portion of the medium intervening between them. This
supposition is, however, directly opposite to that which assigns to the elastic
medium the power of passing freely through all the interstices of the ulti-
mate atoms of matter, since it could never pass between two atoms cohering
in this manner; we cannot therefore, at present, attempt to assert the
identity of the forces of gravitation and cohesion so strongly as this theory
would allow us to do, if it could be established. In short, the whole of our
inquiries respecting the intimate nature of forces of any kind, must be con-
sidered merely as speculative amusements, which are of no further utility
than as they make our views more general, and assist our experimental
investigations.

LECT. XLIX.— ADDITIONAL AUTHORITIES.

Bernoulli, De Gravitate ^Etheris, 12mo, Amst. 1683. Newton's Optics ; Queries.
Huygens, Op. Rel. i. Hambergerus, De Experimento Huygenii, 4to, Jena, 1723.
Hausen, Programmata de Reactione, Leipz. 1740. Richmann on the Force of Water
in Freezing, Nov. Com. Petr. i. 276. Keill's Introd. Lect. viii. Golden on the
Primary Cause acting on Matter, 1745. Knight on Attraction, &c. 4to, Lond. 1748.
Hollmann, Commentationum Sylloge, 4to, Gott. 1764-1784. Bossut sur la Resist-
ance de 1' Ether, 4to, Charleville, 1766. Van Swinden, De Attractione, 4to, Leyd.
1766. Kratzenstein, Amolitio Vis Inertise, Hanov. 1770. Franklin's Miscellanies,
4to, Lond. 1779. Zimmermann, Traitedel'Elasticite de 1'Eau et d'autres Liquides,
Leipz. 1779. Coulomb on the Force of Torsion, Hist, et Mem. 1764, p. 265 ; 1784,
p. 229. Delangez on the Mechanics of Semi-fluids, Mem. della Soc. Ital. iv. 329.
Mossotti on Molecular Action, Scientific Mem. i. 448. Kelland on do. Camb. Tr. vii.

Atomic Theory. — Higgins, Comparative View of the Phlogistic and Antiphlogistic
Theories, 1789. Wenzel, Lehrevon der Verwandschaft derKbrper, 1777. Richter,
Anfangsgriinde der Stochyometrie, 1792. Dalton's New System of Chemical Phi-
losophy, Manch. 1808, 1810. Avogadro, Mem. di Torino, xxvi. 440, xxviii. xxix.
30, xxxi. xxxiii. Mem. della Soc. Ital. 1822. Fisica dei Corpi Ponderabili, 2
vols. Torino, 1837-8. Daubeny's Introduction to the Atomic Theory, Oxf. 1831.
Thomson's, Turner's, and all Treatises on Chemistry.

LECTURE L.

ON COHESION.

THOSE properties of matter, which we have lately examined, if they are
not absolutely inseparable from its constitution, are, at least, always
found attached to such matter as we are able to submit to our experi-
ments. There are, however, many other general affections, to which all
matter appears to be liable, although none is perpetually subjected to
them, and these are principally, if not entirely, dependent on the force of
cohesion.

474 LECTURE L.

In order that any two particles of matter may cohere, it is necessary
that they be within a very small distance of each other, and the density
of any substance, composed of cohesive particles, must probably always
be more than half as great as that of water. There are indeed some solids
apparently a little lighter than this, but they appear to be extremely
porous : and perhaps the solid substances of some of the celestial bodies
may also be a little more rare. It frequently happens, that the compres-
sion of an elastic fluid alone is sufficient to cause the force of cohesion to
take place between its particles ; thus, if common steam be exposed, in a
close vessel, to a pressure greater than that of the atmosphere, it will be
wholly condensed into water, provided that no elevation of temperature be
allowed : and the same has been experimentally shown of many other
aeriform fluids, which may be reduced to liquids by pressure ; but others
of these fluids retain their elasticity, notwithstanding any force which
human art can apply to them.

It is probable that as soon as the particles of any elastic fluid are brought
within the reach of the force of cohesion, it commences at once in its full
extent, so as to cause them to rush together, until it is balanced by that of
repulsion, which continually increases as the particles approach nearer to
each other ; they must then remain, perhaps after some vibrations, in a
state of equilibrium ; and if any cause should tend to separate them, or to
bring them nearer together, they would resist it, in either case, with a force
proportional to the degree of extension or compression. The distance at
which the force of cohesion commences, is not the same for all kinds of
matter, nor even for the same substance at different temperatures ; it is
smaller for vapours of all kinds, in proportion as their temperature is
higher, the cohesion itself being also smaller. If the experiments on the
density of steam have been correct, it follows that the force of repulsion
must increase more rapidly than the distances diminish, for the elasticity
of water is nearly ten times as great as that which would be inferred from
the compression of steam into a substance of equal density : this suppo-
sition agrees also with the experiments on the mean density of the earth,
which is probably not so great as it would be if the force of repulsion
increased in the simple ratio of the density. The law of repulsion appears
also to be in some degree modified by the effect of heat, which increases its
force at greater distances more considerably than at smaller. It appears
indeed, from the diminution of the elasticity of a spring by heating it, that
the repulsive force of the particles of bodies at very small distances is even
diminished by heat, unless the force be again supposed to decrease much
more rapidly than the distance diminishes : thus the diminution of the
elasticity of iron by heat is about thirty times as great as the increase of
the distance of its particles ; so that the original repulsive force must pro-
bably be somewhat diminished, although less than the cohesive force. At
greater distances, however, the force of repulsion is certainly increased ;
for the elasticity of vapours and gases of all kinds is evidently greater as
the temperature is higher. (Plate XXXIX. Fig. 530.)

The cohesion of two or more particles of matter to each other does not
interfere with their power of repelling other particles situated in a different

ON COHESION. 475

direction : thus, two pieces of glass require to be brought together with
considerable force, and generally with some friction, before they can begin
to cohere ; and a small drop of water, falling lightly on the surface of a
pond, may remain for some instants without coming into perfect contact
with it ; the same circumstance is also still more observable in spirit of
wine a little warmed.

The first and simplest effect of cohesion is to produce liquidity. That
all liquids possess some cohesion, is very obvious, from their tendency to
assume a spherical form when they are sufficiently detached from other
substances, and from the suspension of a drop from any solid, to which its
upper surface adheres with sufficient force. Without cohesion, indeed, a
liquid would be only a very fine powder, except that the particles of powders
have not the power of moving with perfect freedom on each other, which
constitutes fluidity. The apparent weakness of the cohesion of liquids is
entirely owing to this mobility, since their form may be changed in any
degree without considerably increasing the distances of their particles, and
it is only under particular circumstances that the effects of their cohesion
can become sensible.

When a liquid is considered as unlimited in its extent, the repulsion of
its particles, situated in all possible directions with regard to each other,
may be supposed in all cases precisely to balance the cohesion, which is
derived from the actions of particles similarly situated ; and this must also
be the state of the internal parts of every detached portion of a liquid,
where they are so remote from the surface as to be beyond the minute
distance which is the limit of the action of these forces. But the external
parts of the drop will not remain in the same kind of equilibrium : they
may be considered as a thin coating of a liquid surrounding a substance
which resists only in a direction perpendicular to its surface, and does not
interfere with the mutual actions of the particles of the liquid. Now since
the repulsive force increases as the distance diminishes, it must be exerted
more powerfully by the nearest particles, while the cohesion is directed
equally towards all the particles within a certain distance, and wherever
the surface is curved, the joint cohesive force will be directed to a remoter
part of the curve than the repulsive force opposed to it, so that each
particle will be urged, by the combination of these forces, towards the
concave side of the curve, and the more as the curvature is greater ;
hence the coating of the liquid, thus constituted, must exert a force on the
parts in contact with it, precisely similar to that of a flexible surface,
which is every where stretched by an equal force ; and from this simple
principle we may derive all the effects produced by a cohesion of this kind,
which, from its being most commonly observed in the ascent of water in
capillary tubes, has been denominated capillary attraction. (Plate XXXIX.
Fig. 531.)

It is, therefore, a general law, that the surface of every detached portion
of a fluid must every where have such a curvature, as to be able to with-
stand the hydrostatical pressure which acts against it ; and hence we may
calculate in many cases the properties of the curve which it must form ;
but in other cases the exact calculation becomes extremely intricate, and

476 LECTURE L.

perhaps impracticable. A drop descending in a vacuum would be perfectly
spherical ; and if its magnitude were inconsiderable, it would be of the
same form when descending through the air ; a small bubble rising in a
liquid must also be spherical ; but where the drop or the bubble is larger,
its curvature will be greatest where the internal pressure is greatest, or
where the external pressure is least, and in different cases this pressure
may be differently distributed. Where a drop is suspended from a solid,
its length may be such that the pressure at its upper part may become
negative, and its surface will then be concave instead of convex : and
when a bubble rises to the surface of a liquid, it often carries with it a film
of the liquid, of which the weight is probably smaller than the contractile
force with which the surface resists the escape of the air, so that, from the
magnitude of the contractile force, we may determine the greatest possible
weight of a bubble of given dimensions. A slight imperfection of fluidity
probably favours the formation of detached bubbles, by retarding the ascent
of the air, but it has a still greater effect in prolonging their duration when
formed. (Plate XXXIX. Fig. 532.)

In order to determine the forms of the surfaces of liquids in the cases
which most commonly occur, it is necessary to examine how they are
affected by the action of other liquids, and of solids of different descrip-
tions. We may form some idea of the effects of this mutual action, by
neglecting the force of repulsion, as Clairaut has done, and attending only
to that of cohesion. Supposing the horizontal surface of a liquid to be in
contact with a vertical plane surface of a solid of half the attractive power,
it will remain at rest in consequence of the equilibrium of attractions ; for
the particles situated exactly at the junction of the surfaces may be con-
sidered as actuated by three forces ; one deduced from the effect of the
liquid, the other two from that of the two equal portions of the solid above
and below the surface of the fluid ; and it may be shown that the combi-
nation of these three forces will produce a joint result in the direction of
gravity ; consequently the direction of the surface must remain the same
as when it is subjected to the force of gravity alone, since the surface of
every fluid at rest must be perpendicular to the joint direction of all the
forces acting on it. But if the attractive power of the solid be more than
half as great as that of the liquid, the result of the forces will be inclined
towards the solid, and the surface of the liquid, in order to be perpendicular
to it, must be more elevated at the side of the vessel than elsewhere, and
therefore concave ; consequently the fluid must ascend until it arrives at a
position capable of affording an equilibrium in this manner: if, on the
contrary, the attractive power of the solid be weaker, the liquid will
descend, and its surface will be convex. (Plate XXXIX. Fig. 533.)

This mode of reasoning is, however, by no means sufficient to explain
all the phenomena, for it may be inferred from it, that when the attractive
power of the solid is greater or less than half that of the liquid, the surface
of the liquid must, at its origin, be in the same direction with that of the
solid, instead of forming an angle with it, as it often does in reality. But
the difficulty may be removed by reverting to the general principle of
superficial cohesion, and by comparing the common surface of the liquid

ON COHESION. 477

and solid with the surface of a single liquid, of which the attractive power
is equal only to the difference of the respective powers of the substances
concerned. In this manner it may be shown, that if the attractive power
of the solid be equal to that of the liquid, or still greater, it will be wetted
by the liquid, which will rise until its surface acquires the same direction
with that of the solid ; and that, in other cases, the angle of contact will be
greater, in proportion as the solid is less attractive. A similar comparison
is also equally applicable to the contact of two liquids of different densities.
The magnitude of the superficial cohesion or contractility of a liquid may
be expressed, for a certain extent, by a certain weight ; thus every inch
of the surface of water is stretched each way by a force equal to the weight
of the hundredth part of a cubic inch of water, or to two grains and a half :
and for each inch of the surface of mercury, the force is equivalent to 17
grains, which is the weight of ^^-y of a cubic inch of mercury. Thus if a
solid of any form, of which the surfaces are vertical, and which is capable
of being wetted by either of these fluids, be immersed into a reservoir con-
taining it, the fluid will be elevated around it to such a height that 2^ or
17 grains [respectively], for each inch of the circumference of the solid,
will remain above the general level of the reservoir, the surface assuming
nearly the same form as a very long and slender elastic rod, fixed horizon-
tally at one end, and bearing a large weight at the other. (Plate XXXIX.
Fig. 534.)

The elevation of the summit of an extended surface of w^ater, in contact
with the flat and upright surface of a solid which is wetted by it, is one
seventh of an inch : but when two such surfaces, for instance, two plates of
glass, are brought near to each other, the elevation of the water between
them must be greater than this, in order that each inch of the line of con-
tact may support its proper weight : thus, if the distance were one fiftieth
of an inch, the elevation would be a whole inch ; and if the distance were
smaller than this, the elevation would be greater in the same proportion ;
so that when two plates are placed in such a manner as to touch each other
at one of their upright edges, the outline of the water raised between them
assumes the form of a hyperbola. (Plate XXXIX. Fig. 535.)

The weight supported by the cohesion of the water in a tube may be
determined, in a similar manner, from the extent of the circumference ; the
height being an inch in a tube one twenty fifth of an inch in diameter, or
as much greater as the diameter of the tube is smaller : and in a tube
wetted with mercury the height would be half as great. It is obvious that
if the lower part of the tube be either contracted or dilated, the height of
the fluid will remain unaltered, while its weight may be varied without
limit ; for the hydrostatical pressure on the surface is the same, in both
these cases, as if the diameter of the tube were equal throughout its length.
(Plate XXXIX. Fig. 536.)

The attractive force of glass to mercury is less than half as great as the
mutual attraction of the particles of mercury, and the surface of mercury
in a dense glass vessel becomes, therefore, convex and depressed ; the angle
of contact being about 140°, and the depression one 17th of an inch.
Between two plates of glass, the depression of mercury is an inch when

478 LECTURE L.

their distance is -r-f,-, and in a tube, when its diameter is 7!T of an inch.
(Plate XXXIX. Fig. 537, 538.)

A liquid may also adhere to a horizontal surface which is gradually
raised from it, until the hydrostatical pressure becomes sufficient to over-
power the cohesion of its superficial parts ; the internal part of the fluid
being usually raised, not immediately by the force of cohesion, but by the
pressure of the atmosphere. The solid bears the whole weight of the
liquid, which is elevated above the surface ; and when the surface is
perfectly wetted, this weight is equal, at the moment of separation, to the
hydrostatical pressure, or rather suction, corresponding to the height ; but
in other cases the weight may be somewhat greater than the hydrostatical
pressure on the surface of the solid, on account of the elevation which
surrounds the body, and which is not compensated by the excavation
immediately under it. A surface thus raised from water will elevate it to
the height of one fifth of an inch, and will require a force of 50| grains
for each square inch, in order to overcome the apparent attraction of the
water ; and for mercury the utmost height is about one seventh of an inch.
(Plate XXXIX. Fig. 539, 540.)

A -detached portion of a liquid may stand on any surface which it is
not capable of wetting, at a height which is different according to its
magnitude and to the attraction of the surface. If the drop is very small,
its form may be nearly spherical ; but when its extent becomes consider-
able, its height must always be less than that at which the liquid would
separate from a horizontal surface ; and it will approach the nearer to this
limit, as its attraction to the surface on which it stands is weaker. Thus
a wide portion of mercury stands on glass at the height of T^ of an inch,
and on paper nearly at -f ; and a portion of water will stand on a cabbage
leaf, or on a table strewed with the seeds of lycopodium, nearly at the
height of one fifth of an inch. (Plate XXXIX. Fig. 541.)

For the operation of a powder like lycopodium, it appears to be only
necessary that it should possess a weaker power of attraction than water,
and should, therefore, be incapable of being readily wetted by it : each
particle of the powder being then but partially in contact with the water,
will project beyond its surface, and prevent its coming into contact with
any of the surrounding bodies, while the surface assumes such a curvature
as is sufficient to withstand the pressure of the internal parts. (Plate
XXXIX. Fig. 542.)

When a dry and light substance of any kind is placed on the surface of
water, its weight is not sufficient to bring it within the distance at which
cohesion commences, and it floats surrounded by a slight depression. Any
substance of this kind, or any other substance surrounded by a depression,
as a ball of glass or iron floating on mercury, appears to be attracted by
another similar substance in its neighbourhood ; for the depression between
the two substances is increased, and the pressure of the fluid on that side
is consequently lessened, so that they are urged together, by a force which
varies inversely as the square of the distance. And in the same manner,
when two bodies, surrounded by an elevation, approach each other, they
exhibit an attractive force of a similar nature, the pressure of the atmo-

ON COHESION. 479

sphere being diminished by the weight of the water, which is raised between
them to a greater height than on the opposite sides. But when a body,
surrounded by a depression, approaches another, which is surrounded by
an elevation, they seem to repel each other, the pressure of the water
urging the one, and that of the atmosphere the other, in opposite directions.
(Plate XXXIX. Fig. 543.)

If two smooth plates of any kind are perfectly wetted by a fluid, and
brought into contact, they exhibit an appearance of cohesion, which is so
much the greater as the quantity of fluid is smaller : if we attempt to
separate them, the fluid is drawn inwards, so as to have its surface made
concave, and it resists the separation of the plates with a certain force,
which acts with a hydrostatic advantage so much the greater, as their
distance is smaller, and hence produces the appearance of a cohesion
varying in proportion to the distance. (Plate XXXIX. Fig. 544.)

Supposing the two plates to be separated at one end, and the fluid
between them to assume the form of a drop, one of the marginal surfaces
of the drop, being narrower than the other, will act. with a greater advan-
tage, like a tube of smaller diameter, and will tend to draw the drop
towards it ; and the apparent attraction towards the line of contact of the
glasses will increase in proportion as the square of the distance decreases.
This result was experimentally observed almost a century ago, but it has
been usually explained on mistaken grounds. (Plate XXXIX. Fig. 545.)

The attractive power of water being greater than that of oils, a small
portion of oil thrown on water is caused to spread on it with great rapidity
by means of the force of cohesion ; for it does not appear that any want
of chemical affinity between the substances concerned, diminishes their
cohesive power ; water readily adheres to tallow when solid, and probably
essential oils would adhere still more readily to ice. There is, however
some difficulty in understanding how these oils can so suddenly come
within the limit of the cohesive force of water, while the drops of water
themselves sometimes remain for a few seconds beyond it.

A sponge affords us a familiar instance of the application of capillary
attraction to useful purposes ; it is well known that in order to its speedy
operation, it requires to be previously moistened, by the assistance of a
little pressure, otherwise it exhibits the same appearance of repulsion that
is observable in many other cases where the contact is imperfect. The
absorption of moisture by sugar depends on the same principle, and
here the tubes are so minute, that the height of ascent appears to be almost
unlimited.

The magnitude of the cohesion between fluids and solids, as well as of
the particles of fluids with each other, is more directly shown by an ex-
periment on the continuance of a column of mercury, in the tube of a
barometer, at a height considerably greater than that at which it usually
stands, on account of the pressure of the atmosphere. If the mercury has
been well boiled in the tube, it may be made to remain in contact with the
closed end, at the height of 70 inches or more ; and by agitation only it
may be made to cohere so strongly as to occupy the whole length of the
tube of a common barometer, which is several inches more than the height

480 LECTURE L.

at which the pressure of the atmosphere sustains it. A small siphon may
also convey mercury from one vessel into another in the vacuum of an
air pump : and in hoth these cases it is ohvious that no other force than
cohesion can retain the upper surface of the mercury in contact with the
glass, or its internal parts in contact with each other.

The force of cohesion may also he exerted by solid substances on other
solids, either of the same kind, or of different kinds. Thus two masses of
lead, when once united by pressure, assisted by a little friction, require a
very considerable force to separate them, and it may be shown either by
measuring this force, or by suspending the lead in the vacuum of the air
pump, that the pressure of the atmosphere is not materially concerned in
producing this appearance of cohesion, since its magnitude much exceeds
that of the atmospherical pressure. A cohesion of this kind is sometimes
of practical utility in the arts ; little ornaments of laminated silver re-
maining attached to iron or steel, with which they have been made to
cohere by the powerful pressure of a blow, so as to form one mass
with it.

The contact of two pieces of lead, although intimate enough to produce
a considerable cohesion, is by no means so complete as to unite the parts
into one mass ; the union, however, appears to be nearly of the same kind as
the common cohesion of aggregation ; and if the lead were softened into
an amalgam by the addition of mercury, the cohesion of the two masses
would become precisely the same as the internal cohesion of each mass.
Harder substances, such as marble or glass, cohere but weakly, perhaps
because their surfaces are never so perfectly adjusted to each other as to
touch throughout. The interposition of a fluid usually increases the
apparent attraction of such substances, but this circumstance has already
been explained from the effect of the capillary contraction of its surface ;
and when the substances are wholly immersed in a fluid, the cohesion is
little, if at all, increased.

The immediate cause of solidity, as distinguished from liquidity, is the
lateral adhesion of the particles to each other, to which the degree of hard-
ness or solidity is always proportional. This adhesion prevents any change
of the relative situation of the particles, so that they cannot be withdrawn
from their places, without experiencing a considerable resistance from the
force of cohesion, while those of liquids may remain equally in contact
with the neighbouring particles, notwithstanding their change of form.
When a perfect solid is extended or compressed, the particles, being retained
in their situations by the force of lateral adhesion, can only approach
directly to each other, or be withdrawn further from each other, and the
resistance is nearly the same as if the same substance, in a fluid state, were
inclosed in an unalterable vessel, and forcibly compressed or dilated. Thus
the resistance of ice to extension or compression is found by experiment
to differ very little from that of water contained in a vessel ; and the same
effect may be produced even when the solidity is not the most perfect
which the substance admits ; for the immediate resistance of iron or steel
to flexure is the same whether it may be harder or softer. It often happens,
however, that the magnitude of the lateral adhesion is so much limited as

ON COHESION. 481

to allow a greater facility of extension or compression, and it may yet
retain a power of restoring the bodies to their original form by its reaction.
This force may even be the principal or perhaps the only source of the
body's elasticity : thus when a piece of elastic gum is extended, the mean
distance of its particles is not materially increased, for it is said to become
rather more than less dense during its extension ; consequently the change
of form is rather to be attributed to a displacement of the particles, than to
their separation to a greater distance from each other, and the resistance
must be derived from the lateral adhesion only : some other substances
also, approaching more nearly to the nature of liquids, may be extended to
many times their original length, with a resistance continually increasing ;
and in such cases there can scarcely be any material change of the specific
gravity of these substances. Professor Robison has mentioned the juice
of bryony as affording a remarkable instance of such a viscidity.

It is probable that the immediate cause of the lateral adhesion of solids
is a symmetrical arrangement of their constituent parts : it is certain that
almost all bodies are disposed, in becoming solid, to assume the form of
crystals, which evidently indicates the existence of such an arrangement ;
and all the hardest bodies in nature are of a crystalline form. It appears,
therefore, consistent both with reason and with experience to suppose that
a crystallization more or less perfect is the universal cause of solidity. We
may imagine that when the particles of matter are disposed without any
order, they can afford no strong resistance to a motion in any direction,
but when they are regularly placed in certain situations with respect to
each other, any change of form must displace them in such a manner, as
to increase the distance of a whole rank at once ; and hence they may be
enabled to cooperate in resisting such a change. Any inequality of tension
in a particular part of a solid is also probably so far the cause of hardness,
as it tends to increase the strength of union of any part of a series of par-
ticles which must be displaced by a change of form.

The immediate resistance of a solid to extension or compression is most
properly called its elasticity ; although this term has sometimes been used
to denote a facility of extension or compression, arising from the weakness
of this resistance. A practical mode of estimating the force of elasticity
has already been explained, and according to the simplest statement of the
nature of cohesion and repulsion, the weight of the modulus of elasticity
is the measure of the actual magnitude of each of these forces ; and it fol-
lows that an additional pressure, equal to that of the modulus, would
double the force of cohesion, and require the particles to be reduced to half
their distance in order that the repulsion might balance it ; and in the
same manner an extending force equal to the weight of half the modulus
would reduce the force of cohesion to one half, and extend the substance to
twice its dimensions. But, if, as there is some reason to suppose, the
mutual repulsion of the particles of solids varies a little more rapidly than
their distance, the modulus of elasticity will be a little greater than the
true measure of the whole.eohesive and repulsive force : this difference will
.not,* however, affect the truth of our calculations respecting the properties

2i

482 LECTURE L.

of elastic bodies, founded on the magnitude of the modulus as already-
determined.

The stiffness of a solid is measured by its immediate resistance to any
force tending to change its form ; in this sense, if the force be applied so as
to extend or to compress it, or to overcome its lateral adhesion by the effect
which we have formerly called detrusion, the primitive elasticity and
rigidity of the substance, together with its magnitude, will determine its
stiffness : but if the force be otherwise applied, so as to produce flexure or
torsion, the form of the body must also be taken into the calculation, in the
manner which has already been explained in the lecture on passive strength.
The stiffness of a body with respect to any longitudinal force is directly
as its transverse section, and inversely as its length ; for the same force
will compress or extend a rod 100 yards long so as to change its length an
inch, that will produce a change of only half an inch in a rod 50 yards
long. We have seen that the space through which a body may be extended
or compressed, without any permanent alteration of form, constitutes its
toughness : that its strength, or the ultimate resistance which it affords,
depends on the joint magnitude of its toughness and elasticity or stiff-
ness, and that its resilience, or the power of overcoming the energy or
impetus of a body in motion, is proportional to the strength and toughness
conjointly.

Softness, or want of solidity, is in general accompanied by a proportional
susceptibility of permanent alteration of form without fracture ; some-
times, however, from a want of cohesion, a soft body is at the same time
brittle. Soft substances, which are capable of direct extension to a consi-
derable degree are called viscous or tenacious ; of these, birdlime, sealing
wax, and glass sufficiently heated, are some of the most remarkable.
Harder substances which have the same property are called ductile, and
when the alteration is made by percussion and compression, they are
termed malleable. Of all substances gold is perhaps the most ductile : the
thinness of leaf gold and of the gilding of silver wire has already been men-
tioned ; and it is said that a single grain of gold has been drawn into a
wire 500 yards in length, and consequently little more than -^Vs- of an
inch in diameter. The ductility or tenacity of a spider's web is of a dif-
ferent kind, it is particularly shown by its capability of being twisted,
almost without limit, and of accommodating itself to its new position with-
out any effort to untwist.

With respect to the ultimate agent by which the effects of cohesion are
produced, if it is allowable to seek for any other agent than a fundamental
property of matter, it has already been observed, that appearances extremely
similar might be derived from the pressure of a universal medium of great
elasticity ; and we see some effects, so nearly resembling them, which are
unquestionably produced by the pressure of the atmosphere, that we can
scarcely avoid suspecting that there must be some analogy in the causes.
Two plates of metal, which cohere enough to support each other in the
open air, will often separate in a vacuum : when a boy draws along a stone
by a piece of wet leather, the pressure of the atmosphere appears to be

ON COHESION. 483

materially concerned. The well known experiment, of the two exhausted
hemispheres of Magdeburg, affords a still more striking instance of appa-
rent cohesion derived from atmospherical pressure ; and if we place between
them a thick ring of elastic gum, we may represent the natural equilibrium
between the forces of cohesion and of repulsion ; for the ring would resist
any small additional pressure with the same force as would be required
for separating the hemispheres so far as to allow it to expand in an equal
degree : and at a certain point the ring would expand no more ; the air
would be admitted, and the cohesion destroyed, in the same manner as
when a solid of any kind is torn asunder. But all suppositions founded on
these analogies must be considered as merely conjectural ; and our know-
ledge of every thing which relates to the intimate constitution of matter,
partly from the intricacy of the subject, and partly for want of sufficient
experiments, is at present in a state of great uncertainty and imperfection.
One of the most powerful agents, in changing and modifying the forms
of matter, is the operation of heat, by which the states of solidity, liquidity,
and elastic fluidity are often produced in succession ; and the investigation
of the nature and effects of heat will constitute the subject of the two next
lectures.

LECT. L.— ADDITIONAL AUTHORITIES.

Cohesion in general. — Desaguliers on the Cohesion of Lead, Ph. Tr. 1725, p. 345.
Hambergus, De Cohesione, 4to, Jena, 1732. Winckler, De Causis Conjunctionis,
4to, Leipz. 1736. Felice, do. 4to, 1757.

Capillary Action.— Fabri, Dialogi Physici, Lyons, 1669. Boyle, Ph. Tr. 1676,

?. 775. Hauksbee on the Effect of Capillary Tubes remaining in a Vacuum, ibid.
706, p. 2223; on Different Points, ibid. 1709, p. 258 ; 1711, p. 395; 1712, pp. 413,
539 ; 1713, p. 151. Taylor on the Ascent of Water between Two Plates, ibid. 1712,
p. 538 ; on Attraction of Wood to Water, ibid. 1721, p. 204. Jurin, ibid. 1718,
p. 739; 1719, p. 1083. Bulfinger, Com. Petr. ii. 233, iii. 281. Musschenbroek,
Diss. Phys. pp. 271, 334. Clairaut, Fig. de la Terre, 1743. GeUert on Melted Lead
in Tubes, Com. Petr. xii. 293 ; on Prismatic Tubes, ibid. xii. 302. Segner on the
Surfaces of Fluids, Com. Gott. 1751, i. 301. Tetens, De Fluxu Siphonis in Vacuo,
4to, Biitzow, 1763. Lalande, sur la Cause de 1'Elevation des Liqueurs, 12mo, Par.
1770. Morveau on the Attraction of Water and Oils, Jour, de Phy. i. 172, 460.
Lord C. Cavendish's Table of the Depression of Mercury, Ph. Tr. 1776, p. 382.
Achard on the Adherence of Solids to Fluids, Hist, et Mem. de Berlin, 1776, p. 149.
Schriften, i. 355. Dutour, Jour, de Physique, xi. 127, xiii. Supp. 357, xiv. 216,
xv. 46, 234, xvi. 85, xix. 137, 287. Besile, ibid, xxviii. 171, xxix. 287, 339, xxx.
125. Monge on Apparent Attractions and Repulsions, Hist, et Mem. 1787, p. 506,
Nich. Jour. iii. 269. Bennet, Manch. Mem. iii. 116. Leslie, Ph. Mag. xiv. 193.
Young on the Cohesion of Fluids, Ph. Tr. 1805, p. 65. Laplace, Mec. Cel. Sup-
plem., and Bullet, de la Soc. Philom. 1819, p. 122. Edin. Encyc. art. Capillary
Attraction. Gauss, Principia Generalia Theorise Figurse Fluid, in Statu ^Equilib.
Gott. 1830. Poisson, Mem. de 1'Acad. ix. Theorie de 1' Action CapUlaire, 4to,
1831. Link, Pogg. Annalen, 1832, xxv. 270, xxvii. 193, xxix. 404.

2i2

484

LECTURE LI.

ON THE SOURCES AND EFFECTS OF HEAT.

IT may appear doubtful to some whether the subject of heat belongs
most properly to mechanical or to chemical philosophy. Its influence in
chemistry is unquestionable and indispensable ; but its mechanical effects
are no less remarkable : it could not therefore with propriety be omitted
either in a course of chemical or of physical lectures, especially by those
who are persuaded that what we call heat is, in its intimate nature, rather
a mechanical affection of matter than a peculiar substance. We shall
first inquire into the nature of the principal sources of heat, and next into
the mode of its communication, and its most common effects, whether
temporary or permanent : the measures of heat, and the most probable
opinions respecting its nature, will afterwards be separately considered.

Heat is an influence capable of affecting our nerves in general with the
peculiar sensation which bears its name, and of which the diminution pro-
duces the sensation denominated cold. Any considerable increase of heat
gives us the idea of positive warmth or hotness, and its diminution excites
the idea of positive cold. Both these ideas are simple, and each of them
might be derived either from an increase or from a diminution of a positive
quality : but there are many reasons for supposing heat to be the positive
quality, and cold the diminution or absence of that quality ; although we
have no more experience of the total absence of heat, than of its greatest
possible accumulation, which might be called the total absence of cold.
Our organs furnish us, in some cases, with very delicate tests of any
increase or diminution of heat ; but it is more usually recognised by the
enlargement of bulk, generally produced in those bodies to which heat is
attached in an increased quantity, and the contraction of those from which
it is subtracted.

The simplest modes of exciting heat appear to be the compression of
elastic fluids, and the collision or friction of solid bodies ; although a more
usual and a more powerful source of heat is found in various chemical
combinations and decompositions, which are produced by the peculiar
elective attractions of different substances for each other, or from the influ-
ence of the solar rays, which are probably emitted in consequence of the
chemical processes that continually take place at the surface of the sun.

The effects of the condensation and rarefaction of elastic fluids are
shewn by the condenser and the air pump ; when an exhaustion is made
with rapidity, the thermometer, placed in the receiver of the air pump,
usually sinks a degree or two ; and when the air is readmitted abruptly
into a partial vacuum, the sudden condensation of the rarefied air raises
the mercury : and a similar elevation of temperature is produced by the
operation of the condenser. Much of this heat is soon dissipated, but by

ON THE SOURCES AND EFFECTS OF HEAT. 485

observing the velocity with which the thermometer rises, Mr. Dalton has
estimated that air, compressed to half its dimensions, has its temperature
elevated about 50 degrees of Fahrenheit ; and some of his experiments
indicate, when accurately examined, a still greater change.* For the
present we may define the sense of the term degree, in Fahrenheit's scale,
as corresponding to an expansion of a portion of mercury amounting to
one ten thousandth part of its bulk ; and a degree of Reaumur originally
corresponded to an expansion of a weak spirit of wine, amounting to one
thousandth part of its bulk. It may be inferred from the velocity of
sound, supposing that the excess of its velocity above the common calcula-
tion is wholly derived from the heat and cold produced by condensation
and expansion, that a condensation amounting to -r^- of the bulk of any
portion of air will raise its temperature one degree of Fahrenheit. When
air is very rapidly compressed in the condenser of an air gun, it is some-
times so much heated as actually to set on fire a small portion of tow,
placed near the end of the barrel, t

The production of heat by friction is too well known to require an
experimental proof; but Count Rumford has taken particular pains to
ascertain every circumstance which can be supposed to be concerned in
the operation of this cause ; and the results of his experiments are so
striking, that they deserve to be briefly related. He took a cannon, not
yet bored, having a projection of two feet beyond its muzzle, a part which
is usually cast with the piece, in order to insure the solidity of the metal
throughout, by the pressure which its weight occasions. This piece was
reduced to the form of a cylinder, joined to the cannon by a smaller neck,
and a large hole was bored in it : the whole cannon was then made to
revolve on its axis by means of the force of horses, while a blunt steel
borer was pressed against the bottom of the hollow cylinder, by a force
equal to about 10,000 pounds avoirdupois ; the surface of contact of the
borer with the bottom of the cylinder being about 2 square inches. This
apparatus was wrapped up in flannel, when its temperature was about 60°.
In half an hour, when the cylinder had made 960 turns, the horses being
stopped, a mercurial thermometer was introduced into a perforation in the
bottom of the cylinder, extending from the side to the axis, and it stood
at 130°, which Count Rumford considers as expressing very nearly the
mean temperature of the cylinder. The dust or scales, abraded by the
borer, weighed only 837 grains, or about -^ of the whole weight of the
cylinder. In another experiment, the cylinder was surrounded by a tight
deal box, fitted writh collars of leather, so as to allow it to revolve freely,
and the interval between the cylinder and the box was filled with 19
pounds of cold water, which was excluded from the bore of the cylinder
by oiled leathers fixed on the borer ; and after two hours and a half, the
water was made to boil. Hence Count Rumford calculates that the heat

* Manch. Mem. v. 515.

f On the production of heat by condensation, and cold by rarefaction, see Dar-
win, Ph. Tr. 1788, p. 43; Pictet, Jour, de Phy. xlvii. 186; Baillet, ibid, xlviii.
166 ; Ph. Mag. xiv. 363.

486 LECTURE LI.

produced in this manner, by the operation of friction, was equal to that of
9 wax candles, each three quarters of an inch in diameter, continuing to
burn for the same time.*

A still more rapid increase of temperature may be obtained, where the
relative velocity of the bodies is more considerable, or where they strike
each other with violence. Thus a soft nail may be so heated, by three or
four blows of a hammer, that we may light a match with it ;t and by
continuing the operation, it may be made red hot : two pieces of wood may
also be set on fire by means of a lathe. When a waggon takes fire, for
want of having its wheels properly greased, the friction is probably
increased by the tenacity of the hardened tar, which perhaps becomes the
more combustible as it dries.

One of the most remarkable circumstances, attending the production of
heat by friction, is the discovery of Professor Pictet, that it is often much
more powerfully excited by soft substances than by harder ones. In
making some experiments in a vacuum, in order to examine how far the
presence of air might be concerned in the effects of friction, he accidentally
interposed some cotton between the bulb of his thermometer and the cup,
which was subjected to the friction of various substances as it revolved ;
and he found that the soft filaments of the cotton excited much more heat,
than any other of the substances employed.^

The chemical production of heat is of greater practical importance than
its mechanical excitation ; but by what means chemical changes operate in
exciting heat, we cannot attempt to determine. There is certainly no
general law of composition or decomposition that can be applied to all such
cases : most commonly heat is produced when oxygen exchanges an aeri-
form for a solid state, or enters into a new combination, and still remains
elastic ; but in the case of gunpowder, heat is disengaged while an elastic
fluid is produced from a solid ; and in some other cases the oxygenous
principle is wholly unconcerned. It appears on the whole, that however
heat may be excited, the corpuscular powers of cohesion and repulsion are
always disturbed and called into action, their equilibrium being destroyed
and again restored, whether by mechanical or by chemical means. A wax
candle, f of an inch in diameter, loses a grain of its weight in 37 seconds,
and consumes about three grains, or 9 cubic inches, of oxygen gas,
producing heat enough to raise the temperature of about 15,000 grains of
water a single degree. According to the experiments of Mr. Lavoisier and
Mr. Laplace, the combustion of ten grains of phosphorus requires the con-
sumption of 15 grains of oxygen, the combustion of ten grains of charcoal
26, and of hydrogen gas 56 ; and by the heat produced during the combus-
tion of a pound of phosphorus, 100 pounds of ice may be melted, during
that of a pound of charcoal 96£, of hydrogen gas 295£, of wax 133, and of
olive oil 149 ; and during the deflagration of a pound of nitre with about

* Ph. Tr. 1798, p. 80. Essays, ii. IX. Nich. Jour. ii. 106. See also Haldot, ibid,
xxvi. 30.

t Mem. d'Arcueil, ii. 441.

J Essais de Physique, Geneve, 1790.

ON THE SOURCES AND EFFECTS OF HEAT. 487

one sixth part of its weight of charcoal, twelve pounds of ice may be
melted.*

The manner in which heat, when excited or extricated by any of these
means, passes from one body to another, requires to be very particularly
examined. We shall find that this communication happens in one or both
of two ways, by contact, or by radiation ; and that it may also differ both
with respect to the quantity of heat concerned, and to the time occupied
by the process. Whatever heat may be, we may safely conclude that in
substances of the same kind, at the same temperature or apparent degree
of warmth or coldness, its quantity must be proportional to the mass or
weight ; for instance, that a quart of the water of a given cistern contains
twice as much heat as a pint ; and where this is true of the different parts
of any substance, they must remain in equilibrium with respect to heat.
But if two equal portions of the same substance, containing different
quantities of heat, be in contact, they will affect each other in such a
manner as to have their temperatures equalised, and the more rapidly as
the contact is more perfect. Thus, if two portions of a fluid at different
temperatures be mixed together, they will acquire immediately an inter-
mediate temperature ; and when two solids are in contact, the quantity of
heat, communicated by the hotter to the colder in a given time, is nearly
proportional to the difference of the temperatures. Hence it would follow,
that they could never become precisely of the same temperature in any
finite time ; but in fact the difference of temperature is rendered, in a mode-
rate time, too small to be perceptible. The nature of the substances
concerned has also a material effect on the velocity with which heat is
communicated through their internal parts ; metallic bodies in general
conduct it the most readily, earthy and vitreous bodies the least ; but the
various metals possess this power in different degrees ; silver and copper
conduct heat more rapidly than iron, and platina transmits it but very
slowly. Professor Pictet supposes that heat ascends within solid bodies
more readily than it descends ; but the effect of the air remaining in the
imperfect vacuum of the air pump may be sufficient to explain his experi-
ments ; the difference of temperature producing an ascending current
in the neighbourhood of the heated body, by means of which the cold ail-
continually approaches its lower parts, and carries the heat upwards : and
it has been found that the rarefaction of air does not by any means
diminish its power of conducting heat, in proportion to the diminution of
its density.

Count Rumford's experiments t have shown that all fluids are very
imperfect conductors of heat by immediate contact, although it is scarcely
credible that they can be absolutely nonconductors ; but heat is usually
communicated between different portions of the same fluid, almost entirely

* On combustion, consult Hooke, Micographia, p. 103. Lavoisier and Laplace, Hist,
et Mem. 1780, p. 355, H. 3. Rumford, Nich. Jour, xxxii. 105 ; xxxiv. 319 ; xxxv.
95. Davy, Ph. Tr. 1817. Sym, Annals of Ph. viii. 321. Davies, ibid. (2nd series),
x. 447. Dobereiner, Schweigger's Jahrbuch, iv. 91 ; viii. 321.

t Ph. Tr. 1786, p. 273 ; 1792, p. 48. Essays, Lond. 1796. See also Dalton,
Manch. Mem. v. 373. Thomson, Nich. Jour. iv. 529 ; 8vo, i. 81. Murray, ibid,
i. 165, 242. Trail, ibid. xii. 133. Despretz, Comptes Rendus, vii. 933.

488 LECTURE LI.

by the mixture of their particles : hence a fluid heated on its surface
transmits the heat very slowly downwards, since the parts which are first
heated, being rendered specifically lighter, retain their situation above the
colder and heavier parts ; while, on the contrary, any cause of heat,
applied at the bottom of a vessel, very soon reduces all its contents to a
uniform temperature. It appears also, from some late experiments, that
the immediate transmission of heat within the internal parts of solids is
much slower than has commonly been supposed ; and it has been found
almost impossible to keep a thermometer, at the centre of a large and solid
globe of metal, at the same temperature with that of its superficial parts.*

Besides the communication of heat by contact, it is usually, if not
always, emitted from the surfaces of bodies in the form of radiant heat,
which is thrown off in all directions, wherever it meets no obstacle from a
substance impervious to it, and is propagated nearly in the same manner
as light, and probably with the same velocity, without producing any
permanent effect on the temperature of the medium transmitting it. Thus,
a thermometer, suspended by a fine thread under the receiver of an air
pump, or in the Torricellian vacuum, will continue to vary its temperature
with that of the surrounding bodies : and in this case the whole of the heat
must be communicated by radiation. Mr. Leslie has discovered that the
quantity of heat thus emitted depends not only on the temperature, but
also on the nature of the surface concerned, a polished surface of tin
emitting only TW> or less than one eighth part as much, as the same surface
blackened. A surface of tin scraped with a file in one direction has its
powers of radiation more than doubled ; but by crossing the scratches,
they are reduced nearly to their original state ; and a coating of isinglass,
resin, or writing paper, or a glassy surface of any kind, produces an effect
nearly approaching to that of black paint. This radiation from a heated
surface, like that of light, takes place in almost equal degrees in every
direction ; and its magnitude is nearly independent of the nature of the
fluid in contact with the surface, provided however that it be an elastic
fluid ; for water does not seem to transmit every kind of radiant heat with
freedom. It appears that the radiant heat emitted by a surface of glass, or
of black paint, is about one third greater than that which is at the same
time carried off by the atmospheric air ; but that the radiation from a
metallic surface is only one sixth of that which the air receives. Mr.
Leslie has also found that the same surfaces which emit heat the most
freely, are also the readiest to receive it from the radiation of other bodies.t

The solar heat radiates freely through air, glass, water, ice, and many
other transparent mediums, without producing any sensible effect on their
temperatures, and even when it is concentrated by the effect of a burning
mirror, it scarcely affects the air through which it passes, and other trans-
parent mediums but little. But the heat of a fire warms a piece of common

* The law of conduction is not yet correctly defined. See Kelland, on the pre-
sent State of our Knowledge of the Laws of Conduction of Heat, Rep. of Brit.
Ass. 1841. The law of radiation in vacua has been determined by MM. Dulong and
Petit ; their experiments will be found in the Annales de Chimie, vii. 225, &c.
Thomson's Annals, vol. xiii. ; or in the art. Heat, in the Encyclop. Metr.

f Inquiry into the Nature and Propagation of Heat, Lond. 1804.

ON THE SOURCES AND EFFECTS OF HEAT. 489

glass very rapidly, and its further progress is almost entirely interrupted
by the glass, although probably a certain portion still continues to accom-
pany the light in all cases. Hence a screen of glass is sometimes practically
convenient for allowing us the sight of a fire, and protecting us at the same
time from its too great heat. Mr. Lambert* showed that culinary heat
was much more strongly reflected by mirrors of metal than of glass, although
little difference was observable in the quantity of light, and he very justly
attributed this difference to the interception of a part of the heat by the glass,
which operated with respect to it like an opaque substance, although it trans-
mitted the light with freedom. Opaque substances in general appear to be
wholly impervious to radiating heat of all kinds ; but Dr. Herschelt has found
that dark red glass, which transmits a very small portion of light only, suf-
fers some kinds of radiant heat to pass through it with very little interruption.

In other respects, radiating heat is subject, in all cases, to the optical
laws which govern the reflection and refraction of light. Dr. Hoffmann
appears to have been the first that collected the invisible heat of a stove
into a focus by the reflection of one or more concave mirrors.^ Buffon,
Saussure, Pictet, and Mr. King, made afterwards similar experiments on
the heat of a plate of iron and of a vessel of boiling water. Mr. Pictet, as
well as Hoffmann, employed two mirrors facing each other ; and by means
of this arrangement the experiment may be performed when the thermo-
meter is placed at a considerable distance from the heated body.

The temperature of the air, not being affected by the radiation of heat,
is probably in all respects indifferent to its emission in this manner ; and
as the rays of light cross each other freely in all possible directions, so it
appears that heat may flow in different directions through the same medium
without being interrupted ; nor does there seem to be any more reason that
a hot body should cease to emit heat while it is receiving heat from another
body, than that a luminous body should cease to afford light when another
body shines on it. This continual interchange of heat, constituting in
common cases a kind of equilibrium of motion, appears to have been first
suggested by Mr. Prevost,§ as an explanation of an experiment on the
reflection of cold, revived by Mr. Pictet, but originally made some centuries
before, by Plempius, and by the Academicians del Cimento. A thermo-
meter, for example, must be supposed to retain its temperature by means
of the continual accession of radiant heat from the surrounding bodies,
supplying the place of that which is continually thrown off in all direc-
tions towards those bodies. Supposing the thermometer to be placed near
the focus of a metallic speculum, not much less than a hemisphere, about
one half of the heat, which the thermometer would otherwise have received
from the surrounding bodies, must be intercepted by the mirror, which,
being metallic, emits itself but little radiant heat, but reflects, notwith-
standing, an equal quantity of heat, from the objects on the opposite side,
so that the temperature of the thermometer remains unaltered. But all
the heat, which falls on the thermometer from the mirror, must have passed

• * Pyrometrie, 4to, Berl. 1779. See Mariotte, Hist, et M&n. i. 223 ; Traite de la
Nat. des Couleurs, 1686.

f Ph. Tr. 1800, p. 255, &c. : Wolfe, Ph. Tr. 1769, p. 4.

§ Sur 1'Equilibre du Feu, Geneve, 1792. Du Calorique Rayonnant, Gen. 1809.

490 LECTURE LI.

through the conjugate or corresponding focus ; and if a body at the same
temperature be placed in that focus, the radiation will still be the same : but
if a substance absolutely cold were placed there, the whole of the heat before
reflected by the mirror would be intercepted, that is, almost half of that
which was received by the thermometer from the surrounding bodies ; and
if a piece of ice be put in the conjugate focus, a delicate thermometer will
instantly show its effect in depressing the temperature ; as if the cold were
absolutely reflected in the same manner as heat or light.

Dr. Herschel's experiments have shown that radiant heat consists of
various parts, which are differently refrangible, and that in general, in-
visible heat is less refrangible than light. This discovery must be allowed
to be one of the greatest that has been made since the days of Newton,
although the theories of some speculative philosophers might have led to it
a few years earlier. Dr. Herschel was occupied in determining the pro-
perties of various kinds of coloured glass, which rendered them more or less
fit for enabling the eye to view the sun through a telescope ; and for this
purpose it was necessary to inquire which of the rays would furnish the
greatest quantity of light, without subjecting the eye to the inconvenience
of unnecessary heat. He first observed that the heat became more and
more considerable as the thermometer approached the extreme red rays
in the prismatic spectrum ; and pursuing the experiment, he found not
only that the heat continued beyond the visible spectrum, but that it was
even more intense when the thermometer was at a little distance without
the limits of the spectrum, than in any point within it.* (Plate XXXIX.
Fig. 546, 547.)

Sir Henry Englefieldt has repeated these experiments with many ad-
ditional precautions, and Mr. Davy was a witness of their perfect accuracy :
the excess of heat beyond the spectrum was even considerable enough to
be ascertained by the sense of warmth occasioned by throwing it on the
hand. The skin appears, when compared with a thermometer, to have its
sensibility more adapted to the perception of radiant heat than to that of
heat imparted by contact, perhaps because a much smaller quantity of
heat is sufficient to raise the temperature of the thin cuticle very consider-
ably, than would be required in order to affect any thermometer in the
same degree.

It was first observed in Germany by Hitter, and soon afterwards hi
England by Dr. Wollaston, that the muriate of silver is blackened by
invisible rays, which extend beyond the prismatic spectrum, on the violet
side. It is therefore probable that these black or invisible rays, the
violet, blue, green, perhaps the yellow, and the red rays of light, and the
rays of invisible heat, constitute seven different degrees of the same scale,
distinguished from each other into this limited number, not by natural

* Herschel, Ph. Tr. 1800, p. 255, &c. Leslie, in Nich. Jour. iv. 244, called in
question this experiment. Landriani (Volta Lettere sull' Aria delle Paludi, 1777,
p. 136) andRochon (Recueil des Mem. 1785, p. 348) had placed the point of
greatest heat near the yellow. The matter was completely investigated by Seebeck,
Abhand. der Akad. Berlin, 1818-19, p. 305, and he found that the difference was
due to the substance of the prism : with water the point of greatest heat is in the
yellow ray ; with crown glass in the red ; and with flint-glass, beyond the red.

f Jour, of the Royal Institution, 1802, p. 202.

ON THE SOURCES AND EFFECTS OF HEAT. 491

divisions, but by their effects on our senses : and we may also conclude
that there is some similar relation between heated and luminous bodies of
different kinds.

The effects of heat, thus originating, and thus communicated, may be
divided into those which are temporary only, and those which are per-
manent. The permanent effects are principally confined to solids, but the
temporary effects are different with respect to substances in different states
of aggregation, and they also frequently comprehend a change from one
of these states to another. The effect of heat on an elastic fluid is the
simplest of all these, being merely an expansion of about one five
hundredth of its bulk for each degree of Fahrenheit that the temperature
is raised ; or an equivalent augmentation of the elasticity when the fluid
is confined to a certain space. This expansion is very nearly the same for
all gases and vapours, amounting to -^ for each degree, at the common
temperature of 56° of Fahrenheit, but at higher temperatures it is less
than 3-j-ff of the bulk of the gas, and at lower temperatures somewhat
more, being nearly the same in quantity for the same portion of the fluid
at all temperatures.

When an elastic fluid is contracted by cold within certain limits, deter-
mined by the degree of pressure to which it is exposed, as well as by the
nature of the fluid, its particles become subjected to the force of cohesion ;
they rush still nearer together, and form a liquid. Thus, when steam,
under the common atmospheric pressure, is cooled below the heat of
boiling water, it is instantly condensed, and becomes water : but with a
pressure of two atmospheres, it wotild be condensed at a temperature 36°
higher, and with the pressure of half our atmosphere only, it might
be cooled without condensation 83° lower than the common temperature
of boiling water. And similar effects take place in vapours of other kinds
at higher or lower temperatures, a double pressure producing in all cases
an equal disposition to condensation, with a depression of temperature of
between 20 and 40 degrees, and most commonly of about 35°, of Fahren-
heit. Thus, the vapour of spirit of wine is usually condensed at 175° of
Fahrenheit ; but with a double pressure it is condensed at a temperature
39° higher ; and with the pressure of half an atmosphere, at a temperature
35° lower ; and the vapour of ether, which is commonly condensed at
102°, requires a temperature 38° higher, with a double pressure, or as
much lower, with half the usual pressure. If the temperature be below
the freezing point of the liquid, the pressure being sufficiently lessened, the
vapour may still retain its elasticity, but a further reduction of temperature
or increase of pressure will convert it immediately into a solid.

The expansion of liquids by the effect of heat is much less uniform and
regular than that of elastic fluids, since it varies considerably, not only in
different liquids, but also in the same liquid at different temperatures,
being in general greater as the temperature is more elevated, and sometimes
almost in proportion to the excess of the temperature above a certain point,
at which it begins. This variation appears to be the least considerable in
mercury, although even this fluid expands a little more rapidly as it
becomes more heated ; but the expansion is always very nearly one ten

492 LECTURE LI.

thousandth for each degree : that of water is equal to this at the tem-
perature 64°, and is greater or less nearly in proportion, to the distance
from 39°, where it begins, but in high temperatures it varies less, since it
is not quite four times as great at the heat of boiling water. The expan-
sion of spirit of wine at 70° is six times as great as that of mercury : its
utmost variation is much less than that of water, although it is at least
twice as great in some parts of the scale as in others.

It has already been observed that an elevation of temperature consider-
ably diminishes the powers of cohesion and of repulsion in solid bodies :
the same is also true of liquids ; for the height to which a liquid rises in a
capillary tube is diminished somewhat less than -njV^ for each degree of
Fahrenheit that the temperature is raised, the force of superficial cohesion
being diminished both by the diminution of the immediate actions of the
particles, and by that of the distances to which they extend.

When the temperature of a liquid is so much elevated as to become
equal to that of its vapour in a state capable of sustaining the atmo-
spherical pressure, or any other pressure which may be substituted for it,
a certain portion of the liquid is converted into vapour, and the heat being
generally applied at the bottom of the vessel, the vapour rises up in
bubbles, and the effect is called boiling. The whole liquid is not converted
at once into vapour, because a certain quantity of heat appears to be
consumed by the change, and a constant supply of heat is necessary, in
order that the operation may be continued.

It is not, however, only at the boiling point that a fluid begins to be
converted into vapour : the pressure of the atmosphere is not sufficient
wholly to prevent the detachment of a certain quantity of vapour from its
surface, at a temperature which is incapable of supporting it separately in
the form of steam in the open air, and it may be thus suspended, when
mixed either with common air, or with any other elastic fluid, at the
ordinary temperature of the atmosphere. And it appears that the
quantity, which is thus suspended, bears very nearly some constant pro-
portion to the density of which the steam is capable at the given tem-
perature in a separate state, the interposition of the air either not affecting
the distance at which the cohesion would take place, or altering it equally
in all cases. It seems to be most probable that the density of vapour,
suspended in this manner in the atmosphere, is always about twice as
great, or at least half as great again, as that of steam existing inde-
pendently at the same temperature. There is, perhaps, no liquid absolutely
free from a disposition to evaporate : even mercury rises in the vacuum of
the barometer, and lines the cavity with small globules ; and it is said that
the effect of light is favourable to this slow evaporation. At whatever
temperature evaporation takes place it is always accompanied by the
production of cold ; hence it is usual in warm climates, to employ various
methods of promoting evaporation, in order to lower the temperature
of the air, to cool liquids for drinking, or even to procure ice for domestic
uses.

It appears that all aqueous fluids are contracted by cold, until we arrive
at a certain point, which is generally about 7 or 8 degrees higher than their

ON THE SOURCES AND EFFECTS OF HEAT. 493

freezing point ;* they then expand again almost in an equal degree as they
are still more cooled ; and provided that they be free from agitation, they
may remain liquid at a temperature considerably below the point at which
they usually freeze, and at which their ice always melts. Water may be
cooled in this manner to about 10° of Fahrenheit, and if it be then agitated,
or especially if a small particle of ice or snow be thrown into it, a certain
part of it will instantly congeal, and its temperature will be raised at once
to 32°, in consequence of the heat which is always produced or extricated
in the act of freezing, f In most cases, although not in all, the solid
occupies more space than the fluid : thus, it is probable that ice, when per-
fectly free from air bubbles, is at least one 16th lighter than water at the
same temperature. A saturated solution of Glauber's salts, or sulfate of
soda, in hot water, may be cooled slowly to the temperature of the atmo-
sphere, when the pressure of the air is excluded, and may be made to crys-
tallize by admitting it suddenly, the liquor becoming at the same time
warm in consequence of the heat which is extricated ; and there is no
doubt but that the congelation of water, and perhaps of all other sub-
stances, is a crystallization of the same kind.

The expansions of solid bodies appear to be more regular than those of
liquids or even of elastic fluids ; they vary little at any temperature,
although it is said that they do not always take place in their full extent
at the instant that the substance has become heated, and that a blow, or
the agitation produced when they are made to sound by the friction of the
bow of a violin, may sometimes be observed to cause them to assume the
state of equilibrium with greater rapidity. Brass expands about one hun-
dred thousandth of its length for each degree of Fahrenheit, copper and
gold a little less ; silver somewhat more ; glass and platina less than half
as much ; iron and steel about two thirds as much ; tin one third more, and
lead and zinc about half as much more. Wood and earthenware are the
least expansible of all known solids. The diminution of the elasticity of
iron and steel by the elevation of their temperature amounts to about 5 0'0 0
of the whole for each degree ; but probably various substances are variously
affected in this respect.

The liquefaction of solids, and their conversion into fluids by the opera-
tion of heat, is liable to fewer irregularities than any other of its effects ;
the change depending only on the temperature, and not being accelerated
or retarded by any accidental circumstances. When the temperature is
too low, or the pressure too small, for the existence of the substance in a
liquid form, it may still be converted into vapour, either mixed with air,
or in a separate state ; thus ice loses weight when it is exposed to a dry
frosty wind ; and camphor, benzoin, and ammonia are sublimed by heat
without being melted, although it is probable that a pressure sufficiently
strong might enable them to exist as liquids in elevated temperatures. In
all changes from solidity to liquidity or to elastic fluidity, a certain quan-

* On the point of maximum density of water, see "Waller's Trans, of the Floren-
tine Exp. p. 77. Blagden on the Congelation of Aqueous Solutions, Ph. Tr. 1788,
p. 277, Hope, Ed. Tr. v. 379. Some substances contract in freezing : see Despretz,
in Pogg. Ann. xli. 498.

f See Blagden, Ph. Tr. 1788, p. 125 ; Walker, ibid. 1788, p. 395.

494 LECTURE LI.

tity of heat disappears, except some cases in which a chemical decomposi-
tion has accompanied the change ; thus, in the detonation of gunpowder, a
large quantity of gas acquires the state of elasticity, but at the same time a
great degree of heat is produced.

The effects of the expansion of bodies by heat, and of their contraction
by cold, are observed in the frequent accidents which happen to glass and
to porcelain from a sudden change of temperature. Glass conducts heat so
slowly, that one side of a vessel may become much heated, and conse-
quently expanded, while the other remains much colder, and if the vessel
cannot readily accommodate its form to this change of proportions, it will
most commonly crack, the colder parts dividing, in consequence of their
being too much stretched by the adjoining hotter parts. Hence the thinner
the glass is, the less liable it is to crack from any sudden expansion ; and
if it be very thick, however simple its form may be, it will still crack ;
for no flexure, which it can assume, can be sufficient for the equilibrium
of the external parts without being too great for that of the parts near the
middle.

When glass in fusion is very suddenly cooled, its external parts become
solid first, and determine the magnitude of the whole piece ; while it still
remains fluid within. The internal part, as it cools, is disposed to contract
still further, but its contraction is prevented by the resistance of the
external parts, which form an arch or vault round it, so that the whole is
left in a state of constraint ; and as soon as the equilibrium is disturbed in
any one part, the whole aggregate is destroyed. Hence it becomes necessary
to anneal all glass, by placing it in an oven, where it is left to cool slowly ;
for, without this precaution, a very slight cause would destroy it. The
Bologna jars, sometimes called proofs, are small thick vessels, made for
the purpose of exhibiting this effect ; they are usually destroyed by the
impulse of a small and sharp body, for instance a single grain of sand,
dropped into them ; and a small body appears to be often more effectual
than a larger one ; perhaps because the larger one is more liable to strike
the glass with an obtuse part of its surface. In the same manner the
glass drops, sometimes called Prince Rupert's drops, which are formed by
suffering a portion of green glass in fusion to fall into water, remain in
equilibrium while they are entire ; but when the small projecting part is
broken off, the whole rushes together with great force, and rebounding by
its elasticity, exhibits the effect of an explosion. The ends of these drops
may, sometimes, but not always, be gradually ground off without destroy-
ing them, so that the concussion produced by breaking the drop seems to
be concerned in the destruction of the equilibrium.*

The tempering of metals appears to bear a considerable analogy to the
annealing of glass ; when they are made red hot, and suddenly cooled, they
acquire a great degree of hardness, which renders them proper for some
purposes, while the brittleness which accompanies it would be inconvenient
for others. By heating them again to a more moderate temperature, and

* Hooke's Microg. Bruni, Ph. Tr. 1745, p. 272. Watson, ibid. 1745, p. 505.
Lecat, ibid. 1749, p. 175. Hanow, Versuche mit den Spring-Kolbchen, 4to, Danz.
1751.

ON THE SOURCES AND EFFECTS OF HEAT. 495

suffering them to cool more gradually, they are rendered softer and more
flexible, and the more as the heat which is thus applied is the more consi-
derable. [The oxid] which forms itself on the surface of polished iron or
steel, serves as a test of the degree of heat which is applied to it, the yellow-
ish colour which it assumes indicating the first stage of tempering, the
violet the second, and the blue the last ; and if the heat be raised till the
surface becomes grey, the steel will be rendered perfectly soft. The density
of metals is also a little increased by tempering them, probably for the same
reason as water is more dense than ice. In what manner the unequal
distribution of the mutual actions of the particles of bodies contributes to
increase their hardness, cannot be very positively ascertained, although
some conjectures might be formed which would, perhaps, be in some mea-
sure explanatory of the facts : but it is safer, in the present state of our
knowledge, to be contented with tracing the analogy between these effects
in substances of different kinds, and under different circumstances, without
attempting to understand completely the immediate operation of the forces
which are concerned.

LECT. LI.— ADDITIONAL AUTHORITIES.

Treatises on Heat.—Royle, De Frigore, 4to, Lond. 1683. Petit sur le Froid et
le Chaud, 1671. Casatus, De Igne,Leipz. 1688. Muller, De Frigore, 4to, Jena,
1698. Winckler, De Frigore, 4to, Leipz. 1737. Chatelet, Diss. sur le Feu, 1744.
Bikker, De Igne, 4to, Utr. 1756. Hillary on Fire, Lond. 1760. Belgrado, Del
Calore, Parma, 1764. Herbert, De Igne, Vienn. 1773. Marat, Decouvertes sur le
Feu, 1779. Recherches sur le Feu, 1780. Magellan sur la NouveUe Theorie du
Feu, 4to, Lond. 1780. Scheele, Traite de 1'Air et du Feu (Jr.), 1781. Hopson on
Fire, 1781. Baader, Vom Warmestoff, Vienn. 1786. Carradori, Teoriadel Calore,
2 vols. Flor. 1787. Berlinghieri, 4to, Pisa, 1787. Marne, Ueber Feuer, 1787.
Weber, do. Landshut, 1788. La Serre, Theorie du Feu, Avignon, 1788. Seguin
sur les Phenomenes du Calorique. De Luc, Lettres Physiques, 5 vols. Leseme-
lier sur 1'Air et le Feu, 2 vols. Paris, 1788. Lorenz Untersuchung des Feuers,
Kopen. 1789. Mayer, Ueber die Gesetze des Warmestoffs, Erlang, 1791. Lampa-
dius, Ueber Electr. und Warme, Berl. 1793. Voigt, Theorie des Feuers, Jena,
1793. Lichtenberg in Erxleben's Naturlehre, 1794 ; Gottling, Weimar, 1794.
Harrington on Fire, 1796. Maugin, Theorie du Feu, 1800. Berthollet, Essai de
Chimie Statique, 1803. Paulet, Diss. sur le Feu, Lausanne, 1807. P. Prevost,
Traite de Calorique Rayonnante, 1809. Oersted, Ansicht der Chemischen Natur-
gesetze, Berl. 1812. Pasley's Treatise on Heat, 1820. Paulsen, De Caloris
Theoria, Gott. 1821. Fourier, Theorie Analytique de la Chaleur, 4to, 1822.
Nobili, Nuovi Trattati sopra il Calorico, &c. Modena, 1822. Bournou, Obs. et
Reflex, sur la Calorique, Paris, 1824. Library of Useful Knowledge, Cab. Cyc. &c.
Peclet, Traite de la Chaleur et de ses Applications, 2 vols. Paris, 1828. Quetelet,
Physique Populaire de la Chaleur, 12mo, Bruxelles, 1832, Bischoff, Warmelehre,
1837. Kelland, Theory of Heat, Camb. 1837.— Thomson on Heat, 1840. Gehler's
Physikalisches Worterbuch, 1841 ; and Gmelin's Handbuch, art. Warme, 1843, are
the most complete treatises on the subject.

Radiation. — Newton, Ph. Tr. 1701, p. 827; Opusc. ii. 422. Martine's Essays,
1740, p. 236. Rickmann, Nov. Com. Petr. i. 174, 195, U. 172. Lambert, Act.
Helv. ii. 172. Erxleben, Nov. Com. Gott. 1777, p. 74, and the authorities given at
p. 1636. Rumford, Ph. Tr. 1804, p. 90. Maycock, Nich. Jour. 1810, vol. xxvi.
Delaroche, Jour, de Phy. Ixxv. 201. Berard, Ann. de Ch. Ixxxv. 309. Powell,
Ph. Tr. 1825, and Report of Br. Ass. vols. i. and ix.

Conducting Powers. — Ingenhousz, Nouvelles Experiences, Par. 1789. Humboldt,
Jour, de Phy. xliii. 304. Meyer, Gren's Jour. iv. 22. Biot, Traite de Phy. vol. iv.
Despretz, Ann. de Ch. xix. 97. Delarive, ibid. xl. 91.

496

LECTURE LII.

ON THE MEASURES AND THE NATURE OF HEAT.

THE principal particulars concerning the origin, the progress, and the
effects of heat, having been noticed in the last lecture, we now proceed to
examine the most usual modes of measuring its degrees and its quantity,
and to inquire into the most probable opinions respecting its intimate
nature and its immediate operation.

The expansion of solids is measured by a pyrometer, which is calculated
for rendering the smallest change of dimensions perceptible either by
mechanical or by optical means. The first of these methods was adopted
by those who first investigated these effects ; a bar of metal being placed
in a vessel of water or of oil, which was heated by lamps, while the extre-
mities of the bar were in contact with a fixed point on one side, and on the
other with a series of levers, which multiplied the expansions so as to
render them easily observable by means of the end of the last lever, serving
as an index. But it is obvious that the expansion of the fixed part of the
instrument, and the irregular changes of temperature of the levers them-
selves, must very much interfere with the accuracy of such an instrument.
A much more correct mode of determination is to employ two microscopes,
fixed to an apparatus, which is always kept, by means of ice, at a constant
temperature, and to observe with a micrometer the change of place of
either end of the heated bar.

For such purposes, the degrees of heat may be ascertained by the
natural measures of the freezing and boiling points of certain liquids, and
of water in particular ; but for subdividing the intervals between these
points, other means must be employed. The most natural mode of deter-
mining the intermediate degrees of heat, which must be considered as the
standard for the comparison of all others, is too laborious and complicated
for common use. If we mix together equal quantities of the same liquid
at two different temperatures, they will obviously acquire an intermediate
temperature, which is the natural mean between the separate temperatures,
provided that no heat be lost or gained during the process ; and provided
that no irregularity be produced from the approach of the liquid to a state
of congelation, the existence of which might be detected by a comparison of
experiments on various liquids at the same temperatures. By repeating
the operation, we may subdivide the intervals as often as we please, or we
may mix the liquids in any other proportion, so as to obtain at once any
other point of the scale, which may afterwards be identified by a thermo-
meter of any description.

There is also another method of comparing the divisions of a thermometer
with those of the natural scale, but it is not wholly free from objections ;
the instrument being placed in a cone of the sun's rays, made to converge
by means of a lens or mirror, the quantity of heat falling on it must be

THE MEASURES AND THE NATURE OF HEAT. 497

nearly in the inverse proportion of the square of its distance from the
focus ; and the elevation of a common thermometer appears to be nearly
proportional to the heat which is thrown on it in this manner.

The expansion of solids probably approaches the nearest to the steps of
the natural scale, although even in this there seems to be some inequality ;
but that of mercury is scarcely less regular, and a portion of mercury
inclosed in a bulb of glass, having a fine tube connected with it, forms a
thermometer the most convenient, and most probably the most accurate, of
any, for common use ; the degrees corresponding very nearly with those
of the natural scale, although, according to the most accurate experiments,
they appear to indicate, towards the middle of the common scale of
Fahrenheit, a temperature 2 or 3 degrees too low. There is an inequality
of the same kind, but still greater, in the degrees of the spirit thermometer ;
and this instrument has also the disadvantage of being liable to burst in a
heat below that of boiling water ; although it is well -calculated for the mea-
surement of very low temperatures, since pure alcohol has never yet been
frozen, while mercury has been reduced to a solid by the cold of Siberia
and of Hudson's Bay ; but both mercury and linseed oil support a heat of
between 5 and 600° without ebullition. For higher temperatures than
this, a thermometer has been made of semitransparent porcelain, containing
a fusible metal, which may be compared with the upper part of the mercu-
rial scale, and then continued further ; and the expansion of such of the
metals, as are difficult of fusion, affords another mode of determining the
highest degrees of heat. Mr. Wedgwood's thermometer* derives its proper-
ties from the contraction of a small brick of prepared clay, which contracts
the more, as the heat to which it is exposed is higher : it may be extremely
useful for identifying the degree of heat which is required for a particular
purpose : but for the comparison of temperatures by an extension of the
numerical scale, we have not sufficient evidence of its accuracy to allow us
to depend on its indications ; and it is scarcely credible that the operation
of furnaces, of any kind, can produce a heat of so many thousand degrees
of a natural scale, as Mr. Wedgwood's experiments have led him to sup-
pose ; nor is the supposition consistent with the observations of other
philosophers.

Mercurial thermometers are in general hermetically sealed, the tube
being perfectly closed at the end, in order to exclude dust, and to prevent
the dissipation of the mercury. When a standard thermometer is to be
adjusted, its freezing point is readily fixed by immersing it wholly in
melting snow or pounded ice ; but for the boiling point, some further pre-
cautions are required ; the easiest method appears to be, to immerse its bulb
in an open vessel of boiling water, to cover it with several folds of cloth,
and to pour hot water continually over it ; for if it were immersed to a
considerable depth, the pressure would raise the temperature of the boiling
point, and if it were not covered, the mercury in the tube would be too
cold. Attention must also be paid to the state of the barometer ; it must
either stand at 80 inches, or the place of the boiling point must be raised,
when the barometer is lower than 30, and lowered when it is higher ; the
* Ph. Tr. 1782, p. 305 ; 1784, p. 358 ; 1786, p. 390.
2 K

498 LECTURE LII.

difference of nine tenths of an inch either way requiring an alteration
amounting to T^- of the interval between freezing and boiling. This
interval is subdivided, in Fahrenheit's thermometer, into 180 degrees ; in
Reaumur's, into 80, and in the centigrade thermometer of Celsius and of the
French, into 100 ; and in making the subdivision, care must be taken to
examine the equality of the bore throughout, by observing the length occu-
pied by a detached portion of mercury, and to allow for any irregularities
which may have been thus detected. The scales of Reaumur and of Celsius
begin at the freezing point of water ; but in that of Fahrenheit the freezing
point stands at 32°, the scale beginning from the cold produced by a
freezing mixture, which was supposed by Fahrenheit to be the greatest
that would ever occur in nature.

The expansion, which is observed in a mercurial thermometer, is in
reality only the difference of the expansions of mercury and of glass ; but
this circumstance produces no difference in the accuracy of the results. The
separate effects of the expansion of glass are, however, sometimes per-
ceptible ; thus, when a thermometer is plunged suddenly into hot water,
the glass, being first heated, expands more rapidly than the mercury, and,
for a moment, the thermometer falls. This circumstance would perhaps
be still more observable in a thermometer of spirit or of water ; for an
equal bulk of these liquids would be much longer in acquiring the tempe-
rature of the surrounding medium than a mercurial thermometer.

The expansion of elastic fluids affords in some cases a test of heat, which
is very convenient from its great delicacy, and because a very small quan-
tity of heat is sufficient to raise their temperature very considerably. The
thermometer first invented by Drebel was an air thermometer ;* but instru-
ments of this kind, when they are subject to the variations of the pressure
of the atmosphere as well as to those of its temperature, are properly called
manometers, and require, for enabling us to employ them as thermometers,
a comparison with the barometer ; while on the other hand, they may be
used as barometers, if the temperature be otherwise ascertained. They are
however, very useful even without this comparison, in delicate experiments
of short duration, since the changes of the barometer are seldom very
rapid ; and they may also be wholly freed from the effects of the pressure
of the atmosphere, in various ways. Bernoulli's method t consists in closing
the bulb of a common barometer, so as to leave the column of mercury in
equilibrium with the air contained in the bulb at its actual temperature,
and capable of indicating, by the changes of its height and of its pressure,
any subsequent changes in the temperature of the air, which must affect
both its bulk and its elasticity. Mr. Leslie's photometer,;}: or differential
thermometer, has some advantages over this instrument, but it can only be
employed where the changes of temperature can be confined to a part only

* The invention is claimed for Drebel, by Boerhaave (Elem. Chimise, 2 vols. 4to,
Lugd. 1732, i. 152), and by Musschenbroek (Elem. Phil. Nat. § 780) ; whereas
Santorio claims it as his own (Comm. in Avicennam, 1626), and his claim is sup-
ported by others. See Martine's Essays, Edin. 1787, and Dr. Traill's Thermome-
ter and Pyrometer, Lib. of Useful Knowledge.

t Segner, De ^Equandis Thermometris Aeris, 4to, Gott. 1739.

t On Heat ; and Nich. Jour. iii. 461, 518.

THE MEASURES AND THE NATURE OF HEAT. 499

of the instrument. The elasticity of the air contained in the bulb is here
counteracted, not by the pressure of a column of mercury, but by the elas-
ticity of another portion of air in a second bulb, which is not to be exposed
to the heat or cold that is to be examined : and the difference between the
temperatures of the two bulbs is indicated by the place of a drop of a liquid,
moving freely in the tube which joins them. (Plate XXXIX. Fig. 548...
550.)

The degree of heat, as ascertained by a thermometer, is only to be con-
sidered as a relation to the surrounding bodies, in virtue of which a body
supports the equilibrium of temperature when it is in the neighbourhood of
bodies equally heated : thus, if a thermometer stands at 60°, both in a
vessel of water, and in another of mercury, we may infer that the water
and the mercury may be mixed without any change of their temperature :
but the absolute quantity of heat, contained in equal weights, or in equal
bulks, of any two bodies at the same temperature, is by no means the same.
Thus, in order to raise the temperature of a pound of wrater from 50° to
60°, we need only to add to it another pound of water at 70°, which while
it loses 10° of its own heat, will communicate 10° to the first pound ; but
the temperature of a pound of mercury at 50° may be raised 10°, by means
of the heat imparted to it, by mixing with it one thirtieth part of a pound
of water, at the same temperature of 70°. Hence we derive the idea of the
capacities of different bodies for heat, which was first suggested by Dr.
Irvine,* the capacity of mercury being only about one thirtieth part as
great as that of water. And by similar experiments it has been ascer-
tained, that the capacity of iron is one eighth of that of water, the capacity
of silver one twelfth, and that of lead one twenty-fourth. But for equal
bulks of these different substances, the disproportion is not quite so great :
thus, copper contains nearly the same quantity of heat in a given bulk as
water ; iron, brass, and gold, a little less, silver £ as much, but lead and
glass each about one half only.

It is obvious that if the capacity of a body for heat, in this sense of the
word, were suddenly changed, it would immediately become hotter or
colder, according to the nature of the change, a diminution of the capacity
producing heat, and an augmentation cold. Such a change of capacity is
often a convenient mode of representation for some of the sources of heat
and cold : thus, when heat is produced by the condensation of a vapour,
or by the congelation of a liquid, we may imagine that the capacity of the
substance is diminished ; and that it overflows, as a vessel would do if its
dimensions were contracted. It appears also from direct experiments, in
some such cases, that the capacity of the same substance is actually greater
in a liquid than in a solid state, and in a state of vapour, than in either ;
and both Dr. Irvine and Dr. Crawford t have attempted to deduce, from a
comparison of the proportional capacities of water and ice, with the quan-
tity of heat extricated during congelation, a measure of the whole heat
which is contained in these substances, and an estimation of the place
which the absolute privation of heat, or the natural zero, ought to occupy
" in the scale of the thermometer. Thus, when a pound of ice, at 32°, is
* Chemical Essays. f On Animal Heat, &c. 2nd edit. 1788.

2 K 2

500 LECTURE LIT.

mixed with a pound of water at 172° of Fahrenheit, the whole excess of
140° is absorbed in the conversion of the ice into water, and the mixture is
reduced to the temperature of 32° ; and, on the other hand, when a pound
of ice freezes, a certain quantity of heat is evolved which is probably
capable of raising the temperature of a pound of water 140°, or that of 140
pounds a single degree. Dr. Crawford found, by means of other experi-
ments, that a quantity of heat capable of raising the temperature of water
9° would raise that of ice as much as 10° : hence he inferred that the capa-
city of ice was T9V as great as that of water, and that if this capacity,
instead of being reduced to -j-9^, had been wholly destroyed, the quantity of
heat extricated would have been 10 times as great, or about 1400°, which
has, therefore, been considered as the whole quantity of heat contained in a
pound of water at 32°, and the beginning of the natural scale has been
placed about 1368° below the zero of Fahrenheit. Dr. Irvine makes the
capacity of ice still less considerable, and places the natural zero about
900 degrees below that of Fahrenheit.

If direct experiments on the quantities of heat, required for producing
certain elevations of temperature, in different states of the same substance,
compared in this manner with the emission or absorption of heat which
takes place while those changes are performed, agreed with similar experi-
ments made on different substances, there could be no objection to the
mode of representation. But if it should appear that such comparisons
frequently present us with contradictory results, we could no longer con-
sider the theory of capacities for heat as sufficient to explain the pheno-
mena. With respect to the simple changes constituting congelation and
liquefaction, condensation and evaporation, and compression and rarefaction,
there appears to be at present no evidence of the insufficiency of this theory ;
it has not perhaps yet been shown that the heat absorbed in any one change
is always precisely equal to that which is emitted in the return of the sub-
stance to its former state, but nothing has yet been advanced which renders
this opinion improbable ; and the estimation of the natural zero, which is
deduced from this doctrine, may at least be considered as a tolerable
approximation.

If, however, we attempt to deduce the heat produced by friction and by
combustion, from changes of the capacities of bodies thus estimated, we shall
find that the comparison of a very few facts is sufficient to demonstrate the
imperfection of such a theory. Count Rumford* found no sensible differ-
ence between the capacities of solid iron and of its chips ; but if we even
suppose, for the sake of the argument, that the pressure and friction of the
borer had lessened the capacity of the iron one twelfth, so as to make it no
greater than that of copper; we shall then find that one twelfth of the
absolute heat of the chips, thus abraded, must have amounted to above
60,000 degrees of Fahrenheit, and consequently that the natural zero ought
to be placed above 700,000 degrees below the freezing point, instead of 14 or
1500 only. It is, therefore, impossible to suppose that any alteration of
capacities can account for the production of heat by friction ; nor is it at
' all easier to apply this theory correctly to the phenomena of combustion.
* Ph. Tr. 1798, p. 80.

THE MEASURES AND THE NATURE OF HEAT. 501

A pound of nitre contains about half its weight of dry acid, and the capa-
city of the acid, when diluted, is little more than half as great as that of
water ; the acid of a pound of nitre must therefore contain less heat than
a quarter of a pound of water : hut Lavoisier and Laplace have found,* that
the deflagration of a pound of nitre produces a quantity of heat sufficient to
melt 12 pounds of ice, consequently the heat extricated by the decomposition
of a pound of dry nitrous acid must be sufficient to melt 24 pounds of ice ; and
even supposing the gases, extricated during the deflagration, to absorb no
more heat than the charcoal contained, which is for several reasons highly
improbable, it follows that a pound of water ought to contain at least as
much heat, as would be sufficient to melt 48 pounds of ice, that is about
6720 degrees of Farenheit.

In short, the further we pursue such calculations, the more we shall be
convinced of the impossibility of applying them to the phenomena. In
such a case as that of the nitrous acid, Dr. Black's term of latent heat t
might be thought applicable, the heat being supposed to be contained in the
substance, without being comprehended in the quantity required for main-
taining its actual temperature. But even this hypothesis is wholly inappli-
cable to the extrication of heat by friction, where all the qualities of the
substances concerned remain precisely the same after the operation as before
it. If any further argument were required in confutation of the opinion,
that the heat excited by friction is derived from a change of capacity, it
might be obtained from Mr. Davy's experiment on the mutual friction of
two pieces of ice, which converted them into water, in a room at the tem-
perature of the freezing point : for in this case it is undeniable that the
capacity of the water must have been increased during the operation ; and
the heat produced could not, therefore, have been occasioned by the dimi-
nution of the capacity of the ice.

This discussion naturally leads us to an examination of the various theories
which have been formed respecting the intimate nature of heat ; a subject
upon which the popular opinion seems to have been lately led away by very
superficial considerations. The facility with which the mind conceives the
existence of an independent substance, liable to no material variations,
except those of its quantity and distribution, especially when an appro-
priate name, and a place in the order of the simplest elements has been
bestowed on it, appears to have caused the most eminent chemical philoso-
phers to overlook some insuperable difficulties attending the hypothesis of
caloric. Caloric has been considered as a peculiar elastic or ethereal fluid,
pervading the substance or the pores of all bodies, in different quan-
tities, according to their different capacities for heat, and according to
their actual temperatures ; and being transferred from one body to another
upon any change of capacity, or upon any other disturbance of the equi-
librium of temperature : it has also been commonly supposed to be the
general principle or cause of repulsion ; and in its passage from one body
to another, by radiation, it has been imagined by some to flow in a con-

. * Hist, et Mem. 1780, p. 355, H. 3.

•f Black's Lectures, 2 vols. 4to, Ed. See Lavoisier, Traite Elem. de Chimie, 1 789.

502 LECTURE LII.

tinued stream, and by others in the form of separate particles, moving, with
inconceivable velocity, at great distances from each other.

The circumstances which have been already stated respecting the produc-
tion of heat by friction, appear to afford an unanswerable confutation of
the whole of this doctrine. If the heat is neither received from the surround-
ing bodies, which it cannot be without a depression of their temperature,
nor derived from the quantity already accumulated in the bodies themselves,
which it could not be, even if their capacities were diminished in any ima-
ginable degree, there is no alternative but to allow that heat must be actu-
ally generated by friction ; and if it is generated out of nothing, it cannot
be matter, nor even an immaterial or semimaterial substance. The colla-
teral parts of the theory have also their separate difficulties : thus, if heat
were the general principle of repulsion, its augmentation could not diminish
the elasticity of solids and of fluids ; if it constituted a continued fluid, it
could not radiate freely through the same space in different directions ;
and if its repulsive particles followed each other at a distance, they would
still approach near enough to each other, in the focus of a burning glass, to
have their motions deflected from a rectilinear direction.

If heat is not a substance it must be a quality ; and this quality can only
be motion. It was Newton's opinion, that heat consists in a minute vibra-
tory motion of the particles of bodies, and that this motion is communicated
through an apparent vacuum, by the undulations of an elastic medium,
which is also concerned in the phenomena of light. If the arguments
which have been lately advanced, in favour of the undulatory nature of
light, be deemed valid, there will be still stronger reasons for admitting this
doctrine respecting heat, and it will only be necessary to suppose the vibra-
tions and undulations, principally constituting it, to be larger and stronger
than those of light, while at the same time the smaller vibrations of light,
and even the blackening rays, derived from still more minute vibrations,
may, perhaps, when sufficiently condensed, concur in producing the effects
of heat. These effects, beginning from the blackening rays, which are in-
visible, are a little more perceptible in the violet, which still possess but a
faint power of illumination ; the yellow green afford the most light ; the
red give less light, but much more heat, while the still larger and less fre-
quent vibrations, which have no effect on the sense of sight, may be sup-
posed to give rise to the least refrangible rays, and to constitute invisible
heat.

It is easy to imagine that such vibrations may be excited in the component
parts of bodies, by percussion, by friction, or by the destruction of the
equilibrium of cohesion and repulsion, and by a change of the conditions
on which it may be restored, in consequence of combustion, or of any other
chemical change. It is remarkable that the particles of fluids, which are
incapable of any material change of temperature from mutual friction,
have also very little power of communicating heat to each other by their
immediate action, so that there may be some analogy, in this respect,
between the communication of heat and its mechanical excitation.

The effects of heat on the cohesive and repulsive powers of bodies have

THE MEASURES AND THE NATURE OF HEAT. 503

sometimes been referred to the centrifugal forces and mutual collisions of
the revolving and vibrating particles : and the increase of the elasticity of
aeriform fluids has been very minutely compared with the force which
would be derived from an acceleration of these internal motions. In solids
and in liquids, however, this increase of elasticity is not observable, and
the immediate effect of heat diminishes not only the force of cohesion, but
also in some degree, that of repulsion, so that these vibrations, if they
exist, must derive their effect on the corpuscular forces from the alterations
which they produce on the situation of the particles, with respect to the
causes of these forces.

The different chemical effects of heat and light are far from furnishing
any objection to this system ; it is extremely easy to imagine the attraction
between two or three bodies to be modified by the agitations, into which
their particles are thrown. If certain undulations be capable of affecting
one of the three bodies only, its cohesion with both the others may be
weakened, and hence their mutual attraction may be comparatively
increased ; and from various combinations of such differences, in the
operation of different kinds of heat and light, a great diversity of effects of
a similar kind may be derived.

If heat, when attached to any substance, be supposed to consist in
minute vibrations, and when propagated from one body to another, to
depend on the undulations of a medium highly elastic, its effects must
strongly resemble those of sound, since every sounding body is in a state
of vibration, and the air, or any other medium, which transmits sound,
conveys its undulation to distant parts by means of its elasticity. And we
shall find that the principal phenomena of heat may actually be illustrated
by a comparison^with those of sound. The excitation of heat and sound
are not only similar, but often identical ; as in the operations of friction
and percussion : they are both communicated sometimes by contact and
sometimes by radiation ; for besides the common radiation of sound
through the air, its effects are communicated by contact, when the end of
a tuning fork is placed on a table, or on the sounding board of an instru-
ment, which receives from the fork an impression that is afterwards propa-
gated as a distinct sound. And the effect of radiant heat, in raising the
temperature of a body upon which it falls, resembles the sympathetic
agitation of a string, when the sound of another string, which is in unison
with it, is transmitted to it through the air. The water, which is dashed
about by the vibrating extremities of a tuning fork dipped into it, may
represent the manner in which the particles at the surface of a liquid are
thrown out of the reach of the force of cohesion, and converted into
vapour ; and the extrication of heat, in consequence of condensation, may
be compared with the increase of sound produced by lightly touching a
long cord which is slowly vibrating, or revolving in such a manner as to
emit little or no audible sound ; while the diminution of heat, by expan-
sion, and the increase of the capacity of a substance for heat, may be
attributed to the greater space afforded to each particle, allowing it to be
equally agitated with a less perceptible effect on the neighbouring particles.
In some cases, indeed, heat and sound not only resemble each other in

504 LECTURE LII.

their operations, but produce precisely the same effects ; thus, an artificial
magnet, the force of which is quickly destroyed by heat, is affected more
slowly in a similar manner, when made to ring for a considerable time ;
and an electrical jar may be discharged, either by heating it, or by causing
it to sound by the friction of the finger.

All these analogies are certainly favourable to the opinion of the
vibratory nature of heat, which has been sufficiently sanctioned by the
authority of the greatest philosophers of past times, and of the most sober
reasoners of the present. Those, however, who look up with unqualified
reverence to the dogmas of the modern schools of chemistry, will probably
long retain a partiality for the convenient, but superficial and inaccurate,
modes of reasoning, which have been founded on the favourite hypothesis
of the existence of caloric as a separate substance ; but it may be presumed
that in the end a careful and repeated examination of the facts, which have
been adduced in confutation of that system, will make a sufficient impres-
sion on the minds of the cultivators of chemistry, to induce them to listen
to a less objectionable theory.

[Considerable advances have been made in our knowledge of the pro-
perties of heat since the first publication of these Lectures. They have
owed their existence, for the most part, to the discovery of a very delicate
measure of variation of temperature by means of its galvanic influence.
It is well known that when a current passes along a wire, a tangential
force is put in play which tends to cause the deflection of a magnetic
needle placed in the neighbourhood of the wire. (See additions to Lect. LV.)
Moreover it has been discovered that heat is capable of producing a gal-
vanic current, the intensity of which is proportional to that of the produ-
cing agent. This is exhibited in the following manner. A number of bars
of antimony and bismuth are arranged so as to lie compactly, whilst they
alternate with each other. They are then soldered together in pairs, so
that each bar of the one metal is connected at both ends with a bar of the
other. The extreme bars are united by means of a wire, and it is along
this that a galvanic current travels, on the application of heat to the sol-
dered ends of the bars. The instrument based on these principles, as sug-
gested by Becquerel and improved by Nobili, Melloni, and others, is called
a thermomultiplier. This name arises from the important fact, that by
crossing the wire so as to cause it to pass several times parallel to the mag-
netic needle, the simple effect may be increased to almost any extent, and
thus the instrument may be made to measure the minutest indications of
heat.* Amongst the earliest results of the use of this instrument was the
discovery of Melloni, that rock salt suffers heat of every kind to pass freely
through it, thus forming for heat a substance analogous to that which
glass constitutes for light.1* The field of discovery of the analogous pro-
perties of heat and light was now thrown open. That the former, as well
as the latter, suffers polarization under certain circumstances had been
conjectured by Berard and others. But the experiment on which the

* For a description of the instrument see the Bibliotheque Universelle (new ser.)
ii. 225. Ann. de Ch. xlviii. 198. Ed. Tr. vol. xiii.

•f Ann. de Ch. liii., or Taylor's Scientific Memoirs, i. 32.

THE MEASURES AND THE NATURE OF HEAT. 505

[belief was founded, failed on repetition by Powell and Nobili. And even
M. Melloni, with his thermomultiplier, was unable to detect the existence
of polarization in 1833. In the following year Prof. Forbes took up the
subject, and completely succeeded. The characteristic of polarization of
light is the exhibition of a reference to sides relative to its path. Thus light
which passes directly through one slice of tourmaline is of the same intensity
in whatever way the tourmaline be presented to it. But if a second tour-
maline be applied, the intensity of the ray which merges from it, depends
on its situation relative to the first ; thus proving that the light which had
passed through the first tourmaline differed from common light in having
acquired a property connected with direction perpendicular to its path.
Prof. Forbes showed that the same is equally true of heat. In the first
place, he found that two tourmalines, with their axes crossed, stop more
heat than when they are parallel, even if the source of heat is brass not
luminous. In the next place he discovered an apparatus which facilitated
the exhibition of the different facts connected with polarization. It con-
sists of plates of mica split very thin by the application of sudden heat.
These placed at an angle of about 45°, formed an excellent polarizer, and
enabled him to detect polarization with the greatest facility. By this means
he established the more delicate facts of depolarization and circular pola-
rization. A plate of mica being placed between the polarizer and ana-
lyzer, when in a crossed position, restored to the heat its capacity of being
transmitted through the latter, so that under certain circumstances, the
interposition of the plate actually increased the quantity of heat which
passed through the apparatus. Prof. Forbes discovered circular polari-
zation in 1836, thus establishing the complete analogy between heat and
light. There are, however, some points which appear to present an obstacle
to basing our theory of heat directly on the corresponding theory of light.
In the first place, Prof. Forbes shows that non-luminous heat is less pola-
rizable at the same angle than luminous. In the next place, most bodies
are found to absorb the less refrangible rays in excess, so that the mean
refrangibility is increased by transmission through them. Smoked rock
salt, or mica, was found by Melloni* not to possess this property, conse-
quently it was argued that the state of the surface produces the effect ;
and accordingly with a roughened surface Prof . Forbes t found the quantity
of dark heat transmitted to be in a threefold proportion to that which was
transmitted from a glass lamp. If, however, the surface was regularly
scratched, or if it had a grating before it, heat of every kind was trans-
mitted in the same proportion. This last fact may probably tend to recon-
cile the theories of heat and light, for I have proved J that interference,
whether by means of a prism or by Franenhofer's gratings, when they
are regular, produces no effect in adding to, or diminishing the quantity of
light. The total amount received on a screen is in exact proportion to the
amount of surface left uncovered, the effect of interference being merely a
displacement of its position.

* Comptes Rendus, Sep. 3, 1839.

f Proceedings of the Royal Soc. of Edin. p. 281. Ed. Tr. vol. xv.
t On the Aggregate Effect of Interference, Camb. Tr. vol. vii. part ii. On the
Absolute Intensity of Interfering Light, Ed. Tr. xv. 315.

506 LECTURE LII.

[The theory of heat may be said to rest where it did at the time these
lectures were written. The facts which have just been mentioned clearly
point out its undulatory character. But in what way the major part of
the ordinary phenomena al-e to be explained by this doctrine does not yet
appear. We have no satisfactory explanation of expansion, of tardy con-
duction, of the change of form of certain crystals, of latent heat, and the
like. Neither will a merely undulatory hypothesis relieve us from some of
the difficulty attendant on certain of the very phenomena which appeared
to suggest it ; such as the difference between solar and terrestrial heat,
which Melloni shows to depend on the different mixture of the different
rays ; the effect of different screens which variously affect different kinds
of heat,* the sifting of heat by a screen, so as to render it more capable of
transmission through another similar screen. These facts appear to
demand a corpuscular theory, wholly or partly accompanied by transverse
vibrations. The hypothesis which I have advancedt is, that heat is due to
the existence of repulsive atoms which penetrate all material substances ;
so that expansion arises from the accumulation of such atoms ; but that
the transmission of heat is partly effected by transverse pulses, very nearly
in the same way as the tidal water is conveyed up a channel, and accumu-
lates at its upper extremity. Solar heat is transmitted altogether by such
transverse pulses, so that its intensity is measured by the intensity of the
pulses, whilst the heat of a fire is perhaps due in part to normal ones,
or, which is the same thing, to a flow of atoms impelling by their repul-
sion those wliich are in advance of them.

The reader will find Professor Forbes's researches in the Edinburgh
Transactions, vols. xiii., xiv., and xv. ; and in the Philosophical Maga-
zine, vol. vi., &c. Melloni' s, in the Annales de Chimie, vol. liii., &c. ; or in
the Scientific Memoirs, vols. i., ii., &c. He may also consult Powell's
Reports on Radiant Heat. Reports of the British Association, 1832, 1840.]

LECT. LII.— ADDITIONAL AUTHORITIES.

Thermometers, fyc. — Fludd, De Philos. Moysiaca, 1638. Boyle's Works. Hooke's
Statical Therm. Birch, ii. 1. Wallis and Beale, Ph. Tr. 1669, p. 1113. Lahire,
Hist, et Mem. 1706, p. 432 ; 1710, p. 546, H. 13 ; 1711, p. 144, H. 40. Amon-
tons (air then), ibid. 1703, p. 101, H. 6. Taylor on the Expansion of Fluids,
Ph. Tr. 1723, p. 291. Bulfinger, Comm. Petr. iii. 196, 242, iv. 216. Reaumur,
Hist, et Mem. 1730, p. 452 ; 1731, p. 250, H. 6. Delisle, Ph. Tr. 1736, p. 221.
Ellicott's Pyrometer, ibid. 1736, p. 297. Weitbrecht, Comm. Petr. viii. 310.
Krafft, ibid. ix. 241. Ludolff, Mis. Berl. 1740, p. 255. Grischow, ibid. 1740,
p. 267. Celsius, Schwed. Abhand. 1742, p. 197. Wargentin, ibid. 1749, p. 167.
Smeaton's Pyrometer, Ph. Tr. 1754, p. 598. Lord Cavendish on Max. and Min.
Ther. ibid. 1757, p. 300. Bergen, DeTher. 4to, Nuremb. 1757. Sulzer, Act. Helv.
iii. 259. Hennert, Traite des Therm. Hague, 1768. Haubold, De Therm. Reaum.
4to, Leipz. 1771. Strohmeyer, Ueber die Ther. Gott. 1775. Roy on Ramsden's
Pyrometer, Ph. Tr. 1785, p. 461. Report on Ther. ibid. 1777, p. 816. Van
Swinden, Comparaison des Ther. Amst. 1778. Schuckburgh, Ph. Tr. 1779. Luz,
Ueber die Ther. Nuremb. 1781. Six's Register, Therm. Ph. Tr. Ixxii. 1794,
Rutherford's Register, Ed. Tr. iii. 247. Rumford's Differential Ther. Ph. Tr. 1804,
p. 77. Dalton, Nich. Jour. 8vo. v. 34. Daniell's Pyrometer, Quarterly Journal of

* Melloni, Annales de Ch. Iv. 337.

f Kelland's Theory of Heat, art. 155, 169, 194.

ON ELECTRICITY IN EQUILIBRIUM. 507

Science, xi. 309. Ph. Tr. 1830, p. 257. Guyton de Morveau's Pyrometer, Ann.
de Ch. xlvi. 276.

On the Expansion of Gases. — Priestley, Experiments and Observations on Air,
3 vols. 1774-7. Gay Lussac, Ann. de Ch. cxxviii (xliii.) 137. Dalton, Manch.
Mem. v. 595. Gilbert, in Gilb. Ann. xiv. 266. Rudberg, Pogg. Ann. xli. 271 ;
xliv. 119.

Expansion of Liquids. — Hallstrb'm, Pogg. Ann. i. 129, xix. 135. De Luc,
Recherches sur les Modifications de 1'Atmosphere. Gay Lussac, Ann. de Ch. ii.
130.

Solids.— Smeaton, Ph. Tr. 1754, p. 598. Errata. Roy, Ph. Tr. 1777, p. 653.
De Luc on Pyrometry and Areometry, ibid. 1778, p. 419. Lavoisier and Laplace,
Ann. de Ch. i. 101. Dulong and Petit, ibid. ii. 240, vii. 113, 225. Daniell, Ph. Tr.
1831, p. 443.

Freezing, Sfc. — Braun, De Frigore Artificial!, 4to, Petersb. 1760 ; on the Freez-
ing of Mercury, Nov. Com. Petr. ii. 268, 302. Hutchins on do. Ph. Tr. 1776,
p. 174 ; 1783, p. 303. Cavendish on Hutchins's Exper. ibid. 1783, p. 303 ; on
M'Nabs, ibid. 1786, p. 241 ; 1788, p. 166. Blagden's Hist, of the Congelation of
Mercury, ibid. 1783, p. 329. Guthrie on do. 4to, Petersb. 1785. Walker on
Freezing Mixtures, Ph. Tr. 1788, p. 395 ; 1795, p. 270 ; 1801, p. 120.

Specific Heat. — Meyer, Ann. de Ch. xxx. 46. De la Roche and Berard, ibid.
Ixxxv. 72. Dulong and Petit, ibid, (new series), vii. 113, 142, x. 395. Ure, Ph. Tr.
1818, p. 378. Haycraft, Ed. Tr. x. 195. Avogadro, Mem. di Torino, xxviii. 1,
xxix. 79. Neumann, Pogg. Ann. xxiii. 1. Thomson's Heat (1840) is very copious.

LECTURE LIII.

ON ELECTRICITY IN EQUILIBRIUM.

THE phenomena of electricity are as amusing and popular in their
external form as they are intricate and abstruse in their intimate nature.
In examining these phenomena a philosophical observer will not he content
with such exhibitions as dazzle the eye for a moment, without leaving
any impression that can be instructive to the mind, hut he will be anxious
to trace the connexion of the facts with their general causes, and to com-
pare them with the theories which have been proposed concerning them :
and although the doctrine of electricity is in many respects yet in its
infancy, we shall find that some hypotheses may be assumed which are
capable of explaining the principal circumstances in a simple and satisfac-
tory manner, and which are extremely useful in connecting a multitude of
detached facts into an intelligible system. These hypotheses, founded on
the discoveries of Franklin, have been gradually formed into a theory, by
the investigations of Aepinus and Mr. Cavendish, combined with the
experiments and inferences of Lord Stanhope, Coulomb, and Robison.

We shall first consider the fundamental hypotheses on which this system
depends, and secondly the conditions of equilibrium of the substances
concerned in it ; determining the mode of distribution of the electric fluid,
and the forces or pressures derived from its action when at rest ; all which
will be found to be deduced from the theory precisely as they are experi-
mentally observable. The motions of the electric fluid will next be noticed,

508 LECTURE LIII.

as far as we can form any general conclusions respecting them ; and the
manner in which the equilibrium of electricity is disturbed, or the excita-
tion of electricity, will also be considered ; and, in the last place, it will be
necessary to take a view of the mechanism or the practical part of elec-
tricity, and to examine the natural and artificial apparatus concerned in
electrical phenomena, as well as in those effects, which have been denomi-
nated galvanic.

It is supposed that a peculiar ethereal fluid pervades the pores, if not
the actual substance, of the earth and of all other material bodies, passing
through them with more or less facility, according to their different powers
of conducting it : that the particles of this fluid repel each other, and are
attracted by the particles of common matter : that the particles of common
matter also repel each other : and that these attractions and repulsions are
equal among themselves, and vary inversely as the squares of the distances
of the particles.

The effects of this fluid are distinguished from those of all other sub-
stances by an attractive or repulsive quality, which it appears to commu-
nicate to different bodies, and which differs in general from other
attractions and repulsions, by its immediate diminution or cessation, when
the bodies, acting on each other, come into contact, or when they are
touched by other bodies. The name electricity is derived from electrum,
amber ; for it was long ago observed that amber, when rubbed, continues
for some time to attract small bodies ; but at present electricity is usually
excited by other means. In general a body is said to be electrified, when
it contains, either as a whole, or in any of its parts, more or less of the
electric fluid than is natural to it ; and it is supposed that what is called
positive electricity depends on a redundancy, and negative electricity on a
deficiency of the fluid.

These repulsions and attractions are supposed to act, not only between
two particles which are either perfectly or very nearly in contact with
each other, but also between all other particles at all distances, whatever
obstacles may be interposed between them. Thus, if two electrified balls
repel each other, the effect is not impeded by the interposition of a plate of
glass : and if any other substance interposed appears to interfere with their
mutual action, it is in consequence of its own electrical affections. In
these respects, as well as in the law of their variation, the electrical forces
differ from the common repulsion which operates between the particles of
elastic fluids, and resemble more nearly that of gravitation. Their
intensity, when separately considered, is much greater than that of
gravitation, and they might be supposed to be materially concerned in the
great phenomena of the universe ; but in the common neutral state of all
bodies, the electrical fluid, which is every where present, is so distributed,
that the various forces hold each other exactly in equilibrium, and the
separate results are destroyed ; unless we choose to consider gravitation
itself as arising from a comparatively slight inequality between the elec-
trical attractions and repulsions.

The attraction of the electric fluid to common matter is shown by its
communication from one body to another, which is less copiously supplied

ON ELECTRICITY IN EQUILIBRIUM. 509

with it, as well as by many other phenomena ; and this attraction of the
fluid of the first body, to the matter of the second, is precisely equal to its
repulsion for the quantity of the fluid which naturally belongs to the
second, so as to saturate the matter. For the excess or deficiency of the
fluid in the first body does not immediately produce either attraction or
repulsion, so long as the natural distribution of the fluid in the second body
remains unaltered.

Since also two neutral bodies, the matter which they contain being
saturated by the electric fluid, exhibit no attraction for each other, the
matter in the first must be repelled by the matter in the second : for
its attraction for the fluid of the second would otherwise remain uncom-
pensated. We are, however, scarcely justified in classing this mutual
repulsion among the fundamental properties of matter ; for useful as these
laws are in explaining electrical appearances, they seem to deviate too far
from the magnificent simplicity of nature's works, to be admitted as
primary consequences of the constitution of matter : they may, however,
be considered as modifications of some other more general laws, which are
yet wholly unknown to us.

When the equilibrium of these forces is destroyed, the electric fluid is
put in motion ; those bodies which allow the fluid a free passage, are called
perfect conductors ; but those which impede its motion more or less, are
nonconductors, or imperfect conductors. For example, while the electric
fluid is received into the metallic cylinder of an electrical machine, its
accumulation may be prevented by the application of the hand to the
cylinder which receives it, and it will pass off through the person of the
operator to the ground ; hence the human body is called a conductor. But
when the metallic cylinder, or conductor, of the machine is surrounded
only by dry air, and supported by glass, the electric fluid is retained, and
its density increased, until it becomes capable of procuring itself a passage
some inches in length, through the air, which is a very imperfect conductor.
If a person, connected with the conductor, be placed on a stool with glass
legs, the electricity will no longer pass through him to the earth, but may
be so accumulated, as to make its way to any neighbouring substance
which is capable of receiving it, exhibiting a luminous appearance called a
spark ; and a person or a substance, so placed as to be in contact with
nonconductors only, is said to be insulated. When electricity is subtracted
from the substance thus insulated, it is said to be negatively electrified, but
the sensible effects are nearly the same, except that in some cases the form
of the spark is a little different.

Perfect conductors, when electrified, are in general either overcharged or
undercharged with electricity in their most distant parts at the same time;
but nonconductors, although they have an equal attraction for the electric
fluid, are often differently affected in different parts of their substance,
even when those parts are similarly situated in every respect, except that
some of them have had their electricity increased or diminished by a
foreign cause. This property of nonconductors may be illustrated by
means of a cake of resin, or a plate of glass, to which a local electricity
n/ay be communicated in any part of its surface, by the contact of an

510 LECTURE LIII.

electrified body ; and the parts thus electrified may afterwards be distin-
guished from the rest, by the attraction which they exert on any small
particles of dust or powder projected near them ; the manner in which
the particles arrange themselves on the surface, indicating also in some
cases the species of electricity, whether positive or negative, that has
been employed ; positive electricity producing an appearance somewhat
resembling feathers ; and negative electricity an arrangement more like
spots. The inequality in the distribution of the electric fluid in a noncon-
ductor may remain for some hours, or even some days, continually
diminishing till it becomes imperceptible.

These are the fundamental properties of the electric fluid, and of the
different kinds of matter as connected with that fluid. We are next to
examine its distribution, and the attractive and repulsive effects exhibited
by it, under different forms. Supposing a quantity of redundant fluid to
exist in a spherical conducting body, it will be almost wholly collected into
a minute space contiguous to the surface, while the internal parts remain
but little overcharged. For we may neglect the actions of the portion of
fluid which is only occupied in saturating the matter, and also the effect of
the matter thus neutralised, since the redundant fluid is repelled as much
by the one as it is attracted by the other : and we need only to consider
the mutual actions of the particles of this superfluous fluid on each other.
It may then be shown, in the same manner as it is demonstrated of the
force of gravitation, that all the spherical strata which are remoter from
the centre than any given particle, will have the whole of their action on it
annihilated by the balance of their forces, and that the effective repulsion
of the interior strata will be the same, as if they were all collected in the
centre. This repulsion will, therefore, impel the particles of the fluid
towards the surface, as long as it exists, and nothing will impede the
condensation of the redundant fluid there, until it is exhausted from the
neighbourhood of the centre. In the same manner it may be shown, that
if there be a deficiency of fluid, it will be only in the external parts, the
central parts remaining always in a state of neutrality : and since the
quantity of electric fluid taken away from a body, in any common experi-
ment, bears but a very small proportion to the whole that it contains, the
deficiency will also be found in a very small portion of the sphere, next to
its surface. And if, instead of being spherical, the body be of any
other form, the effects of electricity will still be principally confined to
its surface. This proposition was very satisfactorily investigated by Mr.
Cavendish ;* and it was afterwards more fully shown, by Dr. Gray'st
experiments, that the capacities of different bodies for receiving electricity,
depend much more on the quantity of their surfaces, than on their solid
contents : thus, the conductor of an electrical machine will contain very
nearly or quite as much electricity if hollow as if solid.

If two spheres be united by a cylindrical conducting substance of small
dimensions, there will be an equilibrium, when the actions of the redundant
fluid in the spheres, on the whole fluid in the cylinder, are equal ; that is,
when both the spheres have their surfaces electrified in an equal degree :

* Ph. Tr. 177G. See also ibid. 1771, p. 584. f Ibid. 1788, p. 121.

ON ELECTRICITY IN EQUILIBRIUM. 511

but if the length of the cylinder is considerable, the fluid within it can
only remain at rest when the quantities of redundant fluid are nearly equal
in both spheres, and consequently when the density is greater in the
smaller. And for a similar reason, in bodies of irregular forms, the fluid
is always most accumulated in the smallest parts ; and when a conducting
substance is pointed, the fluid becomes so dense at its extremity, as easily
to overcome the forces which tend to retain it in its situation. (Plate
XXXIX. Fig. 551.)*

In this distribution we find a very characteristic difference between the
pressure of the electric fluid and the common hydrostatic pressure of
liquids or of simple elastic fluids; for these exert on every surface
similarly situated a pressure proportionate to its magnitude ; but the
electric fluid exerts a pressure on small and angular surfaces, greater, in
proportion to their magnitudes, than the pressure on larger parts : so that
if the electric fluid were in general confined to its situation by the pressure
of the atmosphere, that pressure might easily be too weak to oppose its
escape from any prominent points. It does not appear, however, that this
pressure is the only cause which prevents the escape of the electric fluid ;
nor is it certain that this fluid can pass through a perfect vacuum,
although it has not yet been proved, that a body placed in a vacuum is
perfectly insulated. Whatever the resistance may be, which prevents the
dissipation of electricity, it is always the more easily overcome, as the
electrified substance is more pointed, and as the point is more promi-
nent ; and even the presence of dust is often unfavourable to the success
of electrical experiments, on account of the great number of pointed termi-
nations which it affords.

The general effect of electrified bodies on each other, if their bulk is
small in comparison with their distance, is, that they are mutually repelled
when in similar states of electricity, and attracted when in dissimilar states.
This is a consequence immediately deducible from the mutual attraction of
redundant matter and redundant fluid, and from the repulsion supposed to
exist between any two portions either of matter or of fluid, and it may
also easily be confirmed by experimental proof. A neutral body, if it were
a perfect nonconductor, would not be affected either way by the neighbour-
hood of an electrified body : for while the whole matter contained in it
remains barely saturated with the electric fluid, the attractions and repul-
sions balance each other. But in general, a neutral body appears to be
attracted by an electrified body, on account of a change of the disposition
of the fluid which it contains, upon the approach of a body either
positively or negatively electrified. The electrical affection produced in
this manner, without any actual transfer of the fluid, is called induced
electricity.

When a body positively electrified approaches to a neutral body, the
redundancy of the fluid expels a portion of the natural quantity from the

* On charge and distribution consult Winkler, Electr. Kraft des Wassers, Leipz.
1746. Beccaria, Ph. Tr. 1767, p. 297. Achard, Hist, et Mem. de Berlin, 1780,
* p. 47. Coulomb. Hist, et Mem. 1785, p. 612; 1786, p. 67; 1787, p. 421 ; and
the Analytical investigations of Poisson, Mem. de 1'Institut, 1811, 1, 163, 274 ; and
, Essay, 4to, Netting. 1828.

512 LECTURE LIU.

nearest parts of the neutral body, so that it is accumulated at the opposite
extremity ; while the matter, which is left deficient, attracts the redundant
fluid of the first hody, in such a manner as to cause it to be more con-
densed in the neighbourhood of the second than elsewhere ; and hence the
fluid of this body is driven still further off, and all the effects- are re-
doubled. The attraction of the redundant fluid of the electrified body for
the redundant matter of the neutral body, is stronger than its repulsion
for the fluid which has been expelled from it, in proportion as the square
of the mean distance of the matter is smaller than that of the mean dis-
tance of the fluid : so that in all such cases of induced electricity, an
attraction is produced between the bodies concerned. And a similar
attraction will happen, under contrary circumstances, when a neutral
body and a body negatively electrified, approach each other.

The state of induced electricity may be illustrated by placing a long
conductor at a little distance from an electrified substance, and directed
towards it ; and by suspending pith balls or other light bodies from it, in
pairs, at different parts of its length : these will repel each other, from
being similarly electrified, at the two ends, which are in contrary states
of electricity, while at a certain point towards the middle, they will
remain at rest, the conductor being here perfectly neutral. It was
from the situation of this point that Lord Stanhope* first inferred the true
law of the electric attractions and repulsions, although Mr. Cavendish^
had before suggested the same law as the most probable supposition.

The attraction thus exerted by an electrified body upon neutral sub-
stances, is strong enough, if they are sufficiently light, to overcome their
gravitation, and to draw them up from a table at some little distance :
upon touching the electrified body, if it is a conductor, they receive a
quantity of electricity from it, and are again repelled, until they are
deprived of their electricity by contact with some other substance, which,
if sufficiently near to the first, is usually in a contrary state, and there-
fore renders them still more capable of returning, when they have touched
it, to the first substance, in consequence of an increased attraction, assisted
also by a new repulsion. This alternation has been applied to the con-
struction of several electrical toys ; a little hammer, for example, has been
made to play between two bells ; and this instrument has been employed
for giving notice of any change of the electrical state of the atmosphere.
The repulsion, which takes place between two bodies, in a similar state of
electricity, is the cause of the currents of air which always accompany the
discharge of electricity, whether negative or positive, from pointed sub-
stances ; each particle of air, as soon as it has received its electricity from
the point, being immediately repelled by it ; and this current has also been
suppossd to facilitate the escape of the electricity, by bringing a continual
succession of particles not already overcharged.

If two bodies approach each other, electrified either positively or
negatively in different degrees, they will either repel or attract each other,
according to their distance : when they are very remote, they exhibit a

* Lord Mahon's Principles of Electr. 4to, Lond. 1779. * ,

f On the Principal Phenomena of Electr. Ph. Tr. 1771, p. 584.

ON ELECTRICITY IN EQUILIBRIUM. 513

repulsive force, but when they are within a certain distance, the effects of
induced electricity overcome the repulsion which would necessarily take
place, if the distribution of the fluid remained unaltered by their mutual
influence.

When a quantity of the electric fluid is accumulated on one side of a
non-conducting substance, it tends to drive off the fluid from the other
side ; and if this fluid is suffered to escape, the remaining matter exerts
its attraction on the fluid which has been imparted to the first side, and
allows it to be accumulated in a much greater quantity than could have
existed in an equal surface of a conducting substance. In this state, the
body is said to be charged ; and for producing it the more readily, each
surface is usually coated with a conducting substance, which serves to
convey the fluid to and from its different parts with convenience. The
thinner any substance is, the greater quantity of the fluid is required for
charging it in this manner, so as to produce a given tension, or tendency to
escape : but if it be made too thin, it will be liable to break, the attractive
force of the fluid for the matter on the opposite side overcoming the
cohesion of the substance, and perhaps forcing its way through the tem-
porary vacuum which is formed.

When a communication is made in any manner by a conducting sub-
stance between the two coatings of a charged plate or vessel, the equili-
brium is restored, and the effect is called a shock. If the coatings be
removed, the plate will still remain charged, and it may be gradually dis-
charged by making a communication between its several parts in suc-
cession, but it cannot be discharged at once, for want of a common con-
nexion : so that the presence of the coating is not absolutely essential to
the charge and discharge of the opposite surfaces. Such a coated substance
is most usually employed in the form of a jar. Jars were formerly filled
with water, or with iron filings ; the instrument having been principally
made known from the experiments of Musschenbroek and others at
Leyden, it was called the Leyden phial ; but at present a coating of tin
foil is commonly applied on both sides of the jar, leaving a sufficient space
at its upper part, to avoid the spontaneous discharge, which would often
take place between the coatings, if they approached too near to each other ;
and a ball is fixed to the cover, which has a communication with the
internal coating, and by means of which the jar is charged, while the
external coating is allowed to communicate with the ground. A collection
of such jars is called a battery, and an apparatus of this kind may be
made so powerful, by increasing the number of jars, as to exhibit many
striking effects by the motion of the electric fluid, in its passage from one
to the other of the surfaces.

The conducting powers of different substances are concerned, not only
in the facility with which the motions of the electric fluid are directed into
a particular channel, but also in many cases of its equilibrium, and par-
ticularly in the properties of charged substances, which depend on the
resistance opposed by nonconductors to the ready transmission of the fluid.
These powers may be compared, by ascertaining the greatest length of
of the substances to be examined, through which a spark or a shock

2L

514 LECTURE LIII.

will take its course, in preference to a given length of air, or of any other
standard of comparison. The substances, which conduct electricity the
most readily, are metals, well burnt charcoal, animal bodies, acids, saline
liquors, water, and very rare air. The principal nonconductors are glass,
ice, gems, dry salts, sulfur, amber, resins, silk, dry wood, oils, dry air of
the usual density, and the barometrical vacuum. Heat commonly increases
the conducting powers of bodies ; a jar of glass may be discharged by a
moderate heat, and liquid resins are capable of transmitting shocks,
although they are by no means good conductors : it is remarkable also that
a jar may be discharged by minute agitation, when it is caused to ring by
the friction of the finger. It has been observed that, in a great variety of
cases, those substances, which are the best conductors of heat, afford also
the readiest passage to electricity; thus, copper conducts heat more
rapidly, and electricity more readily, than iron, and platina less than
almost any other metal ; glass also presents a considerable resistance to the
transmission of both these influences. The analogy is, however, in many
respects imperfect, and it affords us but little light, with regard either to
the nature of heat, or to that of the electric fluid.

LECT. LIII.— ADDITIONAL AUTHORITIES.

Treatises on Electricity. — Mortensson, De Electr. 4to, Upsal, 1740. Desa-
guliers, A Dissertation concerning Electr. 1742. Winckler's Gedanke von der
Elektr. Leipz. 1744. Eigenschaften der Elektr. Materie, 1745. Bose, Recherches
sur la Cause de 1'Electr. 4to, Berlin, 1745. Tentamina Electr. 4to, Wittemb. 1747.
Waitz, Abhand. von der Elektr. 4to, Berlin, 1745. Piderit, De Electr. Marb. 1745.
Watson's Exp. and Obs. with Sequel, 1746. Miiller, Ursachund Nutzen der Elektr.
1746. Nollet, Essai sur 1'Electr. 12mo, Paris, 1746. Recherches sur do. 4to,
1749. Lettres sur do. 12mo, 1753. Martin on Electr. Bath, 1748. Jallabert sur
1'Electr. 1749. Boulanger, Traite de I7 Electr. 12mo, 1750. Secondat, Obser-
vations Physiques, 12mo, 1750. Verrati sur 1'Electr. 12mo, Montp. 1750.
Buia, Electr. Effectuum Explicatio, 1751. Franklin, Exp. and Obs. 4to, 1751-
54. Navarro, Physica Electr. Madrid, 1752. Klingenstierna, Electr. Stockholm,
1755. Beccaria, Lettere dell' Elettr. fol. Bolog. 1758 (tr.), Lond. 1776; Ex-
perimenta, 4to, Turin, 1772. Egelin, De Electr. 4to, Utrecht, 1759. Wesley's
Electr. made plain, 12mo, 1760. Saussure, De Electr. Geneve, 1766. Lullin,
De Electr. Geneve, 1766. Hartmann's Versuche in leeren Ratime, Hanov.
1766. Priestley's Introduction to Electr. 1769. Ferguson's Introduction to
Electr. 1771. Sigaud de la Fond, Traite de 1'Electr. 12mo, 1771. Precis des
Phenomenes Electr. 1781. Jacquet, Precis de 1'Electr. Vienna, 1775. Gross,
Elektrische Pauschen, Leipz. 1776. Dubois, Lettres sur 1'Electr. 1776. Bar-
letti (R. S.) 1771. Socin, Anfangsgriinde der Elektr. Hanau, 1777. Gallitzin
sur 1'Electr. 4to, Petersb. 1778. Lord Mahon's Principles of Electr. 4to, 1779.
Wilson's Short View of Electr. 4to, 1780. Lyons's New System of Electr. 4to,

1780. Marat, Recherches sur 1'Electr. 1782. Lacepede sur 1'Electr. 2 vols.

1781. Cuthbertson, Van der Elektr. Amst. 1782-94. D'Inarre, Naturlehre,
Frankf. 1783. Adams, Essay on Electr. 1784. Donndorff's Lehre von der Elektr.
Erf. 1784. Tressan sur la Fluide Electr. 2 vols. 1786. Priestley's Introd. to
Electr. 1787. Beck, Lehre von Electr. 1787. Langenbucher's Elektr. Augs. 1788.
Bennet's Experiments, Derby, 1789. Briefe iiber die Elektr. von C. L. Leipz. 1789.
Brook on Electr. 1790. Peart on do. Gainsborough, 1791. Lampadius, Ueber
Electr. und Warme, Berlin, 1793. Cavallo's Electr. 3 vols. 1795. Morgan's
Lectures on Electr. 2 vols. 12mo. Saxtorph, Darstellung der Elektr. 2 vols.
Copenh. 1803. Deluc, Traite Elementaire sur le Fluide Electro-galvanique, Milan,
1804. Sammlung Elektr. Spielwerke, Niirnb. 1804. Robison's Mechanical Phi-
losophy. Singer's Elements of Electr. 1814. Galle, Beitrage zur Erweiterung der
Elektr. 2 vols. Salzb. 1816. Adams's Electricity, 4to, 1823. Roslin, Priifungvler

ON ELECTRICITY IN EQUILIBRIUM. 515

Electr. Ulm, 1823. Leschan, Grundziige der Reinen Elektr. 1826. Farrar, Ele-
ments of Electr. Camb. N. E. 1826. Fechner, Lehrbuch der Galvanismus, Leipz.
1829. Murphy's (Mathematical) Principles of Electr. 1833. Becquerel, Traite
Experimentale de 1' Electr. et du Magnetisme, 7 vols. Paris, 1834. Nobili, Nuovi
Trattati, Modena, 1838. Roget's Electr. (Lib. of Useful Knowledge). Thomson's
Heat and Electr. 1840. Sturgeon, Lect. on Electr. 1842. Lardner and Walker's
Electricity, in Cab. Cyc. 2 vols. Noad's Electricity (new ed.) 1844.

Memoirs. — Hauksbee's Physico-Mech. Experiments, 4to, 1709, and Ph. Tr. 1706,
p. 2277; 1707, pp.2313, 2372; 1708, pp. 82, 87; 1709, pp. 391, 439; 1711,
p. 328. Gray's Exp. ibid. 1720, p. 104; 1731, p. 18; 1732, p. 397. Dufay's,
Hist, et Mem. 1733, pp. 23, 73, 233, 457 ; 1734, pp. 341, 503 ; 1737, pp. 86, 307.
Schilling, Mis. fieri. 1734, p. 334. Desaguliers, Ph. Tr. 1739, pp. 186, 200 ;
1741, p. 634; 1742, pp. 14, 140. Bose, Hist, et Mem. 1743, H. 45. Nollet,

ibid. 1745 1766 (various memoirs). Watson, Ph. Tr. 1745, p. 481 ; 1747,

pp. 388, 695, 704 ; 1751, pp. 202, 362. Hollmann, ibid. 1745, p. 239. Lemonnier,
ibid. 1746, p. 290. Dutour, Mem. des Sav. Etr. i. 345, ii. 246, 516, 537, iii. 244.
Wilson, Ph. Tr. 1753, p. 347 ; 1763, p. 436. Canton, ibid. 1753, p. 350 ; 1754,
p. 780. Leroy, Hist, et Mem. 1753, p. 447, H. 18 ; 1755, p. 264, H. 20.
Franklin, Ph. Tr. 1755, pp. 300,305 ; 1760, p. 525. Aepinus, Hist, et Mem. de
Berlin, 1756. Nov. Com. Petr. vii. 277. Delaval on the Influence of Temp. Ph.
Tr. 1759, p. 83 ; 1761, 353; with Canton's Remarks, 1762, p. 457. Beccaria,
1760, pp. 514, 525. Priestley, ibid. 1769, pp. 57, 63; 1770, p. 192. Cigna,
Mis. Taur. ii. 31, 77, iii. 31, v. I. 97. Brydone, Ph. Tr. 1773, p. 163. Gallitzin,
Acta Petr. i. II. H. 25. Achard, Jour, de Phy. xix. 417, xxv. 429. Van Marum,
ibid. xxxi. 343; 1788, p. 148; Experiences, 2 vols. 4to, Haarlem, 1787, 1795.
Gilbert's Jour. i. 239, 256, x. 121. Vassali and Zimmermann, Mem. della Soc.
Ital. iv. 264. Nicholson, Ph. Tr. 1789, p. 265. Deluc, Jour, de Phy. xxxvi. 450.
Von Arnim, Gilb. Jour. v. 33, vi. 116. Remer, ibid. viii. 323. Clos, Jour, de
Phy. liv. 316. Wollaston, Ph. Tr. 1801, p. 427. Snow Harris on the Elementary
Laws of Electr. Ph. Tr. 1834 ; 1836, p. 417 ; 1839, p. 215 ; 1842, p. 165. Riess,
Repert. der Phys. ii. Pogg. Ann. xl. 321, xliii. 47, xlv. 1, liii. 1. Pfaff, ibid. xliv.
332. PouiUet sur 1'Electr. des Fluides Elastiques, Ann. de Ch. xxxv. 401.
Knockerhauer, Ueber die Gebundene Elektr. Pogg. Ann. xlvii. 444, Iviii. 211, 391.

Conducting Powers. — Plot's Catalogue of Electrics, Ph. Tr. 1698, p. 384.
Gray on the Electr. of Water, ibid. 1732, p. 227. Desaguliers, ibid. 1741, p. 661.
Watson, ibid. 1746, p. 41. Mazeas, ibid. 1753, p. 377. Ammersin, De Electr.
Lignorum, 24mo, Luzern. 1754. Priestley on the Conducting Power of Charcoal,
ibid. 1770, p. 211. Harley on Vapour, ibid. 1774, p. 389 ; on Glass, 1778, p. 1049.
Cavendish, ibid. 1776. Achard, Jour, de Phy. xv. 117, xxii. 245. Bergman on the
Conducting Power of Water, ibid. xiv. 192. Cavallo, Ph. Tr. 1783, p. 495, and
Electricity. Morgan on a Vacuum, ibid. 1785, p. 272. Volta, Gilb. Ann. xiv. 257.
Tremery, Bulletin de la Soc. Philom. No. 19. Erman, Gilb. Jour. xi. 143. Snow
Harris, Ph. Tr. 1827, p. 18.

Theory of Electricity.— Gordon, Versuch einer Erklarung der Elektr. Erf. 1745.
Rosenberg, Von der Ursachen der Elektr. Bresl. 1745. Werenberg, Gedanken
yon der Elektr. 1745. Kratzenstein, Theoria Electr. 4to, Hal. 1746. Ellicott,
Ph. Tr. 1748, p. 195. J. Euler, De Causa Electr. 4to, Petersb. 1755 ; Hist,
et Mem. de Berlin, 1757, p. 125. Wilcke, Dissertatio Physica de Electr. con-
trariis, 4to, Rostock, 1757 ; Schwed. Abhand. xxxix. 68. Symmer, Ph. Tr.
1759, p. 340. Cigna on the Analogy of Magnetism and Electr. Mis. Taur. i.
Aepinus, Tentamen Theorise Electr. et Magnet. 4to, Petersb. 1759 ; Nov. Com.
Petr. x. 296. Dutour sur la Matiere Electr. 12mo, Paris, 1760. Bergmann
on the Existence of Two Fluids, Ph. Tr. 1764, p. 84. Bauer, Theorie der
Elektr. 1770. Herbert, Theoria Phenom. Electr. Vienna, 1778. Euler's Letters,
ii. 34. Barletti, Mem. de la Soc. Ital. i. 1, ii. 1, iv. 304, vii. 444. Coulomb,
Hist, et Mem. 1784, &c. Haiiy, Exposition Theorique, 1787. Biot, Bulletin
de la Soc. Philom. No. 51. Tremery, ibid. No. 63. Schrader, Versuch einer neuen
Theorie, Altona, 1796. Gren, Grundriss der Naturlehre, Halle, 1797, sec. 1408.
Heidmann, Theorie der Elektr. 2 vols. Wien. 1799. Ritter, Das Elektr. System
der Korper, Leipz. 1805. Winterl, Gehlen Jour. vi. 1, 201. Oersted, Ansichtder
Naturgesetze, Berl. 1802. Parrot, Grundriss der Theoretischen Physik, ii. 3.
Becquerel, Annales de Chimie, xlvi. 265, 337, xlvii. 113, xlix. 131. Avogadro, Jour,
de Phys. Ixiii. 450. Faraday's Experimental Researches in Electricity, Ph. Tr. 1832
..f'f. Republished, 2 vols. 1839, 1844.

2 L2

516

LECTURE LIV.

ON ELECTRICITY IN MOTION.

THE manner in which the electric fluid is transferred from one body to
another, the immediate effects of such a transfer, the causes which origi-
nally disturb the equilibrium of electricity, and the practical methods,
by which all these circumstances are regulated and measured, require to be
considered as belonging to the subject of electricity in motion. Among
the modes of excitation by which the equilibrium is originally disturbed,
one of the most interesting is the galvanic apparatus, which has been of late
years a very favourite subject of popular curiosity, and of which the theory
and operation will be briefly examined, although the subject appears
rather to belong to the chemical than to the mechanical doctrine of
electricity.

The progressive motion of the electric fluid through conducting sub-
stances is so rapid, as to be performed in all cases without a sensible
interval of time. It has indeed been said, that when very weakly excited,
and obliged to pass to a very great distance, a perceptible portion of
time is actually occupied in its passage ; but this fact is somewhat
doubtful, and attempts have been made in vain, to estimate the interval
employed in the transmission of a shock through several miles of wire.
We are not to imagine that the same particles of the fluid, which enter
at one part, pass through the whole conducting substance, any more than
that the same portion of blood, which is thrown out of the heart in each
pulsation, arrives at the wrist, at the instant that the pulse is felt there.
The velocity of the transmission of a spark or shock far exceeds the actual
velocity of each particle, in the same manner as the velocity of a wave
exceeds that of the particles of water concerned in its propagation ; and
this velocity must depend both on the elasticity of the electric fluid, and
on the force with which it is confined to the conducting substance. If this
force were merely derived from the pressure of the atmosphere, we
might infer the density of the fluid from the velocity of a spark or shock,
compared with that of sound ; or we might deduce its velocity from a
determination of its density. It has been supposed, although perhaps
somewhat hastily, that the actual velocity is nearly equal to that of
light*

When a conducting substance approaches another, which is electrified,
the distribution of the electric fluid within it is necessarily altered by
induction, before it receives a spark, so that its remoter extremity is
brought into a state similar to that of the first body : hence it happens
that when the spark passes, it produces less effect at the remoter end of
the substance, while the part presented to the electrified body is most

* Watson's Exp. to determine the Celerity of Electricity, Ph. Tr. 1748, pp. 49,
491. Wheatstone, ibid. 1834, p. 583.

ON ELECTRICITY IN MOTION. 517

affected, on account of its sudden change to an opposite state. But if
both ends approach bodies in opposite states of electricity, they will both
be strongly affected when the shock takes place, while the middle of the
circuit undergoes but little change.

The manner in which the electric fluid makes its way, through a more
or less perfect nonconductor, is not completely understood : it is doubtful
whether the substance is forced away on each side, so as to leave a vacuum
for the passage of the fluid, or whether the newly formed surface helps to
guide it in its way ; and in some cases it has been supposed that the
gradual communication of electricity has rendered the substance more
capable of conducting it, either immediately, or, in the case of the air, by
first rarefying it. However this may be, the perforation of a jar of glass
by an overcharge, and that of a plate of air by a spark, appear to be
effects of the same kind, although the charge of the jar is principally con-
tained in the glass, while the plate of air is perhaps little concerned in the
distribution of the electricity.

The actual direction of the electric current has not in any instance been
fully ascertained, although there are some appearances which seem to
justify the common denominations of positive and negative. Thus, the
fracture of a charged jar of glass, by spontaneous explosion, is well defined
on the positive, and splintered on the negative side, as might be expected
from the passage of a foreign substance from the former side to the latter ;
and a candle, held between a positive and a negative ball, although it
apparently vibrates between them, is found to heat the negative ball much
more than the positive. We cannot, however, place much dependence on
any circumstance of this kind, for it is doubtful whether any current of
the fluid, which we can produce, possesses sufficient momentum to carry
with it a body of sensible magnitude. It is in fact of little consequence to
the theory, whether the terms positive and negative be correctly applied,
provided that their sense remain determined ; and that, like positive and
negative quantities in mathematics, they be always understood of states
which neutralise each other. The original opinion of Dufay,* of the
existence of two distinct fluids, a vitreous and a resinous electricity, has at
present few advocates, although some have thought such a supposition
favoured by the phenomena of the galvanic decomposition of water.

When electricity is simply accumulated without motion, it does not
appear to have any effect, either mechanical, chemical, or physiological, by
which its presence can be discovered ; the acceleration of the pulse, and
the advancement of the growth of plants, which have been sometimes
attributed to it, have not been confirmed by the most accurate experi-
ments.t An uninterrupted current of electricity, through a perfect
conductor, would perhaps be also in every respect imperceptible, since the
best conductors appear to be the least affected by it. Thus, if we place

* Ph. Tr. 1733, xxxviii. 258. Hist, et Mem. 1733, p. 457.
t Consult Kies et Koestlin, DeEffectibus Electr. 4to, Tubing. 1775. Ingenhousz,
Versuche mit Pflanzen, 3 vols. Vienna, 1778-90. Bertholon, do. Leipz. 1785.

k, De 1' Application de 1'Electr. 4to, Amst. 1788. Van Marum, Proefne-

met Teylers Electrisir Maschine, 4to, Haarlem, 1795.

518 LECTURE LIV.

our hand on the conductor of an electrical machine, the electricity will
pass off continually through the body, without exciting any sensation. A
constant stream of galvanic electricity, passing through an iron wire is,
however, capable of exciting a considerable degree of heat, and if it be
transmitted through the hands of the operator, it will produce a slight
numbness, although in general some interruption of the current is neces-
sary in order to furnish an accumulation sufficient to produce sensible
effects ; and such an interruption may even increase the effect of a single
spark or shock ; thus, gunpowder is more readily fired by the discharge of
a battery passing through an interrupted circuit, than through a series of
perfect conductors.

The most common effect of the motion of the electric fluid is the produc-
tion of light. Light is probably never occasioned by the passage of the
fluid through a perfect conductor; for when the discharge of a large
battery renders a small wire luminous, the fluid is not wholly confined to
the wire, but overflows a little into the neighbouring space. There is
always an appearance of light, whenever the path of the fluid is inter-
rupted by an imperfect conductor ; nor is the apparent contact of conduct-
ing substances sufficient to prevent it, unless they are held together by a
considerable force ; thus, a chain, conveying a spark or shock, appears
luminous at each link, and the rapidity of the motion is so great, that we
can never observe any difference in the times of the appearance of the light
in its different parts ; so that a series of luminous points, formed by the
passage of the electric fluid, between a string of conducting bodies, repre-
sents at once a brilliant delineation of the whole figure in which they are
arranged. A lump of sugar, a piece of wood, or an egg, may easily be
made luminous in this manner ; and many substances, by means of their
properties as solar phosphori, retain for some seconds the luminous
appearance thus acquired. Even water is so imperfect a conductor, that a
strong shock may be seen in its passage through it ; and when the air is
sufficiently moistened or rarefied to become a conductor, the track of the
fluid through it is indicated by streams of light, which are perhaps derived
from a series of minute sparks passing between the particles of water or of
rarefied air. When the air is extremely rare, the light is greenish ; as it
becomes more dense, the light becomes blue, and then violet, until it
no longer conducts. The appearance of the electrical light of a point
enables us to distinguish the nature of the electricity with which it is
charged ; a pencil of light, streaming from the point, indicating that its
electricity is positive, while a luminous star, with few diverging rays,
shows that it is negative. The sparks, exhibited by small balls, differently
electrified, have also similar varieties in their forms, according to the
nature of their charges.* (Plate XL. Fig. 552.)

The production of heat by electricity frequently accompanies that of
light, and appears to depend in some measure on the same circumstances.
A fine wire may be fused and dissipated by the discharge of a battery ;
and without being perfectly melted, it may sometimes be shortened or

* Consult Doppelmayer, Ueber das Elektr. Licht. 1749. Nairne, Ph. Tr.JL777,
p. 614. Nicholson, ibid. 1789, p. 265. Davy on a Vacuum, ibid. 1822, p. 64rv

ON ELECTRICITY IN MOTION. 519

lengthened, accordingly as it is loose or stretched during the experiment.
The more readily a metal conducts, the shorter is the portion of it which
the same shock can destroy ; and it has sometimes been found that a
double charge of a battery has been capable of melting a quadruple length
of wire of the same kind.*

The mechanical effects of electricity are probably in many cases the
consequences of the rarefaction produced by the heat which is excited ;
thus, the explosion attending the transmission of a shock or spark through
the air, may easily be supposed to be derived from the expansion caused
by heat ; and the destruction of a glass tube, which contains a fluid in a
capillary bore, when a spark is caused to pass through it, is the natural
consequence of the conversion of some particles of the fluid into vapour.
But when a glass jar is perforated, this rarefaction cannot be supposed to
be adequate to the effect. It is remarkable that such a perforation may be
made by a very moderate discharge, when the glass is in contact with oil
or with sealing wax ; and no sufficient explanation of this circumstance
has yet been given.

A strong current of electricity, or a succession of shocks or sparks,
transmitted through a substance, by means of fine wires, is capable of
producing many chemical combinations and decompositions, some of which
may be attributed merely to the heat which it occasions, but others are
wholly different. Of these the most remarkable is the production of
oxygen and hydrogen gas from common water, which are usually extri-
cated at once, in such quantities, as, when again combined, will reproduce
the water which has disappeared ; but in some cases the oxygen appears to
be disengaged most copiously at the positive wire, and the hydrogen at the
negative.t

When the spark is received by the tongue, it has generally a subacid
taste ; and an explosion of any kind is usually accompanied by a smell
somewhat like that of sulfur, or rather of fired gunpowder. The peculiar
sensation, which the electric fluid occasions in the human frame, appears
in general to be derived from the spasmodic contractions of the muscles
through which it passes ; although in some cases it produces pain of a dif-
ferent kind ; thus, the spark of a conductor occasions a disagreeable
sensation in the skin, and when an excoriated surface is placed in the
galvanic current, a sense of smarting, mixed with burning, is experienced.
Sometimes the effect of a shock is felt most powerfully at the joints, on
account of the difficulty which the fluid finds in passing the articulating
surfaces which form the cavity of the joints. The sudden death of an
animal, in consequence of a violent shock, is probably owing to the im-
mediate exhaustion of the whole energy of the nervous system. It is
remarkable that a very minute tremor, communicated to the most elastic

* Kinnersley on an Electrical Air Thermometer, and on the Extension of Wire,
Ph. Tr. 1763, p. 84. Nairne on Shortening Wires, ibid. 1780, p. 334. Riess,
Fogg. Ann. xl. 321 ; xliii. 47 ; xlv. 1.

t Consult Cavendish, Ph. Tr. 1788, p. 261. Pearson, ibid. 1797, p. 142. Wol-
laston, ibid. 1801, p. 417. Davy, ibid. 1807, p. 1. Van Trostwyk, Gren's Jour,
i*. 130. Schonbein, Pogg. Ann. 1. 616.

620 LECTURE LIV.

parts of the body, in particular to the chest, produces an agitation of the
nerves, which is not wholly unlike the effect of a weak electricity.

The principal modes, in which the electric equilibrium is primarily
destroyed, are simple contact, friction, a change of the form of aggrega-
tion, and chemical combinations and decompositions. The electricity pro-
duced by the simple contact of any two substances is extremely weak, and
can only be detected by very delicate experiments : in general it appears
that the substance which conducts the more readily, acquires a slight
degree of negative electricity, while the other substance is positively
electrified in an equal degree. The same disposition of the fluid is also
usually produced by friction, the one substance always losing as much as
the other gains ; and commonly although not always, the worst conductor
becomes positive. At the instant in which the friction is applied, the
capacities or attractions of the bodies for electricity appear to be altered,
and a greater or less quantity is required for saturating them ; and upon
the cessation of the temporary change, this redundancy or deficiency is
rendered sensible. When two substances of the same kind are rubbed
together, the smaller or the rougher becomes negatively electrified ; perhaps
because the smaller surface is more heated, in consequence of its under-
going more friction than an equal portion of the larger, and hence becomes
a better conductor ; and because the rougher is in itself a better conductor
than the smoother, and may possibly have its conducting powers increased
by the greater agitation of its parts which the friction produces. The back
of a live cat becomes positively electrified, with whatever substance it
is rubbed ; glass is positive in most cases, but not when rubbed with
mercury in a vacuum, although sealing wax, which is generally negative,
is rendered positive by immersion in a trough of mercury. When a white
and a black silk stocking are rubbed together, the white stocking acquires
positive electricity, and the black negative, perhaps because the black dye
renders the silk both rougher and a better conductor.

Those substances, which have very little conducting power, are some-
times called electrics, since they are capable of exhibiting readily the
electricity which friction excites on their surfaces, where it remains accu-
mulated, so that it may be collected into a conductor ; while the surfaces
of such substances, as have greater conducting powers, do not so readily
imbibe the fluid from others with which they are rubbed, since they may
be supplied from the internal parts of the substances themselves, when
their altered capacity requires it; thus, glass, when heated to 110° of
Fahrenheit, can with difficulty be excited, becoming an imperfect con-
ductor : but a thin plate of a conducting substance, when insulated, may
be excited almost as easily as an electric, commonly so called.

Vapours are generally in a negative state, but if they rise from metallic
substances, or even from some kinds of heated glass, the effect is uncertain,
probably on account of some chemical actions which interfere with it.
Sulfur becomes electrical in cooling, and wax candles are said to be some-
times found in a state so electrical, when they are taken out of their
moulds, as to attract the particles of dust which are floating near them.
The tourmalin, and several other crystallized stones, become electrics^

ON ELECTRICITY IN MOTION. 521

when heated or cooled, and it is found that the disposition, assumed by
the fluid, bears a certain relation to the direction in which the stone trans-
mits the light most readily ; some parts of the crystal being rendered
always positively and others negatively electrical, by an increase of
temperature.

The most remarkable of the phenomena, attending the excitation of
electricity by chemical changes, are those which have lately received the
appellation of galvanic. Some of the effects which have been considered
as belonging to galvanism are probably derived from the electrical powers
of the animal body, and the rest have been referred by Mr. Volta, and
many other philosophers on the continent, to the mere mechanical actions
of bodies possessed of different properties with regard to electricity.
Thus, they have supposed that when a circulation of the electric fluid is
produced through a long series of substances in a certain direction, the
differences of their attractions and of their conducting powers, which
must remain the same throughout the process, keep up this perpetual
motion, in defiance of the general laws of mechanical forces. In this
country it has been generally maintained, that no explanation founded on
such principles could be admissible, even if it were in all other respects
sufficient and satisfactory, which the mechanical theory of galvanism cer-
tainly is not.

The phenomena of galvanism appear to be principally derived from an
inequality in the distribution of the electric fluid, originating from
chemical changes, and maintained by means of the resistance opposed to
its motion, by a continued alternation of substances of different kinds,
which furnishes a much stronger obstacle to its transmission than any of
those substances alone would have done. The substances employed must
neither consist wholly of solids nor of fluids, and they must be of three
different kinds, possessed of different powers of conducting electricity;
but whether the difference of their conducting powers is of any other con-
sequence than as it accompanies different chemical properties, is hitherto
undetermined. Of these three substances, two must possess a power of
acting mutually on each other, while the other appears to serve principally
for making a separate connexion between them : and this action may be of
two kinds, or perhaps of more ; the one is oxidation, or the combination
of a metal or an inflammable substance with a portion of oxygen derived
from water or from an acid, the other sulfuration, or a combination with
the sulfur contained in a solution of an alkaline sulfuret.

We may represent the effects of all galvanic combinations, by con-
sidering the oxidation as producing positive electricity in the acting liquid,
and the sulfuration as producing negative electricity, and by imagining
that this electricity is always communicated to the best conductor of the
other substances concerned, so as to produce a circulation in the direction
thus determined. For example, when two wires of zinc and silver,
touching each other, are separately immersed in an acid, the acid, becoming
positively electrical, imparts its electricity to the silver, and hence it flows
back into the zinc : when the ends of a piece of charcoal are dipped into
water and into an acid, connected together by a small tube, the acid,

522 LECTURE LIV.

becoming positive, sends its superfluous fluid through the charcoal into the
water ; and if a wire of copper be dipped into water and a solution of
alkaline sulfuret, connected with each other, the sulfuret, becoming nega-
tive, will draw the fluid from the copper on which it acts ; and in all these
cases the direction of the current is truly determined, as it may be shown by
composing a battery of a number of alternations of this kind, and either
examining the state of its different parts by electrical tests, or connecting
wires with its extremities, which, when immersed into a portion of water,
will exhibit the production of oxygen gas where they emit the electric
fluid, and of hydrogen where they receive it. These processes of oxidation
and of sulfuration may be opposed to each other, or they may be combined
in various ways, the sum or difference of the separate actions being ob-
tained by their union ; thus it usually happens that both the metals
employed are oxidable in some degree, and the oxidation, which takes place
at the surface of the better conductor, tends to impede the whole effect,
perhaps by impeding the passage of the fluid through the surface. The
most oxidable of the metals, and probably the worst conductor, is zinc ;
the next is iron ; then come tin, lead, copper, silver, gold, and platina.
(Plate XL. Fig. 553.. .555.)

In the same manner as a wire charged with positive electricity causes
an extrication of oxygen gas, so the supply of electricity through the more
conducting metal promotes the oxidation of the zinc of a galvanic battery ;
and the effect of this circulation may be readily exhibited, by fixing a wire
of zinc, and another of silver or platina, in an acid, while one end of each
is loose, and may be brought together or separated at pleasure : for at the
moment that the contact takes place, a stream of bubbles rising from the
platina, and a white cloud of oxid falling from the zinc, indicate both the
circulation of the fluid and the increase of the chemical action. But when,
on the other hand, a plate of zinc is made negative by the action of an
acid on the greater part of its surface, a detached drop of water has less
effect on it, than in the natural state : while a plate of iron, which touches
the zinc, and forms a part of the circle with it, is very readily oxidated at a
distant point : such a plate must therefore be considered, with regard to
this effect, as being made positive by the electricity which it receives from
the acid or the water ; unless something like a compensation be supposed
to take place, from the effects of induced electricity. Instead of the ex-
trication of hydrogen, the same causes will sometimes occasion a deposition
of a metal which has been dissolved, will prevent the solution of a metal
which would otherwise have been corroded, or produce some effects which
appear to indicate the presence of an alkali, either volatile or fixed. All
these operations may, however, be very much impeded by the interposition
of any considerable length of water, or of any other imperfect conductor.
(Plate XL. Fig. 556.)

It is obvious, that since the current of electricity, produced by a galvanic
circle, facilitates those actions from which its powers are derived, the effect
of a double series must be more than twice as great as that of a single one :
and hence arises the activity of the pile of Volta, the discovery of which
forms the most important era in the history of this department of natural

ON ELECTRICITY IN MOTION. 523

knowledge. The intensity of the electrical charge, and the chemical and
physiological effects of a pile or battery, seem to depend principally on the
number of alternations of substances ; the light and heat more on the
joint magnitude of the surfaces employed. In common electricity, the
greatest heat appears to be occasioned by a long continuation of a slow
motion of the fluid ; and this is perhaps best furnished in galvanism by a
surface of large extent, while some other effects may very naturally be
expected to depend on the intensity of the charge, independently of the
quantity of charged surface. It may easily be imagined, that the tension
of the fluid must be nearly proportional to the number of surfaces, im-
perfectly conducting, which are interposed between the ends of a pile or
battery, the density of the fluid becoming greater and greater by a limited
quantity at each step ; and it is easily understood, that any point of the
pile may be rendered neutral, by a connexion with the earth, while those
parts which are above it or below it, will still preserve their relations un-
altered with respect to each other : the opposite extremities being, like the
opposite surface of a charged jar, in contrary states, and a partial discharge
being produced, as often as they are connected by a conducting substance.
The various forms in which the piles or troughs are constructed, are of
little consequence to the theory of their operation : the most convenient
are the varnished troughs, in which plates of silvered zinc are arranged
side by side, with intervening spaces for the reception of water, or of an
acid. (Plate XL. Fig. 557.)

It is unquestionable that the torpedo, the gymnotus electricus, and some
other fishes, have organs appropriated to the excitation of electricity, and
that they have a power of communicating this electricity at pleasure to
conducting substances in their neighbourhood. These organs somewhat
resemble in their appearance the plates of the galvanic pile, although we
know nothing of the immediate arrangement, from which their electrical
properties are derived ; but the effect of the shock, which they produce,
resembles in all respects that of the weak charge of a very large battery.
It has also been shown by the experiments of Galvani, Volta, and Aldmi,
that the nerves and muscles of the human body possess some electrical
powers, although they are so much less concerned in the phenomena which
were at first attributed to them by Galvani, than he originally supposed,
that many philosophers have been inclined to consider the excitation of
electricity as always occasioned by the inanimate substances employed, and
the spasmodic contractions of the muscles as merely very delicate tests of
the influence of foreign electricity on the nerves.

Such is the general outline of the principal experiments and conclusions
which the subject of galvanism afforded before Mr. Davy's* late ingenious
and interesting researches, which have thrown much light, not only on the
foundation of the whole of this class of phenomena, but also on the nature
of chemical actions and affinities in general. Mr. Davy is inclined to infer
from his experiments, that all the attractions, which are the causes of
chemical combinations, depend on the opposite natural electricities of the

% Outlines of a view of Galvanism, Jour, of the Roy. Inst. i. 49. Also, Hi. Tr.
1807, p. 1 ; 1808, pp. 1, 333; 1809, p. 1.

524 LECTURE LIV.

bodies concerned ; since such bodies are always found, by delicate tests, to
exhibit, either during their contact or after separation, marks of different
species of electricity ; and their mutual actions may be either augmented
or destroyed, by increasing their natural charges of electricity, or by
electrifying them in a contrary way. Thus, an acid and a metal are found
to be negatively and positively electrical with respect to each other ; and
by further electrifying the acid negatively, and the metal positively, their
combination is accelerated ; but when the acid is positively electrified, or
the metal negatively, they have no effect whatever on each other. The
acid is also attracted, as a negative body, by another positively electrified,
and the metal by a body negatively electrified, so that a metallic salt may
be decomposed in the circuit of Volta, the positive point attracting the
acid, and the negative point the metal ; and these attractions are so strong
as to carry the particles of the respective bodies through any intervening
medium, which is in a fluid state, or even through a moist solid ; nor are
they intercepted in their passage, by substances which, in other cases, have
the strongest elective attractions for them. Alkali, sulfur, and alkaline
sulfurets, are positive with respect to the metals, and much more with
respect to the acids : hence they have a very strong natural tendency to
combine with the acids and with oxygen : and hydrogen must also be con-
sidered as belonging to the same class with the alkalis.

Supposing now a plate of zinc to decompose a portion of water ; the
oxygen, which has a negative property, unites with the zinc, and probably
tends to neutralise it, and to weaken its attractive force ; the hydrogen is
repelled by the zinc, and carries to the opposite plate of silver its natural
positive electricity ; and if the two plates be made to touch, the energy of
the plate of zinc is restored, by the electricity which it receives from the
silver : and it receives it the more readily, as the two metals, in any case
of their contact, have a tendency to become electrical, the zinc positively,
and the silver negatively. Mr. Davy therefore considers this chemical
action as destroying, or at least counteracting, the natural tendency of the
electric fluid to pass from the water to the zinc, and from modifications of
this counteraction he explains the effects of galvanic combinations in all
cases. Thus, in a circle composed of copper, sulfuret, and iron, the fluid
tends to pass from the iron towards the sulfuret, and from the copper to
the iron, in one direction, and in the opposite direction from the copper to
the sulfuret, with a force which must be equal to both the others, since
there would otherwise be a continual motion without any mechanical
cause, and without any chemical change ; but the action of the sulfuret on
the copper tends to destroy its electromotive, or rather electrophoric, power,
of directing the current towards the sulfuret, and its combination with the
sulfur makes it either positively electrical, or negatively electrical in a less
considerable degree ; consequently the fluid passes, according to its natural
tendency, from the copper to the iron, and from the iron to the sulfuret.
In a third case, when copper, an acid, and water, form a circle, the natural
tendency is from the acid to the copper on one side, and from the acid to
the water, and from the water to the copper on the other ; here we niust
suppose the first force to be only a little weakened by the chemical action,

ON ELECTRICITY IN MOTION. 525

while the third is destroyed, so that the first overcomes the second, and the
circulation is determined, although very feebly, in such a direction, that
the fluid passes from the acid to the copper. When, in the fourth place,
the combination consists of copper, sulfuret, and water, the tendencies are,
first, from the copper to the sulfuret, and from the water to the copper ;
and secondly, from the water to the sulfuret : in this instance a chemical
action must be supposed between the oxygen of the water and the sulfuret,
which lessens the electromotive tendency, more than the action that takes
place between the sulfuret and the copper, so that the fluid passes from the
copper to the sulfuret ; and the current has even force enough to prevent
any chemical action between the water and the copper, which would tend
to counteract that force, if it took place.

Mr. Davy has observed that the decomposition of the substances
employed in the battery of Volta, is of much more consequence to their
activity than either their conducting power, or their simple action on the
other elements of the series : thus, the sulfuric acid, which conducts elec-
tricity better, and dissolves the metals more readily, than a neutral
solution, is, notwithstanding, less active in the battery, because it is not
easily decomposed. Mr. Davy has also extended his researches, and the
application of his discoveries, to a variety of natural as well as artificial
phenomena, and there can be no doubt but that he will still make such
additions to his experiments, as will be of the greatest importance to this
branch of science.

The operation of the most usual electrical machines depends first on the
excitation of electricity by the friction of glass on a cushion of leather,
covered with a metallic amalgam, usually made of mercury, zinc, and tin,
which probably, besides being of use in supplying electricity readily to
different parts of the glass, undergoes in general a chemical change, by
means of which some electricity is extricated. The fluid, thus excited, is
received into an insulated conductor by means of points, placed at a small
distance from the surface which has lately undergone the effects of friction,
and from this conductor it is conveyed by wires or chains to any other
parts at pleasure. Sometimes also the cushion, instead of being connected
with the earth, is itself fixed to a second conductor, which becomes nega-
tively electrified ; and either conductor may contain within it a jar, which
may be charged at once by the operation of the machine, when its
internal surface is connected either with the earth, or with that of the
jar contained in the opposite conductor. The glass may be either in the
form of a circular plate or of a cylinder, and it is uncertain which of the
arrangements affords the greatest quantity of electricity from the same
surface ; but the cylinder is cheaper than the plate, and less liable to
accidents, and appears to be at least equally powerful. (Plate XL. Fig.
558, 559.)

The plate machine in the Teylerian museum, employed by Van Marum,
when worked by two men, excited an electricity, of which the attraction
was sensible at the distance of 38 feet, and which made a point luminous
at 27 feet, and afforded sparks nearly 24 inches long. A battery charged
by it, melted at once twenty five feet of fine iron wire. Mr. Wilson had

526 LECTURE LIV.

also a few years ago, in the Pantheon in London, an apparatus of singular
extent; the principal conductor was 150 feet long, and 16 inches in
diameter, and he employed a circuit of 4,800 feet of wire.*

The electrophorus derives its operation from the properties of induced
electricity. A cake of a nonconducting substance, commonly of resin or of
sulfur, is first excited by friction, and becomes negatively electric : an
insulated plate of a conducting substance, being placed on it, does not come
sufficiently into contact with it to receive its electricity, but acquires by
induction an opposite state at its lower surface, and a similar state at its
upper ; so that when this upper and negative surface is touched by a
substance communicating with the earth, it receives enough of the electric
fluid to restore the equilibrium. The plate then being raised, the action of
the cake no longer continues, and the electricity, which the plate has
received from the earth, is imparted to a conductor or to a jar ; and the
operation may be continually repeated, until the jar has received a charge
of an intensity equal to that of the plate when raised. Although the
quantity of electricity, received by the plate, is exactly equal to that
which is emitted from it at each alternation, yet the spark is far less
sensible ; since the effect of the neighbourhood of the cake is to increase
the capacity of the plate, while the tension or force impelling the fluid is
but weak ; and at the same time the quantity received is sufficient, when
the capacity of the plate is again diminished, to produce a much greater
tension, at a distance from the cake. (Plate XL. Fig. 560.)

The condenser acts in some measure on the same principles with the
electrophorus, both instruments deriving their properties from the effects of
induction. The use of the condenser is to collect a weak electricity from
a large substance into a smaller one, so as to make its density or tension
sufficient to be examined. A small plate, connected with the substance, is
brought nearly into contact with another plate communicating with the
earth ; in general a thin stratum of air only is interposed ; but sometimes
a nonconducting varnish is employed ; this method is, however, liable to
some uncertainty, from the permanent electricity which the varnish some-
times contracts by friction. The electricity is accumulated by the attrac-
tion of the plate communicating with the earth, into the plate of the
condenser ; and when this plate is first separated from the substance to be
examined, and then removed from the opposite plate, its electricity is
always of the same kind with that which originally existed in the sub-
stance, but its tension is so much increased as to render it more easily
discoverable. This principle has been variously applied by different
electricians, and the employment of the instrument has been facilitated by
several subordinate arrangements. (Plate XL. Fig. 561.)

Mr. Cavallo's multiplierf is a combination of two condensers ; the
second or auxiliary plate of the first, like the plate of the electrophorus, is
moveable, and carries a charge of electricity, contrary to that of the
substance to be examined, to the first or insulated plate of the second
condenser, which receives it repeatedly, until it has acquired an equal

* Wilson's Account of his Experiments, 4to, 1778. \

t Nich. Jour. i. 394.

ON ELECTRICITY IN MOTION. 527

degree of tension ; and when the two plates of this condenser are sepa-
rated, they both exhibit an electricity much more powerful than that of
the first condenser. The force is, however, still more rapidly augmented
by the instruments of Mr. Bennet* and Mr. Nicholson,t although it has
been supposed that these instruments are more liable to inconvenience
from the attachment of a greater portion of electricity to the first plate
of the instrument, which leaves, for a very considerable time, a certain
quantity of the charge not easily separable from it. Mr. Bennet employs
three varnished plates laid on each other, but Mr. Nicholson has substituted
simple metallic plates, approaching only very near together, so that there
can be no error from any accidental friction. In both of these instruments,
the second plate of a condenser acquires an electricity contrary and nearly
equal to that of the first, by means of which it brings a third plate very
nearly into the same state with the first ; and when the first and third
plates are connected and insulated, they produce a charge nearly twice as
great in the second plate, while the first plate becomes at the same time
doubly charged ; so that by each repetition of this process, the intensity of
the electricity is nearly doubled : it is therefore scarcely possible that any
quantity should be so small as to escape detection by its operation. (Plate
XL. Fig. 562, 563.)

The immediate intensity of the electricity may be measured, and its
character distinguished, by electrical balances, and by electrometers of
different constructions. The electrical balance measures the attraction or
repulsion exerted by two balls at a given distance, by the magnitude of the
force required to counteract it ; and the most convenient manner of apply-
ing this force is by the torsion of a wire, which has been employed for the
purpose by Mr. Coulomb.^ The quadrant electrometer of Henley § ex-
presses the mutual repulsion of a moveable ball and a fixed column, by the
divisions of the arch to which the ball rises. These divisions do not
exactly denote the proportional strength of the action, but they are still of
utility in ascertaining the identity of any two charges, and in informing us
how far we may venture to proceed in our experiments with safety; and
the same purpose is answered, in a manner somewhat less accurate, by the
electrometer, consisting of two pith balls, or of two straws, which are
made to diverge by a smaller degree of electricity. Mr. Bennet's electro-
meter || is still more delicate ; it consists of two small portions of gold leaf,
suspended from a plate, to which the electricity of any substance is com-
municated by contact : a ve*y weak electricity is sufficient to make them
diverge, and it may easily be ascertained whether it is positive or negative,
by bringing an excited stick of sealing wax near the plate, since its
approach tends to produce by induction a state of negative electricity in
the remoter extremities of the leaves, so that their divergence is either
increased or diminished, accordingly as it was derived from negative or
from positive electricity : a strip of gold leaf or tin foil, fixed within the

* Ph. Tr. 1787, pp. 32, 288 ; 1794, p. 266.
t Ibid. 1788, p. 403. Nich. Jour. i. 395 ; ii. 370 ; iv. 95.
% Hist, et Mem. 1785, p. 569. § Ph. Tr. 1772, p. 359.

1 Ph. Tr. 1787, p. 26.

528 LECTURE LIV.

glass which covers the electrometer, opposite to the extremities of the
leaves, prevents the communication of any electricity to the glass, which
might interfere with the action of the instrument. When the balls of an
electrometer stand at the distance of 4 degrees, they appear to indicate a
charge nearly 8 times as great as when they stand at one degree : a charge
8 times as great in each ball producing a mutual action 64 times as great
at any given distance, and at a quadruple distance a quadruple force ; in
the same manner a separation of 9 degrees is probably derived from an
intensity 27 times as great as at 1. In Lane's electrometer* the magnitude
of a shock is determined by the quantity of air through which it is obliged
to pass, between two balls, of which the distance may be varied at plea-
sure ; and the power of the machine may be estimated by the frequency of
the sparks which pass at any given distance. It appears from Mr. Lane's
experiments, that the quantity of electricity required for a discharge is
simply as the distance of the surfaces of the balls, the shocks being twice
as frequent when this distance is only -^ of an inch as when it is TV Mr.
Volta says, that the indications of Lane's and Henley's electrometer agree
immediately with each other ; but it seems difficult to reconcile this result
with the general theory. Sometimes the force of repulsion between two
balls in contact is opposed by a counterpoise of given magnitude, and as
soon as this is overcome, they separate and form a circuit which discharges
a battery ; whence the instrument is called a discharger. (Plate XL. Fig.
564... 568.)

It must be confessed that the whole science of electricity is yet in a very
imperfect state : we know little or nothing of the intimate nature of the
substances and actions concerned in it : and we can never foresee, without
previous experiment, where or how it will be excited. We are wholly
ignorant of the constitution of bodies, by which they become possessed of
different conducting powers ; and we have only been able to draw some
general conclusions respecting the distribution and equilibrium of the sup-
posed electric fluid, from the laws of the attractions and repulsions that it
appears to exert. There seems to be some reason to suspect, from the
phenomena of cohesion and repulsion, that the pressure of an elastic
medium is concerned in the origin of these forces ; and if such a medium
really exists, it is perhaps nearly related to the electric fluid. The identity
of the general causes of electrical and of galvanic effects is now doubted by
few ; and in this country the principal phenomena of galvanism are
universally considered as depending on chemical changes ; perhaps, also,
time may show, that electricity is very materially concerned in the essen-
tial properties, which distinguish the different kinds of natural bodies, as
well as in those minute mechanical actions and affections which are
probably the foundation of all chemical operations ; but at present it is
scarcely safe to hazard a conjecture on a subject so obscure, although Mr.
Davy's experiments have already in some measure justified the boldness of
the suggestion.

* Ph. Tr. 1767, p. 451.

.0*

ON ELECTRICITY IN MOTION. 529

LECT. LIV.— ADDITIONAL AUTHORITIES.

Excitation of Electricity. By simple contact. — Weber, Korper ohne Reiben zu
Elek. 1781. Bennet, Ph. Tr. 1787, p. 26; Nich. Jour. 8vo. i. 144, 184. Haiiy
sur FElect. de la Pression, Ann. de Ch. v. 95. Becquerel, ibid.xxii. 91.

By Friction.— Hauksbee, Ph. Tr. 1705, p. 2165. Gray, ibid. 1735, p. 166.
Symmer, ibid. 1759, p. 308. Beccaria, ibid. 1766, p. 105. Bergmann, Schwed.
Abhand. xxv. p. 344. Henley, Ph. Tr. 1774, p. 389 ; 1777, p. 122. Wilcke, De
Electr. contrariis, Gott. 1790. Ritter, Das Elektr. System der Korper. Pereyro,
Arch, de 1'Electr. ii. 395. Lists of substances which, under certain conditions,
produce certain kinds of electricity, will be found in Erxleben's Naturlehre by Lich-
tenberg, and in Cavallo's Treatise on Electr.

By Steam. — Lavoisier and Laplace sur 1'Electr. qu' absorbent les Corps qui se
reduisent en Vapeurs, Hist, et Mem. 1781, p. 292, H. 6. Volta, Del Modo di
render sensibilissima la piu Debole Elettr. Appendice Ph. Tr. 1782, p. 274. Me-
teorologische Briefe, Leipz. 1793, p. 193. Bennet, Ph. Tr. 1787. Erman, Abhand.
der Ak. zu Berlin, 1814, p. 123. Pouillet sur FElectr. des Fluides Elastiques, Ann.
de Ch.xxxv. 365, 401. Armstrong, Ph. Mag. xvii. 370, 452, xviii. 51, 328, xx. 5,
xxii. 1, xxiii. 194. Pattinson, ibid. xvii. 376, 457. Schafhautl, ibid. xvii. 449,
xviii. 14, 95, 265. Williams, ibid, xviii. 93. Faraday, Ph. Tr. 1843, p. 1.

Electrical Apparatus. — Bohnenberger's Elektrisirmaschinen, Stuttg. 1784.
Guttle, Instrumenten Kabinet, 1790. Kunze, Schauplatz der Gemeinniitzigen Ma-
schinen.

Electrical Machines. Otto v. Guericke, Experimenta Nova de Vacuo Spatio,
Amst. 1672, p. 140. Hauksbee on a Glass Globe lined with Sealing Wax, Ph. Tr.
1708, p. 219. Hausen, Novi Profectus in Historia Electr. 4to, 1743. Winkler,
Descriptio Pyrorgani sui Electr. Ph. Tr. 1747, p. 497. Faure, Congetture intorno
alia Mach. Elettr. 4to, Rome, 1747. Espinasse, Ph. Tr. 1767, p. 186. Leroy, A
Machine for producing both Species of Electr. Hist, et Mem. 1772, I. 499, H. 9.
Nooth on the Cushion and Flap, Ph. Tr. 1773, p. 333. Nairne, ibid. 1774, p. 79,
and Treatise on do. 1787. Planta's Plate Machine, Allg. Deutsche Bibliot. xxiv.
549. Ingenhousz, do. Ph. Tr. 1769, p. 659. Schmidt, Beschreibung einer Elek. 4to,
Berlin, 1778. Langenbiicher, do. Anspach, 1780. Bohnenberger, 1784. Rouland,
Description des Machines a Taffetas, Amst. 1785. Van Marum, Description d'une
tres Grande Machine, 4to, Haarlem, 1785. Nicholson's Exp. Ph. Tr. 1789, p. 265.
Cuthbertson, Beschreibung einer Elektrisirmaschine von Deimann und Trostwyk,
Leipz. 1790. Wolff's, Gilb. Ann. xii. 597. Wolfram's, ibid. Ixxiv. 53. Hare's,
Sturgeon's Ann. i. 487. Page's, Silliman's American Journal, xxvi. 110. Dujar-
din's, Ann. de Ch. N. S. ix. 111.

Jars and Batteries. — Kriiger, Geschichte der Erde, Halle, 1746, p. 177, announces
the discovery of Von Kleist, of the power of charged glass. (Cuneus) Musschen-
broek, Hist, et Mem. 1746, p. 2. Winkler, Die Starke der Elekt. Kraft des
Wassers, Leipz. 1746. Wilcke, Schwed. Abhand. 1758, p. 241 ; 1762, pp. 213,
253. Wilkinson on the Ley den Phial, 1798. Dana, Schweigg. Jour, xxiii. 257.

Electrophorus. — Volta, Lettere sul Elettroforo Perpetuo, Scelta di Opusculi
Interessanti, Milan, viii. 127 (1775), ix. 91, x. 37. Wilcke, Schwed. Abhand. 1777,
pp. 54, 116, 200. Ingenhouss, Exp. and Theory of do. Ph. Tr. 1778, p. 1027.
Henley, ibid. 1778, p. 1049. Kraft, Acta Petr. 1771, p. 154. Achard, Hist, et
Mem. de Berlin, 1766, p. 162. Klindworth, Goth. Mag. i. II. 35. Lichtenberg,
ibid. i. II. 42. Weber, Beschreibung des Luftelektrophorus, Augsb. 1779.

Condensers. — Volta on the Method of rendering very sensible Small Degrees of
Electr. Ph. Tr. 1 782, p. 237. Cavallo on manifesting Small Quantities of Electr. ibid.
1788, p. 1. His Collector, ibid. 1788, p. 255. Cuthbertson, Nich. Jour. ii. 281.
Read, ibid, ii. 495. Bohnenberger, Beschreibung Elektritatsnerdoppeler, Tubing.
1798.

Electrometers, Sfc.— D'Arcy's Electr. Hist, et Mem. 1749, p. 63, H. 7. Rich-
mann's, Nov. Com. Petr. iv. 301. Comus's, Jour, de Phy. vii. 520. Canton's,
Ph. Tr. xlviii. 350, 780. Cavallo's, ibid. 1777, p. 388; 1780, p. 15. Brooke's,
ibid. 1782, p. 384. Saussure's, Voyages, ii. 202. Deluc's, Nouvelles Idees sur la
Meteorologie, p. 397. Lawson's, Ph. Mag. xi. 251. Marechaux's, Gilb. Ann. xv.
93, 99i xvi. 115, xix. 476, xx. 357, xxii. 318, xxv. 4, 18, xxvi. 29, 123. Behrens's,
ibid. <xxiii. 24. Bohnenberger's, ibid. Ii. 390. Oersted's, Pogg. Ann. liii. 612.
Harris's, Ph. Tr. 1836, p. 447.

2 M

530 LECTURE LIV.

Galvanism. — Al. Galvani, De Viribus Electr. inMotu Musculari Commentarius ,
Bolog. 1791 ; Com. Bon. vii. O. 363. Mayer, Abhand. von Galvani und Andern,
2 vols. Prague, 1793. Creve, do. Frankf. 1793. Volta on Galvani's Discoveries,
Ph. Tr. 1793, p. 10 ; Annales de Ch. xxiii. 270, xl. 255 ; Nich. Jour. 8vo. i. 135 ;
on the Electr. excited by contact. Ph. Tr. 1800, p. 403 ; on the Identity of the
Electr. and Galv. Fluids, Brugnatelli's Ann. xix. 38, 163. Collezione dell'-Opere di
Volta, 3 vols. Firenza, 1816. Pfaff on Volta' s Theory, &c. Gilb. Ann. x. 219, Lxviii.
273 ; Schweigger's Jour. xlvi. 129 ; Uebersicht iiber den Voltaismus, Stuttg. 1804 ;
Ann. de Ch. xli. 236. Aldini, De Animali Electr. 4to, Bolog. 1794. Esperienze sul
Galvanismo, Bolog. 1802. Essai Theorique, 2 vols. Paris, 1804. Ritter, Beitragezu
Nahren Kentniss des Galv. Jena, 1800-5. Experiments and Remarks, Gilb. Jour,
ii. vii. viii. 386, ix. 1, 212, xiii. 1, 265, xvi. 293 ; Nich. Jour. vi. 223, vii. 288.
Sue, Hist, de Galvan. 2 vols. Paris, 1802. Fourcroy's Experiments, Ann. de Ch.
xxxix. 103. Deluc, Traite Elem. sur le Fluide Electro-galvanique, 2 vols. Paris,
1804. Wilkinson's Elements of Galv. 2 vols. 1804. Bellinger!, Esp. ed Obs. sul
Galv. 4to, Turin, 1816. Bostock's Hist, of Galv. 1818. Parrot, Handbuch der
Physik, vol. ii. Muller, Elemente der Elek. und Elektro Chemie, Berlin, 1819.
Karsten, Ueber Contact Electr. Berlin, 1836. Negro, Experiment!, 4to, Padova,
1833. Daniell's Introd. to the Study of Chem. Phil. 1843. Golding Bird's Nat. Ph.
184. . Le Journal de Physique, Bulletin de la Soc. Philom., Annales de Ch., Gilbert's
Annalen, Schweigger's do., Poggendorff's do., Brugnatelli's Jour., Nich. Jour., and
the other periodicals of the period, contain numerous accounts of experiments and
remarks by Volta, Nicholson, Carlisle, Biot, Cuvier, Lehof, Berzelius, Fourcroy,
Grimm, Ritter, Hermbstadt, Hebebrand, Pfaff, Bourguet, Davy, Heidemann, Rein-
hold, Curtet, Bouvier, Erman, Aldini, Pepys, Jaeger, Bunzen, Brugnatelli, Gilbert,
Von Arnim, Friedlander, Cruickshank, Eandi, Robertson, Desormes, Cuthbertson,
Garboin, Wilkinson, Rossi, Gautherot, Fabroni, and others ; whilst the more
recent periodicals, such as the Archives de 1'Electr., Mem. de la Soc.de Philos. et
d'Hist. Nat. de Geneve, the Comptes Rendus, Jameson's Jour., the Philosophical
Mag., Quarterly Journal of Science, Silliman's Jour. &c. contain papers by Peclet,
Belli, Fechner, Becquerel, Henrici, Martens, Marianini, Parrot, Draper, Grove,
Daniell, Hare, De la Rive, Oersted, Schonbein, Heer, Andrews, Seebeck, Golding,
Bird, and others.

Mathematical Theory of Galvanism, — Ohm, Die Galvanische Kette Mathema-
tisch bearbeitet, Berlin, 1827; translated in Taylor's Scientific Memoirs, ii. 401 ;
also papers in Schweigg. Jour, xliv, 110, xlix. 1, Iviii. 393, Ixiii. 1, 159, 385, Ixiv.
21, 138, 257, Ixvii. 341 ; Pogg. Ann. vi. 459, vii. 45, 117; Karsten's Archiv, vii.
1, 452, xiv. 475, xvi. 1. Fechner, Maasbestimmungen iiber die Galvanische Kette,
4to, Leipz. 1831; also papers in Schweigg. Jour. Ivii. 9, 291, Ix. 17, Ixiii. 249;
Henrici in Pogg. Ann. liii. 277; Pogg. in Pogg. Ann. liv. 161, Iv. 43, Ivi. 353 ;
Jacobi in do. Ivii. 85 ; Draper in Ph. Mag. xv. 206, 339. Heer in Bullet, des
Sciences Physiques en Neerlande, 1839, p. 319 ; 1840, p. 132 ; Lenz in Pogg. Ann.
xlvii. 584, lix. 203, 407.

Galvanic Apparatus. — Izarn, Manuel du Galvanisme, Paris, 1805. Children,
Ph. Tr. 1809, p. 32. Cruickshank's, Nich. Jour. 4to, 187, 254. Wollaston, Thorn-
son's Annals, vi. 209, and in Children's Account of Exp. with a Large Battery, Ph.
Tr. 1815, p. 363. Pepys. ibid. 1823, p. 187. Hare's New Theory of Galvanism,
Philadelph. 1819. Silliman's Journal, vii. 347. Zamboni, L' Elettromotoro Perpe-
tuo, 2 vols. Verona, 1820. Faraday's Exp. Researches, 10th series, &c. Daniell's
Constant Battery, Ph. Tr. 1836, p. 117 ; 1837, p. 141 ; 1838, p. 41 ; 1839, p. 89 ;
1842, p. 137. Becquerel's Battery, Pogg. Ann. xxxvii. 429. Karsten, Ueber Con-
tactelekt. Berlin, 1838. Grove on a Gas Battery, Ph. Mag. xxi. 417.

Animal Electricity. See also Galvanism. Torpedo.— Redi, Exp. Natur. 1666.
Lorenzi, Obs. intorno alii Torp. 1678. Reaumer, Hist, et Mem. 1714, p. 344,
H. 19. Walsh, Ph. Tr. 1773, pp. 63, 461; 1775, p. 465. Hunter, ibid. 1773,
p. 481, with plates. Pringle's Discourse, 4to, 1775. Cavendish's Imitation of the
Torp. Ph. Tr. 1776, p. 196. Spallanzani, Op. Scelt. di Milano, 1783. Mem.
della Soc. Ital. ii. 603. Girardi, ibid. iii. 553. Langguth, De Torp. 4to, Witt.
1784. Volta, Brugnatelli's Ann. 1805, p. 223. Humboldt, Ann. de Ch. 1. 15.
H. Davy, Ph. Tr. 1829, p. 15 ; 1832, p. 259. J. Davy, ibid. 1834, p. 531. Col-
ladon, Comptes Rendus, 1836, p. 490. Linari, ibid. 1836, p. 46. Matteuci, Arch,
de 1'Electr. iii. 153. Proceedings of the Electrical Society, p. 512.

Gymnotus.— Richer, Hist, et Mem. i. 116, vii. II. 92. Berker, Reisb nach
Rio, 1680. Williamson, Ph.Tr. 1775, p. 94. Garden, ibid. 1775, p. 102. Hunter,

ON MAGNETISM. 531

ibid. 1775, p. 395. Ginsan, De Gym. Tubing. 1819. Schonbein, Arch, de
1'Electr. i. 445. Faraday, Ph. Tr. 1839, p. 1. Letheby, Proceedings of the Electr.
Soc. p. 367.

Other Animals.— Geoffroj ( Anatomy), Bullet, dela Soc. Philom. No. 70 ; Mem.
du Musee d'Hist. Nat. i. 392. Rudolphi, Abh. der Akad. Berl. 1820, p. 223.
Marianini, Soprala Scossa che provano gli Animali, Venice, 1S28. DuBois, Pogg.
Ann. pp. Iviii. 1. Quse apud Veteres exstant Argumenta, Berl. 1843. Nobili, Delia
Rana, Mem. i. 135. Matteuci, Arch, de 1'Electr. ii. 628, 419, iii. 5. Essai sur
les Phe"n. Electr. des Animaux, Paris, 1840.

Mineral Electricity.— Wilson on the Tourmaline, &c. Ph. Tr. 1759, p. 308 ;
1762, p. 443. Due de la Noya Caraffa sur la Tourmeline, 4to, Paris, 1759. Aepinus
sur do. Petersb. 1762. Miiller, Au Born. 4to, Vienna, 1773. Bergmann, Ph. Tr.
1766, p. 236. Zallinger, Vom Tourm. Vienna, 1779. Haiiy, Hist, et Mem. 1785,
p. 206 ; Ann. de Ch. ix. 59 ; Mem. de 1'Inst. i. 49 ; Traite des Characters des
Pierres Precieuses, Paris, 1817, p. 146 ; Traite de Mineralogie, 1822, p. 206.
Becquerel, Ann. de Ch. xxxvii. 1, 355. Brewster, Ed. Jour. ii. 308. Forbes, Ed.
Tr. xiii. 27. Riess and Rose, Pogg. Ann. lix. 553.

Thermo-Electricity.—Seebeck, Abh. der Akad. Berl. 1822 ; Pogg. Ann. vi.
pp. 1, 133, 253. Yelin, Der Thermomagnetismus, Munch. 1823. Gumming,
Camb. Tr. ii. 1. Becquerel, Ann. de Ch. xxiii. 135, xxxi. 371, xli. 353. Sturgeon,
Ph. Mag. 1831, p. 1, 116. Pouillet, Comptes Rendus, v. 785. Prideaux, Ph.
Mag. iii. 205, 262, 398. Andrews, ibid. x. 433. Watkins, ibid. xi. 304. Wheat-
stone on the Thermo-electric Spark, ibid. x. 414. Matteuci, Bibliot. Univ. xv. 187 ;
Arch, de 1'Electr. ii. 227.

LECTURE LV.

ON MAGNETISM.

THE theory of magnetism bears a very strong resemblance to that of
electricity, and it must therefore be placed near it in a system of natural
philosophy. We have seen the electric fluid not only exerting attractions
and repulsions, and causing a peculiar distribution of neighbouring por-
tions of a fluid similar to itself, but also excited in one body, and trans-
ferred to another, in such a manner as to be perceptible to the senses, or at
least to cause sensible effects, in its passage. The attraction and repulsion,
and the peculiar distribution of the neighbouring fluid, are found in the
phenomena of magnetism ; but we do not perceive that there is ever any
actual excitation, or any perceptible transfer of the magnetic fluid from
one body to another distinct body ; and it has also this striking peculiarity,
that metallic iron is very nearly, if not absolutely, the only substance
capable of exhibiting any indications of its presence or activity.

For explaining the phenomena of magnetism, we suppose the particles of
a peculiar fluid to repel each other, and to attract the particles of metallic
iron with equal forces, diminishing as the square of the distance increases ;
and the particles of such iron must also be imagined to repel each other, in
a similar manner. Iron and steel, when soft, are conductors of the mag-
netic fluid, and become less and less pervious to it as their hardness in-
crea^es. The ground work of this theory is due to Mr. Aepinus,* but the

* See p. 515.
2 M 2

532 LECTURE LV.

forces have been more particularly investigated by Coulomb* and others.
There are the same objections to these hypotheses as to those which con-
stitute the theory of electricity, if considered as original and fundamental
properties of matter : and it is additionally difficult to imagine, why iron,
and iron only, whether apparently magnetic or not, should repel 'similar
particles of iron with a peculiar force, which happens to be precisely a
balance to the attraction of the magnetic fluid for iron. This is obviously
improbable ; but the hypotheses are still of great utility in assisting us to
generalise, and to retain in memory, a number of particular facts which
would otherwise be insulated. The doctrine of the circulation of streams
of the magnetic fluid has been justly and universally abandoned, and
some other theories, much more ingenious and more probable, for instance
that of Mr. Prevost,t appear to be too complicated, and too little supported
by facts, to require much of our attention.

The distinction between conductors and nonconductors is, with respect
to the electric fluid, irregular and intricate : but in magnetism, the softness
or hardness of the iron or steel constitutes the only difference. Heat, as
softening iron, must consequently render ,it a conductor : even the heat of
boiling water affects it in a certain degree, although it can scarcely be sup-
posed to alter its temper ; but the effect of a moderate heat is not so con-
siderable in magnetism as in electricity. A strong degree of heat appears,
from the experiments of Gilbert, J and of Mr. Cavallo,§ to destroy com-
pletely all magnetic action.

It is perfectly certain that magnetic effects are produced by quantities
of iron incapable of being detected either by their weight or by any
chemical tests. Mr. Cavallo || found that a few particles of steel, adhering
to a hone, on which the point of a needle was slightly rubbed, imparted to
it magnetic properties ; and Mr. Coulomb^f has observed that there are
scarcely any bodies in nature which do not exhibit some marks of being
subjected to the influence of magnetism, although its force is always pro-
portional to the quantity of iron which they contain, as far as that quan-
tity can be ascertained ; a single grain being sufficient to make 20 pounds
of another metal sensibly magnetic. A combination with a large propor-
tion of oxygen deprives iron of the whole or the greater part of its
magnetic properties ; finery cinder is still considerably magnetic, but the
more perfect oxids and the salts of iron only in a slight degree ; it is also
said that antimony renders iron incapable of being attracted by the magnet.
Nickel, when freed from arsenic and from cobalt, is decidedly magnetic,
and the more so as it contains less iron. Some of the older chemists sup-
posed nickel to be a compound metal containing iron, and we may still
venture to assume this opinion as a magnetical hypothesis. There is in-
deed no way of demonstrating that it is impossible for two substances to
be so united as to be incapable of separation by the art of the chemist ;

* Hist, et Mem. 1785, pp. 569, 578 ; 1789, p. 455. Mem. de 1'Instit. iii. 176.
f De 1'Origine des Forces Magnetiques, Geneve, 1788.
t De Magnete, fol. Lond. 1600.

§ Ph. Tr. 1787, p. 6. || Ibid. 1786, p. 62.

H Bulletin de la Soc. Philom. No. 61, 63. Jour, de Phy. liv. 240, 267, 454; See
Young, Jour, of the Roy. Inst. i. 134, 217.

ON MAGNETISM. 533

had nickel been as dense as platina, or as light as cork, we could not have
supposed that it contained any considerable quantity of iron, but in fact
the specific gravity of these metals is very nearly the same, and nickel is
never found in nature but in the neighbourhood of iron ; we may therefore
suspect, with some reason, that the hypothesis of the existence of iron in
nickel may be even chemically true. The aurora borealis is certainly in
some measure a magnetical phenomenon, and if iron were the only sub-
stance capable of exhibiting magnetic effects, it would follow that some
ferruginous particles must exist in the upper regions of the atmosphere.
The light usually attending this magnetical meteor may possibly be
derived from electricity, which may be the immediate cause of a change
of the distribution of the magnetic fluid, contained in the ferruginous
vapours, that are imagined to float in the air.

We are still less capable of distinguishing with certainty in magnetism,
than in electricity, a positive from a negative state, or a real redundancy of
the fluid from a deficiency. The north pole of a magnet may be con-
sidered as the part in which the magnetic fluid is either redundant or de-
ficient, provided that the south pole be understood in a contrary sense :
thus, if the north pole of a magnet be supposed to be positively charged,
the south pole must be imagined to be negative ; and in hard iron or steel
these poles may be considered as unchangeable.

A north pole, therefore, always repels a north pole, and attracts a south
pole. And in a neutral piece of soft iron, near to the north pole of a
magnet, the fluid becomes so distributed by induction, as to form a tem-
porary south pole next to the magnet, and the whole piece is of course
attracted, from the greater proximity of the attracting pole. If the bar is
sufficiently soft, and not too long, the remoter end becomes a north pole,
and the whole bar a perfect temporary magnet. But when the bar is of
hard steel, the state of induction is imperfect, from the resistance opposed
to the motion of the fluid ; hence the attraction is less powerful, and an
opposite pole is formed, at a certain distance, within the bar ; and beyond
this another pole, similar to the first ; the alternation being sometimes re«
peated more than once. The distribution of the fluid within the magnet is
also affected by the neighbourhood of a piece of soft iron, the north pole
becoming more powerful by the vicinity of the new south pole, and the
south pole being consequently strengthened in a certain degree ; so that the
attractive power of the whole magnet is increased by the proximity of the
iron. A weak magnet is capable of receiving a temporary induction of a
contrary magnetism from the action of a more powerful one, its north pole
becoming a south pole on the approach of a stronger north pole ; but the
original south pole still retains its situation at the opposite end, and
restores the magnet nearly to its original condition, after the removal of
the disturbing cause.

The polarity of magnets, or their disposition to assume a certain direc-
tion, is of still greater importance than their attractive power. If a small
magnet, or simply a soft wire, be poised on a centre, it will arrange itself
in such a direction, as will produce an equilibrium of the attractions and
repulsions of the poles of a larger magnet ; being a tangent to a certain

534 LECTURE LV.

oval figure passing through those poles, of which the properties have been
calculated by various mathematicians. This polarity may easily be
imitated by electricity ; a suspended wire being brought near to the ends
of a positive and negative conductor, which are placed parallel to each
other, as in Nairne's electrical machine, its position is perfectly similar to
that of a needle attracted by a magnet, of which those conductors repre-
sent the poles. (Plate XLI. Fig. 569.)

The same effect is observable in iron filings placed near a magnet, and
they adhere to each other in curved lines, by virtue of their induced mag-
netism, the north pole of each particle being attached to the south pole of
the particle next it. This arrangement may be seen by placing the filings
either on clean mercury, or on any surface that can be agitated ; and it
may be imitated by strewing powder on a plate of glass supported by
two balls, which are contrarily electrified.* (Plate XLI. Fig, 570.)

The polarity of a needle may often be observed when it exhibits no sen-
sible attraction or repulsion as a whole ; and this may easily be understood
by considering that when one end of a needle is repelled from a given point,
and the other is attracted towards it, the two forces, if equal, will tend to
turn it round its centre, but will wholly destroy each other's effects with
respect to any progressive motion of the whole needle. Thus, when the
end of a magnet is placed under a surface on which iron filings are spread,
and the surface is shaken, so as to leave the particles for a moment in the
air, they are not drawn sensibly towards the magnet, but their ends, which
are nearest to the point over the magnet, are turned a little downwards, so
that they strike the paper further and further from the magnet, and then
fall outwards, as if they were repelled by it. (Plate XLI. Fig. 571.)

The magnets, which we have hitherto considered, are such as have a sim-
ple and well determined form ; but the great compound magnet, which
directs the mariner's compass, and which appears to consist principally of
the metallic and slightly oxidated iron, contained in the internal parts of
the earth, is probably of a far more intricate structure, and we can only
judge of its nature from the various phenomena derived from its influence.

The accumulation and the deficiency of the magnetic fluid, which deter-
mine the place of the poles of this magnet, are probably in fact consider-
ably diffused, but they may generally be imagined, without much error in
the result, to centre in two points, one of them nearer to the north pole of
the earth, the other-to the south pole. In consequence of their attractions
and repulsions, a needle, whether previously magnetic or not, assumes
always, if freely poised, the direction necessary for its equilibrium ; which,
in various parts of the globe, is variously inclined to the meridian and to
the horizon. Hence arises the use of the compass in navigation and in
surveying : a needle, which is poised with the liberty of horizontal motion,
assuming the direction of the magnetic meridian, which for a certain time
remains almost invariable for the same place ; and a similar property is
also observable in the dipping needle, which is moveable only in a vertical
plane ; for when this plane is placed in the magnetic meridian, the needle

* Bazin, Descrip. des Courans Mag. en 15 Planches, 4to, Strasb. 1753. Roget,
Jour, of the Roy. Inst. 1831, p. 311.

ON MAGNETISM. 535

acquires an inclination to the horizon, which varies according to the situa-
tion of the place with respect to the magnetic poles. (Plate XLI. Fig. 572,
573.)

The natural polarity of the needle may be in some measure illustrated
by inclosing an artificial magnet in a globe ; the direction of a small needle,
suspended over any part of its surface, being determined by the position of
the poles of the magnet, in the same manner as the direction of the compass
is determined by the magnetical poles of the earth, although with much
more regularity. In either case the whole needle is scarcely more or less
attracted towards the globe than if the influence of magnetism were
removed ; except when the small needle is placed very near to one of the
poles of the artificial magnet, or, on the other hand, when the dipping
needle is employed in the neighbourhood of some strata of ferruginous sub-
stances, which, in particular parts of the earth, interfere materially with
the more general effects, and alter the direction of the magnetic meridian.

A bar of soft iron, placed in the situation of the dipping needle, acquires
from the earth, by induction, a temporary state of magnetism, which may
be reversed at pleasure by reversing its direction ; but bars of iron, which
have remained long in or near this direction, assume a permanent polarity ;
for iron, even when it has been at first quite soft, becomes in time a little
harder. A natural magnet is no more than a heavy iron ore, which, in the
course of ages, has acquired a strong polarity from the great primitive
magnet. It must have lain in some degree detached, and must possess but
little conducting power, in order to have received and to retain its mag-
netism.

We cannot, from any assumed situation of two or more magnetic
poles, calculate the true position of the needle for all places ; and even in the
same place, its direction is observed to change in the course of years, accord-
ing to a law which has never yet been generally determined, although the
variation which has been observed, at any one place, since the discovery of
-the compass, may perhaps be comprehended in some very intricate expres-
sions ; but the less dependence can be placed on any calculations of this
kind, as there is reason to think that the change depends rather on chemical
than on physical causes. Dr. Halley* indeed conjectured that the earth
contained a nucleus, or separate sphere, revolving freely within it, or rather
floating in a fluid contained in the intermediate space, and causing the
variation of the magnetic meridian ; and others have attributed the effect
to the motions of the celestial bodies : but in either case the changes pro-
duced would have been much more regular and universal than those which
have been actually observed. Temporary changes of the terrestrial mag-
netism have certainly been sometimes occasioned by other causes ; such
causes are, therefore, most likely to be concerned in the more permanent
effects. Thus, the eruption of Mount Hecla was found to derange the
position of the needle considerably ; the aurora borealist has been observed
to cause its north pole to move 6 or 7 degrees to the westward of its usual

*.?h.'Tr. 1693, p. 563.

f See Arago, Ann. de Ch. xxxix. 369. Fox, Ph. Tr. 1831, p. 199. Sabine,
Obs. on Days of unusual Mag. Disturbance, 4to, 1843.

536 LECTURE LV.

position ; and a still more remarkable change occurs continually in the
diurnal variation. In these climates the north pole of the needle moves
slowly westwards from about 8 in the morning till 2, and in the evening
returns again ;* a change which has with great probability been attributed
to the temporary elevation of the temperature of the earth, eastwards
of the place of observation, where the sun's action takes place at an earlier
hour in the morning, and to the diminution of the magnetic attraction in
consequence of the heat thus communicated. In winter this variation
amounts to about 7 minutes, in summer to 13 or 14.

Important as the use of the compass is at present to navigation, it would
be still more valuable if its declination from the true meridian were con-
stant for the same place, or even if it varied according to any discoverable
law ; since it would afford a ready mode of determining the longitude of a
place by a comparison of an astronomical observation of its latitude with
another of the magnitude of the declination. And in some cases it may
even now be applied to this purpose, where we have a collection of late and
numerous observations. Such observations have from time to time been
arranged in charts, furnished with lines indicating the magnitude of the
declination or variation at the places through which they pass, beginning
from the line of no variation, and proceeding on the opposite sides of this
line to show the magnitude of the variation east or west. It is obvious
that the intersection of a given parallel of latitude, with the line showing
the magnitude of the variation, will indicate the precise situation of the
place at which the observations have been made.

The line of no variation passed in 1657 through London, and in 1666
through Paris : its northern extremity appears to have moved continually
eastwards, and its southern parts westwards ; and it now passes through
the middle of Asia. The opposite portion seems to have moved more uni-
formly westwards ; it now runs from North America to the middle of the
South Atlantic. On the European side of these lines, the declination is
westerly ; on the South American side, it is easterly. The variation in
London has been for several years a little more than 24°. In the West
Indies it changes but slowly ; for instance it was 5° near the island of
Barbadoes, from 1700 to 1756. (Plate XLI. Fig. 574 . . 576. Plate
XLII. XLIII.)

The dip of the north pole of the needle in the neighbourhood of London
is 72°.t Hence the lower end of a bar standing upright, as a poker, or a
lamp iron, becomes always a north pole, and the temporary south pole of
a piece of soft iron being uppermost, it is somewhat more strongly attracted
by the north pole of a magnet placed over it, than by its south pole ; the
distribution of the fluid in the magnet itself being also a little more favour-
able to the attraction, while its north pole is downwards. It is obvious
that the magnetism of the northern magnetic pole of the earth must
resemble that of the south pole of a magnet, since it attracts the north pole ;

* Graham, Obs. made in 1722. Ph.Tr. xxxiii. 96, 383. The daily variation has
been more accurately observed by Christie, Ph. Tr. 1823-5-7 ; and at Gottinge^ by
Goldschmidt and others, Res. des Mag. Vereins, v. y.

t It is now about 69°.

ON MAGNETISM. 537

so that if we considered the nature of the distribution of the fluid, rather
than its situation in the earth, we should call it a south pole. Although
it is impossible to find any places for two, or even for a greater number of
magnetic poles, which will correctly explain the direction of the needle in
every part of the earth's surface, yet the dip may be determined with
tolerable accuracy, from the supposition of a small magnet placed at the
centre of the earth, and directed towards a point in Baffin's Bay, about 75°
north latitude, and 70° longitude west of London ; and the variation of the
dip is so inconsiderable, that a very slow change of the position of this
supposed magnet would probably be sufficient to produce it ; but the ope-
ration of such a magnet, according to the general laws of the forces con-
cerned, could not possibly account for the very irregular disposition of the
curves indicating the degree of variation or declination ; a general idea of
these might perhaps be obtained from the supposition of two magnetic poles
situated in a line considerably distant from the centre of the earth ; but this
hypothesis is by no means sufficiently accurate to allow us to place any
dependence on it. (Plate XLI. Fig. 577, 578.)

The art of making magnets consists in a proper application of the attrac-
tions and repulsions of the magnetic fluid, by means of the different con-
ducting powers of different kinds of iron and steel, to the production and
preservation of such a distribution of the fluid in a magnet, as is the best
fitted to the exhibition of its peculiar properties.

We may begin with any bar of iron that has long stood in a vertical
position ; but it is more common to employ an artificial magnet of greater
strength. When one pole of such a magnet touches the end of a bar of hard
iron or steel ; that end assumes in some degree the opposite character, and
the opposite end the same character : but in drawing the pole along the
bar, the first end becomes neutral, and afterwards has the opposite polarity ;
while the second end has its force at first a little increased, then becomes
neutral, and afterwards is opposite to what it first was. When the opera-
tion is repeated, the effect is at first in some measure destroyed, and it is
difficult to understand why the repetition adds materially to the inequality
of the distribution of the fluid ; but the fact is certain, and the strength of
the new magnet is for some time increased at each stroke, until it has
acquired all that it is capable of receiving. Several magnets, made in this
manner, may be placed side by side, and each of them being nearly equal
in strength to the first, the whole collection will produce together a much
stronger effect ; and in this manner we may obtain from a weak magnet
others continually stronger, until we arrive at the greatest degree of polarity
of which the metal is capable. It is, however, more usual to employ the
process called the double touch ; placing two magnets, with their opposite
poles near to each other, or the opposite poles of a single magnet, bent into
the form of a horseshoe, in contact with the middle of the bar ; the opposite
actions of these two poles then conspire in their effort to displace the mag-
netic fluid, and the magnets having been drawn backwards and forwards
repeatedly, an equal number of times to and from each end of the bar, with
a considerable pressure, they are at last withdrawn in the middle, in order
to keep the poles at equal distances.

538 LECTURE LV.

Iron filings, or the scoriae from a smith's forge, when finely levigated,
and formed into a paste with linseed oil, are also capable of being
made collectively magnetic. A bar of steel, placed red hot between two
magnets, and suddenly quenched by cold water, becomes in some degree
magnetic, but not so powerfully as it may be rendered by other means.
For preserving magnets, it is usual to place their poles in contact with
the opposite poles of other magnets, or with pieces of soft iron, which, in
consequence of their own induced magnetism, tend to favour the accumu-
lation of the magnetic power in a greater quantity than the metal can
retain after they are removed. Hence the ancients imagined that the mag-
net fed on iron.

A single magnet may be made of two bars of steel, with their ends pressed
into close contact ; and it might be expected that when these bars are
separated, or when a common magnet has been divided in the middle, the
portions should possess the properties of the respective poles only. But in
fact the ends which have been in contact are found to acquire the properties
of the poles opposite to those of their respective pieces, and a certain point
in each piece is neutral, which is at first nearer to the newly formed
pole than to the other end, but is removed by degrees to a more central
situation. In this case we must suppose, contrarily to the general prin-
ciples of the theory, that the magnetic fluid has actually escaped by degrees
from one of the pieces, and has been received from the atmosphere by the
other.

There is no reason to imagine any immediate connexion between mag-
netism and electricity, except that electricity affects the conducting powers
of iron or steel for magnetism, in the same manner as heat or agitation.
In some cases a blow, an increase of temperature, or a shock of electricity,
may expedite a little the acquisition of polarity ; but more commonly any
one of these causes impairs the magnetic power. Professor Robison found,
that when a good magnet was struck for three quarters of an hour, and
allowed in the mean time to ring, its efficacy was destroyed ; although the
same operation had little effect when the ringing was impeded ; so that the
continued exertion of the cohesive and repulsive powers appears to favour
the transmission of the magnetic as well as of the electric fluid. The inter-
nal agitation, produced in bending a magnetic wire round a cylinder, also
destroys its polarity, and the operation of a file has the same effect. Mr.
Cavallo* has found that brass becomes in general much more capable of
being attracted when it has been hammered, even between two flints ; and
that this property is again diminished by fire : in this case it may be con-
jectured that hammering increases the conducting power of the iron con-
tained in the brass, and thus renders it more susceptible of magnetic
action. Mr. Cavallo t also observed that a magnetic needle was more
powerfully attracted by iron filings during their solution in acids, espe-
cially in the sulfuric acid, than either before or after the operation : others
have not always succeeded in the experiment ; but there is nothing impro-
bable in the circumstance, and there may have been some actual difference
in the results, dependent on causes too minute for observation. In subjects
* Ph. Tr. 1786, p. 62. f Ibid. 1787, p. 6.

ON MAGNETISM. 539

so little understood as the theory of magnetism, we are obliged to admit
some paradoxical propositions, which are only surprising on account of
the imperfect state of our knowledge. Yet, little as we can understand
the intimate nature of magnetical actions, they exhibit to us a number
of extremely amusing as well as interesting phenomena ; and the prin-
ciples of crystallization, and even of vital growth and reproduction, are
no where so closely imitated, as in the arrangement of the small particles
of iron in the neighbourhood of a magnet, and in the production of a
multitude of complete magnets, from the influence of a parent of the
same kind.

[Numerous and important as are the additions which have been recently
made to our knowledge of the agencies of electric and magnetic forces, our
limits will merely suffice us to mention those which appear to constitute
new and distinct branches of science.

In 1819, Professor Oersted, of Copenhagen, discovered* that a current
of voltaic electricity exerts an action on the magnetic needle, which differs
in its character from the other forces observed in nature, inasmuch as it is
tangential to the course of the current. This will be best understood from
an inspection of the accompanying figures, in which N and S are the north
and south poles of Tin ±

a magnet, cz is a c
wire, along which
flows a current of
voltaic electricity,
the end c being in
connexion with the
positive or copper
plate of the simple
battery, and the

other end with the

~"c
- negative or zinc

plate. In figure 1,
where the wire is above the needle, it causes the north pole to be deflected
towards the east, as at n ; in figure 2, where it is below, towards the west.
Were the wire placed in the same horizontal plane with the needle, the
poles of the latter would simply suffer elevation or depression. The effect
of this force on the north pole of a magnet (that on the south pole being
of course the reverse) is represented by the following diagram, in which
the repulsion is in the
direction in which the
hands of the watch are ac- •* *•$ °-

customed to move. The
science which is built on
this fact is termed ELEC-
TRO-MAGNETISM.

Fro,m the nature of the ^-j^^

* Thomson's Annals of Philosophy, 1820, xvi. 273. A.nn. de Ch. xxii. 201.
Schweigg. Jour, xxxii. 199 ; xxxiii. 123.

540 LECTURE LV.

action which we have de-
scribed, it is evident that the
effect can he multiplied al-
most indefinitely hy simply
coiling the wire and placing
the needle within the coil ;
for the currents on each side
of the needle all tend to move
it in the same direction.

In this way the galvano-
meter is constructed.* A
small needle is suspended hy
a fibre of silk, and a coil of
wire, coated with sealing wax
or silk, causes the voltaic current to circulate in directions parallel to it.
The tangential action of the current overcomes the magnetic action of the
earth and deflects the needle. The delicacy and value of this instrument
have been greatly increased by the inventions of Gumming t and Nobili.J
Instead of a single needle, two needles are used, which are placed with
their poles opposite ways, so that the directive tendency due to the earth's
action is completely neutralized, and the torsion of the suspending thread
is the sole impediment to motion. In the figure the needles are ordi-
nary sewing needles similarly magnetized, and passed parallel to each
other through a flat bit of straw which is attached to the fibre of silk. The
coil of wire passes about the lower needle, having in its upper part an
opening through which the straw passes freely. The amount of force
exerted is the sum of the actions of the upper and lower currents on the
lower needle, together with the difference of those on the upper wire.
The action of the voltaic current on a magnetic needle is very similar to
the action of one magnetic needle on another, except that it is perpendi-
cular to the direction of the current. Now the action of a magnetic needle
produces the magnetic state in a bar of soft iron, and hence it is natural to
conclude that a voltaic current should produce a similar state. Accord-
ingly, if a considerable quantity of copper wire be twisted round a piece of
soft iron bent into the form of a horseshoe, and a voltaic current be passed
along the wire, the result is the formation of a powerful magnet. On
discontinuing the communication with the voltaic pile, the iron is instantly
reduced to nearly its former state. This presents us with a promising field
of research in its applicability to economical purposes as a moving power.
And although the endeavours of Jacobi § and others have as yet been only
partially successful, there is every reason to suppose that time will

* Schweigger, in his Jour. 1821. f Camb. Tr. 1821, p. 281.

J Memorie ed Osservazioni colla Descrizione de suoi Apparati, 2 vols. Firenze,
1834. See also Melloni, Arch, de 1'Electr. i. 165.

§ Ritchie, Phil. Mag. iv. 13. Dal Negro, Nuova Macchina Elettro-Mag. Ann.
delle Scienze del Regno Lomb. Venet. 1834. Jacobi, Mem. sur 1' Application de
1'Electro-Mag. au Mouvement des Mach. Potsdam, 1835. Sturgeon, in Stuf.-Ann.
i. 75 ; viii. 81. Davenport, ibid, ii. 284. Davidson, Mechanics' Magazine, Nov.
1842.

ON MAGNETISM. 541

develope the means of rendering this agent one of the great assistants to
human power. An attractive force can be created and destroyed at plea-
sure, and thereby an alternation'of action, so necessary to dynamical effects,
can be produced. The electro-magnetic telegraph is based on the same
principles.* A wheel has twenty-four conductors placed on its circum-
ference at equal intervals, so that when it is turned through a complete
revolution, the voltaic circuit is completed and broken twenty-four times.
This wheel is placed at one station, and another wheel, together with
an electro-magnet, at the other; a pair of wires sufficing to effect the
communication between them. When the circuit is complete, the electro-
magnet is in action and causes its accompanying circle to move through
one division ; that is to say, each turn of the one wheel causes a similar
movement in the other. Now to every division is attached a letter of the
alphabet. If then the instrument be standing at C, and it be requi-
site to convey the letter F, the first wheel must be turned through three
divisions, by which D, E, and F are successively presented to the ob-
server at the other station ; the last of which only is suffered to rest.
The close analogy between the agents which produce the varied forms of
electricity and magnetism is rendered still closer by the beautiful dis-
coveries of Faraday and others. When a current is passing along a wire,
it induces a similar current along a wire placed near the first, at the times
of making and of breaking the contact. Now we have seen that a current
of electricity passing round a bar of iron renders it a magnet, and it was
easy to conjecture that, conversely, a magnet should produce a current in
a coil wound about it. Faraday f proved that this is the case at the moment
only of its becoming or ceasing to become a magnet. The coil was
wrapped round a piece of soft iron, the extremities of which could be
brought simultaneously in contact with the ends of a horseshoe magnet.
At the instant of forming this contact a current of electricity was produced
along the coil, the effect of which was sensible to the galvanometer. Soon
after this discovery, all the usual electrical effects were produced in this
way, and in 1832 was constructed, by M. Pixii, J a very powerful magneto-
electric machine. This machine, as improved by Saxton § and Clarke, ||
consists of a compound horseshoe magnet of a large size fixed in a given
position. A piece of soft iron, of much the same shape, has a quantity of
insulated copper wire wound round it, and is so placed as to be capable of
rapidly presenting its ends alternately to the poles of the fixed magnet. By
this means it becomes constantly magnetized, demagnetized, and oppositely
magnetized. Thus the conditions requisite for the development of an

* Wheatstone, Mech. Mag. 1840. Walker's Electr. Mag. vol. ii. Sturgeon's
Annals, v. 337. Steinheil, Ueber Teleg. 4to, Munch, 1838. Morse, Ann. de Ch.
Ixxii. 219. Lenz, Ueber die Praktischen Anwendungen des Galv. Petersb. 1839.
De Heer, Theorie de la Teleg. Electr. Bullet, des Sci. Phys. en Neerland, 1839.
Finlayson, The Application of the Electric Fluid to the Useful Arts. For the
application of galvanism to gilding, &c. see Jacobi, Galvanoplastik, St. Petersb.
1840. Smee's Metallurgy.

t F?*a*day, Ph. Tr. 1832. Experimental Researches in Electricity, 1839.

J Ann. de Ch. 1. 322. § Ph. Mag. ix. 262.

|| Sturgeon's Ann. i. 145.

542 LECTURE LV.

electric current are attained, and very little ingenuity is requisite to
render the machine available as a powerful electrical machine of a peculiar
character.

The last mode of developing a current which we shall mention is that
discovered by M. Arago.* If a plate of copper be made to rotate with
considerable rapidity in a horizontal plane, a magnetic needle placed above
or below it tends to follow its motion, and that quite irrespective of the
motion of the air, as may be proved by interposing a plate of glass or
other substance between them. It is evident that this effect is due to the
evolution of a current of electricity, which travels from the centre to the
circumference of the plate. For further information, the reader is referred
to Faraday's Researches ; he will also find an excellent article on electro-
magnetism, by Roget, in the Library of Useful Knowledge.]

LECT. LV.-ADDITIONAL AUTHORITIES.

Treatises. — Peregrinus, De Magnete, 4to, Augsb. 1558. Norman, The New
Attractive, 4to, 1596. Ridley, On Magn. Bodies and Motions, 4to, Lond. 1613.
Cabseus, Philosophia Mag. fol. Ferrara, 1629. Kircher, Magnes, 4to, Col. 1643.
Lieutaud. Magnetologia, 4to, Lugd. 1668. Dalance, Traite de 1'Aimant, 4to, Liege,
1691. Eberhard's Mag. Theorie, 4to, Leipz. 1720. Becker, Der Mineralische
Mag. Miihlh. 1729. Euler, D. and J. Bernoulli, Dutour, Pieces qui ont rem-
portees la prix de 1'Acad. 4to, Paris, 1748. Du Fay, Amnerkungen, Erf. 1748.
Penrose on Mag. 1753. Adams's Essay, 4to, 1753. Scarella, De Mag. 2 vols.
4to, Brescia, 1759. Cooper's Experimental Mag. 1761. Wilcke, Tal om Mag.
Stock. 1764. Brugmann, De Materia Magnetica, 4to, Franeker, 1765. Lo-
vett's Electr. and Mag. 1766. Van Swinden, De Phsen. Mag. Lugd. 1772.
Recueil de Mem. sur 1' Analogic de 1'Electr. et du Mag. 3 vols. Haag. 1784. Le-
monnier, Loix du Mag. 2 vols. 1776. Gabler, Theoria Mag. Ingolst. 1781.
Cavallo on Mag. 1787. Walker, 1794. Lorimer, 4to, 1795. Haiiy, Expos, de la
Theorie, Altenb. 1801. V. Lowerrorn, Ueber den Magnet. Kopenhag. 1802. Biot,
Traite de Physique, and art. Magnetism in Edin. Encyc. Roucher-Deratte, Traite
sur 1'Electr. &c. 1803. Meissen, Ueber den Mag. 1819. Barlow's Essay on Mag.
Attractions, 1823. Peytavin, Essai sur la Constit. des Fluides Magnetiques, 1830.
Brewster's Magnetism, 1837 ; also an essay in many treatises on electricity.

Memoirs.— Du Fay, Hist, et Mem. 1728, p. 355 ; 1730, p. 142; 1731, p. 417.
Servington Savery's Mag. Obs. Ph. Tr. 1730, p. 295. Lambert, Hist, et Mem. de
Berlin, 1766, pp. 22, 49. Franklin, Am. Tr. iii. 10. Krafft, Com. Petr. xii. 276.
Kirwan, Ir. Tr. vi. 177. Ritter, Gilb. Ann. iv. 1. Kratzenstein, Lichtenb. Mag.
iv. 132. Poisson, Mem. de 1'Inst. 1821, pp. 247, 448. Ampere, ibid. 1823,
p. 175. Harris, Ed. Tr. vol. xi. Christie, Ph. Tr. 1828, p. 325. Blondeau, Mem.
de Brest, i. 385, 401. Haldat, Mem.de 1'Acad. de Nancy, 1830. . 1839 ; Ann.
de Ch. xlii. 53. Scoresby, Jameson's Jour. 1832, p. 319, &c. Kupfer, Ann. de
Ch. xxxvi. 50.

Artificial Magnets. — Leuwenhoek on the Mag. Quality acquired by Iron after
standing a long time in the same Posture, Ph. Tr. xxxiii. 72. Marcel, ibid.
1730, p. 112. Reaumer, Hist, et Mem. 1723. Duhamel, ibid. 1735, 1745,
p. 181. Knight's Method, Ph. Tr. 1744-5, pp. 161, 361 ; 1776, p. 591 ; 1779,
p. 51. Canton, ibid. 1751, p. 31. Michell on Artificial Magnets, Camb. 1751.
Klingenstierna, De Mag. Artif. Stock. 1752. Riviere sur les Aimans Artificiels,
1752. Richmann, Nov. Com. Petr. iv. 235. Nebel, de Mag. Artif. 4to, Utr. 1756.
Antheaulme sur les Aimans Artificiels, 1760. Coulomb, Mem. de 1'Inst. vi. 399.
Barlow, Ed. Jour. i. 344. Scoresby's Exp. on the Development of Magnetical
Properties in Steel and Iron by Percussion, Ph. Tr. 1820, p. 241. Baden Powell,
Ann. of Phil. 1822, p. 92. Weber, Magnetismus des Eisens durch die Erde, Res.
des Mag. Ver. 1841, p. 85.

* Ann. deCh. xxvii. 363 ; xxviii. 325 ; xxxii. 213.

ON MAGNETISM. 543

Compensation for Local Attraction. — Sabine, Ph. Tr. 1819, p. 112. Barlow,
ibid. 1831, p. 215. Airy, ibid. 1839, p. 167.

Instruments. Mariner's Compass. — Lous, Tentamen Exp. ad Compassum per-
ficiendum, 4to, Hafnise, 1734. Knight, Ph. Tr. 1750, pp. 505, 513. Duhamel,
Hist, et Mem. 1750, p. 154. Bain on the Compass, 1817. Gilbert, M'Culloch,
and Alexander's Compass, in Barlow's art. Magnetism in Encyc. Metr. Kater, Ph.
Tr. 1821, p. 104.

Declination.— Lahire, Hist, et Mem. 1716, p. 7. Lemonnier, ibid. 1778, p. 68.
Coulomb, ibid. 1785, p. 560. Wilcke, Schwed. Abhand. 1763, p. 154. Brander,
Beschreibung eines Mag. Dec. und Incl. Augsb. 1779. Cassini, Mem. de 1'Inst. v.
145. Prony, Jour, de Phy. xliv. 474. Troughton, Nich. Jour. 1806, p. 179.
Bidone, Mem. de TurinJSll, p. 141. Bessel, Schumacher's Ast. Nachr. vi. 221.
Weber, Res. des Mag. Ver. 1837, p. 104. Gauss, ibid. 1841, p. 1. Simonoff,
ibid. 62. Report of the Committee of Physics of the Royal Soc. 1840, p. 30.
Lamont, Ann. fur Meteorolog. und Erdmagnet. 1842, ii. 179. Gambey in Pouillet's
Physique, pi. xi. fig. 266. The practical processes of finding the variation at sea will
be found in Raper's Navigation and Nautical Ast. 1840.

Inclination.— Buache, Hist, et Mem. 1732, p. 377. Nairne, Ph. Tr. 1772,
p. 476. Borda, Gilb. Ann. iv. 449. Robinson, in Sabine Ph. Tr. 1822, p. 1.
Weber, Res. der Gott. Ver. 1837, p. 81. Kreil, Die Mag. Apparat. zu Prag. ibid.
1839, p. 91. Lloyd on the Mag. Obs. at Dublin, 4to, 1842.

Intensity. — Coulomb, Mem.de 1'Instit. iii. 176. Hansteen, Beobachtungen iiber
die Intensitat des Mag. in Nord Europa, Pogg. Ann. iii. 225. Christie, Ph. Tr.
1833, p. 343. Gauss, Magnetometer, 4to, Gott. 1833 ; Res. des Mag. Ver. 1837,
p. 1, 20, 58 ; 1840, p. 1. Weber, ibid. 1836, p. 63 ; 1838, p. 68 ; 1841, p. 79;
translated in Sci. Mem. vol. ii. Lamont, Ueber Bestimmung der Horizontal-Inten-
sitat, 4to, Munch. Ueber das Mag. Observatorium bei Miinchen, 4to, 1842.

Theory of Terrestrial Mag. — Whiston, The Latitude and Longitude found by the
Dipping Needle, 1721. Zegollstrom, Theoria Decl. Mag. Upsal, 1755. Dunn's
Mag. Atlas, 1776. Steinhauser, De Mag. Telluris, Wittenb. 1806. Mollweide,
Gilb. Ann. xxix. 1, 251, Ixx. 26. Humboldt and Biot, Jour, de Phy. xlix. 429.
Quinet, Theorie de 1' Aimant, 4to, Paris, 1809 ; Expos£ des Variations, Mag. 1826.
Hansteen, Untersuchungen iiber den Mag. der Erde, 4to, Christiana, 1819 ; Gilb.
Ann. Ixv. 313, Ixx. 36, 110, Ixxi. 273, Ixxv. 145 ; Pogg. Ann. iv. 277, ix. 49, 229,
xxviii. 473, 578. Schumacher's Ast. Nach. vii. 17. Duperrey's Chart, Pogg. Ann.
xxi. 151. Barlow, Ph. Tr. 1831, p. 99. Gauss, Intensitas Vis Mag. Terrest. ad
Mensuram revocata, Gott. 1833 ; Sci. Mem. ii. 184, 313. Res. des Magn. Vereins,
1838, p. 1, 146; 1839, p. 50. Gauss and Weber's Atlas, 4to, Leipz. 1840.
Sabine, Ph. Tr. 1840, p. 129; 1841, p. 11 ; 1842, p. 9. Bessel, Schum. Ast.
Jahrb. 1843, p. 117.

Observations.— See Christie's Report Br. Ass. 1833 ; Sabine's Report, vol. vi. ;
or Dove's Repertor. Band v. ; from which the preceding list is for the most part
extracted : also Kupfer Recueil d'Obs. faites a St. Petersb. 4to, 1837. . . Annu-
aire Magnetique, 5 vols. 4to, St. Petersb. 1836. . . Gauss and Weber, Resultate,
Leipz. 1836.... and Scientific Mem. ii. 20. Kreil, Mag. und Meteor. Obs. zu
Prag. 4to, 1839. Lamont, Ann. der Meteor, und des Erdmagnetismus, Munch.
1842. Quetelet, Mem. de Brux. xvi

Electro-magnetism. — Kastner, Obs. de Electro-mag. Erlang. 1821. Schrader,
Dissertatio de Electro-mag. Halse, 1821. Erman , Umrisse zu Elektr. Magn. Berl.
1821. Seebeck, Ueber den Magn. des Galvan. Kette, 4to, Berl. 1822. Am-
pere, Recueil d'Obs. Electro-magn. 1822 ; Mem. sur la Theorie Math, des Phen.
Electro-dyn. Mem. de 1'Instit. 1823, vi. 175 ; Ann. de Ch. xv. 57. Precis
de la Theorie des Phsen. Electro-dyn. 1824 ; ExposS Methodique des Phaen. 1824 ;
Theorie des do. 4to, 1826. Demonfernand's Manuel, 1823 ; translated by Gumming,
with additions, Camb. 1827. De la Rive, Recherches sur la Distribution de 1'Electr.
Dyn. dans les Corps, Geneve, 1825. Pfaff, Der Electro-mag. Hamburg, 1824 ;
Gilb. Ann. Ixxiv. 249. Guerin, Action Mutuelle des Fils Conducteurs, &c. 1828.
Pohl, Der Electro-mag. Theor. Practisch Dargestellt, Berlin, 1830. Fechner,
Elementarlehrbuch des Elektr. Leipz. 1830. Watkins's Electro-mag. 1832.
Poisson, Theorie du Mag. en Mouvement, Mem. de 1'Inst. vi. 439. Nobili,
Question^ sul Magnetismo, Modena, 1838. Antologia di Firenze, Nos. cxxxi. &c.
Marias4ni, Mem. di Fisica Sperimentale, Modena, 1838. Zantedeschi, Relazione
Storico-critica Sperimentale nell' Elettro-mag. Venice, 1840. Matteuci, Bibliot.
Univ. 1840.

544

LECTURE LVI.

ON CLIMATES AND WINDS.

THE science of meteorology relates principally to the natural history
of the air, and to such temporary changes in the earth and sea as are
produced by causes not mechanical only. The subject is of a very com-
plicated and intricate nature ; it comprehends many effects derived from
such causes as belong separately to every department of physics which we
have hitherto examined ; and although it lias occupied the attention of
several philosophers of considerable eminence, we cannot yet boast of
having made any great advancement in it. Whether we shall ever be able
to carry our theories to so high a degree of perfection, as to furnish us
with much information applicable to the purposes of common life, to agri-
culture, or to medicine, is at present uncertain ; although some advantage
has already been derived from the indications of meteorological instru-
ments ; and the philosophy of the science is in many respects much more
advanced than has commonly been supposed. We shall divide this exten-
sive subject into two parts, the first relating principally to the effects of
heat on the atmosphere, including the phenomena of winds ; the second
to the nature and consequences of evaporation, comprehending atmo-
spherical electricity, and to the effects of subterraneous fires and igneous
meteors.

The variations of temperature in different parts of the earth's surface,
require to be examined in the first place ; since they are not only of con-
siderable importance in themselves, but are also among the principal causes
of other changes in the state of the winds and weather. These changes
are measured by thermometers, of various kinds, which have already been
described ; but, for meteorological purposes, some additions are frequently
made to the simple thermometer. In Six's thermometer,* the tube is
twice bent, so as to return in a parallel direction : the bulb is in the form
of a long cylinder, and is usually filled with spirit of wine, which is in
contact with a portion of mercury occupying the lower part of the tube ;
and this is succeeded by a second portion of spirit. The mercury carries
on each of its surfaces an index, which is retained in its remotest situation
by means of a weak spring ; and consequently shows the greatest degree
of heat or of cold that has happened since the last observation. The
indexes are of iron or steel, and may be brought back to the surface at
pleasure by means of a magnet ; they are carried up by the mercury, more
by its capillary action, than by the difference of the specific gravities. A
similar effect is obtained in Rutherford's t arrangement of a pair of ther-
mometers, one with mercury, the other with spirit of wine, placed in a
horizontal position ; one index being without the surface of the mercury,

* Ph. Tr. Ixxii. Six on Meteorology, Maidst. 1794.
f Ed. Tr. iii. 247.

ON CLIMATES AND WINDS. 545

the other within that of the spirit : the thermometers being in contrary
directions, both indexes may be brought back to their places, by merely
raising the end of the instrument. Self registering thermometers have
also sometimes been constructed, for keeping a still more accurate account
of all the variations of temperature that have occurred, by describing a
line on a revolving barrel, which shows the height for every instant during
the whole time of their operation. (Plate XLI. Fig. 579, 580.)

The climates of different parts of the earth's surface are unquestionably
owing in great measure to their position with respect to the sun. At the
equator, where the sun is always nearly vertical, any given part of the
surface receives a much greater quantity of light and heat, than an equal
portion near the poles ; and it is also still more affected by the sun's verti-
cal rays, because their passage through the atmosphere is shorter than that
of the oblique rays. As far as the sun's mean altitude only is concerned,
it appears from Simpson's calculations, that the heat received at the
equator in the whole year is nearly twice and a half as great as at the
poles ; this proportion being nearly the same as that of the meridian heat
of a vertical sun, to the heat derived, at the altitude 23£°, in the middle
of the long annual day at the poles. But the difference is rendered still
greater, by the effect of the atmosphere, which interrupts a greater portion
of the heat at the poles than elsewhere. Bouguer has calculated, upon the
supposition of the similarity of the affections of heat and light, that in
latitude 45°, 80 parts out of 100 are transmitted at noon in July, and 55
only in December. The heat intercepted by the atmosphere is perhaps
not wholly, but very nearly, lost with respect to the climate of the neigh-
bouring places. It is obvious that, at any individual place, the climate
in summer must approach in some degree to the equatorial climate, the
sun's altitude being greater, and in winter to the climate of the polar
regions.

While the earth is becoming warmer at any particular spot, the heat
thrown off by radiation into the atmosphere, and thence into the empty
space beyond it, together with that which is transmitted to the internal
parts of the earth, must be less than the heat received from the sun ; and
when the earth is growing colder, more heat must pass off than is received :
but whenever the heat of the surface is stationary, neither increasing nor
diminishing, as at the times of the greatest and least heat, it is obvious
that the heat received from the sun must be precisely equal to the heat
which is thrown off. Now this quantity may be estimated by the degree
of refrigeration in the night ; and hence Mr. Prevost* has very ingeniously
deduced the proportion of the sun's heat arriving at the surface of the earth
in the latitude of Geneva, in July, and in December ; which he finds to be
as 7 or 8 to 1 ; and this result agrees very well with a calculation deduced
from the length of the day, the sun's altitude, and the interception of his
rays by the atmosphere.

In London the temperature generally varies, in the course of the day
and night, somewhat more than 5°, and less than 20°. In January, the
mean diffrnal variation of temperature is 6°, in March 20°, in July 10°,
* Jour, de Phy. xlii. 81.
2 N

546 LECTURE LVI.

and in September, 18°. Hence, says Mr. Kirwan,* we may understand
the reason of the great frequency of colds in spring and in autumn.

Some philosophers have supposed the earth to become progressively
warmer in the course of ages, while others have imagined that its heat
is exhausted. Both these opinions appear in general improbable. The
greater heat the earth receives by day, the more it throws off, both by
day and by night ; so that in the course of a few ages the heat must pro-
bably have attained its maximum. Local changes may indeed arise from
local circumstances ; thus, the climate of America is said to have become
considerably warmer, since a large part of its surface has been cleared from
its dense forests by human labour : and to judge from the descriptions of
the ancients, it appears that even in Europe the winters were formerly
much colder than they are at present. If, however, Dr. Herschel's opinion
of the variation of the heat of the sun be confirmed, it will introduce a
great uncertainty into all theories upon the subject : since in these calcula-
tions the original heat of the sun has always been supposed unalterable.

The sea is less heated than the land, partly because a greater quantity of
water evaporates from it, and partly because the sun's rays penetrate to a
considerable depth, and have less effect on the surface, while the water is
also mixed, by the agitation of its waves and currents, with the colder
water below. It is also more slowly cooled than the land, since, when the
temperature of the superficial particles is depressed, they become heavier,
and sink to the bottom. For similar reasons, the sea is colder than the
land in hot climates, and by day, and warmer in cold climates, and by
night. These circumstances, however, nearly balance each other, so that
the mean temperatures of both are equal, that of the sea being only less
variable. Although the process of evaporation must cool the sea, yet when
the vapours are condensed without reaching the land, their condensation
must compensate for this effect by an equal extrication of heat.

There is another cause which perhaps contributes in some degree, in tem-
perate climates, to the production of cold ; that is, the alternation of freez-
ing and thawing. Mr. Prevost observes that congelation takes place much
more suddenly than the opposite process of liquefaction ; and that of
course the same quantity of heat must be more rapidly extricated in freez-
ing than it is absorbed in thawing ; that the heat, thus extricated, being
disposed to fly off in all directions, and little of it being retained by
the neighbouring bodies, more heat is lost than is gained by the alternation :
so that where ice has once been formed, its production is in this manner
redoubled. This circumstance must occur wherever it freezes, that is, on
shore, in latitudes above 35° ; and it appears that from about 30° to the
pole, the land is somewhat colder than the sea, and the more as it is further
distant from it ; and nearer the equator the land is warmer than the sea :
but the process of congelation cannot by any means be the principal cause
of the difference, and it is probable that the different capacity of earth and
water for heat is materially concerned in it.

Since the atmosphere is very little heated by the passage of the sun's rays
through it, it is naturally colder than the earth's surface ; arf&vfor this
* Ph. Mag. xvi. 212.

ON CLIMATES AND WINDS. 547

reason, the most elevated tracts of land, which are the most prominent, and
the most exposed to the effects of the atmosphere, are always colder than
situations nearer the level of the sea. The northern hemisphere is somewhat
warmer than the southern, perhaps because of the greater proportion of
land that it contains, and also in some measure on account of the greater
length of its summer than that of the southern : for although, as it was
long ago observed by Simpson, the different distance of the sun compen-
sates precisely for the different velocity of the earth in its orbit, with respect
to the whole quantity of heat received on either side of the equinoctial
points, yet Mr. Prevost has shown, that in all probability the same quan-
tity of heat must produce a greater effect when it is more slowly applied ;
because the portion lost by radiation from the heated body is greater, as
the temperature is higher. Since, therefore, on account of the eccentricity
of the earth's orbit, the north pole is turned towards the sun 7 or 8 days
longer than the south pole, the northern winters must be milder than the
southern : yet the southern summers, though shorter, ought to be some-
what warmer than the northern : but in fact they are colder, partly per-
haps from the much greater proportion of sea, which in some degree
equalises the temperature, and partly for other reasons. The compara-
tive intensity of southern summer and winter is not exactly known ;
but in the island of New Georgia the summer is said to be extremely
cold.

The northern ice extends about 9° from the pole : the southern 18°
or 20° ; in some parts even 30° ; and floating ice has occasionally been
found in both hemispheres as far as 40° from the poles, and sometimes, as
it has been said, even in latitude 41° or 42°. Between 54° and 60° south
latitude, the snow lies on the ground, at the sea side, throughout the sum-
mer. The line of perpetual congelation is three miles above the surface at
the equator, where the mean heat is 84° ; at Teneriffe, in latitude 28°, two
miles ; in the latitude of London, a little more than a mile ; and in latitude
80° north, only 1200 feet. At the pole, according to the analogy deduced
by Mr. Kirwan,* from a comparison of various observations, the mean
temperature should be 31°. In London the mean temperature is 50° ; at
Rome and at Montpelier, a little more than 60° ; in the island of Madeira,
70° ; and in Jamaica, 80°.f

There are frequently some local causes of heat and cold which are inde-
pendent of the sun's immediate action. Thus, it has been observed, that
when the weather has been clear, and a cloud passes over the place of obser-
vation, the thermometer frequently rises a degree or two almost instanta-
neously. This has been partly explained by considering the cloud as a
vesture, preventing the escape of the heat which is always radiating from
the earth, and reflecting it back to the surface : the cloud may also have
been lately condensed, and may itself be of a higher temperature than
the earth. Mr. SixJ has observed that in clear weather, the air is usually
some degrees colder at night, and warmer by day, close to the ground,

* Anjgstimate of the Temperature of different Latitudes, Lond. 1787.
f O;i Isothermal Lines, see Humboldt, Fragments Asiatiques, ii. 398. Mem.
d'Arcueil, Hi. 462. + Ph. Tr. 1784, p. 428 ; 1788, p. 103.

2 N 2

548 LECTURE LVI.

than a few feet above it ; but that in cloudy weather there is less differ-
ence : and it is possible that this circumstance may be derived from
the difference of the quantity of evaporation from the earth's surface,
which occasions a different degree of cold in different states of the atmo-
sphere.

The motions of the air, which constitute winds, are in general dependent,
in the first instance, on variations of temperature. They are so accidental
and uncertain, as to be subjected to no universal laws ; as far however as
any regularity can be observed in their recurrence, it may in most cases
be sufficiently explained.*

The principal phenomena of the periodical winds may be reduced to six
distinct heads : first the general tendency from north east and south east
towards the equator, in latitudes below 30° ; secondly, the deviation of this
. tendency from the precise situation of the equator ; thirdly, the prevalence
of westerly winds between 30° and 40° or more, especially in the southern
hemisphere ; fourthly, the local modifications to which these general effects
are subjected ; fifthly, the monsoons, which vary every half year ; and
lastly, the diurnal changes of land and sea breezes.

With respect to the general tendency of the trade winds to the west, it
may be sufficiently explained by Hadley's theory t of the difference of the
rotatory motion of different parts of the atmosphere, combined with the
currents occasioned by the greater heat at the equator. For the sun's rays,
expanding the air in the neighbourhood of the equator, and causing it to
ascend, produce a current in the lower parts of the atmosphere, which rush
southwards and northwards towards the equator, in order to occupy the
place of the heated air as it rises : and since the rotatory motion of the
earth is greatest at the equator, and is directed eastwards, the air coming
from the poles has of course a relative motion westwards ; and hence the
joint motion of the current is directed, in the northern hemisphere, from
north east to south west, and in the southern, from south east to north
west. [As the winds on both sides approach the equator, the friction of
the earth's surface is constantly tending to give them an easterly direction ;
and since the lengths of the diurnal circles increase very slowly in the
immediate vicinity of the equator, this friction is even more effective than
the change of latitude ; and the westerly direction of the winds is gra-
dually lessened. Moreover, the northerly and southerly currents, coining
here into opposition, mutually annihilate each other's effects. At the equa-
tor, therefore, the trade winds lose their distinctive character, and consti-
tute only currents which depend on the preponderancy of local causes, and
thus vary in different places.^] Dr. Halley§ supposed that the air was
made in some measure to follow the sun round the earth, simply by means
of the expansion of the atmosphere, which takes place immediately under
him, and accompanies him round the globe ; but it does not seem evident
that the air could have any greater tendency to follow the sun that to meet

* See Dove, Meteorologische Untersuchungen, Berlin, 1837. Fechner's Reper-
torium, vol. Hi. f Hadley, Ph. Tr. 1735, xxxix. $8.

£ See Hall's Fragments of Voyages and Travels, 2nd Series, i. 162.
§ Ph. Tr. xvi. 152.

ON CLIMATES AND WINDS. 549

him. Nor can any sufficient cause be found in the attractions of the celes-
tial bodies, either for the general easterly trade winds, or for the current of
the sea in a similar direction, which appears to be the immediate effect of
their friction on the surface of the water.

The second circumstance is easily explained by the greater heat of the
northern than of the southern hemisphere ; so that instead of coinciding
with the equator, the neutral portion of the atmosphere lies between 3°
and 5° of north latitude ; the north east wind not reaching the equator,
and the south east continuing about 3° beyond it. But the situation of the
neutral portion varies with the sun's decimation, accordingly as different
parallels of latitude become in succession somewhat hotter than the neigh-
bouring parts. Where the northern and southern currents meet, their
joint effect must naturally be to produce a due east wind ; but in some
parts of the ocean, temporary calms and irregular squalls have been ob-
served to take place of this easterly wind, which generally prevails in the
neutral parts near the equator.

The third fact, that is, the frequency of westerly winds between the
latitudes 30° and 40°, has not yet been sufficiently explained. The most
probable cause of this circumstance is, that the current of heated air,
which we have hitherto neglected, and which passes, in the upper parts of
the atmosphere, from the equator each way towards the poles, and which,
being the converse of the trade wind, must be a south west and north
west wind, in the different hemispheres, becomes here sufficiently cool to
descend and mix with the lower parts of the atmosphere, or to carry them
along by its lateral friction : and while it descends to complete the circle,
necessary for supplying the current to the equator, its motion with respect
to the horizon must become at a certain time due west, since the cause
which stops its progress northwards, has no tendency to impede its motion
eastwards. The outward bound East India ships generally make their
easting in about 36° south latitude. It is probably also on account of the
rotatory motion of the earth, that south west winds are more common in
our latitudes than south east, and north east than north west.

Among the local modifications to be considered in the fourth place, we
may reckon the greater indistinctness of the third effect in the northern
than in the southern hemisphere, a circumstance which is explained
from the more irregular distribution of sea and land : for between 30°
and 40° south latitude the ocean is scarcely any where interrupted. In
lower latitudes also, near the west coast of Africa, the winds are so much
deflected towards the land, as to become in general westerly instead of
easterly.

The monsoons, which constitute the fifth remarkable circumstance, are
so called from a Malay word, denoting season. They are occasioned by
the peculiar situation of the continent of Asia, on the north side of the
equator. From April to September, the sun having north decimation, the
heat on this continent, a little north of the tropic, is very intense, and the
genera -^current is consequently towards the north. The air, therefore,
coming from south latitudes towards the equator, becomes, on account of the
deficiency of rotatory motion, a south east wind, as usual, which is found
to prevail between Madagascar and New Holland, as far as the equator.

550 LECTURE LVI.

In consequence perhaps of friction in its passage, it gradually loses its im-
petus towards the west, and at the equator is nearly a south wind ; but in
proceeding north from the equator, it becomes, from an excess of rotatory
motion, a south west wind, which blows into the Arabian gulf and the bay
of Bengal. Both these winds are however variously modified by the par-
ticular situations of the islands and continents. From October to March,
on the contrary, the sun having south declination, the south east trade wind
stops at 10° south latitude ; the trade winds on the north side of the equator
are as usual north east ; and beyond the equator they become for some
degrees north west, the circumstances being the reverse of those which
happen in the summer months, at greater distances, on the other side of
the equator. (Plate XLII. XLIII.)

The last fact is the simplest of all. The land and sea breezes are pro-
duced by the ascent of the air over the land in the day time, while the land
is hotter than the sea ; and its descent at night when the land is become
colder : hence the breeze comes from the sea by day, and from the land by
night.

The violent agitations of the air, which constitute hurricanes and whirl-
winds, occur more commonly in tropical climates than in others. The
causes of these storms are little understood : their course is said to be gene-
rally opposite to that of the trade winds ; but tornados, which are less re-
gular hurricanes, originate indifferently from every quarter.

The variations of the weight of the air, wrhich occasion the winds, and
other changes in its density, which are the effects of the winds themselves,
are indicated by the height of the barometer, which is in general the more
variable as the winds are more liable to sudden changes. Hence in the neigh-
bourhood of the equator the height of the barometer is scarcely ever a quarter
of an inch more or less than 30 inches, which is very nearly its mean height on
the level of the sea in every part of the globe : in Great Britain it is some-
times as low as 28 inches, but never higher than 31. We have already
seen that the elevation of any place above the sea reduces the height of the
barometer according to a law which is determined by the general properties
of elastic fluids : thus, at an elevation of 1 mile above the sea, the mean
height of the barometer is 24% inches, and at 2 miles, 20 inches only. The
use of the barometer, in foretelling variations of weather, is perhaps more
limited than has sometimes been supposed ; but by a careful observation,
conclusions may be drawn from it, which may in many cases be of con-
siderable utility : and it has even been applied with success, by some late
navigators, to the prediction of changes of wind, at times when they could
not have been suspected from any other circumstances.*

LECT. LVI.— ADDITIONAL AUTHORITIES.

Meteorology in general. — Richard, Hist. Naturelle de 1'Air et des Meteores, 10
vols. 12mo, Paris, 1770-1. Toaldo, Saggio Meteorologico, 4to, Padova, 1770.
La Meteor. Applicata, 4to, "Venezia, 1786. Deluc, Idees sur la Metgprologie,
2 vols. 1786-7. Cotte, Traite de Met. 4to, Paris, 1774; Mem. sur IziMeteor.
2 vols. 4to, 1778. Horrebow, Tractatus Historico-meteorologicus, 4to, Copenhag.

See D'Alcmbert, Reflexions sur la Cause Generate des Vents, 4to, 1747.

ON AQUEOUS AND IGNEOUS METEORS. 551

1780. Saussure, Voyages dans les Alpes, 4 vols. 4to, Neuchatel, 1796. . . Dalton's
Essays, 1793. Forster on Atmospheric Phenomena, 1823. Schouw, Beitrage zur
Vergleichenden Klimatologie, Copenhag. 1827; Ed. Jour, of Science, viii.311. Bailly
de Merlieux ; Resume complet de Meteor. 32mo, 1830. Rigaud. Harvey's Meteor,
in Encyc. Metrop. Kamtz, Lehrbuch der Meteor. 3 vols. 1831-6. Course of
Meteorology (trans.}, Lond. 1845. Howard's Climate of London, 3 vols. 1833.
Quetelet, Aper£u Historique des Obs. de Meteorologie faites en Belgique, 4to, Brux.
1834. Forbes's Reports on Meteorology, Br. Ass. 1834, 1840 ; translated into
German, and amplified by Mahlmann, Leipz. 1836. Dove's Repertorium, 1839,
vol. iii. Daniell's Meteor. Essays, var. ed. Front's Bridgwater Treatise, 1834.

Storms. — Redfield, Silliman's Journal, 1831, p. 17. Reid on the Law of Storms,
Lond. 1833; Edin. Review, Ixviii. 406. Espy on do. Dove, Scientific Mem.
part ix.

LECTURE LVIT.

ON AQUEOUS AND IGNEOUS METEORS.

THE phenomena originating from the evaporation of water constitute a
large proportion of the subjects of meteorology: they are materially
influenced by the diversities of climates and winds, which we have lately
considered ; and they appear to contribute to the electrical changes, which
form a principal part of luminous or igneous meteors : nor is the action of
water wholly unconcerned in many of the effects of subterraneous fires,
which have also a slight connexion with atmospherical electricity ; and it
has been conjectured that the only igneous meteors, which appear wholly
independent of any of these phenomena, may originate from volcanic
commotions in other worlds.

The action of heat appears to detach continually from the surface of
water, and perhaps of every other liquid, and even solid, a certain quantity
of vapour, in the form of an invisible gas ; but when the space above the
liquid is already charged with as much vapour as can exist in it at the
actual temperature, the vapour, thus continually thrown off, either remains
suspended in the form of visible particles, or falls back immediately into
the liquid. This is the simplest mode of explaining the continuance of
evaporation, under the pressure of any dry gas, however dense, and its
apparent suppression in the presence of moist air, however rare. Some-
times also, when the temperature of the liquid is elevated, so that minute
globules either of steam or of air rise through it, some visible particles are
projected upwards by each globule, and continue to float in the air ; this
appears, however, to be an irregularity unconnected with the principal
process of slow evaporation. •

The quantity of vapour, which can exist in the space above any portion
of water, has been supposed by Deluc,* Volta,t and Dalton,^ to be wholly
independent of the nature, the density, or even the presence of the air or
gas wMeh that space contains : and we may easily imagine that the
smallest distance at which the particles of water, constituting vapour, can

* Ph. Tr. 1792, p. 400. f Gren's Journal, iii. 479.

t See Ph. Tr. 1826.

552 LECTURE LVII.

exist, without coming within the reach of their mutual cohesion, is the
same, whatever other particles may be scattered through the interven-
ing space. It appears, however, more consistent with some experiments,
to suppose, that the presence of air of the usual density allows the particles
of water to approach a little nearer together without cohering, so that the
utmost quantity of moisture that can be contained in a cubic foot of air
at a given temperature is not exactly the same as would make a cubic foot of
pure vapour, but always in a certain proportion to it ; and it seems to fol-
low,, from the experiments of Saussure, compared with those of Pictet, that
the weight of the vapour contained in a cubic foot of air is about one half
greater than that of a cubic foot of pure vapour at the same temperature.

When the air, in the neighbourhood of the surface of the water, has
become thus saturated with moisture, the evaporation proceeds very slowly,
the vapour being precipitated as soon as it rises : but if the air be continu-
ally changed, so that the moistened portion may be removed, and dry air
substituted for it, the process will be greatly expedited ; and such a change
may be effected either by wind, or by the natural circulation, occasioned
by any elevation of temperature communicated by the water to the neigh-
bouring air ; but when this circulation is prevented, the evaporation is
much diminished, although the temperature may be considerably elevated.
In moderate exposures, the depth of the quantity of water, evaporating in
24 hours from any surface, is expressed, according to Mr. Dalton's experi-
ments, by the height of the column of mercury equivalent to the force of
steam at the given temperature, deducting, however, the effect of the elas-
ticity of the moisture already existing in the air.

Since the quantity of moisture, which the air [or rather a given space^
is capable of receiving, is greater as its temperature is greater, we may
obtain a natural measure of the quantity which it contains by reducing
it to the temperature at which the moisture begins to be deposited. Thus,
if we take a glass of cold water, and add to it some common salt, or some
muriate of lime, we may cool the air near it so much as to cause it to
deposit a part of its moisture on the glass : and by measuring the tempera-
ture of the water when the precipitation begins, Mr. Dalton estimates the
true state of the air with respect to moisture. Thus, if the glass begins to
be moistened when the water is at 40°, he infers from the known elasticity
of steam at that temperature, that the quantity of moisture contained in
the air is equivalent to the pressure of a column of mercury about a
quarter of an inch in height ; and if the actual temperature of the air be
50°, the corresponding elasticity of steam being a little more than one third
of an inch, the daily evaporation in such air will amount to about one
ninth of an inch, making 40 inches in the whole year. In fact, however,
the air is* usually moister than this, and the mean evaporation of all
England is, according to Mr. Dalton,* about 23 inches only.

In hotter climates, and in particular situations, the evaporation may be
considerably greater. The Mediterranean Sea, being surrounded by land,
is more heated than the ocean, and the winds which blow ovens^t are
drier ; consequently its evaporation is greater than that of the Atlantic,
and its specific gravity is increased by the increased proportion of salt ;
* Manch. Mem. v. 346.

ON AQUEOUS AND IGNEOUS METEORS. 553

so that at the straights of Gibraltar, a current runs inwards at the surface
and outwards near the bottom, for the same reason as the air, when it is
denser in a passage than in the adjoining room, blows a candle towards the
room at the lower part of the door, and draws it towards the passage at the
upper. Had there been a continual current inwards through the Straights,
at all parts, the Mediterranean must in the course of ages have become a
rock of salt. It is indeed remarkable that all lakes, into which rivers run
without any further discharge, are more or less salt, as well as lakes in
general near the sea : but where a river runs through a lake into the sea,
it must necessarily, in the course of time, have carried the salt of the lake
with it, if it had ever existed.

Experiments on the deposition of moisture, like those of Mr. Dalton, are
liable to a slight inaccuracy, on account of the effects of an apparent
elective attraction, by means of which, some substances seem to attract
humidity at a temperature a little higher than others. Thus, a surface of
metal often remains dry, in the neighbourhood of a piece of glass which is
covered with moisture. It is certain that some substances attract moisture
from the air, even when the quantity which it contains is incomparably
less than that which would saturate it, since it is on this circumstance
that the construction of hygrometers depends ; and it is probably by a
property somewhat similar, that even surfaces of different kinds possess
different attractive powers for moisture nearly ready to be deposited. It is,
however, only necessary to employ, for Mr. Dalton's experiment, a sub-
stance which has a very weak attraction for moisture ; and any kind of
metal will perhaps be found sufficiently correct in its indications.

It has been observed, that a piece of metal, placed on glass, usually pro-
tects also the opposite side of the glass from the deposition of dew ; and
Mr. Benedict Prevost has shown, that in general, whenever the metal is
placed on the warmer side of the glass, the humidity is deposited more
copiously either on itself, or on the glass near it ; that when it is on the
colder side, it neither receives the humidity, nor permits its deposition on
the glass ; but that the addition of a second piece of glass, over the metal,
destroys the effect, and a second piece of metal restores it. It appears that,
from its properties with respect to radiant heat, the metallic surface pro-
duces these effects, by preventing the ready communication either of heat
or of cold to the glass.*

The quantity of invisible moisture, contained in air, may be, in some
degree, estimated from the indications of hygrometers, although these in-
struments have hitherto remained in a state of great imperfection. A sponge,
a quantity of caustic potash, or of sulfuric acid, or a stone of a peculiar
nature, has sometimes been employed for determining the degree of moisture
of the air, from which it acquires a certain augmentation of its weight.
A cord dipped in brine,t or the beard of an oat, is also often used for the
same purpose : the degree in which it untwists, from the effect of moisture,
being shown by an index. But the extension of a hair, or of a slip of
• -'

* B. Prevost on Dew, Ann. de Chimie, xliv. 75.
t Smeaton, Ph. Tr. 1771, p. 198.

554 LECTURE LVII.

whalebone, which have been employed by Saussure* and Deluc,t appear
to be more certain and accurate in their indications. The hair hygrometer
acquires more speedily the degree corresponding to any given state of the
air, but it seems to reach the utmost extent of its scale before it arrives at
perfect humidity : while the whalebone hygrometer appears to express a
greater change upon immersion in water than from the effect of the moist-
est transparent air, which has also been considered by some as an imper-
fection. Both these instruments are impaired by time, and acquire contrary
errors, so that a mean between both is more likely to be correct than either
separately. Their indications are at all times widely different from each
other, and the mean appears to approach much nearer to a natural scale
than either of them. Mr. Leslie J employs a very delicate thermometer, of
which the bulb is moistened, for measuring the drjoiess of the air, by the
cold produced during evaporation, when the thermometer is exposed to it ;
but this mode of estimating the quantity of moisture appears to be liable
to considerable uncertainty. (Plate XLI. Fig. 581.)

In order that the scale of a hygrometer should be perfectly natural, it
ought to express, at all temperatures, the proportion of the quantity of
moisture in the air to that which is required for its saturation ;§ thus, at
100 degrees, it should imply that the slightest depression of temperature
would produce a deposition ; at 50 degrees, that the air contains only half
as much water as would saturate it, or, supposing the thermometer at 52°,
that a deposition would be produced in it by a depression of 17°. And
if we know the actual temperature, and the temperature at which the depo-
sition takes place, we may find the height of the natural hygrometer, by
the proportion of the corresponding elasticities of steam. The mean
height of the natural hygrometer in London is probably about 80° ; that
of Deluc's hygrometer, with proper corrections, being nearly 70°: so
that a depression of 6° must usually be sufficient to cause a deposition of
moisture.

The quantity of water actually contained in a cubic foot of air, satu-
rated with moisture, appears to be about 2 grains at the freezing point,
4 grains at 48°, 6 at 60°, and 8 at 68° ; and the density of the vapour,
thus mixed with air, is, according to Saussure's experiments, about three
fourths as great as that of the air itself ; so that moist air is always a little
lighter than dry air ; and the more so as the air is warmer, provided that
it be saturated with moisture by means of the presence of water. It follows
from the properties of moisture thus determined, that if any two portions
of perfectly humid air, at different temperatures, be mixed together, there
must be a precipitation : thus, a cubic foot of air at 32° being mixed with
another at 60°, their common temperature must be 46° ; if they are satu-

* Essai sur 1'Hygrometrie, Neuch. 1783. Jour, de Phy. xxxii. 24, 98.

t Ph. Tr. 1791, 1, 389. Jour, de Ph. xxx. 437 ; xxxii. 132.

J Nich. Jour. iii. 401. A short Account of Instruments depending on the Rela-
tions of Air to Heat and Moisture, Edin. 1813. Descrip. of Instrs. for Improving
Meteor. Obs. Edin. 1820. See on this subject Forbes's Supplementary Report on
Meteor. Brit. Ass. 1840, p. 95.

§ This phrase must not be supposed to imply any combination between the air
and vapour.

ON AQUEOUS AND IGNEOUS METEORS. 555

rated with moisture, they must contain 8 grains of water when separate ;
but when mixed they will he too cold by 2° to contain the same quantity ;
since air at 48° can only contain four grains for each foot ; and it has been
supposed that such mixtures frequently occasion a precipitation in nature.
Thus, it often happens that the breath of an animal, which is in itself
transparent, becomes visible when mixed with a cold atmosphere ; and in
such cases the deposition may perhaps be facilitated by the cooling of the
warmer air to a certain degree, even before a perfect mixture has taken
place.*

When visible vapour has been thus deposited from transparent air, by
means either of cold or of mixture, it generally remains for some time sus-
pended, in the form of a mist or of a cloud : sometimes, however, it appears
to be at once deposited 011 the surface of a solid, in the form of dew or of
hoar frost ; for it is not probable that the crystallized form, in which hoar
frost is arranged, can be derived from the union of the particles already
existing in the air as distinct aggregates.t

The dew, which is commonly deposited on vegetables, is partly derived,
in the evening, from the vapours ascending from the heated earth, since it
is then found on the internal surface of a bell glass ; and towards the morn-
ing, from the moisture descending from the air above, as it begins to cool.
Sometimes, however, in warmer weather, the dew begins to descend in the
evening ; this the French call serein : the humidity deposited by mists on
trees, and by moist air on windows, generally within, but sometimes with-
out, they call givre. [The cause of the deposition of dew has been satis-
factorily assigned by Dr. Wells. ^ It is traceable to two circumstances, the
radiation of heat, and the condensation of vapour by cold. Owing to the
former circumstance, different substances on the earth's surface become
cool with different degrees of rapidity, according to their mechanical tex-
ture, or their position, or whatever it may be. When they have cooled
down to such a point that the existing vapour in the atmosphere near them
can no longer be retained in its elastic state, it becomes water, and is depo-
sited on their surface. The cause of deposition is the previous cooling
of the substance on which it takes place. Dr. Wells found, as Mr. Six
had done before him, that a thermometer laid on a grass plot in a clear
night, indicates a cold many degrees lower than a thermometer hung at
some height from the ground. This is owing to the fact that grass radiates
heat well; and accordingly it receives a copious deposition of dew, which
a worse radiator would not do. Moreover, if the sky becomes overcast, or
if any substance be interposed between it and the grass, radiation is checked,
or it may be that the grass receives more heat from the surrounding objects
or clouds than it radiates, and thus its temperature becomes raised. Under
these circumstances the deposition of dew ceases.]

Mists are said to consist sometimes of other particles than pure water :
these are called dry mists, and they have been supposed to blight vege-

* Hatton, Dissertation on various Subjects of Natural Philosophy, 4to, Edin.
1792.

t See Howard's Essay on the Modification of Clouds, 1832.
: Wolls on Dew, 1814.

556 LECTURE LVII.

tables. Such mists are sometimes attended by a smell, resembling that
which is occasioned by an electric spark. Rain falling after a dry season
deposits, when it has been suffered to stand, some particles of foreign
matter which it has brought down from the atmosphere. There must in-
deed frequently be a multiplicity of substances of various kinds floating in
the air ; the wind has been found to carry the farina of plants as far as 30
or 40 miles, and the ashes of a volcano more than 200. It only requires
that the magnitude of the particles of any substance be sufficiently reduced
in size, in order to render them incapable of falling with any given velo-
city ; and when this velocity is very small, it may easily be overpowered
by any accidental motions of the air. The diameter of a sphere of water,
falling at the rate of one inch only in a second, ought to be one six hundred
thousandth of an inch, which is about the thickness of the upper part of
a soap bubble at the instant when it bursts ; but the particles of mists are
incomparably larger than this, since they would otherwise be perfectly
invisible as separate drops : the least particle that could be discovered by
the naked eye, being such as would fall with a velocity of about a foot in
a second, if the air were perfectly at rest. But it is very probable that the
resistance, opposed to the motion of particles so small, may be considerably
greater than would be expected from a calculation derived from experi-
ments made on a much larger scale, and their descent consequently much
slower.

When the particles of a mist are united into drops capable of descending
with a considerable velocity, they constitute rain ; if they are frozen
during their deposition, they exhibit the appearance of a perfect crystal-
lization, and become snow : but if the drops already formed are frozen,
either by means of external cold, or on account of the great evaporation
produced by a rapid descent through very dry air, they acquire the cha-
racter of hail, which is often observed in weather much too hot for the
formation of snow.

It cannot be doubted but that there is a connexion betwen the descent of
the barometer and the fall of rain ; but no satisfactory reason has yet been
assigned for the circumstance ; nor is it possible to foretel, with certainty,
that rain will follow any changes in the height of the barometer that have
been observed. The immediate dependence of rain, or of any other atmo-
spherical phenomena, on the influence of the moon, appears to be rendered
highly improbable, not only by mathematical calculations of the effects of
the moon's attraction, but also by the irregularity of the very observations
which have been adduced in favour of such a connexion. But however
uncertain the ultimate causes of rain may be in general, their effects in
some places are sufficiently constant to be attributed to permanent local
circumstances, and in particular to the periodical recurrence of similar
winds.

In low and level countries, clouds may often begin to descend from the
upper regions of the atmosphere, and may be redissolved by the warmer
air below ; but when they descend in an equal degree among mountains,
they fall on the earth ; and besides the quantity of water which they fur-
nish for vegetation, and that which is carried off by evaporation, they

ON AQUEOUS AND IGNEOUS METEORS. 557

afford, by means of springs and rivers, a constant supply for the use of
man and of other animals in distant parts. The upper regions of the
atmosphere are however by no means the principal sources of rain in ordi-
nary climates, since a gage placed on a very high building seldom collects
more than two thirds as much rain as another standing on the ground
below :* and the effects of mountains in collecting rain are perhaps
chiefly derived from the ascending currents which they occasion, and by
which the air saturated with moisture is carried to a higher and a colder
region.

The Abyssinian rains are the causes of the inundation of the Nile : they
last from April to September ; but for the first three months the rain is
only in the night. The inundation, in Egypt, begins at present about
the 17th of June ; it increases for 40 days, and subsides in the same time ;
but the ancient accounts, as well as some modern ones, assign a longer
duration to it. The river Laplata rises and falls at the same times as the
Nile. The Ganges, the Indus, the Euphrates, the river of Ava or Pegu,
and many other large rivers, have also considerable inundations at regular
periods. In many other countries there are seasons at which the rains
seldom fail to recur ; and sometimes the periodical rains are different in
different parts of the same country. Thus the coast of Malabar, which is
to the west of the Gate mountains, or Gauts, enjoys summer weather,
without rain, from September to April, while that of Coromandel, which
is on the eastern side, experiences all the rigours of its winter ; being at
this time exposed to the influence of the north east trade wind. Vicissi-
tudes of a similar nature are also observed on the north and south sides of
the island of Jamaica. The mean fall of rain in London is about 23
inches ; at Exeter, which is nearer to the Atlantic, 33 ; the average of
England and Wales is 31.

The evaporations and precipitations, and probably also the condensations
and expansions, which take place on a large scale in the atmosphere, and
in the clouds, cannot fail of producing changes in their electrical qualities,
and these changes appear to be the principal sources of the phenomena of
thunder and lightning. The clouds, when electrified, being more or less
insulated by the interposition of the air, exhibit attractive and repulsive
effects, and are discharged by explosions, either among themselves, or
communicating with the earth, in the same manner as bodies which have
been electrified by artificial means ; they also sometimes produce, in the
neighbouring parts of the earth, and in the animals on its surface, a state
of induced electricity ; and in this case the returning stroke, or the sudden
restoration of the equilibrium, when the electricity of the nearest clouds is
imparted to the more remote, may be fatal, without any appearance of an
immediate discharge, at the place where the animal stands.

We can, however, by no means precisely ascertain in what manner all
the electrical phenomena of the atmosphere are produced. It appears from

* From the observations of Prof. Phillips at York, the fall of rain during twelve
months ^as 25'7 in. on the ground, 19'8 in. 44 feet above the ground, and not quite
15 in. 213 feet above it. Rep. of Br. Ass. 1834, p. 560.

558 LECTURE LVII.

the experiments of Beccaria* and Cavallof that the air is in general
positively electrical, and most so in cold and clear weather ; in cloudy-
weather more slightly : and that during rain, the air is generally in a nega-
tive state. Mr. Read J has found that air charged with putrid vapours of
any kind, and in particular the air of close rooms, is almost always nega-
tively electrified. The electricity is more readily communicated to an
electrometer in an elevated situation, and in damp weather, than in other
circumstances ; a candle is also very useful in collecting it. When a wire
is connected with a kite, being continued along the string, we may fre-
quently ohtain from it sparks a quarter of an inch long.

We find a complete and interesting description of the effects of a violent
thunder storm in a paper by Mr. Brereton, inserted in the Philosophical
Transactions^ The circumstance happened in September 1780, at East
Bourn, in a house occupied by Mr. Adair : it was built of stone, and stood
facing the sea. About nine o'clock, in a very stormy morning, a black
cloud approached the house ; several balls of fire were seen to drop from it
successively into the sea, and one in particular, appearing like an immense
sky rocket, broke against the front of the house in different directions.
Mr. Adair was standing at a window on the first floor, with his hands
clasped together, and extended against the middle of the frame : his hands
were forced asunder, he was thrown several yards off on the floor, and
remained for some time speechless and motionless, although not insensible 5
his clothes were much torn ; several articles of metal about his person were
partially melted, while others, apparently in similar circumstances, and in
particular a silver buckle, escaped ; and his skin was in many parts much
scorched and lacerated. The whole of the glass in the window, and a pier
glass near it, were completely destroyed, and scattered about the room ;
most of the furniture was broken to pieces, and all the bell wires were
melted. In the room above this, a lady and her maid were driven to a
distant part, and rendered insensible for some time, but not hurt ; in the
room below, two servants, who were near the windows, were struck dead :
both the bodies were turned black : one of them had a wound near the
heart ; and neither of them became stiff after death ; a third servant, who
was a little behind one of them, escaped with the loss of a telescope, which
he held in his hand, and with the sensation of a violent pressure on his
head and on his back. A large stone was forced out of the wall near them,
and thrown into the room, and some other similar effects were observed,
which marked the progress of the explosion.

For guarding against accidents so dreadful, Dr. Franklin's great invention
of metallic conductors may be very advantageously employed : for, when
properly fixed, they afford a degree of security which leaves very little room
for apprehension. A conductor ought to be continued deep into the earth,

* Delia Elettricita Terrestre Atmosferica, 4to, Torino, 1775.
f Ph. Tr. 1776, p. 407 ; 1777, p. 48.

J Jour, of Electricity, Ph. Tr. 1792, p. 225 ; 1794, pp. 185, 266. Treatise on
Atmospheric Electricity, 1793. See Arago, Annuairefor 1838.
§ 1781, p. 42.

ON AQUEOUS AND IGNEOUS METEORS. 559

or connected with some well or drain : it should be of ample dimensions,
and where smallest, of copper, since copper conducts electricity more readily
than iron. In one instance a conductor of iron, four inches wide and half
an inch thick, appears to have heen made red hot by a stroke of lightning.
It seems to be of some advantage that a conductor should be pointed, but
the circumstance is of less consequence than has often been supposed.*
Mr. Wilson exhibited some experiments in which a point was struck at a
greater distance than a ball, and therefore argued against the employment
of pointed conductors. Mr. Nairne, f on the contrary, showed that a ball
is often struck in preference to a point. But it has been observed, that if a
point attracts the lightning from a greater distance, it must protect a greater
extent of building. It is easy to show, by hanging cotton or wool on a con-
ductor, that a point repels light electrical bodies, and that a pointed con-
ductor may, therefore, drive away some fleecy clouds ; but this effect is
principally derived from a current of air repelled by the point ; and such
a current could scarcely be supposed to have any perceptible effect on clouds
so distant as those which are concerned in thunder storms. In order to
escape personal danger in a thunder storm, the best precautions are, to avoid
eminences, and all exposed situations, as well as a near approach to conduc-
tors. The neighbourhood of windows, looking-glasses, fire-places, and trees,
must always be considered as hazardous.

It has been supposed that a sudden condensation of the air, arising from
cold, accompanied by a deposition of moisture, and propagated by a con-
tinuation of the cause, by means of the cold occasioned by expansion, pro-
duces frequently the noise of thunder, without any lightning, and without
any electrical agitation : but it does not appear that the opinion is well
established. J

The phenomena of waterspouts, if not of electrical origin, appear to have
some connexion with electrical causes. A waterspout generally consists of
large drops like a dense rain, much agitated, and descending or ascending
with a spiral motion, at the same time that the whole spout is carried along
horizontally, accompanied in general by a sound like that of the dashing
of waves. Spouts are sometimes, although rarely, observed on shore, but
generally in the neighbourhood of water. They are commonly largest
above ; sometimes two cones project, the one from a cloud, the other from
the sea below it, to meet each other, the junction being accompanied by a
flash of lightning : and when the whole spout has exhibited a luminous
appearance, it has perhaps served to conduct electricity slowly from the
clouds to the earth. Some of these circumstances may be explained by
considering the spout as a whirlwind, carrying up drops of water, which it
has separated from the surface of the waves ; and the remainder may per-

* See Report of Committee appointed to consider of a Method for securing Pow-
der Magazines ; with Mr. B. Wilson's Dissent, Ph. Tr. 1773, Ixiii. 42. See also
ibid. liv. 247; Ixviii. 999. Cavallo, ibid. 1788, p. 1. Murray's Treatise on At-
mospheric Electr. 1828. Harris, On the Utility of fixing Lightning Conductors in
Ships, Plymouth, 1830. Annals of Electr. iv. 310 ; v. 41.

t Ph?Tr. 1774, p. 79; 1778, p. 823.

J See Harris's Essay on Thunderstorms, 1843.

560 LECTURE LVII.

haps be deduced from the cooperation of electricity, already existing in a
neighbouring cloud.

It is doubtful whether the light of the aurora borealis may not be of an
electrical nature : the phenomenon is certainly connected with the general
cause of magnetism ; the primitive beams of light are supposed to be at an
elevation of at least 50 or 100 miles above the earth, and every where in a
direction parallel to that of the dipping-needle ; but perhaps, although the
substance is magnetical, the illumination, which renders it visible, may still
be derived from the passage of electricity, at too great a distance to be dis-
covered by any other test.

Earthquakes* and volcanos appear to originate in chemical changes,
which take place within the substance of the earth : they have probably
little further connexion with electricity, than as causes which occasionally
destroy the electrical equilibrium ; for although some authors have inferred,
from the great velocity with which the shock of an earthquake is trans-
mitted from place to place, that its nature must be electrical, yet others
have, with greater probability, attributed the rapid succession of the effects
to the operation of a single cause, acting at a great distance below the earth's
surface. There are however some circumstances, which indicate such a
connexion between the state of the atmosphere and the approach of an
earthquake, as cannot easily be explained by any hypothesis.

The shocks of earthquakes and the eruptions of volcanos, are in all
probability modifications of the effects of one common cause : the same
countries are liable to both of them ; and where the agitation produced by
an earthquake extends further than there is any reason to suspect a
subterraneous commotion, it is probably propagated through the earth
nearly in the same manner as a noise is conveyed through the air. Volca-
nos are found in almost all parts of the world, but most commonly in the
neighbourhood of the sea ; and especially in small islands ; for instance, in
Italy, Sicily, Iceland, Japan, the Caribbees, the Cape Verd islands, the
Canaries, and the Azores : there are also numerous volcanos in Mexico and
Peru, especially Pichincha and Cotopaxi. The subterraneous fires, which
are continually kept up in an open volcano, depend perhaps in general on
sulfureous combinations and decompositions, like the heating of a heap of
wet pyrites, or the union of sulfur and iron filings : but in other cases they
may perhaps approach more nearly to the nature of common fires. A
mountain of coal has been burning in Siberia for almost a century, and
must probably have undermined in some degree the neighbouring country.
The immediate cause of an eruption appears to be very frequently an
admission of water from the sea, or from subterraneous reservoirs ; it has
often happened that boiling water has been discharged in great quantities
from a volcano ; and the force of steam is perhaps more adequate to the
production of violent explosions, than any other power in nature. The
consequence of such an admission of water, into an immense collection of

^ * Bertrand, M£moires Historiques et Physiques sur lea Tremblemens de Terre. A
1 Haye, 1757. Michell, Conjectures concerning the Cause of Earthquakes^ Ph. Tr.

ON AQUEOUS AND IGNEOUS METEORS. 561

ignited materials, may in some measure be understood, from the acci-
dents which occasionally happen in founderies ; thus a whole furnace of
melted iron was lately dissipated into the air in Colebrook Dale, by the
effect of a flood, which suddenly overflowed it.

The phenomena of earthquakes and volcanos are amply illustrated by
the particular accounts, transmitted to the Royal Society by Sir William
Hamilton, of those which have happened at different times in Italy.* The
earthquake, which desolated Calabria, in 1783, was fatal to about 40,000
persons, continuing its ravages for more than three months ; it destroyed
the towns and villages occupying a circle of nearly 50 miles in diameter,
lying between 38 and 39 degrees latitude, and extending almost from the
western to the eastern coast of the southernmost point of Italy, besides
doing considerable damage to places at much greater distances from its
origin, which is supposed to have been either immediately under the town
of Oppido, in the centre of this circle ; or under some part of the sea,
between the west of Italy, and the volcanic island of Stromboli. This
island, as well as Mount Etna, had smoked less than usual before the
earthquake, but they both exhibited appearances of an eruption during its
continuance ; Etna towards the beginning, and Stromboli at the end.
Before each shock the clouds were usually motionless for a certain time,
and it rained violently ; frequently also lightning and sudden gusts of
wind accompanied the rain. The principal shocks appeared to consist in a
sudden elevation of the ground to a considerable height, which was propa-
gated somewhat like a wave, from west to east : besides this, the ground
had also a horizontal motion backwards and forwards, and in some mea-
sure in a circular direction. This motion was accompanied by a loud
noise ; it continued in one instance for ten seconds without intermission ;
and it shook the trees so violently that their heads nearly reached the
ground. It affected the plains more strongly than the hills. In some
places luminous exhalations, which Sir William Hamilton thinks rather
electrical than igneous, were emitted by the earth : the sea boiled up near
Messina, and was agitated as if by a copious discharge of vapours from its
bottom ; and in several places water, mixed with sand, was thrown up to
a considerable height. The most general effect of these violent commotions
was the destruction of buildings of all kinds, except the light barracks of
wood or of reeds, into which the inhabitants retreated as soon as they were
aware of their danger : the beds of rivers were often left dry, while the
shock lasted, and the water on its return overflowed their banks : springs
were sometimes dried up, and new ones broke out in other places. The hills
which formed the sides of steep vallies were often divided by deep chasms
parallel to the vallies ; and in many cases large portions of them were sepa-
rated, and removed by the temporary deluge to places half a mile or a mile
off ; with the buildings and trees still standing on them ; and in this
manner hills were levelled, and vallies were filled up. But the most fatal
accident of this kind happened at Scilla, where so large a portion of a cliff

* Ph.Tr. 1767, Ivii. 192 ; 1768, Iviii. 1 ; lix. 18 ; 1780, Ixx. 42 ; 1783, p. 169:
and Count Ippolito, ibid. p. 209, 1795, 73. See also Hamilton's Observations on
the Volcanos of the Two Sicilies, 2 vols. fol. 1776.

2 o

502 LECTURE LVII.

was thrown into the sea, that it raised an immense wave, which carried oft*
more than 2000 inhabitants who were collected on the beach, and even
extended its formidable effects to the opposite coast of Sicily, where several
persons perished by it in a similar manner.

The eruptions of volcanos are usually attended by some shocks like those
of earthquakes, although commonly less violent. Open volcanos con-
tinually throw out, in more or less abundance, smoke, ashes, and pumice
stones, or light cinders ; but their most formidable effects are produced by
a torrent of ignited lava, which, like a vast deluge of liquid or semiliquid
fire, lays waste the country over which it runs, and buries all the works of
human art. In March, 1767, Vesuvius began to throw out a considerable
quantity of ashes and stones, which raised its summit in the course of the
year no less than 200 feet, forming first a little mountain of pumice stones
within the crater, which by degrees became visible above its margin. The
smoke, which was continually emitted, was rendered luminous at night, by
the light derived from the fire burning below it. In August some lava had
broken through this mountain, and in September it had filled the space
left between it and the former crater. On the 13th and 14th of October
there were heavy rains, which perhaps supplied the water concerned in the
eruption that shortly followed. On the morning of the 19th, clouds of
smoke were forced, in continual succession, out of the mouth of the vol-
cano, forming a mass like a large pine tree, which was lengthened into an
arch, and extended to the island of Caprea, 28 miles off ; it was accom-
panied by much lightning, and by an appearance of meteors like shooting
stars. A mouth then opened below the crater, and discharged a stream of
lava, which Sir William Hamilton ventured to approach within a short
distance, imagining that the violence of the confined materials must have
been exhausted ; but on a sudden the mountain opened with a great noise
at a much lower point, about a quarter of a mile from the place where he
stood, and threw out a torrent of lava, which advanced straight towards
him, while he was involved in a shower of small pumice stones and ashes,
and in a cloud of smoke. The force of the explosions was so great, that
doors and windows were thrown open by them at the distance of several
miles : the stream of lava was in some places two miles broad, and 60 or
70 feet deep ; it extended about six miles from the summit of the moun-
tain, and remained hot for several weeks. In 1794 a still more violent
eruption occurred : it was expected by the inhabitants of the neighbour-
hood, the crater being nearly filled, and -the water in the wells having
subsided. Showers of immense stones were projected to a great height ;
and ashes were thrown out so copiously, that they were very thick at
Taranto, 250 miles off ; some of them also were wet with salt water. A
heavy noxious vapour, supposed to be carbonic acid, issued in many places
from the earth, and destroyed the vineyards in which it was suffered to
remain stagnant. A part of the town of Torre del Greco was overwhelmed
by a stream of lava, which ran through it into the sea ; yet notwithstand-
ing the frequency of such accidents, the inhabitants had so strong, a predi-
lection for their native spot, that they refused the offer of a safer situation
for rebuilding their houses.

ON AQUEOUS AND IGNEOUS METEORS. 5G3

Convulsions of these kinds must have very materially influenced the
disposition of the strata of the earth, as well as the form of its surface ;
but it is by no means fully determined how far such causes have been
concerned, or how far the effects are to be attributed to the intermediation
of water only. Mineralogists and geologists have been principally divided
into two classes with respect to their theories of the earth, some maintain-
ing the Vulcanian, and some the Neptunian hypothesis. It appears to be
impossible to decide with any certainty between these opposite opinions ;
nor is it perhaps of much consequence for any purpose of practice, or even
of science. The Neptunians are certainly able to establish their own
theory positively, and to prove that the fluid parts of the earth and sea
must have been very materially concerned in producing the changes which
have happened to the solid parts ; but it may be difficult for them to con-
fute the assertion, that heat, whether caused by volcanos or otherwise, has
also been a very powerful agent in these operations, and in some cases the
joint effects of heat and of increased pressure appear to have been con-
cerned in giving to minerals of different kinds their actual form ; although
on the whole it seems probable that the operation of heat has been much
more limited than that of aqueous solutions and precipitations. Mr.
Davy has also very justly inferred, from his experiments with the battery
of Volta, that the effects of the electricity excited by means of chemical
changes within the earth, have probably been very materially concerned in
the gradual formation of a variety of mineral productions.

The arguments for establishing the general fact, that great convulsions
have actually happened to the earth, are too well known to require minute
examination : the variety of fossil substances, many of them marine pro-
ductions, and some almost preserving a recent appearance, that are found
in mountains remote from the sea, are undeniable proofs that the levels of
the earth's surface must have undergone considerable changes ; although
some philosophers are of opinion, that such of the primary mountains as
are above 6 or 700 feet high, have never been wholly covered by the sea.
It is not at all easy to explain the change of climate, which some of these
circumstances appear to indicate ; the remains of animals inhabiting hot
countries, and the marine productions of hot climates, which are frequently
found in high northern latitudes, would induce us to suspect, that the posi-
tion of the earth's axis was at a former time very different from its pre-
sent position : and we can scarcely assign any other probable cause for
this change, than the casual interference, and perhaps incorporation, of a
comet with the earth. The probabilities of such an event, in the whole
course of time, are however so small, that we have no reason to be appre-
hensive of the chance of its occurring in future, for it is not enough that a
comet should approach so near to the earth as to be very powerfully at-
tracted by it, its motion must also be directed almost in a straight line
towards the earth ; otherwise it might only be inflected into a new orbit,
and go off again, without having caused any other disturbance than a
partial overflow of the sea.

The face of the globe has also been very materially changed in the
course of ages, by the gradual operation of the sea and of rivers. The sea

2 o2

564 LECTURE LVII.

has incroached in particular parts, and retired from others ; and the
mouths of large rivers, running through low countries, have often been
variously modified, by a deposition and transfer of the matter washed
down from the land. At Havre the sea undermines the steep coast, and
recedes at Dunkirk, where the shore is flat : in Holland the Zuyder Zee
was probably formed in the middle ages by continual irruptions of the
sea, where only the small lake Flevo had before existed ; and the mouths
of the Rhine have been considerably altered, both in their dimensions and
in their directions. The mud, deposited by large rivers, generally causes a
Delta, or triangular piece of land, to grow out into the sea ; thus the
mouth of the Mississippi is said to have advanced above 50 miles since the
discovery of America ; and the sea has retired from Rosetta above a mile
in 40 years. The mouths of the Arno and of the Rhone consist also in
great measure of new land.*

The meteors denominated shooting stars are observed to move in all
directions, as well upwards as downwards, although they frequently seem
to have a tendency towards a particular quarter in the course of the same
evening. Their height is seldom less than 20 miles, and sometimes as
much as 100 or 200, but usually about 50 ; their velocity is commonly
about 20 miles in a second, which differs very little from that of the earth
in its orbit. The rapidity of their motion, as well as its occasional devia-
tion from a right line, has generally been considered as a reason for sup-
posing that they depend on electricity ; but the opinion is by no means
fully established.

Other igneous meteors, which nearly resemble in their appearance the
largest of these, are sometimes observed to fall on the earth, either entire
or divided ; and after their fall, certain stones have been found, which have
been supposed to have descended in an ignited state.t Mr. Howard J has
ascertained that almost all these stones agree in their general characters,
and in their chemical analysis, especially in the circumstance of containing
nickel. It has been conjectured, both in this country and on the con-
tinent, that they have been emitted by lunar volcanos, and it has been
observed, that since they would find little or no resistance from the very
rare atmosphere of the moon, they would require a velocity of projection
only four times as great as that which a cannon ball sometimes receives, in
order to rise into the sphere of the earth's attraction. Their heat and
combustion may not improbably be derived from the great condensation
which they must occasion in the air immediately before them, and even
their friction might easily produce enough of electric light, to render them
visible in the dark. Among many such substances projected from the
moon, it is probable that a few only would be directed towards the earth,
and many more would be made to revolve in ellipses round it, and become
little satellites, too small for human observation, except when they enter

* The reader is referred to Lyell's or Ansted's Geology.

t On meteoric stones, see Chladni on the Siberian iron, Riga, 1794 ; Ueber
Feuer-Meteore, Vienna, 1819, with App. by Schreibers ; and art. Stones (Meteor.)*
Encyc. Metrop.

J Ph. Tr. 1802, p. 168.

ON VEGETATION. 665

far enough into the atmosphere to produce an appearance of light, re-
sembling that of a shooting star ; but it is scarcely probable that their
velocity could ever be at all comparable with that which has been attri-
buted to these meteors. There is, however, no difficulty in supposing, on
the other hand, that the wandering substances, which may be moving
through empty space, with a velocity equal to that of the shooting stars,
may be so much retarded, when they penetrate deep into our atmosphere,
as to make but a moderate impression by their fall on the ground ; and if
we suppose the meteors to be of one kind only, they must be referred
rather to the description of shooting stars than to that of the productions
of lunar volcanos; although the undulatory motion, sometimes observed in
these meteors, seems to be in some measure inconsistent with the progress
of a heavy body, moving by means of its natural inertia in a straight line.

LECTURE LVIII.

ON VEGETATION.

IT may appear idle to some persons, to attempt to reduce the outlines
of natural history into so small a compass, as is required for their becom-
ing a part of this course of lectures ; and it would indeed be a fruitless
undertaking to endeavour to communicate a knowledge of the particular
subjects of this science, even in a much longer time than we shall bestow
on it. But many naturalists have spent a great portion of their lives in
learning the names of plants and animals, and have known at last less of
the philosophy of the science, than might have been told them in a few
hours, by persons who had observed with more enlarged views, and who
had reasoned on general principles. And we shall perhaps find it possible
to collect into a small compass the most useful information, that has
hitherto been obtained, respecting the laws of animal and vegetable life, as
well as the foundations of the methods, by which the most received syste-
matical classifications have been regulated.

The surface of the earth, as well sea as land, is occupied by innumerable
individuals, constituting an immense variety of distinct species of animated
and inanimate beings, comprehended in the three grand divisions of natural
bodies. The mineral kingdom consists of such substances as are composed
of particles either united without any regular form, or collected together by
accretion or external growth only. When mineral substances crystallize,
they often imitate the form, and almost assume the external appearance of
vegetables : but their particles are never extended to admit others between
them, aad to be thus enlarged in all their dimensions ; their growth is only
performed by the addition of similar particles, upon the surface of those
that have been already deposited.

566 LECTURE LVIII.

Vegetables derive their existence, by seeds, or otherwise, from a parent
stock, their parts are extended and evolved from within, and they imbibe
their nutriment by superficial absorption only. There is indeed in the
crystallization of minerals a slight resemblance to a reproduction or genera-
tion, when a small portion of the substance serves as a basis for the forma-
tion of subsequent crystals : but this portion becomes a constituent part of
the crystal, while it preserves its original form ; a seed, on the contrary, is
a substance naturally and completely detached from the plant, and con-
taining within itself the simplest rudiments of a new individual, which is
afterwards evolved and enlarged. Sometimes, however, vegetables are pro-
pagated by means of bulbs, or by spreading roots, by slips, or by ingrafted
scions, without a seed detached in the regular manner ; but in these cases
the new plant is much more identical with the old one, than when it is
raised from a seed, being as it were a continuation of the same existence.
Plants are nourished in great measure by means of their roots ; and some-
times, where they are without roots, their nutriment is probably absorbed
by all parts of their surface.

Animals are distinguished from vegetables by the reception of their
food, for digestion and assimilation, into an internal cavity constituting a
stomach. The existence of a stomach, calculated for the digestion of food,
appears to be the best, if not the only criterion of an animal. Some vegeta-
bles, indeed, have a power of catching and detaining animals, by curling
up their leaves so as to cover them, as the drosera or sundew, and the
dionaea muscipula, or catchfly ; but this mechanism can scarcely be
intended for their immediate nutriment, at least the leaf can scarcely be
supposed to assume the character of a stomach. It is true that we imagine
all animals to have sensation, and all plants to be without it ; and if it were
possible to discriminate decisively between sensation and irritation, the dis-
tinction would supersede every other : but in many cases it is extremely
difficult to say where sensation is present, and where irritation only pro-
duces the same apparent effects. We cannot be sure that the hydra, or
fresh water polypus, or the trichurus sol, an animalcule described by
Dr. Shaw, suffers any sensation of pain when it is divided into two parts ;
at least the pain seems to agree remarkably well with its constitution, for
it lives and thrives with increased vigour, as two distinct animals. On the
other hand, many plants are easily stimulated to perform motions, which
have the appearance of muscular actions, influenced by sensation : the sen-
sitive plants close or depress their leaves, in consequence of agitation or of
electricity ; the stamina of the barberry and of the pellitory are thrown
into motion, when touched with a needle, and those of rue, and of the
grass of Parnassus, have at times alternate motions without any apparent
cause. A zoophyte is an animal absolutely fixed to one place ; and the
vallisneria is a vegetable possessed of a certain limited power of locomotion.
A plant chooses in preference to turn towards the light ; and it has been
known that an ash tree on a wall, when incapable of being any longer sup-
ported by the wall only, has concentrated all its force in the production of
one large root, descending to the ground. Some of these circumstances
may be explained without recurring to any thing like volition ; but, as

ON VEGETATION. 567

far as we know, the same explanations might be applied to some animal
motions; and although it is very possihle that there may be a certain
limit, where the influence of mind and sensation terminates, and the laws
of vegetable life only prevail ; yet the place of the division is not strongly
enough marked, to allow it to form a characteristic in an artificial system.
It has been asserted that some worms are nourished by absorption only,
without the assistance of a stomach ; thus hydatids, which are supposed
to be of an animal nature, appear to be simply bags of a fluid without any
visible opening ; but a few doubtful cases of this kind can scarcely be
sufficient to invalidate the general position, that all bodies decidedly
animal have a cavity for the reception of food. There are usually also
some chemical distinctions in the component parts of animals and vegeta-
bles ; animal substances commonly containing greater proportions of azote
or nitrogen, and of phosphoric acid ; but there are some exceptions to this
observation ; thus the carica papaya, or papaw, contains nearly the same
principles as are usually found in substances of animal origin. In general
we may readily distinguish a small portion of an animal from a vegetable
substance, by the smell produced in burning it. According to common
language, we say, that minerals have growth only, but not always ; that
vegetables grow and live also ; and that animals have sensation, as well as
life and increase of magnitude.

Mineralogy is a branch of natural history so nearly allied to chemistry,
that it cannot be completely understood without a previous knowledge of
that science. It may therefore be more properly considered as belonging
to a course of chemical than of physical lectures.

The vegetable kingdom presents to us a spectacle highly interesting by
its variety and by its elegance ; but the economy of vegetation appears to
be little diversified, although little understood. With respect to the appa-
rent perfection of their functions, and the complication of their structure,
we may consider all vegetables as belonging to two principal divisions, in
one of which the seed is prepared with the assistance of a flower, having
its stamina and its pistils, with petals or a calyx ; while in the other, the
preparation of the seed is less regular and conspicuous, and hence such
plants are called cryptogamous. In some of these there is a slight resem-
blance to the flowers of other vegetables, but on the whole, the class
appears to form one of the connecting links between the three kingdoms of
nature ; its physiology is probably simple, but it has been little examined.
The herbs, palms, shrubs, and trees, which constitute the numerous genera
of flowering vegetables, exhibit the greatest diversity in the forms and dis-
positions of the organs of fructification, while they have all a general resem-
blance in their internal economy.

Every vegetable may be considered as a congeries of vessels, in which, by
some unknown means, the aqueous fluids, imbibed by its roots, are sub-
jected to peculiar chemical and vital actions, and exposed in the leaves to
the influence of the light and air ; so as to be rendered fit for becoming
constituent parts of the plant, or of the peculiar substances contained
within it.

The first process in the germination of a seed is its imbibing moisture,

568 LECTURE LVIII.

and undergoing a chemical fermentation, in which oxygen is absorbed,
and a part of the mucilage contained in the seed is converted into sugar ; a
substance probably more nutritive to the young plant. The radicle shoots
downwards, and the seed leaves, or cotyledons, which are generally two,
although sometimes more or less numerous, raise themselves above the
ground, till in a short time they die and drop off, being succeeded by the
regular and more adult leaves.

In every transverse section of a vegetable, we commonly discover at least
four different substances. The parts next to the axis of the tree or branch
consist of medulla or pith, which is supposed by some to be the residence
of the vegetable life of the plant ; but a tree may live for many years after
being in great measure deprived of its medulla. The pith is of a loose and
light spongy texture ; it sends a ramification into each branch and each
leaf, where it appears to serve also as a reservoir of moisture. The pith is
surrounded by the woody part, composed of fibres more or less strongly
compacted together, but not actually ramifying into each other in any
great degree, although there is reason to suspect some lateral communica-
tions between them. They are interrupted, at certain intervals, in many
trees, by fibres, in a radiating direction, forming what is called the silver
grain. Like the bones in animals, the wood constitutes the strongest part
of the vegetable ; and like them too it is in a certain degree furnished with
vessels. It has even been supposed by some, that the fibres themselves are
distinct tubes, and by others, that the interstices between them serve the
purpose of vessels, but neither of these opinions is at present generally
received •. The wood consists of a number of concentric layers or strata,
formed in successive years ; the external part, which is last formed, is
called the alburnum, or white wood, and this part is the most vascular.
The bark encompasses the wood ; and this also consists, in trees, of several
layers, which are produced in as many different years ; the external parts
usually cracking, and allowing us at their divisions to observe their num-
ber, the inner layer only being of immediate use. This layer is called the
liber, and since this material was once used instead of paper, the Romans
called a book also liber. The bark consists of fibres of the same kind as
the wood, but more loosely connected. It is covered by the cuticle, which
extends itself in a very great degree, as the growth of the vegetable
advances, but at last cracks, and has its office supplied by the outer layers
of bark. Between the bark and the cuticle a green pulpy substance, or
parenchyma, is found, which seems to be analogous to the rete mucosum,
interposed between the true skin and the cuticle in animals. Mr. Desfon-
taines* has observed, that in palms, and in several other natural orders of
plants, the annual deposition of new matter is not confined to the external
surface, but that it takes place in various parts of the plant, as if it were
composed of a number of ordinary stems united together.

There are three principal kinds of vessels in the different parts of vege-
tables : the sap vessels, which are found both in the wood and in the bark,
although their nature appears to require further examination : secondly,
the air vessels, or tracheae, which are composed of single threads wound
* Mem. de 1'Instit. i. 478.

ON VEGETATION. 569

into a spiral tube, like the spring of a bell, and capable of being easily
uncoiled ; these, though they have been called air vessels, and supposed by
some to serve the purposes of respiration, are described by others as con-
taining, during the life of the plant, an aqueous fluid : and they are pro-
bably little more than sap vessels, with an additional spiral coat ; they are
not found in the bark, nor in all species of plants ; and it has thence been
inferred that they are not immediately necessary to the growth of the
plant. The third kind are the proper vessels of the plant, which are gene-
rally disposed in concentric circles, and appear to be unconnected with the
sap vessels, and to contain the milky, resinous, and other peculiar juices,
which are found in different kinds of plants ; for the sap is nearly the same
in all, at least it is independent of the gums and resin, which often distin-
guish particular plants ; it contains a certain portion of mucilage, and pro-
bably in some plants, as the sugar maple, a considerable quantity of sugar.
Mr. Mirbel* has also made a number of still more accurate distinctions
respecting the structure of the different kinds of vessels. The circulation
of the sap is not completely understood ; when an orifice is made near the
root of a tree, it flows most copiously from above : when near the summit,
from below. Dr. Hope actually reverted the natural course of the juices
of a tree, without changing its position ; by inoculating a willow with two
others, he completely united its existence with theirs, and then, removing
its roots, he found that its vegetation was supported by the juices of the
two others. A tree may also be actually inverted, and the upper part will
strike root, the lower putting out branches and leaves.

Plants perspire very considerably, and also emit a quantity of gases of
different kinds ; they generate a slight degree of heat, which may be
observed by means of the thermometer, and by the melting of snow in con-
tact with them. The growth of every tree takes place at the internal sur-
face of the bark, not only the bark itself being formed there, but the wood
also being deposited by the bark ; for Dr. Hope separated the whole of the
bark of a branch of willow from the wood, leaving it connected only at the
ends, so as to constitute a hollow cylinder, parallel to the wood ; and he
found that new layers were formed within the bark ; and in another expe-
riment a part of the wood, deprived of the bark, although protected from
the air, was only covered with new bark as it grew over from the old bark
above and below. The layers of wood, which are added in successive
seasons, and keep a register of the age of the tree, are very easily observed
when it is cut across ; sometimes as many as 400 have been found in firs,
and oaks are said to have lived 1000 years.

Mr. Knight f has inferred, from a great variety of experiments, that the
sap, either usually or universally, ascends through the wood into the
leaves, and then descends through the bark to nourish the plant. The
leaves seem to be somewhat analogous to lungs, or rather to the gills of

* Bullet, de la Soc. Philom. No. 60. Journal de Phy. lii. 336. Anatomie et Phy-
siologic Vgget. 2 vols. Paris, 1815.

•f His papers are in the Ph. Tr. 1795, p. 290; 1799, p. 195 ; 1801, p. 333 ;
1803, p. 277; 1804, p. 183.

570 LECTURE LVIII.

fishes : for plants have need of air, and it has been found, that even seeds
will not germinate in a vacuum. As the lungs of animals appear to be
concerned in forming the blood, so it may be inferred from Mr. Knight's
experiments, that the sap first ascends to the leaves through the external
fresh wood of alburnum, and through the central vessels of the young
leaves and branches, derived from the alburnum, and accompanied by the
spiral tubes ; and after being perfected by exposure to light and air in the
leaves, it descends in the bark, and serves for the secretion of the alburnum,
and of the internal layers of the bark, being conveyed probably by two
distinct sets of vessels. The sap, thus prepared by the leaves in the
summer and autumn, is supposed to leave its extractive matter in the tree
throughout the winter, in such a state as to be ready to unite with the
aqueous juices, which ascend from the root in the succeeding spring. The
internal parts of the wood, having served the purposes of vegetation, are har-
dened, and perhaps dried up, so as to be afterwards principally subservient
to strength alone. By subsequent experiments, Mr. Knight has also found,
that when a branch hangs downwards, the sap still appears to proceed
from the part of the bark which is uppermost ; so that the direction of the
force of gravity seems to be concerned in determining that of the motion of
the sap. There appears also to be some reason to suppose that mechanical
means assist in the protrusion of the sap, and the consequent growth of the
tree ; for if a tree be more agitated by the wind in one direction than in
another, its diameter will be greatest in that direction.

The process of grafting depends on a remarkable property of the growth
of vegetables ; if the cut surface of the inner bark of a small branch, or
cutting, be placed in contact with that of the branch of another tree, they
will unite sufficiently for the nourishment of the cutting ; provided, how-
ever, that the nature of the plants be not too different. Something of the
same kind occurs in animal life, where a tooth has been transplanted into
the socket of another, or where the spur of a cock has been inserted into his
comb.

Plants have their natural periods of life, either of a few days, as in the
case of some of the fungi, of a year, of a few years, or of many centuries.
They have also their diseases ; they are often infested by insects, as in the
gall of the oak, and the woodruff of the rose, or by animalcules of a still
lower order, which are either the causes of the smut of corn, or constant
attendants on it. From unnatural and too luxuriant culture, they become
sterile, and produce double flowers instead of fruits and seeds. When
deprived of sufficient moisture, or nipped by frost, their leaves and branches
often die ; and if the plants recover their vigour, a separation is effected
by a natural process, resembling the sloughing of decayed parts of animals ;
but when the whole plant sinks, the dead leaves continue to adhere to it.
The annual fall of leaves in autumn appears to be a natural separation
nearly of the same kind, which takes place when the leaves are no longer
wanted ; the growth of the plant being discontinued, and their functions
being no longer required.

Succulent plants generally die when the cuticle is removed, but not all

ON VEGETATION. 571

other plants. The air appears to be injurious to vegetables where it is not
natural ; hence arises the benefit of Mr. Forsyth's* method of completely
excluding the air from the wounded parts of trees, by means of which
their losses are often in great measure repaired, and they acquire new
strength and vigour. Sometimes a diminution of the magnitude of a tree
immediately increases its fertility ; its force being more concentrated by
lopping off its useless branches and leaves, it produces a larger quantity of
fruit, with the juices which would have been expended in their nourish-
ment.

The Linnean system of vegetables is confessedly rather an artificial than
a natural one ; but it is extremely well adapted for practice, and its uni-
versal adoption has been productive of the most important improvements
in the science of botany. Of the 24 classes into which Linne has divided
the vegetable kingdom, 23 are distinguished by the forms of the flowers and
fruit, and the 24th by the want of a regular florescence. The first 10 are
named from monandria, in order, to decandria ; then follow dodecandria ;
icosandria, and polyandria ; the names expressing the number of the
stamina, or filaments, surrounding the seed vessel ; and the orders are
deduced in a similar manner from the number of pistils or little columns
immediately connected with the seed vessel ; and denominated mono-
gynia, digynia, and so forth, as far as polygynia. These classes differ
little in general with respect to their natural habits, except the twelfth,
icosandria, which is characterized by the attachment of the filaments to
the green cup, surrounding the flower, and which comprehends the most
common fruit trees : this class has, however, been incorporated by some
later botanists with the next. In the third class we find most of the
natural order of grasses ; the fifth, pentandria, is by far the most numerous
of any : the sixth contains the lilies, and many other bulbous plants. The
14th class, didynamia, is known by two longer and two shorter filaments ;
it is perfectly natural, and comprehends flowers similar in their structure
to the foxglove and the deadnettle. The 15th also, tetradynamia, is a class
of plants strongly characterized even by chemical properties ; two of the
filaments are here shorter than the other four : cresses, radishes, and many
other acrid and ammoniacal vegetables, belong to this class, as well as the
turnip and cabbage, which, when cultivated, become mild and nutritious.
The class monadelphia contains a few plants similar to the mallow ; they
are known by the union of the filaments at their bases into a cylinder :
those of the next class have generally nine united, and one separate, whence
the class is named diadelphia ; it contains the papilionaceous flowers, some-
what resembling a butterfly in their form, like the pea, and other legu-
minous plants, the broom, the furze, and the acacia. The 18th class, poly-
adelphia, has the filaments of its flowers united into several masses or
bundles, as the hypericum or tutsan. The next class is perfectly natural,
and contains the composite flowers, which have a peculiar union of the
summits of the filaments ; it is named syngenesia : sunflowers, daisies, and
artichokes, are familiar examples of the plants of this class. The 20th
class, gynandria, though it contains the natural family of the orchides, has
* On the Diseases of Forest Trees, 1791.

572 LECTURE LVIII.

been omitted by some late botanists ; here the filaments are fixed on the
pistil ; or more properly, in the arums, within the pistils. The three fol-
lowing classes, monoecia, dioecia, and polygamia, differ from the rest in
having some flowers with filaments or chives, and some with pistils only,
either on the same plant, or on different plants, or mixed with flowers of
the more common construction. Most of the forest trees belong to these
classes, but the distinctions which separate them from other classes are not
always very uniformly preserved, and, for this reason, many later botanists
have disused them. The plants of the last class, cryptogamia, are exceed-
ingly numerous ; the families of ferns, mosses, algae, or membranous weeds,
and fungi or mushrooms, fill up its extensive departments ; some have also
separated a part of the algae under the name of hepaticae, or gelatinous
weeds. In this class the fructifications are extremely various ; some of
the fuci and confervae approach so much in their general appearance and
mode of growth to corallines and zoophytes, that they seem to form an
obvious connexion between the lowest ranks of the vegetable and animal
kingdoms ; while other plants of the class are scarcely distinguishable by
their appearance from some of the productions of the mineral kingdom.

The French have introduced into very general use the botanical system
of Jussieu. The most prominent feature in this system is the division of
all the genera into a hundred natural orders, which are also arranged in
fifteen classes. Jussieu begins, like Linne, with the separation of crypto-
gamic from phanerogamic plants; the seeds of the cryptogamic plants,
which form the first class, being without cotyledons or seed leaves, and all
other plants being distinguished into such as have seeds with one and with
two cotyledons. Accordingly as the stamina or filaments are inserted
below the pistil, on the calyx, or on the seed vessel, the first description of
seeds affords three distinct classes. The plants which have two cotyledons
follow, and are divided into apetalous, monopetalous, and polypetalous,
from distinctions respecting the corolla or flower leaves, which are some-
what arbitrarily understood ; and lastly diclinous, from the separation of
the stamina and pistils. The three first of these divisions are subdivided
according to the insertion of the stamina, and the union or separation of
the antherae, which they support, into ten classes, making, with the four
already mentioned, fourteen, to which the diclinous plants add a fifteenth.
The orders are determined without any particular limitation of the parts
from which the characters are taken. This system is of acknowledged
merit as a philosophical classification of the natural orders of plants ;
such vegetables as nearly agree in their habits and appearances being
brought more uniformly together than in the system of Linne. Hence,
in the arrangement of a botanical garden, or in a treatise on the chemical
or medical properties of plants, it might be employed with advantage : but
for the practical purposes of botanical investigation it appears to be utterly
unfit, since its author has sacrificed all logical and systematical laws
to the attempt to follow nature, in analogies, which are often discoverable
only with great difficulty, and which are seldom reducible to methodical
definitions.

ON ANIMAL LIFE. 673

LECT. LVIIL— ADDITIONAL AUTHORITIES.

A FEW OF THE MORE IMPORTANT WORKS ONLY ARE GIVEN.

Botany in general. — Smith's Introduction, 1807, &c. Decandolle, Theorie Elem.
de la Bot. 1819. Link, Elementa Philos. Bot. Berol. 1824. Lindley's Introduc-
tion, 1835. Henslow's, 12mo, 1836. Achille Richard, Nouv. Elem. de la Bo-
tanique.

Vegetable Physiology.— Willoughby, Ph. Tr. 1669, p. 963 ; 1670, p, 1165

Malpighius, Anatome Plantarum, fol. Lond. 1675-9. Grew on the Anatomy of
Veget. 12mo, 1671, fol. 1682. Hales, Vegetable Staticks, 1727. Duhamel, Phy-
sique des Arb res, 2 vols. 4to, Paris, 1758. Hedwig, Descrip. Muscorum, fol. Leipz.

1792 and other works. Darwin's Phytologia 4to, Lond. 1800. Senebier,

Physiologic Vegetale, 5 vols. Geneve, 1801. Saint-Hilaire, Mem. sur les Plantes
auxquelles on attribue un Placenta Central Libre, 4to, Paris, 1816. A. Brogniart,
Turpin, and other writers in the Annales des Sciences Naturelles ; Du Petit-Thouars,
Cours de Phytologie. Dutrochet, Recherches sur la Structure intime des Vegdtaux,
Paris, 1824. L' Agent du Mouvement Devoile", 1826. Nouvelles Recherches sur
1'Endosmose et 1'Exosmose, 1828. Cassini, Opuscules Phytologiques, 2 vols. Paris,
1826. Decandolle, Organographie Vegetale, 2 vols. Paris, 1827. Cours de Bota-
nique, 3 vols. 1832. Brown's Microscopic Obs. 1829, and various other works.
Slack, Trans, of the Soc. of Arts, vol. xlix. Viviani, Organi Elementari delle
Piante, Geneva, 1831. Roget's Animal and Vegetable Physiology, 2 vols. 1834.
Liebig's Agricultural Chemistry, 8vo. 1843.

Systematic Botany. — Ray, Historia Generalis Plantarum, fol. Lond. 1686-8.
Tournefort's History of Plants (trans, by Martyn), 2 vols. 1732. Linnaeus Genera
Plantarum, var. ed. Jussieu's Natural System, Hist, et Mem. 1773, p. 214, H. 34 ;
1774, 1775 : Genera Plant. Turici, 1791. English Botany, or Coloured Figures of

British Plants, by Sowerby, 35 vols. Lond. 1790 Withering's British Plants

(Linnsean), 4 vols. 1796. Decandolle, Regni Vegetabilis Systema Naturale, 1818-
21, &c. Smith's English Flor. (Linmean), 4 vols. 1824-8. Vol. v. Crypogamia, by
Hooker. Hooker's British Flora (Linnaean), 1836 ; (Natural) 1844. Lindley's
Synopsis (Nat.) 12mo, 1835. Loudon's Encyclopaedia of Plants.

LECTURE LIX.

ON ANIMAL LIFE.

THE functions of animal life are not only more complicated in the same
individual than those of vegetation, but also more diversified in the different
classes into which animals are divided ; so that the physiology of each
class has its peculiar laws. We are indebted to Linne for the first enlarge-
ment of our views of the different classes of animals, and perhaps for the
most convenient arrangement of the animal kingdom; although his
method has never been universally adopted by our neighbours on the con-
tinent.

A considerable portion of the bulk of all animals is composed of tubular
vessels, which originate in a heart ; the heart propels through the arteries,
with the assistance of their own muscular powers, either a colourless
transparent fluid, or a red blood, into the extremities of the veins ; through
which i* again returns to the origin of its motion. Both insects, and
vermes, or worms, have their circulating fluids a little warmer than the
surrounding medium, and generally colourless ; hut insects have legs fur-

574 LECTURE L1X.

nished with joints, and worms have nothing but simple tentacula at most
in the place of legs. Fishes have cold red blood, which is exposed to the
influence of the air contained in water, by means of their gills. The
amphibia receive the air into their lungs, but their blood is cold, like that
of fishes, and in both these classes the heart has only two regular cavi-
ties, while that of animals with warm blood has four ; the whole contents
of one pair being obliged to pass through the lungs, in order to arrive at
the other pair. Of animals with warm blood, the oviparous are birds, and
are generally covered with feathers, the viviparous are either quadrupeds
or cetaceous animals, and are furnished with organs for suckling their
young.

Each of these classes of animals is subdivided by Linne into different
orders, of which we shall only be able to take a very cursory view. The
first class, denominated mammalia, from the female's suckling its young,
comprehends all viviparous animals with warm blood. These, with very
few exceptions, have teeth fixed in their jaw bones ; and from the form and
number of these teeth, the orders are distinguished, except that of ceta-
ceous fishes, which is known by the fins that are found in the place of feet.
The distinctions of the teeth are somewhat minute, but they appear to
be connected with the mode of life of the animal, and they are tolerably
natural. The first order, primates, contains man, monkeys, and bats ; the
second, bruta, among others, the elephant, the rhinoceros, the ant eater,
and the ornithoryhncus, an extraordinary quadruped, lately discovered in
New Holland, with a bill like a duck, and sometimes teeth inserted behind
it ; but there are some suspicions that the animal is oviparous. The order
ferae contains the seal, the dog, the cat, the lion, the tiger, the weasel, and
the mole, most of them beasts of prey; the opossum and the kangaroo also
belong to this order, and the kangaroo feeds on vegetables, although its
teeth are like those of carnivorous animals. The fourth order, glires, com-
prehends beavers, mice, squirrels, and hares ; the fifth, pecora, camels, goats,
sheep, and horned cattle. The sixth order, belluae, contains the horse, the
hippopotamus, and the hog. The cetaceous fishes, or whales, form the
seventh and last order ; they reside in the water, enveloped in a thick
clothing of fat, that is, of oily matter, deposited in cells, which enables
their blood to retain its temperature, notwithstanding the external contact
of a dense medium considerably colder.

Birds are distinguished from quadrupeds, by their laying eggs ; they
are also generally feathered, although some few are rather hairy ; and
instead of hands or fore legs they have wings. Their eggs are covered by
a calcarious shell ; and they consist of a white, or albumen, which nourishes
the chick during incubation, and a yolk, which is so suspended within it,
as to preserve the side on which the little rudiment of a chicken is situated,
continually uppermost, and next to the mother that is sitting on it. The
yolk is in great measure received into the abdomen of the chicken a little
before the time of its being hatched, and serves for its support, like the
milk of a quadruped, and like the cotyledons of young plants, until the
system is become sufficiently strong for extracting its own food out of the
ordinary nutriment of the species.

ON ANIMAL LIFE. 575

Birds are divided, according to the form of their bills, into six orders :
accipitres, as eagles, vultures, and hawks; picae, as crows, jackdaws,
humming birds, and parrots ; anseres, as ducks, swans, and gulls ; grallae,
as herons, woodcocks, and ostriches ; gallinae, as peacocks, pheasants,
turkies, and common fowls ; and, lastly, passeres, comprehending sparrows,
larks, swallows, thrushes, and doves.

The amphibia are in some respects very nearly allied to birds : but their
blood is little warmer than the surrounding medium. Their respiration is
not necessarily performed in a continual succession of alternations, since
the whole of their blood does not pass through the lungs, and the circula-
tion may continue without interruption in other parts, although it may be
impeded in these organs, for want of the motion of respiration. They are
very tenacious of life ; it has been asserted on good authority that some of
them have lived many years without food, inclosed in hollow trees, and
even in the middle of stones ; and they often retain vestiges of life some
days after the loss of their hearts. Their eggs are generally covered with a
membrane only. They have sometimes an intermediate stage of existence,
in which all their parts are not yet developed, as we observe in the tadpole ;
and in this respect they resemble the class of insects. They are now uni-
versally considered as divided into two orders only ; reptilia, as the tortoise,
the dragon, or flying lizard, the frog and the toad ; all these have four feet ;
but the animals which belong to the order serpentes are without feet.
Most of the serpentes are perfectly innocent, but others have fangs, by
which they instil a poisonous fluid into the wounds that they make. In
England the viper is the only venomous serpent ; it is known by its dark
brown colour, and by a stripe of whitish spots running along its back ; but
to mankind its bite is seldom, if ever, fatal.

The first three classes of animals have lungs, as we have already seen,
for respiration, and receive air by the mouth ; those which have gills, and
red blood, are fishes, residing either in fresh or in salt water, or indif-
ferently in both ; their eggs are involved in a membrane, and have no
albumen. Of the six orders of fishes, four have regular gills, supported
by little bones ; and they are distinguished, according to the place of their
ventral fins, into apodes, as the eel and lamprey; jugulares, as the cod;
thoracici, as the sole and perch, and abdominales, as the salmon and pike ;
distinctions which appear to be perfectly artificial, although useful in a
systematic arrangement. The two remaining orders are without bones in
the gills, those of the one being soft, and of the other cartilaginous or
gristly. These are the branchiostegi and chondropterygii of Artedi, which
Linne, from a mistake, classed among the amphibia. The sun fish, the
lump fish, the fishing frog, and the sea horse, are of the former, and the
sturgeon, the skate, and the shark, of the latter order.

Insects derive their name from being almost always divided into a head,
thorax, and abdomen, with very slender intervening portions : although
these divisions do not exist in all insects. They are usually oviparous :
they respire, but not by the mouth ; they have a number of little orifices
on each side of the abdomen, by which the air is received into their rami-
fied tracheae ; and if these are stopped with oil, they are suffocated. In-

576 LECTURE LIX.

stead of bones, they have a hard integument or shell. Their mouths are
formed on constructions extremely various, but generally very complicated :
Fabricius* has made these parts the basis of his classification ; but from
their minuteness in most species, the method is, in practice, insuperably
inconvenient ; and the only way, in which such characters can be -rendered
really useful, is when they are employed in the subdivision of the genera,
as determined from more conspicuous distinctions. Insects have most fre-
quently jaws, and often several pairs, but they are always so placed as to
open laterally or horizontally. Sometimes, instead of jaws, they have a
trunk, or proboscis. In general, they pass through four stages of
existence, the egg, the larva, or stage of growth, the pupa, or chrysalis,
which is usually in a state of torpor or complete inactivity, and the imago,
or perfect insect, in its nuptial capacity. After the last change, the insect
most frequently takes no food till its death.

The Linnean orders of insects are the coleoptera, with hard sheaths to
their wings, generally called beetles ; the hemiptera, of which the sheaths
are of a softer nature, and cross each other, as grasshoppers, bugs, and
plant lice ; the lepidoptera, with dusty scales on their wings, as butterflies
and moths ; the neuroptera, as the libellula, or dragon fly, the may fly,
and other insects with four transparent wings, but without stings ; the
hymenoptera, which have stings, either poisonous or not, as bees, wasps,
and ichneumons ; the diptera, with two wings, as common flies and gnats,
which have halteres, or balancing rods, instead of the second pair of wings ;
and lastly the aptera, without any wings, which form the seventh order,
comprehending crabs, lobsters, shrimps and prawns, for these are properly
insects ; spiders, scorpions, millepeds, centipeds, mites, and monoculi.
The monoculus is a genus including the little active insects found in pond
water, which are scarcely visible to the naked eye, as well as the Molucca
crab, which is the largest of all insects, being sometimes six feet long.
Besides these there are several genera of apterous insects which are
parasitical, and infest the human race as well as other animals.

The vermes are the last and lowest of animated beings, yet some of them
are not deficient either in magnitude or in beauty. The most natural divi-
sions of vermes is into five orders ; the intestina, as earth worms and
ascarides, which are distinguished by the want of moveable appendages, or
tentacula, from the mollusca ; such as the dew snail, the cuttle fish, the
sea anemone, and the hydra, or fresh water polype. The testacea have
shells of one or more pieces, and most of them inhabit the sea, and are
called shell fish, as the limpet, the periwinkle, the snail, the muscle, the
oyster, and the barnacle. The order zoophyta contains corallines, sponges,
and other compound animals, united by a common habitation, which has
the general appearance of a vegetable, although of animal origin ; each of
the little inhabitants, resembling a hydra, or polype, imitating by its
extended arms the appearance of an imperfect flower. The last order,
infusoria, is scarcely distinguished from the intestina and mollusca by any
other character than the minuteness of the individuals belonging to it, and
their spontaneous appearance in animal and vegetable infusions, where we
* Entomologia Systematica, 6 vols. Hafniae, 1792-8.

ON ANIMAL LIFE. 577

can discover no traces of the manner in which they are produced. The
process, by which their numbers are sometimes increased, is no less astonish-
ing than their first production ; for several of the genera often appear to
divide spontaneously, into two or more parts, which become new and dis-
tinct animals, so that in such a case the question respecting the identity of
an individual would be very difficult to determine. The volvox, and some
of the vorticellae are remarkable for their continual rotatory motion,
probably intended for the purpose of straining their food out of the water :
while some other species of the vorticella resemble fungi or corallines in
miniature.

Among the animals of these different classes, the more perfect are in-
formed of the qualities of external objects by the senses of touch, taste,
smell, hearing, and vision. A few quadrupeds are incapable of seeing :
the mole has an eye so small as to be with difficulty distinguishable ; and
the mus typhlus, supposed to be the aspalax of Aristotle, has its eye com-
pletely covered by the skin and integuments, without any perforation.
Birds have hearing, but no external ears, or auriculae. Insects appear to
want the organs of smell ; but it is not impossible that their antennae may
answer the purpose of hearing. Many of the vermes are totally destitute
of sight, and some of all the organs of sense : none of them have either
ears or nostrils. The external senses of animals with warm blood are
usually liable to a periodical state of inactivity in the night time, denomi-
nated sleep. It is said that fishes never sleep ; and it is well known that
some animals pass the whole of the severest part of the winter in a state
nearly resembling their usual sleep.

In animals which approach, in their economy, to that of the human
system, the process for supporting life by nutrition begins with the masti-
cation of the food, which has been received by the mouth. The food thus
prepared is conveyed into the stomach by the operation of swallowing ;
but in ruminating cattle, it is first lodged in a temporary receptacle, and
more completely masticated at leisure. In the stomach, it undergoes
digestion, and being afterwards mixed with the bile and other fluids,
poured in by the liver and the neighbouring glands, it becomes fit for
affording the chyle, or nutritive juice, which is separated from it by the ab-
sorbents of the intestines, in its passage through the convolutions of a canal
nearly forty feet in length. Together with the chyle, all the aqueous fluids,
which are swallowed, must also be absorbed, and pass through the thoracic
duct into the large veins entering the heart, and thence into the general cir-
culation, before they can arrive at the kidneys, by which the superfluous
parts are rejected. The chyle passes unaltered, with the blood, through
the right auricle and ventricle of the heart, and enters the lungs, to be
there more intimately mixed with it, and perhaps to be rendered animal
and vital ; while the blood receives from the air, in the same place, a
supply of oxygen, with a small portion of nitrogen, and emits some super-
fluous carbonic matter, in the form of carbonic acid. The blood, thus
rendered arterial, returning to the left side of the heart, is distributed
thence to every part of the system, supplying nutriment throughout,
while the glands and arteries secrete from it such fluids, as are become

578 LECTURE LIX.

redundant, and such as are required for particular purposes subservient
to the animal functions. It is probably in these processes that heat is
evolved ; for by experiments on living animals, it has been found, that the
Wood, returning from the lungs, is not warmer than before its entrance
into them : we must therefore suppose, that when the florid arterial blood
is, by some unknown means, converted, in the extreme ramifications of
the arteries, into the purple venous blood, to return to the heart by the
converging branches of the veins, there is a much more considerable extri-
cation of heat, than in the conversion of venous into arterial blood, by the
absorption of oxygen and nitrogen in the lungs. If the chyle is actually
converted into blood in the lungs, it is here that we must look for the
formation of the red globules, those singular corpuscles, to which the blood
owes its colour, as it does its power of coagulation to a glutinous lymph,
mixed with a less coagulable serum. The red particles in the human blood
are about ^Vu- of an inch in diameter, somewhat oblong, and flattened ;
they have usually the appearance of a dark point in the centre ; but there
is some reason to suspect that this is merely an optical deception. In a
few animals they are a little smaller, but in most of the amphibia, much
larger and flatter than in man. While the lymph remains fluid, after the
blood has been withdrawn from the vessels, these globules tend to subside,
and to leave it semitransparent : hence arises the appearance of a buff
coat on blood left to coagulate, which is thinner or thicker, accordingly as
the globules are sooner or later arrested in their descent.

The muscles are probably furnished by the blood with a store of that
unknown principle, by which they are rendered capable of contracting, for
producing locomotion or for other purposes, in obedience to the influence
transmitted by the nerves from the sensorium ; the brain and nervous
system in general are also sustained, by means of the vascular circulation,
in a fit state for transmitting the impressions, made by external objects
on the senses, to the immediate seat of thought and memory, in the
sensorium ; and for conveying the dictates of the will, and the habitual
impulses almost independent of volition, to the muscular parts of the
whole frame.

In what manner these reciprocal impressions are transmitted by the
nerves, has never yet been fully determined : but it has long been conjec-
tured that the medium of communication may bear a considerable analogy
to the electrical fluid ; and the extreme sensibility of the nerves to the
slightest portion of electrical influence, as well as the real and apparently
spontaneous excitation of that influence in animal bodies, which have been
of late years evinced by galvanic experiments, have added very materially
to the probability of the opinion. An extremely slender fibre, of a sub-
stance capable of conducting electricity with perfect freedom, enveloped in
a sheath of a perfect nonconductor, would perhaps serve to communicate
an impulse, very nearly in the same manner, as the nerves appear to do.
Indeed nothing can be more fit to constitute a connecting link between
material and immaterial beings, than some modification of a fluid, which
appears to differ very considerably, in its essential properties, from the
common gross matter of the universe, and to possess a subtility and

ON ANIMAL LIFE. 579

an activity, which entitle it to a superior rank in the order of created
substances.

When all the functions of animal life are carried on in their perfect and
natural manner, the animal is said to be in health : when they are dis-
turbed, a state of disease ensues. The diseases to which the human frame
is liable are so various and irregular, that they cannot easily be reduced
to any systematical order. Dr. Cullen has divided them into four classes.
Febrile diseases, which constitute the first class, consist principally in an
increase of the frequency of the pulsations of the heart and arteries, toge-
ther with an elevation of the temperature, the whole animal economy
being at the same time in some measure impaired : they are often accom-
panied by unnatural or irregular actions of the vessels of particular parts,
constituting local inflammations, which were formerly considered as a
derivation of diseased humours, falling on those parts : thus, a pleurisy
is a fever, with an inflammation of the membrane lining the chest. The
incapacity of a part to perform its functions, upon the application of a
natural stimulus, or perhaps more frequently the incapacity of the nerves
to transmit to it the dictates of the mind, constitutes a palsy : such
derangements, and others, by which the actions of the nervous system are
peculiarly impaired, form the class of neuroses, including spasmodic affec-
tions, madness, melancholy, and epilepsy. A general derangement of the
system, without fever, or any peculiar debility of the nerves, constitutes
the class of cachectic diseases, such as atrophy, consumption, scrofula,
and dropsy. Besides these diseases, we have a fourth class, consisting of
local affections only, such as blindness, deafness, tumours, and luxations.

Notwithstanding the labours of men of the greatest learning and genius,
continued for many centuries, it must be confessed that the art of healing
diseases is still in a state of great imperfection. Happily, however, for
mankind, we may observe in almost all cases, where the offending cause is
discoverable, and where the system is not at once overwhelmed by its mag-
nitude, a wise and wonderful provision for removing it, by a mechanism
admirably simple and efficacious ; and it is reasonable to conclude, where
the cause is more obscure, that the same benevolent Providence has em-
ployed agents equally well adapted for counteracting it, although their
operations are utterly beyond the reach of human penetration.

On the subjects embraced in this Lecture, the reader may consult Roget's Bridg-
water Treatise ; Darwin's Zoonomia, 4 vols. 1804 ; or the great works of Buffon,
Histoire Naturelle, 127 vols. Paris, 1799-1808, and Cuvier, Le Regne Animal,
translated, in 16 vols. Lond. 1824-33. Liebig's Animal Chemistry, 8vo. 1843.

2 P2

580

LECTURE LX.

ON THE HISTORY OF TERRESTRIAL PHYSICS.

THROUGHOUT the whole of nature, we discover a tendency to the mul-
tiplication of life, of activity, and of enjoyment : man is placed at the head
of terrestrial beings, the only one that comprehends, and that can trace, in
a faint outline, the whole plan of the universe. We have seen the innumer-
able luminaries which enliven the widely expanded regions of immeasur-
able space, with their brilliant, but distant emanations of light and heat.
Revolving round them at lesser intervals, and cherished by their fostering
influences, are their planets and their comets ; those preserving their dis-
tances nearly equal, and these, ranging more widely from the upper to the
lower regions, without limits to their numbers or to their motions. Having
conjectured what might possibly exist on other planetary globes, we
descended to our own, and examined its structure and the proportions of
its parts. Next we studied the general properties of the matter within our
reach, and then the particular substances or qualities that are either
not material, or are distinguished by very remarkable properties from
other matter, as we found them concerned in the phenomena of heat, of
electricity, and of magnetism ; and we afterwards examined the combi-
nations of all these, in the great atmospherical apparatus of nature, which
serves for the exhibition of meteorological phenomena. The forms and the
laws of animal and vegetable life have been the last objects of our inqui-
ries ; but the magnitude of some departments of natural history, and the
obscurity of others, have prevented our entering more than superficially
upon any of them.

Of the gradual advancement of astronomy we have already taken a his-
torical view. With respect to the other sciences comprehended under the
denomination of proper physics, the progress of discovery has generally
been slow, and frequently casual. The ancients had little or no substantial
knowledge of any part of physics, except astronomy and natural history :
their opinions were in general mere speculations, derived from fancy, and
inapplicable to the real phenomena of nature. Opinions such as these will
only require to be so far examined, as to enable us to trace the imperfect
rudiments of discoveries, which were only completed after intervals of many
ages.

The Chinese are said to have been acquainted with the use of the com-
pass above 3000 years ago ; but in such accounts, it is impossible to ascer-
tain how far the spirit of national vanity may have induced a historian to
falsify his dates.* It has been conjectured that the death of Numa, like
that of Professor Richmann, was occasioned by some unguarded experi-
ments on the electricity of the atmosphere, which drew on him the, effects

* Consult Davies on the History of Magnetical Discovery, British Annual, 1837.
Klaproth, Lettre a M. de Humboldt sur 1' Invention de la Boussole.

ON THE HISTORY OF TERRESTRIAL PHYSICS. 581

of a thunderstorm that was passing by. If, however, the fact was such,
the experiments must probably have been suggested rather by an accidental
discovery of the light on the point of a spear, than by any rational opinions
respecting the nature of the ethereal fire.

Thales is the most ancient of the Grecian philosophers, who appear to
have seriously studied the phenomena of nature. He supposed water to be
the general principle from which all material things are formed, and into
which they are resolved ; an opinion which was without doubt suggested
to him by the obvious effects of water in the nutrition of plants and of
animals. He particularly noticed the properties of the magnet, which had
been before observed to attract iron, as well as the effect of friction in ex-
citing the electricity of amber ; and he attributed to both of these substances
a certain degree of animation, which he considered as the only original
source of motion of any kind.

Anaximander appears to have paid some attention to meteorology ;
he derived the winds from the rarefaction of the air, produced by the
operation of heat : thunder and lightning he attributed to the violent
explosion or bursting of the clouds, which he seems to have considered as
bags, filled with a mixture of wind and water. The same mistaken notion
was entertained by Anaximenes, who compared the light attending the
explosion, to that which is frequently exhibited by the sea, when struck
with an oar.

Pythagoras, great as he was in some other departments of science, rea-
soned respecting physical effects in a manner too mathematical and vision-
ary, to allow him much claim to be ranked among those who have studied
to investigate the minute operations of nature.

Anaxagoras was so far from confining himself to the supposition of four
elements, which was most generally received by the philosophers of anti-
quity, that he imagined the number of elements nearly if not absolutely
infinite. He conceived that the ultimate atoms, composing every sub-
stance, were of the same kind with that substance, and his system was
thence called the homoeomeria ; it erred perhaps less from the truth than
many of the more prevalent opinions. Democritus, adopting the senti-
ments of Leucippus, proposed a still more correct theory of the consti-
tution of matter, supposing it to be ultimately so far homogeneous, that
the weight of its atoms was proportional to their bulk. He asserted that
the forms of these atoms were different and unalterable ; that they were
always in motion, and that besides their primitive difference of form, they
were also susceptible of a variety in the mode of their arrangement. The
space not occupied by the atoms of matter, he considered as a perfect
vacuum.

As Thales had supposed water to be the first principle of all things,
and Anaximenes air, so Heraclitus fixed on fire as the foundation of his
system, attributing to it the property of constant motion, and deriving
all kinds of grosser matter from its condensation in different degrees.
This Doctrine was wholly unsupported by any thing like reason or obser-
vation.

Plato introduced into philosophy a variety of imaginations, which resem-

682 LECTURE LX.

bled the fictions of poetry much more than the truths of science. He
maintained, for example, that ideas existed independently of the human
mind, and of the external world, and that they composed beings of different
kinds, by their union with an imperfect matter. It is observed by Bacon,
in his essay on the opinions of Parmenides, that the most ancient' philo-
sophers, Empedocles, Anaxagoras, Anaximenes, Heraclitus, and Demo-
critus, submitted their minds to things as they found them ; but that Plato
made the world subject to ideas, and Aristotle made even ideas, as well as
all other things, subservient to words ; the minds of men beginning to be
occupied, in those times, with idle discussions and verbal disputations, and
the correct investigation of nature being wholly neglected. Plato enter-
tained, however, some correct notions respecting the distinction of denser
from rarer matter by its greater inertia ; and it would be extremely unjust
to deny a very high degree of merit to Aristotle's experimental researches,
in various parts of natural philosophy, and in particular to the vast col-
lection of real information contained in his works on natural history.
Aristotle attributed absolute levity to fire, and gravity to the earth, consi-
dering air and water as of an intermediate nature. By gravity the ancients
appear in general to have understood a tendency towards the centre of the
earth, which they considered as identical with that of the universe ; and as
long as they entertained this opinion, it was almost impossible that they
should suspect the operation of a mutual attraction in all matter, as a cause
of gravitation. The first traces of this more correct opinion respecting it
are found in the works of Plutarch.

Epicurus appears to have reasoned as justly respecting many particular
subjects of natural philosophy, as he did absurdly respecting the origin of
the world, and of the animals which inhabit it. He adopted in great mea-
sure the principles of Democritus respecting atoms, but attributed to them
an innate power of affecting each other's motions, and of declining, in such
a manner, as to constitute, by the diversity of their spontaneous arrange-
ments, all the varieties of natural bodies. He considered both heat and
cold as material ; the heat emitted by the sun he thought not absolutely
indentical with light, and even went so far as to conjecture that some of
the sun's rays might possibly possess the power of heating bodies, and yet
not affect the sense of vision. In order to explain the phenomena of
magnetism, he supposed a current of atoms, passing, in certain directions,
through the magnet and through iron, which produced all the effects by
their interference with each other. Earthquakes and volcanos he derived
from the violent explosions of imprisoned air.

Among all these opinions and conjectures, there is scarcely any one
which was scientifically established upon sure foundations. Some insulated
observations had a certain degree of merit ; and we find many interesting
facts relating to different departments of natural knowledge, not only in
Aristotle, but also in Theophrastus, Dioscorides, and Pliny, as well as in
some of the historical writers of antiquity. Protagorides of Cyzicum, who
is quoted by Athenaeus, relates that in the time of king Antiochus, it was
usual, as a luxury, to cool water by evaporation ; and it is not impossible
that the custom may have been introduced from the east, where even ice

ON THE HISTORY OF TERRESTRIAL PHYSICS. 583

is frequently made at present by a similar process ; others of the ancients
had remarked, according to Dr. Falconer,* that water usually froze the
more readily for having been boiled ; and it is possible that some other
detached observations of a similar nature may occur to those who have the
curiosity to make them objects of research.

The thirteenth century may be considered as the date of the revival, if
not of the commencement, of physical discoveries. Our countryman,
Roger Bacon, was one of its principal ornaments : he appears to have
anticipated in his knowledge of chemistry, as well as of many other parts
of natural philosophy, the labours of later times. The polarity of the
magnetic needle is described in some lines which are attributed to Guyot, a
French poet, who lived about 1180 ; but some persons are of opinion that
this description was actually written by Hugo Bertius, in the middle of
the succeeding century ; and it is generally believed that the compass was
first employed in navigation by Gioja of Amain, about the year 1260 ; he is
said to have marked the north with a fleur de lis, in compliment to a
branch of the royal family of France, then reigning at Naples. The
declination of the needle from the true meridian is mentioned by Peter
Adsiger, the author of a manuscript which bears the date 1269. The poet
Dante, who flourished at the close of this century, distinguished himself
not only by his literary, but also by his philosophical pursuits ; and we
find among his numerous works an essay on the nature of the elements.

The learned and voluminous labours, by which Gesner and Aldrovandus
enriched the various departments of natural history, may be considered as
comprehending the greatest part of what had been done by the ancients in
the investigation of the economy of the animal world ; but their works
have too much the appearance of collections of what others had asserted,
rather than of original observations of their own.

The first of the moderns whose discoveries respecting the properties of
natural bodies excite our attention, by their novelty and importance, is
Dr. Gilbert, of Colchester : his work on magnetism, published in 1590, con-
tains a copious collection of valuable facts, and ingenious reasonings. He
also extended his researches to many other branches of science, and in
particular to the subject of electricity. It had been found, in the preced-
ing century, that sulfur, as well as amber, was capable of electric excitation,
and Gilbert made many further experiments on the nature of electric
phenomena. The change or variation of the declination of the needle is
commonly said to have been discovered by Gellibrand, a professor at
Gresham college, in the year 1625 ; but it must have been inferred from
Gunter's observations, made in 1622, if not from those of Mair, or of some
other person, as early as 1612 ; for at this time the declination was con-
siderably less than Burrows had found it in 1580.f

In the beginning of the seventeenth century, Lord Bacon acquired, by
his laudable efforts to explode the incorrect modes of reasoning, which had
occupied the schools, the just character of a reformer of philosophy ; but

* On the Knowledge of the Ancients, Manch. Mem. i. 261 ; iii. 278.
f Burroughs's Dissertation in Norman's New Attractive, 3rd edit. Gellibrand,
on the Variation of the Mag. Needle, 1635. See Robison, Mec. Ph. iv. 354.

584 LECTURE LX.

his immediate discoveries were neither striking nor numerous. In 1620,
he proposed, with respect to heat, an opinion which appears to have been
at that time new, inferring, from a variety of considerations, which he has
very minutely detailed in his Novum Organum, that it consisted in " an
expansive motion, confined and reflected within a body, so as to 'become
alternate and tremulous ; having also a certain tendency to ascend." A
similar opinion, respecting the vibratory nature of heat, was also sug-
gested, about the same time, by David Gorlaeus, and it was afterwards
adopted by Descartes,* as a part of his hypothesis respecting the constitu-
tion of matter ; which he imagined to consist of atoms of different forms,
possessing no property besides extension, and to derive all its other qualities
from the operation of an ethereal and infinitely elastic fluid, continually
revolving in different orders of vortices.

A much more important step, than the proposal of any hypothesis con-
cerning the nature of heat, was also made about the year 1620, by Cornelius
Drebel, who appears to have been the original inventor of the method of
measuring the degrees of heat by a thermometer. The utility of the instru-
ment remained, however, much limited, for want of an accurate method of
adjusting its scale, and it was not till the close of the century, that Dr.
Hooke's discovery, of the permanency of the temperature of boiling
water, afforded a correct and convenient limit to the scale on one side,
while the melting of snow served for fixing a similar point on the other ;
although there would have been no great difficulty in forming a scale suffi-
ciently natural, from the proportion of the expansion of the fluid contained
in the thermometer to its whole bulk.

It was about the year 1628, that Dr. Harvey 1* succeeded in demon-
strating, by a judicious and conclusive train of experiments, the true
course of the circulation of the blood, through the veins and arteries, both
in the perfect state of the animal, and during its existence as an embryo.
Servetus had explicitly asserted, in his work on the Trinity, as early as
the year 1553, that the blood performed, in its passage through the lungs,
a complete revolution, beginning and ending in the heart ; and Cisalpinus
had even expressed, in 1569, some suspicions that the circulation of the
whole body was of a similar nature ; but neither of these authors had
advanced any satisfactory proofs in confirmation of his opinions.

In the middle of the seventeenth century, the barometer was invented by
Torricelli ; the variation of the atmospheric pressure was discovered by
Descartes ; and Pascal made several experiments on the difference of its
magnitude at different places, which tended to illustrate the principles, on
which the method of determining heights by barometrical observations is
founded.

What Gesner and Aldrovandus had before done with regard to the
animal kingdom, was performed, a century later, for the vegetable world
by John and Caspar Bauhin, whose works, as collections of all that was to
be found on record respecting ftie distinctions and properties of plants,

* Princip. PLilos. Part IV. § 29.

f Exercitatio Anatomica de Motu Cordis et Sanguinis, Francof. 1628 : his expe-
riments were made in 1616. Consult Cuvier, Le?ons sur 1'Hist. des Sci. Nat.

ON THE HISTORY OF TERRESTRIAL PHYSICS. 585

have not yet been superseded by the latest publications. Our country-
men, Ray and Willughby, contributed also to add much new matter to
the stores of natural history, in all its departments ; and their labours,
as well as those of Tournefort and Re'aumur, are of the more value, as they
were far more studious than their predecessors to discriminate truth from
fiction.

The foundation of the most celebrated of the philosophical societies of
Europe renders the latter half of the seventeenth century a very interesting
period in the history of natural knowledge. The Royal Society of London,
and the Academy of Sciences of Paris, have always been the most distin-
guished of these : and the Florentine Academy del Cimento, although its
labours were not of long duration, produced at first in a short time
a very copious and interesting collection of experiments, relating to various
subjects of physical research. In the Royal Society, Boyle, Hooke, and
Newton were the most industrious, as well as the most successful inves-
tigators of natural phenomena : the elementary doctrines of chemistry, the
nature of combustion, the effects of heat and cold, and the laws of attrac-
tion, repulsion, and cohesion were attentively examined and discussed.
The expansion of water, by a reduction of its temperature, near the freez-
ing point, was first observed by Dr. Croune ; although his experiments
were considered by Dr. Hooke as inconclusive.* The attention of the
society was directed by Newton to the phenomena of electricity, some of
which had been a short time before particularly noticed by Guericke ; the
mode of making electrical experiments was greatly improved by Hauks-
bee ; this accurate observer investigated also the nature of capillary attrac-
tion with considerable success. Early in the succeeding century, many of
the members of the Academy of Petersburg followed the example of other
societies with great industry ; and the experiments of Richmann on heat
were among the first and best fruits of their researches.

Dr. Halley employed himself, with the most laudable zeal, in procuring
information respecting the variation of the compass ; he undertook a
voyage round the world, for the express purpose of making magnetical
observations ; and he published a chart of variation, adapted to the year
I700.f He also collected many particulars respecting the trade winds and
monsoons, and he endeavoured to explain them by a theory which has
been adopted by some of the latest authors, but which is in reality much
less satisfactory than the hypothesis proposed some time afterwards by
Hadley.^ His magnetical investigations were continued with great dili-
gence by Montaigne and Dodson, who published, at different periods, two
charts representing the successive states of the variation. Euler,§ Mayer, ||
and others have attempted, in later times, to discover such general laws as
might be sufficient to determine the magnitude of the variation for every
part of the globe ; but their success has been very much limited.

The science of electricity was diligently cultivated in the middle of the
last century by Stephen Gray, Dufay, Winkler, Nollet, Musschenbroek,

* Birch, iv. 26, 253. f Ph.Tr.xxiii. 1106. J See p. 681.

§ Hist, et M£m. de Berlin, 1755, p. 107 ; 1757, p. 175 ; 1766, p. 213.
II Gott. Anz. 1760, p. 633 ; 1762, p. 377.

686 LECTURE LX.

and Franklin. As early as 1735 it was remarked by Gray, that "the
electric fire seemed to be of the same nature as lightning,"* and their
identity was afterwards more strongly asserted by Winkler, and experi-
mentally demonstrated by Franklin. The shock of a charged jar was first
discovered by Kleist, in 1745 ; and the experiment was repeated by Lalla-
mand and Musschenbroek, who described its disagreeable effects on the
sensations with an exaggeration not the most philosophical. The theory of
the nature of the charge was the second great improvement made by
Dr. Franklin in this science.

The introduction of the Linnean system of botany and zoology is to be
considered as bringing near to perfection the logic and phraseology of
natural history ; nor has its celebrated author wholly neglected the philo-
sophy of the science. The number and the diligence of his successors have
already furnished to the different departments of natural history a much
ampler store of observations than could easily have been expected from the
short time which their labours have occupied. Buffon had merit of a dif-
ferent kind, and though his fancy was too little regulated by mathematical
accuracy, the elegance of his writings have made their subjects highly
interesting to the general reader. Among other modern naturalists of
great respectability, Spallanzani, Daubenton, Degeer, Geoffrey, Pennant,
the Jussieus, Lacepede and Haiiy, have particularly distinguished them-
selves by the importance of their discoveries, and the accuracy of their
descriptions.

. The absorption of heat, during the conversion of ice into water, appears
to have been separately observed by Deluc, Black, and Wilke, about the
year 1755. On this experiment Dr. Black principally founded his doc-
trine of latent heat, supposed to be retained in chemical combination by the
particles of fluids. Dr. Irvine and Dr. Crawford explained the circum-
stances somewhat differently, by the theory of a change of capacity for
heat only. Bergmann,t Lavoisier, Laplace, Kirwan, Seguin, and many
other philosophers have illustrated, by experiments and calculations, the
various opinions which have been entertained on this subject ; and few
chemists, from the times of Boerhaave, Stahl, and Scheele to those of
Priestley and other later authors, have left the properties of heat wholly
unnoticed.

The elegant hypothesis of Aepinus, respecting magnetism and electricity,
founded in great measure on the theory of Franklin, was advanced in 1759 ;
our venerable countryman, Mr. Cavendish, had invented a similar theory,
and had entered in many respects more minutely into the detail of its con-
sequences, without being acquainted with Aepinus's work ; although the
publication of his paper on the subject was 12 years later. Lambert,
Mayer, Coulomb, and Robison have also pursued inquiries of a similar
nature, both theoretically and experimentally, with great success. The
electrophorus of Wilke, and the condenser of Volta, are among the earliest
fruits of the cultivation of a rational system of electricity, and Mr.
Cavendish's investigation of the properties of the torpedo may 'serve as

* Ph. Tr. xxxix. 24.

t Opuscula Physica et Chemica, 6 vols. Upsalise.

ON THE HISTORY OF TERRESTRIAL PHYSICS. 587

a model of accuracy and precision in the conduct of experimental
researches.

The speculations of Boscovich respecting the fundamental properties of
matter, and the general laws of the mutual action of bodies on each other,
have been considered by some candid judges as deserving the highest com-
mendation ; they remain however almost in all cases speculations only ;
and some of the most intricate of them, being calculated for the explanation
of some facts, which have perhaps been much misunderstood, must con-
sequently be both inaccurate and superfluous.

The attention of several experienced philosophers, who are now living,
has been devoted, with much perseverance, to the difficult subject of
hygrometry. Deluc's experiments have offered us a very useful compari-
son of the hygrometrical qualities of various substances : Saussure has
investigated, with great labour, the indications of the hygrometer and the
thermometer, as connected with the presence of a certain portion of vapour,
contained in air of various densities; and Pictet has ascertained some
similar circumstances respecting vapours of different kinds wholly un-
mixed with any air. The hypotheses, which have usually accompanied
the relation of most of these experiments, have however been in general
too little supported by facts to be entitled to universal adoption.

For some years past, the philosophical, as well as the unphilosophical
world has been much occupied and entertained by the discoveries of Gal-
vani, Volta, and others, respecting the operations of the electric fluid. The
first circumstance that attracted Galvani's attention to the subject of
animal electricity, was the agitation of a frog, that had a nerve armed,
that is, laid bare and covered with a metal, when a spark was taken in
its neighbourhood. A person acquainted with the well known laws of
induced electricity might easily have foreseen this effect : it proved, how-
ever, that a frog so prepared was a very delicate electrometer, and it led
Galvani to further experiments. It has been shown by Volta, that an
entire frog may be convulsed by a degree of electricity which affects
an electrometer but very weakly ; but that when prepared in Galvani's
manner, it will be agitated by an electricity one fiftieth part as great,
which cannot be discovered, by any other means, without the assistance
of a condenser. Galvani, however, found that a communication made
between the armed nerve and its muscle, by means of any conducting
substance, was sufficient to produce a convulsion, without the presence of
foreign electricity : hence he concluded that the nerve and muscle, like
the opposite surfaces of a charged jar, were in contrary states of electri-
city, and that the communication produced a discharge between them.
He observed, however, a considerable difference in the effects, when dif-
ferent metals were employed for forming the circuit ; and this circum-
stance led to the discovery of the excitation of electricity by means of a
combination of different inanimate substances only, which Mr. Davy
attributes to Fabroni, Creve, and Dr. Ash. It was, however, still more
satisfactorily demonstrated by Volta ; and he at first supposed that all
the phenomena observed by Galvani were derived from effects of this
kind, but on further examination he was obliged to allow the independent

588 LECTURE LX.

existence of animal electricity. This industrious and ingenious philo-
sopher has the sole merit of the invention of the pile or battery, which has
rendered every other mode of exciting the galvanic action comparatively
insignificant.

No sooner was Volta's essay communicated to the Royal Society, than
a pile was constructed by Mr. Carlisle, and its singular effects in the
decomposition of water were jointly observed by himself and Mr. Nichol-
son. The original existence of animal electricity, as asserted by Galvani
and Volta, has been in some degree confirmed by the experiments of
Aldini, the nephew of Galvani. A number of detached observations, of
considerable merit, have also been made by Pfaff, Ritter, Cruikshank,
Wollaston, Fourcroy, and many other chemists, both in this country and
on the continent. But Mr. Davy's late experiments must be considered
as exceeding in importance every thing that has been done upon the subject
of electricity, since the discovery of the pile of Volta. The conclusions
which they have enabled him to form respecting the electrical properties
of such bodies as have the strongest tendencies to act chemically on each
other, and the power of modifying and counteracting those tendencies
which the electric fluid possesses, have greatly extended our views of the
minute operations of nature, and have opened a new field for future inves-
tigations. I hope that I shall be pardoned by astronomers for having
inserted, on this occasion, in a vacant space among the constellations, in
the neighbourhood of Pegasus, the figure of a galvanic battery ; which
must now be allowed to have as great pretensions to such a distinction as
the electrical machine and the chemical furnace.

The late experiments and speculations of Mr. Dalton, on various sub-
jects, belonging to different branches of physics, have tended to place some
parts of the science of meteorology in a new light. It is true that many
of his hypotheses are very arbitrarily assumed ; some of them are mani-
festly contrary to experiment, and others to analogy and probability ; at
the same time his remarks appear in some cases to be either perfectly cor-
rect, or to lead to determinations which are sufficiently accurate for every
practical purpose. I have attempted to borrow from Mr. Dalton' s ideas some
hints, which I have incorporated with a less exceptionable system ; and by
a comparison of his experiments with those of many other philosophers, I
have deduced some methods of calculation which may perhaps be practi-
cally useful ; in particular a simple rule for determining the elasticity of
steam, and a mode of reducing the indications of hygrometers of different
kinds to a natural scale.

Count Rumford's establishment of a prize medal, to be given every three
years by the Royal Society to the author of the most valuable discovery
respecting heat or light, forms an era less remarkable, than the first adjudi-
cation of the medal to himself, and the second to Mr. Leslie. Count
Rumford's numerous experiments on the production and communication of
heat are highly important, both for the utility which may be derived from
their economical application, arid for the assistance which they afford us in
the investigation of the intimate nature of heat. Mr. Leslie's discovery of
the different properties possessed by surfaces of different kinds, with regard

ON THE HISTORY OF TERRESTRIAL PHYSICS. 589

to emitting and receiving radiant heat, is in every respect highly interesting ;
and the multiplicity and diversity of his experiments would have entitled
him to still higher commendation than he has obtained, if they had heen
more simply and circumstantially related. Perhaps, however, none of
the modern improvements in speculative science deserves a higher rank
than Dr. Herschel's discovery of the separation of heat from light by re-
fraction. Mr. Prevost has made some just remarks on the experiments of
other philosophers respecting heat ; and his own theory of radiant heat,
and his original investigations, on the effect of the solar heat on the earth,
have tended materially to illustrate the subject of his researches.

The general laws of the ascent and descent of fluids in capillary tubes,
and between plates of different kinds, had long ago been established by
the experiments of Hauksbee, Jurin, and Musschenbroek ; many other
circumstances, depending on the same principles, had been examined by
Taylor, Achard and Guyton ; and some advances towards a theory of the
forms assumed by the surfaces of liquids, had been made by Clairaut,
Segner, and Monge. In an essay on the cohesion of fluids, read before
the Royal Society in the year 1804, I have reduced all effects of this
nature to the joint operation of a cohesive and repulsive force, which
balance each other ; assuming only that the repulsion is more augmented
by the approach of the particles to each other than the cohesion ; and I
have had the satisfaction of discovering in this manner a perfect corre-
spondence between many facts, which had not been supposed to have the
slightest connexion with each other. Almost a year after the publication
of this paper, Mr. Laplace read to the National Institute a memoir on capil-
lary tubes, in which, as far as he has pursued the subject, he has precisely
confirmed the most obvious of my conclusions; although his mode of
calculation appears to be by no means unexceptionable, as it does not in-
clude the consideration of the effects of repulsion. Had my paper been so
fortunate as to attract Mr. Laplace's attention before his memoir was pre-
sented to the Institute, he would perhaps have extended the results of my
theory with the same success, which has uniformly distinguished his
labours in every other department of natural philosophy.

When we reflect on the state of the sciences in general, at the beginning
of the seventeenth century, and compare it with the progress which has
been since made in all of them, we shall be convinced that the last two
hundred years have done much more for the promotion of knowledge, than
the two thousand that preceded them : and we shall be still more encouraged
by the consideration, that perhaps the greater part of these acquisitions
has been made within fifty or sixty years only. We have therefore the
satisfaction of viewing the knowledge of nature not only in a state of ad-
vancement, but even advancing with increasing rapidity ; and the universal
diffusion of a taste for science appears to promise, that, as the number of
its cultivators increases, new facts will be continually discovered, and those,
which are already known, will be better understood, and more beneficially
applied. . The Royal Institution, with other societies of a similar nature,
will have the merit of assisting in the dissemination of knowledge, and in
the cultivation of a taste for its pursuit ; and the advantages arising from

I >

690 LECTURE LX.

the general introduction of philosophical studies, and from the adoption
of the practical improvements depending on them, will amply repay the
labours of those, who have been active in the establishment and support
of associations so truly laudable.

LECT. LX.— ADDITIONAL AUTHORITIES.

History of Electricity.— Hausen, Novi Perfectus in Hist. Electr. 4to, Leipz.
1734. Gralath, Geschichte der Elektr. Abhand der Natur. Gesellsch. in Danzig,
1747, i. 23. Hist. Generate et Particuliere de 1'Electr. Paris, 1752. Dalibard, do.
abrege*e, 2 vols. 1766. Priestley's Hist, of Electr. with Original Experiments, 4to,
1764 ; Additions, 4to, 1770. Kriinitz, VerzeichnissderVornehmsten Schriften von
der Elektr. Leipz. 1769. German trans, of Priestley, 4to, fieri. 1772. Kiibu,
Geschichte der Medicinischen und Phys. Elektr. 2 vols. Leipz. 1783 and 1796.
Eyewater, Essay on the Hist, of Electr. 1810. De la Rive, Esquisse Histor. des
Principals Decouvertes faites dans 1'Electr. depuis quelques Annees, Geneve, 1833.
Ann. de 1'Electr. i. 1. Wartmann, ibid. i. 31.

Galvanism. — Ritter, Beitrage zur Nahern Kenntniss des Galv. Jena, 1800-5.
Sue", Hist, der Galv. 4 vols. Paris, 1802-5. Tromsdorff, Geschichte des Galv.
Erfurt, 1808. Bostock's Hist, of Galv. 1818.

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INDEX.

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•

INDEX.

AARON Reschid, 456.

Aberration from colour, 337.

Aberration of light, 363, PI. 29.

Aberration of the stars, 341, 342, 379.

Abutments, 125.

Academicians, 289.

Academy del Cimento, 489, 585.

Academy of Paris, 191, 192, 279, 280, 585.

Academy of Petersburg, 585.

Accelerated motion, PI. 1.

Accelerating forces, 21.

Acceleration, 22.

Acceleration of the moon's motion, 414.

Acceleration of tides, 447, 448.

Accidental properties of matter, 465.

Accommodation of the eye, 353.

Accompaniment, 308.

Accumulation of electricity, 517«

Achard, 589.

Achernar, 395.

Achromatic eye piece, 337.

Achromatic glasses, 337«

Achromatic telescopes, 379, PI. 28.

Acids, 524.

Acoustics, 195, 287-

Acting pump, PI. 23.

Action of water on lead, 277-

Actual focus, 324.

Adair, 558.

Adhesion, 112, 117, 119,480.

Adsiger, 583.

Advancement of science, 589.

Aeolian harp, 299, 312.

Aepinus, 507, 531, 586.

Aerial perspective, 356.

Africa, 437.

Aggregation, 480.
Agitation, 163, 178.

Agricultural instruments, 175.

Air, 232. Buoyancy of the air, 29. Resistance

of the air, 30, 31, 154, 232, 260, 261, 284.
Air consumed, 486.
Air gun, 173, 269, PI. 24.
Air pump, 205, 260, 278, PL 24.
Air thermometer, 498, PL 39.
Air vents, 241.
Air vessel, 138, 255.
Ajutages, 212, 213.
Albategni, 456.
Albertus, 285.

Albinus, 88, 90; b. 1683, d. 1771.
Alcohol, 497.
Aldebaran, 395-
Aldini, 523, 588.

Aldrovandus, 583, 584; b. 1525, d. 1605.
Alembert. See Dalembert.
Alexander, 182, 454.
Alexandria, 75, 183.
Alexandrian school, 453, 454.
Alfred, 186.
Algenib, 394, 395.
Algol, 393, 394, 395.
Alhazen, 375; fl. 1072.
Alkalis, 524.
Allineations, 394, 395.
Almagest, 456.
Almamoun, 456; 4a33.

Alteration, 105, 109.

Alternate motion, PL 14.

Alternation of motion, 257-

Altitude, PL 35.

Alva. Duke of Alva, 187-

Amontons, 191, 256,278; b. 1663, d. 1705.

Amphibia, 575.

Anatomy of plants, 568.

Anaxagoras, 581, 582.

Anaximander, 581 ; b. 611, d. 547, B. c.

Anaximenes, 581, 582.

Anchor. Weighing an anchor, 158.

Andr6, 93.

Andromeda, 395.

Aneurisms, 227-

Angles, 80.

Angles of incidence and reflection, 322.

Anglonorman architecture, 187-

Animal actions, 98.

Animal electricity, 523.

Animal force, 69.

Animal life, 573.

Animal light, 340.

Animal motions, 49.

Animals, 566.

Anne, 190.

Annealing, 494.

Anoria, PL 22.

Antares, 395.

Anthelia, 347-

Antimony, 532.

Antiochus, 582.

Anvil, 61.

Anvils, 170.

Anvil of the ear, 302.

Apollodorus, 186 ; fl. 120.

Apollonius Pergaeus, 183, 455 ; fl. 242, B.C.

Apparent attractions, repulsions, and cohesions,
PL 39.

Apparent diameter of the sun, 416.

Apparent motion of the sun, 416.

Apparent motions, 425.

Apparent motions of the stars, 416.

Appearances of the celestial bodies, 415.

Apsides, PL 34.

Aquarius, 401.

Aqua tinta, 93.

Aqueous humour, 350.

Aquila, 395.

Arabians, 186, 188, 456.

Arago, 372, 542.

Aratus, 463 ; b. 300, B.C.

Arch, 123, 182, PL 11.

Archimedes, 28, 44, 50, 184, 185, 189, 190, 236,
251, 275, 276, 375, 433, 454, PL 22 ; d. 212, B.C.

Architecture, 121, 182, 187. Hydraulic architec-
ture, 235, 237, 238.

Arch lute, 312.

Archytas, 182 ; b. 442, d. 352, B.C.

Arcs of circles, PL 6.

Arcturus, 392, 395, 397-

Are, 84.

Aretin, the monk, 316.

Argo, 395.

Aries, 395, 401.

Aristarchus, 454 ; fl. 264, B.C.

Aristophanes, 374.
2 Q

r>94

INDEX.

Aristotelians, 12.

Aristotle, 5, 19, 183, 275, 315, 316, 374, 442, 577,

582, PL 22; b. 385, d. 322, B.C.
Aristyllus, 454.
Arkwright, 101, 142, 187-
Arnault, 146.
Arnold, 151, 154, PL 16.
Arrangement of particles, 480.
Arrangement of the stars, 392.
Arrow, 173.
Arrowsmith, PI. 42, 43.
Arsenic, 532.
Artedi, 575.
Arteries, 220.
Artificial globe, 432.
Artificial magnets, 537-
Arytaenoid cartilages, 312,313.
Ascent, 25.

Ascent of a double cone, PI. 3.
Ascent of a loaded cylinder, PI. 3.
Ascent of water, PI. 39.
Ash, 587.

Astronomer Royal, 461.
Astronomical instruments, 426.
Astronomical telescope, 334.
Astronomical time, 429.
Astronomy, 387. Practical Astronomy, 425.
Athenaeus, 185, 582 ; fl. 136.
Atmosphere, 206, 546, PL 19, 24.
Atmosphere of Jupiter, 450, 548.
Atmosphere of the sun, 399, 400.
Atmospherical pressure, 207.
Atmospherical refraction, 345, PL 29.
Atmospheric machine, 258.
Atmospheric tides, 450.
Atoms, 469, 581.
Attachment of horses, 167.
Attalus, 186.

Attraction of a sphere, 410.
Attraction of moisture, 553.
Attractions and repulsions of electrified bodies,

511.

Attractions of floating bodies, 478, 513.
Attractions of solids and fluids, 476.
Attractions of the electric fluid, 508.
Attrition, 120.

Atwpod's machine, 23, 41, 118, PL 1.
Auriga, 394.

Aurora boreal is, 533, 560.
Avicenna, 147.
Avoirdupois, 96.
Axes of rotation, 66.
Axis and wheel, 51.
Axis and winch, 157.
Axles, 116.
Azimuth compass, PI. 41.

.'),'. -

Babylonian observations, 452.

Bacon. Roger Bacon, 188, 227, 375, 583 ; b. 1214,

d. 1292.
Bacon. Lord Verulam, 5, 12, 189, 316, 458, 582,

583; b. 1560, d. 1626.
Bag, 203.

Bag pump, 254, PL 23.
Bailly, 452.
Baily, 440.

Balance, 146, PL 8, 9.
Balance. Hydrostatic balance, 235.
Balances, 96.
Balance spring, 147.
Ballast, 249.
Balloon, 206, 265.
Bank, 200.
Banking, 153.
Banks, 238.
Bark, 569.
Barlow, 95.

Barometer, 550, 556, 584, PI. 19.
Barometers, 208.

Barrel chronometer, 146.
Barrow, 190, 377; b. 1630, d. 1677-
Bartholin, 378 ; b. 1616, d. 1680.
Barton, PL 4.
Baskets, 168.
Bass, 312.
Batsha, 449,464.
Battering ram, 179.

Battery, 513.

Battery of Volta, 563, 588.

Bauhin, 584. J. Bauhin, b. 1541, d. 1613. C.

Bauhin, b. 1560, d. 1624.
Bayer, 394.
Bead pump, 256.
Beads in equilibrium, PL 11.
Beam, 114, 115.
Beam compasses, 78, PL 6.
Beams, PL 10.

Beams in equilibrium, PI. 11.
Bear, 394.
Beat, PL 25.
Beats, 305.
Beccaria, 558.
Becquerel, 504.
Beguelin, 381.

Beighton, 266, 279, 282, PI. 24.
Bell, 314.

Bell. Henry Bell, 271.
Bellows, 200, 262, PL 24.
Benedetti, 189.
Bennet, 109, 110, 527, PL 40.
Bent columns and bars, PL 9.
Bent lever, PL 3.
Bent lever balance, 96, 98, PL 9.
Bent pipes, 227-
Bent straps, PL 13.
Berard, 504.

Bergmann, 586 ; b. 1735, d. 1784.
Bernoulli, 46, 155, 203, 228, 268. D. Bernoulli,

191, 210, 213, 280, 281, 296, 300, 317, 498, PL

20, 22, 39. Ja. Bernoulli, 191, 279 ; I*. 1654, d.

1705. Jo. Bernoulli, 191, 279, 281, 282; b.

1667, d. 1748.
Berthoud, 153, 155.
Bertius, 583.
Bessel, 390, 404.
Beudant, 291.

Bevilled wheels, 136, PL 15.
Bianconi, 290.
Biceps, 99.
Billiard balls, PL 5.
Billiards, 62.
Biot, 285, 371.
Bird, 461.
Birds, 574, 575.
Birmingham, 187.
Biscop, 186.
Bissextile, 427.
Bistre, 73.
Bito, 186.

Black, 285, 501, 586; b. 1728, d. 1799.
Blackening rays, 490.
Blackfriars bridge, 126, PL 12, 14.
Blasting, 180.
Blast of air, 200.
Blocks, 53, 176, PI. 4.
Blood, 573, 574, 578.
Blow, 61.

Board of longitude, 192,461.
Board perforated, 1 12.
Bode, 433.

Body. Moveable body, 38.
Body colours, 73.

Boerhaave, 498, 586 ; b. 1668, d. 1738.
Boiling, 492.
Bolognan jars, 494.

Bolts,

Bones of the ear, PI. 25.

Bootes, 395.

Borda, 84, 85, 282; b. 1733, d. 1797-

Borelli, 99.

Boring, 175.

Boscovich, 359, 360, 3C2, 381, 471, 587; b. 1711,

d. 1787-
Bossut, 284.
Botany, 586.

Bottom of a cistern, PL 19.
Bouguer, 280, 342, 379, 381, 421,545; b. 1698,

d.1758.

Boulton, 103, 117, 259, 267, 2G9. ,
Bovillus, 34.
Bow, 173.
Boyle, 5, 278, 377, 585, PL 1<); b. 1027.

d. 1691.
Braces, 130, PL 13.

INDEX.

595

Bradley, 341, 379, 402, 413, 416, 461 ; b. 1622,

d. 1762.

Brahe. See Tycho.
Bramah, 170, 256.
Bramah's press, 199, 254, PI. 23.
Brass, 538.

Breast wheel, 246, PL 22.
Brereton, 558.
Brewster, 371.
Bridge, 125, PI. 11.
Bridges, PI. 14.
Bridgewater. Duke of Bridgewater, 157, 159,

PL 17.

Briggs, 189; b. 1561, d. 1631.
British manufactures, 186.
Brittleness, 110.
Brouncker, 34.
Brunau, 278.

Buat. Chevalier de Buat, 222, 223, 243, 285.
Bubble, 476.
Buchanan, 101.
Bucket revolving, 198.
Bucket wheel, 250.
Buffon, 375, 489, 586.
Bull, 395.
Bullets, 268.
Buoyancy, 202.
Burg, 461.

Burning glasses, 330, 374,
Burning mirror, 375.
Burning rocks, 179.
Burnisher, 92.
Burrows, 583.

Cabbage:ieaf, 478.

Cable, 114.

Caesar, 186, 277, 427, 428, 442, 455 ; b. 99, d. 43,

Calabria, 561.

Calendar, 427, 456.

Calendering mill, 170.

Calking, 72.

Caloric, 501.

Camden, 187.

Camera lucida, 331.

Camera obscura, 331, PL 28.

Camper, 90.

Canal, PL 21.

Canals, 238, 240, 277.

Cancer, 401.

Candles, 486.

Cnno n, 175,

Cannon ball, 179, 233.

Canopus, 395.

Canterbury cathedral, 188.

Canton, 209, 291.

Capacity for heat, 499.

Capella, 394, 395.

Capes, 437.

Capillary action, 475.

Capillary attraction, 227.

Capillary tubes, 477, 589.

Capstan, 158, PL 3, 4.

Capricornus, 401.

Carbonic acid gas, 289.

Carisbrook castle, 160.

Carlisle, 588.

Carpentry, 121, 128.

Carriages, PI. 18.

Carrying, 102.

Cartes. See Descartes.

Cartesian devils, PL 19.

Cartilages of the larynx, PL 26.

Carts connected, 168.

Cart with a crane, 161.

Caspian Sea, 43?.

Cassegrain, 335, 338.

Cassegrain's telescope, 335, PL 28.

Cassini, 34, 147, 399, 456, 458, PL 33. C. F.
Cassini, b. 1714, d. 1784. D. Cassini, b. 1625,
d. 1712. J. Cassini, b. 1677, d. 1756.

Cassiopeia, 393, 394, 395.

Castelli, 2J7; b. ab. 1575, d. 1644.

Casting, 87-

Castor, 397.

Catalogue of references, 193.

Catchfly, 566.

Catenaria, 124.

Catoptrics, 324.

Causation, 11.

Caustics, PL 28.

Cavalleri, 28, 190, 256.

Cavallo, 526, 532, 538, 558, PL 40.

Cavendish, 348, 440, 507, 510, 512, 586.

Caxton, 189.

Ceiling, 114.

Cellular pump, 256.

Celsius, 498.

Centaur, 395.

Central forces, PL 1, 2.

Centre, 32?.

Centre of a bridge, PI. 14.

Centre of gravity, 40, 47, PL 3.

Centre of gyration, 64.

Centre of inertia, 39, PL 2, 3.

Centre of oscillation, 64.

Centre of percussion, 64.

Centre of position, 40.

Centre of pressure, 202.

Centres of bridges, 131 .

Centrifugal bellows, PI. 24.

Centrifugal force, 26, 417.

Centrifugal pump, 253, PL 23.

Centrifugal regulator, 37.

Cepheus, 395.

Ceres, 404, 424.

Chain, 139, PL 7.

Chain loaded, PL 11.

Chain pump, 256.

Chains, 86.

Chair, 131.

Chaldeans, 452.

Chalk, 112.

Chalks, 72.

Changeable stars, 393.

Change of climate, 546.

Changes of form, 169.

Chapman, 284.

Charcoal, 486.

Charge, 513.

Charles, 294.

Charles II., 460.

Charts, 394.

Chase, 94.

Chemical attractions, 523, 524.

Chemical effects of electricity, 519.

Chemical electricity, 521.

Chemistry, 11.

Chemists, 504.

Chersiphron, 182.

Childers, 102.

Childrey, 399.

Chimney, 265.

Chimney pipes, 314, PI. 26.

Chimnies, 277.

Chinese, 74, 91, 452, 580.

Chinese pumps, 257.

Chiron, 453.

Chladni, 292, 297, 300, 317, 318, PL 25.

Chord, PL 25.

Chords of a circle, 33, PL 2.

Choroid coat, 351.

Christian era, 427.

Chromatic aberration, 337-

Chromatic scale, 307.

Chronology, 427.

Chronology of acoustics, 319.

Chronology of astronomers, 463.

Chronology of authors on hydrodynamics, 286.

Chronology of mathematicians and mechanics.

See 194.

Chronology of optical authors, 385.
Chronology of physical authors. See 590.
Chronometer with a barrel, 146.
Churchman, PL 41.
Chyle, 577, 578.
Cicero, 456.

Cimabue, 188 ; b. 1240, d. 1300.
Cimento. Academicians del Cimento, 290,

489.

Circle. Graduated circle, 81.
Circle in perspective, PL 8.
Circles, 77, PL 6.
Circular pendulum, 36.
Circular slider, PL 7.
Circulation of the blood, 220, 584.
Cisalpinus, 584.
Cisterns, 239.

2 Q 2

.598 INDEX.

Electrical attraction and repulsion, 524.
Electrical balance, 527, PL 40.
Electrical light, 340, 518.
Electrical machines, 525, PL 40.
Electrical pressure, 511.
Electric fluid, 508.
Electricity, 13, 508, 531, 585.
Electricity in equilibrium, 507.
Electricity in motion, 516.
Electrics, 520.
Electrified spheres, PI. 39.
Electro-magnetic telegraph, 541.
Electro-magnetism, 539.
Electrometers, 527, PL 40.
Electrophorus, 526, PL 40.
Elevation of a projectile, 30.
Elevation of liquids, 477, PL 39.
Elevations, 438, PL 38.
Ellicott, 153.

Ellipsis, 29, 37, 89, 293, PL 2.
Ellipticity of the earth, 435.
Elliptic motion of a pendulum, 37-
Elliptic orbits, 401.
Elliptic vibrations, PL 2.
Elongation, 418.
Elongation of Venus, PL 34.
Elvius, 282.

Embankments, 237, PL 21.
Emery, 149.
Emission of light, 341.
Empedocles, 374, 582; b. 473, d. 413, B.C.
Encaustic paintings, 74.
Encroachments of the sea, 563, 564.
Encyclopaedia Britannica, 32, 46.
Encyclopedic, 192.
Energy, 59, 172.
Englefield, 490.
English foot, 85.
English philosophers, 5.
Engraving, 71, 91, 188.
Engymeter, 82.
Epact, 429.
Epicureans, 12, 458.
Epicurus, 183, 582; b. 342, d. 270, B.C.
Epicycles, 455.

Epicycloidal teeth, 135, PI. 15.
Epiglottis, 312.
Eprouvette, 103.
Equal areas, PL 1.
Equalization of force, 148.
Equated clocks, 427.
Equation of time, 426.

Equilibrium, 45, PL 3, 8. Stability of equili-
brium, 198.

Equilibrium of animals, 49.
Equilibrium of electricity, PL 39.
Equilibrium of fluids, PL 19.
Equilibrium of gases, 204.
Equilibrium of radiant heat, 489.
Equinoctial tides, 442.
Equinox, PL 34.
Eratosthenes, 401, 454.
Eridanus, 395.
Eskinard, 378.

Essential properties of matter, 464.
Etching, 92.

Ethereal medium, 362, 472, 482.
Euclid, 183, 375.
Eudoxus, 182.
Euler, 66, 191, 282, 300, 317, 360, 379, 380, 415,

421, 432, 461, 585. L. Euler, b. 1707, d. 1783.
Eumenes, 75.
Evaporation, 492, 551.
Excitation of electricity, 587-
Excitement of heat. 484.
Expanse of the universe, 389.
Expansion, 491.
Expansion of pendulums, 153.
Expansion of the air, PI. 24.
Expansions, 496.
Experiment oh elasticity, 22.
Explosions, 560.
Extension, 105, 106, 170, 465.
Extension of a column, PL 9.
Extinction of light, 364.
Eyck. Van Eyck, 73, 188; b. 1371, d. 1441.
Eye, 350, PL 30.
Eyepiece, 336, PL 28.
Eytelwein, 285.

Fabricius, 5/6.

Fabroni, 587-

Fahrenheit, 485, 498.

Falconer, 582.

Fall in a fluid, 203.

Fall of a heavy body, 23, 85.

Fall of a feather, 44.

Fall of leaves, 571.

Fan for corn, 264.

Faraday, 541, 542.

Fata Morgana, 346.

Faunius, 276.

Felt, 143.

Felting, 143.

Fermat, 376, 458.

Fidler, PI. 8.

Field glass, 336, PL 28.

Figure of the earth, 435, PL 34.

Fire, 488.

Fire engines, 255, 276, PL 23.

Fishes, 231, 395, 575.

Fixed ecliptic, 400, PL 32.

Fixed stars, 387, PL 36, 37.

Flageolet, 314.

Flakes of snow, 348.

Flame, 12.

Flamsteed, 460, 461 ; b.1646, d.1719-

Flax, 141.

Flemish weavers, 187.

Flexible fibres, 138.

Flexible vessels, 203.

Flexure, ]05, 107, 112, 113, 480, 482.

Flexure of columns and bars, PL [).
Floating bodies, 201, 478, PL 19.

Floodgates, 239, PL 21.

Floor, 114.

Flower, 567-

Fluid, 196.

Fluids, 195.

Fluoric acid, 92.

Fluor spar, 341.

Flute, 314.

Flute pipe, PI. 26.

Fluxions, 191.

Fly, 101.

Fly clocks, 145.

Fly wheels, 137.

Focus, 324.

Focus of a lens, 326.

Focus of the eye, 353.

Fomalhaut, 395.

Fondeur, 146.

Fontenelle, 190, 422.

Foot, 85.

Forbes, 505, 506.

Force, 19, 26, 60. Accelerating force, 21. Cen-
trifugal force, 26. Definition of force, 21.
Deflective force, 26. Regulation of force, 69.

Force of electricity, 104.

Force of magnetism, 104.

Forces. Regulation of hydraulic forces, 241.

Forcing pump, 254, 276, PL 23.

Forge hammer, PL 18.

Forges, 171.

Forging, 110.

Form of the sky, 356.

Forms of the planets, 412.
I Formulae for elasticity of steam, 272.
| Forsyth, 571.

Fossils, 563.
I Fourcroy, 588.
I Fracture, 105, 110, 113.

Fracture from heat, 494.

Frame for rectilinear motion, PL 14.

Frame saw, PL 4.

Franc, 96.

Franklin, 507, 558, 586; b. 1706, d. 1791.
Fraunhofer, 343, 505.
Frederick II., 145, 457-
Freezing, 493, 546.
French measures, 85.

French weights, 96.

Fresco, 73.

Friction, 71, 117, 485, 514. Avoiding friction,

156, 163.

Friction of fluids, 222, PI. 20, 21.
Friction of ice, 501.
Friction of scapements, 150.
Friction of sluices, 240.

INDEX.

599

Friction wheels, 164, PI. 14, 18.

Frigid zone, 436.

Fringes of colours, 366, 367-

Frisi, 192.

Fulling, 143.

Fulton, 271.

Furnaces, 265.

Fusee of a watch, 148, PI. 15, 16.

Fusorius, 188 ; fl. 1450.

Fust, 189.

Gages for air pumps, 261, PI. 22.

Galen, 352.

Galilean telescope, 334, 336, PL 28.

Galileo, 24, 31, 34, 147, 189, 190, 207, 277, 316,

376, 458; b. 1562, d. 1642.
Gallon, 85.

Galvani, 523, 587, 588 ; b. 1737, d. 1798.
Galvanic battery, PL 40.
Galvanic circuit, PL 40.
Galvanic electricity, 521.
Galvanism, 587-
Galvanometer, 540.
Ganges, 223, 224.
Garnerin, 233.
Garnet, PI. 17.
Gases, 291, 4?0.
Gasometer, 263, PL 24.
Gates, 131.
Gay Lussac, 285.
Gellibrand, 583; b. 1597, d. 1636-
Gemini, 401.
Gemma, 393.
Geneva, 545.
Gensfleisch, 188.
Geoffrey, 586.
Geography, 435.
Geology, 563.

Geometry. Instrumental geometry, 71.
Geometry of mechanics, /I.
Georgian planet, 398, 405, 424.
Germination, 567.
Gerstner, 227-

Gesner, 583, 584 ; b. 1516, d. 1565.
Gilbert, 532, 583.
Gin, 157-
Gioja, 583.
Givre, 555.
Glass, 477.
Glass blower, 200.
Glass blowing, 171.
Glass drops, 494.
Glass vibrating, 301.
Glauber's salts, 493.
Glazier's vice, 1?!, PL 18.
Globe, 432.
Globes, 394.

Globules for finding specific gravities, 236.
Globules for microscopes, 330.
Glottis, 312, 313, PL 26.
Going fusee, with an intermediate spring, 148,

PL 16.

Golden number, 429.
Gold leaf, 322.
Gong, 314.
Gorlaeus, 584.

Gothic architecture, 127, 187-
Gothic roof, PL 12.
Grafting, 570.
Graham, 85, 150, 153, 461.
Grain, 95.
Gramme, 96.
Granite, 176, 177-
Grave harmonics, 306, 317.
Gravesande. See S'Gravesande.
Gravitation, 23, 29, 409,411, 458, 471, 508, PL 34.
Gravitation of light, 361.
Gravity, 582.

Gray. Stephen Gray, 510, 585, 586 ; d. 1736.
Grecian year, 427-
Greeks, 74, 181, 453.
Greenwich, 150, 460.
Gregorian calendar, 428.
Gregorian telescope, 335, PL 28.
Gregory, 335, 378, 458. Pope Gregory, 316, 428.
Gridiron pendulum, PL 16.
Grimaldi, 342, 366, 377.

Grinding, 176.

Groups of stars, 392.

Growth, 539.

Guericke, 207, 260, 278, 585 ; b. 1602, d. 1686.

Guglielmini, 2785 b. 1655, d. 1710.

Guido of Arezzo, 316; fl. 1026.

Guitar, 311.

Guldinus, 189.

Gulf stream, 449.

Gun, 268.

Gunnery, 31-

Gunpowder, 103, 268, 277-

Gunter, 189, 583 ; b. 1581, d. 1626.

Gunter's scale, 82.

Gutenberg, 188.

Guyot, 583.

Guy ton, 589.

Gwynn, 256, PI. 23.

Gymnotus electricus, 523.

Hadley, 548, 585; d. 1744.

Hartley's quadrant, 80, 82, 430, Pi. 35.

Hair hygrometer, 554.

Hales, 278; b. 1677, d. 1761.

Haley, 151.

Halifax, 190.

Hall, 379.

Halley, 191, 389, 406, 408, 414, 432, 459, 460, 535,

548, 585; b. 1660, d. 1742.
Halos, 347, PL 29.
Hamilton, 50. Captain T. Hamilton's gage, 243,

PL 22. Sir W. Hamilton, 561, 562.
Hammer, 61, 158.
Hammering, 110, 170.
Hammering brass, 538.
Hammer of the ear, 302.
Hanin, 98.
Harbours, 240.
Hard bodies, 289.
Harding, 404, 462.
Harmonica, 314.
Harmonic curve, 289.
Harmonics, 196, 304.
Harmonic sounds, 296, 298.
Harmony, 306.
Harness, PL 18.
Harp, 311.
Harpsichord, 311.

Harrison, 150, 153, 192, 461 ; b. 16iM, d. 1776.
Harvest moon, 421.
Harvey, 584 ; b. 1578, d. 1657-
Hats, 143.
Hatton, 85.

Hauksbee, 225, 227, 278, 585, 589.
Hautboy, 314.
Hauy, 586.
Hawser, 140.
Hearing, 301, 303.
Hearing trumpet, PI. 25.
Heart, 220.

Heat, 474, 484, 514, 532, 582, 584, 589, 589,586,
PL 39.

Effect of heat on sound, 289.

Effect of heat on vibrations, 297-

Nature of heat, 496.
Heat from electricity, 518.
Heat from mirrors, 330.
Heat of different latitudes, 545.
Heat of mixtures, 496.
Heat producing a draught, 264.
Hecla, 535.

Height of mountains, 206.
Height of tides, 448.
Heights, PI. 38.
Helfqstate, 333.

Hemispherical counterpoise, 203.
Hemp, 140.
Henderson, 390.
Henley, 527, 528, PL 40.
Henry the Sixth, 188.
Heraclitus, 581, 582 ; fl. 506, B.C.
Hercules, 395, 397.
Hermann, 191 ; b. 1678, d. 1733.
Hermes, 315, 452.

Hero, 145, 158, 185, 186, 258, 276 ; fl 130, B.C.
Herodotus, 182.
Hero's cupping instrument, PL 24.

600

INDEX.

Hero's fountain, 258, PI. 23.

Herschel, 334, 335, 337, 342, 357, 382. 389, 390,

391, 392, 393, 397, &99, 403, 404, 405, 406, 407,

410, 421, 423, 424,461, 489, 490, 546, 589, PI. 31,

33,39.

Hessian bellows, 264.
Hevelius, 458.
Hiero, 184, 276.
High pressure engine, 2/2.
H igh water, 442.

Hipparchus, 393, 402, 413, 451, 454, 455.
History of astronomy, 451.
History of hydraulics and pneumatics, 275.
History of mechanics, 180.
History of music, 315.
History of optics, 374.
History of terrestrial physics, 580.
Hoar frost, 555.
HOI1, PI. 23.
HOll's machine, 257.
Hoffmann, 489.
Hogshead, 95.
Hofiow beams, 108.
Hollow masts, 115.
Homogeneous medium, 321.
Hooke, 5, 76, 106, 123, 145, 147, 159, 190, 203,

205, 256, 265, 278, 282, 377, 378, 382, 416, 430,

458, 459, 460, 584, 585, PI. 6 ; b. 1635, d. 1703.
Hooke's counterpoise, 237, PI. 19.
Hooke's joint, 133, PI. 14.
Hoop, 26.
Hope, 569.
Horizon, I'l. 35.
Horizontal moon, 356, PI. 3u.
Horizontal range, 30, PI. 2.
Horizontal refraction, 346.
Horizontal scapement, PI. 16.
Horizontal surface, 197.
Horizontal watch, 150.
Horn, 314.
Hornsby, 392.
Horrox, 432.

Horse, 102. Positions of a horse's legs, 37.
Horses, 167, PL 18.
Hour glasses, 144.
Howard, 425, 564.
Huddart, 140, 141, 382.
Human voice, 312.
Humboldt, 395.
Humidity, 553.
Humming top, 314.
Hunter's screw, 55, 160, 169.
Hurdy-gurdy, 312.
Hutton, 284.
Huygens, 34, 50, 146, 147, 190, 210, 279, 280, 347,

349, 360, 361, 363, 3<>8, 371, 378, 379, 380, 382,

391, 411, 433, 458; b. 1629, d. 1695.
Hydra, 566.

Hydraulic air vessels, 257, PI. 23.
Hydraulic architecture, 235, 237, 238.
Hydraulic bellows, PI. 24.
Hydraulic forces, 241.
Hydraulic machines, 250.
Hydraulic measures, 243.
Hydraulicostatics, 228.
Hydraulic pressure, 46, 228, 279.
Hydraulic ram, 259.
Hydraulics, 195, 196, 210.
Hydrodynamics, 195.
Hydrometer, 236, PL 21.
Hydrometrical fly, 243, PL 22.
Hydrostatic balance, 235, PL 21.
Hydrostatic instruments, 235.
Hydrostatic parodox, 199.
Hydrostatic press, 170.
Hydrostatic pressure, 511.
Hydrostatics, 195, 196, 197, PL 19.
Hygrometer, 55.'i, 587, PL 41.
Hygrometry, 587-
Hypatia, 276.
Hyperbola, 477.
Hyperbolic fringes, PL 30.
Hypotheses of electricity, 507-

Ibn Junis, 14?, 456.
Ice, 348, 442, 546, 582.

348, 353, 3n>

Idioelectrics. See electrics.

Igneous meteors, 564.

Ignis fatuus, 340.

Illumination, 330.

Illumination of the planets, PL 34.

Image, 327, 329, PL 27, 28.

Image on the retina, 351, 355, PL 30.

Impenetrability, 467.

Impenetrability of matter, 21.

Impulse of a fluid, 46.

Impulse of a jet, 229.

Inanimate force, 69.

Inclinations of the planetary orbits, 401.

Inclined plane, 33, 54, PI. 4, 5, 17.

Index, 76.

Index of refraction, 323.

Indian ink, 73.

Indians, 452.

Indivisibles, 28.

Induced electricity, 512.

Induction, 12.

Inelastic bodies, 57, 59.

Inertia, 17, 26, 39, 470.

Inferior tides, 446.

Infinites, 28.

Inflammable bodies, 323.

Ink, 76, 93.

Insects, 575.

Instinct, 352.

Instruments. Musical instruments, 310.

tical instruments, 328.
Insulated stars, 392.
Intensity of electricity, 527.
Intensity of light, 328.
Interception of light, 321.
Interception of sound, 294.
Interference of light, 364, 370.
Intermediate spring, 148.
Intermitting springs, 216.
Inundations of large rivers, 557.
Inverted pump, 257.
Inverted tide, 443.
Invisible girl, 294.
Invisible heat, 489.
Involutes of circles, PL 15.
Ionian school, 181, 453.
Ionic column, PL 12.
Iris, 351, 354.
Iron, 117, 174, 187, 531.
Iron filings, 534, 538, PL 41.
Iron wheelways, 167-
Irvine, 499, 500, 586.
Italian school, 181.

Jack, PL 17. Kitchen jack, 137.

Jacobi, 540.

Jamaica, 547-

Janson, 375, 376.

Jars, 517.

Jeaurat, 381 ; b. 1704, d. 1803.

Jet, 211, 212, PL 20.

Jet with a ball, 226.

Jewelling, 148.

Jew's harp, 314.

Joggles, 129, PL 13.

Joint focus, 327-

Joints, 123, 128.

Joints for beams, PL 13.

Joints of stones, PL 11.

Journal de Physique, 192.

Journals, 191, 192.

Juan, 284; b. 1713, d. 1773.

Judgment of distance, 355.

JUrgen, 187-

Julian period, 429.

Juno, 404, 424.

Jupiter, 404, 422, 424, PL 33.

Jupiter's satellites, 421.

Jurin, 379, 589; b. 1680, d. 1750.

Jussieu, 572, 586; b. 1699, d. 1777.

Ka? stner, 282.
Kant, 391.
Keel, 248.
Keir's lamp, 227.
Kclland, 488, 506.
Kcmpe, 187.

Op-

INDEX.

601

Kempelen, 313.

Kepler, 28, 29, 375, 376, 381, 393, 401, 457, 458,

459; b. 1571, d. 1630.
Keplerian laws, 401, 402, 411, PI. 1.
Key note, 306.
King, 182,489.
Kingdoms of nature, 565.
Kingpost, 130, PI. 12.
King's College Chapel, 188, PL 12.
Kirb roof, 130, PI. 13.
Kircher, 316; b. 1601, d. 1680.
Kirwan, 546, 547, 586.
Kite, 247, PL 22.
Kleist, 586.

Klingenstierna, 379, 380.
Klugel, 38).
Kneading, 179.
Knight, 569, 570.
Knives, 174.
Kramp, 191.

Kratzenstein, 313, PI. 26.
Kunze, PL 39.

Labour, 60, 253, 254.

Labour of a man, 101.

Lacepede, 586.

Lafaille, 189.

Lagrange, 5, 121, 192, 218, 317, 401, 415.

Lahire, 100, 191 ; b. 1640, d. 1718.

Lahire's pump, 254, 266, PL 23.

Lake, 444.

Lalande, 398, 428, 458, PL 33.

Lallamand, 586.

Lambert, 293, 318, 380, 381, 390, 391, 418, 421,

489,586; b. 1728, d. 1777.
Laminating machine, 170.
Lamp, PL 21.
Lamps, 237.
Land, 437.
Land breezes, 550.
Landen, 192 ; b. 1719, d. 1790.
Lane, 528, PL 40.
Langsdorf, 285.
Laplace, 5, 46, 83, 84, 85, 191, 192, 289, 318, 345,

361, 382, 402, 404, 414, 415, 429, 432, 444, 450,

451, 452, 486, 501, 586, 589.
Larynx, PL 26.
Latent heat, 501.
Lateral adhesion, 107, 480.
Lateral friction of fluids, 225.
Lathe, 174.
Latitude, 425, 430.
Laurie, PL 24.

Lavoisier, 486, 501, 586; b. 1743, d. 1?94.
Laws of gravitation, 409.
Laws of refraction, 324.
Leaden pipes, 242.
Lean, 269.
Leaves, 570.
Lee, 187.
Lee way, 248.
Legs, PL 9.

Leibnitz, 60, 191,280, 376; b. 1646, d. 1716.
Lens, 326.
Lenses, 331, 374, PL 27- Grinding lenses, 177-

Leslie, 291,488, 498, 554, 588.

Leslie's thermometer, 498, PL 34.

Letherland, 149.

Letterpress, 94.

Leucippus, 581.

Leupold, 191 ; d. 1727-

Level. Spirit level, 237.

Levelling, 81.

Levels, 438.

Lever, 50, PL 3, 4.

Levers, 133,156.

Levigating, 179.

Levity, 207-

Lewis, 161, PL 17.

Lexell, 40a

Leyden phial, 513.

Libra, 3&S, 401.

Libration of the moon, 414, 419.

Life, 567-

Life of plants, 570.

Lifting pump, 255, PL 23.

Light, 320, 359, 389, 502, 503, 582, PL 39.

Light from electricity, 518.

Light from friction, 340.

Light-house, 122, PL 11.

Lightning, 557, 581, 586.

Light of a candle, 344.

Light of the heavenly bodies, 421.

Light of spirits, 344.

Light of the stars, 389.

Lincoln cathedral, 188.

Lines, 77-

Lines or hatches, PL 6.

Linne, 571, 5?2, 573, 575, 586.

Linnean system, 571, 586.

Lintearia, PL 19.

Lion, 395.

Liquefaction, 493.

Liquid, 196.

Liquid adhering to a solid, 478.

Liquidity, 475.

Liquids, 425, 470, PL 39.

Loaded chain, PL 11.

Loaded cylinder, PL 3.

Loaded waggon, PL 3.

Lock filled from a reservoir, 214.

Locomotive engine, 272- -275.

Log, 86, PL 22. Hydraulic log, 243.

Logarithmic circle, PL 7.

Logarithmic curve, PL 10.

Logarithms, 82, 206, 458.

Lohmeier, 285.

London, 356, 547.

Longitude, 192, 425, 431, 461.

Longitudinal sounds, 297.

Looming, 346.

Louis XV., 460.

Lowitz, PL 29.

Low water, 442.

Lucernal microscope, 333.

Lucid disc micrometer, 337-

Luc. See Deluc.

Lucretius, 12, 44, 183, 389, 458.

Luminous bodies, 320.

Lunar globe, 423.

Lunar motions, 413, PL 34.

Lunar observations, 431.

Lunar rainbow, 347.

Lunar volcanos, 565.

Lute, 312.

Lycopodium, 478.

Lyonnet, 466.

Lyra, 395.

Lyre, 311, 315.

Machin, 191.

Machine for measuring strength, 116.

Machinery, 132, PL 14.

Machinery of fluids, 241.

Machines, 68.

Maclaurin, 50, 62, 191, 247, 280, 281 ; b. 1698,
d. 1746.

Macrobius, 442.

Madeira, 547.

Magdeburg hemispheres, 207, 483.

Magic lantern, 333, 375.
I Magnet, 534, 535, 581.
I Magnetical attractions and repulsions, 533.
1 Magnetical curves, PL 41.

Magnetical effects, PI. 41.

Magnetical paste, 538.

Magnetical substances, 532.

Magnet in a globe, 535.

Magnetism, 531, 586.

Magnetism by induction, 535.

Magneto-electric machine, 541.

Magnifier. Double magnifier, 336.

Magnifying powers, 330.

Magnifying powers of telescopes, 334, 336.

Magnitude of the planets, PL 34.

Magnitude of the stars, 389.

Mair, 583.

Mairan, 399.

Maire, 335.

Malebranche, 380.

Malta, 450.

Malus, 371.

Mammalia, 574.

Manchester, 18?.

Mandoline, 312.

G02

INDEX.

Mangles, 170.

Manilius, 463.

Mansard roof, 130, PL 13.

Manufactures, 186.

Map of the world, PI. 42.

Marble, 176, 177-

Marcellus, 45, 184, 185.

Marigni, 312.

Marine engines, 271, 272.

Marine octant, PI. 35.

Mariotte, 278, 284, 347, 348, 353, 379 ; d. 1684.

Marquois's scales, 78, PI. 6.

Mars, 403, 423, PI. 32.

Marum. See Van Marum.

Maskelyne, 392, 440, PI. 28.

Masses, 38.

Masts, 115.

Mathematici veteres, 182, 183, 184, 185.

Matrix, 94.

Matter, 464. Impenetrability of matter, 21 .

Matthesius, 278.

Maupertuis, 16, 393.

Maurolycus, 376.

Mayer, 461, 585, 586 ; T. Mayer, b. 1723, d. 1762.

Mazeas, 381.

M'Culloch, PI. 41.

Measurement of the earth, 454.

Measurements of degrees, 436.

Measure of force, 60.

Measures of heat, 496.

Measuring, 71.

Measuring instruments, 86.

Mechain, 84.

Mechanical force, 60.

Mechanical power, 245.

Mechanics. History of mechanics, 180.

Mediterranean, 449, 552.

Medusa's head, 394.

Meibomius, 316.

Melloni, 504, 505, 506.

Melody, 306.

Membranes. Vibrations of membranes, 297.

Meniscus lens, 326.

Menkar, 395.

Mercurial column, 204.

Mercurial thermometer, 49?.

Mercury, the metal, 208, 209, 477, 497, PL 39.

Pressure of mercury, 201.
Mercury, the planet, 403, 422.
Meridian, 83, 426, PL 35.
Mersenne, 316; b.1588, d.1648.
Messenger, 158.
Messier, PL 31.
Metacentre, 202.
Metallic surface, 553.
Metals, 322, 524, 525.
Meteorology, 544, 586.
Meteors, 564, 565.
Meto, 428, 454.
Metre, 84.
Mexicans, 74.
Mezzotinto, 92.
Michael III., 456.
Michell, 391, 392, 440.
Micrometer, 337, PL 28.
Micrometrical scale, PL 7.
Microscopes, PL 28. Double microscopes, 334.

Simple microscopes, 330. Solar microscopes,

332.

Middle ages, 186.
Milky way, 391, 394, PL 31.
Mills, 177, 178, 244, PL 18.
Mineralogy, 567.
Minerals, 565.
Miniatures, 73.
Mining, 175.
Minor scale, 308.
Mirage, 346.
Mirbel, 569.
Mirror, 325, PL 27.
Mirrors, 331, PL 28.
Mists, 555.
Mixed gases, 470.
Mixed plates, 369, PL 30.
Mixed pump, 254.
Mixture, 236.

Mixture of colours, 344, 345.
Modelling, 87-

Modulus of elasticity, 106, 288.
Moeris, 181.

Moon's age, 429.
Moon's phases, PL 34.

Moisture, 552.
Moivre. See Demoivre.
Momentum, 41, 45, 169, PL 2.
Monge, 589.
Monnier, 392.

Monsoons, 549, 585, PL 42, 43.
Montaigne, 585.
Montbret, 93.

Montgolfier, 259, 285, PL 23.
Montpelier, 547-

Moon, 356, 405, 406, 419, 423, PL 33.
Moon as causing tides, 442.
Moons, 405.
I's age

I's pha ,

Moon's surface, PL 34.

Mortar, 123.

Mortar mill, 1/7.

Mortise, 129.

Mosaic work, 74.

Moses, 74.

Motion, 13, PL 1. Composition of motion, 18.

Confined motion, 32. Measure of motion, 60.

Perpetual motion, 70. Quantity of motion,

40. Resolution of motion, 19.
Motion of light, 321.
Motions of the stars, 411.
Mountainous countries, 556.
Mountains, 438, PL 38.
Mouths of rivers, 564.
Mudge, 150, 151, PL 16.
Multiplier of electricity, 526, PL 40.
Multiplying glass, 326, PL 27.
Mural quadrant, PL 35.
Murray, Lord G., 77-
Muscles, 98, 578.
Music. History of music, 315.
Musical characters, 93.
Musical chord, PL 25.
Musical instruments, 319.
Musical pen, PL 6.
Musical sounds, 295.
Musschenbroek, 117, 191, 498, 513, 585, 586,

589; b.1692, d. 1761.
Myopic sight, 354.

Nail, 119.

Nairne, 559, PL 40.

Nairne's machine, 534.

Napier, 189, 458; b. 1555, d. 1622.

Nativity of Christ, 427.

Natural history, 565, 582, 586.

Natural hygrometer, 554.

Natural orders of plants, 572.

Natural zero, 499, 500.

Nature of light, 359.

Nautical almanac, 461.

Neap tide, 442.

Nebula, 391, 392, 393.

Nebula in Orion, 391, PL 31.

Nebulosity, 393.

Needle, 583.

Negative electricity, 509.

Neptunian theory, 563.

Nerves, 578.

Nettis, PL 29.

Newcomen, 266, 279, PL 24.

Newton, 5, 19, 22, 28, 29, 34, 36, 43, 50, 63, 190,
218, 279, 280, 316, 317, 323, 342, 343, 344, 359,
360, 362, 363, 366, 368, 369 370, 371, 377, 378,
379, 380, 381, 382, 388, 402, 409, 410, 411, 415,
430, 435, 440, 442, 449, 453, 458, 459, 460, 461,
466, 467, 468, 471, 472, 490, 502, 585; b. 1642,
d. 1727.

Newtonian reflector, 335.

Newtonian rules of philosophy, 12.

Newtonian telescope, PL 28.

Nicetas, 453, 456.

Nicholson, 85, 151, 154, 174, 527, 588-

Nicholson's circle, 82.

Nickel, 532.

Night, 417.

Nile, 557.

Nilometer, 454.

Nitocris, 182.

Nitre, 486, 501.

Nobili, 504, 505, 540.

Nodes, 401, PL 34.

Nodes of the planets, PL 32.

INDEX.

603

Nollet, 585 ; b. 1700, d. 1770.

Nonconductors, 513.

Noria, 250, PI. 22.

North, 398.

Northern crown, 395.

Northern hemisphere warmer, 549.

North pole, 533.

Norwood, 460.

Notes of music, 309.

Nucleus of a comet, 407.

Numa, 580.

Number of the stars, 389.

Nut, 55, PI. 5.

Nutation, 402.

Nutation of the earth's axis, 402, 412.

Nutrition of animals, 577.

Oblique float boards, 246, 247.

Oblique forces, PI. 3.

Oblique impulse of fluids, 230.

Oblique reflection, 342.

Obliquity of the ecliptic, 412, 454.

Observatory, 114.

Observatory of Greenwich, 150, 4GO.

Octave, 307.

Octant, PI. 35.

Ocular spectra, 357, PI. 30.

Oersted, 539.

Oil mill, 170.

Oil paintings, 73.

Oil spreading on water, 479.

Oily substances, 163.

Olbers, 404, 462.

Opposition, 418.

Opposition of forces, PI. 3.

Optical centre, 327, 352.

Optical instruments, 328.

Optic nerve, 351.

Optics, 195, 196, 320.

Optometer, 354.

Orbit of the sun, 398.

Orbits of comets, 414.

Orders of architecture, 12?.

Orders of plants, 572.

Organ, 314, 316.

Organ pipes, 301, 313, 314, PI. 26.

Orion, 395.

Orreries, 433.

Orthographical projection, 89, PL 8.

Oscillations of fluids, 217.

Osiris, 315.

Overflowing lamp, PI. 21.

Overshot wheel, 244, PI. 22.

Ovid, 181.

Oxid, 495.

Oxygen, 340.

Oxygen gas, 486.

Painting, 356.

Palladio, PI. 11 ; b. 1508, d. 1580.

Pallas, 404, 424.

Panorama, 356.

Pantheon, 127,526, PI. 12.

Pantograph, 79, PI. 6.

Paper, 75, 144, 186.

Papin, 264, 290-

Pappus, J85, 186; fl. 383.

Papyrus, 75.

Parabola, 31, 293, PI. 2.

Parabolas, PI. 10.

Parabolic jet, 217.

Parabolic orbit, 414.

Parachutes, 233.

Paradox. Hydrostatic paradox, 199.

Parallax, 430.

Parallax of the sun, 431, 432.

Parallel motion, PI. 14.

Parallelogram, 18.

Parallel rulers, 78, PI. 14.

Pardies, J79, 380 ; b. 1636, d. 1673.

Parent, 191, 248 ; b. 1666, d. 1716.

Parent's mill, 248, 253.

Parhelia, 347, PI- 29.

Paris, 356.

Parisian academy, 191, 192, 279, 280.

Parker, 131.

Partial electricity, 509.

Partial reflection, 362.

Pascal, 504 ; b. 1623, d. 1662.

Passive strength, 71, 104, PI. 11.

Paternoster work, 256.

Path of the centre of gravity, PI. 3.

Path of the sun, 417.

Paths of the planets, PI. 34.

Pear gage, 262, PI. 24.

Pearson, 433.

Pedestrian, 100.

Pegasus ,395, 588.

Pemberton, 191,459.

Pen, 72.

Pencil, 72, 73.

Pencil of light, 320, PL 26.

Pendulum, 34, 35, 83, 147, PL 2, 5, Circular

pendulum, 36.
Pendulums, 417, 443.
Penetration, 111, 120, 172.
Pennant, 586 ; b. 1726, d. 1798.
Pens, 75.

Pens for lines, 77.
Penumbra, 419.

Perception of external objects, 351.
Percussion, 171.
Perforation of ajar, 517.
Periodical winds, 548.
Periods of the planets, 402, PL 32.
Periscopic spectacles, 332.
Permeability of matter, 468.
Perpetual motion, 70, PL 6.
Perrault, 157, 191 ; b. 1613, d. 1688.
Perrault's ropes, 164.
Perseus, 394.
Persians, 428, 456.
Perspective, 71, 88, PL 7, 8.
Perturbations, 412.
Petit, 488.
Pfaff, 588.

Phantasmagoria, 333, PI. 28.
Phases of planets, 418.
Phases of the moon, 419.
Phenicians, 74.

Pherecydes, 181; b. 600, d. 515, B.C.
Philip III., 460.
Philo, 183, 185, 276.
Philolaus, 453.
Philosophizing, 12.
Phosphorus, 486.
Phosphorus of Bologna, 341.
Photometers, 329, PI. 27.
Physical astronomy, 387.
Physical optics, 340.
Physics, 387.
Physiology, 5?a
Pianoforte, 311.
Piazzi, 404, 462.
Picard, 34, 435, 460 ; d. 1682.
Pictet, 289, 486, 487> 489, 552, 587-
Piers, 126, 240.
Pile engine, 137, 173, PL 1*
Pile of Volta, 522, 588.
Pin, 119.

Pinion, 136, PL 15.
Pipe. Effect of a short pipe, 212, 213. Vertical

pipe, 215.

Pipes, 222, 277- Musical pipes, 296.
Pipes of lead, 242.
Pipes of pumps, 256.
Pisces, 401.
Pise, 123.
Piston, PL 28.
Pistons, 254.
Pitot, 243.

Pittacus, 181 ; b. 652, d. 570, B.C.
Pixii, 541.

Plain astronomy, 387.
Plane mirror, 325.
Planetarium, 433.
Planetary worlds, 422.
Planets, 400, PL 32.
Planispheres, 433.
Planks, PL 10.
Planoconcave lens, 326.
Planoconvex lens, 326.
Plant, 568.
Plaster of Paris, 87-
Plate machine, 525, PL 40.

604 INDEX.

Platina, 467-

Plato, 183, 581, 682 ; b. 429, d. 348, B.C.

Pleiades, 395.

Plempius, 489.

Pliny, 182, 420, 442, 464, 582 ; b. 24, d. 79.

Plough, PI. 18.

Plungers, 253, PL 23.

Plurality of worlds, 422.

Plutarch, 183, 184, 458, 582.

Pneumatic equilibrium, 204, PI. 19.

Peneumatic machines, 259.

Pneumatics, 275, 276.

Pneumatostatics, 196, PI. 19.

Poetry, 422.

Polar circles, 436.

Polarity, 533.

Polarization of heat, 505.

Polarization of light, 371, 3?2, 505.

Poleni, 280; b. 1683, d. 1761.

Poles, 436.

Pole star, 394.

Polished surface, 322.

Polishing, 176.

Polycrates, 181.

Polygon, 20.

Polygraph, 76.

Pores, 467.

Porosity, 360.

Porterfield, 354, 379.

Porters, 102, 162, PI. 17.

Positive electricity, 509.

Pottery, 171.

Pound, 96.

Powder mill, 177.

Powder proof, 103.

Powell, 505, 506.

Power. Mechanical power, 245.

Practical astronomy, 425.

Precession of the equinoxes, 402, 412.

Preponderance, 66, PI. 5, 6.

Presbyopic sight, 355.

Press. Bramah's press, 199, PI. 23.

Presses, 169, 170.

Pressure, 45.

Pressure of a fluid, 198.

Pressure of earth, 124.

Pressure of fluids, PI. 19.

Pressure of the atmosphere, 207.

Prevost, 489, 532, 545, 546, 547, 589. B. Prevost,

553.

Priestley, 381, 586; b. 1733, d. 1804.
Primary mountains, 439.
Printing, 71, 91, 93, 188.
Printing from stones, 93.
Printing press, 169.
Prism, 324, 326, 343, PL 26, 27-
Prismatic spectrum, PL 29.
Proclus, 186.
Procyon, 395.
Progressive motion, 100.
Projectiles, 17, 26, 29, 217, PI- 2.
Projection of a sphere, 90, PL 8.
Projection of light, 361.
Prony, 284.
Proofs, 494.

Propagation of light, 359.
Proper motions of the stars, 392.
Properties of matter, 464, 509.
Prop or shore, 55, PL 5.
Proportional compasses, 79, PL 6.
Props of reservoirs, 238.
Prosperin, 408.
Protagorides, 582.
Ptolemaic system, PL 38.
Ptolemy, 75, 3?5, 393, 401, 420, 452, 453, 454,

455, 456 ; fl. 160.
Ptolemy Philadelphus, 454.
Ptolemy Soter, 454.
Pullies, 52, 159, PL 4.
Pulse, 220.
Pump, 253.
Pumping, 102.
Pumps, PL 23.
Pupil, 354.
Pyramids, 454.
Pyrometers, 496.
Pythagoras, 181, 182, 315, 316, 453, 458, 581;

b. 568, d. 497, B.C.
Pythagorian system, PL 38.

Quadrant, 80.

Quadrant electrometer, 527.

Quadrants, 429, PL 35.

Quarter, 95.

Quays, 240.

Queen post, 130, PL 12.

Quiescent space, 15.

Radiation of heat, 488.

Rafter, 113.

Rafters in equilibrium, PL 11.

Railroads, 272.

Rain, 556, 557-

Rainbows, 346, 369, PL 29, 30.

Raising weights, 156.

Ramelli, 189, 256, PL 23-

Rammelsberg, 179.

Ramsden, 80, 86, 97, 336, 338, 381, 461, PI. 7, 8,
28 ; b. 1730, d. 1800.

Range of a projectile, 30, 217.

Rarefaction, 484.

Ravenna, 450.

Ray, 585; b. 1628, d. 1705.

Ray of light, 320, PL 26.

Reaction, 42.

Read, 558.

Reaumur, 485, 498, 585; b. 1683, d. 175?.

Reciprocal action, 40, 42, 43.

Reciprocal force, 470, PL 2.

Recorde, 375.

Rectification of motion, 134, PL 14.

Rectilinear motion, PL 1.

Redern, 381.

Red light, 365.

Red Sea, 449.

References, 193.

Reflecting surface, 325.

Reflecting telescopes, 334, 337, 378.

Reflection, 62, 342, 361, 374, PL 5, 26.

Reflection of a stone, 233.

Reflection of cold, 489.

Reflection of light, 321.

Reflection of sound, 293, PL 25.

Reflection of waves, 219, 293.

Refraction, 321,322, 361, 375, 430, 433, PL 26,

29.

Refraction of crystals, 348.
Refraction of the atmosphere, 345.
Refractive densities, 323, 329, 375, PL 27.
Refrangibility of heat, 490.
Refrigeration, 545.
Regaforgan pipe, 314, PL 26.
Regulation of force, 69.
Regulation of hydraulic forces, 241.
Regulator, PL 2.
Regulus, 395.
Reich, 440.

Relative motion, PL 1.
Remote tide, 444.
Removing earth, 168.
Removing weights, 156, 161.
Renaud, 279; b. 1652, d. 1719.
Reproduction, 539, 566.
Republican calendar, 428.
Repulsion, 58, 468, 502, 5<)3, PL 39.
Repulsions of floating bodies, 478.
Repulsions of the electric fluid, 508.
Reservoirs, 238.
Resilience, 110, 114, 482.
Resinous electricity, 517.
Resistance of fluids, 222, 230, PL 21.
Resistance of the air, 30,31,154,232, 260, 261,

284.

Resistance to the tides, 444.
Resolution of motion, 19.
Respiration, 575.
Retardation, 22.
Retina, 351.

Retrograde motions, 418.
Returning stroke, 557-
Return of light, 323.
Revolutions of cords, 299.
Revolving doubler, PL 40,
Revolving pendulums, 36, PL 2.
Rheita, 334, 376, PL 28.
Rhinland foot, 85.
Rhythm, 306.
Ribaucourt, 76.

INDEX.

00,5

Riccati, 300, 317-

Richmann, 278, 580, 585 ; d. 1753.

Rifle barrels, 31, 268.

Right ascension, 425, 430.

Ringing, 101.

Ringing a magnet, 538.

Ring of Saturn, 404, 407, 458.

Ritter, 342, 382, 490, 588.

Rise and fall of the tides, 446, 447.

Rising and setting, 433.

Rivers, 222, 238, 438, 563. Tides of rivers, 446.

Road. Circular road, 37.

Robertson, 334.

Roberval, 458-

Robins, 31, 281, 284, 285 ; b. 1707, d. 1751.

Robison, 22, 32, 101, 112, 192, 223, 253, 264, 285,

291, 469, 481, 507. 538, 586 ; b. 1739, d. 1804.
Rochon, 93, 490.
Rock salt, 504.

Rods, 86, PI. 9, 14. Sounds of rods, 300.
Roemer, 341, 378; b. 1644, d. 1710.
Roget, 542.
Roller, PL 17-
Roller pump, 256, PI. 23.
Rollers, 163.
Rolling, PI. 2.
Romans, 91, 186.
Roman year, 427-
Rome, 547.
Romieu, 317-
Romme, 284.
Roof, 55, 130, PI. 5.
Roofs, PI. 13.
Rope making, 139.
Rope pump, 250, PI. 21.
Ropes, 140.
Rosetta, 564
Rosnier, 285.
Rotation, 32, 61, 65.
Rotation of billiard balls, PI. 5.
Rotation of the earth, 417, 548.
Rotation of the moon, 406, 414.
Rotation of the planets, 402.
Rotation of the sun, 398.
Rotatory motion, PI. 5.
Rotatory power, 63, PI. 2.
Rotatory pump, 253.
Rowing, 101.
Roy, 8<i.
Royal Institution, 131, 192, 589. Objects of the

Royal Institution, 1.
Royal Society, 5, 97, 161, 190, 191, 585.
Rudder, 249.
Rudders, 114.
Rudolph, 457.
Rulers, 77.

Rules of philosophy, 12.
Ruling machine, 91.
Rumford. Count Rumford, 268, 284, 285, 329,

485, 487, 500, 588.
Running, 100, PI. 9.
Rupert. Prince Rupert, 256. Prince Rupert's

drops, 494.
Russell, 423.
Rutherford, 544.
Rutherford's thermometer, PI. 41.

Sagittarius, 401.

Sail, PI. 22.

Saint Paul's cathedral, 127.

Saint Pierre, 460, 461.

Sanctorius, 147,189,498; b. 1561, d. 1636.

Sap, 569.

Saracens, 145, 187, 188.

Saros, 452.

Satellites, 405, 406, PI. 33.

Saturn, 404, 424, PI. 33.

Saturn's ring, 404, 407, 458.

Saussure, 489, 552, 554, 587.

Sauveur, 317-

Savery, 265, 279, PL 24.

Saw, PL 4.

Sawing? 1?5.

Saxton, 541.

Scale of heat, 496.

Scale of musical notes, 307-

Scaliger, 429.

Scapement, 148.

Scapements, PL 16.

Scarfing, 129, PL 13.

Schaeffer, 189.

Scheele, 586; b. 1742, d. 1786.

Scheiner, 376 ; b. 1573, d. 1650.

Schemnitz, 257, 258, PI. 23.

Schott, 278.

Schroeter, 403, PL 33.

Scorpio, 395, 401.

Screen of glass, 489.

Screw, 55, 160, PI. 5.

Screw of Archimedes, 251, 252, PL 22.

Screws, 160.

Sculling, 101.

Sculpture, 71.

Sea, 437, 443, 444, 546, 563.

Sea breezes, 550.

Seamanship, 248.

Seasons, 417, PL 34.

Secants, 82.

Secondary mountains, 439.

Section of a canal, PL 21.

Sector, 79, PL 7.

Seebeck, 371, 490.

Segner, 66, 589.

Seguin, 586.

Self registering thermometers, 545.

Semaphore, 77-

Semimaterial existences, 468.

Semiramis, 182.

Semitone, 307.

Semivowels, 313.

Seneca, 182, 374, 442 ; b. 8, d. 65.

Sensation, 567.

Sensation of colours, 345.

Sensation of light, 320.

Senses, 577.

Sensibility of the retina, 352.

Sensible effects of electricity, 519.

Sensible effects of the celestial motions, 415.

Serein, 555.

Series of eclipses, 420.

Series of rods, PI. 14.

Serpent, 314.

Serpentarius, 393, 395.

Serpentes, 575.

Servetus, 584.

Serviere, PL 23.

S'Gravesande, 191 ; b. 1688, d. 1742.

Shadow, 367, PI. 30.

Shaw, 566.

Sheffield, 187.

Shehallion, 440.

Ship, 114, 248, 249, PI. 22.

Ships, 231.

Ship's sails, 281.

Ship's way, PL 22.

Shooting stars, 564.

Shore, PL 5.

Short, 461.

Shot, 268.

Shower bellows, 263, PL 24.

Showers of stones, 562.

Shroud, 140.

Sidereal day, 426.

Signs of the ecliptic, 401.

Signs of the zodiac, 451.

Silk, 142.

Silkworm's thread, 104.

Simonides, 315; b. 579, d. 469, B.C.

Simple sounds, 295.

Simpson, 191, 379, 545, 547 ; b. 1711, d. 1761.

Sine of an angle, 81.

Single vision, 355.

Siphon, 215.

Siphon of Hero, 145.

Siphons, 241.

Sirius, 391, 392, 395, 397-

Sisson, 461.

Six, 547, 555.

Six's thermometer, 544, PI. 41.

Size, 73, 144.

Sky, 356, PI. 30.

Slider pump, 256, PI. 23.

Sliding rule, PL 7.

Sling, 26, 17a

Slitting mill, 174, PL 18.

Sloughing, 570.

Sluice, PL 21.

Sluices, 239.

Smeaton, 53, 60, 64, 122, 123, 129, 159, KB, 240,

006

INDEX.

245, 247, 260, 262, 278, 282, PI. 5, 11 ; b. 1724,

d. 1792.

Smeaton's blocks, PI. 4.
Smith, 335, 3/9 ; d. l/W.
Smith's microscope, 335, PI. 28.
Smoke, 356.
Smoke jack, 247-
Smoky chimnies, 265.
Snellius, 376 ; b. 1591, d. 1626.
Snow, 348, 556, PI. 29.
Society for the encouragement of arts, 192.
Socrates, 181.
Softness, 482.

Solar and culinary heat, 488.
Solar atmosphere, PI. 31.
Solar day, 426.

Solar microscope, 332, PI. 28.
Solar phosphori, 341.
Solar system, 397, PL 32.
Solar tides, 447-
Solidity, 480.
Solids, 470.

Solution of iron filings, 538.
Sorge, 317.
Sosigenes, 455.
Sothic period, 452.
Sound, 287, 503.
Sounding board, 311.
Sounds of rods, 317.
Sources of heat, 484.
Sources of light, 340.
Sources of motion, 68, 69, 101.
Sources of sound, 295.
South, 398.
South America, 74.
South pole, 533.
Space, 15, 388.

Spallanzani, 586 ; b. 1?29, d. 1799.
Spark, 516, PI. 40.
Speaking trumpet, 293, 294, PI. 25.
Specific gravities, 236.
Specific gravities of gases, 291.
Spectrum, 343, PI. 29.
Sphere, 90.

Sphere charged with electricity, 510.
Spheres, 410.
Spheres connected, 510.
Spheroid, 442.
Spica Virginis, 395.
Spider's thread, 104.
Spider's web, 109.
Spinet, 311.
Spinning, 139.
Spinning wheel, 187-
Spiral pipes, 251.
Spiral pump, 252, PI. 22.
Spirit, 544.

Spirit level, 81,237, PL 21.
Spirit of wine, 291,475.
Spirit thermometer, 497.
Spiritual substances, 468.
Sponge, 479.

Spots of the sun, 398, PL 31.
Spring, PL 2, 10.
Spring of a coach, 114.
Springs, 137, 166.
Springs of water, 216, PL 20.
Spring steelyard, 98, PI. 9.
Spring tides, 442, 447-
Spur wheel, 136, PL 15.
Squares, 78.

Stability of a balance, PL 8.
Stability of a wedge, 119.
Stability of equilibrium, 47, 198, PI. 3.
Stability of floating bodies, 202, PL 19.
Stability of fluids, PL 19.
Stability of ships, 249.
Stacada, 313.
Stadium, 454.

Stahl, 586 ; b. 1660, d. 1734.
Stamping, 172.
Standard measures, 82.
Standard weights, 95.
Stanhope, 51, 252,507, 512.
Star Lyra, PI. 31.
Stars, 356, 387, PI. 36, 37-
Statical baroscopes, PL 19.
Statics, 71, 95.
Statics of fluids, 235.
Stationary planets, PL 34.

Statuary's compass, PL 7-

Steam, 205, 474.

Steam boat, PI. 29.

Steam engine, 37, 103, 265, 282, 283, PI. -24.

Steam vessels, 271.

Steel, 174,531.

Steelyards, 96, 97, PL 8, 9.

Steelyard with a crane, 161.

Stencilling, 72.

Stereographical projection, 90, PL 8.

Stereotype printing, 94.

Stevin, 286.

Stevinus, 189 ; d. 1633.

Stick broken by a blow, 65.

Stiffness, 108, 482.

Stile, 75.

Stirrup, 302.

Stockings, 187.

Stodart, 174.

Stone, 116.

Stone cutting, 176, 177.

Stones fallen, 564.

Stones joined, PI. 11.

Stopcocks, 242, PI. 21.

Strabo, 442.

Strain, 130.

Strand, 140.

Strap, 129.

Straps for beams, PL 13.

Straps for wheels, PL 15.

Stray Park engine, 269.

Stream, 245.

Stream of a fluid, PL 20.

Stream of air, PL 21.

Stream of electricity, 517.

Streams of air from electricity, 512.

Strength, 110, 111, 482.

Strength of a column, PI. 10.

Strength of different substances, 116.

Strength of flood gates, 239.

Strength of joints, PI. 13.

Strength of materials, PI. 11.

Strength of muscles, 98.

Strength of ropes, 141.

Striking a magnet, 538.

Striking part, 155.

String of baskets, 168.

Stripes of colours, 365, PL 30.

Strongest forms, 114, 115, PL 10.

Sturm, 291.

Subdominant, 307.

Subterraneous fires, 560.

Sucking and forcing pump, 255.

Sucking pump, 254, PL 23.

Suction, 207.

Sugar mill, 170, PL 18.

Sulfate of soda, 493.

Sulfuric acid, 525.

Sulfurets, 524, 525.

Summer, 417, 547.

Sun, 356, 397.

Sun and planet wheel, 137, 267-

Sun's motion, 411.

Sun's parallax, 431.

Sun's path, PI. 34.

Sun's rays, 325, 496, 545.

Sun's spots, PL 31.

Superficial cohesion, 475.

Superior tides, 446.

Supernumerary rainbows, 369.

Support, PL 3.

Supports for clocks, 155.

Surface of a fluid, 197-

Surface of a liquid, 476.

Surface of the sea, 435.

Surfaces of fluids, PI. 39.

Surging the messenger, 158.

Suspension, 32.

Suspension of a weight, PL 3.

Swan, 394.

Swig, 53, PL 4.

Swiftest descent, 35.

Symington, 246, 271, PI. 24.

Sympathetic sounds, 301.

Sympathy of clocks, 155. ,

Synchronous tide, 443.

Synthetical order, 7-

Syrinx, 341.

System of Ptolemy, 455.

Systems of the world, PI. 38.

INDEX.

007

Tackle, PI. 4, 17-

Tails of comets, 407, PI. 33.

Tambourine, 313.

Tangent, PL 1.

Tangents, 82.

Tarquin, 181.

Tartalea, 189; d. 1557-

Tartini, 317.

Taurus, 401.

Taylor, 317, 379, 589.

Teeth of wheels, 135, PL 15.

Telegraph, 76, PL 6.

Telescope, 375.

Telescopes, 334- -338, PL 28.

Temper, 110.

Temperament, 309, PI. 25.

Temperate zones, 436.

Temperature, 544.

Temperature of running water, 227.

Tempering of metals, 494.

Temper of iron, 480.

Tenacity, 482.

Tenerifle, 547.

Tenon, 129.

Tertiary mountains, 439.

Terpander, 315.

Terrella, or magnet in a globe, 535.

Terrestrial magnetism, 534.

Terrestrial refraction, 346.

Teylerian machine, 525.

Thales, 181, 182,453, 581 ; b. 636, d. 546, B.C.

Thames tunnel, 271.

Thawing, 546.

Theodolite, 80.

Theon, 463.

Theophrastus, 582; b. 373, d. 288, B.C.

Theories of light, 359.

Theory of electricity, 507-

Theory of optics, 320.

Thermometers, 497. 544, 584.

Thermomultiplier, 504.

Thermoscope, PL 39.

Theuth, 181.

Thick plates, 369, PL 30.

Thin plates, 368, PL 30.

Thoth, 181.

Threshing machines, 178.

Threshing mill, PI. 18.

Throwing a stone, 173.

Throwing wheels, 250, PL 22.

Thunder, 557.

Thunderstorm, 558, 559.

Thyreoid cartilage, 312.

Tibiae, 315.

Tide machine, 258.

Tides, 441, PL 38.

Tides of the Atlantic, 444.

Tie beam, 130.

Time, 16, 144, 426.

Timekeepers, 144, 461.

Timocharis, 402, 454.

Tint, PL 6.

Ton, 95.

Topham, 99.

Torpedo, 523.

Torre del Greco, 562.

Torricelli, 207, 277, 584 ; b. 1608, d. 1647-

Torricellian vacuum, 260, 2/7.

Torrid zone, 436.

Torsion, 105, 108.

Total reflection, 324, 362.

Tottering equilibrium, PL 3.

Toughness, 110, 482.

Tourmalin, 371, 372, 520.

Tournefort, 685 ; b. 1656, d. 1708.

Trachea, 312.

Tracheae of plants, 568.

Trade winds, 548, 585, PL 42, 43.

Transit circle, PL 35.

Transit instruments, 429, PL 35.

Transverse strain, 130.

Trevithick, 257, 267, 272.

Triad, 307.

Triangle, representing forces, PL 3.

Triangular compasses, 78, PL 6.

TrichuVus, 566.

Triple stars, 393.

Trithemius, 145.

Trituration, 177.

1, 18, PL 1.

Trochoid,

Trombone, 314.

Tropical year, 427.

Tropics, 436.

Troughton, PL 7, 8.

Trumpet, 314.

Trumpet Marigni, 312, PL 25.

Tubes, 108.

Tun, 95.

Tuning fork, 314, 503.

Tuscan column, PL 12.

Twilight, 418, 433, PL 34.

Twinkling, 389.

Twins, 395.

Twisted ropes, 53.

Twisting, 104, 108, 139.

Tycho Brahe, 457; b. 1546, d.1601.

Tychonic system, PL 38.

Tympanum, 302.

Type metal, 94.

Ubaldi, 189.

Ulloa, 284, 406 ; b. 1716, d. 1795.

Ulugh Beigh, 456.

Umbrella, 294.

Undershot wheel, 245, PL 22.

Undulations of light, 365.

Undulatory theory of light, 370.

Unequal balance, PL 8.

Union, 480.

Union of flexible fibres, 138.

Union of lights, 364.

Unit of engine power, 103.

Uranus, 405.

Uvea, 351.

Valerius, 189.

Valves, 242, PL 21.

Valves of canals, 240.

Vandelli, 284.

Van Eyck. See Eyck.

Van Marum, 525.

Vapour, 551, 555.

Vapours, 205, 558.

Vapours negatively electrical, 520.

Variation chart, 585, PL 41, 42, 43.

Variation in London and in the West Indies
536.

Variation of the compass, 536, 583, 585, PL 41.

Variations of temperature, 544, 545.

Varro, 75.

Vegetable anatomy, 567-

Vegetables, 566.

Velocities of the planets, PL 32.

Velocity, 22, 244. Effect of velocity in overcom-
ing strength, 111.

Velocity due to a height, 25.

Velocity of a blast, 263.

Velocity of an impulse, 111.

Velocity of descent, 33.

Velocity of electricity, 516.

Velocity of fluids, 211.

Velocity of friction, 486-

Velocity of sound, 289, 291, 292.

Velocity of light, 341.

Vena contracta, PI. 20..

Ventilation, 264.

Venturi, 213,225, 226, 251, PI. 20.

Venus, 403, 418, 419, 422, 431, PL 33.

Vera, PL 22.

Vermes, 576.

Vernier, 81, PL 7.

Vertical pipe, 216, PL 20.

Vessel, 248.

Vessels of plants, 568.

Vesta, 400, 404.

Vestibule, 302.

Vesuvius, 562.

Vibrating cord, PL 25.

Vibrations, 295, 305, PL 25.

Vibrations of cords, 297.

Vibrations of fluids, 217.

Vibrations of heat, 502.

Vibrations of sounding bodies, 147.

Vices, 170.

Vielle, 312.

Vince, 50, 117, 382.

<)08

INDEX.

Vinci. Da Vinci, 189 ; b. U4.r,, d. 1520.

Viola diGamba, 312.

Violin, 312.

Violoncello, 312.

Viper, 575.

Virgo, 401.

Virtual focus, 324.

Virtual image, 329, PI. 27-

Virtual velocities, 56.

Viscosity, 482.

Vision, 350.

Vitellio, 375; fl. 1269.

Vitreous electricity, 517-

Vitreous humour, 351.

Vitruvius, 145, 186, 276, 277, PI. 22; fl. 15, B.C.

Voice, 312, PJ. 26.

Volcanos, 560.

Volcanos in the moon, 423.

Volta, 521, 523, 524, 528, 661, 563, 586, 587, 588,

PI. 40.

Voltaic current, 539, 549.
Voltaire. See 194 ; b. 1694, d. 1778.
Vowels, 313.

Vox humana pipe, 314, PI. 26.
Vulcanian theory, 563.

Waggon overturning, PI. 3.

Wain, 394.

Walking, 100, PI. 9.

Walking wheels, 160, 257-

Wall, 122.

Wallace, 79.

Wallingford, 145,188; fl. 1326.

Wallis, 28, 190, 316 ; b. 1616, d. 1713.

Walter of Coventry, 188; fl. 1213.

Waring, 192.

Watches, 192.

Watch scapements, PI. 26.

Water, 209, 291, 323, 475, 477-

Water colours, 73.

Water in air, 554.

Waterloo Bridge, 271.

Water mill, 103.

Water pipes, 241, PL 21.

Water screw, 251, PI. 22.

Water snail, 251.

Waterspouts, 559.

Water-wheels, 230, 244, 282, PL 22.

Water whimsey, 157.

Watt, 37, 51, 101, 103, 187, 192, 265, 266, 267,

269, 282, 283, PL 24.

Waves, 218, PL 20. Combinations of waves, 220.
Wax candle, 486.
Weather, 550.
Weaving, 142.
Wedge, 54, 119, PL 4.
Wedge moving in water, 230.
Wedges for stones, PL 11.
Wedgwood, 192, 497 ; <L 1795.
Weidler, 406.
Weighing, 95.

Weighing machines, 97, PI. 9.
Weight, 12, PL 3.
Weight of air, 207.
Weight in air, 29.
Weight of animals, 117-
Weights of clocks, 137.
Wells, 655.
Were, 224.
West, 398.
Westerly winds, 549.

Westgarth, 257-

Westminster Abbey, 188.

Wet leather, 482.

Whale, 395.

Whalebone hygrometer, 554.

Wheat, 95.

Wheel, PL 1, 14.

Wheel and axis, 51, PL 3.

Wheel carriages, 164, PL 18.

Wheel cutting machine, 137.

Wheelcutters, PL 15.

Wheel of Orfyreus, PL 6.

Wheels, 164, 165, 166.

Wheels and pinions, 52.

Wheels with straps, PL 15.

Wheelways of iron, 167-

Wheelwork, 134.

Whimsey, 157.

Whip, 174.

Whirling table, 27, 198, PL 1.

Whirlpool, 216.

Whispering gallery, 294.

Whistling, 314.

Whitehurst, 85, 145, 146, 259.

White light, 342.

White's crane, 161, PL 17-

Wilfrid, 186.

Wilke, 586.

Wilkins, 285 ; b. 1614, d. 1672.

William of Sens, 188.

William IV., 457.

Willughby, 585 ; b. 1635, d. 1672.

Wilson, 399, 525, 559, PL 31.

Winch, 101, 134, 157, PL 3.

Wind, 246, 550.

Wind and water, 246.

Wind gages, 243.

Windmills, 103, 246, 247, 277,281.

Winds, 544, 548.

W inkier, 585, 586; b. 1703, d. 1770.

Winter, 417, 547.

Wire, 187.

Wire drawing, 171.

Wirtz, 252, 253, PL 22.

Wollaston. Dr. Wollaston, 329, 331, 332, 342,

343, 346, 348, 349, 382, 490, 588, PL 27. 29, 39.

Rev.F. Wollaston, PL 35.
Woltman, PL 22.
Wood, 116.
Wood cuts, 91.
Wooden bridges, 131, PL 14.
Wool, 142.

Woollen manufactures, 186.
Worcester. Marquis of Worcester, 265, 278, 279 ;

d. 1667.

Work of a labourer, 101, 178, 253.
Wren, 190, 459, 461 ; b. 1632, d. 1723.
Writing, 71, 74, 75.
Wunsch, 292.

York Minster, 188.
Young. M. Young, 213.
Young, Dr. T., 270, 370, 371.

Zenith sectors, 429, PL 35.

Zero, 500.

Zodiacal light, 399, PL 31.

Zones, 436.

Zucchius, 381.

Zuyder Zee, 564.

THE END.

Printed by J. & H. Cox (Brothers), 74 & 75, Great Queen Street,
Lineoln's-Inn Fields.

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