121 223
PROFESSOR ERXST MACH
IS&S lOKi
THE;
SCIENCE OF MECHANICS
A CRITICAL AND HISTORICAL ACCOUNT
OF ITS DEVELOPMENT
DR. ERNST MACH
PROFESSOR OF TE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN
THE UNIVERSITY OF VIENNA
TRANSLATED FROM THE GERMAN
BY
THOMAS J. McCORMACK
\VITir TWO HUNDRED AND FIFTY CUTS AND ILLUSTRATIONS
FOURTH EDITION
CHICAGO LONDON
THE OPEN COURT PUBLISHING CO.
1919
COPYRIGHT 1893, 1902, 1919
BY
Til Li OWN Coi'UT PUBLISH INO COMPANY
TRANSLATOR'S PREFACE TO THE
SECOND ENGLISH EDITION.
SINCE the appearance of the first edition of the
present translation of Mach's Mechanics ',* the views
which Professor Mach has advanced on the philoso
phy of science have found wide and steadily increas
ing acceptance. Many fruitful and elucidative con
troversies have sprung from his discussions of the
historical, logical, and psychological foundations of
physical science, and in consideration of the great
ideal success which his works have latterly met with
in Continental Europe, the time seems ripe for a still
wider dissemination of his views in Englishspeaking
countries. The study of the history and theory of
science is finding fuller and fuller recognition In our
universities, and it Is to be hoped that the present ex
emplary treatment of the simplest and most typical
branch* of physics will stimulate^ further progress in
this direction,
The text of the present edition, which contains
the extensive additions made by the author to the
* Die Mechanik in ihrer Entwickelung historischkritisch dargesiellt. Von
Dr. Ernst Mach, Professor an der Universitat zu Wien. Mit 257 Abbildungen.
First German edition, 1883. Fourth German edition, 1901. First edition of
the English translation, Chicago, The Open Court Publishing Co., 1893.
vi TRANS LA TOR' S PREFA CM.
latest German editions, has been thoroughly revised
by the translator. All errors, either of substance or
typography, so far as they have come to the trans
lator's notice, have been removed, and in many cases
the phraseology has been altered. The subtitle of
the work has, in compliance with certain criticisms,
also been changed, to accord more with the wording
of the original title and to bring out the idea that the
work treats of the principles of mechanics predomi
nantly under the aspect of their development (Entwicke
lung). To avoid confusion in the matter of references,
the main title stands as in the first edition.
The author's additions, which are considerable,
have been relegated to the Appendix. This course
has been deemed preferable to that of incorporating
them in the text, first, because the numerous refer
ences in other works to the pages of the first edition
thus hold good for the present edition also, and sec
ondly, because with few exceptions the additions are
either supplementary in character, or in answer to
criticisms. A list of the subjects treated in these ad
ditions is given in the Table of Contents, under the
heading "Appendix" on page xix.
Special reference, however, must be made to the
additions referring to Hertz's Mechanics (pp. 548555),
and to the history of the development of Professor
Mach's own philosophical and scientific views, notably
to his criticisms of the concepts of mass, inertia, ab
solute motion, etc., on pp. 542547, 555574, and 579
TJtANSLA TOR'S PREFA CE. vii
583. The remarks here made will be found highly
elucidative, while the references given to the rich lit
erature dealing with the history and philosophy of
science will also be found helpful.
As for the rest, the text of the present edition of
the translation is the same as that of the first. It has
had the sanction of the author and the advantage
of revision by Mr. C. S. Peirce, well known for his
studies both of analytical mechanics and of the his
tory and logic of physics. Mr. Peirce read the proofs
of the first edition and rewrote Sec. 8 in the chapter
on Units and Measures, where the original was in
applicable to the system commonly taught in this
county.
THOMAS J. McCoRMACK.
LA SALLE, ILL., February, 1902.
AUTHOR'S PREFACE TO THE TRANS
LATION.
Having read the proofs of the present translation
of my work, Die Mechanik in ihrer JLntwickeliing, I can
testify that the publishers have supplied an excellent,
accurate, and faithful rendering of it, as their previous
translations of essays of mine gave me every reason to
expect. My thanks are due to all concerned, and
especially to Mr. McCormack, whose intelligent care
in the conduct of the translation has led to the dis
covery of many errors, heretofore overlooked. I may,
thus, confidently hope, that the rise and growth of the
ideas of the great inquirers, which it was my task to
portray, will appear to my new public in distinct and
sharp outlines. E. MACH.
PRAGUE, April 8th, 1893.
PREFACE TO THE THIRD EDITION.
THAT the Interest in the foundations of mechanics
is still unimpaired, is shown by the works published
since 1889 by Budde, P. and J. Friedlander, H. Hertz,
P. Johannesson, K. Lasswitz, MacGregor, K. Pearson,
J. Petzoldt, Rosenberger, E. Strauss, Vicaire, P.
Volkmann, E. Wohlwill, and others, many of which
are deserving of consideration, even though briefly.
In Prof. Karl Pearson {Grammar of Science, Lon
don, .1892), I have become acquainted with an inquirer
with whose epistemological views I am in accord at
nearly all essential points, and who has always taken
a frank and courageous stand against all pseudo
scientific tendencies in science. Mechanics appears
at present to be entering on a new relationship to
physics, as is noticeable particularly in the publica
tion of H. Hertz. The nascent transformation in our
conception of forces acting at a distance will perhaps
be influenced also by the interesting investigations of
H. Seeliger ("Ueber das Newton'sche Gravitations
gesetz," Sitzungsbericht der Miinchener Akademie, 1896),
who has shown the incompatibility of a rigorous inter
pretation of Newton's law with the assumption of an
unlimited mass of the universe.
VIENNA, January, 1897. E. Mach.
x PREFACE TO THE FIRST EDITION.
ing these cases must ever remain the method at once
the most effective and the most natural for laying this
gist and kernel bare. Indeed, it is not too much to
say that it is the only way in which a real comprehen
sion of the general upshot of mechanics is to be at
tained.
I have framed my exposition of the subject agree
ably to these views. It is perhaps a little long, but, on
the other hand, I trust that it Is clear. I have not in
every case been able to avoid the use of the abbrevi
ated and precise terminology of mathematics. To do
so would have been to sacrifice matter to form ; for the
language of everyday life has not yet grown to be suf
ficiently accurate for the purposes of so exact a science
as mechanics.
The elucidations which I here offer are, In part,
substantially contained In my treatise, Die Geschichte
und die Wurzel de$ Satzes von dcr Erhaltung der Arbeit
(Prague, Calve, 1872). At a later date nearly the same
views were expressed by KIRCHHOFF ( Vorlcsungen iel>er
mathematische Physik: Mechanik, Leipsic, 1874) an d by
HELMHOLTZ {Die Thatsachen in der Wahrnehmitng,
Berlin, 1879), and have since become commonplace
enough. Still the matter, as I conceive it, does not
seem to have been exhausted, and I cannot deem my
exposition to be at all superfluous.
In my fundamental conception of the nature of sci
ence as Economy of Thought, a view which I In
dicated both in the treatise above cited and In my
PREFACE TO THE FIRST EDITION. xi
pamphlet, Die Gestaltcn der Fliissigkeit (Prague, Calve,
1872), and which I somewhat more extensively devel
oped in my academical memorial address, Die okono
mische Natur der physikalischen Forschung (Vienna, Ge
rold, 1882, I no longer .stand alone. I have been
much gratified to find closely allied ideas developed,
in an original manner, by Dr. R. AVENARIUS {Philoso
phie als Denken der Welt, gemdss dem Princip des klein
sten KraftmaasseS) Leipsic, Fues, 1876). Regard for
the true endeavor of philosophy, that of guiding into
one common stream the many rills of knowledge, will
not be found wanting in my work, although it takes a
determined stand against the encroachments of meta
physical methods.
The questions here dealt with have occupied me
since my earliest youth, when my interest for them was
powerfully stimulated by the beautiful introductions of
LAGRANGE to the chapters of his Analytic Mechanics, as
well as by the lucid and lively tract of JOLLY, Prindpien
der Mechanik (Stuttgart, 1852). If DUEHRING'S esti
mable work, Kritische Geschichte der Principien der Me
chanik (Berlin, 1873), did not particularly influence
me, it was that at the time of its appearance, my ideas
had been not only substantially worked out, but actually
published. Nevertheless, the reader will, at least on
the destructive side, find many points of agreement
between Diihring's criticisms and those here expressed.
The new apparatus for the illustration of the sub
ject, here figured and described, were designed entirely
xii PREFACE TO THE FIRST EDITION.
by me and constructed by Mr. F. Hajek, the mechani
cian of the physical institute under my control.
In less immediate connection with the text stand
the facsimile reproductions of old originals in my pos
session. The quaint and naive traits of the great in
quirers, which find in them their expression, have al
ways exerted upon me a refreshing influence in my
studies, and I have desired that my readers should
share this pleasure with me.
K. MACH.
PRAGUE, May, 1883.
PREFACE TO THE SECOND EDITION.
IN consequence of the kind reception which this
book has met with, a very large edition has been ex
hausted in less than five years. This circumstance and
the treatises that have since then appeared of E. Wohl
will, H. Streintz, L. Lange, J. Epstein, F. A. Muller,
J. Popper, G. Helm, M. Planck, F. Poske, and others
are evidence of the gratifying fact that at the present
day questions relating to the theory of cognition are
pursued with interest, which twenty years ago scarcely
anybody noticed.
As a thoroughgoing revision of my work did not
yet seem to me to be expedient, I have restricted my
self, so far as the text is concerned, to the correction
of typographical errors, and have referred to the works
that have appeared since its original publication, as
far as possible, in a few appendices.
E. MACH.
PRAGUE, June, 1888.
PREFACE TO THE FIRST EDITION.
THE present volume is not a treatise upon the ap
plication of the principles of mechanics. Its aim is
to clear up ideas, expose the real significance of the
matter, and get rid of metaphysical obscurities. The
little mathematics it contains is merely secondary to
this purpose.
Mechanics will here be treated, not as a branch of
mathematics, but as one of the physical sciences. Ii
the reader's interest is in that side of the subject, if
he is curious to know how the principles of mechanics
have been ascertained, from what sources they take
their origin, and how far they can be regarded as
permanent acquisitions, he will find, I hope, in these
pages some enlightenment. All this, the positive and
physical essence of mechanics, which makes its chief
and highest interest for a student of nature, is in ex
isting treatises completely buried and concealed be
neath a mass of technical considerations.
The gist and kernel of mechanical Ideas has in al
most every case grown up In the Investigation of very
simple and special cases of mechanical processes ; and
the analysis of the history of the discussions concern
PREFACE TO THE FOURTH EDITION.
THE number of the friends of this work appears to
have increased in the course of seventeen years, and
the partial consideration which my expositions have
received in the writings of Boltzmann, Foppl, Hertz,
Love, Maggi, Pearson, and Slate, have awakened in
me the hope that my work shall not have been in
vain. Especial gratification has been afforded me by
finding in ]. B. Stallo (The Concepts of Modern Physics}
another staunch ally in my attitude toward mechanics,
and in W. K. Clifford (Lectures and Essays and The
Common Sense of the JZxact Sciences}, a thinker of kin
dred aims and points of view.
New books and criticisms touching on my discus
sions have received attention in special additions,
which in some instances have assumed considerable
proportions. Of these strictures, O. Holder's note on
my criticism of the Archimedean deduction (Denken
nnd Anschauutig in der Geometrie, p. 63, note 62) has
been of special value, inasmuch as it afforded me the
opportunity of establishing my view on still firmer
foundations (see pages 512517). I do not at all dis
pute that rigorous demonstrations are as possible in
mechanics as in mathematics. But with respect to
xv i PREFACE TO THE FOURTH EDITION.
the Archimedean and certain other deductions, I am
still of the opinion that my position is the correct
one.
Other slight corrections in my work may have
been made necessary by detailed historical research,
but upon the whole I am of the opinion that I have
correctly portrayed the picture of the transformations
through which mechanics has passed, and presumably
will pass. The original text, from which the later in
sertions are quite distinct, could therefore remain as
it first stood in the first edition. I also desire that no
changes shall be made in it even if after my death a
new edition should become necessary.
E. MACJL
VIENNA, January, 1901.
TABLE OF CONTENTS.
PAGE
Translator's Preface to the Second Edition v
Author's Preface to the Translation. via
Preface to the First Edition ix
Preface to the Second Edition xiii
Preface to the Third Edition xiv
Preface to the Fourth Edition xv
Table of Contents xvii
Introduction i
CHAPTER I.
THE DEVELOPMENT OF THE PRINCIPLES OF STATICS.
I. The Principle of the Lever 8
II. The Principle of the Inclined Plane 24
III. The Principle of the Composition of Forces .... 33
IV. The Principle of Virtual Velocities 49
V. Retrospect of the Development of Statics 77
VI. The Principles of Statics in Their Application to Fluids 86
VII. The Principles of Statics in Their Application to Gas
eous Bodies no
CHAPTER II.
THE DEVELOPMENT OF THE PRINCIPLES OF DYNAMICS.
I. Galileo's Achievements 128
II. The Achievements of Huygens . 155
III. The Achievements of Newton 187
IV. Discussion and Illustration of the Principle of Reaction 201
V. Criticism of the Principle of Reaction and of the Con
cept of Mass 216
VI. Newton's Views of Time, Space, and Motion .... 222
xviil TJETjS SCIJKATCJZ OF MECHANICS.
PAGE
VII. Synoptical Critique of the Newtonian Enunciations . 238
VIII Retrospect of the Development of Dynamics .... 245
CHAPTER III.
THE EXTENDED APPLICATION OF THE PRINCIPLES OF
MECHANICS AND THE DEDUCTIVE DEVELOP
MENT OF THE SCIENCE.
I. Scope of the Newtonian Principles 256
II. The Formulae and Units of Mechanics 269
III. The Laws of the Conservation of Momentum, of the
Conservation of the Centre of Gravity, and of the
Conservation of Areas 287
IV. The Laws of Impact 305
V. D'Alembert's Principle . , 331
VI. The Principle of Vis Viva 343
VII. The Principle of Least Constraint 350
VIII. The Principle of Least Action 364
IX, Hamilton's Principle 380
X. Some Applications of the Principles of Mechanics to
Hydrostatic and Hydrodynamic Questions . 384
CHAPTER IV.
THE FORMAL DEVELOPMENT OF MECHANICS.
I. The Isoperimetrical Problems 421
II. Theological, Animistic, and Mystical Points of View in
Mechanics 446
III. Analytical Mechanics 465
IV. The Economy of Science 481
CHAPTER V.
THE RELATION OF MECHANICS TO OTHER DEPART
MENTS OF KNOWLEDGE.
I. The Relations of Mechanics to Physics 495
II. The Relations of Mechanics to Physiology 504
TABLE OF CONTENTS. xix
PAGE
Appendix 509
I. The Science of Antiquity, 509. II. Mechanical Researches of
the Greeks, 510. III.. and IV. The Archimedean Deduction of the
Law of the Lever, 512, 514. V. Mode of Procedure of Stevinus, 515.
VI. Ancient Notions of the Nature of the Air, 517, VII. Galileo's
Predecessors, 520 VIII. Galileo on Falling Bodies, 522. IX. Gali
leo on the Law of Inertia, 523. X. Galileo on the Motion of Projec
tiles, 525. XL Deduction of the Expression for Centrifugal Force
(Hamilton's Hodograph), 527. XII. Descartes and Huygens on
Gravitation, 528. XIII. Physical Achievements of Huygens, 530.
XIV. Newton's Predecessors, 531. XV. The Explanations of Gravi
tation, 533 XVI. Mass and Quantity of Matter, 536. XVII. Gali
leo on Tides, 537. XVIII. Mach's Definition of Mass, 539. XIX,
Mach on Physiological Time, 541, XX. Recent Discussions of the
Law of Inertia and Absolute Motion, 542. XXI. Hertz's System of
Mechanics, 548. XXII. History of Mach's Views of Physical Sci
ence (Mass, Inertia, etc.), 555. XXIII. Descartes's Achievements
in Physics, 574. XXIV. Minimum Principles, 575. XXV. Grass
mann's Mechanics, 577. XXVI. Concept of Cause, 579 XXVII.
Mach's Theory of the Economy of Thought, 579. XXVIII. Descrip
tion of Phenomena by Differential Equations, 583 XXIX. Mayer
and the Mechanical Theory of Heat, 584. XXX. Principle of En
ergy, 585
Chronological Table of a Few Kminent Inquirers and of Their
More Important Mechanical Works 589
Index 593
THE SCIENCE OF MECHANICS
INTRODUCTION.
1. THAT branch of physics which is at once the old The science
11 i i i i i r i of mechan
est and the simplest and which is therefore treated ics.
as introductory to other departments of this science,
is concerned with the motions and equilibrium of
masses. It bears the name of mechanics.
2. The history of the development of mechanics,
is quite indispensable to a full comprehension of the
science in its present condition. It also affords a sim
ple and instructive example of the processes by which
natural science generally is developed.
An instinctive, irreflective knowledge of the processes instinctive
knowledge.
of nature will doubtless always precede the scientific,
conscious apprehension, or investigation, of phenom
ena. The former is the outcome of the relation in
which the processes of nature stand to the satisfac
tion of our wants. The acquisition of the most ele
mentary truth does not devolve upon the individual
alone : it is preeffected in the development of the race.
In point of fact, it is necessary to make a dis Mechanical
tinction between mechanical experience and median expenences
ical science, in the sense in which the latter term is at
present employed. Mechanical experiences are, un
questionably, very old. If we carefully examine the
ancient Egyptian and Assyrian monuments, we shall
find there pictorial representations of many kinds of
THE SCIENCE OF MECHANICS.
Theme implements and mechanical contrivances; but ac
knowfe a dge counts of the scientific knowledge of these peoples
antiqulty are either totally lacking, or point conclusively to a
very inferior grade of attainment. By the side of
highly ingenious ap
pliances, we behold
the cru dest and rough
est expedients em
ployedastheuseof
sleds, for instance, for
the transportation of
enormous blocks of
stone. All bears an
instinctive, unperfec
ted, accidental char
acter.
So, too, prehistoric
graves contain imple
ments whose construc
tion and employment
imply no little skill
and much mechanical
experience. Thus, long
before theory was
dreamed of, imple
ments, machines, me
chanical experien
ces, and mechanical
knowledge were abun
dant.
3. The idea often
suggests itself that
perhaps the incom
plete accounts we pos
INTRODUCTION. 3
sess have led us to underrate the science of the ancient
world. Passages occur in ancient authors which seem
to indicate a profounder knowledge than we are wont
to ascribe to those nations. Take, for instance, the
following passage from Vitruvius, De Architectures,
Lib. V, Cap. Ill, 6 :
" The voice is a flowing breath, made sensible to A passage
"the organ of hearing by the movements it produces Jius! Vltm "
"in the air. It is propagated In infinite numbers of
"circular zones exactly as when a stone is thrown
"into a pool of standing water countless circular un
"dulations are generated therein, which, increasing
"as they recede from the centre, spread out over a
"great distance, unless the narrowness of the locality
"or some obstacle prevent their reaching their ter
" mination ; for the first line of waves, when Impeded
"by obstructions, throw by their backward swell the
"succeeding circular lines of waves into confusion.
" Conformably to the very same law, the voice also
"generates circular motions ; but with this distinction,
"that in water the circles, remaining upon the surface,
"are propagated horizontally only, while the voice Is
"propagated both horizontally and vertically."
Does not this sound like the Imperfect exposition Controvert
of a popular author, drawn from more accurate disqui evidence. 61 *
sitions now lost? In what a strange light should we
ourselves appear, centuries hence, If our popular lit
erature, which by reason of its quantity is less easily
destructible, should alone outlive the productions of
science ? This too favorable view, however, Is very
rudely shaken by the multitude of other passages con
taining such crude and patent errors as cannot be con
ceived to exist in any high stage of scientific culture.
(See Appendix, I., p. 509.)
4 THE SCIENCE OF MECHANICS.
The origin 4. When, where, and In what manner the develop
ment of science actually began, is at this day difficult
historically to determine. It appears reasonable to
assume, however, that the instinctive gathering of ex
periential facts preceded the scientific classification of
them. Traces of this process may still be detected in
the science of today; indeed, they are to be met with,
now and then, in ourselves. The experiments that
man heedlessly and instinctively makes in his strug
gles to satisfy his wants, are just as thoughtlessly and
unconsciously applied. Here, for instance, belong the
primitive experiments concerning the application of
the lever in all its manifold forms. But the things
that are thus unthinkingly and instinctively discovered,
can never appear as peculiar, can never strike iS as
surprising, and as a rule therefore will never supply an
impetus to further thought.
Thefunc The transition from this stage to the classified,
cia? S ciasses scientific knowledge and apprehension of facts, first be
veiop^ett comes possible on the rise of special classes and pro
fessions who make the satisfaction of definite social
wants their lifelong vocation. A class of this sort oc
cupies itself with particular kinds of natural processes.
The individuals of the class change ; old members
drop out, and new ones come in. Thus arises a need
of imparting to those who are newly come in, the
stock of experience and knowledge already possessed ;
a need of acquainting them with the conditions of the
The com attainment of a definite end so that the result may be
munication , , . , , _, . .
ofknowi determined beforehand. The communication of knowl
e ge< edge is thus the first occasion that compels distinct re
flection, as everybody can still observe in himself.
Further, that which the old members of a guild me
chanically pursue, strikes a new member as unusual
INTRODUCTION. 5
and strange, and thus an impulse Is given to fresh re
flection and investigation.
When we wish to bring to the knowledge of a per involves
1 description.
son any phenomena or processes of nature, we have
the choice of two methods : we may allow the person to
observe matters for himself, when instruction comes
to an end ; or, we may describe to him the phenomena
in some way, so as to save him the trouble of per
sonally making anew each experiment. Description,
however, is only possible of events that constantly re
cur, or of events that are made up of component
parts that constantly recur. That only can be de
scribed, and conceptually represented which is uniform
and conformable to law ; for description presupposes
the,, employment of names by which to designate its
elements ; and names can acquire meanings only when
applied to elements that constantly reappear.
5. In the infinite variety of nature many ordinary A unitary
_ J conception
events occur; while others appear uncommon, per f nature,
plexing, astonishing, or even contradictory to the or
dinary run of things. As long as this is the case we
do not possess a wellsettled and unitary conception of
nature. Thence is imposed the task of everywhere
seeking out in the natural phenomena those elements
that are the same, and that amid all multiplicity are
ever present. By this means, on the one hand, the
most economical and briefest description and com
munication are rendered possible ; and on the other, The nature
,  i i i 11 r ofknowi
when once a person has acquired the skill of recog edge,
nising these permanent elements throughout the great
est range and variety of phenomena, of seeing them in
the same, this ability leads to a comprehensive, compact,
consistent^ and facile conception of the facts. When once
we have reached the point where we are everywhere
6 THE SCIENCE OF MECHANICS.
The adap able to detect the same few simple elements, combin
tation of . .
thoughts to ing in the ordinary manner, then they appear to us as
things that are familiar; we are no longer surprised,
there is nothing new or strange to us in the phenom
ena, we feel at home with them, they no longer per
plex us, they are explained. It is a process of adaptation
of thoughts to facts with which we are here concerned.
The econ 6. Economy of communication and of apprehen
?hought. sion is of the very essence of science. Herein lies
its pacificatory, its enlightening, its refining element.
Herein, too, we possess an unerring guide to the his
torical origin of science. In the beginning, all economy
had in immediate view the satisfaction simply of bodily
wants. With the artisan, and still more so with the
investigator, the concisest and simplest possible knowl
edge of a given province of natural phenomena a
knowledge that is attained with the least intellectual
expenditure naturally becomes in itself an econom
ical aim ; but though it was at first a means to an end,
when the mental motives connected therewith are once
developed and demand their satisfaction, all thought
of its original purpose, the personal need, disappears.
Further de To find, then, what remains unaltered in the phe
velopment r  . 11 if
of these nomena ot nature, to discover the elements thereof
and the mode of their interconnection and interdepend
ence this is the business of physical science. It en
deavors, by comprehensive and thorough description,
to make the waiting for new experiences unnecessary ;
it seeks to save us the trouble of experimentation, by
making use, for example, of the known interdepend
ence of phenomena, according to which, if one kind of
event occurs, we may be sure beforehand that a certain
other event will occur. Even in the description itself
labor may be saved, by discovering methods of de
INTRODUCTION. 7
scribing the greatest possible number of different ob~ Their pres
jects at once and in the concisest manner. All this will sion merely
be made clearer by the examination of points of detail
than can be done by a general discussion. It is fitting,
however, to prepare the way, at this stage, for the
most important points of outlook which in the course
of our work we shall have occasion to occupy.
7. We now propose to enter more minutely into the Proposed
subject of our inquiries, and, at the same time, without treatment,
making the history of mechanics the chief topic of
discussion, to consider its historical development so
far as this is requisite to an understanding of the pres
ent state of mechanical science, and so far as it does
not conflict with the unity of treatment of our main
subject. Apart from the consideration that we cannot
afford to neglect the great incentives that it is in our
power to derive from the foremost intellects of allTheincen
epochs, incentives which taken as a whole are more rived from
fruitful than the greatest men of the present day are with the
able to offer, there is no grander, no more intellectually lects of the
elevating spectacle than that of the utterances of the
fundamental investigators in their gigantic power.
Possessed as yet of no methods, for these were first
created by their labors, and are only rendered compre
hensible to us by their performances, they grapple with
and subjugate the object of their inquiry, and imprint
upon it the forms of conceptual thought. They that
know the entire course of the development of science,
will, as a matter of course, judge more freely and And the in
i r i r r j. crease of
more correctly of the significance of any present scien power
 , , . . . , . , . which such
tific movement than they, who limited in their views a contact
to the age in which their own lives have been spent,
contemplate merely the momentary trend that the course
of intellectual events takes at the present moment.
CHAPTER I.
THE DEVELOPMENT OF THE PRINCIPLES OF
STATICS.
i,
THE PRINCIPLE OF THE LEVER.
The earliest i. The earliest investigations concerning mechan
mechanical . r 1 . , * . , ,
researches ics of which we have any account, the investigations
stadcs. of the ancient Greeks, related to statics, or to the doc
trine of equilibrium. Likewise, when after the taking
of Constantinople by the Turks in 1453 a fresh impulse
was imparted to the thought of the Occident by the an
cient writings that the fugitive Greeks brought with
them, it was investigations in statics, principally evoked
by the works of Archimedes, that occupied the fore
most investigators of the period. (See p. 510.)
Archimedes 2. ARCHIMEDES of Syracuse (287212 B. C.) left
(287212 E. behind him a number of writings, of which several
have come down to us in complete form. We will
first employ ourselves a moment with his treatise DC,
j$Lquiponderantibus, which contains propositions re
specting the lever and the centre of gravity.
In this treatise Archimedes starts from the follow
ing assumptions, which he regards as selfevident :
Axiomatic a. Magnitudes of equal weight acting at equal
assump . x
dons of Ar distances (from their point of support) are in equi
chimedes. r rr J l
librmm.
THE PRINCIPLES OF STATICS. g
b. Magnitudes of equal weight acting at une Axiomatic
i j* ff assump
qual distances (from their point of support") aretions of Ar
V.  .,,  , . ; , chimedes.
not in equilibrium, but the one acting at the
greater distance sinks.
From these assumptions he deduces the following
proposition :
c. Commensurable magnitudes are in equilib
rium when they are inversely proportional to their
distances (from the point of support).
It would seem as if analysis could hardly go be
hind these assumptions. This Is, however, when we
carefully look Into the matter, not the case.
Imagine (Fig. 2) a bar, the weight of which is
neglected. The bar rests on a fulcrum. At equal dis
tances from the fulcrum we ap
pend two equal weights. That jr r
the two weights, thus circum rh rU
stanced, are in equilibrium, is .
the assumption from which Archi
medes starts. We might suppose that this was self Analysis f of
evident entirely apart from any experience, agreeably to medean as
the socalled principle of sufficient reason ; that in view sumplons<
of the symmetry of the entire arrangement there is no
reason why rotation should occur In the one direction
rather than in the other. But we forget, in this, that
a great multitude of negative and positive experiences
Is implicitly contained in our assumption ; the negative,
for instance, that dissimilar colors of the leverarms,
the position of the spectator, an occurrence in the vi
cinity, and the like, exercise no influence ; the positive,
on the other hand, (as it appears in the second as
sumption,) that not only the weights but also their dis
tances from the supporting point are decisive factors
in the disturbance of equilibrium, that they also are cir
io THE SCIENCE OF MECHANICS.
cumstances determinative of motion. By the aid of
these experiences we do indeed perceive that rest (no
motion) is the only motion which can be uniquely* de
termined, or defined, by the determinative conditions
of the case.f
:haracter Now we are entitled to regard our knowledge of
he Irchi the decisive conditions of any phenomenon as sufficient
.uits. only in the event that such conditions determine the
phenomenon precisely and uniquely. Assuming the
fact of experience referred to, that the weights and
their distances alone are decisive, the first proposition
of Archimedes really possesses a high degree of evi
dence and is eminently qualified to be made the foun
dation of further investigations. If the spectator place
himself in the plane of symmetry of the arrangement
in question, the first proposition manifests itself, more
over, as a highly imperative instinctive perception, a
result determined by the symmetry of our own body.
The pursuit of propositions of this character is, fur
thermore, an excellent means of accustoming ourselves
in thought to the precision that nature reveals in her
processes,
rhegenerai 3. We will now reproduce in general outlines the
proposition , . .  , ,  . , . .. . . .
f the lever tram ot thought by which Archimedes endeavors to re
educed to , , . . . . . t
he simple uuce the general proposition of the lever to the par
ifarcase? ticular and apparently selfevident case. The two
equal weights i suspended at a and b (Fig. 3) are, if
the bar ab be free to rotate about its middle point c 9 in
equilibrium. If the whole be suspended by a cord at
c, the cord, leaving out of account the weight of the
* So as to leave only a single possibility open.
tlf, for example, we were to assume that the weight at the right de
scended, then rotation in the opposite direction also would be determined by
the spectator, whose person exerts no influence on the phenomenon, taking
up his position on the opposite side.
THE PRINCIPLES OF STATICS. n
bar, will have to support the weight 2. The equal The general
weights at the extremities of the bar supply accor of the lever
TII i r . i , , . , , reduced to
dmgly the place of the double weight at the centre.
the simple
and partic
ular case.
m '[I] LLJ
Fig. 3. Fig. 4.
On a lever (Fig. 4), the arms of which are in the
proportion of i to 2, weights are suspended in the pro
portion of 2 to i. The weight 2 we imagine replaced
by two weights i, attached on either side at a distance
i from the point of suspension. Now again we have
complete symmetry about the point of suspension, and
consequently equilibrium.
On the leverarms 3 and 4 (Fig. 5) are suspended
the weights 4 and 3.. The leverarm 3 is prolonged
the distance 4, the arm 4 is prolonged the distance 3,
and the weights 4 and 3 are replaced respectively by
[i] d
LJ
t
]
r^T i" 1 "! r I fi r i
c
LJ
j
J LJ LJ
4
Ill
Fig 5
4 and 3 pairs of symmetrically attached weights ^,
in the manner indicated in the figure. Now again we
have perfect symmetry. The preceding reasoning, The gener
which we have here developed with specific figures, is
easily generalised.
4. It will be of interest to look at the manner in
which Archimedes's mode of view, after the precedent
of Stevinus, was modified by GALILEO.
THE SCIENCE OF. MECHANICS.
Galileo's Galileo imagines (Fig. 6) a heavy horizontal prism,
mode of . . . j a i
treatment, homogeneous in material composition, suspended by
its extremities from a homogeneous bar of the same
length. The bar is provided at its middle point
with asuspensory attach
ment. In this case equi
librium will obtain ; this
we perceive at once. But
in this case is contained
every other case, which
Galileo shows in the
Fig. 6. following manner. Let
us suppose the whole
length of the bar or the prism to be z(m } ;/). Cut
the prism in two, in such a manner that one portion
shall have the length zm and the other the length 2//.
We can effect this without disturbing the equilibrium
by previously fastening to the bar by threads, close to
the point of proposed section, the inside extremities of
the two portions. We may then remove all the threads,
if the two portions of the prism be antecedently at
tached to the bar by their centres. Since the whole
length of the bar is 2,(m + #), the length of each half
is m + n. The distance of the point of suspension of
the righthand portion of the prism from the point of
suspension of the bar is therefore m, and that of the
lefthand portion n. The experience that we have
here to deal with the weight, and not with the form,
of the bodies, is easily made. It is thus manifest, that
equilibrium will still subsist if any weight of the mag
nitude 2m be suspended at the distance n on the one
side and any weight of the magnitude zn be suspended
at the distance m on the other. The instinctive elements
of our perception of this phenomenon are even more
THE PRINCIPLES OF STATICS. 13
prominently displayed in this form of the deduction
than in that of Archimedes.
We may discover, moreover, in this beautiful pre
sentation, a remnant of the ponderousness which was
particularly characteristic of the investigators of an
tiquity.
How a modern physicist conceived the same prob Lagrange's
lem, may be learned from the following presentation of t*ion S . enta
LAGRANGE. Lagrange says : Imagine a horizontal ho
mogeneous prism suspended at its centre. Let this
prism (Fig. 7) be conceived divided into two prisms
of the lengths 2m and 2^. If now we consider the
centres of gravity of these two parts, at which we may
imagine weights to act proportional to im and 272, the
2m
2n
Fig. 7.
two centres thus considered will have the distances n
and m from the point of support. This concise dis
posal of the problem is only possible to the practised
mathematical perception.
5. The object that Archimedes and his successors object of
sought to accomplish in the considerations we have here and his suc
presented, consists in the endeavor to reduce the more
complicated case of the lever to the simpler and ap
parently selfevident case, to discern the simpler in the
more complicated, or vice versa. In fact, we regard
a phenomenon as explained, when we discover in it
known simpler phenomena.
But surprising as the achievement of Archimedes
and his successors may at the first glance appear to
us, doubts as to the correctness of it, on further reflec
i 4 THE SCIENCE OF MECHANICS.
Critique of tion, nevertheless spring up. From the mere assump
ods! r me "tion of the equilibrium of equal weights at equal dis
tances is derived the inverse proportionality of weight
and leverarm ! How is that possible ? If we were
unable philosophically and a priori to excogitate the
simple fact of the dependence of equilibrium on weight
and distance, but were obliged to go for that result to
experience, in how much less a degree shall we be able,
by speculative methods, to discover the form of this
dependence, the proportionality !
Thestaticai As a matter of fact, the assumption that the equi
voivedin librium disturbing effect of a weight P at the distance
all their de . e . . ... .
auctions. L from the axis of rotation is measured by the product
P.L (the socalled statical moment), is more or less
covertly or tacitly introduced by Archimedes and all
his successors. For when Archimedes substitutes for
a Targe weight a series of symmetrically arranged pairs
of small weights, which weights extend beyond the point
of support, he employs in this very act the doctrine of
the centre of gravity in its more general form, which is
itself nothing else than the doctrine of the lever in its
more general form. (See Appendix, III., p. 512.)
without it Without the assumption above mentioned of the im
tiSniffm*" port of the product P.L, no one can prove (Fig. 8)
possible.  . , . , .
that a bar, placed in
any way on the ful
crum S, is supported,
with the help of a
I string attached to its
> centre of gravity and
carried over a pulley,
by a weight equal to its own weight. But this is con
tained in the deductions of Archimedes, Stevinus,
Galileo, and also in that of Lagrange.
THE PRINCIPLES OF STATICS,
6. HUYGENS, indeed, reprehends this method, and Huygens's
i rr TI  * i .,,,,. n criticism.
gives a different deduction, m which he believes he has
avoided the error. If in
the presentation of La
grange we imagine the
two portions into which
the prism is divided
turned ninety degrees
about two vertical axes
passing through the cen
tres of gravity s,s' of the
prismportions (see Fig.
go), and it be shown
that under these circum
stances . equilibrium still D
continues to subsist, we Fig. 9.
shall obtain the Huygenian deduction. Abridged
and simplified, it is as follows. In a rigid weightless
Fig. ga. Fig. ga.
plane (Fig. 9) through the point S we draw a straight
line, on which we cut off on the one side the length i
i6
THE SCIENCE OF MECHANICS.
HIS own de and on the other the length 2, at A and B respectively.
On the extremities, at right angles to this straight
line, we place, with the centres as points of contact, the
heavy, thin, homogeneous prisms CD and EF, of the
lengths and weights 4 and 2. Drawing the straight
line HSG (where AG = %AC) and, parallel to it, the
line CF, and translating the prismportion CG by par
allel displacement to FH, everything about the axis
GH is symmetrical and equilibrium obtains. But
equilibrium also obtains for the axis AB ; obtains con
sequently for every axis through S, and therefore also
for that at right angles to AB : wherewith the new
case of the lever is given.
Apparently Apparently, nothing else is assumed here than that
unimpeach , , i 1 i
able. equal weights /,/ (Fig. 10) m the same plane and at
equal distances /,/ from an axis A A' (in this plane)
equilibrate one another. If we place ourselves in the
plane passing through A A perpendicularly to /,/, say
y
M
Fig. 10. Fig, ii.
at the point M, and look now towards A and now
towards A r , we shall accord to this proposition the
same evidentness as to the first Archimedean proposi
tion. The relation of things is, moreover, not altered if
we institute with the weights parallel displacements
with respect to the axis, as Huygens in fact does.
THE PRINCIPLES OF STATICS. 17
The error first arises in the inference : if equilib Yet invoiv
rium obtains for two axes of the plane, it also obtains final infer
for every other axis passing through the point of inter ror.
section of the first two. This inference (if it is not to
be regarded as a purely instinctive one) can be drawn
only upon the condition that disturbant effects are as
cribed to the weights proportional to their distances
from the axis. But in this is contained the very kernel
of the doctrine of the lever and the centre of gravity.
Let the heavy points of a plane be referred to a
system of rectangular coordinates (Fig. u). The co
ordinates of the centre of gravity of a system of masses
m m m" . . . having the coordinates x x' x" . . . y y' y" . . .
are, as we know,
M a them at
,, ~2.mx lEmy ic 1 discus
C,  ,. 77 rrr:  . sion of
2m 2m Huygens's
Inference.
If we turn the system through the angle a, the new co
ordinates of the masses will be
x^ = x cosa y sina, y\ =y cosa \ x sin/r
and consequently the coordinates of the centre of
gravity
^ 2m (x cosa rsintf") 2mx . "2 my
= 5" cosa 77 sina
and, similarly,
7^ = 77 cosa + <? sina.
We accordingly obtain the coordinates of the new
centre of gravity, by simply transforming the coordi
nates of the first centre to the new axes. The centre
of gravity remains therefore the selfsa7?ie point. If
we select the centre of .gravity itself as origin, then
2mx = 2my=Q. On turning the system of axes, this
relation continues to subsist. If, accordingly, equi
20
THE SCIENCE OF MECHANICS.
eatment
the lever
modern
.ysicists.
In the direction here Indicated the Archimedean
view certainly remained a serviceable one even after
no one longer entertained any doubt of the significance
of the product P.L, and after opinion on this point had
been established historically and by abundant verifica
tion. (See Appendix, IV., p. 514.)
7'. The manner in which the laws of the lever, as
handed down to us from Archimedes in their original
simple form, were further generalised and treated by
modern physicists, is very interesting and instructive.
LEONARDO DA VINCI (14521519), the famous painter
and investigator, appears to have been the first to rec
ognise the importance of the general notion of the so
B
D
onardo
Vinci
c
Fig. 13.
called statical moments. In the manuscripts he has
521519). left us, several passages are found from which this
clearly appears. He says, for example : We have a
bar AD (Fig. 13) free to rotate about A> and suspended
from the bar a weight P, and suspended from a string
which passes over a pulley a second weight Q. What
must be the ratio of the forces that equilibrium may ob
tain? The lever arm for the weight P is not AD, but
the "potential" lever AB. The leverarm for the
weight <2 is not AD, but the "potential" lever AC,
The method by which Leonardo arrived at this view
is difficult to discover. But it is clear that he recog
THE PRINCIPLES OF STATICS, 21
nised the essential circumstances by which the effect
of the weight is determined.
Considerations similar to those of Leonardo da Guido
UbaldJ
Vinci are also found in the writings of GUIDO UBALDI.
8. We will now endeavor to obtain some idea of
the way in which the notion of statical moment, by
which as we know is understood the product of a force
into the perpendicular let fall from the axis of rotation
upon the line of direction of the force, could have been
arrived at, although the way that really led to this
idea is not now fully ascertainable. That equilibrium
exists (Fig. 14) if we lay a ^
cord, subjected at both sides
to equal tensions, over a
pulley, is perceived without
difficulty. We shall always
find a plane of symmetry for
the apparatus the plane
which stands at'right angles
to the plane of the cord and bisects (EE] the angle made
by its two parts. The motion that might be supposed A method
11 i  i 11 . by which
possible cannot in this case be precisely determined or the notion
defined by any rule whatsoever : no motion will there icai mo
ment might
fore take place. If we note, now, further, that the mate have been
. 777 arrived at.
rial of which the pulley is made is essential only to the
extent of determining the form of motion of the points
of application of the strings, we shall likewise readily
perceive that almost any portion of the pulley may
be removed without disturbing the equilibrium of
the machine. The rigid radii that lead out to the tan
gential points of the string, are alone essential. We
see, thus, that the rigid radii, (or the perpendiculars on
the linear directions of the strings) play here a part
similar to the leverarms in the lever of Archimedes.
THE SCIENCE OF MECHANICS.
This notion
derived
from the
considera
tion of a.
wheel and
axle.
Let us examine a socalled wheel and axle (Fig.
15) of wheelradius 2 and axleradius i, provided re
spectively with the cordhung loads i and 2 ; an appa
ratus which corresponds in every respect to the lever
of Archimedes. If now we place about the axle, in
any manner we may choose, a second cord, which we
subject at each side to the tension of a weight 2, the
second cord will not disturb the equilibrium. It is
plain, however, that we are also permitted to regard
Fig. 15. ^ Fig. 16.
the two pulls marked in Fig. 16 as being in equilib
rium, by leaving the two others, as mutually destruc
tive, out of account. But we arrive in so doing, dis
missing from consideration all unessential features, at
the perception that not only the pulls exerted by the
weights but also the perpendiculars let fall from the
axis on the lines of the pulls, are conditions deter
minative of motion. The decisive factors are, then,
the products of the weights into the respective per
pendiculars let fall from the axis on the directions of
the pulls ; in other words, the socalled statical mo
ments.
The princi 9. What we have so far considered, is the devel
FeverVn 6 opment of our knowledge of the principle of the lever,
explain" the Quite independently of this was developed the knowl
chines, a edge of the principle of the inclined plane. It is not
necessary, however, for the comprehension of the ma
THE PRINCIPLES OF STATICS,
chines, to search after a new principle beyond that of
the lever ; for the latter is sufficient by itself. Galileo,
for example, explains the inclined plane from the lever
in the following manner.
We have before us (Fig.
17) an inclined plane, on
which rests the weight
Q, held in equilibrium
by the weight P. Gali
leo, now, points out the Fi s J 7
fact, that it is not requisite that Q should lie directly
upon the inclined plane, but that the essential point
is rather the form, or character, of the motion
of Q. We may, consequently, conceive the weight
attached to the bar AC, perpendicular to the inclined
plane, and rotatable about C. If then we institute a Galileo's
. . explanation
very slight rotation about the point C, the weight will of the in
move in the element of an arc coincident with the ini
clined plane. That the path assumes a curve on the
motion being continued is of no consequence here,
since this further movement does not in the case of
equilibrium take place, and the movement of the in
stant alone is decisive. Reverting, however, to the
observation before mentioned of Leonardo da Vinci,
we readily perceive the validity of the theorem Q. CB
= P.CA or Q/P CA/C = ca/&, and thus reach
the law of equilibrium on the inclined plane. Once we
have reached the principle of the lever, we may, then,
easily apply that principle to the comprehension of
the other machines.
the lever.
THE SCIENCE OF MECHANICS.
THE PRINCIPLE OF THE INCLINED PLANE.
stevinus i. STEViNUS, or STEViN, (15481620) was the first
(15481620) . . , , . \ ,  r LI
first investi who investigated the mechanical properties of the m
mechanics clined plane , and he did so in an eminently original
of the in _. ._,._.
clined manner. If a weight lie (Fig.
pane ' 1 8) on a horizontal table, we
perceive at once, since the
pressure is directly perpendic
ular to the plane of the table,
by the principle of symmetry,
 l8  that equilibrium subsists. On a
vertical wall, on the other hand, a weight is not at all
obstructed in its motion of descent. The inclined plane
accordingly will present an intermediate case between
these two limiting suppositions. Equilibrium will not
exist of itself, as it does on the horizontal support, but
it will be maintained by a less weight than that neces
sary to preserve it on the vertical wall. The ascertain
ment of the statical law that obtains in this case, caused
the earlier inquirers considerable difficulty.
Hismodeof Stevinus's manner of procedure is in substance as
reaching its r . TT . , . , . ^ i
law. follows. .He imagines a triangular prism with horizon
tally placed edges, a crosssection of which AJ3C is
represented in Fig. 19. For the sake of illustration
we will say that AB ' = iBC ; also that AC is, horizon
tal. Over this prism Stevinus lays an endless string
on which 14 balls of equal weight are strung and tied
at equal distances apart. We can advantageously re
place this string by an endless uniform chain or cord.
The chain will either be in equilibrium or it will not.
If we assume the latter to be the case, the chain, since
THE PRINCIPLES OF STATICS.
the conditions of the event are not altered by its mo
tion, must, when once actually in motion, continue to
move for ever, that is, it must present a perpetual mo
tion, which Stevinus deems absurd. Consequently only
the first case is conceivable. The chain remains in equi
librium. The symmetrical portion ADC may, there
fore, without disturbing the equilibrium, be removed.
The portion AB of the chain consequently balances
the portion BC. Hence : on inclined planes of equal
heights equal weights act in the inverse proportion of
the lengths of the planes.
Stevinus's
deduction
of the law
of the in
clined
plane.
B
A
Fig. 19. Fig. 20.
In the crosssection of the prism in Fig. 20 let us
Imagine AC horizontal, BC vertical, and AB = 2.BC;
furthermore, the chainweights Q and P on AB and
BC proportional to the lengths ; it will follow then that
26 THE SCIENCE OF MECHANICS.
= 2. The generalisation is selfevi
dent.
The as 2. Unquestionably in the assumption from which
ofst p eii ns Stevinus starts, that the endless chain does not move.
dScticmT there is contained primarily only a purely instinctive
cognition. He feels at once, and we with him, that
we have never observed anything like a motion of the
kind referred to, that a thing of such a character does
not exist. This conviction has so much logical cogency
that we accept the conclusion drawn from it respecting
the law of equilibrium on the inclined plane without the
thought of an objection, although the law if presented
as the simple result of experiment, or otherwise put,
Their in would appear dubious. We cannot be surprised at this
character, when we reflect that all results of experiment are ob
scured by adventitious circumstances (as friction, etc.),
and that every conjecture as to the conditions which are
determinative in a given case is liable to error. That
Stevinus ascribes to instinctive knowledge of this sort
a higher authority than to simple, manifest, direct ob
servation might excite in us astonishment if \ve did not
ourselves possess the same inclination. The question
accordingly forces itself upon us : Whence does this
higher authority come ? If we remember that scientific
demonstration, and scientific criticism generally can
only have sprung from the consciousness of the individ
ual fallibility of investigators, the explanation is not far
Their cog to seek. We feel clearly, that we ourselves have con
tributed nothing to the creation of instinctive knowl
edge, that we have added to it nothing arbitrarily, but
that it exists in absolute independence of our partici
pation. Our mistrust of our own subjective interpre
tation of the facts observed, is thus dissipated.
Stevinus's deduction is one of the rarest fossil in
THE PRINCIPLES OF STATICS. 27
dications that we possess in the primitive history of Highhistor
. ical value of
mechanics, and throws a wonderful light on the pro stevinus's
r ^ r r ,, . . deduction.
cess 01 the formation of science generally, on its rise
from instinctive knowledge. We will recall to mind
that Archimedes pursued exactly the same tendency
as Stevinus, only with much less good fortune. In
later times, also, instinctive knowledge is very fre
quently taken as the startingpoint of investigations.
Every experimenter can daily observe in his own per
son the guidance that instinctive knowledge furnishes
him. If he succeeds in abstractly formulating what
is contained in it, he will as a rule have made an im
portant advance in science.
Stevinus's procedure is no error. If an error were The trust
,.. T _....,. . _... worthiness
contained in it, we should all share it. Indeed, it isofinstinc
r , . . tive knowl
perfectly certain, that the union of the strongest in edge,
stinct with the greatest power of abstract formulation
alone constitutes the great natural inquirer. This by
no means compels us, however, to create a new mysti
cism out of the instinctive in science and to regard this
factor as infallible. That it is not infallible, we very
easily discover. Even instinctive knowledge of so
great logical force as the principle of symmetry em
ployed by Archimedes, may lead us astray. Many of
my readers will recall to mind, perhaps, the intellectual
shock they experienced when they heard for the first
time that a magnetic needle lying in the magnetic
meridian is deflected in a definite direction away from
the meridian by a wire conducting a current being car
ried along in a parallel direction above it. The instinc
tive is just as fallible as the distinctly conscious. Its only
value is in provinces with which we are very familiar.
Let us rather put to ourselves, in preference to
pursuing mystical speculations on this subject, the
28 THE SCIENCE OF MECHANICS.
The origin question: How does instinctive knowledge originate
of instinc . .
tiveknowi and what are its contents? Everything which we ob
serve in nature imprints itself un 'comprehended and itn
analysed in our percepts and ideas, which, then, in their
turn, mimic the processes of nature in their most gen
eral and most striking features. In these accumulated
experiences we possess a treasurestore which is ever
close at hand and of which only the smallest portion
is embodied in clear articulate thought. The circum
stance that it is far easier to resort to these experi
ences than it is to nature herself, and that they are,
notwithstanding this, free, in the sense indicated, from
all subjectivity, invests them with a high value. It
is a peculiar property of instinctive knowledge that it
is predominantly of a negative nature. We cannot so
well say what must happen as we can what cannot hap
pen, since the latter alone stands in glaring contrast to
the obscure mass of experience in us in which single
characters are not distinguished.
instinctive Still, great as the importance of instinctive knowl
knowledge , . n .
and extern edge may be, lor discovery, we must not, from our
mutually point of view, rest content with the recognition of its
condition , *,,. . .
each other, authority. We must inquire, on the contrary : Under
what conditions could the instinctive knowledge in
question have originated? We then ordinarily find that
the v.ery principle to establish which we had recourse
to instinctive knowledge, constitutes in its turn the fun
damental condition of the origin of that knowledge,
And this is quite obvious and natural. Our instinctive
knowledge leads us to the principle which explains that
knowledge itself, and which is in its turn also corrobo
rated by the existence of that knowledge, which is a
separate fact by itself. This we will find on close ex
amination is the state of things in Stevinus's case.
THE PRINCIPLES OF STATICS. 29
3. The reasoning of Stevlnus impresses us as soThemgen
111 i i i i  , , uityofSte
nignly ingenious because the result at which he arrives vinus's rea
, . , . soning.
apparently contains more than the assumption from
which he starts. While on the one hand, to avoid con
tradictions, we are constrained to let the result pass, on
the other an incentive remains which impels us to seek
further insight. If Stevinus had distinctly set forth
the entire fact in all its aspects, as Galileo subsequently
did, his reasoning would no longer strike us as Ingen
ious ; but we should have obtained a much more satis
factory and clear insight Into the matter. In the
endless chain which does not glide upon the prism, is
contained, in fact, everything. We might say, the
chain does not glide because no sinking of heavy bodies
takes place here. This would not be accurate, how
ever, for when the chain moves many of its links really
do descend, while others rise in their place. We must
say, therefore, more accurately, the chain does not
glide because for everybody that could possibly de critique uf
& J J J Stevmus's
scend an equally heavy body would have to ascend deduction,
equally high, or a body of double the weight half the
height, and so on. This fact was familiar to Stevinus,
who presented it, indeed, in his theory of pulleys ;
but he was plainly too distrustful of himself to lay
down the law, without additional support, as also valid
for the inclined plane. But if such a law did not exist
universally, our instinctive knowledge respecting the
endless chain could never have originated. With this
our minds are completely enlightened. The fact that
Stevinus did not go as far as this in his reasoning and
rested content with bringing his (indirectly discovered)
ideas into agreement with his instinctive thought, need
not further disturb us. (See p. 515.)
The service which Stevinus renders himself and his
30 THE SCIENCE OF MECHANICS.
The merit readers, consists, therefore, in the contrast and com
Su?s e proce parison of knowledge that is instinctive with knowledge
dure ' that is clear, in the bringing the two into connection
and accord with one another, and in the supporting
Fig. 21.
the one upon the other. The strengthening of mental
view which Stevinus acquired by this procedure, we
learn from the fact that a picture of the endless chain
and the prism graces as vignette, with the inscription
"Wonder en is gheen wonder," the titlepage of his
THE PRINCIPLES OF STATICS. 31
work Hypomnemata Mathematica (Leyden, 1605).* As Eniighten
r , . , . ment in
a fact, every enlightening progress made In science is science ai
accompanied with a certain feeling of disillusionment, compamed
TT _ . . withdisillu
We discover that that which appeared wonderful to sionment.
us is no more wonderful than other things which we
know instinctively and regard as selfevident ; nay,
that the contrary would be much more wonderful ; that
everywhere the same fact expresses itself. Our puzzle
turns out then to be a puzzle no more ; It vanishes into
nothingness, and takes its place, among the shadows
of history.
4. After he had arrived at the principle of the inExpiana
,. , ^ tion of the
clmed plane, it was easy for Stevmus to apply that other ma
. , . . , . chines by
principle to the other machines and to explain by it stevinus's
principle.
their action. He makes, for example, the following
application.
We have, let us suppose, an inclined plane (Fig.
22) and on it a load Q. We pass a string over the
pulley A at the summit and imagine the load Q held in
equilibrium by the load P.
Stevinus, now, proceeds by
a method similar to that
later taken by Galileo. He
remarks that It is not ne
cessary that the load Q
should lie directly on the
inclined plane. Provided
Fig. 22
only the form of the machine's motion be preserved, the
proportion between force and load will in all cases re
main the same. We may therefore equally well conceive
the load Q to be attached to a properly weighted string
passing over a pulley D: which string is normal to the
*The title given is that of Willebrord Snell's Latin translation (1608) of
Simon Stevin's Wisconstige Gedachtenissen^ Leyden, 1605. Trans.
THE SCIENCE OF MECHANICS.
The funic li
termachine
And the
special case
of the paral
lelogram of
forces.
The general
form of the
lastmen
tioned prin
ciple also
employed.
inclined plane. If we carry out this alteration, we
shall have a socalled funicular machine. We now
perceive that we can ascertain very easily the portion
of weight with which the body on the inclined plane
tends downwards. We have only to draw a vertical
line and to cut off on it a portion ab corresponding to
the load Q. Then drawing on aA the perpendicular
be, we have P/Q = AC JAB = ac/ab. Therefore ac
represents the tension of the string aA. Nothing pre
vents us, now, from making the two strings change
functions and from imagining the load Q to lie on the
dotted inclined plane EDF. Similarly, here, we ob
tain ad for the tension of the second string. In this
manner, accordingly, Stevinus indirectly arrives at a
knowledge of the statical relations of the funicular
machine and of the socalled parallelogram of forces ; at
first, of course, only for the particular case of strings
(or forces) ac, ad at right angles to one another.
Subsequently, indeed, Stevinus employs the prin
ciple of the composition and resolution of forces in
a more general form ; yet the method by which he
X
Fig. 23. Fig. 24.
reached the principle, is not very clear, or at least is
not obvious. He remarks, for example, that if we
have three strings AB y AC, AD, stretched at any
THE PRINCIPLES OF STATICS.
33
given angles, and the weight P is suspended from the
first, the tensions may be determined in the following
manner. We produce (Fig. 23) AB to X and cut off
on it a portion AE. Drawing from the point E, EF
parallel to AD and EG paral
lel to A C, the tensions of AB,
AC, AD are respectively pro
portional to AE, AJF, A G.
With the assistance of this
principle of construction Ste
vinus solves highly compli
o
o
Fig. 25
O
pr ems<
cated problems. He determines, for instance, the solution of
tensions of a system of ramifying strings like that pH C
illustrated in Fig. 24; in doing which of course he
starts from the given tension of the vertical string.
The relations of the tensions of a funicular polygon
are likewise ascertained by construction, in the man
ner indicated in Fig. 25.
We may therefore, by means of the principle of the General re
inclined plane, seek to elucidate the conditions of op
eration of the other simple machines, in a manner sim
ilar to that which we employed in the case of the prin
ciple of the lever.
THE PRINCIPLE OF THE COMPOSITION OF FORCES.
i. The principle of the parallelogram of forces, at The
which STEVINUS arrived and employed, (yet without ex
pressly formulating it,) consists, as we know, of the
following truth. If a body A (Fig. 26") is acted upon
by two forces whose directions coincide with the lines
AB and A C, and whose magnitudes are proportional to
the lengths AB and AC, these two forces produce the
;ram of
forces.
34
THE SCIENCE OF MECHANICS.
Flg<
same effect as a single force, which acts In the direction
of the diagonal AD of the parallelogram A BCD and is
proportional to that diagonal. For instance, if on the
strings AB, AC weights
exactly proportional to the
lengths AB, AC be sup
posed to act, a single
weight acting on the string
^Z>exactlyproportional to
the length AD will produce the same effect as the first
two. The forces AB and AC are called the compo
nents, the force AD the resultant. It is furthermore
obvious, that conversely, a single force is replaceable
by two or several other forces.
Method by 2. We shall now endeavor, in connection with the
which the . . .
general no investigations of Stevmus, to give ourselves some idea
tion of the . . .....
of the manner in which the
general proposition of the
parallelogram of forces
might have been arrived
at. The relation, dis
covered by Stevinus,
that exists between two
mutually perpendicular
forces and a third force
that equilibrates them, we
shall assume as (indi
rectly) given. We sup
pose now (Fig. 27) that
there act on three strings
Fig ' 27 ' OX, OY, O2, pulls which
balance each other. Let us endeavor to determine the
nature of these pulls. Each pull holds the two remain
ing ones in equilibrium. The pull OYwe will replace
parallelo
gram of
forces
might have
been ar
rived at.
THE PRINCIPLES OF STATICS.
35
(following Stevinus's principle") by two new rectangular The deduc
. ,. . ' don of the
pulls, one in the direction Ou (the prolongation of general
OX}, and one at right angles thereto in the direction from the
Ov, And let us similarly resolve the pull OZ in theofstevinus.
directions Ou and Ow. The sum of the pulls in the di
rection Ou, then, must balance the pull OX, and the
two pulls in the directions Ov and Ow must mutually
destroy each other. Taking the two latter as equal
and opposite, and representing them by Om and On,
we determine coincidently with the operation the com
ponents Op and Oq parallel to Ott, as well also as the
pulls Or, Os. Now the sum Op + Oq is equal and op
posite to the pull in the direction of OX; and if we
draw sf parallel to O Y, or rt parallel to OZ, either line
will cut off the portion Ot = Op + Oq : with which re
sult the general principle of the parallelogram of forces
is reached.
The general case of composition may be deduced A different
.,, . .. , . . . . .mode of the
in still another way from the special composition of same de
rectangular forces. Let OA and OB be the two forces
acting at O. For OB substitute
a force OC acting parallel to
OA and a force OD acting at
right angles to OA. There
then act for OA and OB the
two forces OE = OA + OC
and OD, the resultant of which forces OF is at the same
time the diagonal of the parallelogram OAFB con
structed on OA and OB as sides.
3. The principle of the parallelogram of forces, The prin
^ JT JT j. ^ ^ ciple here
when reached by the method of Stevinus, presents it presents it
... self as an
self as an indirect discovery. It is exhibited as a con indirect
discovery*
sequence and as the condition of known facts. We
perceive, however, merely that it does exist, not, as yet
Fig> 28<
THE SCIENCE OF MECHANICS.
And is first why it exists ; that is, we cannot reduce it (as in dy
en 6 ifnciated namics} to still simpler propositions. In statics, in
by Newton
and Vang deed, the principle was not fully admitted until the
time of Varignon, when dynamics, which leads directly
to the principle, was already so far advanced that its
adoption therefrom presented no difficulties. The prin
ciple of the parallelogram of forces was first clearly
enunciated by NEWTON in his Principles of Natural Phi
losophy. In the same year, VARIGNON, independently of
Newton, also enunciated the principle, in a work sub
mitted to the Paris Academy (but not published un
til after its author's death), and made, by the aid of a
geometrical theorem, extended practical application
of it.*
The geometrical theorem referred to is this. If we
consider (Fig. 29) a parallelogram the sides of which
are/ and q, and the diagonal is r, and from any point m
in the plane of the par
allelogram w r e draw per
pendiculars on these
three straight lines,
which perpendiculars
we will designate as
?/, 7;, w, then p . u \
q . v = r . w. This is
easity proved by draw
ing straight lines from ;//
The geo
metrical
theorem
employed
by Varig
non.
Fig. 29.
Fig, 30.
to the extremities of the diagonal and of the sides of
the parallelogram, and considering the areas of the
triangles thus formed, which are equal to the halves
of the products specified. If the point m be taken
within the parallelogram and perpendiculars then be
to his
* In the same year, 1687, Father Bernard Lami published, a little appendix
lis TraztS de mechanigue> developing the same principle. Trans.
THE PRINCIPLES OF STATICS, 37
drawn, the theorem passes into the form f ,u q . v
= r . w. Finally, if m be taken on the diagonal and
perpendiculars again be drawn, we shall get, since the
perpendicular let fall on the diagonal is now zero,
p. u q . v = or p . u = q . v.
With the assistance of the observation that forces The deduc
are proportional to the motions produced by them in
equal intervals of time, Varignon easily advances from
the composition of motions to the composition of forces.
Forces, which acting at a point are represented in
magnitude and direction by the sides of a parallelo
gram, are replaceable by a single iorce, similarly rep
resented by the diagonal of that parallelogram.
If now, in the parallelogram considered,/ and ^Moments of
represent the concurrent forces (the components) and r forces 
the force competent to take their place (the resultant),
then the products pu, qv, rw are called the moments
of these forces with respect to the point m. If the point
m lie in the direction of the resultant, the two moments
pu and qv are with respect to it equal to each other.
4. With the assistance of this principle Varignon is varig
now in a position to treat
the machines in 'a much
simpler manner than were
his predecessors. Let us
consider, for example,
(Fig. 31) a rigid body
capable of rotation about
an axis passing through
O. Perpendicular to the
axis we conceive a plane,
and select therein two Flg ' 3I "
points A 9 B, on which two forces P and Q in the plane
are supposed to act. We recognise with Varignon
38 THE SCIENCE OF MECHANICS.
The deduc that the effect of the forces is not altered if their points
law of the of application be displaced along their line of action,
lever from . ,, . , i ji
the parai since all points in the same direction are rigidly con
principle, nected with one another and each one presses and pulls
the other. We may, accordingly, suppose P applied
at any point in the direction A X, and Q at any point
in the direction BY, consequently also at their point
of intersection M. With the forces as displaced to Jlf,
then, we construct a parallelogram, and replace the
forces by their resultant. We have now to do only
with the effect of the latter. If it act only on movable
points, equilibrium will not obtain. If, however, the
direction of its action pass through the axis, through
the point (9, which is not movable, no motion can take
place and equilibrium will obtain. In the latter case
O is a point on the resultant, and if we drop the per
pendiculars u and v from O on the directions of the
forces p) q, we shall have, in conformity with the the
orem before mentioned, /  ?/ = q v. With this we
have deduced the law of the lever from the principle
of the parallelogram of forces.
The statics Varignon explains in like manner a number of other
adyn a amicai cases of equilibrium by the equilibration of the result
ant force by some obstacle or restraint. On the in
clined plane, for example, equilibrium exists if the re
sultant is found to be at right angles to the plane. In
fact, Varignon rests statics in its entirety on a dynamic
foundation ; to his mind, it is but a special case of dy
namics. The more general dynamical case constantly
hovers before him and he restricts himself in his inves
tigation voluntarily to the case of equilibrium. We
are confronted here with a dynamical statics, such
as was possible only after the researches of Galileo.
Incidentally, it may be remarked, that from Varignon
THE PRINCIPLES OF STATICS,
39
is derived the majority of the theorems and methods
of presentation which make up the statics of modern
elementary textbooks.
5. As we have already seen, purely statical consid Special
11, i   , , 11 i statical con
erations also lead to the proposition of the parallel siderations
r r . . . . also lead to
ogram of forces. In special cases, in fact, the principle the pnn
admits of being very easily verified. We recognise at
once, for instance, that any number whatsoever of equal
forces acting (by pull or pressure) in the same plane at
a point, around which their suc
cessive lines make equal angles,
are in equilibrium. If, for exam
ple, (Fig. 32) the three equal
forces OA, OB, OC act on the
point O at angles of 120, each
two of the forces holds the third
in equilibrium. We see imme
diately that the resultant of OA
and OB is equal and opposite to OC. It is represented
by OD and is at the same time the diagonal of the
parallelogram OAJDB, which readily follows from the
fact that the radius of a circle is also the side of the
hexagon included by it.
6. If the concurrent forces act in the same or in The case of
opposite directions, the resultant is equal to the sum forces en
or the difference of the mere y a
components. We rec
ognise both cases with
out any difficulty as
particular cases of the
principle of the paral
lelogram of forces. If in the two drawings of Fig. 33
we imagine the angle A OB to be gradually reduced
to the value o, and the angle A' O r B' increased to the
Fig. 32.
particular
caste of the
general
principle.
0'
Fig. 33
40 THE SCIENCE OF MECHANICS.
. value 1 80, we shall perceive that O C passes into OA j
AC = OA + OB and O' C" into O' A' A f C = O f A f
O' B'. The principle of the parallelogram of forces
includes, accordingly, propositions which are generally
made to precede it as independent theorems.
The princi j. The principle of the parallelogram of forces, in
shionde the form in which it was set forth by Newton and
rived from . .
experience. Vangnon, clearly discloses itself as a proposition de
rived from experience. A point acted on by two forces
describes with accelerations proportional to the forces
two mutually independent motions. On this fact the
parallelogram construction is based. DANIEL BER
NOULLI, however, was of opinion that the proposition of
the parallelogram of forces was a geometrical truth, in
dependent of physical experience. And he attempted
to furnish for it a geometrical demonstration, the chief'
features of which we shall here take into consideration,
as the Bernoullian view has not, even at the present
day, entirely disappeared.
Daniel Ber If two equal forces, at right angles to each other
tempted f (Fig. 34), act on a point, there can be no doubt, ac
demonstra' n cording to Bernoulli, that the line
tion of the n+ *& <7 . . . .  , ,
truth. ' /\\\ * bisection of the angle (con
formably to the principle of sym
metry) is the direction of the re
sultant r. To determine geomet
rically also the magnitude of the
resultant, each of the forces p is
Flg 34> decomposed into two equal forces
q, parallel and perpendicular to r. The relation in
respect of magnitude thus produced between / and q
is consequently the same as that between r and /. We
have, accordingly :
p = j2 . q and r = ^ . p\ whence r = ^q.
THE PRINCIPLES OF STA TICS. 41
Since, however, the forces q acting at right angles
to r destroy each other, while those parallel to r con
stitute the resultant, it further follows that
r = Zq\ hence jj. T/2, and r = ]/2~. p.
The resultant, therefore, is represented also in re
spect of magnitude by the diagonal of the square con
structed on p as side.
Similarly, the magnitude maybe determined of the The case of
resultant of unequal rectangular components. Here, rectangular
i , i , IP components
however, nothing is known before *
hand concerning the direction of
the resultant r. If we decompose ^ ' ^ '
the components p, q (Fig. 35),
parallel and perpendicular to the
yet undetermined direction r, into
the forces u 9 s and v, /, the new
forces will form with the compo V
nents p, q the same angles that p, Flg> 35 '
q form with r. From which fact the following relations
in respect of magnitude are determined :
r p r q r p r q
 and = , =" and = ,
p u q v q s p t
from which two latter equations follows s = t =pqjr.
On the other hand, however,
z> 2 <? 2
r = u + V = +  or r* =p* f g*.
The diagonal of the rectangle constructed on p and
q represents accordingly the magnitude of the result
ant.
Therefore, for all rhombs, the direction of the re General re
sultant is determined ; for all rectangles, the magni
tude; and for squares both magnitude and direction.
Bernoulli then solves the problem of substituting for
42 THE SCIENCE OF MECHANICS.
two equal forces acting at one given angle, other equal,
equivalent forces acting at a different angle ; and finally
arrives by circumstantial considerations, not wholly
exempt from mathematical objections, but amended
later by Poisson, at the general principle.
Critique of 8. Let us now examine the physical aspect of this
Bernoulli's . . . . . .
method. question. As a proposition derived irom experience,
the principle of the parallelogram of forces was already
known to Bernoulli. What Bernoulli really does, there
fore, is to simulate towards himself a complete ignorance
of the proposition and then attempt to philosophise
it abstractly out of the fewest possible assumptions.
Such work is by no means devoid of meaning and pur
pose. On the contrary, we discover by such proce
dures, how few and how imperceptible the experiences
are that suffice to supply a principle. Only we must
not deceive ourselves, as Bernoulli did ; we must keep
before our minds all the assumptions, and should over
look no experience which we involuntarily employ.
What are the assumptions, then, contained in Bernoul
li's deduction?
The as 9. Statics, primarily, is acquainted with force only
of w^de 1  8 as a pull or a pressure, that from whatever source it
rivedfrom e ?rnay come always admits of being replaced by the pull
experience, or ^ p ressure of a weight. All forces thus may be re
garded as quantities of the same kind &s\& be measured
by weights. Experience further instructs us, that the
particular factor of a force which is determinative of
equilibrium or determinative of motion, is contained
not only in the magnitude of the force but also in its
direction, which is made known by the direction of the
resulting motion, by the direction of a stretched cord,
or in some like manner. We may ascribe magnitude
indeed to other things given in physical experience,
THE PRINCIPLES OF STATICS. 43
such as temperature, potential function, but not direc
tion. The fact that both magnitude and direction are
determinative in the efficiency of a force impressed on
a point is an important though it may be an unob
trusive experience.
Granting, then, that the magnitude and direction Magnitude
c f , . ,...., and direc
oi forces impressed on a point alone are decisive, it will tion the sole
be perceived that two equal and opposite forces, as they factors.
cannot uniquely and precisely determine any motion,
are in equilibrium. So, also, at
right angles to its direction, a
force/ is unable uniquely to de
termine a motional effect. But 
if a force p is inclined at an an
gle to another direction ss' (Fig.
36), it is able to determine a mo _
tion in that direction. Yet ex s>
perience alone can inform us, lg> 3 '
that the motion is determined in the direction of s's
and not in that of ss' ; that is to say, in the direction
of the side of the acute angle or in the direction of the
projection of/ on s's.
Now this latter experience is made use of by
noulli at the very start. The sense, namely, of the re derivable
sultant of two equal forces acting at right angles to one experience.
another is obtainable only on the ground of this expe
rience. From the principle of symmetry follows only,
that the resultant falls in the plane of the forces and
coincides with the line of bisection of the angle, not
however that it falls in the acute angle. But if we sur
render this latter determination, our whole proof is ex
ploded before it is begun.
10. If, now, we have reached the conviction that
our knowledge of the effect of the direction of a force is
44 THE SCIENCE OF MECHANICS.
So also solely obtainable from experience, still less then shall
must the . . .
form of the we believe it in our power to ascertain by any other way
thusde the/0r/# of this effect. It is utterly out of our power,
to divine, that a force p acts in a direction s that makes
with its own direction the angle a, exactly as a force
/ cos a in the direction s ; a statement equivalent to the
proposition of the parallelogram of forces. Nor was
it in Bernoulli's power to do this. Nevertheless, he
makes use, scarcely perceptible it is true, of expe
riences that involve by implication this very mathe
matical fact.
The man A person already familiar with the composition
which the and resolution of forces is well aware that several forces
assump
tions men acting at a point are. as regards their effect, replaceable,
tionedenter to r ' ,. ...
into Ber m every respect and in every direction, by a single force.
noulli's de J J > J ^
auction. This knowledge, in Bernoulli's mode of proof, is ex
pressed in the fact that the forces p, q are regarded as
absolutely qualified to replace in all respects the forces
s, u and /, v, as well in the direction of r as in every
other direction. Similarly r is regarded as the equiv
alent of / and q. It is further assumed as wholly in
different, whether we estimate s, u, /, v first in the
directions of/, q, and then/, q in the direction of r, or
s, u, t, v be estimated directly and from the outset in
the direction of r. But this is something that a person
only can know who has antecedently acquired a very
extensive experience concerning the composition and
resolution of forces. We reach most simply the knowl
edge of the fact referred to, by starting from the knowl
edge of another fact, namely that a force / acts in a
direction making with its own an angle a, with an effect
equivalent to p cos a. As a fact, this is the way the
perception of the truth was reached.
Let the coplanar forces P 9 P' } P". . . be applied to
THE PRINCIPLES OF STATICS. 45
one and the same point at the angles a, a', a" . . . with Mathemat
.!_. & icalanaly
a given direction A. These forces, let us suppose, are sis of the
; results of
replaceable by a single force 77, which makes with X the true and
an angle /r. By the familiar principle we have then assumption.
If U is still to remain the substitute of this system of
forces, whatever direction X may take on the system
being turned through any angle d, we shall further
have
2P cos (a+ d)=n cos O + tf),
or
(ISPcosa 77cos//) cosd (2Psmtx 77 sin/*) sin# = 0.
If we put
cosar IJcosju = ^f,
71 sin//) = ^,
it follows that
^ cosd + B sind = VA* + B* sin (<? + r) = 0,
which equation can subsist for <?z/^ (5 1 only on the con
dition that
A = ^Pcos^ 77 cos/* =
and
J? = (2P sin 71 sinyu) =
whence results
TTcos/j =
From these equations follow for 71 and /* the deter
minate values
_
and
~P sma
46 THE SCIENCE OF MECHANICS.
The actual Granting, therefore, that the effect of a force In every
results not . 11
deducibie direction can be measured by its projection on that di
on any .
other sup rection, then truly every system of forces acting at a
position. . ' , , r i
point is replaceable by a single force, determinate in
magnitude and direction. This reasoning does not hold,
however, if we put in the place of cos a any general func
tion of an angle, cp (a). Yet if this be done, and we still
regard the resultant as determinate, we shall obtain for
<p(pt), as may be seen, for example, from Poisson's
deduction, the form cosar. The experience that several
forces acting at a point are always, in every respect,
replaceable by a single force, is therefore mathemat
ically equivalent to the principle of the parallelogram
of forces or to the principle of projection. The prin
ciple of the parallelogram or of projection is, how
ever, much easier reached by observation than the
General re more general experience above mentioned by statical
mar s ' observations. And as a fact, the principle of the par
allelogram was reached earlier. It would require in
deed an almost superhuman power of 'perception to
deduce mathematically, without the guidance of any
further knowledge of the actual conditions of the ques
tion, the principle of the parallelogram from the gen
eral principle of the equivalence of several forces to a
single one. We criticise accordingly in the deduction
of Bernoulli this, that that which is easier to observe
is reduced to that which is more difficult to observe.
This is a violation of the economy of science. Bernoulli
is also deceived in imagining that he does not proceed
from any fact whatever of observation.
AH addi We must further remark that the fact that the forces
tional as ...  .  ,.....,.
sumption of are independent of one another, which is involved in
the law of their composition, is another experience
which Bernoulli throughout tacitly employs. As long
THE PRINCIPLES OF STATICS. 47
as we have to do with uniform or symmetrical systems
of forces, all equal in magnitude, each can be affected
by the others, even if they are not independent, only
to the same extent and in the same way. Given but
three forces, however, of which two are symmetrical
to the third, and even then the reasoning, provided
we admit that the forces may not be independent, pre
sents considerable difficulties.
11. Once we have been led, directly or indirectly, Discussion
to the principle of the parallelogram of forces, once weacterof the
have perceived it, the principle is just as much an ob prm lpe '
servation as any other. If the observation is recent, it
of course is not accepted with the same confidence as
old and frequently verified observations. We then seek
to support the new observation by the old, to demon
strate their agreement. By and by the new observa
tion acquires equal standing with the old. It is then
no longer necessary constantly to reduce it to the lat
ter. Deduction of this character is expedient only in
cases in which observations that are difficult directly
to obtain can be reduced to simpler ones more easily
obtained, as is done with the principle of the parallel
ogram of forces in dynamics.
12. The proposition of the parallelogram of forces Experimen
has also been illustrated by experiments especially tion of the
, ^ j. j * LI A . 11 principle by
instituted for the purpose. An apparatus very wellacontriv
adapted to this end was contrived by Cauchy. The Cauchy.
centre of a horizontal divided circle (Fig. 37) is marked
by a pin. Three threads /,/',/", tied together at a
point, are passed over grooved wheels r, /, r" 9 which
can be fixed at any point in the circumference of the
circle, and are loaded by the weights /, /', p" . If three
equal weights be attached, for instance, and the wheels
placed at the marks of division o, 120, 240, the point at
THE SCIENCE OF MECHANICS.
Experimen which the strings are knotted will assume a position
tfonofVhe" just above the centre of the circle. Three equal forces
pnncip e. ac ^ n g at an g} es o f j2o, accordingly, are in equilib
rium.
Fig. 37.
If we wish to represent another and different case,
we may proceed as follows. We imagine any two
forces p, q acting at any angle a, represent (Fig. 38)
them by lines, and construct on them as sides a paral
lelogram. We supply, further, a force
equal and opposite to the resultant ;.
The three forces /, q, r hold each
other in equilibrium, at the angles vis
ible from the construction. We now
place the wheels of the divided circle on
the points of division o, a, a f /I, and
load the appropriate strings with the
weights/, q, r. . The point at which the
strings are knotted will come to a position exactly
above the middle point of the circle.
Fig. 38.
THE PRINCIPLES OF STATICS.
THE PRINCIPLE OF VIRTUAL VELOCITIES.
i. We now pass to the discussion of the principle The truth
of virtual (possible) displacements.* The truth of cipie first
. _ remarked
this principle was first remarked by STEVINUS at the by stevinus
close of the sixteenth century in his investigations on
the equilibrium of pulleys and combinations of pulleys.
Stevinus treats combinations of pulleys in the same
way they are treated at the present day. In the case
* Termed in English the principle of "virtual velocities," this being the
original phrase (vitesse virtuelle) introduced by John Bernoulli. See the
text, page 56. The word virtualis seems to have been the fabrication of Duns
Scotus (see the Century Dictionary, under virtual'] ; but virtualiter was used
by Aquinas, and virtus had been employed for centuries to translate 6'vvajLU t
and therefore as a synonym for potentia. Along with many other scholastic
terms, virtual passed into the ordinary vocabulary of the English language.
Everybody remembers the passage in the third book of Paradise Lost,
" Love not the heav'nly Spirits, and how thir Love
Express they, by looks oncly, or do they mix
Irradiance, virtual or immediate touch ? " Milton,
So, we all remember how it was claimed before our revolution that America
had " virtual representation " in parliament. In these passages, as in Latin,
virtual means : existing in effect, but not actually. In the same sense, the
word passed into French ; and was made pretty common among philosophers
by Leibnitz. Thus, he calls innate ideas in the mind of a child, not yet brought
to consciousness, " des connoissances vtrtuelles."
The principle in question was an extension to the case of more than two
forces of the old rule that "what a machine gains in power, it loses in velocity. ,'
Bernoulli's modification reads that the sum of the products of the forces into
their virtual velocities must vanish to give equilibrium. He says, in effect :
give the system any possible and infinitesimal motion you please, and then
the simultaneous displacements of the points of application of the forces,
resolved in the directions of those forces, though they are not exactly velocities,
since they are only displacements in one time, are, nevertheless, virtually
velocities, for the purpose of applying the rule that what a machine gains in
power, it loses in velocity.
Thomson and Tait say : " If the point of application of a force be dis
placed through a small space, the resolved part of the displacement in the di
rection of the force has been called its Virtual Velocity, This is positive or
negative according as the virtual velocity is in the same, or in the opposite,
direction to that of the force." This agrees with Bernoulli's definition which
may be found in Varignon's Nouvelle mecanique. Vol. II, Chap. is. Trans.
THE SCIENCE OF MECHANICS.
stevinus's a (Fig. 39) equilibrium obtains, when an equal weight P
tions on^the is suspended at each side, for reasons already familiar,
of puiieys. m In &, the weight P is suspended by two parallel cords,
d d
p P
Fig 39
each of which accordingly supports the weight P/2,
with which weight in the case of equilibrium the free
end of the cord must also be loaded. In c, P is sus
pended by six cords, and the weighting of the free ex
tremity with P/6 will accordingly produce equilibrium.
In d, the socalled Archimedean or potential pulley,* P
in the first instance is suspended by two cords, each
of which supports P/2 ; one of these two cords in turn
is suspended by two others, and so on to the end, so
that the free extremity will be held in equilibrium by
the weight P/S. If we impart to these assemblages
of pulleys displacements corresponding to a descent of
the weight P through the distance k, we shall observe
that as a result of the arrangement of the cords
the counterweight P
' P/2
P/6
" " P/8
will ascend
a distance h in a
6/1
8/1
c
d
* These terms are not in use in English. Trans.
THE PRINCIPLES OF STATICS. 51
In a system of pulleys in equilibrium, therefore, His conciu
,, ,  , . ,  sions the
the products of the weights into the displacements germ of the
they sustain are respectively equal. (" Ut spatium P "
agentis ad spatium patientis, sic potentia patientis ad
potentiam agentis." Stevini, Hypomnemata, T. IV,
lib. 3, p. 172.) In this remark is contained the germ
of the principle of virtual displacements.
2. GALILEO recognised the truth of the principle in Galileo's
recognition
another case, and that a somewhat more general one ; of the pnn
. .. . cipleinthe
namely, in its application to the inclined plane. On case of the
. .. j _. inclined
an inclined plane (Fig. 40), plane,
the length of which AB is
double the height BC, a load
Q placed on AB is held in
equilibrium by the load P act
ing along the height BC> if
P = <2/2. If the machine be Fig> 4 '
set in motion, P = Q/2 will descend, say, the vertical
distance 7z, and Q will ascend the same distance h along
the incline AB. Galileo, now, allowing the phenom
enon to exercise its full effect on his mind, perceives,
that equilibrium is determined not by the weights
alone but also by their possible approach to and reces
sion from the centre of the earth. Thus, while Q/2 de
scends along the vertical height the distance h, Q as
cends h along the inclined length, vertically, however,
only h/2 j the result being that the products Q(/i/2)
and (<2/2)/z corae out equal on both sides. The eluci
dation that Galileo's observation affords and the light character
. ... i ji i_ i j , i i of Galileo's
it diffuses, can hardly be emphasised strongly enough, observation
The observation is so natural and unforced, moreover,
that we admit it at once. What can appear simpler
than that no motion takes place in a system of heavy
52 THE SCIENCE OF MECHANICS.
bodies when on the whole no heavy mass can descend.
Such a fact appears instinctively acceptable.
Comparison Galileo's conception of the inclined plane strikes
of it with . .
thatofste us as much less ingenious than that of Stevmus, but
vinus. . 1
we recognise it as more natural and more profound. It
is in this fact that Galileo discloses such scientific great
ness : that he had the intellectual audacity to see, in a
subject long before investigated, more than his prede
cessors had seen, and to trust to his own perceptions.
With the frankness that was characteristic of him he
unreservedly places before the reader his own view,
together with the considerations that led him to it.
The Tom 3. ToRRiCELLi, by the employment of the notion of
form of the "centre of gravity." has put Galileo's principle in a
principle.
form in which it appeals still more to our instincts, but
in which it is also incidentally applied by Galileo him
self. According to Torricelli equilibrium exists in a
machine when, on a displacement being imparted to it,
the centre of gravity of the weights attached thereto
cannot descend. On the supposition of a displacement
in the inclined plane last dealt with, P, let us say, de
scends the distance h, in compensation wherefor Q
vertically ascends h . sin a. Assuming that the centre
of gravity does not descend, we shall have
P./iQ./isma .
 __^_  o, or P. h Q. h sin a = 0,
or
If the weights bear to one another some different pro
portion, then the centre of gravity can descend when a
displacement is made, and equilibrium will not obtain.
We expect the state of equilibrium instinctively* when
the centre of gravity of a system of heavy bodies can
THE PRINCIPLES OF STATICS, 53
not descend. The Torricellian form of expression, how
ever, contains in no respect more than the Galilean.
4. As with systems of pulleys and with the inclined The appii
plane, so also the validity of the principle of virtual the princi
displacements is easily demonstrable for the other ma other ma
chines : for the lever, the wheel and axle, and the rest.
In a wheel and axle, for instance, with the radii R, r
and the respective weights P, Q, equilibrium exists,
as we know, when PR~ Qr. If we turn the wheel
and axle through the angle a, P will descend Ra, and
Q will ascend ra. According to the conception of
Stevinus and Galileo, when equilibrium exists, P. Ra
= Q . ra, which equation expresses the same thing as
the preceding one.
5. When we compare a system of heavy bodies inxhecrite
, . , . , , . , . , ., rion of the
which motion is taking place, with a similar system state of
 . , . . .,., . , .  . equilibrium
which is in equilibrium, the question forces itself upon
us : What constitutes the difference of the two cases?
What is the factor operative here that determines mo
tion, the factor that disturbs equilibrium, the factor
that is present in the one case and absent in the other?
Having put this question to himself, Galileo discovers
that not only the weights, but also the distances of
their vertical descents (the amounts of their vertical
displacements) are the factors that determine motion.
Let us call P, P ', P" . . . the weights of a system of
heavy bodies, and //, ti, h" . . . their respective, simul
taneously possible vertical displacements, where dis
placements downwards are reckoned as positive, and
displacements upwards as negative. Galileo finds
then, that the criterion or test of the state of equilib
rium is contained in the fulfilment of the condition
Ph + P ti + P" h" + . . . = 0. The sum Ph f P'h'
_^_ /> ? '//'__ ... is the factor that destroys equilibrium,
54 THE SCIENCE OF MECHANICS.
the factor that determines motion. Owing to its im
portance this sum has in recent times been character
ised by the special designation work.
There is no 6. Whereas the earlier investigators, in the compari
our choice son of cases of equilibrium and cases of motion, directed
of the cri . . . .,.
teria. their attention to the weights and their distances trom
the axis of rotation and recognised the statical mo
ments as the decisive factors involved, Galileo fixes
his attention on the weights and their distances of de
scent and discerns work as the decisive factor involved.
It cannot of course be prescribed to the inquirer
what mark or criterion of the condition of equilibrium
he shall take account of, when several are present to
choose from. The result alone can determine whether
his choice is the right one. But if we cannot, for rea
Andaiiare sons already stated, regard the significance of the stat
from the ical moments as given independently of experience, as
source. something selfevident, no more can we entertain this
view with respect to the import of work. Pascal errs,
and many modern inquirers share this error with him,
when he says, on the occasion of applying the principle
of virtual displacements to fluids : ' ' Etant clair que c'est
la meme chose de faire faire un pouce de chemin a cent
livres d'eau, que de faire faire cent ponces de chemin
a une livre d'eau." This is correct only on the suppo
sition that we have already come to recognise work as
the decisive factor ; and that it is so is a fact which
experience alone can disclose,
illustration If we have an equal armed, equally weigh ted lever
ofthepre 1 . .... .
Ceding re betore us, we recognise the equilibrium of the lever as
the only effect that is uniquely determined, whether we
regard the weights and the distances or the weights
and the vertical displacements as the conditions that
determine motion. Experimental knowledge of this
THE PRINCIPLES OF STATICS.
55
or a similar character must, however, in the necessity of
the case precede any judgment of ours with regard to
the phenomenon in question. The particular way in
which the disturbance of equilibrium depends on the
conditions mentioned, that is to say, the significance
of the statical moment (PZ) or of the work (P/i), is
even less capable of being philosophically excogitated
than the general fact of the dependence.
7. When two equal weights with equal and op Reduction
posite possible displacements are opposed to each erai case of
, . ,  ...., theprinci
other, we recognise at once the subsistence of equilib pie to the
.  simpler and
num. We might now be tempted to reduce the more special case
general case of the weights P, P' with the capacities of
displacement^,^', where
Ph = P'k' ', to the sim
pler case. Suppose we
have, for example, (Fig.
41) the weights 3 P and
4 P on a wheel and axle
with the radii 4 and 3.
We divide the weights
into equal portions of the
definite magnitude P 9 which we designate by a, I, c,
d> e > f> We then transport a, b, c to the level f 3,
and d, e, f to the level 3. The weights will, of
themselves, neither enter on this displacement nor
will they resist it. We then take simultaneously the
weight g at the level and the weight a at the level
( 3, push the first upwards to i and the second
downwards to f 4, then again, and in the same way,
g to 2 and b to \ 4, g to  3 and c to + 4 To all
these displacements the weights offer no resistance,
nor do they produce them of themselves. Ultimately,
however, a, b, c (or 3/>) appear at the level + 4 and
a b
Q. mw/m
r3
<>
t
I
tsfy>/
^ _
fe
tffiy.
%&'<
^ W%^/fffiM^,w2!f(A ( )
1 * / g
+ 2
a
+ 3 J
+ 4.
56 THE SCIENCE OF MECHANICS.
The gen d, <f, f, g (or 4^) at the level 3. Consequently,
' with respect also to the lastmentioned total displace
ment, the weights neither produce it of themselves
,nor do they resist it ; that is to say, given the ratio of
displacement here specified, and the weights will be
in equilibrium. The equation 4 . $P 3 . ^P = is,
therefore, characteristic of equilibrium in the case as
sumed. The generalisation (Ph P'h' = 0) is ob
vious.
Thecondi If we carefully examine the reasoning of this case,
character we shall quite readily perceive that the inference in
ence. em er ~ volved cannot be drawn unless we take for granted
that the order of the operations performed and the path
by which the transferences are effected, are indifferent,
that is unless we have previously discerned that work
is determinative. We should commit, if we accepted
this inference, the same error that Archimedes com
mitted in his deduction of the law of the lever ; as has
been set forth at length in a preceding section and
need not in the present case be so exhaustively dis
cussed. Nevertheless, the reasoning we have pre
sented is useful, in the respect that it brings palpably
home to the mind the relationship of the simple and
the complicated cases.
Theuniyer 8. The universal applicability of the principle of
bfiifjPot The virtual displacements to all cases of equilibrium, was
firmer 6 perceived by JOHN BERNOULLI ; who communicated his
jolmBe* discovery to Varignon in a letter written in 1717. We
will now enunciate the principle In its most general
form. At the points A, J3, C . . . (Fig. 42) the forces
P, P'j P" . . . are applied. Impart to the points any
Infinitely small displacements v, ?', r" . . . compatible
with the character of the connections of the points (so
called virtual displacements), and construct the pro
THE PRINCIPLES OF STATICS. 57
jections /, /', p" of these displacements on the direc General
p j_i r / i enunciati
tions ot the iorces. These projections we consider of the pri
positive when they fall in
the direction of the force, ^ ^ 
and negative when they fall
in the opposite direction.
The products Pp, P'/,
P"/', . . . are called virtual
moments, and in the two
cases just mentioned have Fig. 42.
contrary signs. Now, the principle asserts, that for the
case of equilibrium Pp + P' p' + P" p" )... 0, or
more briefly ^Pp = 0.
g. Let us now examine a few points more in detail. Detailed
T> . XT r 1 ,1 exarnina
Previous to Newton a force was almost universally tion of the
conceived simply as the pull or the pressure of a heavy pnncip e '
body. The mechanical researches of this period dealt
almost exclusively with heavy bodies. When, now,
in the Newtonian epoch, the generalisation of the idea
of force was effected, all mechanical principles known
to be applicable to heavy bodies could be transferred
at once to any forces whatsoever. It was possible to
replace every force by the pull of a heavy body on a
string. In this sense we may also apply the principle
of virtual displacements, at first discovered only for
heavy bodies, to any forces whatsoever.
Virtual displacements are displacements consistent Definition
. of virtual
with the character of the connections oi a system and dispiace
with one another. If, for example, the two points of
a system, A and B } at which forces act, are connected
(Fig. 43, i) by a rectangularly bent lever, free to re
volve about C, then, if C = zCA, all virtual dis
placements of B and A are elements of the arcs of cir
cles having C as centre ; the displacements of B are
58 THE SCIENCE' OF MECHANICS.
always double the displacements of A, and both are in
every case at right angles to each other. If the points
A, B (Fig. 43, 2) be connected by a thread of the length
n /, adjusted to slip through
to
r stationary rings at C and Z>,
1 / \ then all those displacements
(^ 2 t ^ an< 3 & 2 re virtual in
Fig. 43. which the points referred to
move upon or within two spherical surfaces described
with the radii r^ and r 2 about C and D as centres,
where r 1 + r 2 + CD = L
The reason The use of infinitely small displacements instead of
for the use /..,,., T ^ ,, j
of m&mteiyjimfe displacements, such as Galileo assumed, is justi
piacements. fied by the following consideration. If two weights
are in equilibrium on an inclined plane (Fig. 44), the
equilibrium will not be disturbed if the inclined plane,
at points at which it is not in immediate contact with
the bodies considered, passes into
a surface of a different form. The
essential condition is, therefore,
the momentary possibility of dis
Fig. 44. placement in the momentary con
figuration of the system. To judge of equilibrium we
must assume displacements vanishingly small and such
only ; as otherwise the system might be carried over
into an entirely different adjacent configuration, for
which perhaps equilibrium would not exist.
A Hraita That the displacements themselves are not decisive
but only the extent to which they occur in the direc
tions of the forces, that is only their projections on the
lines of the forces, was, in the case of the inclined plane,
perceived clearly enough by Galileo himself.
With respect to the expression of the principle, it
will be observed, that no problem whatever is presented
THE PRINCIPLES OF STATICS.
59
If all the material points of the system on which forces General re
act, are independent of each other. Each point thus
conditioned can be in equilibrium only in the event
that it is not movable in the direction in which the force
acts. The virtual moment of each such point vanishes
separately. If some of the points be independent of
each other, while others in their displacements are de
pendent on each other, the remark just made holds
good for the former ; and for the latter the fundamental
proposition discovered by Galileo holds, that the sum
of their virtual moments is equal to zero. Hence, the
sumtotal of the virtual moments of all jointly is equal
to zero.
10. Let us now endeavor to get some idea of the Examples.
significance of the principle, by the consideration of a
few simple examples that cannot be
dealt with by the ordinary method
of the lever, the inclined plane, and
the like.
The differential pulley of Wes
ton (Fig. 45) consists of two coax
ial rigidly connected cylinders of
slightly different radii r^ and r 2
<r 1 . A cord or chain is passed
round the cylinders in the manner
indicated in the figure. If we pull
in the direction of the arrow with
the force P 9 and rotation takes place Fi g 45
through the angle cp, the weight Q attached below will
be raised. In the case of equilibrium there will exist
between the two virtual moments involved the equa
tion
Q lL
or P=
6o
THE SCIENCE OF MECHANICS.
A suspend A wheel and axle of weight Q (Fig. 46), which on
and axle, the unrolling of a cord to which the weight P is at
tached rolls itself up on a second cord
wound round the axle and rises, gives
for the virtual moments in the case of
equilibrium the equation
In the particular case R r = 0, we
must also put, for equilibrium, Qr = 0, or,
for finite values of r, Q = 0. In reality the
string behaves in this case like a loop in
which the weight Q Is placed. The lat
ter can, if it be different from zero, continue to roll itself
downwards on the string without moving the weight P.
If, however, when R^=r, we also put Q = 0, the re
sult will be P=$, an indeterminate value. As a mat
ter of fact, every weight P holds the apparatus in equi
librium, since when R = r none can possibly descend.
A double A double cylinder (Fig. 47) of the radii ;*, R lies with
a y horizon n friction on a horizontal surface, and a force Q is brought
to bear on the string at
tached to it. Calling the re
sistance due to friction P 9
equilibrium exists when
2 P = (R r!R } Q. If />>
Fig ' 47  (^" r/V?)<2, the cylinder,
on the application of the force, will roll itself up on
the string.
Roberval's Balance (Fig. 48) consists of a paral
lelogram with variable angles, two opposite sides of
which, the upper and lower, are capable of rotation
about their middle points A, B. To the two remaining
sides, which are always vertical, horizontal rods are
tal surface.
Roberval's
balance.
THE PRINCIPLES OF STATICS.
61
fastened. If from these rods we suspend two equal
weights P, equilibrium will subsist independently of
the position of the points
of suspension, because on
displacement the descent
of the one weight is always
equal to the ascent of the
other.
At three fixed points A,
B, C (Fig. 49) let pulleys
be placed, over which three strings are passed loaded
with equal weights and knotted at O. In what posi
tion of the strings will equilibrium exist? We will call
the lengths of the three strings A0 = $
A
Fi s 48
Discussion
of the case
of equilib
rium of
three knot
ted strings
Fig. 49. Fig. 50.
CO = ^ 3 . To obtain the equation of equilibrium, let
us displace the point O in the directions s 2 and s 3 the
infinitely small distances ds^ and <5V 3 , and note that by
so doing every direction of displacement in the plane
ABC (Fig. 50) can be produced. The sum of the vir
tual moments is
P6s 2 P6s 2 cos a + Pds 2 cos (a f ft)
s cos
cos (a
or
+ cos (a + /?)] &r s = 0.
But since each of the displacements
COS/?
s ar
62 THE SCIENCE OF MECHANICS.
bitrary, and each independent of the other, and may by
themselves be taken = 0, it follows that
1 cos a \ cos (a f /?) =
1 __ cos/3 + cos (a + /f) = 0.
Therefore
cos a = cos /?,
and each of the two equations may be replaced by
1 cos a. + cos 2tx = 0;
or cos a = ,
wherefore ;* + /?= 120.
Remarks on Accordingly, in the case of equilibrium, each of the
the preced . , .,. , i r nii
ing case, strings makes with the others angles of 120 ; which is,
moreover, directly obvious, since three equal forces can
be in equilibrium only when such an arrangement ex
ists. This once known, we may find the position of
the point O with respect to ABC in a number of dif
ferent ways. We may proceed for instance as follows.
'We construct on AB, BC, CA, severally, as sides,
equilateral triangles. If we describe circles about these
triangles, their common point of intersection will be
the point O sought ; a result which easily follows from
the wellknown relation of the angles at the centre and
circumference of circles.
The case of A bar OA (Fig. 51) is revolvable about O in the
voivabie plane of the paper and makes with a fixed straight line
about one , IT i
of its ex >./ OX the variable angle
tremities. </ . .
pa. At A there is ap
plied a force P which
makes with OX the
angle y, and at B, on
Fig 51. a ring displaceable
along the length of the bar, a force <2, making with
OX the angle ft. We impart to the bar an infinitely
THE PRINCIPLES OF STATICS. 63
small rotation, in consequence of which B and A move The case of
forward the distances ds and &s l at right angles to OA 9 voivabfe
and we also displace the ring the distance 6r along the of its e
bar. The variable distance OB we will call r, and we
will let OA = a. For the case of equilibrium we have
then
Qdr cos (j3a') + Q$s sin (/? #) +
JP&SI sin (<a? y} = 0.
As the displacement dr has no effect whatever on
the other displacements, the virtual moment therein
involved must, by itself, = 0, and since dr may be of
any magnitude we please, the coefficient of this virtual
moment must also = 0. We have, therefore,
Q cos (J5 ex) = 0,
or when Q is different from zero,
Further, in view of the fact that ds^ = (afr) ds, we
also have
rQ sin (J3 a) + a P sin (a y} = 0,
or since sin (/? a*) = i,
rQ f aP sin (a y) = ;
wherewith the relation of the two forces is obtained.
ii. An advantage, not to be overlooked, which Every gen
every general principle, and therefore also the prin cipie P in?
i r , i i i r volves an
ciple of virtual displacements, fur  1 ^ /x economy of
nishes, consists in the fact that it
saves us to a great extent the ne
cessity of considering every new par ^ 4
ticular case presented. In the posses
sion of this principle we need not, for Fig. 52.
example, trouble ourselves about the details of a ma
chine. If a new machine say were so enclosed in a
A economy
\k thought.
\
64 THE SCIENCE OF MECHANICS.
box (Fig. 52), that only two levers projected as points
of application for the force P and the weight P', and
we should find the simultaneous displacements of these
levers to be h and //, we should know immediately that
in the case of equilibrium P/i = P'fi f , whatever the
construction of the machine might be. Every principle
of this character possesses therefore a distinct econom
ical value.
Further re 12. We return to the general expression of the prin
marks on
the general ciple of virtual displacements, in order to add a few
expression
oftheprin further remarks. If
Ciple> ^ C  at the points A, B,
C . . . . the forces
P, P',P" . . . . act,
and/, /, /'. . . .
are the projections
of infinitely small
mutually compatible displacements, we shall have for
the case of equilibrium
. . .=0.
If we replace the forces by strings which pass over
pulleys in the directions of the forces and attach thereto
the appropriate weights, this expression simply as
serts that the centre of gravity of the system of weights
as a whole cannot descend. If, however, in certain dis
placements it were possible for the centre of gravit} r
to rise, the system would still be in equilibrium, as the
heavy bodies would not, of themselves, enter on any
Modifica such motion. Ill this case the sum above given would
tion of the , , , rr , 1 .
previous be negative, or less than zero. The general expression
condition. of the condition of equilibrium is, therefore,
[_ pf 4. . . . r; o.
When for every virtual displacement there exists
THE PRINCIPLES OF STATICS. 65
another equal and opposite to it, as is the case for ex
ample in the simple machines, we may restrict ourselves
to the upper sign,, to the equation. For if it were pos
sible for the centre of gravity to ascend in certain
displacements, it would also have to be possible, in
consequence of the assumed reversibility of all the vir
tual displacements, for it to descend. Consequently,
in the present case, a possible rise of the centre of
gravity is incompatible with equilibrium.
The question assumes a different aspect, however, The condi
i 111 11 'i i m tion is > that
when the displacements are not all reversible. Two the sum of
bodies connected together by strings can approach moments
each other but cannot recede from each other beyond equal to or
the length of the strings. A body is able to slide or zero,
roll on the surface of another body ; it can move away
from the surface of the second body, but it cannot
penetrate it. In these cases, therefore, there are dis
placements that cannot be reversed. Consequently,
for certain displacements a rise of the centre of gravity
may take place, while the contrary displacements, to
which the descent of the centre of gravity corresponds,
are impossible. We must therefore hold fast to the
more general condition of equilibrium, and say, the sum
of the virtual moments is equal to or less than zero.
13. LAGRANGE in his Analytical Mechanics attempted The La
. grangian
a deduction of the principle of virtual displacements, deduction
 . oftheprin
which we will now consider. At the points A, B, cipie.
C . . . . (Fig, 54) the forces P,P',P". . . . act. We
imagine rings placed at the points in question, and
other rings A', J3' , C' . . . . fastened to points lying in
the directions of the forces. We seek some common
measure Q/2 of the forces P, P' ', P" . . . . that enables
us to put :
THE SCIENCE OF MECHANICS.
Effected by Q
means of a ?l ~
setofpul
leys and a _
Sin  g1 ^ 9*/ ^
weight. 472 . ~
where , n', n" . . . . are whole numbers. Further, we
make fast to the ring A' a string, carry this string back
&oA forth n times between A' and A, then through B' ,
Fig. 54
n' times back and forth between B* and 2?, then through
C' 9 n" times back and forth between C' and C, and,
finally, let it drop at C', attaching to it there the weight
Q/2. As the string has, now, in all its parts the ten
sion <2/2, we replace by these ideal pulleys all the
forces present in the system by the single force Q/2.
If then the virtual (possible) displacements in any given
configuration of the system are such that, these dis
placements occurring, a descent of the weight Q/2 can
take place, the weight will actually descend and pro
duce those displacements, and equilibrium therefore
will not obtain. But on the other hand, no motion
will ensue, if the displacements leave the weight Q/2
in its original position, or raise it. The expression of
this condition, reckoning the projections of the virtual
displacements in the directions of the forces positive,
THE PRINCIPLES OF STATICS. 67
and having regard for the number of the turns of the
string in each single pulley, is
2np + 2'/ + 2"/' + . . . < 0.
Equivalent to this condition, however, is the ex
pression
2 j + 2' / + 2" /' + . . . < 0,
or
14. The deduction of Lagrange, If stripped of the The con
rather odd fiction of the pulleys, really possesses con tures of La
vincing features, due to the fact that the action of a deduction.
single weight is much more immediate to our expe
rience and is more easily followed than the action of
several weights. Yet it is not proved by the Lagrangian
deduction that work is the factor determinative of the
disturbance of equilibrium, but is, by the employment
of the pulleys, rather assumed by it. As a matter of
fact every pulley involves the fact enunciated and rec
ognised by the principle of virtual displacements. The
replacement of all the forces by a single weight that
does the same work, presupposes a knowledge of the
import of work, and can be proceeded with on this as
sumption alone. The fact that some certain cases are it is not,
_ ... , . , . , however, a
more familiar to us and more immediate to our expe proof.
rience has as a necessary result that we accept them
without analysis and make them the foundation of our
deductions without clearly instructing ourselves as to
their real character.
It often happens in the course of the development
of science that a new principle perceived by some in
quirer in connection with a fact, is not immediately
recognised and rendered familiar in its entire generality.
68 THE SCIENCE OF MECHANICS.
The expe Then, every expedient calculated to promote these
dients em 1 ......
ployed to ends, is, as is proper and natural, called into service.
support all . ., r , . , . , , . ,, , , ,
newprin All manner of facts, in which the principle, although
ciples. .  .
contained in them, has not yet been recognised by in
quirers, but which from other points of view are more
familiar, are called in to furnish a support for the new
conception. It does not, however, beseem mature
science to allow itself to be deceived by procedures of
this sort. If, throughout all facts, we clearly sec and dis
cern a principle which, though not admitting" of proof,
can yet be known to prevail, we have advanced much
farther in the consistent conception of nature than if
we suffered ourselves to be overawed by a specious
Value of the demonstration. If we have reached this point of view,
pro of angian we shall, it is true, regard the Lagrangian deduction
with quite different eyes ; yet it will engage neverthe
less our attention and interest, and excite our satis
faction from the fact that it makes palpable the simi
larity of the simple and complicated cases.
15. MAUPERTUIS discovered an interesting proposi
tion relating to equilibrium, which he communicated
to the Paris Academy in 1740 under the name of the
"Loi de repos." This principle was more fully dis
cussed by EULER in 1751 in the Proceedings of the
Berlin Academy. If we cause infinitely small displace
TheLoide ments in any system, we produce a sum of virtual mo
repos ' ments Pp + P'j>' + P"J>" + . . . ., which only reduces
to zero in the case of equilibrium. This sum is the
work corresponding to the displacements, or since for
infinitely small displacements it is itself infinitely small,
the corresponding element of work. If the displace
ments are continuously increased till a finite displace
ment is produced, the elements of the work will, by
summation, produce a finite amount of work. So, if we
THE PRINCIPLES OF STATICS, 69
start from any given initial configuration of the system statement
. r i r  * oftheprin
and pass to any given final configuration, a certain cipie.
amount of work will have to be done. Now Maupertuis
observed that the work done when a final configura
tion is reached which is a configuration of equilibrium,
is generally a maximum or a minimum ; that is, if we
carry the system through the configuration of equilib
rium the work done is previously and subsequently
less or previously and subsequently greater than at the
configuration of equilibrium itself. For the configura
tion of equilibrium
that is, the element of the work or the differential (more
correctly the variation) of the work is equal to zero.
If the differential of a function can be put equal to
zero, the function has generally a maximum or mini
mum value.
1 6. We can produce a very clear representation to Graphical
,1 r ,1 ^ P TVT , , i illustration
the eye of the import of Maupertuis s principle. oftheim
TTT i r r 111 port of the
We imagine the forces of a system replaced by principle.
Lagrange's pulleys with the weight Q/2. We suppose
that each point of the system is restricted to movement
on a certain curve and that the motion is such that
when one point occupies a definite position on its curve
all the other points assume uniquely determined po
sitions on their respective curves. The simple ma
chines are as a rule systems of this kind. Now, while
imparting displacements to the system, we may carry
a vertical sheet of white paper horizontally over the
weight (?/2, while this is ascending and descending
on a vertical line, so that a pencil which it carries shall
describe a curve upon the paper (Fig. 55). When the
pencil stands at the points a, t, d of the curve, there are,
70 THE SCIENCE OF MECHANICS.
:a we see, adjacent positions in the system of points at
diagram, which the weight Q/2, will stand higher or lower than in
the configuration given. The weight will then, if the
system be left to itself, pass into this lower position and
Fig. 55.
displace the system with it. Accordingly, under condi
tions of this kind, equilibrium does not subsist. If
the pencil stands at e, then there exist only adjacent
configurations for which the weight Q/2 stands higher.
But of itself the system will not pass into the last
named configurations. On the contrary, every dis
placement in such a direction, will, by virtue of the
tendency of the weight to move downwards, be re
versed. Stable equilibrium, therefore,, is the condition,
corresponds to the lowest position of the weight or to
a maximum of work done in the system. If the pencil
stands at b, we see that every appreciable displace
ment brings the weight Q/2, lower, and that the weight
therefore will continue the displacement begun. But,
assuming infinitely small displacements, the pencil
moves in the horizontal tangent at $, in which event
the weight cannot descend. Therefore, unstable equi
Unstabie librium is the state that corresponds to the highest position
equilibrium / , 7 . , _. , . \ 7 *
of the weight Q/2, or to a minimum of work done in the
system. It will be noted, however, that conversely
i1brium equi "
THE PRINCIPLES OF STA TICS. 71
every case of equilibrium is not the correspondent of
a maximum or a minimum of work performed. If the
pencil is at/, at a point of horizontal contrary flexure,
the weight in the case of infinitely small displace
ments neither rises nor falls. Equilibrium exists, al
though the work done is neither a maximum nor a
minimum. The equilibrium of this case is the so
called mixed equilibrium * : for some disturbances it is Mixed equ'
stable, for others unstable. Nothing prevents us from
regarding mixed equilibrium as belonging to the un
stable class. When the pencil stands at g, where the
curve runs along 'horizontally a finite distance, equi
librium likewise exists. Any small displacement, in
the configuration in question, is neither continued nor
reversed. This kind of equilibrium, to which likewise
neither a maximum nor a minimum corresponds, is
termed" [72<?w/r#/ or] indifferent. If the curve described Neutral
by Q/2 has a cusp pointing upwards, this indicates a e( * ulhbriuc
minimum of work done but no equilibrium (not even
unstable equilibrium). To a cusp pointing downwards
a maximum and stable equilibrium correspond. In the
last named case of equilibrium the sum of the virtual
moments is not equal to zero, but is negative.
17. In the reasoning just presented, we have asjhepreced
sumed that the motion of a point of a system on one tufnappiTe!
curve determines the motion of all the other points of ^ mo?edi y f
the system on their respective curves. The movability cultcases
of the system becomes multiplex, however, when each
point is displaceable on a surface, in a manner such
that the position of one point on its surface determines
*This term is not used In English, because our writers hold that no
equilibrium is conceivable which is not stable or neutral for some possible
displacements. Hence what is called mixed equilibrium in the text is called
unstable equilibrium by English writers, who deny the existence of equilibrium
unstable in every respect. Trans.
72 THE SCIENCE OF MECHANICS.
uniquely the position of all the other points on their
surfaces. In this case, we are not permitted to consider
the curve described by Q/2, but are obliged to picture
to ourselves a surface described by Q/2. If, to go a
step further, each point is movable throughout a space,
we can no longer represent to ourselves in a purely geo
metrical manner the circumstances of the motion, by
means of the locus of Q/2. In a correspondingly higher
degree is this the case when the position of one of the
points of the system does not determine conjointly all
the other positions, but the character of the system's
motion is more multiplex still. In all these cases, how
ever, the curve described by Q/2 (Fig. 55) can serve
us as a symbol of the phenomena to be considered. In
these cases also we rediscover the Maupertuisian pro
positions.
urtherex We have also supposed, in our considerations up to
ie S same this point, that constant forces, forces independent of
the position of the points of the system, are the forces
that act in the system. If we assume that the forces
do depend on the position of the points of the system
(but not on the time), we are no longer able to conduct
our operations with simple pulleys, but
must devise apparatus the force active in
which, still exerted by Q/2, varies with the
displacement : the ideas we have reached,
however, still obtain. The depth of the
descent of the weight Q/2 is in every case
the measure of the work performed, which
is always the same in the same configura
s 56. tion of the system and is independent of
the path of transference. A contrivance which would
develop by means of a constant weight a force varying
with the displacement, would be, for example, a wheel
THE PRI.\ 7 CIPLES OF STATICS. 73
and axle (Fig. 56) with a noncircular wheel. It would
not repay the trouble, however, to enter into the de
tails of the reasoning indicated in this case, since we
perceive at a glance its feasibility.
18. If we know the relation that subsists between The prin
the work done and the socalled vis viva of a sys Courtivron.
tern, a relation established in dynamics, we arrive
easily at the principle communicated by COURTIVRON in
1749 to the Paris Academy, which is this: For the
configuration of ,, equilibrium, at which the
i j maximum ,, . r , ,
work done is a .  , the vis viva of the system,
minimum J '
, . i maximum ., . ., ,, ,
in motion, is also a  . in its transit through
minimum G
these configurations.
19. A heavy, homogeneous triaxial ellipsoid resting illustration
. on a horizontal plane is admirably adapted to illustrate ous kinds of
equilibrium
the various classes of equilibrium. When the ellip
soid rests on the extremity of its smallest axis, it is in
stable equilibrium, for any displacement it may suffer
elevates its centre of gravity. If it rest on its longest
axis, it is in unstable equilib
rium. If the ellipsoid stand on
its mean axis, its equilibrium is
mixed. A homogeneous sphere
or a homogeneous right cylin
der on a horizontal plane illus Fi & 57.
trates the case of indifferent equilibrium. In Fig, 57
we have represented the paths of the centre of gravity
of a cube rolling on a horizontal plane about one of its
edges. The position a of the centre of gravity is the
position of stable equilibrium, the position b, the posi
tion of unstable equilibrium.
74 THE SCIENCE OF MECHANICS.
The eaten 2o. We will now consider an example which at
ary
first sight appears very complicated but is elucidated
at once by the principle of virtual displacements. John
and James Bernoulli, on the occasion of a conversa
tion on mathematical topics during a walk in Basel,
lighted on the question of what form a chain would
take that was freely suspended and fastened at both
ends. They soon and easily agreed in the view that
the chain would assume that form of equilibrium at
which its centre of gravity lay in the lowest possible
position. As a matter of fact we really do perceive
that equilibrium subsists when all the links of the chain
have sunk as low as possible, when none can sink lower
without raising in consequence of the connections of
the system an equivalent mass equally high or higher.
When the centre of gravity has sunk as low as it pos
sibly can sink, when all has happened that can happen,
stable equilibrium exists. The physical part of the
problem is disposed of by this consideration. The de
termination of the curve that has the lowest centre of
gravity for a given length between the two points A,
B, is simply a mathematical problem. (See Fig. 58.)
Theprinci 21. Collecting all that has been presented, we see,
pie is sim , . . ...... . .
piytherec that there is contained in the principle of virtual dis
ognition of . . .
a fact. placements simply the recognition of a fact that was
instinctively familiar to us long previcnisly, only that
we had not apprehended it so precisely and clearly.
This fact consists in the circumstance that heavy
bodies, of themselves, move only downwards. If sev
eral such bodies be joined together so that they can
suffer no displacement independently of each other,
they will then move only in the event that some heavy
mass is on the whole able to descend, or as the prin
ciple, with a more perfect adaptation of our ideas to
OF STATICS.
75
Fig. 58.
76 THE SCIENCE OF MECHANICS.
what this the facts, more exactly expresses it, only In the event
that work can be performed. If, extending the notion
of force, we transfer the principle to forces other than
those due to gravity, the recognition is again con
tained therein of the fact that the natural occurrences
in question take place, of themselves, only in a definite
sense and not in the opposite sense. Just as heavy
bodies descend downwards, so differences of tempera
ture and electrical potential cannot increase of their
own accord but only diminish, and so on. If occur
rences of this kind be so connected that they can take
place only in the contrary sense, the principle then es
tablishes, more precisely than our instinctive appre
hension could do this, the factor work as determinative
and decisive of the direction of the occurrences. The
equilibrium equation of the principle may be reduced
In every case to the trivial statement, that when noth
ing can happen not/ling does happen.
Theprin 22. It is important to obtain clearly the perception,
fight o?* e that we hav to deal, in the case of all principles,
vilw SSS merely wi^i the ascertainment and establishment of a
fact. If we neglect this, we shall always be sensible
of some deficiency and will seek a verification of the
principle, that is not to be found. Jacobi states in his
Lectures on Dynamics that Gauss once remarked that
Lagrange's equations of motion had not been proved,
but only historically enunciated. And this view really
seems to us to be the correct one in regard to the prin
ciple of virtual displacements.
The differ The task of the early inquirers, who lay the foun
ent tasks of , . , . . ... .
early and of dations of any department 01 investigation, is entirely
fnqufrershi different from that of those who follow. It is the busi
ment. cpar ness of the former to seek out and to establish the
facts of most cardinal importance only; and, as history
THE PRINCIPLES OF STA TICS. 77
teaches, more brains are required for this than is gen
erally supposed. When the most important facts are
once furnished ; we are then placed in a position to
work them out deductively and logically by the meth
ods of mathematical physics; we can then organise the
department of inquiry In question, and show that In the
acceptance of some one fact a whole series of others Is
included which were not to be immediately discerned
in the first. The one task is as important as the other.
We should not however confound the one with the
other. We cannot prove by mathematics that nature
must be exactly what it is. But we can prove, that
one set of observed properties determines conjointly
another set which often are not directly manifest.
Let it be remarked in conclusion, that the princi Every ? en
i r j i i i 11 , eral princi
ple of virtual displacements, like every general prin pie brings
. ., .. *,i ,, i i i iir .i with it dis
ciple, brings with it, by the insight which it furnishes, niusion
disillusionment as well as elucidation. It brings with well as eiu
it disillusionment to the extent that we recognise In it C1
facts which were long before known anu^ven instinct
ively perceived, our present recognition bhing simply
more distinct and more definite ; and elucidation, in
that it enables us to see everywhere throughout the
most complicated relations the same simple facts.
v.
RETROSPECT OF THE DEVELOPMENT OF STATICS.
i. Having passed successively in review the prin Review of
ciples of statics, we are now in a position to take a whole 3 .
brief supplementary survey of the development of the
principles of the science as a whole. This development,
falling as it does in the earliest period of mechanics,
the period which begins in Grecian antiquity and
78 THE SCIENCE OF MECHANICS.
reaches its close at the time when Galileo and his
younger contemporaries were inaugurating modern me
chanics, illustrates in an excellent manner the pro
cess of the formation of science generally. All con
ceptions, all methods are here found in their simplest
form, and as it were in their infancy. These beginnings
The origin point unmistakably to their origin in the experiences of
the manual arts. To the necessity of putting these ex
periences into communicable form and of disseminating
them beyond the confines of class and craft, science
owes its origin. The collector of experiences of this
kind, who seeks to preserve them in written form, finds
before him many different, or at least supposably differ
ent, experiences. His position is one that enables him
to review these experiences more frequently, more vari
ously, and more impartially than the individual work
ingman, who is always limited to a narrow province.
The facts and their dependent rules are brought into
closer temporal and spatial proximity in his mind and
writings, and thus acquire the opportunity of revealing
The econo their relationship, their connection, and their gradual
municaSon. transition the one into the other. The desire to sim
plify and abridge the labor of communication supplies
a further impulse in the same direction. Thus, from
economical reasons, in such circumstances, great num
bers of facts and the rules that spring from them are
condensed into a system and comprehended in a single
expression.
The gene 2. A collector of this character has, moreover, op
ralcharac . . r
ter of prin portunity to take note of some new aspect of the facts
before him of some aspect which former observers
had not considered. A rule, reached by the observation
of facts, cannot possibly embrace the entire fact, in all
its infinite wealth, in all its inexhaustible manifoldness ;
THE PRINCIPLES OF STATICS, 79
on the contrary, it can furnish only a rough outline of
the fact, onesidedly emphasising the feature that is of
importance for the given technical (or scientific) aim in
view. What aspects of a fact are taken notice of, will
consequently depend upon circumstances, or even on Their form
T r i i TT 11 in many as
the caprice of the observer. Hence there is always op pects, acci
....... i dental.
portunity lor the discovery of new aspects of the fact,
which will lead to the establishment of new rules of
equal validity with, or superior to, the old. So, for in
stance, the weights and the lengths of the leverarms
were regarded at first, by Archimedes, as the conditions
that determined equilibrium. Afterwards, by Da Vinci
and Ubaldi the weights,and the perpendicular distances
from the axis of the lines of force were recognised as
the determinative conditions. Still later, by Galileo,
the weights and the amounts of their displacements,
and finally by Varignon the weights and the directions
of the pulls with respect to the axis were taken as the
elements of equilibrium, and the enunciation of the
rules modified accordingly.
3. Whoever makes a new observation of this kind, puriiabii
and establishes such a new rule, knows, of course, ourin y th e e m en
liability to error in attempting mentally to represent struction of
the fact, whether by concrete images or in abstract con
ceptions, which we must do in order to have the mental
model we have constructed always at hand as a substi
tute for the fact when the latter is partly or wholly in
accessible. The circumstances, indeed, to which we
have to attend, are accompanied by so many other,
collateral circumstances, that it is frequently difficult
to single out and consider those that are essential to the
purpose in view. Just think how the facts of friction,
the rigidity of ropes and cords, 'and like conditions in
machines, obscure and obliterate the pure outlines of
So THE SCIENCE OF MECHANICS.
Thisiiabii the main facts. No wonder, therefore, that the discov
us toseek erer or verifier of a new rule, urged by mistrust of him
ofaiinew S self, seeks after a proof of the rule whose validity he
believes he has discerned. The discoverer or verifier
does not at the outset fully trust in the rule ; or, It may
be, he is confident only of a part of it. So, Archimedes,
for example, doubted whether the effect of the action
of weights on a lever was proportional to the lengths of
the leverarms, but he accepted without hesitation the
fact of their influence in some way, Daniel Bernoulli
does not question the influence of the direction of a
force generally, but only the form of its influence. As
a matter of fact, it is far easier to observe that a circum
stance has influence in a given case, than to determine
what influence it has. In the latter inquiry we are in
much greater degree liable to error. The attitude of the
investigators is therefore perfectly natural and defens
ible.
The natural The proof of the correctness of a new rule can be
proof. attained by the repeated application of it, the frequent
comparison of it with experience, the putting of it to
the test under the most diverse circumstances. This
process would, in the natural course of events, get car
ried out in time. The 1 discoverer, however, hastens to
reach his goal more quickly. He compares the results
that flow from his rule with all the experiences with
which he is familiar, with all older rules, repeatedly
tested in times gone by, and watches to see if he do
not light on contradictions. In this procedure, the
greatest credit is, as it should be, conceded to the oldest
and most familiar experiences, the most thoroughly
tested rules. Our instinctive experiences, those gen
eralisations that are made involuntarily, by the irresist
ible force of the innumerable facts that press in upon
THE PRINCIPLES OF STATICS. ST
us, enjoy a peculiar authority; and this is perfectly
warranted by the consideration that it is precisely the
elimination of subjective caprice and of individual er
ror that is the object aimed at.
In this manner Archimedes proves his law of the illustration
lever, Stevinus his law of inclined pressure, Daniel ceding re
Bernoulli the parallelogram of forces, Lagrange the
principle of virtual displacements. Galileo alone is
perfectly aware, with respect to the lastmentioned
principle, that his new observation and perception are
of equal rank with every former one that it is derived
from the same source of experience. He attempts no
demonstration. Archimedes, in his proof of the prin
ciple of the lever, uses facts concerning the centre of
gravity, which he had probably proved by means of the
very principle now in question ; yet we may suppose
that these facts were otherwise so familiar, as to be un
questioned, so familiar indeed, that it may be doubted
whether he remarked that he had employed them in
demonstrating the principle of the lever. The instinc
tive elements embraced in the views of Archimedes and
Stevinus have been discussed at length in the proper
place.
4. It is quite in order, on the making of a new dis The posi
. tionthatad
covery, to resort to all proper means to bring the new vanced sci
TTTI i i ence should
rule to the test. When, however, after the lapse of a occupy,
reasonable period of time, it has been sufficiently often
subjected to direct testing, it becomes science to recog
nise that any other proof than that has become quite
needless ; that there is no sense in considering a rule
as the better established for being founded on others
that have been reached by the very same method of
observation, only earlier ; that one wellconsidered and
tested observation is as good as another. Today, we
82 THE SCIENCE OF MECHANICS.
should regard the principles of the lever, of statical
moments, of the inclined plane, of virtual displace
ments, and of the parallelogram of forces as discovered
by equivalent observations. It is of no importance now,
that some of these discoveries were made directly, while
others were reached by roundabout ways and as de
pendent upon other observations. It is more in keep
ing, furthermore, with the economy of thought and with
insight bet the aesthetics of science, directly to recognise a principle
tificiaidem (say that of the statical moments) as the key to the un
derstanding of all the facts of a department, and really
see how it pervades all those facts, rather than to hold
ourselves obliged first to make a clumsy and lame de
duction of it from unobvious propositions that involve
the same principle but that happen to have become
earlier familiar to us. This process science and the in
dividual (in historical study) may go through once for
all. But having done so both are free to adopt a more
convenient point of view.
Themis 5. In fact, this mania for demonstration in science
mania for 6 results in a rigor that is false and mistaken. Some pro
dernonstra . ,. iij.v j r , .
tion. positions are held to be possessed of more certainty
than others and even regarded as their necessary and
incontestable foundation ; whereas actually no higher,
or perhaps not even so high, a degree of certainty at
taches to them. Even the rendering clear of the de
gree of certainty which exact science aims at, is not at
tained here. Examples of such mistaken rigor are to
be found in almost every textbook. The deductions
of Archimedes, not considering their historical value,
are infected with this erroneous rigor. But the most
conspicuous example of all Is furnished by Daniel Ber
noulli's deduction of the parallelogram of forces (Com
ment. A cad. Petrop. T. I.).
THE PRINCIPLES OF STATICS. 83
6. As already seen, instinctive knowledge enjoys The char
our exceptional confidence. No longer knowing how stinctive
i j j. ,111 knowledge.
we have acquired it, we cannot criticise the logic by
which it was inferred. We have personally contributed
nothing to its production. It confronts us with a force
and irresistibleness foreign to the products of volun
tary reflective experience. It appears to us as some
thing free from subjectivity, and extraneous to us, al
though we have it constantly at hand so that it is more
ours than afe the Individual facts of nature.
All this has often led men to attribute knowledge of its author
this kind to an entirely different source, namely, to view imeKsV "
it as existing a priori m us (previous to all experience). P
That this opinion is untenable was fully explained in
our discussion of the achievements of Stevinus. Yet
even the authority of instinctive knowledge, however
important it may be for actual processes of develop
ment, must ultimately give place to that of a clearly and
deliberately observed principle. Instinctive knowledge
is, after all, only experimental knowledge, and as such
is liable, we have seen, to prove itself utterly insuffi
cient and powerless, when some new region of expe
rience is suddenly opened up.
7. The true relation and connection of the different The true re
principles is the historical one. The one extends farther principles
in this domain, the other farther in that. Notwith cai one.
standing that some one principle, say the principle of
virtual displacements, may control with facility a
greater number of cases than other principles, still
no assurance can be given that it will always maintain
its supremacy and will not be outstripped by some new
principle. All principles single out, more or less arbi
trarily, now this aspect now that aspect of the same
facts, and contain an abstract summarised rule for the
84 THE SCIENCE OF MECHANICS.
refigurement of the facts in thought. We can never
assert that this process has been definitively completed.
Whosoever holds to this opinion, will not stand in the
way of the advancement of science.
Conception 8. Let us, in conclusion, direct our attention for a
of force in . .
statics. moment to the conception of force in statics. Force is
any circumstance of which the consequence is motion.
Several circumstances of this kind, however, each single
one of which determines motion, may be so conjoined
that in the result there shall be no motion." Now stat
ics investigates what this mode of conjunction, in gen
eral terms, is. Statics does not further concern itself
about the particular character of the motion condi
tioned by the forces. The circumstances determinative
of motion that are best known to us, are our own vo
The origin litional acts our innervations. In the motions which
of the no . . .
tion of we ourselves determine, as well as in those to which
we are forced by external circumstances, we are always
sensible of a pressure. Thence arises our habit of rep
resenting all circumstances determinative of motion as
something akin to volitional acts as pressures. The
attempts we make to set aside this conception, as sub
jective, animistic, and unscientific, fail invariably. It
cannot profit us, surely, to do violence to our own nat
uralborn thoughts and to doom ourselves, in that re
gard, to voluntary mental penury. We shall subse
quently have occasion to observe, that the conception
referred to also plays a part in the foundation of dy
namics.
We are able, in a great many cases, to replace the
circumstances determinative of motion, which occur in
nature, by our innervations, and thus to reach the idea
of a gradation of the intensity of forces. But in the esti
mation of this intensity we are thrown entirely on the
THE PRINCIPLES OF STATICS. 85
resources of our memory, and are also unable to com The com
. .... mon char
municate our' sensations. Since it is possible, now acter of ail
ever, to represent every condition that determines
motion by a weight, we arrive at the perception that
all circumstances determinative of motion (all forces)
are alike in character and may be replaced and meas
ured by quantities that stand for weight. The meas
urable weight serves us, as a certain, convenient, and
communicable index, in mechanical researches, just as
the thermometer in thermal researches is an exacter
substitute for our perceptions of heat. As has pre The idea of
,, 11 , i 1 1 i  i r motion an
viously been remarked, statics cannot wholly rid itself auxiliary
of all knowledge of phenomena of motion. This par stat?2! m
ticularly appears in the determination of the direction
of a force by the direction of the motion which it would
produce if it acted alone. By the point of application
of a force we mean that point of a body whose motion
is still determined by the force when the point is freed
from its connections with the other parts of the body.
Force accordingly is any circumstance that deThegene
termines motion; and its attributes may be stated astmtesof"
follows. The direction of the force is the direction of
motion which is determined by that force, alone. The
point of application is that point whose motion is de
termined independently of its connections with the
system. The magnitude of the force is that weight
which, acting (say, on a string) in the direction deter
mined, and applied at the point in question, determines
the same motion or maintains the same equilibrium.
The other circumstances that modify the determination
of a motion, but by themselves alone are unable to pro
duce it, such as virtual displacements, the arms of
levers, and so forth, may be termed collateral condi
tions determinative of motion and equilibrium.
85 THE SCIENCE OF MECHANICS.
THE PRINCIPLES OF STATICS IN THEIR APPLICATION TO
FLUIDS.
NO essen i. The consideration of fluids has not supplied stat
points^r ics with many essentially new points of view, yet nu
voiveJnn merous applications and confirmations of the principles
hlssubj ' ect ' already known have resulted therefrom, and physical
experience has been greatly enriched by the investiga
tions of this domain. We shall devote, therefore, a few
pages to this subject.
2. To ARCHIMEDES also belongs the honor of found
ing the domain of the statics of liquids. To him we
owe the wellknown proposition concerning the buoy
ancy, or loss of weight, of bodies immersed in liquids,
of the discovery of which Vitruvius, De Architectures,
Lib. IX, gives the following account :
vitruvius's "Though Archimedes discovered many curious
Arcbime? "matters that evince great intelligence, that which I am
cov e S ry. 1S " "about to mention is the most extraordinary. Hiero,
"when he obtained the regal power in Syracuse, hav
"ing, on the fortunate turn of his affairs, decreed a
"votive crown of gold to be placed in a certain temple
"to the immortal gods, commanded it to be made of
"great value, and assigned for this purpose an appro
"priate weight of the metal to the manufacturer. The
"latter, in due time, presented the work to the king,
"beautifully wrought ; and the weight appeared to cor
" respond with that of the gold which had been as
" signed for it.
"But a report having been circulated, that some of
"the gold had been abstracted, and that the deficiency
THE PRINCIPLES OF STATICS. 87
"thus caused had been supplied by silver, HIero was The ac
,. iri . ..,.,_ count, ofVi
"indignant at the rraud, and, unacquainted with the truvius.
"method by which the theft might be detected, re
" quested Archimedes would undertake to give it his
"attention. Charged with this commission, he by
"chance went to a bath, and on jumping into the tub,
"perceived that, just in the proportion that his body
"became immersed, in the same proportion the water
"ran out of the vessel. Whence, catching at the
"method to be adopted for the solution of the proposi
tion, he immediately followed it up, leapt out of the
"vessel in joy, and returning home naked, cried out
"with a loud voice that he had found that of which he
"was in search, for he continued exclaiming, In Greek,
"evpijKa, evprjKQt, (I have found it, I have found It !)"
3. The observation which led Archimedes to his Statement
. .  . , . , . . of the Ar
proposition, was accordingly this, that a body im chimedean
 . . . proposition
mersed in water must raise an equivalent quantity oi
water exactly as if the body lay on one pan of a balance
and the water on the other. This conception, which
at the present day is still the most natural and the
most direct, also appears In Archimedes's treatises On
Floating Bodies, which unfortunately have not been
completely preserved but have in part been restored
by F. Commandinus.
The assumption from which Archimedes starts
reads thus :
"It is assumed as the essential property of a liquid The Archi
 , . .. , , medean as
that in all uniform and continuous positions of its parts sumption,
the portion that suffers the lesser pressure is forced
upwards by that which suffers the greater pressure.
But each part of the liquid suffers pressure from the
portion perpendicularly above It if the latter be sinking
or suffer pressure from another portion."
THE SCIENCE OF MECHANICS.
Analysis of Archimedes now, to present the matter briefly,
P ie. prmci conceives the entire spherical earth as fluid in consti
tution, and cuts out of it pyramids the vertices of
which lie at the centre (Fig. 59). All these pyramids
must, in the case of equilib
rium, have the same weight,
and the similarly situated
parts of the same must all
suffer the same pressure.
If we plunge a body a of
the same specific gravity as
water into one of the pyra
mids, the body will com
pletely submerge, and, in
Fig. 59
the case of equilibrium, will supply by its weight the
pressure of the displaced water. The body ^, of less
specific gravity, can sink, without disturbance of equi
librium, only to the point at which the water beneath
it suffers the same pressure from the weight of the
body as it would if the body were taken out and the
submerged portion replaced by water. The body c,
of a greater specific gravity, sinks as deep as it possibly
can. That its weight is lessened in the water by an
amount equal to the weight of the water displaced,
will be manifest if we imagine the body joined to
another of less specific gravity so that a third body is
formed having the same specific gravity as water,
which just completely submerges.
rhe state of 4. When in the sixteenth century the study of the
m the six works of Archimedes was again taken up, scarcely the
teenth cen .  . , . . , , _,
tury. principles of his researches were understood. The
complete comprehension of his deductions was at that
time impossible.
STEVINUS rediscovered by a method of his own the
THE PRINCIPLES OF STATICS. 89
most important principles of hydrostatics and the de The discov
. ir T "'11 i r eriesofSte
ductions tnereirom. It was principally two ideas from vinus.
which Stevinus derived his fruitful conclusions. The
one is quite similar to that relating to the endless
chain. The other consists in the assumption that the
solidification of a fluid in equilibrium does not disturb
its equilibrium.
Stevinus first lays down this principle. Any given The first
/^ . . fundamen
mass of water A (rig. 60), immersed in water, is mtaiprind
equilibrium in all its parts. If A
were not supported by the sur
rounding water but should, let us
say, descend, then the portion of
water taking the place of A and
placed thus in the same circum
stances, would, on the same as
sumption, also have to descend. Fig. 60.
This assumption leads, therefore, to the establishment
of a perpetual motion, which is contrary to our ex
perience and to our instinctive knowledge of things.
Water immersed In water loses accordingly its The second
fundamen
whole weight. If, now, we imagine the surface of the tai princi
submerged w r ater solidified, the vessel formed by this
surface, the was super ficiarium as Stevinus calls it, will
still be subjected to the same circumstances of pres
sure. If empty, the vessel so formed will surfer an
upward pressure In the liquid equal to the weight of the
water displaced. If we fill the solidified surface with
some other substance of any specific gravity we may
choose, it will be plain that the diminution of the
weight of the body will be equal to the weight of the
fluid displaced on immersion.
In a rectangular, vertically placed parallelepipedal
vessel filled with a liquid, the pressure on the horizontal
THE SCIENCE OF MECHANICS.
stevinus's base Is equal to the weight of the liquid. The pressure
deductions. . 1n . . ,
is equal, also, for all parts of the bottom of the same
area. When now Stevimis imagines portions of the
liquid to be cut out and replaced by rigid immersed
bodies of the same specific gravity, or, what is the
same thing, imagines parts of the liquid to become so
lidified, the relations of pressure in the vessel will not
be altered by the procedure. But we easily obtain in
this way a clear view of the law that the pressure on
the base of a vessel is independent of its form, as well
as of the laws of pressure in communicating vessels,
and so forth.
Galileo, in 5. GALILEO treats the equilibrium of liquids in com
the treat . 1,11 . ,
mentof thismumcating vessels and the problems connected there
subject, em .
p with by the help of the principle of virtual displace
ments. NN (Fig. 61) being the
common level of a liquid in equilib
rium in two communicating vessels,
Galileo explains the equilibrium
here presented by observing that in
the case of any disturbance the dis
placements of the columns are to
each other in the inverse proportion
of the areas of the transverse sec
principle of
virtual dis
placements j
B
= c
I
A
S
'T
3
JV
A/
i"' ~i"=.
 ~
Fig. 61.
tions and of the weights of the columns that is, as
with machines in equilibrium. But this is not quite cor
rect. The case does not exactly correspond to the
cases of equilibrium investigated by Galileo in ma
chines, which present indifferent equilibrium. With
liquids in communicating tubes every disturbance of the
common level of the liquids produces an elevation of
the centre of gravity. In the case represented in Fig.
61, the centre of gravity S of the liquid displaced from
the shaded space in A is elevated to S', and we may
THE PRINCIPLES OF STATICS.
regard the rest of the liquid as not having been moved.
Accordingly, in the case of equilibrium, the centre of
gravity of the liquid lies at its lowest possible point.
6. PASCAL likewise employs the principle of virtual The same
1 . principle
displacements, but in a more correct manner, leaving made use of
..,.,.... by Pascal.
the weight of the liquid out of account and considering
only the pressure at the surface. If we imagine two
communicating vessels to be closed by pistons (Fig.
62), and these pistons loaded with
weights proportional to their surface
areas, equilibrium will obtain, because
in consequence of the invariability of
the volume of the liquid the displace
ments in every disturbance are in
versely proportional to the weights. Fig. 62.
For Pascal, accordingly, it follows, as a necessary con
sequence, from the principle of virtual displacements,
that in the case of equilibrium every pressure on a su
perficial portion of a liquid is propagated with undi
minished effect to every other superficial portion, how
ever and in whatever position it be placed. No objec
tion is to be made to discovering the principle in this
way. Yet we shall see later on that the more natural
and satisfactory conception is to regard the principle as
immediate^ given.
7. We shall now, after this historical sketch, again Detailed
n , ,. . , 11 considera
exammethe most important cases oi liquid equilibrium, tionof the
and from such different points of view as may be con su Jec *
venient.
The fundamental property of liquids given us by
experience consists in the flexure of their parts on the
slightest application of pressure. Let us picture to our
selves an element of volume of a liquid, the gravity of
which we disregardsay a tiny cube. If the slightest
92 THE SCIENCE OF MECHANICS.
The funda excess of pressure be exerted on one of the surfaces of
property of this cube, (which we now conceive, for the moment,
mobility of as a. fixed geometrical locus, containing the fluid but
eir par s. ^^ ^ .^ su ] 3s t aric e) the liquid (supposed to have pre
viously been in equilibrium and at rest) will yield and
pass out in all directions through the other five surfaces
of the cube. A solid cube can stand a pressure on its
upper and lower surfaces different in magnitude from
that on its lateral surfaces ; or vice versa. A fluid cube,
on the other hand, can retain its shape only if the same
perpendicular pressure be exerted on all its sides. A
similar train of reasoning is applicable to all polyhe
drons. In this conception, as thus geometrically eluci
dated, is contained nothing but the crude experience
that the particles of a liquid yield to the slightest pres
sure, and that they retain this property also in the in
terior of the liquid when under a high pressure ; it
being observable, for example, that under the condi
tions cited minute heavy bodies sink in fluids, and so on.
A second With the mobility of their parts liquids combine
Fhe P conv_ still another property, which we will now consider. Li
of thek'voi quids suffer through pressure a diminution of volume
which is proportional to the pressure exerted on unit
of surface. Every alteration of pressure carries along
with it a proportional alteration of volume and density.
If the pressure diminish, the volume becomes greater,
the density less. The volume of a liquid continues to
diminish therefore on the pressure being increased, till
the point is reached at which the elasticity generated
within it equilibrates the increase of the pressure.
8. The earlier inquirers, as for instance those of the
Florentine Academy, were of the opinion that liquids
were incompressible. In 1761, however, JOHN CANTON
performed an experiment by which the compressibility
THE PRINCIPLES OF STATICS.
93
of water was demonstrated. A thermometer glass is The first
filled with water, boiled, and then sealed. (Fig. 63.) tionof tte
The liquid reaches to a. But since the space above a is biiity of
airless, the liquid supports no atmospheric pres
sure. If the sealed end be broken off, the liquid
will sink to b* Only a portion, however, of this
displacement is to be placed to the credit of the c "\h
compression of the liquid by atmospheric pres
sure. For if we place the glass before breaking p^
off the top under an airpump and exhaust the ^^
chamber, the liquid will sink to c. This last phe
nomenon is due to the fact that the pressure that
bears down on the exterior of the glass and diminishes
its capacity, is now removed. On breaking off the top,
this exterior pressure of the atmosphere is compensated
for by the interior pressure then introduced, and an
enlargement of the capacity of the glass again sets in.
The portion cb, therefore, answers to the actual com
pression of the liquid by the pressure of the atmos
phere.
The first to institute exact experiments on the com The experi
ments of
pressibility of water, was OERSTED, who employed to oersted on
r J , r J this subject.
this end a very ingenious method. A
thermometer glass A (Fig. 64) is filled
with boiled water and is inverted, with
open mouth, into a vessel of mercury.
Near it stands a manometer tube B filled
with air and likewise inverted with open
mouth in the mercury. The whole ap (
paratus is then placed in a vessel filled
with water, which is compressed by the
aid of a pump. By this means the water
in A is also compressed, and the filament of quicksilver
which rises in the capillary tube of the thermometer
B
Fig. 64.
94 THE SCIENCE OF MECHANICS.
glass Indicates this compression. The alteration of
capacity which the glass A suffers in the present in
stance, is merely that arising from the pressing to
gether of its walls by forces which are equal on all sides.
The esperi The most delicate experiments on this subject have
Grassi. been conducted by GRASSI with an apparatus con
structed by Regnault, and computed with the assist
ance of Lame's correctionformulae. To give a tan
gible idea of the compressibility of water, we will remark
that Grassi observed for boiled water at under an
Increase of one atmospheric pressure a diminution of
the original volume amounting to 5 in 100,000 parts.
If we imagine, accordingly, the vessel A to have the
capacity of one litre (1000 ccm.), and affix to it a cap
illary tube of i sq. mm. crosssection, the quicksilver
filament will ascend in it 5 cm. under a pressure of
one atmosphere.
Surface 9. Surfacepressure, accordingly, induces a physical
pressure in . . . . .  . . , . .. .
<iuces m alteration in a liquid (an alteration in density), which
liquids an   . rr . 1  1 .
alteration can be detected by sumciently delicate means even
ensi y. are a j wa y s a liberty to think that por
tions of a liquid under a higher pressure are more dense,
though it may be very slightly so, than parts under a
less pressure.
The impii Let us imagine now, we have in a liquid (in the in
this fact, terior of which no forces act and the gravity of which
we accordingly neglect) two portions subjected to un
equal pressures and contiguous to one another. The
portion under the greater pressure, being denser, will
expand, and" press against the portion under the less
pressure, until the forces of elasticity as lessened on the
one side and increased on the other establish equilib
rium at the bounding surface and both portions are
equally compressed.
THE PRINCIPLES OF STATICS. 95
If we endeavor, now, quantitatively to elucidate our The state
mental conception of these two facts, the easy mobility these impii
and the compressibility of the parts of a liquid, so that
they will fit the most diverse classes of experience,
we shall arrive at the following proposition : When
equilibrium subsists in a liquid, in the interior of which
no forces act and the gravity of which we neglect, the
same equal pressure is exerted on each and every equal
surfaceelement of that liquid however and wherever
situated. The pressure, therefore, is the same at all
points and is independent of direction.
Special experiments in demonstration of this prin
ciple have, perhaps, never been instituted with the re
quisite degree of exactitude. But the proposition has
by our experience of liquids been made very familiar,
and readily explains it.
10. If a liquid be enclosed in a vessel (Fig. 65) Preiimi
which is supplied with a piston ^4, the cross section marks to
,... .. .. . _^ . . , the discuss
of which is unit in area, and with a piston J3 which ion of Pas
, T . cal's deduo
for the time being is made station E _; m ^ tion.
ary, and on the piston A a load p
be placed, then the same pressure
p, gravity neglected, will prevail
throughout all the parts of the vessel.
The piston will penetrate inward and
the walls of the vessel will continue
to be deform ed till the point is reached
at which the elastic forces of the rigid and fluid bodies
perfectly equilibrate one another. If then we imagine
the piston B, which has the crosssection/, to be mov
able, a force f. p alone will keep it in equilibrium.
Concerning Pascal's deduction of the proposition
before discussed from the principle of virtual displace
ments, it is to be remarked that the conditions of dis
g6 THE SCIENCE OF MECHANICS,
Criticism of placement w r hich he perceived hinge wholly upon the
auction. fact of the ready mobility of the parts and on the
equality of the pressure throughout every portion of
the liquid. If it were possible for a greater compression
to take place in one part of a liquid than in another,
the ratio of the displacements would be disturbed and
Pascal's deduction would no longer be admissible.
That the property of the equality of the pressure is a
property given in experience, is a fact that cannot be
escaped ; as we shall readily admit if we recall to mind
that the same law that Pascal deduced for liquids also
holds good for gases, where even approximately there
can be no question of a constant volume. This latter
fact does not afford any difficulty to our view ; but to
that of Pascal it does. In the case of the lever also, be
it incidentally remarked, the ratios of the virtual dis
placements are assured by the elastic forces of the
leverbody, which do not permit of any great devia
tion from these relations.
Thebehav u. VVe shall now consider the action of liquids un
lourol'li *
quids under der the influence of gravity. The upper surface of a
the action & J . .
of gravity. liquid in equilibrium is horizontal,
jAW(Fig. 66). This fact is at once
JV
rendered intelligible when we re
flect that every alteration of the sur
face in question elevates the centre
of gravity of the liquid, and pushes
Fig. ee. t j ie liquid rnass resting in the shaded
space beneath NN and having the centre of gravity S
into the shaded space above NN having the centre of
gravity S". Which alteration, of course, is at once re
versed by gravity.
Let there be in equilibrium in a vessel a heavy
liquid with a horizontal upper surface. We consider
S'
THE PRINCIPLES OF STATICS. 97
(Fig. 67) a small rectangular parallelepipedon in the The con
interior. The area of its horizontal base, we will say, is equilibrium
a, and the length of its vertical edges dh. The weight subjected
of this parallelepipedon is therefore adhs, where s is uon of g'rav
its specific gravity. If the paral
lelepipedon do not sink, this is
possible only on the condition that
a greater pressure is exerted on the
lower surface by the fluid than on
the upper. The pressures on the
upper and lower surfaces we will Flg 6?>
respectively designate as ap and a (p + df}. Equi
librium obtains when adh.s = otdp or dp/dh = s,
where h in the downward direction is reckoned as posi
tive. We see from this that for equal increments of h
vertically downwards the pressure/ must, correspond
ingly, also receive equal increments. So that p =
As + q\ and if <7, the pressure at the upper surface,
which is usually the pressure of the atmosphere, be
comes 0, we have, more simply, p = h s, that is, the
pressure is proportional to the depth beneath the sur
face. If we imagine the liquid to be pouring into a ves
sel, and this condition of affairs not vet attained, every
liquid particle will then sink until the compressed par
ticle beneath balances by the elasticity developed in it
the weight of the particle above.
From the view we have here presented it will be fur Different
ther apparent, that the increase of pressure in a liquid t?of exist
takes place solely in the direction in which gravity finJof the 6
acts. Only at the lower surface, at the base, of the gravity
parallelepipedon, is an excess of elastic pressure on the
part of the liquid beneath required to balance the
weight of the parallelepipedon. Along the two sides of
the vertical containing surfaces of the parallelepipedon,
9 8 THE SCIENCE Of MECHANICS.
the liquid is in a state of equal compression, since no
force acts in the vertical containing surfaces that would
determine a greater compression on the one side than
on the other.
Level sur If we picture to ourselves the totality of all the
points of the liquid at which the same pressure p acts,
we shall obtain a surface a socalled level surface. If
we displace a particle in the direction of the action of
gravity, it undergoes a change of pressure. If we dis
place it at right angles to the direction of the action of
gravity, no alteration of pressure takes place. In the
latter case it remains on the same level surface, and
the element of the level surface, accordingly, stands at
right angles to the direction of the force of gravity.
Imagining the earth to be fluid and spherical, the
level surfaces are concentric spheres, and the directions
of the forces of gravity (the radii) stand at right angles
to the elements of the spherical surfaces. Similar ob
servations are admissible if the liquid particles be acted
on by other forces than gravity, magnetic forces, for
example.
Their func The level surfaces afford, in a certain sense, a dia
thought. gram of the forcerelations to which a fluid is subjected;
a view further elaborated by analytical hydrostatics.
12. The increase of the pressure with the depth be
low the surface of a heavy liquid may be illustrated by
a series of experiments which we chiefly owe to Pas
cal. These experiments also well illustrate the fact,
that the pressure is independent of the direction. In
Fig. 68, i, is an empty glass tube g ground off at the
bottom and closed by a metal disc //, to which a
string is attached, and the whole plunged into a vessel
of water. When immersed to a sufficient depth we
may let the string go, without the metal disc, which is
THE PRINCIPLES OF STATICS.
99
supported by the pressure of the liquid, falling. In 2, Pascal's ex
i ..,.., , *i > o penments
the metal disc is replaced by a tiny column of mer on the
cury. If (3) we dip an open siphon tube filled with liquids.
mercury into the water, we shall
see the mercury, in consequence
of the pressure at a, rise into
the longer arm. In 4, we see a
tube, at the lower extremity of
which a leather bag filled with
mercury is tied : continued im
mersion forces the mercury
higher and higher into the tube.
In 5, a piece of wood h is driven
by the pressure of the water into
the small arm of an empty siphon
tube. In 6, a piece of wood H
immersed in mercury adheres to
the bottom of the vessel, and is
pressed firmly against it for as
long a time as the mercury is
kept from working its way un
derneath it.
13. Once we have made quite
clear to ourselves that the pres
sure in the interior of a heavy
liquid increases proportionally to
the depth below the surface, the
law that the pressure at the base
of a vessel is independent of its
form will be readily perceived.
The pressure increases as we de
scend at an equal rate, whether the vessel (Fig. 69)
has the form abed or ebcf. In both cases the walls
of the vessel where they meet the liquid, go on deforming
THE SCIENCE OF MECHANICS.
Elucida
tion of this
fact.
till the point is reached at which they equilibrate by the
elasticity developed in them the pressure exerted by the
fluid, that is, take the place as regards pressure of the
fluid adjoining. This fact is
a direct justification of Ste
vinus's fiction of the solidi
fied fluid supplying the place
of the walls of the vessel.
The pressure on the base
always remains P = Ahs,
where A denotes the area of the base, h the depth of
the horizontal plane base below the level, and s the
specific gravity of the liquid.
The fact that, the walls of the vessel being neg
lected, the vessels i, 2, 3 of Fig. 70 of equal base
area and equal pressureheight weigh differently in the
balance, of course
in no wise con
tradicts the laws
of pressure men
tioned. If we take
into account the
lateral pressure, we shall see that in the case of i we
have left an extra component downwards, and in the
case of 3 an extra component upwards, so that on the
whole the resultant superficial pressure is always equal
to the weight.
The princi 14. The principle of virtual displacements is ad
tuaidis r " mirably adapted to the acquisition of clearness and
placements , . L ,, . i , ,
applied to comprehensiveness in cases of this character, and we
eration S of ~ shall accordingly make use of it. To begin with, how
fhis b dSl ever, let the following be noted. If the weight q (Fig.
71) descend from position i to position 2, and a weight
of exactly the same size move at the same time from
THE PRINCIPLES OF STATICS.
2 to 3, the work performed in this operation is qh^ + Prel i
gh 2 = g (h^ \ /2 2 ), the same, that is, as if the weight marks 
q passed directly from i to 3 and the weight at 2 re
mained in its original position. The observation is
easily generalised.
u/
Fig 71.
Fig 72.
Let us consider a heavy homogeneous rectangular
parallelepipedon, with vertical edges of the length h,
base A, and the specific gravity ^ (Fig. 72). Let this
parallelepipedon (or, what is the same thing, Its centre
of gravity) descend a distance dh^ The work done is
then A /is.d/i, or, also, A dhsJi. In the first expres
sion we conceive the whole weight Ahs displaced the
vertical distance dh in the second we conceive the
weight Adhs as having descended from the upper
shaded space to the lower shaded space the distance h,
and leave out of account
the rest of the body.
Both methods of concep
tion are admissible and
equivalent.
15. With the aid of
this observation we shall
obtain a clear insight into
the paradox of Pascal, which consists of the following.
The vessel g (Fig. 73), fixed to a separate support and
consisting of a narrow upper and a very broad lower
cylinder, is closed at the bottom by a movable piston,
Fig. 73.
102 THE SCIENCE OF MECHANICS.
which, by means of a string passing through the axis
of the cylinders, is independently suspended from the
extremity of one arm of a balance. If g be filled with
water, then, despite the smallness of the quantity of
water used, there will have to be placed on the other
scalepan, to balance it, several weights of consider
able size, the sum of which will be A /is, where A is
the pistonarea, h the height of the liquid, and ^ its
specific gravity. But if the liquid be frozen and the
mass loosened from the walls of the vessel, a very
small weight will be sufficient to preserve equilibrium.
Theexpia Let us look to the virtual displacements of the two
the parados cases (Fig. 74). In the first case, supposing the pis
ton to be lifted a distance dh, the virtual moment is
Adhs.h or Ahs.dh. It thus
comes to the same thing,
_dk whether we consider the mass
that the motion of the piston
displaces to be lifted to the
upper surface of the fluid
Flg< 74 through the entire pressure
height, or consider the entire weight A/is lifted the
distance of the pistondisplacement dh. In the second
case, the mass that the piston displaces is not lifted to
the upper surface of the fluid, but surfers a displace
ment which is much smaller the displacement, namely,
of the piston. If A, a are the sectional areas respect
ively of the greater and the less cylinder, and k and /
their respective heights, then the virtual moment of the
present case is Adh s . k f adh s. / = (A k + a /) s . dh ;
which is equivalent to the lifting of a much smaller
weight (Ak f /)$', the distance dh.
1 6. The laws relating to the lateral pressure of
liquids are but slight modifications of the laws of basal
THE PRINCIPLES OF STATICS.
103
Fig 75
pressure. If we have, for example, a cubical vessel The laws of
of i decimetre on the side, which is a vessel of litre pressure,
capacity, the pressure on any one of the vertical lateral
walls A BCD, when the vessel is filled with water, is
easily determinable. The deeper the migratory element
considered descends beneath the surface, the greater
the pressure will be to which it is subjected. We easily
perceive, thus, that the pressure on a lateral wall is rep
resented by a wedge of water A BCD HI resting upon
the wall horizontally
placed, where ID is at
right angles to BD and
ID = HC=AC. The
lateral pressure accor
dingly is equal to half
a kilogramme.
To determine the
point of application of the resultant pressure, conceive
ABCD again horizontal with the waterwedge resting
upon it. We cut off AX BL = $AC, draw the
straight line KL and bisect it at M; Mis the point of
application sought, for through this point the vertical
line cutting the centre of gravity of the wedge passes.
A plane inclined figure forming the base of a vessel The pres
filled with a liquid, is divided into the elements a, a', p?ae?n
11 11 i r dined base.
a" . . . with the depths k, h , h . . . below the level of
the liquid. The pressure on the base is
(ah _ a 9 Ji + a" h" + . . .) s.
If we call the total basearea A, and the depth of its
centre of gravity below the surface H, then
ah + a'h' + a"h" + . . . _ ah + cth' + _ jy
a + a' + a" + ...."" A
whence the pressure on the base is AHs.
io 4 T H& SCIENCE OF MECHANICS,
The deduc ij. The principle of Archimedes can be deduced in
principle of various ways. After the manner of Stevinus, let us
desmaybe conceive in the interior of the liquid a portion of it
various m solidified. This portion now, as before, will be sup
Way ported by the circumnatant liquid. The resultant of
the forces of pressure acting on the surfaces is accor
dingly applied at the centre of gravity of the liquid dis
placed by the solidified body, and is equal and opposite
to its weight. If now we put in the place of the solid
ified liquid another different body of the same form, but
of a different specific gravity, the forces of pressure at
the surfaces will remain the same. Accordingly, there
now act on the body two forces, the weight of the body,
applied at the centre of gravity of the body, and the up
ward buoyancy, the resultant of the surfacepressures,
applied at the centre of gravity of the displaced liquid.
The two centres of gravity in question coincide only in
the case of homogeneous solid bodies.
onemeth If we immerse a rectangular parallelepipedon of al
titude h and base a, with edges vertically placed, in a
liquid of specific gravity s, then the pressure on the
upper basal surface, when at a depth k below the level
of the liquid is aks, while the pressure on the lower
surface is a (k + Ji) s. As the lateral pressures destroy
each other, an excess of pressure a/is upwards re
mains ; or, where v denotes the volume of the paral
lelepipedon, an excess v . s.
Another We shall approach nearest the fundamental con
method in
volving the ception from which Archimedes started, by recourse to
principle of .
virtual dis the principle of virtual displacements. Let a paral
placements. . r
lelepipedon (Fig. 76) of the specific gravity a, base a,
and height h sink the distance dh. The virtual mo
ment of the transference frofn the upper into the lower
shaded space of the figure will be a dh . oh. But while
THE PRINCIPLES OF STATICS
105
this is done, the liquid rises from the lower into the up
per space, and its moment is adhsh. The total vir
tual moment is therefore ah (G s) dh = (p q) dh,
where/ denotes the weight of the body and q the weight
of the displaced liquid.
B
Fig. 76. Fig. 77
1 8. The question might occur to us, whether the is the buoy
, . ancy of a
upward pressure of a body in a liquid is affected by the body in a
r r . liquid af
immersion of the latter in another liquid. As a fact, fecttd by
the nmaer
this very question has been proposed. Let therefore sion ot that
J  1 AX liquid in a
(Fig. 77) a body K be submerged in a liquid A and the second
liquid with the containing vessel in turn submerged in
another liquid B. If in the determination of the loss
of weight in A it were proper to take account of the
loss of weight of A in B, then K's loss of weight would
necessarily vanish when the fluid B became identical
with A. Therefore, K immersed in A would suffer a
loss of weight and it would suffer none. Such a rule
would be nonsensical.
With the aid of the principle of virtual displace The eiuci
fK dation of
ments, we easily comprehend the more complicated more corn
cases of this character. If a body be first gradually cases of this
immersed in B, then partly* in B and partly in A,
finally in A wholly ; then, in the second case, consider
ing the virtual moments, both liquids are to be taken
into account in the proportion of the volume of the
body immersed in them. But as soon as the body is
wholly immersed in A, the level of A on further dis
io6
THE SCIENCE CF MECHANICS.
The coun
terexperi
ment.
placement no longer rises, and therefore B is no longer
of consequence.
TheArchi 19, Archimedes's principle maybe illustrated by a
princfpie ii pretty experiment. From the one extremity of a scale
arfexpen ^ beam (Fig. 78) we hang a hollow cube H, and beneath
it a solid cube M, which exactly fits into
the first cube. We put weights into the
opposite pan, until the scales are in
equilibrium. If now M be submerged
in water by lifting a vessel which stands
beneath it, the equilibrium will be dis
turbed ; but it will be immediately re
stored if H, the hollow cube, be filled
with water.
A counterexperiment is the follow
ing. H is left suspended alone at the
one extremity of the balance, and into
the opposite pan is placed a vessel of
water, above which on an independent
support ^fhangs by a thin wire. The scales are brought
to equilibrium. If now M be lowered until it is im
mersed in the water, the equilibrium of the scales will
be disturbed ; but on filling H with water, it will be
restored.
Remarks on At first glance this experiment appears a little para
mentf pen ~ doxical. We feel, however, instinctively, that M can
not be immersed in the water without exerting a pres
sure that affects the scales. When we reflect, that the
level of the water in the vessel rises, and that the solid
body M equilibrates the surfacepressure of the water
surrounding it, that is to say represents and takes the
place of an equal volume of water, it will be found
that the paradoxical character of the experiment van
ishes.
/
Fig. 78.
THE PRINCIPLES OF STATICS. 107
20. The most important statical principles have The gene
been reached in the investigation of solid bodies. This ptes P of n stat
course is accidentally the historical one, but it is by no h^bUn
means the only possible and necessary one. The dif the^nvesti
ferent methods that Archimedes, Stevinus, Galileo, and embodies
the rest, pursued, place this idea clearly enough before
the mind. As a matter of fact, general statical princi
ples, might, with the assistance of some very simple
propositions from the statics of rigid bodies, have been
reached in the investigation of liquids. Stevinus cer
tainly came very near such a discovery. We shall stop
a moment to discuss the question.
Let us imagine a liquid, the weight of which we neg The dis
lect. Let this liquid be enclosed in a vessel and sub illustration
jected to a definite pressure. A portion of the liquid, statement.
let us suppose, solidifies. On the closed surface nor
mal forces act proportional to the elements of the area,
and we see without difficulty that their resultant will
always be = 0.
If we mark off by a closed curve a portion of the
closed surface, we obtain, on either side of it, a non
closed surface. All surfaces which are bounded by the
same curve (of double curvature) and on which forces
act normally (in the same sense) pro \ \ 4
portional to the elements of the area,
have lines coincident in position for
the resultants of these forces.
Let us suppose, now, that a fluid
cylinder, determined by any closed
plane curve as the perimeter of its
base, solidifies. We may neglect the two basal sur
faces, perpendicular to the axis. And instead of the
cylindrical surface the closed curve simply may be con
sidered. From this method follow quite analogous
loS THE SCIENCE OF MECHANICS
Thedis propositions for normal forces proportional to the ele
cussion and  ,
illustration ments of a plane curve.
statement If the closed curve pass into a triangle, the con
sideration will shape itself thus. The resultant normal
forces applied at the middle points of the sides of the
triangle, we represent in direction, sense, and magni
tude by straight lines (Fig. 80). The
lines mentioned intersect at a point
the centre of the circle described about
I the triangle. It will further be noted,
Fig. so. t j lat ky t  ie s i m pi e parallel displace
ment of the lines representing the forces a triangle is
constructible which is similar and congruent to the
original triangle.
Thededuc Thence follows this proposition :
tion of the . , r n . , . , . l
triangle of Any three forces, which, acting at a point, are pro
this method portional and parallel in direction to the sides of a tri
angle, and which on meeting by parallel displacement
form a congruent triangle, are in equilibrium. We see
at once that this proposition is simply a different form
of the principle of the parallelogram of forces.
If instead of a triangle we imagine a polygon, we
shall arrive at the familiar proposition of the polygon
of forces.
We conceive now in a heavy liquid of specific gravity
yt a portion solidified. On the element a of the closed
encompassing surface there acts a normal force a n z,
where z is the distance of the element from the level of
the liquid. We know from the outset the result.
similar de If normal forces which are determined by <xxz,
another im where a: denotes an element of area and z it's perpen
portant pro _.,.,. r .
position, dicular distance from a given plane E, act on a closed
surface inwards, the resultant will be V. x, in which ex
pression V represents the enclosed volume. The
THE PRINCIPLES OF STATICS. 109
resultant acts at the centre of gravity of the volume,
is perpendicular to the plane mentioned, and is directed
towards this plane.
Under the same conditions let a rigid curved surface The propo
be bounded by a plane curve, which encloses on the deduced, a
plane the area A. The resultant of the forces acting of Greens
n , . . _ . Theorem.
on the curved surface is K, where
R* = (AZny + ( J/V)2 AZVK* cos v,
in which expression Z denotes the distance of the
centre of gravity of the surface A from E, and v the
normal angle of E and A.
In the proposition of the last paragraph mathe
matically practised readers will have recognised a par
ticular case of Green's Theorem, which consists in the
reduction of surfaceintegrations to volumeintegra
tions or vice versa.
We may, accordingly, see into the forcesystem of
n . , . ..,., . f , f . cations of
fluid in equilibrium, or, if you please, see out of it, sys the view
terns of forces of greater or less complexity, and thus
reach by a short path propositions a posteriori. It is a
mere accident that Stevinus did not light on these
propositions. The method here pursued corresponds
exactly to his. In this manner new discoveries can
still be made.
21. The paradoxical results that were reached in Fruitful re
,. .. ,...., 11 i r sults of the
the investigation of liquids, supplied a stimulus to fur investiga
_ . _ i Y i i i 1 iir tionsof this
ther reflection and research. It should also not be left domain.
unnoticed, that the conception of a physicomechanical
continuum was first formed on the occasion of the in
vestigation of liquids. A much freer and much more
fruitful mathematical mode of view was developed
thereby, than was possible through the study even of
no THE SCIENCE OF MECHANICS.
systems of several solid bodies. The origin, in fact,
of important modern mechanical ideas, as for instance
that of the potential, is traceable to this source.
THE PRINCIPLES OF STATICS IN THEIR APPLICATION TO
GASEOUS BODIES.
character i. The same views that subserve the ends of science
partmentof in the investigation of liquids are applicable with but
inquiry.
slight modifications to the investigation of gaseous
bodies. To this extent, therefore, the investigation of
gases does not afford mechanics any very rich returns.
Nevertheless, the first steps that were taken in this
province possess considerable significance from the
point of view of the progress of civilisation and so
have a high import for science generally.
The eius Although the ordinary man has abundant oppor
its e subject tunity, by his experience of the resistance of the air, by
the action of the wind, and the confinement of air in
bladders, to perceive that air is of the nature of a body,
yet this fact manifests itself infrequently, and never in
the obvious and unmistakable way that it does in the
case of solid bodies and fluids. It is known, to be sure,
but is not sufficiently familiar to be prominent in popu
lar thought. In ordinary life the presence of the air is
scarcely ever thought of. (See p. 517.)
The effect Although the ancients, as we may learn from the
of the first r TT . . . . .
disclosures accounts of Vitruvius, possessed instruments which,
ince, 1Spr v like the socalled hydraulic organs, were based on the
condensation of air, although the invention of the air
gun is traced back to Ctesibius, and this instrument
was also known to Guericke, the notions which people
held with regard to the nature of the air as late even
THE PRINCIPLES OF STATICS.
OTTO BC GUERICKE
ifs:^ Potentifs: Elector^ Branded
Conlia.rius
H2 THE SCIENCE OF MECHANICS.
as the seventeenth century were exceedingly curious
and loose. We must not be surprised, therefore, at the
intellectual commotion Which the first more important
experiments in this direction evoked. The enthusiastic
description which Pascal gives of Boyle's airpump ex
periments is readily comprehended, if we transport our
selves back into the epoch of these discoveries. What
indeed could be more wonderful than the sudden dis
covery that a thing which we do not see, hardly feel,
and take .scarcely any notice of, constantly envelopes
us on all sides, penetrates all things ; that it is the most
important condition of life, of combustion, and of gi
gantic mechanical phenomena. It was on this occa
sion, perhaps, first made manifest by a great and strik
ing disclosure, that physical science is not restricted
to the investigation of palpable and grossly sensible
processes.
The views 2. In Galileo's time philosophers explained the
on this sub phenomenon of suction, the action of syringes and
. 1 " pumps by the socalled horror vacui nature's abhor
rence of a vacuum. Nature was thought to possess
the power of preventing the formation of a vacuum by
laying hold of the first adjacent thing, whatsoever it
was, and immediately filling up with it any empty space
that arose. Apart from the ungrounded speculative
element which this view contains, it must be conceded,
that to a certain extent it really represents the phe
nomenon. The person competent to enunciate it must
actually have discerned some principle in the phenom
enon. This principle, however, does not fit all cases.
Galileo is said to have been greatly surprised at hearing
of a newly constructed pump accidentally supplied
with a very long suctionpipe which was not able to
raise water to a height of more than eighteen Italian
THE PRINCIPLES OF STATICS. 113
ells. His first thought was that the horror vacui (or the
resistenza del vacua) possessed a measurable power. The
greatest height to which water could be raised by suc
tion he called altezza limit atissima. He sought, more
over, to determine directly the weight able to draw out
of a closed pumpbarrel a tightly fitting piston resting
on the bottom.
3. TORRICELLI hit upon the idea of measuring theTorriceiirs
resistance to a vacuum by a column of mercury instead espenment
of a column of water, and he expected to obtain a col
umn of about ^ of the length of the water column.
His expectation was confirmed by the experiment per
formed in 1643 by Viviani in the wellknown manner,
and which bears today the name of the Torricellian
experiment. A glass tube somewhat over a metre in
length; sealed at one end and filled with mercury, is
stopped at the open end with the finger, inverted in a
dish of mercury, and placed in a vertical position. Re
moving the finger, the column of mercury falls and re
mains stationary at a height of about 76 cm. By this
experiment it was rendered quite probable, that some
very definite pressure forced the fluids into the vacuum.
What pressure this was, Torricelli very soon divined.
Galileo had endeavored, some time before this, to Galileo's
determine the weight of the air, by first weighing a we?gh P air?
glass bottle containing nothing but air and then again
weighing the bottle after the air had been partly ex
pelled by heat. It was known, accordingly, that the
air was heavy. But to the majority of men the horror
vacui and the weight of the air were very distantly
connected notions. It is possible that in Torricelli's
case the two ideas came into sufficient proximity to
lead him to the conviction that all phenomena ascribed
to the horror vacui were explicable in a simple and
n 4 THE SCIENCE OF MECHANICS.
Atmospher logical manner by the pressure exerted by the weight
discovered of a fluid column a column of air. Torricelli discov
ceiii. ered, therefore, the pressure of the atmosphere ; he also
first observed by means of his column of mercury the
variations of the pressure of the atmosphere.
4. The news of Torricelli's experiment was circu
lated in France by Mersenne, and came to the knowl
edge of Pascal in the year 1644. The accounts of the
theory of the experiment were presumably so imper
fect that PASCAL found it necessary to reflect indepen
dently thereon. {Pesanteur de I' air. Paris, 1663.)
Pascal's ex He repeated the experiment with mercury and with
periments. ... ,
a tube of water, or rather of red wine, 40 feet in length.
He soon convinced himself by inclining the tube that
the space above the column of fluid was really empty ;
and he found himself obliged to defend this view against
the violent attacks of his countrymen. Pascal pointed
out an easy way of producing the vacuum which they
regarded as impossible, by the use of a glass syringe,
the nozzle of which was closed with the finger under
water and the piston then drawn back without much
difficulty. Pascal showed, in addition, that a curved
siphon 40 feet high filled with water does not flow, but
can be made to do so by a sufficient inclination to the
perpendicular. The same experiment was made on a
smaller scale with mercury. The same siphon flows
or does not flow according as it is placed in an inclined
or a vertical position.
In a later performance, Pascal refers expressly to
the fact of the weight of the atmosphere and to the
pressure due to this weight. He shows, that minute
animals, like flies, are able, without injury to them
selves, to stand a high pressure in fluids, provided only
the pressure is equal on all sides ; and he applies this
THE PRINCIPLES OF STATICS. 115
at once to the case of fishes and of animals that live InTheanai
the air. Pascal's chief merit, indeed, is to have estab HqnteTn
lished a complete analogy between the phenomena con fc^re^sur
ditioned by liquid pressure (waterpressure) and those
conditioned by atmospheric pressure.
5. By a series of experiments Pascal shows that
mercury in consequence of atmospheric pressure rises
into a space containing no air in the same way that,
in consequence of waterpressure, it rises into a space
containing no water. If into a deep ves
sel filled with water (Fig. 81) a tube be
sunk at the lower end of which a bag of
mercury is tied, but so inserted that the
upper end of the tube projects out of the
water and thus contains only air, then
the deeper the tube is sunk into the water
the higher will the mercury, subjected Fig. si.
to the constantly increasing pressure of the water, as
cend into the tube. The experiment can also be made,
with a siphontube, or with a tube open at its lower end.
Undoubtedly it was the attentive consideration of The height
this very phenomenon that led Pascal to the idea that tains deter
, , ,,, mined by
the barometercolumn must necessarily stand lower at the barom
eter.
the summit of a mountain than at its base, and that
it could accordingly be employed to determine the
height of mountains. He communicated this idea to
his brotherinlaw, Perier, who forthwith successfully
performed the experiment on the summit of the Puy
de Dome. (Sept. 19, 1648.)
Pascal referred the phenomena connected with ad Adhesion
hesionplates to the pressure of the atmosphere, and
gave as an illustration of the principle involved the re
sistance experienced when a large hat lying flat on a
table is suddenly lifted. The cleaving of wood to the
n6
THE SCIENCE OF MECHANICS.
A siphon
which acts
by water
pressure.
Pascal's
modifica
tion of the
Torricelli
an experi
ment.
bottom of a vessel of quicksilver is a phenomenon of
the same kind.
Pascal imitated the flow produced in a siphon by
atmospheric pressure, by the use of waterpressure.
The two open unequal arms a and
/; of a threearmed tube a b c (Fig.
82) are dipped into the vessels of
mercury e and d. If the whole
arrangement then be immersed in
a deep vessel of water, yet so that
the long open branch shall always
project above the upper surface,
the mercury will gradually rise in
the branches a and d, the columns
finally unite, and a stream begin to flow from the vessel
d to the vessel e through the siphontube open above
to the air.
The Torricellian experiment was modi
fied by Pascal in a very ingenious manner.
A tube of the form abed (Fig. 83), of
double the length of an ordinary barom
etertube, is filled with mercury. The
openings a and b are closed with the fin
gers and the tube placed in a dish of
mercury with the end a downwards. If
now a be opened, the mercury in cd will
all fall into the expanded portion at c 9 and
the mercury in ab will sink to the height
of the ordinary barometercolumn. A vac
uum is produced at b which presses the
finger closing the hole painfully inwards.
If b also be opened the column in a b will
sink completely, while the mercury in the expanded
portion c, being now exposed to the pressure of the
 83.
THE PRINCIPLES OF STATICS. 117
atmosphere, will rise in c d to the height of the barom
etercolumn. Without an airpump it was hardly pos
sible to combine the experiment and the counter
experiment in a simpler and more ingenious manner
than Pascal thus did.
6. With regard to Pascal's mountainexperiment, Suppie
we shall add the following brief supplementary remarks, marks on
Let be the height of the barometer at the level of mountafn
i i r 11 i r experiment
the sea, and let it tall, say, at an elevation of m metres,
to kb ^ where k is a proper fraction. At a further eleva
tion of m metres, we must expect to obtain the barom
eterheight k.kb^ since we here pass through a stratum
of air the density of which bears to that of the first the
proportion of k : 1. If we pass upwards to the altitude
h = n . m metres, the barometerheight corresponding
thereto will be
The principle of the method is, we see, a very simple
one ; its difficulty arises solely from the multifarious
collateral conditions and corrections that have to be
looked to.
7. The most original and fruitful achievements in The expe
* i . f /\ j^ rnents of
the domain of aerostatics we owe to OTTO VON GUE otto yon
TT . . T , , i Guericke.
RICKE. His experiments appear to have been suggested
in the main by philosophical speculations. He pro
ceeded entirely in his own way ; for he first heard of
the Torricellian experiment from Valerianus Magnus
at the Imperial Diet of Ratisbon in 1654, where he dem
onstrated the experimental discoveries made by him
about 1650. This statement is confirmed by his method
i iS THE SCIENCE OF MECHANICS.
of constructing a waterbarometer which was entirely
different from that of Torricelli.
Thehistori Guericke's book (Experimenta nova, ut vocantur*
cal value of .
Guencke's Mapdeburpica. Amsterdam. 1672) makes us realise
book. oo /
the narrow views men took in his time. The fact that
he was able gradually to abandon these views and to
acquire broader ones by his individual endeavor speaks
favorably for his intellectual powers. We perceive
with astonishment how short a space of time separates
us from the era of scientific barbarism, and can no lon
ger marvel that the barbarism of the social order still
so oppresses us.
its specula In the introduction to this book and in various other
tive charac ., _,.,... ... . . . . .
ter. places, Guencke, m the midst of his experimental in
vestigations, speaks of the various objections to the
Copernican system which had been drawn from the
Bible, (objections which he seeks to invalidate,) and
discusses such subjects as the locality of heaven, the
locality of hell, and the day of judgment. Disquisi
tions on empty space occupy a considerable portion
of the work.
Guericke's Guericke regards the air as the exhalation or odor
the air. of bodies, which we do not perceive because we have
been accustomed to it from childhood. Air, to him,
is not an element He knows that through the effects of
heat and cold it changes its volume, and that it is
compressible in Hero's Ball, or Pila Heronis ; on the
basis of his own experiments he gives its pressure at
20 ells of water, and expressly speaks of its weight, by
which flames are forced upwards.
8. To produce a vacuum, Guericke first employed
a wooden cask filled with water. The pump of a fire
engine was fastened to its lower end. The water, it
was thought, in following the piston and the action of
THE PRINCIPLES OF STA TICS,
119
Guericke's First Experiments, (Exfertm. Magded.)
120 THE SCIENCE OF MECHANICS.
His at gravity, would fall and be pumped out. Guericke ex
tempts to , . nni .
produce a pected that empty space would remain. The fastenings
vacuum. . , 1 1 ,, . . .
of the pump repeatedly proved to be too weak, since m
consequence of the atmospheric pressure that weighed
on the piston considerable force had to be applied to
move it. On strengthening the fastenings three power
ful men finally accomplished the exhaustion. But,
meantime the air poured in through the joints of the
cask with a loud blast, and no vacuum was obtained.
In a subsequent experiment the small cask from which
the water was to be exhausted was immersed in a larger
one, likewise filled with water. But in this case, too, the
water gradually forced its way into the smaller cask.
His final Wood having proved in this way to be an unsuit
able material for the purpose, and Guericke having re
marked in the last experiment indications of success,
the philosopher now took a large hollow sphere of
copper and ventured to exhaust the air directly. At
the start the exhaustion was successfully and easily
conducted. But after a few strokes of the piston, the
pumping became so difficult that four stalwart men
(viri quadrati), putting forth their utmost efforts, could
hardly budge the piston. And when the exhaustion
had gone still further, the sphere suddenly collapsed,
with a violent report. Finally by the aid of a copper
vessel of perfect spherical form, the production of the
vacuum was successfully accomplished. Guericke de
scribes the great force with which the air rushed in on
the opening of the cock.
9. After these experiments Guericke constructed
an independent airpump. A great glass globular re
ceiver was mounted and closed by a large detachable
tap in which was a stopcock. Through this opening
the objects to be subjected to experiment were placed
THE PRINCIPLES OF STATICS. 121
in the receiver. To secure more perfect closure the Guerick
receiver was made to stand, with its stopcock under
water, on a tripod, beneath which the pump proper was
Guericke's Airpump. (Exj>erzm, Magdeb!)
placed. Subsequently, separate receivers, connected
with the exhausted sphere, were also employed in the
experiments.
122 THE SCIENCE OF MECHANICS.
The curious The phenomena which Guericke observed with this
phenomena r 1 i nm i i
observed by apparatus are manifold and various. The noise which
means of . . . . .
the air water in a vacuum makes on striking the sides oi the
glass receiver, the violent rush of air and water into
exhausted vessels suddenly opened, the escape on ex
haustion of gases absorbed in liquids, the liberation of
their fragrance, as Guericke expresses it, were imme
diately remarked. A lighted candle is extinguished
on exhaustion, because, as Guericke conjectures, it
derives its nourishment from the air. Combustion, as
his striking remark is, is not an annihilation, but a
transformation of the air.
A bell does not ring in a vacuum. Birds die in it.
Many fishes swell up, and finally burst. A grape is kept
fresh in vaciw for over half a year.
By connecting with an exhausted cylinder a long
tube dipped in water, a waterbarometer is constructed.
The column raised is 1920 ells high; and Von Guericke
explained all the effects that had been ascribed to the
horror vacui by the principle of atmospheric pressure.
An important experiment consisted in the weighing
of a receiver, first when filled with air and then when
exhausted. The weight of the air was found to" vary
with the circumstances ; namely, with the temperature
and the height of the barometer. According to Gue
ricke a definite ratio of weight between air and water
does not exist.
The experi But the deepest impression on the contemporary
mentsrelat .... .
ing to at world was made by the experiments relating to atmos
mospheric . A i i i r . r
pressure, pheric pressure. An exhausted sphere formed of two
hemispheres tightly adjusted to one another was rent
asunder with a violent report only by the traction of
sixteen horses. The same sphere was suspended from
THE PRINCIPLES OF STATICS. 123
a beam, and a heavily laden scalepan was attached to
the lower half.
The cylinder of a large pump is closed by a piston.
To the piston a rope is tied which leads over a pulley
and is divided into numerous branches on which a
great number of men pull. The moment the cylinder is
connected with an exhausted receiver, the men at the
ropes are thrown to the ground. In a similar manner
a huge weight is lifted.
Guericke mentions the compressedair gun as some Guericke's
thing already known, and constructs independently an
instrument that might appropriately be called a rari
fiedair gun. A bullet is driven b}^ the external atmos
pheric pressure through a suddenly exhausted tube,
forces aside at the end of the tube a leather valve which
closes it, and then continues its flight with a consider
able velocity.
Closed vessels carried to the summit of a mountain
and opened, blow out air ; carried down again in the
same manner, they suck in air. From these and other
experiments Guericke discovers that the air is elastic.
10. The investigations of Guericke were continued The investi
by an Englishman, ROBERT BOYLE.* The new experi Robert
ments which Boyle had to supply were few. He ob
serves the propagation of light in a vacuum and the
action of a magnet through it ; lights tinder by means
of a burning glass ; brings the barometer under the re
ceiver of the airpump, and was the first to construct
a balancemanometer ["the statical manometer"].
The ebullition of heated fluids and the freezing of water
on exhaustion were first observed by him.
Of the airpump experiments common at the present
day may also be mentioned that with falling bodies,
* And published by him In 1660, before the work of Von Guericke. Trans.
I2 4
THE SCIENCE OF MECHANICS.
Quantita
tive data.
The fail of which confirms in a simple manner the view of Galileo
bodies in a, , , . rii. i .
vacuum, that when the resistance of the air has been eliminated
light and heavy bodies both fall with the same velo
city. In an exhausted glass tube a leaden bullet and a
piece of paper are placed. Putting the tube in a ver
tical position and quickly turning it about a horizontal
axis through an angle of 180, both bodies will be seen
to arrive simultaneously at the bottom of the tube.
Of the quantitative data we will mention the fol
lowing. The atmospheric pressure that supports a
column of mercury of 76 cm. is easily calculated from
the specific gravity 1360 of mercury to be 1*0336 kg.
to i sq.cm. The weight of 1000 cu.cm. of pure, dry
air at C. and 760 mm. of pressure at Paris at an ele
vation of 6 metres will be found to be 1293 grams,
and the corresponding specific gravity, referred to
water, to be 0001293.
Thediscov ii. Guericke knew of only one kind of air. We
ery of other . . . .
gaseous may imagine therefore the excitement it created when
substances. . j u j /rjs
in 1755 BLACK discovered carbonic acid gas (fixed air)
and CAVENDISH in 1766 hydrogen (inflammable air),
discoveries which were soon followed by other similar
ones. The dissimilar
physical properties of
gases are very strik
ing. Faraday has il
lustrated their great
inequality of weight
by a beautiful lecture
experiment. If from
a balance in equilib
rium, we suspend (Fig. 84) two beakers A, B, the one
in an upright position and the other with its opening
downwards, we may pour heavy carbonic acid gas from
THE PRINCIPLES OF STATICS. 125
above into the one and light hydrogen from beneath
into the other. In both instances the balance turns in
the direction of the arrow. Today, as we know, the
decanting of gases can be made directly visible by the
optical method of Foucault and Toeppler.
12. Soon after Torricelli's discovery, attempts were The mercu
. . rial air
made to employ practically the vacuum thus produced, pump.
The socalled mercurial airpumps were tried. But no
such instrument was successful until the present cen
tury. The mercurial airpumps now in common use
are really barometers of which the extremities are sup
plied with large expansions and so connected that their
difference of level may be easily varied. The mercury
takes the place of the piston of the ordinary airpump.
13. The expansive force of the air, a property ob Boyle's law.
served by Guericke, was more accurately investigated
by BOYLE, and, later, by MARIOTTE. The law which
both found is as follows. If F"be called the volume of
a given quantity of air and P its pressure on unit area
of the containing vessel, then the product V. P is
always = a constant quantity. If the volume of the
enclosed air be reduced onehalf, the air will exert
double the pressure on unit of area ; if the volume of
the enclosed quantity be doubled, the pressure will
sink to onehalf ; and so on. It is quite correct as a
number of English writers have maintained in recent
times that Boyle and not Mariotte is to be regarded
as the discoverer of the law that usually goes by
Mariotte's name. Not only is this true, but it must
also be added that Boyle knew that the law did not
hold exactly, whereas this fact appears to have escaped
Mariotte.
The method pursued by Mariotte in the ascertain
ment of the law was very simple. He partially filled
126
THE SCIENCE OF MECHANICS.
Mariotte's
experi
ments.
Fig. 85.
His appa
ratus.
Torricellian tubes with mercury, measured the volume
of the air remaining, and then performed the Torricel
lian experiment. The new volume of
air was thus obtained, and by subtract
ing the height of the column of mer
cury from the barometerheight, also
the new pressure to which the same
quantity of air was now subjected.
To condense the air Mariotte em
ployed a siphontube with vertical
arms. The smaller arm in which the
air was contained was sealed at the
upper end ; the longer, into which the
mercury was poured, was open at the
upper end. The volume of the air
was read off on the graduated tube,
and to the difference of level of the
mercury in the two arms the barometer
height was added. At the present day
both sets of experiments are performed
in the simplest manner by fastening a
cylindrical glass tube (Fig. 86) rr,
closed at the top, to a vertical scale
and connecting it by a caoutchouc
tube kk with a second open glass tube
r' r', which is movable up and down
the scale. If the tubes be partly filled
with mercury, any difference of level
whatsoever of the two surfaces of mer
Fig. se. cury may be produced by displacing
r' ;', and the corresponding variations of volume of the
air enclosed in r r observed.
It struck Mariotte on the occasion of his investiga
tions that any small quantity of air cut off completely
THE PRINCIPLES OF STATICS. 127
from the rest of the atmosphere and therefore notTheexpan
i rr 1111 . s ' ve f r ce of
directly anected by the latter s weight, also supported isolated
. , . . portions of
the barometer column j as where, to give an instance, theatmos
the open arm of a barometertube is closed. The simple
explanation of this phenomenon, which, of course,
Mariotte immediately found, is this, that the air before
enclosure must have been compressed to a point at
which its tension balanced the gravitational pressure
of the atmosphere ; that is to say, to a point at which
it exertqd an equivalent elastic pressure.
We shall not enter here into the details of the ar
rangement and use of airpumps, which are readily
understood from the law of Boyle and Mariotte.
14. It simply remains for us to remark, that the dis
coveries of aerostatics furnished so much that was new
and wonderful that a valuable intellectual stimulus pro
ceeded from the science.
CHAPTER II.
THE DEVELOPMENT OF THE PRINCIPLES OF
DYNAMICS.
i.
GALILEO'S ACHIEVEMENTS.
Dynamics i. We now pass to the discussion of the funda
m9defn a mental principles of dynamics. This is entirely a mod
ern science. The mechanical speculations of the an
cients, particularly of the Greeks, related wholly to
statics. Dynamics was founded by GALILEO. We shall
readily recognise the correctness of this assertion if we
but consider a moment a few propositions held by the
Aristotelians of Galileo's time. To explain the descent
of heavy bodies and the rising of light bodies, (in li
quids for instance,) it was assumed that every thing and
object sought its place : the place of heavy bodies was
below, the place of light bodies was above. Motions
were divided into natural motions, as that of descent,
and violent motions, as, for example, that of a pro
jectile. From some few superficial experiments and
observations, philosophers had concluded that heavy
bodies fall more quickly and lighter bodies more slowly,
or, more precisely, that bodies of greater weight fall
more quickly and those of less weight more slowly. It
is sufficiently obvious from this that the dynamical
knowledge of the ancients, particularly of the Greeks,
was very insignificant, and that it was left to modern
THE PRINCIPLES OF DYNAMICS.
129
times to lay the true foundations of this department of
inquiry. (See Appendix, VII., p. 520.)
2. '1 he treatise D is cor si e dimostrazwni matematiche,
in which Galileo communicated to the world the first
130 THE SCIENCE OF MECHANICS.
Galileo's dynamical Investigation of the laws of falling bodies,
tionofthe appeared in 1638. The modern spirit that Galileo dis
lawsoffall rr . , ,  , , , .
ing bodies, covers is evidenced here, at the very outset, by the tact
that he does not ask why heavy bodies fall, but pro
pounds the question, How do heavy bodies fall ? in
agreement with what law do freely falling bodies move?
The method he employs to ascertain this law is this.
He makes certain assumptions. He does not, however,
like Aristotle, rest there, but endeavors to ascertain by
trial whether they are correct or not.
His first, The first theory on which he lights is the following,
theory. It seems In his eyes plausible that a freely falling body,
Inasmuch as it is plain that its velocity is constantly
on the increase, so moves that its velocity is double
after traversing double the distance, and triple after
traversing triple the distance ; in short, that the veloci
ties acquired in the descent increase proportionally
to the distances descended through. Before he pro
ceeds to test experimentally this hypothesis, he reasons
on it logically, Implicates himself, however, in so doing,
in a fallacy. He says, if a body has acquired a certain
velocity in the first distance descended through, double
the velocity in double such distance descended through,
and so on ; that is to say, if the velocity In the second
instance is double what it is in the first, then the double
distance will be traversed in the same time as the origi
nal simple distance. If, accordingly, In the case of
the double distance we conceive the first half trav
ersed, no time will, it would seem, fall to the account
of the second half. The motion of a falling body ap
pears, therefore, to take place instantaneously ; which
not only contradicts the hypothesis but also ocular evi
dence. We shall revert to this peculiar fallacy of
Galileo's later on.
THE PRINCIPLES OF DYNAMICS. 131
3. After Galileo fancied he had discovered this as HIS second,
j. u i_i i i i correct, as
sumption to be untenable, he made a second one, ac sumption.
cording to which the velocity acquired is proportional
to the time of the descent. That Is, if a body fall once,
and then fall again during twice as long an Interval of
time as it first fell, it will attain in the second Instance
double the velocity it acquired in the first He found
no selfcontradiction in this theory, and he accordingly
proceeded to investigate by experiment whether the
assumption accorded with observed facts. It was dif
ficult to prove by any direct means that the velocity
acquired was proportional to the time of descent. It
was easier, however, to investigate by what law the
distance increased with the time ; and he consequently
deduced from his assumption the relation that obtained
between the distance and the time, and tested this by
experiment. The deduction r> Discussion
,  . . . ^P and eluci
is simple, distinct, and per
fectly correct. He draws
(Fig. 87) a straight line, and
on it cuts off successive por O
dation of
the true
theory.
tions that represent to him Fig 87.
the times elapsed. At the extremities of these por
tions he erects perpendiculars (ordinates), and these
represent the velocities acquired. Any portion OG of
the line OA denotes, therefore, the time of descent
elapsed, and the corresponding perpendicular GH* the
velocity acquired in such time.
If, now, we fix our attention on the progress of the
velocities, we shall observe with Galileo the following
fact : namely, that at the instant C } at which onehalf
O C of the time of descent OA has elapsed, the velocity
CD is also onehalf of the final velocity AS.
If now we examine two instants of time, E and G,
1 32 THE SCIENCE OF MECHANICS.
Uniformly equally distant in opposite directions from the instant
motion, C, we shall observe that the velocity HG exceeds the
mean velocity CD by the same amount that EF falls
short of it. For every instant antecedent to C there
exists a corresponding one equally distant from it sub
sequent to C. Whatever loss, therefore, as compared
with uniform motion with half the final velocity, is suf
fered in the first half of the motion, such loss is made
up in the second half. The distance fallen through we
may consequently regard as having been uniformly de
scribed with half the final velocity. If, accordingly,
W 7 e make the final velocity v proportional to the time
of descent /, we shall obtain v=gt 9 where g denotes
the final velocity acquired in unit of time the socalled
acceleration. The space s descended through is there
fore given by the equation s = ("//2) / or s = gt* /z.
Motion of this sort, in which, agreeably to the assump
tion, equal velocities constantly accrue in equal inter
vals of time, we call uniformly accelerated motion.
Tabieofthe If we collect the times of descent, the final veloci
lochies.and ties, and the distances traversed, we shall obtain the
distances of n . u ,
descent, following table :
/. v. s.
i. V i x i . f
2. 2g. 2 X 2 . 
0*
^ 3<r Q V Q <b
o* o x a 2~
<r
4. 4r. 4X4. 4
tg. t X / . 
THE PRINCIPLES OF DYNAMICS. 133
4. The relation obtaining between / and s admits Expenmen
of experimental proof; and this Galileo accomplished tion of the
in the manner which we shall now describe.
We must first remark that no part of the knowledge
and ideas on this subject with which we are now so
familiar, existed in Galileo's time, but that Galileo had
to create these ideas and means for us. Accordingly,
it was impossible for him to proceed as we should do
today, and he was obliged, therefore, to pursue a dif
ferent method. He first sought to retard the motion
of descent, that it might be more accurately observed.
He made observations on balls, which he caused to
roll down inclined planes (grooves); assuming that only
the velocity of the motion would be lessened here, but
that the form of the law of descent would remain un
modified. If, beginning from the upper extremity, theThearti
fices em
distances i, 4, 9, 1 6 ... be notched off on the groove, ployed.
the respective times of descent will be representable,
it w r as assumed, by the numbers i, 2, 3, 4 . . . ; a result
which was, be it added, confirmed. The observation of
the times involved, Galileo accomplished in a very in
genious manner. There were no clocks of the modern
kind in his day : such were first rendered possible by
the dynamical knowledge of which Galileo laid the
foundations. The mechanical clocks w T hich were used
were very inaccurate, and were available only for the
measurement of great spaces of time. Moreover, it
was chiefly waterclocks and sandglasses that were in
use in the form in which they had been handed down
from the ancients. Galileo, now, constructed a very
simple clock of this kind, which he especially adjusted
to the measurement of small spaces of time ; a thing
not customary in those days. It consisted of a vessel of
water of very large transverse dimensions, having in
134 TffE SCIENCE OF MECHANICS.
Galileo's the bottom a minute orifice which was closed with the
finger. As soon as the ball began to roll down the in
clined plane Galileo removed his finger and allowed the
water to flow out on a balance ; when the ball had ar
rived at the terminus of its path he closed the orifice.
As the pressureheight of the fluid did not, owing to
the great transverse dimensions of the vessel, percept
ibly change, the weights of the water discharged from
the orifice were proportional to the times. It was in
this way actually shown that the times increased simply,
while the spaces fallen through increased quadratically.
The inference from Galileo's assumption was thus con
firmed by experiment, and with it the assumption itself.
Thereia c. To form some notion of the relation which sub
tion of mo ., . . . .   "JT
tion^on an sists between motion on an inclined plane and that of
plane to free descent, Galileo made the assumption, that a body
descent, which falls through the height of an inclined plane
attains the same final velocity as a body which falls
through its length. This is an assumption that will
strike us as rather a bold one ; but in the manner in
which it was enunciated and employed by Galileo, it is
quite natural. We shall endeavor to explain the way by
which he was led to it. He says : If a body fall freely
downwards, its velocity increases proportionally to the
time. When, then, the body has arrived at a point be
low, let us imagine its velocity reversed and directed
upwards ; the body then, it is clear, will rise. We make
the observation that its motion in this case is a reflection,
so to speak, of its motion in the first case. As then its
velocity increased proportionally to the time of descent,
it will now, conversely, diminish in that proportion.
When the body has continued to rise for as long a
time as it descended, and has reached the height from
which it originally fell, its velocity will be reduced to
THE PRINCIPLES OF DYNAMICS. 135
zero. We perceive, therefore, that a body will rise, justifica
in virtue of the velocity acquired in its descent, just as assumption
high as it has fallen. If, accordingly, a body falling final veioc
down an inclined plane could acquire a velocity which motions are
would enable it, when placed on a differently inclined
plane, to rise higher than the point from which it had
fallen, we should be able to effect the elevation of
bodies by gravity alone. There is contained, accord
ingly, in this assumption, that the velocity acquired by
a body in descent depends solely on the vertical height
fallen through and is independent of the inclination of
the path, nothing more 'than the uncontradictory ap
prehension and recognition of the fact that heavy bodies
do not possess the tendency to rise, but only the ten
dency to fall. If we should assume that a body fall
ing down the length of an inclined plane in some way
or other attained a greater velocity than a body that
fell through its height, we should only have to let the
body pass with the acquired velocity to another in
clined or vertical plane to make it rise to a greater ver
tical height than it had fallen from. And if the velo
city attained on the inclined plane were less, we should
only have to reverse the process to obtain the same re
sult. In both instances a heavy body could, by an ap
propriate arrangement of inclined planes, be forced
continually upwards solely by its own weight a state
of things which wholly contradicts our instinctive
knowledge of the nature of heavy bodies. (Seep. 522.)
6. Galileo, in this case, again, did not stop with
the mere philosophical and logical discussion of his
assumption, but tested it by comparison with expe
rience.
He took a simple filar pendulum (Fig. 88) with a
heavy ball attached. Lifting the pendulum, while
i 3 6
THE SCIENCE OF MECHANICS.
Galileo's elongated its full length, to the level of a given altitude,
experimen _, ,...,,. , , , 11
tai venfica and then letting it fall, it ascended to the same level
assumption on the opposite side. If it does not do so exactly,
Galileo said, the resistance of the air must be .the cause
of the deficit. This is inferrible from the fact that the
deficiency is greater in the case of a cork ball than it is
Effected by in the case of a heavy metal one. However, this neg
Fmp'eding lected, the body ascends to the same altitude on the
ofa^endu opposite side. Now it is permissible to regard the mo
um smng. ^.^ ^ ^ pendulum in the arc of a circle as a motion
of descent along a series of inclined planes of different
inclinations. This seen, we can, with Galileo, easily
cause the body to rise on a different arc on a different
series of inclined planes. This we accomplish by driv
ing in at one side of the thread, as it vertically hangs,
a nail/ or g, which will prevent any given portion of
the thread from taking part in the second half of the
motion. The moment the thread arrives at the line of
equilibrium and strikes the nail, the ball, which has
fallen through ba, will begin to ascend by a different
series of inclined planes, and describe the arc am or an.
Now if the inclination of the planes had any influence
THE PRINCIPLES OF DYNAMICS. 137
on the velocity of descent, the body could not rise to
the same horizontal level from which it had fallen.
But it does. By driving the nail sufficiently low dow r n,
we may shorten the pendulum for half of an oscillation
as much as we please ; the phenomenon, however, al
ways remains the same. If the nail h be driven so low
down that the remainder of the string cannot reach to
the plane E, the ball will turn completely over and
wind the thread round the nail ; because when it has
attained the greatest height it can reach it still has a
residual velocity left.
7. If we assume thus, that the same final velocity is The as
sumption
attained on an inclined plane whether the body fall leads to the
r . law of rela
through the height or the length of the plane, in which tive accei
assumption nothing more is contained than that a body sought,
rises by virtue of the velocity it has acquired in falling
just as high as it has fallen, we shall easily arrive,
with Galileo, at the perception that the times of the de
scent along the height and the length of an inclined
plane are in the simple proportion of the height and
the length ; or, what is the same, that the accelerations
are inversely proportional to the times of descent.
The acceleration along the height will consequently
bear to the acceleration along
the length the proportion of the
length to the height. Let AB
(Fig. 89) be the height and A C ^
the length of the inclined plane. Fig. 89.
Both will be descended through in uniformly accel
erated motion in the times t and / x with the final ve
locity v. Therefore,
~ V txn&
/ and
1 3 8
THE SCIENCE OF MECHANICS.
If the accelerations along the height and the length be
called respectively g and g^ we also have
v = gt and v =
1
whence =  =
g ^i AC
In this way we are able to deduce from the accel
eration on an inclined plane the acceleration of free
descent.
A corollary From this proposition Galileo deduces several cor
ceding law. ollaries, some of which have passed into our elementary
textbooks. The accelerations along the height and
length are in the inverse proportion of the height and
length. If now we cause one body to fall along the
length of an inclined plane and simultaneously another
to fall freely along its height, and ask what the dis
tances are that are traversed by the two in equal inter
vals of time, the solution of the problem will be readily
found (Fig, 90) by simply letting fall from B a perpen
dicular on the length. The part AD, thus cut off, will
be the distance traversed by the one body on the in
clined plane, while the second body is freely falling
through the height of the plane.
A
D
B
Fig. 90. Fig. 91.
Relative If we describe (Fig. 91) a circle on AB as diame
icription of ter, the circle will pass through D, because D is a
inddiame right angle. It will be seen thus, that we can imagine
lel. ir " any number of inclined planes, AE, AF, of any degree
of inclination, passing through A, and that in every
THE PRINCIPLES OF DYNAMICS.
139
circles 
case the chords AG, A IT drawn In this circle from the
upper extremity of the diameter will be traversed in
the same time by a falling body as the vertical diame
ter itself. Since, obviously, only the lengths and In
clinations are essential here, we may also draw the
chords in question from the lower extremity of the
diameter, and say generally : The vertical diameter
of a circle is described by a falling particle in the same
time that any chord through either extremity is so
described.
We shall present another corollary, which, in the The figure
pretty form in which Galileo gave it, is usually no bodies fZn
longer incorporated in elementary expositions. We chords o!
Imagine gutters radiating in a vertical plane from a c
common point A at a
number of different
degrees of inclination
to the horizon (Fig.
92). We place at their
common extremity A
a like number of heavy
bodies and cause them
to begin simultaneous
ly their motion of des
cent. The bodies will
always form at any one
instant of time a circle. After the lapse of a longer time
they will be found in a circle of larger radius, and the
radii increase proportionally to the squares of the
times. If we imagine the gutters to radiate in a space
instead of a plane, the falling bodies will always form
a sphere, and the radii of the spheres will increase pro
portionally to the squares of the times. This will be
140 THE SCIENCE OF MECHANICS.
perceived by imagining the figure revolved about the
vertical A l r .
character 8. We see thus, as deserves again to be briefly
inquiries, noticed, that Galileo did not supply us with a theory
of the falling of bodies, but investigated and estab
lished, wholly without preformed opinions, the actual
facts of falling.
Gradually adapting, on this occasion, his thoughts
to the facts, and everywhere logically abiding by the
ideas he had reached, he hit on a conception, which to
himself, perhaps less than to his successors, appeared
in trie light of a new law. In all his reasonings, Galileo
f followed, to the greatest advantage of science, a prin
"T ciple which might appropriately be called the principle
The print  of contimiity. Once we have reached a theory that ap
ciple of ; J \ *
continuity plies to a particular case, we proceed gradually to
; modify in thought the conditions of that case, as far
j as it is at all possible, and endeavor in so doing to
:. adhere throughout as closely as we can to the concep
tion originally reached. There is no method of pro
cedure more surely calculated to lead to that compre
: hension of all natural phenomena which is the simplest
and also attainable with the least expenditure of men
tality and feeling. (Compare Appendix, IX., p. 523.)
A particular instance will show more clearly than
any general remarks what we mean. Galileo con
A C D F
B H
Fig. 93
siders (Fig. 93) a body which is falling down the in
clined plane AB, and which, being placed with the
THE PRINCIPLES OF D YXAMICS. 141
velocity thus acquired on a second plane B C, for ex Galileo's
ample, ascends this second plane. On all planes J3C y ofVneso
,5 ' r 1 i  i i called law
BD, and so forth, it ascends to the horizontal plane of mertia.
that passes through A. But, just as it falls on BD
with less acceleration than it does on BC, so similar!}*
it will ascend on BD with less retardation than it will
on BC. The nearer the planes BC, BD, BE, BP^
proach to the horizontal plane BH, the less will the
retardation of the body on those planes be, and the
longer and further will it move on them. On the hori
zontal plane BH the retardation vanishes entirely (that '
is, of course, neglecting friction and the resistance of
the air), and the body will continue to move infinitely
long and infinitely far with constant velocity. Thus ad
vancing to the limiting case of the problem presented,
Galileo discovers the socalled law of inertia, according
to which a body not under the influence of forces, i. e.
of special circumstances that change motion, will re
tain forever its velocity (and direction). We shall
presently revert to this subject.
Q. The motion of falling that Galileo found actually The deduc
,. , . . , . , , , tion of the
to exist, is, accordingly, a motion of which the velocity idea of um
increases proportionally to the time a socalled uni ceierated
P , . , . motion.
formly accelerated motion.
It would be an anachronism and utterly unhistorical
to attempt, as is sometimes done, to derive the uniformly
accelerated motion of falling bodies from the constant
action of the force of gravity. " Gravity is a constant
force consequently it generates in equal elements of
time equal increments of velocity ; thus, the motion
produced is uniformly accelerated." Any exposition
such as this would be unhistorical, and would put the
whole discovery in a false light, for the reason that the
notion of force as we hold it today was first created
142 THE SCIENCE OF MECHANICS.
Forces and by Galileo. Before Galileo force was known solely as
accelera  
tions. pressure. Now, no one can know, who has not learned
it from experience, that generally pressure produces
motion, much less /;/ what manner pressure passes into
motion ; that not position, nor velocity, but accelera
tion, is determined by it. This cannot be philosophi
cally deduced from the conception, itself. Conjectures
may be set up concerning it. But experience alone can
definitively inform us with regard to it.
10. It is not by any means selfevident, therefore,
* that the circumstances which determine motion, that
is, forces, immediately produce accelerations. A glance
at other departments of physics will at once make this
clear. The differences of temperature of bodies also
determine alterations. However, by differences of tem
perature not compensatory accelerations are deter
mined, but compensatory velocities.
The fact That it is accelerations which are the immediate ef
that forces , , , , ,
determine fects of the circumstances that determine motion, that
Sonstslm is, of the forces, is a fact which Galileo fere eived in the
tai P lac? en natural phenomena. Others before him had also per
ceived many things. The assertion that everything seeks
its place also involves a correct observation. The ob
servation, however, does not hold good in all cases,
and it is not exhaustive. If we cast a stone into the
air, for example, it no longer seeks its place ; since its
place is b.elow. But the acceleration towards the earth,
the retardation of the upward motion, the fact that Ga
lileo perceived, is still present. His observation always
remains correct; it holds true more generally; it em
braces in one mental effort much more.
ii. We have already remarked that Galileo dis
covered the socalled law of inertia quite incidentally.
A body on which, as we are wont to say, no force acts,
THE PRINCIPLES OF DYNAMICS. 143
preserves its direction and velocity unaltered. The History of
fortunes of this law of inertia have been strange. It called law
, . , . . of inertia.
appears never to have played a prominent part in Gali
leo's thought. But Galileo's successors, particularly
Huygens and Newton, formulated it as an independent
law. Nay, some have even made of inertia a general
property of matter. We shall readily perceive, how
ever, that the law of inertia is not at all an indepen
dent law, but is contained implicitly in Galileo's per
ception that all circumstances determinative of motion,
or forces, produce accelerations.
In fact, if a force determine, not position, not veloTheiawa
. r i simple in
City, but acceleration, change of velocity, it stands to ference
i . . . . , ... from Gali
reason that where there is no force there will be no leo's fund*.
change of velocity. It is not necessary to enunciate servation.
this in independent form. The embarrassment of the
neophyte, which also overcame the great investigators
in the face of the great mass of new material presented,
alone could have led them to conceive the same fact as
two different facts and to formulate it twice.
In any event, to represent inertia as selfevident, or Erroneous
to derive it from the general proposition that "the ef deducing it.
feet of a cause persists, " is totally wrong. Only a
mistaken straining after rigid logic can lead us so out
of the way. Nothing is to be accomplished in the pres
ent domain with scholastic propositions like the one
just cited. We may easily convince ourselves that the
contrary proposition, "cessante causa cessat effectus,"
is as well supported by reason. If we call the acquired
velocity "the effect," then the first proposition is cor
rect ; if we call the acceleration "effect," then the sec
ond proposition holds.
12. We shall now examine Galileo's researches from
another side. He began his investigations with the
144
THE SCIENCE OF MECHANICS,
Notion of notions familiar to his time notions developed mainly
it e exjsted a ?n in the practical arts. One notion of this kind was that
time. of velocity, which is very readily obtained from the con
sideration of a uniform motion. If a body traverse in
every second of time the same distance c, the distance
traversed at the end of t seconds will be s = ct. The
distance c traversed in a second of time we call the ve
locity, and obtain it from the examination of any por
tion of the distance and the corresponding time by the
help of the equation c s/t t that is, by dividing the
number which is the measure of the distance traversed
by the number which is the measure of the time elapsed.
Now, Galileo could not complete his investigations
without tacitly modifying and extending the traditional
idea of velocity. Let us represent for distinctness sake
Fig 94
in i (Fig. 94) a uniform motion, In 2 a variable motion,
by laying off as abscissae in the direction OA the elapsed
times, and erecting as ordinates in the direction AB the
distances traversed. Now, In i, whatever increment
of the distance we may divide by the corresponding In
crement of the time, in all cases we obtain for the ve
locity c the same value. But If we were thus to proceed
In 2, we should obtain widely differing values, and
therefore the word "velocity " as ordinarily understood,
ceases In this case to be unequivocal. If, however, we
consider the increase of the distance In a sufficiently
THE PRINCIPLES OF D YXAmCS. 145
small element of time, where the element of the curve Galileo's
, , , . , modifaca
in 2 approaches to a straight line, we may regard thetionoftMs
Increase as uniform. The velocity in this element of n lcm "
the motion we may then define as the quotient, A s/A /,
of the element of the time into the corresponding ele
ment of the distance. Still more precisely, the velocity
at any instant is defined as the limiting value which
the ratio A s/A t assumes as the elements become in
finitely small a value designated by ds'dt. This new
notion includes the old one as a particular case, and Is,
moreover, immediately applicable to uniform motion.
Although the express formulation of this idea, as thus
extended, did not take place till long after Galileo, we
see none the less that he made use of It in his reason
ings.
13. An entirely new notion to which Galileo was The notion
i T i T r 7 T ' T i i of accelera
ted is the idea of acceleration. In uniformly acceler tion.
ated motion the velocities increase with the time
agreeably to the same law as in uniform motion the
spaces increase with the times. If w T e call r the velo
city acquired in time /, then v = gt. Here g denotes
the increment of the velocity in unit of time or the ac
celeration, which we also obtain from the equation
g = v/t. When the Investigation of variably accel
erated motions was begun, this notion of accelera
tion had to experience an extension similar to that of
the notion of velocity. If in i and 2 the times be again
drawn as abscissae, but now the velocities as ordinates,
we may go through anew the whole train of the pre
ceding reasoning and define the acceleration as dv/dt,
where dv denotes an infinitely small increment of the
velocity and dt the corresponding increment of the
time. In the notation of the differential calculus we
146 THE SCIENCE OF MECHANICS.
have for the acceleration of a rectilinear motion, cp =
Graphic The ideas here developed are susceptible, moreover,
tion of of graphic representation. If we lay off the times as
these ideas. D r , ,. .. , ,,
abscissae and the distances as ordmates, we shall per
ceive, that the velocity at each instant is measured by
the slope of the curve of the distance. If in a similar
manner we put times and velocities together, we shall
see that the acceleration of the instant is measured by
the slope of the curve of the velocity. The course of
the latter slope is, indeed, also capable of being traced
in the curve of distances, as will be perceived from
the following considerations. Let us imagine, in the
E
D
F
B
Fig. 95.
c d e
Fig. 96.
The curve usual manner (Fig. 95), a uniform motion represented
by a straight line OCD. Let us compare with this a
motion OCE the velocity of which in the second half
of the time is greater, and another motion OCF of
which the velocity is in the same proportion smaller.
In the first case, accordingly, we shall have to erect for
the time OB = 2 OA, an ordinate greater than BD =
2 AC, in the second case, an ordinate less than BD.
We see thus, without difficulty, that a curve of dis
tance convex to the axis of the timeabscissae corre
sponds to accelerated motion, and a curve concave
thereto to retarded motion. If we imagine a leadpen
cil to perform a vertical motion of any kind and in
THE PRINCIPLES OP DYNAMICS. 147
front of it during its motion a piece of paper to be uni
formly drawn along from right to left and the pencil to
thus execute the drawing in Fig. 96, we shall be able to
read off from the drawing the peculiarities of the mo
tion. At a the velocity of the pencil was directed up
wards, at b it was greater, at c it was = 0, at d it was
directed downwards, at e it was again = 0. At a, fi,
d, e, the acceleration was directed upwards, at c down
wards ; at c and e it was greatest.
14. The summary representation of what Galileo Tabular
discovered is best made by a table of times, acquired mem of Ga
J ^ lileo's dis
t . V . S . covery.
1 g if
2 2^ 4f
3 3^ 9f
velocities, and traversed distances. But the numbers The table
follow so simple a law, one immediately recognisable, placed by
. . rules for its
that there is nothing to prevent our replacing the cpnstmc
table by a rule for its construction. If we examine the
relation that connects the first and second columns, we
shall find that it is expressed by the equation v = gf y
which, in its last analysis, is nothing but an abbrevi
ated direction for constructing the first two columns
of the table. The relation connecting the first and third
columns is given by the equation s =g / 2 /2. The con
nection of the second and third columns is represented
by s = 5
148 THE SCIENCE OF MECHANICS.
rhe rules. Of the three relations
S ~  p:
strictly, the first two only were employed by Galileo.
Huygens was the first who evinced a higher apprecia
tion of the third, and laid, in thus doing, the founda
tions of important advances.
A remark 15. We may add a remark in connection with
on the rela .,,.,. .,,_.,,
tionofthe this table that is very valuable. It has been stated
previously that a body, by virtue of the velocity it has
acquired in its fall, is able to rise again to its origi
nal height, in doing which its velocity diminishes in
the same way (with respect to time and space) as it
increased in falling. Now a freely falling body ac
quires in double time of descent double velocity, but
falls in this double time through four times the simple
distance. A body, therefore, to which we impart a ver
tically upward double velocity will ascend twice as
long a time, butfaitr times as high as a body to which
the simple velocity has been imparted.
The dispute It was remarked, very soon after Galileo, that there
of the Car . . , . ' / . , , ' , .
tesians and is inherent in the velocity of a body a something that
Leibnitz , . J  . . . . b , . ,
iansonthe corresponds to a force a something, that is, by which
measure of , ..,,,.,
force. a force can be overcome, a certain "efficacy," as it has
been aptly termed. The only point that was debated
was, whether this efficacy was to be reckoned propor
tional to the velocity or to the square of the velocity.
The Cartesians held the former, the Leibnitzians the
latter. But it will be perceived that the question in
volves no dispute whatever. The body with the double
velocity overcomes a given force through double the
THE PRINCIPLES OP DYNAMICS. 149
time, but through four times the distance. With re
spect to time, therefore, its efficacy Is proportional to
the velocity ; with respect to distance, to the square of
the velocity. D'Alembert drew attention to this mis
understanding, although in not very distinct terms. It
Is to be especially remarked, however, that Huygens's
thoughts on this question were perfect!} 7 clear.
1 6. The experimental procedure by which, at the The present
present day, the laws of falling bodies are verified, lstaime_ansof
somewhat different from that of Galileo. Two methods the jaws of
may be employed. Either the motion of falling, which ies.
from its rapidity is difficult to observe directly, is so
retarded, without altering the law, as to be easily ob
served ; or the motion of falling is not altered at all,
but our means of observation are improved In deli
cacy. On the first principle Galileo's inclined
gutter and Atwood's machine rest. Atwood's
machine consists (Fig. 97) of an easily run
ning pulley, over which is thrown a thread,
to whose extremities two equal weights P are
attached. If upon one of the weights P we
lay a third small weight p, a uniformly accel
erated motion will be set up by the added Fi g 97
weight, having the acceleration (//2 jPj/) g a result
that will be readily obtained when we shall have dis
cussed the notion of "mass." Now by means of a
graduated vertical standard connected with the pulley
it may easily be shown that in the times i, 2, 3, 4 ....
the distances i, 4, 9, 16 , . . . are traversed. The final
velocity corresponding to any given time of descent Is
investigated by catching the small additional weight,/,
which is shaped so as to project beyond the outline of
P, in a ring through which the falling body passes,
after which the motion continues without acceleration.
P+p
150 THE SCIENCE OF MECHANICS.
The appa The apparatus of Morin is based on a different prin
Morin, La ciple. A body to which a writing pencil is attached
borde, Lip , .. . . , r 111
pich, and describes on a vertical sheet of paper, which is drawn
Von Babo, . 11 i i
uniformly across it by a clockwork, a horizontal straight
line. If the body fall while the paper is not in motion,
it will describe a vertical straight line. If the two
motions are combined, a parabola will be produced,
of which the horizontal abscissae correspond to the
elapsed times and the vertical ordinates to the dis
tances of descent described. For the abscissae i, 2,
3, 4 .... we obtain the ordinates i, 4, g, 16 . . . . By
an unessential modification, Morin employed instead of
a plane sheet of paper, a rapidly rotating cylindrical
drum with vertical axis, by the side of which the body
fell down a guiding wire. A different apparatus, based
on the same principle, was invented, independently, by
Laborde, Lippich, and Von Babo. A lampblacked
sheet of glass (Fig. 98^) falls freely, while a horizon
tally vibrating vertical rod, which in its first transit
through the position of equilibrium starts the motion
of descent, traces, by means of a quill, a curve on the
lampblacked surface. Owing to the constancy of the
period of vibration of the rod combined with the in
creasing velocity of the descent, the undulations traced
by the rod become longer and longer. Thus (Fig. 98)
be = $ab, cd$ab, de = *jab, and so forth. The
law of falling bodies is clearly exhibited by this, since
ab \ cb 4#, ab\bc\cd ^ab> and so forth.
The law of the velocity is confirmed by the inclinations
of the tangents at the points a, b, c 9 d, and so forth. If
the time of oscillation of the rod be known, the value
of g is determinable from an experiment of this kind
with considerable exactness.
Wheatstone employed for the measurement of mi
THE PRINCIPLES OF DYNAMICS.
151
nute portions of time a rapidly operating clockwork The de
1111 i i vices of
called a cnronoscope, which is set m motion at the be \vheat
r , , , , , , stone and
ginning of the time to be measured and stopped at the Hipp,
termination of it. Hipp has advantageously modified
Fig. g8a.
this method by simply causing a light indexhand to
be thrown by means of a clutch in and out of gear with
a rapidly moving wheelwork regulated by a vibrating
reed of steel tuned to a high note ; and acting as an es
152
THE SCIENCE OF MECHANICS.
Galileo's
minor in
vestiga
tions.
capement. The throwing in and out of gear is effected
by an electric current. Now if, as soon as the body be
gins to fall, the current be interrupted, that is the hand
thrown into gear, and as soon as the body strikes the
platform below the current is closed, that is the hand
thrown out of gear, we can read by the distance the
indexhand has travelled the time of descent.
17. Among the further achievements of Galileo we
have yet to mention his ideas concerning the motion
of the pendulum, and his refutation of the view that
bodies of greater weight fall faster than bodies of less
weight. We shall revert to both of these points on an
other occasion. It may be stated here, however, that
Galileo, on discovering the constancy of the period of
pendulumoscillations, at once applied the pendulum
to pulsemeasurements at the sickbed, as well as pro
posed its use in astronomical observations and to a cer
tain extent employed it therein himself.
The motion 1 8. Of still greater importance are his investiga
of projec . . .....,.,.,
tiles. tions concerning the motion of projectiles. A free body,
according to Galileo's view, constantly experiences a
vertical acceleration g towards the earth. If at the
beginning of its motion it is affected with a vertical
velocity c 9 its velocity at the
end of the time / will be v =
c + gt. An initial velocity up
wards would have to be reck
oned negative here. The dis
tance described at the end of
time t is represented by the
2 , where ct and \gt* are the
X
Fig 99.
equation s = a ( c t [
portions of the traversed distance that correspond re
spectively to the uniform and the uniformly accelerated
motion. The constant a is to be put = when we reckon
THE PRINCIPLES OF DYXAM/CS. 153
the distance from the point that the body passes at time
/ = 0. When Galileo had once reached his fundamental
conception of dynamics, he easily recognised the case
of horizontal projection as a combination of two inde
pendent motions, a horizontal uniform motion, and a
vertical uniformly accelerated motion. He thus intro
duced into use the principle of the parallelogram of mo
tions. Even oblique projection no longer presented the
slightest difficulty.
If a body receives a horizontal velocity c, it de The curve
scribes in the horizontal direction in time t the distance SonTpar
y = ct, while simultaneously it falls in a vertical direc
tion the distance oc =gt 2 /2. Different motiondeter
minative circumstances exercise no mutual effect on one
another, and the motions determined by them take
place independently of each other. Galileo was led to
this assumption by the attentive observation of the
phenomena ; and the assumption proved itself true.
For the curve which a body describes when the two
motions in question are compounded, we find, by em
ploying the two equations above given, the expression
y = i/ (2 c 2 /g) oc. It is the parabola of Apollonius hav
ing its parameter equal to c 2 /Bandits axis vertical,
as Galileo knew.
We readily perceive with Galileo, that oblique pro oblique
rr^i * projection
jection involves nothing new. The velocity c imparted
to a body at the angle a with the horizon is resolvable
into the horizontal component c . cos a and the vertical
component c . sin a. With the latter velocity the body
ascends during the same interval of time / which it
would take to acquire this velocity in falling vertically
downwards. Therefore, c.sma=gt. When it has
reached its greatest height the vertical component of
its initial velocity has vanished, and from the point S
i 5 4 THE SCIENCE OF MECHANICS.
onward (Fig. 100) it continues its motion as a. horizon
tal projection. If we examine any two epochs equally
distant in time, before and after the transit through S,
c we shall see that the body at
O J
these two epochs is equally
distant from the perpendicu
lar through S and situated the
same distance below the hori
zontal line through . The
Fi s 10  curve is therefore symmet
rical with respect to the vertical line through S. It
is a parabola with vertical axis and the parameter
'L
The range To find the socalled range of projection, we have
ojDrojec s j m p} v to consider the horizontal motion during the
time of the rising and falling of the body. For the ascent
this time is, according to the equations above given,
t = c sin cx/g 9 and the same for the descent. With the
horizontal velocity c . cosar, therefore, the distance is
traversed
_ ^ sin or C* . c 2 .
w = c cos a . 2 2 sm a cos a = sin 2 a.
The range of projection is greatest accordingly
when a = 45, and equally great for any two angles
The mutual jo. The recognition of the mutual independence of
indepen .....
dence of the forces, or motiondeterminative circumstances oc
forces. . .
curring in nature, which was
v reached and found expression
in the investigations relating to
projection, is important. A body
Flg ' 10I< may move (Fig. 101) in the di
rection AB, while the space in which this motion oc
curs is displaced in the direction A C. The body then
THE PRINCIPLES OF DYNAMICS. 155
goes from A to D. Now, this also happens If the two
circumstances that simultaneously determine the mo
tions AB and AC, have no influence on one another.
It is easy to see that we may compound by the paral
lelogram not only displacements that have taken place
but also velocities and accelerations that simultane
ously take place. (See Appendix, X., p. 525.)
ii.
THE ACHIEVEMENTS OF HUYGENS.
i. The next in succession of the great mechanical in Huygens's
quirers is HUYGENS, who in every respect must beas S an r S
ranked as Galileo's peer. If, perhaps, his philosophical qmrer *
endowments were less splendid than those of Galileo,
this deficiency was compensated for by the superiority
of his geometrical powers. Huygens not only continued
the researches which Galileo had begun, but he also
solved the first problems in the dynamics of several
masses, whereas Galileo had throughout restricted him
self to the dynamics of a single body.
The plenitude of Huygens's achievements is bestEnumera
seen in his HorologiumOscillatorium, which appeared in genss
1673. The most important subjects there treated of ferments,
the first time, are : the theory of the centre of oscilla
tion, the invention and construction of the pendulum
clock, the invention of the escapement, the determina
tion of the acceleration of gravity, g, by pendulum
observations, a proposition regarding the employment
of the length of the seconds pendulum as the unit of
length, the theorems respecting centrifugal force, the
mechanical and geometrical properties of cycloids, the
doctrine of evolutes, and the theory of the circle of
curvature.
156
SCIENCE OF MECHANICS.
2. With respect to the form of presentation of his
work,, it is to be remarked that Huygens shares with
CHRIS TIANUS HUGENIUS
14. Aprilis
deiiatus 8 Junii 16*05 .
Galileo, in all its perfection, the latter's exalted and
inimitable candor. He is frank without reserve in the
presentment of the methods that led him to his dis
THE PRINCIPLES OF DYNAMICS.
coveries, and thus always
conducts his reader into the
full comprehension of his
performances. Nor had he
cause to conceal these
methods. If, a thousand
years from now, it shall be
found that he was a man, it
will likewise be seen what
manner of man he was.
In our discussion of the
achievements of Huygens,
however, we shall have to
proceed in a somewhat dif
ferent manner from that
which we pursued in the
case of Galileo. Galileo's
views, in their classical sim
plicity, could be given in an
almost unmodified form.
With Huygens this is not
possible. The latter deals
with more complicated
problems; his mathematical
methods and notations be
come inadequate and cum
brous. For reasons of brev
ity, therefore, we shall re
produce all the conceptions
of which we treat, in mod
ern form, retaining, how
ever, Huygens's essential
and characteristic ideas.
Huygens's Pendulum Clock.
158 THE SCIENCE OF MECHANICS.
Centrifugal 3. We begin with the investigations concerning
petal force, centrifugal force. When once we have recognised with
Galileo that force determines acceleration, we are im
pelled, unavoidably, to ascribe every change of velocity
and consequently also every change in the direction of
a motion (since the direction is determined by three
velocitycomponents perpendicular to one another) to
a force. If, therefore, any body attached to a string,
say a stone, is swung uniformly round in a circle, the
curvilinear motion which it performs is intelligible only
on the supposition of a constant force that deflects the
body from the rectilinear path. The tension of the
string is this force ; by it the body is constantly deflected
from the rectilinear path and made to move towards
the centre of the circle. This tension, accordingly, rep
resents a centripetal force. On the other hand, the axis
also, or the fixed centre, is acted on by the tension of
the string, and in this aspect the tension of the string
appears as a centrifugal force.
G f !^^v
G
Fig. 102. Fig. 103.
Let us suppose that we have a body to which a ve
locity has been imparted and which is maintained in
uniform motion in a circle by an acceleration constantly
directed towards the centre. The conditions on which
this acceleration depends, it is our purpose to investi
gate. We imagine (Fig. 102) two equal circles uni
THE PRINCIPLES OF D YNAMICS. 159
formly travelled round by two bodies ; the velocities in Uniform
the circles I and II bear to each other the proportion ecjuai
i : 2. If in the two circles we consider any same arc
element corresponding to some very small angle a, then
the corresponding element s of the distance that the
bodies in consequence of the centripetal acceleration
have departed from the rectilinear path (the tangent),
will also be the same. If we call (p l and (p 2 the re
spective accelerations, and r and r/2 the timeelements
for the angle a, we find by Galileo's law
2s 2s
cp^ , <p 2 = 4 , that is to say cp^ = 4^> 1 .
Therefore, by generalisation, in equal circles 'the
centripetal acceleration is proportional to the square of
the velocity of the motion.
Let us now consider the motion in the circles I and Uniform
II (Fig. 103), the radii of which are to each other as unequal
, . circles.
i : 2, and let us take for the ratio of the velocities of
the motions also 1:2, so that like arcelements are
travelled through in equal times. <p 1? <? 2 , s, is denote
the accelerations and the elements of the distance trav
ersed ; r is the element of the time, equal for both
cases. Then
2s 4s .
VL = ^><P* = ^ that is to sa y <p 3 = 2 ^i
If now we reduce the velocity of the motion in II
onehalf, so that the velocities in I and II become
equal, <p 2 will thereby be reduced onefourth, that is
to say to <^!/2. Generalising, we get this rule: when
the velocity of the circular motion is the same, the cen
tripetal acceleration is inversely proportional to the
radius of the circle described.
4. The early investigators, owing to their following
i6o
THE SCIENCE OF MECHANICS.
Deduction the conceptions of the ancients, generally obtained their
of the gen .... . f r
eraiiawof propositions in the cumbersome form of proportions.
motion. We shall pursue a different method. On a movable
object having the velocity v let a force act during the
element of time r which imparts to the object perpen
dicularly to the direction of its motion the acceleration
cp. The new velocitycomponent thus becomes cpr,
and its composition with the first velocity produces a
new direction of the motion, making the angle a with
the original direction. From this results, by conceiving
the motion to take place in a circle of radius r, and on
account of the smallness of the angular element putting
The para
doxical
character
of this
problem.
Fig. 104. Fig. 105.
tan of = a, the following, as the complete expression
for the centripetal acceleration of a uniform motion in
a circle,
cpr vr r> 2
 = tan a = a = or <p .
v r ^ r
The idea of uniform motion in a circle conditioned
by a constant centripetal acceleration is a little para
doxical. The paradox lies in the assumption of a con
stant acceleration towards the centre without actual
approach thereto and without increase of velocity. This
is lessened when we reflect that without this centripetal
acceleration the body would be continually moving
away from the centre ; that the direction of the accel
THE PRINCIPLES OF DYNAMICS. 161
eratlon Is constantly changing ; and that a change of
velocity (as will appear in the discussion of the prin
ciple of vis vrrd) is connected with an approach of the
bodies that accelerate each other, which does not take
place here. The more complex case of elliptical cen
tral motion is elucidative in this direction. (See p. 5 27.)
5. The expression for the centripetal or centrifugal A different
. ... '" expression
acceleration, cp = ?> 2 //, can easily be put in a somewhat of the law.
different form. If T denote the periodic time of the
circular motion, the time occupied in describing the
circumference, then vT= 2 r TT, and consequently cp =
^r7T 2 /T 2 , in which form we shall employ the expres
sion later on. If several bodies moving in circles have
the same periodic times, the respective centripetal ac
celerations by which they are held in their paths, as is
apparent from the last expression, are proportional to
the radii.
6. We shall take it for granted that the reader is some phe
familiar with the phenomena that illustrate the con which the
siderations here presented : as the rupture of strings of plains.
* insufficient strength on which bodies are whirled about,
the flattening of soft rotating spheres, and so on. Huy
gens was able, by the aid of his conception, to explain
at once whole series of phenomena. When a pendulum
clock, for example, which had been taken from Paris
to Cayenne by Richer (16711673), showed a retarda
tion of its motion, Huygens deduced the apparent
diminution of the acceleration of gravity g thus estab
lished, from the greater centrifugal acceleration of the
rotating earth at the equator ; an explanation that at
once rendered the observation intelligible.
An experiment instituted by Huygens may here be
noticed, on account of Its historical interest. When
Newton brought out his theory of universal gravitation,
162
THE SCIENCE OF MECHANICS.
ing experi'
ment of
Huygens.
An interest Huygens belonged to the great number of those who
were unable to reconcile themselves to the idea of action
at a distance. He was of the opinion that gravitation
could be explained by a vortical medium. If we enclose
in a vessel filled with a liquid a number of lighter bod
ies, say wooden balls in water, and set the vessel ro
tating about its axis, the balls will at once rapidly move
towards the axis. If for instance (Fig. 106), we place
the glass cylinders RR containing the wooden balls KK
by means of a pivot Z on a rotatory apparatus, and ro
tate the latter about its ver
tical axis, the balls will im
mediately run up the cyl
inders in the direction away
from the axis. But if the
tubes be filled with water,
each rotation will force the
balls floating at the extremities EE towards the axis.
The phenomenon is easily explicable by analogy with
the principle of Archimedes. The wooden balls receive
a centripetal impulsion, comparable to buoyancy,
which is equal and opposite to the centrifugal force
acting on the displaced liquid. (See p. 528.)
7. Before we proceed to Huygens's investigations
on the centre of oscillation, we shall present to the
reader a few considerations concerning pendulous and
oscillatory motion generally, which will make up in ob
viousness for what they lack in rigor.
Many of the properties of pendulum motion were
known to GALILEO. That he had formed the concep
tion which we shall new give, or that at least he was
on the verge of so doing, may be inferred from many
scattered allusions to the subject in his Dialogues. The
bob of a simple pendulum of length / moves in a circle
Oscillatory
motion.
THE PRINCIPLES OF DYXAMICS.
163
(Fig. 107) of radius /. If we give the pendulum a very Galileo's
small excursion, it will travel in Its oscillations over ationoflhe
11 11  * aw ^ t * ie
very small arc which coincides approximately with the pendulum.
chord belonging to it. But this
chord is described by a falling
particle, moving on it as on an
inclined plane (see Sect i of this
Chapter, 7), in the same time
as the vertical diameter BD =
2 /. ' If the time of descent be
called /, we shall have 2/
\gt*> that is /= 21/7/57 But
since the continued movement
from B up the line BC' occupies an equal Interval of
time, we have to put for the time T of an oscillation
from Cto C'j T= \\/ Ijg. It will be seen that even from
so crude a conception as this the correct form of the
pendulumlaws is obtainable. The exact expression
for the time of very "small oscillations Is, as we know,
Fig. 107.
Again, the motion of a pendulum bob may be viewed pendulum
as a motion of descent on a succession of inclined viewed as a
planes. If the string of the pendulum makes the angle downfn
a with the perpendicular, the pendulum bob receives planes.
in the direction of the position of equilibrium the accel
eration g. sin a. When a Is small, g. a Is the expres
sion of this acceleration ; in other words, the accelera
tion is always proportional and oppositely directed to
the excursion. When the excursions are small the
curvature of the path may be neglected.
8. From these preliminaries, we may proceed to
the study of oscillatory motion in a simpler manner. A
body is free to move on a straight line OA (Fig. 108),
and constantly receives in the direction towards the
164
THE SCIENCE OF MECHANICS.
A simpler point O an acceleration proportional to its distance from
and modern ,.,, , i T
view of os O. We will represent these accelerations by ordinates
motion. erected at the positions considered. Ordinates upwards
denote accelerations towards the left ; ordinates down
wards represent accel
erations towards the
D'C
A',
II
Fig. 108.
right. The body, left
to itself at A, will
move towards with
varied acceleration,
pass through OtoA^
where OA^ OA,
come back to } and
so again continue its
motion. It is in the
The period first place easily demonstrable that the period of os
tioninde dilation (the time of the motion through AOA^) is in
the amph dependent of the amplitude of the oscillation (the dis
tance OA). To show this, let us imagine in I and
II the same oscillation performed, with single and
double amplitudes of oscillation. As the acceleration
varies from point to point, we must divide OA and
O'A' = 2 OA into a very large equal number of ele
ments. Each element A' B' of O'A 1 is then twice as
large as the corresponding element AB of OA. The
initial accelerations cp and cp stand in the relation
qj = 2 (p. Accordingly, the elements AB and A' B' =
2 AB are described with their respective accelerations
cp and 2cp im the same time r. The final velocities v
and v' in I and II, for the first element, will be v = cpr
and v' = 2 cpr, that is v' = 2 v. The accelerations and
the initial velocities at B and B' are therefore again as
1:2. Accordingly, the corresponding elements that
next succeed will be described in the same time. And
THE PRINCIPLES OF DYNAMICS.
165
of every succeeding pair of elements the same asser
tion also holds true. Therefore, generalising, it will
be readily perceived that the, period of oscillation is
independent of its amplitude or breadth.
Next, let us conceive two oscillatory motions, I and The time of
** . oscillation
II, that have equal excursions (Fig. TOO); but in II let inversti>
'' , ,. proportiun
a fourfold acceleration correspond to the same distance alto tnc
from O. We divide the amplitudes of
I
BA
/
II
Fig. 109.
the oscillations AO and O'A' = OA
into a very large equal number of
parts. These parts are then equal in
I and II. The initial accelerations at
A and A' are cp and 4 <p ; the ele
ments of the distance described are
AB = A'B' = sy and the times are
respectively rand r'. We obtain, then,
r = }/2s/(p, r' = 1/2 j/4 <p = r/2.
The element A'B' is accordingly trav
elled through in onehalf the time
the element AB is. The final velocities r and v' at
B and B 1 are found by the equations v = cpr and
v r = 4 (>(r/2) = 2 v. Since, therefore, the initial velo
cities at B and B' are to one another as i : 2, and the
accelerations are again as 1:4, the element of II suc
ceeding the first will again be traversed in half the
time of the corresponding one in I. Generalising, we
get : For equal excursions the time of oscillation is in
versely proportional to the square root of the accelera
tions.
9. The considerations last presented may be put in
a very much abbreviated and very obvious form by a
method of conception first employed by Newton. New
ton calls those material systems similar that have geo
metrically similar configurations and whose homolo
of the ac
celeration.
166 THE SCIENCE OF MECHANICS.
The princi gous masses bear to one another the same ratio. He
pleofsimil .......
itude. says further that systems of this kind execute similar
movements when the homologous points describe simi
lar paths in proportional times. Conformably to the
geometrical terminology of the present day we should
not be permitted to call mechanical structures of this
kind (of five dimensions) similar unless their homolo
gous linear dimensions as well as the times and the
masses bore to one another the same ratio. The struc
tures might more appropriately be termed affined to
one another.
We shall retain, however, the name phoronomically
similar structures, and in the consideration that is to
follow leave entirely out of account the masses.
In two such similar motions, then, let
the homologous paths be s and as,
the homologous times be t and /?/; whence
the homologous velo
tx
v =  ana yv =
the homologous accel
cities are v =  and yv = & ,
2s , a 2s
erations ( ? = ~fi and 8( P = 7j2~j2'
Thededuc Now all oscillations which a body performs under
laws of os the conditions above set forth with any two different
this a meSiod amplitudes i and or, will be readily recognised as sim
ilar motions. Noting that the ratio of the homologous
accelerations in this case is e= a, we have a = a /ft 2 .
Wherefore the ratio of the homologous times, that is
to say of the times of oscillation, is ft = d= i. We ob
tain thus the law, that the period of oscillation is inde
pendent of the amplitude.
If in two oscillatory motions we put for the ratio
between the amplitudes i : a, and for the ratio between
the accelerations i : a fa we shall obtain for this case
THE PRINCIPLES OF DYNAMICS. 167
e= aju = a/fi 2 , and therefore ft = i ''zb \/ / ; where
with the second law of oscillating motion is obtained.
Two uniform circular motions are always phoronom
ically similar. Let the ratio of their radii be i : a and
the ratio of their velocities i : y. The ratio of their
accelerations Is then = or/^ 2 , and since y=a/fi,
also=^ 2 /flr; whence the theorems relative to cen
tripetal acceleration are obtained.
It is a pity that Investigations of this kind respect
ing mechanical and phoronomical affinity are not more
extensively cultivated, since they promise the most
beautiful and most elucidative extensions of Insight
imaginable.
10. Between uniform motion In a circle and oscil The con
.... . . nection be
latory motion of the kind just discussed an important tweenoscii
relation exists which we shall now consider. We as tion of this
kind and
sume a system of rectangular co x y uniform_
I motion in a
ordmates, having its origin at the X x ""~s"X circle.
centre, <9, of the circle of Fig. no,
about the circumference of which .
we conceive a body to move uni
formly. The centripetal accelera
tion cp which conditions this mo
tion, we resolve In the directions
of X and F; and observe that the ^components of the
motion are affected only by the X components of the
acceleration. We may regard both the motions and
both the accelerations as Independent of each other.
Now, the two components of the motion are osTheiden
cillatory motions to and fro about O. To the excur two.
sion x the accelerationcomponent cp (x/r) or (cpfr) x
In the direction O, corresponds. The acceleration Is
proportional, therefore, to the excursion. And accord
ingly the motion is of the kind just investigated. The
i68 THE SCIENCE OF MECHANICS.
time T of a complete to and fro movement Is also the
periodic time of the circular motion. With respect to
the latter, however, we know that <p= 4.r?r 2 /T 2 , or,
what is the same, that T= 'ZrtVr/cp. Now cp/r is
the acceleration for x=. i, the acceleration that corre
sponds to unit of excursion, which we shall briefly
designate by f. For the oscillatory motion we may
put, therefore, T= 2?r ]/i//". For a single movement
to, or a single movement fro, the common method of
reckoning the time of oscillation, we get, then, T=
TheappH 1 1 Now this result is directly applicable to pen
the^asfre dulum vibrations of very small excursions, where, ne
duiumvi n ~glectmg the curvature of the path, it is possible to ad
brations. k ere { o t j ie conception developed. For the angle of
elongation a we obtain as the distance of the pendulum
bob from the position of equilibrium, la; and as the
corresponding acceleration, ga\ whence
J la
This formula tells us, that the time of vibration is
directly proportional to the square root of the length
of the pendulum, and inversely proportional to the
square root of the acceleration of gravity. A pendulum
that is four times as long as the seconds pendulum,
therefore, will perform its oscillation in two seconds.
A seconds pendulum removed a distance equal to the
earth's radius from the surface of the earth, and sub
jected therefore to the acceleration g/^, will likewise
perform its oscillation in two seconds.
12. The dependence of the time of oscillation on
the length of the pendulum is very easily verifiable by
experiment. If (Fig. in) the pendulums a, b, c,
THE PRINCIPLES OF DYXAMICS. 169
which to maintain the plane of oscillation invariable Experimen
are suspended by double threads, have the lengths i , ti a on of the
, ... .' ' laws of the
4, 9, then a will execute two oscillations to one oscil pendulum,
lation of fr, and three to one of c.
Fig. in.
The verification of the dependence of the time of
oscillation on the acceleration of gravity g is some
what more difficult ; since the latter cannot be arbi
trarily altered. But the demonstration can be effected
by allowing one component only of g to act on the
pendulum. If we imagine the axis of oscillation of
170
THE SCIENCE OF MECHANICS.
Experimen the pendulum A A fixed in the vertically placed plane
tion of the of the paper. will be the intersection of the plane
laws of the r   i , , r i
pendulum. ^^E * oscillation with the plane ot the paper
and likewise the position of equilibrium
of the pendulum. The axis makes with
the horizontal plane, and the plane of os
cillation makes with the vertical plane, the
angle /?; wherefore the acceleration . cos/5
Fig. 112. ^ the acceleration which acts in this plane.
If the pendulum receive in the plane of its oscillation
the small elongation a, the corresponding acceleration
Fig. 113.
will be (g cos /?) a whence the time of oscillation is
T= n Vllz cos"A"
THE PRINCIPLES OF DYNAMICS.
171
. pendulum.
We see from this result, that as ft Is increased the
acceleration g cos ft diminishes, and consequently the
time of oscillation increases. The experiment may be
easily made with the apparatus represented in Fig. 113.
The frame JRR is free to turn about a hinge at C; it can
be inclined and placed on its side. The angle of in
clination is fixed by a graduated arc G held by a set
screw. Every increase of ft increases the time of oscil
lation. If the plane of oscillation be made horizontal,
in which position R rests on the foot F, the time of
oscillation becomes infinitely great. The pendulum
in this case no longer returns to any definite position
but describes several complete revolutions in the same
direction until its entire velocity has been destroyed
by friction.
13. If the movement of the pendulum do not take Theconicai
place in a plane, but be performed In space, the thread
of the pendulum will describe the surface
of a cone. The motion of the conical pen
dulum was also investigated by Huygens.
We shall examine a simple case of this
motion. We imagine (Fig. 114) a pen
dulum of length / removed from the ver
tical by the angle a, a velocity v imparted
to the bob of the pendulum at right
angles to the plane of elongation, and the pendulum re
leased. The bob of the pendulum will move in a hori
zontal circle if the centrifugal acceleration q> developed
exactly equilibrates the acceleration of gravity g; that
is, if the resultant acceleration falls in the direction of
the pendulum thread. But in that case <pjg=. tan or.
If T stands for the time taken to describe one revolu
tion, the periodic time, then (p = ^r7T 2 /T 2 or T =
2 TC ~\/r/<p. Introducing, now, in the place of rj<p the
"4
i 7 2 THE SCIENCE OF MECHANICS.
value / sin a/g tan a = t cos a/g , we get for the periodic
time of the pendulum, T= 2 TT ]/ / cos a/g. For the ve
locity v of the revolution we find v = I/ rep, and since
cp = gta.na it follows that v= \/gl sin a tana. For
very small elongations of the conical pendulum we may
put T==27t ]///, which coincides with the regular
formula for the pendulum, when we reflect that a single
revolution of the conical pendulum corresponds to two
vibrations of the common pendulum.
The deter 14. Huygens was the first to undertake the exact
the accei determination of the acceleration of gravity g by means
gravity by of pendulum observations. From the formula T=
iumf en u it I/ T/g for a simple pendulum with small bob we ob
tain directly g= 7T 2 l/T%. For latitude 45 we obtain
as the value of g, in metres and seconds, 9 . 806. For
provisional mental calculations it is sufficient to re
member that the acceleration of gravity amounts in
round numbers to 10 metres a second,
A remark 15. Every thinking beginner puts to himself the
uia express question how it is that the duration of an oscillation,
mg e aw. ^^ .^ ^ jj me> can foe found by dividing a number that
is the measure of a length by a number that is the
measure of an acceleration and extracting the square
root of the quotient. But the fact is here to be borne in
mind that g= 2s/t 2 , that is a length divided by the
square of a time. In reality therefore the formula we
have is T= 7t ]/(//2/) / 2 . And since l/2s is the ratio
of two lengths, and therefore a number, what we have
under the radical sign is consequently the square of a
time. It stands to reason that we shall find Tin sec
onds only when, in determining g, we also take the sec
ond as unit of time.
In the formula g 7t 2 l/T* we see directly that g is
THE PRINCIPLES OP DYNAMICS. 173
a length divided by the square of a time, according to
the nature of an acceleration.
1 6. The most important achievement of Huveens Theprob
1 1 r i i 1 i ! lem f the
is his solution of the problem to determine the centre centre of
. oscillation.
of oscillation. So long as we have to deal with the dy
namics of a single body, the Galilean principles amply
suffice. But in the problem just mentioned we have to
determine the motion of several bodies that mutually
influence each other. This cannot be done without
resorting to a new principle. Such a ore Huygens
actually discovered.
We know that loner pendulums perform their oscil statement
. ill! T  oftheprob
lations more slowly than snort ones. Let us imagine a lem.
heavy body, free to rotate about an axis, the centre of
gravity of which lies outside of the axis; such JTTT"'
a body will represent a compound pendulum.
Every material particle of a pendulum of this
kind would, if it were situated alone at the
same distance from the axis, have its own pe
riod of oscillation. But owing to the connec Fig.
tions of the parts the whole body can vibrate with only
a single, determinate period of oscillation. If we pic
ture to ourselves several pendulums of unequal lengths,
the shorter ones will swing quicker, the longer ones
slower. If all be joined together so as to form a single
pendulum, it is to be presumed that the longer ones
will be accelerated, the shorter ones retarded, and that
a sort of mean time of oscillation will result There
must exist therefore a simple pendulum, intermediate
in length between the shortest and the longest, that
has the same time of oscillation as the compound pen
dulum. If we lay off the length of this pendulum on
the compound pendulum, we shall find a point that pre
serves the same period of oscillation in its connection
o
o
174 THE SCIEXCE OF MECHANICS.
with the other points as it would have if detached and
left to itself. This point is the centre of oscillation.
MERSEXXE was the first to propound the problem of
determining the centre of oscillation. The solution of
DESCARTES, who attempted it, was, however, precipi
tate and insufficient.
Huygens's 17. Huvgens was the first who gave a general solu
solution. . r , , , , 
tion. Besides Huygens nearly all the great inquirers
of that time employed themselves on the problem, and
we may say that the most important principles of mod
ern mechanics were developed in connection with it.
The new idea from which Huygens set out, and
which is more important by far than the whole prob
lem, is this. In whatsoever manner the material par
ticles of a pendulum may by mutual interaction modify
each other's motions, in every case the velocities ac
quired in the descent of the pendulum can be such only
that by virtue of them the centre of gravity of the par
ticles, whether still in connection or with their connec
tions dissolved, is able to rise just as high as the point
The new from which it fell. Hu}7gens found himself compelled,
whfch P H e uy by the doubts of his contemporaries as to the correct
duced nr " ness of this principle, to remark, that the only assump
tion implied in the principle is, that heavy bodies of
themselves do not move upwards. If it were possible
for the centre of gravity of a connected system of falling
material particles to rise higher after the dissolution
of its connections than the point from which it had
fallen, then by repeating the process heavy bodies
could, by virtue of their own weights, be made to rise
to any height we wished. If after the dissolution of
the connections the centre of gravity should rise to a
height less than that from which it had fallen, we
should only have to reverse the motion to produce the
THE PRINCIPLES OF DYXAMICS. 175
same result. What Huygens asserted, therefore, no
one had ever really doubted ; on the contrary, every
one had instinctively perceived it. Huygens, however,
gave this instinctive perception an abstract, conceptual
form. He does not omit, moreover, to point out, on the
ground of this view, the fruitlessness of endeavors to
establish a perpetual motion. The principle just devel
oped will be recognised as a generalisation of one of Ga
lileo's ideas.
1 8. Let us now see what the principle accomplishes Huygens's
principle
in the determination of the centre of oscillation. Let applied.
OA (Fig. 116), for simplicity's sake,
be a linear pendulum, made up of a
large number of masses indicated in
the diagram by points. Set free at
OA> it will swing through B to OA',
where AB = BA' . Its centre of
gravity S will ascend just as high
on the second side as it fell on the Fig ' IlS<
first. From this, so far, nothing would follow. But
also, if we should suddenly, at the position OJB, re
lease the individual masses from their connections, the
masses could, by virtue of the velocities impressed on
them by their connections, only attain the same height
with respect to centre of gravity. If we arrest the free
outwardswinging masses at the greatest heights they
severally attain, the shorter pendulums will be found
below the line OA', the longer ones will have passed
beyond it, but the centre of gravity of the system will
be found on OA' in its former position.
Now let us note that the enforced velocities are
proportional to the distances from the axis ; therefore,
one being given, all are determined, and the height of
ascent of the centre of gravity given. Conversely,
176 THE SCIEXCE OF
therefore, the velocity of any material particle also is
determined by the known height of the centre of grav
ity. B:it if we know in a pendulum the velocity cor
responding to a given distance of descent, we know its
whole motion.
Thede IQ. Premising 1 these remarks, we proceed to the
tailed reso , .
iution of the problem itself. On a compound linear pendulum (Fig.
problem. r .
117) w r e cut off, measuring from the axis, the
portion = i. If the pendulum move from its
position of greatest excursion to the position
of equilibrium, the point at the distance = i
from the axis will fall through the height k.
The masses ?//, ;;/', m" ... at the distances
/, /, /' . . . will fall in this case the dis
tances rk, r' k, r" k . . ., and the distance of
the descent of the centre of gravity will be :
?nrk \ m'rk \ m"r"k ) 7 2mr
m ~\ m' j in" (.... 2m
Let the point at the distance i from the axis ac
quire, on passing through the position of equilibrium,
the velocity, as yet unascertained, v. The height of
its ascent, after the dissolution of its connections, will
be z' 2 /2". The corresponding heights of ascent of
the other material particles will then be (rz/) 2 /2^,
(/ r) 2 / 2 ^ (>'" ?OV 2 " * The height of ascent of the
centre of gravity of the liberated masses will be
,
m ~t~ + m ^  \ m
 
9
z/ 2 ^ m r 2
m ~\ m' f m" )... 2g~
By Huygens's fundamental principle,
THE PRINCIPLES OF DYNAMICS, 177
From this a relation is deducible between the distance of
descent k and the velocity v. Since, however, all pen
dulum motions of the same excursion are phoronomi
cally similar, the motion here under consideration is,
In this result, completely determined.
To find the length of the simple pendulum that has The length
. , 7 M1 . , , ofthesira
the same period oi oscillation as the compound pen pie isoch
ronous
dulum considered, be it noted that the same relation pendulum.
must obtain between the distance of its descent and its
velocity, as in the case of its unimpeded fall. If y is
the length of this pendulum, ky is the distance of its
descent, and vy its velocity ; wherefore
>=* .......... w
Multiplying equation (a) by equation (&) we obtain
Employing the principle of phoronomic similitude, solution of
we may also proceed in this way. From (a) we get Lm^the
principle of
2mr similitude.
2 mr 2 '
A simple pendulum of length i, under corresponding
circumstances, has the velocity
Calling the time of oscillation of the compound pendu
lum T, that of the simple pendulum of length i T 1 =
TtV'i/g, we obtain, adhering to the supposition of
equal excursions,
T v
== ; wherefore T= 7t <
7\ v
178 THE SCIENCE OP MECHANICS.
Huygens's 2o, We see without difficulty in the Huygenian
principle .... .. r , ,..
:ioit:cai principle the recognition of work as the condition de
prindpieof terminative of velocity, or, more exactly, the condition
determinative of the socalled vis viva. By the vis
rira or living force of a system of masses m, m n
m, , affected with the velocities v, v,, v n . . . ., we
understand the sum *
_v< 2 + ^^+
o ' o n^
The fundamental principle of Huygens is identical with
the principle of vis viva. The additions of later in
quirers were made not so much to the idea as to the
form of its expression.
If we picture to ourselves generally any system of
weights/,/,,/,, . . . ., which fall connected or uncon
nected through the heights h, h r , h n . . . ., and attain
thereby the velocities v, v l9 v tt . . . ., then, by the Huy
genian conception, a relation of equality exists between,
the distance of descent and the distance of ascent of the
centre of gravity of the system, and, consequently, the
equation holds
v* X 2 , f ,v" 2 ,
"+... ^ + ^27 +/ "27+
If we have reached the concept of "mass," which
Huygens did not yet possess in his investigations, we
may substitute for//^ the mass m and thus obtain the
form 2ph= ^^E ?nv 2 , which is very easily generalised
for nonconstant forces.
* This is not the usual definition of English writers, who follow the older
authorities in making the vis viva twice this quantity. Trans.
THE PRINCIPLES OF D YXAMICS, 179
21. With the aid of the principle of living forces General
. . r  , method of
we can determine the duration of the infinitely small determin
. , ing the pe
oscillations of any pendulum whatso /"~~"~\_ nod of pen
ever. We let fall from the centre of f \ dilations.
gravity s (Fig. 1 18) a perpendicular on
the axis ; the length of the perpendic
ular is, say, a. We lay off on this,
measuring from the axis, the length
= i. Let the distance of descent of
the point in question to the position of F:g  IlS 
equilibrium be k, and v the velocity acquired. Since
the work done in the descent is determined by the
motion of the centre of gravity, we have
work done in descent = vis viva :
M here we call the total mass of the pendulum and
anticipate the expression vis viva. By an inference
similar to that in the preceding case, we obtain T=
22. We see that the duration of infinitely small The two
aeierimna
oscillations of any pendulum is determined by two fac tiv ^ factors,
tors by the value of the expression 2mr 2 , which
Euler called the moment of inertia and which Huygens
had employed without any particular designation, and
by the value of agM. The latter expression, which we
shall briefly term the statical moment^ is the product
a P of the weight of the pendulum into the distance of
its centre of gravity from the axis. If these two values
be given, the length of the simple pendulum of the
same period of oscillation (the isochronous pendulum)
and the position of the centre of oscillation are deter
mined.
i8o
THE SCIENCE OF MECHANICS.
Huygens'
nfetSad^o
solution.
For the determination of the lengths of the pendu
lums referred to, Huygens, in the lack of the analytical
methods later discovered, employed a very ingenious
geometrical procedure, which
we shall illustrate by one or
two examples. Let the prob
lem be to determine the time
of oscillation of a homogene
ous, material, and heavy rec
tangle A BCD, which swings
on the axis AB (Fig. 119).
Dividing the rectangle into
minute elements of area//,
/,.... having the distances
r, r fi r n . . . . from the axis, the expression for the
length of the isochronous simple pendulum, or the dis
tance of the centre of oscillation from the axis, is given
by the equation
/r*+/,r,+f,,r, + . .
Fig. 119.
Let us erect on A BCD at C and D the perpendiculars
CJ5, = DF = A C = BD and picture to ourselves a
homogeneous wedge ABCDEF. Now find the distance
of the centre of gravity of this wedge from the plane
through AB parallel to CDEF. We have to consider,
in so doing, the tiny columns /r,/ r f ,f tf r fl . . . . and
their distances r, r n r lf . . . . from the plane referred
to. Thus proceeding, we obtain for the required dis
tance of the centre of gravity the expression
fr . r + /, r, . r, f / r t , . r tf + . . . .
that is, the same expression as before. The centre of
oscillation of the rectangle and the centre of gravity of
THE PRINCIPLES OF D YNAMICS.
181
the wedge are consequently at the same distance from
the axis, \A C.
Following out this idea, we readily perceive the Analogous
correctness of the following assertions. For a homo tions of the
i r n i , r " i .preceding
geneous rectangle of height / swinging about one of methods,
its sides, the distance of the centre of gravity from the
axis is /z/2, the distance of the centre of oscillation \h.
For a homogeneous triangle of height h, the axis of
which passes through the vertex parallel to the base,
the distance of the centre of gravity from the axis is
7z, the distance of the centre of oscillation 7z. Call
ing the moments of inertia of the rectangle and of the
triangle A^ A^ and their respective masses M^ 9 M 2 ,
we get
2J&__^!_ a i  ^2
2k
r . . h*Mi .
Consequently A x = ^^> A 2 = .
O ^j
By this pretty geometrical conception many prob
lems can be solved that are today treated more con
veniently it is true by routine forms.
Fig. 120.
Fig. 121.
23. We shall now discuss a proposition relating to
moments of inertia, that Huygens made use of in a
somewhat different form. Let O (Fig. 121) be the
centre of gravity of any given body. Make this the
i82 THE SCIENCE OF MECHANICS.
Thereia origin of a system of rectangular coordinates, and sup
tion of mo , _ . . . , r , . , ~
mentsof inpose the moment of inertia with reierence to the ZB.XIS
ferredto determined. If m is the element of mass and r its dis
axes. e tance from the Zaxis, then this moment of inertia is
4 = 2mr 2 . We now displace the axis of rotation
parallel to itself to O r , the distance a in the ^direction.
The distance r is transformed, by this displacement,
into the new distance p, and the new moment of
inertia is
9 =
2a2mx + a*2m, or, since 2
calling the total mass M= 2m, and remembering the
property of the centre of gravity 2 MX = 0,
= A + a*M.
From the moment of inertia for one axis through the
centre of gravity, therefore, that for any other axis
parallel to the first is easily derivable.
An appii 24. An additional observation presents itself here.
ca.ti.on of
thispropo The distance of the centre of oscillation is given by
sition. _  ___ 
the equation 1= A + a 2 M/aJlf, where A, M, and a
have their previous significance. The quantities A and
M are invariable for any one given body. So long
therefore as a retains the same value, / will also remain
Invariable. For all parallel axes situated at the same
distance from the centre of gravity, the same body as
pendulum has the same period of oscillation. If we
put d/M= K, then
'T + 
Now since / denotes the distance of the centre of
oscillation, and a the distance of the centre of gravity
from the axis, therefore the centre of oscillation is
always farther away from the axis than the centre of
THE PRINCIPLES OF DYNAMICS. 183
gravity by the distance K/a. Therefore K/a is the dis
tance of the centre of oscillation from the centre of
gravity. If through the centre of oscillation we place
a second axis parallel to the original axis, a passes
thereby into H/a, and we obtain the new pendulum
length
7' H I K I H 7
/'   a j  /.
K a a
a
The time of oscillation remains the same therefore
for the second parallel axis through the centre of oscil
lation, and consequently the same also for every par
allel axis that is at the same distance K/a from the
centre of gravity as the centre of oscillation.
The totality of all parallel axes corresponding to
the same period of oscillation and having the distances a
and K/a from the centre of gravity, is consequently re
alised in two coaxial cylinders. Each generating line
is interchangeable as axis with every other generating
line without affecting the period of oscillation.
25. To obtain a clear view of the relations subsist The axial
ing between the two axial cylinders, as we shall briefly
call them, let us institute the following considerations.
We put A = k 2 M, and then
If we seek the a that corresponds to a given /, and
therefore to a given time of oscillation, we obtain
Generally therefore to one value of / there correspond
two values of a. Only where l/7 2 /4 ~k* = 0, that is
in cases in which /= 2/, do both values coincide in
a=k.
184
THE SCIENCE OF MECHANICS,
If we designate the two values of a that correspond
to every /, by a and /?, then
or
k* = a. ft.
The deter If, therefore, in any pendulous body we know two par
mination of ......
the prece^allel axes that have the same time of oscillation and
byageo different distances a and ft from the centre of gravity,
metrical . . .
method. as is the case for instance where we are able to give the
centre of oscillation for any point of suspension, we
can construct k. We lay off (Fig. 122) a: and ft con
secutively on a straight line, describe a semicircle on
a f ft as diameter, and erect a perpendicular at the
point of junction of the two divisions a and ft. On this
perpendicular the semicircle cuts off k. If on the other
hand we know k, then for every value of a', say A, a
value yu is obtainable that will give the same period
of oscillation as A. We construct (Fig. 123) with A
and k as sides a right angle, join their extremities by a
straight line on which we erect at the extremity of k a
perpendicular which cuts off on A produced the por
tion }JL.
Now let us imagine any body whatsoever (Fig. 124)
with the centre of gravity 0. We place it in the plane
THE PRINCIPLES OF DYNAMICS. 185
of the drawing, and make it swing about all possible An niusm
parallel axes at right angles to the plane of the paper. ide^.
All the axes that pass through the circle a are, we
find, with respect to period of oscillation, interchange
able with each other and also with those that pass
through the circle /3. If instead of a we take a smaller
circle A, then in the place of /3 we shall get a larger
Fig. 124.
circle ju. Continuing in this manner, both circles ul
timately meet in one with the radius k.
26. We have dwelt at such length on the foregoing ^ a
matters for good reasons. In the first place, they have
served our purpose of displaying in a clear light the
splendid results of the investigations of Huygens. For
all that we have given is virtually contained, though
in somewhat different form, in the writings of Huygens,
i86 THE SCIENCE OF MECHANICS.
or is at least so approximately presented in them that
it can be supplied without the slightest difficulty. Only
a very small portion of it has found its way into our
modern elementary textbooks. One of the proposi
tions that has thus been incorporated in our elemen
tary treatises is that referring to the convertibility of
the point of suspension and the centre of oscillation.
The usual presentation, however, is not exhaustive.
Captain KATER, as we know, employed this principle
for determining the exact length of the seconds pen
dulum.
Function of The points raised in the preceding paragraphs have
of inertia, also rendered us the service of supplying enlighten
ment as to the nature of the conception " moment of
inertia." This notion affords us no insight, in point
of principle, that we could not have obtained without
it. But since we save by its aid the individual con
sideration of the particles that make up a system, or
dispose of them once for all, we arrive by a shorter
and easier way at our goal. This idea, therefore, has
a high import in the economy of mechanics. Poinsot,
after Euler and Segner had attempted a similar object
with less success, further developed the ideas that be
long to this subject, and by his ellipsoid of inertia and
central ellipsoid introduced further simplifications.
The lesser 27. The investigations of Huygens concerning the
Sons of a ~ geometrical and mechanical properties of cycloids are
uygens. ^ j ess importance. The cycloidal pendulum, a contriv
ance in which Huygens realised, not an approximate,
but an exact independence of the time and amplitude
of oscillation, has been dropt from the practice of mod
ern horology as unnecessary. We shall not, therefore,
enter into these investigations here, however much of
the geometrically beautiful they may present.
THE PRINCIPLES OF DYNAMICS. 187
Great as the merits of Huygens are with respect to Huygens's
the most different physical theories, the art of horology, achieve
practical dioptrics, and mechanics in particular, his
chief performance, the one that demanded the greatest
intellectual courage, and that was also accompanied
with the greatest results, remains his enunciation of the
principle by which he solved the problem of the centre
of oscillation. This very principle, however, was the
only one he enunciated that was not adequately appre
ciated by his contemporaries ; nor was it for a long
period thereafter. We hope to have placed this prin
ciple here in its right light as identical with the prin
ciple of vis viva. (See Appendix, XIII., p. 530.)
in.
THE ACHIEVEMENTS OF NEWTON.
1. The merits of NEWTON with respect to our sub Newton's
n _ . merits.
ject were twofold. First, he greatly extended the range
of mechanical physics by his discovery of universal
gravitation. Second, he completed the formal enunciation
of the mechanical principles now generally accepted. Since
his time no essentially new principle has been stated.
All that has been accomplished in mechanics since his
day, has been a deductive, formal, and mathematical
development of mechanics on the basis of Newton's
laws.
2. Let us first cast a glance at Newton's achieved Hj s *
ment in the domain of physics. Kepler had deduced discovery,
from the observations of Tycho Brahe and his own,
three empirical laws for the motion of the planets
about the sun, which Newton by his new view rendered
intelligible. The laws of KEPLER are as follows :
i) The planets move about the sun in ellipses, in
one focus of which the sun is situated.
1 88 THE SCIENCE OF MECHANICS.
Kepler's 2~] The radius vector joining each planet with the
laws. Their } J . , .
part in the sun describes equal areas in equal times.
iscovery. cubes of the mean distances of the planets
from the sun are proportional to the squares of
their times of revolution.
He who clearly understands the doctrine of Galileo
and Huygens, must see that a curvilinear motion im
plies deflective acceleration. Hence, to explain the phe
nomena of planetary motion, an acceleration must be
supposed constantly directed towards the concave side
of the planetary orbits.
Central ac Now Kepler's second law, the law of areas, is ex
expiams plained at once by the assumption of a constant plane
Kepler's ^ J , . . .
second law. tary acceleration towards the sun ; or rather, this ac
celeration is another form of expression for the same
fact. If a radius vector describes
in an element of time the area
ABS (Fig. 125), then in the next
equal element of time, assuming
no acceleration, the area BCS
will be described, where BC =
AB and lies in the prolongation
Fig. 125. of AB. But if the central accel
eration during the first element of time produces a
velocity by virtue of which the distance BD will be
traversed in the same interval, the nextsucceeding
area swept out is not BCS, but BES, where CE is par
allel and equal to BD. But it is evident that BES =
BCS~ ABS. Consequently, the law of the areas con
stitutes, in another aspect, a central acceleration.
Having thus ascertained the fact of a central accel
eration, the third law leads us to the discovery of its
character. Since the planets move in ellipses slightly
different from circles, we may assume, for the sake of
THE PRINCIPLES OF DYNAMICS. 189
simplicity, that their orbits actually are circles. If R^ The formal
R 2 , R z are the radii and T 19 T 2 , T^ the respective of this Be
times of revolution of the planets, Kepler's third law dlludbie
i J.J. r 11 from Kep
may be written as follows : ler's third
law.
o
= ^ == "^ = = a constant.
But we know that the expression for the central accel
eration of motion in a circle is <p= ^.R TC 2 /T 2 , or
T 2 = 4 7t 2 R/q>. Substituting this value we get
cp^R^ = <p 2 2 2 = > 3 ^ 3 2 = constant ; or
cp == constant /R* ;
that is to say, on the assumption of a central accelera
tion inversely proportional to the square of the distance,
we get, from the known laws of central motion, Kep
ler's third law ; and vice versa.
Moreover, though the demonstration is not easily
put in an elementary form, when the idea of a central
acceleration inversely proportional to the square of the
distance has been reached, the demonstration that this
acceleration is another expression for the motion in
conic sections, of which the planetary motion in ellipses
is a particular case, is a mere affair of mathematical
analysis.
3. But in addition to the intellectual performancexheques
,. ,  , . , r n 11 tionof the
just discussed, the way to which was fully prepared by physical
Kepler, Galileo, and Huygens, still another achieve tfcisaccei
eration.
ment of Newton remains to be estimated which in no
respect should be underrated. This is an achievement
of the imagination. We have, indeed, no hesitation
in saying that this last is the most important of all.
Of what nature is the acceleration that conditions the
curvilinear motion of the planets about the sun,
of the satellites aboutjhe planets ?,
igo THE SCIENCE OF MECHANICS.
The steps Newton perceived, with great audacity of thought,
Sally ied g " and first in the instance of the moon, that this accel
the1deaof eration differed in no substantial respect from the ac
universal . .. . T ,
gravitation, celeration of gravity so familiar to us. It was prob
ably the principle of continuity, which accomplished
so much in Galileo's case, that led him to his dis
covery. He was wont and this habit appears to be
common to all truly great investigators to adhere as
closely as possible, even in cases presenting altered
conditions, to a conception once formed, to preserve
the same uniformity in his conceptions that nature
teaches us to see in her processes. That which is a
property of nature at any one time and in any one
place, constantly and everywhere recurs, though it
may not be with the same prominence. If the attrac
tion of gravity is observed to prevail, not only on the
surface of the earth, but also on high mountains and in
deep mines, the physical inquirer, accustomed to con
tinuity in his beliefs, conceives this attraction as also
operative at greater heights and depths than those ac
cessible to us. He asks himself, Where lies the limit
of this action of terrestrial gravity ? Should its action
not extend to the moon ? With this question the great
flight of fancy was taken, of which, with Newton's in
tellectual genius, the great scientific achievement was
but a necessary consequence. (See p. 531.)
The appii' Newton discovered first in the case of the moon that
this idea to the same acceleration that controls the descent of a
of the moon, stone also prevented this heavenly body from moving
away in a rectilinear path from the earth, and that, on
the other hand, its tangential velocity prevented it from
falling towards the earth. The motion of the moon
thus suddenly appeared to him in an entirely new light,
but withal under quite familiar points of view. The
. THE PRINCIPLES OF DYNAMICS. 191
new conception was attractive in that it embraced ob
jects that previously were very remote, and it was con
vincing in that it involved the most familiar elements.
This explains its prompt application in other fields and
the sweeping character of its results.
Newton not only solved by his new conception the its univer
. i i i r T i sa l applica
thousand years puzzle ot the planetary system, buttiontoaii
also furnished by it the key to the explanation of a ma en
number of other important phenomena. In the same
way that the acceleration due to terrestrial gravity ex
tends to the moon and to all other parts of space, so do
the accelerations that are due to the other heavenly
bodies, to which we must, by the principle of contin
uity, ascribe the same properties, extend to all parts
of space, including also the earth. But if gravitation is
not peculiar to the earth, its seat is not exclusively in the
centre of the earth. Every portion of the earth, how
ever small, shares it, Every part of the earth attracts,
or determines an acceleration of, every other part.
Thus an amplitude and freedom of physical view were
reached of which men had no conception previously to
Newton's time.
A long series of propositions respecting the action The sweep
1 .  . Ing charac
of spheres on other bodies situated beyond, upon, or ter of its re
within the spheres ; inquiries as to the shape of the
earth, especially concerning its flattening by rotation,
sprang, as it were, spontaneously from this view. The
riddle of the tides, the connection of which with the
moon had long before been guessed, was suddenly ex
plained as due to the acceleration of the mobile masses
of terrestrial water by the moon. (See p. 533.)
4. The reaction of the, new ideas on mechanics was
a result which speedily followed. The greatly varying
accelerations which by the new view the same body be
192 THE SCIENCE OF MECHANICS.
the effect came affected with according to its position in space,
ideas on suggested at once the idea of variable weight, yet also
mechanics. . , .. CIT IT
pointed to one characteristic property of bodies which
was constant. The notions of mass and weight were
thus first clearly distinguished. The recognised vari
ability of acceleration led Newton to determine by spe
cial experiments the fact that the acceleration of gravity
is independent of the chemical constitution of bodies ;
whereby new positions of vantage were gained for the
elucidation of the relation of mass and weight, as will
presently be shown more in detail. Finally, the uni
versal applicability of Galileo's idea of force was more
palpably impressed on the mind by Newton's perform
ances than it ever had been before. People could no
longer believe that this idea was alone applicable to the
phenomenon of falling bodies and the processes most
immediately connected therewith. The generalisation
was effected as of itself, and without attracting partic
ular attention.
Newton's 5, Let us now discuss, more in detail, the achieve
ments in ments of Newton as they bear upon the principles of
the domain 7 . _ _ . , , t f , ,
of mechan mechanics. In so doing, we shall first devote ourselves
exclusively to Newton's ideas, seek to bring them for
cibly home to the reader's mind, and restrict our criti
cisms wholly to preparatory remarks, reserving the
criticism of details for a subsequent section. On pe
rusing Newton's work (Philosophic Naiuralis Principia
Mathematica. London, 1687), the following things
strike us at once as the chief advances beyond Galileo
and Huygens :
1) The generalisation of the idea of force. .
2) The introduction of the concept of mass.
3) The distinct and general formulation of the prin
ciple of the parallelogram of forces.
THE PRINCIPLES OF DYNAMICS'. 193
4) The statement of the law of action and reaction.
6. With respect to the first point little is to be His attitude
added to what has already been said. Newton con tothJflel
ceives all circumstances determinative of motion,
whether terrestrial gravity or attractions of planets, or
the action of magnets, and so forth, as circumstances
determinative of acceleration. He expressly remarks
on this point that by the words attraction and the like
he does not mean to put forward any theory concern
ing the cause or character of the mutual action referred
to, but simply wishes to express (as modern' writers
say, in a differential form) what is otherwise expressed
(that is ? in an integrateiform) in, the description of the
motion. "^Newton's reiterated and emphatic protesta
tions that he is not concerned with hypotheses as to the
causes of phenomena, but has simply to do with the
investigation and transformed statement of actual facts,
a direction of thought that is distinctly and tersely
uttered in his words "hypotheses non fingo," "I do
not frame hypotheses," stamps him as a philosopher
of the highest rank. He is not desirous to astound and The Regu
. , , .... laePhiloso
startle, or to impress the imagination by the originality phandi.
of his ideas : his aim is to know Natiire. *
* This is conspicuously shown in the rules that Newton formed for the
conduct of natural inquiry (the Regulce Philosofhdndi') :
" Rule I. No more causes of natural things are to be admitted than such
as truly exist and are sufficient to explain the phenomena of these things,
" Rule II. Therefore, to natural effects of the same kind we must, as far
as possible, assign the same causes ; e. g., to respiration in man and animals ;
to the descent of stones in Europe and in America ; to the light of our kitchen
fire and of the sun ; to the reflection of light on the earth and on the planets.
"Rule III. Those qualities of bodies that can be neither increased nor
diminished, and which are found to belong to all bodies within the reach of
our experiments, are to be regarded as the universal qualities of all bodies.
[Here follows the enumeration of the properties of bodies which has been in
corporated in all textbooks.]
" If it universally appear, by experiments and astronomical observations,
that all bodies in the vicinity of the earth are heavy with respect to the earth,
and this in proportion to the quantity of matter which they severally contain ;
194
THE SCIENCE OF MECHANICS.
The New 7. With regard to the concept of " mass/' it is to
cept a of on be observed that the formulation of Newton, which de
fines mass to be the quantity of matter of a body as
measured by the product of its volume and density, is
unfortunate. As we can only define density as the mass
of unit of volume, the circle is manifest. Newton felt
distinctly that in every body there was inherent a prop
erty whereby the amount of its motion was determined
and perceived that this must be different from weight.
He called it, as we still do, mass ; but he did not suc
ceed in correctly stating this perception. We shall re
vert later on to this point, and shall stop here only to
make the following preliminary remarks.
The expe 8. Numerous experiences, of which a sufficient num
which point ber stood at Newton's disposal, point clearly to the ex
to the exist . . . .
enceofsucbistence of a property distinct from weight, whereby the
a physical . .
property. ////, '#/% 'fflfa quantity ot motion of the
body to which it belongs is
determined. If (Fig. 126)
we tie a flywheel to a rope
and attempt to lift it by
means of a pulley, we feel
the weight of the flywheel.
If the wheel be placed
on a perfectly cylindrical axle and well balanced, it
will no longer assume by virtue of its weight any de
terminate position. Nevertheless, we are sensible of
that the moon is heavy with respect to the earth in the proportion of its mass,
and our seas with respect to the moon ; and all the planets with respect to one
another, and the comets also with respect to the sun ; we must, In conformity
with this rule, declare, that all bodies are heavy with respect to one another.
" Rule IV. In experimental physics propositions collected by induction
from phenomena are to be regarded either as accurately true or very nearly
true, notwithstanding any contrary hypotheses, till other phenomena occur, by
which they are made more accurate, or are rendered subject to exceptions.
"This rule must be adhered to, that the results of induction may not be
annulled by hypotheses."
JL. f*~
U U
I2(5 
THE PRINCIPLES OF DYNAMICS. 195
a powerful resistance the moment we endeavor to set Mas
,111 , , tinct from
the wheel in motion or attempt to stop it when in mo weight,
tion. This is the phenomenon that led to the enuncia
tion of a distinct property of matter termed inertia, or
" force" of inertia a step which, as we have already
seen, and shall further explain below is unnecessary.
Two equal loads simultaneously raised, offer resistance
by their weight. Tied to the extremities of a cord that
passes over a pulley, they offer resistance to any mo
tion, or rather to any change of velocity of the pulley,
by their mass. . A large weight hung as a pendulum
on a very long string can be held at an angle of slight
deviation from the line of equilibrium with very little
effort. The weightcomponent that forces the pendu
lum into the position of equilibrium, is very small.
Yet notwithstanding this we shall experience a con
siderable resistance if we suddenly attempt to move or
stop the weight. A weight that is just supported by a
balloon, although we have no longer to overcome its
gravity, opposes a perceptible resistance to motion.
Add to this the fact that the same body experiences in
different geographical latitudes and in different parts
of space very unequal gravitational accelerations and
we shall clearly recognise that mass exists as a property
wholly distinct from weight determining the amount of
acceleration which a given force communicates to the
body to which it belongs. (See p. 536.)
Q. Important is Newton's demonstration that the Mass meas
y r . urable by
mass of a body may, nevertheless, under certain con weight,
ditions, be measured by its weight. Let us suppose a
body to rest on a support, on which it exerts by its weight
a pressure. The obvious inference is that 2 or 3 such
bodies, or onehalf or onethird of such a body, will pro
duce a corresponding pressure 2, 3, J, or % times as
i 9 5 THE SCIENCE OF MECHANICS.
The_prere great. If we imagine the acceleration of descent in
t q be S meas creased, diminished, or wholly removed, we shall ex
mass Ty pect that the pressure also will be increased, dimin
Welght< ished, or wholly removed. We thus see, that the pres
sure attributable to weight increases, decreases, and
vanishes along with the " quan
tity of matter" and the magni
tude of the acceleration of de
scent. In the simplest manner
Flg ' 127> imaginable we conceive the pres
sure/ as quantitatively representable by the product of
the quantity of matter m into the acceleration of descent
g by p = mg. Suppose now we have two bodies that
exert respectively the weight pressures /, /', to which
we ascribe the " quantities of matter ' ' m, m', and which
are subjected to the accelerations of descent g, g'\ then
p = mg and /' = m 'g' . If, now, we were able to prove,
that, independently of the material (chemical) compo
sition of bodies, g=g' at every same point on the
earth's surface, we should obtain m/m' =p/p'\ that is
to say, on the same spot of the earth's surface, it would
be possible to measure mass by weight.
Newton's Now Newton established this fact, that g is inde
ment of pendent of the chemical composition of bodies, by
these pre .
requisites, experiments with pendulums of equal lengths but dif
ferent material, which exhibited equal times of oscilla
tion. He carefully allowed, in these experiments, for
the disturbances due to the resistance of the air ; this
last factor being eliminated by constructing from differ
ent materials spherical pendulumbobs of exactly the
same size, the weights of which were equalised by ap
propriately hollowing the spheres. Accordingly, all
bodies maybe regarded as affected with the same g, and
THE PRINCIPLES OF D YNAMfCS. 197
their quantity of matter or mass can, as Newton pointed
out, be measured by their weight.
If we imagine a rigid partition placed between an Suppie
assemblage of bodies and a magnet, the bodies, if the JSSitera
magnet be powerful enough, or at least the majority 10nS *
of the bodies, will exert a pressure on the partition.
But it would occur to no one to employ this magnetic
pressure, in the manner we employed pressure due to
weight, as a measure of mass. The strikingly notice
able inequality of the accelerations produced in the
different bodies by the magnet excludes any such idea.
The reader will furthermore remark that this whole
argument possesses an additional dubious feature, in
that the concept of mass which up to this point has
simply been named and felt as a necessity, but not de
fined, is assumed by it.
10. To Newton we owe the distinct formulation ofxhedoc
the principle of the composition of forces.* If a body composi
is simultaneously acted on by two forces (Fig. 128), forces,
of which one would produce the
motion AB and the other the ^
motion AC in the same interval
of time, the body, since the two
forces and the motions produced Flg ' I28 '
by them are independent of each other, will move in that
interval of time to AD. This conception is in every
respect natural, and distinctly characterises the essen
tial point involved. It contains none of the artificial
and forced characters that were afterwards imported
into the doctrine of the composition of forces.
We may express the proposition in a somewhat
* Roberval's (1668) achievements with, respect to the doctrine of the com
position of forces are also to be mentioned here, Varignon and Lami have al
ready been referred to. (See the text, page 36.)
I 9 8 THE SCIENCE OF MECHANICS.
Discussion different manner, and thus bring it nearer its modern
rine of the form. The accelerations that different forces impart
;omposi , .
ion of to the same body are at the same time the measure of
these forces. But the paths described in equal times
are proportional to the accelerations. Therefore the
latter also may serve as the measure of the forces. We
may say accordingly : If two forces, which are propor
tional to the lines AB and AC, act on a body A in the
directions AB and A C, a motion will result that could
also be produced by a third force acting alone in the
direction of the diagonal of the parallelogram con
structed on AB and A C and proportional to that di
agonal. The latter force, therefore, may be substituted
for the other two. Thus, if <p and ij> are the two ac
celerations set up in the directions AB and A C, then
for any definite interval of time /, AB = Gpf* /2, AC =
^>/ 2 /2. If, now, we imagine AD produced in the same
interval of time by a single force determining the accel
eration %, we get
AD = X* 2 / 2 > and AB AC: AD = cp : $ : %.
As soon as we have perceived the fact that the forces are
independent of each other, the principle of the paral
lelogram of forces is easily reached from Galileo's no
tion of force. Without the assumption of this inde
pendence any effort to arrive abstractly and philosoph
ically at the principle, is in vain.
he law of IT. Perhaps the most important achievement ci
ction and , _ . . .
action. Newton with respect to the principles is the distinct
and general formulation of the law of the equality of
action and reaction, of pressure and counterpressure.
Questions respecting the motions of bodies that exert
a reciprocal influence on each other, cannot be solved
by Galileo's principles alone. A new principle is ne
cessary that will define this mutual action. Such a
THE 'PRINCIPLES OF DYNAMICS. 199
principle was that resorted to by Huygens in his inves
tigation of the centre of oscillation. Such a principle
also is Newton's law of action and reaction.
A body that presses or pulls another body is, ac Newton's
i VT . , . , deduction
cording to Newton, pressed or pulled in exactly the of the law
same degree by that other body. Pressure and counter and C reac
pressure, force and counterforce, are always equal to
each other. As the measure of force is defined by
Newton to be the quantity of motion or momentum
(mass X velocity) generated in a unit of time, it conse
quently follows that bodies that act on each other com
municate to each other in equal intervals of time equal
and opposite quantities of motion (momenta), or re
ceive contrary velocities reciprocally proportional to
their masses.
Now, although Newton's law, in the form here exThereia
. tive imme
.pressed, appears much more simple, more immediate, diacyo^
and at first glance more admissible than that of Huy aad Huy
11 i gens's prin
gens, it will be found that it by no means contains less cipies.
unanalysed experience or fewer instinctive elements.
Unquestionably the original incitation that prompted
the enunciation of the principle was of a purely instinc
tive nature. We know that we do not experience any
resistance from a body until we seek to set it in motion.
The more swiftly we endeavor to hurl a heavy stone
from us, the more our body is forced back by it. Pres
sure and counterpressure go hand in hand. The as
sumption of the equality of pressure and counterpres
sure is quite immediate if, using Newton's own illus
tration, we imagine a rope stretched between two bod
ies, or a distended or compressed spiral spring between
them.
There exist in the domain of statics very many in
stinctive perceptions that involve the equality of pres
200 THE SCIENCE OF MECHANICS.
Statical ex sure and counterpressure. The trivial experience that
wh^h point one cannot lift one's self by pulling on one's chair is
ence 6 of the" of this character. In a scholium in which he cites the
physicists Wren, Huygens, and Wallis as his prede
cessors in the employment of the principle, Newton
puts forward similar reflections. He imagines the
earth, the single parts of which gravitate towards one
another, divided by a plane. If the pressure of the
one portion on the other were not equal to the counter
pressure, the earth would be compelled to move in the
direction of the greater pressure. But the motion of
a body can, so far as our experience goes, only be de
termined by other bodies external to it. Moreover,
we might place the plane of division referred to at any
point we chose, and the direction of the resulting mo
 tion, therefore, could not be exactly determined.
The con i2. The indistinctness of the concept of mass takes
ceptofmass
in its con a very palpable form when we attempt to employ the
nection . . . "I , .  .  .
with this principle of the equality of action and reaction dynam
ically. Pressure and counterpressure may be equal.
But whence do we know that equal pressures generate
velocities in the inverse ratio of the masses ? Newton,
indeed, actually felt the necessity of an experimental
corroboration of this principle. He cites in a scholium,
in support of his proposition, Wren's experiments on
Impact, and made independent experiments himself.
He enclosed in one sealed vessel a magnet and in an
other a piece of iron, placed both in a tub of water,
and left them to their mutual action. The vessels ap
proached each other, collided, clung together, and af
terwards remained at rest. This result is proof of the
equality of pressure and counterpressure and of equal
and opposite momenta (as we shall learn later on,
when we come to discuss the laws of impact).
THE PRINCIPLES OF DYNAMICS. 201
The reader has already felt that the various ermnci The merits
f XT . , J J , , . and defects
ations of Newton with respect to mass and the pnn of Newton's
ciple of reaction, hang consistently together, and that
they support one another. The experiences that lie at
their foundation are : the instinctive perception of the
connection of pressure and counterpressure ; the dis
cernment that bodies offer resistance to change of ve
locity independently of their weight, but proportion:
ately thereto ; and the observation that bodies of greater
weight receive under equal pressure smaller velocities.
Newton's sense of w/^/ fundamental concepts and prin
ciples were required in mechanics was admirable. The
form of his enunciations, however, as we shall later in
dicate in detail, leaves much to be desired. But we have
no right to underrate on this account the magnitude of
his achievements ; for the difficulties he had to conquer
were of a formidable kind, and he shunned them less
than any other investigator.
IV.
DISCUSSION AND ILLUSTRATION OF THE PRINCIPLE OF
REACTION.
i. We shall now devote ourselves a moment ex The princi
clusively to the Newtonian ideas, and seek to bring the tion.
principle of reaction more clearly home to our mind
a
v i i
 \m\
Fig. 129. Fig. 130.
and feeling. If two masses (Fig. 129) J/and m act on
one another, they impart to each other, according
to Newton, contrary velocities J^and v, which are in
versely proportional to their masses, so that
202 THE SCIENCE OP MECHANICS.
General The appearance of greater evidence may be im
elucidation . . ,
of the pnn parted to this principle by the following consideration.
action. We imagine first (Fig. 130) two absolutely equal bodies
a, also absolutely alike in chemical constitution. We
set these bodies opposite each other and put them in
mutual action ; then, on the supposition that the in
fluences of any third body and of the spectator are ex
cluded, the communication of equal and contrary velo
cities in the direction of the line joining the bodies is
the sole uniquely determined interaction.
Now let us group together in A (Fig. 131) w such
bodies a, and put at B over against them m' such
bodies a. We have then before us bodies whose quan
Fig. 131. Fig. 132.
tities of matter or masses bear to each other the pro
portion m : m'. The distance between the groups we
assume to be so great that we may neglect the exten
sion of the bodies. Let us regard now the accelera
tions a, that every two bodies a impart to each other,
as independent of each other. Every part of A, then,
will receive in consequence of the action of B the ac
celeration m'a, and every part of B in consequence of
the action of A the acceleration m a accelerations
which will therefore be inversely proportional to the
masses.
2. Let us picture to ourselves now a mass M (Fig.
132) joined by some elastic connection with a mass m,
both masses made up of bodies a equal in all respects.
Let the mass m receive from some external source an
acceleration q>. At once a distortion of the connection
is produced, by which on the one hand m is retarded
TUE PRINCIPLES OF D YNAMICS. 203
and on the other M accelerated. When both masses The deduc
T , 11 111 t* 011 * th e
nave begun to move with the same acceleration, all notion of
further distortion of the connection ceases. If we call force.' 1 '
a the acceleration of M and ft the diminution of the
acceleration of m t then a = <p ft, where agreeably
to what precedes a M ==. /3m. From this follows
. ^ , aM mcp
a J ft = a \ = cp, or a =
v 7 ? ^^ txi TIJT ; *
m ^ M+ m
If we were to enter more exhaustively into the de
tails of this last occurrence, we should discover that
the two masses, in addition to their motion of progres
sion, also generally perform with respect to each other
motions of oscillation. If the connection on slight dis
tortion develop a powerful tension, it will be impos
sible for any great amplitude of vibration to be reached,
and we may entirely neglect the oscillatory motions,
as we actually have done.
If the expression <x= m<p/M\ m, which deter
mines the acceleration of the entire system, be ex
amined, it will be seen that the product m cp plays a
decisive part in its determination. Newton therefore
invested this product of the mass into the acceleration
imparted to it, with the name of "moving force."
M \ m y on the other hand, represents the entire mass
of the rigid system. We obtain, accordingly, the accel
eration of any mass m' on which the
moving force / acts, from the expres
sion//^', p^j K] p^j
3. To reach this result, it is not at F] .
all necessary that the two connected
masses should act directly on each other in all their
parts. We have, connected together, let us say, the
three masses m^ m 2 , m%, where m^ is supposed to act
204 THE SCIENCE OF MECHANICS.
A condition only on w , and m^ only on m*. Let the mass ?;/, re
whichdoes . r 3 X 2 1
not affect ceive from some external source the acceleration q>.
vious r re In the distortion that follows, the
suit.
masses  m% m 2 m^
receive the accelerations + + ft + <P
y a.
Here all accelerations to the right are reckoned as
positive, those to the left as negative, and it is obvious
that the distortion ceases to increase
when $ = /3 y, $ = cp a,
where dm z = ym 2 , am^ = /3m 2 .
The resolution of these equations yields the com
mon acceleration that all the masses receive ; namely,
a result of exactly the same form as before. When
therefore a magnet acts on a piece of iron which is
joined to a piece of wood, we need not trouble our
selves about ascertaining what particles of the wood
are distorted directly or indirectly (through other par
ticles of the wood) by the motion of the piece of iron.
The considerations advanced will, in some meas
ure, perhaps, have contributed towards clearly impress
ing on us the great importance for mechanics of the
Newtonian enunciations. They will also serve, in a
I subsequent place, to ren
der more readily obvious
the defects of these enun
I ' ciations.
\ ) 4. Let us now turn to
Fig. 134. a few illustrative physical
examples of the principle of reaction. We consider,
say, a loatl L on a table T. The table is pressed by
THE PRINCIPLES OF DYNAMICS. 205
the load/kr/ so much, and so much only, as it in return Some phys
presses the load, that is prevents the same from falling, pies of a the
If/ is the weight, m the mass, and g the acceleration oPreacti^n.
of gravity, then by Newton's conception/ = mg. If
the table be let fall vertically downwards with the ac
celeration of free descent g, all pressure on it ceases.
We discover thus, that the pressure on the table is de
termined by the relative acceleration of the load with
respect to the table. If the table fall or rise with the
acceleration y, the pressure on it is respectively m (g
y) and m (g + 7). Be it noted, however, that no
change, of the relation is produced by a constant velocity
of ascent or descent. The relative acceleration is de
terminative.
Galileo knew this relation of things very well. The The pres
i r 1 A 1 i IT r SUre f the
doctrine of the Aristotelians, that bodies of greater parts of fail
weight fall faster than bodies of less weight, he not only
refuted by experiments, but cornered his adversaries
by logical arguments. Heavy bodies fall faster than
light bodies, the Aristotelians said, because the upper
parts weigh down on the under parts and accelerate
their descent. In that case, returned Galileo, a small
body tied to a larger body must, if it possesses in se the
property of less rapid descent, retard the larger. There
fore, a larger body falls more slowly than a smaller
body. The entire fundamental assumption is wrong,
Galileo says, because one portion of a falling body can
not by its weight under any circumstances press an
other portion.
A pendulum with the time of oscillation T= n Vtjg, A failing
would acquire, if its axis received the downward accel pen u um "
eration y, the time of oscillation T= nV l/g y~
and if let fall freely would acquire an infinite time of
oscillation, that is, would cease to oscillate.
206
THE SCIENCE OF MECHANICS.
The sensa
tion of fall
Poggen
dorff's ap
paratus.
We ourselves, when we jump or fall from an eleva
tion, experience a peculiar sensation, which must be
due to the discontinuance of the gravitational pressure
of the parts of our body on one another the blood, and
so forth. A similar sensation, as if the ground were
sinking beneath us, we should have on a smaller planet,
to which we were suddenly transported. The sensation
of constant ascent, like that felt in an earthquake,
would be produced on a larger planet.
5. The conditions referred to are very beautifully
illustrated by an apparatus (Fig. 135^) constructed
by Poggendorff. A string loaded at both extremities
Fig. rasa. Fig. issb.
by a weight P (Fig. 1350) is passed over a pulley c,
attached to the end of a scalebeam. A weight p is
laid on one of the weights first mentioned and tied by
a fine thread to the axis of the pulley. The pulley
now supports the weight 2 P  / Burning away the
thread that holds the overweight, a uniformly accel
erated motion begins with the acceleration y, with
which P j p descends and P rises. The load on the
pulley is thus lessened, as the turning of the scales in
dicates. The descending weight P is counterbalanced
by the rising weight P, while the added over weight,
instead of weighing/, now weighs (/*/<" )(<" y)* And
since y = (//2 P + /) g, we have now to regard the
load on the pulley, not as/, but as/( 2 P/2P+^). The
THE PRINCIPLES OF DYNAMICS.
207
descending weight, only partially impeded in its motion
of descent, exerts only a partial pressure on the pulley.
We may vary the experiment,
tded at one extremity with the
pulleys a, b, d, of the apparatus as indicated in Fig.
We pass a thread A variation
loaded at one extremity with the weight P over the experiment
Fig. 1350.
tie the unloaded extremity at m, and equilibrate
the balance. If we pull on the string at m, this can
not directly affect the balance since the direction of the
string passes exactly through its axis. But the side a
immediately falls. The slackening of the string causes
a to rise. An unaccelerated motion of the weights would
208
THE SCIENCE OF MECHANICS.
The suspen
sion of mi
not disturb the equilibrium. But we cannot pass from
rest to motion without acceleration.
6. A phenomenon that strikes us at first glance is,
nute bodies that minute bodies of greater or less specific gravity
different than the liquid in which they are immersed, if suffi
gravhy. ciently small, remain suspended a very long time in the
liquid. We perceive at once that
particles of this kind have to over
come the friction of the liquid. If the
cube of Fig. 136 be divided into 8
parts by the 3 sections indicated,
and the parts be placed in a row,
their mass and overweight will re
Fig. 136. main the same, but their crosssec
tion and superficial area, with which the friction goes
hand in hand, will be doubled.
DO such Now, the opinion has at times been advanced with
suspended ' .
particles af respect to this phenomenon that suspended particles
specific of the kind described have no influence on the specific
gravities of . _. , . . . ,
the support gravity indicated by an areometer immersed in the
' liquid, because these particles are themselves areo
meters. But it will readily be seen that if the sus
pended particles rise or fall with constant velocity, as
in the case of very small particles immediately occurs,
the effect on the balance and the areometer must be
the same. If we imagine the areometer to oscillate
about its position of equilibrium, it will be evident
that the liquid with all its contents will be moved with
it. Applying the principle of virtual displacements,
therefore, we can be no longer in doubt that the areo
meter must indicate the mean specific gravity. We
may convince ourselves of the untenability of the rule
by which the areometer is supposed to indicate only
the specific gravity of the liquid and not, that of the sus
THE PRINCIPLES OF D YNAMICS. 209
psnded particles, by the following consideration. In a
liquid A a smaller quantity of a heavier liquid B is in
troduced and distributed in fine drops. The areometer,
let us assume, indicates only the specific gravity of
A. Now, take more and more of the liquid B^ finally
just as much of it as we have of A\ we can, then, no
longer say which liquid is suspended in the other, and
which specific gravity, therefore, the areometer must
indicate.
7. A phenomenon of an imposing kind, in which The phe
... ... . _ ,  . ".. nom^non of
the relative acceleration of the bodies concerned is the tides,
seen to be determinative of their mutual pressure, is
that of the tides. We will enter into this subject here
only in so far as it may serve to illustrate the point we
are considering. The connection of the phenomenon
of the tides with the motion of the moon asserts itself
in the coincidence of the tidal and lunar periods, in
the augmentation of the tides at the full and new
moons, in the daily retardation of the tides (by about
50 minutes), corresponding to the retardation of the
culmination of the moon, and so forth. As a matter
of fact, the connection of the two occurrences was very
early thought of. In Newton's time people imagined
to themselves a kind of wave of atmospheric pressure,
by means of which the moon in its motion was sup
posed to create the tidal wave.
The phenomenon of the tides makes, on every one its impos
that sees it for the first time in its full proportions, antef. ara "
overpowering impression. We must not be surprised,
therefore, that it is a subject that has actively engaged
the investigators of all times. The warriors of Alex
ander the Great had, from their Mediterranean homes,
scarcely the faintest idea of the phenomenon of the
tides, and they were, therefore, not a little taken aback
2io THE SCIENCE OF MECHANICS.
by the sight of the powerful ebb and flow at the mouth
of the Indus ; as we learn from the account of Curtius
Rufus (De Rebus Gestis Aleocandri Magni}, whose
words we here literally quote :
Extract "34 Proceeding, now, somewhat more slowly in
from Cur . . . .
tins Rufus. " their course, owing to the current of the river being
" slackened by its meeting the waters of the sea, they
" at last reached a second island in the middle of the
"< river. Here they brought the vessels to the shore,
" and, landing, dispersed to seek provisions, wholly
" unconscious of the great misfortune that awaited
"them.
Describing " 35. It was about the third hour, when the ocean,
the effect . .
on the army " m its constant tidal flux and reflux, began to turn
derthe " and press back upon the river. The latter, at first
Great ofthe f t , i 1 , i 11 i
tides at the " merely checked, but then more vehemently repelled,
mouth of J \ . . , P
the Jn<*u*. f < at last set back in the opposite direction with a force
"greater than that 'of a rushing mountain torrent.
"The nature of the ocean was unknown to the multi
"tude, and grave portents and evidences of the wrath
"of the Gods were seen in what happened. With
"ever increasing vehemence the sea poured in, com
"pletely covering the fields which shortly before were
" dry. The vessels were lifted and the entire fleet dis
persed before those who had been set on shore, ter
" rifled and dismayed at this unexpected calamity,
" could return. But the more haste, in times of great
"disturbance, the less speed. Some pushed the ships
" to the shore with poles ; others, not waiting to adjust
"their oars, ran aground. Many, in their great haste
"to get away, had not waited for their companions,
"and were barely able to set in motion the huge, un
"manageable barks; while some of the ships were too
"crowded to receive the multitudes that struggled to
THE PRINCIPLES OF DYNAMICS. 211
"get aboard. The unequal division impeded all. TheThedisas
' ' cries of some clamoring to be taken aboard, of others anders
"crying to put off, and the conflicting commands of
Ci men, all desirous of different ends, deprived every one
"of the possibility of seeing or hearing. Even the
"steersmen were powerless; for neither could their
"cries be heard by the struggling masses nor were their
"orders noticed by the terrified and distracted crews.
"The vessels collided, they broke off each other's oars,
" they plunged against one another. One would think
" it was not the fleet of one and the same army that
"was here in motion, but two hostile fleets in combat.
' < Prow, struck stern ; those that had thrown the f ore
"most in confusion were themselves thrown into con
" fusion by those that followed; and the desperation
"of the struggling mass sometimes culminated in
"handtohand combats.
" 36. Already the tide had overflown the fields sur
" rounding the banks of the river, till only the hillocks
"jutted forth from above the water, like islands.
" These were the point towards which all that had given
"up hope of being taken on the ships, swam. The
"scattered vessels rested in part in deep water, where
"there were depressions in the land, and in part lay
"aground in shallows, according as the waves had
"covered the unequal surface of the country. Then,
"suddenly, a new and greater terror took possession
"of them. The sea began to retreat, and its waters
"flowed back in great long swells, leaving the land
"which shortly before had been immersed by the salt
"waves, uncovered and clear. The ships, thus for
"saken by the water, fell, some on their prows, some
" on their sides. The fields were strewn with luggage,
"arms, and pieces of broken planks and oars. The
212 THE SCIENCE OF MECHANICS.
The dismay "soldiers dared neither to venture on the land nor to
of the army. ....... , ,
"remain m the ships, for every moment they expected
"something new and worse than had yet befallen
"them. They could scarcely believe that that which
"they saw had really happened a shipwreck on dry
"land, an ocean in a river. And of their misfortune
"there seemed no end. For wholly ignorant that the
"tide would shortly bring back the sea and again set
"their vessels afloat, they prophesied hunger and dir
"est distress. On the fields horrible animals crept
"about, which the subsiding floods had left behind.
The efforts "37. ,The night fell, and even the king was sore
and the re "distressed at the slight hope of rescue. But his so
turn of the .
tide. " licitude could not move his unconquerable spirit. He
"remained during the whole night on the watch, and
" despatched horsemen to the mouth of the river, that,
" as soon as they saw the sea turn and flow back, they
"might return and announce its coming. He also
"commanded that the damaged vessels should be re
" paired and that those that had been overturned by
"the tide should be set upright, and ordered all to be
"near at hand when the sea should again inundate the
"land. After he had thus passed the entire night in
"watching and in exhortation, the horsemen came
"back at full speed and the tide as quickly followed.
"At first, the approaching waters, creeping in light
"swells beneath the ships, gently raised them, and,
"inundating the fields, soon set the entire fleet in mo
"tion. The shores resounded with the cheers and
" clappings of the soldiers and sailors, who celebrated
" with immoderate joy their unexpected rescue. ' But
"whence/ they asked, in wonderment, 'had the sea
"so suddenly given back these great masses of water?
"Whither had they, on the day previous, retreated?
THE PRINCIPLES OP DYNAMICS. 213
" And what was the nature of this element, which now
"opposed and now obeyed the dominion of the hours? '
"As the king concluded from what had happened that
"the fixed time for the return of the tide was after
"sunrise, he set out. In order to anticipate it, at mid
" night, and proceeding down the river with a few
"ships he passed the mouth and, finding himself at
"last at the goal of his wishes, sailed out 400 stadia
"into the ocean. He then offered a sacrifice to the
"divinities of the sea, and returned to his fleet."
8. The essential point to be noted in the explana The expia
. ration of
tion of the tides is, that the earth as a rigid body canthephe
. . 11 nomena of
receive but one determinate acceleration towards the the tides.
moon, while the mobile particles of water on the sides
nearest to and remotest from the moon can acquire
various accelerations.
Fig. 137.
Let us consider (Fig. 137) on the earth E 9 opposite
which stands the moon M, three points A, J3, C. The
accelerations of the three points in the direction of the
moon, if we regard them as free points, are respect
ively cp + d <p, <p> <p  A <p The earth as a whole,
however, has, as a rigid body, the acceleration cp. The
acceleration towards the centre of the earth we will
call g. Designating now all accelerations to the left
as negative, and all to the right as positive, we get the
following table :
214 THE SCIENCE OF MECHANICS.
ABC
(99+^9?), <p, (9~
gAcp, 0, (g
where the symbols of the first and second lines repre
sent the accelerations which the free points that head
the columns receive, those of the third line the accel
eration of corresponding rigid points of the earth, and
those of the fourth line, the difference, or the resultant
accelerations of the free points towards the earth. It
will be seen from this result that the weight of the water
at A and C is diminished by exactly the same amount.
The water will rise at A and C (Fig. 137). A tidal
wave will be produced at these points twice every
day.
A variation It is a fact not always sufficiently emphasised, that
nomenon, the phenomenon would be an essentially different one
if the moon and the earth were not affected with ac
celerated motion towards each other but were relatively
fixed and at rest. If we modify the considerations
presented to comprehend this case, we must put for the
rigid earth in the foregoing computation, cp = Q simply.
We then obtain for
the free points .... A C
the accelerations. . (<p + A<p\ (<p
+ g g
or (gdcp) cp, (g
or g' <P> '(#'+<?),
where g' = g A cp. In such case, therefore, the
weight of the water at A would be diminished, and the
weight at C increased ; the height of the water at A
THE PRINCIPLES OF DYNAMICS.
215
would be increased, and the height at C diminished.
The water would be elevated only on the side facing
the moon. (Fig. 138.)
K
Fig. 138.
9. It would hardly be worth while to illustrate An niustra
ij IIIT 11 t' lve experi
propositions best reached deductively, by experiments mem.
that can only be performed with difficulty. But such
experiments are not beyond the limits of possibility.
If we imagine a small iron sphere K to swing as a
conical pendulum about the pole of a
magnet N (Fig. 139), and cover the
sphere with a solution of magnetic sul
phate of iron, the fluid drop should, if
the magnet is sufficiently powerful, rep
resent the phenomenon of the tides. But
if we imagine the sphere to be fixed and
at rest with respect to the pole of the
magnet, the fluid drop will certainly not
be found tapering to a point both on
the side facing and the side opposite to
the pole of the magnet, but will remain suspended only
on the side of the sphere towards the pole of the
magnet.
10. We must not, of course, imagine, that the Some fur
entire tidal wave is produced at once by the action sidemions
of the moon. We have rather to conceive the tide
as an oscillatory movement maintained by the moon.
If, for example, we should sweep a fan uniformly and
Fig. 139
216 THE SCIENCE OF MECHANICS.
continuously along over the surface of the water of a
circular canal, a wave of considerable magnitude fol
lowing in the wake of the fan would by this gentle and
constantly continued impulsion soon be produced. In
like manner the tide is produced. But in trie latter
case the occurrence is greatly complicated by the irreg
ular formation of the continents, by the periodical
variation of the disturbance, and so forth. (See Ap
pendix, XVII., p. 537.)
CRITICISM OF THE PRINCIPLE OF REACTION AND OF THE
CONCEPT OF MASS.
The con i. Now that the preceding discussions have made
mass? us familiar with Newton's ideas, we are sufficiently
prepared to enter on, a critical examination of them.
We shall restrict ourselves primarily in this, to the
consideration of the concept of mass and the principle
of reaction. The two cannot, in 'such an examination,
be separated ; in them is contained the gist of New
ton's achievement.
Theexpres 2. In the first place we do not find the expression
tity^fniat "quantity of matter" adapted to explain and elucidate
the concept of mass, since that expression itself is not
possessed of the requisite clearness. And this is so,
though we go back, as many authors have done, to an
enumeration of the hypothetical atoms. We only com
plicate, in so doing, indefensible conceptions. If we
place together a number of equal, chemically homo
geneous bodies, we can, it may be granted, connect
some clear idea with "quantity of matter," and we per
ceive, also, that the resistance the bodies offer to mo
tion increases with this quantity. But the moment we
suppose chemical heterogeneity, the assumption that
THE PRINCIPLES OF DYNAMICS. 217
there Is still something that is measurable by the same Newton's
j ....  . . . . formulatioi
standard, which something we call quantity of matter, of the con
may be suggested by mechanical experiences, but is an
assumption nevertheless that needs to be justified.
When therefore, with Newton, we make the assump
tions, respecting pressure due to weight, that/ = mg,
p' =in'g, and put in conformity with such assumptions
p/p' = m/m' 9 we have made actual use in the operation
thus performed of the siipposition, yet to be justified,
that different bodies are measurable by the same stand
ard.
We might, indeed, arbitrarily posit, that m/m r ///';
that is, might define the ratio of mass to be the ratio
of pressure due to weight when g was the same. But
we should then have to substantiate the use that is made
of this notion of mass in the principle of reaction and
in other relations.
< n
I I
Fig, 140 a. Fig. 140 b.
3. When two bodies (Fig. 140 a"), perfectly equal Anew form
,, i i  7 . illation of
in all respects, are placed opposite each other, we ex the con
pect, agreeably to the principle of symmetry, that they
will produce in each other in the direction of their line
of junction equal and opposite accelerations. But if
these bodies exhibit any difference, however slight, of
form, of chemical constitution, or are in any other re
spects different, the principle of symmetry forsakes us,
unless we assume or know beforehand that sameness of
form or sameness of chemical constitution, or whatever
else the thing in question may be, is not determina
tive. If, however, mechanical experiences clearly and
indubitably point to the existence in bodies of a special
and distinct property determinative of accelerations,
2i8 THE SCIENCE OF MECHANICS.
nothing stands in the way of our arbitrarily establish
ing the following definition :
Definition All those bodies are bodies of equal mass, which, mu
masses. tually acting on each other, produce in each other equal
and opposite accelerations.
We have, in this, simply designated, or named, an
actual relation of things. In the general case we pro
ceed similarly. The bodies A and B receive respec
tively as the result of their mutual action (Fig. 140 ft)
the accelerations cp and f <>'? where the senses of
the accelerations are indicated by the signs. We say
then, B has <p/<p' times the mass of A. If we take A
as our unit, we assign to that body the mass m which im
parts to A m times the acceleration that A in the reaction
imparts to it. The ratio of the masses is the negative
inverse ratio of the counteraccelerations. That these
accelerations always have opposite signs, that . there
are therefore, by our definition, only positive masses,
Is a point that experience teaches, and experience alone
character can teach. In our concept of, mass no theory is in
nition. * " volved ; ".quantity of matter" is wholly unnecessary in
it ; all it contains is the exact establishment, designa
tion, and denomination of a fact. (Compare Appendix,
XVIII., p. 539.)
4. One difficulty should not remain unmentioned in
this connection, inasmuch as its removal is absolutely
necessary to the formation of a perfectly clear concept
of mass. We consider a set of bodies, A, B, C, D . . .,
and compare them all with A as unit,
A, B, C, D, E, F.
1, m, m', m", ;;/'", in""
We find thus the respective mass values, 1, m, m f ,
m". . . ., and so forth. The question now arises, If we
THE PRINCIPLES OF DYNAMICS. 219
select B as our standard of comparison (as our unit), Discussion
shall we obtain for C the massvalue m' /m, and for D cuity ra
the value m"/m, or will perhaps wholly different values the V preced
i t t i n formu
result? More simply, the question maybe put thus : lation.
Will two bodies J3, C, which in mutual action with A
have acted as equal masses, also act as equal masses
in mutual action with each other? No logical necessity
exists whatsoever, that two masses that are equal to a
third mass should also be equal to each other. For
we are concerned here, not with a mathematical, but
with a physical question. This will be rendered quite
clear by recourse to an analogous relation. We place
by the side of each other the bodies A, B, C in the
proportions of weight a, b, c in which they enter into
the chemical combinations AB and AC. There exists,
now, no logical necessity at all for assuming that the
same proportions of weight b, c of the bodies B, C will
also enter into the chemical combination BC. Expe
rience, however, informs us that they do. If we place
by the side of each other any set of bodies in the pro
portions of weight in which they combine with the
body A 9 they will also unite with each other in the
same proportions of weight. But no one can know
this who has not tried it. And this is precisely the case
with the massvalues of bodies.
If we were to assume that the order of combination The order
r i i i i t'li' t f combi
of the bodies, by which their massvalues are deter nation not
., i n i 11 influential
mined, exerted any influence on the massvalues, the
consequences of such an assumption would, we should
find, lead to conflict with experience. Let us suppose,
for instance (Fig. 141), that we have three elastic
bodies, A, B, C, movable on an absolutely smooth and
rigid ring.. We presuppose that A and B in their
mutual relations comport themselves like equal masses
220 THE SCIENCE OF MECHANICS.
and that B and C do the same. We are then also
obliged to assume, if we wish to avoid conflicts with
experience, that C and A in their mutual relations act
like equal masses. If we impart to A a, velocity, A
will transmit this velocity by impact to B, and B to C.
But if C were to act towards A, say, as a greater mass,
A on impact would acquire a greater
velocity than it originally had while
C would still retain a residue of
what it had. With every revolution
in the direction of the hands of a
watch the vis viva of the system
would be increased. If C were the
Flg * I4I> smaller mass as compared with A,
reversing the motion would produce the same result.
But a constant increase of vis viva of this kind is at
decided variance with our experience.
The new 5. The concept of mass when reached in the man
mass e m ner just developed renders unnecessary the special
pficitfy dbe enunciation of the principle of reaction. In the con
reaction 6 cept of mass and the principle of reaction, as we have
stated in a preceding page, the same fact is twice iorm
ulated; which is redundant. If two masses i and 2
act on each other, our very definition of mass asserts
that they impart to each other contrary accelerations
which are to each other respectively as 2:1.
6. The fact that mass can be measured by weight,
where the acceleration of gravity is invariable, can also
be deduced from our definition of mass. We are
sensible at once of any increase or diminution of a pres
sure, but this feeling affords us only a very inexact and
indefinite measure of magnitudes of pressure. An
exact, serviceable measure of pressure springs from
the observation that every pressure is replaceable by
THE PRINCIPLES OF DYNAMICS. 221
the pressure of a number of like and commensurable it also in
volves the
weights. Every pressure can be counterbalanced by fact that
f , . H. mass can be
the pressure of weights of this kind. Let two bodies measured
m and ;;/ be respectively affected in opposite directions
with the accelerations cp and cp', determined by exter
nal circumstances. And let the bodies be joined by a
string. If equilibrium prevails, the acceleration cp in
m and the acceleration cp' in ;;/' are exactly balanced
by interaction. For this case, ac
cordingly, m<p = m'cp'. . When, < 1^ L/'l >
therefore, cp = cp', as is the case 9* ^
when the bodies are abandoned
to the acceleration of gravity, we have, in the case
of equilibrium, also m = m'. It is obviously imma
terial whether we make the bodies act on each other
directly by means of a string, or by means of a string
passed over a pulley, or by placing them on the two
pans of a balance. The fact that mass can be meas
ured by weight is evident from our definition without
recourse or reference to lt quantity of matter. "
7, As soon therefore as we, our attention being The general
drawn to the fact by experience, have perceived in bod this view,
ies the existence of a special property determinative of
accelerations, our task with .egard to it ends with the
recognition and unequivocal designation of this fact.
Beyond the recognition of this fact we shall not get,
and every venture beyond it will only be productive of
obscurity. All uneasiness will 1 vanish when once we
have made clear to ourselves that in the concept of
mass no theory of any kind whatever is contained, but
simply a fact of experience. The concept has hitherto
held good. It is very improbable', but not impossible,
that it will be shaken in the future, just as the concep
222 THE SCIENCE OF MECHANICS.
tion of a constant quantity of heat, which also rested
on experience, was modified by new experiences.
VI.
NEWTON'S VIEWS OF TIME, SPACE, AND MOTION.
i. In a scholium which he appends immediately to
his definitions, Newton presents his views regarding
time and space views which we shall now proceed to
examine more in detail. We shall literally cite, to this
end, only the passages that are absolutely necessary
to the characterisation of Newton's views.
Newton's << So far, my object has been to explain the senses
views of . .,..,
time, space, m which certain words little known are to be used in
and motion.  ...
" the sequel. Time, space, place, and motion, being
"words well known to everybody, I do not define. Yet 1
"it is to be remarked, that the vulgar conceive these
" quantities only in their relation to sensible objects.
"And hence certain prejudices with respect to them
"have arisen, to remove which it will be convenient to
" distinguish them into absolute and relative, true and
"apparent, mathematical and common, respectively.
Absolute "I. Absolute, true, and mathematical time, of it
and relative . .  , . n
time, " self, and by its own nature, flows uniformly on, with
"out regard to anything external. It is also called
* ' dilation.
"Relative, apparent, and common time, is some
" sensible and external measure of absolute time (dura
"tion), estimated by the motions of bodies, whether
"accurate or inequable, and is commonly employed
"in place of true time; as an hour, a day,, a month,
"a year. . .
"The natural days, which, commonly, for the pur
"pose of the measurement of time, are held as equal.
" are in reality unequal. Astronomers correct this in
THE PRINCIPLES OF D YNAMICS. 223
"equality, in order that they may measure by a truer
"time the celestial motions. It may be that there is
"no equable motion, by which time can accurately be
"measured. All motions can be accelerated and re
" tarded. But the flow of absolute time cannot be
"changed. Duration, or the persistent existence of
"things, is always the same, whether motions be swift
"or slow or null."
2. It would appear as though Newton in the re Discussion
marks here cited still stood under the influence of the view of
time.
mediaeval philosophy, as though he had grown unfaith
ful to his resolve to investigate only actual facts. When
we say a thing A changes with the time, we mean sim
ply that the conditions that determine a thing A depend
on the conditions that determine another thing JE>. The
vibrations of a pendulum take place in time when its
excursion depends on the position of the earth. Since,
however, in the observation of the pendulum, we are
not under the necessity of taking into account its de
pendence on the position of the earth, but may com
pare it with any other thing (the conditions of which
of course also depend on the position of the earth), the
illusory notion easily arises that all the things with
which we compare it are unessential. Nay, we may,
in attending to the motion of a pendulum, neglect en
tirely other external things, and find that for every po
sition of it our thoughts and sensations are different.
Time, accordingly, appears to be some particular and
independent thing, on the progress of which the posi
tion of the pendulum depends, while the things that
we resort to for comparison and choose at random ap
pear to play a wholly collateral part. But we must
not forget that all things in the world are connected
with one another and depend on one another, and that
224 THE SCIENCE OF MECHANICS.
General we ourselves and all our thoughts are also a part of
discussion T . 1 t . , . J _,
of the con nature. It is utterly beyond our power to measure the
tune changes of things by time. Quite the contrary, time
is an abstraction, at which we arrive by means of the
changes of things ; made because we are not restricted
to any one definite measure, all being interconnected. .
A motion is termed uniform in which equal increments
of space described correspond to equal increments of
space described by some motion with which we form a
comparison, as the rotation of the earth, A motion
may, with respect to another motion, be uniform. But
the question whether a motion is in itself uniform, is
senseless. , With just as little justice, also, may we
speak of an "absolute time" of a time independent of
change. This absolute time can be measured by com
parison with no motion ; it has therefore neither a
practical nor a scientific value ; and no one is justified
in saying that he knows aught about it. It is an idle
metaphysical conception.
Further eiu It would not be difficult to show from the points of
cidation of r ,  .. . , 1 . . ,
the idea, view of psychology, history, and the science of lan
guage (by the names of the chronological divisions),
that we reach our ideas of time in and through the in
terdependence of things on one another. In these ideas
the profoundest and most universal connection of things
is expressed. When a motion takes place in time, it
depends on the motion of the earth. This is not refuted
by the fact that mechanical motions can be reversed.
A number of variable quantities may be so related that
one set can suffer a change without the others being
affected by it. Nature behaves like a machine. The
individual parts reciprocally determine one another.
But while in a machine the position of one part de
termines the position of all the other parts, in nature
THE PRINCIPLES OF DYNAMICS. 225
more complicated relations obtain. These relations are
best represented under the conception of a number,
n, of quantities that satisfy a lesser number, ri ', of equa
tions. Were n = n', nature would be invariable. Were
n' = n 1, then with one quantity all the rest would
be controlled. If this latter relation obtained in na
ture, time could be reversed the moment this had been
accomplished with any one single motion. But the
true state of things is represented by a different rela
tion between n and n'. The quantities in question are
partially determined by one another ; but they retain
a greater indeterminateness, or freedom, than in the
case last cited. We ourselves feel that we are such a
partially determined, partially undetermined element
of nature. In so far as a portion only of the changes
of nature depends on us and can be reversed by us,
does time appear to us irreversible, and the time that
is past as irrevocably gone.
We arrive at the idea of time, to express it briefly somepsy
and popularly, by the connection of that which isconsifera
, . , . r . tions.
contained in the province of our memory with that
which is contained in the province of our sensepercep
tion. When we say that time flows on in a definite di
rection or sense, we mean that physical events gene
rally (and therefore also physiological events) take
place only in a definite sense.* Differences of tem
perature, electrical differences, differences of level gen
erally, if left to themselves, all grow less and not
greater. If we contemplate two bodies of different
temperatures, put in contact and left wholly to them
selves, we shall find that it is possible only for greater
differences of temperature in the field of memory to
* Investigations concerning the physiological nature of the sensations of
time and space are here excluded from consideration.
226 THE SCIENCE OF MECHANICS.
exist with lesser ones in the field of senseperception,
and not the reverse. In all this there is simply ex
pressed a peculiar and profound connection of things.
To demand at the present time a full elucidation of this
matter, is to anticipate, in the manner of speculative
philosophy, the results of all future special investiga
tion, that is a perfect physical science. (Compare Ap
pendix, XIX., p. 541.)
Newton's 3. Views similar to those concerning time, are de
views of .
space and veloped by Newton with respect to space and motion.
motion.
We extract here a few passages which characterise his
position.
"II. Absolute space, in its own nature and with
" out regard to anything external, always remains sim
ilar and immovable.
"Relative space is some movable dimension or
"measure of absolute space, which our senses deter
"mine by its position with respect to other bodies,
" and which is commonly taken for immovable [abso
lute] space ....
" IV. Absolute motion is the translation of a body
"from one absolute place* to another absolute place ;
" and relative motion, the translation from one relative
" place to another relative place. . . .
Passages " . . . . And thus we use, in common affairs, instead
works. "of absolute places and motions, relative ones; and
"that without any inconvenience. But in physical
"disquisitions, we should abstract from the senses.
" For it may be that there is no body really at rest, to
"which the places and motions of others can be re
"ferred. . . .
" The effects by which absolute and relative motions
* The place, or locus of a body, according to Newton, is not its position,
but depart of space which it occupies. It is either absolute or relative. Trans
THE PRINCIPLES OF DYNAMICS. 227
" are distinguished from one another, are centrifugal
" forces, or those forces in circular motion which pro
' c duce a tendency of recession from the axis. For in
"a circular motion which is purely relative no such
"forces exist; but in a true and absolute circular mo
"tion they do exist, and are greater or less according
"to the quantity of the [absolute] motion.
"For instance. If a bucket, suspended by a long The rota
" cord, is so often turned about that finally the cord is
"strongly twisted, then is filled with water, and held
"at rest together with the water ; and afterwards by
" the action of a second force, it is suddenly set whirl
" ing about the contrary way, and continues, while the
"cord is untwisting itself, for some time in this mo
"tion ; the surface of the water will at first be level,
"just as it was before the vessel began to move ; but,
"subsequently, the vessel, by gradually communicat
"ing its motion to the water, will make it begin sens
" ibly to rotate, and the water will recede little by little
" from the middle and rise up at the sides of the ves
" sel, its surface assuming a concave form. (This ex
" periment I have made myself.)
" .... At first, when the relative motion of the wa Relative
T 1 . .  . and real
"ter in the vessel was greatest, that motion produced motion,
"no tendency whatever of recession from the axis ; the
"water made no endeavor to move towards the cir
"cumference, by rising at the sides of the vessel, but
"remained level, and for that reason its true circular
"motion had not yet begun. But afterwards, when
"the relative motion of the water had decreased, the
"rising of the water at the sides of the vessel indicated
* ' an endeavor to recede from the axis ; and this en
" deavor revealed the real circular motion of the water,
" continually increasing, till it had reached its greatest
228 THE SCIENCE OF MECHANICS.
11 point, when relatively the water was at rest in the
" vessel ....
li It is indeed a matter of great difficulty to discover
"and effectually to distinguish the true from the ap
" parent motions of particular bodies ; for the parts of
" that immovable space in which bodies actually move,
" do not come under the observation of our senses.
Newton's Yet the case is not altogether desperate : for there
criteria for . . . 
distinguish exist to guide us certain marks, abstracted partly
ing absolute ' , . . . , . ,....
fromreia from the apparent motions, which are the differences
tive motion. .
"of the true motions, and partly from the forces that
"are the causes and effects of the true motions. If,
"for instance, two globes, kept at a fixed distance
"from one another by means of a cord that connects
"them, be revolved about their common centre of
"gravity, one might, from the simple tension of the
"cord, discover the tendency of the globes to recede
"from the axis of their motion, and on this basis the
" quantity of their circular motion might be computed.
"And if any equal forces should be simultaneously
"impressed on alternate faces of the globes to augment
"or diminish their circular motion, we might, from
"the increase or decrease of the tension of the cord,
"deduce the increment or decrement of their motion;
"and it might also be found thence on what faces
"forces would have to 'be impressed, in order that the
"motion of the globes should be most augmented;
"that is, their rear faces, or those which, in the cir
" cular motion, follow. But as soon as we knew which
' ' faces followed, and consequently which preceded, we
"should likewise know the direction of the motion.
" In this way we might find both the quantity and the
"direction of the circular motion, considered even in
"an immense vacuum, where there was nothing ex
THE PRINCIPLES OF DYNAMICS. 229
"ternal or sensible with which the globes could be
" compared . . . ."
4. It is scarcely necessary to remark that in the re The predi
n i >.T i cations of
flections here presented Newton has again acted con Newton
, . . . . , . , are not the
trary to his expressed intention only to investigate actual expression
facts. No one is competent to predicate things about facts.
absolute space and absolute motion ; they are pure
things of thought, pure mental constructs, that cannot
be produced in experience. All our principles of me
chanics are, as we have shown in detail, experimental
knowledge concerning the relative positions and mo
tions of bodies. Even in the provinces in which they
are now recognised as valid, they could not, and were
not, admitted without previously being subjected to
experimental tests. No one is warranted in extending
these principles beyond the boundaries of experience.
In fact, such an extension is meaningless, as no one
possesses the requisite knowledge to make use of it.
Let us look at the matter in detail. When we say that Detailed
a body K alters its direction and velocity solely through matter,
the influence of another body J r , we have asserted
a conception that it Is impossible to come at unless
other bodies A, I>, C . . . . are present with reference
to which the motion of the body K has been estimated.
In reality, therefore, we are simply cognisant of a re
lation of the body K to A, B, C . . . . If now we sud
denly neglect A, B, C. . . . and attempt to speak of
the deportment of the body K in absolute space, we
implicate ourselves in a twofold error. In the first
place, we cannot know how K would act in the ab
sence of A, J3, C . . . . ; and in the second place, every
means would be wanting of forming a judgment of the
behaviour of K and of putting to the test what we had
230 THE SCIENCE OF MECHANICS.
predicated, which latter therefore would be bereft of
all scientific significance.
The part Two bodies K and K' , which gravitate toward each
which the . . ,  . ...
bodies of other, impart to each other m the direction of their
inthede line of junction accelerations inversely proportional to
termination . ... .
of motion, their masses ;;/, ;;/ . In this proposition is contained,
not only a relation of the bodies K and K' to one an
other, but also a relation of them to other bodies. For
the proposition asserts, not only that X and K' suffer
with respect to one another the acceleration designated
by K (in f ;;//r 2 ), but also that JT experiences the ac
celeration Km jr* and K' the acceleration + Km/r*
in the direction of the line of junction ; facts which can
be ascertained only by the presence of other bodies.
The motion of a body K can only be estimated by
reference to other bodies A, I>, C . . . . But since we
always have " at our disposal a sufficient number of
bodies, that are as respects each other relatively fixed,
or only slowly change their positions, we are, in such
reference, restricted to no one definite body and can
alternately leave out of account now this one and now
that one. In this way the conviction arose that these
bodies are indifferent generally.
The hy It might be, indeed, that the isolated bodies A, B.
pothesis of 7
a medium C . . . . play merely a collateral role in the determina
in space de
terminative tion of the motion of the body K, and that this motion
of motion. . J
is determined by a medium in which K exists. In such
a case we should have to substitute this medium for
Newton's absolute space. Newton certainly did not
entertain this idea. Moreover, it is easily demonstrable
that the atmosphere is not this motiondeterminative
medium. We should, therefore, have to picture to
ourselves some other medium, filling, say, all space,
with respect to the constitution of which and its kinetic
THE PRINCIPLES OF DYNAMICS, 231
relations to the bodies placed in it we have at present
no adequate knowledge. In itself such a state of things
would not belong to the impossibilities. It is known,
from recent hydrodynamical investigations, that a rigid
body experiences resistance in a frictionless fluid only
when its velocity changes. True, this result is derived
theoretically from the notion of inertia ; but it might,
conversely, also be regarded as the primitive fact from
which we have to start. Although, practically, and at
present, nothing is to be accomplished with this con
ception, we might still hope to learn more in the future
concerning this hypothetical medium ; and from the
point of view of science it would be in every respect
a more valuable acquisition than the forlorn idea of
absolute space. When we reflect that we cannot abol
ish the isolated bodies A, B, C . . . ., that is, cannot
determine by experiment whether the part they play is
fundamental or collateral, that hitherto they have been
the sole and only competent means of the orientation
of motions and of the description of mechanical facts,
it will be found expedient provisionally to regard all
motions as determined by these bodies.
5. Let us now examine the point on which New Critical
examina
ton, apparently with sound reasons, rests his distinc tion of ?
tion of absolute and relative motion. If the earth is distinction
,. .... of absolute
affected with an absolute rotation about its axis, cen from reia
trifugal forces are set up in the earth : it assumes an
oblate form, the acceleration of gravity is diminished
at the equator, the plane of Foucault's pendulum ro
tates, and so on. All these phenomena disappear if
the earth is at rest and the other heavenly bodies are
affected with absolute motion round it, such that the
same relative rotation is produced. This is, indeed, the
case, if we start ab initio from the idea of absolute space.
232 THE SCIENCE OF MECHANICS.
But if we take our stand on the basis of facts, we shall
find we have knowledge only of relative spaces and mo
tions. Relatively, not considering the unknown and
neglected medium of space, the motions of the uni
verse are the same whether we adopt the Ptolemaic or
the Copernican mode of view. Both views are, indeed,
equally correct ; only the latter is more simple and more
practical. The universe is not twice given, with an
earth at rest and an earth in motion ; but only once,
with its relative motions, alone determinable. It is,
accordingly, not permitted us to say how things would
be if the earth did not rotate. We may interpret the
one case that is given us, in different ways. If, how
ever, we so interpret it that we come into conflict with
experience, our interpretation is simply wrong. The
principles of mechanics can, indeed, be so conceived,
that even for relative rotations centrifugal forces arise,
interprcta Newton's experiment with the rotating vessel of
experiment water simply informs us, that the relative rotation of
rotating the water with respect to the sides of the vessel pro
water, duces no noticeable centrifugal forces, but that such
forces are produced by its relative rotation with respect
to the mass of the earth and the other celestial bodies.
No one is competent to say how the experiment would
turn out if the sides of the vessel increased in thickness
and mass till they were ultimately several leagues thick.
The one experiment only lies before us, and our busi
ness is, to bring it into accord with the other facts
known to us, and not with the arbitrary fictions of our
imagination.
6. We can have no doubts concerning the signifi
cance of the law of inertia if we bear in mind the man
ner in which it was reached. To begin with, Galileo
discovered the constancy of the velocity and direction
THE PRINCIPLES OP DYNAMICS, 233
of a body referred to terrestrial objects. Most terres The law oi
trial motions are of such brief duration and extent, that the light oj
it is wholly unnecessary to take into account the earth's
rotation and the changes of its progressive velocity with
respect to the celestial bodies. This consideration is
found necessary only in the case of projectiles cast
great distances, in the case of the vibrations of Fou
cault's pendulum, and in similar instances. When now
Newton sought to apply the mechanical principles dis
covered since Galileo's time to the planetary system,
he found that, so far as it is possible to form any es
timate at all thereof, the planets, irrespectively of dy
namic effects, appear to preserve their direction and
velocity with respect to bodies of the universe that are
very remote and as regards each other apparently fixed,
the same as bodies moving on the earth do with re
spect to the fixed objects of the earth. The comport
ment of terrestrial bodies with respect to the earth is
reducible to the comportment of the earth with respect
to the remote heavenly bodies. If we were to assert
that we knew more of moving objects than this their
last mentioned, experimentally given comportment
with respect to the celestial bodies, we should render
ourselves culpable of a falsity. When, accordingly, we
say, that a body preserves unchanged its direction and
velocity in space^ our assertion is nothing mor,e or less
than an abbreviated reference to the entire universe.
The use of such an abbreviated expression is permit
ted the original author of the principle, because he
knows, that as things are no difficulties stand in the
way of carrying out its implied directions. But no
remedy lies in his power, if difficulties of the kind men
tioned present themselves ; if, for example, the re
quisite, relatively fixed bodies are wanting.
234 THE SCIENCE OF MECHANICS.
The reia 7. Instead; now, of referring a moving body K to
tion of the , ,
bodies of space, that is to say to a system of coordinates, let us
verse to view directly its relation to the bodies of the universe,
each other. .
by which alone such a system of coordinates can be
determined. Bodies very remote from each other, mov
ing with constant direction and velocity with respect
to other distant fixed bodies, change their mutual dis
tances proportionately to the time. We may also say,
All very remote bodies all mutual or other forces ne
glected alter their mutual distances proportionately
to those distances. Two bodies, which, situated at a
short distance from one another, move with constant
direction and velocity with respect to other fixed bod
ies, exhibit more complicated relations. If we should
regard the two bodies as dependent on one another,
and call r the distance, t the time, and a a constant
dependent on the directions and velocities, the formula
would be obtained: dPr/di* = (1/r) [a* (dr/df)*].
It is manifestly much simpler and clearer to regard the
two bodies as independent of each other and to con
sider the constancy of their direction and velocity with
respect to other bodies.
Instead of saying, the direction and velocity of a
mass yw in space remain constant, we may also employ
the expression, the mean acceleration of the mass ju
with respect to the masses m, m', m". ... at the dis
tances r, /, r". ... is = 0, or d*(2mr/2 m)/dt* = 0.
The latter expression is equivalent to the former, as
soon as we take into consideration a sufficient number
of sufficiently distant and sufficiently large masses.
The mutual influence of more proximate small masses,
which are apparently not concerned about each other,
is eliminated of itself. That the constancy of direction
and velocity is given by the condition adduced, will be
THE PRINCIPLES OF DYNAMICS. 235
seen at once if we construct through a as vertex cones The expres
,.,. . . . . sion of the
that cut out dinerent portions of space, and set up the law of iner
condition with respect to the masses of these separate of this re? s
portions. We may put, indeed, for the entire space
encompassing /*, d* (2 mrj 2ni) jdt^ = 0. But the
equation in this case asserts nothing with respect to the
motion of yu, since it holds good for all species of mo
tion where }JL is uniformly surrounded by an infinite
number of masses. If two masses yU 1? // 2 exert on each
other a force which is dependent on their distance r,
then d^rjdt^ = (/^ + yW 2 )/(r). But, at the same time,
the acceleration of the centre of gravity of the two
masses or the mean acceleration of the masssystem
with respect to the masses of the universe (by the prin
ciple of reaction) remains == ; that is to say,
When we reflect that the timefactor that enters The neces
into the acceleration is nothing more than a quantity ence^ofT
that is the measure of the distances (or angles of rota tion of the
tion) of the bodies of the universe, we see that even in
the simplest case, in which apparently we deal with
the mutual action of only two masses, the neglecting
of the rest of the world is impossible. Nature does not
begin with elements, as we are obliged to begin with
them. It is certainly fortunate for us, that we can,
from time to time, turn aside our eyes from the over
powering unity of the All, and allow them to rest on
individual details. But we should not omit, ultimately
to complete and correct our views by a thorough con
sideration of the things which for the time being we y
left out of account.
8. The considerations just presented show, that it
THE SCIENCE OF MECHANICS.
The law of is not necessary to refer the law of inertia to a special
not involve absolute space. On the contrary, it is perceived that
spa2 e me the masses that in the common phraseology exert forces
on each other as well as those that exert none, stand
with respect to acceleration in quite similar relations.
We may, indeed, regard a// masses as related to each
other. That accelerations play a prominent part in the
relations of the masses, must be accepted as a fact of
experience ; which does not, however, exclude attempts
to elucidate this fact by a comparison of it with other
facts, involving the discovery of new points of view.
In all the processes of nature the differences of certain
I quantities u play a de
terminative role. Differ
ences of temperature, of
potential function, and so
forth, induce the natural
*p processes, which consist
in the equalisation of
The familiar expressions d*it/dx 2 ,
are determinative of the
Fig. 143.
Natural these differences.
consist in d^u/dy*. d 2 zi/dz 2 , which
theequali ' J ' p J '.
sationof character of the equalisation, maybe regarded as the
the differ
ences of
quantities.
measure of the departure of the condition of any point
from the mean of the conditions of its environment
to which mean the point tends. The accelerations of
masses may be analogously conceived. The great dis
tances between masses that stand in no especial force
relation to one another, change proportionately to each
other. If we lay off, therefore, a certain distance p as
abscissa, and another r as ordinate, we obtain a straight
line. (Fig. 143.) Every ;ordinate corresponding to
a definite pvalue represents, accordingly, the mean of
the adjacent ordinates. If a forcerelation exists be
tween the bodies, some value d*r/dt* is determined
THE PRINCIPLES OF D YNAMICS. 237
by it which conformably to the remarks above we may
replace by an expression of the form d^rjdp 1 *. By the
forcerelation, therefore, a departure of the rordinate
from the mean of the adjacent ordinates is produced,
which would not exist if the supposed forcerelation
did not obtain. This intimation will suffice here.
g. We have attempted in the foregoing to give the character
" law of inertia a different expression from that in ordi expression
nary use. This expression will, so long as a suffi oHnertia
cient number of bodies are apparently fixed in space,
accomplish the same as the ordinary one. It is as
easily applied, and it encounters the same difficulties.
In the one case we are unable to come at an absolute
space, in the other a limited number of masses only is
within the reach of our knowledge, and the summation
indicated can consequently not be fully carried out. It
is impossible to say whether the new expression would
still represent the true condition of things if the stars
were to perform rapid movements among one another.
The general experience cannot be constructed from the
particular case given us. We must, on the contrary,
wait until such an experience presents itself. Perhaps
when our physicoastronomical knowledge has been
extended, it will be offered somewhere in celestial
space, where more violent and complicated motions
take place than in our environment. The most impor The sim
tant result of our reflexions is, however, that precisely cipiesof 1
7 . . 7 T . 7 .. 7 . mechanics
the apparently simplest mechanical principles are of a very are of a
complicated character, that these principles are founded on pi:catedna
r . tureandare
uncompleted experiences, nay on experiences that never can ail derived
be fully completed, that practically, indeed, they are suf rience," pe
ficiently sectored, in view of the tolerable stability of our
environment, to serve as the foundation { of mathematical
deduction, but that they can by no means themselves be re
238 THE SCIENCE OF MECHANICS.
gardcd as mathematically established truths but only as
principles that not only admit of constant control by expe
rience but actually require it. This perception is valu
able in that it is propitious to the advancement of
science. (Compare Appendix, XX., p. 542.)
SYNOPTICAL CRITIQUE OF THE NEWTONIAN ENUNCIATIONS.
Newton's i. Now that we have discussed the details with
" sufficient particularity, we may pass again under re
view the form and the disposition of the Newtonian
enunciations. Newton premises to his work several
definitions, following which he gives the laws of mo
tion. We shall take up the former first.
Mass. " Definition I. The quantity of any matter is the
"measure of it by its density and volume conjointly.
" . . . This quantity is what I shall understand by the
" term ?nass or body in the discussions to follow. It is
" ascertainable from the weight of the bpdy in ques
tion. For I have found, by pendulum experiments
"of high precision, that the mass of a body is propor
" tional to its weight ; as will hereafter be shown.
Quantity of "Definition II. Quantity of motion is the measure
inertia,' " of it by the velocity and quantity of matter con
force, and . . ,
accelera "jointly.
< ' Definition III. The resident force \vis insita, i. e.
"the inertia] of matter is a power of resisting, by
. "which every body, so far as in it lies, perseveres in
"its state of rest or of uniform motion in a straight
"line.
"Definition IV. An impressed force is any action
"upon a body which changes, or tends to change, its
"state of rest, or of uniform motion in a straight line.
THE PRINCIPLES OP D YXAMICS. 239
"Definition V. A centripetal force is any force by
'which bodies are drawn or impelled towards, or tend
6 in any way to reach, some point as centre.
"Definition VI. The absolute quantity of a centri Forces cias
, c . f . . . ,  . . sifted as ab
' petal force is a measure of it increasing and dimin solute, ac
. , . . , , ,,,, e , , celerative,
' ishing with the efficacy of the cause that propagates and mov
' it from the centre through the space round about.
"Definition VII. The accelerative quantity of a
' centripetal force is the measure of it proportional to
1 the velocity which it generates in a given time.
"Definition VIII. The moving quantity of a cen
1 tripetal force is the measure of it proportional to the
'motion [See Def. n.] which it generates in a given
' time.
"The three quantities or measures of force thus dis The reia
'tinguished, may, for brevity's sake, be called abso forces thus
 , . , .  . . ,. distin
' lute, accelerative, and moving forces, being, for dis guished.
' tinction's sake, respectively referred to the centre of
' force, to the places of the bodies, and to the bodies
1 that tend to the centre : that is to say, I refer moving
' force to the body, as being an endeavor of the whole
'towards the centre, arising from the collective en
' deavors of the several parts ; accelerative force to the
' place of the body, as being a sort of efficacy originat
' ing in the centre and diffused throughout all the sev
' eral places round about, in moving the bodies that
' are at these places ; and absolute force to the centre,
' as invested with some cause, without which moving
' forces would not be propagated through the space
' round about ; whether this latter cause be some cen
' tral body, (such as is a loadstone in a centre of mag
' netic force, or the earth in the centre of the force of
'gravity,) or anything else not visible. This, at least,
' is the mathematical conception of forces ; for their
2 4 o THE SCIENCE OF MECHANICS.
" physical causes and seats I do not in this place con
" sider.
Thedis "Accelerating force, therefore, is to moving force,
tinction . _^
mathemat " as velocity is to quantity of motion. For quantity
icalandnot p . . , , . . , ,
physical, "of motion arises from the velocity and the quantity
"of matter; and moving force arises from the accel
" erating force and the same quantity of matter ; the
"sum of the effects of the accelerative force on the sev
" eral particles of the body being the motive force of
"the whole. Hence, near the surface of the earth,
"where the accelerative gravity or gravitating force is
"in all bodies the same, the motive force of gravity or
" the weight is as the body [mass]. But if we ascend
"to higher regions, where the accelerative force of
"gravity is less, the weight will be equally diminished,
"always remaining proportional conjointly to the mass
"and the accelerative force of gravity. Thus, in those
"regions where the accelerative force of gravity is half
" as great, the weight of a body will be diminished by
" one half. Further, I apply the terms accelerative and
"motive in one and the same sense to attractions and
"to impulses. I employ the expressions attraction, im
" pulse, or propensity of any kind towards a centre,
' i promiscuously and indifferently, the one for the other;
"considering those forces not in a physical sense, but
"mathematically. The reader, therefore, must not
"infer from any expressions of this kind that I may
"use, that I take upon me to explain the kind or the
"mode of an action, or the causes or the physical rea
"son thereof, or that I attribute forces in a true or
"physical sense, to centres (which are only mathemat
ical points), when at any time I happen to say that
"centres attract or that central forces are in action."
TffE PRINCIPLES OF DYNAMICS. 241
2. Definition i is, as has already been set forth, a Criticism o
pseudodefinition. The concept of mass is not made Definitions
clearer by describing mass as the product of the volume
into the density, as density itself denotes simply the
mass of unit of volume. The true definition of mass
can be deduced only from the dynamical relations of
bodies.
To Definition n, which simply enunciates a mode
of computation, no objection is to be made. Defini
tion in (inertia), however, is rendered superfluous by
Definitions ivvm of force, inertia being included and
given in the fact that forces are accelerative.
Definition TV defines force as the cause of the accel
eration, or tendency to acceleration, of a body. The
latter part of this is justified by the fact that in the
cases also in which accelerations cannot take place,
other attractions that answer thereto, as the compres
sion and distension etc. of bodies occur. The cause
of an acceleration towards a definite centre is defined
in Definition v as centripetal force, and is distinguished
in vi, vn, and vm as absolute, accelerative, and mo
tive. It is, we may say, a matter of taste and of form
whether we shall embody the explication of the idea
of force in one or in several definitions. In point of
principle the Newtonian definitions are open to no ob
jections.
3. The Axioms or Laws of Motion then follow, of Newton's
. .  .. _, . , Laws of
which Newton enunciates three : Motion.
" Law I. Every body perseveres in its state of rest
"or of uniform motion in a straight line, except in so
"far as it is compelled to change that state by im
" pressed forces."
< ' Law IL Change of motion [i. e. of momentum] is
proportional to the moving force impressed, and takes
242 THE SCIENCE OF MECHANICS.
" place in the direction of the straight line In which
"such force is impressed."
"Law III. Reaction is always equal and opposite
"to action; that is to say, the actions of two bodies
" upon each other are always equal and directly op
"posite."
Newton appends to these three laws a number of
Corollaries. The first and second relate to the prin
ciple of the parallelogram of forces ; the third to the
quantity of motion generated in the mutual action of
bodies ; the fourth to the fact that the motion of the
centre of gravity is not changed by the mutual action
of bodies ; the fifth and sixth to relative motion.
criticism of 4. We readily perceive that Laws i and n are con
laws?? 8 tained in the definitions of force that precede. Ac
cording to the latter, without force there is no accel
eration, consequently only rest or uniform motion in a
straight line. Furthermore, it is wholly unnecessary
tautology, after having established acceleration as the
measure of force, to say again that change of motion is
proportional to the force. It would have been enough
to say that the definitions premised were not arbitrary
mathematical ones, but correspond to properties of
bodies experimentally given. The third law apparently
contains something new. But we have seen that it is
unintelligible without the correct idea of mass, which
idea, being itself obtained only from dynamical expe
rience, renders the law unnecessary.
The coroi The first corollary really does contain something
these Jaws. new. But it regards the accelerations determined in
a body K by different bodies M, N, P as self evidently
independent of eacri other, whereas this is precisely .
what should have been explicitly recognised as a fact
of experience. Corollary Second is a simple applica
THE PRINCIPLES OF DYNAMICS. 243
tion of the law enunciated in corollary First. The re
maining corollaries, likewise, are simple deductions,
that is, mathematical consequences, from the concep
tions and laws that precede.
5. Even if we adhere absolutely to the Newtonian
points of view, and disregard the complications and in
definite features mentioned, which are not removed
but merely concealed by the abbreviated designations
"Time" and " Space," it is possible to replace New
ton's enunciations by much more simple, methodically
better arranged, and more satisfactory propositions.
Such, in our estimation, would be the following :
a. Experimental Proposition. Bodies set opposite Proposed
, . , . , , , . . substitu
each other induce in each other, under certain circum tions for
. r , . . , the New
stances to be specified by experimental physics, con tonian laws
' . , , . . r , . , . 1 . and defini
trary accelerations in the direction of their line of junc tions.
tion. (The principle of inertia is included in this.)
b. Definition. The massratio of any two bodies is
the negative inverse ratio of the mutually induced ac
celerations of those bodies.
c. Experimental Proposition. The massratios of
bodies are independent of the character of the physical
states (of the bodies) that condition the mutual accel
erations produced, be those states electrical, magnetic,
or what not ; and they remain, moreover, the same,
whether they are mediately or immediately arrived at.
d. Experimental Proposition. The accelerations
which any number of bodies A, B, C . . . . induce in a
body K, are independent of each other. (The principle
of the parallelogram of forces follows immediately from
this.)
e. Definition. Moving force is the product of the
massvalue of a body into the acceleration induced in
that body.
244 THE SCIENCE OF MECHANICS.
Extent and Then the remaining arbitrary definitions of the al
character _ . , . ,,,,,
of the pro gebraical expressions "momentum, "vis viva/ and
posed sub .
stitutions. the like, might follow. But these are by no means in
dispensable. The propositions above set forth satisfy
the requirements of simplicity , and parsimony which,
on economicoscientific grounds, must be exacted of
them. They are, moreover, obvious and clear ; for no
doubt can exist with respect to any one of them either
concerning its meaning or its source ; and we always
know whether it asserts an experience or an arbitrary
convention.
The 6. Upon the whole, we may say, that Newton dis
ments of cerned in an admirable manner the concepts and princi
from the pies that were sufficiently assured to allow of being fur
view of his ther built upon. It is possible that to some extent he
was forced by the difficulty and novelty of his subject,
in the minds of the contemporary world, to great am
plitude, and, therefore, to a certain disconnectedness
of presentation, in consequence of which one and the
same property of mechanical processes appears several
times formulated. To some extent, however, he was,
as it is possible to prove, not perfectly clear himself
concerning the import and especially concerning the
source of his principles. This cannot, however, ob
scure in the slightest his intellectual ' greatness. He
that has to acquire a new point of view naturally can
not possess it so securely from the beginning as they
that receive it unlaboriously from him. He has done
enough if he has discovered truths on which future
generations can further build. For every new infer
ence therefrom affords at once a new insight, a new
control, an extension of our prospect, and a clarifica
tion of our field of view. Like the commander of an
army, a great discoverer cannot stop to institute petty
THE PRINCIPLES OF D YNAMICS. 245
inquiries regarding the right by which he holds each The
r 11 ATVI r achieve
pOSt of vantage he has won. The magnitude of the ments of
11 ^ 111 r ,  rs Newton in
problem to be solved leaves no time for this. But at the light of
i i ,1 . .... ._ . , subsequent
a later period, the case is different. Newton might research,
well have expected of the two centuries to follow that
they should further examine and confirm the founda
tions of his work, and that, when times of greater scien
tific tranquillity should come, the principles of the sub
ject might acquire an even higher philosophical in
terest than all that is deducible from them. Then prob
lems arise like those just treated of, to the solution of
which, perhaps, a small contribution has here been
made. We join with the eminent physicists Thomson
and Tait, in our reverence and admiration of Newton.
But we can only comprehend with difficulty their opin
ion that the Newtonian doctrines still remain the best
and most philosophical foundation of the science that
can be given.
RETROSPECT OF THE DEVELOPMENT OF DYNAMICS.
i. If we pass in review the period in which the de The chief
f . . j result, the
velopment of dynamics fell, a period inaugurated by discovery
Galileo, continued by Huygens, and brought to a close fact,
by Newton, its main result will be found to be the
perception, that bodies mutually determine in each
other accelerations dependent on definite spatial and
material circumstances, and that there are masses. The
reason the perception of these facts was embodied in
so great a number of principles is wholly an historical
one ; the perception was not reached at once, but slowly
and by degrees. In reality only one great fact was es
tablished. Different pairs of bodies determine, inde
pendently of each other, and mutually, in themselves,
246 THE SCIENCE OF MECHANICS.
pairs of accelerations, whose terms exhibit a constant
ratio, the criterion and characteristic of each pair.
This fact Not even men of the calibre of Galileo, Huygens,
even the , . . . .
greatest in and Newton were able to perceive this tact at once.
could per Even they could only discover it piece by piece, as it
in frag ny 'is expressed in the law of falling bodies, in the special
law of inertia, in the principle of the parallelogram of
forces, in the concept of mass, and so forth. Today,
no difficulty any longer exists in apprehending the unity
of the whole fact. The practical demands of communi
cation alone can justify its piecemeal presentation in
several distinct principles, the number of which is really
only determined by scientific taste. What is more, a
reference to the reflections above set forth respecting
the ideas of time, inertia, and the like, will surely con
vince us that, accurately viewed, the entire fact has,
in all its aspects, not yet been perfectly apprehended.
The results The point of view reached has, as Newton expressly
reached ^ , . , , , < f
havenoth states, nothing to do with the " unknown causes 7 of
with the so natural phenomena. That which in the mechanics of
called _.,..,... . .
"causes" the present day is called force is not a something that
of phenom , . , .  , , , ,
ena. lies latent in the natural processes, but a measurable,
actual circumstance of motion, the product of the mass
into the acceleration. Also when we speak of the at
tractions or repulsions of bodies, it is not necessary to
think of any hidden causes of the motions produced.
We signalise by the term attraction merely an actually
existing resemblance between events determined by con
ditions of motion and the results of our volitional im
pulses. In both cases either actual motion occurs or,
when the motion is counteracted by some other circum
stance of motion, distortion, compression of bodies,
and so forth, are produced.
2. The work which devolved on genius here, was
THE PRINCIPLES OF DYNAMICS. 247
the noting of the connection of certain determinative The form of
elements of the mechanical processes. The precise es chaScai
tablishmenfof the form of this connection was rather a m^^naln
task for plodding research, which created the different ?cai 'origin.
concepts and principles of mechanics. We can de
termine the true value and significance of these prin
ciples and concepts only by the investigation of their
historical origin. In this it appears unmistakable at
times, that accidental circumstances have given to the
course of their development a peculiar direction, which
under other conditions might have been very different.
Of this an example shall be given.
Before Galileo assumed the familiar fact of the deForexam
pendence of the final velocity on the time, and put it to leo's laws
, r  , , of falling
the test of experiment, he essayed, as we have already bodies
,.. n ,. , ,,,...,. might have
seen, a different hypothesis, and made the final velocity taken a dif
., , , ...  __ . ... ferentform.
proportional to thesflace described. He imagined, by a
course of fallacious reasoning, likewise already referred
to, that this assumption involved a selfcontradiction.
His reasoning was, that twice any given distance of de
scent must, by virtue of the double final velocity ac
quired, necessarily be traversed in the same time as the
simple distance of descent. But since the first half is
necessarily traversed first, the remaining half will have
to be traversed instantaneously, that is in an interval
of time not measurable. Whence, it readily follows,
that the descent of bodies generally is instantaneous.
The fallacies involved in this reasoning are manifest. Galileo's
Galileo was, of course, not versed in mental Integra and iS mg
tions, and having at his command no adequate methods
for the solution of problems whose facts were in any
degree complicated, he could not but fall into mistakes
whenever such cases were presented. If we call s the
distance and / the time, the Galilean assumption reads
248 THE SCIENCE OF MECHANICS.
in the language of today dsjdt = as, from which fol
lows s = A " f , where a is a constant of experience and
A a constant of integration. This is an entirely different
conclusion from that drawn by Galileo. It does not
conform, it is true, to experience, and Galileo would
probably have taken exception to a result that, as a
condition of motion generally, made s different from
when t equalled 0. But in itself the assumption is by
no means j^contradictory.
Thesuppo Let us suppose that Kepler had put to himself the
Kepler had same question. Whereas Galileo always sought after
made Gali ,  , i , r ,1 j
leo'sre the very simplest solutions or things, and at once re
jected hypotheses that did not fit, Kepler's mode of pro
cedure was entirely different. He did not quail before
the most complicated assumptions, but worked his way,
by the constant gradual modification of his original
hypothesis, successfully to his goal, as the history of
his discovery of the laws of planetary motion fully
shows. Most likely, Kepler, on finding theassumption
dsjdt = as would not work, would have tried a num
ber of others, and among them probably the correct one
ds/dt = a Vs. But from this would have resulted an
essentially different course of development for the sci
ence of dynamics.
in such a It was only gradually and with great difficulty that
concept the concept of " work" attained its present position
might have of importance ; and in our judgment it is to the above
origina\ e mentioned trifling historical circumstance that the diffi
mcchTnics. culties and obstacles it had to encounter are to be as
cribed. As the interdependence of the velocity and the
time was, as it chanced, first ascertained, it could not
be otherwise than that the relation y = gt should appear
as the original one, the equation s =gt*/z as the next
immediate, and gs = v 2 /2 as a remoter inference. In
THE PRINCIPLES OF D YMAMICS. 249
troducing the concepts mass (;//) and force (/), where
p = mg, we obtain, by multiplying the three equations
by m, the expressions mv=ft, ms=J>t%/2 9 ps =
mv 2 /2 the fundamental equations of mechanics. Of
necessity, therefore, the concepts force and momentum
(in v) appear more primitive than the concepts work (/.$)
and vis viva (mv 2 ). It is not to be wondered at, accord
ingly, that, wherever the idea of work made its appear
ance, it was always sought to replace it by the histor
ically older concepts. The entire dispute of the Leib
nitzians and Cartesians, which was first composed in
a manner by D'Alembert, finds its complete explana
tion in this fact.
From an unbiassed point of view, we have exactly Justifica
 i . .  , . , , , tion of this
the same right to inquire after the interdependence of view.
the final velocity and the time as after the interde
pendence of the final velocity and the distance, and to
answer the question by experiment. The first inquiry
leads us to the experiential truth, that given bodies in
contraposition impart to each other in given times defi
nite increments of velocity. The second informs us,
that given bodies in contraposition impart to each other
for given mutual displacements definite increments of
velocities. Both propositions are equally justified, and
both may be regarded as equally original.
The correctness of this view has been substantiated Exempiifi :
r cation of it
in our own day by the example of J . R. Mayer. Mayer, in modem
a modern mind of the Galilean stamp, a mind wholly
free from the influences of the schools, of his own in
dependent accord actually pursued the lastnamed
method, and produced by it an extension of science
which the schools did not accomplish until later in a
much less complete and less simple form. For Mayer,
work was the original concept. That which is called
250 THE SCIENCE OF MECHANICS.
work in the mechanics of the schools, he calls force.
Mayer's error was, that he regarded his method as the
only correct one.
The results 3. We may, therefore, as it suits us, regard the time
Which flow r  i7' r i i r
from it. of descent or the distance of descent as the factor de
terminative of velocity. If we fix our attention on
the first circumstance, the concept of force appears as
the original notion, the concept of work as the derived
one. If we investigate the influence of the second fact
first, the concept of work is the original notion. In
the transference of the ideas reached in the observation
of the motion of descent to more complicated relations,
force is recognised as dependent on the distance be
tween the bodies that is, as a function of the distance,
f(f). The work done through the element of distance dr
is then/(r) dr. By the second method of investiga
tion work is also obtained as a function of the distance,
F (r) ; but in this case we know force only in the form
d. F (f)jdr that is to say, as the limiting value of the
ratio : (increment of work)/(increment of distance.)
The prefer Galileo cultivated by preference the first of these
different in two methods. Newton likewise preferred it. Huygens
qmrers. p ursuec j the second method, without at all restricting
himself to it. Descartes elaborated Galileo's ideas after
a fashion of his own. But his performances are in
significant compared with those of Newton and Huy
gens, and their influence was soon totally effaced. After
Huygens and Newton, the mingling of the two spheres
of thought, the independence and equivalence of which
are not always noticed, led to various blunders and
confusions, especially in the dispute between the Car
tesians and Leibnitzians, already referred to, concern
ing the measure of force. In recent times, however, in
quirers turn by preference now to the one and now to
THE PRINCIPLES OF DYNAMICS. 251
the other. Thus the Galileo Newtonian ideas are culti
vated with preference by the school of Poinsot, the
GalileoHuygenian by the school of Poncelet.
4. Newton operates almost exclusively with the no Theimpor
tance and
tions of iorce, mass, and momentum. His sense of the history of
value of the concept of mass places him above his prede tonian con
cessors and contemporaries. It did not occur to Galileo mass.
that mass and weight were different things. Huygens,
too, in all his considerations, puts weights for masses ;
as for example in his investigations concerning the
centre of oscillation. Even in the treatise De Percus
sione (On Impact), Huygens always says " corpus ma
jus," the larger body, and " corpus minus, " the smaller
body, when he means the larger or the smaller mass.
Physicists were not led to form the concept mass till
they made the discovery that the same body can by the
action ,of gravity receive different accelerations. The
first occasion of this discovery was the pendulumob
servations of Richer (16711673), from which Huy
gens at once drew the proper inferences,' and the
second was the extension of the dynamical laws to the
heavenly bodies. The importance of the first point may
be inferred from the fact that Newton, to prove the pro
portionality of mass and weight on the same spot of the
earth, personally instituted accurate observations on
pendulums of different materials (Prindpia. Lib. II,
Sect. VI, De Motu et Resistentia Corporum Funependu
lorum). In the case of John Bernoulli, also, the first
distinction between mass and weight (in the Meditatio
de Natura Centri Oscillationis. Opera Omnia, Lausanne
and Geneva, Vol. II, p. 168) was made on the ground
of the fact that the same body can receive different
gravitational accelerations. Newton, accordingly, dis
poses of all dynamical questions involving the relations
252 THE SCIENCE OF MECHANICS.
of several bodies to each other, by the help of the ideas
of force, mass, and momentum.
Themeth 5. Huygens pursued a different method for the so
ods of Huy . . r . , , _ 1M ,  ....
gens. lution of these problems. Galileo had previously dis
covered that a body rises by virtue of the velocity ac
quired in its descent to exactly the same height as that
from which it fell. Huygens, generalising the principle
(in his Horologium Oscillatoriuni) to the effect that the
centre of gravity of any .system of bodies will rise by
virtue of the velocities acquired in its descent to, ex
actly the same height as that from which it fell, reached
the principle of the equivalence of work and vis viva.
The names of the formulae which he obtained, were,
of course, not supplied until long afterwards.
The Huygenian principle of work was received by
the contemporary world with almost universal distrust.
People contented themselves with making use of its
brilliant consequences. It was always their endeavor.
to replace its deductions by others. Even after John
and Daniel Bernoulli had extended the principle, it
was its fruitfulness rather than its evidency that was
valued.
The meth We observe, that the GalileoNewtonian principles
odsofNew r . . , .. . ,
ton and were, on account of their greater simplicity and ap
d. parently greater evidency, invariably preferred to the
GalileoHuygenian. The employment of the latter is
exacted only by necessity in cases in which the em
ployment of the former, owing to the laborious atten
tion to details demanded, is impossible ; as in the case
of John and Daniel Bernoulli's investigations of the
motion of fluids.
If *we look at the matter closely, however, the same
simplicity and evidency will be found to belong tO' the
Huygenian principles as to the Newtonian proposi
THE PRINCIPLES OF DYNAMICS. 253
tions. That the velocity of a body is determined by
the time of descent or determined by the distance of
descent, are assumptions equally natural and equally
simple. The form of the law must in both cases be
supplied by experience. As a startingpoint, therefore,
ft = mv and/j = mv 2 /2 are equally well fitted.
6. When we pass to the investigation of the motion The neces^
of several bodies, we are again compelled, in both cases, universai
to take a second step of an equal degree of certainty, two meth
The Newtonian idea of mass is justified by the fact,
that, if relinquished, all rules of action for events would
have an end ; that we should forthwith have to expect
contradictions of our commonest and crudest" experi
ences y and that the physiognomy of our mechanical
environment would become unintelligible. The same
thing 'must be said of the Huygenian principle of work.
If we surrender the theorem 2ps = 2mv 2 /2, heavy
bodies will, by virtue of their own weights, be able to
ascend higher ; all known rules of mechanical occur
rences will have an end. The instinctive factors which
entered alike into the discovery of the one view and of
the other have been already discussed.
The two spheres of ideas could, of course, have The points
grown up much more independently of each other. But of the two
methods.
in view of the fact that the two were constantly in con
tact, it is no wonder that they have become partially
merged in each other, and that the Huygenian appears
the less complete. Newton is allsufficient with his
forces, masses, and momenta. Huygens would like
wise suffice with work, mass, and vis viva. But since
he did not in his time completely possess the idea of
mass, that idea had in subsequent applications to be
borrowed from the other sphere. Yet this also could
have been avoided. If with Newton the massratio of
254 THE SCIENCE OF MECHANICS.
two bodies can be defined as the inverse ratio of the
velocities generated by the same force, with Huygens
it would be logically and consistently definable as the
inverse ratio of the squares of the velocities generated
by the same work.
The respec The two spheres of ideas consider the mutual de
of each. pendence on each other of entirely different factors of
the same phenomenon. The Newtonian view is in so
far more complete as it gives us information regarding
the motion of each mass. But to do this it is obliged
to descend greatly into details. The Huygenian view
furnishes a rule for the whole system. It is only a con
venience, but it is then a mighty convenience, when
the relative velocities of the masses are previously and
independently known.
The gen 7. Thus we are led to see, that in the develop
eral devel ' . ...
opment of ment of dynamics, just as in the development of statics,
in the light the connection of widely different features of mechanical
ceding re phenomena engrossed at different times the attention
of inquirers. We may regard the momentum of a sys
tem as determined by the forces ; or, on the other
hand, we may regard its vis viva as determined by the
work. In the selection of the criteria in question the
individuality of the inquirers has great scope. It will
be conceived possible, from the arguments above pre
sented, that our system of mechanical ideas might,
perhaps, have been different, had Kepler instituted
the first investigations concerning the motions of fall
ing bodies, or had Galileo not committed an error in
his first speculations. We shall recognise also that not
only a knowledge of the ideas that have been accepted
and cultivated by subsequent teachers is necessary for
the historical understanding of a science, but also that
the rejected and transient thoughts of the inquirers,
TPIE PRINCIPLES OF DYNAMICS. 255
nay even apparently erroneous notions, may be very
important and very instructive. The historical investi
gation of the development of a science is most needful,
lest the principles treasured up in it become a system
of halfunderstood prescripts, or worse, a system of
prejudices. Historical investigation not only promotes
the understanding of that which now is, but also brings
new possibilities before us, by showing that which ex
ists to be in great measure conventional and accidental.
From the higher point of view at which different paths
of thought converge we may look about us with freer
powers of vision and discover routes before unknown. /
In all the dynamical propositions that we have dis The substi
l r . i AI rr^l tUtion Of
cussed, velocity plays a prominent role. The reason "integral"
. . .   , . , for "Sifter
of this, in our view, is, that, accurately considered, entiai"
, ~ laws may
every single body of the universe stands in some den some day
. . , . make the
nite relation with every other body in the universe ; concept of
that any one body, and consequently also any several fluous.
bodies, cannot be regarded as wholly isolated. Our
inability to take in all things at a glance alone compels
us to consider a few bodies and for the time being to
neglect in certain aspects the others j a step accom
plished by the introduction of velocity, and therefore
of 'time. We cannot regard it as impossible that inte
gral laws, to use an expression of C. Neumann, will
some day take the place of the laws of mathematical
elements, or differential laws, that now make up the
science of mechanics, and that we shall have direct
knowledge of the dependence on one another of the
positions of bodies. In such an event, the concept of
force will have become superfluous. (See Appendix,
XXL, p. 548, on Hertz's Mechanics; also Appendix
XXII. , p. 555, in answer to criticisms of the views ex
pressed by the author in Chapters I. and II.)
CHAPTER III.
THE EXTENDED APPLICATION OF THE PRINCIPLES
OF MECHANICS AND THE DEDUCTIVE DE
VELOPMENT OF THE SCIENCE.
SCOPE OF THE NEWTONIAN PRINCIPLES.
Newton's i. The principles of Newton suffice by themselves,
afe n um es without the introduction of any new laws, to explore
scope 1 and thoroughly every mechanical phenomenon practically
power. occurring, whether it belongs to statics or'to dynamics.
If difficulties arise in any such consideration, they are
invariably of a mathematical, or
formal, character, and in no re
spect concerned with questions
of principle. We have given,
let us suppose, a number of mas
ses m , m 2 , m^. ... in space, with
definite initial velocities 27 x , z/ 2 ,
v z . . . . We imagine, further, lines
of junction drawn between every
Fig. 144 two masses. In the directions of
these lines of junction are set up the accelerations and
counteraccelerations, the dependence of which on the
distance it is the business of physics to determine. In
a small element of time r the mass m^ for example,
will traverse in the direction of its initial velocity the
distance 7> 5 r, and in the directions of the lines joining
THE EXTENSION OP THE PRINCIPLES. 257
it with the masses m., m^, m~. . . ., being affected in Schematic
, ,. . . , 1 ' 2 ' . 3 . ' _ * r , illustration
such directions with the accelerations <p*, cp\, cp^. . . ., of thepre
the distances (^f/2)r 2 , (9>/2)r 2 , (^/2)r 2 . . . . If statement.
we imagine all these motions to be performed indepen
dently of each other, we shall obtain the new position
of the mass m 5 after lapse of time r. The composition
of the velocities v 5 and cp\r, q>\i, q>\r . . . . gives the
new initial velocity at the end of time r. We then
allow a second small interval of time r to elapse, and,
making allowance for the new spatial relations of the
masses, continue in the same way the investigation of
the motion. In like manner we may proceed with
every other mass. It will be seen, therefore, that, in
point of principle, no embarrassment can arise ; the
difficulties which occur are solely of a mathematical
character, where an exact solution in concise symbols,
and not a clear insight into the momentary workings
of the phenomenon, is demanded. If the accelerations
of the mass m 5 , or of several masses, collectively neu
tralise each other, the mass m^ or the other masses
mentioned are in equilibrium and will move uniformly
onwards with their initial velocities. If, in addition,
the initial velocities in question are 0, both equilib
rium and rest subsist for these masses.
Nor, where a number of the masses m,, m . . . . The same
... idea ap
have considerable extension, so that it is impossible to plied to ag
gregEtes of
speak of a single line joining every two masses, is the dif material
ficulty, in poin.t of principle, any greater. We divide
the masses into portions sufficiently small for our pur
pose, and draw the lines of junction mentioned between
every two such portions. We, furthermore, take into
account the reciprocal relation of the parts of the
same large mass ; which relation, in the case of rigid
masses for instance, consists in the parts resisting
258 THE SCIENCE ' OF MECHANICS.
every alteration of their distances from one another.
On the alteration of the distance between any two parts
of such a mass an acceleration is observed proportional
to that alteration. Increased distances diminish, and
diminished distances increase in consequence of this
acceleration. By the displacement of the parts with
respect to one another, the familiar forces of elasticity
are aroused. When masses meet in impact, their
forces of elasticity do not come into play until contact
and an incipient alteration of form take place.
A practical 2. If we imagine a heavy perpendicular column
otthe r scope resting on the earth, any particle m in the interior of
principles. 3 the column which we may choose to isolate in thought,
is in equilibrium and at rest. A vertical downward ac
celeration g is produced by the earth in the particle,
which acceleration the particle obeys. But in so doing
it approaches nearer to the particles lying beneath it,
and the elastic forces thus awakened generate in m a
vertical acceleration upwards, which ultimately, when
the particle has approached near enough, becomes
equal to g. The particles lying above m likewise
approach m with the acceleration g. Here, again,
acceleration and counteracceleration are produced,
whereby the particles situated above are brought to
rest, but whereby m continues to be forced nearer and
nearer to the particles beneath it until the acceleration
downwards, which it receives from the particles above
it, increased by g, is equal to the acceleration it re
ceives in the upward direction from the particles be
neath it. We may apply the same reasoning to every"
portion of the column and the earth beneath it, readily
perceiving that the lower portions lie nearer each other
and are more violently pressed together than the parts
above. Every portion lies between a less closely pressed
THE EXTENSION OF THE PRINCIPLES, 259
upper portion and a more closely pressed lower por Rest in the
 . , , . ,. , ; light of
tion ; its downward acceleration g is neutralised by atheseprin
surplus of acceleration upwards, which it experiences pears S as P a
r i it TTT T 11 ii special case
from the parts beneath. We comprehend the equilib of motion.
rium and rest of the parts of the column by imagining
all the accelerated motions which the reciprocal rela
tion of the earth and the parts of the column determine,
as in fact simultaneously performed. The apparent
mathematical sterility of this conception vanishes, and
it assumes at once an animate form, when we reflect
that in reality no body is completely at rest, but that
in all, slight tremors and disturbances are constantly
taking place which now give to the accelerations of de
scent and now to the accelerations of elasticity a slight
preponderance. Rest, therefore, is a case of motion,
very infrequent, and, indeed, never completely realised.
The tremors mentioned are by no means an unfamiliar
phenomenon. When, however, we occupy ourselves
with cases of equilibrium, we are concerned simply with
a schematic reproduction in thought of the mechanical
facts. We then purposely neglect these disturbances,
displacements, bendings, and tremors, as here they
have no interest for us. All cases of this class, which
have a scientific or practical importance, fall within the
province of the socalled theory of elasticity. The whole The unity
^ . and homo
OUtCOme of Newton's achievements is that we every geneity
which these
where reach our goal with one and the same idea, and principles
i introduce
by means of it are able to reproduce and construct be into the
forehand all cases of equilibrium and motion. All
phenomena of a mechanical kind now appear to us
as uniform throughout and as made up of the same
elements.
3. Let us consider another example. Two mas
ses m, m are situated at a distance a from each
260 THE SCIENCE OF MECHANICS.
A general other. (Fig. 145.) When displaced with respect to
cation of each other, elastic forces proportional to the change
the power . . , ,
oftheprin x 2 of distance are supposed to be
ciples. '  "^  \ i i r i i
 B B _ awakened. Let the masses be
x* movable in the Jfdirection par
Fi & ^s allel to a, and their coordinates
be x 13 x 2 . If a force /is applied at the point x 2 , the
following equations obtain :
where p stands for the force that one mass exerts on
the other when their mutual distance is altered by the
value i. All the quantitative properties of the me
chanical process are determined by these equations.
But we obtain these properties in a more comprehensi
ble form by the integration of the equations. The ordi
nary procedure is, to find by the repeated differentia
tion of the equations before us new equations in suffi
cient number to obtain by elimination equations in x^
alone or x 2 alone, which are afterwards integrated. We
shall here pursue a different method. By subtracting
the first equation from the second, we get
/, or
at  
the equa
rth putting *,*!=:*,
this exam T O
pie. d*U O ,/,[%/ /Y! I P /Ov
m ~tfJi ~'*fL u dU TV ('">)
and by the addition of the first and the second equa
tions
.
^ =/ or, putting * 2 + ,r x = ?.,
rHE EXTENSION OF THE PRINCIPLES. 261
The integrals of (3) and (4) are respectively The integ
__ _ _ rals of these
l2rt f develop
 . / + B cos J^ ./+,? + f and menta 
v = .  + C/ + D\ whence
To take a particular case, we will assume that the A particu
action of the force /"begins at /== 0, and that at this the exam
time
that is, the initial positions are given and the initial
velocities are = 0. The constants A, B, C, D being
eliminated by these conditions, we get
/
(5) J , l= ^
(6) . 3 ^_. 2 >
(7) J r s _ Jfl =_
262 THE SCIENCE OF MECHANICS.
The inform We see from (<\ and (6) that the two masses, in addi
ation which . . r wy , v y , _ . . . . 1P .
the result tion to a uniformly accelerated motion with half the
dons give acceleration that the force f would impart to one of
this exam these masses alone, execute an oscillatory motion sym
metrical with respect to their centre of gravity. The
duration of this oscillatory motion, T=2 ft'i/m/ip, is
smaller in proportion as the force that is awakened in
the same massdisplacement is greater (if our attention
is directed to two particles of the same body, in pro
portion as the body is harder). The amplitude of os
cillation of the oscillatory motion //2/ likewise de
creases with the magnitude p of the force of displace
ment generated. Equation (7) exhibits the periodic
change of distance of the two masses during their pro
gressive motion. The motion of an elastic body might
in such case be characterised as vermicular. With hard
bodies, however, the number of the oscillations is so
great and their excursions so small that they remain
unnoticed, and may be left out of account. The oscil
latory motion, furthermore, vanishes, either gradually
through the effect of some resistance, or when the two
masses, at the moment the force /begins to act, are a
distance a f//2/ apart and have equal initial veloci
ties. The distance a + //2/ that the masses are apart
after the vanishing of their vibratory motion, is//2/
greater than the distance of equilibrium a. A tension
j, namely, is set up by the action of/, by which the
acceleration of the foremost mass is reduced to one
half whilst that of the mass following is increased by
the same amount. In this, then, agreeably to our as
This in sumption, py/m //2 m or y =flip. As we see, it is
formation .  .. _ ... ,
is exhaus in our power to determine the minutest details of a
phenomenon of this character by the Newtonian prin
ciples. The investigation becomes (mathematically,
THE EXTENSION OF THE PRINCIPLES. 263
yet not in point of principle) more complicated when
we conceive a body divided up into a great number of
small parts that cohere by elasticity. Here also in the
case of sufficient hardness the vibrations may be neg
lected. Bodies in which we purposely regard the mu
tual displacement of the parts as evanescent, are called
rigid bodies.
4. We will now consider a case that exhibits the The deduc
7 r 7 T*T   i *JT tionof the
schema of a lever. We imagine the masses M, m,, m laws of the
. . lever by
arranged in a triangle and joined by elastic connec Newton's
tions. Every alteration of the sides, and consequently prmcip
also every alteration of the angles, gives rise to accel
erations, as the result of which the triangle endeavors to
assume its previous form and size. By the aid of the
Newtonian principles we can deduce from such a
schema the laws of the lever, and at the same time feel
that the form of the deduction, although it may be
more complicated, still
remains admissible when
we pass from a schematic
lever composed of three
masses to the case of a
real lever. The mass M
we assume either to be in itself very large or conceive
it joined by powerful elastic forces to other very large
masses (the earth for instance). M then represents
an immovable fulcrum.
Let m^j now, receive from the action of some ex The
ternal force an acceleration /perpendicular to the line deduction
of junction Mm 2 = c [ d. Immediately a stretching
of the lines m l m^ = b and m^M=a is produced, and
in the directions in question there are respectively set
up the accelerations, as yet undetermined, s and ff, of
which the components s(e/fy and a(e/a) are directed
j<5 4 THE SCIENCE OF MECII IN1CS.
oppositely to the acceleration/ Here e is the altitude
of the triangle ?n^m 2 M. The mass m 2 receives the
acceleration s', which resolves itself into the two com
ponents s'(d/&) in the direction of M and s(ejb) par
allel to f. The former of these determines a slight ap
proach of ;// 2 to M. The accelerations produced in M
by the reactions of m 1 and m 2 , owing to its great mass,
are imperceptible. We purposely neglect, therefore,
the motion of M.
The deduc The mass m 19 accordingly, receives the accelera
iSe'dby tion / s(ejb) ff(e/a) 9 whilst the 'mass m 2 suffers
ratfono'f ~ the parallel acceleration s'(e/&). Between s and a a
ions. era simple relation obtains. If, by supposition, we have a
very rigid connection, the triangle is only impercept
ibly distorted. The components of s and a perpendicular
to / destroy each other. For if this were at any one
moment not the case, the greater component would
produce a further distortion, which would immediately
counteract its excess. The resultant of s and is
therefore directly contrary to/ and consequently, as is
readily obvious, & (c/a) = j (<//). Between s and /,
further, subsists the familiar relation m i s~m 2 s f or
s = s'(m 2 lm^). Altogether m 2 and /;/ 1 receive re
spectively the accelerations s'(c/&) and / ^'(V^)
C^ 2 A z i)(^"T" dJ), or, introducing in the place of the
variable value s'(e/U) the designation q>, the accelera
tions <p and/ <p(p l %l m \) (c + die).
On die pre At the commencement of the distortion, the accel
ceding sup . . . ......
positions eration of m J3 owing to the increase of <z?, diminishes,
the laws for ... . . Tr ,
the rotation whilst that of m* increases. If we make the altitude e
of the lever .... .. . ...
are easily of the triangle very small, our reasoning still remains
applicable. In this case, however, a becomes = c = r^
and a\b = c\d=r 2 . We see, moreover, that the
distortion must continue, <p increase, and the accelera
THE EXTENSION OF THE PRINCIPLES. 265
tion of m^ diminish until the stage is reached at which
the accelerations of m^ and m 2 bear to each other the
proportion of r : to r 2 . This is equivalent to a rotation
of the whole triangle (without further distortion) about
M, which mass by reason of the vanishing accelera
tions is at rest. As soon as rotation sets in, the rea
son for further alterations of <p ceases. In such a case ;
consequently,
cp = ? J / cp 2. 2 J. or (p = r 2 ^p 1  "o .
; "i I m i r i> * m i r i" ~r M 2 r $~
For the angular acceleration ^ of the lever we get
Nothing prevents us from entering still more into Discussion
^ r fe of the char
the details of this case and determining the distortions acter of the
preceding
and vibrations of the parts with respect to each other, result.
With sufficiently rigid connections, however, these de
tails may be neglected. It will be perceived that we
have arrived, by the employment of the Newtonian prin
ciples, at the same result to which the Huygenian view
also would have led us. This will not appear strange to
us if we bear in mind that the two views are in every re
spect equivalent, and merely start from different aspects
of the same subjectmatter. If we had pursued the
Huygenian method, we should have arrived more
speedily at our goal but with less insight into the de
tails of the phenomenon. We should have employed
the work done in some displacement of m^ to deter
mine the vires viva of m l and m 2 , wherein we should
have assumed that the velocities in question # 1? z' 2
maintained the ratio z^/z^ r iAV ^^ e exampl 6
here treated is very well adapted to illustrate what
such an equation of condition means. The equation
266
THE SCIENCE OF MECHANICS.
simply asserts, that on the slightest deviations of v^/v^
from r : /r 2 powerful forces are set in action which in
point of fact prevent all further deviation. The bodies
obey of course, not the equations, but the forces.
A simple 5. We obtain a very obvious case if we put in the
case of the . ,  ..,.
same exam example just treated ^ 1 =w 2 =;w and # #(Fig.
P e ' I 47) The dynamical state of the system ceases to
change when cp = 2 (/ 2 <
Fig. 147.
that is, when the accel
erations of the masses
at the base and the ver
tex are given by 2//5
and //5* At the com
mencement of the dis
The equi
librium of
the lever
deduced
from the
same con
siderations.
tortion (p increases, and simultaneously the accelera
tion of the mass at the vertex is decreased by double
that amount, until the proportion subsists between the
two of 2 : i.
We have yet to consider the case of equilibrium of
a schematic lever, consisting (Fig. 148) of three masses
;;z 13 ;;/ 2 , and M 9 of which the last is again supposed
to be very large or to be elastically connected with
very large masses. We imagine two equal and oppo
site forces s, s applied to m and m^ in the direction
m^m^, or, what is the same thing, accelerations im
pressed inversely proportional to the masses m lf m 2 .
The stretching of the connection m^ m 2 also generates
THE EXTENSION OF THE PRINCIPLES. 267
accelerations inversely proportional to the masses ;;; 1?
m 2 , which neutralise the first ones and produce equi
librium. Similarly, along m 1 M imagine the equal and
contrary forces /, / operative ; and along m 2 M the
forces u, u. In this case also equilibrium obtains.
If M be elastically connected with masses sufficiently
large, u and t need not be applied, inasmuch
as the lastnamed forces are spontaneously evoked the
moment the distortion begins, and always balance the
forces opposed to them. Equilibrium subsists, accord
ingly, for the two equal and opposite forces s, j as
well as for the wholly arbitrary forces t, u. As a matter
of fact s, s destroy each other and t, u pass through
the fixed mass M, that is, are destroyed on distortion
setting in.
The condition of equilibrium readily reduces itself The reduc
tion of the
to the common form when we reflect that the mo preceding
. i T, f , case to the
ments of t and u, forces passing through M, are with common
respect to M zero, while the moments of s and s are
equal and opposite. If we compound / and s to p, and
u and s to q> then, by Varignon's geometrical principle
of the parallelogram, the moment of/ is equal to the
sum of the moments of s and /, and the moment of q
is equal to the sum of the moments of u and j. The
moments of/ and q are therefore equal and opposite.
Consequently, any two forces / and q will be in equi
librium if they produce in the direction m^ m 2 equal
and opposite components, by which condition the equal
ity of the moments with respect to M is posited. That
then the resultant of / and q also passes through M, is
likewise obvious, for s and s destroy each other and
t and u pass through M.
6. The Newtonian point of view, as the example
just developed shows us, includes that of Varignon.
268 THE SCIENCE OF
Newton's We were right, therefore, when we characterised the
point of .._... . i i
view in statics of Vangnon as a dynamical statics, which, start
varignon's. ing from the fundamental ideas of modern dynamics,
voluntarily restricts itself to the investigation of cases
of equilibrium. Only in the statics of Varignon, owing
to its abstract form, the significance of many opera
tions, as for example that of the translation of the
forces in their own directions, is not so distinctly ex
hibited as in the instance just treated.
The econ The considerations here developed will convince,
omy and ii> T 1
wealth of us that we can dispose by the Newtonian principles
theNewton . , . , . 1 , .  , . ,
ian ideas, of every phenomenon of a mechanical kind which may
arise, provided we only take the pains to enter far
enough into details. We literally see through the cases
of equilibrium and motion which here occur, and be
hold the masses actually impressed with the accelera
tions they determine in one another. It is the same
grand fact, which we recognise in the most various
phenomena, or at least can recognise there if we make
a point of so doing. Thus a unity, homogeneity, and
economy of thought were produced, and a new and
wide domain of physical conception opened which
before Newton's time was unattainable..
The New Mechanics, however, is not altogether an end in it
toman and ....
the modern, self it has also problems to solve that touch the needs
methods, of practical life and affect the furtherance of other sci
ences. Those problems are now for the most part ad
vantageously solved by other methods than the New
tonian, methods whose equivalence to that has already
been demonstrated. It would, therefore, be mere im
practical pedantry to contemn all other advantages and
insist upon always going back to the elementary New
tonian ideas. It is sufficient to have once convinced
ourselves that this is always possible. Yet the New
TUE EXTENSION OF THE PRINCIPLES. 269
tonian conceptions are certainly the most satisfactory
and the most lucid ; 'and Poinsot shows a noble sense
of scientific clearness and simplicity in making these
conceptions the sole foundation of the science.
rr.
THE FORMULAE AND UNITS OF MECHANICS.
1. All the important formulae of modern mechanics History of
were discovered and employed in the period of Galileo i and
* . units of
 and Newton. The particular designations, which, mechanics.
owing to the frequency of their use, it was found con
venient to give them, were for the most part not fixed
upon until long afterwards. The systematical mechan
ical units were not introduced until later still. Indeed,
the last named improvement, cannot be regarded as
having yet reached its completion.
2. Let s denote the distance, / the time, v the in The orig
stantaneous velocity, and <p the acceleration of a uni tionso?*
formly accelerated motion. From the researches ofnuygens"
Galileo and Huygens, we derive the following equa
tions :
v = cpt
Multiplying throughout by the mass m, these equa The intro
duction
tions give the following : of "mas
and k 'mov
m v = m (p t in S force."
mv*
m<ps=^ r .
270 THE SCIENCE OF MECHANICS
Final form and, denoting the moving force m cp by the letter p, we
of the fun
damental obtain
equations.
mv=.pt
pt*
ms= <=
Equations (i) ail contain the quantity cp ; and each
contains in addition two of the quantities s } t, v, as
exhibited in the following table :
f*,/
q> \ s, t
( s, v
Equations (2) contain the quantities m, p, s, ?, v ;
each containing ?;/, p and in addition to m, p two of the
three quantities s, t, v, according to the following table :
s, v
The scope Questions concerning motions due to constant forces
and appli .,, ,. >. N . . . , T rr
cation of are answered by equations (2) m great variety. If, for
ti e ns. equa example, we want to know the velocity v that a mass
m acquires in the time / through the action of a force
/, the first equation gives v =ft/m. If, on the other
hand, the time be sought during which a mass m with
the velocity v can move in opposition to a force p, the
same equation gives us t = m v/p. Again, if we in
quire after the distance through which m will move with
velocity v in opposition to the force p, the third equa
tion gives s = mv 2 /2p. The two last questions illus
trate, also, the futility of the DescartesLeibnitzian dis
pute concerning the measure of force of a body in mo
tion. The use of these equations greatly contributes
THE EXTENSION OF THE PRINCIPLES. 271
to confidence in dealing with mechanical ideas. Sup
pose, for instance, we put to ourselves, the question,
what force f will impart to a given mass m the velocity
v ; we readily see that between ;;/, p, and v alone, no
equation exists, so that either s or / must be supplied,
and consequently the question is an indeterminate one.
We soon learn to recognise and avoid indeterminate
cases of this kind. The distance that a mass ;;/ acted
on by the force / describes in the time /, if moving
with the initial velocity 0, is found by the second equa
tion s =pt* Jim.
3. Several of the formulas in the abovediscussed The names
equations have received particular names. The force formula? of
of a moving body was spoken of by Galileo, who al tions have
ternately calls it "momentum," "impulse," and "en
ergy. " He regards this momentum as proportional to
the product of the mass (or rather the weight, for Gali
leo had no clear idea of mass, and for that matter no
more had Descartes, nor even Leibnitz) into the velo
city of the body. Descartes accepted this view. He put
the force of a moving body = my, called it quantity of
motion, and maintained that the sumtotal of the quan
tity of motion in the universe remained constant, so that
when one body lost momentum the loss was compen
sated for by an increase of momentum in other bodies.
Newton also employed the designation "quantity of
motion " for m v, and this name has been retained to the Momen
i ti tum an< ^
present day. [But momentum is the more usual term.] impulse.
For the second member of the first equation, viz. ft,
Belanger, proposed, as late as 1847, the name impulse.*
The expressions of the second equation have received
* See, also, Maxwell, Matter and Motion^ American edition, page 72. But
this word is commonly used In a different sense, namely, as " the limit of a
force which is infinitely great but acts only during an Infinitely short time."
See Routh, Rigid Dynamics, Part I, pages 6566. Trans.
272 THE SCIENCE OF MECHANICS.
f is awa no particular designations. Leibnitz (1695) called the
and work. . . , ,  i . . _ . .
expression mv* of the third equation vis viva or living
force, and he regarded it, in opposition to Descartes,
as the true measure of the force of a body in motion,
calling the pressure of a body at rest vis mortua, or
dead force. Coriolis found it more appropriate to give
the term \mv* the name vis viva. To avoid confusion,
Belanger proposed to call mv 2 living force and fynv*
living power [now commonly called in English kinetic
energy]. For ps Coriolis employed the name work.
Poncelet confirmed this usage, and adopted the kilo
grammemetre (that is, a force equal to the weight of a
kilogramme acting through the distance of a metre) as
the unit of 'work.
The history A. Concerning the historical details of the origin of
of the ideas ^ ... .
quantity of these notions "quantity of motion" and "vis viva."
motion and
vis viva, a glance may now be cast at the ideas which led Des
cartes and Leibnitz to their opinions. In his Principia
Philosophic, published in 1644, II, 36, DESCARTES ex
pressed himself as follows :
" Now that the nature of motion has been examined,
"we must consider its cause, which maybe conceived
"in two senses : first, as a universal, original cause
" the general cause of all the motion in the world ; and
"second, as a special cause, from which the individual
" parts of matter receive motion which before they did
"not have. As to the universal cause, it can rnani
"festly be none other than God, who in the beginning
" created matter with its motion and rest, and who now
" preserves, by his simple ordinary concurrence, on the
"whole, the same amount of motion and rest as he
"originally created. For though motion is only a con
"dition of moving matter, there yet exists in matter
"a definite quantity of it, which in the world at large
THE EXTENSION OF THE PRINCIPLES. 273
" never increases or diminishes, although in single por Passage
"tions it changes; namely, in this way, that we must cartes' s ts "
" assume, in the case of the motion of apiece of matter rinc * * a
"which is moving twice as fast as another piece, but in
" quantity is only one half of it, that there is the same
" amount of motion in both, and that in the proportion
" as the motion of one part grows less, in the same pro
" portion must the motion of another, equally large
"part grow greater. We recognise it, moreover, as
"a perfection of God, that He is not only in Himself
"unchangeable, but that also his modes of operation
" are most rigorous and constant ; so that, with the ex
"ception of the changes which indubitable experience
" or divine revelation offer, and which happen, as our
"faith or judgment show, without any change in the
"Creator, we are not permitted to assume any others
' { in his works lest inconstancy be in any way pre
dicated of Him. Therefore, it is wholly rational to
"assume that God, since in the creation of matter he
"imparted different motions to its parts, and preserves
"all matter in the same way and conditions in which
"he created it, so he similar \y preserves in it the same
"quantity of motion." (See Appendix, XXIII. , p. 574.)
The merit of having first sought after a more uni ^^^
versal and more fruitful point of view in mechanics, of Descar
Jr f tes's phys
cannot be denied Descartes. This is the peculiar task ! cal inqmr
of the philosopher, and it is an activity which con
stantly exerts a fruitful and stimulating influence on
physical science.
Descartes, however, was infected with all the usual
errors of the philosopher. He places absolute confi
dence in his own ideas. He never troubles himself to
put them to experiential test. On the contrary, a min
imum of experience always suffices him for a maximum
274 THE SCIENCE OF MECHANICS.
of inference. Added to this, is the indistinctness of
his conceptions. Descartes did not possess a clear
idea of mass. It is hardly allowable to say that Des
cartes defined mv as momentum, although Descartes's
scientific successors, feeling the need of more definite
notions., adopted this conception. Descartes's greatest
error, however, and the one that vitiates all his phys
ical inquiries, is this, that many propositions appear
to him selfevident a priori concerning the truth of
which experience alone can decide. Thus, in the two
paragraphs following that cited above (3739) it is
asserted as a selfevident proposition that a body pre
serves unchanged its velocity and direction. The ex
periences cited in 38 should have been employed, not
as a confirmation of an a priori law of inertia, but as a
foundation on which this law in an empirical sense
should be based.
Leibnitz Descartes's view was attacked by LEIBNITZ (1686)
on quantity . ...
of motion, m the Ada Eruditorum, in a little treatise bearing the
title : " A short Demonstration of a Remarkable Error
of Descartes and Others, Concerning the Natural Law
by which they think that the Creator always preserves
the same Quantity of Motion ; by which, however, the
Science of Mechanics is totally perverted."
In machines in equilibrium, Leibnitz remarks, the
loads are inversely proportional to the velocities of dis
placement ; and in this way the idea arose that the
product of a body (" corpus," " moles ") into its velocity
is the measure of force. This product Descartes re
garded as a constant quantity. Leibnitz's opinion,
however, is, that this measure of force is only acci
dentally the correct measure, in the case of the ma
chines. The true measure of force is different, and
must be determined by the method which Galileo and
THE EXTENSION OF THE PRINCIPLES. 275
Huygens pursued. Every body rises by virtue of the Leibnitz on
1 , . . ,  .  .. the meas
velocity acquired in its descent to a height exactly urc of force
equal to that from which it fell. If, therefore, we as
sume, that the same " force 5 ' is requisite to raise a
body m a height 4^ as to raise a body 4/0 a height //,
we must, since we know that in the first case the ve
locity acquired in descent is but twice as great as in
the second, regard the product of a "body" into the
s quart of its velocity as the measure of force.
In a subsequent treatise (1695), Leibnitz reverts to
this subject. He here makes a distinction between
simple pressure (vis mortud) and the force of a moving
body (pis viva}, which latter is made up of the sum of
the pressureimpulses. These impulses produce, in
deed, an " impetus " (mz^ 3 but the impetus produced
is not the true measure of force ; this, since the cause
must be equivalent to the effect, is (in conformity with
the preceding considerations) determined by mv 2 .
Leibnitz remarks further that the possibility of per
petual motion is excluded only by the acceptance of his
measure of force.
Leibnitz, no more than Descartes, possessed a gen The idea of
mass in
uine concept of mass. Where the necessity of such Leibnitz's
an idea occurs, he speaks of a body (corpus}^ of a load
(nwles} 9 of differentsized bodies of the same specific
gravity, and so forth. Only in the second treatise, and
there only once, does the expression "massa " occur,
in all probability borrowed from Newton. Still, to de
rive any definite results from Leibnitz's theory, we must
associate with his expressions the notion of mass, as
his successors actually did. As to the rest, Leibnitz's
procedure is much more in accordance with the meth
ods of science than Descartes's. Two things, however,
are confounded : the question of the measure of force
276 THE SCIENCE OF MECHANICS.
in a sense, and the question of the constancy of the sums 2m v and
and Leib ~2> tn v~ . The two have in reality nothing to do with
each right, each other. With regard to the first question, we now
know that both the Cartesian and the Leibnitzian meas
ure of force, or, rather, the measure of the effective
ness of a body in motion, have, each in a different
sense, their justification. Neither measure, however,
as Leibnitz himself correctly remarked, is to be con
founded with the common, Newtonian, measure of
force.
Tbedis With regard to the second question, the later in
suifof^mfs 6 " vestigations of Newton really proved that tor free ma
standings. terial systems not acted on by external forces the Car
tesian sum 2mv is a constant ; and the investigations
of Huygens showed that also the sum Smv 2 is a con
stant, provided work performed by forces does not alter
it. The dispute raised by Leibnitz rested, therefore,
on various misunderstandings. It lasted fiftyseven
years, till the appearance of D'Alembert's Traite de
dynamiqiiC) in 1743. To the theological ideas of Des
cartes and Leibnitz, we shall revert in another place.
Theappii 5. The three equations above discussed, though
the l funda they are only applicable to rectilinear motions produced
equations by constant forces, may yet be considered thefunda
forces? e mental equations of mechanics. If the motion be recti
linear but the force variable, these equations pass by a
slight, almost selfevident, modification into others,
which we shall here only briefly indicate, since mathe
matical developments in the present treatise are wholly
subsidiary.
From the first equation we get for variable forces
m v = I p dt \ C } where / is the variable force, dt the
timeelement of the action, \pdt the sum of all the
THE EXTENSION' OF THE PRINCIPLES. 277
products p . dt from the beginning to the end of the
action, and C a constant quantity denoting the value
of m v before the force begins to act. The second equa
tion passes in like manner into the forms s= I dt I df
\Ct\D> with two socalled constants of integration,
The third equation must be replaced by
Curvilinear motion may always be conceived as the
product of the simultaneous combination of three rec
tilinear motions, best taken in three mutually perpen
dicular directions. Also for the components of the mo
tion of this very general case, the abovegiven equa
tions retain their significance.
6. The mathematical processes of addition, sub The units of
. . ... ... . 1 mechanics.
traction, and equating possess intelligible meaning only
when applied to quantities of the same kind. We can
not add or equate masses and times, or masses and
velocities, but only masses and masses, and so on.
When, therefore, we have a mechanical equation, the
question immediately presents itself whether the mem
bers of the equation are quantities of the same kind.,
that is, whether they can be measured by the same unit,
or whether, as we usually say, the equation is homo
geneous. The units of the quantities of mechanics will
form, therefore, the next subject of our investigations.
The choice of units, which are, as we know, quan
tities of the same kind as those they serve to measure^
is in many cases arbitrary. Thus, an arbitrary mass is
employed as the unit of mass, an arbitrary length is
employed as the unit of length, an arbitrary time as the
unit of time. The mass and the length employed as
units can be preserved ; the time can be reproduced
278 THK SCIENCE OF MECHANICS.
Aibitrary by pendulumexperiments and astronomical observa
units, and . _. . ,. f . r
derived or tions. But units like a unit of velocity, or a unit of
unitL" acceleration, cannot be preserved, and are much more
difficult to reproduce. These quantities are conse
quently so connected with the arbitrary fundamental
units, mass, length, and time, that they can be easily
and at once derived from them. Units of this class
are called derived or absolute units. This latter desig
nation is due to GAUSS, who first derived the magnetic
units from the mechanical, and thus created the possi
bility of a universal comparison of magnetic measure
ments. The name, therefore, is of historical origin.
Thede As unit of velocity we might choose the velocity
rived units ....... ... n ni
of velocity, with which, say, q units of length are travelled over in
tion, and unit of time. But if we did this, we could not express
the relation between the time /, the distance s, and the
velocity v by the usual simple formula s = vt, but
should have to substitute for it s = q.vt. If, however,
we define the unit of velocity as* the velocity with
which the unit of length is travelled over in unit of
time, we may retain the form $ = vt. Among the de
rived units the simplest possible relations are made
to obtain. Thus, as the unit of area and the unit of vol
ume, the square and cube of the unit of length are al
ways employed.
According to this, we assume then, that by unit ve
locity unit length is described in unit time, that by unit
acceleration unit velocity is gained in unit time, that
by unit force unit acceleration is imparted to unit mass,
and so on.
The derived units depend on the arbitrary funda
mental units ; they are functions of them. The func
tion which corresponds to a given derived unit is called
its dimensions. The theory of dimensions was laid down
THE EXTENSION OF THE PRINCIPLES. 279
by FOURIER, in 1822, in his Theory of Heat. Thus, if /The theory
. ofdiraen
denote a length, / a time, and m a mass, the dimen
sions of a velocity, for instance, are I It or // 1 . After
this explanation, the following table will be readily un
derstood :
NAMES SYMBOLS DIMENSIONS
Velocity ........... v 7/" 1
Acceleration ......... cp lt~*
Force ............ / mlt~*
Momentum ......... mv
Impulse ........... pt
Work ............ ps
Vis viva
Moment of inertia ...... & ml 2
Statical moment ....... D ml*t~ 2
This table shows at once that the abovediscussed equa
tions are homogeneous, that is, contain only members of
the same kind. Every new expression in mechanics
might be investigated in the same manner.
7. The knowledge of the dimensions of a quantity The usefui
is also important for another reason. Namely, if the theory of e
value of a quantity is known for one set of fundamental sions?"
units and we wish to pass to another set, the value of
the quantity in the new units can be easily found from
the dimensions. The dimensions of an acceleration,
which has, say, the numerical value <p, are //~ 2 . If
we pass to a unit of length A. times greater and to a
unit of time r times greater, then a number \ times
smaller must take the place of / in the expression //~ 2 ,
and a number r times smaller the place of /. The
numerical value of the same acceleration referred to
the new units will consequently be (r 2 A) (p. If we
2 8o THE SCIENCE OF MECHANICS,
take the metre as our unit of length, and the second as
our unit of time, the acceleration of a falling body for
example is 981, or as it is customary to write it, in
dicating at once the dimensions and the fundamental
measures : 981 (metre/second 2 ). If we pass now to
the kilometre as our unit of length (A. = 1000), and to
the minute as our unit of time (r 60), the value of the
same acceleration of descent is (60 X 60/1000)981,
or 35*316 (kilometre/minute 2 ).
The inter [8. The following statement of the mechanical units
national L . . .
Bureau of at present in use m the United States and ureat Britain
and Meas is substituted for the statement by Professor Mach of
the units formerly in use on the continent of Europe.
All the civilised governments have united in establish
ing an International Bureau of Weights and Measures
in the Pavilion de Breteuil, in the Pare of St. Cloud,
at Sevres, near Paris. In some countries, the stan
dards emanating from this office are exclusively legal ;
in others, as the United States and Great Britain, they
are optional in contracts, and are usual with physi
cists. These standards are a standard of length and a
standard of mass (not weight.}
The inter The unit of length is the International Metre, which
unitof is defined as the distance at the melting point of ice
eng * between the centres of two lines engraved upon the
polished surface of a platiniridium bar, of a nearly
Xshaped section, called the International Prototype
Metre. Copies of this, called National Prototype Me
tres, are distributed to the different governments. The
international metre is authoritatively declared to be
identical with the former French metre, used until the
adoption of the international standard ; and it is im
possible to ascertain a*iy error in this statement, be
EXTENSION OF THE PRINCIPLES. 281
cause of doubt as to the length of the old metre,
owing partly to the imperfections of the standard, and
partly to obstacles now intentionally put in the way of
such ascertainment. The French metre was defined
as the distance, at the meltingpoint of ice, between
the ends of a platinum bar, called the metre des archives.
It was against the law to touch the ends, which made
it difficult to ascertain the distance between them.
Nevertheless, there was a strong suspicion they had
been dented. The metre des archives was intended to
be one tenmillionth of a quadrant of a terrestrial
meridian. In point of fact such a quadrant is, ac
cording to Clarke, 32814820 feet, which is 10002015
metres.
The international unit of mass is the kilogramme. Theinter
11 r  11 ,,...,. national
which is the mass of a certain cylinder of platimndium unit of
mass.
called the International Prototype Kilogramme. Each
government has copies of it called National Prototype
Kilogrammes. This mass was" intended to be identical
with the former French kilogramme, which was defined
as the mass of a certain platinum cylinder called the
kilogramme des archives. The platinum being" somewhat
spongy contained a variable amount of occluded gases,
and had perhaps suffered some abrasion. The kilo
gramme is 1000 grammes ; and a gramme was intended
to be the mass of a cubic centimetre of water at its
temperature of maximum density, about 3*93 C. It
is not known with a high degree of precision how nearly
this is so, owing to the difficulty of the determination.
The regular British unit of length is the Imperial The British
Yard which is the distance at 62 F. between the cen length.
tres of two lines engraved on gold plugs inserted in a
bronze bar usually kept walled up in the Houses of
Parliament in Westminster". These lines are cut rela
js.j TUE SCIEXCE OF MECHANICS.
'cn.iitinii, lively deep, and the burr is rubbed off and the surface
f ! nct : uif" rendered mat, by rubbing with charcoal. The centre
nflf whii of such a line can easily be displaced by rubbing ; which
P"? :ncav " is probably not true of the lines on the Prototype me
tres. The temperature is, by law, ascertained by a
mercurial thermometer ; but it was not known, at the
time of the construction of the standard, that such
thermometers may give quite different readings, ac
cording to the mode of their manufacture. The quality
of glass makes considerable difference, and the mode
of determining the fixed points makes still more. The
best way of marking these points is first to expose the
thermometer for several hours to wet aqueous vapor at
a known pressure, and mark on its stem the height of
the column of mercury. The thermometer is then
brought down to the temperature of melting ice, as
rapidly as possible, and is immersed in pounded ice
which is melting and from which the water is not
allowed to drain off. The mercury being watched
with a magnifying glass is seen to fall, to come to
rest, and to commence to rise, owing to the lagging
contraction of the glass. Its lowest point is marked
on the stem. The interval between the two marks is
then divided into equal degrees. When such a ther
mometer is used, it is kept at the temperature to be
determined for as long a time as possible, and imme
diately after is cooled as rapidly as it is safe to cool it,
and its zero is redetermined. Thermometers, so made
and treated, will give very constant indications. But
the thermometers made at the Kew observatory, which
are used for determining the temperature of the yard,
are otherwise constructed. Namely the meltingpoint
is determined first and the boilingpoint afterwards ;
and the thermometers are exposed to both tempera
THE EXTENSION OF THE PRINCIPLES. 283
tures for many hours. The point which upon such a Relative
thermometer will appear as 62 will really be consider thelne'tre
ably hotter (perhaps a third of a centigrade degree)
than if its meltingpoint were marked in the other way.
If this circumstance is not attended to in making com
parisons, there is danger of getting the yard too short
by perhaps one twohundredthousandth part. General
Comstock finds the metre equal to 39*36985 inches.
Several less trustworthy determinations give nearly the
same value. This makes the inch 2 540014 centimetres.
At the time the United States separated from Eng The Ameri
r can unit of
land, no precise standard of length was legal*; and length.
none has ever been established. We are, therefore,
without any precise legal yard ; but the United States
office of weights and measures, in the absence of any
legal authorisation, refers standards to the British Im
perial Yard.
The regular British unit of mass is the Pound, de The British
fined as the mass of a certain platinum weight, called SSss?
the Imperial Pound. This was intended to be so con
structed as to be equal to 7000 grains, each the 576oth
part of a former Imperial Troy pound. This would be
within 3 grains, perhaps closer, of the old avoirdupois
pound. The British pound has been determined by
Miller to be 04535926525 kilogramme ; that is the kilo
gramme is 2204621249 pounds.
At the time the United States separated from Great
Britain, there were two incommensurable units of
weight, the av air dzipois pound &.&& the Troy pound. Con
gress has since established a standard Troy pound,
which is kept in the Mint in Philadelphia. It was a
copy of the old Imperial Troy pound which had been
adopted in England after American independence. It
* The socalled standard of 1758 had not been legalised.
2 S 4 THE SCIENCE OF MECHANICS.
TheAnvjri Is a hollow brass weight of unknown volume ; and no
ui!lUV* ' accurate comparisons of it with modern standards have
ever been published. Its mass is, therefore, unknown.
The mint ought by law to use this as the standard of
gold and silver. In fact, they use weights furnished
by the office of weights and measures, and no doubt
derived from the British unit ; though the mint officers
profess to compare these with the Troy pound of the
United States, as well as they are able to do. The old
avoirdupois pound, which is legal for most purposes,
differed without much doubt quite appreciably from
the British Imperial pound ; but as the Office of Weights
and Measures has long been, without warrant of law,
standardising pounds according to this latter, the legal
avoirdupois pound has nearly disappeared from use of
late years. The makers of weights could easily detect
the change of practice of the Washington Office.
Measures of capacity are not spoken of here, be
cause they are not used in mechanics. It may, how
ever, be well to mention that they are defined by the
weight of water at a" given temperature which they
measure.
The unit of The universal unit of time is the mean solar day or
its one 864ooth part, which is called a second. Side
real time is only employed by astronomers for special
purposes.
Whether the International or the British units are
employed, there are two methods of measurement of
mechanical quantities, the absolute and the gravitational.
The absolute is so called because it is not relative to
the acceleration of gravity at any station. This method
was introduced by Gauss.
The special absolute system, widely used by physi
cists in the United States and Great Britain, is called
THE EXTENSION OF THE PRINCIPLES. 285
the CentimetreGramineSecond system. In this sys The abso
. . lute system
tern, writing C for centimetre, G for gramme mass, of the
j o t i United
and b ior second, states and
Great Brit
the unit of length Is C ;
the unit of mass is G ;
the unit of time is S ;
the unit of velocity is C/S
the unit of acceleration (which might
be called a "galileo/' because Gali
leo Galilei first measured an accele
ration) is C/S 2 ;
the unit of density is G/C 3 ;
the unit of momentum is . G C/S ;
the unit of force (called a dyne) is . . . G C/S 2 ;
the unit of pressure (called one mil
lionth of an absolute atmosphere) is . . G/C S 2 ;
the unit of energy (vis viva, or work,
called an erg) is ^GC 2 /S 2 ;
etc.
The gravitational system of measurement of me The Gravi
chanical quantities, takes the kilogramme or pound, or system,
rather the attraction of these towards the earth, com
pounded with the centrifugal force,, which is the ac
celeration called gravity, and denoted by g, and is dif
ferent at different places, as the Unit of force, and
the footpound or kilogrammemetre, being the amount
of gravitational energy transformed in the descent of a
pound through a foot or of a kilogramme through a
metre, as the unit of energy. Two ways of reconciling
these convenient units with the adherence to the usual
standard of length naturally suggest themselves, namely,
first, to use the pound weight or the kilogramme weight
divided by g as the unit of mass, and, second, to adopt
srfu THE SCIEXCE OF MECHANICS.
such a unit of time as will make the acceleration of g,
at an Initial station, unity. Thus, at Washington, the
acceleration of gravity is 980 05 galileos. If, then,
we take the centimetre as the unit of length, and the
0*031943 second as the unit of time, the acceleration
of gravity will be i centimetre for such unit of time
squared. The latter system would be for most pur
poses the more convenient ; but the former is the more
familiar,
oiupari In either system, the formula p = mg is retained ;
sonoi the . ,  ,. , , , ,...'
absolute but in the former g retains its absolute value, wjiile in
tationai the latter it becomes unity for the initial station. In
Paris, g is 98096 galileos ; in Washington it is 98005
galileos. Adopting the more familiar system, and
taking Paris for the initial station, if the unit of force
is a kilogramme's weight, the unit of length a centi
metre, and the unit of time a second, then the unit of
mass will be 1/9810 kilogramme, and the unit of
energy will be a kilogrammecentimetre, or (1/2)
(1000/9810) GC 2 /S 2 . Then, at Washington the
gravity of a kilogramme will be, not i, as at Paris,
but 980 1/981 0 = 099907 units or Paris kilogramme
weights. Consequently, to produce a force of one Paris
kilogrammeweight we must allow Washington gravity
to act upon 981 0/9801 = i 00092 kilogrammes.]*
In mechanics/ as in some other branches of physics
closely allied to it, our calculations involve but three
fundamental quantities, quantities of space, quantities
of time, and quantities of mass. This circumstance is
a source of simplification and power in the science
which should not be underestimated.
* For some critical remarks on the preceding method of exposition, see
Nature, in the issue for November 15, 1894.
THE EXTENSION OP THE PRINCIPLES. 287
THE LAWS OF THE CONSERVATION OF MOMENTUM, OF THE
CONSERVATION OF THE CENTRE OF GRAVITY, AND
OF THE CONSERVATION OF AREAS.
1. Although Newton's principles are fully adequate Speciaiisa
. , , . , , , , , tion of the
to deal with any mechanical problem that may arise, mechanical
it is yet convenient to contrive for cases more frequently
occurring, particular rules, which will enable us to treat
problems of this kind by routine forms and to dis
pense with the minute discussion of them. Newton
and his successors developed several such principles.
Our first subject will be NEWTON'S doctrines concern
ing freely movable material systems.
2. If two free masses ;// and m 1 are subjected in Mutual ac
the direction of their line of junction to the action of masses,
forces that proceed from other masses, then, in the in
terval of time /, the velocities v, if will be generated,
and the equation (/ + /') t = m v + m'v' will subsist.
This follows from the equations // = mv and /'/' =
rn'if . The sum mv \ m'v' is called the momentum of
the system, and in its computation oppositely directed
forces and velocities are regarded as having opposite
signs. If, now, the masses m, m' in addition to being
subjected to the action of the external forces /, f' are
also acted upon by internal forces, that is by such as
are mutually exerted by the masses on one another, these
forces will, by Newton's third law, be equal and op
posite, ^, q. The sum of the impressed impulses
is, then, (/+/' + q f) t = 0> + /') t, the same as
before ; and, consequently, also, the total momentum
of the system will be the same. The momentum of a
2SS THE SCIEXCE OF MECHANICS.
system Is thus determined exclusively by externalioicces,
that is, by forces which masses outside of the system
exert on its parts.
Law of the Imagine a number of free masses m, m', m". . . .
tion^Mo distributed in any manner in space and acted on by
mcntum. , . , ,, , 1*1 i
external forces /, /, / . . . . whose lines have any di
rections. These forces produce in the masses in the
interval of time / the velocities v, v' 9 v". . . . Resolve
all the forces in three directions x, }>, z at right angles
to each other, and do the same with the velocities.
The sum of the Impulses in the ^direction will be equal
to the momentum generated In the ^direction ; and
so with the rest. If we imagine additionally in action
between the masses m, m', m". . . ., pairs of equal and
opposite internal forces q, q, r, r, s, s, etc. ,
these forces, resolved, will also give in every direction
pairs of equal and opposite components, and will con
sequently have on the sumtotal of the impulses no in
fluence. Once more the momentum is exclusively de
termined by external forces. The law which states
this fact is called the law of the conservation of momen
tum*
Law of the 3. Another form of the same principle, which New
tionofthe ton likewise discovered, is called the law of the conser
Centre of .....
Gravity. vation of the centre of grav
M e in A and B
(Fig. 149) two masses, zm
Flg " 149 ' and m, in mutual action,
say that of electrical repulsion ; their centre of gravity
Is situated at S, where J5S 2 AS. The accelerations
they impart to each other are oppositely directed and
in the inverse proportion of the masses. If, then, in
consequence of the mutual action, 2 m describes a dis
tance AD, m will necessarily describe a distance BC =
THE EXTEXS10X OF THE PRIXCJPLRS. 289
The point S will still remain the position of the
centre of gravity, as CS = zDS. Therefore, two masses
cannot, by mutual action, displace their common centre
of gravity.
If our considerations involve several masses, dis This law
., i .  i i >ii 1 applied to
tributed in any way in space, the same .result will also systems of
be found to hold good for this case. For as no two of
the masses can displace their centre of gravity by mu
tual action, the centre of gravity of the system as a
whole cannot be displaced by the mutual action of its
parts.
Imagine freely placed in space a system of masses
/;/, ;;/', m" . . . . acted on by external forces of any kind.
We refer the forces to a system of rectangular coordi
nates and call the coordinates respectively ,Y, y, s, x' 9
y, z', and so forth. The coordinates of the centre of
gravity are then
IS my
'/  =%~T
in which expressions #, y, z may change either by uni
form motion or by uniform acceleration or by any other
law, according as the mass in question is acted on by
no external force, by a constant external force, or by a
variable external force. The centre of gravity will have
in all these cases a different motion, and in the first
may even be at rest. If now internal forces, acting be
tween every two masses, m' and /?/", come into play in
the system, opposite displacements w', w" will thereby
be produced in the direction of the lines of junction
of the masses, such that, allowing for signs, m'w' +
m r 'w" = 0. Also with respect to the components x^
and cc 2 of these displacements the equation m'x^ +
m"x 2 = will hold. The internal forces consequently
290 THE SCJEXCE OF MECHANICS.
produce In the expressions <?, 7;, 3 only such additions
as mutually destroy each other. Consequently, the
motion of the centre of gravity of a system is determined
by external forces only.
Acceiera If ive wish to know the acceleration of the centre of
centre ct gravity of the system, the accelerations of the system's
, ., , J , , Ti . , ,, ,
must be similarly treated, it cp, cp , cp . . . , de
note the accelerations of ;//, m', m". ... in any direc
tion, and cp the acceleration of the centre of gravity in
the same direction, cp = 2mcp/^m, or putting the
total mass 2w = M, cp = 2m <p/M. Accordingly, we
obtain the acceleration of the centre of gravity of a
system in any direction by taking the sum of all the
forces in that direction and dividing the result by the
total mass. The centre of gravity of a system moves
exactly as if all the masses and all the forces of the
system were concentrated at that centre. Just as a
single mass can acquire no acceleration without the
action of some external force, so the centre of gravity
of a system can acquire no acceleration without the
action of external forces.
4. A few examples may now be given in illustra
tion of the principle of the conservation of the centre
of gravity.
Movement Imagine an animal free in space. If the animal
maHree in move in one direction a portion m of its mass, the re
space. m ainder of it J/will be moved in the opposite direction,
always so that its centre of gravity retains its original
position. If the animal draw back the mass m, the
motion of M also will be reversed. The animal is un
able, without external supports or forces, to move itself
from the spot which it occupies, or to alter motions im
pressed upon it from without.
A lightly running vehicle A is placed on rails and
THE EXTENSION OF THE PRINCIPLES. 291
loaded with stones. A man stationed in the vehicle of a ve
. hicle, from
casts out the stones one after another, m the same di which
.. 11  r stones are
rection. The vehicle, supposing the friction to be suf cast.
ficiently slight, will at once be set in motion in the op
posite direction. The centre of gravity of the S3 7 stem
as a whole (of the vehicle + tne stones) will, so far as
its motion is not destroyed by external obstacles, con
tinue to remain in its original spot. If the same man
were to pick up the stones from without and place
them in the vehicle, the vehicle in this case would also
be set in motion ; but not to the same extent as before,
as the following example will render evident.
A projectile of mass m is thrown with a velocity v Motion of a
, cannon and
from a cannon of mass Jlf. In the reaction. JkT also re its projec
5 tile.
ceives a velocity, V, such that, making allowance for
the signs, MV \ mv = 0. This explains the socalled
recoil. The relation here is V= (injM^ v ; or, for
equal velocities of flight, the recoil is less according as
the mass of the cannon is greater than the mass of the
projectile. If the work done by the powder be expressed
by A, the vires viucz will be determined by the equation
J/F" 2 /2  mv*/2 = A ; and, the sum of the momenta
being by the firstcited equation = 0, we readily obtain
F= \/ Q.A m/M(M~\ m}. Consequently, neglecting
the mass of the exploded powder, the recoil vanishes
when the mass of the projectile vanishes. If the mass
m were not expelled from the cannon but sucked into
it, the recoil would take place in the opposite direc
tion. But it would have no time to make itself visible
since before any perceptible distance had been trav
ersed, m would have reached the bottom of the bore.
As soon, however, as $/"and m are in rigid connection
with each other, as soon, that is, as they are relatively
at rest to each other, they must be absolutely at rest,
292 TIIK SCIENCE OF MECHANICS.
for the centre of gravity of the system as a whole has
no motion. For the same reason no considerable mo
tion can take place when the stones in the preceding
example are taken into the vehicle, because on the
establishment of rigid connections between the vehicle
and the stones the opposite momenta generated are
destroyed. A cannon sucking in a projectile would
experience a perceptible recoil only if the sucked in
projectile could fly through it.
osciiia Imagine a locomotive freely suspended in the air,
body of a & or, what will subserve the same purpose, at rest with
insufficient friction on the rails. By the law of the
conservation of" the centre of gravity, as soon as the
heavy masses of iron in connection with the piston
rods begin to oscillate, the body of the locomotive will
be set in oscillation in a contrary direction a motion
which may greatly disturb its uniform progress. To
eliminate this oscillation, the motion of the masses of
iron worked by the pistonrods must be so compensated
for by the contrary motion of other masses that the
centre of gravity of the system as a whole will remain
in one position. In this way no motion of the body of
the locomotive will take place. This is done by affix
ing masses of iron to the drivingwheels.
illustration The facts of this case may be very prettily shown
case. by Page's electromotor (Fig. 150). When the iron
core in the bobbin AB is projected by the internal forces
acting between bobbin and core to the right, the body
of the motor, supposing it to rest on lightly movable
wheels rr, will move to the left But if to a spoke of
the flywheel R we affix an appropriate balanceweight
a, which always moves in the contrary direction to the
iron core, the sideward movement of the body of the
motor may be made totally to vanish.
THE EXTEXSIO.V OF THE PRINCIPLES,
293
Of the motion of the fragments oE a bursting bomb A bursting
we know nothing. But it is plain, by the law of the
conservation of the centre of gravity, that, making al
lowance for the resistance of the air and the obstacles
the individual parts may meet, the centre of gravity of
the system will continue after the bursting to describe
the parabolic path of its original projection.
5. A law closely allied to the law of the centre of Law of the
i 1 1 ,. , ,  . , Conserva
gravity, and similarly applicable to free systems, is thetionof
principle of the conservation of areas. Although Newton
Fig. 150.
had, so to say, this principle within his very grasp, it
was nevertheless not enunciated until a long time after
wards by EULER, D'ARCY, and DANIEL BERNOULLI.
Euler and Daniel Bernoulli discovered the law almost
simultaneously (1746), on the occasion of treating a
problem proposed by Euler concerning the motion of
balls in rotatable tubes, being led to It by the consider
ation of the action and reaction of the balls and the
tubes. D'Arcy (1747) started from Newton's Investiga
tions, and generalised the law of sectors which the
latter had employed to explain Kepler's laws.
294
THE SCIENCE OF MECHANICS.
Deduction
of the law.
Two masses ;;/, m' (Fig. 151) are in mutual action.
jgy v  rtue O f this action the masses describe the dis
tances AJB, CD in the direction of their line of junction.
Allowing for the signs, then, m . AB f ;;/'. CD = 0.
Drawing radii vectores to the moving masses from any
point O, and regarding
the areas described in
opposite senses by the
radii as having opposite
signs, we further obtain
m. OAB + m. OCD = 0.
Which is to say, if two
masses mutually act on
each other, and radii vec
tores be drawn to these
masses from any point,
the sum of the areas
described by the radii
multiplied by the respec
tive masses Is = 0. If the masses are also acted on
by external forces and as the effect of these the areas
OAE and OCJF&re described, trie joint action of the
internal and external forces, during any very small
period of time, will produce the areas OA G and OCH.
But it follows from Varignon's theorem that
=m OAE + m'OCJ? +
mOAB + m'OCD = mOAE + m'OCF;
in other words, the sum of the products of the areas so de
scribed into the respective masses which compose a system
is unaltered by the action of internal forces.
If we have several masses, the same thing may be
asserted, for every two masses, of the projection on any
given plane of the motion. If we draw radii from
THE EXTENSION OF THE PRINCIPLES. 295
any point to the several masses, and project on any
plane the areas the radii describe, the sum of the
products of these areas into the respective masses will
be independent of the action of internal forces. This
is the law of the conservation of areas.
If a single mass not acted on by forces is moving interpreta
i ?   i , , . tionofthe
uniformly forward in a straight line and we draw a law.
radius vector to the mass from any point O, the area
described by the radius increases proportionally to the
time. The same law holds for 2m/, in cases in which
several masses not acted on by forces are moving,
where we signify by the summation the algebraic sum
of all the products of the areas (/) into the moving
masses a sum which we shall hereafter briefly refer
to as the sum of the mass areas. If internal forces
come into play between the masses of the s} r stem, this
relation will remain unaltered. It will still subsist,
also, if external forces be applied whose lines of action
pass through the jfar^v/ point O, as we know from the
researches of Newton.
If the mass be acted on by an external force, the
area / described by its radius vector will increase in
time by the law/" at* fa j bt + c, where a depends
on the accelerative force, b on the initial velocity, and
c on the initial position. The sum 2mf increases by
the same law, where several masses are acted upon by
external accelerative forces, provided these may be re
garded as constant, which for sufficiently small inter
vals of time is always the case. The law of areas in
this case states that the internal forces of the system
have no influence on the increase of the sum of the mass
areas.
A free rigid body may be regarded as a system
whose parts are maintained in their relative positions
296 THE SCIENCE OP MECHANICS.
Uniform r<> by internal forces. The law of areas is applicable there
irelTigid* fore to this case also. A simple instance is afforded
b d> * by the uniform rotation of a rigid body about an axis
passing through its centre of gravity. If we call m a
portion of its mass, r the distance of the portion from
the axis, and a its angular velocity, the sum of the
massareas produced in unit of time will be 2m
(r/2) ra = (a/2) 2mr* 9 or, the product of the moment
of inertia of the system into half its angular velocity.
This product can be altered only by external forces.
illustrative 6. A few examples may now be cited in illustration
examples. r ,t i
of the law.
If two rigid bodies K and K' are connected, and K
is brought by the action of internal forces into rotation
relatively to K' , immediately K' also will be set in ro
tation, in the opposite direction. The rotation of K
generates a sum of massareas which, by the law, must
be compensated for by the production of an equal, but
opposite, sum by K f .
Opposite This is very prettily exhibited by the electromotor
the a wheei of Fig. 152. The fly wheel of the motor is placed in
a n freee1ec f a horizon taF plane, and the motor thus attached to a
tromotor. ,1  11 r i ^ ATM
vertical axis, on which it can freely turn. The wires
conducting the current dip, in order to prevent their
interference with the rotation, into two conaxial gutters
of mercury fixed on the axis. The body of the motor
(jST') is tied by a thread to the stand supporting the
axis and the current is turned on. As soon as the fly
wheel (K} 9 viewed from above, begins to rotate in the
direction of the hands of a watch, the string is drawn
taut and the body of the motor exhibits the tendency
to rotate in the opposite direction a rotation which im
mediately takes place when the thread is burnt away.
The motor is, with respect to rotation about its
THE EXTENSION OF THE PRINCIPLES. 297
axis, a free system. The sum of the massareas gen itsexpiana
erated, for the case of rest, is = 0. But the wheel of law. 3
the motor being set in rotation by the action of the in
ternal electromagnetic forces, a sum of massareas is
produced which, as the total sum must remain = 0, is
compensated for by the rotation in the opposite direc
tion of the body of the motor. If an index be attached
to the body of the motor and kept in a fixed position
2 gS THE SCIENCE OF MECHANICS.
by an elastic spring, the rotation of the body of the
motor cannot 'take place. Yet every acceleration of
the wheel in the direction of the hands of a watch (pro
duced by a deeper immersion of the battery) causes
the index to swerve in the opposite direction, and every
retardation produces the contrary effect,
A variation A beautiful but curious phenomenon presents itself
phenome 16 when the current to the motor is interrupted. Wheel
non ' and motor continue at first their 'movements in oppo
site directions. But the effect of the friction of the
axes soon becomes apparent and the parts gradually
assume with respect to each other relative rest. The
motion of the body of the motor is seen to dimmish ;
for a moment it ceases ; and, finally, when the state of
relative rest is reached, it is reversed and assumes the
direction of the original motion of the wheel. The
whole motor now rotates in the direction the wheel did
at the start. The explanation of the phenomenon is
obvious. The motor is not a perfectly free system. It
is impeded by the friction of the axes. In a perfectly
free system the sum of the massareas, the moment
the parts reentered the state of relative rest, would
again necessarily be = 0. But in the present instance,
an external force is introduced the friction of the
axes. The friction on the axis of the wheel diminishes
the massareas generated by the wheel and body of
the motor alike. But the friction on the axis of the
body of the motor only diminishes the sum of the mass
areas generated by the body. The wheel retains, thus,
an excess of massarea, which when the parts are rela
tively at rest is rendered apparent in the motion of the
entire motor. The phenomenon subsequent to the in
terruption of the current supplies us with a model of
what according to the hypothesis of astronomers has
THE EXTENSION OF THE PRINCIPLES. 299
taken place on the moon. The tidal wave created by its niustra
the earth has reduced to such an extent by friction the case of the
velocity of rotation of the moon that the lunar day has
grown to a month. The flywheel represents the fluid
mass moved by the tide.
Another example of this law is furnished by reac Reaction
t ionwheels. If air or gas be emitted from the wheel
(Fig. 1530) in the direction of the short arrows, the
whole wheel will be set in rotation in the direction of
the large arrow. In Fig. 153^, another simple reac
tionwheel is represented. A brass tube rr plugged at
both ends and appropriately perforated, is placed on a
second brass tube R, supplied with a thin steel pivot
through which air can be blown ; the air escapes at
the apertures O, O r .
It might be supposed that sucking on the reaction variation
. . ofthephe
wheels would produce the opposite motion to that renomenaof
suiting from blowing. Yet this does not usually take wheels,
place, and the reason is obvious. The air that is
sucked into the spokes of the wheel must take part
immediately in the motion of the wheel, must enter
the condition of relative rest with respect to the wheel ;
and when the system is completely at rest, the sum of
its massareas must be 0. Generally, no perceptible
rotation takes place on the sucking in of the air. The
circumstances are similar to those of the recoil of a
cannon which sucks in a projectile. If, therefore, an
elastic ball, which has but one escapetube, be attached
to the reactionwheel, in the manner represented in
Fig. 1530, and be alternately squeezed so that the
same quantity of air is by turns blown out and sucked
in,, the wheel will continue rapidly to revolve in the
same direction as it did in the case in which we blew
into it. This is partly due to the fact that the air
300
TJfK SCUK+VCE OS' MECHANICS.
Fig. 153 b.
THE EXTENSION OF THE PRINCIPLES.
301
sucked into the spokes must participate in the motion Expiana
of the latter and therefore can produce no reactional variations,
rotation, but it also partly results from the difference
of the motion which the air outside the tube assumes
in the two cases. In blowing, the air flows out in jets,
and performs rotations. In sucking, the air comes in
from all sides, and has no distinct rotation.
The correctness of this view is easily demonstrated.
If we perforate the bottom of a hollow cylinder, a closed
bandbox for instance, and
place the cylinder on the steel
pivot of the tube R, after the
side has, been slit and bent in
the manner indicated in Fig.
154, the box will turn in the
direction of the long arrow
when blown into and in the Fig. 154.
direction of the short arrow when sucked on. The air,
here, on entering the cylinder, can continue its rotation
unimpeded) and this motion is accordingly compensated
for by a rotation in the opposite direction.
7. The following case also exhibits similar condi Reaction
tions. Imagine a tube (Fig. 155 a) which, running
straight from a to b, turns at right angles
to itself at the latter point, passes to *r,
describes the circle cdef, whose plane
is at right angles to ab, and whose cen
tre is at b, then proceeds from f to g,
and, finally, continuing the straight line
ab, runs from g to h. The entire tube
is free to turn on an axis ah. If we
pour into this tube, in the manner in
dicated in Fig. 155^, a liquid, which flows in the di
rection cdef, the tube will immediately begin to turn
302
THE SCIENCE OF MECHANICS.
In the direction fedc. This impulse, however, ceases,
the moment the liquid reaches the point/ and flowing
out into the radius/^ is obliged to join in the motion
of the latter. By the use of a constant stream of liquid,
therefore, the rotation
of the tube may soon
be stopped. But if the
stream be interrupted,
the fluid, in flowing off
through the radius fg,
will impart to the tube
a motional impulse in
the direction of its own
motion, cdef, and the
tube will turn in this di
rection. All these phe
nomena are easily ex
plained by the law of
areas.
The tradewinds, the
deviation of the oceanic
currents and of rivers,
Foucault's pendulum
experiment, and the
like, may also be treated
Fig  I55b  as examples of the law
Another pretty illustration is afforded by
Let a body
with the moment of inertia & rotate with the angular
velocity a and, during the motion, let its moment
of inertia be transformed by internal forces, say by
springs, into &, a will then pass into a', where a =
a'Q', that is a' = a (/'). On any considerable dimi
nution of the moment of inertia, a great increase of
Additional of areas.
tions. ra ~ bodies with variable moments of inertia.
THE EXTEXSIO.V OF THE PRINCIPLES.
3C3
angular velocity ensues. The principle might con
ceivably be employed, instead of Foucault's method,
to demonstrate the rotation of the earth, [in fact, some
attempts at this have been made, with no very marked
success].
A phenomenon which substantially embodies the Rotating
conditions last suggested is the following. A glass funnel" 1 a
funnel, with its axis placed in a vertical position, is
rapidly filled with a liquid in such a manner that the
stream does not enter in the direction of the axis but
strikes the sides. A slow rotatory motion is thereby
set up in the liquid which as long as the funnel is full, is
not noticed. But when the fluid retreats into the neck
of the funnel a its moment of inertia is so diminished
and its angular velocity so increased that a violent
eddy with considerable axial depression is created.
Frequently the entire effluent jet is penetrated by an
axial thread of air.
8. If we carefully examine the principles of the Both prin
i cipiesare
centre of gravity and of the areas, we shall discover in simply spe
cial cases of
the law of
i<P_ ^jP> action and
IV v .,/ 2w^ reaction.
Fig. 156.
.
both simply convenient
f r J . f
modes of expression, for
practical purposes, of
a wellknown property
of mechanical phenom
ena. To the accelera
tion cp of one mass m
there always corresponds a contrary acceleration (p f of
a second mass m', where allowing for the signs m (p +
;;/ cp' = 0. To the force m <p corresponds the equal
and opposite force m'<p\ When any masses m and
2 m describe with the contrary accelerations 2 cp and cp
the distances vw and w (Fig. 156), the position of
their centre of gravity S remains unchanged, and the
3 o 4 THE SCIENCE OF MECHANICS.
sum of their massareas with respect to any point O
is, allowing for the signs, 2 ///./+ m . 2/ 0. This
simple exposition shows us, that the principle of the
centre of gravity expresses the same thing with respect
to parallel coordinates that the principle of areas ex
presses with respect to polar coordinates. Both contain
simply the fact of reaction.
But they The principles in question admit of still another
construed e simple construction. Just as a single body cannot,
sat?ons e S h " without the influence of external forces, that is, without
inertfa! the aid of a second body, alter its uniform motion of
progression or rotation, so also a system of bodies can
not, without the aid of a second system, on which it
can, so to speak, brace and support itself, alter what
may properly and briefly be called its mean velocity of
progression or rotation. Both principles contain, thus,
a generalised statement of the law of inertia, the correct
ness of which in the present form we not only see but
fed.
importance This feeling is not unscientific ; much less is it
stinctive detrimental. Where it does not replace conceptual in
mechanicai sight but exists by the side of it, it is really the funda
facts. , . . n . . _ . ,
mental requisite and sole evidence of a complete mastery
of mechanical facts. We are ourselves a fragment of
mechanics, and this fact profoundly modifies our mental
life.* No one will convince us that the consideration
of mechanicophysiological processes, and of the feel
ings and instincts here involved, must be excluded from
scientific mechanics. If we know principles like those 
of the centre of gravity and of areas only in their ab
stract mathematical form, without having dealt with the
palpable simple facts, which are at once their applica
* For the development of this view, see E. Mach, Grundlinien der Lehre
von den Bewegungsempfindungen, (Leipsic : Engelmann, 1875.)
THE EXTEXSIOX OF THE PRIXCIPLES. 305
tion and their source, we only half comprehend them,
and shall scarcely recognise actual phenomena as ex
amples of the theory. We are In a position like that
of a person who is suddenly placed on a high tower
but has not previously travelled in the district round
about, and who therefore does not know how to inter
pret the objects he sees.
THE LAWS OF IMPACT.
i. The laws of Impact were the occasion of the Historical
position of
enunciation of the most important principles of me the Laws of
chanics, and furnished also the first examples of the
application of such principles. As early as 1639, a
contemporary of Galileo, the Prague professor, MARCUS
MARCI (born in 1595), published In his treatise De Pro
portions Mot us (Prague) a few results of his Investiga
tions on Impact. He knew that a body striking In
elastic percussion another of the same size at rest, loses
Its own motion and communicates an equal quantity
to the other. He also enunciates, though not always
with the requisite precision, and frequently mingled
with what is false, other propositions which still hold
good. Marcus Marci was a remarkable man. He pos
sessed for his time very creditable conceptions regard
Ing the composition of motions and "impulses." In
the formation of these ideas he pursued a method sim
ilar to that which Roberval later employed. He speaks
of partially equal and opposite motions, and of wholly r
opposite motions, gives parallelogram constructions,
and the like, but Is unable, although he speaks of an
accelerated motion of descent, to reach perfect clear
ness with regard to the idea of force and consequently
also with regard to the composition of forces. In spite
306
77//i SCIENCE OF MECHANICS.
There of this, however, he discovers Galileo's theorem re
garding the descent of bodies in the chords of circles,
f
Marci.
IQANNESMARCVS MAR.CI PHIL: O MEDIC: DOCTOR
$>r tiatus Laru&frena? fferrmme&irorumxn Bocrma.
also a few propositions relating to the motion of the
pendulum, and has knowledge of centrifugal force and
so on. Although Galileo's Discourses had appeared a
THE EXTE^SIOX OF THE PRINCIPLES.
337
year previously, we cannot, in view of the condition of
things produced in Central Kurope by the Thirty Years'
War, assume that Marci was acquainted with them.
Not only would the many errors in Marci's book thus
be rendered unintelligible, but it would also have to
3 oS THE SCIENCE OF MECHANICS.
Thesourcesbe explained how Marci, as late as 1648, In a continu
knowiedge. atioii of his treatise, could have found it necessary to
defend the theorem of the chords of circles against the
Jesuit Balthasar Conradus. An imperfect oral com
munication of Galileo's researches is the more reason
able conjecture.* When we add to all this that Marci
was on the very verge of anticipating Newton in the
discovery of the composition of light, we shall recog
nise in him a man of very considerable parts. His
writings are a worthy and as yet but slightly noticed
object of research for the historian of physics. Though
Galileo, as the clearestminded and most able of his
contemporaries, boie away in this province the palm,
we nevertheless see from writings of this class that he
was not by any means alone in his thought and ways
of thinking.
There 2. GALILEO himself made several experimental at
Galileo. tempts to ascertain the laws of impact ; but he was not
in these endeavors wholly successful. He principally
busied himself with the force of a body in motion, or
with the li force of percussion," as he expressed it,
and endeavored to compare this force with the pressure
of a weight at rest, hoping thus to measure it. To this
end he instituted an extremely ingenious experiment,
which we shall now describe.
A vessel I (Fig. 157) in whose base is a plugged
orifice, is filled with water, and a second vessel II is
hung beneath it by strings ; the whole is fastened to
the beam of an equilibrated balance. If the plug is
removed from the orifice of vessel I, the fluid will fall
* I have been convinced, since the publication of the first edition of this
work, (see E. Wohlwill's researches, Die Entdeckung des Beharrungsgesetzec,
in the Zeitschriftfli* Volkerpsyckologie, 1884, XV, page 387,) that Marcus Marci
derived his information concerning the motion of falling bodies, from Galileo's
earlier Dialogues* He may also have known the works of BenedettL
THE EXTENSION OF THE PRINCIPLES.
309
in a jet into vessel II. A portion of the pressure due Galileo's
to the resting weight of the water in I is lost and re ment.
placed by an action of impact on vessel II. Galileo
expected a depression of the whole scale, by which he
hoped with the assistance of a counterweight to de
termine the effect of the impact He was to some ex
tent surprised to obtain no depression, and he was un
able, it appears, perfectly to clear up the matter in his
mind.
3. Today, of course, the explanation is not diffi
cult. By the removal of the plug there is produced,
1
I
Fig. 157.
first, a diminution of the pressure. This consists of Expiana
two factors : (i) The weight of the jet suspended inexperi
the air is lost ; and (2) A reactionpressure upwards is
exerted by the effluent jet on vessel I (which acts like
a Segner's wheel). Then there is an Increase of pres
sure (Factor 3) produced by the action of the jet on the
bottom of vessel II. Before the first drop has reached
the bottom of II, we have only to deal with a diminu
tion of pressure, which, when the apparatus is in full
operation, Is immediately compensated for. This initial
3 io THE SCIENCE OF MECHANICS.
Determine depression was, In fact, all that Galileo could observe.
tioii of the _ , .  j j
mechanical Let us imagine the apparatus in operation, and denote
viUedV 11 " the height the fluid reaches in vessel I by h, the corre
sponding velocity of efflux by v 9 the distance of the
bottom of I from the surface of the fluid in II by k, the
velocity of the jet at this surface by w, the area of the
basal orifice by a, the acceleration of gravity by g, and
the specific gravity of the fluid by s. To determine
Factor (i) we may observe that v is the velocity ac
quired in descent through the distance k. We have,
then, simply to picture to ourselves this motion of de
scent continued through k. The time of descent of
the jet from I to II is therefore the time of descent
through h f k less the time of descent through h.
During this time a cylinder of base a is discharged
with the velocity v. Factor (i), or the weight of the
jet suspended in the air, accordingly amounts to
I g ^ g
To determine Factor (2) we employ the familiar
equation mv = pt. If we put t = i, then m v = p, that
is the pressure of reaction upwards on I is equal to the
momentum imparted to the fluid jet in unit of time.
We will select here the unit of weight as our unit of
force, that is, use gravitation measure. We obtain for
Factor (2) the expression [av(s/gy]v =#, (where the
expression in brackets denotes the mass which flows
out in unit of time,) or
Similarly we find the pressure on II to be
a v . !L } w = q, or factor 3 :
S)
THE EXTENSION OF THE PRINCIPLES. 311
Mathemat
j. _ _
a  \/"lgh VZgUi + k). ica! devel
<r d 6 v ' '
The total variation of the pressure Is accordingly
opment of
the result.
as
g
or, abridged,
which three factors completely destroy each other. In
the very necessity of the case, therefore, Galileo could
only have obtained a negative result.
We must supply a brief comment respecting Fac A comment
tor (2). It might be supposed that the pressure on the by the ex
basal orifice which is lost, Is a hs and not 2 a /is. But
this statical conception would be totally Inadmissible
in the present, dynamical case. The velocity # Is not
generated by gravity Instantaneously in the effluent
particles, but is the outcome of the mutual pressure
between the particles flowing out and the particles left
behind ; and pressure can only be determined by the
momentum generated. The erroneous introduction of
the value a /is would at once betray itself by selfcon
tradictions.
If Galileo's mode of experimentation had been less
elegant, he would have determined without much diffi
culty the pressure which a continuous fluid jet exerts.
But he could never, as he soon became convinced,
have counteracted by a pressure the effect of an Instan
taneous impact. Take and this is the supposition of
3 i2 THE SCIENCE OF MECHANICS.
Galileo's Galileo a freely falling, heavy body. Its final veloc
reasomng. .^ we ^^ j ncreases proportionately to the time.
The very smallest velocity requires a definite portion
of time to be produced in (a principle which even Mari
otte contested). If we picture to ourselves a body
moving vertically upwards with a definite velocity, the
body will, according to the amount of this velocity,
ascend a definite time, and consequently also a definite
distance. The heaviest imaginable body impressed
in the vertical upward direction with the smallest im
aginable velocity will ascend, be it only a little, in
opposition to the force of gravity. If, therefore, a
heavy body, be it ever so heavy, receive an instan
taneous upward impact from a body in motion, be the
mass and velocity of that body ever so small, and such
impact impart to the heavier body the smallest imagin
able velocity, that body will, nevertheless, yield and
Compan move somewhat in the upward direction. The slightest
son of the . , ,
ideas im impact, therefore, is able to overcome the greatest pres
pressure. sure ; or, as Galileo says, the force of percussion com
pared with the force of pressure is infinitely great. This
result, which is sometimes attributed to intellectual ob
scurity on Galileo's part, is, on the contrary, a bril
liant proof of his intellectual acumen. . We should say
today, that the force of percussion, the momentum,
the impulse, the quantity of motion m v, is a quantity
of different dimensions from the pressure /. The dimen
sions of the former are m!t~ l , those of the latter;?/// 2 .
In reality, therefore, pressure is related to momentum
of impact as a line is to a surface. Pressure is /, the
momentum of impact is/ /. Without employing mathe
matical terminology it is hardly possible to express the
fact better than Galileo did. We now also see why it
is possible to measure the impact of a continuous fluid
THE EXTEXSIOX OF THE PRINCIPLES. 313
jet by a pressure. We compare the momentum de
stroyed per second of time with the pressure acting
per second of time, that is, homogeneous quantities of
the form //.
4. The first systematic treatment of the laws ofThesyste
impact was evoked in the year 1668 by a request of the mentof the
Royal Society of London. Three eminent physicists pact.
WALLIS (Nov. 26, 1668), WREN (Dec. 17, 1668), and
HUYGENS (Jan. 4, 1669) complied with the invitation of
the society, and communicated to it papers in which,
independently of each other, they stated, without de
ductions, the laws of impact. Wallis treated only of
the impact of inelastic bodies, Wren and Huygens only
of the impact of elastic bodies. Wren, previously to
publication, had tested by experiments his theorems,
which, in the main, agreed with those of Huygens.
These are the experiments to which Newton refers in
the Principia. The same experiments were, soon after
this, also described, in a more developed form, by Ma
riotte, in a special treatise, Sur le Choc des Corps. Ma
riotte also gave the apparatus now known in physical
collections as the percussionmachine.
According to Wallis, the decisive factor in impact waiiis's re
is moment um > or the product of the mass {pondus} into
the velocity (ccleritas). By this momentum the force
of percussion is determined. If two inelastic bodies
which have equal momenta strike each other, rest will
ensue after impact If 'their momenta are unequal,
the difference of the momenta will be the momentum
after impact. If we divide this momentum by the sum
of the masses, we shall obtain the velocity of the mo
tion after the impact. Wallis subsequently presented
his theory of impact in another treatise, Me chant ca sive
de Mot it, London, 1671. All his theorems may be
314 THE SCIENCE OF MECHANICS,
brought together in the formula now in common use,
// = (;;/r + in'v')j(t}i + ^')< in which ;;/, m' denote the
masses, ?*, v' the velocities before impact, and u the
velocity after impact.
Huygens's 5. The ideas which led Huygens to his results, are
and results, to be found in a posthumous treatise of his, De Motu
Corporum ex Percussions^ 1703. We shall examine these
in some detail. The assumptions from which Huygens
m
o
m
O
V V
Fig. 158.
Fig. 159.
An Illustration from De Percussione (Huygens).
proceeds are : (i) the law of inertia ; (2) that elastic
bodies of equal mass, colliding with equal and oppo
site velocities, separate after impact with the same ve
locities ; (3) that all velocities are relatively estimated ;
(4) that a larger body striking a smaller one at rest
imparts to the latter velocity, and loses a part of its
own ; and finally (5) that when one of the colliding
bodies preserves its velocity, this also is the case with
the other*
THE EXTENSION OF THE PRINCIPLES. 315
Huygens, now, imagines two equal elastic masses. First, equal
, . , . , IT i AT elastic
which meet with equal and opposite velocities v. After masses ex
the impact they rebound from each other with exactly locitiec
the same velocities. Huygens is right in assuming and
not deducing this. That elastic bodies exist which re
cover their form after impact, that in such a transac
tion no perceptible vis viva is lost, are facts which ex
perience alone can teach us. Huygens, now, conceives
the occurrence just described, to take place on a boat
which is moving with the velocity v. For the specta
tor in the boat the previous case still subsists ; but for
the spectator on the shore the velocities of the spheres
before impact are respectively 2 z? and 0, and after im
pact and 2 v. An elastic body, therefore, impinging
on another of equal mass at rest, communicates to the
latter its entire velocity and remains after the impact
itself at rest. If we suppose the boat affected with any
imaginable velocity, u, then for the spectator on the
shore the velocities before impact will be respectively
// + ' and u r, and after impact // v and // j~ v.
But since u j v and u v may have any values what
soever, it may be asserted as a principle that equal
elastic masses exchange in impact their velocities.
A body at rest, however great, is set in motion Second, the
, 07 relativeve
by a body which strikes it, however small: as Gaiocityofap
. proach and
lileo pointed out. Huygens, now, recession is
shows, that the approach of the *
bodies before impact and their :
recession after impact take place
with the same relative velocity. A Flg ' I6 " '
body m impinges on a body of mass M at rest, to which
it imparts in impact the velocity, as yet undetermined,
w. Huygens, in the demonstration of this proposition,
supposes that the event takes place on a boat moving
316 THE SCIENCE OF MECHANICS,
from M towards m with the velocity w/2. The initial
velocities are, then, v w/2 and w/2, ; and the final
velocities, x and f tu/2. But as M has not altered
the value, but only the sign, of its velocity, so m, If a
loss of vis viva is not to be sustained in elastic impact,
can only alter the sign of its velocity. Hence, the final
velocities are (v w/2) and + w/2. As a fact,
then, the relative velocity of approach before impact
Is equal to the relative velocity of separation after im
pact. Whatever change of velocity a body may suffer,
in every case, we can, by the fiction of a boat in mo
tion, and apart from the algebraical signs, keep the
value of the velocity the same before and after impact.
The proposition holds, therefore, generally.
ThircUfthe If two masses M and ;;/ collide, with velocities V
of approach and v inversely proportional to the masses, Rafter im
?y proper 6 " pact will rebound with the velocity F"and m with the
massesso 6 velocity v. Let us suppose that the velocities after
focitieVoT impact are F t and v^ ; then by the preceding proposi
tion we must have V \ v = V^ f v v and by the prin
ciple of vis viva
MV* mv* _MV^ mv^
_ ] _ _ _____  _ .
Let us assume, now, that ^ 1 =v{ w, then, neces
sarily, V l = V w ; but on this supposition
MV* , mv* MV* , mv*
_ . _ _. _.
And this equality can, in the conditions of the case,
only subsist if w = ; wherewith the proposition above
stated is established.
Huygens demonstrates this by a comparison, con
structively reached, of the possible heights of ascent
of the bodies prior and subsequently to impact. If
THE EXTENSION OF THE PRINCIPLES. 317
the velocities of the impinging bodies are not inversely This propo
proportional to the masses, they may be made such by thetiction
i r r i rr^t t Of a moving
the fiction ot a boat in motion. The proposition thus boat, made
,,,..., to apply to
includes all imaginable cases. an cases.
The conservation of vis viva in impact is asserted
by Huygens in one of his last theorems (n), which he
subsequently also handed in to the London Society.
But the principle is unmistakably at the foundation of
the previous theorems.
6. In taking up the study of any event or phenom Typical
. modes of
enon A, we may acquire a knowledge of its component natural in
elements by approaching it from the point of view of a
different phenomenon 2?, which we already know \ in
which case our investigation of A will appear as the
application of principles before familiar to us. Or, we
may begin our investigation with A itself, and, as na
ture is throughout uniform, reach the same principles
originally in the contemplation of A. The investiga
tion of the phenomena of impact was pursued simul
taneously with that of various other mechanical pro
cesses, and both modes of analysis were really pre
sented to the inquirer.
To begin with, we may convince ourselves that the impact in
the New
problems of impact can be disposed of by the New toman
tonian principles, with the help of only a minimum of view.
new experiences. The investigation of the laws of im
pact contributed, it is true, to the discovery of New
ton's laws, but the latter do not rest solely on this foun
dation. The requisite new experiences, not contained
in the Newtonian principles, are simply the informa
tion that there are elastic and inelastic bodies. Inelastic
bodies subjected to pressure alter their form without
recovering it ; elastic bodies possess for all their forms
definite systems of pressures, so that every alteration
3 i8 THE SCIENCE OF MECHANICS.
of form is associated with an alteration of pressure, and
vice versa. Elastic bodies recover their form ; and the
forces that induce the form alterations of bodies do not
come into play until the bodies are in contact.
First, in Let us consider two inelastic masses M and m mov
nusses. ing respectively with the velocities F"and v. If these
masses come in contact while possessed of these un
equal velocities, internal formaltering forces will be
set up in the system M, m. These forces do not alter
the quantity of motion of the system, neither do they
displace its centre of gravity. With the restitution of
equal velocities, the formalterations cease and in in
elastic bodies the forces which produce the alterations
vanish. Calling the common velocity of motion after
Impact ?/, it follows that Mu f mu = MV 'j Mv, or
v = (MF+ mv}/(M + m), the rule of Wallis.
impact in Now let us assume that we are investigating the
iTnt Joint phenomena of impact without a previous knowledge of
Newton's principles. We very soon discover, when
we so proceed, that velocity is not the sole determina
tive factor of impact; still another physical quality is
decisive weight, load, mass, pondus, moles, massa. The
moment we have noted this fact, the simplest case is
easily dealt with. If two bodies of equal weight or
equal mass collide with equal and
:Sr *~2: opposite velocities ; if, further, the
vv \~J bodies do not separate after impact
* m * , \
but retain some common velocity,
plainly the sole uniquely deter
mined velocity after the collision is the velocity 0. If,
further, we make the observation that only the dif
ference of the velocities, that is only relative velocity,
determines the phenomenon of impact, we shall, by
imagining the environment to move, (which experience
77IE EXTENSION OF THE PRIXCIPLES.
3*9
tells us has no influence on the occurrence,) also readily
perceive additional cases. For equal inelastic masses
with velocities v and or v and v' the velocity after
impact is #/2 or ( v + '')/ 2  ^ stan< 3s to reason that
we can pursue such a line of reflection only after ex
perience has informed us what the essential and de
cisive features of the phenomena are.
If we pass to unequal masses, we must not only The expe
i i r riential
know from experience that mass generally is of conse conditions
.f f ...... of this
quence, but also in what manner its influence is effec method,
tive. If, for example, two bodies of masses i and 3
with the velocities v and F collide, we might reason
V
Fig. 162.
Fig. 163.
thus. We cut out of the mass 3 the mass i (Fig. 162),
and first make the masses i and i collide : the result
ant velocity is (v + ^0/2. There are now left, to
equalise the velocities (v + ^")/2 and V 9 the masses
i j i == 2 and 2, which applying the same principle
gives
2
4 """ 1 + 3 *
Let us now consider, more generally, the masses
m and m' 9 which we represent in Fig. 163 as suitably
proportioned horizontal lines. These masses are af
fected with the velocities v and v', which we represent
by ordinates erected on the masslines. Assuming that
3 2o THE SCIENCE OF MECHANICS.
its points of w < ///, we cut off from m' a portion m. The offsetting
wffih'e of ;;/ and m gives the mass 2m with the velocity (v f
Newtonian. , dotted line indicates this relation. We
proceed similarly with the remainder m m. We cut
off from 2m a portion m' m, and obtain the mass
2 m (m m) with the velocity (v + ?'')/ 2 an< ^ tne
mass 2 (m' Hi) with the velocity [(# + p')/a + v'~\/2.
In this manner we may proceed till we have obtained
for the whole mass m + m' the same velocity u. The
constructive method indicated in the figure shows very
plainly that here the surface equation (m + m'} u =
mv \ mv subsists. We readily perceive, however,
that we cannot pursue this line of reasoning except the
sum m v + m't>', that is the form of the influence of m
and v 9 has through some experience or other been pre
viously suggested to us as the determinative and de
cisive factor. If we renpunce the use of the Newtonian
principles, then some other specific experiences con
cerning the import of m v which are equivalent to those
principles, are indispensable.
Second, the 7. The impact of elastic masses may also be treated
impact of . . 1
elastic by the Newtonian principles. The sole observation
masses in . . .  , r . ri .,_.
Newton's here required is, that a deformation of elastic bodies
calls into play forces of restitution, which directly de
pend on the deformation. Furthermore, bodies pos
sess impenetrability; that is to say, when bodies af
fected with unequal velocities meet in impact, forces
which equalise these velocities are produced. If two
elastic masses M } m with the velocities C, c collide, a
deformation will be effected, and this deformation will
not cease until the velocities of the two bodies are
equalised. At this instant, inasmuch as only internal
forces are involved and therefore the momentum and
THE EXTEXS10X OF THE PRINCIPLES. 321
the motion of the centre of gravity of the system re
main unchanged, the common equalised velocity will be
MC + m c
u = .
J/J m
Consequently, up to this time, M's velocity has suf
fered a diminution C // ; and M'S an increase // r.
But elastic bodies being bodies that recover their
forms, in perfectly elastic bodies the very same forces
that produced the deformation, will, only in the in
verse order, again be brought into play, through the
very same elements of time and space. Consequently,
on the supposition that m is overtaken by M, M will a
second time sustain a diminution of velocity C //, and
;;/ will a second time receive an increase of velocity
n c. Hence, we obtain for the velocities V, v after
impact the expressions V= 2u C and v = 2 u c, or
_
If in these formulae we put M=m, it will f ollow T be deduc
c tion by this
that V= c and v = C : or, if the impinging masses are x ie T of ali
. . . the laws.
equal, the velocities which they have will be inter
changed. Again, since in the particular case Jlf/m =
c/C or MC+mc = Q also u = 0, it follows that
V= 2u C= C and v = Zu c = c; that is,
the masses recede from each other in this case with the
same velocities (only oppositely directed) with which
they approached. The approach of any two masses
M" 9 m affected with the velocities C, c 9 estimated as
positive when in the same direction, takes place with
the velocity C c\ their separation with the velocity
V v. But it follows at once from V= 2u C,
v = Zuc, that V v = (C~ c); that is, the rela
tive velocity of approach and recession is the same.
322 THE SCIRXCR OF MECHANICS.
By the use of the expressions V=2u C and v =
2 # ^ W e also very readily find the two theorems
m i) = MC + m c and
+ me 2 ,
which assert that the quantity of motion before and
after impact, estimated in the same direction, is the
same, and that also the vis viva of the system before
and after impact is the same. We have reached, thus,
by the use of the Newtonian principles, all of Huy
gens's results.
The impii 8. If we consider the laws of impact from Huygens's
H a u% g ens's point of view, the following reflections immediately
claim our attention. The height of ascent which the
centre of gravity of any system of masses can reach is
given by its vis viva, JJSV/2Z/ 2 . In every case in which
work is done by forces, and in such cases ihe masses
follow the forces, this sum is increased by an amount
equal to the work done. On the other hand, in every
case in which the system moves in opposition to forces,
that is, when work, as we may say, is done ^lpon the
system, this sum is diminished by the amount of work
done. As long, therefore, as the algebraical sum of
the work done on the system and the work done by the
system is not changed, whatever other alterations may
take place, the sum.20ze' 2 also remains unchanged.
Huygens now, observing that this first property of ma
terial systems, discovered by him in his investigations
on the pendulum, also obtained in the case of impact,
could not help remarking that also the sum of the
vires mvce must be the same before and after im
pact. For in the mutually effected alteration of the
forms of the colliding bodies the material system con
sidered has the same amount of work done on it as, on
THE EXTEXSIQX OF THE PRINCIPLES. 323
the reversal of the alteration, is done by it, provided al
ways the bodies develop forces wholly determined by
the shapes they assume, and that they regain their
original form by means of the same forces employed to
effect its alteration. That the latter process takes
place, definite experience alone can inform us. This law
obtains, furthermore, only in the case of socalled per
fectly elastic bodies.
Contemplated from this point of view, the majority The deduc
_ . . ^ i tion f the
of the Huygeman laws of impact follow at once. Equal laws of im
, . , ,11 P acr by the
masses, which strike each other with equal but oppo notion of
i 111 *TM "'^" r 'and
site velocities, rebound with the same velocities. The work.
velocities are uniquely determined only when they are
equal, and they conform to the principle of vis viva
only by being the same before and after impact. Fur
ther it is evident, that if one of the unequal masses in
impact change only the sign and not the magnitude of
its velocity, this must also be the case with the other.
On this supposition, however, the relative velocity of
separation after impact is the same as the velocity of
approach before impact. Every imaginable case can
be reduced to this one. Let c and c' be the velocities
of the mass ;;/ before and after impact, and let them be
of any value and have any sign. We imagine the whole
system to receive a velocity // of such magnitude that
ti \ c = (?i + O r = (V O/2 I* will b e seen
thus that it is always possible to discover a velocity of
transportation for the system such that the velocity of
one of the masses will only change its sign. And so
the proposition concerning the velocities of approach
and recession holds generally good.
As Huygens's peculiar group of ideas was not fully
perfected, he was compelled, in cases in which the ve
locityratios of the impinging masses were not origin
THE SCIENCE OF MECHANICS.
mass. ea
Construc
the special
and general
case of im
pact.
ally known, to draw on the GalileoNewtonian system
for certain conceptions, as was pointed out above.
Such an appropriation of the concepts mass and mo
mentum, is contained, although not explicitly ex
pressed, in the proposition according to which the ve
locity of each impinging mass simply changes its sign
when before impact M/m = c/C. If Huygens had
wholly restricted himself to his own point of view, he
would scarcely have discovered this proposition, al
though, once discovered, he was able, after his own
fashion, to supply its deduction. Here, owing to the
fact that the momenta produced are equal and oppo
site, the equalised velocity of the masses on the com
pletion of the change of form will be u == 0. When the
alteration of form is reversed, and the same amount of
work is performed that the system originally suffered,
the same velocities with opposite signs will be restored.
If we imagine the entire system affected with a ve
locity of translation, this particular case will simulta
neously present ther#<?r0/case.
Let the impinging masses be
represented in the figure by
MjBC and m = AC (Fig.
164), and their respective velo
cities by C = AD and c = BE.
On AB erect the perpendicular
CF 9 and through F draw IK
""""^J
F,*"""'
H
K
^\^
F
C
R
Fig. 164.
parallel to AB.
Then ID = (m. C c)/(M+ vi) and
z )* O n ^ e supposition now
KE = (M . C
that we make the masses M and m collide with the
velocities ID and KE, while we simultaneously impart
to the system as a whole the velocity
u = AI=KB = C (m . Cc~)/(M + m) =
~~ ) = (MC+
THE EXTENSION OF THE PRINCIPLES. 325
the spectator who is moving forwards with the velocity
u will see the particular case presented, and the spec
tator who is at rest will see the general case, be the
velocities what they may. The general formulae of im
pact, above deduced, follow at once from this concep
tion. We obtain :
V=AG=C 2 ^
M + m M + m
1 M \ m J/4 m
Huygen's successful employment of the fictitious signifi
. . . , canceofthe
motions is the outcome of the simple perception that fictitious
bodies not affected with differences of velocities do not
act on one another in impact. All forces of impact are
determined by differences of velocity (as all thermal
effects are determined by differences of temperature).
And since forces generally determine, not velocities,
but only changes of velocities, or, again, differences of
velocities, consequently, in every aspect of impact the
sole decisive factor is differences of velocity. With re
spect to which bodies the velocities are estimated, is
indifferent. In fact, many cases of impact which from
lack of practice appear to us as different cases, turn
out on close examination to be one and the same.
Similarly, the capacity of a moving body for work, velocity, a
. . . . physical
whether we measure it with respect to the time of its level.
action by its momentum or with respect to the distance
through which it acts by its vis viva, has no signifi
cance referred to' a single body. It is invested with
such, only when a second body is Introduced, and, in
the first case, then, it is the difference of the veloci
ties, and in the second the square of the difference that
is decisive. Velocity is a physical level, like tempera
ture, potential function, and the like.
3 26 THE SCIENCE OF MECHANICS.
Possible It remains to be remarked, that Huygens could
origin 6 ^ have reached, originally, In the investigation of the
ideis. enss phenomena of impact, the same results that he pre
viously reached by his Investigations of the pendulum.
In every case there is one thing and one thing only to
be done, and that Is, to discover in all the facts the same
elements^ or, if we will, to rediscover in one fact the
elements of another which we already know. From
which facts the investigation starts, is, however, a
matter of historical accident.
Conserva 9. Let us close our examination of this part of the
mentiim 1 ?^ subject with a few general remarks. The sum of the
terpreted.  , P i j j
momenta of a system of moving bodies is preserved in
impact, both In the case of inelastic and elastic bodies.
But this preservation does not take place precisely in
the sense of Descartes. The momentum of a body is
not diminished in proportion as that of another is in
creased ; a fact which Huygens was the first to note.
If, for example, two equal Inelastic masses, possessed
of equal and opposite velocities, meet in Impact, the
two bodies lose in the Cartesian sense their entire mo
mentum. If, however, we reckon all velocities in a
given direction as positive, and all in the opposite as
negative, the sum of the momenta is preserved. Quan
tity of motion, conceived in this sense, is always pre
served.
The vis viva of a system of inelastic masses is al
tered in impact ; that of a system of perfectly elastic
masses is preserved. The diminution of vis viva pro
duced in the impact of inelastic masses, or produced
generally when the impinging bodies move with a com
mon velocity, after impact, is easily determined. Let
M, m be the masses, C, c their respective velocities be
THE EXTENSIO.V OF THE PRINCIPLES. 327
fore Impact, and // their common velocity after impact ; Conserva
. T a.1 1 f tl0n C>f 7"V
then the loSS Of VIS Viva IS svra in 5m
pact inter
mc  .l(JJ/ m\lP ....... fl preted.
which in view of the fact that it = (J/C+ /;/ f },'(M+ //:)
may be expressed in the form (J/w/J/+l//) (C <r) 2 ,
Carnot has put this loss in the form
^M(C //) 2 f ///(// <) * .......... (2)
If we select the latter form, the expressions J J/(C // ; 2
and ;;/(// r) 2 will be recognised as the vis viva gen
erated by the work of the internal forces. The loss of
vis viva in impact is equivalent, therefore, to the work
done by the internal or socalled molecular forces. If
we equate the two expressions (i) and (2), remember
ing that (M + ?//) // = MC j /// c, we shall obtain an
identical equation. Carnot's expression Is Important
for the estimation of losses due to the Impact of parts
of machines.
In all the preceding expositions we have treated oblique
. . impact.
the impinging masses as points which moved only In the
direction of the lines joining them. This simplifica
tion is admissible when the centres of gravity and the
point of contact of the Impinging masses He in one
straight line, that Is, in the case of socalled direct Im
pact. The investigation of what Is called oblique Im
pact Is somewhat more complicated, but presents no
especial Interest In point of principle.
A question of a different character was treated by The centre
WALLIS. If a body rotate about an axis and Its motion sion ercus
be suddenly checked by the retention of one of its
points, the force of the percussion will vary with the
position (the distance* from, the axis) of the point ar
rested. The point at which the intensity of the impact
is greatest is called by Wallis the centre of percussion.
3 28 THE SCIENCE OF MECHANICS.
If this point be checked, the axis will sustain no pres
sure. We have no occasion here to enter in detail
into these investigations ; they were extended and de
veloped by Wallis's contemporaries and successors in
many ways.
aiiis io. We will now briefly examine, before concluding
n u tfojg section, an interesting application of the laws of
impact ; namely, the determination of the velocities of
projectiles by the ballistic pendulum. A mass M is sus
pended by a weightless and massless
string (Fig. 165), so as to oscillate as a
pendulum. While in the position of
equilibrium it suddenly receives the hori
zontal velocity V. It ascends by virtue
of this velocity to an altitude h = (/)
(1 cos a) = V*/Zg, where /denotes the
length of the pendulum, a the angle of
elongation, and g the acceleration of
gravity. As the relation T=. Ttv'l/g subsists between
the time of oscillation T and the quantities /, g, we
easily obtain V= (gT/n) 1/2 (1 cos a), and by the
use of a familiar trigonometrical formula, also
its formula. If now the velocity V is produced by a projectile of
the mass m which being hurled with a velocity v and
sinking in M is arrested in its progress, so that whether
the impact is elastic or inelastic, in any case the two
masses acquire after impact the common velocity V 9 it
follows that m?; = (M{ m) V\ or, if m be sufficiently
small compared with M, also v = (Mjm) V\ whence
finally
2 M _ . a
THE EXTENSION OF THE PRINCIPLES. 329
If it Is not permissible to regard the ballistic pen A different
dulum as a simple pendulum, our reasoning, In con
formity with principles before employed,, will take the
following shape. The projectile m with the velocity v
has the momentum mv, which is diminished by the
pressure/ due to impact In a very short interval of
time r to mV. Here, then, m (v F) =/r, or, if V
compared with v Is very small, mv =ff. With Pon
celet, we reject the assumption of anything like in
stantaneous forces, which generate instanter velocities.
There are no instantaneous forces. What has been
called such are very great forces that produce per
ceptible velocities In very short intervals of time, but
which In other respects do not differ from forces that
act continuously. If the force active In impact cannot
be regarded as constant during its entire period of ac
tion, we have only to put in the place of the expression
ft the expression Cpdt. In other respects the reason
ing is the same.
A force equal to that which destroys the momentum The vis
viva and
of the projectile, acts in reaction on the pendulum. If work of the
. pendulum.
we take the line of projection of the shot, and conse
quently also the line of the force, perpendicular to the
axis of the pendulum and at the distance b from It, the
moment of this force will be bp^ the angular accelera
tion generated bpf2mr^^ and the angular velocity pro
duced in time r
b . p r bmv
The vis viva which the pendulum has at the end of
time r is therefore
330 THE SCIENCE OF MECHANICS.
The result, By virtue of this vis viva the pendulum performs
the excursion a, and its weight Mg, (a being the dis
tance of the centre of gravity from the axis,) is lifted
the distance a (I cos a). The work performed here
isMga(l costf), which is equal to the abovemen
tioned vis viva. Equating the two expressions we
readily obtain
I/ 2 M?a 2m r 2 (1 cos a)
v = 7 ,
mb
and remembering that the time of oscillation is
\2rnr*
T =7t \
\ Mga '
and employing the trigonometrical reduction which
was resorted to immediately above, also
2 M a _ .' a
v = . gT . sm .
TC m b 2
interpreta This formula is in every respect similar to that ob
tion of the J r
result. tamed for the simple case. The observations requisite
for the determination of z/, are the mass of the pendu
lum and the mass of the projectile, the distances of
the centre of gravity and point of percussion from the
axis, and the time and extent of oscillation. The form
ula also clearly exhibits the dimensions of a velocity.
The expressions 2/?r and sin (or/2) are simple num
bers, as are also Mjm and a/b, where both numerators
and denominators are expressed in units of the same
kind. But the f actor gT has the dimensions //"*, and
Is consequently a velocity. The ballistic pendulum
was invented by ROBINS and described by him at length
in a treatise entitled New Principles of Gunnery, pub
lished in 1742.
TUE EXTENSION Of THE PRINCIPLES. 331
D ALEMBERT'S PRINCIPLE.
1. One of the most important principles for the History of
r f r the prin
rapid and convenient solution of the problems of me cipie.
chanics is the principle of D' A I ember t. The researches
concerning the centre of oscillation on which almost all
prominent contemporaries and successors of Huygens
had employed themselves, led directly to a series of
simple observations which D' ALEMBERT ultimately gen
eralised and embodied in the principle which goes by
his name. We will first cast a glance at these prelim
inary performances. They were almost without excep
tion evoked by the  desire to replace the deduction of
Huygens, which did not appear sufficiently obvious, by
one that was more convincing. Although this desire was
founded, as we have already seen, on a miscompre
hension due to historical circumstances, we have, of
course, no occasion to regret the new points of view
which were thus reached.
2. The first in importance of the founders of the James Ber
theory of the centre of oscillation, after Huygens, iscomribu
T. , , , rir tionstothe
JAMES BERNOULLI, who sought as early as 1686 to ex theory of
plain the compound pendulum by the lever. He ar of osciiia
rived, however, at results which not only were obscure
but also were at variance with the conceptions of Huy
gens. The errors of Bernoulli were animadverted on
by the Marquis de L'HOPITAL in the Journal de Rotter
dam, in 1690. The consideration of velocities acquired
in infinitely small intervals of time in place of velocities
acquired infinite times a consideration which the last
named mathematician suggested led to the removal
332 THE SCIENCE OF MECHANICS.
of the main difficulties that beset this problem ; and in
1691, in \hQActaEruditorum, and, later, in 1703, in the
Proceedings of the Paris Academy James Bernoulli cor
rected his error and presented his results in a final and
complete form. We shall here reproduce the essential
points of his final deduction.
james Ber A horizontal, massless bar AB (Fig. 166) is free to
duction of rotate about A ; and at the distances r, r' from A the
the law of ,..,,,
the com masses ;;/, m 1 are attached. The accelerations with which
pound pen
duiumfrom these masses as thus connected
^4 will fall must be different from
the accelerations which they
T ,. ff would assume if their connec
Fig. 166.
tions were severed and they fell
freely. There will be one point and one only, at the
distance x, as yet unknown, from A which will fall
with the same acceleration as it would have if it were
free, that is, with the acceleration g. This point is
termed the centre of oscillation.
If m and ;;/ were to be attracted to the earth, not
proportionally to their masses, but m so as to fall when
free with the acceleration cp = grjx and m' with the
acceleration cp' = gr' /x, that is to say, if the natural
accelerations of the masses were proportional to their
distances from A, these masses would not interfere with
one another when connected. In reality, however, m
sustains, in consequence of the connection, an upward
component acceleration g cp, and m' receives in virtue
of the same fact a downward component acceleration
cp' g; that is to say, the former suffers an upward
force of m(g cp)=g(x r/x)m and the latter a
downward force of m' (cp f g) = g (r' xjx) m'.
Since, however, the masses exert what influence
they have on each other solely through the medium of
THE EXTENSION OF THE PRINCIPLES. 333
the lever by which they are joined, the upward force The law of
J J J r the distn
upon the one and the downward force upon the other bution of
r r the effects
must satisfy the law of the lever. If m in conse of the im
pressed
quence of its being connected with the lever is held f ^^ r
back by a force /from the motion which it would take, nouiii's ex
ample.
if free, it will also exert the same force /on the lever
arm r by reaction. It is this reaction pull alone that
can be transferred to m and be balanced there by a
pressure /'== (r/r'}f, and is therefore equivalent to the
latter pressure. There subsists, therefore, agreeably
to what has been above said, the relation g (r' x /x)
m' = r /r' . g (x r/x} m or, (x r) m r = (r r x) m'r',
from which we obtain x = (mr 2 + m'r'^/(inr + m'r'^) 9
exactly as Huygens found it. The generalisation of
this reasoning, for any number of masses, which need
not lie in a single straight line, is obvious.
3. JOHN BERNOULLI (in 1712) attacked in a different The prm
manner the problem of the centre of oscillation. His John Ber
performances are easiest consulted in his Collected lution of
Works {Opera, Lausanne and Geneva, 1762, Vols. Iloftnecen
and IV). We shall examine in detail here the main lation.
ideas of this physicist. Bernoulli reaches his goal by
conceiving the masses and forces separated.
First, let us consider two simple pendulums of dif The first
ferent lengths /, /' whose bobs are affected with gravi Bernoulli's
tational accelerations proportional to the lengths of the
pendulums, that is, let us put ///' g/g'. As the time
of oscillation of a pendulum is T= nV Ijg, it follows
that the times of oscillation of these pendulums will be
the same. Doubling the length of a pendulum, ac
cordingly, while at the same time doubling the accel
eration of gravity does not alter the period of oscilla
tion.
Second, though we cannot directly alter the accel
334
THE SCIEXCE OF MECHANICS.
The second eration of gravity at any one spot on the earth, we
1 can do what amounts virtually to this. Thus, imagine
a straight massless bar of length 2a, free to rotate about
its middle point; and attach to the one ex
' tremity of it the mass m and to the other the
mass m. Then the total mass is m + m' at
j the distance a from the axis. But the force
\ a which acts on it is (in ?;/) g, and the ac
! m celeration, consequently, (m m' /m \ ;;/) g.
Fig. 167. Hence, to find the length of the simple pen
dulum, having the ordinary acceleration of
gravity g, which is isochronous with the present pen
dulum of the length a, we put, employing the preced
ing theorem,
m m
m m
The third
determina
centre of
Third, we imagine a simple pendulum of length i
with the mass m at its extremity. The weight of m
produces, by the principle of the lever, the same ac
celeration as half this force at a distance 2 from the
point of suspension. Half the mass m placed at the
distance 2, therefore, would surfer by the action of the
force impressed at i the same acceleration, and a fourth
of the mass m would surfer double the acceleration ; so
that a simple pendulum of the length 2 having the orig
inal force at distance i from the point of suspension
and onefourth the original mass at its extremity would
be isochronous with the original one. Generalising
this reasoning, it is evident that we may transfer any
force / acting on a compound pendulum at any dis
tance r, to the distance i by making its value rf 9 and
any and every mass placed at the distance r to the
distance i by making its value r*m, without changing
TJ1E EXTENSION OF THE PRINCIPLES. 335
the time of oscillation of the pendulum. If a force /
act on a lever arm a (Fig. 168) while at the distance r
from the axis a mass m is attached, f will be equiva
lent to a force af/r impressed on
m and will impart to it the linear
acceleration af/m r and the angu
lar acceleration af/mr 2 . Hence,
to find the angular acceleration
r J j i Fi S l68 
of a compound pendulum, we
divide the sum of the statical moments by the sum of
the moments of inertia.
BROOK TAYLOR, an Englishman,* also developed The re
. . searches of
this idea, on substantially the same principles, but Brook Tay
quite independently of John Bernoulli. His solution,
however, was not published until some time later, in
1715, in his work, Methodus Incrementorum.
The above are the most important attempts to solve
the problem of the centre of oscillation. We shall see
that they contain the very same ideas that D'Alembert
enunciated in a generalised form.
4. On a system of points M, M', M". . . . connected Motion of a
with one another in any way,f the forces P, P', P". . . . polntssub
are impressed. (Fig. 169.) These forces would im straints. n
part to the free points of the system certain determinate
motions. To the connected points, however, different
motions are usually imparted motions which could
be produced by the forces W, W, W". . . . These
last are the motions which we shall study.
Conceive the force P resolved into W and V, the
force P' into W and V [ ', and the force P" into W"
* Author of Taylor's theorem, and also of a remarkable work on perspec
tive. Trans.
t In precise technical language, they are subject to constraints, that is,
forces regarded as infinite, which compel a certain relation between their
motions, Trans.
33 6 THE SCIENCE OF MECHANICS.
statement and F", and so on. Since, owing to the connections,
of D'Alem . , rrr rrrf r rru rr
berr&prin only the components 17, W , ///.... are effective,
lpe ' therefore, the forces V, V, V" . . . . must be equilib
rated by the connections. We will call the forces P, P',
P" the impressed forces,
the forces W 9 W, W" ,
which produce the ac
tual motions, the effective
forces, and the forces V,
V, V" . . . . the forces
gained and lost, or the
FI i6 equilibrated forces. We
perceive, thus, that if we
resolve the impressed forces into the effective forces
and the equilibrated forces, the latter form a system
balanced by the connections. This is the principle of
D'Alembert. We have allowed ourselves, in its expo
sition, only the unessential modification of putting
forces for the momenta generated by the forces. In this
form the principle was stated by D'ALEMBERT in his
Traite de dynamique, published in 1743.
Various As the'system V, V , V" . ... is in equilibrium, the
which the principle of virtital disp lac erne jits is applicable thereto,
may be ex This gives a second form of D'Alembert's principle.
pressed. ...
A third form is obtained as follows : The forces P, P'. . . .
are the resultants of the components W, W . . . . and
F, V. . . . If, therefore, we combine with the forces
W, W and F, W the forces P, P' ,
equilibrium will obtain. The forcesystem P, W, V
is in equilibrium. But the system Vis independently
in equilibrium. Therefore, also the system P, Wis
in equilibrium, or, what is the same thing, the system
P, Wis in equilibrium. Accordingly, if the effective
forces with opposite signs be joined to the impressed
THE EXTENSION OF TPIE PRINCIPLES.
337
Fig. 170.
forces, the two, owing to the connections, will balance.
The principle of virtual displacements may also be ap
plied to the system P, W. This LAGRANGE did in his
Mccaniqite analytique, 1 788.
The fact that equilibrium subsists between the sys An equiva
tem P and the system W, may be expressed in still pie em
i TTT i ployed by
another way. We may say that Hermann
,. . T J . ,,, ^ andEuler.
the system W is equivalent to the
system P. In this form HER
MANN (Phoronomia, 1716) and
EULER {Comment A cad. Petrop. ,
Old Series, Vol. VII, 1740) employed the principle.
It is substantially not different from that of D' Alembert.
5. We will now illustrate D' Alembert' s principle by
one or two examples.
On a massless wheel and axle with the radii R, r the illustration
loads P and Q are hung, which are not in equilibrium, bert'sprm
ciple by the
We resolve the force P into (i) W '
(the force which would produce the
actual motion of the mass if this were
freej and (2) V, that is, we put
P = W+ ^and also Q = W"+ V'\
it being evident that we may here
disregard all motions that are not
in the vertical. We have, accord
ingly, V= P W and V'= Q W,
and, since the forces V, V are in equilibrium, also
V. R = V. r. Substituting for V, V in the last equa
tion their values in the former, we get
motion of a
wheel and
axle.
Fig. 171.
(i)
which may also be directly obtained by the employ
ment of the second form of D'Alembert's principle.
From the conditions of the problem we readily perceive
33 8 THE SCIENCE OF MECHANICS.
that we have here to deal with a uniformly accelerated
motion, and that all that is therefore necessary is to
ascertain the acceleration. Adopting gravitation meas
ure., we have the forces W and W 9 which produce in
the masses Pjg and Q/g the accelerations y and y' 9
wherefore, W=(P/g)y and W=(Q/g)y f . But we
also know that y'= y(rjR}. Accordingly, equation
(i) passes into the form
whence the values of the two accelerations are ob
tained
PRQr _ , , PR Or
____ 
____
These last determine the motion.
Employ It will be seen at a glance that the same result can
inent of the ., f , . ,
ideas stat be obtained by the employment of the ideas of statical
meat and moment and moment of inertia. We get by this method
moment of .
inertia, to for the angular acceleration
obtain this
8
and as y = R cp and y'= r cp we reobtain the pre
ceding expressions.
When the masses and forces are given, the problem
of finding the motion of a system is determinate. Sup
pose, however, only the acceleration y is given with
which P moves, and that the problem is to find the loads
P and Q that produce this acceleration. We obtain
easily from equation (2) the result P == Q (JR. g + r y)
r /(g y}R z , that is, a relation between P and Q.
One of the two loads therefore is arbitrary. The prob
THE EXTENSION OF THE PRINCIPLES.
339
lem in this form is an indeterminate one, and may be
solved in an infinite number of different ways.
The following may serve as a second example.
A weight P (Fig. 172") free to move on a vertical A second it
. _ lustration
straight line AB, is attached to a cord of the prin
passing over a pulley and carrying a ~
weight Q at the other end. The cord
makes with the line AB the variable
angle a. The motion of the present
case cannot be uniformly accelerated.
But if we consider only vertical mo
tions we can easily give for every
value of a the momentary accelera
tion (y and ;/') of P and Q. Proceeding exactly as
we did in the last case, we obtain
P= W + V,
Q = W + V
also
y cos a = V, or, since y' = y cos a,
Q \ P
Q _[_ j C os a y 1 cos a = P y; whence
> / &
P Qcosa
Fig. 172.
PQcosa
Again the same result may be easily reached by the solution of
employment of the ideas of statical moment and mo also bythe
ment of inertia in a more generalised form. The fol statical mo
lowing reflexion will render this clear. The force, or
statical moment, that acts on .Pis P Q cos a.
the weight Q moves cos a times as fast as P; conse
quently its mass is to be taken cos 2 <# times. The ac
celeration which P receives, accordingly is,
But eraiised! en ~
340 TV/A SCIENCE OF MECHANICS.
P Qcosa P Q cos a
y = Q ^'^p g
 cos 2 tfH v ~
^
In like manner the corresponding expression for y' may
be found.
The foregoing procedure rests on the simple re
mark, that not the circular path of the motion of the
masses is of consequence, but only the relative veloci
ties or relative displacements. This extension of the
concept moment of inertia may often be employed to
advantage.
import and 6. Now that the application of D'Alembert's prin
of ETAiem ciple has been sufficiently illustrated, it will not be diffi
cipie. cult to obtain a clear idea of its significance. Problems
relating to the motion of connected points are here dis
posed of by recourse to experiences concerning the
mutual actions of connected bodies reached in the in
vestigation of problems of equilibrium. Where the last
mentioned experiences do not suffice, D'Alembert's
principle also can accomplish nothing, as the examples
adduced will amply indicate. We should, therefore,
carefully avoid the notion that D'Alembert's principle
is a general one which renders special experiences su
perfluous. Its conciseness and apparent simplicity are
wholly due to the fact that it refers us to experiences
.already in our possession. Detailed knowledge of the
subject under consideration founded on exact and mi
nute experience, cannot be dispensed with. This knowl
edge we must obtain either from the case presented,
by a direct investigation, or we must previously have
obtained it, in the investigation of some other subject,
and carry it with us to the problem in hand. We learn,
in fact, from D'Alembert's principle, as our examples
show, nothing that we could not also have learned by
THE EXTENSION OF THE PRINCIPLES. 341
other methods. The principle fulfils in the solution
of problems, the office of a routineform which, to a
certain extent, spares us the trouble of thinking out
each new case, by supplying directions for the employ
ment of experiences before known and familiar to us.
The principle does not so much promote our insight
into the processes as it secures us a practical mastery of
them. The value of the principle is of an economical
character.
When we have solved a problem by D'Alembert'sThereia
. . tlon *
principle, we may rest satisfied with the experiences D'Aiem^
r c J bert's pnn
previously made concerning equilibrium, the applica cipie to .the
tion of which the principle implies. But if we wish cipies of
r * mechanics.
clearly and thoroughly to apprehend the phenomenon,
that is, to rediscover In it the simplest mechanical ele
ments with which we are familiar, we are obliged to
push our researches further, and to replace our expe
riences concerning equilibrium either by the Newtonian
or by the Huygenian conceptions, in some way similar
to that pursued on page 266. If we adopt the former
alternative, we shall mentally see the accelerated mo
tions enacted which the mutual action of bodies on one
another produces ; if we adopt the second, we shall di
rectly contemplate the work done, on which, in the
Huygenian conception, the vis viva depends. The latter
point of view is particularly convenient if we employ
the principle of virtual displacements to express the
conditions of equilibrium of the system V or P W~
D'Alembert's principle then asserts, that the sum of
the virtual moments of the system V 9 o"r of the system
P W, is equal to zero. The elementary work o' ihe
equilibrated forces, If we leave out of account the strain
ing of the connections, is equal to zero. The total
work done, then, is performed solely by the system P,
342 THE SCIENCE OF MECHANICS.
and the work performed by the system ?F"must, accord
ingly, be equal to the work done by the system P. All
the work that can possibly be done is due, neglecting
the strains of the connections, 'to the impressed forces.
As will be seen, D'Alembert's principle in this form is
not essentially different from the principle of vis viva.
Form of ap 7. In practical applications of the principle of
D'AJem^ D'Alembert it is convenient to resolve every force P .
cfpietand 1 impressed on a mass m of the system into the mutually
ingequa perpendicular components X, Y 9 Z parallel to the axes
tions of mo . , ..  . ....
tion. of a system of rectangular coordinates ; every effective
force J^into corresponding components m% 9 mrj, m2,,
where <?, 77, 5 denote accelerations in the directions of
the coordinates ; and every displacement, in a similar
manner, into three displacements dx, $y, dz. As the
work done by each component force is effective only in
displacements parallel to the directions in which the
components act, the equilibrium of the system (P, W}
is given by the equation
2\ (X m <?) 6x + ( Y mrj) dy + (Zmg) dz\ = $ (1)
or
2(Xdx + Ydy + Z8s) = 2m(gdx+riSy + Zdz). . (2)
These two equations are the direct expression of the
proposition above enunciated respecting the possible
work of the impressed forces. If this work be = 0, the
particular case of equilibrium results. The principle
of virtual displacements flows as a special case from
this expression of D'Alembert's principle ; and this is
quite in conformity with reason* since in the general
as well as in the particular case the experimental per
ception of the import of work is the sole thing of con
sequence.
Equation (i) gives the requisite equations of mo
THE EXTENSION OF THE PRINCIPLES. 343
tion ' } we have simply to express as many as possible
of the displacements 6x, d}', ds by the others in terms
of their relations to the latter, and put the coefficients
of the remaining arbitrary displacements = 0, as was
illustrated in our applications of the principle of vir
tual displacements*
The solution of a very few problems by D'Alem Conve
, ......_ . i r ^nienceand
bert s principle will suffice to impress us with a full utility of
, . . ... f , D'Alem
sense of its convenience. It will also give us the con bert's prin
viction that it is possible, in every case in which it may
be found necessary, to solve directly and with perfect
insight the very same problem by a consideration of
elementary mechanical processes, and to arrive thereby
at exactly the same results. Our conviction of the
feasibility of this operation renders the performance of
it, In cases in which purely practical ends are In view,
unnecessary.
THE PRINCIPLE OF VIS VIVA.
T. The principle of vis viva, as we know, was first The orig
employed by HUYGENS. JOHN and DANIEL BERNOULLI iSaYfqrm of
had simply to provide for a greater generality of ex C ipie. nn
pression ; they added little. If/, p', p". . . . are weights,
m, m', m" . . . . their respective masses, k, h' 9 h". . . . the
distances of descent of the free or connected masses,
and v, v 1 ', v" . . . . the velocities acquired, the relation
obtains
If the initial velocities are not = 0, but are # , v Q ' 9
?/ ". . . ., the theorem will refer to the increment of the
vis viva by the work and read
344
THE SCIENCE OF MECHANICS.
pie appli
tc forces
i The principle still remains applicable when p . . . .
ed . r
of are, not weights, but any constant forces, and h . . .
not the vertical spaces fallen through, but any paths in
the lines of the forces. If the forces considered are
variable, the expressions//;, /'//. . , . must be replaced
by the expressions Cpds, Cj>' ds f , . . ., in which J> de
notes the variable forces and ds the elements of dis
tance described in the lines of the forces'. Then
or
The princi
ple illus
trated by
the motion
of a wheel
and axle.
v Q ^ ......... (1)
2. In illustration of the principle of vis viva we
shall first consider the simple problem which we treated
by the principle of D'Alembert. On
a wheel and axle with the radii R, r
hang the weights P, Q. When this
machine is set in motion, work is per
formed by which the acquired vis viva
is fully determined. For a rotation of
the machine through the angle a, the
work is
P. Ra Q. ra a(PR Qr).
Calling the angular velocity which
corresponds to this angle of rotation, cp, the vis viva
generated will be
Fig. 173
<rvr = g_
% 2g J
Consequently, the equation obtains
a (PR Qr) =
(1)
Now the motion of this case is a uniformly accelerated
motion ; consequently, the same relation obtains here
between the angle a, the angular velocity cp, and the
THE EXTENSION OF THE PRINCIPLES.
345
angular acceleration if;, as obtains in free descent be
tween s, ?>, g. If in free descent s = v 2 /2g, then here
a = (p 2 /2tfj.
Introducing this value of a in equation (i), we get
for the angular acceleration of P, ip = (PR Qr/
PR* f Qr 2 )g, and, consequently, foruts absolute ac
celeration y = (Pit Qr/P~R 2 ~^~Qr 2 ] Rg, exactly as
in the previous treatment of the problem.
As a second example let us consider the case of a A roiling
cylinder on
massless cylinder of radius r. m the surface of which, an inclined
plane.
diametrically opposite each other, are fixed two equal
masses m, and which in consequence of the weight of
Fig. 174.
 375
these masses rolls without sliding down an inclined
plane of the elevation a. First, we must convince our
selves, that in order to represent the total vis viva of
the system we have simply to sum up the vis viva of
the motions of rotation and progression. The axis of
the cylinder has acquired, we will say, the velocity u
' in the direction of the length of the inclined plane, and
we will denote by v the absolute velocity of rotation of
the surface of the cylinder. The velocities of rotation v
of the two masses m make with the velocity of progres
sion u the angles 6 and Q' (Fig. 175), where 6 \~ 6'
= 180. The compound velocities w and z satisfy
therefore the equations
w 2 = u 2 \ v 2 2 uv cos
z 2 =u 2 + v 2
34 6 THE SCIENCE OF MECHANICS.
The law of But since cos ft = cos 6', it follows that
motion of
su f h i l iv* + z* ='2 // 2 4 2e/ 2 or.
cylinder. ' ' ' '
}>mw j \mz = \m'lu^ + J/tf2z' 2 =;//// 2  772 z/ 2 .
If the cylinder moves through the angle <?, ;;/ describes
in consequence of the rotation the space r cp, and the
axis of the cylinder is likewise displaced a distance rep.
As the spaces traversed are to each other, so also
are the velocities v and ?/, which therefore are equal.
The total vis viva may accordingly be expressed by
2m if 2 . If /is the distance the cylinder travels along
the length of the inclined plane, the work done is
2?ng. /sin a = 2mu 2 ; whence u = V gl* sin a. If we
compare with this result the velocity acquired by a body
in sliding down an inclined plane, namely, the velocity
j/'a^/sina, it will be observed that the contrivance we
are here considering moves with only onehalf the ac
celeration of descent that (friction neglected) a sliding
body would under the same circumstances. The rea
soning of this case is not altered if the mass be uni
formly distributed over the entire surface of the cylin
der. Similar considerations are applicable to the case
of a sphere rolling down an inclined plane. It will be
seen, therefore, that Galileo's experiment on falling
bodies is in need of a quantitative correction.
A modifica Next, let us distribute the mass ;;/ uniformly over
preceding the surface of a cylinder of radius R> which is coaxal
with and rigidly joined to a massless cylinder of radius
r, and let the latter roll down the inclined plane. Since
here v/u = jR./r, the principle of vis viva gives mgl
~f j^ 2 /r 2 ), whence
THE EXTENSION OF TFIE PRINCIPLES. 347
For Rjr = i the acceleration of descent assumes its
previous value g/z. For very large values of R/r the
acceleration of descent is very small. When R/r = oo
it will be impossible for the machine to roll down the
inclined plane at all.
As a third example, we will consider the case of a The motion
chain, whose total length is /, and which lies partly on on an in
a horizontal plane and partly on a plane having the plane,
angle of elevation a. If we imagine the surface on
which the chain
rests to be very
smooth, any very
small portion of
the chain left hang
, . Fig. 176.
ing over on the in
clined plane will draw the remainder after it. If /* is
the mass of unit of length of the chain and a portion x
is hanging over, the principle of vis viva will give for
the velocity v acquired the equation
IJilv^ X . X 2 .
^ 2  = pxg 2" sm a = ^ Y sm a >
or v = x ~\/g sin a /I. In the present case, therefore,
the velocity acquired is proportional to the space de
scribed. The very law holds that Galileo first con
jectured was the law of freely falling bodies. The
same reflexions, accordingly, are admissible here as at
page 248.
3. Equation (i), the equation of vis viva, can always Extension
be employed, to solve problems of moving bodies, cipufo?"*
when the total distance traversed and the force that vzva '
acts in each element of the distance are known. It was
disclosed, however, by the labors of Euler, Daniel Ber
noulli, and Lagrange, that cases occur in which the
34S
THE SCIENCE, OF MECHANICS.
principle of vis viva can be employed without a knowl
edge of the actual path of the motion. We shall see
later on that Clairaut also rendered important services
in this field.
There Galileo, even, knew that the velocity of a heavy
E e ui r e C r. ieS falling body depended solely on the vertical height de
scended through, and not on the length or form of the
path traversed. Similarly, Huygens finds that the vis
viva of a heavy material system is dependent on the
vertical heights of the masses of
the system. Euler was able to
make a further step in advance.
If a body K (Fig. 177). is at
tracted towards a fixed centre
C in obedience to some given
law, the increase of the vis viva
in the case of rectilinear ap
proach is calculable from the
initial and terminal distances
( r o> r i} But tne increase is the
same, if ^"passes at all from the
position r to the position r f , independently of the
form of its path, KB. For the elements of the work
done must be calculated from the projections on the
radius of the actual displacements, and are thus ulti
mately the same as before.
The re If K is attracted towards several fixed centres C,
searches of .
Daniel Ber C , C . . . ., the increase of its vis viva depends on the
noulli and ...
Lagrange. initial distances r , r Q , r Q . . , . and on the terminal
distances r lt r f ', r t ". . . ., that is on the initial and ter
minal positions of K. Daniel Bernoulli extended this
idea, and showed further that where movable bodies
are in a state of mutual attraction the change of vis viva
is determined solely by their initial and terminal dis
Fig. 177
THE EXTENSION OF THE PRINCIPLES. 349
tances from one another. The analytical treatment of
these problems was perfected by Lagrange. If we join
a point having the coordinates a, &, c with a point hav
ing the coordinates oc, y, z, and denote by r the length
of the line of junction and by a, fi, y the angles that
line makes with the axes of x, y, z, then, according to
Lagrange, because
r * = ( x _ fl )3 + (y _ J)2 + ( Z _ ^2,
a; # <^r /, v dr
 ~   
=   r,
r dy
z c dr
cos y =  = .
7 r dz
Accordingly, if f(r) = ~ is the repulsive force, or The force
J dr compo
. . . nents, par
the negative of the attractive force acting between the tiai differ
ential coef
two points, the components will be ficientsof
^ , the same
d r dP(r\ function of
A' cosrdi
^. ~
ir ^ N /?  
Y=f(r) cos/3 == ~~ T = ~ ,
y dr dy dy
7 , //^(r) ^/r dF(f}
Z =f(r) cos y = ^+ = ,~^
J ^ J y dr dz dz
The forcecomponents, therefore, are the partial
differential coefficients of one and the same function of
r, or of the coordinates of the repelling or attracting
points. Similarly, if several points are in mutual ac
tion, the result will be
TTOL
dx
dy'
350 THE SCIENCE OF MECHANICS*
The force where U is a function of the coordinates of the points.
one ion. 'jr^g f unc ti on was subsequently called by Hamilton*
the forcefunction.
Transforming, by means of the conceptions here
reached, and under the suppositions given, equation
(i) into a form applicable to rectangular coordinates,
we obtain
2J(Xdx + Ydy + Zdz) = 2%m (v* z> *) or,
since the expression to the left is a complete differen
tial,
, dU dU .
x + _ _ dy + jdz =
dx dy ' dz
where U^ is a function of the terminal values and U Q
the same function of the initial values of the coordi
nates. This equation has received extensive applica
tions, but it simply expresses the knowledge that under
the conditions designated the work done and therefore
also the ins viva of a system is dependent on the posi
tions, or the coordinates, of the bodies constituting it.
If we imagine all masses fixed and only a single
one in motion, the work changes only as U changes.
The equation U= constant defines a socalled level
surface, or surface of equal work. Movement upon
such a surface produces no work. U increases in the
direction in which the forces tend to move the bodies.
VII.
THE PRINCIPLE OF LEAST CONSTRAINT.
i. GAUSS enunciated (in Crell&s Journal fur Mathe
matik, Vol. IV, 1829, p. 233) a new law of mechanics,
the principle of least constraint. He observes, that, in
* On a General Method in Dynamics, Phil. Trans, for 1834. See also C. G.
J. Jacobi, Vorlesungen liber Dynimik, edited by Clebsch, 1866.
THE EXTENSION OF THE PRINCIPLES. 351
the form which mechanics has historically assumed, dy History of
namics is founded upon statics, (for example, D'Alem pieofieas<
. . ..,.,, constraint.
bert's principle on the principle of virtual displace
ments,) whereas one naturally would expect that in
the highest stage of the science statics would appear 
as a particular case of dynamics. Now, the principle
which Gauss supplied, and which we shall discuss in
this section, includes both dynamical and statical cases.
It meets, therefore, the requirements of scientific and
logical aesthetics. We have already pointed out that this
is also true of D'Alembert's principle in its Lagrangian
form and the mode of expression above adopted.
No essentially new principle, Gauss remarks, can now be
established in mechanics ; but this does not exclude
the discovery of new points of view, from which mechan
ical phenomena may be fruitfully contemplated. Such
a new point of view is afforded by the principle of
Gauss.
2. Let m, m. .... be masses, connected in any man statement
i r c i i i . f the P rin ~
ner with one another. These masses, iijree, would, under cipie.
the action of the forces im
pressed on them, describe in a
very short element of time the
spaces a b, a f & r . . . . ; but in
consequence of their connec
tions they describe in the same
element of time the spaces a c,
a f c f .... Now, Gauss's principle asserts, that the mo
rion of the connected points is such that, for the motion
actually taken, the sum of the products of the mass of
each material particle into the square of the distance of
its deviation from the position it would have reached if
free, namely m(bc}* + m, (V/) 2 + = 2m(6c)*, is
a minimum, that is, is smaller for the actual motion
352 THE SCIENCE OF MECHANICS.
than for any other conceivable motion in the sa?jie con
nections. If this sum, ^///(Z'r) 2 , is less for rest than
for any motion, equilibrium will obtain. The principle
includes, thus, both statical and dynamical cases.
Definition  The sum 2?Ji(fic') 2 is called the "constraint."* In
of "con ...... . .
straim." forming this sum it is plain that the velocities present
in the system may be neglected, as the relative posi
tions of a, b, c are not altered by them.
3. The new principle is equivalent to that of
D'Alembert ; it may be used in place of the latter ; and,
as Gauss has shown, can also be deduced from it. The
impressed forces carry the free mass ;;/ in an element of
time through the space ab 9 the effective forces carry the
same mass in the same time in consequence of the con
nections through the space ac. We resolve ab into ac
and cb\ and do the same for all the
masses. It is thus evident that
forces corresponding to the dis
tances c by c, b f . . . . and propor
tional to mc& 9 m t c f b f ... 9 do not,
owing to the connections, become
effective, but form with the connections an equilibrat
ing system. If, therefore, we erect at the terminal posi
tions c t c n c,, the virtual displacements cy 9 c, y ,....,
forming with cb y c r ,.... the angles 9 0,.... we may
apply, since by D'Alembert's principle forces propor
tional to mcb, m, c f b r ... are here in equilibrium, the
principle of virtual velocities. Doing so, we shall have
* Professor Mach's term is Abweickungssumme. The Abweichung is the
declination or departure from free motion, called by Gauss the Ablenkung.
(See Duhring, Principien dcr Mechanik, 168, 169 ; Routh, Rigid Dynamics,
Part I, 390394.) The quantity 2 ? (b c Y is called by Gauss the Zwang\ and
German mathematicians usually follow this practice. In English, the term
constraint is established in this sense, although it is also used with another*
hardly quantitative meaning, for the force which restricts a body absolutely
to moving in a certain way. Trans.
TUE EXTE.VSIOX OF THE PRINCIPLES. 353
. . . (1) The deduc
tiou of the
But principle
(by)* = (b)* + (cy)*1bc.Cytt*e, constraint.
9, and
ycosO (2)
Accordingly, since by (i) the second member of
the righthand side of (2) can only be = or negative,
that is to say, as the sum 2m(cy)* can never be dimin
ished by the subtraction, but only increased, therefore
the lefthand side of (2) must also always be positive
and consequently 2m(by)* always greater than 2m
(be)*> which is to say, every conceivable constraint
from unhindered motion is greater than the constraint
for the actual motion.
4. The declination, be, for the very small element various
, _ . , , forms in
of time r, may, for purposes of practical treatment, be which the
designated by s, and following SchefHer (Schlomilch's may be ex
Zeitschrift fur Mathematik und Physik, 1858, Vol. Ill,
p. 197), we may remark that s = yr 2 /2, where y de
notes acceleration. Consequently, 2ms 2 may also be
expressed in the forms
r 2 r 2 r 4
i 4:
where/ denotes the force that produces the declination
from free motion. As the constant factor in no wise
affects the minimum condition, we may say, the actual
motion is always such that
2ms 2 (1)
or
2 ps (2)
or
2my 2 (3)
is a minimum.
354
THE SCIENCE OF MECHANICS.
The motion 5. We will first employ, m our illustrations, the
of a wheel , . , , TT . r r
and axle, third form. Here again, as our first example, we se
lect the motion of a wheel and axle by
the overweight of one of its parts
and shall use the designations above
frequently employed. Our problem
is, to so determine the actual accel
erations y of P and y, of Q, that
GP/O (g rY + (Q/g) (> ri) 2
shall be a minimum, or, since y f =
y(r/fy, so that P (g 7)2 f
= N shall assume its smallest value.
Putting, to this end,
exactly as in
an inclined
plane.
we get y = (PR Qr/PR* +
the previous treatments of the problem.
Descent on As our second example, the motion of descent on
inclined plane may be taken. In this case we shall
employ the first form, 2ms 2 .
Since we have here only to
deal with one mass, our in
quiry will be directed to find
ing that acceleration of de*
scent y for the plane by
which the square of the de
clination (V 2 ) is made a minimum. By Fig. 181 we
have
Fig. 181.
and putting d(s *}/dy = 0, we obtain, omitting all
constant factors, 2y 2^sinaf = or y = g. sintf, the
familiar result of Galileo's researches.
THE EXTENSION OF THE PRINCIPLES.
355
The following example will show that Gauss's prin A case of
ciple also embraces cases of equilibrium. On the arms rimn. 1
a, a' of a lever (Fig. 182) are hung the heavy masses
;;/,;;/. The principle requires that m(g ^) 2 +
m'(g /') 2 shall be a minimum. But y' =. y(a /a).
Further, if the masses are in
. , & &'
versely proportional to the J~ ZS
lengths of the leverarms, that ?[_]
is to say, if mjm' = a' /a, then
, , i , N n Fig. 182.
y = y{mjm ). Conse
quently, m (^ y) 2  m\g + y  ;////#') 2 = JW must
be made a minimum. Putting dNjdy = 0, we get
;// (i + m/m'}y = or y= 0. Accordingly, in this case
equilibrium presents the least constraint from free mo
tion.
Every new cause of constraint, or restriction upon New causes
i r i f  i .  of con
the freedom of motion, increases the quantity of con straint in
. . crease the
straint. but the increase is always the least possible, departure 1
, , . .from free
If two or more systems be connected, the motion of motion,
least constraint from the motions of the unconnected
systems is the actual motion.
If, for example, we join together several simple
pendulums so as to form a compound linear pendulum,
the latter will oscillate with the motion
of least constraint from the motion of the
single pendulums. The simple pendulum,
for any excursion a, receives, in the di
rection of its path, the acceleration g
sin a. Denoting, therefore, by y sin a the
acceleration corresponding to this excur
sion at the axial distance i on the com
pound pendulum, ISm (g sin a rysin a} 2 or 2m (g
r y] 2 will be the quantity to be made a minimum. Conse
quently, 2m(g ry)r= 0, and y =
Fig. 183.
356
THE SCIENCE OF MECHANICS.
The problem Is thus disposed of in the simplest man
ner. But this simple solution is possible only because
the experiences that Huygens, the Bernoullis, and oth
ers long before collected, are implicitly contained in
Gauss's principle,
iiiustra 6. The increase of the quantity of constraint, or
preceding declination, from free motion by new causes of con
statement. . i i i i i r 11 i
stramt may be exhibited by the following examples.
Over two stationary pulleys A, B, and beneath a
movable pulley C (Fig. 184), a cord is passed, each
Fig. 184.
Fig. 185.
extremity of which is weighted with a load P; and on
C a load zP (/is placed. The movable pulley will
now descend with the acceleration (p/^P + /) g. But
if we make the pulley A fast, we impose upon the
system a new cause of constraint, and the quantity of
constraint, or declination, from free motion will be in
creased. The load suspended from B, since it now
moves with double the velocity, must be reckoned as
possessing four times its original mass. The mova
ble pulley accordingly sinks with the acceleration
A simple calculation will show that the
constraint in the latter case is greater than in the former.
THE EXTENSION OP THE PRINCIPLES.
357
A number, n, of equal weights, /, lying on a smooth
horizontal surface, are attached to n small movable
pulleys through which a cord is drawn in the manner
Indicated in the figure and loaded at its free extremity
with f. According as all the pulleys are movable or all
except one axejixed, we obtain for the motive weight/,
allowing for the relative velocities of the masses as re
ferred to j>, respectively, the accelerations (4*3/1 + q.n)g
and (4/5) g If all the n + i masses are movable, the
deviation assumes the value/ g/^n f i, which Increases
as n, the number of the movable masses, is decreased.
Fig. 186.
7. Imagine a body of weight O, movable on rollers Treatment
of a me
on a horizontal surface, and having an inclined plane chanicai
problem by
face. On this, inclined face a body of weight P is different
mechanical
placed. W e now perceive instinctively that P will de principles.
scend with quicker acceleration when Q is movable
and can give way, than it will when Q is fixed and P's
descent more hindered. To any distance of descent h
of P a horizontal velocity v and a vertical velocity u of
P and a horizontal velocity w of Q correspond. Owing
to the conservation of the quantity of horizontal mo
tion, (for here only internal forces act,) we have Pv =
Qw, and for obvious geometrical reasons (Fig. 186)
also
u = (v \~ w) tan a
The velocities, consequently, are
358 THE SCIENCE OF MECHANICS.
First, by the V = *
principles P 4 O
of the con
servation of jp
fno Q KT W=j r QCOt< X .U.
For the work Ph performed, the principle of vis
viva gives
_. Pu* Pi Q ^ \22
Ph= + =? = cot or .T +
^ 2 ^^v^p+e ' 2
^P \ 2 2/ 2
; CO H y
c
Multiplying by ^>, we obtain
/ A , g
^  j 
2 *
To find the vertical acceleration y with which the
space h is described, be it noted that h = & 2 /2 y. In
troducing this value in the last equation, we get
7
Q ' 6 '
For Q oo, y = g sin 2 a, the same as on a sta
tionary inclined plane. For Q = 0, ^ = g, as in free
descent. For finite values of Q = mP, we get,
1 + *
since T    > 1,
2 m
f ;;/) sin 2
The making of <2 stationary, being a newly imposed
cause of constraint, accordingly increases the quantity
of constraint, or declination, from free motion.
To obtain y, in this case, we have employed the
principle of the conservation of momentum and the
THE EXTENSION OF THE PRINCIPLES. 359
principle of vis viva. Employing Gauss's principle, Second, by
we should proceed as follows. To the velocities de cipie of
. Gauss,
noted as u, v, w the accelerations y, o, correspond.
Remarking that in the free state the only acceleration
is the vertical acceleration of P, the others vanishing,
the procedure required is, to make
a minimum. As the problem possesses significance
only when the bodies P and Q touch, that is only when
y = (d f ) tan a, therefore, also
Forming the differential coefficients of this expression
with respect to the two remaining independent vari
ables d and , and putting each equal to zero, we ob
tain
_ [>_ (tf _j_ e) tan or] Pt&n a + PS = and
[f ($ + )tan#] 7>tan<*i Qe = 0.
From these two equations follows immediately
Pd Qs = 0, and, ultimately^ the same value for y
that we obtained before.
We will now look at this problem from another
point of view. The body P describes at an angle ft
with the horizon the space s, of which the horizontal
and vertical components are v and ?/, while simulta
neously Q describes the horizontal distance w. The
forcecomponent that acts in the direction of s is Psm ft,
consequently the acceleration in this direction, allow
ing for the relative velocities of P and Q, is
P.s'm/3
360 THE SCIENCE OF MECHANICS.
Third, by Employing the following equations which are di
tended con rectly deducible,
cept of mo ,_
inent of In Q W = P V
ertia.
n = v tan ft.
the acceleration in the direction of s becomes
and the vertical acceleration corresponding thereto is
an expression, which as soon as we introduce by means
of the equation u = (v + ?/) tan a:, the anglefunc
tions of a for those of ft, again assumes the form above
given. By means of our extended conception of mo
ment of inertia we reach, accordingly, the same result
as before.
Fourth, by Finally we will deal with this problem in a direct
cip?es. prm manner. The body P does not descend on the mova
ble inclined plane with the vertical acceleration g, with
which it would fall if free, but with a different vertical
acceleration, y. It sustains, therefore, a vertical coun
terforce (P/g} (g 7). But as P and Q, friction
neglected, can only act on each other by means of a
pressure S, normal to the inclined plane, therefore
P f NO
(g y} = S cos a
t = 2 _
From this is obtained
THE EXTENSION OF THE PRINCIPLES. 361
and by means of the equation y ~ (d \ e) tana:, ulti
mately, as before,
Q
*
If we put P= O and or = 45, we obtain for this Discussion
^ ^ ^ J of the re
particular case y = %g 9 d = ^, = *. For /*/f = suits.
Q/g= i we find the "constraint,'* or decimation from
free motion, to be g 2/3. If we make the inclined plane
stationary, the constraint will be g 2 /2. If /Amoved on
a stationary inclined plane of elevation ft, where
tan/5? = y/d, 'that is to say, in the same path in which
it moves on the movable inclined plane, the constraint
would only be g 2 /$. And, in that case it would, in
reality, be less impeded than if it attained the same
acceleration by the displacement of Q.
8. The examples treated will have convinced us that Gauss's
no substantially new insight or perception is afforded by affords no
. . . newinsight
Gauss's principle. Employing form (3) of the prin
ciple and resolving all the forces and accelerations in
the mutually perpendicular coordinatedirections, giv
ing here the letters the same significations as in equa
tion (i) on page 342, we get in place of the declination,
or constraint, 2 my 2 , the expression
'^ IZ \ 2 1
77 +P5 (
/ \ m J J
and by virtue of the minimum condition
362 THE SCIENCE OF MECHANICS.
,;//
or2[(X ;
Gauss's and If no connections exist, the coefficients of the (in
D'Alem ^ T > T 7 o n i ^
bert's prin that case arbitrary) dE,, dij, dc severally made = 0,
mutabie m give the equations of motion. But if connections do
exist, we have the same relations between dg, drj, d2,
as above in equation (i), at page 342, between d '#, dy,
dz. The equations of motion come out the same ; as
the treatment of the same example by D'Alem bert's
principle and by Gauss's principle fully demonstrates.
The first principle, however, gives the equations of
motion directly, the second only after differentiation.
If we seek an expression that shall give by differentia
tion D'Alembert's equations, we are led perforce to the
principle of Gauss. The principle, therefore, is new
only in form and not in matter. Nor does it, further,
possess any advantage over the Lagrangian form of
D'Alembert's principle in respect of competency to com
prehend both statical and dynamical problems, as has
been before pointed out (page 342).
Thephys There is no need of seeking a mystical or metaphys
ical basis p ^ . . , _, . \ J
of the prin /<rtf/reason for Gauss s principle. The expression ' ' least
ciple.
constraint" may seem to promise something of the
sort ; but the name proves nothing. The answer to the
question, "In what does this constraint consist ? " can
not be derived from metaphysics, but must be sought
in the facts. The expression (2) of page 353, or (4) of
page 361, which is made a minimum, represents the
work done in an element of time by the deviation of the
constrained motion from the free motion. This work,
the work due to the constraint, is less for the motion
actually performed than for any other possible motion.
THE EXTENSION OF THE PRINCIPLES.
363
/ \
Fig. 187.
nisable in
the sim
plest cases.
Once we have recognised work as the factor deter Role of the
factor work.
minative of motion, once we have grasped the mean
ing of the principle of virtual displacements to be, that
motion can never take place except where work can be
performed, the following converse truth also will in
volve no difficulty, namely, that all the work that can
be performed in an element of time actually is per
formed. Consequently, the total diminution of work
due in an element of time to the connections of the
system's parts is restricted to the portion annulled by
the counterwork of those parts. It is again merely a
new aspect of a familiar fact with which we have here
to deal.
This relation is displayed in the very simplest cases. The foun
_ * J * * i dationsof
Let there be two masses m and m at A, the one im the princi
ple recog
pressed with a force/, the other with t,.
the force q. If we connect the two, we . *B
shall have the mass 27/2 acted on by a
resultant force r. Supposing the spaces
described in an element of time by the
free masses to be represented by AC,
AB, the space described by the con
joint, or double, mass will be AO =
^AD. The deviation, or constraint,
is m(OB* + OC*}. It is less than
it would be if the mass arrived at the end of the ele
ment of time in M or indeed in any point lying out
side of B C 9 say N, as the simplest geometrical con
siderations will show. The deviation is proportional
to the expression / 2 f ^ 3 + dpq cos 8/2, which in the
case of equal and opposite forces becomes 2/ 2 , and in
the case of equal and likedirected forces zero.
Two forces p and q act on the same mass. The
force q we resolve parallel and at right angles to the
364 THE SCIENCE OF MECHANICS.
Even in the direction of / in r and s. The work done in an element
the compo of time is proportional to the squares of the forces, and
forces its if there be no connections is expressible by/ 2 + ^ 2 =
are?ound? / 3 ~ r + s 2 . If now r act directly counter to the
force /, a diminution of work will be effected and the
sum mentioned becomes (/ r) 2 f s 2 . Even in the
principle of the composition of forces, or of the mutual
independence of forces, the properties are contained
which Gauss's principle makes use of. This will best
be perceived by Imagining all the accelerations simul
taneously performed. If we discard the obscure verbal
form in which the principle is clothed, the metaphysical
impression which it gives also vanishes. We see the
simple fact ; we are disillusioned, but also enlightened.
The elucidations of Gauss's principle here presented
are in great part derived from the p'aper of SchefHer
cited above. Some of his opinions which I have been
unable to share I have modified.. We cannot, for ex
ample, accept as new the principle which he himself
propounds, for both in form and in import it is identical
with the D'AlembertLagrangian.
VIII.
THE PRINCIPLE' OF LEAST ACTION.
Theorig i. MAUPERTUIS enunciated, in 1747, a principle
scure form which he called * ' le firincife de la moindre quantite d'ac
oftheprin ,.,,, ... ,., . __., , , .
cipieof tton," the principle of least action. He declared this
least action. . . , ,  , . , . , i .,..,,
principle to be one which eminently accorded with the
wisdom of the Creator. He took as the measure of
the "action" the product of the mass, the velocity,
and the space described, or mvs. Why, it must be
confessed, is not clear. By mass and velocity definite
quantities may be understood ; not so, however, by
THE EXTENSION OF THE PRINCIPLES. 365
space, when the time Is not stated in which the space
is described. If, however, unit of time be meant, the
distinction of space and velocity in the examples treated
by Maupertuis is, to say the least, peculiar. It appears
that Maupertuis reached this obscure expression by an
unclear mingling of his ideas of vis viva and the prin
ciple of virtual velocities. Its indistinctness will be
more saliently displayed by the details.
2. Let us see how Maupertuis applies his principle. Determina
r . tion of the
If M, m be two inelastic masses, Cand c their velocities laws of im
pact by this
before impact, and u their common velocity after im principle.
pact, Maupertuis requires, (putting here velocities for
spaces,) that the " action 77 expended in the change of
the velocities in Impact shall be a minimum. Hence,
M(C #) 2 + m (c ?/) 2 is a minimum ; that is,
M(C */) + m (c w) = ; or
_ __
M f m
For the impact of elastic masses, retaining the same
designations, only substituting Kand v for the two ve
locities after Impact, the expression M(C F) 2 f
m(c z>) 2 is a minimum that is to say,
M(cr)<tr+ffi(t v)ttv = o ..... (i)
In consideration of the fact that the velocity of ap
proach before impact is equal to the velocity of reces
sion after impact, we have
C c = (y ?;) or
C + V (,+ p)=0. ........... (2)
and
d V <tv = Q ................ (3)
The combination of equations (i), (2), and (3)
readily gives the familiar expressions for V and z>.
These two cases may, as we see, be viewed as pro
366 THE SCIENCE OF MECHANICS.
cesses in which the least change of vis viva by reaction
takes place, that is, in which the least counterwork is
done. They fall, therefore, under the principle of
Gauss.
Matiper 3. Peculiar is Maupertuis's deduction of the law of
auction of the lever. Two masses M and ;;/ (Fig. 188) rest on a
the lever by bar a, which the fulcrum divides into the portions
cipie. x and a x. If the bar be set in rotation, the veloci
ties and the spaces described will be proportional to
the lengths of the leverarms, and Mx 2 + m(a ^) 2
is the quantity to be made a minimum, that is MX
m (a #) = ; whence x = ma/M + ?# a condition
that in the case of equilib
t t rium is actually fulfilled. In
llf. m criticism of this, it is to be
V lc ' ' ~a^x ' remarked, first, that masses
Fig. 188. n t subject to gravity or
other forces, as Maupertuis
here tacitly assumes, are always in equilibrium, and,
secondly, that the inference from Maupertuis's deduc
tion is that the principle of least action is fulfilled
only in the case of equilibrium, a conclusion which it
was certainly not the author's intention to demonstrate.
The correc If it were sought to bring this treatment into ap
tion of Man ., i ,i i i 1111
pertuis's proximate accord with the preceding, we should have
deduction. ,, , , 7 * * \ i
to assume that the heavy masses M and m constantly
produced in each other during the process the least
possible change of vis viva. On that supposition, we
should get, designating the arms of the lever briefly by
a, b, the velocities acquired in unit of time by u 9 v, and
the acceleration of gravity by g, as our minimum ex
pression, M(g u) 2 f m(g ^) 2 ; whence M(g u)
du ( m(g v}dv = 0. But in view of the connection
of the masses as lever,
THE EXTENSION OF THE PRINCIPLES.
367
V
_,
du = T */z; ;
#
whence these equations correctly follow
mb T Ma m b
u = a
"Ma* 4 mb*** Ma 2 \ mb 2 &J
and for the case of equilibrium, where u=v = 0,
Thus, this deduction also, when we come to rectify
it, leads to Gauss's principle.
4. Following the precedent of Fermat and Leib Treatment
^ r ofthemo
nitz, Maupertuis also treats by his method the
of light. Here again, however,
he employs the notion " least ac
tion " in a totally different sense.
The expression which for the
case of refraction shall be a min
imum, is m . A R \~ n . RB,
where AR and RB denote the
paths described by the light in
the first and second media re
spectively, and m and n the corresponding velo
cities. True, we really do obtain here, if R be de
termined in conformity with the minimum condition,
the result sin ar /sin/? = n/m = const. But before, the
* ' action " consisted in the change of the expressions
mass X velocity X distance ; now, however, it is con
stituted of the sum of these expressions. Before, the
spaces described in unit of time were considered ; in
the present case the total spaces traversed are taken.
Should not m. AR n. RB or \jn ri)(AR R&)
be taken as a minimum, and if not, why not ? But
Fig. 189.
3 68 THE SCIENCE OF MECHANICS.
even if we accept Maupertuis's conception, the recip
rocal values of the velocities of the light are obtained,
and not the actual values.
character! It will thus be seen that Maupertuis really had no
sation of . .  .  1 .
Mauper principle, properly speaking, but only a vague form
tuis's prin r , . , r i 11 1
cipie. ula, which was forced to do duty as the expression of
different familiar phenomena not really brought under
one conception. I have found it necessary to enter
into some detail in this matter, since Maupertuis's per
formance, though it has been unfavorably criticised by
all mathematicians, is, nevertheless, still invested with
a sort of historical halo. It would seem almost as if
something of the pious faith of the church had crept
into mechanics. However, the mere endeavor to gain
a more extensive view, although beyond the powers of
the author, was not altogether without results. Euler,
at least, if not also Gauss, was stimulated by the at
tempt of Maupertuis.
Euler'scon c Euler's view is, that the purposes of the phe
tributions J p ,. , ' f . . r ,
to this sub nomena of nature afford as good a basis of explana
tion as their causes. If this position be taken, it will
be presumed a priori that all natural phenomena pre
sent a maximum or minimum. Of what character this
maximum or minimum is, can hardly be ascertained
by metaphysical speculations. But in the solution of
mechanical problems by the ordinary methods, it is
possible, if the requisite attention be bestowed on the
matter, to find the expression which in all cases is
made a maximum or a minimum. Euler is thus not
led astray by any metaphysical propensities, and pro
ceeds much more scientifically than Maupertuis. He
seeks an expression whose variation put = gives the
ordinary equations of mechanics.
For a single body moving under the action of forces
THE EXTEA"SIO.V OF THE PRINCIPLES.
369
hands.
Euler finds the requisite expression in the formula The form
/ / i 71 11 PI 11 which the
/ v as, where ds denotes the element of the path and principle,
*J , . . m , . . assumed in
v the corresponding velocity. This expression is smaller Eui<
for the path actually taken than for any other infinitely
adjacent neighboring path between the same initial
and terminal points, which the body ma}^ be constrained
to take. Conversely, therefore, by seeking the path that
makes Cv ds a minimum, we can also determine the
path. The problem of minimising Cv ds is, of course,
as Euler assumed, a permissible one, only when v de
pends on the position of the elements ds, that is to
say, when the principle of vis viva holds for the forces,
or a forcefunction exists, or what is the same thing,
when v is a simple function of coordinates. For a mo
tion in a plane the expression would accordingly as
sume the form
JVC
>/J)\
1 4X
,dx
In the simplest cases Euier's principle is easily veri
fied. If no forces act, v is constant, and the curve of
motion becomes a straight line, for which Cvds^z
v C ds is unquestionably shorter than for any other
curve between the same terminal points.
Also, a body moving on a curved surface
without the action of forces or friction,
preserves its velocity, and describes on
the surface a shortest line.
The consideration of the motion of a
projectile in a parabola ABC (Fig. 190)
will also show that the quantity Cv ds
is smaller for the parabola than for any
other neighboring curve ; smaller, even,
than for the straight line ABC between the same ter
minal points. The velocity, here, depends solely on the
Euier's
principle
applied to
/" the motion
of a projec
tile.
Fig. 190
370 THE SCIENCE OF MECHANICS.
Mathemat vertical space described by the body, and is therefore
ojnnent'of the same for all curves whose altitude above OC is the
same. If we divide the curves by a system of horizontal
straight lines into elements which severally correspond,
the elements to be multiplied by the same v's, though
in the upper portions smaller for the straight line AD
than for A JE>, are in the lower portions just the reverse ;
and as it is here that the larger z>'s come into play, the
sum upon the whole is smaller for ABC than for the
straight line.
Putting the origin of the coordinates at A 9 reckon
ing the abscissas x vertically downwards as positive,
and calling the ordinates perpendicular thereto y 9 we
obtain for the expression to be minimised
1
where g denotes the acceleration of gravity and a the
distance of descent corresponding to the initial velocity.
As the condition of minimum the calculus of variations
gives
= Cor
dy C
~ ' or
y  Cd *
J
V%
and, ultimately,
THE EXTENSION OF TPIE PRINCIPLES, 371
where C and ' denote constants of Integration that
pass into C= V zga and C'= 0, If for x = 0, dx/dy =
and 7 be taken. Therefore, j = 2/W. By this
method, accordingly, the path of a projectile is shown
to be of parabolic form.
6. Subsequently. Lagrange drew express attention The addi
^ J ' b & ^ tions of La
to the fact that Euler's principle is applicable only in grange and
cases in which the principle of vis viva holds. Jacobi
pointed out that we cannot assert that Cv ds for the ac
tual motion is zmtmrnum, but simply that the variation of
this expression, In its passage to an Infinitely adjacent
neighboring path, Is 0. Generally, indeed, this con
dition coincides with a maximum or minimum, but it
is possible that it should occur without such; and the
minimum property in particular Is subject to certain
limitations. For example, If a body, constrained to
move on a spherical surface, is set in motion by some
impulse, it will describe a great circle, generally a
shortest line. But If the length of the arc described
exceeds 180, It is easily demonstrated that there exist
shorter Infinitely adjacent neighboring paths between
the terminal points.
7.. So far, then, this fact only has been pointed out, Euler's _
that the ordinary equations of motion are obtained by but one of
equating the variation of Cv ds to zero. But since the give y the 1C
f + . 4 i i r T T equations
properties of the motion of bodies or of their paths may of motion,
always be defined by differential expressions equated
to zero, and since furthermore the condition that the
variation of an integral expression shall be equal to
zero is likewise given by differential expressions equated
to zero, unquestionably varioiis other Integral expres
sions may be devised that give by variation the ordi
nary equations of motion, without its following that the
372 THE SCIENCE OF MECHANICS,
integral expressions in question must possess on that
account any particular physical significance.
Yet the ex 8. The striking fact remains, however, that so simple
must pos an expression as Cv ds does possess the property men
sessaphys . J ... j . .
icai import, tioned, and we will now endeavor to ascertain its phys
ical import. To this end the analogies that exist be
tween the motion of masses and the motion of light, as
well as between the motion of masses and the equilib
rium of strings analogies noted by John Bernoulli
and by Mobius will stand us in stead.
A body on which no forces act, and which there
fore preserves its velocity and direction constant, de
scribes a straight line. A ray of light passing through
a homogeneous medium (one having everywhere the
same index of refraction) describes a straight line. A
string, acted on by forces at its extremities only, as
sumes the shape of a straight line.
Elucidation A body that moves in a curved path from a point
otthisim . ... r
port by the A to a point B and whose velocity v = (p(x, y, z) is a
motion of a . /
mass, the function of coordinates, describes between A and B a
motion of a . >, . . .
ray of Hght, curve for which generally J ?/ </j is a minimum. A ray
equilibrium of light passing from A to B describes the same curve,
if the refractive index of its medium, n= <p(x, y, z^) }
is the same function of coordinates ; and in this case
Cnds is a minimum. Finally, a string passing from
A to B will assume this curve, if its tension S =
<p (x 9 y, z) is the same abovementioned function of co
ordinates ; and for this case, also, CSds is a minimum.
The motion of a mass may be readily deduced from
the equilibrium of a string, as follows. On an element
ds of a string, at its two extremities, the tensions S, S f
act, and supposing the force on unit of length to be jp,
in addition a force JP. ds. These three forces, which
we shall represent in magnitude and direction by J5A 9
THE EXTENSION OF THE PRINCIPLES.
373
BC, BD (Fig. 191), are in equilibrium. If now, a body, The motion
with a velocity v represented in magnitude and direc deduced
tion by AB, enter the element of the path ds, and re equilibrium
J of a string.
ceive within the same the velocity component J3F =
BD, the body will proceed on
ward with the velocity v f = BC.
Let Q be an accelerating force
whose action is directly opposite
to that of P] then for unit of time
the acceleration of this force will
be Q, for unit of length of the
string Q/Z', and for the element
of the string (Q/ztyds. The body will move, therefore,
in the curve of the string, if we establish between the
forces P and the tensions S, in the case of the string,
and the accelerating forces Q and the velocity v in the
case of the mass, the relation
D
Fig. 191.
The minus sign indicates that the directions of P and
<2 are opposite.
A closed circular string is in equilibrium when be
^
tween the tension S of the string, everywhere constant,
and the force P falling radially outwards on unit of
length, the relation P = S/r obtains, where r is the
radius of the circle. A body will move with the con
stant velocity v in a circle, when between the velocity
and the accelerating force Q acting radially inwards
the relation
O V 7 r2
^ = or Q = obtains.
v r ^ r
A body will move with constant velocity v in any curve
when an accelerating force Q=v*/r constantly acts
The equi
libnum of
closed
374
THE SCIENCE OF MECHANICS.
on it in the direction of the centre of curvature of each
element. A string will lie under a constant tension *$
in any curve if a force P = S/r acting outwardly from
the centre of curvature of the element is impressed on
unit of length of the string.
The deduc No concept analogous to that of force is applicable
motion of to the motion of light. Consequently, the deduction of
th g e moSSis the motion of light from the equilibrium of a string or
and 1 ?]^ 8 the motion of a mass must be differently effected. A
equilibrium . . .  
of strings, mass, let us say, is moving with the velocity AB = v.
(Fig. 192.) A force in the direction
BD is impressed on the mass which
produces an increase of velocity BE,
so that by the composition of the ve
locities BC AB and BE the new
velocity BF= v' is produced. If we
resolve the velocities v, v f into com
ponents parallel and perpendicular to
the force in question, we shall per
ceive that the parallel components alone
are changed by the action of the force.
This being the case, we get, denoting
by k the perpendicular component, and by a and a 1
the angles v and v' make with the direction of the
force,
k = v sin a
k = v 1 sin a' or
sma v f
sin a' v *
If, now, we picture to ourselves a ray of light that
penetrates in the direction of v a refracting plane at
right angles to the direction of action of the force, and
thus passes from a medium having the index of refrac
D
Fig. 192.
THE EXTEXSroX OF THE PRINCIPLES. 375
tion n into a medium having the index of refraction ', Develop
mentofthi*
where >n/ri = r/v', this ray of light will describe the illustration
< an:e path as the body in the case above. If, there
fore, we wish to imitate the motion of a mass by the
motion of a ray of light (in the same curve), we must
everywhere put the indices of refraction, n, proportional
to the velocities. To deduce the indices of refraction
from the forces, we obtain for the velocity
d IT) ~ Pdq * and
for the index of refraction, by analogy,
where P denotes the force and dq a distanceelement
in the direction of the force. If ds is the element of
*the path and a the angle made by it with the direction
of the force, we have then
d\ ~\ = P cos a . ds
(n*\
d\\ = Pcosa . ds.
For the path of a projectile, under the conditions above
assumed, we obtained the expression y = 2 V ax. This
same parabolic path will be described by a ray of light,
if the law n = V%g(a + x) be taken as the index of
refraction of the medium in which it travels.
9. We will now more accurately investigate the Relation of
, . 1  . . . . , , the mini
manner in which this minimum property is related to mum prop
\heform of the curve. Let us take, first, (Fig. 193) a format e
broken straight line ABC, which intersects the straight
line MN, put AB = s, BC= /, and seek the. condition
that makes vs f v's' a minimum for the line that passes
376
THE SCIENCE OF MECHANICS.
First, do through the fixed points A and B, where v and v r are
the mini?. supposed to have different, though constant, values
mum condi , , , , * ^ , 7 r r 11  i T~>
tion. above and below J/YV'. If we displace the point B an
Infinitely small distance to JO, the new line through A
and 6* will remain parallel to the original one, as the
drawing symbolically shows. The expression vs f v's'
is increased hereby by an amount
v m sin a f v m sin a /f ,
where m = DB. The alteration is accordingly propor
tional to #sin ar+p'sinor', and the condition of
minimum is that
, ' f A
z/sin a =0, or . 7 = .
sin a v
M
Fig. 193.
If the expression s/v f* s '/v' is to be made a minimum,
we have, in a similar way,
sin a' v''
Second, the If, next, we consider the case of a string stretched
ofthis^on" in the direction ABC, the tensions of which >S and S'
equilibrium are different above and below MN, in this case it is
o as rmg. ^^ minimum of S s f S f s ' that is to be dealt with. To
obtain a distinct idea of this case, we may imagine the
THE EXTENSION OF THE PRINCIPLES.
377
string stretched once between A and B and thrice be
tween B and C, and finally a weight P attached. Then
S = P and S' = 3 P. If we displace the point B a dis
tance m, 'any diminution of the expression Ss \ S's'
thus effected, will express the increase of work which
the attached weight P performs. If Sm$ma\
S'm sin a' = 0, no work is performed. Hence, the mini
mum of Ss j S's' corresponds to a maximum of work.
In the present case the principle of least action is sim
ply a different form of the principle of virtual displace
ments.
Now suppose that ABC is a ray of light, whose ve Third, the
r o > application
locities z f and v' above and below MN are to each other of this con
dition to the
as 3 to i. The motion of light be
tween two points A and B is such
that the light reaches B in a mini
mum of time. The physical reason
of this is simple. The light travels
from A to JB, in the form of ele
mentary waves, by different routes.
Owing to the periodicity of the light,
the waves generally destroy each
other, and only those that reach the
designated point in equal times, that is, in equal phases,
produce a result. But this is true only of the waves
that arrive by the minimum path and its adjacent neigh
boring paths. Hence, for the path actually taken by
the light s/v + J /v is a minimum. And since the in
dices of refraction n are inversely proportional to the
velocities v of the light, therefore also ns \ n's' is a
minimum.
In the consideration of the motion of a mass the con
dition that vs + v's' shall be a minimum, strikes us as
something novel. (Fig. 195.) If a mass, in its passage
k
A
Dk i
motion of a
ray of light.
V
\
\
f
a\
\
B
Fig.
\
E
195
378 THE SCIENCE OP MECHANICS.
Fourth, it a through a plane MN, receive, as the result of the action
application .,. . i i T ,, TN r> /*
to the mo of a force impressed in the direction DJb y an increase of
mabs a velocity, by which v, its original velocity, is made v', we
have for the path actually taken by the mass the equa
tion r sin ex = v' sin a ' k. This equation, which is also
the condition of minimum, simply states that only the ve
locitycomponent parallel to the direction of the force is
altered, but that the component k at right angles thereto re
mains unchanged. Thus, here also, Euler's principle
simply states a familiar fact i a a new form. (See p. 5 75. )
Form of the i o. The minimum condition v sin a + ?/ sin a'=
minimum it *t r .c *j. i 1
condition may also be written, it we pass trom a nmte broken
applicable .... , , r ,1 r
to curves, straight line to the elements of curves, in the form
v sin a  (v f dv} sin(# f da) =
or
d(v sin oi) =
or, finally,
v sin (x = const.
In agreement with this, we obtain for the motion
of light
d (n sin a) = 0, n sin a = const,
sinarN ' sin or
= 0, = r^?^/.
v ) v
and for the equilibrium of a string
^(6* sin a) = 0, ,Ssina: = const.
To illustrate the preceding remarks by an ex
ample, let us take the parabolic path of a projectile,
where a always denotes the angle that the element of
the path makes with the perpendicular. Let the ve
locity be v l/2(# + #), and let the axis of thejyor
dinates be horizontal. The condition v . sin a. = const,
or V zg(a + x*) . dy/ds const, is identical with that
which the calculus of variation gives, and we now know
THE EXTEXSIOX OF THE PRINCIPLES.
379
Fig. 196.
its simple physical significance. If we picture to ourselves illustration
a string whose tension is S = V 2 ?(a + #). an arrange topical
v y . . cases by
ment which might be effected by fixing frictionless curvilinear
J motions.
pulleys on horizontal parallel rods placed in a vertical
plane, then passing the string through these a sufficient
number of times, and finally attaching
a weight to the extremity of the string,
we shall obtain again, for equilibrium,
the preceding condition, the phys
ical significance of which is now ob
vious. When the distances between
the rods are made infinitely small the
string assumes the parabolic form.
In a medium, the refractive index of
which varies in the vertical direction
by the law n = I/ '2 g(a f .#), or the velocity of light in
which similarly varies by the law v = 1/V zg(a f #),
a ray of light will describe a path which is a parabola.
If we should make the velocity in such a medium
v l/zg^a^x*), the ray would describe a cycloidal path,
for which, not CV zg(a + #) . ds, but the expression
Cds/V?>g(a\x) would be a minimum.
ii. In comparing the equilibrium of a string with
the motion of a mass, we may employ in place of a
string wound round pulleys,
a simple homogeneous cord,
provided we subject the cord
to an appropriate system of
forces. We readily observe
that the systems of forces
that make the tension, or,
as the case may be, the ve
locity, the same function of coordinates, are differ
ent. If we consider, for example, the force of gravity,
Fig. 197.
THE SCIENCE OF MECHANICS.
?he condi v = 1/2^(0 + x). A string, however, subjected to the
conse action of gravity, forms a catenary, the tension of
thLiM<icd which is given by the formula S = vi nx, where ;//
uigjuiaio ^^ ^ ^^ constants. The analogy subsisting between
the equilibrium of a string and the motion of a mass is
substantially conditioned by the fact that for a string
subjected to the action of forces possessing a force
function U, there obtains in the case of equilibrium
the easily demonstrable equation 7+ 5 = const. This
physical interpretation of the principle of least action
is here illustrated only for simple cases ; but it may
also be applied to cases of greater complexity, by
imagining groups of surfaces of equal tension, of equal
velocity, of equally refractive indices constructed which
divide the string, the path of the motion, or the path
of the light into elements, and by making a in such a
case represent the angle which these elements make
with the respective surfacenormals. The principle of
least action was extended to s}^stems of masses by La
grange, who presented it in the form
d "2m Cvds = 0.
If we reflect that the principle of vis viva, which is the
real foundation of the principle of least action, is not
annulled by the connection of the masses, we shall
comprehend that the latter principle is in this case also
valid and physically intelligible.
IX,
HAMILTON'S PRINCIPLE.
i. It was above remarked that various expressions
can be devised whose variations equated to zero give
the ordinary equations of motion. An expression of
this kind is contained in Hamilton's principle
THE EXTEXSIOX OF THE PRINCIPLES.
381
= , or
fou+
The points
of identity
of Hamil
ton's and
D'Alem
bert's prin
ciples.
of a wheel
and axle.
where 6U and dT denote the variations of the work
and the vis viva, vanishing for the initial and terminal
epochs. Hamilton's principle is easily deduced from
D'Alembert's, and, conversely, D'Alemberfs from
Hamilton's ; the two are in fact identical, their differ
ence being merely that of form.*
2. We shall not enter here into any extended in Hamilton':
J m principle
vestigation of this subject, but simply exhibit the iden applied to
tity of the two principles by an example
the same that served to illustrate the prin
ciple of D'Alembert: the motion of awheel
and axle by the overweight of one of its
parts. In place of the actual motion, we
may imagine, performed in the same inter
val of time, a different motion, varying in
finitely little from the actual motion, but
coinciding exactly with it at the beginning
and end. There are thus produced in every element
of time dt, variations of the work (#7) and of the vis
viva (tfT 7 ); variations, that is, of the values U and T
realised in the actual motion. But for the actual mo
tion, the integral expression, above stated, is = 0, and
may be employed, therefore, to determine the actual
motion. If the angle of rotation performed varies in
the element of time dt an amount a from the angle of
the actual motion, the variation of the work corre
sponding to such an alteration will be
dU= (PJ? Qr} a = Ma.
* Compare, for example, Kirchhoff, Vorlesungen ilber mathematische Phy
sik, Mecha.n.ik, p. 25 et seqq.* and JacobI, VorUsungsn uber Dynamik, p, 58.
382 THE SCIENCE OF MECHANICS.
Mathemat The vis viva, for any given angular velocity GO, is
ical devel _. Q
opmentof 1 , p;?0 /O,^ ^
this case. 1 =  (/</C f <2 r )~e>'>
and for a variation #<# of this velocity the variation of
the vis viva is
But if the angle of rotation varies in the element dt an
amount a,
* da ,
$G0 ~~ and
dt
The form of the integral expression, accordingly, is
dt
But as
d^ N_^
therefore,

dt
The second term of the lefthand member, though,
drops out, because, by hypothesis, at the beginning
and end of the motion a = 0. Accordingly, we have
dt
an expression which, since a in every element of time
is arbitrary, cannot subsist unless generally
THE EXTENSION OF TPIE PRINCIPLES. 383
Substituting for the symbols the values they represent,
we obtain the familiar equation
doo __ PR Qr
~Tt ~ ~
D'Alembert s principle gives the equation The same
r r & ^ results ob
d N\ tained by
r \fy A the use of
/// I ' D'AIem.
/ bert's prin
which holds for every possible displacement. We might, Clple '
in the converse order, have started from this equation,
have thence passed to the expression
and, finally, from the latter proceeded to the same re
sult
dt
dt
"i
/
dt
3. As a second and more simple example let us illustration
j , , ,. r , , ,^ ofthispoint
consider the motion of vertical descent. For every by the mo
infmitely small displacement s the equation subsists ticaide^ er "
[mg m(dv/dty]s = b, in which the letters retain scent '
their conventional significance. Consequently, this
equation obtains
C( mv _ m <*v\ s dt=
*' \ dt I
which, as the result of the relations
v (mvs} dv , ds
384 THE SCIENCE OF MECHANICS.
\
d
, J , n
 ' dt (m v s) = 0,
tit g
provided s vanishes at both limits, passes into the form
i
J (mgs + m v ~ j dt= 0,
that is, into the form of Hamilton's principle.
Thus, through all the apparent differences of the
mechanical principles a common fundamental same
ness is seen. These principles are not the expression
of different facts, but, in a measure, are simply views
of different aspects of the same fact.
x.
SOME APPLICATIONS OF THE PRINCIPLES OF MECHANICS TO
HYDROSTATIC AND HYDRODYNAMIC QUESTIONS.
Method of ' i. We will now supplement the examples which
eliminating . . r , , t . r . , . . .
the action we have given of the application ot the principles
oiuiqui/ of mechanics, as they applied to rigid bodies, by a
m sses * f ew hydrostatic and hydrodynamic illustrations. We
shall first discuss the laws of equilibrium of a weightless
liquid subjected exclusively to the action of socalled
molecular forces. The forces of gravity we neglect in
our considerations. A liquid may, in fact, be placed
in circumstances in which it will behave as if no forces
of gravity acted. The method of this is due to PLA
TEAU.* It is effected by immersing olive oil in a mix
ture of water and alcohol of the same density as the
oil. By the principle of Archimedes the gravity of the
masses of oil in such a mixture is exactly counterbal
anced, and the liquid really acts as if it were devoid of
weight.
* Statique exfiirimentale et tktorzque des liquides, 1873.
THE EXTENSION OF THE PRINCIPLES.
385
2. First, let us imagine a weightless liquid mass The work ot
A molecular
free in space. Its molecular forces, we know, act only forces de
. . . pendent on
at very small distances. Taking as our radius the dis a change in
J to the liquid's
tance at which the molecular forces cease to exert a superficial
area.
measurable influence, let us describe about a particle
#, b) c in the interior of the mass a sphere the so
called sphere of action. This sphere of action is regu
larly and uniformly filled with other particles. The
resultant force on the central particles a, b, c is there
fore zero. Those parts only that lie at a distance from
the bounding surface less than the radius of the sphere
of action are in different dynamic conditions from the
particles in the interior. If the radii of curvature of
Fig. 199. Fig. 200.
the surfaceelements of the liquid mass be all regarded
as very great compared with the radius of the sphere
of action, we may cut off from the mass a superficial
stratum of the thickness of the radius of the sphere of
action in which the particles are in different physical
conditions from those in the interior. If we convey
a particle a in the interior of the liquid from the posi
tion a to the position b or c, the physical condition
of this particle, as well as that of the particles which
take its place, will remain unchanged. No work can
be done in this way. Work can be done only when a
particle is conveyed from the superficial stratum into
the interior, or/ from the interior into the superficial
stratum. That is to say, work can be done only by a
386
THE SCIENCE OP MECHANICS.
change of size of the surface. The consideration whether
the density of the superficial stratum is the same as
that of the Interior, or whether It is constant through
out the entire thickness of the stratum, is not primarily
essential. As will readily be seen, the variation of the
surfacearea is equally the condition of the perform
ance of work when the liquid mass Is immersed in a
second liquid, as in Plateau's experiments.
Diminution We now Inquire whether the work which by the
ficiTurea^ transportation of particles into the interior effects a
tivl work? 1 " diminution of the surfacearea Is positive or negative,
that is, whether work is performed or work is ex
pended. If we put two fluid drops in contact, they
will coalesce of their own accord ;
and as by this action the area
of the surface is diminished, it
follows that the work that pro
duces a diminution of superfi
cial area in a liquid mass is posi
tive. Van der Mensbrugghe has
demonstrated this by a very pretty experiment. A
square wire frame is dipped into a solution of soap and
water, and on the soapfilm formed a loop of moistened
thread is placed. If the film within the loop be punc
tured, the film outside the loop will contract till the
thread bounds a circle in the middle of the liquid sur
face. But the circle, of all plane figures of the same
circumference, has the greatest area ; consequently,
the liquid film has contracted to a minimum.
Consequent The following will now be clear. A weightless
condition ,..,, . , . , 1 1 r
of liquid liquid, the forces acting on which are molecular forces,
equilibrium . . . . ...
will be in equilibrium in all forms in which a system of
virtual displacements produces no alteration of the
liquid's superficial area. But all infinitely small changes
Fig. 201.
THE EXTENSION OF THE PRINCIPLES 387
of form may be regarded as virtual which the liquid
admits without alteration of its volume. Consequently,
equilibrium subsists for all liquid forms for which an
infinitely small deformation produces a superficial va
riation = 0. For a given volume a minimum of super
ficial area gives stable equilibrium ; a maximum un
stable equilibrium.
Among all solids of the same volume, the sphere
has the least superficial area. Hence, the form which
a free liquid mass will assume, the form of stable equi
librium, is the sphere. For this form a maximum of
work is done ; for it, no more can be done If the
. liquid adheres to rigid bodies, the form assumed is de
pendent on various collateral conditions, which render
the problem more complicated.
3. The connection between the size and the farm of Mode of de
the liquid surface may be investigated as follows. We the connec
. J & tionofthe
imagine the closed outer sur size and
face of the liquid to receive
without alteration of the li
quid's volume an infinitely
small variation. By two sets of
mutually perpendicular lines
,
of curvature, we cut up the
original surface into infinitely small rectangular ele
ments. At the angles of these elements, on the original
surface, we erect normals to the surface, and determine
thus the angles of the corresponding elements of the
varied surface. To every element dO of the original
surface there now corresponds an element dO' of the
varied surface ; by an infinitely small displacement, dn,
along the normal, outwards or inwards, dO passes into
dO' and into a corresponding variation of magnitude.
Let dp, dq be the sides of the element dO. For the
388
THE SCIENCE 0/<* MECHANICS.
The mathe sides dp', dq of the element dO', then, these relations
matical de
velopment obtain
of this
method. , ,
din
r
where r and / are the radii of curvature of the princi
pal sections touching the elements of the lines of cur
vature/, q, or the socalled principal radii of curva
ture. 15 ' The radius of curvature of an outwardly convex
element is reckoned as positive, that of an outwardly
concave element as negative, in the usual manner. For
the variation of the element we obtain, accordingly,
#. dO^=dO' dO dpdq(\ +
Neglecting the higher powers of dn we
get
1 ' " ' ~ i.dO.
The variation of the whole surface,
then, is expressed by
^j\dn.dO .... (1)
Furthermore, the normal displacements
must be so chosen that
that is, they must be such that the sum of the spaces
produced by the outward and inward displacements of
* The normal at any point of a surface is cut by normals at infinitely neigh
boring points that lie in two directions on the surface from the original point,
these two directions being at right angles to each other ; and the distances
from the surface at which these normals cut are the two principal, or extreme,
radii of curvature of the surface. Trans.
Fig. 203.
THE EXTENSION OF THE PRINCIPLES. 389
the superficial elements (in the latter case reckoned as
negative) shall be equal to zero, or the volume remain
constant.
Accordingly, expressions fi") and (2) can be putAconditi<
s J * ' on which
simultaneously = only if i/r f~ I / r ' ^ as the same value thegenen
for all points of the surface. This will be readily seen pressions
_ , ,, . . , . T , , obtained,
from the following consideration. Let the elements depends.
dO of the original surface be symbolically represented
by the elements of the line AX (Fig. 204) and let the
normal displacements Sn be erected as ordinates
thereon in the plane JS, the outward displacements up
wards as positive and the inward displacements down
wards as negative.
Join the extremities E
of these ordinates so
as to form a curve,
and take the quadra ^^ n
ture of the curve,
 Fig. 204.
reckoning the sur
face above A X as positive and that below It as nega
tive. For all systems of dn for which this quadra
ture 0, the expression (2) also = 0, and all such
systems of displacements are admissible, that is, are
virtual displacements.
Now let us erect as ordinates, in the plane E' , the
values of i/r + i/r that belong to the elements dO. A
case may be easily Imagined in which the expressions
(i) and (2) assume coincidently the value zero. Should,
however, i/r+ i/^ have different values for different
elements, it will always be possible without altering
the zerovalue of the expression (2), so to distribute
the displacements d n that the expression (i) shall be
different from zero. Only on the condition that i/r +
i/r' has the same value for all the elements, is expres
39 o THE SCIENCE OF MECHANICS.
sion (i) necessarily and universally equated to zero
with expression (2).
The sum Accordingly, from the two conditions (i) and (2) it
eq^fiibdum follows that I//' + l/r'= const ; that is to say, the sum
S,nstantforof the reciprocal values of the principal radii of curva
su?fa*ce le ture, or of the radii of curvature of the principal nor
mal sections, is, in the case of equilibrium, constant
for the whole surface. By this theorem the dependence
of the area of a liquid surface on its superficial /?r;;z is
defined. The train of reasoning here pursued was
first developed by GAUSS,* in a much fuller and more
special form. It is not difficult, however, to present
its essential points in the foregoing simple manner.
Application 4. A liquid mass, left wholly to itself, assumes, as
erai cond?" we have seen, the spherical form, and presents an ab
Smemipted solute minimum of superficial area. The equation
liquid mas 1 ^ + ^, = onst  s here visibly fulfilled in the form
2/J? = const, JR. being the radius of the sphere. If the
free surface of the liquid mass be bounded by two solid
circular rings, the planes of which are parallel to each
other and perpendicular to the line joining their mid
dle points, the surface of the liquid mass will assume
the form of a surface of revolution. The nature of the
meridian curve and the volume of the enclosed mass
are determined by the radius of the rings J?, by the
distance between the circular planes, and by the value
of the expression \/r \\/r' for the surface of revolu
tion. When
1 U 1  1 4 1 I
7" t V~"7~ h co ~~ '&'
the surface of revolution becomes a cylindrical surface.
For 1/r + l/r'= 0, where one normal section is'con
* Principia Generalia, Theories Figures Fluidorum in Statu JEquilibrii
Gottingen, 1830; Werke, Vol. V, 29, GSttingen, 1867.
THE EXTENSION OF THE PRINCIPLES. 391
vex and the other concave, the meridian curve assumes
the form of the catenary. Plateau visibly demonstrated
these cases by pouring oil on two circular rings of wire
fixed in the mixture of alcohol and water above men
tioned.
Now let us picture to ourselves a liquid mass Liquidmas
1 tit r r .  , , . ses whose
bounded by surfaceparts for which the expression surfaces are
1/r 4 1/V' has a positive value, and by other parts cave and
. , partly con
for which the same expression has a negative value, vex
or, more briefly expressed, by convex and concave sur
faces. It will be readily seen that any displacement
of the superficial elements outwards along the normal
will produce in the concave parts a diminution of the
superficial area and in the convex parts an increase.
Consequently, work is performed when concave surfaces
move outwards and convex surfaces inwards. Work
also is performed when a superficial portion moves
outwards for which 1/r + 1/r' ) a, while simulta
neously an equal superficial portion for which 1 f r f
1/r' > a moves inwards.
Hence, when differently curved surfaces bound a
liquid mass, the convex parts are forced inwards and
the concave outwards till the condition 1/r + 1/r' =
const is fulfilled for the entire surface. Similarly, when
a connected liquid mass has several isolated surface
parts, bounded by rigid bodies, the value of the ex
pression 1/r + 1/r' must, for the state of equilibrium
be the same for all free portions of the surface.
For example, if the space between the two circular Experi
mental
rings in the mixture of alcohol and water above re illustration
ferred to, be filled with oil, it is possible, by the use conditions,
of a sufficient quantity of oil, to obtain a cylindrical
surface whose two bases are spherical segments. The
curvatures of the lateral and basal surfaces will accord
392 THE SCIENCE OF MECHANICS.
ingly fulfil the condition l/R + l/oo = l/p + I/A or
p = 2^?, where p is the radius of the sphere and R that
of the circular rings. Plateau verified this conclusion
by experiment.
Uqdm 5 . Let us now study a weightless liquid mass which
iSfhiT encloses a hollow space. The condition that 1/r + 1/r'
low space. shal][ haye the same value f or the interior and exterior
surfaces, is here not realisable. On the contrary, as
this sum has always a greater positive value for the
closed exterior surface than for the closed interior sur
face, the liquid will perform work, and, flowing from
the outer to the inner surface, cause the hollow space
to disappear. If, however, the hollow space be occu
pied by a fluid or gaseous substance subjected to a de
terminate pressure, the work done in the lastmen
tioned process can be counteracted by the work ^ex
pended to produce the compression, and thus equilib
rium may be produced.
Theme Let us picture to ourselves a liquid mass confined
properties between two 'similar and similarly situated surfaces
of bubbles. very near eacri other. A bubble is such
a system. Its primary condition of equi
librium is the exertion of an excess of
pressure by the inclosed gaseous con
tents. If the sum 1/r + 1/r' has the
value + a for the exterior surface, it will
Fig. 205. haye for the interior surface very nearly
the value a. A bubble, left wholly to itself, will al
ways assume the spherical form. If we conceive such
a spherical bubble, the thickness of which we neglect,
the total diminution of its superficial area, on the
shortening of the radius r by dr, will be ibrxdr. If,
therefore, in the diminution of the surface by unit
of area the work A is performed, then A.rfrndr will
THE EXTENSION OP THE PRINCIPLES. 393
be the total amount of work to be compensated for
by the work of compression p.^r^ndr expended by
the pressure f on the inclosed contents. From this
follows ^A/r =p \ from which A may be easily calcu
lated if the measure of r is obtained and p is found by
means of a manometer introduced in the bubble.
An open spherical bubble cannot subsist. If an Open
. bubbles.
open bubble is to become a figure of equilibrium, the
sum 1/r + V'"' must not on ly be constant for each of
the two bounding surfaces, but must also be equal for
both. Owing to the opposite curvatures of the sur
faces, then, l/r\ 1 //' = (). Consequently, r = r'
for all points. Such a surface is called a minimal sur
face ; that is, it has the smallest area consistent with
its containing certain closed contours. It is also a sur
face of zerosum of principal curvatures ; and its ele
ments, as we readily see, are saddleshaped. Surfaces
of this kind are obtained by constructing closed space
curves of wire and dipping the wire into a solution of
soap and water.* The soapfilm assumes of its own
accord the form of the curve mentioned.
6. Liquid figures of equilibrium, made up of thin Plateau's
films, possess a peculiar property. The work of the u?e?of equi
forces of gravity affects the entire mass of a liquid ;
that of the molecular forces is restricted to its super
ficial film. Generally, the work of the forces of grav
ity preponderates. But in thin films the molecular
forces come into very favorable conditions, and it is
possible to produce the figures in question without
difficulty in the open air. Plateau obtained them by
dipping wire polyhedrons into solutions of soap and
water. Plane liquid films are thus formed, which meet
* The mathematical problem of determining such a surface, when the
forms of the wires are given, is called Plateau's Problem. Trans.
394 THE SCIENCE OF MECHANICS.
one another at the edges of the framework. When
thin plane films are so joined that they meet at a hol
low edge, the law l/r + l/r' ~ const no longer holds
for the liquid surface, as this sum has the value zero
for plane surfaces and for the hollow edge a very large
negative value. Conformably, therefore, to the views
above reached, the liquid should run out of the films,
the thickness of which would constantly decrease, and
escape at the edges. This is, in fact, what happens.
But when the thickness of the films has decreased to a
certain point, then, for physical reasons, which are, as
it appears, not yet perfectly known, a state of equilib
rium is effected.
Yet, notwithstanding the fact that the fundamental
equation 1/r+l/V' = const is not fulfilled in these fig
ures, because very thin liquid films, especially films of
viscous liquids, present physical conditions somewhat
different from those on which our original suppositions
were based, these figures present, nevertheless, in all
cases a minimum of superficial area. The liquid films,
connected with the wire edges and with one another,
always meet at the edges by threes at approximately
equal angles of 120, and by fours in corners at approxi
mately equal angles. And it is geometrically demon
strable that these relations correspond to a minimum
of superficial area. In the great diversity of phenom
ena here discussed but one fact is expressed, namely
that the molecular forces do work, positive work, when
the superficial area is diminished.
The reason 7 The figures of equilibrium which Plateau ob
eiuiHbrium tained by dipping wire polyhedrons in solutions of
metrical, soap, form systems of liquid films presenting a re
markable symmetry. The question accordingly forces
itself upon us, What has equilibrium to do with sym
THE EXTENSION OF THE PRINCIPLES. 395
metry and regularity ? The explanation is obvious.
In every symmetrical system every deformation that
tends to destroy the symmetry is complemented by an
equal and opposite deformation that tends to restore it.
In each deformation positive or negative work is done.
One condition, therefore, though not an absolutely
sufficient one, that a maximum or minimum of work
corresponds to the form of equilibrium, is thus sup
plied by symmetry. Regularity is successive symme
try. There is no reason, therefore, to be astonished
that the forms of equilibrium are often symmetrical
and regular.
8. The science of mathematical hydrostatics arose The figure
. , . . ., . , _ 7 _ oftheearth
in connection with a special problem that of the figure
\\ilk
3
Fig. 206.
of the earth. Physical and astronomical data had led
Newton and Huygens to the view that the earth is an
oblate ellipsoid of revolution. NEWTON attempted to
calculate this oblateness by conceiving the rotating
earth as a fluid mass, and assuming that all fluid fila
ments drawn from the surface to the centre exert the
same pressure on the centre. HUYGENS'S assumption
was that the directions of the forces are perpendicular
to the superficial elements. BOUGUER combined both
assumptions. CLAIRAUT, finally (Theorie de la figure
de la terre, Paris, 1743), pointed out that the fulfilment
of both conditions does not assure the subsistence of
equilibrium.
396
THE SCIENCE OF MECHANICS.
Clairaut's
point of
view.
Conditions
of equilib
rium of
Clairaut's
canals.
Clairaut's startingpoint is this. If the fluid earth
is in equilibrium, we may, without disturbing its equi
librium, imagine any portion of it solidified. Accord
ingly, let all of it be solidified but a canal AB, of any
form. The liquid in this canal must also be in equilib
rium. But now the conditions which control equilib
rium are more easily investigated. If equilibrium exists
in every imaginable canal of this kind, then the entire
mass will be in equilibrium. Incidentally Clairaut re
marks, that the Newtonian assumption is realised when
the canal passes through the centre (illustrated in Fig.
206, cut 2), and the Huygenian when the canal passes
along the surface (Fig. 206, cut 3).
But the kernel of the problem, according to Clai
raut, lies in a different view. In all imaginable canals,
Z
Fig. 207. Fi & 2o8 
even in one which returns into itself, the fluid must be
in equilibrium. Hence, if crosssections be made at
any two points M and N of the canal of Fig. 207, the
two fluid columns MPN and MQN must exert on the
surfaces of section at M and N equal pressures. The
terminal pressure of a fluid column of any such canal
cannot, therefore, depend on the length and the form
of the fluid column, but must depend solely on the po
sition of its terminal points.
Imagine in the fluid in question a canal MN of any
form (Fig. 208) referred to a system of rectangular co
THE EXTENSION OF THE PRINCIPLES. 397
ordinates. Let the fluid have the constant density p Mathemat
and let the forcecomponents X, Y, Z acting on unit of sion of
mass of the fluid in the coordinate directions, be f unc ditions, and
 r *ke conse
tions of the coordinates x, y, z of this mass. Let the quent gen
. eral condi
element of length of the canal be called ds, and let its tjon of
liquid equi
projections on the axes be dx, dy, dz. The forcecorn librium.
ponents acting on unit of mass in the direction of the
canal are then X(dxjds\ Y(dyjds\ Z(dzjds\ Let
q be the crosssection ; then, the total force Impelling
the element of mass pqds in the direction ds, is
This force must be balanced by the Increment of pres
sure through the element of length, and consequently
must be put equal to q . dp. We obtain, accordingly,
dp = p (Xdx + Ydy + Zdz]. The difference of pres
sure (/) between the two extremities M and N is found
by integrating this expression from Mto N. But as this
difference is not dependent on the form of the canal
but solely on the position of the extremities M and N,
it follows that p (Xdx + Ydy + Zdz), or, the density
being constant, Xdx + Ydy + Zdz, must be a com
plete differential. For this it Is necessary that
dU dU dU
where 7 is a function of coordinates. Hence, according
to Clairaut, the general condition of liqitid equilibrium is,
that the liquid be controlled by forces 'which can be ex
pressed as the partial differential coefficients of one and
the same fimction of coordinates.
9. The Newtonian forces of gravity, and in fact all
central forces, forces that masses exert in the direc
tions of their lines of junction and which are functions
39 8 THE SCIENCE OF MECHANICS.
character of the distances between these masses, possess this
foVc h e e s property. Under the action of forces of this character
produce 6 10 the equilibrium of fluids is possible. If we know U,
eqm i num replace the first equation by
tdU , >dU. <dUj '
v^ .. dx + T ay + T  dz
up P \^dx dy dz
or
dp = p dU and / = p U\ const.
The totality of all the points for which U = const
is a surface, a socalled /<??'<?/ surface. For this surface
also / = *#//. As all the forcerelations, and, as we
now see, all the pressurerelations, are determined by
the nature of the function U, the pressurerelations,
accordingly, supply a diagram of the forcerelations,
as was before remarked in page 98.
ciairaufs In the theory of Clairaut, here presented, is con
fem of the rained, beyond all doubt, the idea that underlies the
potential. doctrine of forcefunction or potential, which was after
wards developed with such splendid results by La
place, Poisson, Green, Gauss, and others. As soon
as our attention has been directed to this property of
certain forces, namely, that they can be expressed as
derivatives of the same function U", it is at once recog
nised as a highly convenient and economical course to
investigate in the place of the forces themselves the
function U.
If the equation
df = p (Xdx + Ydy + Zdz) = pdU
be examined, it will be seen that Xdx{ Ydy \ Zdz
is the element of the work performed by the forces on
unit of mass of the fluid in the displacement d 's, whose
projections are dx, dy, dz. Consequently, if we trans
port unit mass from a point for which &= C : to an
THE EXTENSION" OF THE PRINCIPLES, 399
other point, indifferently chosen, for which U = C<>, character
,1 r , r i isticsof the
or, more generally, from the surface u=C l to the forcefunc
surface U=^ C 2 , we perform, no matter by what path
the conveyance has been effected, the same amount of
work. All the points of the first surface present, with
respect to those of the second, the same difference of
pressure \ the relation always being such, that
where the quantities designated by the same indices
belong to the same surface.
10. Let us picture to ourselves a group of such character
very closely adjacent surfaces, of which every two sue level, or
cessive ones differ from each other by the same, very til?! sur n
small, amount of work required to transfer a mass from
one to the other ; in other words, imagine the surfaces
U= C, U^= C+ dC, U= C+zdC, and so forth.
A mass moving on a level surface evidently per
forms no work. Hence, every component force in a
direction tangential to the
surface is = ; and the di
rection of the resultant
forceis everywhere normal
to the surface. If we call dn
the element of the normal
intercepted between two
consecutive surf aces, and/
the force requisite to con
vey unit mass from the
one surface to the other pjg. 209.
through this element, the
work done is/, dn = d C. As d C is by hypothesis every
where constant, the force /== dCjdn is inversely pro
portional to the distance between the surfaces consid
4 oo THE SCIENCE OF MECHANICS.
ered. If, therefore, the surfaces U are known, the
directions of the forces are given by the elements of a
system of curves everywhere at right angles to these
surfaces, and the inverse distances between, the sur
faces measure the magnitude of the forces.* These sur
faces and curves also confront us in the other depart
ments of physics.  We meet them as equipotential
surfaces and lines of force in electrostatics and mag
netism, as isothermal surfaces and lines of flow in the
theory'of the conduction of heat, and as equipotential
surfaces and lines of flow in the treatment of electrical
and liquid currents.
illustration ii. We will now illustrate the fundamental idea of
?a;u?s a doc Clairaut's doctrine by another, very simple example.
s?mpie ya Imagine two mutually perpendicular planes to cut the
example. paper at ^^ angles j n the straight lines OX and OY
(Fig. 210). We assume that a forcefunction exists
jj = X y } where x and y are the distances from the
two planes. The forcecomponents parallel to OX and
>Fare ihen respectively
^7___
Jx ~ y
and
Y = x
~~ dy ~
* The same conclusion may be reached as follows. Imagine a wr.ter pipe
laid from New York to Key West, with its ends turning up vertically, and of
glass Let a quantity of water be poured into it, and when equilibrium is
attained, let its height be marked on the glass at both ends. These two marks
will be on one level surface. Now pour in a little more water and again mark
the heights at both ends. The additional water in New York balances the
additional water in Key West. The gravities of the two are equal. But their
quantities are proportional to the vertical distances between the marks. '
Hence the force of gravity on a fixed quantity of water is inversely as those
vertical distances, that is, inversely as the distances between consecutive
level surfaces. Trans.
THE EXTENSION OF THE PRINCIPLES.
401
The level surfaces are cylindrical surfaces, whose
generating lines are at right angles to the plane of the
paper, and whose directrices, xy = const, are equi
lateral hyperbolas. The lines of force are obtained by
turning the first mentioned system of curves through
an angle of 45 in the plane of the paper about O. If
a unit of mass pass
from the point r to
by the route rp'O, or
rqO } or by any other
route, the work done
is always Op X O q.
If we imagine a
closed canal OprqO
filled with a liquid,
the liquid in the ca
nal will be in equi
librium. If transverse
sections be made at
any two points, each
section will sustain
Fig. 210.
at both its surfaces the same
pressure.
We will now modify the example slightly. Let the A modifica
r , _, _.,, i . tion of this
forces be X = j>, Y= a, where a has a constant example,
value. There exists now no function U so constituted
that X= dUjdx and Y= dU/dy ; for in such a case it
would be necessary that dX/dy = d Y/dx, which is ob
viously not true. There is therefore no forcefunction,
and consequently no level surfaces. If unit of mass
be transported from r to O by the way of p, the work
done is a X Oq. If the transportation be effected by
the route rqO, the work done is a X O q + Op X O q.
If the canal OprqO were filled with a liquid, the liquid
could not be in equilibrium, but would be forced to
02 THE SCIENCE OF MECHANICS.
rotate constantly in the direction OfrgO. Currents of
this character, which revert into themselves but con
tinue their motion indefinitely, strike us as something
quite foreign to our experience. Our attention, how
ever, is directed by this to an important property of
the forces of nature, to the property, namely, that the
work of such forces may be expressed as a function of
coordinates. Whenever exceptions to this principle
are observed, we are disposed to regard them as appa
rent, and seek to clear up the difficulties involved,
's is. We shall now examine a few problems of liquid
o motion. The founder of the theory of hydrodynamics is
. TORRICELLI. Torricelli,* by observations on liquids dis
charged through orifices in the bottom of vessels, dis
covered the following law. If the time occupied in the
complete discharge of a vessel be divided into * equal
intervals, and the quantity discharged in the last, the
th ; interval be taken as the unit, there will be dis
charged in the ( l) th , the ( a)* , the ( 3 ) th
interval, respectively, the quantities 3, 5, 7 . . .  and
so forth. An analogy between the motion of falling
bodies and the motion of liquids is thus clearly sug
gested. Further, the perception is an immediate^ one,
that the most curious consequences would ensue if the
liquid, by its reversed velocity of efflux, could rise
higher than its original level. Torricelli remarked,
in fact, that it can rise at the utmost to this height,
and assumed that it would rise exactly as high if all
resistances could be removed. Hence, neglecting all
resistances, the velocity of efflux, v, of a liquid dis
charged through an orifice in the bottom of a vessel is
connected with the height h of the surface of the liquid
by the equation v = V*gh ; that is to say, the velocity
* De Motu Gravium Projectorum, 1643.
THE EXTENSION OF THE PRINCIPLES, 403
of efflux is the final velocity of a body freely falling
through the height //, or liquidhead ; for only with
this velocity can the liquid just rise again to the sur
face. *
Torricelli's theorem consorts excellently with the Varignon's
i i i r i i deduction
rest of our knowledge of natural processes: but weoftheveio
. city of
feel, nevertheless, the need of a more exact insight, efflux.
VARIGNON attempted to deduce the principle from the
relation between force and the momentum generated by
force. The familiar equation pt=mv gives, if by a
we designate the area of the basal orifice, by h the
pressurehead of the liquid, by s its specific gravity,
by g the acceleration of a freely falling body, by v the
velocity of efflux, and by r a small interval of time,
this result
. avrs _
ahs . r = . v or v 2 =gh.
&
Here ahs represents the pressure acting during the
time r on the liquid mass avrs/g. Remembering that
v is a final velocity, we get, more exactly,
if
. arr . rs
ahs . r = 2 . v*
and thence the correct formula
v* = 2gh.
13. DANIEL BERNOULLI investigated the motions of
fluids by the principle of vis viva. We w?ll now treat
the preceding case from this point of view, only ren
dering the idea more modern. The equation which we
employ isjfs = mv 2 /2. In a vessel of transverse sec
tion q (Fig. 211), into which a liquid of the specific
* The early inquirers deduce cheir propositions in, the incomplete form of
proportions, and therefore usually put V proportional to ^gh or ^~h.
404
THE SCIENCE OF MECHANICS.
Daniel Ber gravity s is poured till the head h is reached, the surface
treatment sinks, say, the small distance dh, and the liquid mass
problem, q . dh . s/g is discharged with the velocity p. The work
done is the same as though the weight q . dh . s had
descended the distance h. The path of the motion in
the vessel is not of consequence here. It makes no
difference whether the stratum q . dh
is discharged directly through the
basal orifice, or passes, say, to a
position a, while the liquid at a is
displaced to I, that at b displaced to
c, and that at c discharged. The work
done is in each case q . dh , s . h.
Equating this work to the vis viva of the discharged
liquid, we get
q . dh . s
dk
Fig. air.
.. .
q . dh . s . h =
, or
v = i/2g7i.
The sole assumption of this argument is that all
the work done in the vessel appears as vis viva in the
liquid discharged, that is to say, that the velocities
within the vessel and the work spent in overcoming
friction therein may be neglected. This assumption is
not very far from the truth if vessels of sufficient width
are employed, and no violent rotatory t motion is set up.
The law of Let us neglect the gravity of the liquid in the ves
liquid efflux . , . ...... , . .
when pro sel, and imagine it loaded by a movable piston, on
whose surfaceunit the pressure p falls. If the piston
be displaced a distance dh, the liquid volume q . dh
will be discharged. Denoting the density of the liquid
by p and its velocity by v, we then shall have
sure of
pistons.
V 2 \2j>
q . p . dh = q . dh . p ^ , or v = vl .
*J if /w/
THE EXTENSION OF THE PRINCIPLES.
405
Wherefore, under the same pressure, different liquids
are discharged with velocities inversely proportional to
the square root of their density. It is generally sup
posed that this theorem is directly applicable to gases.
Its form, indeed, is correct ; but the deduction fre
quently employed involves an error, which we shall
now expose.
14. Two vessels (Fig. 212) of equal crosssections The a
are placed side by side and connected with each other SSsla
by a small aperture in the base of their dividing walls, lov^of
For the velocity of flow through this aperture we ob gases '
tain, under the same suppositions as before,
dh s
or v = V i g ( Ji , k. ^
j. ^ ~ 0" 21 " <> ^ j. ^y
If we neglect the gravity of the liquid and imagine
the pressures p^ and p z produced by pistons, we shall
similarly have v=V 2(/ 1 / 2 )/P ^ or example, if the
pistons employed be loaded with the weights P and
P/2, the weight P will sink the distance h and P/%
will rise the distance h. The work (P/z)/i is thus left,
to generate the vis viva of the effluent fluid.
A gas under such circumstances would behave dif
ferently. Supposing the gas to flow from the vessel
containing the load Pinto that contain
ing the load P/2, the first weight will
fall a distance //, the second, however,
since under half the pressure a gas dou
bles its volume, will rise a distance 2/1,
so that the work P/i (P/2) 2/^ =
would be performed. In the case of
gases, accordingly, some additional
work, competent to produce the flow between the vessels
must be performed. This work the gas itself performs,
by expanding, and by overcoming by its force of expan
dh
The behav
iour of a
gas under
the as
sumed con
ditions.
Fig. 212.
406 ' THE SCIENCE OF MECHANICS.
The result sion a pressure. The expansive force/ and the volume
ibrrrTbut^ 1 w of a gas stand to each other in the familiar relation
magnftude./2/ = ^, where k, so long as the temperature of the
gas remains unchanged, is a constant. Supposing the
volume of the gas to expand under the pressure f by
an amount dw, the work done is
For an expansion from W Q to w, or for an increase of
pressure from / to /, we get for the work
Conceiving by this work a volume of gas ze/ of
density p, moved with the velocity v, we obtain
._
V
The velocity of efflux is, accordingly, in this case also
inversely proportional to the square root of the density ;
Its magnitude, however, is not the same as in the case
of a liquid,
incom But even this last view is very defective. Rapid
pletenessof _ r i 1 r ,
this view, changes of the volumes of gases are always accom
panied with changes of temperature, and, consequently
also with changes of expansive force. For this reason,
questions concerning the motion of gases cannot be
dealt with as questions of pure mechanics, but always
involve questions of heat. [Nor can even a thermo
dynamical treatment always suffice : it is sometimes
necessary to go back to the consideration of molecular
motions.]
15. The knowledge that a compressed gas contains
stored up work, naturally suggests the inquiry, whether
THE EXTENSION OF THE PRINCIPLES. 407
this is not also true of compressed liquids. As a mat Relative
r ^ volumes of
ter of fact, every liquid under pressure ts compressed, compressed
3 J ^ r gases and
To effect compression work is requisite, which reap liquids.
pears the moment the liquid expands. But this work,
in the case of the mobile liquids, is very small. Imag
ine, in Fig. 213, a gas and a mobile liquid of the same
volume, measured by OA, subjected to the same pres
sure, a pressure of one atmosphere, designated b} r AB.
If the pressure be reduced to onehalf an atmosphere,
the volume of the gas will be doubled, while that of
the liquid will be increased by only about 25 millionths.
The expansive work of the gas is represented by the
surface ABDC, that of the liquid by ABLK, where
H
A J^ =000002 5 OA. If the pressure decrease till it
become zero, the total work of the liquid is represented
by the surface ABI, where AI = 000005 OA, and the
total work of the gas by the surface contained between
AB, the infinite straight line ACEG . . . ., and the
infinite hyperbola branch BDFH . . . . Ordinarily,
therefore, the work of expansion of liquids may be
neglected. There are however phenomena, for ex
ample, the soniferous vibrations of liquids, in which
work of this very order plays a principal part. In such
cases, the changes of temperature the liquids undergo
must also be considered. We thus see that it is only
by a fortunate concatenation of circumstances that we
are at liberty to consider a phenomenon with any close
4 o8 THE SCIENCE OF MECHANICS.
approximation to the truth as a mere matter of molar
mechanics.
The hydro 1 6. We now come to the idea which DANIEL BER
principie NOULLi sought to apply in his work Hydro dynamic a, sive
Bernoulli, de Viribus et Motibus Fluidorum Comment arii (1738).
When a liquid sinks, the space through which its cen
tre of gravity actually descends (des census actualis} is
equal to the space through which the centre of gravity
of the separated parts affected with the velocities ac
quired in the fall can ascend (as census fotentialis}. This
Idea, we see at once, is identical with that employed
by Huygens. Imagine a vessel filled with a liquid
(Fig. 214) \ and let its horizontal cross
section at the distance x from the plane
of the basal orifice, be called f(x\ Let
the liquid move and its surface descend
a distance dx. The centre of gravity,
then, descends the distance xf(x) . dx/M,
where JkT= Cf(x) dx. If k is the space of
potential ascent of the liquid In a cross
lg ' 2I4 ' section equal to unity, the space of po
tential ascent in the crosssection /(#) will be k/f(x} 2 ,
and the space of potential ascent of the centre of
gravity will be
r dx
} 7(^\ N
J Jv*) k~
M ~ M*
where ,
For the displacement of the liquid's surface through a
distance dx, we get, by the principle assumed, both
JV and k changing, the equation
#/(#) dx = Ndk + kdN.
THE EXTENSION OF THE PRINCIPLES. 409
This equation was employed by Bernoulli in the solu The parai
. ... .. i lelism of
tion of various problems. It will be easily seen, that strata.
Bernoulli's principle can be employed with success
only when the relative velocities of the single parts of
the liquid are known. Bernoulli assumes, an assump
tion apparent in the formulas, that all particles once
situated in a horizontal plane, continue their motion
in a horizontal plane, and that the velocities in the
different horizontal planes are to each other in the in
verse ratio of the sections of the planes. This is the
assumption of the parallelism of strata. It does not, in
many cases, agree with the facts, and in others its
agreement is incidental. When the vessel as compared
with the orifice of efflux is very wide, no assumption
concerning the motions within the vessel is necessary,
as we saw in the development of Torricelli's theorem.
17. A few isolated cases of liquid motion were The water
treated by NEWTON and JOHN BERNOULLI. We shall of Newton,
consider here one to which a
familiar law is directly applic
able. A cylindrical Utube with
vertical branches is filled with
a liquid (Fig. 215). The length
of the entire liquid column is /,
If in one of the branches the
column be forced a distance x
below the level, the column in lg ' 2I5 '
the other branch will rise the distance x, and the
difference of level corresponding to the excursion x
will be 2 x. If a is the transverse section of the tube
and s the liquid's specific gravity, the force brought
into play when the excursion x is made, will be zasx,
which, since it must move a mass al$/gv?i\l determine
the acceleration (2 asx}/(als/g) = (2^//) x, or, for unit
4io THE SCIENCE OF MECHANICS.
excursion, the acceleration 2/7. We perceive that
pendulum vibrations of the duration
will take place. The liquid column, accordingly, vi
brates the same as a simple pendulum of half the length
of the column.
The liquid A similar, but somewhat more general, problem was
of joh n um treated by John Bernoulli. The two branches of a
Bernoulli. . , . , J , ._.  , . ,
cylindrical tube (Fig. 216), curved in any manner, make
with the horizon, at the
points at which the
surfaces of the liquid
move, the angles a
and ft. Displacing one
of the surfaces the dis
tance x 9 the other sur
r rr
lace suners an equal
displacement. A difference of level is thus produced
x (sin a f sin/?), and we obtain, by a course of reason
ing similar to that of the preceding case, employing
the same symbols, the formula
r= l
(sin or f sin/?) '
The laws of the pendulum hold true exactly for the
liquid pendulum of Fig. 215 (viscosity neglected), even
for vibrations of great amplitude ; while for the filar
pendulum the law holds only approximately true for
small excursions.
1 8* The centre of gravity of a liquid as a whole can
rise only as high as it would have to fall to produce its
velocities. In every case in which this principle appears
to present an exception, it can be shown that the excep
FHE EXTENSION OF THE PRINCIPLES. 411
tion is only apparent. One example is Hero's fountain. Hero's
J * r . r . fountain.
This apparatus, as we know, consists of three vessels,
which may be designated in the descending order as
A 9 B, C. The water in the open vessel A
falls through a tube into the closed vessel
C ; the air displaced in C exerts a pressure
on the water in the closed vessel B, and
this pressure forces the water in B in a
jet above A whence it falls back to its
original level. The water in B rises, it is
true, considerably above the level of B,
but in actuality it merely flows by the
circuitous route of the fountain and the
vessel A to the much lower level of C.
Another ap
parent exception
to the principle
in question is
that of Montgol
fi e r ' s hydra ulic
ram, in which the
liquid by its own
gravitational
work appears to
rise considerably
above its original
level. The liquid
flows (Fig. 217)
from a cistern A
through a long
pipe RR and a valve V 9 which opens inwards, into a
vessel B. When the current becomes rapid enough, the
valve V is forced shut, and a liquid mass m affected with
the velocity v is suddenly arrested in RR, which must
4 I2
THE SCIENCE OF MECHANICS.
be deprived of its momentum. If this be done in the
time /, the liquid can exert during this time a pressure
q = mv/t, to which must be added its hydrostatical
pressure /. The liquid, therefore, will be able, during
this interval of time, to penetrate with a pressure/ f ^
through a second valve into a pila Heronis, H, and in
consequence of the circumstances there existing will
rise" to a higher level in the ascensiontube SS than
that corresponding to its simple pressure p. It is
to be observed here, that a considerable portion of the
liquid must first flow off into JB, before a velocity requi
site to close Fis produced by the liquid's work in JZR.
A small portion only rises above the original level ;
the greater portion flows from A into B. If the liquid
discharged from 5* were collected, it could be easily
proved that the centre of gravity of the quantity thus
discharged and of that received in B lay, as the result
of various losses, actually below the level of A.
The principle of the hydraulic ram, that of the
An illustra
tion, which . . , , , ., , . . .,
elucidates transference of work done by a large liquid mass to a
the action
of the hy
draulic ram ,
smaller one, which
thus acquires a great
vis viva, may be illus
trated in the following
very simple manner.
Close the narrow
opening O of a funnel
and plunge it, with its
wide opening down
wards, deep into a
large vessel of water. If the finger closing the upper
opening be quickly removed, the space inside the
funnel will rapidly fill with water, and the surface of the
water outside the funnel will sink. The work performed
Fig. 218.
THE EXTENSION OF THE PRINCIPLES. 413
Is equivalent to the descent of the contents of the funnel
from the centre of gravity ^ of the superficial stratum
to the centre of gravity S' of the contents of the fun
nel. If the vessel is sufficiently wide the velocities in
it are all very small, and almost the entire vis viva is
concentrated in the contents of the funnel. If all the
parts of the contents had the same velocities, they
could all rise to the original level, or the mass as a
whole could rise to the height at which its centre of
gravity was coincident with S. But in the narrower
sections of the funnel the velocity of the parts is
greater than in the wider sections, and the former
therefore contain by far the greater part of the vis
viva. Consequently, the liquid parts above are vio
lently separated from the parts below and thrown
out through the neck of the funnel high above the
original surface. The remainder, however, are left
considerably below that point, and the centre of grav
ity of the whole never as much as reaches the original
level of S.
19. One of the most important achievements of Hydrostatic
Daniel Bernoulli is his distinction of hydrostatic and dynamic
777. roi pressure.
tiydrodynamic pressure. The pressure
which liquids exert is altered by motion ;
and the pressure of a liquid in motion
may, according to the circumstances, be
greater or less than that of the liquid at rest
with the same arrangement of parts. We
will illustrate this by a simple example.
The vessel A, which has the form of a body
of revolution with vertical axis, is kept Flg ' 2I9>
constantly filled with a frictionless liquid, so that its
surface at mn does not change during the discharge
at kl. We will reckon the vertical distance of a particle
4 i4 THE SCIENCE OF MECHANICS.
Determina from the surface /// n downwards as positive and call
tion of the _ .. P . .
pressures it z. Let us follow the course of a prismatic element of
generally ... . . . .
acting in li volume, whose horizontal basearea is a and height 6,
quids in . . . . , _
motion. m its downward motion, neglecting, on the assump
tion of the parallelism of strata, all velocities at right
angles to z. Let the density of the liquid be p, the
velocity of the element v, and the pressure, which is
dependent 'on z, p. If the particle descend the dis
tance dz, we have by the principle of vis viva
(1)
that is, the increase of the vis viva of the element is
equal to the work of gravity for the displacement in
question, less the work of the forces of pressure of the
liquid. The pressure on the upper surface of the element
is ap y that on the lower surface is a \_p r (dp/dz}fl~\.
The element sustains, therefore, if the pressure in
crease downwards, an upward pressure a (dp/dz)p ;
and for any displacement dz of the element, the work
a(dp/dz)/3dz must be deducted. Reduced, equation
(i) assumes the form
;2\ dp
and, integrated, gives
*
.......... (2)
If we express the velocities in two different hori
zontal crosssections <z 1 and a 2 at the depths z^ and z 2
below the surface, by v^ v 2 , and the corresponding
pressures by/ 1? / 2 , we may write equation (2) in the
form
THE EXTENSION' OF THE PRINCIPLES. 415
Taking for our crosssection a l the surface, z 1 = 0, J
p^=. ; and as the same quantity of liquid flows through P^?^ ith
all crosssections in the same interval of time, a l v l =the^ircum
a 2 v 2 . Whence, finally, the motion.
The pressure / 2 of the liquid in motion (the hydro
dynamic pressure) consists of the pressure pg% 2 of the
liquid at rest (the hydrostatic pressure) and of a pres
sure (p/2)z;2 [(#2 #f )/# 1] dependent on the density,
the velocity of flow, and the crosssectional areas. In
cross sections larger than the surface of the liquid, the
hydrodynamic pressure is greater than the hydrostatic,
and vice versa.
A clearer idea of the significance of Bernoulli's illustration
of these re
principle may be obtained by imagining the liquid m suits by the
the vessel A unacted on by gravity, and its outflow quids under
. r pressures
produced by a constant pressure p. on the surface, produced
* * A by pistons.
Equation (3) then takes the form
If we follow the course of a particle thus moving, it
will be found that to every increase of the velocity of
flow (in the narrower crosssections) a decrease of
pressure corresponds, and to every decrease of the ve
locity of flow (in the wider crosssections) an increase
of pressure. This, indeed, is evident, wholly aside
from mathematical considerations. In the present case
every change of the velocity of a liquid element must be
exclusively produced by the work of the liquid' s forces
of pressure. When, therefore, an element enters into
a narrower crosssection, in which a greater velocity
of flow prevails, it can acquire this higher velocity only
4 i6
THE SCIENCE OF MECHANICS.
on the condition that a greater pressure acts on its rear
surface than on. its front surface, that is to say, only
when it moves from points of higher to points of lower
pressure, or when the pressure decreases in the direc
tion of the motion. If we imagine the pressures in
a wide section and in a succeeding narrower section
to be for a moment equal, the acceleration of the ele
ments in the narrower section will not take place the
elements will not escape fast enough ; they will accumu
late before the narrower section ; and at the entrance
to it the requisite augmentation of pressure will be im
mediately produced. The converse case is obvious.
Treatment 2o. In dealing with more complicated cases, the
of a liquid ! . ,
problem in problems of liquid motion, even though viscosity be
which vis r J
cosily and
friction are
considered.
Fig. 220.
neglected, present great difficulties ; and when the
enormous effects of viscosity are taken into account,
anything like a dynamical solution of almost every
problem is out of the question. So much so, that al
though these investigations were begun by Newton,
we have, up to the present time, only been able to
master a very few of the simplest problems of this class,
and that but imperfectly. We shall content ourselves
with a simple example. If we cause a liquid contained
in a vessel of the pressurehead h to flow, not through
an orifice in its base, but through a long cylindrical
tube fixed in its side (Fig. 220), the velocity of efflux
THE EXTENSION' OF THE PRINCIPLES. 417
v will be less than that deducible from Torricelli's law,
as a portion of the work is consumed by resistances
due to viscosity and perhaps to friction. We find, in
fact, that v 1/2 g~h^, where 7^ < h. Expressing by // 1
the z/<r/^#yhead, and by 7i 2 the resistanceheadi, we may
put h=:h l j ^2 ^ to ^6 ma i n cylindrical tube we
affix vertical lateral tubes, the liquid will rise in the
latter tubes to the heights at which it equilibrates the
pressures in the main tube, and will thus indicate at all
points the pressures of the main tube. The noticeable
fact here is, that the liquidheight at the point of influx
of the tube is = 7i 2 , and that it diminishes in the direc
tion of the point of outflow, by the law of a straight
line, to zero. The elucidation of this phenomenon is
the question now presented.
Gravity here does not act directly on the liquid in The condi*
the horizontal tube, but all effects are transmitted to it perform
by the pressure of the surrounding parts. If we imagwork in
1 J 1 T 1 J SUCh CaSGS 
me a prismatic liquid element of basal area a. and
length ft to be displaced in the direction of its length
a distance dz, the work done, as in the previous case, is
= 
dz ' ' dz
For a finite displacement we have
Work is done when the element of volume is displaced
from a place of higher to a place of lower pressure.
The amount of the work done depends on the size of
the element of volume and on the difference of pressure
at the initial and terminal points of the motion, and
not on the length and the form of the path traversed.
418
THE SCIENCE OF MECHANICS.
The conse
quences of
these con
ditions.
If the diminution ot pressure were twice as rapid in
one case as in another, the difference of the pressures
on the front and rear surfaces, or \h& force of the work,
would be doubled, but the space through which the
work was done would be halved. The work done would
remain the same, whether done through the space ab
or ac of Fig. 221.
Through every crosssection q of the horizontal tube
the liquid' flows with the same velocity v. If, neglect
ing the differences of velocity in the same crosssection,
we consider a liquid element which exactly fills the
section q and has the length ft, the vis viva qfip(z&/2)
of such an element will persist unchanged throughout
its entire course in the tube.
This is possible only provided
the vis viva consumed by friction
is replaced by the work of the
liquids forces of pressure. Hence,
in the direction of the motion
of the element the pressure
must diminish, and for equal distances, to which the
same work of friction corresponds, by equal amounts.
The total work of gravity on a liquid element q ft p
issuing from the vessel, is q fi pgk. Of this the portion
#/?p(# 2 /2) is the vis viva of the element discharged
with the velocity v into the mouth of the tube, or, as
v = l/2g& 19 the portion qfipgh r The remainder of
the work, therefore, q j3 pg/i 2 , is consumed in the tube,
if owing to the slowness of the motion we neglect the
losses within the vessel.
If the pressureheads respectively obtaining in the
vessel, at the mouth, and at the extremity of the tube,
are h, A 2 , 0, or the pressures are/ = hgp 9 / 2 h^g p } 0,
then by equation (i) of page 417 the work requisite to
THE EXTENSION OF THE PRINCIPLES. 419
generate the vis viva of the element discharged into
the mouth of the tube is
and the work transmitted by the pressure of the liquid
to the element traversing the length of the tube, is
or the exact amount consumed in the tube.
Let us assume, for the sake of argument, that the indirect
demonstra
pressure does not decrease from p 9 at the mouth to tion of
1 11 p 1 these con
Zero at the extremity of the tube by the law of a straight sequences.
line, but that the distribution of the pressure is differ
ent, say, constant throughout the entire tube. The
parts in advance then will at once suffer a loss of ve
locity from the friction, the parts which follow will
crowd upon them, and there will thus be produced at
the mouth of the tube an augmentation of pressure
conditioning a constant velocity throughout its entire
length. The pressure at the end of the tube can only
be = because the liquid at that point is not prevented
from yielding to any pressure impressed upon it.
If we imagine the liquid to be a mass of smooth A simile
elastic balls, the balls will be most compressed at the which these
i r.,1 1,1 ii i i phenomena
bottom or the vessel, they will enter the tube in a state may be
r . i MI i 11 1 i easily con
01 compression, and will gradually lose that state in ceived.
the course of their motion. We leave the further de
velopment of this simile to the reader.
It is evident, from a previous remark, that the work
stored up in the compression of the liquid itself, is very
small. The motion of the liquid is due to the work of
gravity in the vessel, which by means of the pressure
of the compressed liquid is transmitted to the parts in
the tube.
4 2
THE SCIENCE OF MECHANICS.
A partial An Interesting modification of the case just dis
cation of cussed is obtained by causing the liquid to flow through
discussed, a tube composed of a number of shorter cylindrical
tubes of varying widths. The pressure In the direction
of outflow then diminishes (Fig. 222) more rapidly in
the narrower tubes, in which a greater consumption of
work by friction takes place, than in the wider ones.
We further note, in every passage of the liquid into a
Fig, 222.
wider tube, that is to a smaller velocity of flow, an in
crease of pressure (a positive congestion) ; in every
passage into a narrower tube, that is to a greater velo
city of flow, an abrupt diminution of pressure (a nega
tive congestion). The velocity of a liquid element on
which no direct forces act can be diminished or In
creased only by its passing to points of higher or lower
pressure.
CHAPTER IV.
THE FORMAL DEVELOPMENT OF MECHANICS.
i.
THE ISOPERIMETRICAL PROBLEMS.
i. When the chief facts of a physical science have The formal
c . asdistin
once been fixed by observation, a new period of its guished
. from the de
development begins the dediictive, which we treatedductive.de
.._. velopment
in the previous chapter. In this period, the facts are of physical
, ., , . ... ., science.
reproducible in the mind without constant recourse to
observation. Facts of a more general and complex
character are mimicked in thought on the theory that
they are made up of simpler and more familiar obser
vational elements. But even after we have deduced
from our expressions for the most elementary facts
(the principles) expressions for more common and more
complex facts (the theorems) and have discovered in
all phenomena the same elements, the developmental
process of the science is not yet completed. The de
ductive development of the science is followed by its
formal development. Here it is sought to put in a clear
compendious form, or system, the facts to be repro
duced, so that each can be reached and mentally pic
tured with the least intellectual effort. Into our rules
for the mental reconstruction of facts we strive to in
corporate the greatest possible uniformity, so that these
rules shall be easy of acquisition. It is to be remarked,
that the three periods distinguished are not sharply
422 THE SCIENCE OF MECHANICS.
separated from one another, but that the processes of
development referred to frequently go hand in hand,
although on the whole the order designated is unmis
takable.
Theisoperi 2. A powerful influence was exerted on the formal
metrical r 1 i 11
problems, development of mechanics by a particular class of
tk>ns q o! s mathematical problems, which, at the close of the
maxima ..,.. . ., ,
and minima seventeenth and the beginning ot the eighteenth cen
turies, engaged the deepest attention of inquirers.
These problems, the socalled isoperimetrical problems,
will now form the subject of our remarks. Certain
questions of the greatest and least values of quanti
ties, questions of maxima and minima, were treated by
the Greek mathemati
cians. Pythagoras is
said to have taught that
iw v , _~ the circle, of all plane
"Xj' figures of a given peri
p. 223 m.eter, has the greatest
area. The idea, too, of a
certain economy in the processes of nature was not
foreign to the ancients. Hero deduced the law of the
reflection of light from the theory that light emitted
from a point A (Fig. 223) and reflected at M will travel
to B by the shortest route. Making the plane of the
paper the plane of reflection, SS the intersection of
the reflecting surface, A the point of departure, B the
point of arrival, and M the point of reflection of the
ray of light, it will be seen at once that the line AMB* ,
where B' is the reflection of B, is a straight line. The
line AMB f is shorter than the line ANB', and there
fore also AMB is shorter than ANB. Pappus held
similar notions concerning organic nature ; he ex
FORMAL DEVELOPMENT. 423
plained, for example, the form of the cells of the honey
comb by the bees' efforts to economise in materials.
These ideas fell, at the time of the revival of the The re
searches or
sciences, on not unfruitful soil. They were first taken Kepler, Per
up by FERMAT and ROBERVAL, who developed a method Robervai.
applicable to such problems. These inquirers ob
served, as Kepler had already done, that a magni
tude y which depends on another magnitude x, gen
erally possesses in the vicinity of its greatest and least
values a peculiar property. Let x (Fig. 224) denote
abscissas and y ordinates. If, while x increases, y pass
through a maximum value, its increase, or rise, will
be changed into a decrease, or
fall ; and if it pass through a
minimum value its fall will be
changed into a rise. The neigh
boring values of the maximum
\
or minimum value, consequently, pi gi 224 .
will lie very near each other, and
the tangents to the curve at the points in question will
generally be parallel to the axis of abscissas. Hence,
to find the maximum or minimum values of a quan
tity, we seek the parallel tangents of its curve.
The method of tangents may be put in analytical The
P r, , , . _ method of
iorm. For example, it is required to cut off from a tangents,
given line a a portion x such that the product of the
two segments x and a x shall be as great as possible.
Here, the product x (a x) must be regarded as the
quantity y dependent on x. At the maximum value of
y any infinitely small variation of x 9 say a variation ,
will produce no change in y. Accordingly, the required
value of x will be found, by putting
x(a x) = (x f <?) (a x <?)
or
424
THE SCIENCE OP MECHANICS.
or
__ ^ 2x S
As S may be made as small as we please, we also get
mal effect.
whence x a/2.
In this way, the concrete idea of the method of
tangents may be translated into the language of alge
bra ; the procedure also contains, as we see, the germ
of the differential calculus.
The refrac Fermat sought to find for the law of the refraction
aamjnf * of light an expression analogous to that of Hero for
law of reflection. He remarked
that light, proceeding from a
point A, and refracted at a
point jM~ 9 travels to J3, not by
the shortest route, but in the
shortest time. If the path AMB
is performed in the shortest
time, then a neighboring path
ANB, infinitely near the real
path, will be described in the
same time. If we draw from JV on AM and from M on
NB the perpendiculars NP and MQ, then the second
route, before refraction, is less than the first route by a
distance MP=NM sin a, but is larger than it after
refraction by the distance NQ = NM sin/3. On the
supposition, therefore, that the velocities in the first
and second media are respectively v^ and v 2 , the time
required for the path AMB will be a minimum when
Fig. 225.
JVMsiua
=
or
FORMAL DEVELOPMENT. 425
v. sin a:
1 == . 7j = n }
v 2 smp
where n stands for the index of refraction. Hero's law
of reflection, remarks Leibnitz, is thus a special case
of the law of refraction. For equal velocities (7^ = ?; 2 ),
the condition of a minimum of time is identical with
the condition of a minimum of space.
Huygens. in his optical investigations, applied and
 f . ._ . f . completion
further perfected the ideas of Fermat, considering, not of Format's
...... researches.
only rectilinear, but also curvilinear motions of light,
in media in which the velocity of the light varied con
tinuously from place to place. For these, also, he
found that Fermat's law obtained. Accordingly, in all
motions of light, an endeavor, so to speak, to produce
results in a minimum of time appeared to be the funda
mental tendency.
3. Similar maximal or minimal properties were The prob
brought out in the study of mechanical phenomena. brachisto
As we have already noticed, John Bernoulli knew that
a freely suspended chain assumes the form for which
its centre of gravity lies lowest. This idea was, of
course, a simple one for the investigator who first rec
ognised the general import of the principle of virtual
velocities. Stimulated by these observations, inquir
ers now began generally to investigate, maximal and
minimal characters. The movement received its most
powerful impulse from a problem propounded by John
Bernoulli, in June, 1696* the problem of the brackis
tochrone. In a vertical plane two points are situated,
A and B. It is required to assign in this plane the
curve by which a falling body will travel from A to B
in the shortest time. The problem was very ingeniously
* Ada, Erudztorum, Leipsic.
426 THE SCIENCE OF MECHANICS.
solved by John Bernoulli himself ; and solutions were
also supplied by Leibnitz, L'Hopital, Newton, and
James Bernoulli.
johnBer The most remarkable solution was JOHN BER
geniousso NOULLi's own. This inquirer remarks that problems
problem of of this class have already been solved, not for the mo
tochrone. tion of falling bodies, but for the motion of light. He
accordingly imagines the motion of a falling body re
placed by the motion of
^\ a ray of light. (Comp.
\ p. 379) The two points
N_ A and B are supposed
"^p to be fixed in a medium
in which the velocity of
Fig. 226. , . , , . ,
light increases in the
vertical downward direction by the same law as the
velocity of a falling body. The medium is supposed
to be constructed of horizontal layers of downwardly
decreasing density, such that v = Vigh denotes the
velocity of the light in any layer at the distance h be
low A. A. ray of light which travels from A to B un
der such conditions will describe this distance in the
shortest time, and simultaneously trace out the curve
of quickest descent.
Calling the angles made by the element of the
curve with the perpendicular, or the normal of the
layers, a, a', a". . . ., and the respective velocities
v, v' 9 v". . . ., we have
sine/ s'ma"
= ,  = T^ = ....= k = const.
. v v
or, designating the perpendicular distances below A
by x, the horizontal distances from A by y, and the arc
of the curve by s 9
FORMAL DEVELOPMENT. 4 2 7
The bra
chisto
j 1 chrone a
_/ZjV __ ^ cycloid.
V
whence follows
dy* = k 2 v 2 ds 2 = k* v 2 (dx* + ^y 2 )
and because v = "\/igx also
1
dy = dx<*\ , where a =
J \a x
This is the differential equation of a cycloid, or curve
described by a point in the circumference of a circle of
radius r = #/2 = i/4v 2 , rolling on a straight line.
To find the cycloid that passes through A and B, The con
^ * struction ot
it is to be noted that all cycloids, inasmuch as they
produced by similar con
structions, are similar ;
and that if generated by
the rolling of circles on
AD from the point A as
origin, are also similarly
, . , Fig. 227.
situated with respect to
the point A. Accordingly, we draw through AB a
straight line, and construct any cycloid, cutting the
straight line in B*. The radius of the generating
circle is, say, r' . Then the radius of the generating
circle of the cycloid sought is r= r\AB '/AJ3'}.
This solution of John Bernoulli's, achieved entirely
without a method, the outcome of pure geometrical
fancy and a skilful use of such knowledge as happened
to be at his command, is one of the most remarkable
and beautiful performances in the history of physical
science. John Bernoulli was an aesthetic genius in this
field. His brother James's character was entirely differ
ent. James was the superior of John in critical power,
428
THE SCIENCE OF MECHANICS.
Compari but in originality and imagination was surpassed by the
scientific latter. James Bernoulli likewise solved this problem,
of John and though in less felicitous form. But, on the other hand,,
aoui!?. er ~ he did not fail to develop, with great thoroughness, a
general method applicable to such problems. Thus,
in these two brothers we find the two fundamental
traits of high scientific talent separated from one
another, traits, which in the very greatest natural
inquirers, in Newton, for example, are combined to
gether. We shall soon see those two tendencies, which
within one bosom might have fought their battles un
noticed, clashing in open conflict, in the persons of
these two brothers.
Vignette to Leibnitzii et Johannis Bernoulli} comercium efiistolicum.
Lausanne and Geneva, Bousquet, 1745.
James Ber 4. James Bernoulli finds that the chief object of
noulli'sre .._ 11, .,
marks on research hitherto had been to find the values of a vari
the general
nature of able quantity, for which a second variable quantity,
problem, which is a function of the first, assumes its greatest or
its least value. The present problem, however, is to find
FORMAL DEVELOPMENT. 429
from among an infinite number of curves one which pos
sesses a certain maximal or minimal property. This, as
he correctly remarks, is a problem of an entirely dif
ferent character from the other and demands a new
method.
The principles that Tames Bernoulli employed in The princi
pies em
the solution of this problem (A eta Eruditorum, May, ployed in
James Ber
1607}* are as follows: nouiii'sso
*' ' . lution.
(1) If a curve has a certain property of maximum
or minimum, every portion or element of the curve has
the same property.
(2) Just as the infinitely adjacent values of the
maxima or minima of a quantity in the ordinary prob
lems, for infinitely small changes of the independent
variables, are constant, so also is the quantity here to
be made a maximum or minimum for the curve sought,
for infinitely contiguous curves, constant.
(3) It is finally assumed, for the case of the brachis
tochrone, that the velocity is v = '}/2g/i, where h de
notes the height fallen through.
If we picture to ourselves a very small portion
of the curve (Fig. 228), and, imagining a horizontal tlfres^f
,. , , n L James Ber
ime drawn through JB, cause nouiii's so
the portion taken to pass into
the infinitely contiguous por
tion ADC, we shall obtain, by _
considerations exactly similar ^
to those employed in the treat
ment of Fermat's law, the well
known relation between the lg ' 2 * '
sines of the angles made by the curveelements with
the perpendicular and the velocities of descent. In
this deduction the following assumptions are made,
* See also his works, Vol. II, p. 768.
43=>
THE SCIENCE OF MECHANICS.
cal prob
lem.
(i), that the fart, or element, ABC is brachistochro
nous, and (2), that ADC is described in the same time
as ABC. Bernoulli's calculation is very prolix ; but
its essential features are obvious, and the problem is
solved * by the abovestated principles.
The Pro With the solution of the problem of the brachisto
jamesBer chrone, Tames Bernoulli, in accordance with the prac
noulli, or ... . .
thepropositice then prevailing among mathematicians, proposed
tion of the . . . i
general iso the following more general "isopenmetncal problem ":
" Of all isoperimetrical curves (that is, curves of equal
"perimeters or equal lengths) between the same two
"fixed points, to find the curve such that the space
"included (i) by a second curve, each of whose ordi
" nates is a given function of the corresponding ordi
"nate or the corresponding arc of the one sought, (2)
"by the ordinates of its extreme points, and (3) by the
"part of the axis of abscissas lying between those ordi
" nates, shall be a maximum or minimum."
For example. It is required to find the curve BFN,
described on the base BN such, that of all curves of
the same length on BN>
this particular one shall make
the area BZN a minimum,
where PZ=.{PFJ\ LM =
(LKy i , and so on. Let the
relation between the ordi
nates of BZN and the cor
responding ordinates oiBFN
be given by the curve BIT. To obtain PZ from PF 9
draw FGH at right angles to BG, where BG is at right
angles to BN. By hypothesis, then, PZ= GH, and
* For the details of this solution and for information generally on the his
tory of this subject, see Woodhouse's Treatise on Isoperimetrical Problems
and the Calculus of Variations, Cambridge, 1810. Trans.
FORMAL DEVELOPMENT. 43 *
for the other ordinates. Further, we put BP = y,
= x, PZ = x n .
John Bernoulli gave, forthwith, a solution of this John
problem, in the form Si
lem '
where a is an arbitrary constant. For n = 1,
that is, JST^/V is a semicircle on J5W" as diameter, and
the area ZNis equal to the are&JBFN. For this par
ticular case, the solution, in fact, is correct. But the
general formula is not universally valid.
On the publication of John Bernoulli's solution,
James Bernoulli openly engaged to do three things :
first, to discover his brother's method ; second, to point
out its contradictions and errors ; and, third, to give the
true solution. The jealousy and animosity of the two
brothers culminated, on this occasion, in a violent and
acrimonious controversy, which lasted till James's
death. After James's death, John virtually confessed
his error and adopted the correct method of his brother.
James Bernoulli surmised, and in all probability James Ber
correctly, that John, misled by the results of his re criticism of
searches on the catenary and the curve of a sail filled nouiH's so
.,.,,,. , , lution.
with wind, had again attempted an indirect solution,
imagining BJ?N filled with a liquid of variable density
and taking the lowest position of the centre of gravity
as determinative of the curve required. Making the
ordinate PZ=p, the specific gravity of the liquid in
the ordinate PF = x must be f/x, and similarly in
every other ordinate. The weight of a vertical fila
432 THE SCIENCE OF MECJIAXICS.
ment is then / . dy/x 9 and its moment with respect to
1 pdv 1
noulli's
general so
lution.
Hence, for the lowest position of the centre of gravity,
J. Cp dy, or Cp dy = BZN, is a maximum. But the
fact is here overlooked, remarks James Bernoulli, that
with the variation of the curve BFN the weight of the
liquid also is varied. Consequently, in this simple
form the deduction is not admissible.
The funds In the solution which he himself gives, James Ber
principie of noulli once more assumes that the small portion F F ltt
James Ber . ,
of the curve possesses the prop
erty which the whole curve pos
sesses. And then taking the four
successive points F F, F n F fft ,
of which the two extreme ones
are fixed, he so varies F t and
F f , that the length of the arc F
F, F rl F lft remains unchanged,
which is possible, of course, only by a displacement
of two points. We shall not follow his involved and
unwieldy calculations. The principle of the process is
clearly indicated in our remarks. Retaining the des
ignations above employed, James Bernoulli, in sub
stance, states that when
d $ doc
"'
Cpdy is a maximum, and when
v.._ c jx*
1/2 / j
Jpdyi
is a minimum.
FORMAL DEVELOPMENT. 433
The dissensions between the two brothers were, we
may admit, greatly to be deplored. Yet the genius of
the one and the profundity of the other have borne, in
the stimulus which Euler and Lagrange received from
their several investigations, splendid fruits.
5. Euler (Problcmatis IsoperimetriciSolutio Gcneralis^ Euier's
Com. Acad. Petr. T. VI, for 1733, published in 1738)* ciassifica
3 ' 3J* r . tionoi the
was the first to give a more general method of treating isoperimet
b & . rical prob
these questions of maxima and minima, or isopenmetri lems.
cal problems. But even his results were based on
prolix geometrical considerations, and not possessed of
analytical generality. Euler divides problems of this
category, with a clear perception and grasp of their
differences, into the following classes :
(1) Required, of all curves, that for which a prop
erty A is a maximum or minimum.
(2) Required, of all curves, equally possessing a
property A, that for which B is a maximum or mini
mum.
(3) Required, of all curves, equally possessing two
properties, A and B, that for which C is a maximum
or minimum. And so on.
A problem of the first class is (Fig. 231) the finding Example*
of the shortest curve through M and N. A problem of
the second class is the finding of a curve through M
and N, which, having the given length A, makes the
area MPN "a maximum. A problem of the third class
would be : of all curves of the given length A, which
pass through M, N and contain the same area
MPN~B, to find one which describes when rotated
about J/TVthe least surface of revolution. And so on.
* Euier's principal contributions to this subject are contained in three
memoirs, published in the Commentaries of Petersburg f or the years 1733, J 736i
and 1766, and in the tract Methodus inveniendi Lineas Curvas Proprietate
Maximi Minimive gaudentes, Lausanne and Geneva, 1744. Trans.
434 THE SCIENCE OF MECHANICS.
We may observe here, that the finding of an abso
lute maximum or minimum, without collateral condi
tions, is meaningless. Thus, all the curves of which in
the first example the shortest is sought
possess the common property of pas
sing through the points M and N.
The solution of problems of the
first class requires the variation of two
elements of the curve or of one point.
This is also sufficient. In problems
of the second class three elements or
Fig. 231. iwo points must be varied ; the reason
being, that the varied portion must
possess in common with the unvaried portion the prop
erty A, and, as B is to be made a maximum or mini
mum, also the property,^, that is, must satisfy two con
ditions. Similarly, the solution of problems of the third
class, requires the variation of four elements. And
so on.
The com The solution of a problem of a higher class involves,
mutability ,.,. . 1 . . .,,.
of the isq by implication, the solution of its converse, in all its
caPproper forms. Thus, in the third class, we vary four elements
Ruler's in of the curve, so, that the varied portion of the curve
shall share equally with the original portion the values
A and B and, as C is to be made a maximum or a
minimum, also the value C. But the same conditions
must be satisfied, if of all curves possessing equally B
and C that for which A Is a maximum or minimum is
sought, or of all curves possessing A and C that for
which B is a maximum or minimum is sought. Thus
a circle, to take an example from the second class, con
tains, of all lines of the same length A, the greatest
area B, and the circle, also, of all curves containing
the same area B, has the shortest length A. As the
FORMAL DEVELOPMENT. 435
condition that the property A shall be possessed in
common or shall be a maximum, is expressed in the
same manner, Euler saw the possibility of reducing the
problems of the higher classes to problems of the first
class. If, for example, it is required to find, of all
curves having the common property A, that which
makes B a maximum, the curve is sought for which
A ~\~ mB is a maximum, where m is an arbitrary con
stant. If on any change of the curve, A f mB, for any
value of m, does not change, this is generally possible
only provided the change of A, considered by itself,
and that of J3, considered by itself, are = 0.
6. Euler was the originator of still another impor The funda
. . mental
tant advance. In treating the problem of finding the principle of
. . . .  . , . . James Ber
brachistochrone in a resisting medium, which was in nouiii's
111 method
vestierated by Herrmann and him, the existing mem shown not
i ^ , , , . , . to be uni
ods proved incompetent. For the brachistochrone in versaiiy
a vacuum, the velocity depends solely on the vertical
height fallen through. The velocity In one portion of
the curve is in no wise dependent on the other por
tions. In this case, 'then, we can indeed say, that if
the whole curve is brachistochronous, every element
of it is also brachistochronous. But in a resisting
medium the case is different. The entire length and
form of the preceding path enters into the determina
tion of the velocity in the element. The whole curve
can be brachistochronous without the separate ele
ments necessarily exhibiting this property. By con
siderations of this character, Euler perceived, that the
principle introduced by James Bernoulli did not hold
universally good, but that in cases of the kind referred
to, a more detailed treatment was required.
7. The methodical arrangement and the great num
ber of the problems solved, gradually led Euler to sub
43 6 THE SCIENCE OF MECHANICS.
Lagrange's stantially the same methods that Lagrange afterwards
histo e ry n of e developed in a somewhat different form, and which
ins ofVari now go by the name of the Calculus of Variations. First,
a ions. jQ^n Bernoulli lighted on an accidental solution of a
problem, by analogy. James Bernoulli developed, for
the solution of such problems, a geometrical method.
Euler generalised the problems and the geometrical
method. And finally, Lagrange, entirely emancipating
himself from the consideration of geometrical figures,
gave an analytical method. Lagrange remarked, that
the increments which functions receive in consequence
of a change in their form are quite analogous to the in
crements they receive in consequence of a change of
their independent variables. To distinguish the two
species of increments, Lagrange denoted the former
by d, the latter by d. By the observation of this anal
ogy Lagrange was enabled to write down at once the
equations which solve problems of maxima and minima.
Of this idea, which has proved itself a very fertile one,
Lagrange never gave a verification ; in fact, did not
even attempt it. His achievement is in every respect
a peculiar one. He saw, with great economical in
sight, the foundations which in his judgment were suf
ficiently secure and serviceable to build upon. But
the acceptance of these fundamental principles them
selves was vindicated only by its results. Instead of
employing himself on the demonstration of these prin
ciples, he showed with what success they could be em
ployed. (J&ssai d'une nouvelle methods poitr determiner
les maxima et minima des formules integralcs indcfimes.
Misc. Taur. 1762.)
The difficulty which Lagrange's contemporaries and
successors experienced in clearly grasping his idea, is
quite intelligible. Euler sought in vain to clear up the
FORMAL f DE VEL OPMENT. 437
difference between a variation and a differential byThemis
concep
imaginin: constants contained in the function, with tions of La
grange s
the change of which the form of the function changed, idea.
The increments of the value of the function arising
from the increments of these constants were regarded
by him as the variations, while the increments of the
function springing from the increments of the indepen
dent variables were the differentials. The conception
of the Calculus of Variations that springs from such a
view is singularly timid, narrow, and illogical, and does
not compare with that of Lagrange. Even Lindelof's
modern work, so excellent in other respects, is marred
by this defect. The first really competent presenta
tion of Lagrange's idea is, in our opinion, that of JEL
LETT.* Jellett appears to have said what Lagrange per
haps was unable fully to say, perhaps did not deem it
necessary to say.
8. Tellett's view is, in substance, this. Quantities Joiiett'sex
....... ^ position of
generally are divisible into constant and variable quan theprmci
. . , pies of the
titles ; the latter being subdivided into independent Calculus of
.... Variations.
and dependent variables, or such as may be arbitrarily
changed, and such whose change depends on the
change of other, independent, variables, in some way 
connected with them. The latter are called functions
of the former, and the nature of the relation that con
nects them is termed the form of the function. Now,
quite analogous to this division of quantities into con
stant and variable, is the division of the forms of func
tions into determinate (constant) and indeterminate (vari
able). If the form of a function, y = <p(x) t is inde
terminate, or variable, the value of the function y can
change in two ways : (i) by an increment dx of the
* An Elementary Treatise on the Calculus of Variations, By the Rev.
John Hewitt Jellett. Dublin, 1850.
43 8 THE SCIENCE OF MECHANICS.
independent variable x, or (2) by a change oiform, by
a passage from cp to cp^ The first change is the dif
ferential dy, the second, the variation dy. Accord
ingly,
dy = cp (x j dx) cp (V), and
The object The change of value of an indeterminate function
of the cal .
cuius of va due to a mere change of form involves no problem,
unrated, just as the change of value of an independent variable
involves none. We may assume any change of form
we please, and so produce any change of value we
please. A problem is not presented till the change in
value of a determinate function (F} of an indetermi
nate function <p, due to a change of form of the included
indeterminate function, is required. For example, if
we have a plane curve of the indeterminate form y==
cp (V), the length of its arc between the abscissae X Q
and x l is
= PJi+few) 2 . dx = r J
J > \ dx j J \
a determinate function of an indeterminate function.
The moment a definite form of curve is fixed upon, the
value of can be given. For any change of form of
the curve, the change in value of the length of the arc,
SS, is determinate. In the example given, the func
tion S does not contain the function y directly, but
through its first differential coefficient dy/dx, which is
itself dependent on y. Let u = J?(y) be a determinate
function of an indeterminate function y = <p (V) ; then
du=F(y + Sy}  F(y) = ~ Sy.
Again, let u =F(y, dyjdx) be a determinate function
FORMAL DEVELOPMENT. 439
of an indeterminate function, y = cp(x). For a change
of form of <>, the value of y changes by dy and the
value of dy/dx by $(dy/dx). The corresponding change
in the value of u is
s.
dy J ^ dy_ dx
doc
The expression d ~ is obtained by our definition from Expres
F X sions tor
dy __ __
dx~ dx dx~~~ dx coefficients,
Similarly, the following results are found :
^__^cty
~ ~ '
and so forth.
We now proceed to a problem, namely, the d e A problem.
termination of the form of the function jy= <p(x) that
will render
where
V
r
a maximum or minimum ; cp denoting an indetermi
nate, and F a determinate function. The value of U
may be varied (i) by a change of the limits, x Q9 x.
Outside of the limits, the change of the independent
variables x, as such, does not affect U\ accordingly,
if we regard the limits as fixed, this is the only respect
in which we need attend to x. The only other way
(2) in which the value of U is susceptible of variation
44 o THE SCIENCE OF MECHANICS.
is by a change of the form of y = ^?(V). This produces
a change of value in
amounting to
dy i* j
$y $ $
dx d x 2
and so forth. The total change in U, which we shall
call DU, and to express the maximumminimum con
dition put =0, consists of the differential ^/7and the
variation dU. Accordingly,
Expression Denoting by V^dx^ and V^doc^ the increments of
variation of U due to the change of the limits, we then have
the func
tion in  r i
question. DU= V^ dx^ F Q dx Q + d Vdx =
d V. dx = 0.
But by the principles stated on page 439 we further get
4.
<7 y + <7 7^"
For, the sake of brevity we put
d _T N dV p *L P
7  "* 9 7  1 J 7 o  9 J
Then
FORMAL DEVELOPMENT.
x \.
J
y gratioi
/ the thi
term o
One difficulty here is, that not only dy, but also the f *Pg^
terms ddy/dx, d* dy/dx 2 .... occur in this equation, variati
terms which are dependent on one another, but not
in a directly obvious manner. This drawback can be
removed by successive integration by parts, by means
of the formula
Cu dv = u v Cv du.
By this method
dP^ . , /* . , ,
~d2 Sy +J j^ Sydx > and so on 
Performing all these integrations between the limits,
we obtain for the condition DU= the expression
~dx
/>
+
which now contains only dy under the integral sign.
The terms in the first line of this expression are
independent of any change in the form of the function
and depend solely upon the variation of the limits.
44 2 THE SCIENCE OF MECHANICS.
The inter The terms of the two following lines depend on the
Sie resSts. change in the form of the function, for the limiting
values of x only ; and the indices i and 2 state that
the actual limiting values are to be put in the place of
the general expressions. The terms of the last line,
finally, depend on the general change in the form of
the function. Collecting all the terms, except those in
the last line, under one designation a^ <X Q , and calling
the expression in parentheses in the last line /?, we
have
But this equation can be satisfied only if
,a = o ................ a)
and
ty&y<fx = Q ................ (2)
XQ
For if each of the members were not equal to zero,
each would be determined by the other. But the in
tegral of an indeterminate function cannot be expressed
in terms of its limiting values only. Assuming, there
fore, that the equation
The equa holds generally good, its conditions can be satisfied,
tion which . .
solves the since ov is throughout arbitrary and its generality of
problem, or
makes the form cannot be restricted, only by making ft = 0. By
function in > J J t> / J
question a the
maximum
or mini
mum.
therefore, the form of the function y <p(x) that makes
the expression U a maximum or minimum is defined.
FORMAL DEVELOPMENT. 443
Equation (3) was found by Euler. But Lagrange first
showed the application of equation (i), for the deter
mination of a function by the conditions at its limits.
By equation (3), which it must satisfy, the form of the
function y = <p(x) is generally determined ; but this
equation contains a number of arbitrary constants,
whose values are determined solely by the conditions
at the limits. With respect to notation, Jellett rightly
remarks, that the employment of the symbol d in the
first two terms V^dx^ =V Q dx Q of equation (i), (the
form used by Lagrange,) is illogical, and he correctly
puts for the increments of the independent variables
the usual symbols dx^ dx^.
o. To illustrate the use to which these equations A practical
_ _ , . ...... illustration
may be put, let us seek the form of the function that of the use
J r of these
makes equations.
a minimum the shortest line. Here
V=f( d J\
\dxj
All expressions except
d_y_
dx
p _____ .
'~'~r
vanish in equation (3)^ and that equation becomes
dP^/dx = ; which means that P 19 and consequently
its only variable, dy/dx, is independent of x. Hence,
dy/dx = a, and y  ax f , where a and ^ are con
stants.
The constants a, b are determined by the values of
444 THE SCIENCE OF MECHANICS.
Develop the limits. If the straight line passes through the
ment of the . .. .
illustration, points X O ,}'Q and # 1? y l9 then
and as dx^ = dx l = 0, <5j> = Sy^ = 0, equation (i)
vanishes. The coefficients $ (dy/dx} 9 d (dtyjdx*^ ....
independently vanish. Hence, the values of a and b
are determined by the equations (;;z) alone.
If the limits X Q , x l only are given, butj/ , j^ are
indeterminate, we have dx^ = dx^ = 0, and equation
(i) takes the form
which, since &y Q and dy^ are arbitrary, can only be
satisfied if a = 0. The straight line is in this case
y 3, parallel to the axis of abscissas, and as b is inde
terminate, at any distance from it.
It will be ' noticed, that equation (i) and the sub
sidiary conditions expressed in equation (#2), with re
spect to the determination of the constants, generally
complement each other.
If
is to be made a minimum, the integration of the appro
priate form of (3) will give
x c' x ~
e
If Z is a minimum, then ^nZ also is a minimum, and
the curve found will give, by rotation about the axis
of abscissae, the least surface of revolution. Further,
FORMAL DEVELOPMENT. 445
to a minimum of Z the lowest position of the centre of
gravity of a homogeneously heavy curve of this kind
corresponds ; the curve is therefore a catenary. The
determination of the constants c, c' is effected by means
of the limiting conditions, as above.
In the treatment of mechanical problems, a dis variations
r m and virtual
tinction is made between the increments of coordinates displace
ments dis
that actually take place in time, namely, dx, dy, dz, tinguished.
and the possible displacements dx, dy, dz, considered,
for instance, in the application of the principle of vir
tual velocities. The latter, as a rule, are not varia
tions ; that is, are not changes of value that spring
from changes in the form of a function. Only when
we consider a mechanical system that is a continuum,
as for example a string, a flexible surface, an elastic
body, or a liquid, are we at liberty to regard d x, $y,
dz as indeterminate functions of the coordinates x, y,
z, and are we concerned with variations.
It is not our purpose in this work, to develop math importance
r * 9 r of the cal
ematical theories, but simply to treat the purely phys cuius of va
' r J r J r J riations for
ical part of mechanics. But the history of the isopen mechanics,
metrical problems and of the calculus of variations had
to be touched upon, because these researches have ex
ercised a very considerable influence on the develop
ment of mechanics. Our sense of the general prop
erties of systems, and .of properties of maxima and
minima in particular, was much sharpened by these
investigations, and properties of the kind referred to
were subsequently discovered in mechanical systems
with great facility. As a fact, physicists, since La
grange's time, usually express mechanical principles
in a maximal or minimal form. This predilection
would be unintelligible without a knowledge of the
historical development.
446 THE SCIENCE OF MECHANICS.
n.
THEOLOGICAL. ANIMISTIC, AND MYSTICAL POINTS OF VIEW
IN MECHANICS.
i. If, in entering a parlor in Germany, we happen
to hear something said about some man being very
pious, without having caught the name, we may fancy
that Privy Counsellor X was spoken of, or Herr von
Y ; we should hardly think of a scientific man of our
acquaintance. It would, however, be a mistake to sup
pose that the want of cordiality, occasionally rising to
embittered controversy, which has existed in our day
between the scientific and the theological faculties,
always separated them. A glance at the history of
science suffices to prove the contrary.
The con People talk of the " conflict " of science and the
fhe e church olo y> or better of scienc e and the church. It is in
' truth a prolific theme. On the one hand, we have the
long catalogue of the sins of the church against pro
gress, on the other side a " noble army of martyrs,"
among them no less distinguished figures than Galileo
and Giordano Bruno. It was only by good luck that
Descartes, pious as he was, escaped the same fate.
These things are the commonplaces of history ; but it
would be a great mistake to suppose that the phrase
" warfare of science" is a correct description of its
general historic attitude toward religion, that the only
repression of intellectual development has come from
priests, and that if their hands had been held off, grow
ing science would have shot up with stupendous velo
city. No doubt, external opposition did have to be
fought j and the battle with it was no child's play.
FORMAL DEVELOPMENT 447
Nor was any engine too base for the church to handle The stnier
J & pleofscien
in this struggle. She considered nothing but how to tists with
00 their own
conquer ; and no temporal policy ever was conducted P^on
so selfishly, so unscrupulously, or so cruelly. But in ideas,
vestigators have had another struggle on their hands,
and by no means an easy one, the struggle with their
own preconceived ideas, and especially with the notion
that philosophy and science must be founded on the
ology. It was but slowly that this prejudice little by
little was erased.
2. But let the facts speak for themselves, while we Historical
. examples.
introduce the reader to a few historical personages.
Napier, the inventor of logarithms, an austere Puri
tan, who lived in the sixteenth century, was, in addi
tion to his scientific avocations, a zealous theologian.
Napier applied himself to some extremely curious
speculations. He wrote an exegetical commentary on
the Book of Revelation, with propositions and mathe
matical demonstrations. Proposition XXVI, for ex
ample, maintains that the pope is the Antichrist ; propo
sition XXXVI declares that the locusts are the Turks
and Mohammedans ; and so forth.
Blaise Pascal (16231662), one of the most rounded
geniuses to be found among mathematicians and phys
icists, was extremely orthodox and ascetical. So deep
were the convictions of his heart, that despite the gen
tleness of his character, he once openly denounced at
Rouen an instructor in philosophy as a heretic. The
healing of his sister by contact with a relic most seri
ously impressed him, and he regarded her cure as a
miracle. On these facts taken by themselves it might
be wrong to lay great stress ; for his whole family were
much inclined to religious fanaticism. But there are
plenty of other instances of his religiosity. Such was
44 8 THE SCIENCE OF MECHANICS.
Pascal. his resolve, which was carried out, too, to abandon
altogether the pursuits of science and to devote his life
. solely to the cause of Christianity. Consolation, he
used to say, he could find nowhere but in the teachings
of Christianity ; and all the wisdom of the world availed
him not a whit. The sincerity of his desire for the
conversion of heretics is shown in his Lettres provin
ciates, where he vigorously declaims against the dread
ful subtleties that the doctors of the Sorbonne had
devised, expressly to persecute the Jansenists. Very
remarkable is Pascal's correspondence with the theo
logians of his time ; and a modern reader is not a little
surprised at finding this great "scientist" seriously
discussing in one of his letters whether or not the Devil
was able to work miracles.
otto yon Otto von Guericke, the inventor of the airpump,
Guericke. .,. 1f . , , . . r i i i
occupies himself, at the beginning of his book, now
little over two hundred years old, with the miracle of
Joshua, which he seeks to harmonise with the ideas
of Copernicus. In like manner, we find his researches
on the vacuum and the nature of the atmosphere in
troduced by disquisitions concerning the location of
heaven, the location of hell, and so forth. Although
Guericke really strives to answer these questions as ra
tionally as he can, still we notice that they give him
considerable trouble, questions, be it remembered,
that today the theologians themselves would consider
absurd. Yet Guericke was a man who lived after the
Reformation !
The giant mind of Newton did not disdain to employ
itself on the interpretation of the Apocalypse. On such
subjects it was difficult for a sceptic to converse with
him. When Halley once indulged in a jest concerning
theological questions, he is said to have curtly repulsed
FORMAL DEVELOPMENT. 449
him with the remark : "I have studied these things ; Newtonand
Leibnitz.
you have not ! "
We need not tarry by Leibnitz, the inventor of the
be'st of all possible worlds and of preestablished har
mony inventions which Voltaire disposed of in Can
dide, a humorous novel with a deeply philosophical pur
pose. But everybody knows that Leibnitz was almost
if not quite as much a theologian, as a man of science.
Let us turn, however, to the last century. Euler, in Euier.
his Letters to a German Princess, deals with theologico
philosophical problems in the midst of scientific ques
tions. He speaks of the difficulty involved in explaining
the interaction of body and mind, due to the total
diversity of these two phenomena, a diversity to his
mind undoubted. The system of occasionalism, devel
oped by Descartes and his followers, agreeably to which
God executes for every purpose of the soul, (the soul it
self not being able to do so,) a corresponding movement
of the body, does not quite satisfy him. He derides,
also, and not without humor, the doctrine of pre
established harmony, according to which perfect agree
ment was established from the beginning between the
movements of the body and the volitions of the soul,
although neither is in any way connected with the
other, just as there is harmony between two different
but likeconstructed clocks. He remarks, that in this
view his own body is as foreign to him as that of a
rhinoceros in the midst of Africa, which might just as
well be in preestablished harmony with his soul as
its own. Let us hear his own words. In his day, Latin
was almost universally written. When a German
scholar wished to be especially condescending, he
wrote in French : " Si dans le cas d'un d®lement
"de mon corps Dieu ajustait celui d'un rhinoceros,
450 THE SCIENCE OF MECHANICS.
"en sorte que ses mouvements fussent tellement d'ac
" cord avec les ordres de mon ame, qu'il levat la pattc
" au moment que je voudrais lever la main, et ains
" des autres operations, ce serait alors mon corps. Je
"me trouverais subitement dans la forme d'un rhino
<e ceros au milieu de 1'Afrique, mais non obstant cek
"mon ame continuerait les meme operations. J'aurais
"^galement 1'honneur d'^crire a V. A., mais je ne sais
"pas comment elle recevrait mes lettres. "
Euier's One would almost imagine that Euler, here, had been
theological . __ . . 1
proclivities tempted to play Voltaire. And yet, apposite as was
his criticism in this vital point, the mutual action oi
body and soul remained a miracle to him, still. But he
extricates himself, however, from the question of the
freedom of the will, very sophistically. To give some
idea of the kind of questions which a scientist was per
mitted to treat in those days, it may be remarked that
Euler institutes in his physical "Letters" investiga
tions concerning the nature of spirits, the connection
between body and soul, the freedom of the will, the
influence of that freedom on physical occurrences,
prayer, physical and moral evils, the conversion of sin
ners, and such like topics ; and this in a treatise full
of clear physical ideas and not devoid of philosophical
ones, where the wellknown circlediagrams of logic
have their birthplace.
character 3. Let these examples of religious physicists suffice.
logical We have selected them intentionally from among the
the great in foremost of scientific discoverers. The theological pro
clivities which these men followed, belong wholly to
their innermost private life. They tell us openly things
which they are not compelled to tell us, things about
which they might have remained silent. What they
litter are not opinions forced upon them from without ;
FORMAL DEVELOPMENT. 451
they are their own sincere views. They were not con
scious of any theological constraint. In a court which
harbored a Lamettrie and a Voltaire, Eulerhad no rea
son to conceal his real convictions.
According: to the modern notion, these men should character
& ' ot their age.
at least have seen that the questions they discussed
did not belong under the heads where they put them,
that they were not questions of science. Still, odd as
this contradiction between inherited theological beliefs
and "independently created scientific convictions seems
to us, it is no reason for a diminished admiration of
those leaders of scientific thought. Nay, this very fact
is a proof of their stupendous mental power : they were
able, in spite of the contracted horizon of their age, to
which even their own aper$us were chiefly limited, to
point out the path to an elevation, where our genera
tion has attained a freer point of view.
Every unbiassed mind must admit that the age in
which the chief development of thascience of mechan
ics took place, was an age of predominantly theological
cast. Theological questions were excited by everything,
and modified everything. No wonder, then, that me
chanics took the contagion. But the thoroughness with
which theological thought thus permeated scientific
inquiry, will best be seen by an examination of details.
4. The impulse imparted in antiquity to this direc Galileo's
tion of thought by Hero and Pappus has been alluded on the
, strength of
to m the preceding chapter. At the beginning of the materials.
seventeenth century we find Galileo occupied with prob
lems concerning the strength of materials. He shows
that hollow tubes offer a greater resistance to flexure
than solid rods of the same length and the same quantity
of material, and at once applies this discovery to the
explanation of the forms of the bones of animals, which
452 'I HE SCIENCE OF MECHANICS.
are usually hollow and cylindrical in shape. The phe
nomenon is easily illustrated by the comparison of a
flatly folded and a rolled sheet of paper. A horizontal
beam fastened at one extremityand loaded at the other
may be remodelled so as to be thinner at the loaded
end without any loss of stiffness and with a consider
able saving of material. Galileo determined the form of
a beam of equal resistance at each crosssection. He
also remarked that animals of similar geometrical con
struction but of considerable difference of size would
comply in very unequal proportions with the laws of
resistance.
Evidences The forms of bones, feathers, stalks, and other or
of design  .
in nature, game structures, adapted, as they are, in their minut
est details to the purposes they serve, are highly cal
culated to make a profound impression on the thinking
beholder, and this fact has again and again been ad
duced in proof of a supreme wisdom ruling in nature.
Let us examine, for instance, the pinionfeather of a
bird, The quill is a hollow tube diminishing in thick
ness as we go towards the end, that is, is a body of
equal resistance. Each little blade of the vane re
peats in miniature the same construction. It would
require considerable technical knowledge even to imi
tate a feather of this kind, let alone invent it. We
should not forget, however, that investigation, and
not mere admiration, is the office of science. We
know how Darwin sought to solve these problems, by
the theory of natural selection. That Darwin's solution
is a complete one, may fairly be doubted ; Darwin him
self questioned it. All external conditions would be
powerless if something were not present that admitted
of variation. But there can be no question that his
theory is the first serious attempt to replace mere ad
FORMAL DEVELOPMENT. 453
miration of the adaptations of organic nature by seri
ous inquiry into the mode of their origin.
Pappus's ideas concerning the cells of honeycombs The ceils c
rj: & J the honey
Were the subject of animated discussion as late as the comb.
eighteenth century. In a treatise, published in 1865,
entitled Homes Without Hands (p. 428), Wood substan
tially relates the following : " Maraldi had been struck
with the great regularity of the cells of the honey
comb. He measured the angles of the lozengeshaped
plates, or rhombs, that form the terminal walls of the
cells, and found them to be respectively 109 28' and
70 32'. Rdaumur, convinced that these angles were in
some way connected with the economy of the cells,
requested the mathematician Konig to calculate the
form of a hexagonal prism terminated by a pyramid
composed of three equal and similar rhombs, which
would give the greatest amount of space with a given
amount of material. The answer was, that the angles
should be 109 26' and 70 34'. The difference, accord
ingly, was two minutes. Maclaurin,* dissatisfied with
this agreement, repeated Maraldi's measurements, found
them correct, and discovered, in going over the calcu
lation, an error in the logarithmic table employed by
Konig. Not the bees, but the mathematicians were
* wrong, and the bees had helped to detect the error ! "
Any one who is acquainted with the methods of meas
uring crystals and has seen the cell of a honeycomb,
with its rough and nonreflective surfaces, will question
whether the measurement of such cells can be executed
with a probable error of only two minutes, f So, we
must take this story as a sort of pious mathematical
* Philosophical Transactions for 1743. Trans.
t But see G. F. Maraldi in the M&moires de Vacadimie for 1712. It is, how
ever, now well known the cells vary considerably. See Chauncey Wright,
Philosophical Discussions, 1877, p. 311. Trans.
454 TJIE SCIENCE OF MECHANICS.
fairytale, quite apart from the consideration that noth
ing would follow from it even were it true. Besides,
from a mathematical point of view, the problem is too
imperfectly formulated to enable us to decide the ex
tent to which the bees have solved it.
other The ideas of Hero and Fermat, referred to in the
instances. . ,
previous chapter, concerning the motion of light, at
once received from the hands of Leibnitz a theolog
ical coloring, and played, as has been before mentioned,
a predominant role in the development of the calculus
of variations. In Leibnitz's correspondence with John
Bernoulli, theological questions are repeatedly dis
cussed in the very midst of mathematical disquisitions.
Their language is not unfrequently couched in biblical
pictures. Leibnitz, for example, says that the problem
of the brachistochrone lured him as the apple had lured
Eve.
Thetheo Maupertuis, the famous president of the Berlin
neiofthe Academy, and a friend of Frederick the Great, gave
feasfac 6 a new impulse to the theologising bent of physics by
the enunciation of his principle of least action. In the
treatise which formulated this obscure principle, and
which betrayed in Maupertuis a woeful lack of mathe
matical accuracy, the author declared his principle to be
the one which best accorded with the wisdom of the
Creator. Maupertuis was an ingenious man, but not a
man of strong, practical sense. This is evidenced by
the schemes he was incessantly devising : his bold prop
ositions to found a city in which only Latin should be
spoken, to dig a deep hole in the earth to find new
substances, to institute psychological investigations by
means of opium and by the dissection of monkeys, to
explain the formation of the embryo by gravitation, and
so forth. He was sharply satirised by Voltaire in the
FORMAL DEVELOPMENT. 455
Histoire du docteur Akakia, a work which led, as we
know, to the rupture between Frederick and Voltaire.
Maupertuis's principle would in all probability soon Euier'sre
^ r r r J tention of
have been forgotten, had Euler not taken up the sug the theoiog
b ' .... ical basis of
gestion. Euler magnanimously left the principle its this prin
name, Maupertuis the glory of the invention, and con
verted it into something new and really serviceable.
What Maupertuis meant to convey is very difficult to
ascertain. What Euler meant may be easily shown by
simple examples. If a body is constrained to move on a
rigid surface, for instance, on the surface of the earth, it
will describe when an impulse is imparted to it, the
shortest path between its initial and terminal positions.
Any other path that might be prescribed it, would be
longer or would require a greater time. This principle
finds an application in the theory of atmospheric and
oceanic currents. The theological point of view, Euler
retained. He claims it is possible to explain phenomena,
not only from their physical causes, but also from their
purposes. " As the construction of the universe is the
"most perfect possible, being the handiwork of an
" allwise Maker, nothing can be met with in the world
"in which some maximal or minimal property is not
"displayed. There is, consequently, no doubt but
"that all the effects of the world can be derived by
"the method of maxima and minima from their final
"causes as well as from their efficient: ones."*
5. Similarly, the notions of the constancy of the
quantity of matter, of the constancy of the quantity of
* " Quum enim mundi nniversi fabrica sit perfectissima, atque a creatore
sapientissimo absoluta, nihil omnino in nmndo contingit, in quo non uiaxhm
minimive ratio quaepiam eluceat; quam ob rein dubium prorsus est nullum,
quin omnes mundi effectus ex causis finalibus, ope rnethodi maximorum et
minimorum, aeqxie feliciter determinari quaeant, atque ex ipsis causis efficicn
tibus." (Methodus inveniendi linens curvas maximi minimive proprietate
gaudentes. Lausanne, 1744.)
45 6 THE SCIENCE OF MECHANICS.
The central motion, of the Indestructibility of work or energy, con
modern ceptions which completely dominate modern physics,
ma } iniy S of all arose under the influence of theological ideas. The
orfgin. gl a notions in question had their origin in an utterance of
Descartes, before mentioned, in the Principles of 'Philos
ophy ', agreeably to which the quantity of matter and mo
tion originally created in the world, such being the
only course compatible with the constancy of the Crea
tor, is always preserved unchanged. The conception
of the manner in which this quantity of motion should
be calculated was very considerably modified in the
progress of the idea from Descartes to Leibnitz, and to
their successors, and as the outcome of these modifi
cations the doctrine gradually and slowly arose which
is now called the "law of the conservation of energy."
But the theological background of these ideas only
slowly vanished. In fact, at the present day, we still
meet with scientists who indulge in selfcreated mys
ticisms concerning this law.
Gradual During the entire sixteenth and seventeenth centu*
transition . ..... . ,
from the nes, down to the close of the eighteenth, the prevail
point of ing inclination of inquirers was, to find in all physical
laws some particular disposition of the Creator. But
a gradual transformation of these views must strike
the attentive observer. Whereas with Descartes and
Leibnitz physics and theology were still greatly inter
mingled, in the subsequent period a distinct endeavor
is noticeable, not indeed wholly to discard theology,
yet to separate it from purely physical questions. Theo
logical disquisitions were put at the beginning or rele
gated to the end of physical treatises. Theological
speculations were restricted, as much as possible, to
the question of creation, that, from this point onward,
the way might be cleared for physics.
FORMAL DEVELOPMENT. 457
Towards the close of the eighteenth century a re ultimate
. complete
markable change took place, a change which was emancipa
apparently an abrupt departure from the current trend physics
F.f J r f from t heo]
of thought, but in reality was the logical outcome of ogy.
the development indicated. After an attempt in a
youthful work to found mechanics on Euler' s principle
of least action, Lagrange, in a subsequent treatment
of the subject, declared his intention of utterly disre
garding theological and metaphysical speculations, as
in their nature precarious and foreign to science. He
erected a new mechanical system on entirely different
foundations, and no ane conversant with the subject
will dispute its excellencies. All subsequent scientists
of eminence accepted Lagrange's view, and the pres
ent attitude of physics to theology was thus substan
tially determined.
6. The idea that theology and physics are two dis The mod
C3  7 r J era ideal
tinct branches of knowledge, thus took, from its first always the
' ' attitude of
germination in Copernicus till its final promulgation * he greatest
* r f inquirers.
by Lagrange, almost two centuries to attain clearness
in the minds of investigators. At the same time it
cannot be denied that this truth was always clear to
the greatest minds, like Newton. Newton never, de
spite his profound religiosity, mingled theology with
the questions of science. True, even he concludes his
Optics > whilst on its last pages his clear and luminous
intellect still shines, with an exclamation of humble
contrition at the vanity of all earthly things. But his
optical researches proper, in contrast to those of Leib
nitz, contain not a trace of theology. The same may
be said of Galileo and Huygens. Their writings con
form almost absolutely to the point of view of La
grange, and may be accepted in this respect as class
ical. But the general views and tendencies of an age
458 THE SCIENCE OF MECHANICS.
must not be judged by Its greatest, but by its average,
minds.
The theo To comprehend the process here portrayed, the gen
logical con i r rr t
ception of eral condition of affairs in these times must be consid
naturai and ered. It stands to reason that in a stage of civilisation
a'bie. in which religion is almost the sole education, and the
only theory of the world, people would naturally look
at things in a theological point of view, and that they
would believe that this view was possessed of compe
tency in all fields of research. If we transport ourselves
back to the time when people played the organ with
their fists, when they had to have .the multiplication table
visibly before them to calculate, when they did so much
with their hands that people nowadays do with their
heads, we shall not demand of such a time that it
should critically put to the test its own views and the
ories. With the widening of the intellectual horizon
through the great geographical, technical, and scien
tific discoveries and inventions of the fifteenth and six
teenth centuries, with the opening up of provinces in
which it was impossible to make any progress with the
old conception of things, simply because it had been
formed prior to the knowledge of these provinces, this
bias of the mind gradually and slowly vanished. The
great freedom of thought which appears in isolated
cases in the early middle ages, first in poets and then
in scientists, will always be hard to understand. The en
lightenment of those days must have been the work of a
few very extraordinary minds, and can have been bound
to the views of the people at large by but very slender
threads, more fitted to disturb those views than to re
form them. Rationalism does not seem to have gained
a broad theatre of action till the literature of the eigh
teenth century. Humanistic, philosophical, historical,
FORMAL DEVELOPMENT. 459
and physical science here met and gave each other
mutual encouragement. All who have experienced, in
part, in its literature, this wonderful emancipation of
the human intellect, will feel during their whole lives a
deep, elegiacal regret for the eighteenth century.
7. The old point of view, then, is abandoned. Its The en
history is now detectible only in the form of the mementofthe
new views.
.
chanical principles. And this form will remain strange
to us as long as we neglect its origin. The theological
conception of things gradually gave way to a more
rigid conception ; and this was accompanied with a
considerable gain in enlightenment, as we shall now
briefly indicate.
When we say light travels by the paths of shortest
time, we grasp by such an expression many things.
But we do not know as yet why light prefers paths of
shortest time. We forego all further knowledge of the
phenomenon, if we find the reason in the Creator's wis
dom. We of today know, that light travels by all
paths, but that only on the paths of shortest time do
the waves of light so intensify each other that a per
ceptible result is produced. Light, accordingly, only
appears to travel by the paths of shortest time. After Extrava
ficince as
the prejudice which prevailed on these questions had well as
.  .,.,,. . economy in
been removed, cases were immediately discovered in nature.
which by the side of the supposed economy of nature
the most striking extravagance was displayed. Cases
of this kind have, for example, been pointed out by
Jacob! In connection with Euler's principle of least ac
tion. A great many natural phenomena accordingly
produce the impression of economy, simply because
they visibly appear only when by accident an econom
ical accumulation of effects take place. This is the
same idea in the province of inorganic nature that Dar
460 THE SCIENCE OF MECHANICS.
win worked out in the domain of organic nature. We
facilitate instinctively our comprehension of nature by
applying to it the economical ideas with which we are
familiar.
Expiana Often the phenomena of nature exhibit maximal
imaiand * or minimal properties because when these greatest or
effects. least properties have been established the causes of all
further alteration are removed. The catenary gives
the lowest point of the centre of gravity for the simple
reason that when that point has been reached all fur
ther descent of the system's parts is impossible. Li
quids exclusively subjected to the action of molecular
forces exhibit a minimum of superficial area, because
stable equilibrium can only subsist when the molecular
forces are able to effect no further diminution of super
ficial area. The important thing, therefore, is not the
maximum or minimum, but the removal of work ; work
being the factor determinative of the alteration. It
sounds much less imposing but is much more elucida
tory, much more correct and comprehensive, instead
of speaking of the economical tendencies of nature, to
say : " So much and so much only occurs as in virtue
of the forces and circumstances involved can occur. "
Points of The question may now justly be asked, If the point
identity in. rit 1*111 i
thetheoiogof view of theology which led to the enunciation of the
scientific principles of mechanics was utterly wrong, how comes
t?ons ep it that the principles themselves are in all substantial
points correct ? The answer is e^sy. In the first place,
the theological view did not supply the contents of the
principles, but simply determined their guise \ their mat
ter was derived from experience. A similar influence
would have been exercised by any other dominant type
of thought, by a commercial attitude, for instance, such
as presumably had its effect on Stevinus's thinking. In
FORMAL DEVELOPMENT. 461
the second "place, the theological conception of nature
itself owes its origin to an endeavor to obtain a more
comprehensive view of the world ; the very same en
deavor that is at the bottom of physical science. Hence,
even admitting that the physical philosophy of theology
is a fruitless achievement, a reversion to a lower state of
scientific culture, we still need not repudiate the sound
root from which it has sprung and which is not differ
ent from that of true physical inquiry.
In fact, science can accomplish nothing by the con Necessity
* ^ of a con
sideration of mdividualia.cts : from time to time it muststant con
sideration
cast its .glance at the world as a whole. Galileo's of the AH.
laws of falling bodies, Huygens's principle of vis viva,
the principle of virtual velocities, nay, even the con
cept of mass, could not, as we saw, be obtained, ex
cept by the alternate consideration of individual facts
and of nature as a totality. We may, in our men
tal reconstruction of mechanical processes, start from
the properties of isolated masses (from the elementary
or differential laws), and so compose our pictures of
the processes ; or, we may hold fast to the properties
of the system as a whole (abide by the integral laws).
Since, however, the properties of one mass always in
clude relations to other masses, (for instance, in ve
locity and acceleration a relation of time is involved,
that is, a connection with the whole world,) it is mani
fest that purely differential, or elementary, laws do not
exist. It would be illogical, accordingly, to exclude
as less certain this necessary view of the All, or of the
more general properties of nature, from our studies.
The more general a new principle is and the wider its
scope, the more perfect tests will, in view of the possi
bility of error, be demanded of it.
The conception of a will and intelligence active in
462 THE SCIENCE OF MECHANICS.
Pagan ideas nature is by no means the exclusive property of Chris
dces P rife in tian monotheism. On the contrary, this idea is a quite
the modern .... . ,... _
world, familiar one to paganism and fetishism. Paganism,
however, finds this will and intelligence entirely in in
dividual phenomena, while monotheism seeks it in the
All. Moreover, a pure monotheism does not exist.
The Jewish monotheism of the Bible is by no means
free from belief in demons, sorcerers, and witches ;
and the Christian monotheism of mediaeval times is
even richer in these pagan conceptions. We shall not
speak of the brutal amusement in which church and
state indulged in the torture and burning of witches,
and which was undoubtedly provoked, in the majority
of cases, not by avarice but by the prevalence of the
ideas mentioned. In his instructive work on Primitive
Culture Tylor has studied the sorcery, superstitions,
and miraclebelief of savage peoples, and compared
them with the opinions current in mediaeval times con
cerning witchcraft. The similarity is indeed striking.
The burning of witches, which was so frequent in
Europe in the sixteenth and seventeenth centuries, is
today vigorously conducted in Central Africa. Even
now and in civilised countries and among cultivated
people traces of these conditions, as Tylor shows, still
exist in a multitude of usages, the sense of which, with
our altered point of view, has been forever lost.
8. Physical S9ience rid itself only very slowly of
these conceptions.' The celebrated work of Giambatista
della Porta, Magia naturalis, which appeared in 1558,
though it announces important physical discoveries, is
yet filled with stuff about magic practices and demono
logical arts of all kinds little better than those of a red
skin medicineman. Not till the appearance of Gil
bert's work, De magnete (in 1600), was any kind of re
FORMAL DEVELOPMENT. 4^3
striction placed on this tendency of thought. When we Amm^sti
reflect that even Luther is said to have had personal science,
encounters with the Devil, that Kepler, whose aunt had
been burned as a witch and whose mother came near
meeting the same fate, said that witchcraft could not
be denied, and dreaded to express his real opinion of
astrology, we can vividly picture to ourselves the
thought of less enlightened minds of those ages.
Modern physical science also shows traces of fetish
ism, as Tylor well remarks, in its " forces." And the
hobgoblin practices of modern spiritualism are ample
evidence that the conceptions of paganism have not
been overcome even by the cultured society of today.
It is natural that these ideas so obstinately assert
themselves. Of the many impulses that rule man
with demoniacal power, that nourish, preserve, and
propagate him, without his knowledge or supervision,
of these impulses of which the middle ages present
such great pathological excesses, only the smallest
part is accessible to scientific analysis and conceptual
knowledge. The fundamental character of all these
instincts is the feeling of our oneness and sameness
with nature ; a feeling that at times can be silenced
but never eradicated by absorbing intellectual occupa
tions, and which certainly has a sound basis, no matter
to what religious absurdities it may have given rise.
9. The French encyclopaedists of the eighteenth
century imagined they were not far from a final ex
planation of the world by physical and mechanical prin
ciples ; Laplace even conceived a mind competent to
foretell the progress of 'nature for all eternity, if but the
masses, their positions, and initial velocities were given.
In the eighteenth century, this joyful overestimation of
the scope of the new physicomechanical ideas is par
464 THE SCIENCE OF MECHANICS.
oveesti donable. Indeed, it is a refreshing, noble, and ele
theme vating spectacle ; and we can deeply sympathise with
vie a w. 1Ca this expression of intellectual joy, so unique in history.
But now, after a century has elapsed, after our judg
ment has grown more sober, the worldconception of the
encyclopaedists appears to us as a mechanical mythology
in contrast to the animistic of the old religions. Both
views contain undue and fantastical exaggerations of
an incomplete perception. Careful physical research
will lead, however, to an analysis of our sensations.
We shall then discover that our hunger is not so essen
tially different from the tendency of sulphuric acid for
zinc, and our will not so greatly different from the
pressure of a stone, as now appears. We shall again
feel ourselves nearer nature, without its being neces
sary that we should resolve ourselves into a nebulous
and mystical mass of molecules, or make nature a
haunt of hobgoblins. The direction in which this en
lightenment is to be looked for, as the result of long
and painstaking research, can of course only be sur
mised. To anticipate the result, or even to attempt to
introduce it into any scientific investigation of today,
would be mythology, not science.
Pretensions Physical science does not pretend to be a complete
and atti . r _..... ..
tude^of view of the world ; it simply claims that it is working
science, toward such a complete view in the future. The high
est philosophy of the scientific investigator is precisely
this toleration of an incomplete conception of the world
and the preference for it rather than an apparently per
fect, but inadequate conception. Our religious opin
ions are always our own private affair, as long as we do
not obtrude them upon others and do not apply them
to things which come under the jurisdiction of a differ
ent tribunal. Physical inquirers themselves entertain
FORMAL DEVELOPMENT. 465
the most diverse opinions on this subject, according to
the range of their intellects and their estimation of the
consequences.
Physical science makes no investigation at all into
things that are absolutely inaccessible to exact investi
gation, or as yet inaccessible to it. But should prov
inces ever be thrown open to exact research which are
now closed to it, no wellorganised man, no one who
cherishes honest intentions towards himself and others,
will any longer then hesitate to countenance inquiry
with a view to exchanging his opinion regarding such
provinces for positive knowledge of them.
When, today, we see society waver, see it change Results of
the incom
its views on the same question according to its mood and pieteness of
our view of
the events of the week, like the register of an organ, when the world.
we behold the profound mental anguish which is thus
prpduced, we should know that this is the natural and
necessary outcome of the incompleteness and transi
tional character of our philosophy. A competent view
of the world can never be got as a gift ; we must ac
quire it by hard work. And only by granting free sway
to reason and experience in the provinces in which they
alone are determinative, shall we, to the weal of man
kind, approach, slowly, gradually, but surely, to that
ideal of a monistic view of the world which is alone
compatible with the economy of a sound mind.
in.
ANALYTICAL MECHANICS.
i . The mechanics of Newton are purely geometrical. The geo
TT i i i i r i..  metrical
He deduces his theorems from his initial assumptions mechanics
., , ,. . of Newton.
entirely by means of geometrical constructions. His
procedure is frequently so artificial that, as Laplace
466 THE SCIENCE OF MECHANICS.
remarked, it is unlikely that the propositions were dis
covered in that way. We notice, moreover, that the
expositions of Newton are not as candid as those of
Galileo and Huygens. Newton's is the socalled syn
thetic method of the ancient geometers.
Analytic When we deduce results from given suppositions,
mechanics. . nj , 7 j TTTI it
the procedure is called synthetic. When we seek the
conditions of a proposition or of the properties of a fig
ure, the procedure is analytic. The practice of the latter
method became usual largely in consequence of the
application of algebra to geometry. It has become
customary, therefore, to call the algebraical method
generally, the analytical. The term " analytical me
chanics," which is contrasted with the synthetical, or
geometrical, mechanics of Newton, is the exact equiva
lent of the phrase " algebraical mechanics."
Euierand 2. The foundations of analytical mechanics were
n'n's con laid by EULER (Mechanic a, sive Motus Scientia Analytice
Exposita, St. Petersburg, 1736). But while Euler's
method, in its resolution of curvilinear forces into tan
gential and normal components, still bears a trace of
the old geometrical modes, the procedure of MACLAURIN
(A Complete System of Fluxions, Edinburgh, 1742) marks
a very important advance. This author resolves all
forces in three fixed directions, and thus invests the
computations of this subject with a high degree of
symmetry and perspicuity.
3. Analytical mechanics, however, was brought to
its highest degree of perfection by LAGRANGE. La
grange's aim is (Mecanique analytique, Paris, 1788) to
dispose once for all oi the reasoning necessary to resolve
mechanical problems, by embodying as much as pos
sible of it in a single formula. This he did. Every case
that presents itself can now be dealt with by a very
FORMAL DEVELOPMENT. 467
simple, highly symmetrical and perspicuous schema;
and whatever reasoning is left is performed by purely
mechanical methods. The mechanics of Lagrange
is a stupendous contribution to the economy of
thought.
In statics, Lagrange starts from the principle of statics
500 r m r m founded on
virtual velocities. On a number of material points the princi
ples of vir
;;/ 1? ;;/ 2 , ;# 3 . . . ., definitely connected with one another, tuai veioci
are impressed the forces P^ P 2 , P% . . . . If these
points receive any infinitely small displacements / 1?
/ 2 , / 3 . . . . compatible with the connections of the sys
tem, then for equilibrium 2 ' Pp = ; where the well
known exception in which the equality passes into an
inequality is left out of account.
Now refer the whole system to a set of rectangular
coordinates. Let the coordinates of the material points
be x^, j/ 1 , z 19 x 2 , y 2 , z 2 . . . . Resolve the forces into
the components X lf V 1} Z 19 X 2 , Y 2 , Z 2 . . . . parallel
to the axes of coordinates, and the displacements into
the displacements $x l} 6y I9 dz^, dx^, $y 2 > $z 2 '  
also parallel to the axes. In the determination of the
work done only the displacements of the point of appli
cation in the direction of each forcecomponent need
be considered for that component, and the expression
of the principle accordingly is
+ Ydy \Z6z) = ........ (1)
where the appropriate indices are to be inserted for
the points, and the final expressions summed.
The fundamental formula of dynamics is derived Dynamics
. on the prin
from D'Alembert's principle. On the material points cipie of
. . r r D'Alem
m v m 2 , ;;/ 3 . . . ., having the coordinates # 1 ,jy 1 , # 15 x t> , ben.
j; 2 , z 2 . . . . the forcecomponents X lt F x , Z 19 X 2 , Y 2)
Z 2 . . . . act. But, owing to the connections of the
468 THE SCIENCE OF MECHANICS.
system's parts, the masses undergo accelerations, which
are those of the forces.
These are called the effective forces. But the impressed
forces, that is, the forces which exist by virtue of the
laws of physics, X, Y, Z. . . . and the negative of these
effective forces are, owing to the connections of the
system, in equilibrium. Applying, accordingly, the
principle of virtual velocities, we get
?\(Y
2{ \X ;
Discussion 4. Thus, Lagrange conforms to tradition in making
grange's statics precede dynamics. He was by no means com
pelled to do so. On the contrary, he might, with equal
propriety, have started from the proposition that the
connections, neglecting their straining, perform no
work, or that all the possible work of the system is due
to the impressed forces. In the latter case he would
have begun with equation (2), which expresses this
fact, and which, for equilibrium (or nonaccelerated
motion) reduces itself to (i) as a particular case. This
would have made analytical mechanics, as a system,
even more logical.
Equation (i), which for the case of equilibrium
makes the element of the work corresponding to the
assumed displacement = 0, gives readily the results
discussed in page 69. If
dV dV dV
dot dy> dz !
FORMAL DEVELOPMENT. 4 6 9
that is to say, If X, Y t Z are the partial differential co
efficients of one and the same function of the coordi
nates of position, the whole expression under the sign
of summation is the total variation, 6 V, of V. If the
latter is = 0, J^is in general a maximum or a minimum.
5. We will now illustrate the use of equation (i) by indication
J u ^ / J of the gen
a simple example. If all the points of application of the g*{ h s ^ u .
forces are independent of each other, no problem is tion of stat
* ical prot>
presented. Each point is then In equilibrium only Jems.
when the forces impressed on it, and consequently
their components, are = 0. All the displacements d x,
dy, dz. . . . are then wholly arbitrary, and equation
(i) can subsist only provided the coefficients of all the
displacements doc, dy, dz. . . . are equal to zero.
But if equations obtain between the coordinates of
the several points, that is to say, if the points are sub
ject to mutual constraints, the equations so obtaining
will be of the form J?(jx^ y 19 z^ x 2 , y 2 , z 2 . . . .) = 0,
or, more briefly, of the form F= 0. Then equations
also obtain between the displacements, of the form
<?jr dF dF dF
? _ i _ i _ i _ 8 .... 
which we shall briefly designate as.>^ 0. If the
system consist of n points, we shall have 372 coordi
nates, and equation (i) will contain 372 magnitudes
d,r, dy, dz. . . . If, further, between the coordinates
m equations of the form ^=0 subsist, then m equa
tions of the form DP= will be simultaneously given
between the variations dx, dy, #.... By these
equations m variations can be expressed in terms of the
remainder, and so inserted in equation (i). In (i),
therefore, there are left 3 n m arbitrary displace
ments,, whose coefficients are put = 0. There are thus
470
THE SCIENCE OF MECHANICS.
A statical
example.
obtained between the forces and the coordinates 3 n m
equations, to which the m equations (F = 0) must be
added. We have, accordingly, in all, 3/2 equations,
which are sufficient to determine the $n coordinates of
the position of equilibrium, provided the forces are
given and only the form of the system's equilibrium is
sought.
But if the form of the system is given and the forces
are sought that maintain equilibrium, the question is
indeterminate. We have then, to determine 3 n force
components, only 372 m equa
tions ; the m equations (F = 0)
not containing the forcecompo
nents.
As an example of this man
ner of treatment we shall select
a lever OM= a, free to rotate
about the origin of coordinates
in the plane XY, and having at its end a second, simi
lar lever MN=b. At M and N, the coordinates of
which we shall call x, y and x i9 j/ 1 , the forces X, Fand
X 19 YI are applied. Equation (i), then, has the form
Xdx + X^d*! + Ydy + Y^dy^ = ... (3)
Of the form F= two equations here exist ; namely,
X
Fig. 232.
1
j)*t*=Q f
The equations DF= 0, accordingly, are
l y) dy^ . (5)
Here, two of the variations in (5) can be expressed
in terms of the others and introduced in (3). Also for
FORMAL DEVELOPMENT. 47 1
purposes of elimination Lagrange employed a per .^^an^e's
fectly uniform and systematic procedure, which may nate coeffi
be pursued quite mechanically, without reflection. We
shall use it here. It consists in multiplying each of the
equations (5) by an indeterminate coefficient A, /*, and
adding each, in this form to (3). So doing, we obtain
The coefficients of the four displacements may now
be put directly = 0. For two displacements are ar
bitrary, and the two remaining coefficients may be
made equal to zero by the appropriate choice of A and
yu which is tantamount to an elimination of the two
remaining displacements.
We have, therefore, the four equations
X l
(6)
We shall first assume that the coordinates are given,
and seek the forces that maintain equilibrium. The
values of A and JJL are each determined by equating to
zero two coefficients. We get from the second and
fourth equations,
X, A
 L anc i JK
whence
that is to say, the total component force impressed at
N has the direction MN. From the first and third
equations we get
472 THE SCIENCE OF MECHANICS.
,
Their em A =
ployment in
the deter
f an< ^ fr m ^ese by simple reduction
y
that is to say, the resultant of the forces applied at M
and N acts in the direction OM. *
The four forcecomponents are accordingly subject
to only two conditions, (7) and (8). The problem, con
sequently, is an indeterminate one ; as it must be from
the nature of the case ; for equilibrium does not depend
upon the absolute magnitudes of the forces, but upon
their directions and relations.
If we assume that the forces are given and seek the
four coordinates, we treat equations (6) in exactly the
same manner. Only, we can now make use, in addi
tion, of equations (4). Accordingly, we have, upon the
elimination of A. and JJL, equations (7) and (8) and two
equations (4). From these the following, which fully
solve the problem, are readily deduced
* The mechanical interpretation of the indeterminate coefficients A ft may
be shown as follows. Equations (6) express the equilibrium of two free points
on which in addition to X, V, X, Y other forces act which answer to the re
maining expressions and just destroy^", Y,X"i, Y^. The point N, for example,
is in equilibrium if XL is destroyed by a force n(xix} t undetermined as yet
in magnitude, and YI by a force /J, (y y}. This supplementary force is due
to the constraints. Its direction is determined ; though its magnitude is not,
If we call the angle which it makes with the axis of abscissas a, we shall have
that is to say, the force due to the connections acts in the direction of .
FORMAL DEVELOPMENT. 473
^ Character
Yl entpror"
Simple as this example is, it is yet sufficient to give
us a distinct idea of the character and significance of
Lagrange's method. The mechanism of this method is
excogitated once for all, and in its application to par
ticular cases scarcely any additional thinking is re
quired. The simplicity of the example here selected
being such that it can be solved by a mere glance at
the figure, we have, in our study of the method, the
advantage of a ready verification at every step.
6. We will now illustrate the application of equa General
tion (2), which is Lagrange's form of statement of the P s S oiution
D'Alembert's principle. There is no problem when Seal "pro?
the masses move quite independently of one another.
Each mass yields to the forces applied to it ; the va
riations dx, dy, 3 z . . . . are wholly arbitrary, and each
coefficient may be singly put = 0. For the motion of
n masses we thus obtain 3 n simul
taneous differential equations.
But if equations of condition
(]?=. 0) obtain between the coordi
nates, these equations will lead to
others (DF= 0) between the dis
placements or variations. With the
latter we proceed exactly as in the
application of equation (i). Only it must be noted
here that the equations F= must eventually be em
ployed in their undifferentiated as well as in their dif
ferentiated form, as will best be seen from the follow
ing example.
474 THE SCIENCE OF MECHANICS.
A dynam A heavy material point /#, lying in a vertical plane
ica exam ^^ .^ ^^ to move on a straight line, y = ax, inclined
at an angle to the horizon. (Fig. 233.) Here equa
tion (2) becomes
and, since X= 0, and Y= mg, also
^* + (*+S)*= ........ ()
The place of ^~ is taken by
jy = tf.# ................... (10)
and for DF = we have
Equation (9), accordingly, since dy drops out and
dx is arbitrary, passes into the form
By the differentiation of (10), or (J^= 0), we have
d*y __
~dt*~
and, consequently,
Then, by the integration of (n), we obtain
a t*
and
where b and c are constants of integration, determined
by the initial position and velocity of m. This result
can also be easily found by the direct method.
FORMAL DEVELOPMENT. 475
Some care is necessary In the application of equa A modifica
J r * ^ . tionofthis
tion (i) if F = contains the time. The procedure in example.
such cases maybe illustrated by the following example.
Imagine in the preceding case the straight line on
which m descends to move vertically upwards with the
acceleration y. We start again from equation (9)
F= is here replaced by
(12)
To form DJF= 0, we vary (12) only with respect to x
and y, for we are concerned here only with the possible
displacement of the system in its position at any given
instant, and not with the displacement that actually
takes place in time. We put, therefore, as in the pre
vious case,
and obtain, as before,
But to get an equation In x alone, we have, since x
and y are connected in (13) by the actual motion, to
differentiate (12) with respect to t and employ the re
sulting equation
for substitution in (13). In this way the equation
d * x , ( , , <l* x \
_+( ;f+ y + tf _J tf=
is obtained, which, integrated, gives
47 6 THE SCIENCE OF MECHANICS.
If a weightless body ;;/ lie on the moving straight
line, we obtain these equations
results which are readily understood, when we re
flect that, on a straight line moving upwards with the
acceleration y, m behaves as if it were affected with a
downward acceleration y on the straight line at rest.
Discussion 7 The procedure with equation (12) in the preced
i ? Ied h ^m^~ ing example may be rendered somewhat clearer by the
ple * following consideration. Equation (2), D'Alembert's
principle, asserts, that all the work
that can be done in the displacement
of a system is done by the impressed
forces and not by the connections. This
jl is evident, since the rigidity of the con
nections allows no changes in the rela
Fig. 234.
tive positions which would be neces
sary for any alteration in the potentials of the elastic
forces. But this ceases to be true when the connec
tions undergo changes in time. In this case, the changes
of the connections perform work, and we can then ap
ply equation (2) to the displacements that actually take
place only provided we add to the impressed forces the
forces that produce the changes of the connections.
A heavy mass m is free to move on a straight line
parallel to 6>F(Fig. 234.) Let this line be subject to
FORMAL DEVELOPMENT. 477
a forced acceleration In the direction of x, suc ^ t ^ at I ^ u h S g r ^ d 1 ]
the equation J? = becomes if jed exam
D'Alembert's principle again gives equation (9).
But since from DF= it follows here that doc 0,
this equation reduces itself to
,= ............ (15)
in which 6y Is wholly arbitrary. Wherefore,
and
to which must be supplied (14) or
It is patent that (15) does not assign the total work
of the displacement that actually takes place, but only
that of Qmz possible displacement on the straight line
conceived, for the moment, as fixed.
If we imagine the straight line massless, and cause
it to travel parallel to itself in some guiding mechan
ism moved by a force my, equation (2) will be re
placed by
and since dx, dy are wholly arbitrary here, we obtain
the two equations
_
47 8 THE SCIENCE OF MECHANICS.
which give the same results as before. The apparently
different mode of treatment of these cases is simply the
result of a slight inconsistency, springing from the fact
that all the forces involved are, for reasons facilitating
calculation, not included in the consideration at the
outset, but a portion is left to be dealt with subse
quently.
Deduction 8. As the different mechanical principles only ex
i f p ie e o?"* press different aspects of the same fact, any one of
grange's them is easily deducible from any other \ as we shall
tSPdySam now illustrate by developing the principle of vis viva
icai equa frQm e q. uat ; on ( 2 ) o f pa g e ^53. Equation (2) refers to
instantaneously possible displacements, that is, to "vir
tual" displacements. But when the connections of a
system are independent of the time, the motions that
actually take place are "virtual " displacements. Conse
quently the principle may be applied to actual motions.
For doc, dy, $z, we may, accordingly, write dx, dy,
dz> the displacements which take place in time, and
put
2 (Xdx + Ydy + Zdz) =
The expression to the right may, by introducing for
dx, (dx/df) dt and so forth, and by denoting the velo
city by v, also be written
!d*x dx d*y dy d* z dz
\dW dt dt + d* Tt dt + fi* Tt dt =
FORMAL DEVELOPMENT. 479
Also in the expression to the left, (dx/df] dt may be Force
r . 9 \ f J J function.
written for dx. Bat this gives
J2 (Xdx + Ydy + Zdz) =
where V Q denotes the velocity at the beginning and v
the velocity at the end of the motion. The integral to the
left can always be found if we can reduce it to a single
variable, that is to say, if we know the course of the
motion in time or the paths which the movable points
describe. If, however, X, Y, Z are the partial differ
ential coefficients of the same function Uoi coordinates,
if, that is to say,
dU
^_
<p
as is always the case when only central forces are in
volved, this reduction is unnecessary. The entire ex
pression to the left is then a complete differential. And
we have
which is to say, the difference of the forcefunctions
(or work) at the beginning and the end of the motion
is equal to the difference of the vires iriva at the be
ginning and the end of the motion. The vires invce are
in such case also functions of the coordinates.
In the case of a body movable in the plane of X
and Y suppose, for example, X = y, Y= x ; we
then have
J(ydx xdy} =
But if X= a, Y= x, the integral to the left is
f(a dx + x dy). This integral can be assigned the
moment we know the path the body has traversed, that
4 So THE SCIENCE OF MECHANICS.
is, if y is determined a function of x. If, for example,
y =jfix 2 , the integral would become
_ J(a
dx = a
The difference of these two cases is, that in the first
the work is simply a function of coordinates, that a
forcefunction exists, that the element of the work is a
complete differential, and the work consequently is de
termined by the initial and final values of the coordi
nates, while in the second case it is dependent on the
entire path described.
Essential Q. These simple examples, in themselves present
of an^t ing no difficulties, will doubtless suffice to illustrate the
ch^nTcs". general nature of the operations of analytical mechan
ics. No fundamental light can be expected from this
branch of mechanics. On the contrary, the discovery
of matters of principle must be substantially completed
before we can think of framing analytical mechanics ;
the sole aim of which is a perfect practical mastery of
problems. Whosoever mistakes this situation, will
never comprehend Lagrang'e's great performance, which
here too is essentially of an economical character. Poin
sot did not altogether escape this error.
It remains to be mentioned that as the result of the
labors of Mobius, Hamilton, Grassmann, and others, a
new transformation of mechanics is preparing. These
inquirers have developed mathematical conceptions
that conform more exactly and directly to our geomet
rical ideas than do the conceptions of common analyt
ical geometry ; and the advantages of analytical gene
rality and direct geometrical insight are thus united.
But this transformation, of course, lies, as yet, beyond
the limits of an historical exposition. (See p. 577.)
FORMAL DEVELOPMENT. 4*1
THE ECONOMY OF SCIENCE.
i. It is the object of science to replace, or save, ex The basrf
periences, by the reproduction and anticipation of facts economy
in thought. Memory is handier than experience, and
often answers the same purpose. This economical
office of science, which fills its whole life, is apparent
at first glance ; and with its full recognition all mys
ticism in science disappears.
Science is communicated by instruction, in order
that one man may profit by the experience of another
and be spared the trouble of accumulating it for him
self ; and thus, to spare posterity, the experiences of
whole generations are stored up in libraries.
Language, the instrument of this communication, The eco
 . , nomical
is itself an economical contrivance. Experiences are characte
analysed, or broken up, into simpler and more familiar "
experiences, and then symbolised at some sacrifice of
precision. The symbols of speech are as yet restricted
in their use within national boundaries, and doubtless
will long remain so. But written language is gradually
being metamorphosed into an ideal universal character.
It is certainly no longer a mere transcript of speech.
Numerals, algebraic signs, chemical symbols, musical
notes, phonetic alphabets, may be regarded as parts
already formed of this universal character of the fu
ture ; they are, to some extent, decidedly conceptual,
and of almost general international use. The analysis
of colors, physical and physiological, is already far
enough advanced to render an international system of
colorsigns perfectly practical. In Chinese writing,
4 g 2 THE SCIENCE OF MECHANICS.
Possibility we have an actual example of a true ideographic Ian
sanan 1 " 61 guage, pronounced diversely in different provinces, yet
guage. everywhere carrying 'the same meaning. Were the
system and its signs only of a simpler character, the
use of Chinese writing might become universal. The
dropping of unmeaning and needless accidents of gram
mar, as English mostly drops them, would be quite
requisite to the adoption of such a system. But uni
versality would not be the sole merit of such a char
acter ; since to read it would be to understand it. Our
children often read what they do not understand \ but
that which a Chinaman cannot understand, he is pre
cluded from reading.
Econom 2. In the reproduction of facts in thought, we
iero c f h a a n ac " never reproduce the facts in full, but .only that side of
sentatFona them which is important to us, moved to this directly
worM. or indirectly by a practical interest. Our reproductions
are invariably abstractions. Here again is an econom
ical tendency.
Nature is composed of sensations as its elements.
Primitive man, however, first picks out certain com
pounds of these elements those namely that are re
latively permanent and of greater importance to him.
The first and oldest words are names of " things. 37
Even here, there is an abstractive process, an abstrac
tion from the surroundings of the things, and from the
continual small changes which these compound sensa
tions undergo, which being practically unimportant are
not noticed. No inalterable thing exists. The thing
is an abstraction, the name a symbol, for a compound
of elements from whose changes we abstract. The
reason we assign a single word to a whole compound is
that we need to suggest all the constituent sensations
at once. When, later, we come to remark the change
FORMAL DEVELOPMENT, 483
ableness, we cannot at the same time hold fast to the
idea of the thing's permanence, unless we have recourse
to the conception of a thingiriitself, or other such like
absurdity. Sensations are not signs of things ; but, on
the contrary, a thing is a thoughtsymbol for a com
pound sensation of relative fixedness. Properly speak
ing the world is not composed of "things" as its ele
ments, but of colors, tones, pressures, spaces, times,
in short what we ordinarily call individual sensations.
The whole operation is a mere affair of economy.
In the reproduction of facts, we begin with the more
durable and familiar compounds, and supplement these
later with the unusual by way of corrections. Thus,
we speak of a perforated cylinder, of a cube with bev
eled edges, expressions involving contradictions, un
less we accept the view here taken. All judgments are
such amplifications and corrections of ideas already
admitted.
3. In speaking of cause and effect we arbitrarily The ideas
, . r , , , . cause and
give relief to those elements to whose connection we effect,
have to attend in the reproduction of a fact in the re
spect in which it is important to us. There is no cause
nor effect in nature ; nature has but an individual exis
tence \ nature simply is. Recurrences of like cases in
which A is always connected with B, that is, like results
under like circumstances, that is again, the essence of the
connection of cause and effect, exist but in the abstrac
tion which we perform for the purpose of mentally re
producing the facts. Let a fact become familiar, and
we no longer require this putting into relief of its con
necting marks, oux" attention is no longer attracted to
the new and surprising, and we cease to speak of cause
and effect. Heat is said to be the cause of the tension
of steam ; but when the phenomenon becomes familiar .
484 TUE SCIENCE OF MECHANICS.
we think of the steam at once with the tension proper
to its temperature. Acid is said to be the cause of the
reddening of tincture of litmus ; but later we think of
the reddening as a property of the acid.
Hume, Hume first propounded the question, How can a
Schopen thing A act on another thing B ? Hume, in fact, re
hauer's ex . , . ,
pianadons iects causality and recognises only a wonted succes
of cause ... " '_ , , , ,
and effect, sion in time. Kant correctly remarked that a necessary
connection between A and B could not be disclosed by
simple observation. He assumes an innate idea or
category of the mind, a Verstandesbe griff, under which
the cases of experience are subsumed. Schopenhauerj
who adopts substantially the same position, distin
guishes four forms of the "principle of sufficient rea
son 3 ' the logical, physical, and mathematical form,
and the law of motivation. But these forms differ only
as regards the matter to which they are applied, which
may belong either to outward or inward experience.
Cause and The natural and commonsense explanation is ap
economicai parently this. The ideas of cause and effect originally
implements r j , j r , . , 
of thought, sprang from an endeavor to reproduce facts in thought.
At first, the connection of A and J3, of C and D, of E
and Fj and so forth, is regarded as familiar. But after
a greater range of experience is acquired and a con
nection between M and N is observed, it often turns
out that we recognise M as made iip of A, C, E, and N
of B, D 7 J?, the connection of which was before a fa
miliar fact and accordingly possesses with us a higher
authority. This explains why a person of experience
regards a new event with different eyes than the nov
ice. The new experience is illuminated by the mass
of old experience. As a fact, then, there really does
exist in the mind an "idea" under which fresh experi
ences are subsumed ; but that idea has itself been de
FORMAL DEVELOPMENT. 485
veloped from experience. The notion of the necessity
of the causal connection is probably created by our
voluntary movements in the world and by the changes
which these indirectly produce, as Hume supposed but
Schopenhauer contested. Much of the authority of
the ideas of cause and effect is due to the fact that they
are developed instinctively and involuntarily, and that
we are distinctly sensible of having personally con
tributed nothing to their formation. We may, indeed,
say, that our sense of causality is not acquired by the
individual, but has been perfected in the develop
ment of the race. Cause and effect, therefore, are
things of thought, having an economical office. It can
not be said why they arise. For it is precisely by the
abstraction of uniformities that we know the question
"why." (See Appendix, XXVI, p. 579.)
4. In the details of science, its economical character Econom
is still more apparent. The socalled descriptive sci tures of
1 . n . . , .all laws of
ences must chiefly remain content with reconstructing nature,
individual facts. Where it is possible, the common fea
tures of many facts are once for all placed in relief. But
in sciences that are more highly developed, rules for the
reconstruction of great numbers of facts maybe embod
ied in a single expression. Thus, instead of noting indi
vidual cases of lightrefraction, we can mentally recon
struct all present and future cases, if we know that the
incident ray, the refracted ray, and the perpendicular
lie in the same plane and that sin a/sm /3 = n. Here,
instead of the numberless cases of refraction in different
combinations of matter and under all different angles
of incidence, we have simply to note the rule above
stated and the values of n, which is much easier. The
economical purpose is here unmistakable. In nature
there is no law of refraction, only different cases of re
486 THE SCIENCE OF MECHANICS.
fraction. The law of refraction is a concise compen
dious rule, devised by us for the mental reconstruction
of a fact, and only for its reconstruction in part, that
is, on its geometrical side.
The econ 5. The sciences most highly developed economically
nSthemaS are those whose facts are reducible to a few numerable
ences. elements of like nature. Such is the science of mechan
ics, in which we deal exclusively with spaces, times,
and masses. The whole previously established econ
omy of mathematics stands these sciences in stead.
Mathematics may be defined as the economy of count
ing. Numbers are arrangementsigns which, for the
sake of perspicuity and economy, are themselves ar
ranged in a simple system. Numerical operations, it
is found, are independent of the kind of objects operated
on, and are consequently mastered once for all. When,
for the first time, I have occasion to add five objects to
seven others, I count the whole collection through, at
once ; but when I afterwards discover that I can start
counting from 5, I save myself part of the trouble ;
and still later, remembering that 5 and 7 always count
up to 12, I dispense with the numeration entirely.
Arithmetic The object of all arithmetical operations is to save
andaige ^^^ nume ration, by utilising the results of our old
operations of counting. Our endeavor is, having done
a sum once, to preserve the answer for future use. The
first four rules of arithmetic well illustrate this view.
Such, too, is the purpose of algebra, which, substitut
ing relations for values, symbolises and definitively
fixes all numerical operations that follow the same rule.
For example, we learn from the equation
FORMAL DEVELOPMENT. 487
that the more complicated numerical operation at the '
left may always be replaced by the simpler one at the
right, whatever numbers x and y stand for. We thus
save ourselves the labor of performing In future cases
the more complicated operation. Mathematics is the
method of replacing in the most comprehensive and
economical manner possible, new numerical operations
by old ones done already with known results. It may
happen in this procedure that the results of operations
are employed which were originally performed centu
ries ago.
Often operations involving intense mental effort The theory
. . . ofdeter
may be replaced by the action of semimechanical minants.
routine, with great saving of time and avoidance of
fatigue. For example, the theory of determinants
owes its origin to the remark, that it is not necessary
to solve each time anew equations of the form
from which result
but that the solution may be effected by means of the
coefficients, by writing down the coefficients according
to a prescribed scheme and operating with them me
chemically. Thus,
and similarly
'' *\ = P, and ^ ^ !== a
/ /} yT x I *
4 88 THE SCIENCE OF MECHANICS.
Calculating Even a total disburdening of the mind can be ef
fected in mathematical operations. This happens where
operations of counting hitherto performed are symbol
ised by mechanical operations with signs, and our brain
energy, instead of being wasted on the repetition of
old operations, is spared for more important tasks.
The merchant pursues a like economy, when, instead
of directly handling his bales of goods, he operates
with bills of lading or assignments of them. The
drudgery of computation may even be relegated to a
machine. Several different types of calculating ma
chines are actually in practical use. The earliest of
these (of any complexity) was the differenceengine of
Babbage, who was familiar with the ideas here pre
sented.
other ab A numerical result is not always reached by the
methods of actual solution of the problem \ it may also be reached
results? 6 indirectly. It is easy to ascertain, for example, that a
curve whose quadrature for the abscissa x has the value
#'", gives an increment mx m ~*dx of the quadrature for
the increment dx of the abscissa. But we then also know
that Cnix m ' t dx = x m \ that is, we recognise the quan
tity x' n from the increment mx m ~~ l dx as unmistakably
as we recognise a fruit by its rind. Results of this
kind, accidentally found by simple inversion, or by
processes more or less analogous, are very extensively
employed in mathematics.
That scientific work should be more useful the more
it has been used, while mechanical work is expended in
use, may seem strange to us. When a person who
daily takes the same walk accidentally finds a shorter
cut, and thereafter, remembering that it is shorter, al
ways goes that way, he undoubtedly saves himself the
difference of the work. But memory is really not work.
FORMAL DEVELOPMENT. 489
It only places at our disposal energy within our present
or future possession, which the circumstance of igno
rance prevented us from availing ourselves of. This
is precisely the case with the application of scientific
ideas.
The mathematician who pursues his studies with Necessity
c of clear
out clear views of this matter, must often have the views on
7 this sub
uncomfortable feeling that his paper and pencil sur ject.
pass him in intelligence. Mathematics, thus pursued
as an object of instruction, is scarcely of more educa
tional value than busying oneself with the Cabala. On
the contrary, it induces a tendency toward mystery,
which is pretty sure to bear its fruits.
6. The science of physics also furnishes examples Examples
of this economy of thought, altogether similar to those omy of ec
we have just examined. A brief reference here will suf physics,
fice. The moment of Inertia saves us the separate con
sideration of the individual particles of masses. By
the forcefunction we dispense with the separate in
vestigation of individual forcecomponents. The sim
plicity of reasonings Involving forcefunctions springs
from the fact that a great amount of mental work had
to be performed before the discovery of the properties
of the forcefunctions was possible. Gauss's dioptrics
dispenses us from the separate consideration of the
single refracting surfaces of a dioptrical system and
substitutes for it the principal and nodal points. But
a careful consideration of the single surfaces had to
precede the discovery of the principal and nodal points.
Gauss's dioptrics simply saves us the necessity of often
repeating this consideration.
We must admit, therefore, that there is no result of
science which In point of principle could not have been
arrived at wholly without methods. But, as a matter
490 THE SCIENCE OF MECHANICS.
science a of fact, within the short span of a human life and with
problem, man's limited powers of memory, any stock of knowl
edge worthy of the name is unattainable except by the
greatest mental economy. Science itself, therefore,
may be regarded as a minimal problem, consisting of
the completest possible presentment of facts with the
least possible expenditure of thought.
7. The function of science, as we take it, is to re
place experience. Thus, on the one hand, science
must remain in the province of experience, but, on the
other, must hasten beyond it, constantly expecting con
firmation, constantly expecting the reverse. Where
neither confirmation nor refutation is possible, science
is not concerned. Science acts and only acts in the
domain of uncompleted experience. Exemplars of such
branches of science are the theories of elasticity and
of the conduction of heat, both of which ascrioe to the
smallest particles of matter only such properties as ob
servation supplies in the study of the larger portions.
The comparison of theory and experience may be far
ther and farther extended, as our means of observation
increase 1 in refinement.
The princi Experience alone, without the ideas that are asso
nu?ty?the ciated with it, would forever remain strange to us.
entific Those ideas that hold good throughout the widest do
mains of research and that supplement the greatest
amount of experience, are the most scientific. The prin
ciple of continuity, the use of which everywhere per
vades modern inquiry, simply prescribes a mode of
* conception which conduces in the highest degree to the
economy of thought.
8. If a long elastic rod be fastened in a vise, the
rod may be made to execute slow vibrations. These
are directly observable, can be seen, touched, and
FORMAL DEVELOPMENT. 491
graphically recorded. If the rod be shortened, the Example n
r J , IT lustrative
vibrations will increase in rapidity and cannot be diof the
1111 j method of
rectly seen ; the rod will present to the sight a blurred science,
image. This is a new phenomenon. But the sensa
tion of touch is still like that of the previous case \ we
can still make the rod record its movements ; and if
we mentally retain the conception of vibrations, we can
still anticipate the results of experiments. On further
shortening the rod the sensation of touch is altered ;
the rod begins to sound ; again a new phenomenon is
presented. But the phenomena do not all change at
once ; only this or that phenomenon changes ; conse
quently the accompanying notion of vibration, which
is not confined to any single one, is still serviceable,
still economical. Even when the sound has reached
so high a pitch and the vibrations ' have become so
small that the previous means of observation are not
of avail, we still advantageously imagine the sounding
rod to perform vibrations, and can predict the vibra
tions of the dark lines in the spectrum of the polarised
light of a rod of glass. If on the rod being further
shortened all the phenomena suddenly passed into new
phenomena, the conception of vibration would no
longer be serviceable because it would no longer afford
us a means of supplementing the new experiences by
the previous ones.
When we mentally add to those actions of a human
being which we can perceive, sensations and ideas like
our own which we cannot perceive, the object of the
idea we so form is economical. The idea makes ex
perience intelligible to us ; it supplements and sup
plants experience. This idea is not regarded as a great
scientific discovery, only because its formation is so
natural that every child conceives it. Now, .this is
4 Q2 THE SCIENCE OF MECHANICS
exactly what we do when we imagine a moving body
which has just disappeared behind a pillar, or a comet
at the moment invisible, as continuing its motion and
retaining its previously observed properties. We do
this that we may not be surprised by its reappearance.
We fill out the gaps in experience by the ideas that
experience suggests.
AH scien 9 Yet not all the prevalent scientific theories origi
odes'no't nated so naturally and artlessly. Thus, chemical, elec
fhe^nc? 11 trical, and optical phenomena are explained by atoms.
Snuity? n " But the mental artifice 'atom was not formed by the
principle of continuity ; on the contrary, it is a pro
duct especially devised for the purpose in view. Atoms
cannot be perceived by the senses ; like all substances,
they are things of thought. Furthermore, the atoms
are invested with properties that absolutely contradict
the attributes hitherto observed in bodies. However
well fitted atomic theories may be to reproduce certain
groups of facts, the physical inquirer who has laid to
heart Newton's rules will only admit those theories as
provisional helps, and will strive to attain, in some more
natural way, a satisfactory substitute.
Atoms and The atomic theory plays a part in physics similar
tai artifices, to that of certain auxiliary concepts in mathematics;
it is a mathematical model for facilitating the mental
reproduction of facts. Although we represent vibra
tions by the harmonic formula, the phenomena of cool
ing by exponentials, falls by squares of times, etc., no
one will fancy that vibrations in themselves have any
thing to do with the circular functions, or the motion
of falling bodies with squares. It has simply been ob
served that the relations between the quantities inves
tigated were similar to certain relations obtaining be
tween familiar mathematical functions, and these more
FORMAL DEVELOPMENT. 493
familiar ideas are employed as an easy means of sup
plementing experience. Natural phenomena whose re
lations are not similar to those of functions with which
we are familiar, are at present very difficult to recon
struct. But the progress of mathematics may facilitate
the matter.
As mathematical helps of this kind, spaces of more Muiti
than three dimensions may be used, as I have else sio a ^
where shown. But it is not necessary to regard these,
on this account, as anything more than mental arti
fices. *
*As the outcome of the labors of Lobatchevski, Bolyai, Gauss, and Rie
mann, the view has gradually obtained currency in the mathematical world,
that that which we call space is a particular, actual case of a more general,
conceivable case of multiple quantitative manifoldness. The space of sight
and touch is a threefold manifoldness; it possesses three dimensions ; and
every point in it can be defined by three distinct and independent data. But
it is possible to conceive of a quadruple or even multiple spacelike manifold
ness. And the character of the manifoldness may also be differently conceived
from the manifoldness of actual space. We regard this discovery, which is
chiefly due to the labors of Riemann, as a very important one. The properties
of actual space are here directly exhibited as objects of experience, and the
pseudotheories of geometry that seek to excogitate these properties by meta
physical arguments are overthrown.
A thinking being is supposed to live in the surface of a sphere, with no
other kind of space to institute comparisons with. His space will appear to
him similarly constituted throughout. He might regard _it as infinite, and
could only be convinced of the contrary by experience. Starting from any two
points of a great circle of the sphere and proceeding at right angles thereto on
other great circles, he could hardly expect that the circles last mentioned
would intersect. So, also, with respect to the space in which we live, only ex
perience can decide whether it is finite, whether parallel lines intersect in it,
or the like. The significance of this elucidation can scarcely be overrated.
An enlightenment similar to that which Riemann inaugurated in science was
produced in the rnind of humanity at large, as regards the surface of the earth,
by the discoveries of the first circumnavigators.
The theoretical investigation of the mathematical possibilities above re
ferred to, has, primarily, nothing to do with the question whether things really
exist which correspond to these possibilities; and we must not hold mathe
maticians responsible for the popular absurdities which their investigations
have given rise to. The space of sight and touch is 2?/zr<?<?dimensional ; that,
no one ever yet doubted. If, now, it should be found that bodies vanish from
this space, or new bodies get into it, the question might scientifically be dis
cussed whether it would facilitate and promote our insight into things to con
ceive experiential space as part of a fourdimensional or multidimensional
494 THE SCIENCE OF MECHANICS.
Hypotheses This is the case, too, with all hypothesis formed
and facts . . . r ,>.
for trie explanation of new phenomena. Our concep
tions of electricity fit in at once with the electrical phe
nomena, and take almost spontaneously the familiar
course, the moment we note that things take place as
if attracting and repelling fluids moved on the surface
of the conductors. But these mental expedients have
nothing whatever to do with the phenomenon itself.
(See Appendix, XXVII, p. 579.)
space. Yet in such a case, this fourth dimension would, none the less, remain
a pure thing of thought a inenta'l fiction.
But this is not the way matters stand. The phenomena mentioned were
not forthcoming until after the new views were published, and were then ex
hibited in the presence of certain persons at spiritualistic seances. The fourth
dimension was a very opportune discover^ for the spiritualists and for theo
logians who were in a quandary about the location of hell. The use the spiri
tualist makes of the fourth dimension is this, 'it is possible to move out of a
finite straight line, without passing the extremities, through the second dimen
sion ; out of a finite closed surface through the third ; and, analogously, out
of a finite closed space, without passing through the enclosing boundaries,
through the fourth dimension. Even the tricks that prestidigitateurs, in the
old days, harmlessly executed in three dimensions, are now invested with a
new halo by the fourth. But the tricks of the spiritualists, the tying or untying
of knots in endless strings, the removing of bodies from closed spaces, are all
performed in cases where there is absolutely nothing at stake. All is purpose
less jugglery. We have not yet found &n accoucheur who has accomplished,
parturition through the fourth dimension. If we should, the question would
at once become a serious one. Professor Simony's beautiful tricks in rope
tying, which, as the performance of a prestidigitateur, are very admirable,
speak against, not for, the spiritualists.
Everyone is free to set up an opinion and to adduce proofs in support of
it. Whether, though, a scientist shall find it worth his while to enter into
serious investigations of opinions so advanced, is a question which his reason
and instinct alone can decide, if these things, in the end, should turn out to
be true, I shall not be ashamed of being the last to believe them. What I have
seen of them was not calculated to make me less sceptical.
I myself regarded multidimensioned spa^e as a mathematicophysical
help even, prior to the appearance of Riemann's memoir. But 1 trust tuat
no one will employ what I have thought, said, and written on this subject as a
basis for the fabrication of ghost stories. (Compare Mach, Die Geschi>.hte und
die Wurzel des Satzes von, der Er ka.it ting der Arbeit.}
CHAPTER V.
THE RELATIONS OF MECHANICS TO OTHER DE
PARTMENTS OF KNOWLEDGE,
i.
THE RELATIONS OF MECHANICS TO PHYSICS.
1. Purely mechanical phenomena do not exist. The J t ^ a
production of mutual accelerations in masses is, to all J^
appearances, a purely dynamical phenomenon. Butbeion
with these dynamical results are always associated ence 
thermal, magnetic, electrical, and chemical phenom
ena, and the former are always modified in proportion
as the latter are asserted. On the other hand, thermal,
magnetic, electrical, and chemical conditions also can
produce motions. Purely mechanical phenomena, ac
cordingly, are abstractions, made, either intentionally
or from necessity, for facilitating our comprehension of
things. The same thing is true of the other classes of
physical phenomena. Every event belongs, in a strict
sense, to all the departments of physics, the latter be
ing separated only by an artificial classification, which
is partly conventional, partly physiological, and partly
historical.
2. The view that makes mechanics the basis of the
remaining branches of physics, and explains all physical
phenomena by mechanical ideas, is in our judgment a
prejudice. Knowledge which is historically first, is
not necessarily the foundation of all that is subsequently
496 THE SCIENCE OF MECHANICS.
The me gained. As more and more facts are discovered and
aspects of classified, entirely new ideas of general scope can be
necessarily formed. We have no means of knowing, as yet, which
its funda .
mental of the physical phenomena go deepest, whether the
mechanical phenomena are perhaps not the most super
ficial of all, or whether all do not go equally deep. Even
in mechanics we no longer regard the oldest law, the
law of the lever, as the foundation of all the other
principles. ^
Artificiality The mechanical theory of nature, is, undoubtedly,
chanicai in an historical view, both intelligible and pardonable :
conception .
of the and it may also, for a time, have been of much value.
world. J . . .
But, upon the whole, it is an artificial conception.
Faithful adherence to the method that led the greatest
investigators of nature, Galileo, Newton, Sadi Carnot,
Faraday, and J. R. Mayer, to their great results, re
stricts physics to the expression of actual facts, and
forbids the construction of hypotheses behind the facts,
where nothing tangible and verifiable is found. *If this
is done, only the simple connection of the motions of
masses, of changes of temperature, of changes in the
values of the potential function, of chemical changes,
and so forth is to be ascertained, and nothing is to be
imagined along with these elements except the physical
attributes or characteristics directly or indirectly given
by observation.
This idea was elsewhere * developed by the author
with respect to the phenomena of heat, and indicated,
in the same place, with respect to electricity. All hy
potheses of fluids or media are eliminated from the
theory of electricity as entirely superfluous, when we
reflect that electrical conditions are all given by the
* Mach, Die Geschichte und die Wurzel des Satzes von der Erhaltung dsr
Arbeit.
ITS RELATIONS TO OTHER SCIENCES. 497
values of. the potential function V and the dielectric science
" should be
constants. If we assume the differences of the values based on
facts, not
of Kto be measured (on the electrometer) by the forces, on hypoth
and regard Kand not the quantity of electricity Q as
the primary notion, or measurable physical attribute,,
we shall have, for any simple insulator, for our quan
tity of electricity
ay d*V d*V
(where x, y, z denote the coordinates and dv the ele
ment of volume,) and for our potential*
Here Q and F appear as derived notions, in which no
conception of fluid or medium is contained. If we
work over in a similar manner the entire domain of
physics, we shall restrict ourselves wholly to the quan
titative conceptual expression of actual facts. All su
perfluous and futile notions are eliminated, and the
imaginary problems to which they have given rise fore
stalled. (See Appendix XXVIII, p. 583.)
The removal of notions whose foundations are his
torical, conventional, or accidental, can best be fur
thered by a comparison of the conceptions obtaining
in the different departments, and by finding for the
conceptions of every department the corresponding
conceptions of others. We discover, thus, that tem
peratures and potential functions correspond to the
velocities of massmotions. A single velocityvalue, a
single temperaturevalue, or a single value of potential
function, never changes alone. But whilst in the case
of velocities and potential functions, so far as we yet
* Using the terminology of Clausius.
498 THE SCIENCE OF MECHANICS,
pesirabii know, only differences come into consideration, the
compare significance of temperature is not only contained in its
tive pfays
ic*. difference with respect to other temperatures. Thermal
capacities correspond to masses, the potential of an
electric charge to quantity of heat, quantity of elec
tricity to entropy, and so on. The pursuit of such re
semblances and differences lays the foundation of a
comparative physics, which shall ultimately render pos
sible the concise expression of extensive groups of facts,
without arbitrary additions. We shall then possess a
homogeneous physics, unmingled with artificial atomic
theories.
It will also be perceived, that a real economy of
scientific thought cannot be attained by mechanical
hypotheses. Even if an hypothesis were fully com
petent to reproduce a given department of natural phe
nomena, say, the phenomena of heat, we should, by
accepting it, only substitute for the actual relations be
tween the mechanical and thermal processes, the hy
pothesis. The real fundamental facts are replaced by
an equally large number of hypotheses, which is cer
tainly no gain. Once an hypothesis has facilitated,
as best it can, our view of new facts, by the substitu
tion of more familiar ideas, its powers are exhausted.
We err when we expect more enlightenment from an
hypothesis than from the facts themselves.
circum 3* The development of the mechanical view was
whicif fa favored by many circumstances. In the first place, a
deve*io]> e connection of all natural events with mechanical pro
mechanical cesses is unmistakable, and it is natural, therefore, that
view  we should be. led to explain less known phenomena by
better known mechanical events. Then again, it was
first in the department of mechanics that laws of gen
eral and extensive scope were discovered. A law of
ITS RELATIONS TO OTHER SCIENCES. 499
this kind is the principle of vis viva 2 (17^ 7 ) =
2%m(v\ ?'), which states that the increase of the
vis viva of a system in its passage from one position to
another is equal to the increment of the forcefunction,
or work, which is expressed as a function of the final
and initial positions. If we fix our attention on the
work a system can perform and call it with Helmholtz
the Spannkraft, *$*,* then the work actually performed,
U, will appear as a diminution of the Spannkraft^ K,
initially present; accordingly, S=K 7, and the
principle of vis viva takes the form
^ __ iy 2 /// 1] 2 const,
that is to say. every diminution of the Spannkraft, is The con
J J 2 y servation of
compensated for by an increase of the vis viva. In this Energy,
form the principle is also called the law of the Conser
vation of Energy ', in that the sum of the Spannkraft (the
potential energy) and the vis viva (the kinetic energy)
remains constant in the system. But since, in nature,
it is possible that not only vis viva should appear as the
consequence of work performed, but also quantities of
heat, or the potential of an electric charge, and so forth,
scientists saw in this law the expression of a mechanical
action as the basis of all natural actions. However,
nothing is contained in the expression but the fact of
an invariable quantitative connection between mechani
cal and other kinds of phenomena.
4. It would be a mistake to suppose that a wide
and extensive view of things was first introduced into
physical science by mechanics. On the contrary, this
* Helmhohz tisnd this tarni in 1847; but it is not found in his Hubsoqwnt
papers; and in 1882 (Wisstnschaftliche Abhandlttngen, II, 965) he expressly
discards it in favor of the English "potential energy," Ho even (p. <}6B) pre
fers Clausius'K word Ergal to Spannkr&ft, which is quite out of agreement
with modern terminology. Trans,
500 THE SCIENCE OF MECHANICS.
Compre insight was possessed at all times by the foremost
ness of inquirers and even entered into the construction of
view the ,..,.. 11 r
condition, mechanics itself, and was, accordingly, not first created
suit, of me by the latter. Galileo and Huygens constantly alter
nated the consideration of particular details with the
consideration of universal aspects, and reached their
results only by a persistent effort after a simple and
consistent view. The fact that the velocities of indi
vidual bodies and systems are dependent on the spaces
descended through, was perceived by Galileo and
Huygens only by a very detailed investigation of the
motion of descent in particular cases, combined with
the consideration of the circumstance that bodies gen
erally, of their own accord, only sink. Huygens
especially speaks, on the occasion of this inquiry, of
the impossibility of a mechanical perpetual motion ;
he possessed, therefore, the modern point of view. He
felt the incompatibility of the idea of a perpetual motion
with the notions of the natural mechanical processes
with which he was familiar.
Exempiifi Take the fictions of Stevinus say, that of the end
this in ste less chain on the prism. Here, too, a deep, broad
searches!" insight is displayed. We have here a mind, disciplined
by a multitude of experiences, brought to bear on an
individual case. The moving endless chain is to Ste
vinus a motion of descent that Is not a descent, a mo
tion without a purpose, an intentional act that does
not answer to the intention, an endeavor for a change
which does not produce the change. If motion, gener
ally, is the result of descent, then in the particular case
descent is the result of motion. It is a sense of the
mutual Interdependence of v and h in the equation
v = ]/2g/i that is here displayed, though of course in
not so definite a form. A contradiction exists in this
ITS RELATIONS TO OTHER SCIENCES. 501
fiction for Stevinus's exquisite investigative sense that
would escape less profound thinkers.
This same breadth of view, which alternates the Also, in the
researches
individual with the universal, is also displayed, only in of QU not
this instance not restricted to mechanics, in the per Mayer.
formances of Sadi Carnot. When Carnot finds that
the quantity of heat Q which, for a given amount of
work Z, has flowed from a higher temperature / to a
lower temperature /, can only depend on the tempera
tures and not on the material constitution of the bodies,
he reasons in exact conformity with the method of
Galileo. Similarly does J. R. Mayer proceed in the
enunciation of the principle of the equivalence of heat
and work. In this achievement the mechanical view
was quite remote from Mayer's mind ; nor had he need
of it. They who require the crutch of the mechanical
philosophy to understand the doctrine of the equiva
lence of heat and work, have only half comprehended
the progress which it signalises. Yet, high as we may
place Mayer's original achievement, it is not on that
account necessary to depreciate the merits of the pro
fessional physicists Joule, Helmholtz, Clausius, and
Thomson, who have done very much, perhaps all, to
wards the detailed estaltlishmcnt and fcrfcctlon of the
new view. The assumption of a plagiarism of Mayer's
ideas is in our opinion gratuitous. They who advance
it, are under the obligation to fnwe it. The repeated
appearance of the same idea is not new in history. We
shall not take up here the discussion of purely personal
questions, which thirty years from now will no longer
Interest students. But It is unfair, from a pretense of
justice, to insult men, who If they had accomplished
but a third of their actual services, would have lived
highly honored and unmolested lives, (Seep. 584.)
5 o2 THE SCIENCE OF MECHANICS.
The inter 5. We shall now attempt to show that the broad
enceof the view expressed in the principle of the conservation
ture. of energy, is not peculiar to mechanics, but is a condi
tion of logical and sound scientific thought generally.
The business of physical science is the reconstruction
of facts in thought, or the abstract quantitative expres
sion of facts. The rules which we form for these recon
structions are the laws of nature. In the conviction that
such rules are possible lies the law of causality. The
law of causality simply asserts that the pheno'mena of
nature are dependent on one another. The special em
phasis put on space and time in the expression of the
law of causality is unnecessary, since the relations of
space and time themselves implicitly express that phe
nomena are dependent on one another.
The laws of nature are equations between the meas
urable elements a ft yd . . . . coof phenomena. As na
ture is variable, the number of these equations is al
ways less than the number of the elements.
If we know all the values of a /3yd . . ., by which,
for example, the values of A JJLV . . . are given, we may
call the group afiyS . . . the cause and the group
"kjjLv . ... the effect. In this sense we may say that the
effect is uniquely determined by the cause. The prin
ciple of sufficient reason, in the form, for instance, in
which Archimedes employed it in the development of
the laws of the lever, consequently asserts nothing
more than that the effect cannot by any given set of
circumstances be at once determined and undetermined.
If two circumstances a and A are connected, then,
supposing all others are constant, a change of A will
be accompanied by a change of a, and as a general
rule a change of a by a change of A. The constant
observance of this mutual interdependence is met with
RELATIONS TO OTHER SCIE.\ 7 CES. 503
in Stevinus, Galileo, Huygens, 'and other great inquir j
ers. The idea is also at the basis of the discovery of
^/////^/'phenomena. Thus, a change in the volume of basis of ail
" ' tfroaf dis
a gas due to a change of temperature is supplemented cm 
by the counterphenomenon of a change of tempera
ture on an alteration of volume ; Seebeck's phenome
non by Peltier's effect, and so forth.
Care must, of course, be exercised, in
such inversions, respecting the form
of the dependence. Figure 235 will
render clear how a perceptible altera
tion of a may always be produced by r
an alteration of A, but a change of A
not necessarily by a change of a. The relations be
tween electromagnetic and induction phenomena, dis
covered by Faraday, are a good instance of this truth.
If a set of circumstances aftyfi. . ., by which a various
' ' ' J iormsof <x
second set \iiv . . . is determined, be made to pass i; 1 "}^ " ot
r J * f this truth.
from its initial values to the terminal values oe' fi'y
d' . . ., then TLJJ.V . . . also will pass into A'/fV. . .
If the first set be brought back to its initial state, also
the second set will be brought back to its initial state.
This is the meaning of the "equivalence of cause and
effect/' which Mayer again and again emphasizes.
If the first group suffer only periodical changes, the
second group also can suffer only periodical changes,
not continuous permanent ones. The fertile methods
of thought of Galileo, Huygens, S. Carnot, Mayer,
and their peers ? are all reducible to .the simple but sig
nificant perception, that purely periodical ultfratwns <>f
one set of circumstances can only constitute the source of
similarly periodical alterations of a second se/ of ci re it in
stances, not of continuous and permanent alterations. Such
maxims, as "the effect is equivalent to the cause/ 1
5 o 4 THE SCIENCE OF MECHANICS.
"work cannot be created out of nothing," "a. per
petual motion is impossible/' are particular, less defi
nite, and less evident forms of this perception, which
in itself is not especially concerned with mechanics, but
is a constituent of scientific thought generally. With
the perception of this truth, any metaphysical mystic
ism that may still adhere to the principle of the con
servation of energy* is dissipated. (See p. 585.)
Purpose of All ideas of conservation, like the notion of sub
the ideas of
conserva stance, have a solid foundation in the economy of
thought. A mere unrelated change, without fixed point
of support, or reference, is not comprehensible, not
mentally reconstructible. We always inquire, accord
ingly, what idea can be retained amid all variations as
permanent, what law prevails, what equation remains
fulfilled, what quantitative values remain constant ?
When we say the refractive index remains constant in
all cases of refraction, ^remains = 9810;^ in all cases
of the motion of heavy bodies, the energy remains con
stant in every isolated system, all our assertions have
one and the same economical function, namely that of
facilitating our mental reconstruction of facts.
THE RELATIONS OF MECHANICS TO PHYSIOLOGY.
Conditions i All science has its origin in the needs of life.
dlvefo 6 However minutely it may be subdivided by particular
science! vocations or by the restricted tempers and capacities of
those who foster it, each branch can attain its full and
best development only by a living connection with the
whole. Through such a union alone can it approach
* When we reflect that the principles of science are all abstractions that
presuppose repetitions of similar cases, the absurd applications of the law of
the conservation of forces to the universe as a whole fall to the ground.
ITS RELATIONS TO OTHER SCIENCES. 505
its true maturity, and be insured against lopsided and
monstrous growths.
The division of labor, the restriction of individual Confusion
. ot the
inquirers to limited provinces, the investigation of moans and
those provinces as a lifework, are the fundamental science,
conditions of a fruitful development of science. Only
by such specialisation and restriction of work can the
economical instruments of thought requisite for the
mastery of a special field be perfected. But just here
lies a danger the danger of our overestimating the in
struments, with which we are so constantly employed,
or even of regarding them as the objective point of
science.
2. Now, such a state of affairs has, in our opinion, physics
actually been produced by the disproportionate formal mmufthe
development of physics. The majority of natural in physiology.
quirers ascribe to the intellectual implements of physics,
to the concepts mass, force, atom, and so forth, whose
sole office is to revive economically arranged expe
riences, a reality beyond and independent of thought.
Not only so, but it has even been held that fliese forces
and masses are the real objects of inquiry, and, if once
they were fully explored, all the rest would follow from
the equilibrium and motion of these masses. A person
who knew the world only through the theatre, if brought
behind the scenes and permitted to view the mechan
ism of the stage's action, might possibly believe that
the real world also was in need of a machineroom, and
that if this were once thoroughly explored, we should
know all. Similarly, we, too, should beware lest the
intellectual machinery, employed in the representation
of the world on the stage of thought, be regarded as the
basis of the real world.
3. A philosophy is involved in any correct view of
5 o6 THE SCIENCE OF MECHANICS.
The at the relations of special knowledge to the great body of
pT"mVeef x " knowledge at large, a philosophy that must be de
motions, manded of every special investigator. The lack of it
is asserted in the formulation of imaginary problems,
in the very enunciation of which, whether regarded as
soluble or insoluble, flagrant absurdity is involved.
Such an overestimation of physics, in contrast to physi
ology, such a mistaken conception of the true relations
of the two sciences, is displayed in the inquiry whether
it is possible to explain feelings by the motions of
atoms?
Explication Let us seek the conditions that could have impelled
of this . .
anomaly, the mind to formulate so curious a question. We find
in the first place that greater confidence is placed in our
experiences concerning relations of time and space ;
that we attribute to them a more objective, a moremz/
character than to our experiences of colors, sounds,
temperatures, and so forth. Yet, if we investigate the
matter accurately, we must surely admit that our sen
sations of time and space are just as much sensations
as are our sensations of colors, sounds, and odors, only
that in our knowledge of the former we are surer and
clearer than in that of the latter. Space and time are
wellordered systems of sets of sensations. The quan
tities stated in mechanical equations are simply ordinal
symbols, representing those members of these sets
that are to be mentally isolated and emphasised. The
equations express the form of interdependence of these
ordinal symbols.
A body is a relatively constant sum of touch and
sight sensations associated with the same space and
time sensations. Mechanical principles, like that, for
instance, of the mutually induced accelerations of two
masses, give, either directly or indirectly, only some
ITS RELATIONS TO OTHER SCIENCES. 507
combination of touch, sight, light, and time sensations.
They possess intelligible meaning only by virtue of
the sensations they involve, the contents of which may
of course be very complicated.
It would be equivalent, accordingly, to explaining MorfcMjf
the more simple and immediate by the more compli sncher
r J . rors.
cated and remote, if we were to attempt to derive sen
sations from the motions of masses, wholly aside from
the consideration that the notions of mechanics are
economical implements or expedients perfected to
represent mechanical and not physiological or psycho
logical facts. If the means and aims of research were
properly distinguished, and our expositions were re
stricted to the presentation of actual facts, false prob
lems of this kind could not arise.
4. All physical knowledge can only mentally repre The prmci
sent and anticipate compounds of those elements we dumjW mt
call sensations. It is concerned with the connection of ti/m but
these elements. Such an element, say the heat of a body aspect o"
A, is connected, not only with other elements, say with
such whose aggregate makes up the flame J3, but also
with the aggregate of certain elements of our body, say
with the aggregate of the elements of a nerve J\T. As
simple object and element JVis not essentially, but only
conventionally, different from A and B, The connection
of A and B is a problem of physics^ that of A and N a
problem of physiology. Neither is alone existent; both
exist at once. Only provisionally can we neglect
either. Processes, thus, that in appearance arc purely
mechanical, are, in addition to their evident mechani
cal features, always physiological, and, consequently,
also electrical, chemical, and so forth. The science of
mechanics does not comprise the foundations, no, nor
even a part of the world, but only an aspect of it.
APPENDIX.
I.
(See page 3.)
Recent research has contributed greatly to our
knowledge of the scientific literature of antiquity, and
our opinion of the achievements of the ancient world
in science has been correspondingly increased. Schia
parelli has done much to place the work of the
Greeks in astronomy in its right light, and Govi has
disclosed many precious treasures in his edition of
the Optics of Ptolemy. The view that the Greeks
were especially neglectful of experiment can no longer
be maintained unqualifiedly. The most ancient ex
periments are doubtless those of the Pythagoreans,
who employed a monochord with moveable bridge for
determining the lengths of strings emitting harmonic
notes. Anaxagoras's demonstration of the corporeal
ity of the air by means of closed inflated tubes, and
that of Empedocles by means of a vessel having its
orifice inverted in water (Aristotle, Physics} are both
primitive experiments. Ptolemy instituted systematic
experiments on the refraction of light, while his ob
servations in physiological optics are still full of in
terest today. Aristotle {Meteorology) describes phe
nomena that go to explain the rainbow. The absurd
stories which tend to arouse our mistrust, like that of
Pythagoras and the anvil which emitted harmonic
5 io THE SCIENCE OF MECHANICS.
notes when struck by hammers of different weights,
probably sprang from the fanciful brains of ignorant
reporters. Pliny abounds in such vagaries. But they
are not, as a matter of fact, a whit more incorrect or
nonsensical than the stories of Newton's falling apple
and of Watts's teakettle. The situation is, more
over, rendered quite intelligible when we consider the
difficulties and the expense attending the production
of ancient books and their consequent limited circula
tion. The conditions here involved are concisely dis
cussed by J. Mueller in his paper, "Ueber das Ex
periment in den physikalischen Studien der Grie
chen," Naturwiss. Verein zu Innsbruck, XXIII. , 1896
1897.
11.
(See page 8 J
Researches in mechanics were not begun by the
Greeks until a late date, and in no wise keep pace
with the rapid advancement of the race in the domain
of mathematics, and notably in geometry. Reports
of mechanical inventions, so far as they relate to the
early inquirers, are extremely meager. Archytas, a
distinguished citizen of Tarentum {circa 400 B. C.),
famed as a geometer and for his employment with the
problem of the duplication of the cube, devised me
chanical instruments for the description of various
curves. As an astronomer he taught that the earth
was spherical and that it rotated upon its axis once a
day. As a mechanician he founded the theory of pul
leys. He is also said to have applied geometry to
mechanics in a treatise on this latter science, but all
information as to details is lacking. We are told,
though, by Aulus Gellius (X. 12) that Archytas con
APPENDIX, 51 1
structed an automaton consisting of a flying dove of
wood and presumably operated by compressed air,
which created a great sensation. It is, in fact, char
acteristic of the early history of mechanics that atten
tion should have been first directed to its practical
advantages and to the construction of automata de
signed to excite wonder in ignorant people.
Even in the days of Ctesibius (285247 B. C.) and
Hero (first century A. D.) the situation had not ma
terially changed. So, too, during the decadence of
civilisation in the Middle Ages, the same tendency as
serts itself. The artificial automata and clocks of this
period, the construction of which popular fancy as
cribed to the machinations of the Devil, are well
known. It was hoped, by imitating life outwardl}', to
apprehend it from its inward side also. In intimate
connexion with the resultant 'misconception of life
stands also the singular belief in the possibility of a
perpetual motion. Only gradually and slowly, and in
indistinct forms, did the genuine problems of mechan
ics loom up before the minds of inquirers. Aristotle/s
tract, Mechanical Problems (German trans, by Poselger,
Hannover, 1881) is characteristic in this regard. Aris
totle is quite adept in detecting and in formulating
problems ; he perceived the principle of the parallel
ogram of motions, and was on the verge of discover
ing centrifugal force; but in the actual solution of
problems he was infelicitous. The entire tract par
takes more of the character of a dialectic than of a
scientific treatise, and rests content with enunciating
the "apories," or contradictions, involved in the prob
lems. But the tract upon the whole very well illus
trates the intellectual situation that is characteristic
of the beginnings of scientific investigation.
5 i2 THE SCIENCE OF MECHANICS.
"If a thing take place whereof the cause be not
apparent, even though it be in accordance with na
ture, it appears wonderful. . . . Such are the instances
in which small things overcome great things, small
weights heavy weights, and incidentally all the prob
lems that go by the name of 'mechanical.' . . . To
the apories (contradictions) of this character belong
those that appertain to the lever. For it appears con
trary to reason that a large weight should be set in
motion by a small force, particularly when that weight
is in addition combined with a larger weight. A weight
that cannot be moved without the aid of a lever can be
moved easily with that of a lever added. The pri
mordial cause of all this is inherent in the nature of
the circle, which is as one should naturally expect :
for if is not contrary to reason that something won
derful should proceed out 'of something else that is
wonderful. The combination of contradictory prop
erties, however, into a single unitary product is the
most wonderful of all things. Now, the circle is ac
tually composed of just such contradictory properties.
For it is generated by a thing that is in motion and
by a thing that is stationary at a fixed point. "
In a subsequent passage of the same treatise there
is a very dim presentiment of the principle of virtual
velocities.
Considerations of the kind here adduced give evi
dence of a capacity for detecting and enunciating prob
lems, but are far from conducting the investigator to
their solution.
in.
(See page 14.)
It may be remarked in further substantiation of
the criticisms advanced at pages 1314, that it is very
APPENDIX. 513
obvious that if the arrangement is absolutely sym
metrical in every respect, equilibrium obtains on the
assumption of any form of dependence whatever of
the disturbing factor on .Z, or, generally, on the as
sumption P.f{L}\ and that consequently t\ie. particular
form of dependence PL cannot possibly be inferred
from the equilibrium. The fallacy of the deduction
must accordingly be sought in the transformation to
which the arrangement is subjected. Archimedes
makes the action of two equal weights to be the same
under all circumstances as that of the combined
weights acting at the middle point of their line of
junction. But, seeing that he both knows and as
sumes that distance from the fulcrum is determina
tive, this procedure is by the premises impermissible,,
if the two weights are situated at unequal distances
from the fulcrum. If a weight situated at a distance
from the fulcrum is divided into two equal parts, and
these parts are moved in contrary directions symmet
rically to their original point of support ; one of the
equal weights will be carried as near to the fulcrum
as the other weight is carried from it. If it is assumed
that the action remains constant during such proce
dure, then, the particular form of dependence of the
moment on L is implicitly determined by what has
been done, inasmuch as the result is only possible
provided the form be PL, or be proportional 'to L* But
in such an event all further deduction is superfluous.
The entire deduction contains the proposition to be
demonstrated, by assumption if not explicitly.
5 r 4 THE SCIENCE OF MECHANICS.
IV.
(See page 20.)
Experiments are never absolutely exact, but they
at least may lead the inquiring mind to conjecture that
the key which will clear up the connexion of all the
facts is contained in the exact metrical expression
PL. On no other hypothesis are the deductions of
Archimedes, Galileo, and the rest Intelligible. The
required transformations, extensions, and compres
sions of the prisms may now be carried out with per
fect certainty.
A knife edge may be introduced at any point un
der a prism suspended from its center without dis
turbing the equilib
rium (see Fig 236),
and several such ar
rangements may be
rigidly combined to
1 zsg gether so as to form
Fig. 236. apparently new cases
of equilibrium. The
conversion and disintegration of the case of equi
librium into several other cases (Galileo) is possible
only by taking into account the value of PL. I can
not agree with O. Holder who upholds the correct
ness of the Archimedean deductions against my criti
cisms in his essay Denken und Anschauung in dcr Gco
metrie, although I am greatly pleased with the extent
of our agreement as to the nature of the exact sci
ences and their foundations. It would seem as if
Archimedes (JDe cequiponderantibus) regarded it as a
general experience that two equal weights may under
all circumstances be replaced by one equal to their
APPENDIX. 5*5
combined weight at the center (Theorem 5, Corrol
ary 2). In such an event, his long deduction (Theo
rem 6) would be necessary, for the reason sought fol
lows immediately (see pp. 14, 513). Archimedes's
mode of expression is not in favor of this view.
Nevertheless, a theorem of this kind cannot be re
garded as a priori evident; and the views advanced
on pp. 14, 513 appear to me to be still uncontro
verted.
(See page 29.)
Stevinus's procedure may be looked at from still
another point of view. If it is a fact, for our mechan
ical instinct, that a heavy endless chain will not ro
tate, then the individual simple cases of equilibrium
on an inclined plane which Stevinus devised and
which are readily controlled quantitatively, may be
regarded as so many special experiences. For It Is
not essential that the experiments should have been
actually carried out, if the result is beyond question
of doubt. As a matter of fact, Stevinus experiments
in thought. Stevinus's result could actually have
been deduced from the corresponding physical exper
iments, with friction reduced to a minimum. In an
analogous manner, Archimedes's considerations with
respect to the lever might be conceived after the
fashion of Galileo's procedure. If the various mental
experiments had been executed physically, the linear
dependence of the static moment on the distance of
the weight from the axis could be deduced with per
fect rigor. We shall have still many instances to acl
duce, among the foremost Inquirers in the domain of
mechanics, of this tentative adaptation of special
5 i6 THE SCIENCE OF MECHANICS.
quantitative conceptions to general instinctive im
pressions. The same phenomena are presented in
other domains also. I may be permitted to refer in
this connexion to the expositions which I have given
in my Principles of Heat, page 151. It may be said
that the most significant and most important advances
in science have been made in this manner. The habit
which great inquirers have of bringing their single
conceptions into agreement with the general concep
tion or ideal of an entire province of phenomena., their
constant consideration of the whole in their treatment
of parts, may be characterised as a genuinely philo
sophical procedure. A truly philosophical treatment
of any special science will always consist in bringing
the results into relationship and harmony with the
established knowledge of the whole. The fanciful
extravagances of philosophy, as well as infelicitous
and abortive special theories, will be eliminated in
this manner.
It will be worth while to review again the points
of agreement and difference in the mental procedures
of Stevinus and Archimedes. Stevinus reached the
very general view that a mobile, heavy, endless chain
of any form stays at rest. He is able to deduce from
this general view, without difficulty, special cases,
which are quantitatively easily controlled. The case
from which Archimedes starts, on the other hand, is
the most special conceivable. He cannot possibly
deduce from his special case in an unassailable man
ner the behavior which may be expected under more
general conditions. If he apparently succeeds in so
doing, the reason is that he already knows the result
which he is seeking, whilst Stevinus, although he too
doubtless knows, approximately at least, what he is
APPENDIX. 517
In search of, nevertheless could have found it directly
by his manner of procedure, even if he had not known
it. When the static relationship is rediscovered in
such a manner it has a higher value than the result of
a metrical experiment would have, which always de
viates somewhat from the theoretical truth. The de
viation increases with the disturbing circumstances,
as with friction, and decreases with the diminution of
these difficulties. The exact static relationship is
reached by idealisation and disregard of these dis
turbing elements. It appears in the Archimedean and
Stevinian procedures as an hypothesis without which
the individual facts of experience would at once be
come involved in logical contradictions. Not until
we have possessed this hypothesis can we by operat
ing with the exact concepts reconstruct the facts and
acquire a scientific and logical mastery of them. The
lever and the inclined plane are selfcreated ideal ob
jects of mechanics. These objects alone completely
satisfy the logical demands which we make of them ;
the physical lever satisfies these conditions only in
measure in which it approaches the ideal lever. The
natural inquirer strives to adapt his ideals to reality,
VI.
(See page no }
Our modern notions with regard to the nature of
air are a direct continuation of the ancient ideas. An
axagoras proves the corporeality of air from its resist
ance to compression in closed bags of skin, and from
the gathering up of the expelled air (in the form of
bubbles?) by water (Aristotle, Phystcs, IV., 9). Ac
cording to Empedoclcs, the air prevents the water
5 i8 THE SCIENCE OF MECHANICS.
from penetrating into the interior of a vessel immersed
with its aperture downwards (Gomperz, Griechische
Denker, L, p. 191). Philo of Byzantium employs for
the same purpose an inverted vessel having in its bot
tom an orifice closed with wax. The water will not
penetrate into the submerged vessel until the wax
cork is removed, wherupon the air escapes in bubbles.
An entire series of experiments of this kind is per
formed, in almost the precise form customary in the
schools today (Philonis lib. de ingeniis spiritualibus, in
V. Rose's Anecdota gr&ca et latino^. Hero describes
in his Pneumatics many of the experiments of his
predecessors, with additions of his own ; in theory he
is an adherent of Strato, who occupied an intermedi
ate position between Aristotle and Democritus. An
absolute and continuous vacuum, he says, can be
produced only artificially, although numberless tiny
vacua exist between the particles of bodies, including
air, just as air does among grains of sand. This is
proved*, in quite the same ingenuous fashion as in our
present elementary books, from the possibility of rare
fying and compressing bodies, including air (inrush
ing and outrushing of the air in Hero's ball). An ar
gument of Hero's for the existence of vacua (pores)
between corporeal particles rests on the fact that rays
of light penetrate water. The result of artificial^ in
creasing a vacuum, according to Hero and his prede
cessors, is always the attraction and solicitation of
adjacent bodies. A light vessel with a narrow aper
ture remains hanging to the lips after the air has been
exhausted. The orifice may be closed with the finger
and the vessel submerged in water. "If the finger
be released, the water will rise in the vacuum created,
although the movement of the liquid upward is not
APPENDIX. 519
according to nature. The phenomenon of the cup
pingglass is the same ; these glasses, when placed
on the body, not only do not fall off, although they
are heavy enough, but they also draw out adjacent
particles through the pores of the body.' ' The bent
siphon is also treated at length. "The filling of the
siphon on exhaustion of the air is accomplished by
the liquid's closely following the exhausted air, for
the reason that a continuous vacuum is inconceiv
able." If the two arms of the siphon are of the same
length, nothing flows out. "The water is held in
equilibrium as in a balance." Hero accordingly ccn
ceives of the flow of water as analogous to the move
ment of a chain hanging with unequal lengths over a
pulley. The union of the two columns, which for us
is preserved by the pressure of the atmosphere, is
cared for in his case by the ** inconceivability of a
continuous vacuum." It is shown at length, not that
the smaller mass of water is attracted and drawn
along by the greater mass, and that conformably to
this principle water cannot flow upwards, but rather
that the phenomenon is in harmony with the principle
of communicating vessels. The many pretty and in
genious tricks which Hero describes in his Pneumatics
and in his Automata, and which were designed partly
to entertain and partly to excite wonder, offer a
charming picture of the material civilisation of the clay
rather than excite our scientific interest. The auto
matic sounding of trumpets and the opening of tem
ple doors, with the thunder simultaneously produced,
are not matters which interest science properly so
called. Yet Hero's writings and notions contributed
much toward the diffusion of physical knowledge
(compare W. Schmidt, If<n?$ Wtrkt, Leipslc, 1899,
5 20 THE SCIENCE OF MECHANICS.
and Diels, System des Strato, Sitzungsberichte der J3er~
liner Akademie, 1893).
VII,
(See page 129.)
It has often been asserted that Galileo had prede
cessors of great prominence in his method of think
ing, and while it is far from our purpose to gainsay
this, we have still to emphasise the fact that Galileo
overtowered them all. The greatest predecessor of
Galileo, to whom we have already referred in another
place, was Leonardo da Vinci, 14521519; now, it
was impossible for Leonardo's achievements to have
influenced the development of science at the time, for
the reason that they were not made known in their
entirety until the publication of Venturi in 1797. Leo
nardo knew the ratio of the times of descent down
the slope and the height of an inclined plane. Fre
quently also a knowledge of the law of inertia is at
tributed to him. Indeed, some sort of instinctive
knowledge of the persistence of motion once begun
will not be gainsaid to any normal man. But Leo
nardo seems to have gone much farther than this.
He knows that from a column of checkers one of the
pieces may be knocked out without disturbing the
others; he knows that a body in motion will move
longer according as the resistance is less, but he be,
lieves that the body will move a distance proportional
to the impulse, and nowhere expressly speaks of the
persistence of the motion when the resistance is alto
gether removed. (Compare Wohlwill, Bibliotkeca Ma
thematica, Stockholm, 1888, p. 19). Benedetti (1530
1590) knows that falling bodies are accelerated, and
explains the acceleration as due to the summation
APPENDIX. 521
of the impulses of gravity (Divers, speculat. math, et
physic, liber, Taurini, 1585). He ascribes the progres
sive motion of a projectile, not as the Peripatetics
did, to the agency of the medium, but to the virtus
imflressa, though without attaining perfect clearness
with regard to these problems. Galileo seems actu
ally to have proceeded from Benedetti's point of view,
for his youthful productions are allied to those of
Benedetti. Galileo also assumes a virtus zmfiressa,
which he conceives to decrease in efficiency, and ac
cording to Wohlwill it appears that it was not until
1604 that he came into full possession of the laws of
falling bodies.
G. Vailati, who has devoted much attention to Be
nedetti's investigations (Atti della 7?. Acad* di Torino,
Vol. XXXIII., 1898), finds the chief merit of Bene
detti to be that he subjected the Aristotelian views to
mathematical and critical scrutiny and correction, and
endeavored to lay bare their inherent contradictions,
thus preparing the way for further progress. He
knows that the assumption of the Aristotelians, that
the velocity of falling bodies is inversely proportional
to the density of the surrounding medium, is un
tenable and possible only in special cases. Let the
velocity of descent be proportional to f t?, where/
is the weight of the body and q the upward impulsion
due to the medium. If only half the velocity of de
scent is set up in a medium of double the density,
the equation/ ^ = 2 (/ 2^) must exist, a relation
which is possible only in case j* = 3#. Light bodies
per se do not exist for Benedetti; he ascribes weight
and upward impulsion even to air, Differentsixed
bodies of the same material fall, in his opinion, with
the same velocity. Benedetti reaches this result by
522 THE SCIENCE OF MECHANICS
conceiving equal bodies falling alongside each other
first disconnected and then connected, where the con
nexion cannot alter the motion. In this he approaches
to the conception of Galileo, with the exception that
the latter takes a profotmder view of the matter.
Nevertheless, Benedetti also falls into many errors;
he believes, for example, that the velocity of descent
of bodies of the same size and of the same shape is
proportional to their weight, that is, to their density.
His reflexions on catapults, no less than his views on
the' oscillation of a body about the center of the earth
in a canal bored through the earth, are interesting,
and contain little to be criticised. Bodies projected
horizontally appear to approach the earth more slowly,
Benedetti is accordingly of the opinion that the force
of gravity is diminished also in the case of a top rotat
ing with its axis in a vertical position. He thus does
not solve the riddle fully, but prepares the way for
the solution.
vin.
(See page 134.)
If we are to understand Galileo's train of thought,
we must bear in mind that he was already in posses
sion of instinctive experiences prior to his resorting
to experiment.
Freely falling bodies are followed with more diffi
culty by the eye the longer and the farther they have
fallen ; their impact on the hand receiving them is in
like measure sharper ; the sound of their striking
louder. The velocity accordingly increases with the
time elapsed and the space traversed. But for scien
tific purposes our mental representations of the facts
of sensual experience must be submitted to conceptual
APPENDIX. 523
formulation. Only thus may they be used for discov
ering by abstract mathematical rules unknown prop
erties conceived to be dependent on certain initial
properties having definite and assignable arithmetic
values; or, for completing what has been only parti}'
given. This formulation is effected by Isolating and
emphasising what is deemed of Importance, by neg
lecting what Is subsidiary, by abstracting, by idealis
ing. The experiment determines whether the form
chosen is adequate to the facts. Without some pre
conceived opinion the experiment is Impossible, be
cause its form is determined by the opinion. For
how and on what could we experiment if we did not
previously have some suspicion of what we were
about ? The complemental function which the experi
ment Is to fulfil is determined entirely by our prior
experience. The experiment confirms, modifies, or
overthrows our suspicion. The modern inquirer would
ask In a similar predicament : Of what Is v a function?
What function of / Is ?>? Galileo asks, In his Ingenu
ous and primitive way : Is v proportional to s f Is y
proportional to /? Galileo, thus, gropes his way along
synthetically, but reaches his goal nevertheless. Sys
tematic, routine methods are the final outcome of re
search, and do not stand perfectly developed at the
disposal of genius in the first steps it takes, (Com
pare the article "Ueber Gedankenexperlmente," Zett
schrift fur denfhys. und chew. Unterricht, 1897, I.)
ix.
(See page 140.)
In an exhaustive study m the Ztitschrift ffir Vol&cr
psychologic, 1884, Vol. XIV., pp. 365410, and Vol.
5 2 4 THE SCIENCE OF MECHANICS.
XV., pp. 70135, 337387, entitled " Die Entdeckung
des Beharrungsgesetzes," E. Wohlwill has shown that
the predecessors and contemporaries of Galileo, nay,
even Galileo himself, only very gradually abandoned
the Aristotelian conceptions for the acceptance of the
law of inertia. Even in Galileo's mind uniform cir
cular motion and uniform horizontal motion occupy
distinct places. Wohlwill's researches are very ac
ceptable and show that Galileo had not attained per
fect clearness in his own new ideas and was liable to
frequent reversion to the old views, as might have
been expected.
Indeed, from my own exposition the reader will
have inferred that the law of inertia did not possess
in Galileo's mind the degree of clearness and univer
sality that it subsequently acquired. (See pp. 140 and
143.) With regard to my exposition at pages 140
141, however, I still believe, in spite of the opinions
of Wohlwill and Poske, that I have indicated the
point which both for Galileo and his successors must
have placed in the most favorable light the transition
from the old conception to the new. How much was
wanting to absolute comprehension, may be gathered
from the fact that Baliani was able without difficulty
to infer from Galileo's statement that acquired velo
city could not be destroyed, a fact which Wohlwill
himself points out (p. 112). It is not at all surpris
ing that in treating of the motion of heavy bodies,
Galileo applies his law of inertia almost exclusively
to horizontal movements. Yet he knows that a mus
ketball possessing no weight would continue rectiline
arly on its path in the direction of the barrel, {Dia
logues on the two World Systems, German translation,
Leipsic, 1891, p. 184.) His hesitation in enunciating
APPENDfX. 525
in Its most general terms a law that at first blush ap
pears so startling, is not surprising.
x.
(See page 155.)
We cannot adequately appreciate the extent of
Galileo's achievement in the analysis of the motion of
projectiles until we examine his predecessors' endeav
ors in this field. Santbach (1561) is of opinion, that a
cannonball speeds onward in a straight line until Its
velocity is exhausted and then drops to the ground in
a vertical direction. Tartaglia (1537) compounds the
path of a projectile out of a straight line, the arc of a
circle, and lastly the vertical tangent to the arc. He
Is perfectly aware, as Rivius later (1582) more dis
tinctly states, that accurately viewed the path Is
curved at all points, since the deflective action of
gravity never ceases ; but he is yet unable to arrive
at a complete analysis. The Initial portion of the
path Is well calculated to arouse the Illusive impres
sion that the action of gravity has been annulled by
the velocity of the projection, an illusion to which
even Benedetti fell a victim. (See Appendix, vn,, p.
129.) We fail to observe any descent in the Initial
part of the curve, and forget to take into account the
shortness of the corresponding time of the descent.
By a similar oversight a jet of water may assume the
appearance of a solid body suspended in the air, if
one Is unmindful of the fact that It is made up of a
mass of rapidly alternating minute particles. The
same Illusion Is met with in the centrifugal pendulum,
In the top, in Aitken's flexible chain rendered rigid
by rapid rotation {Philosophical Maga&iue, 1878), in
the locomotive which rushes safely across a defective
5 26 THE SCIENCE OF MECHANICS.
bridge, through which it would have crashed if at
rest, but which, owing to the insufficient time of des
cent and of the period in which it can do work, leaves
the "bridge intact. On thorough analysis none of
these phenomena are more surprising than the most
ordiriar}' events. As Vailati remarks, the rapid spread
of firearms in the fourteenth century gave a distinct
impulse to the study of the motion of projectiles, and
indirectly to that of mechanics generally. Essentially
the same conditions occur in the case of {lie ancient
catapults and in the hurling of missiles by the hand,
but the new and imposing form of the phenomenon
doubtless exercised a great fascination on the curios
ity of people.
So much for history. And now a word as to the
notion of "composition." Galileo's conception of the
motion of a projectile as a process compounded of
two distinct and independent motions, is suggestive
of an entire group of similar important epi^temologi
cal processes. We may say that it is as Important to
perceive the non dependence of two circumstances A
and J3 on each other, as it is to perceive the dependence
of two circumstances A and C on each other. For
the first perception alone enables us to pursue the
second relation with composure. Think only of how
serious an obstacle the assumption of nonexisting
causal relations constituted to the research of the
Middle Ages. Similar to Galileo's discovery is that
of the parallelogram of forces by Newton, the compo
sition of the vibrations of strings by Sauveur, the com
position of thermal disturbances by Fourier. Through
this latter inquirer the method of compounding a phe
nomenon out of mutually independent partial phe
nomena by means of representing a general integral
APPENDIX. 527
as the sum of particular integrals has penetrated into
every nook and corner of physics. The decomposi
tion of phenomena into mutually independent parts
has been aptly characterised by P. Volkmann as iso
lation, and the composition of a phenomenon out of
such parts, superposition. The two processes combined
enable us to comprehend, or "reconstruct In thought,
piecemeal, what, as a whole, it would be impossible
for us to grasp.
"Nature with its myriad phenomena assumes a
unified aspect only in the rarest cases; in the major
ity of instances it exhibits a thoroughly composite
character . . . ; it is accordingly one of the duties of
science to conceive phenomena as made up of sets of
partial phenomena, and at first to study these partial
phenomena in their purity. Not until we know to
what extent each circumstance shares in the phenom
enon as an entirety do we acquire a command over
the whole. . . ." (Cf. P. Volkmann, Erkenntnisstheo
retische Grundziige der Naturwissenschaft, 1896, p. 70.
Cf. also my Principles of Heat, German edition, pp,
123, 151, 452).
XI.
(See page 161.)
The perspicuous deduction of the expression for
centrifugal force based on the principle of Hamilton's
hodograph may also be mentioned. If a body move
uniformly in a circle of radius r (Fig, 237), the velo
city 7/ at the point A of the path is transformed by
the traction of the string into the velocity v of like
magnitude but different direction at the point JB. If
from O as centre (Fig, 238) we lay off as to magni
tude and direction all the velocities the body succcs
528
THE SCIENCE OF MECHANICS.
sively acquires, these lines will represent the sum
of the radii v of the circle. For OM to be trans
formed into ON y the perpendicular component to it,
MJV, must be added. During the period of revolu
tion T the velocity is uniformly increased in the direc
tions of the radii r by an amount ZTTV. The numeri
M N
Fig. 237.
Fig. 238.
Fig. 23?.
cal measure of the radial acceleration Is therefore
cp , and since vT=2rtr, therefore also <p = .
If to OM=u the very small component w is added
(Fig. 239), the resultant will strictly be a greater
velocity
= #_ } as the approximate ex
traction of the square root will show. But on contin
uous deflection r vanishes with respect to v ; hence,
2v
only the direction, but not the magnitude, of the
velocity changes.
XII.
(See page 162.)
Even Descartes thought of explaining the centri
petal impulsion of floating bodies in a vortical me
dium, after this manner. But Huygens correctly re
APPENDIX. 529
marked that on this hypothesis we should have to
assume that the lightest bodies received the greatest
centripetal impulsion, and that all heavy bodies would
without exception have to be lighter than the vortical
medium. Huygens observes further that like phe
nomena are also necessarily presented in the case of
bodies, be they what they may, that do not participate
in the whirling movement, that is to say, such as
might exist without centrifugal force in a vortical
medium affected with centrifugal force. For exam
ple, a sphere composed of any material whatsoever
but moveable only along a stationary axis, say a wire,
is impelled toward the axis of rotation in a whirling
medium.
In a closed vessel containing water Huygens
placed small particles of sealing wax which are
slightly heavier than water and hence touch the bot
tom of the vessel. If the vessel be rotated, the par
ticles of sealing wax will flock toward the outer rim
of the vessel. If the vessel be then suddenly brought
to rest, the water will continue to rotate while the
particles of sealing wax which touch the bottom and
are therefore more rapidly arrested in their move
ment, will now be impelled toward the axis of the
vessel. In this process Huygens saw an exact replica
of gravity. An ether whirling in one direction only,
did not appear to fulfil his requirements. Ultimately,
he thought, it would sweep everything with it* He
accordingly assumed etherparticles that sped rapidly
about in all directions, it being his theory that In a
closed space, circular, as contrasted with radial, mo
tions would of themselves preponderate. This ether
appeared to him adequate to explain gravity. The
detailed exposition of this kinetic theory of gravity is
53 o THE SCIENCE OF MECHANICS.
found in Huygens's tract On the Cause of Gravitation
(German trans, by Mewes, Berlin, 1893). See also
Lasswitz, Gcschichte der Atomistik, 1890, Vol. II., p.
344
XITI.
(See page 187 )
It has been impossible for us to enter upon the
signal achievements of Huygens in physics proper.
But a few points may be briefly indicated. He is the
creator of the wavetheory of light, which ultimately
overthrew the emission theory of Newton. His at
tention was drawn, in fact, to precisely those features
of luminous phenomena that had escaped Newton.
With respect to physics he took up with great enthu
siasm the idea of Descartes that all things were to be
explained mechanically, though without being blind
to its errors, which he acutely and correctly criticised.
His predilection for mechanical explanations rendered
him also an opponent of Newton's action at a distance,
which he wished to replace by pressures and impacts,
that is, by action due to contact. In his endeavor to
do so he lighted upon some peculiar conceptions, like
that of magnetic currents, which at first could not
compete with the influential theory of Newton, but
has recently been reinstated in its full rights in the
unbiassed efforts of Faraday and Maxwell. As a
geometer and mathematician also Huygens is to be
ranked high, and in this connexion reference need be
made "only to his theory of games of chance. His
astronomical observations, his achievements in theo
retical and practical dioptrics advanced these depart
ments very considerably. As a technicist he is the
inventor of the powdermachine, the idea of which
APPENDIX. 53 1
has found actualisation in the modern gasmachine.
As a physiologist he surmised the accommodation of
the eye by deformation of the lens. All these things
can scarcely be mentioned here. Our opinion of Huy
gens grows as his labors are made better known by the
complete edition of his works. A brief and reveren
tial sketch of his scientific career in all its phases is
given by J. Bosscha in a pamphlet entitled Christian
Huyghens, Rede am 200. Gedachtnisstage seines Lehens
endes, German trans, by Engelmann, Leipsic, 1895.
XIV.
(See page 190.)
Rosenberger is correct in his statement (Newton
und seine physikalischen Principien, 1895) that the Idea
of universal gravitation did not originate with New
ton, but that Newton had many highly deserving pred
ecessors. But it may be safely asserted that it was,
with all of them, a question of conjecture, of a groping
and imperfect grasp of the problem, and that no one
before Newton grappled with the notion so compre
hensively and energetically; so that above and beyond
the great mathematical problem, which Rosenberger
concedes, there still remains to Newton the credit of
a colossal feat of the imagination.
Among Newton's forerunners may first be men
tioned Copernicus, who (in 1543) says: t l am at
least of opinion that gravity is nothing" more than a
natural tendency implanted in particles by the divine
providence of the Master of the Universe, by virtue of
which, they, collecting together in the shape of a
sphere, do form their own proper unity and integrity.
And it is to be assumed that this propensity is in*
herent also in the sun, the moon, and the other plan
53 2 THE SCIENCE OF MECHANICS.
ets." Similarly, Kepler (1609), like Gilbert before
him (1600), conceives of gravity as the analogue of
magnetic attraction. By this analogy, Hooke, it seems,
is led to the notion of a diminution of gravity with the
distance ; and in picturing its action as due to a kind
of radiation, he even hits upon the idea of its acting
inversely as the square of the distance. He even
sought to determine the diminution of its effect (1686)
by weighing bodies hung at different heights from the
top of Westminster Abbey (precisely after the more
modern method of Jolly), by means of spring balances
and pendulum clocks, but of course without results.
The conical pendulum appeared to him admirably
adapted for illustrating the motion of the planets. .
Thus Hooke really approached nearest to Newton's
conception, though he never completely reached the
latter's altitude of view.
In two instructive writings {Kepler'' $ Lehre von der
Gravitation, Halle, 1896: Die Gravitation bei Galileo
u. Horelli, Berlin, 1897) E. Gold beck investigates the
early history of the doctrine of gravitation with Kepler
on the one hand and Galileo and Borelli on the other.
Despite his adherence to scholastic, Aristotelian no
tions, Kepler has sufficient insight to see that there is
a real physical problem presented by the phenomena
of the planetary system; the moon, in his view, is
swept along with the earth in its motion round the
sun, and in its turn drags the tidal wave along with
it, just as the earth attracts heavy bodies. Also, for
the planets the source of motion is sought in the sun,
from which immaterial levers extend that rotate with
the sun and carry the distant planets around more
slowly than the near ones. By this view, Kepler was
enabled to guess that the period of rotation of the sun
APPENDIX. 533
was less than eightyeight days, the period of revolu
tion of Mercury. At times, the sun Is also conceived
as a revolving magnet, over against which are placed
the magnetic planets. In Galileo's conception of the
universe, the formal, mathematical, and esthetical
point of view predominates. He rejects each and
every assumption of attraction, and even scouted the
idea as childish in Kepler. The planetary system had
not yet taken the shape of a genuine physical problem
for him. Yet he assumed with Gilbert that an imma
terial geometric point can exercise no physical action,
and he did very much toward demonstrating the ter
restrial nature of the heavenly bodies. Borelli (in his'
work on the satellites of the Jupiter) conceives the
planets as floating between layers of ether of differing
densities. They have a natural tendency to approach
their central body, (the term attraction is avoided,)
' which is offset by the centrifugal force set up by the
revolution. Borelli illustrates his theory by an experi
ment very similar to that described by us in Fig. 106,
p. 162. As will be seen, he approaches very closely
to Newton. His theory is, though, a combination of
Descartes's and Newton's.
xv.
(Sec page 191.)
Newton illustrated the identity of terrestrial grav
ity with the universal gravitation that determined the
motions of the celestial bodies, as follows. He con
ceived a stone to be hurled with successive increases
of horizontal velocity from the top of a high moun
tain. , Neglecting the resistance of the air, the para
bolas successively described by the stone will increase
in length until finally they will fall clear of the earth
534 THE SCIENCE OF MECHANICS.
altogether, and the stone will be converted into a
satellite circling round the earth. Newton begins with
the fact of universal gravity. An explanation of the
phenomenon was not forthcoming, and it was not his
wont, he says, to frame hypotheses. Nevertheless he
could not set his thoughts at rest so easily, as is ap
parent from his wellknown letter to Bentley. That
gravity was immanent and innate in matter, so that
one body could act on another directly through empty
space, appeared to him absurd. But he is unable to
decide whether the intermediary agency is material or
immaterial (spiritual?). Like all his predecessors and
successors, Newton felt the need of explaining gravi
tation, by some such means as actions of contact. Yet
the great success which Newton achieved in astron
omy with forces acting at a distance as the basis of
deduction, soon changed the situation very consider
ably. Inquirers accustomed themselves to these forces
as points of departure for their explanations and the
impulse to inquire after their origin soon disappeared
almost completely. The attempt was now made to
introduce' these forces into all the departments of
physics, by conceiving bodies to be composed of par
ticles separated by vacuous interstices and thus acting
on one another at a distance. Finally even, the re
sistance of bodies to pressure and impact, this is to
say, even forces of contact, were explained by forces
acting at a distance between particles. As a fact, the
functions representing the former are more compli
cated than those representing the latter.
The doctrine of forces acting at a distance doubt
less stood in highest esteem with Laplace and his
contemporaries. Faraday's unbiassed and ingenious
conceptions and Maxwell's mathematical formulation
APPENDIX. 535
of them again turned the tide in favor of the forces
of contact. Divers difficulties had raised doubts in
the minds of astronomers as to the exactitude of New
ton's law, and slight quantitative variations of it were
looked for. After it had been demonstrated, however,
that electricity travelled with finite velocity, the ques
tion of a like state of affairs in connexion with the
analogous action of gravitation again naturally arose.
As a fact, gravitation bears a close resemblance to
electrical forces acting at a distance, save in the single
respect that so far as we know, attraction only and
not repulsion takes place in the case of gravitation.
Foppl ("Ueber eine Erweiterung des Gravitations
gesetzes," Sitzungsber. d. Munch. Akad., 1897, p. 6 et
seq.) is of opinion, that we may, without becoming
involved in contradictions, assume also with respect
to gravitation negative masses, which attract one an
other but repel positive masses, and assume therefore
also finite fields of gravitation, similar to the electric
fields. Drude (in his report on actions at a distance
made for the German Naturforscherversammhmg of
1897) enumerates many experiments for establishing
a velocity of propagation for gravitation, which go
back as far as Laplace. The result is to be regarded
as a negative one, for the velocities which it is at all
possible to consider as such, do not accord with one
another, though they are all very large multiples of
the velocity of light. Paul Gerber alone ("Ueber die*
raumliche u. zeitliche Ausbreitung der Gravitation/'
Zeitschrtft f. Math. u. P/tys., 1898, IL), from the peri
helial motion of Mercury, forty one seconds in a cen
tury, finds the velocity of propagation of gravitation
to be the same as that of light. This would speak in
favor of the ether as the medium of gravitation. (Com
536 THE SCIENCE OF MECHANICS.
pareW. Wien, " Ueber die Moglichkeit einer elektro
magnetischen Begriindung der Mechanik," Archives
Nterlandaises, The Hague, 1900, V., p. 96.)
XVI.
(See page 195.)
It should be observed that the notion of mass as
quantity of matter was psychologically a very natural
conception for Newton, with his peculiar develop
ment. Critical inquiries as to the origin of the con
cept of matter could not possibly be expected of a
scientist in Newton's day. The concept developed
quite instinctively , it is discovered as a datum per
fectly complete, and is adopted with absolute ingenu
ousness. The same is the case with the concept of
force. But force appears conjoined with matter. And,
inasmuch as Newton invested all material particles
with precisely identical gravitational forces, inasmuch
as he regarded the forces exerted by the heavenly
bodies on one another as the sum of the forces of the
individual particles composing them, naturally these
forces appear to be inseparably conjoined with the
quantity of matter. Rosenberger has called attention
to this fact in his book, Newton und seine physikalischen
Principien (Leipzig, 1895, especially page 192).
I have endeavored to show elsewhere (Analysis of
the Sensations, Chicago, 1897) how starting from the
constancy of the connexion between different sensa
tions we have been led to the assumption of an abso
lute constanc}^ which we call substajice, the most ob
vio'tis and prominent example being that of a moveable
body distinguishable from its environment. And see
ing that such bodies are divisible into homogeneous
parts, of which each presents a constant complexus
A PPENDIX. 537
of properties, we are induced to form the notion of a
substantial something that is quantitatively variable,
which we call matter. But that which we take away
from one body, makes its appearance again at some
other place. The quantity of matter in its entirety,
thus, proves to be constant. Strictly viewed, how
ever, we are concerned with precisely as many sub
stantial quantities as bodies have properties, and
there is no other function left for matter save that of
representing the constancy of connexion of the several
properties of bodies, of which man is one only. (Com
pare my Principles of Heat, German edition, 1896,
page 425.)
xvn.
(See page 216.)
Of the theories of the tides enunciated before
Newton, that of Galileo alone may be briefly men
tioned. Galileo explains the tides as due to the rela
tive motion of the solid and liquid parts of the earth,
and regards this fact as direct evidence of the motion
of the earth and as a cardinal
argument .in favor of the Co
pernican system. If the earth
(Fig. 240) rotates from the
west to the east, and is affected
at the same time with a pro
gressional motion, the parts of
the earth at a will move with the sum, and the parts
at b with the difference, of the two velocities. The
water in the bed of the ocean, which is unable to fol
low this change in velocity quickly enough, behaves
like the water in a plate swung rapidly back and forth,
or like that in the bottom of a skiff which is rowctl
538 THE SCIENCE Of MECHANICS.
with rapid alterations of speed : it piles up now in
the front and now at the back. This is substantially
the view that Galileo set forth in the Dialogue on the
Two World Systems. Kepler's view, which supposes
attraction by the moon, appears to him mystical and
childish. He is of the opinion that it should be rele
gated to the category of explanations by "sympathy"
and "antipathy," and that it admits as easily of refu
tation as the doctrine according to which the tides
are created by radiation and the consequent expansion
of the water. That on his theory the tides rise only
once a day, did not, of course, escape Galileo's atten
tion. But he deceived himself with regard to the
difficulties involved, believing himself able to explain
the daily, monthly, and yearly periods by considering
the natural oscillations of the water and the altera
tions to which its motions are subject. The principle
of relative motion is a correct feature of this theory,
but it is so infelicitously applied that only an ex
tremely illusive theory could result. We will first
convince ourselves that the conditions supposed to
be involved would not have the effect ascribed to
them. Conceive a homogeneous sphere of water;
any other effect due to rotation than that of a corres
ponding oblateness we should not expect. Now, sup
pose the ball to acquire in addition a uniform motion
of progression. Its various parts will now as before
remain at relative rest with respect to one another.
For the case in question does not differ, according to
our view, in any essential respect from the preceding,
inasmuch as the progressive motion of the sphere
may be conceived to be replaced by a motion in the
opposite direction of all surrounding bodies. Even
for the person who is inclined to regard the motion
APPENDIX. 539
as an " absolute " motion, no change Is produced in
the relation of the parts to one another by uniform
motion of progression. Now, let us cause the sphere,
the parts of which have no tendency to move with re
spect to one another, to congeal at certain points, so
that seabeds with liquid water in them are produced.
The undisturbed uniform rotation will continue, and
consequently Galileo's theory Is erroneous.
But Galileo's idea appears at first blush to be ex
tremely plausible; how is the paradox explained? It
is due entirely to a negative conception of the law of
Inertia. If we ask what acceleration the water expe
riences, everything is clear. Water having no weight
would be hurled off at the beginning of rotation ;
water having weight, on the other hand, would de
scribe a central motion around the center of the earth
With its slight velocity of rotation It would be forced
more and more toward the center of the earth, with
just enough of Its centripetal acceleration counter
acted by the resistance of the mass lying beneath,
as to make the remainder, conjointly with the given
tangential velocity, sufficient for motion in a circle*
Looking at it from this point of view, all doubt and
obscurity vanishes. But It must In justice be added
that it was almost Impossible for Galileo, unless his
genius were supernatural, to have gone to the bottom
of the matter. He would have been obliged to antici
pate the great intellectual achievements of Htiygens
and Newton.
xvin.
(See pae 2x8.)
H. Streintz's objection (Die fhysikalisehen Grtmd*
lagcn dcr Mechanik, Lelpsic, 1883, p, 117), that a com
54 o THE SCIENCE OF MECHANICS,
parison of masses satisfying my definition can be ef
fected only by astronomical means, I am unable to
admit. The expositions on pages 202, 218221 amply
refute this. Masses produce in each other accelera
tions in impact, as well as when subject to electric
and magnetic forces, and when connected by a string
on At wood's machine. In my Elements of Physics
(second German edition, 1891, page 27) I have shown
how massratios can be experimentally determined on
a centrifugal machine, in a very elementary and pop
ular manner. The criticism in question, therefore,
may be regarded as refuted.
My definition is the outcome of an endeavor to
establish the interdependence of phenomena and to re
move all metaphysical obscurity, without accomplish
ing on this account less than other definitions have
done. I have pursued exactly the same course with
respect to the ideas, " quantity of electricity " (" On
the Fundamental Concepts of Electrostatics," 1883,
Popular Scientific Lectures, Open Court Pub. Co., Chi
cago, 1898), " temperature,' 7 "quantity of heat" {Zcit
schrift fur den physikalischen u?id chemischen Unterricht,
Berlin, 1888, No. i), and so forth. With the view
here taken of the concept of mass is associated, how
ever, another difficulty, which must also be carefully
noted, if we would be rigorously critical in our analy
sis of other concepts of physics, as for example the
concepts of the theory of heat. Maxwell made refer
ence to this point in his investigations of the concept
of temperature, about the same time as I did with re
spect to the concept of heat. I would refer here to
the discussions on this subject in my Principles of
Heat (German edition, Leipsic, 1896), particularly
page 41 and page 190.
APPENDIX. 541
XIX.
(See page 226.)
My views concerning physiological time, the sen
sation of time, and partly also concerning physical
time, I have expressed elsewhere (see Analysis of the
Sensations, 1886, Chicago, Open Court Pub. Co., 1897,
pp. 109118, 179181). As in trie study of thermal
phenomena we take as our measure of temperature an
arbitrarily chosen indicator of volume, which varies in
almost parallel correspondence with our sensation of
heat, and which is not liable to the uncontrollable
disturbances of our organs of sensation, so, for simi
lar reasons, we select as our measure of time an arbi
trarily chosen motion, (the angle of the earth's rotation,
or path of a free body,) which proceeds in almost
parallel correspondence with our sensation of time.
If we have once made clear to ourselves that we arc
concerned only with the ascertainment of the inter
dependence of phenomena, as I pointed out as early as
1865 (J[7ebcr den Zeitsinn dcs Ohres, Sitzuwgsberichte der
Wiener Akademie) and 1866 (Fichte's Zeits thrift flir
Philosophic), all metaphysical obscurities disappear.
(Compare J Epstein, Die logischen Principien der Zeit
messung, Berlin, 1887.)
I have endeavored also (Principles of Heat ^ German
edition, page 51) to point out the reason for the natu
ral tendency of man to hypostatise the concepts which
have great value for him, particularly those at which
he arrives instinctively, without a knowledge of their
development. The considerations which I there* ad
duced for the concept of temperature may be easily
applied to the concept of time, and render the origin
542 THE SCIENCE OF MECHANICS.
of Newton's concept of "absolute" time intelligible.
Mention is also made there (page 338) of the connex
ion obtaining between the concept of energy and the
irreversibility of time, and the view is advanced that
the entropy of the universe, if it could ever possibly
be determined, would actually represent a species of
absolute measure of time. I have finally to refer here
also to the discussions of Petzoldt ("Das Gesetz der
Eindeutigkeit," Vierteljahrsschrift fur wissenschaftliche
Philosophic, 1894, page 146), to which I shall reply in
another place.
(See page 238.)
Of the treatises which have appeared since 1883
on the law of inertia, w all of which furnish welcome
evidence of a heightened interest in this question, I
can here only briefly mention that of Streintz (Physi
kalische Grundlagen der Mechanik, Leipsic, 1883) and
that of L. Lange {Die geschichtliche JZntwicklung des
Bewegungsbegriffes, Leipsic, 1886).
The expression " absolute motion of translation"
Streintz correctly pronounces as devoid of meaning
and consequently declares certain analytical deduc
tions, to which he refers, superfluous. On the other
hand, with respect to rotation, Streintz accepts New
ton's position, that absolute rotation can be distin
guished from relative rotation. In this point of view,
therefore, one can select every body not affected with
absolute rotation as a body of reference for the ex
pression of the law of inertia.
I cannot share this view. For me, only relative
motions exist (Erhaltung der Arbeit, p. 48 ; Science of
Mechanics, p. 229), and I can see, in this regard, no
APPENDIX. 543
distinction between rotation and translation. When a
body moves relatively to the fixed stars, centrifugal
forces are produced ; when it moves relatively to some
different body, and not relatively to the fixed stars,
no centrifugal forces are produced. I have no objec
tion to calling the first rotation "absolute" rotation,
if it be remembered that nothing is meant by such a
designation except relative rotation with respect to the
fixed stars. Can we fix Newton's bucket of water,
rotate the fixed stars, and then prove the absence of
centrifugal forces?
The experiment is impossible, the idea is mean
ingless, for the two cases are not, in senseperception,
distinguishable from each other. I accordingly re
gard these two cases as the same case and Newton's
distinction as an illusion (Science of Mechanics, page
232).
But the statement is correct that it is possible to
find one's bearings in a balloon shrouded in fog, by
means of a body which does not rotate with respect
to the fixed stars. But this is nothing more than an
indirect orientation with respect to the fixed stars ; it
is a mechanical, substituted for an optical, orienta
tion.
I wish to add the following remarks in answer to
Streintz's criticism of my view. My opinion is not to
be confounded with that of Euler (Streintz, pp. 7, 50),,
who, as L/ange has clearly shown, never arrived at
any settled and intelligible opinion on the subject.
Again, I never assumed that remote masses only, and
not near ones, determine the velocity of a body
(Streintz, p. 7); I simply spoke of an influence inde
pendent of distance. In the light of my expositions
at pages 222245, the unprejudiced and careful reader
5 4 4 THE SCIENCE OF MECHANICS.
will scarcely maintain with Streintz (p. 50), that after
so long a period of time, without a knowledge of
Newton and Euler, I have only been led to views
which these inquirers so lorg ago held, but were
afterwards, partly by them and partly by others, re
jected. Even my remarks of 1872, which were all
that Streintz knew, cannot justify this criticism. These
were, for good reasons, concisely stated, but they are
by no means so meagre as they must appear to one
who knows them only from Streintz's criticism. The
point of view which Streintz occupies, I at that time
expressly rejected.
Lange's treatise is, in my opinion, one of the best
that have been written on this subject. Its methodi
cal movement wins at once the reader's sympathy. Its
careful analysis and study, from historical and criti
cal points of view, of the concept of motion, have
produced, it seems to me, results of permanent value.
I also regard its clear emphasis and apt designation
of the principle of "particular determination" as a
point of much merit, although the principle itself, as
well as its application, is not new. The principle is
really at the basis of all measurement. The choice of
the unit of measurement is convention ; the number
of measurement is a result of inquiry. Every natural
inquirer who is clearly conscious that his business is
simply the investigation of the interdependence of
phenomena, as I formulated the point at issue a long
time ago (18651866), employs this principle. When,
for example {Mechanics , p. 218 et seq.), the negative
inverse ratio of the mutually induced accelerations of
two bodies is called the massratio of these bodies,
this is a convention, expressly acknowledged as arbi
trary ; but that these ratios are independent of the
APPENDIX. 545
kind^and of the order of combination of the bodies is
a result of inquiry. I might adduce numerous similar
nstances from the theories of heat and electricity as
well as from other provinces. Compare Appendix II.
Taking it in its simplest and most perspicuous
form, the law of inertia, in Lange's view, would read
as follows :
Three material points, JPi, jP 2 , P 8 , are simultane
ously hurled from the same point in space and then
left to themselves. The moment we are certain that
the points are not situated in the same straight line,
we join each separately with any fourth point in space,
<2 These lines of junction, which we may respec
tively call 6*1, G^ G$, form, at their point of meeting,
a threefaced solid angle. If now we make this solid
angle preserve, with unaltered rigidity, its form, and
constantly determine In such a manner Its position,
that PI shall always move on the line G\> P<z on the
line G^ P% on the line G^ these lines may be regarded
as the axis of a coordinate or Inertial system, with
respect to which every other material point, left to It
self, will move In a straight line. The spaces de
scribed by the free points in the paths so determined
will be proportional to one another.
A system of coordinates with respect to which
three material points move in a straight line Is, ac
cording to Lange, under the assumed limitations, a
simple convention. That with respect to such a system
also a fourth or other free material point will move in
a straight line, and that the paths of the different
points will all be proportional to one another, arc re
sults of inquiry.
In the first place, we shall not dispute the fact
that the law of inertia can be referred to such a system
546 THE SCIENCE OF MECHANICS.
of time and space coordinates and expressed HI this
form. Such an expression is less fit than Strelntz's
for practical purposes, but, on the other hand, Is, for
its methodical advantages, more attractive. It espe
cially appeals to my mind, as a number of years ago I
was engaged with similar attempts, of which not the
beginnings but only a few remnants {Mechanics, pp.
234235) are left. I abandoned these attempts, be
cause I was convinced that we only apparently evade
by such expressions references to the fixed stars and
the angular rotation of the earth. This, in my opin
ion, is also true of the forms in which Streintz and
L,ange express the law.
In point of fact, it was precisely by the considera
tion of the fixed stars and the rotation of the earth
that we arrived at a knowledge of the law of inertia
as It at present stands, and without these foundations
we should never have thought of the explanations
here discussed (Mechanics, 232233). The considera
tion of a small number of isolated points, to the ex
clusion of the rest of the world, is in my judgment In
admissible {Mechanics, pp. 229235).
It Is quite questionable, whether a fourth material
point, left to itself, would, with respect to Lange's
"inertial system," uniformly describe a straight line,
if the fixed stars were absent, or not invariable, or
could not be regarded with sufficient approximation
as Invariable.
The most natural point of view for the candid In
quirer must still be, to regard the law of inertia pri
marily as a tolerably accurate approximation, to refer
it, with respect to space, to the fixed stars, and, with
respect to time, to the rotation of the earth, and to
await the correction, or more precise definition, of
APPENDIX. 547
our knowledge from future experience, as I have ex
plained on page 237 of this book.
I have still to mention the discussions of the law
of inertia which have appeared since 1889. Reference
may first be made to the expositions of Karl Pearson
{Grammar of Science, 1892, page 477), which agree
with my own, save in terminology. P. and J. Fried
lander {Absolute und relative Bewcgung, Berlin, 1896)
have endeavored to determine the question by means
of an experiment based on the suggestions made by
me at pages 217218; I have grave doubts, however,
whether the experiment will be successful from the
quantitative side. I can quite freely give my assent
to the discussions of Johannesson {Das J3eharrung$~
gesetz, Berlin, 1896), although the question, remains
unsettled as to the means by which the motion of A
body not perceptibly accelerated by other bodies Is to
be determined. For the sake of completeness, the
predominantly dialectic treatment by M. E. Vicaire,
Soring scientifique de Bruxelles, Jrf>5 3 as well as the in
vestigations of J. G. MacGregor, Royal Society of Can
ada, Jr8(?5, which are only remotely connected with
the question at issue; remain to be mentioned. I have
no objections to Budde's conception of space as a sort
of medium (compare page 230), although I think that
the properties of this medium should be demonstrable
physically in some other manner, and that they should
not be assumed ad hoc. If all apparent actions at a
distance, all accelerations, turned out to be effected
through the agency of a medium, then the question
would appear in a different light, and the solution is
to be sought perhaps in the view set forth on page
230.
54 8 THE SCIENCE OF MECHANICS.
XXI.
(See page 255.)
Section VIII., "Retrospect of the Development
of Dynamics," was written in the year 1883. It con
tains, especially in paragraph 7, on pages 254 and
255, an extremely general programme of a future sys
tem of mechanics, and it is to be remarked that the
Mechanics .of Hertz, which appeared in the year 1894,*
marks a distinct advance in the direction indicated.
It is impossible in the limited space at our disposal
to give any adequate conception of the copious ma
terial contained in this book, and besides it is not our
purpose to expound new systems of mechanics, but
merely to trace the development of ideas relating to
mechanics. Hertz's book must, in fact, be read by
every one interested in mechanical problems.
Hertz's criticisms of prior s}^stems of mechanics,
with which he opens his work, contains some very
noteworthy epistemological considerations, which from
our point of view (not to be confounded either "with
the Kantian or with the atomistic mechanical concepts
of the majority of physicists), stand in need of certain
modifications. The constructive images f (or better,
perhaps, the concepts), which we consciously and
purposely form of objects, are to be so chosen that
the " consequences which necessarily follow from them
in thought " agree with the " consequences which nec
essarily follow from them in nature." It is demanded
of these images or concepts that they shall be logically
*H. Hertz, Die Princijtzen der Mechanik zn neuem Zusam?nenhange dar~
gestellt. Leipzig, 1894.
t Hertz uses the term Bild (image or picture) in the sense of the old Eng
lish philosophical use of zVfta, and applies it to systems of ideas or concepts
relating to any province.
APPENDIX. 549
admissible, that is to say, free from all selfcontradic
tions ; that they shall be correct, that Is, shall con
form to the relations obtaining between objects; and
finally that they shall be appropriate, and contain the
least possible superfluous features. Our concepts, it
is true, are formed consciously and purposely by us,
but they are nevertheless not formed altogether arbi
trarily, but are the outcome of an endeavor on our
part to adapt our ideas to our sensuous environment.
The agreement of the concepts with one another is a
requirement which is logically necessary, and this
logical necessity, furthermore, is the only necessity
that we have knowledge of. The belief in a necessity
obtaining in nature arises only in cases where our
concepts are closely enough adapted to nature to
ensure a correspondence between the logical infer
ence and the fact. But the assumption of an adequate
adaptation of our ideas can be refuted at any moment
by experience. Hertz's criterion of appropriateness
coincides with our criterion of economy.
Hertz's criticism that the GalileoNewtonian sys
tem of mechanics, particularly the notion of force, lacks
clearness (pages 7, 14, 15) appears to us justified only
in the case of logically defective expositions, such as
Hertz doubtless had in mind from his student days*
He himself partly retracts his criticism in another
place (pages 9, 47) \ or at any rate, he qualifies it. But
the logical defects of some individiial interpretation
cannot be imputed to systems as such* To be sure,
it is not permissible today (page 7) "to speak of a
force acting in one aspect only, or, in the case of cen
tripetal force, to take account of the action of inertia
twice, once as a mass and again as a force.** But
neither is this necessary, since Huygens and Newton
55 o THE SCIENCE OF MECHANICS,
were perfectly clear on this point. To characterise
forces as being frequently "emptyrunning wheels/'
as being frequently not demonstrable to the senses,
can scarcely be permissible. In any event, "forces"
are decidedly in the advantage on this score as com
pared with "hidden masses " and "hidden motions."
In the case of a piece of iron lying at rest on a table,
both the forces in equilibrium, the weight of the iron
and the elasticity of the table, are very easily demon
strable.
Neither is the case with energic mechanics so bad
as Hertz would have It, and as to his criticism against
the employment of minimum principles, that it in
volves the assumption of purpose and presupposes
tendencies directed to the future, the present work
shows in another passage quite distinctly that the
simple import of minimum principles Is contained In
an entirely different property from that of purpose.
Every system of mechanics contains references to the
future, since all must employ the concepts of time,
velocity, etc.
Nevertheless, though Hertz's criticism of existing
systems of mechanics cannot be accepted in all their
severity, his own novel views must be regarded as a
great step in advance. Hertz, after eliminating the
concept of force, starts from the concepts of time,
space, and mass alone, with the idea In view of giv
ing expression only to that which can actually be ob
served. The sole principle which he employs may
be conceived as a combination of the law of Inertia
and Gauss's principle of least constraint. Free masses
move uniformly in straight lines. If they are put In
connexion In any manner they deviate, In accordance
with Gauss's principle, as little as possible from this
APPENDIX. 551
motion; their actual motion is more nearly that of
free motion than any other conceivable motion. Hertz
says the masses move as a result of their connexion
in a straightcst path. Every deviation of the motion
of a mass from uniformity and rectilinearity is due, in
his system, not to a force but to rigid connexion with
other masses. And \\here such matters are not vis
ible, he conceives hidden masses with hidden motions.
All physical forces are conceived as the effect of such
actions. Force, forcefunction, energy, in his system,
are secondary and auxiliary concepts only. Let us
now look at the most important points singly, and
ask to what extent was the way prepared for them.
The notion of eliminating force may be reached in
the following manner. It is part of the general idea
of the GalileoNewtonian system of mechanics to
conceive of all connexions as replaced by forces which
determine the motions required by the connexions ;
converse^'', everything that appears as force may be
conceived to be due to a connexion. If the first idea
frequently appears in the older systems, as being his
torically simpler and more immediate, in the case of
Hertz the latter is the more prominent. If we reflect
that in both cases, whether forces or connexions be
presupposed, the actual dependence of the motions
of the masses on one another is given for every in
stantaneous conformation of the system by linear dif
ferential equations between the co animates of the
masses, then the existence of these equations may be
considered the essential thing, the thing established
by experience. Physics indeed gradually accustoms
itself to look upon the description of the facts by dif
ferential equations as its proper aim, a point of "view
which was taken also in Chapter V. of the present
55 2 TJIE SCIENCE OF MECHANICS.
work (1883). But with these the general applicabil
ity of Hertz's mathematical formulations is recognised
without our being obliged to enter upon any further
interpretation of the forces or connexions.
Hertz's fundamental law may be described as a
sort of generalised law of Inertia, modified by connex
ions of the masses. For the simpler cases, this view
was a natural one, and doubtless often forced Itself
upon the attention. In fact, the principle of the con
servation of the center of gravity and of the conserva
tion of areas was actually described in the present
work (Chapter III.) as a generalised law of inertia.
If we reflect that by Gauss's principle the connexion
of the masses determines a minimum of deviation from
those motions which it would describe for Itself, we
shall arrive at Hertz's fundamental law the moment we
consider all the forces as due to the connexions. For
on severing all connexions, only isolated masses mov
ing by the law of inertia are left as ultimate elements.
Gauss very distinctly asserted that no substantially
new principle of mechanics could ever be discovered.
And Hertz's principle also is only new In form, for it
is identical with Lagrange's equations. The minimum
condition which the principle involves does not refer
to any enigmatic purpose, but its Import Is the same
as that of all minimum laws. That alone takes place
which is dynamically determined (Chapter III.). The
deviation from the actual motion Is dynamically not
determined ; this deviation is not present ; the actual
motion is therefore unique.*
*See Petzoldt's excellent article "Das Gesetz der Eindeutigkeit " (Vftr
teljahrsschrift fur vuissenschaftliche Philosophie, XIX., page 146, especially
page 186). R. Henke is also mentioned in this article as having approached
Hertz's view in his tract Ueber die Methode der kle fnsten Quadrate (Lfclpsic t
1894).
APPENDIX. 553
It is hardly necessary to remark that the physical
side of mechanical problems is not only not disposed
of, but is not even so much as touched, by the elabo
ration of such a formal mathematical system of me
chanics. Free masses move uniformly in straight
lines. Masses having different velocities and direc
tions if connected mutually affect each other as to
velocity, that is, determine in each other accelera
tions. These physical experiences enter along with
purely geometrical and arithmetical theorems into the
formulation, for which the latter alone would In no
wise be adequate ; for that which is uniquely deter
mined mathematically and geometrically only, is for
that reason not also uniquely determined mechani
cally. But we discussed at considerable length in
Chapter II., that the physical principles in question
were not at all selfevident, and that even their exact
significance was by no means easy to establish*
In the beautiful ideal form which Hertz has given
to mechanics, its physical contents have shrunk to
an apparently almost imperceptible residue. It is
scarcely to be doubted that Descartes if he lived to
day would have seen in Hertz's mechanics, far more
than in Lagrange's " analytic geometry of four dimen
sions," his own ideal. For Descartes, who in his op
position to the occult qualities of Scholasticism would
grant no other properties to matter than extension
and motion, sought to reduce all mechanics and phys
ics to a geometry of motions, on the assumption of a
motion indestructible at the start
It is not difficult to analyse the psychological cir
cumstances which led Hertz to his system. After in
quirers had succeeded in representing electric and
magnetic forces that act at a distance as the results
554 THE SCIENCE OF MECHANICS.
of motions in a medium, the desire must again have
awakened to accomplish the same result with respect
to the forces of gravitation, and if possible for all
forces whatsoever. The Idea was therefore very nat
ural to discover whether the concept of force generally
could not be eliminated. It cannot be denied that
when we can command all the phenomena taking
place in a medium, together with the large masses
contained in it, by means of a single complete pic
ture, our concepts are on an entirely different plane
from what they are when only the relations of these
isolated masses as regards acceleration are known.
This will be willingly granted even by those \\ho are
convinced that the interaction of parts in contact is
not more intelligible than action at a distance. The
, present tendencies in the development of physics are
entirely in this direction.
If we are not content to leave the assumption of
occult masses and motions in its general form, but
should endeavor to investigate them singly and in de
tail, we should be obliged, at least in the present state
of our physical knowledge, to resort, even in the sim
plest cases, to fantastic and even frequently question
able fictions, to which the given accelerations would
be far preferable. For example, if a mass m is mov
ing uniformly in a circle of radius r, with a velocity
z>, which we are accustomed to refer to a centripetal
force proceeding from the center of the circle,
we might instead of this conceive the mass to be
rigidly connected at the distance 2r with one of the
same size having a contrary velocity. Huygens 7 s cen
tripetal impulsion would be another , example of a
force replaced by a connexion. As an ideal program
APPENDIX. 535
Hertz's mechanics Is simpler and more beautiful, but
for practical purposes our present S3 T stem of mechan
ics is preferable, as Hertz himself (page 47), with his
characteristic candor, admits.*
XXII.
(See page 255.)
The views put forward in the first two chapters of
this book were worked out by me a long time ago.
At the start they were almost without exception coolly
rejected, and only gradually gained friends. All the
essential features of my Mechanics I stated originally
In a brief communication of five octavo pages entitled
On the Definition of Mass. These were the theorems
now given at page 243 of the present book. The
communication was rejected by Poggendorf's Anna
len, and it did not appear until a year later (1868), In
Carl's Rcpertorium. In a lecture delivered in 1871, I
outlined my epistemological point of view in natural
science generally, and with special exactness for phys
ics. The concept of cause is replaced there by the
concept of function; the determining of the depend
ence of phenomena on one another, the economic ex
position of actual facts, is proclaimed as the object,
and physical concepts as a means to an end solely.
I did not care now to impose upon any editor the re
sponsibility for the publication of the contents of this
lecture, and the same was published as a separate
tract in i872.f In 1874, when Kirchhoff In his Me
chanics came out with his theory of "description" and
""Compare J, Classen, "Die Principien der Mochanik hen flrt/ tint!
Boltzniann " (Jnhrlnich tier Ifambuvgischtit itoi&sM$chaftlfch<*M v?//AiW/f>r ,
XV., p. i, Hamburg, 1898).
f J&rhaltung der Arbeit^ Prague, 1872.
55 6 THE SCIENCE OF MECHANICS.
other doctrines, which were analogous in part only to
my views, and still aroused the "universal astonish
ment " of his colleagues, I became resigned to my
fate. But the great authority of Kirchhoff gradually
made itself felt, and the consequence of this also
doubtless was that on its appearance in 1883 my Me
chanics did not evoke so much surprise. In view of
the great assistance afforded by Kirchhoff, it is alto
gether a matter of indifference with me that the pub
lic should have regarded, and partly does so still, my
interpretation of the principles of physics as a contin
uation and elaboration of Kirchhoff's views ; whilst in
fact mine were not only older as to date of publica
tion, but also more radical.*
The agreement with my point of view appears
upon the whole to be increasing, and gradually to ex
tend over more extensive portions of my work. It
would be more in accord with my aversion for polem
ical discussions to wait quietly and merely observe
what part of the ideas enunciated may be found ac
ceptable. But I cannot suffer my readers to remain
in obscurity with regard to the existing disagreements,
and I have also to point out to them the way in which
they can find their intellectual bearings outside of
this book, quite apart from the fact that esteem for
my opponents also demands a consideration of their
criticisms. These opponents are numerous and of all
kinds: historians, philosophers, metaphysicians, logi
cians, educators, mathematicians, and physicists, I
can make no pretence to any of these qualifications in
any superior degree. I can only select here the most
important criticisms, and answer them in the capacity
of a man who has the liveliest and most ingenuous in
*See the preface to the first edition.
APPENDIX. 557
terest in understanding the growth of physical ideas.
I hope that this will also make it easy for others to
find their way in this field and to form their own judg
ment.
P. Volkmann in his writings on the epistemology*
of physics appears as my opponent only In certain
criticisms on individual points, and particularly by
his adherence to the old systems and by his predilec
tion for them. It is the latter trait, in fact, that sepa
rates us ; for otherwise Volkmann's views have much
affinity with my own. He accepts my adaptation of
ideas, the principle of economy and of comparison,
even though his expositions differ from mine in indi
vidual features and tary in terminology. I, for my
part, find in his writings the important principle of
isolation and superposition, appropriately emphasised
and admirably described, and I willingly accept them.
I am also willing to admit that concepts which at the
start are not very definite must acquire their "retro
active consolidation" by a "circulation of knowl
edge/' by an "oscillation ' T of attention. I also agree
with Volkmann that from this last point of view New
ton accomplished in his day nearly the best that it
was possible to do; but I cannot agree with Volk
mann when he shares the opinion of Thomson and
Tait, that even in the face of the substantially differ
ent epistemological needs of the present day, New
ton's achievement is definitive and exemplary. On
the contrary, it appears to me that if Volkmann's pro
cess of "consolidation" be allowed complete sway, it
must necessarily lead to enunciations not differing in
* Erkenntnisstheoretische Grund&ngederNaturwfsstinsfhttft, Loipadg, t8}6
Ueber Newton's Phflosojhia Natural/*. Kttnigshntx, i$tfi<>J5fwfnJtrui9ff fn
das Studium dur theoretzschen Physik, Leipsie, igco. Our references ar to
the lastnamed work.
55 8 TfJ SCIENCE OF MECHANICS.
any essential point from my own. I follow with gen
uine pleasure the clear and objective discussions of
G. Heymans.* The differences which I have with
Hoflerf and Poske J relate in the main to individual
points. So far as principles are concerned, I take
precisely the same point of view as Petzoldt, and we
differ only on questions of minor importance. The
numerous criticisms of others, which either refer to
the arguments of the writers just mentioned, or are
supported by analogous grounds, cannot out of regard
for the reader be treated at length. It will be suffi
cient to describe the character of these differences by
selecting a few individual, but important, points.
A special difficulty seems to be still found in ac
cepting my definition of mass. Streintz (compare p.
540) has remarked in criticism of it that it is based
solely.upon gravity, although this was expressly ex
cluded in rny first formulation of the definition (1868).
Nevertheless, this criticism is again and again put
forward, and quite recently even by Volkmann (/t><r.
ctt., p. 1 8). My definition simply takes note of the
fact that bodies in mutual relationship, whether it be
that of action at a distance, so called, or whether rigid
or elastic connexions be considered, determine in one
another changes of velocity (accelerations). More than
this, one does not need to know in order to be able to
form a definition with perfect assurance and without
the fear of building on sand. It is not correct as
*Die Gesetze ^lnd Elements des 'wisscnschaftlicken Denkcns, II., Leipzig
1894.
\Studien zur gegen'wi.irt/g'en Philoso$hle der mathematischen dlechetnik,
Leipzig, 1900.
J Vierteljahrsschrift fur vuzssenschaftliche Philosophic^ Leipzig, 1884* P a # *
385.
" Das Gesetz der Eindeutigkeit ( Vierteljahrsschriftfnr ivfssenschaftltche
e^ XIX. , page 146).
APPENDIX. 559
Hofler asserts (Joe. cit., p. 77), that this definition
tacitly assumes one and the same force acting on both
masses. It does not assume even the notion of force,
since the latter is built up subsequently upon the no
tion of mass, and gives then the principle of action
and reaction quite independently and without falling
into Newton's logical error. In this arrangement one
concept is not misplaced and made to rest on another
which threatens to give way under it. This is, as I
take it, the only really serviceable aim of Volkrnann's
"circulation" and "oscillation." After we have de
fined mass by means of accelerations, it is not difficult
to obtain from our definition apparently new variant
concepts like "capacity for acceleration, 7 ' "capacity
for energy of motion" (Hofler, loc. cit., page 70). To
accomplish anything dynamically with the concept of
mass, the concept in question must, as I most em
phatically insist, be a dynamical concept. Dynamics
cannot be constructed with quantity of matter by it
self, but the same can at most be artificially and arbi
trarily attached to it (loc, cit., pages 71, 72). Quantity
of matter by itself is never mass, neither is it thermal
capacity, nor heat of combustion, nor nutritive value,
nor anything of the kind. Neither does "mass"
play a thermal, but only a dynamical r61e (compare
Hofler, loc. cit., pages 71, 72). On the other hand,
the different physical quantities are proportional to
one another, and two or three bodies of unit mass
form, by virtue of the dynamic definition, a body of
twice or three times the mass, as is analogously the
case also with thermal capacity by virtue of the ther
mal definition. Our instinctive craving for concepts
involving quantities of things, to which Hdfler (&>r.
cit , page 72) is doubtless seeking to give expression,
5 6o THE SCIENCE OF MECJIAJ\ JCS.
and which amply suffices for everyday purposes, is
something that no one will think of denying. But
a scientific concept of "quantity of matter" should
properly be deduced from the proportionality of the
single physical quantities mentioned, instead of, con
trariwise, building up the concept of mass upon
" quantity of matter." The measurement of mass by
means of weight results from my definition quite nat
urally, whereas in the ordinary conception the meas
urability of quantity of matter by one and the same
dynamic measure is either taken for granted outright,
or proof must be given beforehand by special experi
ments, that equal weights act under all circumstances
as equal masses. In my opinion, the concept of mass
has here been subjected to thorough analysis for the
first time since Newton. For historians, mathema
ticians, and physicists appear to have all treated the
question as an easy and almost selfevident one* It
is, on the contrary, of fundamental significance and
is deserving of the attention of my opponents*
Many criticisms have been made of my treatment
of the law of inertia. I believe I have shown (1868),
somewhat as Poske has done (1884), that any deduc
tion of this law from a general principle, like the law
of causality, is inadmissible, and this view has now
won some support (compare Heymans, loc. cif., page
432). Certainly, a principle that has been universally
recognised for so short a time only cannot be regarded
as a fHori selfevident. Heymans {loc. cit., p. 427) cor
rectly remarks that axiomatic certainty was ascribed
a few centuries ago to a diametrically opposite form
of the law. Heymans sees a supraempirical element
only in the fact that the law of inertia is referred to
absolute space, and in the further fact that both in
APPENDIX. 561
the law of Inertia and in its ancient diametrically op
posite form something constant is assumed in the con
dition of the body that is left to itself (loc. cit*, page
433) We shall have something to say further on re
garding the first point, and as for the latter it is psy
chologically intelligible without the aid of metaphys
ics, because constant features alone have the power
to satisfy us either intellectually or practically,
which is the reason that we are constantly seeking for
them. Now, looking at the matter from an entirely
unprejudiced point of view, the case of these axio
matic certainties will be found to be a very peculiar
one. One will strive in vain with Aristotle to con
vince the common man that a stone hurled from the
hand would be necessarily brought to rest at once after
its release, were it not for the air which rushed in be
hind and forced 1 it forwards. But he would put just
as little credence in Galileo's theory of infinite uni
form motion. On the other hand, Benedetti's theory
of the gradual diminution of the vis imprcssa, which
belongs to the period of unprejudiced thought and of
liberation from ancient preconceptions, will be ac
cepted by the common man without contradiction.
This theory, in fact, Is an immediate reflexion of ex
perience, while the firstmentioned theories, which
Idealise experience in contrary directions, arc a pro
duct of technical professional reasoning. They exer
cise the illusion of axiomatic certainty only upon the
mind of the scholar whose entire customary train of
thought would be thrown out of gear by a disturbance
of these elements of his thinking. The behavior of
inquirers toward the law of inertia seems to me from
a psychological point of view to be adequately ex
plained by this circumstance, and I am inclined to
562 THE SCIENCE OF MECHANICS.
allow the question of whether the principle is to be
called an axiom, a postulate, or a maxim, to rest in
abeyance for the time being. Heymans, Poske, and
Petzoldt concur in finding an empirical and a supra
empirical element in the law of inertia. According to
Heymans (Joe. cit., p. 438) experience simply afforded
the opportunity for applying an a priori valid prin
ciple. Poske thinks that the empirical origin of the
principle does not exclude its a priori validity (loc.
czt.y pp. 401 and 402). Petzoldt also deduces the law
of inertia in part only from experience, and regards it
in its remaining part as given by the law of unique
determination. I believe I am not at variance with
Petzoldt in formulating the issue here at stake as fol
lows : It first devolves on experience to inform us
what particular dependence of phenomena on one an
other actually exists, what the thing to be determined
is, and experience alone can instruct us on this
point. If we are convinced that we have been suffi
ciently instructed in this regard, then when adequate
data are at hand we regard it as unnecessary to keep
on waiting for further experiences ; the phenomenon
is determined for us, and since this alone is determi
nation, it is uniquely determined. In other words, if
I have discovered by experience that bodies determine
accelerations in one another, then in all circumstances
where such determinative bodies are lacking I shall
expect with unique determination uniform motion in
a straight line. The law of inertia thus results imme
diately in all its generality, without our being obliged
to specialise with Petzoldt ; for every deviation from
uniformity and rectilinearity takes acceleration for
granted. I believe I am right in saying that the same
fact is twice formulated in the law of inertia and in
APPENDIX. 563
the statement that forces determine accelerations (p.
143). If this be granted, then an end is also put to
the discussion as to whether a vicious circle is or is
not contained in the application of the law of inertia
(Poske, Hofler).
My inference as to the probable manner in which
Galileo reached clearness regarding the law of inertia
was drawn from a passage in his third Dialogue,*
which was literally transcribed from the Paduan edi
tion of 1744, Vol. III., page 124, in my tract on The
Conservation of Energy (Kng. Trans., in part, in my
Popular Scientific Lectures, third edition, Chicago,
The Open Court Publishing Co.). Conceiving a body
which is rolling down an inclined plane to be con
ducted upon rising inclined planes of varying slopes,
the slight retardation which it suffers on absolutely
smooth rising planes of small inclination, and the re
tardation zero, or unending uniform motion on a hori
zontal plane, must have occurred to him. Wohlwill
was the first to object to this way of looking at the
matter (see page 524), and others have since joined
* " Constat jam, quod mobile ex quiete In A descendens per AB, pjnulus
acquirit velocitatis juxta temporis ipsius inerementnm : gradum vero in B
esse maximum, acquisitonnn, et stiapte natura immntnbil tor improssuw,
sublatis scilicet causis accelerationis novae, aut retardationis : accclcrationis
Fig. 241.
Inquam, si adhuc super extenso piano tiltcmis progrcdorottu" ; nstnrdationis
vero, dum super planum acclive /?C fit reflexio : in hori/.ontuli antum (>"//
nequabilis motus juxta gradum valocitatis ex A in /;? acquisttac In infunttnu
^ixtenderetxir/'
" It is plain now that a movable body, starting from rest at A and da
564 THE SCIENCE OF MECHANICS.
him. He asserts that uniform motion in a circle and
horizontal motion still occupied distinct places in Ga
lileo's thought, and that Galileo started from the an
cient concepts and freed himself only very gradual!}
from them. It is not to be denied that the different
phases in the intellectual development of the great in
quirers have much interest for the historian, and sonic
one phase may, in its importance in this respect, be
relegated into the background by the others. One
must needs be a poor psychologist and have little
knowledge of oneself not to know how difficult it is
to liberate oneself from traditional views, and how
even after that is done the remnants of the old ideas
still hover in consciousness and are the cause of occa
sional backslidings even after the victory has been
practically won. Galileo's experience cannot have
been different. But with the physicist it is the in
stant in which a new view flashes forth that is of
greatest interest, and it is this instant for which he
will always seek. I have sought for it, I believe I
have found it; and I am of the opinion that it left its
unmistakable traces in the passage in question. Poske
(Joe. cit., page 393) and Hofler (Joe. cit., pages in,
112) are unable to give their assent to my interpreta
tion of this passage, for the reason that Galileo does
not expressly refer to the limiting case of transition
from the inclined to the horizontal plane ; although
scending down the inclined plane AR, acquires a velocity proportional to
the increment of its time : the velocity possessed at B is the greatest of the
velocities acquired, and by its nature immutably impressed, provided all
"uses of new acceleration or retardation are taken away: I say accelcru
m, having in view its possible further progress along the plane extended ;
APPENDIX. 565
Poske grants that the consideration of limiting cases
was frequently employed by Galileo, and although
Hofler admits having actually tested the educational
efficacy of this device with students. It would indeed
be a matter of surprise if Galileo, who may be re
garded as the inventor of the principle of continuity,
should not in his long intellectual career have applied
the principle to this most important case of all for
him. It is also to be considered that the passage
does not form part of the broad and general discus
sions of the Italian dialogue, but is tersely couched, in
the dogmatic form of a result, in Latin. And in this
way also the "velocity immutably impressed'* may
have crept in.*
*Even granting that Galileo reached his knowledge of the law of inertia
only gradually, and that it was presented to him merely as an accidental dis
covery, nevertheless the following passages which arc taken from the Paduau
edition of 1744 will show that his limitation of the law to horizontal iioti
was justified by the inherent nature of the subject treated ; and the assump
tion that Galileo toward the end of his scientific career did not possess a full
knowledge of the law, can hardly he maintained.
"Sagr. Ma quando I'artigliena si piantasse non a perpendicolo, ma in
clinata verso qualche parte, qual dovrehbe e&ser 1 il ruoto delhi pa'la ? and
rebbe ella forse, come nel 1'altro tiro, per la linea porpemlteolant, o ritor
nando anco poi per 1'istessa ? "
'Simpl. Questo non farebbe ella, ma uscita del pe.tf/o segmterebbe il
sno moto per la linea retta, rho contmua la dirittura dellu cunna, notion in
quanto il proprio peso la farebhe declinar da tal dirittura verso ten a,"
"Sagr. Talche la dirittura della canna e la regolatrico del moto della
palla : ne. fuori di tal linea si rmiove, o muoverebbe, so 'I peso proprio mm
la facesse delinare in gift. . . ," J3t\tfaje> sojfrnt t dut r MUSSIM* sisf^tnt t/t'l
innndo,
<( Sagr. But if the gtm were not placed in the prpmidietilui\ Inn were in
clined in some direction; what then would bes the motion of the ball ? Would
it follow, perhaps, as in the other case, the perpendicular, and in rrttmuMg
fall also by the same line ? "
*' Simpl. This it will not do, but having left the cannon it will follow Jt
own motion in the straight line which in a continuation of ilm axis of dm
barrel, save in so far as its own weight shall cause it 10 dovtaiu from that
direction toward the earth."
* Sagr. So that the axis of the barrel is the regulator f 'he itmtiim of the
l>all : and it ne'ther does nor will move outside of that line unless, it** own ,
weight causes it to drop downwards, , . .'*
566 THE SCIENCE OF MECHANICS.
The physical instruction which I enjoyed was in
all probability just as bad and just as dogmatic as it
was the fortune of my older critics and colleagues to
enjoy. The principle of inertia was then enunciated
as a dogma which accorded perfectly with the system.
I could understand very well that disregard of all ob
stacles to motion led to the principle, or that it must
be discovered, as Appelt says, by abstraction ; never
theless, it always remained remote and within the
comprehension of supernatural genius only. And
where was the guarantee that with the removal of all
obstacles the diminution of the velocity also ceased?
Poske (loc. cit., p. 395) is of the opinion that Galileo,
to use a phrase which I have repeatedly employed,
" disce?'ned" or "perceived" the principle immediately.
But what is this discerning? Enquiring man looks
here and looks there, and suddenly catches a glimpse
of something he has been seeking or even of some
thing quite unexpected, that rivets his attention.
Now, I have shown how this "discerning" came
about and in what it consisted. Galileo runs his eye
over several different uniformly retarded motions, and
suddenly picks out from among them a uniform, in
"Attendere insuper licet, quod velocltatis gradus, quicunque in mobill
reperiatur, est in illo suapte natura indelebiliter irnpressus, dum externae
causae accelerations, aut retardationis tollantur, quod in solo horizontal!
piano contingit : nam in planis declivibus ad est jam causa accelerationis
majoris, in acclivibus vero retardationis. Ex quo pariter sequitur, matum in
horizontal! esse quoque aeternum : si enim est aequabllis, nori debiliatur,
aut remittitur, et multo minus tollitur," Discorsi e ditnastrazioni inatema,
ticke. Dialogo terzo.
" Moreover, it is to be* remarked that the degree of velocity a body has is
indestructibly impressed in it by its own nature, provided external causes of
acceleration or retardation are wanting, which happens only on. horizontal
planes: for on descending planes there is greater acceleration, and on as
cending planes retardation. Whence it follows that motion in a horizontal
plane is perpetual : for if it remains the same, it is not diminished, or
abated, much less abolished."
APPENDIX. 567
finitely continued motion, of so peculiar a character
that if it occurred by itself alone it would certainly
be regarded as something altogether different in kind.
But a very minute variation of the Inclination trans
forms this motion into a finite retarded motion, such
as we have frequently met with in our lives. And
now, no more difficulty is experienced in recognising
the identity between all obstacles to motion and re
tardation by gravity, wherewith the ideal type of un
influenced, infinite, uniform motion is gained. As I
read this passage of Galileo's while still a young man,
a new light concerning the necessity of this ideal link
in our mechanics, entirely different from that of the
dogmatic exposition, flashed upon me. I believe that
every one will have the same experience who will ap
proach this passage without prior bias. I have not
the least doubt that Galileo above all others experi
enced that light. May my critics see to it how their
assent also is to be avoided.
I have now another important point to discuss in
opposition to C. Neumann, * whose wellknown publi
cation on this topic preceded minef shortly. I con
tended that the direction and velocity which Is taken
into account in the law of inertia had no comprehen
sible meaning if the law was referred to cc absolute
space." As a matter of fact, we can metrically deter
mine direction and velocity only in a space of which
the points are marked directly or indirectly by given
bodies. Neumann's treatise and my own were suc
cessful in directing attention anew to this point, which
*Dze PHncipien d?r GalileiNewton* schen Theorie^ Leipzig, 1870.
* Erhaltung der Arbeit , Prague, 18721, (Translated in part in the article
on "The Conservation of Energy," Popular Scientific Lectures, third edition,
Chicago, 1898.
568 THE SCIENCE OF MECHANICS.
had already caused Newton and Euler much Intellec
tual discomfort; yet nothing more than partial at
tempts at solution, like that of Streintz, have resulted.
I have remained to the present day the only one who
insists upon referring the law of inertia to the earth,
and in the case of motions of great spatial and tempo
ral extent, to the fixed stars. Any prospect of com
ing to an understanding with the great number of my
critics is, in consideration of the profound differences
of our points of view, very slight. Rut so far as I
have been able to understand the criticisms to which
my view has been subjected, I shall endeavor to an
swer them.
Hofler is of the opinion that the existence of "ab
solute motion" is denied, because it is held to be
"inconceivable." But it is a fact of "more painstak
ing selfobservation" that conceptions of absolute
motion do exist. Conceivability and knowledge of
absolute motion are not to be confounded. Only the
latter is wanting here (Joe. cit., pages 120, 164). . . .
Now, it is precisely with knowledge that the natural
inquirer is concerned. A .thing that is beyond the
ken of knowledge, a thing that cannot be exhibited to
the senses, has no meaning in natural science. I have
not the remotest desire of setting limits to the Imagi
nation of men, but I have a faint suspicion that the
persons who imagine they have conceptions of Cf ab
solute motions," in the majority of cases have in mind
the memory pictures of some actually experienced
relative motion; but let that be as it may, for It Is in
any event of no consequence. I maintain even more
than Hofler, viz., that there exist sensory illusions of
absolute motions, which can subsequently be repro
duced at any time. Every one that has repeated my
.//7Y:'.V/;/A'. 569
experiments on the sensations of movement has ex
perience d the full sensory power of such illusions.
One imagines one is flying off with one's entire en
vironment, which remains at relative rest with respect
to the body; or that one is rotating in a space that is
distinguished by nothing that is tangible. But no
measure can be applied to this space of illusion ; its
existence cannot be proved to another person, and it
cannot be employed for the metrical and conceptual
description of the facts of mechanics; it has nothing
to do with the space of geometr}'.* Finally, when
Holler (loc. cit., p. 133) brings forward the argument
that "in every relative motion one at least of the
bodies moving with reference to each other must be
affected with absolute motion," I can only say that
for the person who considers absolute motion as mean
ingless in physics, this argument has 110 force what
ever. But I have no further concern here with philo
sophical questions. To go into details as Hofler has
in some places {loc. cit., pp. 124126) would serve
no purpose before an understanding had been reached
on the main question.
Heymans {loc. cit., pp. 412, 448) remarks that an
inductive, empirical mechanics could have arisen, but
that as a matter of fact a different mechanics, based
on the nonempirical concept of absolute motion, has
arisen. The fact that the principle of inertia has
always been suffered to hold for absolute motion
which is nowhere demonstrable, instead of being re
*I flatter myself on being able to resist the temptation to infustj li^htrnwn
into a serious discussion by showing its ridiculous sido, but in reflecting on
these problems I was involuntarily forced to think of the question which n
very estimable but eccentric man once debated with in as to whether a yard
of cloth in one's dreams is as long: as a real yard of cloth, Is the dreamyard
to be really introduced into mechanics as a standard of mtmsuremtnt ?
57 o THE SCIENCE OF MECHANICS.
garded as holding good for motion with respect to
some actually demonstrable system of coordinates, is
a problem which is almost beyond power of solution
by the empirical theory. Heymans regards this as a
problem that can have a metaphysical solution only.
In this I cannot agree with Heymans. He admits
that relative motions only are given in experience.
With this admission, as with that of the possibility of
an empirical mechanics, I am perfectly content. The
rest, I believe, can be explained simply and without
the aid of metaphysics. The first dynamic principles
were unquestionably built up on empirical founda
tions. The earth was the body of reference ; the tran
sition to the other coordinate systems took place
very gradually. Huygens saw that he could refer the
motion of impinging bodies just as easily to a boat on
which they were placed, as to the earth. The devel
opment of astronomy preceded that of mechanics con
siderably. When motions were observed that were
at variance with known mechanical laws when re
ferred to the earth, it was not necessary immediately
to abandon these laws again. The fixed stars were
present and ready to restore harmony as a new sys
tem of reference with the least amount of changes in
the concepts. Think only of the oddities and difficul
ties which would have resulted if in a period of great
mechanical and physical advancement the Ptolemaic
system had been still in vogue, a thing not at all in
conceivable.
But Newton referred all of mechanics to absolute
space! Newton is indeed a gigantic personality; little
worship of authority is needed to succumb to his in
fluence. Yet even his achievements are not exempt
from criticism. It appears to be pretty much one and
APPENDIX. 57*
the same thing whether we refer the laws of motion to
absolute space, or enunciate them in a perfectly ab
stract form ;. that is to say, without specific mention
of any system of reference. The latter course is un
precarious and even practical; for in treating special
cases every student of mechanics looks for some ser
viceable system of reference. But owing to the fact
that the first course, wherever there was any real
issue at stake, was nearly always interpreted as having
the same meaning as the latter, Newton's error was
fraught with much less danger than it would other
wise have been, and has for that reason maintained
itself so long. It is psychologically and historically
intelligible that in an age deficient in epistemological
critique empirical laws should at times have been
elaborated to a point where they had no meaning. It
cannot therefore be deemed advisable to make meta
physical problems out of the errors and oversights of
our scientific forefathers, but it is rather our duty to
correct them, be they small people or great. I would
not be understood as saying that this has never hap
pened.
Petzoldt (loc. cit, pp. 192 et seq.), who is in ac
cord with me in my rejection of absolute motion, ap
peals to a principle of Avenarius,* by a consideration
of which he proposes to remove the difficulties in
volved in the problem of relative motion. I am per
fectly familiar with the principle of Avenarius, but I
cannot understand how all the physical difficxiltics in
volved in the present problem can be avoided by re
ferring motions to one's own body. On the contrary,
in considering physical dependencies abstraction must
*Dcr menschttchc Wvltbegr?j[f, Leipzig, 1891, p. 130.
572 THE SCIENCE OF MECHANICS.
be made from one's own body, so far as it exercises
any influence.*
The most captivating reasons for the assumption
of absolute motion were given thirty years ago by C.
Neumann (loc. cit., p. 27). If a heavenly body be
conceived rotating about its axis and consequently
subject to centrifugal forces and therefore oblate,
nothing, so far as we can judge, can possibly be
altered in its condition by the removal of all the re
maining heavenly bodies. The body in question will
continue to rotate and will continue to remain oblate.
But if the motion be relative only, then the case of
rotation will not be distinguishable from that of rest.
All the parts of the heavenly body are at rest with re
spect to one another, and the oblateness would neces
sarily also disappear with the disappearance of the
rest of the universe. I have two objections to make
here. Nothing appears to me to be gained by making
a meaningless assumption for the purpose of eliminat
ing a contradiction. Secondly, the celebrated mathe
matician appears to me to have made here too free a
use of intellectual experiment, the fruitfulness and
value of which cannot be denied. When experiment
ing in thought, it is permissible to modify unimportant
circumstances in order to bring out new features in a
given case; but it is not to be antecedently assumed
that the universe is without influence on the phenom
enon here in question. If it is eliminated and contra
dictions still result, certainly this speaks in favor of
the importance of relative motion, which; if it involves
difficulties, is at least free from contradictions.
* Analyse der E,mfindit<ngeiii zweite Anflage, Jena. 1900, pp, ir, 12 33, 38
208; English translation, Chicago, The Open Court Pub, Co., 1897, pp. 13 et
seq.
APPENDIX. 573
Volkmann (loc. cit., p. 53) advocates an absolute
orientation by means of the ether. I have already
spoken on this point (comp. pp. 230, 547), but I am
extremely curious to know how one ether particle is to
be distinguished from another. Until some means of
distinguishing these particles is found, it will be pref
erable to abide by the fixed stars, and where these
forsake us to confess that the true means of orienta
tion is still to be found.
Taking everything together, I can only say that I
cannot well see what is to be altered in my exposi
tions The various points stand in necessary connex
ion. After it has been discovered that the behavior
of bodies toward one another is one in which acceler
ations are determined, a discovery which was twice
formulated by Galileo and Newton, once in a general
and again in a special form as a law of inertia, it is
possible to give only one rational definition of mass,
and that a purely dynamical definition. It is not at
all, in my judgment, a matter of taste.* The concept
of force and the principle of action and reaction fol
low of themselves. And the elimination of absolute
motion is equivalent to the elimination of what is
physically meaningless.
It would be not only taking a very subjective and
shortsighted view of science, but it would also be
foolhardy in the extreme, were I to expect that my
views in their precise individual form should be in
corporated without opposition into the intellectual
systems of my contemporaries. The history of sci
ence teaches that the subjective, scientific philoso
*My definition of mass takes a more organic and more natural placet i i
Hertz's mechanics than his own, for it contains implicitly the (gorm of hin
** fundamental law."
574 THE SCIENCE OF MECHANICS.
phies of individuals are constantly being corrected
and obscured, and in the philosophy or constructive
image of the universe which humanity gradually
adopts, only the very strongest features of the
thoughts of the greatest men are, after some lapse of
time, recognisable. It is merely incumbent on the in
dividual to outline as distinctly as possible the main
features of his own view of the world.
XXIII.
(See page 273.)
Although signal individual performances in sci
ence cannot be gainsaid to Descartes, as his studies
on the rainbow and his enunciation of the law of re
fraction show, his importance nevertheless is con
tained rather in the great general and revolutionary
ideas which he promulgated in philosophy, mathe
matics, and the natural sciences. The maxim of
doubting everything that has hitherto passed f.or es
tablished truth cannot be rated too high ; although it
was more observed and exploited by his followers
than by himself. Analytical geometry with its mod
ern methods is the outcome of his idea to dispense
with the consideration of all the details of geometrical
figures by the application of algebra, and to reduce
everything to the consideration of distances. He was
a pronounced enemy of occult qualities in physics,
and strove to base all physics on mechanics, which
he conceived as a pure geometry of motion. He has
shown by his experiments that he regarded no physi
cal problem as Insoluble by this method. He took
too little note of the fact that mechanics is possible
only on the condition that the positions of the bodies
are determined in their dependence on one another by
APPENDIX. 575
a relation of force, by a function of time; and Leibnitz
frequently referred to this deficiency. The mechani
cal concepts which Descartes developed with scanty
and vague materials could not possibly pass as copies
of nature, and were pronounced to be phantasies even
by Pascal, Huygens, and Leibnitz. It has been re
marked, however, in a former place, how strongly
Descartes's ideas, in spite of these facts, have per
sisted to the present day. He also exercises a power
ful influence upon physiology by his theory of vision,
and by his contention that animals were machines,
a theory which he naturally had not the courage to
extend to human beings, but by which he anticipated
the idea of reflex motion (compare Duhem, IS Evolution
des theories physiques, Louvain, 1896).
XXIV,
(See page 378.)
To the exposition given on pages 377 and 378, in
the year 1883, I have the following remarks to add.
It will be seen that the principle of least action, like
all other minimum principles in mechanics, is a sim
ple expression of the fact that in the instances in
question precisely so much happens as possibly can
happen under the circumstances, or as is determined,
viz., uniquely determined, by them. The deduction
of cases of equilibrium from unique determination has
already been discussed, and the same question will be
considered in a later place. With respect to dynamic
questions, the import of the principle of unique de
termination has been better and more perspicuously
elucidated than in my case by J. Petzoldt in a work
entitled Maxima, Minima und Oekonomic (Altenburg,
1891). He says (loc. cit., page u): "In the case of
57 6 THE SCIENCE OF MECHANICS.
all motions, the paths actually traversed admit of be
ing interpreted as signal instances chosen from an in
finite number of conceivable instances. Analytically,
this has no other meaning than that expressions may
always be found which yield the differential equations
of the motion when their variation is equated to zero,
for the variation vanishes only when the integral
assumes a unique value."
As a fact, it will be seen that in the instances
treated at pages 377 and 378 an increment of velocity
is uniquely determined only in the direction of the
force, while an infinite number of equally legitimate
incremental components of velocity at right angles to
the force are conceivable, which are, however, for the
reason given, excluded by the principle of unique de
termination. I am in entire accord with Petzoldt
when he says : " The theorems of Euler and Hamilton,
and not less that of Gauss, are thus nothing more
than analytic expressions for the fact of experience
that the phenomena of nature are uniquely deter
mined." The uniqueness of the minimum is determi
native.
I should like to quote here, from a note which I
published in the November number of the Prague
Lotos for 1873, the following passage: "The static
and dynamical principles of mechanics may be ex
pressed as isoperimetrical laws. The anthropomor
phic conception is, however, by no means essential,
as may be seen, for example, in the principle of vir
tual velocities. If we have once perceived that the
work A determines velocity, it will readily be seen
that where work is not done when the system passes
into all adjacent positions, no velocity can be ac
quired, and consequently that equilibrium obtains.
APPENDIX. 577
The condition of equilibrium will therefore be 8A ^=0;
where A need not necessarily be exactly a maximum
or minimum. These laws are not absolutely restricted
to mechanics; they may be of very general scope.
If the change in the form of a phenomenon J3 be de
pendent on a phenomenon A, the condition that JB
shall pass over into a certain form will be 34 = 0."
As will be seen, I grant in the foregoing passage
that it is possible to discover analogies for the prin
ciple of least action in the most various departments
of physics without reaching them through the circuit
ous course of mechanics. I look upon mechanics, not
as the ultimate explanatory foundation of all the other
provinces, but rather, owing to its superior formal
development, as an admirable prototype of such an
explanation. In this respect, my view differs appar
ently little from that of the majority of physicists, but
the difference is an essential one after all. In further
elucidation of my meaning, I should like to refer to
the discussions which I have given in my Principles
of Heat (particularly pages 192, 318, and 356, German
edition), and also to my article " On Comparison in
Physics" {Popular Scientific Lectures , Knglish trans
lation, page 236). Noteworthy articles touching on
this point are: C. Neumann, "Das Ostwald'sche
Axiom des Energieurnsatzes" (JBerichte dcr k. sacks*
Geselhchctft, 1892, p. 184), and Ostwald, " Ueber das
Princip des ausgezeichneten Falles" {l&c. cit.> 1893,
p. 600).
XXV.
(See page 480.)
The Ausdehnungslchre of 1844, in which Grassmarm
expounded his ideas for the first time, is in many re
57 8 TJJE SCIENCE OF MECHANICS.
spects remarkable. The Introduction to it contains
epistemological remarks of value. The theory of spa
tial extension is here developed as a general science,
of which geometry is a special tridimensional case ;
and the opportunity is taken on this occasion of sub
mitting the foundations of geometry to a rigorous cri
tique. The new and fruitful concepts of the addition
of linesegments, multiplication of linesegments, etc.,
have also proved to be applicable in mechanics.
Grassmann likewise submits the Newtonian principles
to criticism, and believes he is able to enunciate them
in a single expression as follows: "The total force
(or total motion) which is inherent in an aggregate of
material particles at any one time is the sum of the
total force (or total motion) which has inhered in it
at any former time, and all the forces that have been
imparted to it from without in the intervening time;
provided all forces be conceived as linesegments
constant in direction and in length, and be referred
to points which have equal masses." By force Grass
mann understands here the indestructibly impressed
velocity. The entire conception is much akin to that
of Hertz. The forces (velocities) are represented as
linesegments, the moments as surfaces enumerated
in definite directions, etc., a device by means of
which every development takes a very concise and
perspicuous form. But Grassmann finds the main
advantage of his procedure in the fact that every step
in the calculation is at the same time the clear ex
pression of every step taken in the thought ; whereas,
in the common method, the latter is forced entirely
into the background by the introduction of three arbi
trary coordinates. The difference between the ana
lytic and the synthetic method is again done away
APPENDIX. 579
with, and the advantages of the two are combined.
The kindred procedure of Hamilton, which has been
illustrated by an example on page 528, will give some
idea of these advantages.
XXVI.
(See page 485.)
In the text I have employed the term "cause" in
the sense in which it Is ordinarily used. I may add
that with Dr. Carus,* following the practice of the
German philosophers, I d^sting^cish " cause, " or jR.ea.l
grund, from Erkenntnissgrund* I also agree with Dr.
Carus in the statement that "the signification of cause
and effect is to a great extent arbitrary and depends
rquch upon the proper tact of the observer. ""f
The notion of cause possesses significance only as
a means of provisional knowledge or orientation. In
any exact and profound investigation of an event the
inquirer must regard the phenomena as dependent on
one another in the same way that the geometer regards
the sides and angles of a triangle as dependent on one
another. He will constantly keep before his mind, in
this way, all the conditions of fact.
xxvn.
(See pa& 494)
My conception of economy of thought was devel
oped out of my experience as a teacher, out of the
work of practical instruction. I possessed this con
ception as early as 1861, when. I began my lectures
as PrivatDocent, and at the time believed that I was
* See his Grund, Ursache und Zvtjt>c& t R. v. Grumbkow, Dresden, x88*
and his Fundamental Problems^ pp. 7991, Chicago: The Open Court Publish
ing Co., 1891.
t Fundamental Problems, p. 84,
5 8o THE SCIENCE OF MECHANICS.
in exclusive possession of the principle, a conviction
which will, I think, be found pardonable. I am now,
on the contrary, convinced that at least some presenti
ment of this idea has always, and necessarily must
have, been a common possession of all inquirers who
have ever made the nature of scientific investigation
the subject of their thoughts. The expression of this
opinion may assume the most diverse forms ; for ex
ample, I should most certainly characterise the guid
ing theme of simplicity and beauty which so distinctly
marks the work of Copernicus and Galileo, not only
as sesthetical, but also as economical. So, too, New
ton's Regulcz philosophandi are substantially influenced
by economical considerations, although the economi
cal principle as such is not explicitly mentioned. In
an interesting article, " An Episode in the History of
Philosophy," published In The Open Court for April
4> 1^95, Mr. Thomas J. McCormack has shown that
the idea of the economy of science was very near to
the thought of Adam Smith (Essays}. In recent times
the view in question has been repeatedly though di
versely expressed, first by myself in my lecture Ueber
die ILrhaltzmg der Arbeit (1875), then by Clifford in
his Lectures and ILssays (1872), by Kirchhoff in his
Mechanics (1874), and by Avenarius (1876). To an
oral utterance of the political economist A. Herrmann
I have already made reference in my Erhaltung der
Arbeit (p. 55, note 5); but no work by this author
treating especially of this subject is known to me.
I should also like to make reference here to the
supplementary expositions given in my Popular Scien
tific Lectures (English edition, pages 186 et seq.) and
in my Principles of Heat (German edition, page 294).
In the latter work, the criticisms of Petzoldt (Vicrtel
APPENDIX. 581
jahrsschrift fur wissenschaftliche Philosophic, 1891) are
considered. Husserl, in the first part of his work,
Logische Untersuchungen (1900), has recently made
some new animadversions on my theory of mental
economy; these are in part answered in my reply to
Petzoldt. I believe that the best course is to post
pone an exhaustive reply until the work of Husserl is
completed, and then see whether some understanding
cannot be reached. For the present, however, I
should like to premise certain remarks. As a natural
inquirer, I am accustomed to begin with some special
and definite inquiry, and allow the same to act upon
me in all its phases, and to ascend from the special
aspects to more general points of view. I followed
this custom also in the investigation of the develop
ment of physical knowledge. I was obliged to pro
ceed in this manner for the reason that a theory of
theory was too difficult a task for me, being doubly
difficult in a province in which a minimum of indis
putable, general, and independent truths from which
everything can be deduced is not furnished at the
start, but must first be sought for. An undertaking
of this character would doubtless have more prospect
of being successful if one took mathematics as one's
subjectmatter. I accordingly directed my attention
to individual phenomena : the adaptation of ideas to
facts, the adaptation of ideas to one another,* mental
* Popular Scientific Lectures^ English edition, pp 244 tit seq.> whore the
adaptation of thoughts to one another is described as the object of theory
proper. Grasstnann appears to me to say pretty much the same In tho intro
duction to his Ausdehnungslehre of 3:844, page xix: **Tho firnt division of all
the sciences is that into real and formal, of which the real sciences depict
reality in thought as something independent of thought, and find their truth
in the agreement of thought with that reality ; the formal sciences, ot tho
other hand, have as their object that which has been posited by thought
and itself, find their truth in the agreement of the mental processes with one
another."
582 THE SCIENCE OF MECHANICS.
economy, comparison, Intellectual experiment, the
constancy and continuity of thought, etc. In this in
quiry, I found it helpful and restraining to look upon
everyday thinking and science in general, as a bio
logical and organic phenomenon, in which logical
thinking assumed the position of an ideal limiting
case. I do not doubt for a moment that the investi
gation can be begun at both ends. I have also de
scribed my efforts as epistemological sketches.* It
may be seen from this that I am perfectly able to dis
tinguish between psychological and logical questions,
as I believe every one else is who has ever felt the
necessity of examining logical processes from the
psychological side. But it is doubtful if anyone who
has carefully read even so much as the logical analysis
of Newton's enunciations in my Mechanics, will have
the temerity to say that I have endeavored to erase
all distinctions between the < blind " natural thinking
of everyday life and logical thinking. Even if the
logical analysis of all the sciences were complete, the
biologicopsychological investigation of their develop
ment would continue to remain a necessity for me,
which would not exclude our making a new logical
analysis of this last investigation. If my theory of
mental economy be conceived merely as a teleological
and provisional theme for guidance, such a concep
tion does not exclude its being based on deeper foun
dations,")" but goes toward making it so. Mental econ
omy is, however, quite apart from this, a very clear
logical ideal which retains its value even after its logi
cal analysis has been completed. The systematic
form of a science can be deduced from the same prin
* Principles of Heat, Preface to the first German edition.
^Analysis of the Sensations, second German edition, pages 6465.
APPENDIX. 583
ciples in many different manners, but some one of
these deductions will answer to the principle of econ
omy better than the rest, as I have shown in the case
of Gauss's dioptrics.* So far as I can now see, I do
not think that the investigations of Husserl have
.affected the results of my inquiries. As for the rest,
I must wait until the remainder of his work is pub
lished, for which I sincerely wish him the best suc
cess.
When I discovered that the idea of mental econ
omy had been so frequently emphasised before and
after my enunciation of it, my estimation of my per
sonal achievement was necessarily lowered, but the
idea itself appeared to me rather to gain in value on
this account; and what appears to Husserl as a de
gradation of scientific thought, the association of it
with vulgar or " blind" (?) thinking, seemed to me to
be precisely 'an exaltation of it. It has outgrown the
scholar's study, being deeply rooted in the life of hu
manity and reacting powerfully upon it.
xxvin.
(See page 4970
The paragraph on page 497, which was written In
1883, met with little response from the majority of
physicists, but it will be noticed that physical exposi
tions have since then closely approached to the ideal
there indicated. Hertz's "Investigations on the Prop
agation of Electric Force" (1892) affords a good in
stance of this description of phenomena by simple
differential equations.
* Principles of Heat, German edJtim, page 394,
584 THE SCIENCE OF MECHANICS.
XXIX.
(See page 501.)
In Germany, Mayer's works at first met "with a very
cool, and in part hostile, reception ; even difficulties
of publication were encountered; but in England
they found more speedy recognition. After they had
been almost forgotten there, amid the wealth of new
facts being brought to light, attention was again
called to them by the lavish praise of Tyndall in his
book Heat a Mode of Motion (1863). The consequence
of this was a pronounced reaction in Germany, which
reached its culminating point in Diihring's work Robert
Mayer, the Galileo of the Nineteenth Century (1878). It
almost appeared as if the injustice that had been
done to Mayer was now to be atoned for by injustice
towards others. But as in criminal law, so here, the
sum of the injustice is only increased in this way, for
no algebraic cancelation takes place. An enthusiastic
and thoroughly satisfactory estimate of Mayer's per
formances was given by Popper in an article in A.US
land (1876, No. 35), which is also very readable from
the many interesting epistemological apergus that it
contains. I have endeavored (Principles of If eat ^ to
give a thoroughly just and sober presentation of the
achievements of the different inquirers in the domain
of the mechanical theory of heat. It appears from
this that each one of the inquirers concerned made
some distinctive contribution which expressed their
respective intellectual peculiarities. Mayer may be
regarded as the philosopher of the theory of heat and
energy; Joule, who was also conducted to the prin
ciple of energy by philosophical considerations, fur
APPENDIX. 585
nishes the experimental foundation ; and Helmholtz
gave to it its theoretical physical form. Helmholtz,
Clausius, and Thomson form a transition to the views
of Carnot, who stands alone in his ideas. Each one
of the firstmentioned inquirers could be eliminated.
The progress of the development would have been
retarded thereby, but it would not have been checked
(compare the edition of Mayer's works by Weyrauch,
Stuttgart, 1893).
xxx.
(See page 504.)
The principle of energy is only briefly treated in
the text, and I should like to add here a few remarks
on the following four treatises, discussing this subject,
which have appeared since 1883 : Die physikalischen
Grundsatze der elektrischen Kraftubcrtragung, by J.
Popper, Vienna, 1883; Die JLehre von der Mnergie, by
G. Helm, Leipsic, 1887; Das Princlf der J&rhaltung
der Energie, by M. Planck, Leipsic, 1887 ; and Das
Problem der Continuitdt in der Mathematik und J^fccha
nik, by F. A. Muller, Marburg, 1886.
The independent works of Popper and Helm are,
in the aim they pursue, in perfect accord, and they
quite agree in this respect with my own researches,
so much so in fact that I have seldom read anything
that, without the obliteration of individual differences,
appealed in an equal degree to my mind. These two
authors especially meet in their attempt to enunciate
a general science of energetics ; and a suggestion of
this kind is also found in a note to my treatise Ueher
die Erhaltung der Arbeit, page 54. Since then ** ener
getics " has been exhaustively treated by Helm, Ost
wald, and others.
5 86 THE SCIENCE OF MECHANICS.
In 1872, in this same treatise (pp. 42 et seqq.), I
showed that our belief in the principle of excluded
perpetual motion is founded on a more general belief
in the unique determination of one group of (mechani
cal) elements, a /By . . ., by a group of different ele
ments, xyz . . . Planck's remarks at pages 99, 133,
and 139 of his treatise essentially agree with this;
they are different only in form. Again, I have re
peatedly remarked that all forms of the law of causal
ity spring from subjective impulses, which nature is
by no means compelled to satisfy. In this respect
my conception is allied to that of Popper and Helm.
Planck (pp. 21 et seqq., 135) and Helm (p. 25 et
seqq.) mention the 6C metaphysical" points of view
by which Mayer was controlled, and both remark
(Planck, p. 25 et seqq., and Helm, p. 28) that also
Joule, though there are no direct expressions to justify
the conclusion, must have been guided by similar
ideas. To this last I fully assent.
With respect to the socalled " metaphysical"
points of view of Mayer, which, according to Helm
holtz, are extolled by the devotees of metaphysical
speculation as Mayer's highest achievement, but which
appear to Helmholtz as the weakest feature of his
expositions, I have the following remarks to make.
With maxims, such as, "Out of nothing, nothing
comes/* "The effect is equivalent to the cause/ 1 and
so forth, one can never convince another of anything.
How little such empty maxims, which until recently
were admitted in science, can accomplish, I have
illustrated by examples in my treatise Die jKrhaltung
der Arbeit. But in Mayer's case these maxims are, in
my judgment, not weaknesses. On the contrary, they
are with him the expression of a powerful instinctive
APPENDIX. 587
yearning, as yet unsettled and unclarified, after a
sound, substantial conception of what is now called
energy. This desire I should not exactly call meta
physical. We now know that Mayer was not wanting
in the conceptual power to give to this desire clear
ness. Mayer's attitude in this point was in no respect
different from that of Galileo, Black, Faraday, and
other great inquirers, although perhaps many were
more taciturn and cautious than he.
I have touched upon this point before in my Analy
sis of the Sensations, Jena, 1886, English translation,
Chicago, 1897, p. 174 et seqq. Aside from the fact
r that I do not share the Kantian point of view, in fact,
occupy no metaphysical point of view, not even that
of Berkeley, as hasty readers of my lastmentioned
treatise have assumed, I agree with F. A, Miiller's
remarks on this question (p. 104 et seqq.). For a
more exhaustive discussion of the principle of energy
see my Principles of Heat.
CHRONOLOGICAL TABLE
OF A FEW
EMINENT INQUIRERS
AND OP
THEIR MORE IMPORTANT MECHANICAL WORKS.
ARCHIMEDES (287212 B. C.). A complete edition of his works was
published, with the commentaries of Eutocius, at Oxford, in
1792; a French translation by F. Peyrard (Paris, 1808); a Ger
man translation by Ernst Nizze (Stralsund, 1824).
LEONARDO DA VINCI (14521519). Leonardo's scientific manuscripts
are substantially embodied in H. Grothe's work, ' ' Leonardo
da Vinci als Ingenieur und Philosoph " (Berlin, 1874).
GUIDO UBALDI(O) e Marchionibus Montis (15451607). Afechani
corum Liber (Pesaro, 1577).
S. STEVINUS (i548i620). Beghinseltn der Weegkonst (Leyden,
1585); Ifypojfintwiata Mathematica (Leyden, 1608).
GALILEO (15641642). Discorsi e dimostraziotti matematiche (Ley
den, 1638). The first complete edition of Galileo's writings
was published at Florence (18421856), in fifteen volumes 8vo.
KEPLER (15711630). Astronomia Nova (Prague, 1609) ; Ifdrrtw
nice Mundi (Linz, 1619); Stereometria Dolionun (Linz, 1615).
Complete edition by Frisch (Frankfort, 1858),
MARCUS MARCI (15951667). De Proportions Mot us (Prague, 1639).
DESCARTES (15961650). Principia Philosophic (Amsterdam, 1644).
ROBERVAL (16021675). Sitr la composition des mouvemeuts. Atu\
Mdm. de r Acad. de Paris, T. VI.
GUERICKE (16021686). jRjcperimenfa JViyva, ut Vocantur
burgica (Amsterdam, 1672).
59 o THE SCIENCE OF MECHANICS.
FERMAT (16011665). Varia Opera (Toulouse, 1679).
TORRICELLI (16081647). Opera Geometrica (Florence, 1644).
WALLIS (16161703). Mechanica Sive de Motu (London, 1670).
MARIOTTE (16201684). (Euvres (Leyden, 1717).
PASCAL, (16231662). Rdcit de la grande experience de req
des liqueurs (Paris, 1648); Traite de I'eqTiilibre des liqueurs et
de la pesanteur de la masse de I* air. (Paris, 1662).
BOYLE (16271691). Experimenta Phvsico Mechanica (London,
1660).
HUYGENS (16291695). A StMimary Account of the JLa*vs of Mo
tion. Philos. Trans. 1669 ; Horologiitm Oscillatarium (Paris,
1673); Opuscula Posthuma (Leyden, 1703).
WREN (16321723). Lex Naturee de Collisions Corporum* Philos.
Trans. 1669.
LAMI (16401715). Nouvelle manikre de ddvwntrer les prindpaux
theor&mes des Siemens des mecaniques (Paris, 1687).
NEWTON (16421726). Philosophies Naturalis Principia Mathenia
tica (London, 1686).
LEIBNITZ (16461716). Acta Eruditorum, 1686, 1695 ; JLeibnitzii
et Joh. Bernoullii Comerciunt Epistolicuw (Lausanne and Ge
neva, 1745).
JAMES BERNOULLI (16541705). Opera Omnia ("Geneva, 1744).
VARIGNON (16541722). Projet d^une nouvelle mecaniquc (Paris,
1687).
JOHN BERNOULLI (16671748). Acta Erudit. 1693; Opera Omnia
(Lausanne, 1742).
MAUPERTUIS (16981759). Mem. de t'Acad. de Paris, 1740 ; /l//w,
de P Acad. de JBerZin, 1745, I 747 "> GZuvres (Paris, 1752).
MACL.AURIN (16981746). A Complete System of fluxions (Edin
burgh, 1742).
DANIEL BERNOULLI (17001782). Comment. Acad. Petrop*^ T. X.
Hydrodynamica (Strassburg, 1738).
KULER (17071783). Mechanica sive Mo tits Scientifi (Petersburg,
1736) ; Methodus Inveniendi Lineas Curvets (Lausanne, 1744).
CHRONOLOGICAL TABLE. 591
Numerous articles in the volumes of the Berlin and St. Peters
burg academies.
CLAIRAUT (17131765). Theorie de la Jigztre de la terre (Paris,
1743)
D'ALEMBERT (17171783). Traite de dynaunque (Paris, 1743).
LAGRANGE (17361813). Essai d^une nottvelle method? pour deter
miner les maxima et minima. Misc. Taurin. 1762 ; Alecanique
analytique (Paris, 1788).
LAPLACE (17491827). Met unique celeste (Paris, 1799).
FOURIER (17681830). Theorie analytique de la chaleur (Paris,
1822).
GAUSS (17771855). De Figura Fluidorum in Statit A^qiti'ibriL
Comment. Societ. Gottin^., 1828 ; Weues Princip der Mechanik
(Crelle's Journal, IV, 1829); hitensitas Vis Magnetic^ Ter res iris
ad Memuram Absolutam Revocata (1833). Complete works
(Gottingen, 1863).
POINSOT (17771859). Elements de statiqite (Paris, 1804).
PONCELET (17881867). Cours de mecanique (Metz, 1826),
BELANGER (17901874). ' Cours de me'cftnique (Paris, 1847).
MOBIUS (1790 1867). Statik (Leipsic, 1837).
CORIOLIS (1792 1843). Traitl de meamiqiu (Paris, 1829).
C. G. J. JACOBI (18041851). Vorfesungen Uber Dynamik, heraus
gegeben von Clebsch (Berlin, 1866).
W. R. HAMILTON (18051865). Lectures on Quaternions, 1853,
Essays.
GRASSMANN (18091877). Ausdehnutigslehre (Leipsic, 1844).
H. HERTZ (^571894). Princi'pien dcr Afcchartik (T^eipsi<\ 1804)
INDEX.
Absolute, space, time, etc. (See the
nouns.)
Absolute units, 278, 284.
Abstractions, 482.
Acceleration, Galileo on, 131 et seq.;
Newton on, 238; also 218, 230, 236,
243, 245,
Action and reaction, 198201, 242.
Action, least, principle of, 364380,
454 ; sphere of, 385.
Adaptation, in nature, 452; of thoughts
to facts, 6, 515 et seq., 581 et seq.
Adhesion plates, 515
Aerometer, effect of suspended par
ticles on, 208.
Aerostatics. See Air.
Affined, 166.
Air, expansive force of, 127; quanti
tative data of, 124; weight of, 113;
pressure of, 114 et seq.; nature of,
517 et seq.
Airpump, 122 et seq.
Aitken, 525.
Alcohol and water, mixture of, 384 et
seq.
Algebra, economy of, 486.
Algebraical mechanics, 466.
All, The, necessity of its considera
tion in research, 235, 461, 516.
Analytical mechanics, 465480.
Analytic method, 466.
Anaxogoras, 509, 517.
Animal free in space, 290.
Animistic points of view in mechan
ics, 461 et seq.
Appelt, 566.
Archimedes, on the lever and the
centre of gravity, 8ix; critique of
his deduction, 1314, 513 et seq,;
illustration of its value, 10 ; on hy
drostatics, 8688; various modes of
deduction of his hydrostatic prin
ciple, 104 ; illustration of his prin
ciple, 106.
Archytas, 510.
Areas, law of the conservation of,
293305.
Aristotle, 509, 511, 517, 518.
Artifices, mental, 492 et seq*
Assyrian monuments, i.
Atmosphere. See Air.
Atoms, mental artifices, 492.
Attraction, 246.
Atwood's machine, 149.
Automata, 511.
Avenarius, R., x, 571, 580.
Axiomatic certainties, 561.
Babbage, on calculating machines,
488.
Babo, Von, 150.
Baliani, 524.
Ballistic pendulum, 328.
Balls, elastic, symbolising pressures
in liquid, 419.
Bandbox, rotation of, 301.
Barometer, height of raotintaiaH de
termined by, 115, 117.
Base, pressure of liquids on, 90, 99,
Belangcr ( on impulse, 37*.
Benedetti, 520 et seq,, 561,
Berkeley, 387.
Bernoulli, Daniel, his geometrical
dernw strauon of the parallelo
gram of forces, 40*42; criticiHin of
Bernoulli s demonstration, 4246;
on the law of areas, 393 ; on the
principle of vis vtoa, 343, 348; on
594
THE SCIENCE OF MECHANICS.
the velocity of liquid efflux, 403 ;
his hydro dynamic principle, 408;
on the parallelism of strata, 409;
his distinction of hydrostatic and
hydrodynamic pressure, 413.
Bernoulli, James, on the catenary,
74 ; on the centre of oscillation, 311
et seq ; on the brachistochrone,
426; on the isoperimetrical prob
lems, 428 et seq.; hfs character, 428;
his quarrel with John, 431 ; his Pro
gramrna, 430.
Bernoulli, John, his generalisation of
the principle of virtual velocities,
56; on the catenary, 74; on centre
of oscillation, 333, 335 ; on the prin
ciple otuzsvzva, 343 ; on the anal
ogies between motions of masses
and light, 372; his liquid pendulum
410 ; on the brachistochrone, 425 et
seq.; his character, 427; his quar
rel with James, 431 ; his solution of
the isoperimetrical problem, 431.
Black, 124, 587.
Boat in motion, Huygen's fiction of
a, 315. 325. 570.
Body, definition of, 506.
Bolyai, 493.
Bomb, a. bursting, 293.
Borelii, 533.
Bosscha, J., 531.
Bouguer, on the figure of the earth,
395
Boyle, his law, 125 et seq.; his inves
tigations in aerostatics, 123.
Brachistochrone, problem of the, 425
et seq.
Brahe, Tycho, on planetary motion,
187.
Bruno, Giordano, his martyrdom,
446.
Bubbles, 392.
Bucket of water, Newton's rotating,
327, 232, 543.
Budde, 547.
Cabala, 489.
Calculating machines, 488.
Calculus, differential, 424 ; of varia
tions, 436 et seq.'
Canal, fluid, equilibrium of, 396 et
seq.
Cannon and projectile, motion of,
291.
Canton, on compressibility of liquids,
92.
Carnot, his performances, 501, 585;
his formula, 327.
Carus, P., on cause, 516.
Catenary, The, 74, 379, 425.
Cauchy, 47.
Causality, 483 et seq., 502.
Cause and effect, economical charac
ter of the ideas, 485; equivalence
of, 502, 503 ; Mach on, 555 ; Carus
on, 579
Causes, efficient and final, 368.
Cavendish, 124.
Cells of the honeycomb, 453.
Centimetre grammesecond system,
285.
Central, centrifugal, and centripetal
force, See Force.
Centre of gravity, 14 et seq, descent
of, 52; descent and ascent of, 174
et seq,, 408 ; the law of the conser
vation of the, 287305.
Centre of gyration, 334,
Centre of oscillation, 173 et seq., 331
335 ; Mersenne, Descartes, and
Huygens on, 174 et seq.; relations
of, to centre of gravity, 180185;
convertibility of, with point of sus
pension, 186.
Centre of percussion, 327.
Centripetal impulsion, 528 et seq.
Chain, Stevinus's endless, 25 et seq.>
500; motion of, on inclined plane,
347
Change, unrelated, 504.
Character, an ideal universal, 481.
Chinese language, 482.
Church, conflict of science and, 446.
Circular motion, law of, ifx> 161.
Clairaut, on vis viva, work, etc., 348;
on the figure of the earth, 395 ; on
liquid equilibrium, 396 et seq.; oa
level surfaces, etc., 398,
Classen, J,, 555
Classes and trades, the function af,
in the development of science, 4.
INDEX.
595
Claasius, 497, 499, 501, 585.
Clifford, 580.
Coefficients, indeterminate, La
grange's, 471 et seq.
Collision of bodies. See Impact.
Colors, analysis of, 481.
Column, a heavy, at rest, 258.
Commandinus, 87.
Communication, the economy of, 78.
Comparative physics, necessity of,
498.
Component of force, 34.
Composition, of forces, see Forces;
Gauss's principle and the, 364 ; no
tion of, 526.
Compression of liquids and gases,
407.
Conradus, Balthazar, 308.
Conservation, of energy, 499 et seq.
585 et seq.; of quantity of motion,
Descartes and Leibnitz on, 272, 274,
purpose of the ideas of, 504.
Conservation of momentum, of the
centre of gravity, and of areas,
laws of, 287305 ; these laws, the
expression of the laws of action
and reaction and inertia, 303.
Conservation of momentum and ?'/$
viva interpreted, 326 et seq.
Constancy of quantity of matter, mo
tion, and energy, theological basis
of, 456.
Constraint,,335, 352 ; least, principle
of, 350364, 550, 5/6.
Continuity, the principle of, 140, 490
et seq , 565,
Continuum, physicomechanical, 109.
Coordinates, forces a function of,
397. See ForceJunction,
Copernicus, 232, 457, 531, 580.
Coriolis, on vis wiva arid work, 272.
Counterphenomena, 503.
Counterwork, 363, 366,
Counting, economy of, 486.
Courtivron, his law of equilibrium,
73
CteIbius, his airgun, no, 511.
Currents, oceanic, 302.
Curtius Rufus, 210,
Curveelements, variation of, 432,
Curves, maxima and minima of, 429.
Cycloids, 143, 186, 379, 427.
Cylinder, double, on a horizontal
surface, 60; rolling on an inclined
plane, 345
Cylinders, axal, symbolising the re
lations of the centres of gravity and
oscillation, 183.
D'Alembert, his settlement of the
dispute concerning the measure of
force, 149, 276; his principle, 331
343
D'Arcy, on the law of areas, 293.
Darwin, his theories, 452, 459.
Declination from free motion, 352
356.
Deductive development of science,
421.
Democntus, 518.
Demonstration, the mania for, 18,82;
artificial, 82.
Departure from free motion, 355,
Derived units, 278.
Descartes, on the measure of force,
148, 250, 270, 272276 ; on quantity of
motion, conservation of momen
tum* etc., 272 et seq.; character of
his physical inquiries, 273, 538, 553,
574 ; his mechanical ideas, 250,
Descent, en inclined planes, 134 et
seq.; law of, 137; in chords of cir
cles, 138; vertical, motion of,
treated by Hamilton's principle,
383; quickest, curve of, 436; of
centre of gravity, 52, 174 et seq.,
408.
Description, a fundamental feature
of science, 5, 555,
Design, evidences of, in natures* 452.
Determinants, economy of, 487.
Determination, particular, 544.
Determinative factors of physical
processes, 76.
Diels, 520.
Differences, of quantities, thwir rftle
in nature, 336; of velacUiK 333.
Differential calculua, 424.
Differential laws, 355, 461.
Dimensions, theory of, 279,
Dioptrics, Gauss's economy of, 4$*;,
Disillusionment, due to insight, 77
596
THE SCIENCE OF MECHANICS.
Dreamyard, 569.
Dub em, 575.
Diihring, x, 352, 584.
Dynamics, the development of the
principles of, 128255 ; retrospect
of the development of, 245255;
founded by Galileo, 128 ; proposed
new foundations for, 243 ; chief re
sults of the development of, 245,
246; analytical, founded, by La
grange on the principle of virtual
velocities, 467.
Earth, figure of, 395 et seq.
Economical character of analytical
mechanics, 480.
Economy in nature, 459.
Economy of description, 5.
Economy of science, 481494.
Economy of thought, the basis and
essence of science, ix, 6, 481 ; of
language, 481 ; of all ideas, 482 ; of
the ideas cause and effect, 484; of
the laws of nature, 485 ; of the law
of refraction, 485 ; of mathematics,
486; of determinants, 487; of cal
culating machines, 488; of Gauss's
dioptrics, moment of inertia, force
function, 489; history of Mach's
conception of, 579.
Efflux, velocity of liquid, 402 et seq.
Egyptian monuments, i.
Eighteenth century, character of, 458.
Elastic bodies, 315, 317, 320.
Elastic rod, vibrations of, 490.
Elasticity, revision of the theory of,
496.
Electromotbr, Page's, 262 ; motion of
a free, 296, et seq.
Elementary laws, see Differential
la<w$,
Ellipsoid, triaxal, 73 ; of inertia, 186;
central, 186.
Empedocles, 509, 517,
Encyclopaedists, French, 463.
Energetics, the science of, 585.
Energy, Galileo's use of the word,
271; conservation of, 499 et seq.;
potential and kinetic, 272, 499;
principle of, 585 et seq. See Vis
viva..
Enlightenment, the age of, 458.
Epstein, 541.
Equations, of motion, 342; of me
chanics, fundamental, 270.
Equilibrium, the decisive conditions
of, 53; dependence of ; on a maxi
mum or minimum of work, 69 ; sta
ble, unstable, mixed, and neutral
equilibrium, 7071 ; treated by
Gauss's principle, 355 ; figures of,
393; liquid, conditions of, 386 et
seq.
Equipotential surfaces. See Level
Surfaces.
Ei gal, 499.
Error, our liability to, in the recon
struction of facts, 79.
Ether, orientation by means of the,
573
Euler, on the loz de rcpos^ 68 ; on
moment of inertia, 179, 182, 186; on
the law of areas, 393 ; his form of
D' Alembert's principle, 337 ; on vis
viva, 348; on the principle of least
action, 368, 543, 576; on the jsoperi
metrical problems and the calculus
of variations, 433 et seq ; his theo
logical proclivities, 449, 455; his
contributions to analytical me
chanics, 466 ; on absolute motion,
543, 568.
Exchange of velocities Jn impact, 315,
Experience, i et seq., 481, 490.
Experimenting in thought, 523, 582,
Experiments, 509, 514.
Explanation, 6.
Extravagance in nature, 459,
Facts and hypotheses, 494, 496, 498,
Falling bodies, early views of, 128;
investigation of the laws of, 130 at
seq,, 520 etseq.; laws of, accident
of their form, 247 et seq.; see Des
cent.
Falling, sensation of, 206.
Faraday, 124, 503, 530, 534, 5^7
Feelings, the attempt to explain them
by motion, 506.
Fermat, on the method of tangents,
423
Fetishism, in modern ideas, 463,
INDEX.
597
Fiction of a boat in motion, Huy
gens's, 315, 325.
Figure of the earth, 395 et seq.
Films, liquid, 386, 392 et seq.
Fixed stars, 543 et seq., 568.
Flow, lines of, 400 ; of liquids, 416 et
seq.
Fluids, the principles of statics ap
plied to, 86r 10. See Liquids.
Fluid hypotheses, 496.
Foppl, 535
Force, moment of, 37; the experien
tial nature of, 4244 ; conception of;
in statics, 84 ; general attributes
of, 85; the Galilean notion of, 142,
dispute concerning the measure of
148, 250, 270, 274276; centrifugal
and centripetal, 158 et seq.; New
ton on, 192, 197, 238,239; moving,
203, 243 ; resident, impressed, cen
tripetal, accelerative, moving, 238^
239 ; the Newtonian measure of,
203, 239, 276 ; lines of. 400.
Force function, 398 et seq., 479, 489;
Hamilton on, 350.
Forcerelations, character of, 237.
Forces, the parallelogram of, 32, 33
48, 243 ; principle of the composi
tion and resolution of, 3348, 197 et
seq.; triangle of, 108 ; mutual inde
pendence of, 154; living, see Vts
viva*, Newton on the parallelogram
of, 192, 197; impressed, equili
brated, effective, gained and lost,
336; molecular, 384 et seq,; fxinc
tions of co5rdinates, 397, 402; cen
tral, 397; at a distance, 534 et seq*
Formal development of science, 423,
Formulae, mechanical, 269286.
Foucault and Toepler, optical method
of, 125.
Foucault's pendulum, 302.
Fourier, 270, 526.
Free rigid body, rotation of, 295.
Free systems, irmtulal action of, 287.
Friction, of xnmtite bodies in liquids,
208 ; motion of liquids tinder, 416
et seq.
Friedlander, P. And J., 547.
Functions, mathematical, their office
in science, 492.
Fundamental equations of mechan
ics, 270.
Funicular machine, 32.
Funnel, plunged in water, 412; rotat
ing liquid in, 303.
" Galileo," name for unit of acceler
ation, 285.
Galileo, his dynamical achievements,
128155 ; his deduction of the law
of the lever, 12, 514; his explana
tion of the inclined plane by the
lever, 23 ; his recognition of the
principle of virtual velocities, 51;
his researches in hydrostatics, 90;
his theory of the vacuum, 112 et
seq ; his discovery of the laws of
falling bodies, 130 et seq , 522; his
clock, 133 ; character of his in
quiries, 140; his foundation of the
law of inertia, 143, 524 et seq., 563
et seq ; on the notion of accelera
tion, 145; tabular presentment of
his discoveries, 147; on the pendu
lum and the motion of projectiles,
152 et seq., 525 et seq.; founds dy
namics, 128; his pendulum, 162;
his reasoning on the laws of falling
bodies, 130, 131, 247; his favorite
concepts, 250; on impact, 308312;
his struggle with the Church, 446;
on the strength of materials, 451;
does not mingle science with the
ology, 457; on inertia, 509; his
predecessors, 520 et seq.; on gravi
tation, 533; on the tides, 537 et
seq., 580, 587.
Gaseous bodies, th principles of
statics applied to, 1x0127.
Gases, flow of, 405 ; compression of,
407.
Gauss, his view of the principle of
virtual velopities, 76; on absolute
units, 278 ; his principle of louat
constraint, 350364, 550 et seq., 576;
on the statics of liquids, 390; his
dioptrics, 489,
Gerbcr, Paul, 535.
Gilbert, 462, 532, 533,
Goldbeck, B, 532,
Compere, 518,
598
THE SCIENCE OF MECHANICS.
Grassi, 94.
Grassmann, 480, 577 et seq., 581.
Gravitation, universal, 190, 531 et
seq., 533
Gravitational system of measures,
284286.
Gravity, centre of. See Centre of
gravity.
Greeks, science of, 509 et seq.
Green's Theorem, 109.
Guericke, his theological specula
tions, 448 ; his experiments in aero
statics, 117 et seq.; his notion of
air, 118 ; his airpump, 120: his air
gun, 123.
Gyration, centre of, 334.
Hal ley, 448.
Hamilton, on forcefunction, 350; his
hodograph, 527; his principle, 380
384, 480, 576
Heat, revision of the theory of, 496.
Helm, 585 et seq.
Helmholtz, ix ; on the conservation
of energy, 499, 501. 585.
Hemispheres, the Magdeburg, 122.
Henke, R., 552.
Hermann, employs a form of D'Alem
bert's principle, 337; on motion in
a resisting medium, 435.
Hero, his fountain, 411 ; on the mo
tion of light, 422 ; on maxima and
minima, 451, 511, 518 et seq.
Herrmann, A., 580.
Hertz's system of Mechanics, 548 et
seq., 583.
Heymans, 558 et seq., 569 et seq.
Hiero, 86.
Hipp, chronoscope of, 151.
Hodograph, Hamilton's, 527.
Hciner, 558 et seq., 568.
Holder, O., 514.
Hollow space, liquids .enclosing, 392.
Homogeneous, 279.
Hooke, 532.
Hopital, L' on the centre of oscilla
tion, 331 ; on the brachistochrone,
426.
Horror vacui, 112.
Hume, on causality, 484.
Husserl, 581 et seq.
Huygens, dynamical achievements
of, 155187; his deduction of the
law of the lever, 1516; criticism
of his deduction, 1718 ; his rank as
an inquirer, 155 ; character of his
researches, 156 et seq.; on centri
fugal and centripetal force, 158 et
seq ; his experiment with light
balls in rotating fluids and his ex
planation of gravity, 162, 528 et
seq.; on the pendulum and oscilla
tory motion, 162 et seq.; on the
centre of oscillation, 173 et seq.;
his principle of the descent and
rise of the centre of gravity, 174;
his achievements in physics, 186,
187, 530; his favorite concepts, 251;
on impact, 313327, 570; on the
principle of t>zs viva, 343, 348; on
the figure of the earth, 395 ; his op
tical researches, 425; does not min
gle science and theology, 457, 575.
Hydraulic ram, Montgolfier's, 411.
Hydrodynamic pressure, 413.
Hydrodynamics, 402420.
Hydrostatic pressure, 413.
Hydrostatics, 384402.
Hypotheses and facts, 494.
Images, constructive, 548 et seq.
Inclined plane, the principle of the,
2433 5i5 et seq,; Galileo's deduc
tion of its laws, 151; descent on t
354 ; movable on rollers, 357 et seq.
Indeterminate coefficients, La
grange's, 471 et seq.
Inelastic bodies, 317, 318.
Inertia, history and criticism of the
law of, 141, 143, 232, 238, 520* 524 et
seq., 542 et seq,, 560 et stq , 563 et
seq., 567 et seq.; moment of, 179,
182,186,489; bodies with variable
moments of, 302 ; Newton on, 238,
243.
Inertial system, 5x5.
Impact, the Jaws of, 305330; force
of, compared with pressure, 312;
in the Newtonian view, 317 et seq.;
oblique, 327; Maupertuis's treat
ment of, 365.
Impetus, 275,
INDEX.
599
Impulse, 271.
Inquirers, the grsat, character and
x'alue of their performances, 7 ;
their different tasks, 76 ; their atti
tude towards religion, 457 ; their
philosophy, 516.
Inquiry, typical modes of, 317.
Instinct, mechanical, importance of,
304.
Instinctive knowledge, its cogency,
origin, and character, i, 2628, 83.
Instincts, our animal, 463.
Instruction, various methods of, 5.
Integral laws, 255, 461.
Intelligence, conception of, in na
ture, 461.
Interdependence of the facts of na
ture, 502 et passim.
Internal forces, action of, on free
systems, 289, 295.
International language, 481.
Isolation, 527.
Isoperimetrical problems, 421446 ;
Euler's classification of, 433,
Isothermal surfaces, 400.
Jacobi, 76, 381,459; on principle of
least action, 371.
Jellett, on the calculus of variations,
437 et seq.
Johannesson, 547.
Jolly, x, 532.
Joule, 501, 584.
Judgments, economical character of
all, 483.
Kant, on causality, 484.
Kater, 186.
Kepler, his laws of planetary motion,
187; possibility of his discovery of
the laws of falling bodies, 248 ; on
maxima and minima, 423 ; on as
trology, 463; on gravity, 532; on
tides, 538,
Kilogramme, 281.
Kilogrammemetre, 272.
Kinetic energy, 272, 499.
KirchholF, ix, 381, 555 et seq., 580.
Knowledge, instinctive, I, 2628, 83 ;
the communication of, the founda
tion of science, 4 ; the nature of, 5;
the necessary and sufficient condi
tions of, 10.
Kb'nig, on the cells of the honey
comb, 453.
Laborde, apparatus of, 150.
Lagrange, his deduction of the law of
the lever, 13 ; his deduction of the
principle of virtual velocities, 65
67; criticism of this last deduction,
6768; his form of D'AIembert's
principle, 337; on zrzs viva, 349 ; on
the principle of least action, 371 ;
on the calculus of variations, 436 et
seq.; emancipates physics from the
ology, 457; his analytical mechan
ics, x, 466, 553 ; his indeterminate
coefficients, 471 et seq.
Larni, on the composition of forces.
36.
Lange, 542 et seq.
Language, economical character of,
481 ; possibility of a universal, 482;
the Chinese, 482.
Laplace, 463 534.
Lasswitz, 530.
Lateral pressure, 103.
Laws of nature, 502.
Laws, rules for the mental recon
struction of facts, 8384, 485.
Least action, principle of, 364380;
its theological kernel, 454 ; anal
ogies of, 577
Least constraint, principle of, 350
Leibnitz, on the measure of force,
148, 250, 270, 274276, 575; on quan
tity of motion, 274; on the motion
of light, 425, 454; on the brachisto
chrone, 426; as a theologian, 449,
Level surfaces, 98, 398 ct seq,
Lever, the principle of tho, 825, 5x3
et soq; "potential," sso ; applica
tion of its principles to tho expla
nation of the other machines* ass;
its laws deduced by Newton's prin
ciples, 263267; conditions of its
rigidity, 96 ; Mauportuis's treat
ment of, 366.
Libraries, stored up experience, 48*.
Light, motion of, 422, 424, 426 j Man
6oo
THE SCIENCE OF MECHANICS.
pertuis on motion of, 367 ; motion
of, in refracting media, 374376, 377~
379 ; its minimal action explained,
459
Limiting cases, 565.
Lindeloff, 437.
Lippich, ap