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^^P^'>X

^^/^^^

':IU^7:> i^y^^ ^ ^^y^

MISCELLANEOUS ^WORKS//

OF THE LATE

THOMAS YOUNG,/ M.D., F.R.S., &o.,

AND ONE OF THE EIGHT FOREIGN ASSOCIATES OF THE
NATIONAL INSTITUTE OF FRANCE.

Vols. I. & IL,
INCLUDING HIS SCIENTIFIC MEMOIRS, <fcc.,

ET>1TED BV

GEORGE PEACOCK, D.D.,

F.R.a., r.GJB., F.B.AJB., F.C.PA, ETC.,

^^ DEAN OF ELY,

LOWRDVAK PROFSBBOR OF AflTROllOMT TK TITO UMIVER8ITT OF CAMBKIDOK,
AHD TORMBRLT FELLOW AND TITTOR OF TKIinTT OOLLKGB.

VOLUME J.

LONDON:
JOHN MURRAY, ALBEMARLE STREET.

1855.

3 iqq.i

HARVARD COLLEGE LIBRAftT

Uniform with Dr. Young's MlBcellaneoufl Works.
This Day, with Portrait^ 8vo. 15/,

THE LIFE OF THOMAS YOUNG, M.D., F.R.S.

By GEORGE PEACOCK, D.D., Dean of Elt.

■' ^^-

LONDON : PBIMTRD BT V. GLOWaS AND 80NB, BTAMVOBD ffHSBT,
AND CBAKINO CROSS.

ADVEETISEMENT.

The following edition of the Miscellaneous Scientific Works of
Dr. Young contains all his contributions to the Transactions of
the Royal Society ; the principal Articles prepared for the Sup-
plement of the Encyclopaedia Brltannica, including a selection
of the more elaborate of his Scientific Biographies ; many
Essays from Nicholson's and Brande's Journals ; some Reviews
on scientific subjects from the Quarterly Review ; and several
Essays, either separately published or dispersed in different
publications.

An Essay entitled ' Remarks on the Measurement of
Minute Particles, especially those of Blood and of Pus,' as
illustrating the applications of an optical instrument invented
by him called the Eriometer, has been taken from Dr. Young's
'Introduction to Medical Literature.' It was inserted with
the view of completing the series of his optical writings, which
are contained in the first volume. An Essay ' On the Cohesion
of Fluids' is taken from an Appendix to his 'Elementary
Illustrations of the Celestial Mechanics of La Place,' with a
similar view of completing the series of his Memoirs on this
difficult subject. And two sections or chapters from the Mathe-
matical Elements of Natural Philosophy, one of them ' On the
Equilibrium and Strength of Elastic Substances,' and the other
containing 'Some Propositions on Waves and Sound,' have
been inserted, partly on account of their great intrinsic value,
and partly because they are not included in the new edition of
Dr. Young's Lectures which has been edited by Professor

«2

1 V ADVEHTISEMENT.

Kelland. The preceding are the only Articles which formed
parts of separate publications of Dr. Young.

It is hardly necessary to observe, that many of his earlier,
and not a few of his later, Articles and Essays have been
omitted, as being frequently of a controversial nature merely,
or as not containing matter which was considered sufficiently
new or important to be reprinted ; they could not, in fact, have
been inserted without very unduly swelling the bulk of this
publication.

The Medical Works of Dr. Young, with the exception of a
few Reviews, an Essay on Palpitations inserted in the fifth
volume of the Medical Transactions of the College of London,
and an Essay on Bathing in the Supplement of the Encyclo-
paedia Britannica, are all contained in separate publications. It
was at one time proposed to add a volume containing these
Essays and some selections from his Medical Works, but the
scheme was afterwards abandoned.

A selection from Dr. Young's Optical Correspondence has
been subjoined to his Optical Memoirs, as containing materials
of no small importance, for illustrating the history of the pro-
gress of optical discovery at the memoi:able period at which
they were written.

The notes were added by the Editor, with a view of occasion-
ally illustrating the subjects considered in the text, and of
pointing out their bearing upon the researches of other authors.
Very careful references have also been generally added, for
which the editor is greatly indebted to the assistance of his
friend, Mr. Leitch.

CONTENTS OF VOL. I.

Niimber. Page

I. — Observations on Vision 1

til. — On the Mechanism of the Eye 12
III. — Outlines of Experiments and Inquiries respecting Sound and

Light 64

IV. — An Essay on Cycloidal Curves 99

V. — An Essay on Music 115

VL— A Letter to Mr. Nicholson, respecting Sound and Light . 131

II. — On the Theory of Light and Colours 140

VIII. — An Account of some Cases of the Production of Colours not

hitherto described 170

IX. — ^Experiments and Calculations relative to Physical Optics . 179

X. — ^Reply to the Animadversions of the Edinburgh Reviewers . 1 92

XL— Harmonic Sliders . 216

XII. — Review of Laplace's Memoir, ' Sur la Loi de la R6fraction

extraordinaire dans les Cristaux diaphanes * . . 220

Xlll. — Review of the * M6moires de Physique et de Chimie de la

Society d'Arcueil ' 234

XIV. — Review of Mains, Biot, Seebeck, and Brewster, on Light . 260

XV. — The Article * Chromatics,* from the Supplement to the Ency-

clopcBdia Britanhica 279

XVI. — Remarks on the Measurement of Minute Particles, especially

those of the Blood and of Pus 343

XV IL — Selections from Correspondence relating to Optical Subjects . 359

XVIII.— Theoretical Investigations, intended to illustrate the Pheno-
mena of Polarisation 412

XIX. — An Essay on the Cohesion of Fluids 418

XX. — The Article • Cohesion,' from the Supplement to the Kiicy-

clopajdia Britannica 454

XXL — On the Cohesion of Fluids 485

XXII. — Hydraulic Investigations 491

XXIII. — On the Functions of the Heart and Arteries . . . .511

XXIV.— Remarks on the Emj)loymcnt of Oblique Riders anil on other

Alterations in the Construction of Shi[»s . . . 535
XXV.— A Numerical Table of Elective Attractions . . . .563

XXVk — A Review of Sir Humphry Davy's Elements of Chemical Phi-
losophy 575

CONTKN'fS OF VOIi. I.

LIST OB' PLATES.

Dirtcti(/m to Binder.

Number.

I. — Observations on Vision

II. — Mechanism of the Eye

III. — Sound and Light

IV. — (yycloidal Curves
VI I. — Light and Colours
XXIV.— -Oblique Riders

Figs.

Fig.

Figs.

. 1—3
. 4—9'
10—18
19
20—30
31—54
55—60.
61—83'
84-88
89—100
101—112
. 113.
114—127
128—131
132—140

to fcboe Ffjufe 11

114
169
562

ERKATA.

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',,125, „ 2 „ bottom, /or ** -ft-" read "tV- /
bottom, for "part" read " path.'v
bottom, for " 92 " read " cos. V*^ /
„ 357, „ 12 „ top, /or "supra" read " and supni.**^
„ 392, „ 12 „ top, for «au" read "en."v'
„ 422, „ 2 „ top, /o/- "4ar"' read " 4r.'\'
„ 486, „ 9 „ bottom, for •« c Mf" read " c* Mf:' ^
,,489, „ 7 „ bottom, /or "r*ie+c*C" read "r*jB = c*C.^

„ 158, „ 8
»» 268, ti 5

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Co^'

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No. L

OBSERVATIONS ON VISION.

From the Philosophical Transactions for 1793, vol. Ixxziii., p. 160.
Read May 30, 1793.*

It is well known that the eye, when not acted upon by any
exertion of the mindy conveys a distinct impression of those
objects only which are situated at a certain distance from
itself; that this distance is difierent in different persons, and
that the eye can, by the volition of the mind, be accommo-
dated to view other objects at a much less distance : but how
this accommodation is effected, has long been a matter of dis-
pute, and has not yet been satisfactorily explained It is
equally true, though not commonly observed, that no exer-
tion of the mind can accommodate the eye to view objects at
a distance greater than that of indolent vision, as may easily

* This memoir was written by Dr. Young in his twentieth year, and led to his
election as a Fellow of the Royal Society in the year following. It excited more atten*
tion at the time of its publication than its intrinsic importance probably deserved,
chiefly in consequence of the discovery announced in it, of the muscularity of the crys-
talline lens, being claimed by the celebrated John Hunter, whose lectures he was at that
time attending. A chaige, in iSict, was indirectly insinuated in some quarters, though
afterwards fully explain^ and abandoned, that he had derived it from his teacher.
Mr. Hunter died in the following October, before he had completed the researches upon
which his views had been originally founded : they were resumed by his successor,
Mr., afterwards Sir Everard Home, who made, in conjunction with Mr. Ramsden, a
series of experiments recorded in the Philosophical Transactions for 1795, which ap-
peared to Dr. Toung to negative his conclusion so decisively, that he formally aban-
doned it, in the following words at the end (p. 68) of his Gottiogen Dissertation. '< De
Corporis humani Viribus conservatricibus," pubUshed in 1796 : **8entmiia nuper d4
lerUiB cnf8taUina*usu in ooulo ad divenaa renan videHdarum distantiaa accommodando
proponict^ neque nova erai tuque vera videtur.*' He reiterated the same opinion at
the end of his memoir in the Philosophical Ti^ansactions, read 16th Nov., 1800,
entitled ^ Outlines of Experiments and Inquiries respecting Sound and Light," which
forma No. III. of this volume. The circumstances and experiments which induced him
to resume his fonner views are recorded in the *' Memoir on the Mechanism of the
Eye," which immediately succeeds the one given in the text, as well a^ in his reply to
the animadversions of the Edinbuigh Reviewers, which appears in a subsequent part,
Ho. X., of this volume. — Note by the Editor,

VOL. L * B

2 OBSERVATIONS ON VISION. No. I.

be experienced by any person to whom this distance of indo-
lent vision is less than infinite.

The principal parts of the eye, and of its appurtenances,
have been described by various authors. Winslow is gene-
rally very accurate ; but Albinus, in Musschenbroek's Intro-
duction has represented several particulars more correctly. I
shall suppose their account complete, except where I mention
or delineate the contrary.

The first theory that I find of the accommodation of the
eye is Kepler's. He supposes the ciliary processes to contract
the diameter of the eye, and lengthen its axis, by a muscular
power. But the ciliary processes neither appear to contain
any mnscular fibres, nor have they any attachment by which
they can be capable of performing this action.

Descartes imagined the same contraction and elongation
to be effected by a muscularity of the crystalline, of which he
supposed the ciliary processes to be the tendons. He did not
attempt to demonstrate this muscularity, nor did he enough
consider the connection with the ciliary processes. He says,
that the lens in the mean time becomes more convex, but attri-
butes very little to this drcumstance.

De la Hire maintains that the eye undergoes no change, ex-
cept the contraction and dilatation of the pupil. He does not
attempt to confirm this opinion by mathematical demonstration ;
he solely rests it on an experiment which has been shown by
Dr. Smith to be fallacious. Haller too has adopted this
opinion, however inconsistent it seems with the known principles
of optics, and with the slightest regard to hourly experience.

Dr. Pemberton supposes the crystalline to contain muscular
fibres, by which one of its surfaces is flattened while the other
is made more convex. But, besides that he has demonstrated
no such fibres, Dr. Jurin has proved that a change like this is
inadequate to the effect.

Dr. Porterfield conceives that the ciliary processes draw for-
ward the crystalline, and make the cornea more convex. The
ciliary processes are, from their structure, attachment, and
direction, utterly incapable of this action ; and by Dr. Jurin's
calculations, there is not room for a sufiicient motion of this

No. I. OBSERVATIONS ON VISION.

kind, withoat a very visible increase in the length of the eye's
axis : such an increase we cannot observe.

Dr. Jurin's hypothesis is, that the uvea, at its attachment to
the cornea, is muscular, and that the contraction of this ring
makes the cornea more convex. He says, that the fibres of
this muscle may as well escape our observation, as those of the
muscle of the interior ring. But if such a muscle existed, it
must, to overcome the resistance of the coats, be far stronger
than that which is only destined to the uvea itself ; and the
uvea, at this part, exhibits nothing but radiated fibres, losing
themselves, before the circle of adherence to the sclerotica, in
a brownish granulated substance, not unlike in appearance to
capsular ligament, common to the uvea and ciliary processes,
but which may be traced separately from them both. Now at
the interior ring of the uvea, the appearance is not absolutely
inconsistent with an annular muscle. His theory of accommo-
dation to distant objects is ingenious, but no such accommodation
takes place.

Musschenbroek conjectures that the relaxation of the ciliary
zone, which appears to be nothing but the capsule of the
vitreous humour where it receives the impression of the ciliary
processes, permits the coats of the eye to push forwards the
crystalline and cornea. Such a voluntary relaxation is wholly
without example in the animal economy, and were it to take
place, the coats of the eye would not act as he imagines, nor
could they so act unobserved. The contraction of the ciliary
zone is equally inadequate and unnecessary.

Some have supposed the pressure of the. external muscles,
especially the two oblique muscles, to elongate the axis of
the eye.* But their action would not be sufficiently regular,
nor sufficiently strong ; for when a much greater pressure is
made cm the eye, than they can be supposed capable of
effin^ting, no*sensible difference is produced in the distinctness
of vision.

• Dr. Hosack (Phil. Trans., 1794, vol. Imiv. p. 196) has asserted that external
pranare can alter the focus of the eje, but I have applied it under the most favour-
able circumstaDces without this effect. Mr. Ramsden's microscopical obserrationi
have broof^t new arguments of great strength in fiivoor of this opinion. — MS, Not§
by Dr. Tcmg.

4 OBSERVATIONB ON VISION. No. I.

Others say that the muscles shorten the axis : these have still
less reason on their side. ,

Those who maintain that the ciliary processes flatten the
crystalline, are ignorant of their structure, and of the effect re-
quired : these processes are yet more incapable of drawing back
the crystalline, and such an action is equally inconsistent with
obsenratioa

ProbaUy other suppositions may have been formed, liable
to as strong objections as those opinions which I have enume-
rated.

From these considerations, and from the observation of Dr.
Porterfield and others, that those who have been couched have
no longer the power of accommodating the eye to different dis-
tances, I had concluded that the rays of light, emitted by ob-
jects at a small distance, could only be brought to foci on the
retina by a nearer approach of the crystalline to a spherical
form ; and I could imagine no other power capable of producing
this change than a muscularity of a part, or the whole, of its
capsule.*

But in closely examining with the naked eye, in a strong
light, the crystalline from an ox, turned out of its capsule, I
discovered a structure which appears to remove all the di£S-
culties with which this branch q{ optics has long been obscured.
On viewing it with a magnifier, this structure became more
evident

The crystalline lens of the ox is an orbicular, convex, trans-
parent body, composed of a considerable number of similar
coats, of which the exterior closely adhere to the interior.
Each of these coats consists of six muscles, intermixed with a
gelatinous substance, and attached to six membranous tendons.
Three of the tendons are anterior, three posterior ; their length
is about two-thirds of the semi-diameter of the coat; their
arrangement is that of three equal and equidistant rays, meet-
ing in the axis of the crystalline ; one of the anterior is directed
towards the outer angle of the eye, and one of the posterior

* The late experimentg of Mr.- Home and Mr. Ramsden (Phil. Trana. 1795, toI.
Izzzy. p. 453^ have io far controverted this received opinion, that the whole theorjr
of the moaciilarity of the crystalline lens now stands on very weak foundations. —
MS. Note hy Dr. Toung,

No. L OBSEBVATIONS ON VISIOK* 5

towards the inner angle, so that the posterior are placed oppo-
site to the middle of the interstices of the anterior ; and planes
passing through each of the six, and through the axis, would
mark on either surface six regular equidistant rays. The
muscular fibres arise from both sides of each tendon ; they
diyerge till they reach the greatest circumference of the coat,
and, baring passed it, they again converge, till they are
attached respectively to the sides of the nearest tendons of the
oj^KMdte surface. The anterior or posterior portion of the six
viewed together exhibits the appearance of three penniformi-
radiated muscles. The anterior tendons of all the coats are
situated in the same planes, and the posterior ones in the con-
tinuations of these planes beyond the axis. Such an arrange-
ment of fibres can be accounted for on no other supposition than
that of muscularity. This mass is inclosed in a strong mem-
branous capsule, to which it is loosely connected by minute
Vessels and nerves ; and the connection is more observable near
its greatest circumference. Between the mass and its capsule
is found a considerable quantity of an aqueous fluid, the liquid
of the crystalline.

I conceive, therefore, that when the will is exerted to riew
an object at a small distance, the influence of the mind is con-
veyed through the lenticular ganglion, formed from branches of
the third and fifth pairs of nerves, by the filaments perforating
the sclerotica, to the orbiculus ciliaris, which may be considered
as an annular plexus of nerves and vessels ; and thence by
the dliary processes to the muscle of the crystalline, which,
by die contraction of its fibres, becomes more convex, and
collects the diverging rays to a focus on the retina. The
disposition of fibres in each coat is admirably adapted to pro-
duce this change ; for, since the least surface that can contain
a given bulk is that of a sphere (Simpson's Fluxions, p. 486),
the contraction of any surface must bring its contents nearer
to a spherical form. The liquid of the crystalline seems to
serve as a synovia in facilitating the motion, and to admit
a sufficient change of the muscular part, with a smaller motion
of the capsule.

It remains to be inquired whether these fibres can produce

6 OBSERVATIONS ON VISION. No. !•

an alteration in the form of the lens sufficiently great to aooonnt
for tlie known effects.

In the ox's eye, the diameter of the crystalline is 700 thou*
sandths of an inch, the axis of its anterior segment 225, of its
posterior 350. In the atmosphere it collects parallel rays at
the distance of 235 thousandths. From these data we fiiKl, by
means of Smith's Optics, Art. 366, and a quadratic, that its
ratio of refraction is as 10000 to 6574.* Hauksbee makes it
only as 10000 to 6832,7, but we cannot depend on his experi-
ment, since he says that the image of the candle which he
viewed was enlarged and distorted ; a circumstance that he
does not explain, but which was evidently occasioned by the
greater density of the central parts. Supposing, with Hauksbee
and others, the refraction of the aqueous and vitreous humours
equal to that of water, viz. as 10000 to 7465, the ratio of
refraction of the crystalline in the eye will be as 10000 to 8806,
and it would collect parallel rays at the distance of 1226 thou-
sandths of an inch : but the distance of the retina from the
crystalline is 550 thousandths, and that of the anterior surface
of the cornea 250 ; hence (by Smith, Art. 367), the focal
distance of the cornea and aqueous humour alone must be
2329. Now, supposing the crystalline to assume a spherical
form, its diameter will be 642 thousandths, and its focal dis-
tance in the eye 926. Then disregarding the thickness of the
cornea, we find (by Smith, Art. 370), that such -an eye will
collect those rays on the retina, which diverge from a point at
the distance of 12 inches and 8 tenths. This is a greater change
than is necessary for an ox's eye, for if it be supposed capable
of distinct vision at a distance somewhat less than 12 inches,
yet it probably is far short of being able to collect parallel
rays. The human crystalline is susceptible of a much greater
change of form.

The ciliary zone may admit of as much extension as this
diminution of the diameter of the crystalline will require ; and

* Without doubt this refractive power is greater than the truth on account of the
much less density of the superficial parts, by means of which the efficient part is
smaller than the measure here stated, which was taken from the outside of the
capsule. The conclusion of the calculation is not however materially affected by this
difference.— iTS. IM0 by Dr. Tcmg.

Ko. I. OBSERVATIONS ON VISION. 7

its elasticity will assist the cellular texture of the vitreous
humour, and perhaps the gelatinous part of the crystalline, in
restoring the indolent form.*

It may be questioned whether the retina takes any part in
supplying the lens with nerves ; but, from the analogy of the
olfactory and auditory nerves, it seems more reasonable to sup-
pose that the optic nerve serves no other purpose than that of
conveying sensation to the brain.

Although a strong light and dose examination are required,
in order to see the fibres of the crystalline in its entire state, yet
their direction may be demonstrated, and their attachment
shown, without much diflSculty. In a dead eye the tendons are
discernible through the capsule, and sometimes the anterior
ones even through the cornea and aqueous humour. When the
crystalline falls, it veryfrequentiy separates as far as the centre
into three portions, each baring a tendon in its middle. If
it be carefully stripped of its capsule, and the smart blast of a
fine blow^pipe be applied close to its surface in different parts, it
will be found to crack exactly in the direction of the fibres above
described, and all these cracks wiU be stopped as soon as they
reach either of the tendons. The application of a littie ink to
the crystalline is of great use in showing the course of the fibres.

When first I observed the structure of the crystalline, I was
not aware that its muscularity had ever been suspected. We
have, however, seen that Descartes supposed it to be of this
nature ; but he seems to think that the accommodation of the
eye to a small distance is principally performed by the elonga-
tion of the eye's axis. Indeed, as a bell shakes a steeple, so
must the coats of the eye be affected by any change in the
crystalline ; but the effect of this will be very inconsiderable ;
yet, as far as it does take place, it will co-operate with the
other change.

* It hai bMn olifecUd tliat tht cryBtaUiDe will not reMsume its flat form without
an antagonintic mude, but the contrary is demonstrable by taking it out in its capsnla
and sqnaeiing it between the tiogen ; for being let go it restores itselC Another

azperimant may be adduced : close the teeth gently, then make a strong effort to
bite; a considerable swell of the msMeter muscle may be externally felt without any
nearer approach of the jaws to eadi other : this swelling subsides immediately as the

mind relaxes the muscle. No doubt the cross connection of cellular membraae acts in
these cases, and Ztam has described cross threads eren in the crystalline.— JfS. Not$
by Dr. Tcumg.

8 0BSEBVATI0N8 ON VISION. No. 1.

But the laborious and accurate Leeuwenhoek, by the help of
his powerful microscopes, has described the course of the fibres
of the crystalline, in a variety of animals ; and he haa even
gone so far as to call it a muscle ;* but no one has pursued
the hint, and probably for this reason, that from examining only
dried preparations, he has imagined that each coat consists of
drcumvolutions of a single fibre, and has entirely overlooked
the attachment of the fibres to tendons ; and if the fibres were
continued into each other in the manner that he describes, the
strict analogy to muscle would be lost, and their contraction
could not, conveniently, have that efiect on the figure of the
lens, which is produced by help of the tendons. Yet notwith-
standing neither be, nor any other physiologist, has attempted
to explain the accommodation of the eye to different distances
by means of these fibres, still much anatomical merit must be
allowed to the &ithful description, and elegant delineation, of
the crystallines of various animals, which he has given in the
Philosophical Transactions, Vol. XIV. p. 780, and Vol. XXIV.
p. 1723. It appears, from his descriptions and figures, that
the crystalline of hogs, dogs, and cats, resembles what I have
observed in oxen, sheep, and horses ; that in hares and rabbits
the tendons on each side are only two, meeting in a straight
line in the axis ; and that in whales they are five, radiated in
the same manner as where there are three. It is evident that
this variety will make no material difierence in the action of
the muscle. I have not yet had an opportunity of examining
the human crystalline, but from its readily dividing into three
parts, we may infer that it is similar to that of the ox. The
crystalline in fishesf being spherical, such a change as I attribute
to the lens in quadrupeds cannot take place in that class of
animals.

It has been observed that the central part of the crystalline
becomes rigid by age, and this is sufficient to account for

* '* Now if the crystalliDe humour (which I hare sometimes called the crist.
mittcle) in our eyes," &c. PhiL Trans, vol. mit. p. 1729. — " Orystallimm muscu-
Item, ams kumorem cryttallinum dictum** &c. Leeuweoh. op. omn. I. p. 102.

f The lens of fishes may be easily understood by considering it as of the nature of
ligament, though more homy, for whatever Hunter has said of the cuttle-fish, I
cannot conceive what could be the action of such a spherical muscle. — M8. Note by
Dr, Young,

No. L OBSERVATIONS ON VISION. 9

presbyopia, without any diminution of the humours ; although
I do not deny the existence of this diminution, as a concomitant
circumstance.

I shall here beg leave to attempt the solution of some
optical queries, which have not been much considered by
authors.

1. Musschenbroek asks, What is tiie cause of the lateral
radiations which seem to adhere to a candle viewed witii
winking eyes ? I answer, the most conspicuous radiations are
those whidi, diverging from below, form, each with a vertical
line, an angle of about seven degrees ; this angle is equal to
that which the edges of the eyelids when closed make with a
horizontal line ; and the radiations are evidentiy caused by the
reflection of light from those flattened edges. The lateral radi-
ations are produced by the light reflected from the edges of
the lateral parts of the pupillary margin of the uvea, while its
superior and inferior portions are covered by the eyelids.
The whole uvea being hidden before the total close of the
eyelids, these horizontal radiations vanish before the perpen-
dicular ones.

2. Some have inquired. Whence arises that luminous cross,
which seems to proceed from the image of a candle in a looking-
glass ? This is produced by the direction of the friction by
which the glass is commonly polished : the scratches placed in
a horizontal direction, exhibiting the perpendicular part of the
cross, and the vertical scratches the horizontal part, in a manner
that may easily be conceived.

8. Why do sparks appear to be emitted when the eye is
rubbed or compressed in the dark ? This is Musschenbroek's
fourth query. When a broadish pressure, as that of the finger,
is made on the opaque part of the eye in the dark, an orbicular
spectrum appears on the part opposite to that which is pressed :
the Hght of the disc is faint, that of the circumference much
stronger; but when a narrow surface is applied, as that of a
pin's head, or of the nail, the image is narrow and bright.
This is evidentiy occasioned by the irritation of the retina at
tiie part touched, referred by the mind to the place from whence
light coming through the pupil would fall on this $pot; the

10 OBSERVATIONS ON VIBIOK/ No. I.

irritatioii is greatest where the flexure is greatest, viz. at the
circumferencey and sometimes at the centre, of the depressed
part. But in the presence of light, whether the eye be open or
closed, the circumference only will be luminous, and the disc
dark ; and if the eye be viewing any object at the part where
the image appears, that object will be totally invisible. Henoe
it follows, that the tension and compression of the retina
destroy all the irritation, except that which is produced by its
flexure ; and this is so slight on the disc, that the apparent light
there is fainter than that of the rays arriving at all other parts
through the eyelids. This experiment demonstrates a truth,
which may be inferred from many other arguments, rur., that
the supposed rectification of the inverted image on the retina
does not depend on the direction of the incident rays. Newton,
in his sixteenth query, has described this phantom as of pavo-
nian colours, but I can distinguish no other than white ; and
it seems most natural that this, being the compound or average
of all existing sensations of light, should be produced when
nothing determines to any particular colour. This average
seems to resemble the middle form, which ISr Joshua Reynolds
has el^antly insisted on in his discourses ; so that perhaps
some principles of beautiful contrast of colours may be drawn
from hence, it being probable that those colours which together
approach near to while light will have the most pleasing eflect
in apposition. It must be observed, that the sensation of light
from pressure of the eye subsides almost instantly after the
motion of pressure has ceased, so that the cause of Uie irritation
of the retina is a change, and not a difference, of form ; and
therefore the sensation of light appears to depend immediately
on a minute motion of some part of the optic nerve.

If the anterior part of the eye be repeatedly pressed so as
to occasion some degree of pain, and a continued pressure be
then made on the sclerotica^ while an interrupted pressure is
made on the cornea ; we shall frequentiy be able to observe
an appearance of luminous lines, branched, and somewhat con-
nected with each other, darting from every part of the field of
view, towards a centre a little exterior and superior to the axis
of the eye. This centre corresponds to the insertion of the

N?L

OBSERVATIONS OS VISION.

Jiff f J- 3.

To f'arf piuit IL Vol. I.

No. I. OBSERVATIONS ON YIBIOK* 11

optic nerve, and the appearance of lines is probably occasioned
by that motion of the retina which is produced by the sudden
return of the circulating fluid, into the veins accompanying the
ramifications of the arteria centralis, after having been detained
by the pressure which is now intermitted. As such an ob-
struction and such a re-admission must require particular
circumstances, in order to be efiected in a sensible degree, it
may naturally be supposed that this experiment will not always
easily succeed.

EoepUinatian of the Figvrea.

Fig. 1. A vertical section of the ox's eye, of twice the natand size.

A. The cornea, covered by the tunica conjunctiva.

BCB. The sclerotica, covered at BB by the tunica albuginea,
and tunica conjunctiva.

DD. The choroid, consisting of two laminas.

£E. The circle of adherence of the choroid and sclerotica.

FG, FG. The orbicnlus ciliaris.

HI, HK. The uvea: its anterior surface the iris ; its posterior
sur&ce lined with pigmentum nigrum.

IK. The pupil.

HL, HL. The ciliary processes, covered with pigmentum
nigrum.

MM. The retina.

N. The aqueous humour.

O. The crystalline lens.

P. The vitreous humour.

QR, QR. The zona ciliaris.

RS, RS. The annulus mucosus.
Fig. 2. The structure of the ciystalline lens, as viewed in front.
Fig. 3. A side view of the crystalline.

12 HECHANiaU OF THE ETE. Ito. II.

No.n.
ON THE MECHANISM OF THE ETE.

From the Philofiophical Transactions for 1801, toI. xdH p. 23.
Read Koyember 27, 1800. \c i

I. — In the year 1793, I had the honour of laying before the
Royal Society, some obserrations on the faculty by which the
eye accommodates itself to the perception of objects at diffisrent
distances.* The opinion which I then entertained, although it
had never been placed exactly in the same light, was neither
80 new, nor so much forgotten, as was supposed by myself, and
by most of those with whom I had any intercourse on the sub-
ject. Mr. Himter, who had long before formed a similar
opinion, was still less aware of having been anticipated in it,
and was engaged, at the time of his death, in an investigation
of the facts relative to it ; f an investigation for which, as far
as physiology was concerned, he was undoubtedly well qualified.
Mr. Home, with the assistance of Mr. Bamsden, whose recent
loss this Society cannot but lament, continued the inquiry which
Mr. Hunter had begun ; and the results of his experiments
appeared very satisfactorily to confute the hypotheas of the
muscularity of the crystalline lens, t I therefore thought it
incumbent on me to take the earliest opportunity of testifying
my persuasion of the justice of Mr. Home's conclusions, which
I accordingly mentioned in a Dissertation published at Gottin-
gen in 1796, § and also in an Essay presented last year to this
Society. II About three months ago, I was induced to resume

♦ Supra, No. I. t PWl- Trans, for 1794, p. 21.

: Phil. Trans, for 1795, p. 1.

§ De Corporis hnmani Yiribus conservatridbns, p. 68.

li Infra, No. III. p. 96.

No. XL MECHANISM OF THE ETE. 13

the subject, by perusing Dr. Porterfield's paper on the internal
motions of the eye ;* and I have very unexpectedly made some
observations, which I think I may venture to say, appear
to be finally conclusive in favour of my former opinion, as hr
as that opinion attributed to the lens a power of changing its
figure. At the same Hme, I must remark, that every person
who has been engaged in experiments of this nature, will be
aware of the extreme delicacy and precaution requisite, both
in conducting them, and in drawing inferences from them ; and
will also readily allow, that no apology is necessary for the
fiillacies which have misled many others, as well as myself, in
the application of those experiments to optical and physiological
determinations.

11. — Besides the inquiry respecting the accommodation of the
eye to different distances, I shall have occasion to notice some
other particulars relative to its functions ; and I shall begin
with a general consideration of the sense of vision. I shall
then enumerate some dioptrical propositions subservient to my
purposes, and describe an instrument for readily ascertaining
the fi)cal distance of the eye. On these foundations I shall
investigate the dimensions and refractive powers of the human
eye in its quiescent state ; and the form and magnitude of the
picture which is delineated on the retina. I shall next inquire,
how great are the changed which the eye admits, and what
degree of alteration in its proportions will be necessary for
these changes, on the various suppositions that are principally
deserving of comparison. I shall proceed to relate a variety of
experiments which appear to be tiie most proper to decide on
the trudi of each of these suppositions, and to examine such
arguments as have been brought forwards, against the opinion
which I shall endeavour to maintain ; and I shall conclude with
some anatomical illustrations of the capacity of the organs of
various classes of animals, for the functions attributed to them.

m. — Of all the external senses, the eye is generally supposed
to be by far the best understood ; yet so complicated and so

• Edinb. Med. Essays, toI. it. p. 124.

1 4 MBCHANISH OF THE EYE. No. II.

divergified are its powers, that many of them haye been hitherto
uninvestigated ; and on others, much laborious research has
been spent in vain. It cannot indeed be denied, that we are
capable of explaining the use and operation of its different
parts, in a far more satisfactory and interesting manner than
those of the ear, which is the only organ that can be strictly
compared with it ; since, in smelling, tasting, and feeling, the
objects to be examined come almost unprepared into immediate
contact with the extremities of the nerves ; and the only diffi-
culty is, in conceiving the nature of the effect produced by
them, and its communication to the sensorium. But the eye
and the ear are merely preparatory organs, calculated for
transmitting the impressions of light and somid to the retina,
and to the termination of the soft auditory nerve. In the eye,
light is conveyed to the retina, without any change of the nature
of its propagation : in the ear, it is very probable, that instead
of the successive motion of different parts of the same elastic
medium, the small bones transmit the vibrations of sound, as
passive inelastic hard bodies, obeying the motions of the air in
their whole extent at the same instant In the eye, we judge
very precisely of the direction of light, from the part of the
retina on which it impinges ; in the ear, we have no other cri-
terion than the slight difference of motion in the small bones,
according to the part of the tympanum on which the sound, con*
oentrated by different reflections, first strikes ; hence, the idea
of direction is necessarily very indistinct, and there is no reason
to suppose, that different parts of the auditory nerve are exclu-
sively affected by sounds in different directions. Each sensitive
point of the retina is capable of receiving ilistinct impressi<xie,
as well of the colour as of the strength of light ; but it is not
absolutely certain, that every part of the auditory nerve is
capable of receiving the impression of each of the much greater
diversity of tones that we can distinguish ; although it is ex-
tremely probable that all the different parts of the surface
exposed to the fluid of the vestibule, are more or less affected by
every sound, but in different degrees and succession, according
to the direction and quality of the vibration. Whether or no,
strictiy speaking, we can hear two sounds, or see two objects,

No. n. MECHAKISM OF THE ETE. 15

in the same instant, cannot easily be determined : but it is
sufficient that we can do both, without the intervention of any
intenral pf time perceptible to the mind ; and indeed we could
form no idea of magnitude, without a comparative and there-
fore nearly contemporary perception of two or more parts of the
same object. The extent of the field of perfect vision for each
position of the eye is certainly not very great ; but it will
appear hereafter, that its refractive powers are calculated
to take in a moderately distinct view of a whole hemisphere :
the sense of hearing is equally perfect in almost every direc-
tion.

IV. — Dioptrical Propositions.
Proposition I. — Phenomenon,

In all refractions, the ratio of the sine of the angle of inci-
dence to the sine of the angle of refraction is constant. (New-
ton's Opt I. Ax. 5 ; Smitii's Opt. 13 ; Wood's Opt. 24.)

Scholium 1. We shall call it the ratio of m to m 7 1, and
m H^ 1, n. In refractions out of air into water, m = 4 and
n « 3, very nearly ; out of air into glass, the ratio is nearly
that of 3 to 2.

Scholium 2. According to Barrow {LecL Opt. ii. 4), Huy-
gens, Euler {Conject, phys, circa prop, soni et luminis. Opusc.
t u.), and the opinion which I lately submitted to the Royal
Society,* the velocity of light is the greater the rarer the me-
dium : according to Newton (Schol. Prop. 96. 1. i. Princip.
Prop. 10. p. 3. 1. ii. Opt), and the doctrine more generally
received, the reverse. On both suppositions, it is always the
same in the same medium, and varies in the ratio of the sines
of the angles. This circumstance is of use in facilitating the
computation of some very complicated refractions.

Proposition II. — Phenomenon.

If between two refracting mediums, a third medium, termi-
nated by parallel surfaces, be interposed, the whole refraction
wiU remain Unchanged. (Newton's Opt L i. p. 2. Prop. 3 ;
Smitii r. 399 ; Wood, 105.)

' * See p. 81 of this yolume.
VOL. I. * B 3

16 MECHANISM OF THE EYE. JSTo. II.

Corollary. Hence, when the refractions out of two mediums
into a third are given, the refraction at the common surface of
these mediums may be thus found. Let the refractions given
be as m : n, and as m^ : n^ ; then the ratio sought will be that
of m nM m^n. For instance, let the three mediums be glass,
water, and air; then m = 3, n = 2, m'=4, n^=3, mn's9,
and m^ n = 8. If the ratios be 4 : 3, and 13 : 14, we have
m vi} : m^n : : 39 : 56 ; and dividing by 56 * 39 we obtain 2.3
and 3.3 nearly for m and m + 1, in Schol. 1. Prop. 1.

Proposition III. — Problem. (Rg. 4.)

At the vertex of a given triangle (CBA), to place a given
refracting surface (B), so that the incident and refracted rays
may coincide with the sides of the triangle (A6 and BC).

Let the sides be called d and e, the base being unity ; then
in the base take, next to d (or AB), a portion (AE) equal

to ;r5in^' ">• ^^^ = )dTir5; *^™* * "°« (^BorDB) to
the vertex, and the surface must be perpendicular to this line,
whenever the problem is physically possible. When e becomes

infinite, and parallel to the base, take — or — next to rf, for

the intersection of the radius of curvature.

Proposition IV. — Theorem. (Fig. 5.)

In oblique refractions at spherical surfaces, the line (AI,
KL), joining the conjugate foci (A, I ; K, L), passes through
the point (G), where a perpendicular from the centre (H) falls
on the line (EF), bisecting the chords (BC, BD), cut off from
the incident and refracted rays.

Corollary 1. Let t and u be die cosines of incidence and
refraction, the radius being 1, and d and e the respective
distances of the foci of incident and refracted rays ; then e^

mduu
mdu^ndt-^ntt*

Corollary 2. For a plane surface, e :« — " "

ntt

Corollary 3. For parallel rays, <f = oo , and e =

m »— II f

No. n. MECHANISM OF THE EYE. 17

Scholium 1. It may be observed, that the caustic by refrac-
tion stops short at its cusp, not geometrically, but physically,
the total reflection interfering.

Corollary 4. Call """ , b. and — -, c ; then e = ^ — >

and e ^ b =i t-— ; or, in words, the rectangle contained by the

focal lengths of parallel rays, passing and repassing any surface
in the same lines, is equal to the rectangle contained by the
diflerences ^tween these lengths and the distances of any
conjugate foci.

Corollary 5. For perpendicular rays, e = -^— - = wi + -j^ >

or, if the radius be a, c = ^^^^ ; and if d and e be given

to find the radius, a = —-jt — .

Corollary 6. For rays perpendicular and parallel, e = m, or
e = m a.

Corollary 7. For a double convex lens, neglecting the thick-
ness, call the first radius y, the second A, and e = , ^„^ ^.

Hence n = j^ • ^—^ ; and for parallel rays, e = ^5-~^, and

n = tf • ^— T— . If y = A = a, e = J*.^^ ; and*for parallel rays

« = —^ : calling this principal focal length i, ^ = j^-j , aa in
Cor. 4 ; whence we have the joint focus of two lenses ; also,

Corollary 8. In a sphere, e := m a ' ^ rf - (m - 2^ o » ^°^ *'^®

distance from the centre, and b = ^.

Scholium 2. In all these cases, if the rays converge, d must
be negative. For instance, to find the joint focus of two con-
vex or concave lenses, the expression becomes, = e . :.

Corollary 9. In Cor. 3, the divisor becomes ultimately con-
stant ; and, when the inclination is small, the focus varies aauu.

VOL. I. 0

18 MECHAKISM OF THE EYE. No. II.

CoroUary 10. Fop parallel rays falling obliquely on a double
oonrex, or double concave lens, of inconriderable thickness, the

radius being 1, « = ^, \..ng> * ^^ich varies ultimately as the
square of the cosine of incidence, or as 5-i^ t + f.

Scholium 3. In the double convex lens, the thickness dimi-
nishes the efiect of the obliquity near the axis ; in the double
concave, it increases it.

Scholium 4. No spherical surface, excepting in one particular
case, (Wood, 155,) can collect an oblique pencil of rays, even
to a physical point. The oblique rays which we have hitherto
considered are only such as lie in that section of the pencil
which is made by a plane passing through the centre and the
radiant point. They continue in this plane notwithstanding
the refraction, and therefore will not meet the rays of the col-
lateral sections, till they arrive at tlie axis. The remark was
made by Sir Isaac Newton, and extended by Dr. Smith
(Smith r. 493, 494); it appears, however, to have been too
little noticed, (Wood, 362.) The geometrical focus thus
becomes a line, a circle, an oval, or other figure, according to
the form of the pencil, the nature of the surface, and the place
of the plane receiving the image. Some of the varieties of the
focal image of a cylindrical pencil obliquely refracted are shown
in Fig. 31.

CoroUary 11. Hence the line joining the remoter conjugate
foci will always pass through the centre. The distance of the

remoter focus of parallel rays will be expressed by/ = — —j;
and the least circle of aberration will be at the distance
(1 +im)^Cm"tt-«o* ^^^^^*°8 ^ length of aberration in the
ratio of the distance of its limits from the surface. In the case
ofCor.lO/--— ^

Corollary 12. This proposition extends also to reflected rays ;
and in that case, the line from the centre passes dirough the
point of incidence.

No. II. MECHAiaSM OP THE EYB. 19

Proposition V. — Problem.

To find the place and magnitude of the image of a small
object, after refraction at any number of spherical surfaces.

Construction. (Fig. 6.) From any point (B) in the object
(AB), draw lines to (G), the centre of the first surface,
and to (D), the focus of parallel rays coming in a contrary
direction : irom the intersection of the second line (BD) with
flie tangent (EF) at the vertex, draw a line (EH) parallel to
the axis, and it will cut the first line (BC) in (H), the first
image of the point (B). Proceed with this image as a new
objeet, and repeat the operation for each surface, and the last
point will be in the image required. For calculation, find the
place of the image by Cor. 5. Prop. IV., and its magnitude
will be to that of the object, as their respective distances from
the centre.

Corollary. If a confused image be received on any given
plane, it will be necessary, in order to determine its magnitude,
to advert to the aperture admitting the rays. If the aperture
be supposed to be infinitely small, it may be considered as a
radiant point in order to find the direction of the emergent
rays.

Proposition VI. — Problem.

To determine the law by which the refraction at a spherical
suriace must vary, so as to collect parallel rays to a perfect
focus.

Solvtioru Let v be the versed sine to the radius 1 ; then, at
each point without the axis, n remaining the same, m must
become ^mm ±2/1 1;; and all the rays will be collected in the
principal focus.

Corollary. The same law will serve for a double convex lens,
in the case of equidistant conjugate foci, substituting n for m.

Proposition VII. — Problem.

To find the principal focus of a sphere, or lens, of which the
internal parts are more dense than the external.

c 2

20 MECHANISM OF THE EYE. No. 11.

Solution. In order that the focal distance may be finite, the
density of a finite portion about the centre must be equable :

call the radius of this portion y , that of the sphere being unity;

let the whole refraction out of the surrounding medium into

this central part, he asmton; take r = -, ^^ — -, and let

* ' log. m — log. n

the density be supposed to vary everywhere inversely as the
power — of the distance from the centre : then the principal

focal distance from the centre will be ^^-^ • —r- — . When
r = 1, it becomes ^,„ r if-t — r • For ^ 1©ds> deduct one

' 2 (H. L. m - H. L. n) '

fourth of the difference between its axis and the diameter of
the sphere of which its surfaces are portions.

Corollary. If the density be supposed to vary suddenly at
the surface, m must express the difference of the refractions at
the centre and at the surface ; and the focal distance, thus
determined, must be diminished according to the refiraction at
the surfsu^.

Proposition VIII.* — Problem.

To find the path of a ray of light falling obliquely on a
sphere, of a reft'active density varying as any power of the
distance from the centre.

The refractive density, in the sense of these propositions,
varies as the ratio of the sines, and as the velocity of light in
the medium. (Schol. 2. Prop. I.) Let the velocity at the

_ j__
distance xhe x ^ ; then, considering the refractive force as a
species of attraction, we have, in Prop. 41. Lib. 1, Princip.

_ j_

VABFD = x '•,Q=«, the sine of incidence, the radius

being unity, Z=*ar^^,Dc = -

/zr~

X X^ X r ^

* The enandatioQ and demoDstratioD of this Proposition, as printed in the Trans-
actions, were erroneous : their correct forms, as printed in the text, were given in
Nicholson's Journal for Aug. 1802, p. 262, at the end of* An Answer to Mr. Oough'a
£8sa7 on the Theory of Compound Sounds."— JVo/tf by the Editor.

No. II. MECHAKISH OF THE ETE. 21

-i

i * / ^ 1 « ^ X '^ ^

, and the fluxion of the area de-

1 . 2 . IZ^-i

scribed by the radius = —45x»' ar.l — «*a:r | . Let

the sine of the inclination to the radius at each point be called

2—1 i.— 2

y ; then y = sx ^ > y "= —1-1" $ x^ x, and the fluxion

of the area = j^z^ if '^ " V y\ ' ^^ which the fluent is
j^Z^ Y, y being the sine of the arc Y ; and the angle corre-
sponding is — ^ Y. The value of that angle being found for

any two values of x or y, the diflerence is the intervening angle
described by the radius. This angle is therefore always to the
diflerence of the inclinations as r to r — 1, and the deviation is
to that difference as 1 to r — 1.

Corollary. Hence in the passage to the apsis, and the
return to the surface, the deviation is always proportionate to
the arc cut off by the incident ray produced : therefore such a
sphere could never collect parallel rays to any focus, the lateral
density being too small towards the surface.

General Sc/tolium, The two first propositions relate to well-
known phenomena ; the third can hardly be new ; the fourth
approaches the nearest to Maclaurin's construction, but is
far more simple and convenient ; the fifth and sixth have no
difficulty ; the seventh may either be deduced from the eighth,
or may be demonstrated independently of it The one is
abridged by a property of logarithms ; the other is derived
from the laws of centripetal forces, on the supposition of
velocities directly as the refractive densities, correcting the
series for the place of the apsis, and making the sine of inci-
dence variable, to determine the fluxion of the angle of
deviation.

V. — Dr. Porterfield has employed an experiment, first
made by Scheiner, to the determination of the focal distance
of the eye ; and has described, under the name of an optometer.

22 MJSCHANISM OF THE EYE. No. 11.

a very excellent instrument founded on the principle of the
phenomenon.* But the apparatus is capable of considerable
improvement ; and I shall beg leave to describe an optometer,
simple in its construction, and equally convenient and accurate
in its application.

Let an obstacle be interposed between a radiant point (B,
Fig. 7), and any refracting surface, or lens (C D), and
let this obstacle be perforated at two points (A and B)
only. Let the refracted rays be intercepted by a plane, so as
to form an image on it. Then it is evident, that when this
plane (EF) passes through the focus of refracted rays, the
image formed on it will be a single point. But if the plane
be advanced forwards (to GH), or removed backwards (to
IK), the small pencils passing through the perforations will no
longer meet in a single point, but will fall on two distinct spots
of the plane (G, H; I, K); and, iu either case, form a
double image of the object.

Let us now add two more radiating points, (S and T, Fig.
8,) the one nearer to the lens than the first point, the other
more remote ; and when the plane which receives the images
passes through the focus of rays coming from the first point,
the images of the second and third points must both be double
{s 8y t t)} since the plane (EF) is without the focal distance
of rays coming from the farthest point, and within that of rays
coming from the nearest Upon this principle, Dr. Porter-
field*s optometer was founded.

But, if the three points be supposed to be joined by a line,
and this line to be somewhat inclined to the axis of the lens,
each point of the line, except the first point (R, Tig. 9), will
have 'a double image ; and each pair of images, being con-
tiguous to those of the neighbouring radiant points, will form
with them two continued lines, and the images being more
widely separated as the point which they represent is further
from the first radiant point, the lines (s t^ s t) will converge
on each side towards (r) the image of this point, and there
will intersect each other.

The same happens when we look at any object through two

* EdiDb. Med. Efisays, toI. ir. p. 185.

No. 11. MECHANISM OF THE KTE. 23

pin-boles, within the limits of the pupil. If the object be at
the point of perfect vision, the image on the retina will be
single ; but, in every other case, the image being double, we
shall appear to see a double object : and, if we look at a line
pointed nearly to the eye, it will appear as two lines, crossing
each other in the point of perfect vision. For this purpose, the
holes may be converted into slits, which render the images
nearly as distinct, at the same time that they admit more
light The number may be increased from two to four, or
more, whenever particular investigations render it necessary.

The optometer may be made of a slip of card-paper, or of
ivory, about eight inches in length, and one in breadth, divided
longitudinally by a black line, which must not be too strong.
The end of the card must be cut as is shown in Fig. 10,
in order that it may be turned up, and fixed in an in-
clined position by means of the shoulders: or a detached
piece, nearly of this form, may be applied to the optometer,
as it b here engraved. A hole about half an inch square must
be made in thb part i and the sides so cut as to receive a
slider of thick paper, with slits of different sizes, from a
fortieth to a tenth of an inch in breadth, divided by spaces
somewhat broader: so that each observer may choose that
which best stuts the aperture of his pupiL In order to adapt
the instrument to the use of presbyopic eyes, tlie other end
must be furnished with a lens of four inches focal length ; and
a scale must be made near the line on each side of it, divided
from one end into inches, and from the other according to the
table here calculated from Cor. 7. Prop. IV., by means of
which, not only diverging, but also parallel and converging
rays from the lens are referred to their virtual focus. The
instrument is easily applicable to the purpose of ascertaining
the focal length of spectacles required for myopic or presby-
opic eyes. Mr« Gary has been so good as to furnish me with
the numbers and focal lengths of the glasses commonly made ;
and I have calculated the distances at which those numbers
must be placed on the scale of the optometer, so that a presby-
opic eye may be enabled to see at eight inches distance, by
using the glasses of the focal length placed opposite to the

MECHANISM OF THE EYfi.

No. II.

nearest crossing of the lines ; and a myopic eye with parallel
rays, by using the glasses indicated by the number that stands
opposite their furthest crossing. To facilitate the observation,
I have also placed these numbers opposite that point which
will be the nearest crossing to myopic eyes ; but this, upon the
arbitrary supposition of an equal capability of change of focus
in every eye, which I must confess is often far from the truth.
It cannot be expected, that every person, on the first trial,
will fix precisely upon that power which best suits the defect
of his sight. Few can bring their eyes at pleasure to the state
of full action, or of perfect relaxation ; and a power two or
three degrees lower than that which is thus ascertained, will be
found sufficient for ordinary purposes. I have also added to
the second table, such numbers as will point out the spectacles
necessary for a presbyopic eye, to see at twelve and at eighteen
inches respectively : the middle series will perhaps be the most
proper for placing the numbers on the scale. The optometer
should be applied to each eye ; and, at the time of observing,
the opposite eye should not be shut, but the instrument should
be screened from its view. The place of intersection may be
accurately ascertained, by means of an index sliding along the
scale.

The optometer is represented in Fig. 11 and 12 ; and the
manner in which the lines appear, in Fig. 13.

Table I.

For extending the scale by a lens of 4 inches focus.

2.00

2.Q3

3.59

200

3.92

4.51

-12

6.00

2.22

3.U0

3.64

4.00

—30

4.62

—11

6.29

2.40

3.06

3.70

—200

4.08

—25

4.76

—10

6.67

2.55

3.11

3.75

—100

4.17

—20

5.00

—9.6

H.90

2.67

3.16

3.78

-^0

4.35

-15

5.45

-9.0

7.20

2.77

3.83

3.81

—45

4.39

—14

5.60

-8.5

7.56

2.86

8.45

100

3.85

—40

4.44

-13

5.78

—8.0

8.00

Table II. For placing the numbers indicating the focal length of convex

glasses.

Foe

VIII.

XII.

XVIII.

Foe.

VIII,

XII.

XVIII.

Foe.

VIII.

xn.

XVIII.

8.00

12.00

18.00

13.33

30.00

180.00

—24.00

—14.40

10.00

17.14

32.73

14.40

36.00

-^.00

—16.80

—11.45

10.28

18.00

36.00

16.00

48.00

—144.00

—24.00

— 12.00

— 9.00

10.91

20.00

45.00

18.67

84.00

— 63.0n

-13.33

— 8.57

— 6.92

11.20

21.00

50.40

24.00

— 36.00

4.5

-10.29

— 7.$0

— 6.00

11.56

22.29

58.50

29.33

—132.00

— 28.29

4.0

— 8.00

— 6.00

— 5.14

12.00

S4.00

72.00

40.00

- 60.00

— 22.50

3.5

-6.22

— 4.94

— 4.34

12.77

.26.40

99.00

72.00

— 36.00

— 18.00 3.0

-4.80

— 4.00

-3.6

No. ir.

MECHANISM OF THE EYE.

Table III. For concave glasses.

Nombcr.

Focus and v^*— ♦ Focuiand
furthcBt „itl? Number, furthest
place. P"**- • 1 place.

Nearest
place.

Number.

Focus and
furthest
place.

Nearest
place.

0
1
2
3
4
5
6

1 4.00 ■ 7 ' 8
24 1 3.43 ,87
18 3.27 9 1 6
16 3.20 10 5
12 3 00 11 4.5
10 2.86 12 4.0
9 1 2.77 13 3.5

2.67
2.54
2.40
2.22
2.12
2.00
1 87

14
15
16
17
18
19
20

3.00
2.':5
2.50
2.25
2.00
1.75
1.50

1.71
1.63
1.54
1.44
1.33
1.22
1.02

VI. — Being convinced of the advantage of making every
observation with as little assistance as possible, I have endea-
voured to confine most of my experiments to my own eyes ;
and I shall, in general, ground my calculations on the suppo-
sition of an eye nearly similar to my own. I shall therefore
first endeavour to ascertain all its dimensions, and all its
faculties.

For measuring the diameters, I fix a small key on each point
of a pair of compasses ; and I can venture to bring the rings
into immediate contact with the sclerotica. I'he transverse
diameter is externally 98 hundredths of an inch.

To find the axis, I turn the eye as much inwards as possible,
and press one of the keys close to the sclerotica, at the exter-
nal angle, till it arrives at the spot where the spectrum formed
by its pressure coincides with the direction of the visual axis,
and, looking in a glass, I bring the other key to the cornea.
The optical axis of the eye, making allowance of three hun-
dredths for the coats, is thus found to be 91 hundredths of an
inch, firom the external surface of the cornea to the retina.
With an eye less prominent, this method might not have
succeeded.

The vertical diameter, or rather chord, of the cornea, is 45
hundredths ; its versed sine 11 hundredths. To ascertain the
versed sine, I looked with the right eye at the image of the
left, in a small speculum held close to the nose, while the left
eye was so averted that the margin of the cornea jappeared as a
straight line, and compared the projection of the cornea with
the image of a cancellated scale held in a proper direction be-
hind the left eye, and close to the left temple. The horizontal
chord of the cornea is nearly 49 hundredths.

26 MECHAKISM OF THE BYE. No. II.

Hence the radius of the cornea is 31 hundredths. It may
be thought that I assign too great a convexity to the cornea ;
but I have corrected it by a number of concurrent observations,
which will be enumerated hereafter.

The eye being directed towards its image, the projection of
the margin of the sclerotica is 22 hundredths from the margin
of the cornea, towards the external angle, and 27 towards the
internal angle of the eye : so that the cornea has an eccen-
tricity of one fortieth of an inch, with respect to the sectiou of
the eye perpendicular to the visual axis.

The aperture of the pupil varies from 27 to 13 hundredths ;
at least this is its apparent size, which must be somewhat dimi-
nished, on account of the magnifying power of the cornea,
perhaps to 25 and 12. When dilated, it is nearly as eccentric
as the cornea ; but, when most contracted, its centre coincides
with the reflection of an image from an object held immediately
before the eye ; and this image very nearly with the centre of
the whole apparent margin of the sclerotica : so that the cornea
is perpendicularly intersected by the visual axis*

My eye, in a state of relaxation, collects to a focus on the
retina, those rays which diverge vertically from an object at the
distance of ten inches from the cornea, and the rays which
diverge horizontally from an object at seven inches distance.
For, if I hold the plane of the optometer vertically, the images
of the line appear to cross at ten inches ; if horizontally, at
seven. The difference is expressed by a focal length of 23
inches. I have never experienced any inconvenience from this
imperfection, nor did I ever discover it till I made these
experiments ; and I believe I can examine minute objects with
as much accuracy as most of those whose eyes are differently
formed. On mentioning it to Mr. Gary, he informed me, that
he had frequently taken notice of a similar circumstance : that
many persons were obliged to hold a concave glass obliquely,
in order to see with distinctness, counterbalancing, by the
inclination of the glass, the too great refractive power of the
eye in the direction of that inclination, (Cor. 10, Prop. IV.)
and finding but little assistance from spectacles of the same
focal length. The difference is not in the cornea, for it exists

No. II. MECHANISM OF THE EYK. 27

when the effect of the cornea is removed by a method to be
described hereafter. The cause is, without doubt, the obliquity
of the uvea, and of the crystalline lens, which is nearly parallel
to it, with respect to the visual axis : this obliquity will appear,
from the dimensions already given, to be about 10 degrees.
Without entering into a very accurate calculation, the difference
observed is found (by the same corollary) to require an inclination
of about 13 degrees; and the remaining three degrees may easily
be added, by the greater obliquity of the posterior surface of
the crystalline opposite the pupil. There would be no difficulty
in fixing the glasses of spectacles, or the concave eye-glass of a
telescope, in such a position as to remedy the defect.

In order to ascertain the focal distance of the lens, we must
assign its probable distance irom the cornea. Now the versed
sine of the cornea being 1 1 hundredths, and the uvea being
nearly flat, the anterior surface of the lens must probably be
somewhat behind the chord of the cornea; but by a very incon-
siderable distance, for the uvea has the substance of a thin
membrane, and the lens approaches very near to it : we will
therefore call this distance 12 hundredths. The axis and pro-
portions of the lens must be estimated by comparison with
anatomical observations ; since they affect, in a small degree,
the determination of its focal distance. M. Petit found the
axis almost always about two lines, or 18 hundredths of an inch.
The radius of the anterior surface was in the greatest number
3 lines, but oftener more than less. We will suppose mine
to be 3i, or nearly A of an inch. The radius of the posterior
surface was most frequently 2i lines, or I of an inch.* The

optical centre will be therefore (^^-r^ = ) about one-tenth of
an inch from the anterior surface : hence we have 22 hundredths,
for the distance of the centre from the cornea. Now, taking
10 inches as the distance of the radiant point, the focus of tlie
cornea will be 115 hundredths behind the centre of the lens.
(Cor. 5, Prop. IV.) But the actual joint focus is (91 - 22 = )
69 behind the centre : hence, disregarding the thickness of the
lens, its principal focal distance is 173 hundredths. (Cor. 7,

• M^m. de rAcad. de Paris, 1730, p. 6. Ed. Amst.

28 MECHANISM OF THE EYE. No. IL

Prop. IV.) For its refractive power in the eye, we hare
(by Cor. 7, Prop. IV.) n = 13,5, and to = 14,5. Calculating
upon this refractive power, with the consideration of the thick-
ness also, we find that it requires a correction, and comes near
to the ratio of 14 to 13 for the sines. It is well known that the
refractive powers of the humours are equal to that of water ;
and, that the thickness of the cornea is too equable to produce
any efiect on the focal distance.

For determining the refractive power of the crystalline lens
by a direct experiment, I made use of a method suggested to
me by Dr. WoUaston. I found the refractive power of the
centre of the recent human crystalline to that of water, as 21
to 20. The difference of this ratio from the ratio of 14 to 13,
ascertained from calculation, is probably owing to two circum-
stances. The first is, that the substance of the lens being in
some degree soluble in water, a portion of the aqueous fluid
within its capsule penetrates after death, so as somewhat to
lessen the density. When dry, the refractive power is little
inferior to that of crown glass. The second circumstance is, the
unequal density of the lens. The ratio of 14 to 13 is founded
on the supposition of an equable density : but the central part
being the most dense, the whole acts as a lens of smaller dimen-
sions ; and it may be found by Prop. VII. that if the central
portion of a sphere be supposed of uniform density, refracting
as 21 to 20, to the distance of one half of the radius, and
the density of the external parts to decrease gradually, and at
the surface to become equal to that of the surrounding medium,
the sphere thus constituted will be equal in focal length to a
uniform sphere of the same size, with a refraction of 16 to 15
nearly. And the effect will be nearly the same, if the centi'al
portion be supposed to be smaller than this, but the density
to be somewhat greater at the surface than that of the sur-
rounding medium, or to vary more rapidly externally than in-
ternally ; or, if a lens of equal mean dimensions, and equal
focal length, with the crystalline, be supposed to consist of two
segments of the external portion of such a sphere, the refractive
density at the centre of this lens must be as 18 to 17. On the
whole, it is probable that the refractive power of the centre of the

No. II. MECHANISM OF THE EYE. 29

human crystalline, in its living state, is to that of water nearly
as 18 to 17 ; that the water imbibed after death reduces it to
the ratio of 21 to 20 ; but that, on account of the unequable
density of the lens, its effect in the eye is equivalent to a re-
fraction of 14 to 13 for its whole size. Dr. WoUaston has
ascertained the refraction out of air, into the centre of the
recent crystalline of oxen and sheep, to be nearly as 143 to
100 ; into the centre of the crystalline of fish, and into the dried
crystalline of sheep, as 152 to 100. Hence the refraction of
the crystalline of oxen in water should be as 15 to 14 : but
the human crystalline, when recent, is decidedly less refractive.

These considerations will explain the inconsistency of dif-
ferent observations on the refractive power of the crystalline ;
and, in particular, how the refraction which I formerly calcu-
lated, from measuring the focal length of the lens,* is so much
greater than that which is determined by other means. But,
for direct experiments. Dr. Wollaston^s method is exceedingly
accurate.

When I look at a minute lucid point, such as the image of
a candle in a small concave speculum, it appears as a radiated
star^ as a cross, or as an unequal line, and never as a perfect
point, xmless I apply a concave lens inclined at a proper angle,
to correct the unequal refraction of my eye. If I bring the
point very near, it spreads into a surface nearly circular, and
almost equably illuminated, except some faint lines, nearly in a
radiating direction. For this purpose, the best image is a candle,
or a small speculum, viewed through a minute lens at some
little distance, or seen by reflection in a larger lens. If any
pressure has been applied to the eye, such as that of the finger
keeping it shut, the sight is often confused for a short time after
the removal of the finger, and the image is in this case spotty
or curdled. The radiating lines are probably occasioned by
some slight inequalities in the surface of the lens, which is very
superficially furrowed in the direction of its fibres : the curdled
appearance will be explained hereafter. When the point is
further removed, the image becomes evidently oval, the vertical
diameter being longest, and the lines a little more distinct than

♦ Supra, p. 6.

30 MECHANISM OF THE EYE. No. 11.

before, the light being strongest in the neighbourhood of the
centre ; but immediately at the centre there is a darker spot,
owing to such a slight depression at the vertex as is often
observable in examining the lens after death. The situation of
the rays is constant, though not regular ; the most conspicuous
are seven or eight in number ; sometimes about twenty fainter
ones may be counted. Removing the "point a little further, the
image becomes a short vertical line ; the rays that diverged
horizontally being perfectly collected, while the vertical rays are
still separate. In the next stage, which is the most perfect
focus, the line spreads in the middle, and approaches nearly to
a square, with projecting angles, but is marked with some
darker lines towards the diagonals. The square then flattens
into a rhombus, and the rhombus into a horizontal line un-
equally bright. At every greater distance, the line lengthens,
and acquires also breadth, by radiations shooting out from it, but
does not become a uniform surface, the central part remaining
always considerably brightest, in consequence of the same flat-
tening of the vertex which before made it fainter. Some of
these figures bear a considerable analogy to the images derived
from the refraction of oblique rays, (Schol. 4, Prop. IV.) and
still more strongly resemble a combination of two of them in
opposite directions ; so as to leave no doubt, but that both sur-
faces of the lens are oblique to the visual axis, and co-operate
in distorting the focal point. This may also be verified, by
observing the image delineated by a common glass lens, when
inclined to the incident rays. (See Fig. 31 — 43.)

The visual axis being fixed in any direction, I can at the
same time see a luminous object placed laterally at a consi-
derable distance from it ; but in various directions the angle
is very different. Upwards it extends to 50 degrees, inwards
to 60, downwards to 70, and outwards to 90 degrees These
internal limits of the field of view nearly correspond with
the external limits formed by the different parts of the fece,
when the eye is directed forwards and somewhat downwards,
which is its most natural position ; although the internal limits
are a little more extensive than the external; and both are well
calculated for enabling us to perceive the most readily, such

No. II. MECHANISM OF THS EYE. 31

objects as are the most likely to concern us. Dr. Wollaston's
eye has a larger field of view, both vertically and horizontally,
but nearly in the same proportions, except that it extends further
upwards. It is well known, that the retina advances further
forwards towards the internal angle of the eye, than towards
the external angle ; but upwards and downwards its extent is
nearly equal, and is indeed every way greater than the limits of
the field of view, even if allowance is made for the refraction
of the cornea only. The sensible portion seems to coincide
more nearly with the painted choroid of quadrupeds : but the
whole extent of perfect vision is little more than 10 degrees ; or,
more strictly speaking, the imperfection begins within a degree
or two of the visual axis, and at the distance of 5 or 6 degrees
becomes nearly stationary, until, at a still greater distance, vision
is wholly extinguished. The imperfection is partly owing to
the unavoidable aberration of oblique rays^ but principally to
the insensibility of the retina : for, if the image of the sun
itself be received on a part of the retina remote from the axis,
the impression will not be suSiciently strong to form a perma-
nent spectrum, although an object of very moderate brightness
will produce this effect when directly viewed. It would probably
have been inconsistent with the economy of nature, to bestow a
larger ^are of sensibility on the retina. The optic nerve is at
present very large ; and the delicacy of the oi^an renders it,
even at present, very susceptible of injury from slight irritation,
and very liable to inflammatory afiections ; and, in order to make
the sight so perfect as it is, it was necessary to confine that perfec-
tion within narrow limits. The motion of the eye has a range of
about 55 degrees in every direction ; so that the field of perfect
vision, in succession, is by this motion extended to 110 degrees.
But the whole of the retina is of such a form as to receive
the most perfect image, on every part of its surface, that the
atate of each refracted pencil will admit ; and the varying den«-
aity of the crystalline renders that state more capable of deline-
ating such a picture, than any other imaginable contrivance could
have done. To illustrate this, I have constructed a diagram,
representing the successive images of a distant object filling the
whole extent of view, as they would be formed by the succes-

32 MECHANISM OF THE EYE. No. 11.

sive refractions of the different surfaces. Taking the scale of
my own eye, I am obliged to substitute, for a series of objects
at any indefinitely great distance, a circle of 10 inches radius ;
and it is most convenient to consider only those rays which pass
through the anterior vertex of the lens ; since the actual centre
of each pencil must be in the ray which passes through the
centre of the pupil, and the short distance of the vertex of the
lens from this point will always tend to correct the unequal
refraction of oblique rays. The first curve (Fig. 19) is the
image formed by the furthest intersection of rays refracted
at the cornea : the second, the image formed by the nearest in-
tersection ; the distance between these, shows the degree of con-
fusion in the image , and the third curve, its brightest part. Such
must be the form of the image which the cornea tends to deli-
neate in an eye deprived of the crystalline lens ; nor can any
external remedy properly correct the imperfection of lateral
vision. The next three curves show the images formed after
the refraction at the anterior surface of the lens, distinguished in
the same manner ; and the three following, the result of all the
successive refractions. The tenth curve is a repetition of the
ninth, with a slight correction near the axis, at F, where, from
the breadth of the pupil, some perpendicular rays must fall. By
comparing this with the eleventh, which is the form of the re-
tina, it will appear that nothing more is wanting for their perfect
coincidence, than a moderate diminution of density in the lateral
parts of the lens. If the law, by which this density varies, were
more accurately ascertained, its effect on the image might be
estimated by means of the eighth proposition ; and probably
the image, thus corrected, would approach very nearly to the
form of the twelfth curve.

To find the place of the entrance of the optic nerve, I fix
two candles at ten inches distance, retire sixteen feet, and direct
my eye to a point four feet to the right or left of the middle of
the space between them : they are then lost in a confused spot
of light ; but any inclination of the eye brings one or the other
of them into the field of view. In Bernoulli's eye, a greater
deviation was required for the direction of the axis ;* and the

♦ Comm. Petrop. J. p. 314.

No. II. MECHANISM OF THE EYE. 33

obscured part appeared to be of greater extent From the
experiment here related, the distance of the centre of the optic
nerve from the visual axis is found (by Prop. V.) to be 16 hun-
dredths of an inch ; and the diameter of the most insensible part
of the retina, one-thirtieth of an inch. In order to ascertain the
distance of the optic nerve from the point opposite to the pupil,
I took the sclerotica of the human eye, divided it into segments,
from the centre of the cornea towards the optic nerve, and ex-
tended it on a plane. I then measured the longest and shortest
distances from the cornea to the perforation made by the nerve,
and their difference was exactly one-fifth of an inch. To this
we must add a fiftieth, on account of the eccentricity of the
pupil in the uvea, which in the eye that I measured was not
great, and the distance of the centre of the nerve from the
point opposite the pupil will be 11 hundredths. Hence it ap-
pears that the visual axis is five hundredths, or one-twentieth of
an inch, further from tlie optic nerve than the point opposite the
pupil. It is possible that this distance may be different in dif-
ferent eyes ; in mine, the obliquity of the lens, and the eccen-
tricity of the pupil with respect to it, will tend to throw a direct
ray upon it, without much inclination of the whole eye ; and it
is not improbable, that the eye is also turned slightly outwards,
if looking at any object before it, although the inclination is
too small to be subjected to measurement

It must also be observed, that it is very difficult to ascertain
the proportions of the eye so exactly as to determine, with cer-
tainty, the size of an image on the retina ; the situation, curva-
ture, and constitution of the lens, make so material a difference
in the result, that there may possibly be an error of almost one-
tenth of the whole. In order, therefore, to obtain some confir-
mation firom experiment, I placed two candles at a small dis-
tance fi^m each other, turned the eye inwards, and applied the
ring of a key so as to produce a spectrum, of which the edge
coincided with the inner candle ; then, fixing my eye on the out-
ward one, I found that the spectrum advanced over two-sevenths
of the distance between them. Hence, the same portion of the
retina that subtended an angle of seven parts at the centre of
motion of the eye, subtended an angle of five at the supposed

VOL. I. D

84 MECHANISM OP THE EYE. No. II.

intersection of the principal rays (Fig. 14); and the distance
of this intersection from the retina was 637 thousandtlis. This
nearly corresponds with the former calculation; nor can the
distance of the centre of the optic nenre from the point of most
perfect vision be, on any supposition, much less than that which
is here assigned. And, in the eyes of quadrupeds, the most
strongly painted part of the choroid is further from the nerve
than the real axis of the eye.

I have endeavoured to express, in four figures, the form of
every part of my eye, as nearly as I have been able to ascertain
it ; the first (Fig. 20) is a vertical section ; the second (Fig. 21)
a horizontal section ; the third and fourth are front views, in
different states of the pupil. (Fig. 22 and 23.)

Considering how little inconvenience is experienced from so
material an inequality in the refraction of the lens as I have
described, we have no reason to expect a very accurate provi-
sion for correcting the aberration of the lateral rays. But, as
far as can be ascertained by the optometer, the aberration
arising from figure is completely corrected ; since four or more
images of the same line appear to meet exactly in the same
point, which they would not do if the lateral rays were
materially more refracted than the rays near the axis. The
figure of the surfaces is sometimes, and perhaps always, more
or less hyperbolical • or elliptical : in the interior laminae,
indeed, the solid angle of the margin is somewhat rounded off;
but the weaker refractive power of the external parts must
greatly tend to correct the aberration arising from the too
great curvature towards the margin of the disc. Had the
refractive power been uniform, it might have collected the
lateral rays of a direct pencil nearly as well ; but it would have
been less adapted to oblique pencils of rays ; and the eye must
also have been encumbered with a mass of much greater
density than is now required, even for the central parts : and,
if the whole lens had been smaller, it would also have admitted
too little light. It is possible too, that Mr. Ramsden's observa-
tion,t on the advantage of having no reflecting surface, may
be well founded : but it has not been demonstrated, that less

♦ PeUt, M^m. de PAcad. 1725, p. 20. f Phil. Tnms. for 1795, p. 2.

Ko. II. MECHANISM OF THE EYE. 35

light is lost in passing through a medium of variable density,
than in a sudden transition from one part of that medium to
another ; nor are we yet sufficiently acqumnted with the cause
of this reflection, to be enabled to reason satisfactorily on the
subject. But neither this gradation, nor any other provision, has
the effect of rendering the eye perfectly achromatic. Dr. Jurin
had remarked this long ago,* from observing the colour border-
ing the image of an object seen indistinctly. Dr. Wollaston
pointed out to me on the optometer, the red and blue appear-
ance of the opposite internal angles of the crossing lines ; and
mentioned, at the same time, a very elegant experiment for
proving the dispersive power of the eye. He looks through a
prism at a small lucid point, which of course becomes a linear
spectrum. But the eye cannot so adapt itself as to make the
whole spectrum appear a line ; for, if the focus be adapted to
collect the red rays to a point, the blue will be too much
refracted, and expand into a surface ; and the reverse will
happen if the eye be adapted to the blue rays ; so that, in
either case, the line will be seen as a triangular space. The
observation is confirmed by placing a small concave speculum
in different parts of a prismatic spectrum, and ascertaining
the utmost distances at which the eye can collect the rays of
different* colours to a focus. By these means I find, that the
red rays, from a point at 12 inches distance, are as much
refracted as white or yellow light at 11. Tlie difference is
equal to the refraction of a lens 132 inches in focus. But the
aberration of the red rays in a lens of crown glass, of equal
mean refractive power with the eye, would be equivalent to the
efiect of a lens 44 inches in focus. If, therefore, we can
depend upon this calculation, the dispersive power of the eye
collectively is one-third of the dispersive power of crown glass,
at an equal angle of deviation. I cannot observe much aberra-
tion in the violet rays. This may be, in part, owing to their
faintness ; but yet I think their aberration must be less than
that of the red rays. I believe it was Mr. Ramsden's opinion,
that since the separation of coloured rays is only observed
where there is a sudden change of density, such a body as the

♦ Smith, e. 96.

D 2

36 MEC3HANISM OF THE EYE. No. II.

lens, of a density gradually varying, would have no effect
whatever in separating the rays of different colours. If this
hypothesis should appear to be well-founded, we must attribute
the whole dispersion to the aqueous humour ; and its dispersive
power will be half that of crown glass, at the same deviation.
But we have an instance in the atmosphere of a very gradual
change of density ; and yet Mr. Gilpin informs me that the stars,
when near the horizon, appear very evidently coloured. At a
more favourable season of the year, it would not be difficult to
ascertain, by means of the optometer, the dispersive power of the
eye, and of its different parts, with greater accuracy than by
the experiment here related. Had the dispersive power of the
whole eye been equal to that of flint glass, the distance of per-
fect vision would have varied from 12 inches to 7 for different
rays, in the same state of the mean refractive powers.

VII. — The faculty of accommodating the eye to various dis-
tances, appears to exist in very different degrees in different
individuals. The shortest distance of perfect vision in my eye
is 26 tenths of an inch for horizontal, and 29 for vertical rays.
This power is equivalent to the addition of a lens of 4 inches
focus. Dr. WoUaston can see at 7 inches, and with converging
rays ; the difference answering to 6 inches focal lengA. Mr.
Abernethy has perfect vision from 3 inches to 30, or a
power equal to that of a lens 3J inches in focus. A young
lady of my acquaintance can see at 2 inches and at 4 ; the
difference being equivalent to 4 inches focus. A middle-aged
lady at 3 and at 4 ; the power of accommodation being only
equal to the effect of a lens of 12 mches focus. In general I
have reason to think that the faculty diminishes in some degree
as persons advance in life; but some also of a middle age
appear to possess it in a very small degree. I shall take the
range of my own eye, as being probably about the medium,
and inquire what changes will be necessary in order to produce
it ; whether we suppose the radius of the cornea to be dimi-
nished, or the distance of the lens from the retina to be increased,
or these two causes to act conjointly, or the figure of the lens
itself to undergo an alteration.

No. II. MECHANISM OF THE EYE. 37

1. We have calculated that when the eye is in a state of
relaxation, the refraction of the cornea is such as to collect
rays diverging from a point ten inches distant, to a focus at
the distance of 13| tenths. In order that it may bring to the
same focus rays diverging from a point distant 29 tenths, we
find (by Cor. 5. Prop. IV.) that its radius must be diminished from
31 to 25 hundredths, or very nearly in the ratio of five to four.

2. Supposing the change from perfect vision at 10 inches to
29 tenths, to be efiected by a removal of the retina to a greater
distance from the lens, this will require (by the same Corollary)
an elongation of 135 thousandths, or more than one-seventh of
the diameter of the eye. In Mr. Abernethy's eye an elonga-
tion of 17 hundredths, or more than one-sixth, is requisite^

3. If the radius of the cornea be diminished one-sixteenth^
or to 29 hundredths, the eye must at the same time be elongated
97 thousandths, or about one-ninth of its diameter.

4. Supposing the crystalline lens to change its form ; if it
became a sphere, its diameter would be 28 hundredths, and, its
anterior surface retaining its situation, the eye would have per-
fect vision at the distance of an inch and a half. (Cor. 5 and
8. Prop. IV.) This is more than double the actual change.
But it is impossible to determine precisely how great an alter-
ation of form is necessary, without ascertaining the nature of
the curves into which its surfaces may be changed. If it were
always a spheroid more or less oblate, the focal length of each
surface would vary inversely as the square of the axis : but, if
the surfaces became, from spherical, portions of hyperbolic
conoids, or of oblong spheroids, or changed from more obtuse
to more acute figures of this kind, the focal length would vary
more rapidly. Disregarding the elongation of the axis, and
supposing the curvature of each surface to be changed propor-
tionally, the radius of the anterior must become about 21, and
that of the posterior 15 hundredths.

VIII. — I shall now proceed to inquire, which of these changes
takes place in nature ; and I shall begin with a relation of ex-
periments made in order to ascertain the curvature of the cor-
nea in all circumstances.

38 MECHANISM OF THE EYE. Ko. 11.

The method described in Mr. Home's Croonian Lecture for
1795,* appears to be far preferable to the apparatus of the
preceding year :f for a difference in the distance of two images
seen in the cornea, would be far greater and more conspicuous,
than a change of its prominency, and far less liable to be
disturbed by accidental causes. It is nearly, and perhaps
totally impossible to change the focus of the eye, witliout some
motion of its axis. The eyes sympathize perfectly with each
other ; and the change of focus is almost inseparable from a
change of the relative situation of the optic axes ; so much,
that if I direct both my eyes at an object beyond their furthest
focus, I cannot avoid bringing that focus a little nearer : while
one axis moves, it is not easy to keep the other perfectly at rest ;
and it is not impossible, that a change in the proportions of some
eyes may render a slight alteration of the position of the axis
absolutely necessary. These considerations may partly explain
the trifling difference in the place of the cornea that was observed
in 1794. It appears that the experiments of 1795 were made
with considerable accuracy, and no doubt with excellent instru-
ments ; and their failing to ascertain the existence of any change,
induced Mr. Home and Mr. Ramsden to abandon, in great
measiu-e, the opinion which suggested them, and to suppose,
that a change of the cornea produces only one-third of the
effect Dr. Olbers of Bremen, who in the year 1780 published
a most elaborate dissertation on the interaal changes of the
eye,} which he lately presented to the Royal Society, had been
equally unsuccessful in his attempts to measure this change of
the cornea, at the same time that his opinion was in favour of
its existence.

Room was however still left for a repetition of the experi-
ments ; and I began with an apparatus nearly resembling that
which Mr. Home has described. I had an excellent achromatic
microscope, made by Mr. Ramsden for my friend Mr. John
Ellis, of five inches focal length, magnifying about 20 times.
To this I adapted a cancellated micrometfer, in the focus of the
eye not employed in looking through the microscope : it was

♦ Phil. Trails, for 1796, p. 2. f Phil. Trans, for 1795, p. 13.

X De Ociili Mutationjbus intcrnis. Getting. 1780, 4'^,

No. II. MECHANISM OF THE EYE. 39

a large card divided by horizontal and vertical lines into for-
tieths of an inch. When the image in the microscope was com-
pared with this scale, care was taken to place the head so that
the relative motion of the images on the micrometer, caused by
the unsteadiness of the optic axis, should always be in the direc-
tion of the horizontal lines, and that there could be no error,
from this motion, in the dimensions of the image taken verti-
cally. I placed two candles so as to exhibit images in a vertical
position in the eye of Mr. Konig, who had the goodness to assist
me ; and, having brought them into the field of the microscope,
where they occupied 35 of the small divisions, I desired him to
fix his eye on objects at different distances in the same direc-
tion : but I could not perceive the least variation in the distance
of the images.

Knding a considerable difficulty in a proper adjustment of
the microscope, and being able to depend on my naked eye in
measuring distances, without an error of one 500th of an
inch, I determined to make a similar experiment without any
magnifying power. I constructed a divided eye»glass of two
portions of a lens, so small, that they passed between two images
reflected from my own eye ; and, looking in a glass, I brought
the apparent places of the images to coincide, and then made
the change requisite for viewing nearer objects : but the images
still coincided. Neither could I observe any change in the
images reflected from the other eye, where they could be
viewed with greater convenience, as they did not interfere
with the eye-glass. But, nut being at that time aware of the
perfect sympathy of the eyes^ I thought it most certain to con-
fine my observation to the one with which I saw. I must re-
markf that by a little habit, I have acquired a very ready
command over the accommodation of my eye, so as to be able
to view an object with attention, without adjusting my eye to
its distance.

I also stretched two threads, a little inclined to each other,
across a ring, and divided them by spots of ink into equal
spaces. I then fixed the ring, applied my eye close behind it,
and placed two candles in proper situations before me, and
a third on one side, to illuminate the threads. Then setting a

40 MECHANISM OF THE EYE. No. II.

small looking-glass, first at four inches distance and next at
two, I looked at the images reflected in it, and observed at what
part of the threads they exactly reached across in each case ;
and with the same result as before.

I next fixed the cancellated micrometer at a proper distance,
illuminated it strongly, and viewed it through a pin-hole, by
which means it became distinct in every state of the eye ; and,
looking with the other eye into a small glass, I compared the
image with the micrometer, in the manner already described.
I then changed the focal distance of the eye, so that the lucid
points appeared to spread into surfaces, from being too remote
for perfect vision; and I noted on the scale the distance of their
centres ; but that distance was invariable.

Lastly, I drew a diagonal scale, with a diamond, on a look-
ing-glass (Fig. 15), and brought the images into contact
with the lines of the scale. Then, since the image of the
eye occupies on the surface of a glass half its real dimen-
sions, at whatever distance it is viewed, its true size is always
double the measure thus obtained. I illuminated the glass
strongly, and made a perforation in a narrow slip of black card,
which I held between the images ; and was thus enabled to
compare them with the scale, although their apparent distance
was double that of the scale. I viewed them in all states of
the eye ; but I could perceive no variation in the interval
between them.

The sufficiency of these methods may be thus demonstrated.
Make a pressure along the edge of the upper eyelid with any
small cylinder, for instance a pencil, and the optometer will
show that the focus of horizontal rays is a little elongated,
while that of vertical rays is shortened ; an effect which can
only be owing to a change of curvature in the cornea. Not
only the apparatus here described, but even the eye unassisted,
will be capable of discovering a considerable change in the
images reflected from the cornea, although the change be much
smaller than that which is requisite for the accommodation of
the eye to different distances. On the whole, I cannot hesitate
to conclude, that if the radius of the cornea were diminished
but one-twentietli, the change would be very readily percep-

Ko. II. MECHAKISM OF THE EYE. 41

tible by some of the experiments related ; and the whole altera-
tion of the eye requires one-fifth.

But a much more accurate and decisive experiment remains.
I take out of a small botanical microscope, a double convex
lens, of eight-tenths radius and focal distance, fixed in a socket
one-fifth of an inch in depth ; securing its edges with wax, I
drop into it a little water, nearly cold, till it is three-fourths
full, and then apply it to my eye, so that the cornea enters half
way into the socket, and is everywhere in contact with the
water. (Fig. 16.) My eye immediately becomes pres-
byopic, and the refractive power of the lens, which is reduced
by the water to a focal length of about 16 tenths (Cor. 5.
Prop. IV.), is not sufficient to supply the place of the cornea,
rendered inefficacious by the intervention of the water ; but the
addition of another lens, of five inches and a half focus, restores
my eye to its natural state, and somewhat more. I then apply
the optometer, and I find the same inequality in the horizontal
and vertical refractions as without the water ; and I have, in
both directions, a power of accommodation equivalent to a focal
length of four inches, as before. At first sight indeed, the ac-
commodation appears to be somewhat less, and only able to
bring the eye from the state fitted for parallel rays to a focus at
five inches distance ; and this made me once imagine, that the
cornea might have some slight efiect in the natural state ; but,
considering that the artificial cornea was about i9i tenth of an
inch before the place of the natural cornea, I calculated the
efiect of this difference, and found it exactly sufficient to account
for the diminution of the range of vision. I cannot ascertain
the distance of the glass lens from the cornea to the hundredth
of an inch ; but the error cannot be much greater, and it may
be on either side.

After this it is almost necessary to apologize for having
stated the former experiments ; but, in so delicate a subje<ft, we
cannot have too great a variety of concurring evidence.

IX. — Having satisfied myself that the cornea is not concerned
in the accommodation of the eye, my next object was to inquire
if any alteration in the length of its axis could be discovered ;

42 MECHANISM OF THE EYE. No. II.

for this appeared to be the only possible alternative : and, con-
sidering that such a change must amount to one-seventh of the
diameter of the eye, I flattered myself with the expectation of
submitting it to measurement. Now, if the axis of the eye
were elongated one-seventh, its transverse diameter must be
diminished one-fourteenth, and the semi-diameter would be
shortened a thirtieth of an inch.

I therefore placed two candles so that when the eye was
turned inwards, and directed towards its own image in a glass,
the light reflected from one of the candles by the sclerotica ap-
peared upon its external margin, so as to define it distinctly
by a bright line ; and the image of the other candle was seen
in the centre of the cornea. I then applied the double eye-
glass, and the scale of the looking-glass, in the manner already
described; but neither of them indicated any diminution of the
distance, when the focal length of the eye was changed.

Another test, and a much more delicate one, was the appli-
cation of the ring of a key at the external angle, when the eye
was turned as much inwards as possible, and confined at the
same time by a strong oval iron ring, pressed against it at the
internal angle. The key was forced in as far as the sensibility
of the integuments would admit, and was wedged, by a mode-
rate pressure, between the eye and the bone. In this situation
the phantom caused by the pressure extended within the field
of perfect vision, and was very accurately defined ; nor did it,
as I formerly imagined, by any means prevent a distinct per-
ception of the objects actually seen in that direction; and
a straight line coming within the field of this oval phantom,
appeared somewhat inflected towards its centre (Fig. 17) ;
a distortion easily understood by considering the eflfect of the
pressure on the form of the retina. Supposing now, the dis-
tance between the key and the iron ring to have been, as it
really was, invariable, the elongation of the eye must have been
either totally or very nearly prevented; and, instead of an
increase of the length of the eye's axis, the oval spot caused
by the pressure would have spread over a space at least ten
times as large as the most sensible part of the retina. But no
such circumstance took place : the power of accommodation

No. II. MECHANISM OP THE EYE. 43

was as extensive as ever ; and there was no perceptible change
either in the size or in the figure of the oval spot.

Again, since the rays which pass through the centre of the
pupil, or rather the anterior vertex of the lens, may, as already
observed, be considered as delineating the image ; and, since
the divergence of these rays with respect to each other, is but
little afiected by the refraction of the lens, they may still be
said to diverge from the centre of the pupil ; and the image of
a given object on the retina must be very considerably en-
larged, by the removal of the retina to a greater distance from
the pupil and lens. (Cor. Prop. V.) To ascertain the real
magnitude of the image with accuracy is not so easy as it at
first sight appears ; but, besides the experiment last related,
which might be employed as an argument to this purpose, there
are two other methods of estimating it. The first is too hazard-
ous to be of much use ; but, with proper precautions, it may be
attempted. I fix my eye on a brass circle placed in the rays of
the sun, and, after some time, remove it to the cancellated mi-
crometer ; then changing the focus of my eye, while the micro-
meter remains at a given distance^ I endeavour to discover
whether there is any difference in the apparent magnitude of
the spectrum on the scale ; but I can discern none. I have
not inisisted on the attempt ; especially as I have not been able
to make the spectrum distinct enough without inconvenience ;
and no light is sufficiently strong to cause a permanent impres-
sion on any part of the retina remote from the visual axis. I
therefore had recourse to another experiment. I placed two
candles so as exactly to answer to the extent of the termination
of the optic nerve, and, marking accurately the point to which
my eye was directed, I made the utmost change in its focal
length ; expecting that, if there were any elongation of the axis,
the external candle would appear to recede outwards upon
the visible space. (Fig. 18.) But this did not happen ; the
apparent place of the obscure part was precisely the same
as before. I will not undertake to say, that I could have ob-
served a very minute difference either way: but I am per-
suaded, that I should have discovered an alteration of less than
a tenth part of the whole.

44 MECHANISM OF THE KYE. No. II.

It may be inquired if no change in the magnitude of the
image is to be expected on any other supposition ; and it will
appear to be possible, that the changes of curvature may be so
adapted, that the magnitude of the confused image may remain
perfectly constant. Indeed, to calculate from the dimensions
which we have hitherto used, it would be expected that the
image should be diminished about one-fortieth, by the utmost
increase of the convexity of the lens. But the whole depends
on the situation of the refracting surfaces, and the respective
increase of their curvature, which, on account of the variable
density of the lens, can scarcely be estimated with sufficient
accuracy. Had the pupil been placed before the cornea, the
magnitude of the image must, on any supposition, have been
very variable : at present, this inconvenience is avoided by the
situation of the pupil ; so that we have here an additional
instance of the perfection of this admirable organ.

From the experiments related, it appears to be highly im-
probable that any material change in the length of the axis
actually takes place : and it is almost impossible to conceive by
what power such a change could be effected. The straight
muscles, with the adipose substance lying under them, would
certainly, when acting independently of the socket, tend to
flatten the eye : for, since their contraction would necessarily
lessen the circumference or superficies of the mass that they
contain, and round off all its prominences, their attachment
about the nerve and the anterior part of the eye must there-
fore be brought nearer together. (Fig. 24, 25.) Dr. Olbers
compares the muscles and the eye to a cone of which the
sides are protruded, and would by contraction be brought
into a straight line. But tliis would require a force to preserve
the cornea as a fixed point, at a given distance from the origin
of the muscles ; a force which certainly does not exist. In the
natural situation of the visual axis, the orbit being conical, the
eye might be somewhat lengthened, although irregularly, by
being forced further into it ; but, when turned towards either
side, the same action would rather shorten its axis ; nor is there
anything about the human eye that could supply its place.
In quadrupeds the oblique muscles are wider than in man ;

No. II. MECHANISM OF THE EYE. 45

and in many situations might assist in the effect. Indeed a
portion of the orbicular muscle of the globe is attached so near
to the nerye, that it might also co-operate in the action : and I
have no reason to doubt the accuracy of Dr. Olbers, who
states, that he effected a considerable elongation, by tying threads
to the muscles, in the eyes of hogs and of calves ; yet he does
not say in what position the axis was fixed ; and the flaccidity
of the eye after death might render such a change very easy as >
would be impossible in a living eye. Dr. Olbers also mentions
an observation of Professor Wrisberg, on the eye of a man
whom he believed to be destitute of the power of accommoda-
tion in his life-time, and whom he found, after death, to have
wanted one or more of the muscles : but this want of accom-
modation was not at all accurately ascertained. I measured, in
the human eye, the distance of the attachment of the inferior
oblique muscle from the insertion of the nerve : it was one-fifth
of an inch ; and from the centre of vision not a tenth of an
inch ; so that, although the oblique muscles do in some positions
nearly form a part of a great circle round the eye, their action
would be more fitted to flatten than to elongate it We have
therefore i-eason to agree with Winslow, in attributing to them
the office of helping to support the eye on that side where the
bones are most deficient: they seem also well calculated to
prevent its being drawn too much backwards by the action of
the straight muscles. And, even if there were no difficulty in
supposing the muscles to elongate the eye in every portion, yet
at least some small difference would be expected in the extent
of the change, when the eye is in different situations, at an
interval of more than a right angle from each other ; but the
optometer shows that there is none.

Dr. Hosack alleges that he was able, by making a pressure
on the eye, to accommodate it to a nearer object :* it does not
appear that he made use of very accurate means of ascertaining
the fact ; but, if such an effect took place, the cause must have
been an inflection of the cornea.

It is unnecessary to dwell on the opinion which supposes a
joint operation, of changes in the curvature of the cornea and

♦ Phil. Trana. for 1794, p. 212.

46 MECHANISM OF THE EYE. No. II.

in the length of the axis. This opinion had derived very great
respectability, from the most ingenious and elegant manner in
which Dr. Olbers had treated it, and from being the last result
of the investigation of Mr. Home and Mr. Ramsden. But
either of the series of experiments which have been related,
appears to be suflScient to confute it.

X. — It now remains to inquire into the pretensions of the
crystalline lens to the power of altering the focal length of the
eye. The grand objection to the efficacy of a change of figure
in the lens was derived from the experiments in which those
who have been deprived of it have appeared to possess the
faculty of accommodation.

My friend Mr. Ware, convinced as he was of the neatness
and accuracy of the experiments related in the Croonian Lee*
ture for 1795, yet could not still help imagining, from the ob-
vious advantage all his patients found, after the extraction of
the lens, in using two kinds of spectacles, that there must, in
such cases, be a deficiency in that faculty. This circumstance,
combined with a consideration of the directions very judiciously
given by Dr. Porterfield, for ascertaining the point in question,
first made me wish to repeat the experiments upon various
individuals, and with the instrument which I have above de-
scribed as an improvement of Dr. Porterfield's optometer : and
I must here acknowledge my great obligation to Mr. Ware,
for the readiness and liberality with which he introduced me to
such of his numerous patients as he thought most likely to
furnish a satisfactory determination. It is unnecessary to
enumerate every particular experiment; but the universal
result is, contrary to the expectation with which I entered on
the inquiry, that in an eye deprived of the crystalline lens, the
actual focal distance is totally unchangeable. This will appear
from a selection of the most decisive observations.

1. Mr. R. can read at four inches, and at six only, with the
same glass. He saw the double lines meeting at three inches,
and always at the same point ; but the cornea was somewhat
irregularly prominent, and his vision not very distinct ; nor had
I, at the time I saw him, a convenient apparatus.

No. 11. MECHANISM OP THE EYE. ' 47

I afterwards provided a small optometer, with a lens of less
than two inches focus, adding a series of letters, not in alpha-
betical order, and projected into such a form as to be most
legible at a small inclination. The excess of the magnifying
power had the advantage of making the lines more divergent,
and their crossing more conspicuous ; and the letters served for
more readily naming the distance of the intersection, and, at
the same time, for jud^ng of the extent of the power of distin-
guishing objects too near or too remote for perfect vision.
(Fig. 26.)

2. Mr. J. had not an eye very proper for the experiment ;
but he appeared to distinguish the letters at 2^ inches, and at
less than an inch. This at first persuaded me, that he must
have a power of changing the ibcal distance : but I afterwards
recollected that he had withdrawn his eye considerably, to look
at the nearer letters, and had also partly closed his eyelids, no.
doubt contracting at the same time the aperture of the pupil ;
an action which, even in a perfect eye, always accompanies the
change of focus. The slider was not applied.

3. Miss H., a young lady of about twenty, had a very narrow
pupil, and I had not an opportunity of trying the small opto-
meter : but, when she once saw an object double through the
slits, no exertion could make it appear single at the same dis-
tance. She used for distant objects a glass of 4^ inches focus ;
with this she could read as far off as 12 inches, and as near as
5 : for nearer objects she added another of equal focus, and
could then read at 7 inches, and at 2j^.

4. Hanson, a carpenter, aged 63, had a cataract extracted
a few years since from one eye : the pupil was clear and large,
and he saw well to work with a lens of 2| inches focus ; and
could read at 8 and at 15 inches, but most conveniently at
11; With the same glass, the lines of the optometer appeared
always to meet at 11 inches; but he could not perceive that
they crossed, the line being too strong, and the intersection too
distant. The experiment was afterwards repeated with the
small optometer : he read the letters from 2 to 3 inches ; but
the intersection was always at 2i inches. He now fully under-
stood the circumstances that were to be noticed, and saw the

48 MECHANISM OF THE EYE. No. II.

crossing with perfect distinctness : at one time, he said it was a
tenth of an inch nearer ; hut I observed that he had remoyed
his eye two or three tenths from the glass, a circumstance
which accounted for this small difference.

5. Notwithstanding Hanson's age, I consider him as a very
fair subject for the experiment. But a still more unexception-
able eye was that of Mrs. Maberly. She is about 30, and
had the crystalline of both eyes extracted a few years since,
but sees best with her right. She walks without glasses ; and,
with the assistance of a lens, of about four inches focus, can
read and work with ease. She could distinguish the letters of
the small optometer irom an inch to 2i inches ; but the inter-
section was invariably at the same point, about 1 9 tenths of an
inch distant. A portion of the capsule is stretched across the
pupil, and causes her to see remote objects double, when with-
put her glasses ; nor can she, by any exertion, bring the two
images nearer together, although the exertion makes them more
distinct, no doubt by contracting the pupil. The experiment
with the optometer was conducted, in the presence of Mr. Ware,
with patience and perseverance ; nor was any opinion given to
make her report partial.

Considering the diflSculty of finding an eye perfectly suitable
for the experiments, these proofe may be deemed tolerably
satisfactory. But, since one positive argument will counter-
balance many negative ones, provided it be equally grounded
on fact^ it becomes necessary to inquire into the competency of
the evidence employed to ascertain the power of accommodation
attributed, in the Croonian Lecture for 1794, to the eye of
Benjamin Clerk. And it appears, that the distinction long
since very properly made by Dr. Jurin, between distinct vision
and perfect vision, will readily explain away the whole of that
evidence.

It is obvious that vision may be made distinct to any given
extent, by means of an aperture, sufficiently small, provided, at
the same time, that a sufficient quantity of light be left, while
the refractive powers of the eye remain unchanged. And it is
remarkable, that in those experiments, when the comparison
with the perfect eye was made, the aperture of the imperfect

No. II. MECHANISM OF THE EYE. 49

eye only was very considerably reduced. Benjamin Clerk,
with an aperture of -A- of an inch, could read with the same
glass at 1| inch, and at 7 inches.* With an equal aperture, I
can read at 1^ inch and at 30 inches; and I can retain the
state of perfect relaxation, and read with the same aperture at
2^ inches ; and this is as great a difference as was observed in
Benjamin Clerk's eye. It is also a fact of no small impor-
tance, that Sir Henry Englefield was much astonished, as
well as the other observers, at the accuracy with which the
man's eye was adjusted to the same distance, in the repeated
trials tliat were made with itf This circumstance alone makes
it highly probable, that its perfect vision was confined within
veiy narrow limits.

Hitherto I have endeavoured to show the inconveniences
attending other suppositions, and to remove the objections to
the opinion of an internal change of the figure of the lens. I
shall now state two experiments, which, in the first place, come
very near to a mathematical demonstration of the existence of
such a change, and, in the second, explain in great measure its
origin, and the manner in which it is effected.

I have already described the appearances of the imperfect
image of a minute point at different distances from the eye, in
a state of relaxation. For the present purpose, I will only
repeat, that if the point is beyond the furthest focal distance of
the eye, it assumes that appearance which is generally described
by the name of a star, the central part being considerably
the brightest. (Fig. 39 — 42.) But, when the focal distance
of the eye is shortened, the imperfect image is of course
enlarged ; and, besides this necessary consequence, the light is
also very differently distributed ; Uie central part becomes faint,
and the margin strongly illuminated, so as to have almost the
appearance of an oval ring. (Fig. 44.) If I apply the slider
of the optometer, the shadows of the slits, while the eye is re-
laxed, are perfectly straight, dividing the oval either way into
parallel segments (Fig. 45, 47): but when the accommodation
takes place, they immediately become curved, and the more so
the further they are from the centre of the image, to which

* Phil. Trans, for 1795, p. 9. t Phil. Trans, for 1795, p. 8.

VOL. I. E

50 MECHANISM OF THE EYE. No. II.

their concavity is directed. (Fig. 46, 48.) If the point be
brought much within the focal distance, the change of the eye
will increase the illumination of the centre, at the expense of
the margin. The same appearances are equally observable,
when the eflect of the cornea is removed by immersion in water;
and the only imagiuable way of accounting for the diversity, is
to suppose the central parts of the lens to acquire a greater
degree of curvature than the marginal parts. If the refraction
of the lens remained the same, it is absolutely impossible that
any change of the distance of the retina should produce a cur-
vature in those shadows, which, in the relaxed state of the eye,
are found to be in all parts straight ; and that neither the form
nor the relative situation of the cornea is concerned, appears
from the application of water already mentioned.

The truth of this explanation is fully confirmed by the opto-
meter. When I look through four narrow slits, without exer-
tion, the lines always appear to meet in one point : but when
I make the intersection approach me, the two outer lines meet
considerably beyond the inner ones, and the two lines of the
same side cross each other at a still greater distance. (Fig. 27.)

The experiment will not succeed with every eye : nor can it
be expected that such an imperfection should be universal : but
one case is sufficient to esUblish the argument, even if no other
were found. I do not however doubt, that in those who have a
large pupil, the aberration may be very frequently observable.
In Dr. WoUaston's eye, the diversity of appearance is imper-
ceptible ; but Mr. Konig described the intersections exactly as
they appear to me, although he had received no hint of what I
had observed. The lateral refraction is the most easily ascer*
tained, by substituting for the slits a tapering piece of card, so
as to cover all the central parts of the pupil, and thus deter-
mining the nearest crossing of the shadows transmitted through
the marginal parts only. When the furthest intersection was at
38^ I could bring it to 22 parts with two narrow slits ; but with
the tapered card only to 2K From these data we may deter-
mine pretty nearly into what form the lens must be changed,
supposing both the surfaces to undergo proportional alterations
of curvature, and taking for granted the dimensions already

No. II. MECHANISM OF THE EYE. 51

laid down : for, from the lateral aberration tlius given, we may
find (by Prop. III.) the subtangents at about one-tenth of an
inch from the axis ; and the radius of currature at each vertex
is already determined to be about 21 and 15 hundredths of an
inch. Hence the anterior surface must be a portion of a hyper-
boloid, of which the greater axis is about 50 ; and the posterior
surface will be nearly parabolical. In this manner the change
will be effected, without any diminution of the transverse dia-
meter of the lens. The elongation of its axis will not exceed
the fiftieth of an inch ; and on the supposition with which we
set out, the protrusion will be chiefly at the posterior vertex.
The form of the lens thus changed will be nearly that of Fig.
29 ; the relaxed state being nearly as represented in Fig. 28.
Should, however, the rigidity of the internal parts, or any other
considerations, render it convenient to suppose the anterior
surface more changed, it would still have room, without inter-
fering with the uvea ; or it might even force the uvea a little
forwards, without any visible alteration of the external appear-
ance of the eye.

From this investigation of the change of the figure of the
lens, it appears that the action which I formerly attributed to
the external coats, cannot afford an explanation of the pheno-
menon. The necessary effect of such an action would be. to
produce a figure approaching to that of an oblate spheroid;
and, to say nothing of the inconvenience attending a dimi-
nution of the diameter of the lens, the lateral refraction would
be much more increased than the central; nor would the
slight change of density, at an equal distance from the axis,
be at all equivalent to the increase of curvature : we must
therefore suppose some different mode of action in the power
producing the change. Now, whether we call the lens a
muscle or not, it seems demonstrable, that such a change of
figure takes place as can be produced by no external cause ;
and we may at least illustrate it by a comparison with the
usual action of muscular fibres. A muscle never contracts,
without at the same time swelling laterally, and it is of no
consequence which of the effects we consider as primary. I
was induced, by an occasional opacity, to give the name of

E 2

52 MECHANISM OF THE EYE. No. II.

membranous tendons to the radiations from the centre of the
lens ; but, on a more accurate examination, nothing really ana-
logous to tendon can be discovered. And, if it were supposed
that the parts next the axis were throughout of a tendinous, and
therefore unchangeable nature, the contraction must be princi-
pally effected by the lateral parts of the iBbres ; so that the coats
would become thicker towards the margin, by their contraction,
while the general alteration of form would require them to be
thinner ; and there would be a contrariety in the actions of the
various parts. But, if we compare the. central parts of each
surface to the belly of the muscle, there is no difficulty in
conceiving their thickness to be immediately increased, and to
produce an immediate elongation of the axis, and an increase
of the central curvature ; while the lateral parts co-operate
more or less, according to their distance from the centre, and
in different individuals in somewhat different proportions. On
this supposition, we have no longer any difficulty in attributing
a power of change to the crystalline of fishes. M. Petit, in
a great nv Jiber of observations, uniformly found the lens of
fishes more or less flattened : but even if it were not, a slight
extension of the lateral part of the superficial fibres would allow
those softer coats to become thicker at each vertex, and to form
the whole lens into a spheroid somewhat oblong ; and here, the
lens being the only agent in refraction, a less alteration than in
other animals would be sufficient. It is also worthy of inquiry,
whether the state of contraction may not immediately add to
the refractive power. According to the old experiment, by
which Dr. Goddard attempted to show that muscles become
more dense as they contract, such an effect might naturally be
expected. That experiment is, however, very indecisive, and the
opinion is, indeed, generally exploded, but perhaps too hastily ;
and whoever shall ascertain the existence or non-existence of
such a condensation, will render essential service to physiology
in general.

Dr. Pemberton, in the year 1719, first systematically dis-
cussed the opinion of the muscularity of the crystalline lens.*

* De Facilitate Ocali qua ad di^ersas Rerum distantias se accommodat. L.B. 1719.
Ap. Hall. Disp. Anat. ir. p. 301.

No. IJ. MECHANISM OF THE EYE. 53

He referred to Leeuwenhoek's microscopical observations ; but
be so overwhelmed his subject with intricate calculations, that
few have attempted to develop it : and he grounded the whole
on an experiment borrowed from Barrow, which with me
has totally failed ; and I cannot but agree with Dr. Olbers in
the remark, that it is easier to confute him than to understand
him. He argued for a partial change o) the figure of the lens ;
and perhaps the opinion was more just than the reasons adduced
for its support. Lobe, or rather Albinus,* decidedly favours
a similar theory ; and suggests the analogy of the lens to the
muscular parts of pellucid animals, in which even the best
microscopes can discover no fibres. Camper also mentions
the hypothesis with considerable approbation.! Professor Reil
published, in 1793, a Dissertation on the Structure of the Lens ;
and, in a subsequent paper, annexed to the translation of my
former Essay in Professor Gren's Journal,} he discussed the
question of its muscularity. I regret that I have not now an
opportunity of referring to this publication ; but I do not recol-
lect that Professor Reil's objections are differentr.from those
which I have already noticed.

Considering the sympathy of the crystalline lens with the
uvea, and the delicate nature of the change of its figure, there
is little reason to expect that any artificial stimulus would be
more successful in exciting a contractive action in the lens, than
it has hitherto been in the uvea ; much less would that contrac-
tion be visible without art. Soon after Mr. Hunter's death, I
pursued the experiment, which he had suggested, for ascertain-
ing how far such a contraction might be observable. My
apparatus (Fig. 30) was executed by Mr. Jones. It con-
sisted of a wooden vessel blacked within, which was to be
filled with cool, and then with wanner water : a plane speculum
was placed under it ; a perforation in the bottom was filled with
a plate of glass ; proper rings were fixed for the reception of
the lens, or of the whole eye, and also wires for transmitting
electricity : above these, a piece of ground and painted glass,

• De quibuBdam Oculi Paitibus. L. B. 1746. Ap. Hall. Disp. Anat. iv. p. 301.
t De Oculo Humane. L. B. 1742. Ap. Hall. Disp. Anat vii. 2. p. 108, 109.
t 1794, p. a52, 354.

54 MECHAinSM OF THE EYE. No. II.

for receiving the image, was supported by a* bracket, which
moved by a pivot, in connection with a scale divided into fif-
tieths of an inch. With this apparatus I made some experi-
ments, assisted by Mr. AVilkinson, whose residence was near
a slaughter-house: but we could obtain by this method no
satisfactory evidence of the change ; nor was our expectation
much disappointed. I understand also, that another member
of this Society was equally unsuccessful, in attempting to pro-
duce a conspicuous change in the lens by electricity.

XL — In man and in the most common quadrupeds, the struc-
ture of the lens is nearly similar. The number of radiations is
of little consequence ; but I find that in the human crystalline
there are ten on each side (Fig. 49), not three, as I once,
from a hasty observation, concluded.* Those who find any
difficulty in discovering the fibres, must have a sight very ill
adapted to microscopical researches. I have laboured with the
most obstinate perseverance to trace nerves into the lens, and
I have sometimes imagined that I had succeeded ; but I cannot
positively go further than to state my full conviction of their
existence, and of the precipitancy of those who have absolutely
denied it. Tlie long nerves, which are very conspicuous be-
tween the choroid and sclerotic coats, divide each into two,
three, or more branches, at the spot where the ciliary zone
begins, and seem indeed to furnish the choroid with some fine
filaments at the same place. The branches often re -unite, with
a slight protuberance, that scarcely deserves the name of a
ganglion ; here they are tied down, and mixed with the hard
whitish-brown membrane that covers the compact spongy sub-
stance in which the vessels of the ciliary processes anas-
tomose and subdivide. (Fig. 50.) The quantity of the
nerves which proceeds to the iris appears to be considerably
smaller than that which arrives at the place of division : hence
there can be little doubt that the division is calculated to supply
the lens with some minute branches ; and it is not improbable,
from the appearance of the parts, that some fibres may pass to
the cornea ; although it might more naturally be expected, that

♦ Dc Corp. Hum. Vir. Cons., p. 68.

No. II. MECHANISM OF THE EYE. 55

the tunica conjunctiva would be supplied from without. But
the subdivisions which probably pass to the lens, enter imme-
diately into a mixture of ligamentous substance and of a tough
brownish membrane; and I have not hitherto been able to
develop them. Perhaps animals may be found in which this
substance is of a different nature ; and I do not despair that,
with the assistance of injections, for more readily distinguishing
the blood-vessels, it may still be possible to trace them in
quadrupeds. Our inability to discover them is scarcely an
argument against their existence : they must -naturally be deli-
cate and transparent ; and we have an instance, in the cornea,
of considerable sensibility, where no nerve has yet been traced.
The capsule adheres to the ciliary substance, and the lens to
the capsule, principally in two or three points ; but I confess, I
have not been able to observe that these points are exactly
opposite to the trunks of nerves ; so that, probably, the adhesion
is chiefly caused by those vessels which are sometimes seen
passing to the capsule in injected eyes. We may, however,
discover ramifications from some of these points, upon and
within the substance of the lens (Fig. 51), generally follow-
ing a direction near to that of the fibres, and sometimes pro-
ceeding from a point opposite to one of the radiating lines of
the same surface. £ut the principal vessels of the lens appear
to be derived from the central artery, by two or three branches
at some little distance from the posterior vertex ; which I
conceive to be the cause of the frequent adhesion of a portion
of a cataract to the capsule, about this point : they follow
nearly the course of the radiations, and then of the fibres ;
but there is often a superficial subdivision of one of the radii
at the spot where one of them enters. The vessels coming
from the choroid appear principally to supply a substance,
hitherto unobserved, which fills up the marginal part of the
capsule of the crystalline, in the form of a thin zone, and
makes a slight elevation, visible even through the capsule
(Fig. 52 — 54). It consists of coarser fibres than the lens, but
in a direction nearly similar ; they are often intermixed with
small globules. In some animals, the margin of the zone is
crenated, especially behind, where it is shorter : this is observ-

56 MECHANISM OF THE EYE. No. II.

able in the partridge ; and, in the same bird, the whole sur-
face of the lens is seen to be covered with points, or rather
globules, arranged in regular lines (Fig. 55), so as to have
somewhat the appearance of a honeycomb, but towards the
vertex less uniformly disposed. This regularity is a sufficient
proof that there could be no optical deception in the appear-
ance ; although it requires a good microscope to discover it dis-
tinctly : but the zone may be easily peeled off under water, and
hardened in spirits. Its use is uncertain ; but it may possibly
secrete the liquid bf the crystalline ; and it as much deserves the
name of a gland, as the greater part of the substances usually
80 denominated. In peeling it off, I have very distinctly observed
ramifications, which were passing through it into the lens
(Fig. 53) ; and indeed it is not at all difficult to detect
the vessels connecting the margin of the lens with its cap-
sule ; and it is surprising that M. Petit should have doubted
of their existence. I have not yet clearly discerned this
crystalline gland in the human eye ; but I infer the existence
of something similar to the globules, from the spotted appear-
ance of the image of a lucid point already mentioned; for
which I can no otherwise account, than by attributing it to a
derangement of these particles, produced by the external force,
and to an uqequal impression made by them on the surface of
the lens.

In birds and in fishes, the fibres of the crystalline radiate
equally, becoming finer as they approach the vertex, till they
are lost in a uniform substance, of the same degree of firmness,
which appears to be perforated in the centre by a blood-
vessel. (Fig. 56.) In quadrupeds, the fibres at their angular
meeting are certainly not continued, as Leeuwenhoek imagined,
across the line of division ; but there does not appear to be any
dissimilar substance interposed between them, except tliat very
minute trunks of vessels often mark that line. But since the
whole mass of the lens, as far as it is moveable, is probably
endued with a power of changing its figure, there is no need
of any strength of union, or place of attachment, for the fibres,
since the motion meets with little or no resistance. Every
common muscle, as soon as its contraction ceases, returns to

No. II. MECHAKISM OF THE EYE. 57

its natural form, even without the assistance of an antagonist ;
and the lens itself, when taken out of the eye, in its capsule,
has elasticity enough to reassume its proper figure, on the
removal of a force that has compressed it. The capsule is
highly elastic ; and, since it is laterally fixed to the ciliary zone,
it must CO operate in restoring the lens to its flattest form. If
it be inquired why the lens is not capable of becoming less
convex, as well as more so, it may be answered, that the lateral
parts have probably little contractive power ; and, if they had
more, they would have no room to increase the size of the disc,
which they must do, in order to shorten the axis ; and the parts
about the axis have no fibres so arranged as to shorten it by
their own contraction.

I consider myself as being partly repaid for the labour lost in
search of tlie nerves of the lens, by having acquired a more
accurate conception of the nature and situation of the ciliary
substance. It had already been observed, that in the hare and
in the wolf, the ciliary processes are not attached to the cap-
sule of the lens ; and if by the ciliary processes we understand
those filaments which are seen detached after tearing away
the capsule, and consist of ramifying vessels, the observation
is equally true of the common quadrupeds, and, I will venture
to say, of the human eye.* Perhaps this remark has been
made by others, but the circumstance is not generally under-
stood. It is so difficult to obtain a distinct view of these
bodies, undisturbed, that I am partly indebted to accident for
having been undeceived respecting them: but, having once
made the observation, I have learnt to show it in an unques-
tionable manner. I remove the posterior hemisphere of the
sclerotica, or somewhat more, and also as much as possible of
the vitreous humour, introduce the point of a pair of scissors
into the capsule, turn out the lens, and cut off the greater part
of the posterior portion of the capsule^ and of the rest of the
vitreous humoiu*. I next dissect the choroid and uvea from
the sclerotica ; and, dividing the anterior part of the capsule
into segments from its centre, I turn them back upon the
ciliary zone.' The ciliary processes then appear, covered with

• Vid. Hall. Physiol, v. p. 432, ct Duvcrncj, ibi clUt.

58 MECHAinSM OF THE ETE. No. II.

their pigment, and perfectly distinct both from the capsule
and from the uvea (Fig. 57) ; and the surface of the capsule
is seen shining, and evidently natural, close to the base of
these substances. I do not deny that the separation between
the uvea and the processes extends somewhat further back
than the separation between the processes and the capsule ; but
the di£Perence is inconsiderable, and, in the calf, does not amount
to above half the length of the detached part. The appearance
of the processes is wholly irreconcileable with muscularity ; and
their being considered as muscles attached to the capsule is
therefore doubly inadmissible. Their lateral union with the
capsule commences at the base of their posterior smooth sur-
face, and is continued nearly to the point where they are more
intimately united with the termination of the uvea ; so that,
however this portion of the base of the processes were disposed
to contract, it would be much too short to produce any sensible
effect. What their use may be, cannot easily be determined ;
if it were necessary to have any peculiar organs for secretion,
we might call them glands, for the percolation of the aqueous
humour ; but there is no reason to think them requisite for this
purpose.

The marsupium nigrum of birds, and the horseshoe-like
appearance of the choroid of fishes, are two substances which
have sometimes, with equal injustice, been termed muscular.
All the apparent fibres of the marsupium nigrum are, as
Haller had very truly asserted, merely duplicatures of a
membrane, which, when its ends are cut off, may easily be
unfolded under the microscope, with the assistance of a fine
hair pencil, so as to leave no longer any suspicion of a muscular
texture. The experiment related by Mr. Home* can scarcely
be deemed a very strong argument for attributing to this sub-
stance a faculty which its appearance so little authorises us to
expect in it. The red substance in the choroid of fishes
(Fig. 58) is more capable of deceiving the observer; its
colour gives it some little pretension, and I began to examine
it with a prepossession in favour of its muscular nature. But,
when we recollect the general colour of the muscles of fishes,

• Phil. Trans, for 1796, p. 18.

No. II. MECHANISM OF THE EYE. 59

the consideration of its redness will no longer have any weight.
Stripped of the membrane which loosely covers its internal
surface (Fig. 59), it seems to have transverse divisions, some-
what resembling those of muscles, and to terminate in a
manner somewhat similar (Fig. 60); but, when viewed in a
microscope, the transverse divisions appear to be cracks, and
the whole mass is evidently of a uniform texture, without the
least fibrous appearance ; and, if a particle of any kind of
muscle is compared with it, the contrast becomes very striking.
Besides, it is fixed down, throughout its extent, to the posterior
lamina of the choroid, and has no attachment capable of direct-
ing its efiect ; to say nothing of the difficulty of conceiving what
that efiect could be. Its use must remain, in common with
that of many other parts of the animal frame, entirely concealed
from our curiosity.

The bony scales of the eyes of birds, which were long ago
described in the Philosophical Transactions by Mr. Ranby,*
and by Mr. Warren, f afterwards in two excellent Memoirs of
M. Petit on the eye of the turkey and of the owl,t and lately
by Mr Pierce Smith, § and Mr. Home,|| can, on any suppo-
sition, have but little concern in the accommodation of the eye
to different distances : they rather seem to be necessary for the
protection of that organ, large and prominent as it is, and un-
supported by any strength in the orbit, against the various
accidents to which the mode of life and rapid motion of those
animals must expose it; and they are much less liable to
fracture than an entire bony ring of the same thickness would
have been. The marsupium nigrum appears to be intended to
assist in giving strength to the eye, to prevent any change in
the place of the lens by external force ; it is so situated as to
intercept but little light, and that little is principally what
would have fallen on the insertion of the optic nerve ; and it
seems to be too firmly tied to the lens, even to admit any con-
siderable elongation of the axis of the eye, although it certainly
would not impede a protrusion of the cornea.

♦ Phil. Trans, vol. xxxiii. p. 223, Abr. vol. vii. p. 435.

t Phil. Trans, vol. xxxiv. p. 113, Abr. vol. vii. p. 437.

t Mem. de TAcad. 1735, p. 163; 1736, p. 166, Ed. Ainst.

§ Phil. Trans, for 1795, p. 263. || Phil. Trans, for 1796, p. 14.

60 MECHANISM OP THE EYE. No. II.

With respect to the eyes of insects, an observation of Poupart
deserves to be repeated here. He remarks that the eye of the
libellula is hollow; that it communicates with an air-vessel
placed longitudinally in the trunk of the body; and that it
is capable of being inflated from this cavity : he supposes that
the insect is provided with this apparatus, in order for the
accommodation of its eye to the perception of objects at dif-
ferent distances.* I have not yet had an opportunity of ex-
amining the eye of the libellula ; but there is no difficulty in
supposing that the means of producing the change of the
refractive powers of the oye may be, in different classes of
animals, as diversified as their habits and the general conforma-
tion of their organs.

I beg leave to correct here an observation in my former paper,
relative to the faint lateral radiations, which I supposed to pro-
ceed from the margin of the iris.f I find, on further exami-
nation, that they are occasioned by reflections from the eye-
lashes.

XII. —I shall now finally recapitulate the principal objects
and results of the investigation which I have taken the liberty
of detailing so fully to the Royal Society. First, the deter-
mination of the refractive power of a variable medium, and
its application to the constitution of the crystalline lens.
Secondly, the construction of an instrument for ascertaining,
upon inspection, the exact focal distance of every eye, and the
remedy for its imperfections. Thirdly, to show the accurate
adjustment of every part of the eye, for seeing with distinctness
the greatest possible extent of objects at the same instant.
Fourthly, to measure the collective dispersion of coloured rays
in the eye. Fifthly, by immerging the eye in water, to demon-
strate that its accommodation does not depend on any change
in the curvature of the cornea. Sixthly, by confining the eye
at the extremities of its axis, to prove that no material altera-
tion of its length can take place. Seventhly, to examine what
inference can be drawn from the experiments hitherto made on
persons deprived of the lens ; to pursue the inquiry on the

* Phil. Trans, vol. xxu. p. 673, Abr. ii.p. 762.
t Supra, p. 9.

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No. II. MECHANISM OF THE EYE. 61

principles suggested by Dr. Porterfield ; and to confirm his
opinion of the utter inability of such persons to change the
refractive state of the organ. Eighthly, to deduce, from the
aberration of the lateral rays, a decisive argument in favour of
a change in the figure of the crystalline ; to ascertain, from
the quantity of this aberration, the form into which the lens
appears to be thrown in my own eye, and the mode by which
the change must be produced in that of every other person.
And I flatter myself that I shall not be deemed too precipitate
in denominating this series of experiments satisfactorily demon-
strative.

Explanation of the Figures,

Fig. 4. See page 16. Prop. III.

Fig. 6. See page 16. Prop. IV.

Fig. 6. See p^^e 19. Prop. V.

Fig. 7 — 9. Relating to the optometer. See page 21.

Fig. 10. The form of the ends of the optometer, when made of
card. The apertures in the shoulders are for holding a lens : the square
ends turn under, and are fastened together.

Fig. 11. The scale of the optometer. The middle line is divided,
from the lower end, into inches. The next column shows the number
of a concave lens requisite for a short-sighted eye ; by looking through
the slider and observing the number opposite to which the intersection
appears wheq most remote. By observing the place of apparent
intersection when nearest, the number requisite will be found in the
other column, provided that the eye have the average power of accom-
modation. At the other end, the middle line is graduated for ex-
tending the scale of inches by means of a lens four inches in focus :
the negative numbers implying that such rays as proceed from them
are made to converge towards a point on the other side of the lens.
The other column shows the focal length of convex glasses required by
those eyes to which the intersection appears, when nearest, opposite the
respective places of the numbers.

Fig. 12. A side view of the optometer, half its size.

Fig. 13. The appearance of the lines through the slider.

Fig. 14. Method of measuring the magnitude of an image on the
retina. See page 33.

Fig. 15. Diagonal scale drawn on a looking-glass.

62 MECHANISM OF THE EYE. No. II.

Fig. 16. The method of applying a lens with water to the cornea.

Fig. 17. The appearance of a spectrum occasioned by pressure;
and the inflection of straight lines seen within the limits of the spec-
trum.

Fig. 18. An illustration of the enlargement of the image, which
would be the consequence of an elongation of the eye : the images of
the candles, which, in one instance, fall on the insertion of the nerve,
felling, In the other instance, beyond it.

Fig. 19. The successive forms of the image of a large distant
object, as it would be delineated by each refractive surface in the
eye ; to show how that form at last coincides with the retina. EG is
the distance between the foci of horizontal and vertical rays in my eye.

Fig. 20. Vertical section of my right eye, seen from without ; twice
the natural size.

Fig. 21. Horizontal section, seen from above.

Fig. 22. Front view of my lefl eye when the pupil is contracted ; of
the natural size.

Fig. 23. The same view when the pupil is dilated.

Fig. 24. Outline of the eye and its straight muscles when at rest.

Fig. 25. Change of figure which would be the consequence of the
action of those muscles u)x>n the eye, and upon the adipose substance
behind it.

Fig. 26. Scale of the small optometer.

Fig. 27. Appearance of four images of a line seen by my eye when
its focus is shortest

Fig. 28. Outline of the lens when relaxed ; from a comparison of
M. Petit's measures with the phenomena of my own eye, and on the
supposition that it is found in a relaxed state afler death.

Fig. 29. Outline of the lens sufficiently changed to produce the
shortest focal distance.

Fig. 30. Apparatus for ascertaining the focal length of the leus in
water.

Fig. 31. Various forms of the image depicted by a cylindrical pencil
of rays obliquely refracted by a spherical surface, when received on planes
at distances progressively greater.

Fig. 32. Image of a minute lucid object held very near to my eye.
. Fig. 33. The same appearance when the eye has been rubbed.

Fig. 34 — 40. Different forms of the image of a lucid point at greater
and greater distances ; the most perfect focus being like Fig. 36, but
much smaller.

Fig. 41. Image of a very remote point seen by my right eye.

Fig. 42. Image of a remote jx)int seen by my lefl eye; being more

No. II. MECHANISM OF THE EYE. 63

obtuse at one end, probably from a less obliquity of the posterior surface
of the crystalline lens.

Fig. 43. Combination of two figures similar to the fifth variety of
Fig. 31 ; to imitate Fig. 41.

Fig. 44. Appearance of a distant lucid point when the eye is adapted
to a very near object.

Fig. 45, 47. Shadow of parallel wires in the image of a distant point
when the eye is relaxed.

Fig. 46, 48. The same shadows rendered carved by a change in the
figure of the crystalline lens.

Fig. 49. The order of the fibres of the human crystalline.

Fig. 50. The division of the nerves at the ciliary zone ; the sclerotica
being removed. One of the nerves of the uvea is seen passing forwards
and subdividing. From the calf.

Fig. 51. Ramifications from the margin of the crystalline lens.

Fig. 52. The zone of the crystalline faintly seen tlirough the capsule.

Fig. 53. The zone raised from its situation, with the ramifications
passing through it into the lens.

Fig. 54. The zone of the crystalline detached.

Fig. 55. The crenated zone, and the globules regularly arranged on
the crystalline of the partridge.

Fig. 56. The order of the fibres in the lens of birds and fishes.

Fig. 57. The segments of the capsule of the crystalline turned back,
to show the detached ciliary processes. From the calf.

Fig. 58. Part of the choroid of the cod-fish, with its red substance.
The central artery hangs loose firom the insertion of the nerve.

Fig. 59. The membrane covering this substance internally, raised by
the blow-pipe.

Fig. 60. The appearance of the red substance, after the removal of
the membrane.

64 EXPERIMENTS AND INQUIRIES No. III.

No. III.

OUTLINES OF EXPERIMENTS AND INQUIRIES RESPECTING

SOUND AND LIGHT.

From the Philosophical Transactioiu.

In a Letter addressed to Edward Wiiittaker Grev, M.D., SkcR.S,

Read January 16th, 1800.

Dear Sir,

It has long been my intention to lay before the Royal
Society a few observations on the subject of sound ; and I have
endeavoured to collect as much information, and to make as
many experiments, connected with this inquiry, as circumstances
enabled me to do ; but the further I have proceeded, the more
widely the prospect of what lay before me has been extended ;
and, as I find, that the investigation, in all its magnitude, will
occupy the leisure hours of some years, or perhaps of a life, I
am determined, in the mean time, lest any unforeseen circum-
stances should prevent my continuing the pursuit, to submit to
the Society some conclusions which I have already formed from
the results of various experiments. Their subjects are, I. The
measurement of the quantity of air discharged through an aper-
ture. II. The determination of the direction and velocity of a
stream of air proceeding from an orifice. III. Ocular evidence
of the nature of sound. IV. The velocity of sound. V. Sonorous
cavities. VI. The degree of divergence of sound. VII. The
decay of sound. VIII. The harmonic sounds of pipes. IX. The
vibrations of different elastic fluids. X. The analogy between
light and sound. XL The coalescence of musical sounds. XII.
The frequency of vibrations constituting a given note. XIII.
The vibrations of chords. XIV. The vibrations of rods and

No. III.

RESPECTING SOUND AND LIGHT.

plates. XV. The human voice. XVI. The temperament of
musical intervals.

I. — Of the Qimntity of Air discharged through an Aperture.

A piece of bladder was tied over the end of the tube of a
large glass funnel, and punctured with a hot needle. The
funnel was inverted in a vessel of water ; and a gage, with a
graduated glass tube, was so placed as to measure the pressure
occasioned by the different levels of the surfaces of the water.
As the air escaped through the puncture, it was supplied by a
phial of known dimensions, at equal intervals of time ; and,
according to the frequency of this supply, the average height
of the gage was such as is expressed in the first Table. It
appears, that the quantity of air discharged by a given aperture
was nearly in the subduplicate ratio of the pressure ; and that
the ratio of the expenditures by different apertures, with the
same pressure, lay between the ratio of their diameters and that
of their arean. The second, third, and fourth Tables show the
result of similar experiments, made with some variations in the
apparatus. It maybe inferred, from comparing the experiments
on a tube with those on a simple perforation, that the expendi-
ture is increased, as in water, by the application of a short pipe.

Table I. Table II.

.00018

.25

3.9

.00018

.58

11.7

.00018

15.6

.001

.045

7.8

.001

15.6

.001

31.2

.004

.35

46.8

.07
.07

1.
2.

2000.
2900.

I. A is the area, in square inches, of an aperture nearly circular. B,
the pressure in inches. C, the number of cubic inches discharged in one
minute.

All numbers throughout this paper, where the contrary is not
expressed, are to be understood of inches, linear, square, or cubic.

n. A is the area of the section of a tube about two inches long. B,
the pressure. C, the quantity of air discharged in a minute, by esti-
mation.

VOL. I. F

KZFERIMENTS AND INQOIRIES
Table III.

No. III.

.0064

1.15

46.8

.0064

10.

.45

46.8

.0064

13.5

.35

31.2

.0064

13.6

46.8

Table IV.

.003

.88

4«.8

m. A is the area of the section of a ttrbe. B, its I^gth. C, the
pressure. D, the dischoige in a minute.

TV. A is the area of an oval sq^ertare, formed by flattening a glass
tube at the end: its diameters were .025 and .152. B, the pressure.
C, the discharge.

II. — Of the Direction and Vdodty of a Stream of Air.

An apparatus was contrived for measuring, bj means of a
water-gage communicating with a reservoir of air, the pressure
by which a current was forced from the reservoir through a
cylindrical tube ; and the gage was so sensible, that, a regular
blast being supplied from the lungs, it showed the slight varia-
tion produced by every pulsation of the heart. The current of
air issuing from the tube was directed downwards, upon a
wlnte plate, on which a scale of equal parts was engraved, and
which was thinly covered with a coloured liquid : the breadth
of the surface of the plate laid bare was observed at different
distances from the tube, and with different degrees of pressure,
care being taken that the liquid should be so shallow as to
yield to the slightest impression of air. The results are
collected in Tables V. and VI., and are exhibited to the
eye in Figs. 61 — 72. In order to measure, with greater
certainty and precision, the velocity of every part of the current,
a second cavity, furnished with a gage, was provided, and
pieces perforated with apertures of different sizes were adapted
to its orifice : the axis of the current was directed as accurately
as possible to the centres of these apertures, and the results of
the experiments* with various pressures and distances, are in-
serted in Tables VII., VIII., and IX. The velocity of a stream
being, both according to the commonly received opinion and
to the experiments already related, nearly in the subduplicate

No. in. RESPEcrma sound and light. 67

ratio of the pressure occasioning it, it was inferred, that an
equal pressure would be required to stop its progress, and that
the Telocity of the current, where it struck against the aperture,
must be in the subduplicate ratio of the pressure marked by
the gage. The ordinates of the curves in Figs. 73 — 83 were
therefore taken reciprocally in the subduplicate ratio of the
pressure marked by the second gage to that indicated by the
first, at the yarious distances represented by the absdsses.
Each figure represents a difierent degree of pressure in the
first cavity. The curve nearest the axis is deduced from
observations in which the aperture opposed to the tube was
not greater than that of the tube itself; and shows what would
be the diameter of the current, if the velocities of every one
of its particles in the same circular section, including those of
the contiguous air, which must have acquired as much motion
as the current has lost, were equal among themselves. As
the central particles must be supposed to be less impeded in
their motion than the superficial ones, of course the smaller
the aperture opposed to the centre of the current, the greater
the velocity ought to come out, and the ordinate of the curve
the smaller; but, where the aperture was not greater than
that of the tube, the difference of the velocities at the same
distance was scarcely perceptible. When the aperture was
larger than that of the tube, if the distance was very small, of
course the average velocity came out much smaller than that
which was inferred from a smaller aperture : but, where the
ordinate of the internal curve became nearly equal to this
aperture, there was but little difierence between the velocities
indicated with difierent apertures. Indeed, in some cases, a
larger aperture seemed to indicate a greater velocity; this
might have arisen in some degree from the smaller aperture
not having been exactly in the centre of the current; but
there is greater reason to suppose, that it was occasioned by
some resistance derived from the air returning between the
sides of the aperture and the current entering it Where
this took place, the external curves, which are so constructed
as that their ordinates are reciprocally in the subduplicate
ratio of the pressure observed in the second cavity, with

68 EXPERIMENTS AND INQUIRIES No. III.

apertures equal in semidiameter to their initial ordinate,
approach, for a short distance, nearer to the axis than the
internal curve : after this, they continue their course very near
to this curve. Hence it appears that no observable part of
b the motion diverged beyond the limits of the solid which would
be formed by the revolution of the internal curve, which is
seldom inclined to the axis in an angle so great as ten degrees.
A similar conclusion may be made, from observing the flame
of a candle subjected to the action of a blowpipe : there is no
divergency beyond the narrow limits of the current ; the flame,
on the contrary, is everywhere forced by the ambient air
towards the current, to supply the place of that which it has
carried away by its friction. The lateral communication of
motion, very ingeniously and accurately observed in water by
Professor Venturi, is exactly similar to the motion here
fid^iown to take place in air ; and these experiments fully justify
him in rejecting the tenacity of water as its cause : no doubt
it arises from the relative situation of tiie particles of the fluid,
in the line of the current, to that of the particles in the con-
tiguous strata, which is such as naturally to lead to a com-
munication of motion nearly in a parallel direction ; and this
may properly be termed friction. The lateral pressure, which
urges the flame of a candle towards the stream of air from a
blowpipe, is probably exactly similar to that pressure which
causes the inflection of a current of air near an obstacle.
Mark the dimple which a slender stream of air makes on the
surface of water ; bring a convex body into contact with the
side of the stream, and the place of the dimple will imme-
diately show that the current is inflected towards the body,
and, if the body be at liberty to move in every direction, it will
be urged towards the current, in the same manner as, in
Venturi's experiments, a fluid was forced up a tube inserted
into the side of a pipe through which water was flowing. A
similar interposition of an obstacle in the course of the wind is
probably often the cause of smoky chimneys. One circum-
stance was observed in these experiments, which it is extremely
difficult to explain, and which yet leads to very important
consequences : it may be made distinctly perceptible to the

No. III. RESPECTING SOUND AND LIGHT. 69

eye, by forcing a current of smoke very gently through a fine
tube. When the velocity is as small as possible, the stream
proceeds for many inches without any observable dilatation ;
it then immediately diverges at a considerable angle into a
cone, Fig. 84 ; and, at the point of divergency, there is an ^
audible and even visible vibration. The blowpipe also affords
a method of observing this phsenomenon: as far as can be
judged from the motion of the flame, the current seems to
make something like a revolution in the surface of the cone,
but this motion is too rapid to be distinctly discerned. Wlien
the pressure is increased, the apex of the cone approaches
nearer to the orifice of the tube, Figs. 85, 86 ; but no degree
of pressure seems materially to alter its divergency. The
distance of the apex from the orifice is not proportional to the
diameter of the current ; it rather appears to be the greater
the smaller the current, and is much better defined in a small
current than in a large one. Its distance in one experiment
is expressed in Table X., from observations on the surface of
a liquid ; in other experiments, its respective distances were
sometimes considerably less with the same degrees of pressure.
It may be inferred, from the numbers of Tables VII. and VIIL,
that in several instances a greater height of the first gage pro-
duced a less height of the second : this arose from the nearer
approach of the apex of the cone to the orifice of the tube, the
stream losing a greater portion of its velocity by this divergence
than it gained by the increase of pressure. At first sight,
the form of the current bears some resemblance to the vena
cantracta of a jet of water ; but Venturi has observed, that in
water an increase of pressure increases, instead of diminishing,
the distance of the contracted section from the orifice. Is it
not possible, that the facility with which some spiders are said
to project their fine threads to a great distance, may depend
upon the small degree of velocity with which they are thrown
out, so that, like a minute current^ meeting with little inter-
ruption from the neighbouring air, they easily continue their
course for a considerable time ?

EXPEBTMENTO ASD INQUIRIES
Table V.

No. III.

3.8

.12

.17

.25

.30

.52

.35

.54

.37

.56

.39

,58

10.

.40

15.

18.

.50

20.

V. The diameter of the tube .07. A* is the distance of the liquid
firom the oriiice. B, the pressure. C, the diameter of the sur&ce of the
Lquid displaced.

Table VI. TaWe VII.

.13

.25

.35

10.

.85

15.

.35

20.

.35

.06

.16

.083

.16

.25

.35

.45

.53

1.2

1.5

.67

.55

1.3

14.

VI. Diameter of the tube .1. A, B, and C, as in Table V.

Vn. Diameter of the tube .06. A is the distance of the opposite
aperture from the orifice of the tube. B, the diameter of the aperture.
C, the pressure, indicated by the first gage. D, the height of the second

gage-

No. III.

RESPXCTING aaUSD AND LIGHT.

Table VIII.

.§

'• 1

.06

.15

.06

.16

.06

.15

.06

.16

.03

.05

.03

.017

.12

.08

.02

.034

.22

.00

.82

.86

.17

.06

.04

.62

.28

.22

.21

.08

.07

.36

.32

.12

1.1

1.2

.68

.52

.42

.16

.18

.15

.14

1.5

.68

.52

.23

.18

.04

.05

1.7

.83

.63

.25

.22

.05

.06

l.»

1.8

.75

.85

.26

.06

.07

2.1

1.5

1.2

.88

.34

.07

)».

2.3

1.7

1.4

.37

.46

.anr

.84

.08

.06

.08

10.

2.6

*•

1.9

1.6

1.1

•*

.37

.09

Vm. Diameter of the tube .1. A, B, C, and D, as in Table Vn.
Table IX. Tabl« X.

3.3

1.15

.15

.06

.15

.06

1.2

3.
1.5

1.8

.35

.34

.13

.125

.15

• 18

• 1

IX. Diameter of the tube .3. A, B, C, and D, as in Table VII.

X. A is the pressure. B, the distance of the apex of the cone from
the orifice of a tube .1 in diameter.

III. — Ocular Evidence of the Nature of Sound.
A tube about the tenth of an inch in diameter, with a lateral
orifice half an inch from its end, filed rather deeper than the
axis of the tube, Ilg. 87, was inserted at the apex of a conical
cavity containing about twenty cubic inches of air, and luted
perfectly tight : by blowing through the tube a soimd nearly
in unison with the tenor C was produced. By gradually
increasing the capacity of the cavity as far as several gallons,
with the same mouth-piece, the sound, although faint, became
more and more grave, till it was no longer a musical note.
Even hetore this period a kind of trembling was distinguishable ;
and this» as the cavity was still further increased, was changed

72 EXPERDIENTS AND INQUIRIES No. III.

into a succession of distinct puflfe, like the sound produced by
an explosion of air from the lips ; as slow, in some instances, as
4 or 3 in a second. These were undoubtedly the single vibra-
tions, which, when repeated with sufficient frequency, impress
on the auditory nerve the sensation of a continued soimd. On
forcing a current of smoke through the tube, the vibratory
motion of the stream, as it passed out at the lateral orifice, was
evident to the eye : although, from various circumstances, the
quantity and direction of its motion could not be subjected to
exact mensuration. This species of sonorous cavity seems
susceptible of but few harmonic sounds. It was observed that
a faint blast produced a much greater frequency of vibrations
than that which was appropriate to the cavity : a circumstance
similar to this obtains also in large organ-pipes ; but, several
minute observations of this kind, although they might assist in
forming a theory of the origin of vibrations, or in confirming
such a theory drawn from other sources, yet, as they are not
alone sufficient to afibrd any general conclusions, are omitted
at present, for the sake of brevity.

lY.— Ofthe Velocity of Sound.

It has been demonstrated by M. De la Grange and others,
that any impression whatever communicated to one particle of
an elastic fluid, will be transmitted through that fluid with an
uniform velocity, depending on the constitution of the fluid,
without reference to any supposed laws of the continuation of
that impression. Their theorem for ascertaining this velocity
is the same as Newton has deduced from the hypothesis of a
particular law of continuation : but it must be confessed, that
the result diflers somewhat too widely from experiment, to give
^ /(s fiill confidence in the perfection of the theory. Corrected
by the experiments of various observers, the velocity of any
impression transmitted by the common air, may, at an average,
be reckoned 1130 feet in a second.

V. — Of sonorous Cavities.

M. De la Grange has also demonstrated, that all impressions
are reflected by an obstacle terminating an elastic fluid, with

No. III. RESPECTING SOtTND AND LIGHT. 73

the same velocity with which they arrived at that obstacle.
When the walls of a passage, or of an unfurnished room, are
smootib and perfectly parallel, any explosion, or a stamping with
the foot, communicates an impression to the air, which is
reflected from one wall to the other, and from the second again
towards the ear, nearly in the same direction with the primitive
impulse : this takes place as frequently in a second, as double
the breadth of the passage is contained in 1130 feet; and the
ear receives a perception of a musical sound, thus determined
in its pitch by the breadth of the passage. On making the ex-
periment, the result will be found accurately to agree with this
explanation. If the sound is predetermined, and the frequency
of vibrations such as that each pulse, when doubly reflected,
may coincide with the subsequent pulse proceeding directly
from the sounding body, the intensity of the sound will be
much increased by the reflection ; and also, in a less degree, if
the reflected pulse coincides with the next but one, the next but
two, or more, of the direct pulses. The appropriate notes of a
room may readily be discovered by sin^ng the scale in it ; and
they will be found to depend on the proportion of its length or
breadth to 1130 feet. The sound of the stopped diapason pipes
of an organ is produced in a manner somewhat similar to the
note from an explosion in a passage ; and that of its reed pipes
to the resonance of the voice in a room : the length of the pipe
In one case determining the sound, in the other increasing its
strength. The frequency of the vibrations does not at all im-
mediately depend on the diameter of the pipe. It must be con-
fessed that much remains to be done in explaining the precise
manner in which the vibration of the air in an organ-pipe is
generated. M. Daniel Bernoulli has solved several difiicult
problems relating to the subject ; yet some of his assumptions
are not only gratuitous, but contrary to matter of fact.

VI. — Of the Ditergence of Sound.

It has been generally asserted, chiefly on the authority of
Newton, that if any sound be admitted through an aperture
into a chamber, it will diverge from that aperture equally in all
directions. The chief arguments in favour of this opinion are

74 EXPERIMENTS AND INQUIRIES No. III.

deduced from considering the phaenomena of the pressure of
fluids, and the motion of waves excited in a pool of water. But
the inference seems to be too hastily drawn : there is a very
material difference between impulse and pressure ; and, in the
case of waves of water, the moving force at each point is the
power of gravity, which, acting primarily in a perpendicular
direction, is only secondarily converted into a horizontal force,
in the direction of the progress of the waves, being at each step
disposed to spread equally in every direction : but the impulse
transmitted by an elastic fluid, acts primarily in the direction of
its progress. It is well known, that if a person calls to another
with a speaking trumpet, he points it towards the place where
his hearer stands : and I am assured by a very respectable
Member of the Royal Society, that the report of a cannon
appears many times louder to a person towards whom it is
fired, than to one placed in a contrary direction. It must have
occurred to every one's observation, that a sound such as that
of a mill^ or a fall of water^ has appeared much louder after
turning a comer, when the house or other obstacle no longer
intervened; and it has been already remarked by Euler, on tim
head, that we are not acquainted with any substance perfectly
impervious to sound. Indeed, as M. Lambert has very truly
asserted, the whole theory of the speaking trumpet, supported
as it is by practical experience, would fall to the ground, if it
were demonstrable that sound spreads .equally in every direc-
tion. In windy weather it may often be observed, ihat the
sound of a distant bell varies almost instantaneously in its
strength, so as to appear at least twice as remote at one time as
at another ; an observation which has also occurred to another
gentleman, who is uncommonly accurate in examining the
phaenomena of nature. Now, if sound diverged equally in all
directions, the variation produced by the wind could never
exceed one-tenth of the apparent distance ; but, on the suppo-
sition of a motion nearly rectilinear, it may easily happen that
a slight change in the direction of the wind may convey the
sound, either directly or after reflection, in very different de-
grees of strength, to the same spot. From the experiments on
the motion of a current of air, already related, it would be

No. III. RESPECTING SOUND AND LIGHT. . 75

expected tliat a sound, admitted at a considerable distance from
its origin through an aperture, would proceed, with an ahnost
imperceptible increase of divergence, in the same direction ; for,
the actual Telocity of the particles of air, in the strongest sound,
is incomparably less than that of the slowest of the currents in
the experiments related, where the beginning of the conical
divergence took place at the greatest distance. Dr. Matthew
Young has objected, not without reason, to M. Hube, that the
existence of a condensation will cause a divergence in sound :
but a much greater degree of condensation must have existed
in the currents described than in any sound. There is indeed
one difference between a stream of air and a sound ; that, in
sound, the motions of different particles of air are not synchro-
nous ; but it is not demonstrable that this circumstance would
affect the divergency of the motion, except at the instant of
its commencement, and perhaps not even then in a material
degree ; for, in general, the motion is commuDicated with a
very gradual increase of intensity. The subject, however*
deserves a more particular investigation ; and, in order to obtain
a more solid foundation for the argument, it is proposed, as
soon as circumstances permit, to institute a course of experi-
ments for ascertaining as accurately as possible the different
strength of a sound once projected in a given direction, at dif-
ferent distances from the axis of its motion.

Vn.— Of the Decay of Sound.

Various opinions have been entertained respecting the decay
of sound. M. De la Grange has published a calculation, by
which its force is shown to decay nearly in the simple ratio of
the distances; and M. Daniel Bernoulli's equations for the
sounds of conical pipes lead to a similar conclusion. The same
inference would follow from a completion of the reasoning of
Dr. Helsham, Dr. Matthew Young, and Professor VenturL
It has been very elegantly demonstrated by Maclaurin, and
may also be proved in a much more simple manner, that when
motion is communicated through a series of elastic bodies
increasing in magnitude, if the number of bodies be supposed
infinitely great, and their difference infinitely small, the motion

76 EXPERIMENTS AND INQTHRIES No. III.

of the last will be to that of the first in the subdnplicate ratio of
their respective magnitudes ; and since, in the case of concentric
spherical laminse of air, the bulk increases in the duplicate ratio
of the distance, the motion will in this case be directly, and the
velocity inversely, as the distance. But, however true this may
be of the first impulse, it will appear, by pursuing the calcula-
tion a little further, that every one of the elastic bodies, except
the last, receives an impulse in a retrograde direction, which
ultimately impedes the effect of the succeeding impulse, as much
as a similar cause promoted that of the preceding one : and
thus, as sound must be conceived to consist of an infinite
number of impulses, the motion of the last lamina will be pre*
cisely equal to that of the first } and, as far as this mode of
reasoning goes, sound must decay in the duplicate ratio of the
distance. Hence it appears, that the proposal for adopting the
logarithmic curve for the form of the speaking trumpet, was
founded on fallacious reasoning. The calculation of M. De la
Grange is left for fiiture examination ; and it is intended^ in the
mean time, to attempt to ascertain the decay of sound as nearly
as possible by experiment : should the result favour the con-
clusions from that calculation, it would establish a marked
difierence between the propagation of sound and of light

yUL—Ofthe Harmonic Sounds of Pipes.

In order to ascertain the velocity with which organ-pipes of
difierent lengths require to be supplied with air, according to
the various appropriate sounds which they produce, a set of
experiments was made, with the same mouthpiece, on pipes
of the same bore, and of different lengths, both stopped and
open. The general result was, that a similar blast produced as
nearly the same sound as the length of the pipes would permit ;
or at least that the exceptions, though very numerous, lay
equally on each side of this conclusion. The particular results
are expressed in Table XI. and in Fig. 88. They explain
how a note may be made much louder on a wind instrument
by a swell, than it can possibly be by a sudden impression of the
blast. It is proposed, at a future time, to ascertain, byexperiment,
the actual compression of the air within the pipe under different

[

No. III.

RBSPECriNG SOUND AND LIGHT.

circumstances : from some very slight trials, it seemed to be
nearly in the ratio of the frequency of vibrations of each
harmonic.

Table XJ.

OPEN.

STOPPED.

4.5

0.7

8.8

4.5

0.3

1-8

4.1

6.8

1.2
5.0

1.7
9.0

lO'O

3
5

9.4

0.3

0.9

0.8

8.0

9.4

0.2

0.4

2.0

18.0

0.45

1.6

5.0

8.0

20.0

1.1

1.6

8.5

16.5

18.0

7.0

8.0

19.0

20.0

16.1

0.4

0.6

16.1

0.4

1.0

0.6

0.65

1.1

0.8

1.0

2.2

0.9

1.1

2.4

1.2

2.2

4.7

1.6

2.4

4.9

2.2

4.7

11.5

2.5

4.8

9.0

3.4

13.5

6.0

7.0

4 0

15 0

7
8

6.5

10.0

20.5

0.8
1.1

1.1
3.8

«!

7
9

1.0

20.5

0.6

0.8

1.8

3.8

0.8

1.9

3.2

3.8

12.

1.1

1.9

5.7

12.

4.5

5.7

XI. A is the length of the pipe from the lateral orifice to the end. C,
the pressure at which the soond began. B, its termination, by lessening
the pressure ; D, by increasing it. E, the note answering to the first
soimd of each pipe, according to the German method of notation. F,
the number showing the place of each note in the regular series of
harmonics. The diameter of the pipe was .35 ; the air-duct of the
mouth-piece measured, where smallest, .25 by .035 ; the lateral orifice
.25 by .125. The apparatus was not calculated to apply a pressure of
above 22 inches. Where no number stands under C, a sudden blast was
required to produce the note.

IX.— 0/ the Vibrations of different Elastic Fluids.

All the methods of finding the velocity of sound agree in
determining it to be, in fluids of a given elasticity, reciprocally

78 EXPERIMENTS AND INQtJiRIES No. III.

•s

in the subduplicate ratio of the density : hence, in pure hydro-
gen gas it should be V 13 ~ 3 . 6 times as great as in common
air ; and the pitch of a pipe should be a minor fourteenth •>

higher in this fluid than in the common air. It is therefore
probable that the hydrogen gas used in Professor Chladni's *

late experiments was not quite pure. It must be observed, «s

that in an accurate experiment of this nature, the pressure
causing the blast ought to be carefully ascertained. There
can be no doubt but that, in the observations of the French •^

Academicians on the velocity of sound, which appear to have
been conducted with all possible attention, the dampness and
coldness of the night air must have considerably increased its ^

density : hence, the velocity was found to be only 1109 feet in
a second ; while Derham's experiments, which have an equal
appearance of accuracy, make it amount to 1 142. Perhaps ^

the average may, as has been already mentioned, be safely
estimated at 1130. It may be here remarked, that the well-
known elevation of the pitch of wind instruments, in the course *^
of playing, sometimes amounting to half a note, is not, as is
commonly supposed, owing to any expansion of the instru-
ment, for this should produce a contrary effect, but to the •^
increased warmth of the air in the tube. Dr. Smith has made
a similar observation, on the pitch of an organ in summer and
winter, which he found to differ more than twice as much as the *^
English and French experiments on the velocity of soimd.
Bianconi found the velocity of sound, at Bologna, to differ, at
different times, in the ratio of 152 to 157. ^^

X. — Of the Analogy between LiglU and Sound.

j Ever since the publication of Sir Isaac Newton's incom- '^

parable writings, his doctrines of the emanation of particles of
light from lucid substances, and of the formal pre-existence
of coloured rays in white light, have been almost universally *^

admitted in this country, and but little opposed in others.
Leonard Euler indeed, in several of his works, has advanced
some powerful objections against them, but not sufficiently
powerful to justify the dogmatical reprobation with which he
treats them ; and he has left that system of an ethereal vibration,

No. III. RESPECmNG SOUND AND LIGHT. 79

which after Huygens and some others he adopted, equally

' liable to be attacked on many weak sides. Without pretending

^ to decide positively on the controversy, it is conceived that some

considerations may be brought forwards, which may tend to

diminish the weight of objections to a theory similar to the

f Huygenian. There are also one or two diflSculties in the

^ Newtonian system, which have been little observed. The first

is, the uniform velocity with which light is supposed to be

0 projected from all luminous bodies, in consequence of heat, or

otherwise. How happens it that, whether the projecting force

is the slightest transmission of electricity, the friction of two

1^ pebbles, the lowest degree of visible ignition, the white heat of

a wind furnace, or the intense heat of the sun itself, these

wonderful corpuscles are always propelled with one uniform

m velocity? For, if they differed in velocity, that difference ought

to produce a different refraction. But a still more insuperable

difficulty seems to occur, in the partial reflection from every

^ refracting surface. Why, of the same kind of rays, in every

circumstance precisely similar, some should always be reflected,

and others transmitted, appears in this system to be wholly

0 inexplicable. That a medium resembling, in many properties,
that which has been denominated ether, does i eally exist, is
undeniably proved by the phaenomena of electricity ; and the

^ arguments against the existence of such an ether throughout

the universe, have been pretty sufficiently answered by Euler.
The rapid transmission of the electrical shock shows that the ,

f electric medium is possessed of an elasticity as great as is

1 necessary to be supposed for the propagation of light. Whether
the electric ether is to be considered as the same with the lumi-

f nous ether, if such a fluid exists, may perhaps at some future

. time be discovered by experiment ; hitherto I have not been

^ able to observe that the refractive power of a fluid undergoes

any change by electricity. The uniformity of the motion of
li^t in the same medium, which is a difficulty in the Newtonian
theory, favours the admission of the Huygenian ; as all impres-
sions are known to be transmitted through an elastic fluid with
the same velocity. It has been already shown, that sound, in
all probability, has very little tendency to diverge : in a medium

80 EXPERIMENTS AND INQUIRIES No. Til.

V }

80 highly elastic as the luminous ether must be supposed to be, (

the tendency to diverge may be considered as infinitely small,
and the grand objection to the system of vibration will be
removed. It is not absolutely certain, that the white line visible
in all directions on the edge of a knife, in the experiments of J

Newton and] of Mr. Jordan, was not partly occasioned by the
tendency of light to diverge. Euler's hypothesis, of the trans- *

mission of light by an agitation of the particles of the refract-
ing media themselves, is liable to strong objections ; according ^
to this supposition, the refraction of the rays of light, on entering
the atmosphere from the pure ether which he describes, ought
to be a million times greater than it is. For explaining the ^
phenomena of partial and total reflection, refraction, and inflec-
tion, nothing more is necessary than to suppose all refracting
media to retain, by their attraction, a greater or less quantity t^
of the luminous ether, so as to make its density greater than
that which it possesses in a vacuum, without increasing its elasti-
city ; and that light is a propagation of an impulse communi- ^
cated to this ether by luminous bodies : whether this impulse is
produced by a partial emanation of the ether, or by vibrations
of the particles of the body, and whether these vibrations are, ^
as Euler supposed, of various and irregular magnitudes, or
whether they are uniform, and comparatively large, remains to
be hereafter determined. Now, as the direction of an impulse,
transmitted through a fluid, depends on that of the particles in
synchronous motion, to which it is always perpendicular, whatr
ever alters the direction of the pulse, will inflect the ray of ^
light. If a smaller elastic body strike against a larger one, it is
well known that the smaller is reflected more or less powerfully,
according to the difierence of their magnitudes : thus, there is >
always a reflection when the rays of light pass from a rarer to .
a denser stratum of ether ; and frequently an echo when a sound J|
strikes against a cloud. A greater body, striking a smaller one, ^ i
propels it, without losing all its motion : thus, the particles of a |
denser stratum of ether do not impart the whole of their motion |
to a rarer, but, in their efibrt to proceed, they are recalled by ^
the attraction of the refracting substance with equal force ; and
thus a reflection is always secondarily produced, when the rays

Ko. III. RESPECTING SOUND AND LIGHT. 81

of light pass from a denser to a rarer stratum. Let AB,
Fig. 89, be a ray of light falling on the reflecting surface
FG ; cd the direction of the vibration, * pulse, impression,
or condensation. When d comes to H, the impression will be,
either wholly or partly, reflected with the same velocity as it
arrived, and EH will be equal to DH ; the angle EIH to DIH
or GIF ; and the angle of reflection to that of incidence. Let
FG, Fig. 90, be a refracting surface. The portion of the
pulse IE, which is travelling through the refracting medium,
will move with a greater or less velocity in the subduplicate
ratio of the densities, and HE will be to K I in that ratio.
But HE is, to the radius I H, the sine of the angle of refraction ;
and KI that of the angle of incidence. This explanation of
refraction is nearly the same as that of Euler. The total
reflection of a ray of light by. a refracting surface is explicable
in the same manner as it^ simple refraction ; HE, Fig. 91,
being so much longer than KI, tliat the ray first becomes
parallel to FG, and then, having to return through an equal
diversity of media, is reflected in an equal angle. When
a ray of light passes near an inflecting body, surrounded,
as all bodies are supposed to be, with an atmosphere of ether
denser than the ether of the ambient air, the part of the ray
nearest the body is retarded, and of course the whole ray
inflected towards the body, Fig. 92. The repulsion of inflected
rays has been very ably controverted by Mr. Jordan, the
ingenious author of a late publication on the Inflection of
Light It has already been conjectured by Euler, that the
colours of light consist in the diflferent frequency of the vibra-
tions of the luminous ether : it does not appear that he has sup-
ported this opinion by any argument ; but it is strongly con-
firmed, by the analogy between the colours of a thin plate and
the sounds of a series of organ-pipes.* The phsenomena of the
colours of thin plates require, in the Newtonian system, a
very complicated supposition, of an ether, anticipating by its
motion the velocity of the corpuscles of light, and thus pro-
ducing the fits of transmission and reflection ; and even this
supposition does not much assist the explanation. It appears

* This analogy is fanciful and altogether unfounded. — Note by the Editor,

VOL, I. a

82 EXPERIMENTa AND INQUIRIES No. III.

from the accurate analysis of the phsenomena which Newton
has gi^en, and which has by no means been superseded by
any later observations, that the same colour recurs wheneyer
the thickness answers to the terms of an arithmetical progres-
sion. Now this is precisely similar to the production of the
same sound, by means of an uniform blast, from organ-pipes
which are different multiples of the same length. Supposing
white light to be a continued impulse or stream of luminous
ether, it may be conceived to act on the plates as a blast of air
does on the organ-pipes, and to produce vibrations regulated
in frequency by the length of the lines which are terminated
by the two refracting surfaces. It may be objected that, to
complete the analogy^ there should be tubes to answer to the
organ-pipes : but the tube of an organ-pipe is only necessary to
prevent the divergence of the impression, and in light there is
little or no tendency to diverge ; and indeed, in the case of a
resonant passage, the air is not prevented from becoming sono-
rous by the liberty of lateral motion. It would seem that the
determination of a portion of the track of a ray of light through
any homogeneous stratum of ether is sufficient to establish a
length as a basis for colorific vibrations. In inflections the length
of the track of a ray of light through the inflecting atmosphere
may deteimine its vibrations : but, in this case, as it is probable
that there is a reflection from every part of the surface of the
surrounding atmosphere, contributing to the appearance of the
white line in every direction, in the experiments already men-
tioned, so it is possible that there may be some second reflection
at the immediate surface of the body itself, and that, by mutual
reflections between these two surfaces, something like the
anguiform motion suspected by Newton may really take place ;
and then the analogy to the colours of thin plates wiU be still
stronger. A mixture of vibrations, of all possible frequencies,
may easily destroy the peculiar nature of each, and concur in a
general effect of white light. The greatest difficulty in this sys-
tem is, to explain the different degree of refraction of differentiy
coloured light, and the separation of white light in refraction ;
yet, considering how imperfect the theory of elastic fluids still
remains, it cannot be expected that every circumstance should

f 1

No. III. RESPECTING SOUND AND LIGHT. 83

at once be clearly elucidated. It may hereafter be considered
how far the excellent experiments of Count Rumford, which
tend very greatly to weaken the evidence of the modern doc-
trine of heat, may be more or less favourable to one or the other
system of light and colours. It does not appear that any com-
parative experiments have been made on the inflection of light
by substances possessed of different refractive powers ; un-
doubtedly some very interesting conclusions might be expected
from the inquiry.

XL — Of the Coalescence of Musical Sounds.

It is surprising that so great a mathematician as Dr. Smith
could have entertained for a moment, an idea tliat the vibra-
tions constituting different sounds should be able to cross each
other in all directions, without affecting the same individual
particles of air by their joint forces : undoubtedly they cross,
without disturbing each other's progress ; but this can be no
otherwise effected than by each particle's partaking of both
motions. If this assertion stood in need of any proof, it might
be amply furnished by the phaenomena of beats, and of the
grave harmonics observed by Romieu and Tartini; which
M. De la Grange has already considered in the same point of
view. In the first place, to simplify the statement, let us
suppose, what probably never precisely happens, that the par-
ticles of air, in transmitting the pulses, proceed and return with
uniform motions ; and in order to represent their position to
the eye, let the uniform progress of time be represented by the
increase of the absciss, and the distance of the particle from its
original podtion, by the ordinate. Fig. 93 — 98. Then, by
supposing any two or more vibrations in the same direction to
be combined, the joint motion will be represented by the sum
or difference of the ordinates. When two sounds are of equal
strength, and nearly of the same pitch, as in Fig. 96^ the joint
vibration is alternately very weak and very strong, pro-
ducing the effect denominated a beat. Fig. 103, B and C ;
which is slower and more marked, as the sounds approach
nearer to each other in frequency of vibrations ; and, of these
beats there may happen to be several orders, according to the

84 EXPERIMENTS AND INQUIRIES Ko. III.

periodical approximations of the numbers expressing the pro-
portions of the vibrations. The strength Df the joint sound is
double that of the simple sound only at the middle of the beat,
but not throughout its duration ; and it may be inferred, that
the strength of sound in a concert will not be in exact proportion
to the numberof instruments composing it. Could any method
be devised for ascertaining this by experiment, it would assist
in the comparison of sound with light. In Fig. 93, let P
and Q be the middle points of the progress or regress of
a particle in two successive compound vibrations ; then, CP
being = PD, KR = RN, GQ = QH, and MS = SO, twice their
distance, 2RS = 2RN+2NM+2MS = KHr-hNM+NM-hMO
-= KM+NO, is equal to the sum of the distances of the cor-
responding parts of the simple vibrations. For instance, if the
two sounds be as 80 : 81, the joint vibration will be as 80.5 ;
the arithmetical mean between the periods of the single vibra-
tions. The greater the difference in the pitch of two sounds,
the more rapid the beats, till at last, like the distinct pufis of
air in the experiments already related, they communicate the
idea of a continued sound ; and this is the fundamental har-
monic described by Tartini. For instance, in Fig. 94 — ^97, the
vibrations of sounds related as 1 : 2, 4 : 5, 9 : 10, and
5 : 8, are represented ; where the beats, if the sounds be not
taken too grave, constitute a distinct sound, which corresponds
with the time elapsing between two successive coincidences, or
near approaches to coincidence : for, that such a tempered
interval still produces a harmonic, appears from Fig. 98.
But, besides this primary harmonic, a secondary note is
sometimes heard, where the intermediate compound vibrations
occur at a certain interval, though interruptedly ; for instance,
in the coalescence of two sounds related to each other as 7 : 8,
5 : 7, or 4 : 5, there is a recurrence of a similar state of the
joint motion, nearly at the interval of tt) tV^ or i of the whole
period : hence in the concord of a major third, the fourth
below the key note is heard as distinctly as the double
octave, as is seen in some degree in Fig. 95 ; AB being
nearly two- thirds of CD. The same sound is sometimes pro-
duced by taking the minor sixth below the key note ; probably

No. III. RESPECTING SOUND AND LIGHT. 85

because this sixth, like every other note, is almost always
attended by an octave, as a harmonic. If the angles of all the
figures resulting from the motion thus assumed be rounded off,
they will approach more nearly to a representation of the actual
circumstances ; but, as the laws by which the motion of the
particles of air is regulated, differ according to the different
origin and nature of the sound, it is impossible to adapt a
demonstration to them all : if, however, the particles be supposed
to follow the law of the harmonic curve, derived from uniform
circular motion, the compound vibration will be the harmonic
instead of the arithmetical mean ; and the secondary sound of
the interrupted vibrations will be more accurately formed, and
more strongly marked. Figs. 101, 102 : the demonstration
is deducible from the properties of the circle. It is remark-
able, that the law by which the motion of the particles is
governed, is capable of some singular alterations by a combina-
tion of vibrations. By adding to a given sound other similar
sounds, related to it in frequency, as the series of odd numbers,
and in strength inversely in the same ratios, the right lines
indicating an uniform motion may be converted very nearly
into figures of sines, and the figures of sines into right lines^ as
in Figs. 99, 100.

XII. — Of the Frequency of Vibrations constituting a given

Note.

The number of vibrations performed by a given sound in a
second, has been variously ascertained ; first, by Sauveur, by a
very ingenious inference from the beats of two sounds ; and
since, by the same observer and several others, by calculation
from the weight and tension of a chord. It was thought worth
while, as a confirmation, to make an experiment suggested,
but coarsely conducted, by Mersennus, on a chord 200 inches
in length, stretched so loosely as to have its single vibrations
visible ; and, by holding a quill nearly in contact with the
chord, they were made audible, and were found, in one experi-
ment, to recur 8.3 times in a second. By lightly pressing the
chord at one-eighth of its length from the end, and at other
shorter aliquot distances, the fundamental note was found to be

86 EXPERIMENTS AND INQUIBIES No. III.

one-sixth of a tone higher than the respective octave of a
tuning-fork marked C: hence, the fork was a comma and a
half above the pitch assumed by Sauveur, of an imaginary C,
consisting of one vibration in a second.

XIII.— Cy the VibratioM of Chords.

By a singular oversight in the demonstration of Dr. Brook
Taylor, adopted as it has been by a number of later authors,
it is asserted, that if a chord be once inflected into any other
form than that of the harmonic curve, it will, since those parts
which are without this figure are impelled towards it by an
excess of force, and those within it by a deficiency, in a very
short time arrive at or very near the form of this precise curve.
It would be easy to prove, if this reasoning were allowed, that
the form of the curve can be no other than that of the axis,
since the tending force is continually impelling the chord
towards this line. The case is very similar to that of the New-
tonian proposition respecting sound. It may be proved, that
every impulse is communicated along a tended chord with an
uniform velocity ; and this velocity is the same which is inferred
from Dr. Taylor's theorem ; just as that of sound, determined
by other methods, coincides with the Newtonian result. But,
although several late mathematicians have ^ven admirable
solutions of all possible cases of the problem, yet it has still
been supposed, that the distinctions were too minute to be
actually observed ; especially, as it might have been added,
since the inflexibility of a wire would dispose it, according to the
doctrine of elastic rods, to assume the form of the harmonic
curve. The theorem of Euler and De la Grange, in the case
where the chord is supposed to be at first at rest, is in effect
this : continue the figure each way, alternately on difierent
ffldes of the axis, and in contrary positions; then, from any
point of the curve, take an absciss each way, in the same pro-
portion to the length of the chord as any given portion of time
bears to the time of one semivibration, and the half sum of the
ordinates will be the distance of that point of the chord from the
axis, at the expiration of the time given. If the initial figure
of the chord be composed of two right lines, as generally hap-

No. III. RESPECTING SOUND AND LIGHT. 87

pens in musical instruments and experiments, its successive
forms will be such as are represented in Figs. 107, 108 : and
this result is folly confirmed by experiment. Take one of
the lowest strings of a square piano forte, round which a fine
silvered wire is wound in a spiral form : contract the light of a
window, so that, when the eye is placed in a proper position,
the image of the light may appear small, bright, and well
defined, on each of the 04>nvolutions of the wire. Let the chord
be now made to vibrate, and the luminous point will delineate
its path^ like a burning coal whirled round, and will present to
the eye a line of light, which by the assistance of a microscope,
may be very accurately observed. According to the difierent
ways by which the wire is put in motion, the form of this path
is no less diversified and amusing, than the multifarious forms
of the quiescent lines of vibrating plates, discovered by Professor
Chladni ; and is indeed in one respect even more interesting,
as it appears to be more within the reach of mathematical cal-
culation to determine it; although hitherto, excepting some
slight observations of Busse and Chladni, principally on the
motion of rods, nothing has been attempted on the subject.
For the present purpose, the motion of the chord may be
simplified, by tying a long fine thread to any part of it, and
fixing this thread in a direction perpendicular to that of the
chord, without drawing it so tight as to increase the tension :
by these means, the vibrations are confined nearly to one plane,
which scarcely ever happens when the chord vibrates at liberty.
If the chord be now inflected in the middle, it will be found,
by comparison with an object which marked its quiescent
position, to make equal excursions on each side of the axis ;
and the figiu-e which it apparently occupies will be terminated
by two lines, the more luminous as they are nearer the
ends. Fig. 109. But, if the chord be inflected near one of
its extremities, Fig. 110, it will proceed but a very small dis-
tance on the opposite side of the axis, and will there form a
very bright line, indicating its longer continuance in that
place; yet it will return on the former side nearly to the
point firom whence it was let go, but will be there very faintly
visible, on account of its short delay. In the middle of the

88 EXPERIMENTS AND INQUIRIES No. 111.

diord, the excursions on eax^Ii side the axis are always equal ;
and, beyond the middle, the same circumstances take place as
in the half where it was inflected, but on the opposite side of
the axis; and this appearance continues unaltered in its pro-
portions, as long as the chord vibrates at all : fully confirming
the non-existence of the harmonic curve, and the accuracy of
the construction of Euler and De la Grange. At the same
time, as M. Bernoulli has justly observed, since every figure
may be infinitely approximated, by considering its ordinates as
composed of the ordinates of an infinite number of trochoids of
different magnitudes, it may be demonstrated that all these
constituent curves would revert to their initial state, in the
same time that a similar chord bent into a trochoidal curve
would perform a single vibration ; and this is in some respects
a convenient and compendious method of considering the pro*
blem. But, when a chord vibrates freely, it never remains long
in motion, without a very evident departure from the plane of
the vibration ; and, whether from the original obliquity of the
impulse, or from an interference with the reflected vibrations of
the air, or from the inequability of its own weight or flexibility,
or from the immediate resistance of the particles of air in con*
tact with it, it is thrown into a very evident rotatory motion,
more or less simple and uniform according to circumstances.
Some specimens of the figures of the orbits of chords are
exhibited in Fig. 104. At the middle of the chord, its orbit
has always two equal halves, but seldom at any other
point. The curves of Fig. 106 are described by combining
together various circular motions, supposed to be performed in
aliquot parts of the primitive orbit : and some of them approach
nearly to the figures actually observed. When the chord is of
unequal tliickness, or when it is loosely tended and forcibly
inflected, the apsides and double points of the orbits have a very
evident rotatory motion. The compound rotations seem to
demonstrate to the eye the existence of secondary vibrations,
and to account for the acute harmonic sounds which generally
attend the fundamental sound. There is one fact respecting
these secondary notes, which seems entirely to have escaped
observation. If a chord be inflected at one-half, one-third, or

No. III. RESPECTING SOUND AND LIGHT. 89

any other aliquot part of its length, and then suddenly left at
liberty, the harmonic note which would be produced by divid-
ing the chord at that point is entirely lost, and is not to be dis-
tinguished during any part of the continuance of the sound.
This demonstrates, that the secondary notes do not depend
upon any interference of the vibrations of the air with each
other, nor upon any sympathetic agitation of auditory fibres,
nor upon any effect of reflected sound upon the chord, but
merely upon its initial figure and motion. If it were supposed
that the chord when inflected into right lines, resolved itself
necessarily into a number of secondary vibrations, according to
some curves which, when properly combined, would approxi-
mate to the figure given, the supposition would indeed in some
respects correspond with the phsenomenon related ; as the co-
efiicients of all the curves supposed to end at the angle of inflec-
tion would vanis$h. But, whether we trace the constituent curves
of such a figure through the various stages of their vibrations,
or whether we follow the more compendious method of Euler
to the same purpose, the figures resulting from this series of
vibrations are in fact so simple, that it seems inconceivable how
the ear should deduce the complicated idea of a number of
heterogeneous vibrations, from a motion of the particles of air
which must be extremely regular, and almost uniform ; an uni-
formity which, when proper precautions are taken, is not con-
tradicted by examining the motion of the chord with the assist-
ance of a powerful magnifier. Tliis diflSculty occurred very
strongly to Euler; and De la Grange even suspects some
fallacy in the experiment, and that a musical ear judges from
previous association. But, besides that these sounds are dis-
ooyerable to an ear destitute of such associations, and, when
the sound is produced by two strings in imperfect unison, may
be verified by counting the number of their beats, the experi-
ment already related is an undeniable proof that no fallacy
of this kind exists. It must be confessed, that nothing fully
satisfactory has yet occurred to account for the phaenomena :
but it is highly probable that the slight increase of tension pro-
duced by flexure, which is omitted in the calculations, and the
unavoidable inequality of thickness or flexibility of different

90 EXPERIMENTS AND INQUrRIES No. III.

parts of the same chord, may, by disturbing the isochronism of
the subordinate vibrations^ cause all that variety of sounds
which is so inexplicable without them. For, when the slightest
difference is introduced in the periods, there is no diflBculty iu
conceiving how the sounds may be distinguished ; and indeed,
in some cases, a nice ear will discover a slight imperfection in
the time of harmonic notes : it is also often observed, in tuning
an instrument, that some of the single chords produce beating
sounds, which undoubtedly arise from their want of perfect
uniformity. It may be perceived that any particular harmonic
is loudest, when the chord is inflected at about one-third of the
corresponding aliquot part firom one of the extremities of that
part An observation of Dr. Wallis seems to have passed
unnoticed by later writers on harmonics. If the string of a violin
be struck in the middle, or at any other aliquot part, it will
give either no sound at all, or a very obscure one. This is true,
not of inflection, but of the motion communicated by a bow ;
and may be explained from the circumstance of the successive
impulses, reflected from the fixed points at each end, destroying
each other : an explanation nearly analogous to some observa-
tions of Dr. Matthew Young on the motion of chords. When
the bow is applied not exactly at the aliquot point, but very
near it, the corresponding harmonic is extremely loud ; and the
fundamental note, especially in the lowest harmonics, scarcely
audible: the chord assumes the appearance, at the aliquot
points, of as many lucid lines as correspond to the number of
the harmonic, more nearly approaching to each other as
the bow approaches more nearly to the point, Kg. 111.
According to the various modes of applying the bow, an
immense variety of figures of the orbits are produced, Fig.
105, more than enough to account for all the difference of tone
in different performers. In observations of this kind, a series
of harmonics is frequently heard in drawing the bow across
the same part of the chord: these are produced by the
bow ; they are however not proportionate to the whole length
of the bow, but depend on the capability of the portion of
the bowstring, intercepted between its end and the chord, of
performing its vibrations in times which are aliquot parts of the

No, in. RESPECTING SOUND AND LIGHT. 91

vibration of the chord : hence it would seem, that the bow takes
effect on the chord but at one instant during each fundamental
vibration. In these experiments, the bow was strung with the*
second string of a violin : and, in the preparatory application
of resin, the longitudinal sound of Chladni was sometimes
heard ; but it was observed to differ at least a note in different
parts of the string.

XIY.—Ofthe VibraHons of Rods and Plates.

Some experiments were made, with the assistance of a most
excellent practical musician, on the various notes produced by
a glass tube^ an iron rod, and a wooden ruler ; and, in a case
where the tube was as much at liberty as possible, all the
harmonics corresponding to the numbers from 1 to 13, were
distinctly observed ; several of them at the same time, and
others by means of different blows. This result seems to differ
from the calculations of Euler and Count Riccati, confirmed as
they are by the repeated experiments of Professor Chladni ; it
is not therefore brought forward as sufficiently controverting
those calculations, but as showing the necessity of a revision of
the experiments. Scarcely any note could ever be heard when
a rod was loosely held at its extremity ; nor when it was held
in the middle, and struck one-seventh of the length from one
end. The very ingenious method of Professor Chladni, of
observing the vibrations of plates by strewing fine sand over
them, and discovering the quiescent lines by the figures into
which it is thrown, has hitherto been little known in this
country ; his treatise on the phsenomena is so complete^ that no
other experiments of the kind were thought necessary. Glass
vessels of various descriptions, whether made to sound by
percussion or friction, were found to be almost entirely free
from harmonic notes ; and this observation coincides with the
experiments of Chladni.

XV.— Of the Human Voice.

The human voice, which was the object originally proposed
to be illustrated by these researches, is of so complicated a
nature, and so imperfectly understood, that it can be on this

92 EXPERIMENTS AND INQUIRIES No. III.

occasion but superficially considered. No person, unless we
except *M. Ferrein, has published anything very important on
the subject of the formation of the voice, before or since Dodart;
his reasoning has fully shown the analogy between the voice and
the voix humaine and regal organ-pipes ; but his comparison
with the whistle is unfortunate ; nor is he more happy in his
account of the falsetto. A kind of experimental analysis of the
voice may be thus exhibited. By drawing in the breath, and at
the same time properly contracting the larynx, a slow vibration
of the ligaments of the glottis may be produced, making a
distinct clicking sound : upon increasing the tension, and the
velocity of the breath, this clicking is lost, and the sound
becomes continuous, but of an extremely grave pitch : it may,
by a good ear, be distinguished two octaves below the lowest
A of a common bass voice, consisting in that case of about 26
vibrations in a second. The same sound may be raised nearly
to the pitch of the common voice ; but it is never smooth and
clear, except perhaps in some of those persons called ventrilo-
quists. When the pitch is raised still higher, the upper orifice
of the larynx, formed by the summits of the ary taenoid cartilages
and the epiglottis, seems to succeed to the office of the ligaments
of the glottis, and to produce a retrograde falsetto, which is
capable of a very great degree of acuteness. The same difier-
ence probably takes place between the natural voice and the
common falsetto : the rimula glottidis being too long to admit
of a sufficient degree of tension for very acute sounds, the upper
orifice of the larynx supplies its place ; hence, taking a note
within the compass of either voice, it may be held, with the
same expanse of air, two or three times as long in a falsetto as
in a natural voice ; hence, too, the difficulty of pajt^ing smoothly
from the one voice to the other. It has been remarked that the
larynx is always elevated when the sound is acute : but this
elevation is only necessary in rapid transitions, as in a shake ;
and then probably because, by the contraction of the capacity
of the trachea, an increase of the pressure of the breath can be
more rapidly effected this way, than by the action of the abdo-
minal muscles alone. The reflection of the sound thus produced
from the various parts of the cavity of the mouth and nostrils.

No. III. RESPECriNG SOUND AND LIGHT. 93

mixing at various intervals with the portions of the vibrations
directly proceeding from the larynx, must, according to the
temporary form of the parts, variously affect the laws of the
motion of the air in each vibration, or, according to Euler's
expression, the equation of the curve conceived to correspond
with this motion, and thus produce the various characters of the
vowels and semi-vowels. The principal sounding board seems
to be the bony palate ; the nose, except in nasal letters, affords
but little resonance; for the nasal passage maybe closed by
applying the finger to the soft palate, without much altering
the sound of vowels not nasal. A good ear may distinctly
observe, especially in a loud bass voice, besides the fundamental
note, at least four harmonic sounds, in the order of the natural
numbers ; and, the more reedy the tone of the voice, the more
easily they are heard. Faint as they are, their origin is by
no means easy to be explained. This observation is precisely
confirmed in a late dissertation of M. Knecht, published in the
musical newspaper of Leipsic. Perhaps, by a close attention
to the harmonics entering into the constitution of various
sounds, more may be done in their analysis than could otherwise
be expected.

XVL — Of the Temperament of Musical Intervals*

It would have been extremely convenient for practical musi-
cians, and would have saved many warm controversies among
theoretical ones, if three times the ratio of 4 to 5, or four times
that of 5 to 6, had been equal to the ratio of 1 to 2. As it
happens to be otherwise, it has been much disputed in what
intervals the imperfection should be placed. The Aristoxe-
nians and Pythagoreans were in some sense the beginners of
the controversy. Sauveur has given very comprehensive tables
of a great number of systems of temperament ; and his own
now ranks among the many that are rejected. Dr. Smith* has
written a large and obscure volume, which, for every purpose
but for the use of an impracticable instrument, leaves the
whole subject precisely where it found it. Kirnberger, Mar-
purg, and other German writers, have disputed with great

* See infra, p. 134.

94 EXPERIMENTS AND INQUIRIES No. HI.

bitterness, almost every one for a particular method of tuning.
It is not with any confidence of success that one more attempt
is made, which rests its chief claim to preference on the
similarity of its theory to the actual practice of the best
instrument-makers. However we estimate the degi'ee of im-
perfection of two tempered concords of the same nature, it
will appear that the manner of dividing the temperament
between them does not materially alter its aggregate sum;
for instance, the imperfection of a comma, in a major third,
occasions it to beat very nearly twice as fast as that of half a
comma. If, indeed, the imperfection were great, it might
afiect an interval so materially as to destroy its character ; as,
in some methods of temperament, a minor third diminished by
two commas approaches more nearly to the ratio 6 : 7, than
to 5 : 6 ; but, with this limitation, the sum of harmony is nearly
equal in all systems. Hence, if every one of the twelve major
and minor thirds occurred equally often in the compositions
which are to be performed on an instrument, it would be of no
great consequence, to the sum of the imperfections, among
which of the thirds they were divided : and, even in this case,
the opinion of the best practical authors is, that the difierence
of character produced by a difference of proportions in various
keys, would be of considerable advantage in the general effect
of modulation. But, when it is considered, that upon an
average, of all tiie mumc ever composed, some particular keys
occur at least twice as often as others, there seems to be a very
strong additional reason for making the harmony the most
perfect in those keys which are the most frequently used ; since
the aggregate sum of all the imperfections which occur in
playing, must by this means be diminished in the greatest
possible degree, and the diversity of character at the same time
preserved. Indeed, in practice, this method, under different
modifications, has been almost universal ; for, although many
have pretended to an equal temperament, yet the methods
which they have employed to attain it have been evidently
defective. It appears to me, that every purpose may be
answered, by making C : £ too sharp by a quarter of a comma,
which will not offend the nicest ear ; E : Gj, and Ab : C,

No.m.

RESFECriNG SOUND AND UGHT.

equal j F| : A{ too sharp by a comma ; and the major thirds
of all the intermediate keys more or less perfect, as they
approach more or less to C in the order of modulation. The
fifths are perfect enough in every system. The results of this
method are shown in Table XII. In practice, nearly the same
efiect may be very simply produced, by tuning from C to F,
Bb, Eb, Gj, C}, Fjl, six perfect fourtiis ; and C, G, D, A, E, B,
F}, six equally imperfect fifths, Fig. 112. If the unavoidable
imperfections of the fourths be such as to incline them to
sharpness, the temperament will approach more nearly to
equality, which is preferable to an inaccuracy on the other side.
An easy method of comparing difierent systems of temperament
is exhibited in Fig. 113, which may easily be extended to all
the systems that have ever been invented.

Table XII.

50000

1 C

+ .0013487

1 A, E -

.0023603

53224

2G,F

.0019006

2D,B

.0029192

56131

3D,Bt7

.0024525

3G,Fj

.0034641

59676

4A,E|;

.0034641

4C,Cj

.0044756

63148

5E,A|7

.0044756

5F,Gj

.0049353

66822

6B,qJ

.0049553

6b. E|,

•0053950

71041
74921

71^

.0053950

79752
84197

89304

1 Eb, GJ, C|, FJ - -0000000

94723

2 F, B|?, E, B .0004597

100000

8 C, G, D, A .0010116

A shows the division of a monochord correspoDding to each note
in the system proposed. B, the logarithm of the temperament of each
of the major thirds. C, of the minor thirds. D, of the fifths ; C and
D being both negative.

96 EXPERIMENTS AND INQUIRIES No. III.

Thus, Sir, I have endeavoured to advance a few steps only,
in the investigation of some very obscure but interesting sub-
jects. As far as I know, most of these observations are new ;
but, if they should be found to have been already, made by any
other person, their repetition in a connected chfun of inference
may still be excusable. I am persuaded also, that at least
some of the positions maintained are incontrovertibly consistent
with truth and nature ; but, should further experiments tend to
confute any opinions that I have suggested, I shall relinquish
them with as much readiness as I have long since abandoned
the hypothesis which I once took the liberty of submitting to
the Royal Society, on the functions of the crystalline lens.*

I am, &c.

Thomas Young.

Emanuel Coll^, Cambridge,
Slh Jaly, 1799..

Explanation of tJie Figures.

Figs. 61 — 66. The section of a stream of air from a tube of .07 inch
in diameter, as ascertained by measuring the breadth of the impression
on the surface of a liquid. The pressure impelling the current, was
in Fig. 61, 1 inch. Fig. 62, 2. Fig. 63, 3. Fig. 64, 4. Fig. 66, 7.
Fig. 66, 10.

Figs. 67 — 72. A similar section, where the tube was .1 in diameter,
compared with the section as inferred from the experiments with two
gauges, which is represented by a dotted line. From this comparison it
appears, that where the velocity of the current was small, its central
parts only displaced the liquid ; and that, where it was great, it dis-
placed, on meeting with resistance, a surface somewhat greater than its
own section. The pressure was in Fig. 67, 1. Fig. 68, 2. Fig. 69, 3.
Fig. 70, 4. Fig. 71, 7. Fig. 72, 10.

Figs. 73 — 80. A, the half section of a stream of air from a tube .1
in diameter, as inferred from experiments with two water gauges. The
pressure was in Fig. 73, .1. Fig. 74, .2. Fig. 75, .5. Fig. 76, 1.
Fig. 77, 3. Fig. 78, 5. Fig. 79. 7. Fig. 80, 10. The fine lines,
marked B show the result of the observations with an aperture .15 in
diameter opposed to the stream • C with .3 ; and D with .5.

Figs. 81 — 83. A, the half section of a current from a tube .3 in
diameter, with a pressure of .5 of 1, and of 3. B shows the course
of a portion next the axis of the current, equal in diameter to those
represented by the last figures.

_. - ., * Supra, p. 12.

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SOITXD AND LIGHT.

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A I

N(A III. RESPECTINa SOUND AND LIGHT. 97

Fig. 84. The appefiraDCC of a stream of smoke forced very gently
from a fine tabe. Figs. 85 and 86, the same appearance when the
pressure is gradually increased.

Fig. 87. See Section III.

Fig. 88. The perpendicular lines over each division of the horizontal
line show, by their length and distance from ihat Une, the extent of
pressure capable of producing, from the respective pipes, the hannonic
notes indicated by the figures placed opposite the beginning of each,
according to the scale of 22 inches parallel to them. The lai^r
numbers, opposite the middle of each of these lines, show the number
of vibrations of the corresponding sound in a second.

Figs. 89—93. See Section X,

Fig. 94. The combination of two equal sounds constituting the
interval of an octave, supposing the progress and regress of the particles
of air equable. Figs. 95, 96, 97, a similar representation of a major
third, major tone, and minor sixth.

Fig. 98. A fourth, tempered about two commas.

Fig. 99. A vibration of a similar nature, combined with subordinate
vibrations of the same kind in the ratios of 8, 5, and 7.

Fig. 100. A vibration represented by a curve of which the ordinates
are the sines of circular arcs increasing uniformly, corresponding with
the motion of a cycloidal pendulum, combined with similar subordinate
vibrations in the ratios of 3, 5, and 7.

Figs. 101 and 102. Two different positions of a major third, com-
posed of similar vibrations, as represented by figures of sines.

Fig. 103. A contracted representation of a series of vibrations. A,
a simple uniform sound. B, the beating of two equal sounds nearly in
unison, as derived from rectilinear figures. 0, the beats of two equal
sounds, derived from figures of sines. D, a musical consonance making
by its frequent beats a fundamental harmonic. £, the imperfect beats
of two unequal sounds.

Fig. 104. Various forms of the orbit of a musical chord, when
inflected, and when struck.

Fig. 105. Forms of the orbit, when the sound is produced by means
of a bow. ^

Fig. 106. Epitrocboidal curves, formed by combining a simple
rotation or vibration with other subordinate rotations or vibrations.

Figs. 107 and 108. The successive forms of a tended chord, when
inflected and let go, according to the construction of De la Grange and
Euler.

VOL. I. H

98 EXPERIMENTS RESPBCTINa SOUND AND UGHT. No. III.

Fig. 109. The appeannce of a vibrating diord which had been
inflected in the middle^ the strongest lines representing the most
luminous parts.

Fig. 110. The appearance of a vibrating chord, when inflected at any
other point than the middle.

Fig. 111. The appearance of a chord, when put in motion By a bow
applied nearly at one third of the length from its end.

Fig. 112. The method of tuning recommended for common use.

Fig. 118. A comparative view of different systems of temperament.
The whole circomference represents an octave. The inner circle L is
divided into 30103 parts, corresponding with the logarithmical parts
of an octave. The next circle R shows the magnitude of the simplest
musical and other ratios. Q is divided into twelve equal parts, repre-
senting the semitones of the equal temperament described by Zarhno,
differing but little from the system of Aristoxeuus, and warmly recom-
mended by Marpuig and other late writers. Y exhibits the system
proposed in this paper as the most desirable; and P the practical
method nearly approaching to it, which corresponds with the eleventh
method in Marpurg^s enumeration, except that, by beginning with G
instead of B, the practical effect of the temperament is precisely
inverted. E is the system of Kimbeiger and Sulzer ; which is derived
from one perfect third, ten perfect and two equally imperfect fifths.
M is the system of mean tones, the sistema partidpato of the old
Italian writers, still frequently used in tuning organs, approved also by
Dr. Smith for common use. S shows the result of all the calculations
in Dr. Smith's harmonics, the system proposed for his changeable
harpsichord, but neith^ in lihat nor any other form capable of practical
application.

No. IV. AN ESSAY ON CnTCLOIDAL CURVES. 99

No. IV.

AN BSSAT ON

CYCLOIDAL CURVES.'

FelioM aninue, qnibus b«c cognonere primis,
Inqne domos saperam scaadere, cura fiiit.

On Mathematical Stmbojjb.

Many of the most celebrated modern mathematiciaiis bare been
disposed to pride themselves on the very great superiority wliich
they attribute to the modem methods of calculation over those
which were known to the ancients. Tliat, in the course of so
many centuries, mathematical sciences, like all others, should
have been very considerably advanced, is no more than must
have been expected, from the great number of persons who have
employed their talents in the cultivation of those sciences. But
if we examine the matter impartially, we shall have reason to

* This Eflsay was the third of a Beries which appeared in the British Magazine for
leOO, nnkr the title of <" The Uplologiet," on thesahjecte indicated in fiie foUowing
motto:

'< Grammatlcus, rhetor, geometres, pictor, aliptes,
Angnr, schcenohaies, medico^ magus^ omnia novit
GrKcnlus esuriens."

The introductory ohseryations afford a good insight into his peculiar views of the
superiority of geometrical to analytical processes of ravestigation, on the grounds of
their greater simplicity, clearness, and brevity. There is manifested in the criticisms
which are intermixed with them a certain flippancy and bitterness of expression,
which too frequently appeared in his replies, at least in early life, to those who
controverted his opinions : the character of -the introductory remarks upon some pro-
positions centred in a paper by Lord Brougham, printed in the Philosophical
Transactions for 1798, was probably not without its influaoce in proToking those
persevering attacks upon his optical researches, which appeared in some of the earlier
Nnmbtsrs of the Edinburgh Review.

The investigations contained in this Essay, though neither yery important nor
original, are characteristic of Dr. Young's remarkable aptitude for applying a some-
what impure geometry to the solution of problems, to which an approach is usually
required to be made by processes of a much more regular and elaborate character. —
Note hjf th0 Editor,

100 AN ESSAY ON No. IV,

believe not only that mathematics have been as slow in their
real advancement as any other part of philosophy, but that the
modems have very frequently neglected the more essential, for
frivolous and superficial advantages. To say nothing of the
needless incumbrances of new methods of variations, of combi*
natorial analyses, and of many other similar innovations, the
strong inclination which has been shown, especially on the Con-
tinent, to prefer the algebraical to the geometrical form of re-
presentation, is a sufiicient proof, that, instead of endeavouring
to strengthen and enlighten the reasoning faculties, by accus-
toming them to such a consecutive train of argument as can be
fully conceived by the mind, and represented with all its links
by the recollection, they have only been desirous of sparing
themselves as much as possible the pains of thought and labour
by a kind of mechanical abridgment, which, at best, only serves
the office of a book of tables in facilitating computations, but
which very often fails even of this end, and is at the same
time the most circuitous and the least intelligible. These phi-
losophers are like the young Englishman on his travels, who
visits a country by driving with all possible speed from place to
place by night, and refreshing his fatigues in the daytime by
lounging half asleep at his hotel. Undoubtedly there are some
countries through which one may reasonably wish to travel by
night ; and undoubtedly there are some cases where algebrdcal
symbols are more convenient than geometrical ones : but when
we see an author exerting all his ingenuity in order to avoid
every idea that has the least tincture of geometry, when he
obliges us to toil through immense volumes filled with all man-
ner of literal characters, without a single diagram to diversify
the prospect, we may observe with the less surprise that sudi
an author appears to be confused in his conception of the most
elementary doctrines, and that he fancies he has made an im-
provement of consequence, when, in fact, he is only viewing an
old object in a new disguise. It happens frequently in the de-
scription of curves, and in the solution of problems, that tiie
geometrical construction is very simple and easy, while it
almost exceeds the powers of calculus to express the curve or
the locus of the equation in a manner strictly algebraical ; and^

No. IV. CYCLOIDAL CURVES. 101

indeed, the astonishiiig advances that were made in a compa-
ratively short time by Euclid, by Apollonius, and, above all, by
Archimedes, are sofficient to prove that the method of repre-
sentation which they employed could not be very limited in its
application ; and the precision and elegance with which tiie
method of geometrical fluxions is treated by Newton and
Maclaurin form a strong contrast to the tedious afiectation of
abstraction and obscurity which unfortunately pervades the
writings of many great mathematicians of a later date. It
would be of inestimable advantage to the progress of all the
sdences if some diligent and judicious collector would under-
take to compile a complete system of mathematics ; not as an
elementary treatise, nor as a mere index of reference, but to
contain every proposition, with a concise demonstration, that
has ever yet been communicated to the public. Until this is
done nothing is left but for every individual who is curious in
the search of geometrical knowledge to look overall the mathe-
matical authors and all the literary memoirs of the last and
present centuries : for without this he may very easily fancy he
has made discoveries, when the same facts had been known and
forgotten long before he existed. An instance of this has
lately occurred to a young gentleman in Edinburgh, a man who
certainly promises, in the course of time, to add considerably to
our knowledge of the laws of nature. The tractory, tractrix,
or equitangential curve, was first described by Huygens, and
afterwards more fully by M. Bomie, M^nu Acad. 1712, and
by Mr. Perks, Ph. Tr. v. xix. n. 345, Abr. v. iv. p. 466. Bo-
mie and Perks have shown many remarkable properties belong-
ing to it ; and one in particular, which may be briefly demon-
strated, that it is die involute of the catenaria : for since the
equation of the catenaria is zz = iax + xx^ we have zz ^ ax
+ xx^ and x : z: : z : a+x, therefore the vertex of the right-
angled triangle, of which the base is the evolved radius, and
tiie hypotenuse a line parallel to the axis of the curve, describes
a right line ; and the perpendicular of this triangle is always
=3 a, and is the constant tangent of the curve described by the
evolution. Cotes has also, in his Logometria, investigated the
properties of the tractrix of the circle. Bernoulli observed, in

102 AN ESSAY ON No. IV.

1730^ that the tractrix was one of the tautochrooons curves in
a i^esisting medium. In 1786 it was the subject of a dispute
between MM. Clairaut and Fontaine: it is not yet entirely
forgotten cm tliat spot of academic ground which gave birth to
the discoYeries of Newton ; and its equation is to be found in a
work no less common than £mer9on*8 Fluxions, nearly in the
same form as that which is published as new in the Philosophi-
cal Transactions for 1798. We find in the same paper a new
method of dividing an elliptic area in a given ratio ; but the
curve which the author calls a cycloid is the companion of a
trochoid, and is only a distortion of the figure by which Newton
had very simply and elegantly solved the same problem. It is
unnecessary to compare the attempt to demonstrate the incom-
mensurability of an oval with the Newtonian method : since
Dr. Waring's proof, deduced from the nature of the equation
of limits, is decidedly more satisfactory than any other hitherto
made known. On the whole it appears that this ingenious gen-
tleman has becD somewhat unfortunate in the dioioe of those
problems which he has selected as specimens of the elegance of
the modem mode of demonstration ; whether those which he
has brought forward without proof would have furnished him
with a more favourable opportunity for the display of neatness
and accuracy, may be more easily determined whenever he may
think proper to lay before the public their analysis, construction,
and demonstration, at full length. But, allowing the superi-
ority of the modem calcidations in many cases, their great ad-
vantage appears to be derived from the methods of series and
approximations ; indeed, however we may wish to adhere to the
rigour of the ancient demonstrations, it is absolutely necessary,
for the purposes of the higher geometry, to extend, in some mea-
sure, the foundations which the ancients laid in thdr postulates.
Perhaps the most material addition may be comprehended in
this form: ^^Let it be granted that any curve line may be
drawn whenever an indefinitely great number of points may be
geometrically found in, or indefinitely near to, that line." No
doubt it is mathematically impossible to comply with this pos-
tulate ; but it must be remembered that it is also impossible to
draw, with strictly mathematical accuracy, a right line or a

No, IV. CYCLOIDAL CURVES. 103

drcle ; bat in both cases we can approach snffidently near to
the truth for practice : and it appears to be more convenient to
consider snch carves as are thus described as belonging to geo-
metry, than to limit the number of geometrical curves, accord-
ing to Descartes, to those of which the ordinate and absciss are
comparable by an algebraical equation. This postulate forms
the connecting link between rational and irrational quantities,
between the infinite and the indefinite, between perfect resem-
blanoe and identity ; and the irrational geometry which has
long been tacitly built on it, exhibits the principal advantages of
analytical calculas in a more elegant form. The ground-work
of this irrational geometry is found in the method of exhaustions
of Euclid and Ardumedes ; and it has been employed more or
less generally by Descartes, Newton, Cotes, Boberyal, Vari-
gnon, Delahire, MacUuirin, and many other mathematicians. In
the annexed essay on cycloidal curves the geometrical form of
flaxions, or, more properly speaking, the Newtonian method of
ultimate ratios, has principally been adopted ; and it is pre*
sumed that by a comparison with algebraical calculations on
the same subjects, the superior perspicuity and conciseness of
this method will readily appear.

On Ctcloidal Cttrves.

«« n semble qu'ane destinee particalii^re attacbee k la cycloKde
loi donne preferaUement aux autres ooorbes an plus grand
nomlffe de propriety remarquables."

D^MiHon J.

When a circle is made to rotate on a rectilinear basis, the
figure described on the plane of the basis by any point in the
plane of the circle, is called a trochoid. A circle concentric
with the generating circle, and passing through the describing
point, may be called the describing circle.

Definition II.
If the describing point is in the circumference of the rotating

104 AN ESSAY ON No. IV. .

circle, the two circles ooiacide, and the curve is called a
cycloid.

Definition III.

If a circular basis be substituted for a rectilinear one, the
trochoid will become an epitrochoid, and the cycloid an epicy-
cloid.

Scholium 1. These terms have hitherto been too promiscu-
ously employed; the terms cycloid and trochoid have been
used indifierently ; and the term epicycloid has comprehended
£he epitrochoid, the terms prolate and contracted being some-
times added to imply that the describing point is within or
without the generating circle. The interior epicycloid and
epitrochoid may very properly be distinguished by the names
hypocycloid and hypotrochoid^ whenever they are the separate
objects of consideration. The different species of epicycloids
may be denominated according to the number of their cusps,
combined with that of the entire revolutions which they com-
prehend ; for instance, the epicycloid described by a circle on
an equal basis is a simple unicuspidate epicycloid ; and if the
diameter of the generating circle be to that of the baas as 5 to
2, the figure will be a quintuple bicuspidate epicycloid. If the
describing circle of a trochoid or cycloid be so placed as to
touch the middle of the basis, and each of the ordinates parallel
to the basis be diminished by the corresponding ordinate of the
circle, the curve thus generated has been denominated the
companion of the trochoid or cycloid, the figure of sines, and
the harmonic curve.

Scholium 2. Tlie invention of the cycloid has been attributed
by Wallis, Ph. Tr. for 1697, n. 229, to Cardinal Cusanus. who
wrote about the year 1450 : but it seems to be at least as pro-
bable that the curve which appears in Cusanus's figure was
meant for the semicircle employed in finding a mean propor-
tional. Bovil]us, in 1501, has a juster claim to the merit of
the invention of the cycloid and trochoid, if it can be any merit
to have merely imagined such curves to exist In 1599 Galileo
gave a name to the common cycloid, and attempted its quadra-
ture ; but having been accidentally misled by repeated ezperi-

No* IV. CYGLOroAL CURVES. 105

menta on the weight of a flat subfitaDoe cut into a cycloidal
form, he fancied that the area bore an incommensurable ratio
to that of the circle, and desisted from the investigation. Mer*
sennus described the cycloid, in 1615, under the name of la
trocholde, or la roulette, but he went no further. Roberval
seems to have first discovered the comparative quadrature and
rectification of the cycloid, and the content of a cycloidal solid,
about the year 1635, but his treatise was not printed until 1695.
Torricelli, in 1644, first published the quadrature and the
method of drawing a tangent Wallis gave, in 1670, a perfect
quadrature of a portion of the cycloid. The ejucycloid is said
to have been invented by RcBmer ; its rectification and evolute
were investigated by Newton in the Principia, published in
1687. In 1695 Mr. Caswell showed the perfect quadrability
of a portion of the epicycloid, and Dr.Halley immediately
published an extension of Caswell's discovery, together with a
comparison of all epitrochoidal with circular areas. M. Vari-
gnon is also said to have reduced the rectification of the
epitrodioid to that of the ellipsis, in the same year. Nicole,
Delahire, Pascal, Reaumur, Maclaurin, the Bernoullis, the
commentators on Newton, and many others, have contributed
to the examination of cycloidal curves, both in planes and in
curved surfaces ; and Waring, th^ most profound of modern
algebraists, has considerably extended his researches upon the
nature of those lines which are generated by a rotatory pro-
gression of other curves. In the present essay the most
remarkable properties of cycloidal curves are deduced in a
simpler and more general manner than appears to have been
hitherto done, the equations of several species are investigated,
a singular property of the quadricuspidate hypocycloid is
demonstrated, and the peculiarities of the spiral of Archimedes
are derived from its generation as an epitrochoid.

Pboposition I. — Theorem, (Kg. 1 1 4.)

In any curve generated by the rotation of another on any
basis, the right line joining the describing point, and the point
of contact of the generating curve and the basis, is always per-
pendicular to the curve described.

106 AN ESSAY ON No. IV.

It may by some be deemed sufficient to consider the gene-
rating cunre as a rectilinear polygon of an infinite number of
sides ; since, in this point of view, the proposition requires no
further demonstration ; and, indeed, Newton and others hare
not scrupled to take it for granted : but it is presumed that a
more rigid proof will not be conadered as superfluous. Let M
be the describing point, and P the point of contact; and let
LO, MP, and NQ, be successive positions of ihe same chord of
the generating cunre at infinitely small distances ; then it is
obvious, and easily demonstrable, that the arcs OP and PQ,
described by the point P of the generating curve in its passage
from O to P, and from P to Q, will be perpendicular to the
basis at P, and will, therefore, touch each other. Let the arcs
L, IM K, and N, be described with the radius PM, on the
centres O, P, and Q. Then the curve described by M will
touch IMK ; for since O and Q lie ultimately in the same
direction from P, if L be above IMK, N will also be above it,
since these points must be in the circles L and N, and infinitely
near to M ; and if L is below IMK, N, for the same reason,
must be below it ; and M is common to the circle and the curve,
therefore the curve touches the circle IMK at M, and is per-
pendicular to the radius PM.

•
Proposition U. — Problem.

To draw a tangent to a cycloidal curve at any given point.

On the pven point as a centre describe a circle equal to the
describing circle of the curve ; and from the intersection of this
circle, with the line described by the centre of the generating
circle, let fall a perpendicular on the basis ; the point thus
found will be the point of contact, and the tangent will be per-
pendicular to the right line joining this point of contact and the
given point, by the first proposition. It will be obvious, from
inspection, which of the two intersections of the circle to be
described, with the track of the centre, is to be taken as the
place of that centre corresponding to the given point.

Na IV. CYCLOIDAL CURVES. 107

Proposition III. — Problem. (Kg. 115.)

To find the length of an epitrochmd.

Let C be the centre of the basis V P, K that of the rotating
drcle PR, and of the describing circle 6L, P the point of
contact, and M the describing point. Then joining MXC,
and sapposing VX to be an element representing the motion
of the point P in either the basis or the generating circle, draw
the arc MN on the centre C, and join CVN, then NM will
represent the motion of the point M as far as it is produced by
the revolution round the centre C : take MO to VX as GE to
PEL, then MO will be the motion of M arising firom the revo-
lution round El, and NO will be the element of the curve
produced by the joint motion. Let CH be parallel to PM,
then CX or CP : CM : : VX : MN, and PK:MK::CP2
HM : : VX : MO, therefore CM : HM : : MN : MO, and
these lines being perpendicular to CM, HM, the triangle
NMO is similar to CMH, and MN : NO : : CM : CH, hence,
CP ; CH ; : VX: NO. Take PY to CP as PK to CK,
then CH : CP : : PM : PY : : NO : VX. On L describe
tlie circle PFB, and draw IMLF : let FD be perpendicular to
PRB, take D£ to DF as PG to PL, and £ will be always in
the ellipsis BEP: let AE and AF be tangents to the ellipsis
and circle at E and F ; then the increment of the arc BF will
be to MO as PL to GL, and to VX as PL to PR. Join
GM, and parallel to it draw PI ; then FIL is a right angle,
and ILP = AED, and IM : IL : : PG : PL : : DE : DF, by
construction ; therefore the figure IPML is similar to D AEF,
as PL to PM so is AF to AE, and so is the increment of the
arc BF to that of BE ; but the increment of BF is to VX
as PL to PR, therefore the increment of BE b to VX as PM
to PR. Now, it was proved that NO : VX : 2 PM : PY;
therefore the increment of BE is to NO as P Y to PR, or as
CP to 8CK ; and the whole elliptic arc BE is to the whole
SM as the radius of the basis to twice the distance of the
centres*

CcroOary 1. llie fluxion of every cycloidal arc is pro-

108 AN ESSAY ox No. IV;

portional to the distance of the describing point from the point
of contact.

Corollary 2. In the epicycloid the ellipsis coincides with its
axis BP^ and the arc BE with BD, which is double* the versed
une of half the arc 6M, in the describing or generating
circle : therefore the length of the curve is to this versed sine
as four times the distance of the centres to the radius of the

Propobition IV.— Problem. (Figs. 115, 116.)

To find the centre of curvature of an epitrochoid.

Let PY be, as in the last Proposition, to CP as PK to CK,
and on the diameter PY describe the circle PZY, cutting PO
in Z : take OW a third proportional to OZ and OP, and W
will be the centre of curvature. For, let QP = VX be the
space described by P, while NO is described by O ; it is
obvious from Prop. I. that the intersection of NQ and OP
must be the centre of curvature. Let QF be perpendicular to
PO, and FA parallel to QN ; then, by Prop. III. NO : VX
or QP : : PO : PY, but by similar triangles QP : QF
PY : PZ ; therefore NO : QF : : PO : PZ, and by division,
NO : AO : : OP : OZ, and by similar triangles : : OW
OF or OP.

Corollary. When Z coincides with O or M, OW is infinite
therefore whenever PZY intersects the describing circle, the
epitrochoid will have a point of contrary flexure at the same
distance from C as this intersection : and the circle PZY is
given when the basis and generating circle are given, whatever
the magnitude of the describing circle may be.

Proposition V. — Problem. (Rg. 117.)

To find the evolute of an epicycloid.

In the epicycloid SM, the point M being in the circum-
ference of PMR, PZ will be to PM in the constant ratio of
PY to PK, and MZ to PM as BY to PR, and PM to MW
in tfie same ratio ; hence PM : P W : : RY : PY : : CR : CP,
therefore the point W is always in a circle PWS of which the

No. IV. CTGLOmAL CURVES. 109

radius is to PK in that proportion; and which touches SP in P.
Ou the centre C describe a circle AB© touching PWE in H ;
then, since CR : CP : : PR : PH, we have by division OR :
CP : : CP : P, and the circle PWE being to AB© as PMR
to SPy the arc PM being equal to SP, the rimilar arc PW
will be equal to AB, and, taking AB® := PWB, B© will be
always equal to BW, and W in a curve ©WS ^milar to SM,
of which it is the evolute.

Proposition YL— Problem. (Fig. 118.)

To find the area of an epitrochoid.

On the centre C describe a circle touching the epitrochoid
in S, take Gil to GC as PR to PC, and let the circle G^Tl
describe on the basis SG the epicycloid S0. Then taking GM
always to G4> as GL to GIT, M will be in the epitrochoid SM;
for the angular motion of the chord G<I>, is the same as that of
GM in the primary epitrochoid. Let SA be the evolute of
S<&, and GWB its generating circle. On diameters equal to
BG, BL, and BIT, describe three circles, AD, AE, and AF,
touching the right line AB in A; let the angle BAD be
always equal to GIK^, and it is evident that AD, AE, and AF,
mil be equal respectively to WG, WM, and W*. But the
angular motion of WG on W being equal to the sum of the
angular motions of GM on G and CG on C, is to that of AF,
or of GM, or half that of KM, in the ratio of CIl to CG, or
CR to CP; therefore the fluxions of the areas SWG, SWM,
and SW^ are to those of the segments AD, AE, and AF,
in the same ratio; and that ratio being constant, the whole
areas, and their differences, are also respectively to each other
as CR to CP.

Scholium. The quadrable spaces of Halley are those which
are comprehended between the arc of the epitrochoid, that of
the describing circle, and that of a circle concentric with the
basis and cutting the describing circle at the extremities of its
diameter.

110 AN ESSAY OBf No. IV.

PROPO0ITION VIL— ProWew. (Kg. 119.)

To find a central equation for the epicycloid.

Let CT be perpendicular to RT, the tangent at the point
M, then PMS will be a right angle, and PM parallel to CT.
On the centre C describe through M the circle MNO, and let
MQ be perpendicular to RO. Then the rectangle OQN = PQR,
OQ : PQ : : QR : QN, by addition OQ : PQ : : OR : PN ;

henceby divirion OP : PQ:: m: PN,ajidPQ«55^^^^^

But PM^sPBxPQ "TrX INP : and by amUar triangles

CT : CR : : PM : PR, whence CT? = J§ x PMy = CB^ x

j^ I«tMZandRYbetaDgeDtBtoSP,thaiINPBMZjr,aiid

IRP = RYy, CT = CR x 5^* ^^^ CT will be to MZ in the

constant ratio of CR to RY. Putting CP = a, CR = &, CM^s,
CT « u, thentttt « bb ^z^*

Peoposition VIIL— Ph^few. (Fig. 120.)

To find a geometrical equation for the oonchoidal epitrochoid.

Let OP = PK On the centre C dedcribe a circle equal to
OM, cutting SC m Z. Join MZF, then the arc DZ » GM,
and MZ is parallel to CE, therefore £F ig also equal to
DZ GOT GM, CF is parallel to KM^ and MF^CE : therefore
this epitrodx)id is the curve named by Delahire the conc]K»d
of a circular baos^ as was first observed by Reaumur in 1708,
and afterwards by Maclaurin in 1720. Call CE^ a, DE, i,
ZHy Xf HM, y, ZM, s ; and let ZI be perpendicular to CE ;

then FZ ss a -«9 CI = ^ , and, CIZ and ZHM being shnilar,

CZ : CI : : ZM : ZH, or y : ^ :is : x; hence bx^as-8i
bx+ 88»a$j Vof + 2&cj*+«* = aV, and by substituting for <•,
** + Uof^ifaf + 4^;r»+2ay+2iay+y*-a«y« = 0.

CcroUary I. Join FN, and complete the parallelogram
MFNL, then since £F=DZ = £N, FN is perpendicular to

No. IV. CTCLOIOAIi CURVES. Ill

EK. and ML to NL, and, NL being always equal to FM or
CK, L is always in a cirele described on the centre N, LM a
tangent to that circle, and ZM a perpendicular to that tangent
drawn from the point Z.

CcroUary 3. Fig. 121. The unicuspidate epicycloid admits of
a peculiar central equation, with respect to the point SL Call
SM, tj and let ST«tt be perpendicular to the tangent MT,

then u = 2^ *. For the triangles SIP and MTS being similar,

and IP being half of SM, or *, SP = ^f , SPy : SMy : : IPy

: STy or f : !• : : ^ : u«, and %aif = «».

CaroUatyd. Iig.l2L The unicuspidate epicycloid is one of
the caustics of a circle. For making the angle CRY « MRC
^i MKP » I SGP, the triangle CRY is isosceles, and CY is
constant; so that all rays in the direction of the tamgent MR
will be reflected by the circle QR towards Y, and consequently
SM will be the caustic of a radiant point at Y.

Proposition lX.—FrMem. (Fig 122.)

To find a geometrical equation for the tricuspidato hypo-
cycloid.

Let PA and MF be perpendicular to CS. Join PMB, KM,
RMG, and PD. Then the angle APR is equal to the differ-
ence of APC and MPR, or to that of their complements PRM,
PCA: but PRM = J PKM = i-.PCA, therefore APB = i
PCA = ADP=APS, and the triangles APS and APR are
similar and equal. LetSC = a, SF = x» FM^y, and SB=^r.
ThenSA :SP : : SP : SD,andSP = Var. Draw PE perpen-
dicular to BP; then BE = SD = 2a, BC=a-r, EC = 3a-r,
and by similar triangles, CP : CR : : EC : CG =+ EC = a-+r;
therefore GB« ir ; but BE : BG : : BP : BM, or 2a : |r : :

^^: ^ Var'=BM ; again BP : BM : : BA : BF, or a/otx ^
Vlw-:: -J- : -^t and SFe««r-^, 6aa:=6ar-rr, aiid r = 3a

T»

± V9aa-6ax. But MFj«BMj-BFy, or 3^=5^-55^, and

112 AN ESSAY ON No. IV.

36ay = 4ar*-r*. By adding to this the square of the former
equation, and proceeding in the same manner to exterminate r,
we obtain an equation of the value of x and y, which, when
the surds are brought to the same side, and the square of the
whole is taken, is at last reduced to or*— 4aa:*+2a:y — 12ajry'
+.y*+12^y'=0, a regular equation of the fourth order.

Scholium. The equation of the corresponding hypotrochoids
may be investigated nearly in the same manner, by dividing
PR and PM in a given ratio, but the process will be somewhat
more tedious.

Proposition X.— Problem. (Fig. 123.)

To find a geometrical equation for the bicuspidate epicycloid.

Let CP = PR. Join RMT, PM, PD ; draw CT perpendi-
cular to RT, TE to CR, and EG, MB, RA, to SC. Then the
angle DRP = i MKP = SCP, and by equal triangles, RA=
CT, and RD=CD, and by similar triangles RM : RP : : RE :
RT,and RP : RD : : RT : RC ; therefore RM : RD : : RE :
RC, and ME is parallel to SC, and EG = BM. Put CP a,
BC = a;, BM=y, CM = «, CT^u; then by Prop. Vll. u« = |
71^; or J u« = «« -a«; butRC : CT :: CT : CE :: CE :

i£* 27 27

EG, or y, hence y = j^ m* = 16a* y" -gj-tt* = — a*i^=ss -^cuf^

XX +yy -^ cui ; whence by involution the equation of the sixth
order may be had at length.

Corollary. Since CRM = SCR, a ray in the direction of the
tangent MR will be reflected by a circle FR always parallel to
SC : therefore SM is the caustic of the circle FR when the
incident rays are parallel to CS.

Proposition XL — Problem, (Fig. 124)

To find a geometrical equation for the quadricuspidate hy-
pocycloid.

Let CR = PR, then the angle PRM = iPKM = 2PCS, RAC
ACR, RA = RC = RB = RP, AB = SC, and drawing the
perpendiculars CT, TD, TE, and MF, RM = RT, AM =BT,
AF = EC, FC = AE, and FM =BD. Let SC =a, FC =a:, FM

No. IV. CYCLOIDAL CURVES. 113

=y, CM=*, CT = M;then AB: AC:: AC : AT:: AT: AE,

whence AT = axx^j and in the same manner BT=ayyJ ; and
CT being a mean proportional between AT and TB, tt* =

c^a^f^y and u* = aV^. But by Prop. VII., 3u« = a* - A
therefore 27aV^=a*— 5^ =a*— j:*— y*^; whence the equation
may be had at length by involution. The same result may be
obtained by Dr. Waring's method of reduction, from axx\ +

Corollary. Since the portion of the tangent AB intercepted
between the perpendiculars AC, BC, is a constant quantity,
this hypocycloid may in that sense be called an equitangential
curve ; and the rectangular comer of a passage must be rounded
off into the form of this curve in order to admit a beam of a
given length to be carried round it.

Proposition XII. — Problem,

To investigate those cases in which the general propositions
either fail or require peculiar modifications.

Ceue 1. Fig. 125. If the generating circle be considered as
infinitely sma11> or the basis as infinitely large, so as to become
a straight line, the epicycloid will become a common cycloid,
and the ratio of CP to CK in Prop. III. cor. 2, becoming that
of equality, the length of the arc SM will be four times the
versed sine of half PM, and VM twice the chord BM or VX :
therefore the square of the arc VM is always as the absciss
VZ. The evolute is an equal cycloid, and the circles in Prop.
VI. being as 1 to 4, the area of the cycloid is to that of its
generating circle as 3 to 1. The properties of the cycloid as
an isochronous and as a brachistochronous curve belong to
mechanics, and it is demonstrated by writers on optics that its
caustic is composed of two cycloids.

Case 2. Fig. 126. If the generating circle be supposed to
become infinite while the base remains finite, the epicycloid will
become tiie involute of a circle ; and the fluxion of the curve
being always, by Prop. III. cor. 1, to that of PM as PM to CP,
its length SM will be a third proportional to IP and PM. Call

CP, a, and PM, x, then the fluxion of SM is -^5 but the

VOL. I. I

114 AN ESSAY ON CYCLOIDAL CURVES. No. IV.

rectangle contained by half PM and the fluxion of SM Is the
fluxion of the area PSM, or PSM ==/'^ = ^. The epitro-
choid described by the point C of the generating plane will be
the spiral of Archimedes, since CN is always equal to PM =
PS = QV; and since the angular motion of CN and PM are

also equal, the area CON = PSM = ^ . Instead of the ellipsis
of Prop. III., let PX be a parabola of which IP is the parameter,
and contmuing NM to X, the arc PX will be equal to CON.
For making LQ= CP, it is well known that the fluxion of PX
yaiies as XQ, or as PN, which represents the fluxion of CON.
For the curvature, PY in Prop. IV. becomes = CP, and the
radius is a third proportional to NZ and NP.

Case 3. Supposing now the generating circle to become
again fimte, but to have its concavity turned towards the basis,
the same ciu've will be described as would be described by the
rotation of a third circle on the same basis in a contrary direc-
tion, equal in diameter to the difference of those of the two first
circles

Case 4. If the circles be of the same size, with their con-
cavities turned the same way, no curve can be described ; but
if the generating circle be still further lessened, a hypocycloid
will be produced, of the same figure as tiiat which would be
described by a third circle equal in diameter to the difference
of the two first. All the general propositions are equally
applicable to hypocycloids with other epicycloids, as might
easily have been understood from an inspection of the figures,
if there had been room for a double series.

Case 5. Fig. 127. If the diameter of the generating drcle
be half that of the basis, the hypocycloid will become a right
line, and the hypotrochoid an ellipsis. For since the angle
PKM = 2PCS, PCM, being half PKM, coincides with PCS,
and M is always in CS. Let 6NL be the describing circle of
the hypotrochoid, and join GNO, then NL is parallel, and ON
perpendicular, to SC, and ON = HL, which is always to GO as
CL to CG ; therefore AN is an ellipsis : and the centre C will
evidently describe a circle.

N?1V.

CYCLOIDAL CURVES.

Fi0f Wk .117.

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BgW.

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Sinndui^ dCo .luAff.Undan

Tc fa4>epa4fe 114: Toll.

No. y. Air E8SAT ON MUSIC. 115

No. V.

AN ESSAY ON MUSIC.

From the British Magazine for October, 1800.

I. Of Music ik general.

The agreeable effect of melodious sounds, not only on the human
ear, bat on the feelings and on the passions, is so unirersal and
80 powerful, as deservedly to excite the attention of the psycho-
logical philosopher. For what ultimate end a susceptibility
tor this peculiar pleasure has been implanted by nature in the
mind, is not easy to be ascertained ; but setting aside the well
known pleasing sensation of a delicate titillation wherever the
nerves are possessed of great sensibility, and the associations of
an interesting voice, giving expression to poetical and impas-
aonied diction, it is probable that the taste for all complicated
and scientific music is wholly acquired.

Music may be considered as consisting of three component
parts, liiydim, melody, and harmony. Rhythm is an agreeable
succession of sounds considered with respect to the time of their
whole duration. Melody is an agreeable succession in respect
to the pitch, or the frequency of vibrations of each sound.
Harmony is an agreeable combination of several sounds at the
same dme. It is evident that rhythm and melody are almost
inseparaUe ; but that harmony is by no means necessary to the
existence of music. In the first place, it is easy to conceive
that a love of rhythm, or of the periodical recurrence of the
same or similar sensations at equal intervals of time, may be
derived firom the habit of a certain equality and recurrence in
the motions of the body, such as walking, or in children who
cannot yet walk, firom the passive motion of gestation; this
predilectioQ for the return of customary sensations appears to

I 2

116 AN ESSAY ON MUSIC. No. V.

be an innate and ixindamental tendency of the human system,
to which physiologists and metaphysicians have been obliged
ultimately to refer many properties, both of body and mind.
But be this as it may, the love of rhythm, which is, perhaps,
the lowest ingredient in musical taste, is, if possible, still more
universal than the love of harmony and melody. Poetry, or
rather metrical composition, is distinguished from prose only by
the regularity of its rhythm ; and the knowledge of metre and
prosody, however high it may rank in the critic's e8timation,*ls
a subordinate and comparatively insignifi^cant branch of musical
science. The natural fondness for rhythm is the principal
foundation of the pleasure of dancing, an amusement inti-
mately connected with music, and no less popular. The rhythm
of a musical composition is almost always at least twofold, often
three or fourfold, consisting of subordinate divisions or bars,
and periodical returns of larger members, either phrases or
strains, containing equal numbers of those divisions. All this
is perfectly natural^ but perhaps, not so necessary to music as
Mr. Walter Young, in his excellent essay, printed in the
Edinburgh Transactions, appears to imagine ; for those who
are already experienced musicians are generally observed to
delight in recitative, where the rhythm is almost entirely lost ;
and still more in fugues, where two or three series of rhythms,
almost independent of each other, are carried on at the same
time, one part beginning its subdivisions when another has
made some progress, and a third is still to follow. But the
pleasure derived from such compositions is, as Kimberger has
observed, more intellectual than sensual, arising in a great
measure from the consciousness of being able to comprehend
that which is " caviare to the general." Rhythm is generally
marked in performance by a slight increase of force at the
beginning of each subdivision or bar ; sometimes, and in some
instruments always, the change of sounds, in point of acutenesa
and gravity, or the interruption of the same sound, is a suffi-
cient distinction ; and sometimes, after the rhythm has already
been firmly impressed on the mind, neither change of sound nor
of strength is perpetually repeated ; the imagination alone
being sufficient to conceive the continuation of the rhythm : but

No, V. AN ESSAY (W MUSIC. 117

this constitutes a kind of tempo rvbato^ where the perception of
measure is intentionally weakened or suspended. The Aeolian
harp pleases indeed without rhythm, but the pleasure would
soon be exhausted by repetition.

The next constituent part of music is melody. Melody naay
in some sense be said to please on the same principle as rhythm,
the partiality of the mind to a regular recurrence of intervals :
for though we have it not in our power to count the single
vibrations of musical sounds numerically, yet we are evidently
able to compare with ease such sounds as are related to each
other in the simplest numerical ratios. For instance, if a
treble and a tenor voice sing the same part, there is scarcely an
ear so inaccurate as not to perceive their resemblance, which is
produced by the recurrence of two vibrations of the treble note
at the same interval of time with one of the tenor. The same
love of order may easily be extended to the comparison of fifths
and fourths, where the proportions are as two to three, and as
three to four. This is enough to account in some degree for
the pleasure derived from melody, or the succession of sounds
bearing certain proportions to each other, in respect to gravity
and acuteness ; besides that the same intervals, which are most
melodious in succession^ are found also to form the most pleasing
combination of harmony when cotemporary ; for since the pre-
ceding sound is very frequently continued by reflection from
surrounding objects so as to become cotemporary with the
succeeding, and perhaps always remains fixed in the imagina-
tion, it is obvious that sounds, in order to be perfectly melodious,
must also be harmonious. Add to this the impressioii generally
made in infancy by the more or less melodious ditties of the
nurse's voice, and the connexion of refined and chromatic melo-
dies with the natural expression of the moans of grief, or the
exclamations of joy : and from the union of all these causes it
may easily be conceived from whence the love of melody, as
an acquired faculty, may, without much difficulty, be derived.

The pleasure arising from harmony is not so simple and uni-
versal as that which is produced by a combination of rhythm
and melody. Harmony requires for its execution and per-
ception a greater degree of cultivation both in the performer

118 AN ESSAY ON MUSIC. Ko. V.

and in the bearer than melody alone. Cotemporary sonndft
may, from the dne proportion of the times of their vibrations,
give a similar pleasure to that of melody, when the miiid^ con-
sidering them in succession, finds thettk capable of a ready
comparison. But the characteristic of hannony is the regular,
and at the same time dirersified, motion of the air, which arise*
from the combined yibrations, and which appears to be well
calculated to produce the pleasure that the mind derives from
the perception of symmetry. In this point of view, a concord
may be considered as a eingle sound, distinguished from a dis*
cord by a superior quality of tone ; in the same manner as the
tone of the harmonica is more agreeable than that of a
newsman's horn, as the note of a nightingale is sweeter than
that of a frog, and a smooth rich voice more pleasing than a
hoarse and nasal croaking. Thus the harshness and dis-
agreeable quality of a single sound may also, on a more nioei
examination, be sometimes found to consist in a want of har^
monious proportion in those secondary notes, which generally
enter into its composition. This quality of sound, sometimes
called its tone, register, colour, or timbre, might be considered
as a fourth component part of music ; it depends on the law by
which the parts of the sounding body, and the particles of the
air, are governed with respect to the velocity of their progress
and regress in each vibration, or in different successive vibra-
tions. No doubt, much of the pleasure derived from music
depends on it ; but as it is capable of little difersity on the same
instrument, it is seldom considered in treating of the theory of
music. The various combinations of the stops of the organ and
harpsichord, the use of the harmonics of the harp and violin,
the bowing nearer to or farther from the bridge, the application
of muffles of various kinds, the change of the aperture of the
lips in wind instruments, the choice of vowels and consonants
more or less adapted to the powers of the voice ; and in full
pieces, the judicious introduction of different voices and instru-
ments, as subservient to the general effect ; all this relates to
the quality of sound, and whoever adequately relishes the works
of the great modem masters, will be folly competent to judge
of its practical importance.

No. V. AN ESSAY ON MUSIC. 119

Be the causes what they may, natural or habitual, simple or
complicated, it is certain that a very great majority of mankind
experiences pleasure from music : this pleasure is a social
pleasure, and, connected as it is with sentiment and passion, it
is a rational pleasure. The pursuit of musical excellence, if
properly conducted, amply exercises the faculties, at the same
time that it forms a desirable variety, when intermixed with
literary or professional employments. To call it an amusement
only, betrays an ignorance of the nature and difficulty of the
study ; so far is the science of music from being of a light and
superficial nature, that, in its whole extent, it is scarcely less
intricate or more easily acquired than the most profound of the
more regular occupations of the schools; and even practical
perfection in music requires so much intense and laborious
application, such a minute accuracy of perception, and so
rapid an association of various sensitive ideas, with other ideas
and mechanical motions, that it is inconceivable how men, who
have no appearance of superior brilliancy in any other accom-
plishment, should be able to attain a conception and execution
in music, which seem almost to require the faculties of a
superior order of beings. An intemperate and dissipated
attachment to music may indeed often be productive of evils ;
but probably the same individuals, who have I)een its victims,
would have been equally idle and irregular if they had been
destitute of this accomplishment. A considerable share of the
pleasure of practical music arises from causes perfectly distinct
from the sensual perceptions : the consciousness of having over-
come difficulties, the laudable satisfaction of entertaining others,
and the interest and emulation produced by a concurrence of
others in the same pursuits; all these entirely outweigh the
temporary amusement of the ear, and wholly remove the ob-
jection, which might be made, to the enervating effect of a
continued devotion to pleasurable sensations. The ancient
philosophers, with all the manliness and dignity of character to
which they aspired, were not ashamed to consider music as an
indispensable part of a liberal education ; and Plato devotes
three of the earlier years of his young citizens entirely to tiie
study of the lyre : nor are we without examples in modem

120 AN ESSAY ON MUSIC, No. V,

times, of philosophers, and princes, and heroes, who have ex-
celled as much in musical performances, as in literature and in
arms,

11. Of the Origin of the Scale.

The first lyre, with three strings, is said to have been in-
vented in Egypt by Hermes, under Osiris, between the years
1800 and 1500 before Christ. The second and third string
were, perhaps, the octave and fifth of the first, or more probably
its fifth and fourth ; as it would be easy to sing the octave with
the accompaniment of the primitive note only. The melody
might be either always in unison with one of the strings, re-
sembling a very simple modem bass part ; or the intervals
might be occasionally filled up by the voice, without accom-
paniment. We have, in modem music, a specimen of a
pleasing air, by Rousseau, formed on three notes alone, the
key-note with its second and third; but there can be little
doubt that the earliest melodies must have had a greater com-
pass than this ; although some suppose the three strings of the
oldest lyre to have been successive notes of the scale. The
trumpet is said to have been invented about the same time : a
little experience might have taught the Egyptians to produce
from it the octaves, the 12th, 17th, 23rd, and other harmonica
of the primitive sound, which are related to it in the ratio of
the integers from 1 to 9, and the same sounds might have
been observed by a delicate ear among the secondary notes of
a long chord ; and then, by descending three octaves from the
23rd, and two from the 17th, they might have added to their
lyre the second and major third of the principal note. But it
does not appear that this method ever occurred to the ancients :
they seem rather to have attended to the intervals of the notes
within the octave, than to the union of similar notes in the
natural harmonics : and, besides, the series of natural har-
monics would never have furnished a true fourth or sixth. It is
uncertain when, or by whom, the fourth string was added :
but the merit of increasing the number to seven is attributed to
Terpander, about the year 700 before Christ, two centuries
after Homer : although some persons have asserted that he

No. V. AN ESSAY ON MUSIC. 121

only brought the improvement from Egypt, and that Hermes
was also the inventor of the lyre with seven strings. Pytha-
goras, or Simonldes, about the year 500 added an eighth, and
Timotheus a ninth string : the number was afterwards extended
to two octaves ; and Epigonus is said to have used a lyre of
forty strings, or rather a harp, as he played without a plectrum :
but the theory of the ancient music soon became more intricate
than interesting. The lyre of eight strings comprehended an
octave, corresponding pretty accurately with the notes of our
natural scale, beginning with e : the key-note was a, so that the
melody appears to have borne usually a minor third, which has
also been observed to be the case in the airs of most unculti-
vated nations ; but there was a considerable diversity in the
manner of tuning the lyre, according to the great variety
of modes and genera that were introduced. These modes
were of a nature totally different from the modem modulations
into various keys, but they must have afforded a more copious
fund of striking, if not of pleasing melodies than we have at
present. In some of the genera, intervals of about a quarter
tone were employed ; but this practice, on account of its
difficulty, was soon abandoned ; a difficulty which is not easily
overcome by the most experienced of modem singers ; although
some great masters have been said to introduce a progression
of quarter tones, in pathetic passages, with surprising effect.
The tibia of the ancients, as it appears evidently from Theo-
phrastus, although not from the misinterpretations of his
commentators and of Pliny, had a reed mouth-piece about
three inches long, and therefore was more properly a clarinet
than a flute ; and the same performer generally played on two
at once, and not in unison. Pollux, in the time of Commodus,
describes, under the name of the Tyrrhene pipe, exactly such
an organ as is figured by Hawkins, composed of brass tubes,
and blown by bellows: nor does he mention it as a new
discovery : it appears, from other authors, to have been often
furnished with several registers of pipes ; and it is scarcely
possible that the performer, who is represented by Julian
as having considerable execution, should have been contented
without occasionally adding harmony to his melody. That the

122 AN BaSAT ON MUSIC, No. V.

Yoice was accompanied by thorough bass on the lyre^ is undeni-
ably proved by a passage of Plato : and that the ancients had
some knowledge of singing in three parts, is evident from
Macrobius. Martini, who is one of the strongest opponents
of that opinion, which attributes to the ancients a knowledge
of counterpoint, observes that ^^ they allowed no concords but
the octave, fourth, and fifth, or at most very rarely the third ;
yet they were not without a knowledge of concord of har-
monious parts. It is known with certainty, that two parts,
whether vocal or instrumental, or mixed, besides unison,
performed at the same time the same melody, either always
in octaves, or probably always in fifths, or always in fourths ;
which was called a symphony : perhaps also, they changed in
the course of the performance from one interval to another,
and this might be done by more than two parts at the same
time/' It is not improbable that this statement may be accu-
rate : nor is it necessary to suppose a velry exquisite and refined
skill in the intlicades of composition, to produce all the efiects
that have with any probability been attributed to music. It is
well known that Rousseau and others have maintained, that
harmony is rather detrimental than advantageous to an in-
teresting melody, in which true music consists; and it may
easily be observed, that im absolute solo, whether a passage or
a cadence, is universally received, even by cultivated hearers,
with more attention and applause, than the richest modulations
of a powerfril harmony.

The nunor scale being the most commonly used by the
ancients, it was natural for Pope Gregory, who in the year
600 is said to have marked the notes by the Roman lettere, to
begin with A, the key-note of that scale : although if, as there
is some reason to suppose, the B was originally flat, A was not
the key-note, but its fiflth, until the B natural was introduced,
and denoted by a square b instead of a round one. By degrees
the chromatic scale was filled up, and the five added intervals
were denoted by the letter belonging to the note above them,
with the addition of the round b, or by the note below, with
the addition of four lines crossing each other, implying a half
note, as composed of four commas. A simple cross would,

No. v. AN ESSAT OR MUSIC. l23

however, at present, be much more convenient, as more readily
distinguishable from the square b, which is used to signify a
natural note, in opposition to these flats and sharps. This is
the historical account of the origin of the scale ; but, according
to the modem theory and practice of music, the subject may be
more easily understood, by beginning with an explanation of
the major scale.

III. Pbactical Apfucation of the Scales.

llie simplest proportions of two sounds to elu^h other, next to
unison, is when the frequency of their vibrations is related as
one to two : such sounds bear a very strong resemblance to
each other, and when named, they are denoted by the same
letter, and are only distinguished by the appellations in alt, in
altissimo, on the one side, and double, and double double, on
the other. The Germans, with great propriety, make use of
small letters or capitals, with one, two, or more lines over or
under them. The note marked by the tenor cleff is called

c, the octaves above c, c, as far as six lines, which is, perhaps^

the highest note used in music : the octaves below c, are c, C,
C, C : C is probably not audible, vibrating but eight times in

a second. C with six lines below it, would denote a sound, of
which the complete vibrations should last precisely a second*
1 he series of natural notes is this. A, B, C, D, E, F, G, A, B|

c, d. .b,*c, J . .The subjoined table will show the absolute fre-
quency and the dimensions of each vibration of the octaves of
c, and the length of the simplest organ pipe that produces it :
but, according to the different temperature of the air, and the
pitch of the instruments, these numbers may vary somewhat
from perfect accuracy : and it must be observed, that the usual
pitch of concerts, in London, is somewhat higher than this
standard ; and in Germany, perhaps a little lower.

124

AN ESSAY ON MUSIC.

No.Vi

Sound moYes in a second 1130 feet

Note.

Yibrations in a aeoond.

Length of open pipe
in feet

565.00

282.50

141.25

70.62

^2 audible

35.31

17.66

"c"

8.83

c
c

128
256

512

4.41
2.21
1.10

c»

1024

.65

2048

•28

4096

.14

c« ,

8192

.07

Any soimd may be assumed at pleasure for the primitive or
standard note of a piece of music, and is then denominated the
key note ; and the idea of this note is perpetually impressed on
the mind in all simple compositions, both from its frequent
recurrence, and from the relation that all the other sounds bear
to it C being the key note of the scale called natural, we
shall consider it as the foundation of the scale. The next in
importance is the fifth, G, wliich, for various reasons, is inti-
mately connected with the key note. The first reason is, that it
constitutes the most perfect melody and harmony with C, since
every alternate vibration of C coincides with every third of G ;
the second is^ that an attentive ear may almost always distin-
guish the fifth, at least its octave, the 12th, whenever any
instrument sounds C ; it being one of those secondary sounds
which are called natural harmonics, and which may generally
be observed, in the proportion of the natural numbers, as far as

No.V.

AN ESSAY ON MUSIC.

125

twenty or more, but which have not hitherto been completely
explained : thirdly, a stopped pipe, if blown forcibly, springs
immediately from C to g, and an open pipe first to c, and then
to g. The interval, between C and G, is most naturally divided
by the note E, which answers to the number 5, when C and G
are represented by 4 and 6, and which is found among the
natural harmonics both of chords and pipes. These three notes
constitute the harmonic triad, or common accord, in the major
scale, which is the most perfect, or rather the only perfect har-
mony. But the intervals are still much too large for melody,
and require a further subdivision ; we now therefore take the
fifth below instead of above the key, or its octave, the fourth
above, F, which is to C as 4 to 3 : this sound is nowhere found
among the natural harmonics of C, but C is the most distin-
guishable of its harmonics, and therefore the relation is nearly
the same. The scale is completed by filling up the perfect
triads of G and F : the fifth of G furnishing D, the second of
the key, which is also the ninth natural harmonic of C ; the
third of G, the seventh, B, which is the fifteenth harmonic of C ;
and the third of F being the sixth of the key, A, which is
neither among the harmonics of C, nor has c among its harmo-
nicsL Hence we have a second table, in which the proportions
of the length of a chord, or pipe, producing the various sounds,
are detailed, and the place among the principal natural harmo-
nics of the key annexed.

Notes,

Proportions.

Natural Harmony.

Key C
2d D

1
9

dd £

5, 10

4th F

0,(0

6th 6

3,6, 12

6th A
7th B
8th c

2, 4, 8, 16

126 AK JSSSAY ON MUSIC. No. V*

Now, when twp or more perfectly harmonious parts are per-*
fonuQjd togeth^^^ they must i^ecessarily he found all in the
saine triad, C, E, G ; G, B, D ; or F, A, C ; and the succes-
;sioa of these triads, in various forms, is sufficient for the acoom-
paniment of any simple melody. A regular melody always
tenoinates by an ascent or descent of one degree to ^e key
note ; the last note but one must Iheprefore be always B or D :
aiiid both of these being in the triad of G, G is called the
governing note, or the dominant of C; and F, being in the
fijBime manner governed by C, is called its subdominant And
it is usual, in all regular compositions of any length, to depart
for a short dme from the principal harmony of the key note,
9nd to modulate into the key of the dominant, then to return,
and to modulate for a still shorter time into the subdominant,
before the final close in the tonic or key note. It is necessary,
tjaerefpre, for greater variety, to complete the scale of the
dominant, as well as that of every other note which may be
oQcasionally introduced as a principal key note ; but to do this
with mathematical accuracy, in the same proportions as have
]been explained, would be practically impossible, and even
theoretically inconvenient : hence arises the necessity of temper*-
ing some intervals, to make the ptliers more tolerable, without
too much increasing the number of sounds. It has been found
suiScient in [Nractice, to add five notes to the seven which have
been enumerated ; but the best proportions of these have not yet
been absolutely determined : some have made all the twelve
intervals equal : others have left the whole scale of C perfect :
others again have taken a middle path, and have introduced a
slight imperfection into thb key, in order to make the neigh-
bouring ones the less disagreeable. The least circuitous intro-
duction of these notes is shown in the third table, together with
the proportion that they bear to C when thus considered. They
are denominated nearly in the German manner, the addition of
the syllable " is" signifying what the English call sharp, and the
French diSsej and that of ^^ es," flat or Mmol

No.V.

AN ES8AT ON MUSIC.

127

Notes.

BelatioDS.

ProportioDS.

as 7th of G

Bee

as 4th of F

as 7th of D

Ees

as 4th of Bes

Gi8

as 3rd of £

Bat a still greater variety being required than these major
scales afford, it has been found, that the interval of a fifth may
be agreeably, though somewhat less harmoniously, divided, by
placing the minor third below, instead of above« the major ; so
that C maybe to £ as £ was to G, and consequently £ to G as
C was to £. llie £, thus depressed to t, differs but by a
comma, or in the ratio of 80 : 81, from the £es found above, as
the fourth of Bes ; therefore the same string serves for both
notes; and the scale becomes C, D^ £e8, F, G, A, B, C;
which is the ascending minor scale, the A and B being rejtained
as leading best towards the key note, and the major triad of the
dominant being therefore necessary to the cadoice. But in
descending, the triad of tlie aubdominant F may conform to the
character of the minor mode, and Aes is substituted for A ; and
most frequently Bes for B, as dividing the interval from C to
Aes more equally and more melodiously.

Thus we have a pretty coipprehensive view of the most usual
practical relations of all the notes to each other. Their use as
discords is somewhat more complicated, and would lead furtbar
into the science of music than is conustent with the nature of so
summary a view. But it may be remarked in general, that by
far the most common discord is the note which constitutes the
distinction of the scale of the key from that of its dominant ;
for instance, F with the triad of G, which is called the accord
of the flat seventh of G ; and F, not being in the scale of G, is
considered as a regular preparative to the final accord of C ; in
which that part or instrument by which the F is introduced,
must necessarily descend to £, the third of the key. The
second kind of discords are suspended discordsy wb^ one or

128 AN ESSAY ON MUSIC. No. V.

more notes of any preceding accord are continued after the
commencement of a different harmony in other parts of the
composition. The third, which is rare, and less universally
adopted, consists in an anticipation of a subordinate note of an
accord which is to follow, as in the case of the added sixth of
the French school. The fourth kind are passing discords, where
a note, forming only a melodious step between two others, is
inserted without any regard to its harmonious relations.

IV. Of the Terms expressive of Time.

The notation of music, as it has been established for more
than two centuries, is in general admirably adapted for its pur-
pose : but there is one great deficiency, which might very easily
be remedied, and that is^ the total omission of any character
expressive of the absolute duration of each note, however accu-
rately the relative value of the notes may be prescribed. - It is
true, some little allowance must be made for the execution of
the performer, and for the habits of the audience ; but this is
no reason why time might not be much more accurately noted,
than by the vague terms which are usually adopted. It
would be easy to prefix to each movement a number, signifying
how many bars are to be performed in a minute, which might
at first be ascertained by the help of a stop watch, and would
soon become perfectly familiar both to composers and performers,
even without this assistance. According to Quanz, the number
which should be substituted for Allegro assai, in common time,
is about 40; for Allegretto^ 20; for Larghetto^ 10; and for
Adagio assaiy 5. But it is usual to perform modem music much
more rapidly than this ; or at least the style of composition is so
changed, that the terms are very differently applied. An
allegro, or even an allegretto, in common time, without semi-
quavers, is often performed as fast as 60; seldom slower
than 30.

A very superficial attempt to affix a determinate meaning to
the words denoting musical time, may be seen in the table sub-
joined ; which, if it were more completely and accurately filled
up, might be of considerable use to young musicians ; although
it will appear, from inspection of this table, that composers have

No.V.

AN ESSAY ON MUSIC.

129

VOL. L

130

AN ESSAY ON MUSia

No. V.

hitherto employed those terms in very indefinite ^gnifications.
But it must he confessed, that much latitude must necessarily
be left for the ear and taste of a judicious performer^ and that
it is impossible for human art to describe on paper every deli-
cacy of finished execution.

If wc choose to compare the time, occupied either by a bar,
or by any of its parts, with the vibrations of a pendulum, we
may easily do it by means of the following table, which shows
the number of vibrations in a minute, corresponding to pendu-
lums of difierent lengths, expressed in inches.

Length.

Vibrations.

1
Length.

Vibrations.

liength.

Vibrations.

187

118

167

107

153

142

132

125

No. VL REPLY TO DR. ROBISON. 131

No. VI.

A LETTER TO MR. NICHOLSON, PROFESSOR OF NATURAL PHILO-
SOPHY IN THE ROYAL INSTITUTION, RESPECTING

SOUND AND LIGHT,

AND IN REPLY TO SOME OBSERVATIONS OF PROFESSOR ROBISON.
From Nicholson's Journal for Angtist, 1801.

Sir,
In the Supplement to the Encyclopaedia Britanmca are inserted
several excellent articles by Professor Robison of Edinburgh ;
one of them appears to require some public notice on my part,
and I consider your yaluable Journal as the most eligible
channel for such a communication, especially as you have lately
done me the honour of reprinting the paper which gave rise to
the Professor's animadversions. But in the first place I shall
beg leave to recall the attention of your readers, by a summary
enumeration, to the principal positions which I have in that
paper endeavoured to establish.

1. Sound, as transmitted through the atmosphere, consists in
an undulatory motion of the particles of the air, Sect. III.*
This is generally admitted ; but as the contrary has even very
lately been asserted, it is not superfluous to have decisive
evidence of the &ct. Professor Robison's experiment with a
stopcock furnishes an argument nearly similar.

2. A current of air, forced by a moderate pressure through
a cylindrical pipe, diverges the less as its velocity is less,
Sectll.f

3. At a certain point the divergency of such a current
increases suddenly, and the current mixes with the surrounding
air. Sect II.t

• Supra, p. 71. t Supra, p. 69.

K 2

132 ON SOUND AND LIGHT, No. VI.

4. So far is such a motion 'from spreading equally in all
directions, that on every side of the current the air is urged
more towards it than from it, Sect II.

5. Sound, admitted through an aperture, does not by any
means diyerge equally in all directions, and is probably very
weak except in directions nearly rectilinear. From position
2 and 4, and from experience Sect. V I.

6. Sound probably decays in the duplicate ratio of the dis*
tance. Sect. Vll.f

7. A similar blast of lur produces nearly a similar sound, in
organ pipes properly commensurate. Sect. VIII. J

8. Light is probably the undulation of an elastic medium>
Sect X.§

A. Because its velocity in the same medium is always
equal.

B. Because all refractions are attended with a partial reflec-
tion.

C. Because there is no reason to expect that such a vibra-
tion should diverge equally in all directions, and because it
is probable that it does diverge in a small degree in every
direction.

D. Because the dispersion of differently coloured rays is no
more incompatible with this system than with the common
opinion, which only assigns for it the nominal cause of different
elective attractions.

E. Because refraction and reflection in general are equally
explicable on both suppositions.

F. Because inflection is as well, and it may be added, even
much better explained by this theory.

G. Because all tiie phenomena of the colours of thin plates,
which are in reality totally unintelligible on the common hypo-
thesis, admit a very complete and simple explanation by this
supposition. The analogy which is here superficially indicated,
will probably soon be made public more in detail ; and will
also be extended to the colours of tiiick plates, and to the
fringes produced by inflection, affording, fh)m Newton's own

• Supra, p. 73. t Snpra, p. 75. t Supra, p. 76.

§ Supra, p. 78.

No. VI. IN REPLY TO DR. ROBISON. 133

elaborate experiments, a most convincing argument in fiivonr
of this system.*

9. The particles of air may be jointly actuated by two or
more sounds ; and in this case, the seyeral motions are to be
added or subtracted, in order to find the actual joint motion,
Sect. XI. t

10. The grave harmonic produced by a major third is accom-
panied by a very audible twelfth. This circumstance is ex-
plained> and the effect of subordinate notes and subaltern stops
on the quality of sounds is shown by figures, Sect XI.

11. A noise returning every second, if audible, would be a
C. From Sauveur ; with an experiment, Sect. XII. j;

12. A chord retains always the form of its initial vibration.
From experiments, in favour of Euler's theorem, against the
simple harmonic curve, Sect. XIII.§

13. The vibration of a chord is scarcely ever performed in
the same plane. Its revolutions and its subordinate vibrations
may be rendered distinctly visible under the microscope. Sect.
XIII.

14. If a chord be inflected at any point of aliquot division,
the harmonic secondary note, corresponding to that division, will
not be audible ; an experiment contradictory to some theories
of the origin and of the inseparable nature of harmonic sounds,
Sect. XIII.

15. The human voice is analogous to the organ pipe denomi-
nated from it, which consists of a tongue piece without any
commensurate tube : and the falsetto is probably formed by the
upper orifice of the trachea, assuming the functions of the glot-
tis, Sect. XV. II

16. A temperament of progressive imperfection is the most
convenient for practical music, and is easily approximated by
tuning six perfect, and six equally imperfect fifths, Sect. XVI.If

From the detached nature of the subjects which I have here

* It appears from the enunciation of the last of these propositions that Dr.
Young was abeadj in possession of the prineiple of interference, and of some of its
most important applications ; thej were announced in his celebrated paper '* On the
Theoiy of Light and Colours/' (Infra, No. VII., p. 140,) which was read before the
Royal Society on the 12th of November following.— ^oto by the Editor.
. T Supra, p. 83. t Supra, p. 85. ( Supra, p. 86. || Supra, p. 91.

H Supra, p. 93.

134 ON SOUND AND LIGHT, No, VL

eaumeratedy and the imperfect state of those branches of the
mathematics to which they refer, it would have been in vain to
attempt a very perspicuous and detailed discussion of them.
Mj researches on these subjects have been much interrupted,
and probably will not be very shortly resumed ; but if they be
of no further use to any person, I shall not tliink my labour
lost ; for I flatter myself that the inferences which, they have
led me to draw respecting the theory of colours will throw new
light on all the most interesting parts of optics, while, by a
comparison with the obvious inferences from Dr. HerscheFs
important discoveries, they will also lead to some material
illustrations of the phenomena of heat.

I shall now trouble you with some remarks in reply to Pro-
fessor Robison : the passage to which I allude is this : —

'^ We are surprised to see this work of Dr. Smith greatly
undervalued by a most ingenious gentleman in the Philosophi-
cal Transactions for 1800,* and called a large and obscure
volume, which leaves the matter just as it was, and its results
useless and impracticable. We are sorry to see this, because
we have great expectations from the future labours of this
gentleman in the field of harmonics ; and his late work is rich
in refined and valuable matter. We presume humbly to
recommend to liim attention to his own admonitions to a very
young and ingenious gentleman,t who, he thinks, proceeded too
far in animadverting on the writings of Newton, Barrow, and
other eminent mathematicians." Encyclop. Brit., Suppl., Art.
Temperament, p. 652. (Works, Vol. iv. p. 429.) {

* Supra, p. 93. t Supra, p. 102.

Z This admirable Essay, containing a most luminous exposition of the Theory of
Music, is given in the fourth volume of Sir David Brewster's edition of Dr. Robison's
Works, p. 376. Similar remonstrances against Dr. Young's estimate of the cha-
racter of Dr. Smith's Harmonics were addressed to hun by Mr. John Gough (Man-
chester Memoirs, Vol. V ., '* On the Theory of Compound Sounds "), to which he
replied in Nicholson's Journal for Aug. 1802. The passage in Dr. Smith's work

which gave rise to his observations was the following : ** Different particles of the
Air at Uie ear will keep moving oonstantly opposite ways at the same time. And in
so rare a fluid as air is, where the intervals of the particles are eight or nine times
their diameters, there seems to be room enough for such opposite motions without
impediment : especially as we see the like motions are refUiy performed in water,
which in an equal space contains eight or nine hundred times as many such particles
as air does. For when it rains upon stagnating water, the circular waves propagated
from different centres appear to intersect and pass through or over each other, even
in opposite directions, without any visible alteration in their circular flgure, and
therefore without any sensible alteration of their motions." — Harmonics, 1759,
p. 105,'^Note by the EdUor,

. No. VI. IN REPLY TO DR. ROBISON. 135

According, therefore, to the author of this article, I have, m
the first place, taken the liberty of giving severe advice to a
yoimg mathematician who had never asked it ; secondly, this
advice is equally applicable to my own presumption; and
thirdly, Dr. Smith's Treatise on Harmonics is a work entitled
to the highest praise.

I did, in fact, endeavour to show that the gentleman in ques-
tion had overlooked the labours of some former authors relative
to his subject, but I accompanied my remarks with nothing
like admonition. I have read Dr. Smith's work with attention,
and I imagine, from the polite manner in which Professor Ro-
bison is pleased to speak of my essay, he will not hesitate to
allow that I have understood it. I took it up with great ex-
pectations : those expectations having been completely disap-
pointed, I thought it right to state my cool and unprejudiced
opinion of its merits, in order to prevent a similar disappoint-
ment in others. It is impossible^ therefore, that an ^* attention''
to any ^' admonitions" of a general nature, wherever they may
be found, can influence such an opinion ; and so far only as I
am supposed to be an incompetent judge on the subject of har-
monics can it be asserted that it was either blameable or super-
fluous for me to express that opinion. As a mathematician, and
an optician, I value Dr. Smith highly ; but I must still beg
leave to affirm that his whole book of harmonics contains far,
far less information than either of the articles Temperament
and Trumpet, in the Supplement to the Encyclopaedia.

I do not mean it to be understood that this work is so con-
temptible as not to contain the least particle of important
matter ; but it appears to me that its errors counterbalance its
merits. The only improvement on which Professor Robison
himself seems to set a high value, b the application of the phe-
nomena of beats to tuning an instrument ; on the other hand^
I conceive that the misstatement, relative to the non-interference
of difierent sounds, is an inaccuracy which far outweighs the
merit of Dr. Smith's share of that improvement I have asserted
that Dr. Smith has written a large and obscure volume, which,
for every '^ purpose, but for the use of an impracticable instru-
ment, leaves the whole subject of temperament precisely where

136 ON SOUND AND LIGHT, No. VI.

it found it ;" and that ^* the system proposed for his changeable
harpsichord is neither in that, nor in any other form, capable
of practical application." Professor Robison, on the contrary,
says^ «< We do not see how it can be disputed that Dr. Smith's
theory of the beating of imperfect consonances is one of the most
important discoveries, both for the practice and the science of
music, that have been offered to the public. We are inclined
to consider it as the most important that has been made since
the days of Galileo. We are obliged to call it his discovery.
Mersennus, indeed, had taken particular notice of this undu-
lation of imperfect consonances, and had offered conjectures as
to their causes ; conjectures not unworthy of his great ingenuity.
Mr. Sauveur also takes a still more particular notice of this
phenomenon, and makes a most ingenious use of it for the
solution of a very important musical problem." P. 652 and
651. (Works, p. 429.)

Why then are we obliged to call it Dr. Smith's discovery,
or indeed any discovery at all ? Sauveur had already given
directions for tuning an organ pipe, by means of the rapidity of
its beating with others. Mem. de I'Ac. 1701, 475, ed. Amst
Dr. Smith ingeniously enough extended the method ; but it
appears to me that the extension was perfectly obvious, and
wholly undeserving of the name either of a discovery or of a
theory. If Professor Robison thinks otherwise, there is nothing
further to be said ; but, in all probability, Dr. Smith considered
this improvement as constituting a very small part of the merit
of his treatise. No doubt an organ may be more accurately
tuned by counting the beats than by any other method, although
it may be questioned whether the advantage of counting the
absolute frequency of the beats will ever practically compensate
the tediousness of the process.

It remains to be considered whether Dr. Smith's changeable
harpsichord is, or is not, an impracticable instrument; for,
whatever Signer Doria might exclaim. Dr. Smith himself does
not recommend his scale for common use. It is the opinion
of many unprejudiced practical persons that all occasional
introduction of different semitones is perfectly impracti-
cable ; and some who have heard the effect of Dr. Smith's

No. VI. IN REPLY TO DR, ROBISON. 137

instrument have declared that to them it was by no means
agreeable. And, indeed, if we pay sufficient attention to the
passages and modulations of the greatest composers, we shall be
convinced that, grantmg all possible dexterity in the performer,
it would be absolutely impracticable to adapt them to an
instrument so different from that for which they were composed
as Dr. Smith *8 is from the common harpsichord. It may easily
be conceived that an organ, very correctly tuned, as Mr. Watt's
probably was, for a particular key, might appear ^^ sopra modo
bellissimo" in that key; but the sequel of the story shows
literally what Dr. Smith has allowed, that his temperament is
inapplicable to our instruments, since it was utterly impossible
to sing with it in the key of Ees or E fiat, a key of exceedingly
frequent occurrence. I have been informed, on the best authority,
that Dr. Smith restricted the organist of Trinity College to such
keys and modulations as were best suited to the system by wliich
the organ was tuned ; and that organ, as well as the instruments
which were made for Dr. Smith, has long been tuned according
to the more common method.

I spoke of Dr. Smith's system with flattened major thirds as
of no value, not with regard to its intrinsic merits, but because
it was not intended for any instrument in common use ; since,
in these instruments, the difficulty is not so. much how to divide
the imperfection among the thirds and fifths of the same scale
as to proportion properly the imperfections of the thirds of
different keys. Yet I do not mean it to be understooa that I
can agree to the solidity of those foundations on which Dr.
Smith has built his system for a single scale ; although to
Stanley and to Doria it might be pleasing, because its imper-
fections are far too small to offend the ear. Professor Robison
justly observes that different persons differ exceedingly in their
estimation of the effect of the same temperament on different
concords, and that much of this arises from their different dis-
positions ; it appears, therefore, that Dr. Smith was too preci-
pitate in laying down his principle for the comparison of the
effects of temperament.

With respect to the system which I have proposed, Professor
Robison thinks that the temperaments of several of the thirds

138 ON SOUND AND LIGHT, No. VL

which occur frequently are much too great. If we wish to form
a judgment of any system of temperament, it must be by com-
parison with some other. It does not appear with what system
Professor Robison would wish the comparison to be made, but he
rather seems to incline to the equal temperament^ although he
^ves directions for tuning by another. At any rate, no tem-
perament of an interval can be said to be much too great, unless
it be greater than that of the same interval in the system of
equal temperament ; for if any interval be made more perfect
than this, some other similar interval must be as much less per-
fect. In my system the only thirds, perceptibly greater than
those of the equal temperament, are the major thirds on £,
Aes, B, C sharp, or Cis, and Fis, and the minor on C, Cis,
F, Gis, Bes, and Ees. Of these none can be said to
occur frequently except the major third on E, and the
minor on C. The sixths require no separate consideration.
Now, since the minor chord is intended to be less completely
harmonious than the major, its character will be by no means
materially impaired by this imperfection, which it would be
somewhat difficult to remove^ The third on E is not sharp
enough to be very offensive ; but, in compliance with the usual
practice of making this third somewhat more perfect than the
intervals of Aes and C, I have, in the method recommended for
common use, made it equal to the third of the equal tempera-
ment The directions given for tuning in § 68 and in § 80 of
the article are liable to far greater objections. For instance,
the temperament of the Illds on Aes and Fis, in the latter, is
about .00880, or more than a comma and a half, which Professor
Robison. will readily allow to be ^' much too great " for any
thirds ; since he has asserted, with Dr. Smith and others, that
the error of a comma would be intolerable. Mr. Maxwell has,
however, very decidedly proved, in his Essay on Tune, that the
greatest harmonists, Corelli, Tartini, and Giardini,. have ad-
mitted very frequently the error of a comma in their most refined
compositions. And I have the authority of several celebrated
performers on stringed and wind instruments for asserting that
they take of choice the characteristic semitone, leading into the
key note, considerably sharper than the same note is tuned on

No. VI. m REPLY TO DR. ROBISON. 139

any keyed mstruments, making an imperfection of nearly two
commas, in the relation as third of the dominant, which is the
fundamental note of the chord ; while, in the mean time, our
theorists have been labouring, by the most complicated con-
trivances, to introduce notes into keyed instruments, which shall
have exactly a contrary effect, by making the ascending semi-
tone as wide a step as possible. On asking very lately the
opinion of a practical musician of great eminence, and one who,
in every respect, does honour to his profession, he decidedly
agreed in the superiority of such a diminished semitone, and
observed that the key of £ derived a very elegant character
from the usual method of tuning Dis as Ees, a minor third to
C : hence the Illds on Ees and G being very Uttle tempered,
the Illd on the dominant B must be about a comma and half
too sharp. The fiict is, that in this case the harmony is some-
what impaired, in order to improve the melody. The semitone
is considered only in its relation to the key note : the interval of
15 to 16 is far too small to be distinctly conceived as commensu-
rate, it possesses, therefore, no melody in virtue of the perfection
of its ratio ; and a certain elegance of expression is added, by
approaching to the natural and colloquial ascent of a voice by
imperceptible degrees. It must, however, be confessed that
some excellent musicians prefer a purer harmony ; and in this,
as in all other matters of taste, considerable latitude must be
allowed for the habits and predilections of individuals.
I am. Sir,
With great respect,
Yoxu* obedient humble servant,

Thomas Young.

No. 48, Welbeck-street,
13 July, 1801.

140 THEORY OF LIGHT AND COLOURS, No. VII.

No. VII.

ON THE THEORY OF

LIGHT AND COLOURS.

From the Philosophical Transactions for 1802, p. 12.

A BAKERIAN LECTURE.

Read Nov. 12, 1801.

Although the invention of plausible hypotheses, independent
of any connexion with experimental observations, can be of
very little use in the promotion of natural knowledge ; yet the
discovery of simple and uniform principles, by which a great
number of apparently heterogeneous phenomena are reduced
to coherent and universal laws, must ever be allowed to be of
considerable importance towards the improvement of the human
intellect.

The object of the present dissertation is not so much to pro-
pose any opinions which are absolutely new, as to refer some
theories, which have been already advanced, to their original
inventors, to support them by additional evidence, and to apply
them to a great number of diversified facts, which have hitherto
been buried in obscurity. Nor is it absolutely necessary in
this instance to produce a single new experiment ; for of experi-
ments there is already an ample store, which are so much the
more unexceptionable, as they must have been conducted with-
out the least partiality for the system by which they will be
explained ; yet some facts, hitherto unobserved, will be brought
forwards, in order to show the perfect agreement of that system
with the multifarious phenomena of nature.

The optical observations of Newton are yet unrivalled ; and,
excepting some casual inaccuracies, they only rise in our esti-
mation as we compare them with later attempts to improve

No. VII. THEORY OF LIGHT AND COLOURS. 141

on them. A further consideration of the colonrs of thin plates,
as they are described in the second book of Newton's Optics,
has converted that prepossession which I before entertained for
the undulatory system of light, into a very strong conviction of
its truth and sufficiency; a conviction which has been since
most strikingly confirmed by an analysis of the colours of striated
substances. The phenomena of thin plates are indeed so sin-
gular, that their general complexion is not without great diffi-
culty reconcileable to any theory, however complicated, that
has hitherto been applied to them ; and some of the principal
circumstances have never been explained by the most gratuitous
assumptions ; but it will appear, that the minutest particulars
of these phenomena are not only perfectly consistent with the
theory wUch will now be detsuled, but that they are all the
necessary consequences of that theory, without any auxiliary
suppoations ; and this by inferences so simple, that they be-
come particular corollaries, which scarcely require a distinct
enumeration.

A more extensive examination of Newton*s various writings
has shown me that he was in reality the first that suggested
such a theory as I shall endeavour to maintain ; that his own
opinions varied less from ^his theory than is now almost uni-
versally supposed ; and that a variety of arguments have been
advanced, as if to confute him, which may be found nearly in a
similar form in his own works ; and this by no less a mathe-
matician than Leonard Euler, whose system of light, as far
as it is worthy of notice, either was, or might have been,
wholly borrowed from Newton, Hooke, Huygens, and Male-
branche.

Those who are attached, as they may be with the greatest
justice, to every doctrine which is stamped with the Newtonian
approbation, will probably be disposed to bestow on these con-
siderations so much the more of Uieir attention, as they appear
to coincide more nearly with Newton's own opinions. For
this reason, after having briefly stated each particular position
of my theory, I shall collect, from Newton's various writings,
such passages as seem to be the most &vourable to its admis-
sion ; and although I shall. quote some papers which may be

142 THEORY OF LIGHT AND COLOURS. No. VII.

thought to have been partly retracted at the publication of the
Optics, yet I shall borrow nothing from them that can be sup-
posed to militate against his maturer judgment.

Htpothbsis I.

A luminiferoui ether pervades the univereCy rare and elastic in a
high degree.

Passages from Newton.

" Tne h]rpothesis certainly has a much greater affinity with
his own," that is, Dr. Hooke's, ^* hypothesis^ than he seems to
be aware of ; the vibrations of the ether being as useful and
necessary in this as in his." (Phil. Trans., Vol. VII. p. 5087.
Abr., Vol. 1. p. 145. Not. 1672.)

^^ To proceed to the hypothesis : first, it is to be supposed
therein, that there is an ethereal medium, much of the same
constitution with air, but far rarer, subtler, and more strongly
elastic. It is not to be supposed that this medium is one uniform
matter, but compounded, partly of the main phlegmatic. body
of ether, partly of other various etiiereal spirits, much afber the
manner that air is compounded of the phlegmatic body of air,
intermixed with various vapours and exhalations: for the electric
and magnetic effluvia, and gravitating prijiciple, seem to argue
such variety." (Birch, Hist, of R. S- Vol. III. p. 249, Dec.
1675.)

*^ Is not the heat (of the warm room) conveyed through the
vacuum by the vibrations of a much subtler medium than air ?
— And is not this medium the same with that medium by which
light is refracted and reflected, and by whose vibrations light
communicates heat to bodies, and is put into fits of easy
reflection, and easy transmission? And do not the vibrations
of this medium in hot bodies contribute to the intenseness and
duration of their heat ? And do not hot bodies communicate
their heat to contiguous cold ones, by the vibrations of this
medium propagated from them into the cold ones? And is not
this medium exceedingly more rare and subtle than the air,
and exceedingly more elastic and active? And doth it not
readily pervade all bodies ? And is it not, by its elastic force.

No. Vll. THEORY OF LIOHT AND COLOURS. 143

expanded through all the heavens? — ^May not planets and
comets, and all gross bodies, perform their motions in this
ethereal medium ? — And may not its resistance be so small as
to be inconsiderable ? For instance, if this ether (for so I will
call it) should be supposed 700,000 times more elastic than
our air, and above 700,000 times more rare, its resistance
would be about 600,000,000 times less than that of water.
And so small a resistance would scarce make any sensible
alteration in the motions of the planets in ten thousand years.
If any one would ask how a medium can be so rare, let him
tell me how an electric body can by friction emit an ex-
halation so rare and subtle, and yet so potent?— And how the
efBuvia of a magnet can pass through a plate of glass without
resistance, and yet turn a magnetic needle beyond the glass?"
(Optics, Qu. 18, 22.)

Hypothesis II.

Undulations are excited in this etiier whenever a body becomes

luminous.

Scholium. I use the word undulation, in preference to vibra-
tion, because vibration is generally understood as implying a
motion which is continued alternately backwards and forwards,
by a combination of the momentum of the body with an ac-
celerating force, and which is naturally more or less permanent ;
but an undulation is supposed to consist in a vibratory motion,
transmitted successively through different parts of a medium,
without any tendency in each particle to continue its motion,
except in consequence of the transmission of succeeding undu-
lations, from a distinct vibrating body ; as, in the air, the vibra-
tions of a chord produce the undulations constituting sound.

Passages from Newton.

" Were I to assume an hypothesis, it should be this, if pro-
pounded more generally, so as not to determine what light
is further than that it is something or other capable of
exciting vibrations in the ether : for thus it will become sb
general and comprehensive of other hypotheses, as to leave

144 THEORY OF LIGHT AND COLOURS. No. VII.

little room for new ones to be invented." (Birch, Vol. III.
p. 249, Dec. 1675.)

^* In the second place, it is to be supposed that the ether is
a vibrating medium like air, only the vibrations far more swift
and minute ; those of air, made by a man's ordinary voice,
succeeding one another at more than half a foot, or a foot
distance ; but those of ether at a less distance than the hundred
thousandth part of an inch. And, as in air, the vibrations
are some larger than others, but yet all equally swift, (for in a
ring of bells the sound of every tone is heard at two or three
miles distance in the same order that the bells are struck,)
so, I suppose, the ethereal vibrations differ in bigness, but not
in swiftness. Now, these vibrations, beside their use in re-
flection and refraction, may be supposed the chief means by
which the parts of fermenting or putrefying substances, fluid
liquors, or melted, burning, or other hot bodies, continue in
motion.'' (Birch, Vol. III. p. 251, Dec. 1675.)

" When a ray of light falls upon the surface of any pellucid
body, and is there refracted or reflected, may not waves of
vibrations, or tremors, be thereby excited in the refracting or
reflecting medium ? And are not these vibrations propagated
from the point of incidence to great distances ? And do they
not overtake the rays of light, and by overtaking them suc-
cessively, do not they put them into the fits of easy reflection
and easy transmission described above ?" (Optics, Qu. 17.)

^* Light is in fits of easy reflection and easy transmission,
before its incidence on transparent bodies. And probably it is
put into such fits at its first emission from luminous bodies,
and continues in them during all its progress." (Optics,
Second Book, Part III. Prop. 13.)

Hypothesis III.

Tlie Sensation of different Colours depends an the different fre-
quency of Vibrations excited by Light in the Retina.

Passages from Newton.

" The objector's hypothesis, as to the fundamental part of it,
is not against me. That fundamental supposition is, that the

No. VII. THEORY OF LIGHT AND COLOURS. ' 145

parts of bodies, when briskly agitated, do excite vibrations in
the ether, which are propagated every way from those bodies
in straight lines, and cause a sensation of light by beating
and dashing against the bottom of the eye, something after
the manner that vibrations in the air cause a sensation of
sound by beating against the organs of hearing. Now, the
most free and natural application of this hypothesis to the
solution of phenomena I take to be this — ^that the agitated
parts of bodies, according to their several sizes, figures, and
motions, do excite vibrations in the ether of various depths or
bignesses, which, being promiscuously propagated through that
medium to our eyes, effect in us a sensation of light of a white
colour ; but if by any means those of unequal bignesses be sepa-
rated from one another, the largest beget a sensation of a red
colour, the least or shortest of a deep violet, and the interme-
diate ones of intermediate colours ; much after the manner that
bodies, according to their several sizes, shapes, and motions,
excite vibrations in the air of various bignesses, which, according
to those bignesses, make several tones in sound: that the
largest vibrations are best able to overcome the resistance of a
refracting superficies, and so break through it with least refrac-
tion ; whence the vibrations of several bignesses, that is, the
rays of several colours, which are blended together in light,
must be parted from one another by reft*action, and so cause
the phenomena of prisms and other reft*acting substances ; and
that it depends on the thickness of a thin transparent plate or
bubble, whether a vibration shall be reflected at its further
superficies, of transmitted ; so that, according to the number of
vibrations, interceding the two superficies, they may be reflected
or transmitted for many successive thicknesses. And, since the
vibrations which make blue and violet are supposed shorter than
those which make red and yellow, they must be reflected at a
less thickness of the plate ; which is sufficient to explicate all
the ordinary phenomena of those plates or bubbles, and also of
all natural bodies, whose parts are like so many fragments of
such plates. These seem to be the most plain, genuine, and
necessary conditions of this hypothesis ; and they agree so
justly with my theory, that, if the animadversor think fit to
VOL. L L

146 THEORY OP LIGHT AND COLOURS. No. VII.

apply them, he need not, on that account, apprehend a divorce
from it ; but yet, how he will defend it from other difficulties I
know not." (Phil. Trans. Vol. VII. p. 5088. Abr. Vol. I.
p. 145. Nov. 1672.)

** To explain colours, I suppose, that as bodies of various
sizes, densities, or sensations, do by percussion or other action
excite sounds of various tones, and consequently vibrations in
the air of different bigness ; so the rays of light, by impinging
on the stiff refracting superficies, excite vibrations in the ether,
of various bigness ; the biggest, strongest, or most potent rays,
the largest vibrations ; and others shorter, according to their
bigness, strength, or power : and therefore the ends of the
capillamenta of the optic nerve, which pave or face the retina,
being such refracting superficies, when the rays impinge upon
them, they must there excite these vibrations, which vibrations
(like those of sound in a trunk or trumpet) will run along the
aqueous pores or crystalline pith of the capillamenta, through
the optic nerves, into the sensorium ; and there, I suppose,
affect the sense with various colours, according to their bigness
and mixture ; the biggest with the strongest colours, reds and
yellows ; the least with the weakest, blues and violets ; the
middle with green, and a confusion of all with white — much
after the manner that, in the sense of hearing, nature makes use
of aerial vibrations of several bignesses to generate sounds of
divers tones, for the analogy of nature is to be observed."
(Birch, Vol. III. p. 262. Dec. 1675.)

^'Considering the lastingness of the motions excited in the
bottom of the eye by light, are they not of a vibrating nature ?
Do not the most refrangible rays excite the shortest vibrations,
the least refrangible the largest ? May not the harmony and
discord of colours arise from the proportions of the vibra-
tions propagated through the fibres of the optic nerve into the
brain, as the harmony and discord of sounds arise from the
proportions of the vibrations of the air?" (Optics, Qu. 16,
13, 14.)

Scholium. Since, for the reason here assigned by Newton,
it is probable that the motion of the retina is rather of a vibra-
tory than of an undulatory nature, the frequency of the vibra-

No. Vn. THEOBY OF LIGHT AND COLOURS. 147

tions must be dependent on the constitution of this substance.
Now, as it is almost impossible to conceive each sensitive point
of the retina to contain an infinite number of particles, each
capable of vibrating in perfect unison with every possible undu-
lation, it becomes necessary to suppose the number limited, for
instance, to the three principal colours, red, yellow, and blue,
of which' the undulations are related in magnitude nearly as the
numbers 8, 7, and 6 ; and that each of the particles is capable of
being put in motion less or more forcibly by undulations differing
less or more from a perfect unison ; for instance, the undula-
tions of green light being nearly in the ratio of 6^, will affect
equally the particles in unison with yellow and blue, and pro-
duce die same effect as a light composed of those two species ;
and each sensitive filament of the nerve may consist of three
portions, one for each principal colour. Allowing this st&ter
ment, it appears that any attempt to produce a musical effect
from colours must be unsuccessful, or at least that nothing more
than a very simple melody could be imitated by them ; for the
period, wluch in fact constitutes the harmony of any concord,
being a multiple of the periods of the single undulations, would
in this case be wholly without the limits of sympathy of the
retina, and would lose its effect, in the same manner as the
harmony of a third or fourth is destroyed by depressing it to
the lowest notes of the audible scale. In hearing, there seems
to be no permanent vibration of any part of the organ.

Hypothesis IV.

All material Bodies have an Attraction for the ethereal Medium^
hy means of which it is accumulated within their Substance^
and for a small Distance around them^ in a state of greater
Density i but not of greater Elasticity.

It has been shown that the three former hypotheses, which
may be called essential, are literally parts of the more compli-
cated Newtonian system. This fourth hypothesis differs perhaps
in some degree from any that have been proposed by former
authors, and is diametrically opposite to that of Newton ; but
both being in themselves equally probable, the opposition is

L 2

148 THEORY OF LIGHT AND COLOURS. No. VII.

merely accidental, and it is only to be inquired which is the
best capable of explaining the phenomena. Other suppositions
might perhaps be substituted for this, and therefore I do not
consider it as fundamental, yet it appears to be the simplest and
best of any that have occurred to me.

Proposition I.

AH impubes are propagated in a homogeneous elastic Medium
with an equable Velocity.

Every experiment relative to soimd coincides with the obser-
vation already quoted from Newton, that all undulations are
propagated through the air with equal velocity ; and this is
further confirmed by calculations. (Lagrange. Misc. Taur.
Vol. I. p. 91. Also, much more concisely, in my Syllabus of
a course of Lectures on Natural and Experimental Philosophy,
about to be published. Article 289.) If the impulse be so great
as materially to disturb the density of the medium, it will be
no longer homogeneous ; but, as far as concerns our senses, the
quantity of motion may be considered as infinitely small. It is
surprising that Euler, although aware of the matter of fact,
should still have maintained that the more frequent imdula-
tions are more rapidly propagated. (Theor. mus. and Conject.
phys.) It is possible that the actual velodty of the particles
of the luminiferous ether may bear a much less proportion to
the velocity of the undulations than in soimd, for light may be
excited by the motion of a body movmg at the rate of only one
mile in the time that light moves a hundred millions.

Scholium 1. It has been demonstrated that in different me-
diums the velocity varies in the subduplicate ratio of the force
directly, and of the density inversely. (Misc. Taur. Vol. I.
p. 91. Young's Syllabus. Art. 294.)

Scholium 2. It is obvious, from the phenomena of elastic
bodies, and of sounds, that the undulations may cross each
other without interruption ; but there is no necessity that the
various colours of white light should intermix their undulations,
for, supposing the vibrations of the retina to continue but a
five hundredth of a second after their excitement, a million

No. VIL THEORY OF LIGHT AND COLOURS, 149

undulations of each of a million colours may arrive in distinct
succession within this interval of time, and produce the same
sensible effect, as if all the colours arrived precisely at the same
instant

Proposition II.

An Undulation conceived to originate from the Vibration of a
Singh Particle, must expand through a homogeneous Medium
in a spherical Form^ but toitk different quantities of Motion in
different Parts.

For, since every impulse, contndered as positive or negative,
is propagated with a constant velocity, each part of the undu-
lation must in equal times have passed through equal distances
from the vibrating point. And, supposing the vibrating par-
ticle, in the course of its motion, to proceed forwards to a
small distance in a given direction, the principal strength of
the undulation will naturally be straight before it; behind
it, the motion will be equal, in a contrary direction ; and, at
right angles to the line of vibration, the undulation will be
evanescent.

Now, in order that such an undulation may continue its pro-
gress to any considerable distance, there must be in each part
of it a tendency to preserve its own motion in a right line from
the centre ; for if the excess of force at any part were commu-
nicated to the neighbouring particles, there can be no reason
why it should not very soon be equalized throughout, or, in
other words, become wholly extinct, since the motions in con-
trary directions would naturally destroy each other. The
origin of sound from the vibration of a chord is evidently of
this nature ; on the contrary, in a circular wave of water, every
part is at the same instant either elevated or depressed. It
may be difficult to show mathematically the mode in which
this inequality of force is preserved, but the inference from the
matter of fact appears to be unavoidable; and while the science
of hydrodynamics is so imperfect that we cannot even solve the
simple problem of the time required to empty a vessel by a
given aperture, it cannot be expected that we should be able to
account perfectly for so complicated a series of phenomena as

150 THEORY OF LIGHT AND COLOURS. ""No. VII.

those of elastic fluids. The theory of Huygens^ indeed, explains
the circumstance in a manner tolerahly satisfactory. He sup-
poses eyery particle of the medium to propagate a distinct un-
dulation in all directions, and that the general effect is only
perceptible where a portion of each undulation conspires in
direction at the same instant; and it is easy to show that such a
general undulation would in all cases proceed rectilinearly, with
proportionate force; hut, upon this supposition, it seems to
follow, that a greater quantity of force must be lost by the
divergence of the partial undulations than appears to be con-
sistent with the propagation of the effect to any considerable
distance ; yet it is obvious that some such limitation of the
motion must naturally be expected to take place, for, if the
intensity of the motion of any particular part, instead of conti-
nuing to be propagated straight forwards, were supposed to
affect the intensity of a neighbouring part of the undulation, an
impulse must then have travelled from an internal to an exter-
nal circle in an ohlique direction, in the same time as in the
direction of the radius, and consequently with a greater velo-
city, against the first proposition. In the case of water, the
velocity is by no means so ri^dly limited as in that of an
elastic medium. Yet it is not necessary to suppose, nor is it
indeed probable, that there is absolutely not the least lateral
communication of the force of the undulation, but that, in highly
elastic mediums, this communication is almost insensible. In
the air, if a chord be perfectly insulated, so as to propagate
exactly sneh vibrations as have been described, they will in
fact be much less forcible than if the chord be placed in the
neighbourhood of a sounding-board, and probably in some mea-
sure because of this lateral communication of motions of an op-
posite tendency. And the different intensity of different parts
of the same circular undulation may be observed, by holding a
common tuning-fork at arm's length, while sounding, and turning
it, from a plane directed to the ear, into a position perpendicular
to that plane.

No. VII. THEOBY OF LIGHT AND COLOURS. 151

PaorosiTioN III.

A Portion of a spherical Undulation^ admitted through an

Aperture into a quiescent Medium^ will proceed to be farther

propagated rectilinearly in concentric Superficies^ terminated

laterally by weak and irregular Portions of newly diverging

Undulations,

At the instant of admission, the circumference of each of the
undulations may he supposed to generate a partial undulation,
filling up the nascent angle between the radii and the surface
terminating the medium ; but no sensible addition will be made
to its strength by a divergence of motion from any other parts
of the undulation, for want of a coincidence in time, as has
already been explained with respect to the various force of a
spherical undulation. If indeed the aperture bear but a small
proportion to the breadth of an undulation, the newly generated
undulation may nearly absorb the whole force of the portion
admitted ; and this is the case considered by Newton in the
Principia. But no experiment can be made under these cir-
cumstances with light, on account of the minuteness of its
undulations, and the interference of inflection ; and yet some
faint radiations do actually diverge beyond any probable limits
of inflection, rendering the margin of the aperture distinctly
visible in all directions. These are attributed by Newton to
some unknown cause, distinct from inflection (Optics, Third
Book, Obs. 5) ; and they fully answer the description of this
proposition.

Let the concentric lines in Kg. 128 represent the con-
temporaneous situation of similar parts of a number of suc-
cessive undulations diverging from the point A ; they will also
represent the successive situations of each individual undulation:
let the force of each undulation be represented by the breadth of
the line, and let the cone of light ABC be admitted through
tlie aperture BC ; then the principal undulations will proceed
in a rectilinear direction towards GH, and the faint radiations
on each side will diverge from B and C as centres, without
receiving any additicHial force from any intermediate point D

152 THEORY OF LIOHT AND COLOURS. No. VIL

of the nndulation, on account of the inequality of the lines D£
and DF. But if we allow some little lateral divergence from
the extremities of the undulations, it must diminish their force,
without adding materially to that of the dissipated light ; and
their termination, instead of the right line BG, will assume the
form CH, since the loss of force must be more considerable
near to C than at greater distances. This line corresponds
with the boundary of the shadow in Newton's first observa-
tion, Fig. 128 ; and it is much more probable that such a
dissipation of light was the cause of the increase of the shadow
in that observation, than that it was owing to the action of the
inflecting atmosphere, which must have extended a thirtieth
of an inch each way in order to produce it ; especially when
it is considered that the shadow was not diminished by sur-
rounding the hair with a denser medium than air, which must
in all probability have weakened and contracted its inflecting
atmosphere. In other circumstances the lateral divergence
might appear to increase, instead of dinunishing, the breadth of
the beam.

As the subject of this proposition has always been esteemed
the most difficult part of the undulatory system, it will be
proper to examine here the objections which Newton has
grounded upon it.

^' To me the fundamental supposition itself seems impossible,
namely, that the waves or vibrations of any fluid can, like the
rays of light, be propagated in straight lines, without a con-
tinual and very extravagant spreading and bending every way
into the quiescent medium, where they are terminated by it.
I mistake if there be not both experiment and demonstration
to the contrary." (Phil. Trans. VII. 5089. Abr. I. 146.
Nov. 1672.)

^' Motus omnis per fluidum propagatus diver^t a recto tra-
mite in spatia immota."

^^ Quoniam medium ibi," in the middle of an undulation
admitted, ^' densius est, quam in spatiis hinc inde, dilatabit sese
tarn versus spatia utrinque sita, quam versus pulsuum rariora
intervalla ; eoque pacto — pulsus eadem fere celeritate sese in
medii partes quiescentes hinc inde relaxare debent ; — ideoque

No. VII. THEORY OF LIGHT AND COLOURS. 153

spatium totum occupabunt. — Hoc experimur in sonis." (Prin-
cip. Lib. II. Prop. 42.)

^' Are not all hypotheses erroneous, in which light is supposed
to consist in pression or motion, propagated through a fluid
medium ? — If it consisted in pression or motion, propagated
either in an instant5 or in time, it would bend into the shadow.
For pression or motion cannot be propagated in a fluid in right
lines beyond an obstacle which stops part of the motion, but
will bend and spread -eTery way into the quiescent medium
which lies beyond the obstacle. The waves on the surface of
stagnating water, passing by the sides of a broad obstacle
which stops part of them, bend afterwards, and dilate them-
selves gradually into the quiet water behind the obstacle.
The waves, pulses, or vibrations of the air, wherein sounds
consist, bend manifestly, though not so much as the waves of
water. For a bell or a cannon may be heard beyond a hill,
which intercepts the sight of the sounding body ; and sounds
are propagated as readily through crooked pipes as straight
ones. But light is never known to follow crooked passages
nor to bend into the shadow. For the fixed stars, by the
interposition of any of the planets, cease to be seen. And so
do the parts of the sun, by the interposition of the moon,
Mercury, or Venus. The rays which pass very near to the
edges of any body, are bent a little by the action of the body ;
but this bending is not towards but from the shadow, and is
performed only in the passage of the ray by the body, and
at a very small distance from it So soon as the ray is past
the body, it goes right on." (Optics, Qu, 28.)

Now the proposition quoted from the Principia does not
directly contradict this proposition ; for it does not assert that
such a motion must diverge equally in all directions ; neither
can it with truth be maintained, that the parts of an elastic
medium communicating any motion, must propagate that motion
equally in all directions.* All that can be inferred by reasoning
is, that the marginal part^ of the undulation must be somewhat
weakened, and that there must be a faint divergence in every
direction ; but whether either of these efiects might be of
suflbnent magnitude to be sensible, could not have been inferred

Supra, p. 66-69.

154 THEORY OF LiaHT AND COLOURS. No. VII.

from argnment, if the affirmative bad not been rendered pro-
bable by experiment.

As to the analogy with other fluids, the most natural inference
from it is this : '^ Tiie waves of the air, wherein sounds consist,
bend manifestly, though not so much as the waves of water ;''
water being an inelastic, and air a moderately elastic medium ;
but ether being most highly elastic, its waves bend very far less
than those of the air, and therefore almost imperceptibly. Sounds
are propagated through crooked passages, because their sides
are capable of reflecting sound, just as light would be propa-
gated through a bent tube, if perfectly polished within.

The light of a star is by far too weak to produce, by its faint
divergence, any visible illumination of the margin of a planet
eclipsing it ; and the interception of the sun's light by the moon,
is as foreign to the question, as the statement of inflection is
inaccurate.

To the argument adduced by Huygens, in favour of the
rectilinear propagation of imdulations, Newton has made no
reply ; perhaps because of his own misconception of the nature
of the motions of elastic mediums, as dependent on a peculiar
law of vibration, which has been corrected by later mathe-
maticians.* On the whole, it is presumed, that this proposition
may be safely admitted as perfectly consistent with analogy
and with experiment.

Proposition IV.

When an Undulation arrives at a Surface which is the Limit
of Mediums of different Densities, a partial Reflection takes
place J proportionate in Force to tlie Difference of the Densities.

T}ii6 may be illustrated, if not demonstrated, by the analogy
of elastic ^odies of different sizes. '' If a smaller elastic body
strikes against a larger one, it is well known that the smaller
is reflected more or less powerfully, according to the difference
of their magnitudes : thus, there is always a reflection when
the rays of light pass from a rarer to a denser stratum of
ether ; and frequently an echo when a sound strikes against
a cloud. A greater body striking a smaller one propels it,

♦ Supra, p. 72.

No. VII. THEORY OF LIGHT AND COLOURS, 155

without losing all its motion : thus, the particles of a denser
stratum of ether, do not impart the whole of their motion to a
rarer, but, in their effort to proceed, they are recalled by the
attraction of the refracting substance with equal. force; and
thus a reflection is always secondarily produced, when the rays
of light pass from a denser to a rarer stratum/'* But it is not
absolutely necessary to suppose an attraction in the latter case,
since the effort to proceed would be propagated backwards,
without it, and the undulation would be reversed, a rarefaction
returning in place of a condensation ; and this will perhaps be
found most consistent with the phenomena.

Proposition V.

When an Undulation is transmitted through a surface tenni"
nating different Mediums^ it proceeds in such a direction^ that
the Sines of the Angles of Incidence and Refraction are in the
constant ratio of the Velocity of Propagation in the two
Mediums.

(Barrow, Lect. Opt. 11. p. 4. Huygens, de la Lum, cap. 3.
Euler, ConJ. Phys, Young's Syllabus. Art. 382.)t

Corollary 1. Tlie same demonstrations prove the equality
of the angles of reflection and incidence.

Corollary 2. It appears from experiments cm the refraction
of condensed air, that the ratio of the difference of the sines
varies simply as the density. Hence it follows, by Schol. L
Prop. I. that the excess of the density of the ethereal medium
is in the duplicate ratio of the density of the air ; each particle
co-operating with its neighbours in attracting a greater portion
of it.

Proposition VI.

When an Undulation falls on the Surface of a rarer Medium,
so obliquely that it cannot be regularly refracted, it is totally
reflected^ at an Angle equal to that of its Incidence.X

Corollary. This phenomenon tends to prove the gradual in-
crease and diminution of density at the surface terminating two

• Supra, p. SO. t Supra, p. Si. t IW<i-

156 THEORY OF LIGHT AND COLOUES, No. VII.

mediums^ as supposed in Hypothesis 4 ; although Huygens has
attempted to explain it somewhat differently.

Proposition VI I.
If equidistant Undulations be supposed to pass through a Medium^
of which the Parts are susceptible of permanent Vibrations
somewhat slower than the UndulatianSy their velocity will be
somewhat lessened by this vibratory Tendency; and^ in the
same Medium^ the more, as the Undulations are more frequent.

For, as often as the state of the undulation requires a change
in the actual motion of the particle which transmits it, that
change will be retarded by the propensity of the particle to
continue its motion somewhat longer : and this retardation will
be more frequent and more considerable, as the difference
between the periods of the undulation and of the natural
vibration is greater.

Corollary. It was long an established opinion, that heat con-
sists in vibrations of the particles of bodies, and is capable of
being transmitted by undulations through an apparent vacuum.
(Newt Opt. Qu. 18.) This opinion has been of late very
much abandoned. Count Rumford, Professor Pictet, and Mr.
Davy, are almost the only authors who have appeared to &vour
it ; but it seems to have been rejected without any good grounds,
and will probably very soon recover its popularity.

Let us suppose that these vibrations are less frequent than
those of light; all bodies therefore are liable to permanent
vibrations slower than those of light ; and indeed almost all are
liable to luminous vibrations, either when in a state of ignition,
or in the circumstances of solar phosphori ; but much less easily,
and in a much less degree, than to the vibrations of heat. It
will follow from these suppositions, that the more frequent
luminous undulations will be more retarded than the less
frequent ; and consequently, that blue light will be more re-
frangible than red, and radiant heat least of all ; a consequence
which coincides exactly with the highly interesting experiments
of Dr. Herschel. (Phil. Trans, for 1800, p. 284) It may
also be easily conceived^ that the actual existence of a state of
slower vibration may tend still more to retard the more fre-

No. Vn. THEORY OP UGHT AND COLOURS. ' 157

quent undulations, and that the refractive power of solid bodies
may be sensibly increased by an increase of temperature, as it
actually appears to have been in Euler's experiments. (Acad
de Berlin, 1762, p. 328.)

Scholium. If, notwithstanding, this proposition should appear
to be insufficiently demonstrated, it must be allowed to be at
least equally explanatory of the phenomena with anything that
can be advanced on the other side, from the doctrine of pro-
jectiles ; since a supposed accelerating force must act in some
other proportion than that of the bulk of the particles; and, if
we call this an elective attraction, it is only veiling under a
chemical term, our incapacity of assigning a mechanical cause.
Mr. Short, when he found by observation the equality of the
velocity of light of all colours, felt the objection so forcibly, that
he immediately drew an inference from it in favour of the un-
dulatory system. It is assumed in the proposition, that when
light is dispersed by refraction, the corpuscles of the refracting
substance are in a state of actual alternate motion, and con-
tribute to its transmission ; but it must be confessed that we
cannot at present form a very decided and accurate conception
of the forces concerned in maintaining these corpuscular vibra-
tions.

Proposition VIII.

JFhen two Undulations, from different Origins, coincide either
perfectly or very nearly in Direction^ their joint effe«A is a
Combination of the Motions belonging to each.

Since every particle of the medium is affected by each undu-
lation, wherever the directions coincide, the undulations can
proceed no otherwise than by uniting their motions, so that the
joint motion may be the sum or difference of the separate
motions, accordingly as similar or dissimilar parts of the undu-
lations are coincident

I have, on a former occasion, insisted at large on the
application of tins principle to harmonics ;* and it will appear
to be of still more extensive utility in explaining the phenomena
of colours. The undulations which are now to be compared are
those of equal firequency. When the two series coincide exactly

* Sapra, p. 88.

158 THEORY OF LIGHT AND COLOURS. No. VII.

in point of time, it is obvious that the united velocity of the
particular motions must be greatest, and, in effect at least,
double the separate velocities; and also, that it must be
smallest, and if the undulations are of equal strength, totally
destroyed, when the time of the greatest direct motion belong*
ing to one undulation coincides with that of the greatest
retrograde motion of the other. In intermediate states, the
joint undulation will be of intermediate strength ; but by what
laws this intermediate strength must vary, cannot be deter-
mined without further data. It is well known that a similar
cause produces in sound, that effect which ia called a beat ;
two series of undulations of nearly equal magnitude co-ope-
rating and destroying each other alternately, as they coincide
more or less perfectly in the times of performing their respective
motions.

COROLLABY I. — Of the Colours of striated Surfaces.

Boyle appears to hare been the first that observed the colours
of scratches on polished sur&ces. Newton has not noticed
them. Mazeas and Mr. Brougham* have made some experi-
ments on the subject, yet without deriving any satisfactory
conclusion. But all the varieties of these colours are very
easily deduced from this proposition.

Let there be in a given plane two reflecting points very near
each other, and let the plane be so situated that the reflected
image of a luminous object seen in it may appear to coincide
with the points ; then it is obvious that the length of the inci-
dent and reflected ray, taken together, is equal with respect to
both points^ considering them as capable of reflecting in all
directions. Let one of the pointe be now depressed below the
bOAM^ given plane ; then the wholelfwnof the light reflected from it,

* will be lengthened by a line which is to the depression of the

point as twice the cosine of incidence to the radius. Fig. 129.

If, therefore, equal undulations of given dimensions be
reflected from two points, situated near enough to appear to the
eye but as one, wherever this line is equal to half the breadth
of a whole undulation, the reflection from the depressed point
will so interfere with the reflection from the fixed point, that the

• Phil. Trans, for 1797, vol. liixvii. p. 352.

No. VII. THEORY OF LIGHT AND COLOURS. 159

progressive motion of the one will coincide with the retrograde
motion of the other, and they will both be destroyed ; but, when
this line is equal to the whole breadth of an undulation, the
effect will be doubled ; and when to a breadth and a half, again
destroyed ; and thus for a considerable number of alternations :
and, if the reflected undulations be of different kinds, they will
be variously affected, according to their proportions to the various
length of the line which is the difference between the lengths of
their two paths, and which may be denominated the interval of
retardation.

. In order that the effect may be the more perceptible, a num-
ber of pairs of points must be united into two parallel lines; and,
if several such pairs of lines be placed near each other, they will
facilitate the observation. If one of the lines be made to revolve
round the other as an axis, the depression below the given plane
will be as the sine of the inclination ; and, while the eye and
luminous object remain fixed, the difference of the length of
the paths will vary as this sine.

The best subjects for the experiment are Mr. Coventry's
exquisite micrometers ; such of them as consist of parallel lines
drawn on glass, at the distance of one five hundredth of an
inch, are the most convenient Each of &ese lines appears
under a microscope to consist of two or more finer lines, exactly
parallel, and at the distance of somewhat more than a twentieth
of that of the adjacent lines. I placed one of these so as to
reflect the sun's light at an angle of 45^, and fixed it ui such a
manner, that while it revolved round one of the lines as an axis,
I could measure its angular motion; and I found that the
brightest red colour occurred at the inclinations 10 J, 20| °, 32°,
and 45^ ; of which the sines are as the numbers 1, 2, 3, and 4.
At all other angles also, when the sun's light was reflected from
the surface, the colour vanished with the inclination, and was
equal at equal inclinations on either side.

This experiment affords a very strong confirmation of the
theory. It is impossible to deduce any explanation of it from
any hypothesis hitherto advanced ; and I believe it would be
difficult to invent any other that would account for it. There
is a striking analogy between this separation of colours, and the
production of a musical note by successive echoes from equi-

160 THEORY OF LIGHT AND CX)LOURS. No. VIL

distant iron palisades ; which I have found to correspond pretty
accurately with the known velocity of sound, and the distances
of the surfaces.

It is not improhable that the colours of the integuments of
some insects, and of some other natural bodies, exhibiting in
different lights the most beautiful versatility, may be found to
be of this description, and not to be derived from thin plates.
In some cases, a single scratch or furrow may produce similar
effects^ by the reflection of its opposite edges.

Corollary II. — Of the Colours of thin Plates,

When a beam of light falls on two parallel refracting surfaces,
the partial reflections coincide perfectly in direction; and,
in this case, the interval of retardation, taken between the sur-
faces, is to their distance as twice the cosine of the angle of
refraction to the radius. For, in Kg. 130, drawing AB and CD
perpendicular to the rays, the times of passing through BC and
AD will be equal, and DE will be half the interval of retarda-
tion ; but D£ is to C£ as the sine of DCE to the radius. Hence,
that DE may be constant, or that the same colour may be re-
flected, the thickness CE must vary as the secant of the angle
of refraction CED ; which agrees exactly with Newton's expe-
riments ; for the correction is perfectly inconsiderable.

Let the medium between the surfaces be rarer than the sur-
rounding mediums ; then the impulse reflected at the second
surface, meeting a subsequent undulation at tli& first, will render
the particles of the rarer medium capable of wholly stopping
the motion of the denser, and destroying the reflection (Prop.
IV.), while they themselves will be more strongly propelled
than if they had been at rest ; and the transmitted light will be
increased. So that the colours by reflection will be destroyed,
and those by transmission rendered more vivid, when the double
thicknesses, or intervals of retardation, are any multiples of the
whole breadths of the undulations ; and, at intermediate thick-
nesses, the effects will be reversed ; according to the Newtonian
observations.

If the same proportions be found to hold good with respect
to thin plates of a denser medium, which is indeed not impro-
bable, it will be necessary to adopt the corrected demonstration

No. VII.

THEOBV OF LIGHT AND COLOURS.

161

of Prop, iv., but, at any rate, if a thin plate be interposed between
a rarer and a denser medium, the colours by reflection and
transmission may be expected to change places.

From Newton's measures of the thicknesses reflecting the
different colours, the breadth and duration of their respective
undulations may be very accurately determined ; although it is
not improbable, tliat when the glasses approach very near, the
atmosphere of ether may produce some little irregularity. The
whole visible spectrum appears to be comprised within the ratio
of three to five, or a major sixth in music ; and the undulations
of red, yellow, and blue, to be related in magnitude as the
numbers 7, 8, and 6 ; so that the interval from red to blue*
is a fourth. ' The absolute frequency expressed in numbers is
too great to be distinctly conceived, but it may be better
imagined by a comparison with sound. If a chord sounding the

tenor c, could be continually bisected 40 times, and should
then vibrate, it would afford a yellow green light : this being

4) 40 41

denoted by c, the extreme red would be a, and the blue d.
The absolute length and frequency of each vibration is ex-
pressed in the table ; supposing light to travel in 8i minutes
500,000,000,000 feet.

Goloan.

Extreme
Red . . .

loteraiediate
Orange .

Intermediate.
Yellow . . .

Intermediate
Green • • .

Intermediate
Blue . . .

Intermediate
Indigo .

Intermediate
Violet . . •

Extreme .

I^ni^h of an

Undulation in

parts of an Inch,

in Air.

. 0000266
.0000256
,0000246
.0000240
.00002:^5
. 0000227
.0000219
.0000211
.0000203
.0000196
.0000189
.0000185
.0000181
.0000174
.0000167

NumTier of

Undalationsin

an Inch.

37640
39180
40720
41610
42510
44000
45600
47460
49320
51110
52910
54O70
55240
57490
59750

Number of Undulationa
in a Second.

463 millions of millions.

482

501

512

523

542

561 (= 2«» nearly)

584

607

629

652

6G5

680

707

735

• Sec the correction given in No. VIII., p. 177: for **re'I, yellow, and bli
Bubetitute "red, green, and violet;" and for the numbers **8, 7, and 6" substi
« 7, 6, and hr—Note by the Kditor.

VOL. I. ^

blue "
;titiitn

162 THEORY OF LIGHT AND COLOURS. No. VII.

Scliolium. Tt was not till I had satisfied myself respecting
all these phenomena, that I found in Hooke's Micrographia, a
passage which might have led me earlier to a similar conclusion.
" It is most evident that the reflection from the under or fui*-
ther side of the body, is the principal cause of the production of
tliese colours. — Let the ray fall obliquely on the thin plate,
part therefore is reflected back by the first superficies, — part
refracted to the second surface, — whence it is reflected and
refracted again. So that, after two refractions and one reflec-
tion, there is propagated a kind of fainter ray, — " and, " by
reason of the time spent in passing and repassing, — this fainter
pulse comes behind the" former reflected ** pulse; so that
hereby, (the surfaces being so near together that the eye cannot
discriminate tliem from one,) this confused or duplicated pulse,
whose strongest part precedes, and whose weakest follows, does
produce on the retina the sensation of a yellow. If these
surfaces are further removed as under, the weaker pulse may
become coincident with the" reflection of the "second," or next
following pulse, from the first surface, " and lag behind that
also, and be coincident with the third, fourth, fifth, sixth,
seventh, or eighth — ; so that if there be a thin transparent
body, that from the greatest thinness requisite to produce
colours, does by degrees grow to the greatest thickness,— the
colours, shall be so often repeated, as the weaker pidse does
lose paces with its primary or first pulse, and is coincident with
a" subsequent "pulse. And this, as it is coincident, or
follows from the first hypothesis I took of colours, so upon ex-
periment have I found it in multitudes of instances that seem
to prove it." (P. 65 — 67.) This was printed about seven years
before any of Newton's experiments were made. We are
informed by Newton, that Hooke was afterwards disposed to
adopt his "suggestion" of the nature of colours; and yet it
does not appear that Hooke ever applied that improvement to
his explanation of these phenomena, or inquired into the neces-
sary consequence of a. change of obliquity, upon his original
supposition, otherwise he could not but have discovered a
striking coincidence with the measures laid down by Newton
from experiment- All former attempts to explain the colours

No. VII. THEORY OF LWHT AND COLOURS. 163

of thin plates, have either proceeded on suppositions, which,
like Newton's, would lead us to expect the greatest irregularities
in the direction of the refracted rays ; or, like Mr. Michell's,
would require such efiects from the change of the angle of
incidence, as are contrary to the effects observed ; or, they are
equally deficient with respect to both these circumstances, and
are inconsistent with the most moderate attention to the principal
phenomena.

Corollary III.— Qf ^^ Cohmr9 of thick Plates.

When a be&m of light passes through a refracting surface,
especially if imperfectly polished, a portion of it is irregularly
scattered, and makes the surface visible in all directions, but
most conspicuously in directions not far distant from that of
the light itself; and, if a reflecting surface be placed parallel
to the refracting surface, this scattered light, as well as the
principal beam, will be reflected, and there will also be a new
dissipation of light, at the return of the beam through the
refracting surface. These two portions of scattered light will
coincide in direction ; and if the surfaces be of such a form as
to collect the similar effects, will exhibit rings of colours. Tlie
interval of retardation is here, the difference between the paths
of the principal beam and of the scattered light between the two
surfaces : of course, wherever the inclination of the scattered
light is equal to that of the beam, although in different planes,
the interval will vanish, and all the undulations will conspire.
At other inclinations, the interval will be the difference of the
secants from the secant of the inclination or angle of refraction
of the principal beam. From these causes, all the colours of
concave mirrore observed by Newton and others are necessiiry
consequences: and it appears that their production, thougli
somewhat similar, is by no means, as Newton imagined, iden-
tical with the production of those of thin plates.

Corollary IV. — Of Blackness,

In the three preceding corollaries, we have considered the
refracting and reflecting substances as limited by a mathema*

M 2

164 THKOHY OF LIGHT AXD COLOURS. No. VII.

tical surface : but this is perhaps never T)hysically true. The
ethereal atmospheres may extend on each side the surface as
far as the breadth of one or more undulations ; and, if they be
supposed to vary equally in density at every part, the partial
reflections from each of the infinite number of surfaces, where
the density changes, will very much interfere with each other,
and destroy a considerable portion of the reflected light, so that
the substance may become positively black ; and this effect may
take place in a greater or less degree, as the density of the
ethereal atmosphere varies more or lees equably ; and, in some
cases, particular undulations being more affected than others,
a tinge of colour may be produced. Accordingly, M. Bouguer
has observed a considerable loss of light, and in some instances
a tinge of colour, in total reflections at the surface of a rarer
medium.

Corollary V. — Of Colours by Inflection.

Whatever may be the cause of the inflection of light passing
through a small aperture, the light nearest its centre must be
the least diverted, and the nearest to its sides the most : another
portion of light falling very obliquely on the margin of the
aperture, will be copiously reflected in various directions ; some
of which will either perfectly or very nearly coincide in direc-
tion witli the unreflected light, and, having taken a circuitous
route, will so interfere with it, as to cause an appearance of
colours. The length of the two tracko will differ the less, as
the direction of the reflected light has been less changed by its
reflection, that is, in the light passing nearest to the margin ; so
that the blues will appear in the light nearest the shadow. The
effect will be increased and modified when the reflected light
falls within the influence of the opposite edge, so as to interfere
with the light simply inflected by that also.

But in order to examine the consequences more minutely, it
will be convenient to suppose the inflection caused by an ethe-
real atmosphere, of a density varying as a given power of the
distance from a centre, as in the eighth proposition of the last
Bakerian Lecture.* Putting r = 3, and x = 4, 1 have constructed

* Supra, p. 20.

No. VII. THEORY OF LIOHT AND COLOURS. 105

a diagram, (Fig. 131,) which shows by the two pairs of curves

the relative position of the reflected and unreflected portions

of any one undulation at two successive times, and also, by

shaded lines drawn across the parts where the intervals of

^_ retardation are in arithmetical progression, and where similar

colours will be exhibited at different distances from the in-

I fleeting substance. The result fully agrees with the observations

I of Newton's third book, and with tha«e of later writers. But I

I do not consider it as quit*; cerfciin, until further experiments

I have been made on the inflecting power of different substances,

that Dr. Hooke s explanation of inflection, by the tendency of

light to diverge, may not have some pretensions to truth. I am

sorry to be obliged to recall here the assent which, at first

sight, I was induced to give to a supposed improvement of a

! late author.*

Scholium. In the construction of the diagram, it be-
comes necessary to find the time spent by each ray in its

^ ' 1^

passage. Since the velocity was denoted by x % on the sup-

position of a projectile, it will be as x\ on the contrary
supposition! (Schol. 2, Prop. I.), and the fluxion of the distance

described being , , that of the time will be or

° Vi-yy V 1 - yy

— ^ of which the fluent is r-^ • — • Vl - v(/. There-

^-r yyVl-yy l-r y

fore, with the radius x^ " ~, describe a circle concentric with
the surfaces of the inflecting atmosphere, then the angle de-
scribed by the ray during its passage through the atmosphere,
will always be to the angle subtended by the line cut ofl^ by
this circle from the incident ray produced, in the ratio of r to
r— 1 ; and the time spent in this passage will be in the same
ratio to the time that would have been spent in describing this
intercepted portion with the initial velocity. For y, being equal

to « a: 7 ~i , is the sine of the inclination of the incident ray
j to the radius, where it meets this circle ; therefore, by the

' ♦. Supra, p. 81. + Supra, p^ 15.

166 THEORY OF LIGHT AND COLOURS. No. VII.

proposition quoted, the angle described is in a given ratio to
the angle at the centre, which is the difference of the incli-
nations. Making jr "T or— radius, the sine, instead of y,

becomes 5, and the cosine /y/-~ — *«> 0^*7 \/l - yj/y a^<J>

when y ^ 8Sy 1 — «s ; therefore the line intercepted is to the
difference of the fluents as r to r — 1. (See also Young's
Syllabus, Art. 372.)

Proposition IX.
Radiant LiglU consists in undulations of the luminiferous Ether,

This proposition is the general conclusion from all the pre-
ceding, and it is conci'ived that they conspire to prove it in as
.satisfactory a manner as can possibly be expected from die
nature of the subject. It is clearly granted by Newton, that
there arc undulations, yet he denies that they constitute light ;
but it is sliown in the three first corollaries of the last propo-
sition, that all cases of the increase or diminution of light are
i*eferable to an increase or diminution of such undulations, and
that all the aflecti(ms to which the undulations would be liable,
are distinctly visible in the phenomena of light ; it may there-
fore be very logically inferred, that the undulations are light.

A few detached remarks will serve to obviate some objections
which may be raised against this theory.

1. Newton has advanced the singular refraction of the Ice-
land crystal^ as an argument that the particles of light must
be projected corpuscles ; since he thinks it probable that the
different sides of these particles must be difierently attracted
by the crystal, and since Huygens has confessed his inability
to account in a satisfactory manner for all the phenomena.
But, contrarily to what might have been expected from New-
ton's usual accuracy and candour, he has laid down a new law
for the refraction, without giving a reason for rejecting that of
Huygens, which Mr. Ilaiiy has found to be more accurate
than Newton's; and, without attempting to deduce from his
own system any explanation of the more universal and striking

No. VII. THEORY OF LIGHT AND COLOURS. 167

effects of doubling spars, he has omitted to observe that Huy-
geos's most elegant and ingenious theory perfectly accords with
these general effscts, in all particulars, and of course derives from
them additional pretensions to truth ; this he omits, in order to
point out a difficulty, for which only a verbal solution can be
found in bis own theory, and which will probably long remain
unexplained by any other.

2. Mr. Michell has made some experiments, which appear '^
to show that the rays of light have an actual momentum, by
means of which a motion is produced when they fall on a thin
plate of copper delicately suspended. (Priestley's Optics.) '^
But, taking for granted the exact perpendicularity of the plate,
and the absence of any ascending current of air, yet since, in
every such experiment^ a greater quantity of heat must be com-
municated to the air at the surface on which the light falls than
at the opposite surface, the excess of expansion must necessarily
produce an excess of pressure on the first surface, and a very
perceptible recession of the plate in the direction of the light.
Mr. Bennet has repeated the experiment, with a much more
sensible apparatus, and also in the absence of air ; and very
justly infers from its total failure, an argument in favour of
the undulatory system of light. (Phil. Trans, for 1 792, p. 87.) ^
For, granting the utmost imaginable subtility of the corpuscles
of light, their effects might naturally be expected to bear some
proportion to the effects of the much less rapid motions of the
electrical fluid, which are so very easily perceptible, even in
their weakest states.

3. There are some phenomena of the light of solar phosphori,
which at first sight might seem to fiivour the corpuscular sys-
tem ; for instance, its remaining many months as if in a latent
state, and its subsequent re-emission by the action of heat.
But, on further consideration, there is no difficulty in supposing
the particles of the phosphori which have been made to vibrate
by tiie action of light, to have this action abruptly suspended
by the intervention of cold, whether as contracting the bulk of
the substance or otherwise ; and again, after the restraint is
removed, to proceed in their motion, as a spring would do
which had been held fast for a time in an intermediate stage of

168 TUEOliY OF LIGHT AND COLOURS. No. VII.

its vibration : nor is it impossible that heat itself may, in some
cireumstancel, become in a similar manner latent. (Nichol-
son's Jouraal, vol. ii. p. **99.) But the affections of heat may
perhaps hereafter be rendered more intelligible to us ; at
present, it seems highly probable that light differs from heat
only in the frequency of its undulations or vibrations ; those
undulations which are within certain limits, with respect to fre-
quency, being capable of affecting the optic nerve, and consti-
tuting light ; and those which are slower and probably stronger,
constituting heat only ; that light and bo«it occur to us, each
in two predicaments, the vibratory or permanent, and the undu-
latory or transient state ; vibratory light being the minute
motion of ignited bodies, or of solar phopphori, and undulatory
or radiant light the motion of the ethereal medium excited by
these vibrations; vibratory heat being a motion to which all
material substances are liable, and which is ui<re or less per-
manent ; and undulatory heat that motion of the same ethereal
medium, which has been shown by Mr. King (Morsels of
Criticism* 1786, p. 99), and Mr. Pictet {Essais de Physique,
1790), to be as Ciipable of reflection as light, and by Dr.
Herschel to be capable of separate refraction. (Phil. Trans,
for 1800, p. 284.; How much more readily heat .is commu-
nicated by the free aL'cess of colder substances, than either by
radiation or by transmission through a quiescent ^medium, has
been shown by the valuable experiments of Count Rumford.
It is easy to conceive that some substances permeable to light,
may be unfit for the transmission of heat, in the same manner
as particular substances may transmit some kinds of light, while
they are opaque with respect to others.

On the whole, it appears, that the few optical phenomena
which admit of explanation by the corpuscular system, are
equally consistent with this theory ; that many others, which
have long been known, but never understood, become by these
means perfectly intelligible ; and that several new facts are
found to be thus only reducible to a perfect analogy with other
fact^?, and to the simple priucipleti of the undulatory system.
It is presumed, that henceforth the bccond and third books of
New ton V Optics will be considered as more fully understood

NVVU,

LIGHT JLND COXOURS.

i^'.mm

I'lp an

^. f

^"..v

% y.v

Ftif m.

■' 1 / >

// '

Tr fitrffnnn Ui,'? Fr/. /.

Mo. VII. THEORY OF LIGHT AND COLOURS. 169

than the first has hitherto been ; but, if it should appear to
impartial judges, that additional evidence is wanting for the esta-
blishment of the theory^ it will be easy to enter more minutely
into the details of various experiments, and to show the insuper-
able difficulties attending the Newtonian doctrines, which,
without necessity, it would be tedious and invidious to enume-
rate. The merits of their author in natural philosophy are
great beyond all contest or comparison : his optical discovery
of the composition of white light would alone have immor*
talized his name ; and the very arguments which tend to over-
throw his system, give the strongest proofs of die admirable
accuracy of his experiments.

Sufficient and decisive as these arguments appear, it cannot
be superfluous to seek for further confirmation; which may
with considerable confidence be expected, from an experiment
very ingeniously suggested by Professor Robison, on the re-
fraction of the light returning to us from the opposite margins
of Saturn's ring : for, on the corpuscular theory, the ring must
be considerably distorted when viewed through an achromatic
prism : a similar distortion ought also to be observed in the
disc of Jupiter ; but, if it be found that an equal deviation is
produced in the whole light reflected from these planets, there
can scarcely be any remaining hope to explain the aflections of
light by a comparison with the motions of projectiles.

170 PKODUCTION OF OOLOUKS No. Vlll.

No. VIII.

AN ACCOUNT OF SOME CASES OF THE

PRODUCTION OF COLOURS

NOT HITHERTO DESCRIBED.

From the Philosophical Transactions for 1802, p. 387.

Read July 1, 1802.

Whatever opinion may be entertained of the theory of light
and colours which I have lately had the honour of submitting
to the Royal Society, it must at any rate be allowed that it has
given birth to the discovery of a simple and general law, capable
of explaining a number of the phenomena of coloured light,
which, without this law, would remain insulated and unintelli-
gible. The law is, that ** wherever two portions of the same
light an'ive at the eye by different routes, eitlier exactly or very
nearly in the same direction, the light becomes most intense
when the difference of the routes is any multiple of a certain
length, and least intense in the intermediate state of the inter-
fering portions ; and this length is different for light of different
colours."

I have already shown in detail, the sufficiency of this law for
explaining all the phenomena described in the second and third
books of Newton's Optics, as well as some others not mentioned
by Newton. But it is still more satisfactory to observe its
conformity to other facts, which constitute new and distinct
classes of phenomena, and which could scarcely have agreed so
well with any anterior law, if that law had been erroneous or
imaginary : these are the colours of fibres, and the colours of
mixed plates.

As I was observing the appearance of the fine parallel lines
of light which are seen upon the margin of an object held near

No. VIII. NOT HITHERTO DESCRIBED. 171

the eye, so as to intercept the greater part of the light of a
distant luminous object, and which are produced by the fringes
caused by the inflection of light already known, I observed that
they were sonietimes accompanied by coloured fiinges, much
broader and more distinct ; and I soon found that these broader
fringes were occasioned by the accidental interposition of a hair.
In order to make them more distinct, I employed a horse-hair,
but they were then no longer visible. With a fibre of wool, on
the contrary, they became very large and conspicuous ; and,
with a single silk-worm's tliread, tlieir magnitude was so much
increased, that two or three of them seemed to occupy the
whole field of view. They appeared to extend on each side of
the candle, in the same order as the colours of thin plates, seen
by transmitted light. It occurred to me that their cause must
be sought in the interference of two portions of light, one
reflected from the fibre, the other bending round its opposite
side, and at last coinciding nearly in direction with the former
portion ; that, accordingly as both portions deviated more from
a rectilinear direction, the difference of the length of their paths
would become gradually greater and greater, and would conse-
quently produce the appearances of colour usual in such cases ;
that supposing them to be inflected at right angles, the differ-
ence would amount nearly to the diameter of the fibre, and,
that this difference must consequently be smaller as the fibre
became smaller ; and, the number of fringes in a right angle
becoming smaller, that their angular distances would conse-
quently become greater, and the whole appearance would be
dilated. It was easy to calculate, that for the light least in-
flected, the difference of die paths would be to the diameter
of the fibre, very nearly as the deviation of the ray, at any
point, from the rectilinear direction, to its distance from the
fibre. '^

I therefore made a rectangular hole in a card, and bent
its ends so as to support a hair parallel to the sides of the hole ;
then, upon applying the eye near the hole, the hair of course
appeared dilated by indistinct vision into a surface, of which
the breadth was determined by the distance of the Iiair and the
magnitude of the hole, independently of the temporary aperture

172 PRODUCTION OF COLOURS No. VIII.

of the pupil. When the hair approached so near to the direction
of the margin of a candle that the inflected light was sufficiently
copious to produce a sensible effect, the fringes began to appear ;
and it was easy to estimate the proportion of their breadth to
the apparent breadth of the hair, across the image of which
they extended. I found that six of the brightest red fringes,
nearly at equal distances, occupied the whole of that image.
The breadth of the aperture was f4f]r» cmd its distance from
the hair iV of an inch : the diameter of the hair was less than
^7 of an inch ; as nearly as I could ascertain, it was ^ j^^.
Hence, we have tHtt for the deviation of the first red fringe at
the distance ^^ ; aud, as A •* tHjt : : jiv • -rBihr^i or Tii*sT
for the difference of the routes of the red light where it was
most intense. The measure deduced from Newton's experi-
ments is TT^inr* I thought this coincidence, with only an error
of one-ninth of so minute a quantity, sufficiently perfect to
warrant completely the explanation of the phenomenon, and
even to render a repetition of the experiment unnecessary : for
there are several circumstances which make it difficult to calcu-
late much more precisely what ought to be the result of the
measurement.

When a number of fibres of the same kind, for instance, a
uniform lock of wool, are held near to the eye, we see an
appearance of halos surrounding a distant candle ; but their
brilliancy, and even their existence, depends on the uniformity
of the dimensions of the fibres ; and they are larger as the fibres
are smaller. It is obvious that they are the immediate conse-
quences of the coincidence of a number of fringes of the same
size, which, as the fibres are arranged in all imaginable direc-
tions, must necessarily surround the luminous object at equal
distances on all sides, and constitute circular fringes.

There can be little doubt that the coloured atmospherical
halos are of the same kind : their appearance must depend on
the existence of a number of particles of water, of equal dimen-
sions, and in a proper position, with respect to the luminary and
to the eye. As there is no natural limit to the magnitude of the
spherules of water, we may exj)ect these halos to vary without
limit in their diameters ; and accordingly, Mr. Jordan has

No. VIII. NOT HITHERTO DESCRIBED. 173

observed that their dimensions are exceedingly various, and
has remarked that they frequently change during the time of
observation.

I first noticed the colours of mixed plates, in looking at a
candle through two pieces of plate-glass, with a little moisture
between them. I observed an appearance of fringes resembling
the common coloui*s of thin plates ; and, upon looking for the
fringes by reflection, I found that these new fringes were always
in the same direction as the other fringes, but many times
larger. By examining the glasses with a magnifier, I perceived
that wherever the^e fringes were visible, the moisture was
intermixed with portions of air, producing an appearance similar
to dew. I then supposed that the origin of the coloiu*s was the
same as that of the colours of halos ; but, on a more minute
examination, I found that the magnitude of the portions of air
and water was by no means uniform, and that the explanation
was therefore inadmissible. It was, however, easy to find two
portions of light sufficient for the production of these fringes ;
for, the light transmitted through the water, moving in it with
a velocity different from that of the light passing through the
interstices filled only with air, the two portions would interfere
with each other, and produce efiects of colour according to the
general law. The ratio of the velocities in water and in air is
that of 3 to 4 ; the fringes ought therefore to appear where the
thickness is 6 times as great as that which corresponds to the
same colour in the common case of thin plates ; and, upon
making the experiment with a plane glass and a lens slightly
convex, I found the sixth dark circle actually of the same
diameter as the first in the new fringes. The colours are also
very e^ly produced, when butter or tallow is substituted for
water ; and the rings then become smaller, on account of the
greater refractive density of the oils : but, when water is added,
so as to fill up the interstices of the oil, the rings are very much
enlarged ; for here the difference only of the velocities in water
and in oil is to be considered, and this is much smaller than the
difference between air and water. All these circumstances are
sufficient to satisfy us with respect to the truth of the explana-
tion ; and it is still more confirmed by the effect of inclining the

174 PRODUCTION OF COLOURS No. VIII.

plates to the direction of the light ; for then, instead of dilating,
like the colours of thin plates, these rings contract : and this is
the obvious consequence of an increase of the length of the
paths of the light, which now traverses both mediums obliquely ;
and the etkct is every where the same as that of a thicker
plate.

It must, however, be observed, that the colours are not pro-
duced in the whole light that is transmitted through the me-
diums : a small portion only of each pencil, passing through the
water contiguous to the edges of the particle, is sufficiently
coincident with the light transmitted by the neighbouring por-
tions of air, to produce the necessary interference ; and it is
easy to show that, on account of the natural concavity of the
surface of each portion of the fluid adhering to the two pieces
of glass, a considerable portion of the light which is beginning
to pass through the water will be dissipated laterally by reflec-
tion at its entrance, and that much of die light passing through
the air will be scattered by refraction at the second surface.
For these reasons, the fringes are seen when the plates are not
directly interposed between the eye and the luminous object ;
and on account of the absence of foreign light, even more dis-
tinctly than when they are in the same right line with that
object And if we remove the plates to a considerable distance
out of this line, the rings are still visible, and become larger
than before; for here the actual route of the light passing
through the air, is longer than that of the light passing more
obliquely through the water, and the difference in the times of
passage is lessened. It is however impossible to be quite con-
fident with respect to the causes of these minute variations,
without some means of ascertaining accurately tlie forms of the
dissipating surfaces.

In applying the genial law of interference to these colours,
as well as to those of thin plates already known, i must confess
that it is impossible to avoid another supposition, which is a
part of the undulatory theory, that is, that the velocity of light
is the greater, the rarer the medium ; and that there is also
a condition annexed to the explanation of the colours of thin
plates, which involves another part of the same theory, that is,

No. VIII. NOT HITHERTO DESCRIBED. 175

that where one of the portions of light has been reflected at the
surface of a rarer medium^ it must be supposed to be retarded
one half of the appropriate interval, for instance in the cen-
tral black spot of a soap-bubble, where the actual lengths of
the paths very nearly coincide, but the effect is the same as if
one of the portions had been so retarded as to destroy the other.
From considering the nature of this circumstance, I ventured to
predict, that if the two reflections were of the same kind, made
at the sur&ces of a thin plate, of a density intermediate between
the densities of the mediums containing it, the effect would be
reversed, and the central spot, instead of black, would become
white ; and I have now the pleasure of stating, that I have fully
verified this prediction, by interposing a drop of oil of sassafras
between a prism of flint-glass and a lens of crown-glass : the
central spot seen by reflected light was white, and surrounded
by a dark ring. It was however necessary to use some force,
in oi-der to produce a contact suflKciently intimate ; and the
white spot differed, even at last, in the same degree from per-
fect whiteness, as the black spot usually does from perfect
blackness.

The colours of mixed plates suggested to me an idea which
appears to lead to an explanation of the dispersion of colours by
refraction, more simple and satisfactory than that which I ad-
vanced in the last Bakerian Lecture. We may suppose that
every refractive medium transmits the undulations constituting
light in two separate portions, one passing through its ultimate
particles, and the other through its pores ; and that these por-
tions re-unite continually, after each successive separation, the
one having preceded the other by a very minute but constant
interval, depending on the regular arrangement of the particles
of a homogeneous medium. Now, if these two portions were
always equal, each point of the undulations resulting from their
re-union would always be found half-way between the places
of the corresponding point in the separate portions; but sup-
posing the preceding portion to be the smaller, the newly
combined undulation will be less advanced than if both had
been equal, and the difference of its place will depend, not only
on the difference of the length of the two routes, which will be

176 PRODUcrrioN of colours No. VIII-

constant for all the undulations ; but also on the law and mag-
nitude of those undulations ; so that the larger undulations will
be somewhat further advanced after each re-union Uian the
smaller ones, and, the same operation recurring at every par-
ticle of the medium^ the whole progress of the larger undula-
tions will be more rapid than that of the smaller ; hence the
deviation, in consequence of the retardation of the motion of
light in a denser medium, will of course be greater for the
smaller than for the larger undulations. Assuming the law of
the harmonic curve for the motions of the particles, we might
without much difficulty reduce this conjecture to a comparison
with experiment; but it would be necessary, in order to warrant
our conclusions, to be provided with very accurate measures of
the refractive and dispersive powers of various substances, for
rays of all descriptions.

Dr. Wollaston's very interesting observations would furnish
great assistance in this inquiry, when compared with the sepa-
ration of colours by thin plates. I have repeated his experi-
ments on the spectrum vrith perfect success, and have made
some attempts to procure comparative measures from thin
plates i and I have found that, as Sir Isaac Newton has already
observed, the blue and violet light is more dispersed by refrac-
tion, than in proportion to the difference of the appropriate
dimensions deduced from the phenomena of thin plates. Hence
it happens, that when a line of the light proceeding to form an
image of the rings of colours of thin plates, is intercepted by a
prism, and an actual picture is formed, resembling the scale de-
lineated by Newton from theory, for estimating the colours of
particles of given dimensions, the oblique spectrums, formed by
the different colours of each series, are not straight, but curved,
the lateral refiraction of the prism separating the violet end
more widely than the red. The thickness corresponding to Uio
extreme red, the line of yellow, bright green, bright blue, and
extreme violet, I found to be inversely as the numbers 27, 30,
35, 40, and 45, respectively. In consequence of Dr. Wollas-
ton's correction of the description of the prismatic spectrum,
compared with these observations, it becomes necessary to mo-
dify the supposition that I advanced in the last Bakerian Lee-

No. Vni. NOT HITHERTO DESCRIBED. 177

ture, respecting the proportioiis of the sympathetic fibres of the
retana ; substitutiDg red* green, and violet, for red, yellow, and
blue, and the numbers 7, 6, and 5, for 8, 7, and 6.

The same prismatic analysis of the colours of thin plates,
appears to fiimish a satisfactory explanation of the subdivision
of the light of the lower part of a candle ; for, in fact, the light
transmitted through every part of a thin plate, is divided in a
similar manner into distinct portions, increasing in number with
the thickness of the plate, until they become too minute to be
visible. At the thickness corresponding to the ninth or tenth
portion of red light, the number of portions of different colours
is five ; and their proportions, as exhibited by refraction, are
nearly the same as in the light of a candle, the violet being the
broadest We have only to suppose each particle of tallow to
be, at its first evaporation, of such dimensions as to produce the
same effect as the thin plate of air at this point, where it is
about ifllofl of an inch in thickness, and to reflect, or perhaps
rather to transmit, the mixed light produced by the incipient
combustion around it, and we shall have a light completely
resembling that which Dr. Wollaston has observed. There
appears to be also a fine line of strong yellow light, separate
from the general spectrum, principally derived from the most
superficial combustion at the margin of the flame, and increas-
ing in quantity as the flame ascends. Similar circumstances
might undoubtedly be found in other cases of the production or
modification of light; and experiments upon this subject might
tend greatly to establish the Newtonian opinion, that the colours
of all natural bodies are similar in their origin to those of thin
plates ; an opinion which appears to do the highest honour to
the sagacity of its author, and indeed to form a very consider-
able step in our advances towards an acquaintance with the in-
timate constitution and arrangement of material substances.

I have lately had an opportunity of confirming my former
observations on the dispersive powers of the eye. I find that,
at the respective distances of 10 and 15 inches, the extreme red
and extreme violet rays are similarly refracted, the difference
being expressed by a focal length of 30 inches. Now the in-
terval between red and yellow is about one-fourth of the whole

VOL. I. N

178 ON COLOURS NOT HTTHERTO DESCRIBED. No. VIII.

spectrum ; consequently, a focal length of 120 inches expresses
a power equivalent to the dispersion of the red and yellow, and
this differs but little from 132, which was the result of the
observation already described. I do not know that these expe-
riments are more accurate than the former one ; but I have
repeated them several times under different circumstances, and
I have no doubt that the dispersion of coloured light in the
human eye is nearly such as I have stated it. How it happens
to be no greater, 1 cannot at present undertake to explain.

No. IX. PHT8ICAL opncss. 179

No. IX.

EXPERIMENTS AND CALCULATIONS RELATIVE TO

PHYSICAL OPTICS.

From the PhUosophical TnuMactions for 1804.

A BAKERIAN LECTURE.

Read Nov. 24, 1803.

I. — Experimental Demonstration of the general Law of the
Interference of Light,

In making some experiments on the fringes of colours accom-
panying shadows, I have found so simple and so demonstrative
a proof of the general law of the interference of two portions of
light, which I have already endeavoured to establish, that I
think it right to lay before the Royal Society a short statement
of the facts which appear to me so decisive. The proposition
on which I mean to insist, at present, is simply this — that fringes
of colours are produced by the interference of two portions of
light ; and I think it will not be denied by the most prejudiced,
thai the assertion is proved by the experiments I am about
to relate, which may be repeated with great ease whenever the
sun shines, and without any other apparatus than is at hand to
every one.

Exper. 1. I made a small hole in a window-shutter, and
covered it with a piece of thick paper, which I perforated with
a fine needle. For greater convenience of observation I placed
a small looking-glass without the window-shutter, in such a
position as to reflect the sun's light, in a direction nearly hori-
zontal, upon the opposite wall, and to cause the cone of diverging
light to pass over a table on which were several little screens of

180 EXPERIMENTS AND CALCULATIOSTS No. IX.

card-paper. I brought into the sunbeam a slip of card, about
one-thirtieth of an inch in breadth, and observed its shadow,
either on the wall or on other cards held at different distances.
Besides the fringes of colour on each side of the shadow, the
shadoMT itself was divided by similar parallel fringes, of smaller
dimensions, differing in number, according to the distance at
which the shadow was observed, but leaving the middle of the
shadow always white. Now these fringes were the joint effects
of the portions of light passing on each side of the slip of card,
and inflected, or rather diffracted, into the shadow. For, a little
screen being placed a few inches from the card, so as to receive
either edge of the shadow on its margin, all the fringes which
had before been observed in the shadow on the wall, immediately
disappeared, although the light inflected on the otlier side was
allowed to retain its course, and although this light must have
undergone any modification that the proximity of the other edge
of the slip of card might have been capable of occasioning.
When the interposed screen was more remote from the narrow
card, it was necessary to plunge it more deeply into the shadow,
in order to extinguish the parallel lines ; for here the light,
diffracted from the edge of the object, had entered further into
the shadow in its way towards the fringes. Nor was it for want
of a sufficient intensity of light that one of the two portions was
incapable of producing the fringes alone ; for, when they were
both uninterrupted, the lines appeared, even if the intensity was
reduced to one-tenth or one-twentieth.

Exper. 2. The crested fringes described by the ingenious and
accurate Grimaldi, afford an elegant variation of the preceding
experiment, and an interesting example of a calculation grounded
on it When a shadow is formed by an object which has a rect-
angular termination, besides the usual external fringes, there
are two or three alternations of colours, beginning from the line
which bisects the angle, disposed on each side of it in curves,
which are convex towards the bisecting line, and which con-
verge in some degree towards it, as they become more remote
from the angular point. These fringes are also the joint effect
of the light which is inflected directly towards the shadow from
each of the two outlines of the object ; for if a screen be placed

No. IX. RELATIVE TO PHYSICAL OPTICS. 181

within a few inches of the object, so as to receive only one of
the edges of the shadow, the whole of the fringes disappear : if,
on the contrary, the rectangular point of the screen be opposed
to the point of the shadow, so as barely to receive the angle
of the shadow on its extremity, the fringes will remain undis*
turbed.

II. — Comparison of Measures deducedfrom various Experiments.

If we now proceed to examine the dimensions of the fringes,
under different circumstances, we may calculate the differences
of the lengths of the paths described by the portions of light
which have thus been proved to be concerned in producing those
fringes ; and we shall find that, where the lengths are equal, the
light always remains white; but that, where either the brightest
light, or the light of any given colour, disappears and reappears,
a first, a second, or a third time, the differences of the lengths
of the paths of the two portions are in arithmetical progression,
as nearly as we can expect experiments of this kind to agree
with each other. I shall compare, in this point of view., the
measures deduced from several experiments of Newton, and
from some of my own.

In the eighth and ninth observations of the third book of
Newton's Optics, some experiments are related, which, together
with the third observation, will furnish us with the data neces-
sary for the calculation. Two knives were placed, with their
edges meeting at a very acute angle, in a beam of the sun's
light, admitted through a small aperture, and the point of con-
course of the two first dark lines bordering the shadows of the
respective knives was observed at various distances. The results
of six observations are expressed in the first three lines of the
first Table. On the supposition that the dark line is produced
by the first interference of the light reflected from the edges of
the knives, with the light passing in a straight line between them,
we may assign, by calculating the difference of the two paths,
the interval for the first disappearance of the brightest light, as
it is expressed in the fourth line. The second Table contains the
results of a similar calculation, from Newton's observations on
the shadow of a hair ; and the third, from some experiments of

182

EXPERIMENTS AND CALCULATIONS

No. IX.

luy own, of the same nature ; the second bright line being sup-
posed to correspond to a double interval, the second dark line
to a triple interval, and the succeeding lines to depend on a
continuation of the progression. The unit of all the tables is
an inch.

Table I. Ob§. 9. N.

DUtanoe of the knives from the aperture •
Distance of

the paper

from the

kniyea I4, 3), 8{,

Distances be-
tween the

edges of the

kniyes, op>

posite to

the point of

oonoonrse. .013, .020, .034,

Interval of

disappear-

32,

.057,

96,

.061,

101

131

.0S7

•0000122, .0000155, .0000182, .0000167, .0000166, .0000166

150,

Table II. Oba. 8. N.

Breadth of the hair

Distance of the hair from the apertore
Distances of the scale from the aperture .
^readthsof the shadow . • • • .
Breadth between the second pair of bri^t lines
Interval of disappearance, or half the difference of the

paths O0OO151,

Breadth between the third pair of bright lines . . ^

Interval of disappearance, one-fourth of the difference .0000130,

144
252

.0000173

i
.0000143

.434

125

250

1.167

•0000149

Tabue III.

Breadth of the object

Distance of the object from the aperture .

Distance of the wall from the aperture

Distance of the second pur of dark lines from each other

Interval of disi^pearanee, one-third of the difference .

Exper, 4.

Breadthof the wire . . . .083

Distance of the wire from the aperture 32

Distance of the wall from the aperture 250
(Breadth of the shadow, by three

measurements 815, .826, or .827; mean, .823)

Distance of the first pair of dark lines 1 . 165, 1 . 170, or 1 . 160; mean, 1 . 165

Interval of disappearance . . • .0000194
Distance of the second pair of dark

lines 1.402, 1.395, or 1,400; mean, 1.399

Interval of disappearance . . • .0000187
Distance of the third pair of dark

tines 1.594, 1.580, or 1.585; mean, 1.586

Interval of disappearance . • . .0000128

No. IX. RELATIVE TO PHYSICAL OPTICS. 183

It appears, from five of the six observatioDS of the first Table,
in which the distance of the shadow was varied from about
3 inches to 11 feet, and the breadth of the fringes was increased
in tlie ratio of 7 to 1, that the difference of the routes constituting
the interval of disappearance, varied but one-eleventh at most ;
and that, in three out of the five, it agreed with the mean, either
exactly, or within rir part* Hence we are warranted in in-
ferring that the interval appropriate to the extinction of the
brightest light, is either accurately or very nearly constant

But it may be inferred, from a comparison of all the other
observations, that when the obliquity of the reflection is very
great, some circumstance takes place, which causes the interval
thus calculated to be somewhat greater : thus, in the eleventli
line of the third Table, it comes out one-sixth greater than the
mean of the five already mentioned. On the other hand, the
mean of two of Newton's experiments and one of mine, is a
result about one-fourth less than the former. With respect to
the nature of this circumstance, I cannot at present form a
decided opinion ; but I conjecture that it is a deviation of some
of the light concerned, from the rectilinear direction assigned to
it, arising either from its natural diflraction, by which the mag-
nitude of the shadow is also enlarged, or from some other
unknown cause. If we imagined the shadow of the wire, and
the fringes nearest it, to be so contracted that the motion of the
light bounding the shadow might be rectilinear, we should thus
make a sufficient compensation for this deviation ; but it is dif-
ficult to point out what precise track of the light would cause it
to require this correction.

The mean of the three experiments which appear to have
been least afiected by this unknown deviation, gives .0000127
for the interval appropriate to the disappearance of the brightest
light ; and it may be inferred that if they had been wholly
exempted from its effects, the measure would have been some-
what fflooaller. Now the analogous interval, deduced from the
experiments of Newton on thin plates, is .0000112, which is
about one-eighth less than the former result ; and this spears
to be a coincidence fully sufficient to authorise us to attribute
these two classes of phenomena to the same cause. It is very

184 EXPERIMENTS AND CALCULATIONS No. IX.

easily shown, with respect to the colours of thin plates, that
each kind of light disappears and reappears where the dif-
ferences of the routes of two of its portions are in arithmetical
progression ; and we have seen that the same law may be in
general inferred from the phenomena of diffracted light, even
independently of the analogy.

The distribution of the colours is also so similar in both cases,
as to point immediately to a similarity in the causes. In the
thirteenth obsenration of the second part of the first book,
Newton relates, that the interval of the glasses where the rings
appeared in red light, was to the interval where they appeared
in yiolet light, as 14 to 9 ; and, in the eleyenth observation
of the third book, that the distances between the fringes,
under the same circumstances, were the 22d and 27th of an
inch. Hence, deducting the breadth of the hair and taking
the squares, in order to find the relation of the difference of
the routes, we have the proportion of 14 to 9^, which scarcely
differs from the proportion observed in the colours of the thin
plate.

We may readily detemune, from this general principle, the
form of the crested fringes of Grimaldi, already described ; for
it will appear that, under the circumstances of the experiment
related, the points in which the differences of the lengths of the
paths described by the two portions of light are equal to a con-
stant quantity, and in which, therefore, the same kinds of light
ought to appear or disappear, are always found in equilateral
hyperbolas, of which the axes coincide with the outlines of the
shadow, and the asymptotes nearly with the diagonal line.
Such, therefore, must be the direction of the fringes ; and this
conclusion agrees perfectly with the observation. But it must
be remarked, that the parts near the outlines of the shadow are
so much shaded off, as to render the character of the curve
somewhat less decidedly marked where it approaches to its
axis. These fringes have a slight resemblance to the hyper-
bolic fringes observed by Newton ; but the analogy is only
distant.

No. IX. RELATIVE TO PHYSICAL OPTICS, 185

III. — Application to the Supernumerary Rainbows.

The repetitions of colours sometimes observed within the
common rainbow, and described in the Philosophical Transac-
tions, by Dr. Langwith and Mr. Daval, admit also a very easy
and complete explanation from the same principles, Dr. Pem-
berton has attempted to point out an analogy between these
colours and those of thin plates ; but the irregular reflection
from the posterior surface of the drop, to which alone he attri-
butes the appearance, must be far too weak to produce any visible
effects. In order to understand the phenomenon, we have only
to attend to the two portions of light which are exhibited in the
common diagrams explanatory of the rainbow, regularly reflected
from the posterior surface of the drop, and crossing each other
in various directions, till, at the angle of the greatest deviation,
they coincide with each other, so as to produce, by the greater
intensity of this redoubled light, the common rainbow of 41
degrees. Other parts of these two portions will quit the drop
in directions parallel to each other ; and these would exhibit a
continued diffusion of fainter light, for 25° within the bright
termination which forms the rainbow, but for the general law
of interference, which, as in other similar cases, divides the light
into concentric rings ; the magnitude of these rings depending
on that of the drop, according to the difference of time occupied
in the passage of the two portions, which thus proceed in parallel
directions to the spectator's eye, after having been differently
refracted and reflected within the drop. This difference varies,
at first, nearly as the square of the angular distance from the
primitive rainbow ; and, if the first additional red be at the dis-
tance of 2® from the red of the rainbow, so as to interfere a little
with the primitive violet, the fourth additional red will be at a
distance of nearly 2® more ; and the intermediate colours will
occupy a space nearly equal to the original rainbow. In order
to produce this effect, the drops must be about y^ ^^ ^^ inch, or
.OlSy in diameter : it would be sufficient if they were between
^ and ^. The reason that such supernumerary colours are
not often seen, must be, that it does not often happen that drops
so nearly equal arc found together ; but, that this may some-

186 EXPERIMENTS AND CALCULATIONS No. IX.

times happen, is not in itself at all improbable : we measure
even medicines Jby dropping them from a phial, and it may easily
be conceived that the drops formed by natural operations may
sometimes be as uniform as any that can be produced by art
How accurately this theory coincides with the observation, may
best be determined from Dr. Langwith*s own words.

" August the 21st, 1722, about half an hour past five in the
evening, weather temperate, wind at north-east, the appearance
was as follows : — ^The colours of the primary rainbow were as
usual, only the purple very much inclining to red, and well
defined : under this was an arch of green, the upper part of
which inclined to a bright yellow, the lower to a more dusky
green: under this were alternately two arches of reddish
purple, and two of green : under all, a faint appearance of
another arch of purple, which vanished and returned several
times so quick, that we could not readily fix our eyes upon it
Thxxa the order of the colours was, I. Red, orange-colour, yel-
low, green, light-blue, deep blue, purple. 11. Tight green, dark
green, purple. III. Green, purple. IV. Green, fidnt vanish-
ing purple. You see we had here four orders of colours, and
perhaps the beginning of a fifth : for I make no question but
that what I call the purple, is a mixture of the purple of each
of the upper series with the red of the next below it, and the
green a mixture of the intermediate colours. I send you not
diis account barely upon the credit of my own eyes ; fix* there
was a clergyman and four other gentlemen in company, whom
I desired to view the colours attentively, who all agreed that
they appear in the manner that I have now described. There
are two things which well deserve to be taken notice of, as they
may perhaps direct us, in some measure, to the solution of this
curious phenomenon. The first is, that the breadth of the first
series so far exceeded that of any of the rest, that, as near as I
could judge, it was equal to them all taken together. The
second is, that I have never observed ^hese inner orders of
colours in the lower parts of the rainbow, though they have
often been incomparably more vivid than the upper parts, under
which the colours have appeared. I have taken notice of this
so very often, that I can hardly look upon it to be accidental ;

No. IX. RELATIVE TO PHYSICAL OPTICS. 187

and, if it should prove trae in general, it will bring the disqui-
sition into a narrow compass ; for it will show that this effect
depends upon some property which the drops retain, whilst they
are in the upper part of the air, but lose as they come lower,
and are more mixed with one another." Phil. Trans., Vol.
XXXII. p. 243.

From a consideration of the nature of the secondary rainbow,
of 54^, it may be inferred, that if any such supernumerary
colours were seen attending this rainbow, they would necessarily
be external to it, instead of internal. The circles sometimes
seen encompassing the observer's shadow in a mist, are perhaps
more nearly related to the common colours of thin plates as
seen by reflection*

IV, — Argumentative Inference retpecUng the Nature of Light.

The experiment of Grimaldi, on the crested fringes within
the shadow, together with several others of his observations,
equally importanty has been left unnoticed by Newton. Those
who are attached to the Newtonian theory of light, or to the
hypothesis of modem opticians, founded on views still less en-
larged, would do well to endeavour to imagine any thing like
an explanation of these experiments, derived from their own
doctrines ; and, if they fail in the attempt, to refrain at least
fi*om idle declamation against a system which is founded on the
accuracy of its application to all these facts, and to a thousand
others of a similar nature.

From the experiments and calculations which have been pre-
mised, we may be aUowed to infer, that homogeneous light, at
certain equal distances in the direction of its motion, is possessed
of opposite qualities, capable of neutralising or destroying each
other, and of extinguishing the light, where they happen to be
united ; that these qualities succeed each other alternately in
successive concentric superficies, at distances which are constant
for the same light, passing through the same medium. From the
agreement of the measures, and from the similarity of the phe-
nomena, we may conclude, that these intervals are the same as
are concerned in the production of the colours of thin plates ;

188 EXPERIMENTS AND CALCULATIONS No. IX.

but these are shown, by the experiments of Newton, to be the
smaller, the denser the medium ; and, since it may be presumed *

that their number must necessarily remain unaltered in a ^ven j

quantity of light, it follows of course, that light moves more t

slowly in a denser, than in a rarer medium ; and this being
granted, it must be allowed, that refraction is not the effect of
an attractive force directed to a denser medium. The advocates
for the projectile hjrpothesis of light, must consider which link
in this chain of reasoning they may judge to be the most feeble ;
for, hitherto, I have advanced in this Paper no general h]rpo-
thesis whatever. But, since we know that sound diverges in
concentric superficies, and that musical sounds consist of oppo-
site qualities, capable of neutralising each other, and succeeding
at certain equal intervals, which are different according to the
difference of the note, we are fully authorized to conclude, that
there must be some strong resemblance between the nature of
sound and that of light.

I have not, in the course of these investigations, found any
reason to suppose the presence of such an inflecting medium in
the neighbourhood of dense substances as I was formerly in-
clined to attribute to them ; and, upon considering the pheno- ,
mena of the aberration of the stars, T am disposed to believe
that the luminiferous ether pervades the substance of all mate-
rial bodies with little or no resistance, as freely perhaps as the
wind passes through a grove of trees. '

The observations on the effects of diffiraction and interference
may perhaps sometimes be applied to a practical purpose, in
making us cautious in our conclusions respecting the appear-
ances of minute bodies viewed in a microscope. The shadow of
a fibre, however opaque, placed in a pencil of light admitted
through a small aperture, is always somewhat less dark in the
middle of its breadth than in the parts on each side. A similar J

effect may also take place, in some degree, with respect to the
image on the retina, and impress the sense with an idea of a
transparency which has no real existence : and if a small por-
tion of light be really transmitted through the substance, this
may again be destroyed by its interference with the diffracted
light, and produce an appearance of partial opacity, instead of

No. IX. RELATIVE TO PHYSICAL 01 TICS. 189

uniform semi-transparency. Thus a central dark spot and a
light spot, surrounded by a darker circle, may respectively be
produced in the images of a semi-transparent and an opaque
corpuscle ; and impress us with an idea of a complication of
structure which does not exist. In order to detect the fallacy,
we may make two or three fibres cross each other, and view
a number of globules contiguous to each other ; or we may
obtain a still more effectual remedy by changing the magnifying
power ; and then, if the appearance remain constant in kind and
in degree, we may be assured that it truly represents the nature
of the substance to be examined. It is natural to inquire
whether or no the figures of the globules of blood, delineated
by Mr. Hewson in the Pliil. Trans., Vol. LXIU. for 1773,
might not in some measure have been influenced by a decep-
tion of this kind : but, as far as I have hitherto been able to
examine the globules, with a lens of one-fiftieth of an inch
focus, I have found them nearly such as Mr. Hewson has
described them.

y.-- Remarks on the Colours of Natural Bodies.

Exper. 5. I have already adduced, in illustration of New-
ton's comparison of the colours of natural bodies with those of
thin plates. Dr. Wollaston's observations on the blue light
of the lower part of a candle, which appears, when viewed
through a prism, to be divided into five portions. I have lately
observed a similar instance, still more strongly marked, in the
light transmitted by the blue glass sold by the opticians. This
light is separated by the prism into seven distinct portions,
nearly equal in magnitude, but somewhat broader, and less
accurately defined, towards the violet end of the spectrum.
The first two are red, the third is yellowish green, the fourth
green, the fifth blue, the sixth bluish violet, and the seventh
violet. This division agrees very nearly with that of the light
reflected by a plate of air tVtt of an inch in thickness, cor-
responding to the 11th series of red, and the 18th of violet.
A similar plate of a metallic oxide would perhaps be about
TT^vT of an inch in thickness. But it must be confessed that
there are strong reasons for believing the colouring particles of

190 EXPERtMENTB AND CALCULATIONS No. IX.

natural bodies in general to be incomparably smaller than this ;
and it is probable that the analogy suggested by Newton is
somewhat less close than he imagined. The light reflected
by a plate of air, at any thickness nearly corresponding to the
11th red, appears to the eye to be very nearly white; but, ""

under favourable circumstances, the 11th red and the neigh-
bouring colours may still be distinguished. The light of some
kinds of coloured glass is pure red ; that of others red with a |

little green : some intercept all the light, except the extreme j

red and the blue. In the blue light of a candle, expanded by !

the prism, the portions of each colour appear to be narrower, |

and the intervening dark spaces wider than in the analogous
spectrum derived from the light reflected from a thin plate.
The light of burning alcohol appears to be green and violet
only. The pink dye sold in the shops, which is a preparation
of the carthamus, affords a good specimen of a yellow green
light regularly reflected, and a crimson probably produced by ^

transmission.

VI. — Experiment on the Dark Rays of Bitter.

Exper. 6. The existence of solar rays accompanying light,
more refrangible than the violet rays, and cognisable by their
chemical effects, was first ascertained by Mr. Ritter ; but Dr.
WoUaston made the same experiments a very short time after-
wards, without having been informed of what had been done
on the Continent. These rays appear to extend beyond the
violet rays of the .prismatic spectrum, through a space nearly
equal to that which is occupied by the violet. In order to
complete the comparison of their properties with those of visible i

light, I was desirous of examining the effect of their reflection
from a thin plate of air, capable of producing the well-known
rings of colours. For this purpose I formed an image of the
rings, by means of the solar microscope, with the apparatus
which I have described in the Journals of the Royal Institution,
and I threw this image on paper dipped in a solution of nitrate
of silver, placed at the distance of about nine inches from the
microscope. In the course of an hour portions of three dark
rings were very distinctly visible, much smaller than the

No. IX. RELATIVE TO PHYSICAL OPTICS. 191

brightest rings of the coloured image, and coinciding very
nearly, in their dimensions, with the rings of violet light that
appeared upon the interposition of violet glass. I thought the
dark rings were a little smaller than the violet rings, but the
difference was not sufficiently great to be accurately ascer-
tained ; it might be as much as -sV or iV of the diameters, but
not greater. It is the less surprising that the difference should
be so small, as the dimensions of the coloured rings do not by
any means vary at the violet end of the spectrum so rapidly as
at the red end. For performing this experiment with very
great accuracy a hellostate would be necessary, since the
motion of the sun causes a slight change in the place of the
image ; and leather, impregnated with the muriate of silver,
would indicate the effect with greater delicacy. The experi-
ment, however, in its present state, is sufficient to complete the
analogy of the invisible with the visible rays, and to show that
they are equally liable to the general law which is the principal
subject of this Paper. If we had thermometers sufficiently
delicate, it is probable that we might acquire, by similar means,
information still more interesting, with respect to the rays of
invisible heat discovered by Dr. Herschel ; but at present
there is great reason to doubt of the practicability of such an
experiment.

192 REPLY TO THE EDINBURGH REVIEWERS. No. X.

No. X.
DR. YOUNG'S REPLY TO THE ANIMADVERSIONS OF THE

EDINBURGH REVIEWERS,

ON SOME PAPERS PUBLISHED IN THE PHILOSOPHICAL TRANS-
ACTIONS.*

Iliad oro : si meam in omui vita, turn in dicendo, inoderationem modestiamque
cognostis, ne me hodie, com isti, at prorocavit^ respondero, oblitom esse putetis
mei. — Cic.

A MAN who has a proper regard for the dignity of his own
character, although his sensibility may sometimes bfe awakened
by the unjust attacks of interested malevolence, will esteem it
in general more advisable to bear, in silence, the temporary
effects of a short-lived injury, than to suffer his own pursuits to

• The three preceding Memoirs, Nos. VII., VIII., and IX. which established the
bases of the most important advancement which the science of Physical Optics had
made since the time of Newton, were attacked with great yiolence, soon after their
appearance, in Nos. II. and IX. of the Edinbnrgh Review. These criticisms haye
been commonly attributed to Lord Brougham, and are probably not surpassed in wit,
sarcasm, and power by any other productions of that distinguished writer. It was
unfortunate, however, that they should have been devoted to the support of views
which have been proved to be, nearly in every instance, erroneous. Their influence,
however, upon public opinion was more remarkable than could reasonably have Iteen
expected, even from the great authority of the publication in which they appeared,
and the unquestionable ability with which they were written. They not only
seriously damaged, for the time, the estimation of the scjentific character of Dr.
Young, but diverted public attention from the examination of the truth of his theories,
at least amongst his own countrymen, for nearly twenty years. It was to M. Arago
that the credit is due of having first fully recognised and proclaimed their value, in
connection with his own researches and those of his illustrious friend, M. Frcsnel.

The following passages of these Reviews will sufficiently show the spirit in which
they were written : —

** As this paper (the Bakerian Lecture on the Theory of Light and Colours, No.
Vn.) contains nothing which deserves the name, either of experiment or discovery,
and as it is in fact destitute of every species of merit, we should have allowed it to
pass among the multitude of tliose articles which must always find admittance into
the collections of a Society which is pledged to publish two or three volumes every
year. The dignities of the author, and &e title of Bakerian Lecture, which is pre-
fixed to these lucubrations, should not have saved tliem from a place in the ignoble
crowd. But we have of late observ^ed in the physical world a most unaccountable
predilection for vague hypothesis daily gaining ground ; and we are mortified to see
that the Royal Society, forgetful of those improvements in science to which it owea

No. X. REPLY TO THE EDINBURGH REVIEWERS. 193

be interrupted, in making an effort to repel the invective, and
to punish the duressor. But it is possible that art and malice
may be so insidiously combined, as to give to the grossest mis-
representations the semblance of justice and candour ; and>

its origin, and neglecting the precepts of its most illastrious members, is now, by the
publication of such papers, giving Uie countenance of its highest authority to dangerous
relaiBtions in the principles of physical logic. We wish to raise our feeble voioe
against innovations that can have no other effect than to checlc the progress of Science,
and renew all those wild phantoms of the imagination which Bacon and Newton put
to flight from her temple. We wish to recall philosophers to the strict and severe
methods of investigation pointed out by the transcendent talents of those illustrious
men, and consecrated by their astonishing success; and, for this purpose, we take the
first opportunity that has been presented to us, of calling our readers' attention to
this mode of philosophising, which seems, by the title of the paper now before us, to
have been honoured with more than the ordinary approbation of the Council."

The reviewer, in noticing the second paper (An Account of some Cases of the Pro-
daction of Colours, No.Vlfl.), after giving his own explanation of a special case of the
law of interference, in opposition to that of Dr. Young, concludes with the following
admonition to the Royal Society : —

*'We cannot conclude our review of these articles without entreating, for a
moment, the attention of that illustrious body, which has admitted of late years so
many paltry and unsubstantial papers into its Transactions. Oreat as the services are
whidi the Royal Society has rendered to the world, and valuable as the papers have
been in every volume (not less valuable, surely, since the accession of the present
excellent President), we think on the benefits which it has conferred with feelings of
the warmest gratitude. We only wish that those feelings should be unmingled by
any ideas of regret, from the want of selection to which we are adverting ; and that
it should cease to give its countenance to such vain theories as those which we find
mingled, in this volume, with a vast body of important information. The Society
has, indeed, been in the habit of stating, that the truth and other merits of the specu-
lations which it publishes must rest with their respective authore; but we are afraid
this is not suflSclent. The Society publishes these papers — meets for the purpose of
reading them — calls them its Transactions ; and, in fact, exercises, in many cases, the
power of rejecting the papen which are ofiered. It is in vain, therefore, to disavow
a responsibility which so many circumstances concur in fixing. The public will
always look upon the Society as immediately responsible for the papers which com-
pose its Transactions, unless, indeed, it shall wish to be demded into the rank of a
mere mechanical contrivance for the printing of miscellanies. We implore the
Council, therefore, if they will deign to cast their eyes upon our humble page, to
prevent a degradation of the Institution which has so long held the first rank among
scientific bodies. Let them reflect on the mighty name which has been transmitted
to them —

' Clarum et venerabile nomen

Gentibus, et multum nostne quod proderat urbi.

Such a name may indeed shelter them in their weakness, and make us venerate,
even in the frailty of old age, an institution illustrious for its ancient virtue. But is
it impossible to ward off the encroachments of time, and to renovate, in new achieve-
ments, the vigour of former years ? It is more honourable to support an illustrious
character, than to appeal to it for shelter and protection."

In the third and last of these Reviews, after further exposing " the law of inter-
ference," as absurd and illogical, and endeavouring to explain away the more important
and decisive experiments upon whidi it was founded, he thus closes the whole contro-
veny : " We now dismiss, for the present, the feeble lucubrations of this author, in
which we have sean'Jied without success for some traces of learning, acuteness, and
ingenuity, that might compensate his evident deficiency in the powers of solid thinking,
calm and patient investigation, and successful development of the laws of Natnre, by

VOL. I. O

194 BEPLT TO THE EDINBI7RQH REVIEWERS. No. X.

especially where the subject of the discussion is (Sf a nature
little adapted to the comprehensiofitif the generality of readers,
even a man's firiends may be so far misled by a garbled extract
from his own works, and by the specious mixture of partial
truth with essential falsehood, that they may not only be unable
to defend him irom the unfavourable opinion of others, but may
themselves be disposed to suspect, in spite of their partiality,
that he has been hasty and inconsiderate at least, if not radi-
cally weak and mistaken. In such a case, he owes to his
friends such explanations as will enable them to see clearly the
injustice of the accusation, and the iniquity of its author : and,
if he is in a situation which requires that he should in a certain
degree possess the public confidence, he owes to himself and to
the public to prove, that the charges of imbecility of mind and
perversity of disposition are not more founded with regard to
him, than with regard to all who are partakers with him in the
unavoidable imperfections of human nature.

Precisely such is my situation. I have at various times
communicated to the Royal Society, in a very abridged form,
the results of my experiments and investigations, relating to
different branches of natural philosophy : and the Council of the
Society, with a view perhaps of encouraging patient diligence,
has honoured my essays with a place in their Transactions.
Several of these essays have been singled out, in an unprece-
dented manner, from the volumes in which they were printed,
and have been made the subjects, in the second and ninth
numbers of the Edinburgh Review, not of criticism, but of
ridicule and invective ; of an attack, not only upon my writings
and my literary pursuits, but almost on my moral character.
The peculiarity of the style and tendency of this attack led me

steady and modest observatioQ of her operations. We came to the eiaminatfon with
no ower prejudice than the very allowable prepossession against rague hypothesis, by
which all true loyera of* science haye for aboye a century and a half been swayed.
We pursued it, both on the present and on a former occasion, without any feelings
except those of regret at the abuse of that time and opportunity which no greater
share of talents than Dr. Young's are sufficient to render fruitful by mere diligence
and moderation. From us, howeyer, he cannot claim any portion of respect until he
shall alter his mode of proceeding, or change the subject of his lucubrations ; and we
feel onrselyes more particularly odled upon to express our disapprobation, because, as
distinction has been unwarily bestowed on his labours by the most illustrious of
scientific bodies, it is the more necessary that a free protest should be recorded before
the more humble tribonab of literature." — Note by the Editor,

No. X. REPLY TO THE EDINBUEOH REVIEWERS. 195

at once to suspect, that it must have been suggested by some
other motive than the love of truth ; and I have both internal
and external evidence for believing, that the articles in question
are, either wholly, or in great measure, the productions of an
individual, upon whose mathematical works I had formerly
thought it necessary to make some remarks, which, though not
favourable, were far from being severe;* and whose optical
speculations, partly confuted before, and already forgotten,
appeared, to their fond parent, to be in danger of a still more
complete- rejection from the establishment of my opinions. As
far as my reputation in natural philosophy is concerned, I
should consider a libel of this kind as neither requiring nor
deserving an answer ; but I cannot help feeling the propriety of
endeavouring to defend myself from the more pernicious influ-
ence of those imputations, which might tend to lessen the con-
fidence of tiie public in the professional qualifications of a man,
whose abilities have been thus contemptuously and repeatedly
depreciated. The practice of physic has always been, either
immediately or remotely, the object of my pursuits, and I can
affirm, without fear of contradiction, that I have never neglected
any opportunity either of improving myself in it^i study, or of
being useful to the humblest of those who have committed
themselves to my care, in its application. But I have no right
to expect that any degree of industry that I may have employed,
should encourage a man to intrust me with the management
of that which so nearly concerns his happiness and prosperity,
if he has reason to think me rash, and vain, and wavering in
my opinions, and that even upon subjects which are gene-
rally supposed to admit of proofe perfectiy decisive and satis-
factory.

My Bakerian lecture on the theory of light and colours, and
another paper published in the same volume of the Philoso-
phical Transactions, are the subjects of two of the most scur-
rilous articles in the second number of the Edinburgh Review.
The writer of these articles has, as a prelude to his imputation
of a "vibratory and undulatory mode of reasoning,*' very
unnecessarily recurred to the first essay that I presented to the

* Snpra, p. 101 ; see also note at the foot of p. 99,^Note by the Editor.

196 REPLY TO THE EDINBURGH REVIEWERS. No. X.

Royal Society, as long ago as the year 1793; I am there-
fore obliged to explain the circumstances which led me to the
subject of that essay, and to relate the history of my opinions
concerning it : and as he has thought proper to insinuate, in
the form of insolent admonition, that I have never studied even
" the plainer parts " of the works of Newton, I must state when
and why I actually read those admirable productions; and
I shall think it right to account, at the same time, for the
manner in which, as a medical man, I have been led, for a time, '
into the extensive regions of natural philosophy.

It is now more than fourteen years since I first resolved to
devote my life to the profession of physic. I continued for two
years the pursuit of those attainments, in mathematics and in
general literature, which had before constituted my sole occu-
pation, and which, by the express sentiment of the father of the
medical sciences, and by the universal sufirages of the more
liberal part of mankind, have been allowed to be the surest
and best foundations for the superstructure of the requisite
qualifications of a physician. The causes of disease, obscure in
their nature, and hidden in their operation, elude but too fre-
quently the most diligent researches of the strongest and most
experienced minds : they afibrd ample scope to the most minute
investigation, and the most sagacious discernment; but they
require that the faculties of the observer should have been
sufiiciently prepared, by being employed on subjects of a na-
ture more certainly definable, and more perfectly intelligible.
Classical literature, mathematical philosophy, chemistry and
natural history, a knowledge of difierent countries, and an
acquaintance with difierent languages, are as necessary to the
melioration of those powers of reasoning which are to be called
into activity in the pursuit of a profession, as they are essential
to the perfection of the character of a general scholar, and an
accomplished man. This must be my excuse for having devoted
a considerable portion of my attention to the study of the
classics, on my success in which the Edinburgh Reviewers have,
with an insulting afiectation of candour, thought fit, on another
occasion, to compliment me. I pursued the study of mathe-
matics and natural philosophy as far only as I esteemed them

No. X. REPLY TO THE EDINBURGH REVIEWERS. 197

subseryieDt to other objects : not that I preferred philology to
sdence, but because I thought myself obliged to sacrifice both
to physic. After having rendered myself familiar with many
other mathematical works, I read, in the autumn of 1790, both
the Prindpia of Newton and his Optics. I read not the
" plainest parts of the Principia *' only, but the whole ; and all
that the illustrious author meant to be understood by a reader,
I understood and admired: where he purposely omitted a
demonstration, 1 did not at that time attempt to investigate it.
That I was then satisfied with some few parts which I do not
now think unexceptionable, might easily have happened, even if
I had felt less reverence than I have uniformly done for the
character of the unrivalled author. The Optics too I read with
attention and delight, yet by no means with the same satisfac-
tion that I had derived from the perusal of the Principia.

My attention to optical subjects was not revived till the year
1793, when, in the course of my anatomical studies, the theory
of vision was necessarily to be reconsidered. I saw, what I
then thought none had seen before, that the crystalline lens was
of a fibrous structure ; and I could find no other satisfactory
mode of explaining the phenomena of vision than by attributing
to it muscular powers. On this subject I presented a short
paper to the Royal Society,* to which, from the circumstance
of the late Mr. Hunter's reclamation of the discovery as his
own, a greater degree of novelty was imputed than it perhaps
deserved. Mr. Home too attributed to Mr. Hunter the merit
of a discovery ^not small nor unimportant,'* that of an animal
in which the fibrous structure of the lens was easily traced. I
liad however foimd no difficulty in observing, in the eye of a
quadruped, the arrangement which had been the basis of my
speculations.

It was in the course of the winter which I spent in pursuing
my medical studies at Edinburgh, that I first read Mr. Home's
account of his experiments on vision."!* This investigation con-
vinced Mr. Home that Mr. Hunter, whose sentiments he had

• No. I., p. 1.

t The Oroonian l«ctiu-e OD Muscular Motion. FhiL Trans, for 1794, vol. Ixzjuv.,
p. I.

198 REPLY TO THE EDINBURGH REVIEWERS. No. X.

before adopted, was mistaken in his opinion ; and when I had
afterward^) seen at Gottingen Dr. Olbers' elegant dissertation
on the same subject, I found it impossible to resist, without
making further experiments of my own, the appearance of
evidence which was brought against my favourite opinion. I
had not then learned of the Edinburgh Reviewers how much
easier it was to deny the accuracy of the experiments of my
adversaries, than to oppose them by arguments, or to allow due
weight to their apparent consequences ; and I thought it more
honourable to acknowledge my conviction of their importance,
than to persist either in error or in silence. I judged, with
respect to the matter of fact, perhaps erroneously, but with
regard to all the evidence that was then in existence,* I j.udged
as every unprejudiced mind must have been inclined to do. It
was only in the year 1800 that I was induced to resume the
investigation, in consequence of reading, in the Medical essays
of a Society in Edinburgh, Dr. Porterfield's valuable paper
" On the Internal Changes of the Eye." I improved on his ideas
of the construction of an optometer, and I obtained, by nume-
rous and diversified experiments, such accumulated evidence of
the truth of my original opinion, that I was obliged to submit
to the unexpected necessity of recurring to it once more. Those
who have read my paper, not as a modem reviewer reads, but
with patience and attention, will not, I imagine, think that any
apology is required for this second change of sentiments. I
cannot, however, refuse myself the pleasure of inserting here
a passage from a letter which I have lately received from Dr.
Olbers, the discoverer of the planet PaUas, the same whose
dissertation on vision I have oflen quoted with applause. "You
may easily suppose," says Dr. Olbers, " that your celebrated
essay ^ On the Mechanism of the Eye,'* must have interested me
very particularly. I saw indeed that it completely refuted my
own theory respecting the changes of the eye ; but my object is
to discover truths and not to support my opinion/^ With such a
man as Dr. Olbers, my reviewer would say again, as he has
said of me, it would be " difficult to argue : were we to

• No. II., p. 12.

No. X. REPLY TO THE EDINBURGH REVIEWERS. 199

take the trouble of refuting him, he might tell us, My opinion
is chanffetV*

I have now, I trust, yindicated myself from the charge of any
unwarrantable inconstancy in the changes which my opinions on
the subject of vision have undergone. I shall next enter into
a similar explanation of my motives for applying myself to the
study of the phenomena of sound and light, and of the progress
of my ideas respecting their nature. When I took a d^ree in
physic at Gottingen, it was necessary, besides publishing a
medical dissertation, to deliver a lecture upon some subject
connected with medical studies: and I chose for this, the
formation of the human voice. A few pages, containing a
table of articulate sounds, were printed at the end of my disser-
tation " On the Preservative Powers of the Animal Economy :"
my uncle. Dr. Brocklesby, at the instance of the late most
respectable Dr. Heberden, repeatedly urged me to give some
further explanation of the subject to which these characters
related. When I began the outline of an essay on the human
voice, I found myself at a loss for a perfect conception of what
sound was, and during the three years that I passed at Emma-
nuel College, Cambridge, I collected all the information re-
lating to it that I could procure from books, and I made a
variety of original experiments on sounds of all kinds, and on
the motions of fluids in general. In the course of these inquiries,
I learned, to my surprise, how much further our neighbours on
the continent were advanced in the investigation of the motions of
sounding bodies and of elastic fluids, than any of our own country-
men : and in making some experiments on the production of
sounds, I was so forcibly impressed with the resemblance of the
phenomena that I saw, to those of the colours of thin plates,
with which I was already acquainted, that I began to suspect
the existence of a closer analogy between them than I could
before have easily believed. On further reflection and examina-
tion my opinion was confirmed, and as I thought I could
place the question in a clearer light than that in which it had
generally been viewed, I was induced to insert my observations
in a paper,' which I presented soon afler to the Royal Society,
under the name of ** Outlines of Experiments and Inquiries

200 KBPLY TO THE EDINBURGH REVIEWERS. No. X.

respecting Sound and Light." * A determination to confine my
studies as much as possible to physic was my motive for laying
them before the Society iu a state of confessed imperfection. I
am not disposed to overrate their value ; the compliment which
was paid to them by an experienced veteran in philosophy, who
wrote the best articles of the Encyclopaedia Britannica, is fully
as much as I can flatter myself that they deserve.*!' The motions
of a stream of air, rendered visible by means of smoke, the
diversified rotations of musical chords, the influence of the mode
of agitation on the natural harmonics of strings, the phenomena
of beats, and of grave harmonics, were exanuned in a maimer
which tended to place in a new point of view a subject certainly
curious, and not wholly luimportant

The opinion respecting light, which I first suggested in this
paper as the most probable, was neither the same with £uler*s,
nor, as the reviewer falsely asserts, in any degree borrowed
from him. It was precisely the theory of Hooke and of Huygens,
with the adoption of some suggestions made by Newton himself
as not in themselves improbable. The only objection which
Newton makes to the hypothesis thus modified, is this : — flight
could not be propagated, solely by the undulations of a fluid,
without spreading almost equally in all directions ; and for this
assertion he thinks that there is both experiment and demon-
stration. His arguments from experiment appear to me to
have been sufliciently obviated by what Lambert has advanced
in the Memoirs of Berlin, and by Professor Robison's remarks
on echos in the Encyclopaedia, as well as by many observations
which I have myself made, at different times, on the waves of
water. The demonstration is attempted in the Principia : to
me it appears to be defective ; if I am not allowed to be a com-
petent judge, I can quote others, whose authority will not be
denied. Euler has been called by some an indiflerent philoso-
pher, but he must at least be allowed to have been perfectly
capable of judging of mathematical evidence : he had certainly
read the Principia, and he utterly denied the conclusiveness of the
argument. D' Alembert was a mathematician of acknowledged
eminence, and Lalande*s approbation of his sentiments must

♦ No. Ill,, p. 64. f Supra, p. 134.

No. X. REPLY TO THE EDINBURGH REVIEWERS. 201

give them additional weight : both these mathematiciaDs assert,
as it appears from Lalande's edition of Montucia, that the argu-
ments are so balanced in &vour of the different systems of light,
that the safest way is to confess '* our utter ignorance of the
manner of its propagation." The celebrated Laplace, in com*
paring the opinions respecting light, is contented to call the
Newtonian doctrine a hypothesis, which, on account of the
facility of its application to the phenomena, is extremely pro-
bable. If he had considered the undulatory system as demon-
strably absurd, he certainly would not have expressed himself
in so undedded a manner. The opinion of Franklin adds
perhaps little weight to a mathematical question, but it may
tend to assist b lessening the repugnance which every true phi-
losopher must feel, to the necessity of embracing a physical
theory different from that of Newton.

I have indeed been accused of insinuating ^^ that Sir Isaac
Newton was but a sorry philosopher." But it is impossible that
an impartial person should read my essays on the subject of
light without being sensible that I have as high a respect for
his unparalleled talents and acquirements as the blindest jof his
followers, and the most parasitical of his defenders. I have
acknowledged that ^* his merits are great beyond all contest or
comparison ;" that ** his discovery of the composition of white
light would alone have immortalised his name ;" that the very
arguments which tend to overthrow his hypothesis respecting
the emanation of light, ** give the strongest proofe of the admi-
rable accuracy of his experiments ;" and that a person may,
" with the greatest justice, be attached to every doctrine which
is stamped with the Newtonian approbation." The printer of
the Review, feeling perhaps that the last expressions would
militate too much in my favour, has thought fit to plunder me
of them, by omitting the marks of quotation, and to attribute
them to my antagonist But, much as I venerate the name of
Newton, I am not therefore obliged to believe that he was in-
iallible. I see, not with exultation, but with regret, that he
was liable to err, and that his authority has, perhaps, sometimes
even retarded tiie progress of science. It is now no longer
denied that he was mistaken in an optical experiment respecting

202 REPLY TO THE EDINBUIMIH REVIEWEBS. No. X.

the dispersion of light ; and the only attempt that has been made
to explain the mistake merely shows that there was a possibility
of his being misled by a singular combination of circumstances:
in a case of mathematical optics he was certainly mistaken, as
Dr. Smith has shown, when he asserted that a sphere of water
produces a maximum of density in the light refracted at an
angle of about 26^ : in the mechanical estimation of force he
erred when he calculated the precession of the equinoxes, and
estimated the rotatory power of each particle of the earth's sub-
stance as simply proportional to its distance from the axis.
These mistakes, and perhaps some others, have been acknow-
ledged and corrected by later writers: other persons, less
considerate, have attacked him where he was invulnerable.
One of these is the gentleman whom I have reason to think the
author of the remarks to which I am replying, and who, having
first accused Newton of a palpable and fundamental blunder,
appears now to be desirous of securing to himself the exclusive
privilege of questioning his authority.

What I have hitherto said relates to the state of the question
respecting the nature of light, as it stood before the publication
of the first of the papers which have excited so much virulence.
But I assert that this paper contains an argument, sufficient to
convert that which before was doubt and conjecture, into pro-
bability and conviction. It was in May 1801 that I discovered,
by reflecting on the beautiful experiments of Newton, a law
which appears to me to account for a greater variety of interest-
ing phenomena than any other optical principle that has yet
been made known. I shall endeavour to explain this law by a
comparison.

Suppose a number of equal waves of water to move upon the
surface of a stagnant lake, with a certain constant velocity, and
to enter a narrow channel leading out of the lake. Suppose
then another similar cause to have excited another equal series
of waves, which arrive at the same channel, with the same
velocity, and at the same time with the first Neither series of
waves will destroy the other, but their effects will be combined :
if they enter the channel in such a manner that the elevations
of one series coincide with those of the other, they must together

No. X. REPLY TO THE EDINBUBGH REVIEWERS. 203

produce a series of greater joint elevations ; but if the elevations
of one series are so situated as to correspond to the depressions
of the other, they must exactly fill up those depressions, and the
surface of the water must remain smooth ; at least I can dis-
cover no alternative, either from theory or from experiment

Now I maintain that similar efiects take place whenever two
portions of light are thus mixed ; and this I call the general
law of the interference of light. I have shown that this law
agrees, most accurately, with the measures recorded in New-
ton's Optics, relative to the colours of transparent substances,
observed under circumstances which had never before been
subjected to calculation, and with a great diversity of other ex-
periments never before explained. This, I assert, is a most
powerful argument in £Givour of the theory which I had before
revived : there was nothing that could have led to it in any
author with whom I am acquainted, except some imperfect
hints in those inexhaustible but neglected muies-of nascent
inventions, the works of the great Dr. Robert Hooke, which had
never occurred to me at the time that I discovered the law ; and
except the Newtonian explanation of the combinations of tides
in the Port of Batsha.

It is unnecessary, on this occasion, to enter minutely into the
consequences of the law of the interference of light : they have
been the prindpal subjects of the three papers which have
drawn down upon me the repeated anathemas of the self-
erected Inquisition of the North. Not a single argument has
been produced to invalidate it The reviewer has cursorily
observed that if the law were true, every surfiice opposed to the
light of two candles would appear to be covered with fringes of
colours. Let us suppose the assertion true — what will be the
consequence ? In all common cases the fringes will demon*
strably be invisible; since, if we calculate the length and
breadth of each fringe, we shall find that a hundred such
fringes would not cover the point of a needle ; and an optician
does not require to be told that a mixture like this constitutes
a white light, not distinguishable by the senses from that which
is supposed to have formed them.

In order to answer the charge of inconsistency in my opinions

204 REPLY TO THE EDINBURGH REVIEWERS. No. X.

reapectdng the nature of light, I must begin by observing that

there are two general methods of communicating knowledge ;

the one analytical, where we proceed from the examination of

efiects to the investigation of causes; the other syntheticaL

where we first lay down the causes, and deduce from them the ''}

particular effects. In the synthetical manner of explaining a

new theory we necessarily begin by assuming principles, which

ought, in such a case, to bear the modest name of hypotheses ;

and when we have compared their consequences with all the

phenomena, and have shown that the agreement is perfect,

we may justly change the temporary term hypothesis into

theory. This mode of reasoning is suflBcient to attach a value

and importance to our theory, but it is not fully decisive

with respect to its exclusive truth, since it has not been proved

that no other hypothesis will agree with the facts.

It is exactly in this manner that I have endeavoured to pro-
ceed in my researches. By analysing the experiments of New- ,
ton, and comparing them with my own, I had arrived at prin- ]
ciples, to which I gave, in my paper on the theory of light, the
unassuming title of hypotheses; after comparing these principles
with all the phenomena of light, and showing their perfect con-
sistency, I thought myself authorised to make a conclusion, in
my ninth proposition, which converts the hypothesis into a theory.
I was justified in doing this, because no man had ever attempted
to advance a theory which would bear to be compared mathe-
matically with the phenomena that I enumerated. But, ac-
cording to the nature of the only mode of reasoning which the
circumstances allowed me, it was impossible to infer, from this
synthetical comparison, that no other suppositions would agree
with the phenomena ; and / expressly remarked^ with respect
to one of the four hypotheses which I laid down, that it was
possible to find others which might be substituted for t/. It is "
in this hypothesis and its consequences only, that I have since
attempted to make any improvements. And such improve-
ments I shall ever admit with pleasure, whether they arise from
my own experiments, or firom those of others. One immaterial
correction of this kind I was obliged to make in consequence of
Dr. Wollaston's most interesting observations upon the true

No. X. REPLY TO THE EDINBURGH REVIEWERS. 205

division of the prismatic spectram, which afford an additional
proof that even Newton's experiments, frequently as they have
been repeated by others, may sometimes stand in need of a
more careful examination. And this modification, which has, in
fact, little or no connexion with the essential parts of my theory,
has been adduced as a proof of the ^^ fickle and vibratory nature
of the medium that fills my mind." The reviewer has indeed
in another place denied Uie accuracy of Dr. Wollaston's ex-
periment, but his objections are too futile to deserve notice.*

Respecting another trifling change of sentiment, to which the
reviewer has thought proper to attach great importance, I have
hitherto abstained from explanation, in delicacy towards the
gentleman whose observations were concerned ; I wish to avoid
insisting on his inaccuracy in a very easy calculation ; and for
the same reason, I shall say nothing further on the subject at
present

When the reviewer asserts that ^^ a hypothesis is a work of
fancy, useless in science," it must be supposed that he is speak*-
ing of such hypotheses as have neither been originally deduced
from experiments, nor afterwards compared with them: but
when, in another of his articles, he condemns, as having impeded

* Id the following notice in the Edinbuigh Review for April, 1803, of his paper in
the Philosophical Transactions for 1802, * Chi the oblique Reflection of Iceland Crystal.*
— *' We were much disappointed to find, that so acute and ingenious on experimentalist
had adopted the wild optical theory of vibrations. After stating it, however, chiefly
from Huygens, and applying it to explain the properties of the spar, he goes on to
examine, by accurate experiments, whether the undtilatory system agrees with the
facts. The hvpothesis is, that the different undulations of the elastic medium are
spherical in almost all casee, but that, in the Iceland crystal, those undulations are
spheroidal; and it must be acknowledged, the near coincidence of the experiments,
which are extremely well contrived, and appear to be accurately conducted, give this
theory a plausibility which it did not before possess. We would, however, remark,
that the hypothesis of Aepinus hioiself, by fiur the most consistent, simple, and
universally applicable, of any that has ever been proposed, is still only a gratuitous
hypothesis ; has acquired to its author only the praise of fanciful ingenuity ; and has
perhaps done more harm than good to the science of magnetism, bj withdrawing the
attention of philosophers from the patient and diflicult, but profitable observation of
nature, to the more easy but empty amusement of indulging iheir fancy.

" The hypothesis of Huygens is not, as Dr. WoUaston seems to think, the same with
that of Euler and other unphilosophical inquirers. It appVoaches more nearly to that
of Newton, and assumes the existence of an elastic medium, acting upon and influenced
by the rays of light. These authors, misled by the nature of sound, do not admit the
materiality of light, but assert that it is a vibration propagated throu^ the medium.
But, short as these remarks are, we are loth to waste any more time on such a feeble
and ill-conducted defence of an untenable and useless hypothesis." Vol. ii. p. 99. —
Note by the Editor,

206 REPLY TO THE EDINBURGH REVIEWERS. No. X.

the progress of discovery, tlie beautiful hypothesis which has
been applied, with the greatest success, by Aepinus, by Mr.
Cavendish, and by Professor Robison, to the phenomena of
electricity and magnetism ; we can only regret that a person so
void of a sense of phy»cal elegance should have an opportunity
of obtruding opinions like these on the public ; and we may ex-
pect that he would say, if he dared, that even the hypothesis of
universal gravitation has presented an insuperable barrier to the
advancement of experimental knowledge. He is at least deter-
mined to show that every hypothec must be the work either of
infancy or of dotage; and insinuates that the speculations
which I have extracted from Newton's writings were merely the
amusements of some vacant hours at the close of his scientific
career. It is very true that the queries of Newton were given
^<to the world" at a time when bis brilliant tod solid dis-
coveries were fully established ; but the papers which explain
all his hypotheses concerning light the most at large, and to
which I have had the most frequent occasion to refer, were read
to the Royal Society more than ten years before he began to
write his Frincipia ; and the principal reason that delayed their
publication, appears to have been the apprehension of disputes
with Dr. Hooke. Some were published in the Optics, soon
after Dr. Hooke's death; others are only to be found in
Birch's History of the Royal Society. Had I not taken care to
annex the dates to my quotations, the reviewer might easily
have pleaded his ignorance in excuse for his misrepresentations.

The same plea of ignorance would be but an inadequate
apology for the assertion of a positive falsehood, where he
accuses me of referring to an unpublished work of my own.
The reference could only be intended for the readers of the
essay as a printed paper; my Syllabus was published in
January, 1802; the Transactions not till late m the spring;
and if he had either sent to the publisher for this syllabus, or
made mquiry for it among his literary friends even in Edin-
burgh, he might have found in it some information, on subjects
which he appears to understand but imperfectly.

In the first paragraph of the review of my paper on the pro-
duction of colours, the writer confesses that he has not ^* suffi-

No. X. REPLY TO THE EDINBURQH REVIEWERS. 207

cieni fancy to discover " how the " interference of two portions
of light " could ever produce an appearance of colour. The
poverty of his fancy may indeed easily be admitted, but it is
unfortunate that he either has not patience enough to read, or
intellect enough to understand, the very papers that he is criti-
cising ; for, if he had perused with common attention my
Bakerian lecture on light, he might have understood such a
production of colour without any exertion of fancy at all. He
then quotes firom me the assertion, that a *^ black hair " does
not produce the appearance of fringes^ and he has even the
modesty to refer to a certain page of my paper. I have there
said, that a ^^ horse hair" did not produce that appearance;
and I have left it for the reviewer to decide whether the horse
should be white or black. The truth is, that a fine wire, or a
small hair, whether black or white, exhibits equally well the
colours which I have described. If the fact were otherwise, it
would be utterly unintelligible ; for there is absolutely no foun*
dation for the reviewer's insinuation, that any theory of these
colours was deduced by De Dominis, or can be deduced by any
other person, from the laws of refrttction. He asserts that it is
mathematically impossible for the light to bend round a hair ;
Grimaldi has long ago experimentally dem(»)strated this flexion,
and called it difiraction; an eflect which furnishes the most
striking analogy between the motions of light and those of the
waves of water.

The reviewer next complains of his utter ttant of eomprehenr
sion of the diflerence between the colours of mixed plates, and
those of the plates which have been described by Newton. Had
he sufficiently studied the Optics of Newton, he would have seen
that the thickness of a simple plate of water must be only three
fourths as great as that of a plate of air, in order to produce
similar effects: in the colours which I have described, the
thickness of tiie mixed plate was six times as great as that of the
plate of air : the one series of rings expanded^ upon inclining the
plates, the other contracted. These distinctions are plain enough
for any personofonfiTiary comprehension ; and I was not aware
that it was necessary to provide for extraordinary cases.

We are induced to suppose, from the page which immediately

208 REPLY TO THE EDINBURGH REVIEWERS. No. X.

follows, that^ to speak without a metaphor, neither the fancy nor
the comprehension of the reviewer could enable him to distin-
guish a black spot from a white one. I have said, that when
two glasses are brought into the most intimate contact possible,
with the interposition of a certain fluid, the central spot of the
rings of colours is nearly white : it was before known that,
without any such interposition, the central spot, in similar cir-
cumstances, would be nearly black : and the critic sagaciously
pronounces, that these effects are precisely the same. He quotes
from Newton the expression of the ^^ pellucid central spot,"
meaning the spot which reflected no lights and then explains it,
as if it were exactly similar to that which, in my experiment,
reflected nearly all the light that fell on it, and was therefore
white.

That the lines which are quoted in the same page, from my
paper, present, when thus insulated, an appearance of confusion
and of vague reasoning, is perhaps undeniable, and is perfectly
excusable. The reviewer has not understood the paper in its
entire state, and he might be sufficiently secure, that his readers
would never be able to extricate an intelligible sense from an
arbitrary quotation of a few lines, taken out of the middle of a
paragraph of connected reasoning. He misapprehends and
misrepresents completely the whole subject of the explanation ;
he says that its object is to explain the blue colour of the lower
part of the flame of a candle. Nothing was further from my
thoughts than to assign any reason for this blueness : what I
attempted to illustrate, was an original and important observa-
tion inade by Dr. Wollaston, that a portion of the blue flame of
a candle appeared, when viewed though a prism, to be divided
into a number of distinct masses or images. My illustration of
this phenomenon has not the slightest connexion with what the
reviewer calls his solution of the appearance of different colours
in different flames, which he so humbly intreats his readers to
compare with it. I am not therefore obliged to give an opinion
of any kind respecting this pretended explanation of a phe-
nomenon foreign to the subject ; if I were, it would be sufficient
to say, that no such laws could be supposed to operate, upon tiie
principles of mechanical forces, without producing different

No. X. RBPI.Y TO THE EDINTBURGH REVIEWERS. 209

velocities in light of diflferent colours. But the passage fortu-
nately affords me a most convincing proof of the nature of the
source from which this torrent of invective has originated. We
are here told, that the doctrine of the different flexibility of
light is now universally admitted. I have searched into all the
works that I could find in the libraries to which I have had
access, for opinions respecting the nature of light, and, as far as
I have discovered, the different flexibility of light is admitted^
in the absurd and unwarrantable sense in which it is here em-
ployed, by three writers - only. The first is Mr. Henry
Brougham, the second the anonymous author of an article
in the Encyclopadia Britannica, and the third the assailant
whose injurious attacks I am now repelling. From so remark-
able a coincidence, I think myself authorised to conclude, that
these three writers are one and the same. I have before
hinted that Mr. Brougham's doctrines have been sufficiently
confuted, by Professor Prevost of Geneva.* Mr. Prevost has
satisfactorily defended the experiments of Newton from the
imputations of Mr. Brougham ; but in other respects he has
perhaps treated the young theorist with too much lenity.

I have now answered everything that was intended as an
argument, in the articles published in the second number of the
Review. This constitutes, in fact, but a small part of those
articles : they have much less the appearance of the impartial
discussion of a long disputed question in natural philosophy,
than of the buffoonery of a theatrical entertainment, or of the
jests of a pert advocate, endeavouring to place in a ridiculous
light the evidence of his adversary. To answer such an attack
in similar language would be degrading ; to attempt to oppose
it by argument would be futile. I shall refrain, therefore, from
noticing any of the additional scurrilities which have been
copiously intermixed by the same writer with his remarks on
my last paper. I say the same, because I am unwilling to
suppose that this island has produced two persons capable of so
stupidly misunderstanding, and so wilfully misrepresenting.

♦ In the Philosophical Transactions for 1798, vol. hxjcriii., p. 321, in a paper
entitled ' Some Optical Remarks chiefly relatiye to the Reflcxibility of the Rays of
Light.'— ^bto by the Editor.

VOL. L P

210 REPLY TO THE EDINBURGH REVIEWERS. No. X.

But their identity is of no consequence to the discussion^ and it
is unnecessary to inquire for proofs of it. The whole purpose of
the paper inserted in the ninth number of the Review might be
supposed to have been, not to confute the principles which the
writer attacks, but to show that he is incapable of understanding
even the simplest of them.

I have asserted, that two series of undulations^ interfering
with each other at certain relative intervals^ necessarily produce
certain modifications in their joint effects. These terms not
only belong to the same theory, but are parts of the same posi-
tion which I have already illustrated by a &miliar comparison
in these remarks. The author of the critique has sagaciously
observed, that ** they who object to the theory of interference,
have only to turn a page^ and they find the theory of intervals,
and they need but go on a section further^ and the vibrations
and undulations are very much at their service."

This specimen is su£Scient to explain how naturally it must
appear to him "unaccountable," that the process of interference
should produce certain effects, some of which I never supposed
that it could produce, and others which none who rightly under-
stood my theory could ever doubt that it must produce. He
asks, "on .what known principle" can the production of.
coloured fringes from two beams of white light be explained ?
I answer, certainly on no principle that was known before ; but
upon consideration of the law which I have discovered, most
simply and unavoidably.

The reviewer has afforded me, in the next observation, an
opportunity for a triumph as gratifying as any triumph can be
where the enemy is so contemptible. Conscious of inability to
explain the experiment which I have advanced, too ungenerous
to confess that inability, and too idle to repeat the experiment,
he is compelled to advance the supposition that it was incorrect,
and to insinuate that my hand may easily have erred through a
space so narrow as one-thirtieth of an inch. But the truth is,
that my hand was not concerned : the screen was placed on a
table, and moved mechanically forwards with the utmost cau-
tion ; the experiment succeeded in some circumstances where
the breadth of the object was doubled or tripled ; and I assert

No. X. REPLY TO THE EDINBURGH REVIEWERS. 211

that it was as easy to me to estimate an interval of one-thirtieth
of an inch, as an interval a hundred or a thousand times as
great Let him make the experiment, and then deny the result
if he can.

With equal pertinacity of blnndering, he has remarked that
the interference of light, inflected by two contiguous edges,
ought, upon my principles, to produce, not continued fringes,
but only ^' square or rectangular spots of fringe." Was it not
enough to have demonstrated the weakness of his powers with
regard to physical laws ? And was it necessary to induce his
readers to suppose him incapable of going through a little alge-
braical calculation leading to the properties of the hyperbola?
Let right lines be inflected from the edges of a rectangular
object into its shadow, so as to cut off portions from the oppo-
ate lines, exceeding their own length by a given interval, and I
maintain that the intersections will form continued curves, and
that those curves will be hyperbolas : the shape of the fringes
ought n^, therefore, to be that of detached spots, but of hyper-
bolical curves.

It is " a metaphysical absurdity," says the reviewer, to assert
that qualities can ^^ move " in concentric surfaces. I have not
said that the qualities of light ^* move " in concentric surfaces,
but that they ^^ succeed each other" in concentric surfaces;
and in this there is certainly no metaphysical absurdity. Con-
densation and rarefaction are qualities of the air, and it will not
be denied that, in every musical sound, condensation and rare-
fisustion continually succeed each other in concentric surfaces.

Upon my train of argument respecting the nature of light,
the reviewer observes, first, that an analogy is made the ground
of an inference. I answer, that when the analogy is sufficiently
close, it is a most satisfactory ground of physical inference.
Secondly, he says' that a gratuitous assumption is set down as a
necessary truth. I reply, that the assumption is not gratuitous ;
that nobody, except for the sake of argument, will deny, or can
deny it ; should it be denied, it would be perfectly easy to sub-
stantiate it by showing the unavoidable contradictions that
would result from any alternative that could be substituted for
it. The remaining part of the paragraph is as correctly quoted

P 2

212 REPLY TO THE EDINBURGH REVIEWERS. No. X.

as that edition of the Bible was printed, in which the only error
was the omission of the word not in the seventh Commandment :
here the monosyllable 6trf, which completely inverts the sense of
the passage, and which would have entirely destroyed the force
of the criticism, is therefore very prudently omitted.

I have inserted a caution relating to deceptions in the ex-
amination of microscopical objects, not in order to attach any
additional merit to my own explanations, but as a hint natu-
rally arising out of the subject. The same caution might
perhaps have been suggested by the results of some former^
experiments, but the particular appearances that would be pro-
duced by such fallacies could never before have been so
minutely indicated. That the images of very small objects on
the retina may possibly be affected by such causes, is the natural
inference from my principles ; and it is of no consequence to this
position whether the reviewer can or cannot explain them from
his own.

My comparison of a grove of trees pervaded by the wind,
with the particles of a material body, separated, as all modem
philosophers have supposed them to be, by intervals incompa-
rably greater than their diameters, and allowing an inconceiv-
ably rare medium to penetrate with perfect freedom every
interstice, could scarcely have appeared obscure or inapplicable
to any man unblinded by prejudice or unbiassed by male-
volence.

I have already said enough of Newton to show how I vene-
rate his character, as the first of mathematicians and the
greatest of philosophers. Perhaps, however, the mention of
persons whose views are '' still less enlarged " than his own,
may imply in some measure what I never intended, and may
therefore require some little apology, especially as the expres-
sions will bear to be applied to the objections which I am now
endeavouring to refute. It was, indeed, a want of respect to his
illustrious memory to place the superficial and dogmatical
fancies of a writer in the Edinburgh Review in any kind of
comparison with the deep and refined imaginations of a
Newton. Instead of ^^ still less enlarged " and enlightened, I

No. X. REPLY TO THE EDINBURGH REVIEWERS. 213

ought to have called them narrow and confused, selfish and
interested, puerile and ostentatious.

The indignation of the same violent and arbitrary tribunal
has been excited and called forth by a declaration from a man
whose approbation is so much the more valuable as it is always
bestowed with the most cautious regard to experimental accu-
racy and logical induction. Dr. WoUaston has observed that
** the theory of Huygens affords, as has lately been shown by
Dr. Young, a simple explanation of several phenomena not yet
''accounted for by any other hypothesis.^^ His own observations
on Iceland crystal accord throughout, he says, with this hypo-
thesis of Huygens ; the measures that he has taken ^* correspond
more nearly than could well happen to a false theory." But
he contents himself with stating these undeniable facts ; and
the reviewer goes too far when he asserts that Dr. Wollaston
** has adopted the wild optical theory of vibrations." If Dr.
Wollaston had then been acquainted with the experiments and
calculations which I have made since that time, it is possible
that his assent might have been much more complete and un-
reserved. But while I allow to his experiments all the merit
that a clear conception, a vigorous mind, a steady hand, and an
accurate eye can bestow on them, it must not be said by the
Edinburgh Reviewers that his experiments have given the theory
** a plausibility which it did not before possess." As experi-
ments, they have all the merit of originality, for the autlior,
when he made them, was unacquainted with those of Huygens ;
and his most ingenious invention of %n instrument for measuring
refractive powers enabled him, with great ease, to improve and
extend them. But the experiments of Huygens were elaborate
and diversified, and every argument that can be inferred from
Dr. Wollaston's observations had been anticipated by this great
philosopher upon the ground of his own. It is true that our
reviewer was not likely to have troubled himself with Huygens's
treatise of light ; his business is to censure others, and not to
inform himself ; it was easier for him to call this doctrine ^* a
clumsy hypothesis," and ^^ a dull invention," than to investigate
its truth, and to admire its elegance. He has indeed made
distinctions between Huygens's doctrine and mine, which serve

214 REPLY TO THE EDINBURGH REVIEWERS. No. X.

but to prove still more strongly that he was acquainted with
neither ; I shall only answer his epithets by a quotation from a
writer, whose merits the testimony of Newton is well known to
have raised far above the ordinary rank of his contemporaries.
In Cotes's lectures on hydrostadcs, where he is speaking of the
velocity of light, and takes occasion to mention the hypothesis
of Huygens, the following passage occurs : — ^^ When we take a
particular view of the several parts of this hypothesis, it appears
to be so very ingeniously contrived, and so handsomely put
together, that one can hardly forbear to wish it were true." -
The evidence was at that time imperfect, but the symmetry was
complete.

The reviewer has thought proper to unite, in several instances,
with his invectives against me, some ridicule of the objects of
the Boyal Institution of Great Britain ; an institution in which
its Managers have studied to concentrate all that is useful in
science, or elegant in literature. This connexion appears to
him to add so much weight to his arguments, that be has chosen,
without further provocation, to insinuate its existence more than
a year after it had been dissolved. I accepted the appointment
of Professor of Natural Philosophy in the Royal Institution as
an occupation which would fill up agreeably and advantageously
such leisure hours as a young practitioner of physic must expect
to be left free from professional cares. I was led to hope that
I should be able to impress an audience formed of the most
respectable inhabitants of the metropolis, with such a partiality
as the moderately well-informed are inclined to entertain, for
those who appear to know even a little more than them-
selves of matters of science; that I might be of use to
the public in disseminating the true prindples of natural phi-
losophy ; and that I might in future be remunerated by the
enjoyment of a more extensive confidence in my professional
abilities than could have been granted to a person less generally
known. While I held the situation, I wished to make my
lectures as intelligible as the nature of the subjects permitted ;
but I must confess that it was not my ambition to render them
a substitute for those of any superficial experimenter, that was
in the habit of delivering courses of natural philosophy for the

No. X. REPLY TO THE EDINBURGH REVIEWERS. 215

amusement of boarding schools. Whatever may have been the
imperfections of my lectures, it cannot be asserted, except per-
haps in the Edinburgh Review, that they were fit for audiences
of ladies of fashion only. After fulfilling, for two years, the
duties of the Professorship, I found them so incompatible with
the pursuits of a practical physician, that, in compliance with the
advice of my friends, I gave notice of my wish to resign the
office. I think it, however, just to the Institution, to the public,
and to myself, that the result of my labours, throughout the
• whole extent of natural philosophy and the mechanical arts,
should be rendered of some permanent utility; and I have since
collected such a mass of further references to works of all ages
and of all nations, accompanied by many notes and extracts
from them, that it will henceforwards be easy for every student
and every author to know at once what has been done, and what
remains to be done, in the subject of his particular researches ;
and to what books he must apply for the best information ;
where further information is required, and can be obtained.
Considering how widely this information is at present scattered,
I trust that I shall have rendered a service of some importance
to every department of the sciences, and I am now on the point
of preparing my book for immediate publication. With this
work my pursuit of general science will terminate: henceforwards
I have resolved to confine my studies and my pen to medical
subjects only. For the talents which God has not given me, I
am not responsible, but those which I possess, I have hitherto
cultivated and employed as diligently as my opportunities have
allowed me to do ; and I shall continue to apply them with
assiduity, and in tranquillity, to that profession which has con-
stantly been the ultimate object of all my labours.*

Welbeck Street^ 30th Not. 1804.

* Of the preceding most masterly Reply, which was pahlished hi .the form of
a pamphlet, it was stated by its author, that one copy only teat sold: it consequently
produced no elTi>ct in vindicating his scientific character, or in turning the current
of public opinion m fiiTour of his theory. — Note by the Editcr.

216 HARMONIC SLIDERS. No. XT.

No. XI.

AN ACCOUNT OF DR. YOUNG'S

HARMONIC SLIDERS.

From the Journals of the Royal Institatton of Great Britain for 1802, toI. i., p. 261.

The combinatioii of undulations, howeyer cautiously the
world may adopt its application to the explanation of optical
phenomena, is of acknowledged utility in illustrating the phe-
nomena of musical consonances and dissonances, and of undeni-
able importance in accoimting for many of the phenomena of
the tides. Each tide is an undulation on a large scale ; and,
supposing the general form of the ocean, in consequence of the
attraction of a distant body, to coincide with that of an oblong
spheroid, as it is found by calculation to do, the section of the
surface of each tide, if conceived to be imbent from the circular
form and extended on a plane, would form the harmonic curve.
(Young's Syllabus IV. 151. 155.) It is remarkable that the
motions of the particles of the mr in sound, have been generally
supposed in theory to correspond with the ordinates of this same
curve, and that there is also experimental reason to believe, that
the purest and most homogeneous sounds do in fiict agree very
nearly with the law of this curve. It is therefore by far the
most natural as well as the most convenient to be assumed, as
representing the state of an undulation in general ; and the
name of these harmonic sliders is very properly deduced from
the harmonic curve.

By means of this instrument, the process of nature, in the
combinations of motion which take place in various cases of the

No. XL

HARMONIC SLIDERS.

217

janction of undulations, is rendered visible and intelligible,
with great ease, in the most complicated cases. It is unneces-
sary to explain here, how accurately both the situations and
motions of the particles of air, in sound, may be represented by
the ordinates of the curve at different points ; it is sufficient to
consider them as merely indicating the height of the water con-
stituting a tide, or a wave of any kind, which exists at once in
its whole extent, and of which each point passes also in succes-
sion through any given place of observation. We have then to
examine what will be the effect of two tides, produced by diffe-
rent causes, when united. In order to represent this effect, we
must add to the elevations or depressions in consequence of the
first tide, the elevations or depressions in consequence of the
second, and subtract them when they counteract the effect of
the first : or we may add the whole height of the second above
any given point or line, and then subtract, from all the sums,
the distance of the point assumed below the medium.

To do this mechanically is the object of the harmonic sliders.
The surface of the first tide is represented by the curvilinear
termination of a single board. The second tide is also repre-
sented by the termination of another surface ; but, in order
that the height at each point may be added to the height of the
first tide, the surface is cut transversely into a number of sepa-

inm

218

HARMONIC SLIDERS.

No. XI.

rate pieces or sliders, which are confined within
a groove or frame, and tightened by a screw.
Their lower ends are situated originally in a
right line; but, by loosening the screw and
moying the sliders, they may be made to assume
any otber form : thus they may be applied to the
surface representing the first tide ; and, if the
similar parts of each correspond, the combination
will represent a tide of twice the magnitude of
the simple tides. The more the corresponding

parts are separated, the weaker will
be the joint effect; and, when they
are furthest removed, the whole tides,
if equal, will be annihilated. Thus,
when the general tide of the ocean
arrives by two different channels at
the same port, at such intervals of
time that the high water of one would
happen at the same instant with the
low water of the other, the whole
effect is destroyed, except so far as
the partial tides differ in magnitude.
The principle being once understood,
it may easily be applied to a multi-
plicity of cases : for instance, where
the undulations differ in their dimen-
sions with regard to extent Thus,
the series of sliders being extended to
three or four alternations, the effect
of combining undulations in the ratio of 2 to 1, of 3 to 1, of 2
to 3, of 3 to 4, may be ascertained, by making a fixed surface,

No. XI. HARMONIC SLIDERS. 219

terminatiDg in a series of curres, that bear to those of the
sliding surface the ratio required : and, by making them differ
but slightly, the phenomenon of the beating of an imperfect
unison in music may be imitated, where the joint undulation
becomes alternately redoubled and evanescent. In the last
figure here inserted, the proportion is that of 17 to 18, and
the curvilinear outline represents the progress of the joint
sound from the greatest degree of intensity to the least, and
a little beyond it.

There is no conception more difficnlt, and few more important^ than that of the
transmission of wares and the effects of their interference with each other. The
contriv^ance explained in the text exhibits those effects in wares of water and air, and
even of lights assuming that the vibrations which produce them are in the direction of
the ware's motion. Other contriyances, somewhat analogous to it, have been made by
the present Astronomer Royal and Mr. Wheatstone, whidi exhibit the movements and
in^rferences of waves upon any hypothesis of vibration, which the phenomena of light,'
more especially those of its polarization and transmission through crystallized m<^a,
may render necessary. — Note by the Editor.

220 REVIEW OF LAPLACE No. XIL

No. XIL

REVIEW OF LAPLACE'S MEMOIR " SUR LA LOI DE

LA RteACTION EXTRAORDINAIRE DANS
LES CRISTAUX DIAPHANES.

Luiila premi^ Chsse de Vlnstitta, dans sa stance du 30 Jano, 1809.

Journal de Physique, Janv. 1809."

From the Quarterly Reriew for Nov., 1809, vol. ii., p. 337.

The few who have arrived, in the different departments of
learning and science, at such a degree of eminence as to be
almost " without a second, and without a judge," have not only
the advantage of being able to propagate real knowledge with
uncontrolled authority, but also the less enviable prerogative of
^ving to error the semblance of truth, whenever accidental
haste or inattention may have led them into those inaccuracies,
from which no human intelligence can be wholly exempt* It
is necessary, therefore, for a critic, who undertakes to make a
faithful report of the progressive advancement of the sciences,
to watch with redoubled care the steps of those, who are the most
likely fo lead others astray, if they happen to follow a wrong
path : and while . the ultimate decision always remains with
the public, as with a jury, the judge is bound to state, as fully

* Sir David Brewster, in a letter to Dr. Young, dated January, 1818, mentions a
very striking example of the implicit deference paid to the authority of this great

philosopher by one of the most distinguished of his followers . — " did not

scruple to declare, when I saw him in Edinbui^ last summer, that any experimental
laws of double refraction which I might have discovered must be erroneous, unless
they agreed with those given by Laplace ; and that I ought to compare them with his
before I published them« My reply was, that iis my laws were deduced from experi-
ment, it was rather Laplace's affair to see that his tneoretical ones agreed with mine."
It is proper to add, however, that the confidence formerly reposed in the correctness
of Sir Isaac Newton's optical theories and experiments was almost equally uncon-
ditional, and still more injurious to the progress of the science. — I^'^ote by the Editor,

No. XII. ON EXTRAORDINARY REFRACTION. 221

and impartially as possible, the whole mass of the evidence be-
fore him ; not fearing to adduce all such reasoning as can tend
to the support of the weaker side, when there is any danger of
its being oppressed by the authority and respectability of the
stronger.

These reflections have been suggested to us by an essay, for
which we are indebted to a very celebrated continental mathe-
matician— a man of whom we willingly say, with Heraclitus,
bIs efA,oi Mpomros rqia-fjuvqioiy but on whose works we thought it
necessary, on a former occasion, to make some free remarks.
We then objected to him a want of address or of perseverance
. in the management of his calculations, presuming that the
principles, on which they were founded, were capable of being
applied, with much greater precision, to the phenomena in
question : our suspicion has since that time been justified by an
essay of an anonymous author in this country, who, without any
great parade of calculation, appears to have afforded us a
general and complete solution of the problem, which Mr.
Laplace had examined in particular cases only. We have now
to accuse him of an offence of a different complexion — that is,
the hasty adoption of a general law, without sufficient evidence ;
and an inversion of the method of induction equally unwarrant-
able with any of the paralo^sms of the Aristotelian school.
We complain also, on national grounds, of an unjustifiable
want of candour, in not allotting to the observations of different ,
authors their proper share of originality. What has a man of
science to expect from the public, as a reward for his labours,
but the satisfaction of having it acknowledged, that he has done
something of importance towards extending the sphere of intel-
lectual acquirements ? And who is so capable of directing
public opinion, on subjects respecting which very few will form
an opinion of their own, as a philosopher like Mr. Laplace,
whose works are sure of commanding universal attention, and
almost sure of inforcing implicit belief? The Huygenian law,
of the extraordinary refraction of Iceland crystal, has lately, he
says, been confirmed by ^* Mr. Malus." We know nothing of
the extent of Mr. Malus's researches, but we know that Mr.
Laplace sometimes reads the Philosophical Transactions, and

222 REVIEW OF LAPLACE No. XII.

he either must have seen, or ought to have seen, a paper pub-
lished in them by Dr. WoUaston, as long ago as the year 1802,
which completely establishes the truth of the law in question,
on the most unexceptionable evidence, and by the most accu-
rate experiments. But it seems to be one of the attributes of
a great nation to disregard, on all convenient occasions, the
rights of its neighbours ; we might have made the same remark
in our former criticism* on Mr. Laplace, but there is so little
novelty in the circumstance, that it is unnecessary to dwell any
further on it at present.

It has long been known to opticians, that many crystals, of
different kinds, have the remarkable property of making sub-
stances, viewed through them in certain directions, appear
double ; the effect of their refraction being almost the same as
if a rarer and a denser medium existed together in the same
space ; some part of the light passing through them being
refracted in the same manner as if the denser medium alone
were present, and some as if the rarer only were concerned.
The reason of this double refraction is wholly unknown, nor has
any attempt been hitherto made to discover it. The crystals
of carbonate of lime, in their primitive form, have also a further
peculiarity : they afford a double image, even when the object
is viewed perpendicularly through the two opposite and parallel
sides of a crystal ; an effect which could never arise from the
combination of any two mediums acting in the ordinary manner;
in fact, one of the images only is seen according to tlie laws of
ordinary refraction, and the place of the other is determined by
a law, which is the subject of the present paper. This law was
experimentally demonstrated, and very elegantly applied to the
phenomena by its first discoverer Huygens ; but having been
suggested to him by an hypothesis which was not universally
adopted, it was rejected or neglected by his antagonists, without
any accurate investigation ; and the testimony of the greatest
philosophers of that age, or of any age, having been opposed to
it, it remained forgotten for almost a century. Nor is this the
only instance in which, even within the limits of the physical

♦ Quarterly Review for February, 1809, vol. i., p. 107. Theorie de V Action Ca-
pHlairef par M. Laplace. — Note by the Editor.

No. XII. ON EXTRAORDINARY RBFRACTION. 223

sciences, high authority has been suffered to prevml against
unassuming truth. Mr. Haiiy is the first of the later observers,
who remarked that the true law of extraordinary refraction was
much nearer to the Huygenian law, than to that which had
been substituted for it by Newton. Some time afterwards. Dr.
Wollaston had made a number of very accurate experiments,
with an apparatus singularly well calculated to examine the
phenomena ; but he could find no general principle to connect
them, until the work of Huygens was pointed out to him : he
was then enabled, by means of the Huygenian law, to reduce
his experiments to a comparison with each other ; and in com-
municating them to the Royal Society, he remarked that ** the
oblique refraction, when considered alone, seemed to be nearly
as well explained as any other optical phenomena." Here the
matter rested, until Mr. Mains made the experiments whidi
have led to the present paper.

'^Mr. Mains has lately compared the Huygenian law," says Mr.
Laplace, '' with a very great number of experiments, made with ex-
treme precision, on the natural and artificial sur&ces of the crystal,
and has fomid that the law agrees exactly with his experiments, so
that it must be placed among the most certain, as well as amcxig the
most striking results of physical observation. Huygens bad deduced
it, in a very ingenious manner, from his hypothesis respecting the
propagation of light, which he imagined to consist in the undulations
of an ethereal fluid. This great geometrician supposed the velocity of
the undulations in the ordinary transparent mediums to be smaller than
in a vacuum, and to be equal in every direction : in the Iceland crystal,
he imagined two distinct species of undulations ; the velocity of the one
being the same in a]l directions, in the other variable, and represented
by the radii of an elliptic spheroid, having the point of incidence for its
centre, and its axis being parallel to that of the crystal ; that is, to the
right line which joins the two obtuse solid angles of the rhomboid.
Huygens does not assign any cause for this variety of undulations ; and
the singular phenomena, exhibited by the light which passes from one
portion of the crystal into another, are inexpUoabk upon this hypothesis.
This circumstance, together with the great difficulties presented by the
undulatory theory of light in general, has induced the greater number of
natural philosophers to reject the law of refraction founded on the Huy-
genian system. But since experiments have demonstrated the accuracy
of this remarkable law, it must be entirely separated from the hypothesis

224 REVIEW OP LAPLACE No. XII-

which originally led to its discovery. It would be extremely interesting
to reduce it, as Newton has reduced the law of ordinary refraction, to
the action of attractive or repubi ve forces, of which the effects are only
sensible at insensible distances ; it is indeed very probable that it de-
pends on such an action, as / Jiave satisfied no/self by the following
considerations.

** It is well known that the principle of the least possible action takes
place, in general^ with respect to the motion of a material point actuated
by forces of this kind. In applying this principle to the motion of
light, we may omit the consideration of the minute curve which It
describes, in its passage firom a vacuum into the ti*ansparent medium, and
suppose its velocity constant, when it has anived at a sensible depth.
The principle of the least action is then reduced to the passage of the
light from a point without to a point within the crystal, in such a man-
ner, that if we add the product of the right line described without into
its primitive velocity, to the product of the right line described within
the cr}'stal, into its corresponding velocity, the sum may be a minimum.
This principle always gives the velocity of light in a transparent
medium, when the law of refraction is known, and on the other hand,
gives this law, when we know the velocity. But there is a condition^
which becomes necessary in the case of extraordinary refraction, which
is, that the velocity of the ray of light in the medium must be inde-
pendent of the manner in which it has entered it, and must be determined
only by its situation with respect to the axis of the crystal, that is, by
the angle which the ray forms with a line parallel to the axis." ^' I have
found that the law of extraordinary refraction, laid down by Huygens,
satisfies this condition, and agrees at the same time with the principle
of the least action ; so that there is no reason to doubt that it is derived
from the operation of attractive and repulsive forces, of which the action
is only sensible at insensible distances. The expression of the velocity
to which my analysis has conducted me affords a valuable datum for de-
termining the nature of these forces ; this velocity being measured by a
fraction, of which the numerator is unity, and the denominator the radius
of the spheroid which is described by the light, the velocity in a vacuum
being considered as unity. The velocity of the ordinary ray, in the
crystal, is equal to unify divided by the principal axis of the spheroid,
and is consequently greater than that of the extraordinary ray : the dif-
ference of the squares of the two velocities being proportional to the
square of the sine of the angle which the latter ray makes with the axis;
and this difference represents that of the actions of the crystal on the two
kinds of rays. According to Huygens, the velocity of the extraordinary
ray, in the crystal, is simply expressed by the radius of the spheroid ;
consequently, his hypothesis dcjesno^ agree with the principle of the least

No. XII. ON EXTRAORDINARY REFRACTION. 225

action ; but U is remarkable that it agrees with the principle of Fermat,
which is, that light passes, from a given point without the crystal, to a
given point within it, in the least possible time ; for it is easy to see that
this principle coincides with that of the least action, if we invert the ex-
pression of the velocity. Thus both of these principles lead us to the
law of extraordinaiy refraction discovered by Huygens, provided that,
for Fermat's principle, we take, with Huygens, the radius of the spheroid
as representing the velocity, and, for the principle of the least action,
this radius be made to represent the time employed by the light in
passing through a given space." ** If the diameters of the spheroid are
equal, the figure becomes a sphere, and the refraction resembles ordi*
nary refraction : so that in these phenomena, nature, in proceeding from
what is simple to that which is more complex, takes the form of the
eliq>sis next to that of the circle, as in the motions and the figures of the
heavenly bodies."

Mr. Laplace then gives an account of the controversy be-
tween I>escarte8 and Fermat vespecting the velocity of lights
and concludes his abstract with the following remarks : —

" Mauperiuis, convinced by the arguments of Newton, of the truth of
the suppositions of Descartes, found that the function which is a mini-
mum in the motion of light, is not, as Fermat supposed, the sum of the
quotients, but that of the products of the spaces described, by the corre-
sponding velocities. This result, extended to the fluent of the product of
the fldxion of the space into the velocity, where the motion is variable,
suggested to Euler the principle of the least action, which Mr. de La-
grange afterwards deduced from the primitive laws of motion. The use
which I have now made of this principle, first in order to discover
whether or no the law of extraordinary refraction laid down by Huygens
depends on attractive or repulsive forces^ and thus to raise it into the
rank of those laws which are mathematically accurate ; and, secondly, to
deduce mutually /rom aooA other the laws of refiraction and of the velo-
city of light in transparent mediums, appears to me to be worthy the
attention both of natural philosophers and of mathematicians."

Such is Mr. Laplace's own account of the investigations, into
which he has been led by Mr. Malus's experiments ; and we
shall give him full credit for having demonstrated, in the origi-
nal memoir, everything which he has here asserted.

The principle of Fermat^ although it was assumed by that
mathematician on hypothetical, or even imaginary grounds, is
in fict a fundamental law with respect to undulatory motion,

VOL, I. Q

226 REVIEW OF LAPLACE No. XII.

and is explicitly the basis of every determination in the Huyge-.
nian theory. The motion of every undulation must necessarily
be in a direction perpendicular to its surface ; and this condition
universally includes the law, that the time occupied in its propa-
gation between two given points must be a minimum ; or rather,
more generally, the effects of the collateral undulations must
always conspire the most completely, where the time occupied
in their arrival at two neighbouring points in the direction of
the undulations is equal, which is necessarily a condition of a
minimum. Mr. Laplace seems to be unacquainted with this
most essential principle of one of the two theories which he com-
pares ; for he says, that " it is remarkable," that the Huygenian
law of extraordinary refraction agrees with the principle of
Fermat ; which he would scarcely have observed, if he had
been aware that the lawVas an immediate consequence of the
principle.

In the second place, the law of the least action is precisely
the same with the law of Format, excepting the difference of
the interpretation of the symbols. In the law of Fermat, the
space is divided by the velocity, to find the time : in the prin-
ciple of the least action, instead of dividing by the velocity,
estimated in the Huygenian manner, we multiply by its reci-
procal, to which we give the name of velocity, upon a different
supposition ; but the mathematical conditions of the two deter-
minations are always necessarily identical ; and the law of the
least ax^on must always be applicable to the motions of light,
as determined by the Huygenian theory, supposing only the
prqx)rtion of the velocities to be simply inverted.

Mr. Laplace has therefore given himself much trouble to
prove that coincidence in a particular case, which must neces-
sarily be true in all possible cases. In a person who seems to
delight in long calculations, this waste of labour may easily be
excused. A Turk laughs at an Englishman for walking up
and down & room when he could sit still ; but Mr. Laplace
may walk about, and even dance, as much as he pleases, in the
flowery regions of algebra, without exciting our smiles, pro-
vided that he does no worse than return to the spot from which
he set out : but when, in the rapidity of his motion, his bead

No. XII. ON EXTRAORDINARY REFRACTION. ^ 227

begins to tnrn, it is time for the spectators to think of their
own safety.

Not satisfied with this important discovery, that the extraor-
dinary refraction is consistent with the principle of the least
action, he proceeds to infer, that ** there is no reason to doubt"
that this refraction depends on the immediate operation of at-
tractive and repulsive forces. With as much reason might it
be asserted, that because the sound produced by a submarine
explosion is, in all probability, regularly refracted at its pas-
sage into the air, the sound must be attracted by the air, or
repelled by the water.

Nor would such a conclusion be by any means equally un-
warrantable with that which Mr. Laplace has drawn ; a simple
attractive or repulsive force, acting on a projected corpuscle in
a direction perpendicular to the surface of the water, would be
sufficient to explain such a refraction ; but Mr. Laplace has not
attempted to describe tiie kind of force which would be capable
of produdng the efiects in question with respect to light. He
contents himself with saying, that the velocity within the crystal
must depend only on the situation of the ray of light with re^
spect to the axis, and that this is a necessary ** condition " of the
refraction. The musician celebrated by the epigrammatist,
thought it '* a necessary condition " that a string and its octave
should vibrate together, because the materials of both strings
were taken from the same sheep ; and he applauded himself on
the sufficiency of his explanation with about as much justice as
our author. In feet, the deduction of this " condition," from
any assignable laws of attraction, is the only difficulty in the
question ; and tins is the " dark passage " which the " commen-
tators " have shunned.

But the insertion of such a condition seems even to exempt
the problem from being directiy amenable to the law of the
least action. We apprehend that this law is only demonstrable,
from mechanical principles, in cases of the operation pf attrao
tive forces directed to a certain point, whether fixed or variable,
or acting in parallel lines, so tiiat the velocity, between the
same parallel or concentric surfaces, may be always die same,
whatever its direction may be ; it cannot therefore be applied,

Q 2

228 REVIEW OF LAPLACE No. XII.

without the most unjustifiable violence, to cases in which the
velocity deviates most essentially from this description.

When we consider that, upon such grounds as these, a mathe-
matician of the first celebrity professes to have elevated tiie
principle of Huygens " to the dignity of a rigorous law," we
cannot help being reminded of his Egyptian predecessor, who
had ^' spent forty years in unwearied attention to the motions
and appearances of the celestial bodies, and had drawn out his
soul in endless calculations/' in order to be persuaded at last,
that '^ the sun had listened to his dictates, and had passed from
tropic to tropic by his direction; that the clouds, at his call,
had poured their waters, and the Nile overflowed at his com-
mand."

Mr. Laplace very justly remarks, that nature in these phe-
nomena, as well as in those of astronomy, has tcAen the form of
the ellipsis next to that of the drcle. But in astronomy, we
know why nature *' has taken the form of the ellipsis," since the
elliptic form depends on the simple law of the variation of the
force of gravitation : in these phenomena of extraordinary re-
fraction, on the contrary, no satisfactory attempt has been made
to obtain any such simplification. A solution of this difficulty
might, however, be deduced, upon the Huygenian principles,
fi^m the simplest posable supposition, that of a medium more
easily compressible in one direction than in any direction per-
pendicular to it, as if it consisted of an infinite number of
parallel plates connected by a substance somewhat less elastic
Such a structure of the elementary atoms of the crystal may be
understood by comparing them to a block of wood or of mica.
Mr. Cbladni found that the mere obliquity of tiie fibres of a
rod of Scotch fir reduced the velocity, with which it transmitted
sound, in tiie proportion of 4 to 5. Jt is, therefore, obvious
that a block of such wood must transmit every impulse in sphe-
roidal, that is oval, undulations : and it may also be demon-
strated, as we shall show at the conclusion of this article, that
the spheroid will be truly elliptical, when the body consists either
of plane and parallel strata, or of equidistant fibres, supposing
both to be extremely thin, and to be connected by a less highly
elastic substance ; the spheroid being in the former case oblate

No. XII. ON EXTRAORDINABY REFRACnON. 229

and in the latter oblong. It may also be proved, that while a
complete spheroidal undulation is everywhere propagated by
the motion of the particles in a direction perpendicular to its
sur&ce, a detached portion, like a beam of light or of sound,
will proceed obliquely, in the rectilinear direction of a diameter.

It has often been asserted, and Mr. Laplace repeats the
charge, that the phenomena, which are obsenrable upon the
transmission of light through a second portion of the crystal,
are ^inexplicable'* upon the Huygenian theory. It is true
that they have not yet been explained ; but what right has Mr.
Laplace to suppose, tiiat thb theory has yet attained to its
utmost degree of perfection in every other respect, under all
the obloquy with which it has been loaded ? Had the more
prevailing system afforded anything like an explanation of the
perfect ellipticity of the undulations, it would have been oppro-
briously objected to the Huygenian system, that it was incapable
of accounting for this circumstance ; and the reproach would
have remained hitherto unanswered. It may, however, be
observed, that an undulation, which has passed through a crystal,
is not, as some authors have taken for granted, alike on all sides ;
nor can it be proved, that the difference of its curvature, in its
diflerent sections, may not be sufficient to produce all the ob-
servable modifications of its subsequent subdivision.

lliese considerations, we trust, will amply justify us in
giving it as our opinion, that Mr. Laplace has, in this memoir,
been not a littie superficial in his arguments, and extremely
precipitate in his conclusions. We must again lament the
serious evils which are likely to arise, and which in this case
have actually arisen, firom that unfortunate ^* rage for abstrac-
tion,** which we have already noticed as too universally preva-
lent. ** To avoid such paralogisms and such whims," said the
late Professor Robison on a similar occasion, *' we are convbced
that it is prudent to deviate as little as possible, in our discus-
sions, from THB OEOMETRICAL METHOD."

The proposition, which we left to be demonstrated, was this :
<< an impulse is propagated through every perpendicular section
of a lamellar elastic substance in the form of an elliptic undu-
lation." 'ITie want of figures will, perhaps, render the demor-

230 REVIEW OF LAPLACE Ho. Xll.

stration tomewhat obficure, but the deficiency may easily be
supplied by those who think it worth their while to consider the
subject attentively.*

<< When a particle of the elastic mediom is displaced in an oblique
direction, the resistance, produced by the compression, is the joint result
of the forces arising from the elasticity in the direction of the laminae^
and in a transverse direction : and if the elasticities in these two direo-
tions were equal, the joint result would remain proportional to the
displacement of the particle, being expressed, as well in magnitude as
in direction, by the diagonal of the parallelogram, of which the sides
measure the relative displacements, reduced to their proper directions^
and express the forces which are proportional to them. But when
the elasticity is less in one direction than in the other, the cor-
responding side of the parallelogram expressing the forces must be
diminished, in the ratio which we shall tall that of 1 to m ; and the
diagonal of the parallelogram will no longer coincide in direction with
the line of actual displacement, so that the particle displaced will also
produce a lateral pressure on the neighbouring particle of the medium,
and will itself be urged by a lateral force. This force will, however,
have no effect in promoting the direct propagation of the undulation,
being probably employed in gradually changing the direction of the
actual motions of the successive particles ; and the only efficient force
of elasticity will be that which acts in the direction in which the undu-
lation is advancing, and which is expressed by the portion of the line
of displacement, cut off by a perpendicular felling on it from the end
of the diagonal of the parallelogram of forces ; and the comparative
elasticity will be measured by this portion, divided by the whole line
of displacement. Galling the tangent of the angle formed by the line
of displacement with the line of greatest elasticity t, the radius being
1, the force in this line being also 1, the transverse force will be
expressed by m t, the line of displacement by

V (1 +« 0 its diminution by ^r+Vo* ^^ ^^^^^ed portion, which
measures the force, by ^ (1 + « 0 - ^jn^uy ^^ *^® elasticity, in the
given direction, by -p~^. Hence it follows, that the velocity of an
impulse, moving in that du^tion, will be expressed by V ^-^^-^,

♦ The principle involved in this demonBtration fornifl a capital step in the undn-
latory theory. See Dr. Whewell'a * History of the Inductive Science?.' vol. ii.,
^. in.-^Noie by the Editor.

No. XII. ON EXTRAORDINABY BEFBACnON. 231

"It is next to be proved, that the velocity of an elliptical nndn-
lation, increasing so as to remain always similar, by means of an
impulse propagated always in a direction perpendlcolar to the cir-
comferenoe, is such as woold take place in a medimn thus consti-
tuted. It is obvious that the increment of each of the diameters of
the increasii^ figure must be proportional to the whole diameter;
and this increment, reduced to a direction perpendicular to the curve,
will be {Moportional to the perpendicular &lling on the conji^te dia-
met^, whidi will measure the velocity. We are llierefore to find
the expression for this perpendicular, when it forms an angle with the
greater axis, of which the tangent is t Let the greater semi-axis
be ly and the smaUer n : then the tangent of the angle, formed with

tihe greater axis by lihe conjugate diameter, being -j ; the tangent of
the angle subtoided by the corresponding ordinate of the dream-
scribing circle is found -r^ and the semi-diameter itself equal to
unity, reduced in the ratio of the secants of these angles, that is, to
n V j-T t: ; but, by the known property of the ellipsis, the per-
pendicular required is equal to the product of the semi-axes divided by
this semi-diameter, that is, to V-j-xTT" • ^^ ^^®» therefore, only to

make -n ns:my and the velocity in the given medium will always be
such as is required for the propagation of an undulation, preserving the
form of similar and concentric spheroids, of which the given ellipsis
represents any principal section.

'* If the whole of the undulation were of equal force, this reason-
ing would be sufficient for determining its motion ; but when one
part of it is stronger than another, the superiority of pressure and
motion will obviously be propagated in the direction of the actual
resistance produced by the displacement of the particles, since it is
this resistance which carries on die pressure, and consequently pro-
pagates the motion. It is very remarkable, that the direction of
the resistance will be found, on the supposition which we have
advanced respecting the constitution of the medium, to coincide
everywhere with the diameters of the ellipsis, when the displace-
ment is perpendicular to the sur&ce. For it is proved by authors
on conic sections, that the subnormal of the ellipsis is to the
absciss, as the square of the lesser axis is to the square of the
greater, that is, in this case, as nntol, orasmtol; but if we
divide the ordinate in the same ratio of m to 1, and join the point
of division with the extremity of the subnormal, this line, which

232 REVIEW OF LAPLACE No. XII.

wiU evidently be parallel to the diameter, will express, as we have
already seen, the directioo of the force, when the normal re-
presents that of the displacement An immediate displacement in
the direction of any diameter, making an angle with the axis of

which the tangent is ^, would give a velocity of V'yXTT' ^^^ ^

increment of the diameter would require a velociljr of V — TTTf which

does not vary in the same proportion. It must, however, be re-
membered, that the rectilinear direction of the beam is not supposed
to depend on this circumstance alone : Hnygens considers each point of
the sur&ce of the crystal, on which a beam of light impinges, as the
centre of a new undulation, which spreads, in some measure, in every
direction, but produces no perceptible effect, except where it is sup-
ported by, and co-operates with, the neighbouring undulations ; that is,
in the surface which is a common tangent of the collateral tmdulations ;
but if this principle were applied without the assistance of the obliquity
of force, which we have deduced from the supposition of a stratified
medium, it would lead us to expect that the elementary impulses, being
propagated in a curvilinear trajectory, might be intercepted by an object
not situated in the rectilinear path of the beam ; a conclusion which is
not warranted by experiment."

It is not probable that any other suppofiition respecting the
constitution of the medium, in the Huygenian theory, could
afford a result so strikingly coincident with the phenomena of
extraordinary refraction; and the most decided advocates of
the projectile system must allow, that there is scarcely a
chance, especially after Mr. Laplace's fruitless researches, of
its being capable of an application by any means comparable to
this for predfflon and simplicity. But it must be remembered,
that we have been considering a single class of phenomena
only ; the two rival theories must be viewed in a multiplicity
of various lights, before a fair estimate can be candidly formed
of their comparative merits ; and we are not arguing for a
decision in favour of either, but for a temperate suspension of
judgment, until more complete and more satisfactory evidence
can be obtained.*

* We must do Mr. Laplace the justice to obsei-ve. that since this article wa»
written, he has published, in Uie Memoirs of Arcaeil, another pnper on this subject,
in which the name of Dr. WollaDton is mentioned with due respect. Tlie same
Tohunc contains also an account of some highly intei eating and important experimentii

No. XII. ON EXTRAORDINART REFRACTION. 233

of Mr. Mains, on the apparent polaritj of li^t, aa exhibited by obliqne reflexion,
which present greater dmcnlties to the advocates of the nndalatory theory than any
other facts with whidi we are acqoamted.*— ^oto by Dr. Young.

* It would appear that Dr. Toung upplied to Dr. Wollaston for his permission to
introduce his name in this criticism. The following is his replj : —

" Mj dear Sir,

** I cannot, withoat a most egregious share oftncaioai$e honte^ object to jour
asserting mj claims to origiualitj in the Terification of the Huygenian law : but I
certainly must object to your burying your own claims to an original investigation in
a shroud of anonymous criticism.

** For thoueh Laplace may be answered with most efect m ihi$ country by such a
critique, the ^nis nemMa whidi follows in its suite may be long before it receives
the respect due to your legitimate ofispring.

•' Surely this ought to be published &8t, not by X. T., but by T. T., pro bono
pmblico; and then Laplace mi^t be attacked with at least as good effect with the
same weapoos. It is possible thai Malus may have improved upon my experiments,
and mav nave given Laplaoe results that have greater pretensions to accuracy than
mme : but he is probably not so correct in the angle of the crystal. If he declines
adopting mv measures, and if Mdns suppresses &e real origin of his experiments^
Lim>laoe s silence is more excusable.

"Younever,

Note by the Editor. " W. H. Wollastov."

234 BEVIKW OF THE No. XIIL

No. xm.

REVIEW OF THE

"MliMOIRES DE PHYSIQUE ET DE CHIMIE
DE LA SOClM D'ARCUEIL."

Vols. I. Ain> II.
From the Quarterly Reyiew for May, 1810, toI. iii., p. 462.

These volumes are composed exclusiyely of. the producdons of
a select decad of the most celebrated men of science resident
at Paris, who meet once a fortnight to pass the day together,
in making and discussing philosophical experiments, at the
house of the elder BerthoUet, now a count of the French empire,
situated at Arcueil, in the neighbourhood of a villa which has
lately been purchased by Count Laplace. These two gentle-
men may be considered as the fathers of the society : the other
members are Biot, Gay Lussac, Humboldt, Thenard, Decan-
doUe, Collet Descotils, A. B. BerthoUet, and Malus.

The formation of private associations of this kind seems to be
a natural step in the division of literary labour. In this coun-
try we have had abundant instances both of independent and of
affiliated sodeties, for the cultivation of particular departments
of science, all of which had remained for many years united, as
objects of the attention of the Royal Society alone : and several
of these associations have already been productive of no con-
temptible contributions to the advancement of the several
sciences to which they have been respectively devoted.

The researches of the Society of Arcueil extend to the most
important and interesting of the topics which constitute the
occupation of the first class of the Institute of France : the in-
dividuals who compose it being the most eminent membera of
the Institute in their different departments, they must naturally
have the same facts and opinions to produce and to compare in

No. XnL MEMOIRS OF ABCUEIL. 235

both capadtiee. Indeed a great part of the essays, which are
presented to the public in abstract in the M^moires d* Arcueil,
has been read in a more extended form to the National Insti*
tute ; nor is it likely that any jealousy will be excited in this
celebrated body from the competition : it has always shown a
laudable liberality with respect to the publication of the papers
laid before it ; rightly judging that the paltry consideration of
copyright, and the reservation of the earliest notification of its
discoveries, is wholly unworthy the care of a body devoted to
the cultivation and at the same time to the general dissemina-
tion of sdence.

There is not uncommonly a degree of zeal and emulation at-
tending the pursuits of a private association, which cannot always
be obtained in an equal dcf^ree by any public encouragement
held out to science. Thus the stipends of the academicians of
the Institute, which are sufficient to induce men of small for-
tunes and moderate wishes to devote their attention to science,
are by no means calculated to call the most brilliant powers
into the strongest action ; and a society so constituted is more
likely to do a great deal tolerably, than a little admirably. In
this country we cannot boast of any very high encouragements
directly held out to genius; but th^re is always a prospect,
often indeed delusive, that talents may raise their possessor to
situations of eminence and dignity, in whatever profession they
may be exhibited ; and the remote chance of a high prize seems
to be more likely to produce extraordinary exertions, than a
greater certainty of an inferior one. The advantages which
are derived, in some of our colleges^ from a moderate degree
of success in mathematical and classical pursuits, are some-
what analogous, in their effects, to the encouragements which
have been granted to scientific bodies on the continent, by their
respective governments : but, including all the remote prospects
of promotion, the prizes may on the whole be considered as
much higher; they are, however^ in general adjudged at so
early an age, that their influence as a stimulus to application is
but of short duration.

We do not intend, by this remark, to imply a censure of the
system adopted by our universities in the adjudication of their

236 REVIEW OF THE No. XIIL

honours and rewards ; for it must be remembered, that the ad^
vancement of learning is by no means the principal object of an
academical institution : the diffusion of a respectable share of
instruction in literature and in the sciences, among those classes
which hold the highest situations, and haye the most extensive
influence in the state, is an object of more importance to the
public than the discovery of new truths, or the invention of new
modes of illustrating those which are already established ; and
this object appears to require, for its attainment, a continued
succession of instructors, possessing precisely those qualifications
which are most immediately encouraged by the present system.
We might, perhaps, even venture to assert, that in almost all
departments of learning, the ehmentary doctrines are of far
more practical utility than the more abstruse investigations ;
and that with respect to the general improvement of the talents,
an intense application to a particular branch of study is as often
prejudicial as advantageous. We think that we have observed
numerous instances, both in public life, and in the pursuit of
natural knowledge, in which great scholars and great mathe-
maticians have reasoned less soundly, although more ingeni-
ously, and written less elegantly, although more elaborately,
than others, who, being somewhat more completely in the pos*
session of common sense, at the same time tiiat they had not
neglected those pursuits, which are very properly considered as
essential to the education of a gentleman^ were still far inferior
to them in the refinements of learning or of science.

The two volumes of the Memoirs of the Society of Arcueil
are particularly interesting, as they contain, besides some ori-
ginal articles of high importance, a summary view of the princi-
pal investigations which have, during the last two or three
years, employed the most celebrated of the philosophers of
France. Our attention has also been more irresistibly directed
to them by the manner in which they have been noticed in
a well-known periodical publication, which has acquired no in-
considerable reputation in this country, even with regard to
matters of science.* We are not very ambitious of obtaining
the approbation of those readers, who can have attentively con-

♦ Edinburgh Review for February and May, 1810, vol. xv., pp. 142 and 41b.

No. XIII. MEMOIRS OF ARCUEIL. 237

sidered the articles to which we alluded, without discovering
some of their numerous errors ; yet we have known instances
in which the minds of some well-disposed and candid persons
have been led astray, by the specious and ostentatious perform-
ances of the same school : and we think the present a £Bivourable
opportunity for examining into the validity of its pretensions
to the dictatorial character which it has assumed. The humi-
liating confessions of our national inferiority as mathema-
ticians, which these too liberal critics have lately held forth to
the world, have not escaped the vigilance of our hereditary
rivals on the continent ; a translation of their reflections upon
this subject has been distinguished, in an unusual manner, with
a place in the Journal de Physique. In the present instance,
we must do them the justice to say, that they have not been
deficient in their contributions towards the support and illustra-
tion of their own propositions respecting the actual state of the
sciences in Great Britain. Nor have they been altogether de-
ficient in aflbrding occasional opportunities of triumph to the
philologists, as well as to the mathematicians of the continent :
we shall not enlarge at present on this subject ; but we may
perhaps have occasion to meet them at a Aiture time on the
'* Phoenician plains," which they have very lately introduced to
our acquaintance, in defiance of Laporte du TheU and Coray, as
well as in opposition to all lexicographers and grammarians.*

The contents of the first volume of the Memoirs of Arcueil,
which has been published about three years* have already be-
come generally known through various channels. The mag^
netieal observations of Biot and Humboldt, which stand first on
the list, are so far important as they relate to the intensity of
the magnetic forces acting on the compass, which these philoso-
phers have found to be 137 at Berlin, and 125 at Rome, calling
it 100 at the magnetic equator : but the position, which they
have asrigned to this imaginary line, seems to be less accurate
than that which it occupies in Mr. Churchman's chart. Mr.
Th^nard^s various papers on the U&, and on ethers of difierent

* This refers to a singular mistraoslaiion of a passage from a fngment of
iEachylos, sanetioned 1^ the Reyiewer of the Traduction de Strabon hj Coray and
Laporte da Theil : 'Edinburgh Review' for April, 1810, p. ei.^Note £y the
Editor.

238 REVIEW OF THE No. XIII.

kindfly contain a number of remarkable results relating to the
chemical constitution of these substances. Mr. Bertiiollet has
particularly examined the combination of sulfur with the muri-
atic add, discovered by Dr. Thomson, and thinks tiiat it ought
to be conadered simply as an oxystdfureted muriatic acid. Mr.
/ Gay Luasac describes some interesting experiments on the
expamion of gase$. When two equal balloons were employed,
and one of tiiem being exhausted, a communication was opened
with the other, the heat observed in the first was always nearly
equal to the cold produced in the second, and both were nearly
proportional to the density of the air concerned : but the pro-
portion by no means held good for gases of different kinds ;
hydrogen, for instance, exhibiting a greater change of tem*
perature than common lur. The same gentleman has also
made some observations on evaporation^ and on the decomposi-
tion of the sulfates by heat. Mr. Biot finds that the air in the
bladders of fishes is the purer in proportion as they occupy
deeper parts of the ocean, consisting, in fishes which are found
at great depths^ of much more than half its bulk of oxygen.
Mr. Berthollet describes a useful manometer^ or rather gazo-
meter. Mr. A. B. Berthollet shows that the liqiuyr of Lampor
dins is, as that chemist supposed, a hydruret of sulfur, and not
a carburet Mr. Berthollet gives an account of a cheesy sub-
stance^ obtained from muscular flesh. A short note by Gay
Lussac, on the capacity of different bodies, with respect to
chemical saturation, closes the first volume.

The second volume appears far to exceed the first in the im-
portance of its contents. Besides those articles which we shall
more particularly examine, it* contains a continuation of Mr.
Tb^nard's researches respecting the action of acids on alcohol,
and an extension of the results of the investigation to the
neutral compounds formed by acids with other vegetable and
with some animal substances. In Mr. BerthoUet's observations
on the proportions of the elements of some combinations, allow-
ance is particularly made for the quantity of water which has
often adhered to some of these elements, when they have been
supposed to be pure. This celebrated chemist has also directed
his attention anew to the gases composed principally of hydrogen

No. Xin. MEMOIRS OF ARCITEIL. 239

and earion, and agrees with some of our countrymen in the
opinion, that they all contain oxygen : he is also persuaded that
their composition is not limited to any fixed proportions. Mr.
Decandolle gives a very simple explanation of Ihe well-known
tendency of plants to approach the light; ohsenring that the
calorific effects of light, by which also carbonic acid is decom-
posed, are accompanied by a contraction of the fibres on the
side most affected, which naturally bends the young shoots;
and that those plants which are not coloured by light, for in-
stance the cuscuta, have no disposition to approach it. Mr. Gay
Lussac has presented us with three memoirs, on the relation
between the oxidation of metals and their saturation with acids,
on the mutual combination of gasesy and on the employment of
nitrous gas or nitric oxid in eudiometry, for the foundation of
all of which he seems to be wholly indebted to the ingenious
theories of Mr. Dalton : he has also described a eudiometer in
which an excess of niUic aiad is added, with as little agitation
as possible, to the mixture to be examined ; and one fourth of
the diminution, produced by the formation of nitrous acid,
represents very accurately the quantity of oxygen contained in
it MM. Thenard and Biot biye very carefully analysed the
arragonite^ which they find to be perfectiy identical, in its che-
mical constitution, with the common rhomboidal subcarbonate
of lime, although its refractive powers are considerably greater,
but not in the proportion that might be expected from the still
greater excess of its specific gravity. Mr. A. B. Berthollet
has entered into some elaborate researches on the composition
of ammonia, which, although not perfectly conclusive, yet
appear on the whole to be unfavourable to Mr. Davy*s opinion,
that this substance contains an appreciable portion of oxygen.
MM. Provencal and Humboldt have made a great number of
very accurate experiments on the respiration of fishes^ showing
that a supply of oxygen is absolutely necessary to their exist-
ence, although a very small quantity is sufficient; that the
bulk of the carbonic acid produced is considerably less than
that of the -oxygen absorbed, and that there is some deficiency
of nitrogen. Mr. Descotils makes some practical remarks on
the operation of procuring lead from its sulfuret: he finds that

240 MEMOIRS OF ABCUEIL. No. XIII.

there is considerable loss wherever any gaseous substance is
present, and recommends that it be smelted by fusion with iron
only, or with some of the most metallic of its ores, where it b
possible. Mr. BerthoUet has related, in some short notes, the
results of several very interesting experiments of a miscellaneous
y nature. The first is on the heat produced by percussion, which
he considers as proportional only to the permanent condensa-
tion of the substance compressed : in some cases he found that
no heat was produced, and observing that the apparatus was
half a degree colder than the surrounding objects, he con-
cludes that the agitation of percussion must increase the con-
ducting power for heat; but this can scarcely be deemed a
justifiable inference, since the elevation of temperature gene-
rally observed was 10 or 12 degrees, in comparison with which
the difference of half a degree must have been wholly incon-
siderable, especially as the time of contact was extremely
short The oriental hezoar he finds to be a concretion, pro-
bably of woody fibres. His experiments on the respiration of
small animals agree with those of MM. Allen and Pepys, in
exhibiting an evolution of nitrogen. In order to examine the
truth of Mr. Dalton's hypothesis conceniing the constitution of
mixed gases, he left several combinations for some days in bottles,
which communicated by a narrow tube, and in some cases there
remained to the last very well marked differences in their
respective contents : but he found that hydrogen mixed more
readily with every other gas than any third species would do.
He has confirmed the general result of the doctrines of Dalton,
Wollaston, and Thomson, respecting the proportion of conUn-
nations, but thinks that it admits of many exceptions and modi-
fications: and lastly, he has ascertained, that a portion of
nitrogen adheres so strongly to charcoal, as always to form a
part of the gas which is expelled from it by a strong heat in
coated glass vessels ; so that we can by no means consider
charcoal in its common state, as at all approaching to a simple
elementary substance. Such are the outlines of the results of
the principal investigations related in this volume, besides those
which we are now to proceed to notice somewhat more in
detail.

No. XIII. BIOT ON SOUND IN VAPOURS. 241

Experiments on the Propagation of Sound in Vapours,
By Mr. Biot.

When a liquid of any kind is introduced into the vacuum of a
barometer, the mercury is more or less depressed, according to
the nature of the liquid, and to the temperature of the atmos-
phere, the elasticity of the vapour, which rises firom the liquid,
assisting the weight of the mercury in counteracting the atmos-
pherical pressure : and if we cause the space occupied by the
va})our to be diminished or increased, by adding to or taking
fipom the quantity of mercury in the bason of the barometer,
or by altering the inclination of the tube to the horizon, the
effective height of the mercury will remain in all cases the
same, provided that there be an excess of the liquid in the tube.
Under these circumstances, therefore,~the elasticity of the vapour
is not increased by compression, nor diminished by rarefaction ;
a deposition of a part of the vapour taking place in the one
case, and an additional evaporation in the other. Hence Mr.
Biot argues, tliat a vapour simply so constituted could not
transmit sound, since its elasticity would not be increased at
the part which receives the positive impulse of the vibrating
body, nor diminished where the body is retreating: and the
only way in which he thinks that sound could be conveyed by
such a medium, is by means of the heat evolved by its com-
pression, which must enable it to retain the elastic form with a
temporary increase of density, where the podtive impulse is to
be transmitted. Hence, finding that vapour does actually
transmit sound very perceptibly, that of ether indeed almost as
well as atmospherical air, he infers that such an elevation of
temperature must be produced by the compression of vapour
in general, and he concludes also that a similar effect must
take place in the ordinary transmission of sound through the
atmosphere, according to the explanation which Mr. Laplace
has given of the difference between the observed velocity of
sound, and the velocity calculated from the simple elasticity of
the air, as exhibited by slow compression.

We do not wish to withhold our approbation of Mr. Biot's
diligence in attempting to reduce the ingenious theory of Laplace

VOL. I. R

242 MEMOIRS OF ARCUEIL. No. XIII.

to the test of experiments ; but we must confess that, in tlie
present instance, the experiments, however interesting in them-
selves, appear to be both inconclusive and superfluous, as
applied to the theory in question.

^ We think them inconclusive, because it seems manifest to us
that sound might be transmitted by a vapour, without the pro-

^ perty of the evolution of heat by simple compression. The first
stroke of the vibrating body would cause a slight depontion of

^ the liquid, and the portion thus deposited would by no means
be instantaneously converted into vapour, upon the retreat of
the body. The space would therefore be left a little under-
saturated, and the sound would be transmitted without further
impediment : for a vapour> below the point of saturation, pos-
sesses all the properties of a permanent gas. Besides, the con-

/version of a part of the vapour into a liquid would unavoidably
^ be attended by the extrication of a certain portion of heat, which
would increase the elasticity of the remaining gas, without any
immediate evolution of heat by its compression. It may also
be shown, that even in the actual circumstances of a vapour
capable, in all probability, of being heated by compression and
cooled by expansion, the space must inevitably be somewhlit
undersaturated during the transmission of every sound through
it. Whenever a gas, nearly saturated with humidity, is ex-
panded, there is a deposition of visible moisture ; and we have
every reason to believe, according to the experiments of Mr.
Dalton, that the same must happen to a vapour unmixed with
a more permanent gas: consequently the expansion of the
vapour, where it has followed the receding particles of the
vibrating body, roust necessarily be attended by a deposition of
a minute portion of the liquid, which will not instantly evapo-
rate; so that the vapour will never remain precisely at the
utmost pmnt of elasticity which the general temperature is
capable of suj^orting, and will therefore never be, mathemati-
cally speaking, in the circumstances which Mr. Biot supposes.
But even if it be granted that these experiments have a ten-
dency to support Mr. Laplace's theory, we cannot help thinking
that their support is perfectly unnecessary. The velocity of
sound must obviously depend on the temporary elasticity of the

No. XIIL BIOT ON SOUND IN VAPOURS. 243

medium at the respective points concerned, which is only re-
quired to continue for a time almost inconceivably small, much
smaller than that which would be sufficient to allow the diffusion
of the heat and cold produced by compression and expansion.
We have no instruments delicate enough to measure very pre-
cisely the magnitude of the changes of temperature produced
in such cases ; but, in the first place, it is easily proved from
the well-known circumstance of the constant appearance of
vapour in the receiver of the air-pump^ that at least 20 or 30
degrees of cold must be produced by the expansion of a portion
of air to twice its bulk, since a depression of temperature equal
to this is required, as appears from Mr. Dalton's tables, in
order to produce such a deposition of moisture in a portion
of air thus expanded, even when it has been previously in the
utmost possible state of humidity : and secondly, Mr. Dalton
has very ingeniously inferred, from the rapidity with which a
thermometer begins to sink in the first instance, that about 50^
are actually exhibited; and Dr. Young has sho^n that the
results of some of Mr. Dalton^s experiments make it probable
that the efiect is still greater than he has supposed ; so that it
may be considered as strictly demonstrable, that the velocity of
sound must be increased about one-seventh from this cause,
while the observed increase is about one-fifth. Since therefore
it may be proved, that at least f of the increase must arise
from tiie cause so happily suggested by Mr. Laplace> it appears
to be more natural to suppose, that the whole difference arises
from the same cause, the operation of which cannot be so
accurately traced under any other circumstances, than to
imagine that any second mystery still remains, to be unveiled
by fixture conjectures.

Since the velocity of sound is not in the least affected by any
alteration in the density of the air, it follows that equal degrees
of compression must produce equal elevations of temperature in
all cases ; and that in Gay Lussac's experiments on portions of
air of different densities, the apparent differences must have
arisen principally fit)m the imperfection of the indications of the
thermometers.

We cannot avoid noticing here the utter darkness that seems

244 MEMOIRS OF ARCUEIL. No. XIII.

to have enveloped the secret tribunal, which lately passed its
^ ^ sentence of condemnation on the theory of its own idol, Laplace.
It is asserted in support of this sentence, that in order to have
the elasticity of the aerial wave augmented in the proportion of
two to three, it would be necessary that the temperature should
be raised " 125° of Fahrenheit's scale ;" and that for this pur-
pose the successive portions of air must be compressed into
"one-fifth of their usual space," by means of a velocity of
" impact equal to 3350 feet in a second." We maintain, that
instead of 125% Mr. Laplace's theory does not require, in any
common case, an elevation of one single degree, or even of half
a degree of Fahrenheit. It is not " the elasticity " which is to
be augmented one half, but its excess above the mean pressure ;
and this excess, or the actual condensation, is probably seldom
so great as one thousandth of the whole density ; and it will be
sufficient if such a condensation be accompanied by an elevation
of (me-fowrth of a degree, in order to justify the opinion of this
celebrated t>hilo6opher.

We could easily pardon a mistake of this kind in a hasty
opinion expressed privately by an individual ; although from an
author of any description, however inexperienced and unas-
suming, we should expect a greater degree of attention : but
when such errors are dictatorially proclaimed by an arbiter of
science, as the ultimate deci»ons of critical accuracy, and in
defiance of the authority of a mathematician, who, as we are
taught to believe, at the distance of a few pages only, has so
" few rivals," that " Lagrange is the only man now living who
may be fairly placed by his side," we cannot help feeling the
truth of the observation, " that the foolhardy proceed boldly^
because blindly.*^*

An attempt, equally futile, has been made by the same critic,
where he endeavours to improve on the refined calculations of
Mr. Lagrange respecting the velocity of sound. The chain of
reasoning, by which these calculations are established, is unim-
peachable in the circumstances to which it is applied : the
observations, which the critic has made, on the initial motion of

♦ Edinburgh Review, vol. xv., p. 432. The Reviewer was said to he Professor
Leslie.— J\ro<« by the Editor,

Na XIII. BIOT ON SOUND IN VAPOURa 245

the separate particles of the medium, are verbally true, but
efiectively fallaciou8» since the contemporaneous motion, to
which they relate, although it might take place in the last of a
system of a very small number of atoms, " A, B, C, D, and E,"
yet would become absolutely imperceptible, if their number
were only increased to as many particles of air as would stand
on the point of a needle.

" Professor Leslie '* is certainly much obliged to his kind
countryman who has endeavoured, in his account of this paper,
to support the tottering hypothesis of, aerial undulations, as
contributing to the transfer of radiant heat These undulations
are supposed to be " gentle," and not to excite in the air " the
tremor which causes noise :'* yet they are imagined to be violent
enough to transfer so much heat, as will elevate the tempe-
rature of a body several hundred degrees, by the simple effect
of the progressive condensation, as producing a change of
capacity in the air, which gives out this heat to the air in
contact with it, prepared for its reception by a favourable and
apparently spontaneous dilatation, while the condensation seems
only to be produced by the pressure of the heat, first thrusting
the air before it, and then penetrating it without resistance. If
anything is necessary for the confutation of so unintelligible
and so unprofitable a speculation, after t&e full establishment
of Dr. Ilerschers discovery of invisible solar heat, and after
some late observations on the actual transmission of some portion
of the heat of a fire by radiation through lenses^ it may be
found in Mr. Davy's elegant experiment on the radiation of
the heat excited by galvanism, in the vacuum of an air-pump,
where the effect of reflection is not only not inferior to that
which takes place in the open air, but incomparably greater and
more rapid.

It is indeed remarkable that so much ingenuity and happy
invention, as are exhibited in Mr. Leslie's work on heat, should
be alloyed by so much inaccuracy of reasoning, and so much
want of mathematical preci&ion. Among many instances of this
kind, we will only adduce one passage, p. 127. " If," says this
author, " an ivory ball strikes against another of equal weight,
Uiere should, according to the common theory, be an exact
transfer of motion. But if the velocity of the impinging ball

246 MEMOIRS OF ARCUBIL. No. XIII.

be very considerable, ao far from stopping suddenly, it will re-
coil back again with the same force, while the ball which is
struck will remain at rest/' In other words the comm<Hi
centre of inertia, which was moving forwards before the collision,
will be made to move backwards after it. Now we have been
taught by the laws of motion, Itud down in the Principia, that
^' Quantitas motus, quae coUigitur capiendo summam motuum
factorum ad eandem partem, et difierentiam factorum ad con-
trarias, non mutatur ab actione corporum inter se :" and we
must unavoidably deny the truth either of this fundamental law
of motion, or of the observation recorded by Professor Leslie.
In fact it is perfectly obvious that the experiment has never
been made, and never can be made, with either of the balls
absolutely at rest.

On the Motion of Light in Transparent Mediums. By Mr-

LAPUkCE.

We should have had very little to say of this essay. In
addition to the remarks inserted in our 4th number, p. 337,*
on the abstract of it before published in the Journal de Phy-
sique ; since the farther details of calculation, which it con-
tains, present no difficulties, and consequently display no
ingenuity : but here again our attention has been particularly
excited by some supposed improvements on the theory of extra-
ordinary refraction, which have been suggested in this country,
and we cannot refrain from inquiring how far these improve-
ments are real.

Entertaining the opinion which we have already ventured to
express on the subject, we cannot hesitate to agree in the
sentence, that the *' present memoir is grounded on assumptions,"
at least ^' as gratuitous and arbitrary, as those involved in the
hypothesis with which it is contrasted." But we were not a
little surprised in reading that the phenomena in question might
^^ admit of a very simple investigation, from the frindamental
principle of accelerating or retarding forces ;" and we were
utterly confounded, at first sight, with the next sixteen lines of
the paragraph,! in which, as we are told, the law of extraordinary

* Supra, p. 220. t In the Review quoted above, p. 426.

No. XIII. MALUS ON KEFLECTED LIGHT. 247

refraction is at once deduced firom that principle, ^* without re-
quiring any more aid of the integral calculus." It is charac-
teristic of a great master to obtain the most striking results by
the most simple means : in the present instance, the result is
far more satisfactory than that of the original memoir ; and we
were flattering ourselves for a moment with the idea, that at
least one of our countrymen, who had thus happily succeeded
where Laplace had failed, would deserve to be placed between
him and his great *' rival,'* in that seat which the ^* Emperor of
half of Europe " was once delighted to occupy. But our exul-
tation was of short duration : we soon perceived that the mode
of reasoning employed would serve equally well for any other
imaginable purpose, and that the apparent brevity of the state-
ment could not be considered as surprising, since a demonstra-
tion which proves nothing may easily be concise. It is advanced
as one step of this argument, that the extent of the action of the
extraordinary force, exhibited by the Iceland crystal, " is re-
duced^* in the ratio of the cosine of the inclination of the ray.
But why is the space of action thus reduced ? Only because it
is necessary for the success of the demonstration that it should
be so ; for haw it should become reduced in this ratio, rather
than in any otlier, the critic does not inform us, nor have we any
means of discovering ; and it appears to be as unwarrantable to
assume such a reduction, as it would be to take for granted the
original proposition as self-evident. It is merely the desire of
pointing out one more of a multitude of errors that has led us
to make this objection ; for the question implicates a material
point in the comparison of the two theories of light, which the
next paper will require us to institute.

On a Property of reflected Light; and On a Property of the
repulsive Farces which act on Liglit. By Mr. Malus.

The discovery related in these papers, appears to us to be by
far the most important and interesting that has been made in
France, concerning the properties of light, at least since the
time of Huygens; and it is so much the more deserving of
notice, as it greatly influences the general balance of evidence.

248 MEMOIRS OF ARCUEIL. Na XIII.

in the comparison of the undulatory and the projectile theories
of the nature of light.*

It was known to Huygens and to Newton, that a ray of light,
transmitted and divided by one piece of Iceland crystal, or
rhomboidal subcarbonate of lime, was either subdivided, or not,
by a second piece, according to the relative position of the two
crystals; so that if we looked down through both of them, and
the obtuse angle of one was situated on the north side of the
ray, and that of the other on the north-east side, four images of
any object would be seen ; and only two if the obtuse angle of
the second, was either on the north side or on the east. Now in
the simple Huygenian theory of an undulation resembling that
of sound, the ray must be alike on every side, as well after as
before its passage through the first crystal; nor can it be
imagined how its afiections can be difierent with respect to
north and north-east, or to any other points of the compass ;
and this was advanced by Newton as an objection, which Huy-
gens had not been able to overcome. We ventured to suggest,
on a former occasion, f that the curvature of the undulation, or
in other words, the divergence of the light, might possibly be
different in difierent directions : now Mr. Malus's experiments
are precisely such as to afibrd an answer to this suggestion ;
since they show that the divergence is absolutely unconcerned
in the phenomena* and that a similar division of the light may
be produced by simple refiection from a plane surface where no
change of divergence takes place in any direction.

This statement appears to us to be conclusive with respect
to the insufficiency of the undulatory theory, in its present state,

* In a letter addressed by Malus to Dr. Young fu Foreign Secretary of the Royal
Society, returning thanks for the award of the Ramford medal for his discovery
of the Polarization of Light^ in which he gives an account of some further observa-
tions, he adds, at the conclusion of them : ^* Je ne regarde pas la connaissance dc ces
phenom^nes comme plus favorable aa systfeme de remission qu'i celni des ondula-
tiona. lis de'montrent egalement I'insuffisance des deux hypotheses ; en effet, com-
ment expliquer dans Tune ou (^ans I'autre pourquoi un rayon polarise pent traverser
sous une certaine inclinaison un corps diaphane, en se de'iobant totalement i la
reflexion partielle qui a lien k la surface de ces corps dans les ca& ordlnaires ? '* ^ It
was after the difficulties, which the discovery of polarization created, had come into
view/* says Dr. Whewell, •* and before their solution had been discovered, that we
may place the darkest time of the history of the undulatory theoiT." It was
reserved for Dr. Young, at a later perioil, to sujrgest the hypothesis, by* which those
difficulties have been overcome. — Note by the Editor,

t Supra, p. 229.

Ko. XIII. MALUS ON REFLECTED LIGHT. 249

for explaining all the phenomena of light. But we are not
therefore by any means persuaded of the perfect sufficiency of
the projectile system : and all the satisfaction that we have
derived from an attentive consideration of the accumulated
evidence, which has been brought forward, within the last ten
years, on both sides of the question, is that of being convinced
that much more evidence is still wanting before it can be posi-
tively decided. In the progress of scientific investigation, we
must frequently travel by rugged paths, and through valleys as
well as over mountains. Doubt must necessarily succeed often
to apparent certainty, and must ajgain give place to a certainty
of a higher order ; such is the imperfection of our faculties^
that the descent from conviction to hesitation is not uncommonly
as salutary, as the more agreeable elevation from uncertainty to
demonstration. An example of such alternations may easily be
adduced from the history of chemistry. How universally had
phlogiston once expelled the aerial acid of Hooke and Mayow I
How much more completely had phlogiston given way to
oxygen I And how much have some of oui* best chemists been
lately inclined to restore the same phlogiston to its lost honours !
although now again they are be^nning to apprehend that they
have already done too much in its favour. In the mean time,
the true science of chemistry, as the most positive dogmatist
will not' hesitate to allow, has l)een very rapidly advancing
towards ultimate perfection.

The outline of Mr. Malus's important discovery may be thus
familiarly represented. Suppose the altitude of the sun on the
meridian to be 19^ 10', and a plate of glass, not silvered, to be
so placed, as to reflect a ray of his light directly downwards :
then if a second plate be fixed below and parallel to it, this
plate will again reflect the descending ray into a direction
parallel to the original one, and nothing remarkable will hap-
pen. But if we turn round this second plate, without altering
its inclination to tiie horizon, as soon as it faces the east or the
west, it will no longer reflect any part of the light, either from
its anterior or from its posterior surface : when, however, it has
made half a revolution, and fronts the south, it will again reflect
the usual proportion of the incident light ; and in the inter-

250 MEMOIRS OF ABCUEIL. Ko. XIII.

mediate positions, the reflection will be more or less perfect, as
the reflected ray approaches more or less to the plane of the
meridian. If now, instead of the second plate, we place a piece
of Iceland crystal with its principal section in the plane of the
meridian, the whole of the reflected ray will be transmitted by
the ordinary refraction ; but if we turn round the crystal till the
direction of its principal section become east and west, the ray
will now be subject to the extraordinary refraction only ; and
in all intermediate situations of the crystal, it will be divided
into two portions. Mr. Mains has entered into several more
particular details^ respecting the results of similar experiments
under various circumstances ; but they do not add materially
to the interest of the facts as thus simply stated.

The angle of incidence at which this modification takes place
the most completely, is difierent for substances of different
densities : for water it is 52° 45' ; for glass 54^ 35', and for Ice-
land crystal 56^ 30'. Black substances, such as polished ebony,
have a similar property ; but metals are entirely destitute of it.
When a modified ray is reflected by a metallic mirror, so as to
continue in its principal section, or to proceed in a plane per-
pendicular to it, it still retains its properties : but if its new
direction be equally inclined to both these planes, its modifica-
tion will be destroyed.

Mr. Mains has discovered, that in all doubling crystals, one
of the refractions is always of the extraordinary kind, and tiiat
whether we employ carbonate of lead, sulfate of barita, crys-
tallized sulfur, or rock crystal, the modifications which take
place are precisely of the same nature. He has also ascertained
that the internal reflection of the doubling crystals causes, in
general, a further subdivision of the light reflected.

We are perfectly satisfied, from our own observation, of the
accuracy of Mr. Malus's account of his principal experiment ;
but we are by no means disposed to agree with him in believing
that the modified light is wholly transmitted by the surface,
where it is in no degree reflected : on tiie contrary, we are in-
clined to think, that the portion usuaUy reflected is in this case
wholly absorbed, if not destroyed. We will not presume to
oppose our authority to that of Mr. Malus ; but he has been so

No. XIII. MALUS ON REFLECTED LIOHT. 251

little particular in the detail of bis experiments, that we are at
liberty to doubt of the validity of some of his oonclu8ion& By
employing six or eight successive transmissions through as
many parallel plates, the question might be easily decided ; and
as far as we have examined the phenomena, our results have
difiered in this respect from Mr. Malus's statements, which he
appears rather to have set down as the natural consequences of
other facts^ than from direct experiment.

Mr. Malus observes very truly, that the ordinary phenomena
of optics may be explained, either according to the hypothesis
of Huygens, or by the doctrine of Newton : but he thinks that
those properties, which he has discovered or confirmed, are only
capable of an explanation from a polarity, such as was attri-
buted by Newton to the particles of light ; and for this purpose
he lays down a law respecting the position of their supposed
axes, which he appears to consider as satisfactory, but which we
cannot help thinking manifestiy and utterly inadequate to the
solution of any of the difficulties.

It seems to be undeniable that the general tenor of these
phenomena is such, as obviously to point at some property re-
sembling polarity, which appears to be much more easily
reconcileable witii the Newtonian ideas than with those of
Huygens. We must, however, observe, not only tiiat the
admission of the projectile tiieory is by no means sufficient for
the explanation of Mr. Malus's experiments ; but also, on the
other hand, that there is another very extensive class of pheno-
mena, which seems to lead us even more directly to the doctrines
of the Huygenian school, tiian those which Mr. Malus has dis-
covered, appear to divert us from them. We allude to the
multiplicity of facts, which are referable to the general law
of the mutual destruction of two portions of light, some slight
rudiments of which are to be found in the works of Grimaldi,
and which has been particularly investigated and extended by
our countryman. Dr. Young. It has been justiy conceded,
that " we should not hastily reject even the wildest hypothesis ;"
for, " if a hypothesis be not allowed to warp the understanding,
it may serve at least usefully to connect certain insulated facts."
The truth of this observation is shown in a remarkable manner

252 MEMoms OF abcueil. No. XIII.

by the asffistanoe which Dr. Young has derived from the Huy-
genian theory, in the discovery and establishment of a law which
reduces to a single principle, and explains with a degree of
accuracy, in general perfectly mathematical, and always within
the probable limits of the errors of observation, the phenomena,
before insulated, of the colours of thin plates, of thick plates, of
mixed plates, and of inflected or diffracted light, in an infinite
variety of forms. This law is not only the necessary conse-
quence of a doctrine like that which has been founded on t^ie
theory of Huygens, but is also accompanied by some other
conditions immediately connected with that theory; and it is
rendered still more inseparable from it, by its extension to the
chemical phenomena of the invisible blackening rays, which
could not be explained, upon the Newtonian doctrine of the
undulatory nature of the .sensation only of light, as transmitted
by the optic nerve.

Of the phenomena of light which are more commonly ob-
served, the greater part will admit an explanation equally satis-
factory from either of the theories ; others, although perhaps
not absolutely incompatible with either, appear to us to be more
naturally referable to the Huygenian than to the Newtonian.
The effects of simple reflection and refraction belong to the
former of these divisions : tho^ of the dispersion of the rays
of different colours may also be compared, either with the dif-
ferent velocities, acquired by projectiles of different magnitudes,
in a resisting medium, or with those of waves of different
breadths, spreading on the surface of an imperfectly elastic
liquid. The transmission of light, with little interruption,
through the densest transparent substances, afibrds a difficulty
of the same kind in the Newtonian theory, as the aberration of
the stars in the Huygenian: in the first instance, the ultimate
atoms of matter must either be supposed permeable to light,
or to be scattered at distances inconceivably great, in compari-
son to their own magnitudes, through the apparent dimensions
of the solid bodies : in the second, the porosity of the sub-
stances concerned needs not by any means to be supposed so
excessive ; but there is some difficulty in conceiving tlie free
and rapid passage of the ethereal mediiun through the densest

No. XIII. MALUS ON REFLECTED LIGHT. 253

bodies, at the same time that it must remain in some measure
accumulated within them.

Among the fects which appear favourable to the Huygenian
theory, we must first enumerate the uniformity of the velocity
of light in any one medium, under all circumstances that have
hitherto been observed ; since it is a fundamental law of this
system, while it cannot easily be explwned from any probable
mode of operation of repulsive forces ; and in the second place,
the precise agreement of the hypothecs of Huygens, respecting
spheroidal undulations, with the phenomena of extraordinary
refraction, and the immediate connexion, which we have
pointed out, in a former article,* between this hypothesis and
the simplest possible supposition respecting the constitution of a
stratified medium ; while on the other hand we ima^ne that we
have said enough to make it evident, that neither Mr. Laplace,
nor his critic, has succeeded in deducing any explanation of the
facts fix)m the ordinary laws of accelerative forces. The recti-
linear motion of the light, passing near a material substance,
has often been adduced as an argument in favour of the projec-
tile system ; but we are inclined to class the phenomena, which
occur under such circumstances, with those which are most con-
veniently explained by the undulatory theory, llie dimensions
of the shadow of a hair, as observed by Newton and other
authors, are such as to show undeniably, that light passing at a
distance of one tenth of an inch, or more, from an opaque body,
is -deflected in its course, and at length dissipated into the sur-
rounding space ; now it is contrary to all probability, and even
to direct experiment, to maintain, that any repulsive force can
act on light at such a distance : indeed, if we judged of the ex-
tent of the supposed repulsive force by that which is exhibited
on the approach of two hard bodies, we should not expect it to
act beyond the distance of one ten thousandth of an inch. It
has also been ascertained, that the phenomena of light, inflected
in this manner, are totally independent of the refractive density
of the bodies concerned, which they could not well be, if the
same forces were employed in them as are the immediate
agents in reflection and refraction. We do not know that any

♦ Sapra, No. XII.

254 MEMOIRS OF ARCITEIL. No. XIII.

attempt has been made to assign the precise magnitude of the
addition to the breadth of the shadow from this diflhiction, at
different distances, but we believe it will always be nearly repre-
sented by ri-o x'\ x being the distance from the object in inches.
There are also many other cases in which it is absolutely neces-
sary to suppose such a difiraction in order to reconcile the phe-
nomena with the results of calculation.

Having thus endeavoured to state the arguments on both
sides, in the most impartial manner, we must leave our readers
to satisfy themselves, if they can, with the theory to which they
may be most inclined : for ourselves, we confess that we are
compelled to remain for the present undecided, and we can only
look forwards for further information to the discoveries which
may result from future experiments.

Abstract of Memoirs read to the Institute from the 7 th Marchy
1808, to the 27th February, 1809. By MM. Gay Lussac
and Thenard.

The principal part of these eight memoirs relates to the
brilliant and important discoveries with which our countryman
Mr. Davy has enriched the science of chemistry. It is true that
the authors have confined themselves principally to tl)e relation
of their own experiments, many of which are certainly in some
degree original, and possess great merit : but in other instances,
they have not been so accurate in avoiding the appearance of
laying claim to the discoveries of another, as might have been
consistent with perfect liberality of sentiment We are also
scMTy to observe in various parts of these volumes that the obli-
gations of several authors to the theories and experiments of
Mr. Dalton have not been so distinctly acknowledged, as
candour might have required. We have heard, indeed, that
the successes of the chemists of other nations have sometimes been
held up as reproaches to the members of the National Institute
by a powerful protector ; and that these reproaches have even
been accompanied by threats of abandonment. Supposing this
to have really happened, we can readily make allowances fbr
the substantia] causes, which may have contributed to make the
sight a little dim in reading across the Channel.

No. XIII. GAT LU8SAC*S CHEMICAL EXPERIMENTS. 255

But we are not disposed to be quite so indulgent to that
imperfection of the organs, which obscures all objects that are
merely seen across the Tweed. We think that no impartial
judge, exempt from the influence of an Antianglican spirit,
would have professed to believe that MM. Thenard and Gay
Lussac have established ^*most convincingly, that the new
metals are not simple substances, but really compounds of the
several bases with hydrogen.'* Mr. Davy has most abundantly
confuted this rash and ill-supported opinion, derived from the
accidental result of a single experiment, and incapable of being
reconciled with the opinions professed even by its authors in
other memoirs. And who in this island has a right to expect
that his cursory adoption of a foreign hypothesis shall be put in
competition with the deliberate judgment of a chemical philo-
sopher like Mr. Davy ? — a man whose candour is equal to liis
ingenuity, and whose uncommon talents have been seconded by
the most ardent zeal for the acquirement of knowledge, and
have been crowned by a good fortune commensurate to his
exertions and his opportunities I The coherence of the analo-
gical argument, which is ofiered in support of the hypothesis
of the French chemists, is well calculated to accompany the
modesty with which the truth of the opinion is asserted. ** Every
compound must have the intermediate density of its distinct
ingredients." This observation is in no sense universally true :
but let it pass. Now *^the specific gravity of the alkaline
metals is far less than that of the substances from which they
are derived.'' If these words convey any ideas at all, they are
certainly not such as are applicable to the very simple case in
question. Potassium is very light: when combined with
fixed oxygen, it forms potass, which is heavier than po-
tassium, but may, for aught we know, be &r lighter than fixed
oxygen ; nor, if it were otherwise, would the case be absolutely
unique.

The difficulty, which has given rise to this unwarrantable
opinion respecting the metals of the alkalis, originated in an
experiment on the decomposition of ammonia, in which a con-
siderable portion of hydrogen appeared to be set at liberty.
But according to Mr. Davy's latest repetition of this experi-

256 MEMOIRS OF ARCUEIL. No. XII [.

ment, in a tabe bored out of solid platina, there seems to be
very little mystery in the process : the hydrogen and nitrogen
are both recovered in their proper proportions, except that there
is rather a deficiency of hydrogen than an excess, this substance
appearing partly to enter into combination with the platina.
If, however, in other circumstances, the results should appear
to be more complicated, we shall be much more willing to
admit Mr. Davy's modest conjecture respecting the constitution
of nitrogen, than the singular hypothesis of the French chemists
respecting that of potassium.

In the fluoboric gas, discovered by the author of these
memoirs, there seems to be a singular exception to Mr.
Dalton's general laws of hygrometry ; for this substance does
not appear to be capable of containing any aqueous vapour ;
while Mr. Dalton maintains that the quantity of aqueous
vapour,, which may be present in any space, is nearly inde*
pendent of the nature of the gas that occupies it. The contra-
diction is, however, perhaps more apparent than real, since the
condensation of the vapour is owing to the formation of a
new substance, in consequence of the strong chemical attrac-
tion of the gas for water; and this new fluids which is a
most corrosive acid, follows its own particular laws with respect
to evaporation, being extremely little disposed to assume a
gaseous form.

Experiments on the Propagation of Sound through solid Bodies^
and through the Air contained in very long Pipes, By Mr.

BlOT.

From a number of very accurate experiments on the trans-
mission of the sound of a bell, fixed to one end of a series of
pipes of cast iron, 3121 feet in length, Mr. Biot has inferred
that its velocity, in passing through the substance of the pipes,
was between 10 and 1 1 times as great as in the air which they
contained. A whisper was easily heard at night through the
whole of this length, but in the day the words spoken by the
loudest voice could not be distinctly understood at a much
shorter distance. In speaking through the whole of the pipe,

No. XIII. BIOT OK PBOPAGATION OF SOUND. 257

it was observed that several repetitions of an echo returned to
the speaker at intervals of half a second each. This circum-
stance is not. explained: perhaps it arose from some accidental
projections within the pipes ; but it is singular that these
should have been at equal distances. This difficulty may
indeed be avoided by attributing the echoes to the return of the
soimd from the opposite extremity of the whole pipe, through
its substance, which, by tlie former observation, ought to have
occupied exactly .52" : but on this supposition an equal number
of repetitions should have been heard at the other end of the
pipe; while in fact one sound only was heard, and this was
conveyed through the air.

Mr. Biot's determination of the velocity of the transmission
of sound, through the substance of the pipes, is so far interesting,
as it tends to the confirmatiou of other experiments, which are
in their nature susceptible of more accuracy : but the precise
results, which he has obtained, are of no value whatever. The
ends of the separate pieces of pipe, the shoulders of which were
screwed together, with the interposition of wadding, must have
materially retarded the transmission of the sounds, by the
increase of their bulk, in the same manner as any dilatation or
contraction of a cylindrical cavity, for instance that of a chim-
ney pipe, retards the vibration of the medium contained in it.
Mr. Chladni's experiments, which are exempt from this cause of
error, make the velocity 16 or 17 times as great in iron as in air.

Before we take our leave of the contents of these volumes,
and of the remarks which have been made on them, we must
submit to our readers one more specimen of inaccuracy, which
appears to us to be sufficient of itself to determine the degree
of confidence which ought to be placed in these remarks.
^^Chladni," says the critic, **had assigned the celerity of
vibration through iron and glass at 17,500 feet in a second ;
and Leslie had ehown^ in one of the curious notes annexed to
his book on heat, tiiat through a fir board the velocity of
impulsion, which he proved to be the same as that of vibration,
is 1 7,300 /e0^ in a second J^ Now, having referred to Professor
Leslie's note relating to his experiment, p. 519, we find that
the height of the column, measuring th^ elasticity of fir, is there

VOL. L s

258 MEMOIRS OF ARCUEtt. No. XIII,

calculated to be only "671,625 feet," which corresponds to
a velocity of 4640 feet in a second : and in the text it is as-
serted, probably on the ground of an earlier and still more
hasty estimate, that motion is conveyed through deal " with 5i
times the velocity of sound," that is, with a velocity of about
6220 feet in a second. It appears, therefore, that " the ve-
locity of impulsion," as really calculated by Professor Leslie,
is less than one-third of that which Chladni had assigned from
more direct experiments. Where then can the critic have
found a number approaching so much more nearly to the
truth ? We can only answer, that we have found it by looking
into the index of Dr. Young's Natural Philosophy, for the
article '* sound in wood :" we are there referred to a passage
in which it is said expressly, vol. ii. p. 266, that "according to
the elasticity of fir, as inferred yrom an experiment of Mr. Leslie,
the velocity of an impulse should be 17,300 :" and it appears,
from the same volume, p. 49, that Dr. Young's calculation
was the result of a series of original investigations, applied,
in this as well as in several other cases, to circumstances which
had not before been sufficiently examined. Perhaps the critic
had long ago consulted the same index, and found the same
passages ; perhaps, considering it as of no importance to the
establishment of a point of calculation, to recollect from what
work it was borrowed, he has unintentionally substituted the
name of a Tyrian for that of a Trojan, rmUo discrimine. We
confess, however, that we think a censor ought to be xpore rigidly
correct.

We have perhaps detained our readers too long with the cor-
rection of errors which may be thought incapable of misleading
those who reason at all for themselves : but the work, in which
they are contained, has long assumed an air of* authority, which
may have imposed on the timid, and satisfied the superficial
student ; and it was time that some attempt should be made to
reduce its pretensions to their natural level. We trust that
our remarks may have a pnrapective as well as a retrospective
efiect ; and that, without being again obliged to undertake the
disagreeable task of controversial discussion, we shall have
inspired the candid lovers of science \rith a salutary distrust,

No. XIII. BIOT ON PROPAOATION OF SOUND. 259

which will prevent their acceding unguardedly to all the dogmas
that may hereafter be dictatorially proclaimed through the
same channel, in conformity with the system which seems to
have been adopted, of the uniform discouragement of all do-
mestic pretensions to scientific merit, beyond the limits of a
particular school.

s 2

260 REVIEW OF No. XIV.

No. XIV.

REVIEW OF

MALUS, BIOT, SEEBECK, AND BREWSTER
ON LIGHT.*

From the Qoarterlj Reriew for April, 1814, toL xi., p. 42.

The iDtimate connexion of the subjects of these works with
each other renders it unnecessary to make any apology for in-
cluding our account of them in one article ; since the greater
number of the observations which they contain have arisen more
or less immediately from the prosecution in the different parts
of Europe, of the important discoveries of Mr. Malus, respect-
ing the properties exhibited by light which has been subjected
to oblique reflection or refraction. Of these discoveries we
have already given some account {tujnuy p. 247); and the
honourable testimonials of public approbation^ which their
author has since received, in particular from the Royal Society
of London, as well as ftx>m the Institute of France, sufficiently
show that our estimate of his merits was not exaggerated.
Most unfortunately for the sciences, his' career has been cut short
by an early death, in the midst of his researches and improve-
ments ; but this event did not take place, as Dr. Brewster seems
to imagine, so immediately after the adjudication of Count
Rumford's medal, as to have rendered him incapable of being
informed of the honour that was conferred on him, and of
appreciating its value.

• 1. Tbbobie de la double Rxfbaotion de la. Lumi]^!. Par E. L. Malus.
4to. Paris, 1810. pp. 302 ; with 8 Plates.

2. MCMO^EUI SUE DE MOUVBAnx RaPFORTB EETRE LA REFLEXION ET LA FOLABI-

8ATI0N DE LA LoMiiRE. Par M. BiOT. Lu i riDstitat le 1 Join, 1812.
pp.152; with 1 PUte.

3. Vbbsuche uebeb Spibqbluno und Bbechuno. [Experiments on the Be-

FLECriON AND RErRACTION OF LlOHT.] Bj Dr. SEEBEGC.

4. A Treatise on new Philosophical Instruments, with Experiments on

LtoHT AND Colours. By David Brewster, LL.D. 8vo. Edinburrii, 1813.
1^.442; with 12 Plat«s.

No. XIV. MALU8 ON LIGHT. 261

In the present work of Mr. Mains, there is less of absolute
DOYeHy than of minute and interesting research, upon a point
respecting which some doubt was perhaps entertained, by those
who were not sufficiently acquainted with the few, but satisfac-
tory, experiments relating to it, which before had been made in
this country ; that is, upon tne perfect accuracy of the Huyge-
nian laws of refraction in the Iceland crystal, and other doubling
substances; which, considered in itself, he thinks **one of the
finest discoyeries of this celebrated geometrician.*'

** Newton," he observes, <* was acquainted with the investigatioDS of
Huygens ; yet he attempted to snbstitate for the Huygenian law another,
apparently more simple, bat absolutely contrary to the phenomena, as
Mr. Haiiy first observed and demonstrated. It is difficult to explain the
disr^ard that Newton showed, in this case, to a law which Huygens
had declared to be conformable to his ezpenments.**

'^ Wollaston has examined the refractive power of the crystal, by a
very ingenious method of his own invention, and has shown that the
law is perfectly true in all cases of rays passing in the direction of any
surface of the crystal. He is ^e first that, siber the oblivion of a cen«
tury, thought of verifying, by direct experiments, a law which Huygens
had considered as incontestable, and which Newton had rejected without
examination.'

It may not, however, be altogether superfluous to observe,
that as Dr. WoUaston's experiments seem to have led to Mr.
Malus's researches and discoveries, so Dr. WoUaston's thoug/Us
were in all probability directed to the Huygenian theory by
an earlier paper published in the same volume of the Philo-
sophical Transactions with his own, in which it is asserted,
almost in the terms that Mr. Malus has employed, that Newton,
*' without attempting to deduce from his own system any expla-
nation of the more universal and striking effects of doubling
spars, has omitted to observe that Huygens's most elegant and
ingenious theory peifectly accords with these general effects in
all particulars."* PL Tr. 1802. 45.t In short, whoever reads

* In a note from Dr. Wollaston, dated August 26th, 1801, addressed to Dr. Toung>
and KfeiTittg to his Bakerian Lecture, No.yil., whid^ had been submitted to him, lie
says : **1 like your Bakerian very much, but I cannot say that I have yet inserted the
undulatory doctrine into my creed, and it may be some time before I repeat it with
fluency." He was constitutionally reluctant to acknowledge, or to publish, any
theory or obserration, of the entire correctness of which he was not himself fully
sattsfied.'—^ole 6^ the Editor, t Supra, No. VII^ p. 166.

262 REVIEW OF No. XIV

the account which Huygens pvea of his own examination of
these Bubetances, can scarcely fail to be convinced that this law
must be extremely near the truth : Dr. Wollaston's experiments
affi>rded additional eiddence of its accuracy ; and Mr. Mains,
haying diversified his calculations and observations in a still
greater variety of forms, has left nothing further to be desired
for the complete re-establishment of this remarkable result of
hypothetical theory, the ground-work of which is still by no
means unexceptional, notwithstanding the wonderful simplicity
to which, as we have shown on a former occasion (supra, p.
228), it is capable of being easily reduced.

Mr. Malus has prefixed to his experimental investigations an
analytical treatise on optical phenomena in general, which will
probably be thought, by most English readers, unnecessarily
intricate, and which does not appear to contain any material
novelty. He has examined the forms of the principal refracting
substances by means of a reflective goniometer, resembling Dn
Wollaston's ; and he has taken the mean of a considerable num-
ber of successive repetitions of the measurement For the angle
of the Iceland spar he obtains in this manner, 74^ 55' 2"i ; and
contents himself with 74° 55', which is precisely Dr. Wol-
laston's measure : for the indices of refraction he gives 1.6543
and 1.4833, instead of Dr. Wollaston's 1.657 and 1.488,
although the experiments on some specimens go as far as
1.658. For quartz crystal we have J. 5582 and 1.5484;
for the sulfate of barita, 1.6468 and 1.6352; and for the
arragonite, another form of the carbonate of lime, which some
have suspected to omtain strontia, 1.6931 and 1.5348. From
the laws of extraordinary reflection within a crystal of doubling
spar, Mr. Malus has very ingeniously deduced an explanation
of a reduplication of images, long since observed by Martin, in
particular specimens, which appear to have been interrupted
by fissures of sudi a nature, as to be capable of producing a sub-
division of the rays, like that which takes place in the internal
reflections : the colours observable in these images he refers to
the thickness of the fissures, although it seems at least equally
probable that they are more nearly related to the colours of
crystallized substances, since described by Biot and others.

No, XIV. MALU8 ON LIQHT. 263

Mr. Malus's calculationa of the particular cases of refraction
are founded on the Huygenian method of drawing a tangent
plane to the supposed spheroid, from a point in the surrounding
medium, at which the supposed original undulation would haye
ajrrived while the spheroid is generated. The steps of this
mode of calculation are, however, extremely intricate ; and it
has occurred to us that the problem may be solved in a much
more simple manner by equating the velocities with which the
supposed undulations must advance upon the common surface
of the respective mediums— a condition which is obviously
sufficient for the determination of tiie angular directions of the
actual undulations — just as the velocity with which a bird
swims on the sur&oe of a piece of water, is sufficient for deter*
mining the direction of the wave which follows it. Considering
the velocity of the advance of the undulation with regard to the
spheroid, it ^rnust evidently be identical with the' velocity of
increase or decrease of the perpendicular to the circumference
of the section cut off by the refracting surface ; and with regard
to the surrounding space, it must be to the direct velocity, as
the radius to the sine of the angle of incidence or refraction in
that space. Hence, if r be tiie index of the greatest refractive
density of the substance, a the sine of incidence or refraction
without the crystal, x the semiaxis of the spheroid, and y the
perpendicular falling from the point of incidence on the conju-
gate diameter of the section, we have the equation ri i 8^y\
which determines the physical conditions of the problem, and
reduces it to a mathematical investigation.

Now if the ratio of the greatest and least refractive densities,
or of the equatorial diameter of the spheroid to the axis, be
that of n to 1, and the tangent of the angle formed by the axis
with the refracting surface, p, it may readily be inferred, from
comparing the ordinates of the ellipsis with tiiose of the in-
scribed circle, and from the properties of similar triangles,
that the semidiameter parallel to the given surface will be
n V ^ \J^ ^9 the tangent of the angle formed by the conjugate
semidiameter with the axis, rm : p, and the length of this semi-
diameter V ^^ x. From the known equality of all paral-

264 REVIEW OF No, XIV.

lelograms described about an ellipsis, we have, for the per-
pendicular ialling from the end of this semidiameter on the

former, V ^ . x ; and taking the difference of the squares,

V ^ -Hffp) ij^^vp? ^^^ ^^ distance fix)m the centre ; and for the
sine of the included angle j ^^^Z^ ^l\ ^ jH) = ^- The per-
pendicular falling from the same point on the axis will be
a/ («« + IV) " '^^ *"^ ^*® distance from the centre, jj^j^pp\

Proceeding now to the section formed by the given refracting
surface, let y be the cotangent of the angle comprehended by
its lesser axis and the plane of the ray's motion without the
crystal, and let z be the distance of its centre from that of the
spheroid : we shall then have, for the lesser semiaxis of the

section, V (iJTjr^ **""«')» reduced in the ratio of the con-
jugate diameters of the spheroid, that is, V (~^ ^ - «*) X
'^ "^ JJ24. g4' The ratio of the axes of the section, from the
known similarity of parallel sections of a spheroid, will be that of
" ^ ^^1^ a; to /la: ; or if we call this the ratio of 1 to jw, we

have m = V^TT«» * Hence, in order to find y, we must sub-
stitute these values in the expression for the perpendicular
V^., whenceweharey = V ^=V^ ^ (^ ^_ ^) ^
» ^ ^T?; and taking the fluxion, ri : s = y ^ ^ ^jm+gq^
St^^^ ^y^Z • ^ (^Jt£^-^*) ' ^'""S supposed

to remain constant ; consequently V \ t "* ar* --?*) = — V
TTqq ' ^Jm^ ^^ ^ p» + «4 * *"d the semiaxis of the section,
«. / ^ "^ PP J ( P^ '^ ^ o o\ « /*»»»+ w « 1 + pp

No, XIV. MALUS ON LIGHT. 265

and the semidiameter of the section ending at the point of
incidence, -7 V\ . ^ ifx „ . ^ : whence it is obvious that
this semidiameter, which may be considered as an ordinate
in an elliptic section passing through the centre of the
spheroid, is proportional to the sine of incidence, as Huygens
has demonstrated ; and it will appear that tlie tangent of the
angle formed by the plane of this section with the plane of

But in order to determine more directly the inclination of
the ray within the crystal, we must find tc, the perpendicular
falling from the point of incidence on the lesser semiaxis, before

expressed by j^;^^ and now^^^^J^^^ x ^±^ and its

distance from the centre of the section v = — ?!!!! — -a; Ij^JS

and from the point nearest to the centre of the spheroid, tz-^v;
whence the distance of the point of incidence fit)m this last
point must be tj ([tz-'vy+u*): and adding to the square of
this that of the perpendicular falling on the section fi*om the
centre of the spheroid, or «*— f*^:*, we have V(w*+ti^+2:*— 2rt«)
for the semidiameter at the point of incidence, expressing the
velocity : and the sine of the correspcHiding angle of incidence
or refraction will be j/{[tz^vy+t^) divided by this radius,
while that of the inclination of the plane to the axis will, be

«:V([<^-«?+«»);. being = xV(^-^-.f^.

. n" — T^y It is also evident that the velocity, re-

duced to the direction of a perpendicular to the surface, will

vary as -^ V (1 — <*)•

These expressions may be much simplified by further re-
duction, especially where they are to be applied to surfaces
either parallel or perpendicular to the axis: since in these
cases p^O and m=n, and p = (X> and m = l respectively, and

^ = 0 in both. Hence, in the first case, z =«jr J C^"TTw*

266 REVIEW OF Na XIV.

V " . ^ 9 whence the sine of the angle may be found, di-
viding it by ^ (m" + »* + 2*) I and the tangent will be

the second case, z - x V (1 '- ff —) » ^ = r V (i -f w)

It is not merely with a view of exhibiting a more convenient
mode of solving a problem which Mr. Mains had solved before,
that we have introduced this calculation, but in order to apply
it to the explanation, which we shall attempt to give, of the
very interesting phenomena described at large in Mr. Biot's
memoir.

Mr. Arago had discovered, in 1811, that polarised light was
resolved, by passing through thin plates of mica or sulfate of
lime, or thicker plates of rock crystal, and of some kinds of flint
glass, into two portions differently coloured.! Mr. Biot has
experimentally investigated the law according to wliich these
phenomena take place, and has reduced the results of his
experiments into such a form as to enable us to calculate from
them what colours will be exhibited by a plate of sulfate of
lime, of a given thickness, and in a given situation with respect
to the incident light.

The axis of the crystals of sulfate of lime is, either accurately
or very nearly, in the plane of the plates which they afford, and
makes an angle of 16*^ 13' with one of the natural lines of frac-
ture of the plates ; while that of rock crystal is nearly parallel
to the longitudinal surfiEices of the crystal. Mr. Biot's method
of exhibiting the colours in question is to take a thin and smooth
plate of sulfate of lime, or Muscovy talc, or a well-polished
plate of rock crystal, cut as thin as possible, which affords no
appearance of colour in the open air, except when some of the

* The same myestigation, very slightlj modified, is giyen in the article Chro-
matics, Section XIV., which immediately follows in this volnme. The snbstanoe of
that given by Malus, founded upon the principle of least action, may be seen in Sir J.
Herschers well-known Treatise on Light in the EncyclopsBdia Metropolitana, Art.
7BB.—Note by the Editor.

t The Memoir announcing this important discovery was read to the Institut in
1811, and published in the volume of Memoirs for that year. — Note by the Editor.

No. XIV. BIOT ON LIGHT. 267

incident light has been polarized by reflection from the blue
atmosphere, and to place it horizontally on a black substance ;
then, allowing the white light of the clouds to &11 on it, at an
inclination of about 35°, to receive this light when reflected
from it, on a black glass, making an equal angle with the
reflected rays, in a plane perpendicular to the first plane of
reflection ; so that the plate may be visible by reflection in the
black glass. In this manner the plate appears to be very bril-
liantly illuminated by the light of the colour which it is calcu-
lated to exhibit : when its axis coincides with the plane of ,
incidence, no colour is visible ; and the appearance becomes
most distinct when the axis makes an angle of 45° with that
plane. In this situation of the axis Mr. Biot finds that the
colour reflected by talc and by rock crystal is precisely the same
as if the incidence were perpendicular, and the same as is trans-
mitted by the extraordinary refraction ; while the light trans-
mitted by the ordinary refraction exhibits the complementary
colour, as in the case of the ordinary colours of thin plates : these
transmitted colours being separable, as Mr. Arago had found, by
means of any doubly refractive substance, or by oblique reflection.
In Mr. Biot's arrangement, the light reflected from the upper
8urfisu» of the plate is polarized according to the general laW|
and is, therefore, not reflected by the black glass, but absorbed ;
and the same is true of the light reflected from the lower sur-
face of the plate, and then transmitted back by the ordinary
refraction : but that which has been transmitted [back from] the
lower surface by the extraordinary refraction [not to it, as Mr.
Biot's words imply], has acquired a contrary character: and
when it arrives at the black glass, it is partially reflected. On
the other hand, a black glass, of which the plane of incidence
coincides with that of the plate, reflects the complementary
tint afforded by the light which had been reflected by the lower
surface of the plate, and transmitted back by the ordinary re^
fraction, but exhibits the colour more faintly, because it is mixed
with the white light reflected from the upper surface. A
similar arrangement may also be very conveniently applied to
the observation of the colours of natural bodies, independently
of the glare occasioned by their superfidal reflection.

268 REVIEW OF No. XIV.

The colonrs dependent on the extraordinary refraction Mr*
Biot found to agree exactly with the colours of thin plates of
glass as seen by reflection, and those which are deriyed from the
ordinary refraction with the colours seen by transmission in the
Newtonian experiments, suppodng the thickness of the plate to
be reduced in the ratio of 360 to 1 ; this ratio being constant
for the same specimen of the talc, although the number varied
in different specimens fi*om 383 to 895. For mica, it api>eared
to be 450, but' was liable to still greater variation : for rock
crystal, it was exactly 360, at least in several plates cut out of
the same piece* The measurements of the thickness of the
plates were executed with the greatest care by Mr. Cauchoix's
spherometer, whidi appears to be capable of great precision,
although the pressure exerted by a fine screw, which is the
immediate instrument of examination, must be a cause of con-
siderable uncertainty, where the objects to be measured are
extremely minute.

Mr. Biot observed that when the axis of the crystal approached
to the plane of inddence, the colours ascended in the scale of
Newton's measures, as if the thickness were diminshed ; and
that they descended when the plate was turned in a contrary
direction. The difference thus produced appeared to be greater
in plates of rock crystal and of mica than in those of talc ; but
the comparative measures have not been detailed ; and it may
be remarked that the greater thickness of the plates of rock
crystal employed may possibly have made the difference more
apparent. When the axis made an angle of 45^ with the plane
of incidence, the change of the inclination of the incident light
had no effect on the colour exhibited either by talc or by rock
crystal ; but mica, probably fr*om the oblique situation of the
axis of refraction, did not observe the same law. Mr. Biot has
expressed the thickness corresponding to the tint, exhibited
under these different circumstances, by the formula 1 + (,065
^ H — .195y^H) «■ ; while, in another series of experiments,
the coefficients appeared to be .00959 and .1428 ; H being the
angle formed by the axis with the plane of incidence, and $ the
sine of the angle of incidence ; so that the greatest possible
variation must have been from .87 to 1.26, or from .867 to 1.152.

No. XIV. BIOT ON LIGHT. 269

Mr. Biot has also improved Mr. Malus's expressions for the
intensity of the light under different circumstanoes ; bnt as the
colour is wholly independent of the intensity, we omit to men-
tion these expressions more particularly.

This intricate and laborious investigation appears to have
been conducted with much patience, and with minute attention
to the strictest accuracy ; nor does the present memoir by any
means exhaust the whole of the experiments which Mr. Biot
has promised to the public. Dr. Brewster has remarked that
he has ''the undivided merit of having generalized the facts,''
and of having '' discovered the law of these remarkable pheno-
mena.** This '' law/' however, is merely an expression of the
facts considered as insulated from all others ; and not an ex-
planation by which they are reduced to an analogy with any
more extensive class of phenomena ; and we have no doubt that
the surprise of these gentlemen will be as great as our own satis-
£EU^on in finding that they are perfectly reduceable, like all
other cases of recurrent colours, to the general laws of the inter-
ference of light, which haye been established in this country, and
of which we have given an account in our sixth number {mpr€i9
p. 251) ; and that all their apparent intricacies and capricious
variations are only the necessary consequences of the simplest
application of these laws. They are, in fact, merely varieties
of the colours of '' mixed plates,'* in which the appearances are
found to resemble the colours of simple thin plates, when the
thickness b increased in the same proportion, as tiie difference
of the refractive densities is less than twice the whole density :
the colours exhibited by '' direct transmission," corresponding to
the colours of thin plates seen by reflection, and to the extra-
ordmary refraction of the crystalline substances, and the
colours of mixed plates exhibited by " indirect li^f' to the
colours transmitted through common thin plates, and to tiiose
produced by the ordinary refraction of the polarizing substances.*
The measures which Mr. Biot has obtained differ much less
from the calculation derived from these principles only, than

* This important step in the theoiy, though correct in principle, b incomplete,
inasmnch as the eflfects of polarisation in modifying the phenomena of inteHenooep
bad not yet heen ascertained: they were annoon^ by Arago and Fresnel in the
taith Tolune of the * AonalM de Cfaimie' for 1819.*^o«» hy ths EdUor.

270 REVIEW OF No. XIV.

they differ among themselves ; and we cannot help thinking such
a coincidence sufficient to remove all doubts (if any existed)
of the unirersality of the law on which that calculation is founded ;
notwithstanding the difficulty of explaining the production of
the diflSsrent series of colours by the different refractions. (See
Qu. Rev., No. XVII. p. 124.)»

In the first place, it appears from Mr. Malus*8 experiment,
that the extraordinary and ordinary refractive densities of the
rock crystal, in a plane perpendicular to the axis, are in the
ratio of 159 to 160 ; consequently the difference of the times is
to twice the whole time in the ordinary refraction as 1 to 320,
and to the time in a plate of glass of which the refractive den-
sity is 155, as 1 to 318. In Mr. Biotas experiments on this
substance, the proportion of the thicknesses appeared to be 1 to
360, while in the sulfate of lime, the number varied from 333 to
395 : and it must be observed that any accidental irregularities,
or foreign substances adhering to the plate, would tend, in Mr.
Biot's mode of measurement, to make the thickness appear
greater : while, on the other hand, an error of a single unit in
the third place of decimals of the index of refractive density, as
determined by Mr. Mains, would be sufficient to make the
coincidence perfect : and a greater degree of accuracy can
scarcely be expected in experiments of this kind.

We, have next to inquire what must be the effect of the
obliquity of the incident light according to the general law of
periodical colours ; and we shall here find the agreement of
the experiments with the theory equally striking. We must
compare the excesses of the times occupied in the trans-
misnon of light by the respective refractions, above the time
required for its simple reflection from a point in the upper sur-
fiice, exactly opposite to the respective point of reflection in the
lower ; and the difference between these excesses will give the
interval required for determining the colour. Calling the
thidcness unity, and the sine of incidence «, the excess for the
ordinary refraction will be represented by the time within the
plate, which is as the secant of refraction, diminished by the
difference of the times without the plate, which is as its tangent,

^ la a ReWew of Dr. Toung's Introduotion to Medical Litflratve.— Abte by the E4iJLot,

No. XIV. BIOT ON UQHT. 271

and as the sine of inddence jointly, (see fig. 130,) or by r :
^ (^ - ^) - « : ^ V (1 - ^) = V (r» - i«) ; and for
the extraordinary refraction, when the axis is parallel to the
surface, the former part will be inversely as ~ , and will be
expressedbyr :n V (1 - -~ TT^) ^^ ^^ ^^^r by -^
^ iTw ' "^ ^^ TTot**^' whence the whole becomes (r*

-^TT|**-«^('^-'TTf'0- Now since, in the sub-
stances which we are considering, n is little more than 1, we
may put n = 1 + /, i^ = 1 + 2/, and n* = 1 + 4/;'then V

1 +gq = V (1 + f^:^) = 1 + YTqq ' ^^^^ ^^ slso be the

value of "y—; and if for y^^ ^e write A^ the excess will

»->- IvV^'ctr^;. = V(^ - (1 + 2*-0/) : n.
Now the difierence between sj (r* ^ i) asxAV (f - (I + 2«)

**) M jfrr-u)* *^^ ^^ difierence between the latter root,
and the same quantity divided by n, is / V (r* — (1 + 2 A*//),
or very nearly / V (r* — J*) = / J^Crr^u) * *"^ *^® ^^™ ^'
these differences is / "^ ", . " ,\ » or if 1 — A' = A", Z

jZr^uf * ^"^8 *^® cosine of the inclination of the plane of

incidence to the axis ; nor will the result be sensibly affected
by taking into account the deviation of the refracted ray from
this plane in oblique situations.

This expression will be found to include all the effscts of
a change of inclination observed by Mr. Biot, and to agree
sufficiently .well with the formula which he has deduced from
his measurements. When the light falls perpendicularly on
the sur&ce, « = 0, and the difference becomes Ir ; when its
obliquity is the utmost possible, s being 1, the expression

becomes / "yr- — r^ and its value varies in the ratio of r' to

272 REVIEW OP No. XrV.

r*— 19 according to the position of the axis. ' Thus in the
sulfate of lime, r being 1.525, according to Dr. Wollaston's
table, the utmost possible Tariation is in the ratio of 2.326 to
1.326, and the equivalent thickness for perpendicular rays
being called 1, the extremes will become .755 and 1.325,
instead of .87, and 1.26 or 1.152, which are the results of Mr.
Biot*s different formulas ; and the difference between these is
as great as the variation of the first of them from our calcu-
lation. TVith respect to the singular fact of the indifference of
the angle of incidence, when the inclination of the plane of
incidence to the axis is 45*", our expression agrees exactly with

Mr. Biot's observations : for when A* = i, w"^*Vv r= r,

very nearly: thus if « s= 1, it only becomes 1.586 in-
stead of 1.525, and does not vary sensibly whiles remains
small.

In a similar manner the result may be determined for any
other relative situations of the axis and the refracting surface :

if, for instance, they are perpendicular to each other, —
being V (1 "- «* ^ )> ^^d the tangent of refraction .. ^_^^;^x>
the expression for the excess of time becomes r : V (i ^ n*
•^) — / {^nnu) = ^ (^^ "" *** ^)» ^hil® the excess for the
ordinary refraction is V (r^ — ^) as before ; and the difference
becomes j. ^ ^y which vanisbes with the angle of incidence,

and becomes ultimately .^ . ^. We cannot help thinking

ourselves justified in looking forwards to a perfect coincidence
between this formula and the promised experiments of Mr.
Biot on substances placed in tiiese circumstances. We under-
stand that Dr. Brewster has lately made some observations of
a nature nearly similar ; but we doubt whether he has deter-
mined the refi*active powers of his crystals with suflScient
accuracy to allow of the application of our calculations with
perfect precision.

A sin^ar confirmation of the mode of explaining the colours
of tiiin platesi which we have adopted, is afforded by the expe-

No» XIV. SEEBECK ON LiaHT. . 273

riments of Mr. Arago^ who found that the light forming the
transmitted rings appeared to be polarized in the same direction
with the reflected light, while the rest of the transmitted light
was polarized in a contrary direction. It is a necessary assump-
tion in the theory of periodical colours, that the rings seen by
transmission actually depend on light twice reflected within the
plate, and which must Uierefore be polarized like the rest of
the reflected light ; although without these experiments of Mr.
AragOy it would have been difficult to obtain so direct a demon-
stration of the fact.

The colours exhibited by thick pieces of rock crystal, cut as
in Mr. Biot's unpublished experiments, perpendicularly to the
axis, might be expected to afford some explanation of those
which Dr. Seebeck has observed in large cubes or cylinders of
glass placed between two oblique reflecting swfaces, or be-
tween two piles composed of thirty pieces of glass each, which
produced the eflect of complete polarization on light transmitted
at the appropriate angle. If, however, Dr. Seebeck*8 obser-
vations are correct, the analogy can be only superficial 5 for
the efiects of these pieces of glass seem to depend on their entire
magnitude and outward form, without any particular relation
to an axis of extraordinary refraction. Thus iu the perpen-
dicular transmission of the polarized light through any poiqts
in the diagonals of the surfaces of the cubes, or in the diameters
parallel to their sides, the rays of different colours appeared to
be differently affected according to the part of the glass on
which tliey fell, and to exhibit one or the other only of the two
images, which would have been visible through a piece of
doubling spar, if the glass had not been interposed ; so that
when the whole cube was viewed at once under these circum-
stances, it afforded an appearance of diversified colours ar-
ranged in very singular forms, which Dr. Seebeck compares to
the figures assumed by sand on vibrating pieces of glass, and
discovered some time since by Professor Chladni; but which
appear to have a still nearer resemblance to those which Com-
paretti has described, as produced by the admission of a beam
of light into a dark room, through apertures of different forms ;
and we are much inclined to suspect that they depend on the

VOL. L T

274 REVIEW OF No. XIV.

twofold transmissioD of the light to the eye, perhaps after re-
peated internal reflections, from the different points in the lateral
surfaces of the substances employed. The effects were most
conveniently observed in cubes of If inch, and better in white
than in yellowish glass : in cubes of an inch only, they were in-
distinct : nor were they produced by fluor spar, rock salt, or by
any kind of liquids : they were modified, and sometimes in-
verted, by the interposition of a plate of mica : and ice acted
in a similar manner, in depolarizing the light transmitted
through it. We find in these researches a full confirmation of
the experiments which Mr. Malus had made some time before
his death, to show, that the polarized light, which falls on a
transparent medium at such an angle, as not to be reflected, is
transmitted, with no material diminution of its intensity. Dr.
Seebeck's language is a little enveloped in the mysticism of the
school to which, by some singular caprice of fancy, he has
thought proper to attach himself: but we cannot hesitate to
believe, that as he continues his examination of the phenomena
of nature he will by degrees be persuaded of the futility of the
objections, which Mr. von Goethe has advanced against the
Newt(mian doctrine of the composition of white light, and of the
inaccuracy of the assertions on which some of those objections
are grounded.*

While the optical philosophers of France and Germany have
been engaged in these researches, Dr. Brewster has been very
laudably employed, in this country, in experimental investiga-
tions relating to the same interesting department of physical
science. He has found that the agate cut by a plane perpen-
dicular to its lamina?, transmits one only of the polarised por-
tions of light ; that the polarity of light may be destroyed by
transmitting it in a certain direction through almost all mineral
substances, and through hofn, tortoise-shell, and gum-arabic :
while in certain other directions its properties remain unaltered,
whence he has distinguished, in these substances, different
depolarizing and neutral axes ; and that the light reflected from
the oxydated surface of polished steel is so modified, as to

* There ia an Article by Dr. Young in the Quarterly Review for the preceding
January, Vol. X., p. 427, on Gothe*8 Treatise Zur Farhenlehre. As it contains very
few original observations, it has not been reprinted. — Note by the Editor,

No. XIV. BREWSTER ON LIGHT. 275

prove, in his opinion, that the oxyd is a thin transparent sub-
stance. His observations on the colours, sometimes exhibited
by crystals of Iceland spar, seem to be identical with those of
Martin and Malus.

Dr. Brewster has very ingeniously exercised his inventive
powers in the contrivance of a variety of micrometers, goniome-
ters, microscopes, and telescopes, several of which may very
possibly be found useful in particular circumstances, although
to others there appear to us to be many material objections :
but, without referring to the test of experience, it would be of
little utility for us to discuss their particular merits. Some
detached remarks, however, we shall take the liberty of submit-
ting to our readers, on passages of the work which appear to
require correction. The advantage which Dr. Bi^ewster attri-
butes to the use of a transparent fibre for a micrometer, (p. 71,)
is merely imaginary ; since, although it is true that the cen-
tral rays '^ sufier no inflection," this circumstance afibrds us no
assistance whatever in judging when the rays are actually ^^ cen-
tral ;" and the light transmitted by such a fibre, whenever the
luminous object is in its neighbourhood, could only create con-
fusion. In speaking of a telescope for the measurement of
angular positions, Dr. Brewster observes that ^^ the line, which
joins any two stars, forms every possible angle with the horizon
in the course of 23 hours and 56 minutes ;" (p. 128 ;) but this is
obviously a mistake ; for at the poles of the earth the angle
would not vary; and in other latitudes only within certain
limits. .The table of the variation of the focal length of a tele-
scope (p. 218) is wholly erroneous, from the employment of
linear feet and square inches in diflerent parts of the same
formula. Dr. Brewster has misunderstood Professor Bobison
and Mr. Wilson, where they observe that the focal length of
an achromatic telescope must be lengthened, when it is directed
to a star towards which the earth is moving (p. 221) : it was
not from the different distances of the stars, but from the differ-
ence of the relative velocities of light, that they argued, accord-
ing to the general opinions respecting light, tiie necessity of the
occurrence of such a minute variation. In p. 424, 5, the mag-
nifying power is miscalculated, and we must read 4.9 for 5.6.

T 2

276 REVIEW OF No. XIV.

The most useful part of the whole work appears to be the
series of experiments on the refractive powers of fluid and soft
substances, performed by interposing them between the object
glass of a microscope, and a plane glass nearly in contact with
it, and then measuring the joint focal length of the combination.
The comparative distances, thus obtained, are exhibited in
several extensive tables : but we cannot help feeling some sur-
prise, that the author has not attempted to deduce, from any
one of his numbers, the direct refractive power of the substance
concerned, as he certainly would have done if he had been
aware how easily it might have been accomplished, afi;er a pre-
paratory investigation, dependent on the common laws of diop-
trics. From such an investigation we have obtained formulae
for each of the two series of experiments ; for the first (pp. 258,
268, 270,) / being the focal length expressed by the number

in the table, and r the index of refraction, r = 1 . 887 — '^.

1 .SI

and for the second, (p. 264) r = 2 . 31 '-j . Thus we obtain

for phosphorus 2.125, sulfur 2.008, aloes 1.643, balsam of
Tolu 1.636, oil of cassia 1.625, guaiacum 1.609, and pitch
1.589. Dr. Wollaston's Table gives for phosphorus 1.579, and
for pitch 1.53 ; and there can be no doubt that the accidental
presence of some phosphoric acid, and some oil of turpentine,
on the surfaces of these substances occasioned an error, in these
instances, in Dr. Wollaston*s determinations, however excellent
his method may be in other cases ; for we cannot agree with
Dr. Brewster, in thinking that the acknowledged exhibition of
the index appropriate to the extreme red ray is an objection
to the method. It is remarkable, as our author has justly
observed, that the assignment of so high a refractive density to
phosphorus restores the inference of Newton, respecting the
relation between refractive powers and inflammability, to its
original universality and importance.

Dr. Brewster's mode of ascertaining the refractive powers of
solids, by immersing them in a mixture of fluids of equal refrac-
tive density, is perfectly unobjectionable ; and he olwcrves that
it is easy to discover, in this manner, the internal flaws and other
irregularities of gems, without the labour of polishing any part

No. XIV. BREWSTER ON LIGHT. 277

of their surfiioe. He does, not, however, appear to have fol-
lowed this method in determining the indices of refraction which
are contained in his table, (p. 283,) having employed for this
pnrpose "the same prisms in which the dispersion was cor-
rected," and probably in the same manner: hence from an
erroneous mode of computation his numbers are almost imi-
formly too large: thus we have phosphorus 2.224, sulfur
2.115, carbonate of lime 1 665 and 1.519, oil of cassia 1.641,
and guaiacum 1.619, all of which exceed the more accurate
determinations which we have already mentioned. In the same
manner we find, for diamond 2.487 to 2.470, instead of 2.439,
the density assigned by Newton ; and it is probable that the
chromate of lead and realgar, both of which Dr. Brewster finds
more dense than the diamond, are also rated somewhat too high
at 2.974 . . 2.503, and .2.549 : the former appears to have a
double refraction more distinct than any other known substance.
For a similar reason we can place no dependence whatever
on the table of dispersive powers, which is calculated, accord-
ing to a coai*se approximation, wholly inapplicable to the cir-
cumstances of the experiments. The mode of inclining a prism
of a greater density, until it caused the image of a right lincj
viewed through it and in conjunction with a prism of smaller
density, to be colourless, would be a very good one, provided
that the apparatus were so arranged, that the rays should be
perpendicular to the common ^surface of the prisms ; but even
then Dr. Brewster's mode of calculation would be only appli-
cable to prisms with very small refracting angles. In the only
experiment which is related with precision, (p. 306,) the result
implies an impossibility : for if we trace a ray of light through
its intricate progress from the water to the glass, the angle
of incidence upon the last surface will come out 4P 5', while
the utmost obliquity, at which it could have been transmitted,
is 38^ 14', consequently tlie index of refi^ustion assigned to the
prism, 1.616, must be extremely erroneous if the angular mea-
surements were correct. And since various errors.of this kind
may have afiected the different results in different degrees, we
cannot depend on the tables, even for the order of the differen
dispersive powers.

278 REVIEW OF BREWSTER ON LIGHT. Ko. XIV.

Dr. Brewster appears, however, to have been more suGcessfiil
in confirming and extending the observations of Dr. Blair on
the different proportions in which the prismatic spectrum is
divided, according to the diversity of the substances which
afford it He has shown very clearly, both from theory and by
experiment, that the violet rays must be proportionally more
expanded by a prism with a large angle than by a smaller one
of the same substance ; while he has found, on the other hand,
that a smaller prism of a more dispersive substance almost
always expands the violet rays more than a large prism of a
less dispersive substance ; and that when two such prisms are
combined, they exhibit a green fringe in the usual place of the
red, and a " wine-coloured" fringe in that of the violet The
substances most expansive of the violet are oil of cassia and
suUur; the least expansive, sulfuric acid and water, although
water has not quite so low a dispersive power as fluor spar. It
seems to follow from Dr. Brewster's estimate, that the propor-
tions of 2 red, 3 green, 4 blue, and 3 violet, which are nearly
those of Dr. WoUaston's determination, are changed when sul-
furic acid is employed, at least as much as to 4 red, 3 green,
3 blue, and 2 violet ; but we feel great difficulty in believing
that so great a variation as this could have escaped the notice
of any attentive observer. We have no doubt, however, that if
Dr. Brewster continues to pursue his ingenious investigations,
he will by degrees acquire a habit of introducing greater accu-
racy into his measurements, and, what is of still more import-
ance, more mathematical neatness into his calculations ; and,
with these improvements, we doubt not that his future labours
may be productive of material benefit to those departments of
physical science which have engaged his attention.

No. XV. CHROMATICS. 279

No. XV.

CHROMATICS.

Fruni the Supplement to the Encyclopedia Britanuic-a.
Written ik the Year 1817.

The gradual progress of scientific investigation has con-
tinued to add, from year to year, a multitude of new disco-
veries to our knowledge of experimental and physical optics :
and no department of this subject has received additions so
diversified and so important, as those which relate to the
phenomena of colours, which have been displayed with a thou-
sand brilliant and unexpected transformations, under circum-
stances that, in foi-mer times, could never have been suspected
of exhibiting anything resembling them. The successive expe*
riments and calculations of Dr. Thomas Young (1801), Dr.
Wollaaton (1802) , Mr. Mains (1810), Mr. Arago, Mr. Biot,
Dr. Brewster, Dr. Seebeck, and Mr. Fresnel, have all contri-
buted very essentially to the extension and illustration of this
interesting branch of science. But notwithstanding all that has
hitherto been done, it appears to be utterly impracticable, in
the present state of our knowledge, to obtain a satisfactory
explanation of all the phenomena of optics, considered as
mechanical operations, upon any hypothesis respecting the
nature of light that has hitherto been advanced : it will there-
fore be desirable to consider the facts which have been discovered,
with as little reference as possible to any general theory ; at
the same time, it will be absolutely necessary, as a temporary
expedient, to borrow firom the undulatory system Dr. Young's
law of the interference of light, as afibrding the only practicable
mode of connecting an immense variety of facts with each
other, and of enabling the memory to retain them ; and this
adoption will be the more unexceptionable, as many of the most

280 CHROMATICS. No. XV-

strenuous advocates for the projectile theory have been disposed*
especially since the experiments of Mr. Arago and Mr. Fresnel,
to admit the truth of the results of all the calculations, in which
this law has been employed. The details of its application to
particular cases, together with an examination of the phenomena
of polarisation and of oblique refracticm, will occupy the prin*
cipal part of this article; but it will also be necessary to premise
an account of the few cases of the exhibition of colours which
appear to be independent of its operation.

Section I. — Of the Separation of Colours by Refraction.

The separation of white light into different colours, as its
component parts, by refraction, though firmly established as an
optical fact by Newton, had been in general somewhat neg-
ligently examined as to its details, until Dr. Wollaston pointed
out the inaccm*acy of the common subdivision of the colours of
the prismatic spectrum into seven difierent species. There is
little reason to doubt, that white light consists of an infinite
number of rays, varying gradually among each other, without
any marked distinctions, and continued on the one hand into
the dark chemical rays, and on the other into the rays of invi-
sible heat ; and that all these varieties are separable from each
other by refraction, and preserve always a distinct and constant
refrangibility. The species of homogeneous light, however,
distinguishable from each other by the eye, are only five ; red,
yellow, green, blue, and violet; which are uniform in their
appearance, and well defined in their limits, whenever a perfect
spectrum is correctly exhibited ; whether obtained by interposing
a prism, between the eye and a small or rather narrow bright
object, or between a lens and the image of such an object
formed in its focus : while, in the common method of admitting
a beam of the sun's light through a prism, without either
employing a lens, or previously limiting the angular extent of
the beam, it is obvious that there must be a double source of
the mixture of colours ; and hence has arisen the Newtonian
division of the spectrum into seven parts, which were somewhat
fancifully compared, witb respect to their extent, to the intervals
of the minor diatonic scale in music; although it has been

No, XV. CHROMATICS, 281

Rhown by Dr. Blair, and still more fully by Dr. Brewster, that
their proportions are liable to very great variations according to
the nature of the refracting substances employed.
' Dr. Brewster has remarked, that as, according to the funda-
mental law of refraction, a prism with a large angle must
occasion a dispersion of the several colours somewhat greater
than two smaller prisms of the same substance, having together
an equal mean refractive power ; so also the dispersion of the
most refrangible or violet rays amongst themselves will be
always somewhat greater in a prism with a larger angle, than in
two smaller prisms having an equal mean dispersive power;
hence the green and blue will be less removed from the- red
towards the violet by the single prism, the refraction of the
green remaining in defect when compared with the mean of the
whole ; so that if the two prisms be employed to correct the
mean dispersion of the single one, and the extreme rays of the
spectrum be brought to a perfect coincidence, the refraction of
the green by these prisms being comparatively in excess, the
green rays will be found on the side towards which their re-
fraction tends to carry them ; and the two extreme portions of
red and violet will be left together, forming a crimson, on the
side towards which the refraction of the larger prism is directed.
Tt is obvious also that if, instead, of the two smaller prisms, a
single one of an equal angle, but of twice the dispersive power,
were substituted, the joint effect would be nearly the same : Dr.
Brewster has, however, observed, that in almost all such com-
binations of different substances, the green is on the side
towards which the refraction of the larger prism is directed ; so
that the original proportion of the space occupied by the dif-
ferent rays in the spectrum must be different for difierent
substances. Dr. Brewster has found that the violet is the
most dispersed by oil of cassia and by sulphur, and least by
sulphuric add and by water : the distribution afforded by these
substances appearing to vary from 2 parts of red, 3 green, 4
blue, and 3 riolet, to about 4 red, 3 green, 3 blue, and 2 violet;
while the yellow is always confined to a narrow line.

The immediate effects of the combinations of the primitive
colours on the sense of sight afford an illustration of some of the

282 CHROMATICS. No. XV.

physiological characters of sensation in general. It b well
known that a mixture of red and green light produces a simple
sensation, perfectly identical with that which belongs to the
minute portion of yellow light originally found in the spectrum r
and that a mixture of green and violet makes a perfect blue.
The blue colour of the flame of spirit of wine, for example, is
derived entirely from a mixture of green and violet rays ; while
the blue light of the lower part of the flame of a candle is shown
by the prism to consist of five difierent portions, belonging to
diflerent parts of the spectrum, nearly resembling those which
would be distinguished, if we looked through a prism at a small
portion of a transparent plate, of a certain minute thickness. It
is obvious, therefore, that the eye has no immediate power of
analysing such light ; and if we seek for the simplest arrange-
ment, which would enable it to receive and discriminate the im-
pressions of the difierent parts of the spectrum, we may suppose
three distinct sensations only to be excited by the rays of the
three principal pure colours, falling on any ^ven point of the
retina, the red, the green, and the violet ; while the rays occu-
pying the intermediate spaces are capable of producing mixed
sensations, the yellow those which belong to the red and green,
and the blue those which belong to the green and violet ; the
mixed excitement producing in this case, as well as in tliat of
mixed light, a simple idea only : although it must be observed,
that no homogeneous light can extend its action so far as to
excite at once the sensations of the fibres belonging to the red
and the violet : so that every crimson must necessarily be a
compound colour. A mixture of red and blue light exhibits an
eflfect which appears unintelligible, upon the supposition that a
compound light ought to produce a colour intermediate between
those of its constituent parts ; but this difficulty will vanish, if
we assume that the blue of the spectrum contains a greater
proportion of violet than of green ; so that the green is neutral-
ised into a white by a mixture with the red and part of the
violet, and the remaining violet ^ves its character to the whole,
either alone, or with a mixture of red, according to the propor-
tions employed.

When we look through a prism at a luminous object of con-

No. XV. CHROMATICS. 283

siderable extent, surrounded by a dark space, the spectra
belonging to the several parts of the object are mixed with
each other, so as to produce a light perfectly white, except
towards the ends of the object, where the extreme parts project
beyond each other. At the red end of the spectrum, the whole
of the red belonging to the extreme point retains its place
unaltered, and the green and blue become a greenish yellow,
nearly uniform in its appearance, throughout the space which
belongs to them, while the place of the violet is scarcely
distinguishable from the neighbouring white light ; but at the
opposite end, the violet retains its place and appearance, and
the remainder of the length of the spectrum becomes of a
green, inclining more or less to blue, and continuing to be very
distinctly visibly throughout the extent of the simple spectrum,
the place of the red included ; so that the illuminating power
of the red end of the spectrum must be incomparably greater
than that of the violet end : as may also be inferred by a direct
comparison of the distinctness of objects viewed in these differ-
ent lights. The portion of light totally reflected at the internal
surface of a dense medium, on account of the obliquity of its
incidence, is bounded by a fringe or bow resembling the red
end of the luminous object viewed through a prism ; and the
transmitted portion is bounded by the violet and blue fringe :
but it requires some caution, in observing these colours, to
avoid the optical deception, which causes the neighbouring
space to appear of the complementary colour, especially when
the eye is turned towards it immediately after having received
the impression of the colours actually exhibited.

Section II. — Of the Colours of Halos and Parhelia.

The immediate effect of the different refrangibility of light,
in the production of colours, is sometimes spontaneously exhi-
bited, in the atmospherical phenomena of halos and parhelia, or
paraselenes, attending the sun or moon; the edge nearest to
the luminary being generally reddish, and the remoter parts
more or less green and blue, although without any well marked
separation of the different tints. These appearances have been

284 CHROMATICS. No. XV.

long ago referred by Mariotte to the refraction of the prismatic
crystals of snow, floating in the atmosphere, and descending
through it, in all possible positions, but more especially in a
vertical or horizontal direction, on account of the effect of
gravity, combined with that of the resistance of the air ; and
sometimes, perhaps, from their connexion with other crystals,
making angles of 60^ with either of these positions. This
theory, however simple and satisfactory, had been very unac-
countably neglected for more than a century, and even supers
seded by the awkward and unsupported conjectures of Huygens,
respecting the existence of spherical or cylindrical particles of
hail, including opaque nodules, related to them in a certain
constant ratio; or by the equally inadmissible calculation of
Newton, which assigns a partial maximum to the density of the
light simply refracted through a spherical drop of water, when
the deviation is about 26^ ; and it is only a few years since,
that the doctrine of Mariotte was revived and extended by Dr.
Young* and approved by Mr. Cavendish and Mr. Arago.

In some of the highest northern latitudes, these appearances
of halos and parhelia are almost constant ; and in warmer
countries they are confined to the light clouds which occupy'
the higher and colder regions of the atmosphere. The halos
are broad circles, with their interior margin tolerably well
defined, and about the distance of 22 and 46 degrees from the
sun or moon, but less distinctly terminated externally. Now
the angle of 22^ exactly corresponds to the deviation produced
by a prism of ice, with a refracting angle of 60% when it becomes
a minimum from the equality of the angles of incidence and
emergence ; and in other positions of the prism, the deviation
increases very slowly till it becomes a few degrees greater:
hence the breadth of the circles of each colour being consi-
derable, the colours must fall principally on each other, and
become very indistinctly separated. The external circle may be
referred to the effect of two such refractions in succession : Mr.
Cavendish seems to have thought the angle somewhat too great
to be derived from this source ; and he suggested that it might
depend on a single refraction by the rectangular terminations

• Lectures^ vol. i., p. 442.

No. XV. CHROMATICS. 285

of the crystals : but it does not appear that such terminations
are very commonly observable ; and it may easily be shown,
that the greatest intensity of the light of a halo, formed by two
refractions, must be at more than twice the distance of the edge
of the inner halo, derived from one only.

These halos are commonly accompanied by a white horizontal
circle passing through the sun, derived from the reflection of
the vertical fru;es of the crystals, which are scattered equally
throughout all possible azimuths. There are also generally
coloured parhelia on each side, depending on the refraction of
these vertical prisms ; they are commonly a little without the
halos, because the deviation of the light passing obliquely
through Ihese crystals is somewhat greater than that of the
light transmitted by the crystals which have their axes perpen-
dicular to the plane of incidence and refraction. For a similar
reason, the light passing through the crystals situated horizon-
tally, in various azimuths, is variously modified, so as to produce
the appearance of inverted arches, touching the halos at their
highest points, and sometimes expanding in the form of a pair
of wings, with a point of contrary flexure on each side.

The anthelia seem to be referable to two refractions and an
intermediate reflection, within the same crystal, causing a devi*
ation of about 120 + 22 = 142^ ; and sometimes with two inter-
mediate reflections, produdng an angle of 60 + 22 = 82° only.
It is not, however, very easy to assign a reason for the appear-
ance of an anthelion exactly opposite to the sun, which is said
to have been sometimes seen in the horizontal circle : but it has
been delineated with tiie accompaniment of an oblique cross,
and of other unusual appearances, which must have been derived
fit>m some extraordinary forms of the compound crystals of
snow, existing, at tiie time of the observation, in the atmos-
phere.

Section HI. — Of the Colmrs of the Rainbow.

The general nature of the primary rainbow was cursorily
explained by De Dominis ; but Descartes first applied the trfte
law of refraction, which has lately been discovered, to the
determination of the angular magnitude both of this and of the

286 CHROMATICS. Ko. X^.

secondary rainbow ; although no sufficient reason oould be
assigned for the appearance of colours in either of them, until
Newton ascertained the different refrangibilities of the different
kinds of rays : but as soon as this discovery was established, the
method of fluxions at once enabled him to determine precisely
the limit, at which the broad expanse of light, belonging to
each colour, must necessarily terminate in an edge of greater
brilliancy ; the bright edges of the different colours projecting
gradually beyond each other, so as to form a spectrum some-
what mixed, but still approaching to the common appearance of
a spectrum obtained by the refraction of a prism : and in fact,
the angular distances of the exterior termination of the primary
rainbow and of the interior of the secondary from th% sun are
found to agree very accurately with the calculation of the ex-
treme deviations of the red rays reflected once and twice respec-
tively within the spherical drops of rain ; although the whole
breadth of the coloured appearances is liable to variations depen-
dent on the magnitude of the drops, and belonging to the phe-
nomena of supernumerary rainbows, to be described hereafter.
The light reflected from very small portions of water appears
to be incapable of producing a regular rainbow; thus we
scarcely ever see a rainbow in a cloud, unless it has united its
drops, so that they begin to descend in the form of rain. Dr.
Smith has observed this circumstance, and has attributed it to
a tendency of the bright edge of the expanse of light to lose its
intensity, by being gradually dissipated into the neighbouring
dark space : a tendency which he would probably have been
much at a loss to explain from any of the received doctrines of
optics, but which bears some analogy to the efiects more com-
monly observed in beams of light admitted into dark spaces, and
sometimes designated by the term difiraction.

Section IV. — Of Periodical Cohura in general.

By &r the greatest part of the phenomena of colours, except
their separation by simple refraction, are referable to the
description of periodical or recurrent colours : being character-
ised by an alternation, which is generally repeated, where the
observation is sufficienUy extensive, several times in succession,

No, XV. CHROMATica 287

while the circumstances, on which they depend, are varied uni-
formly and by slow degrees. The number of these alternations,
when light perfectly homogeneous is employed, appears to be
continued without any discoverable limit, although it is always
smaller, for any given change of circumstances, when the least
refrangible or red light is employed, than when the observation
is made on the most refrangible or violet ; so that mixed or
white light always produces a combination of alternations
arranged according to a series of diflerent intervals, which are
at first more or less distinct, but by degrees are so mixed with
each other, as again to be lost in the general effect of white
light. In all these cases, the appearances may be reduced to
calculation by means of the general law of the interference of
two porti<Mis of light, with its appropriate modifications and cor-
rections.

A. — T/<6 law iSf that when two equal partiaiu cf lights in
circumstances exactly similar^ have been separated and caitunde
againy in nearly the same direction^ they will either co-operate^
or destroy each other ^ accordingly as the difference of the timesy
occupied in their separate paths^ is an even or an odd multiple of
a certain half interval^ which is different for the different colours^
but constaritfor the same kind of light.

B. — In live application of this law to different mediums, the
veloaty must be supposed to be inversely as the refractive density.

C. — In reflections at the surface of a rarer medium^ and of
some metalsy in aU very oblique refUctumSj in diffractions,'* and in
some extraordinary refractions^ a half interval appears to be lost.

D. — It is said thaty according to some late observations of Mr.
AragOy two portions of lights polarised in transverse directionsy
do not interfere with each other.

K-*- The principal intervals in air are for the

Extreme Red 0000266 =^7^tf

YeOow 0000235 =Trfnr

Green 0000211 ^^{^

Blue 0000189 = Ti+r>

ExtremViOet .... .0000167 « yyjry
Meany or White 0000225 = ttttt inch.

• See No. XVII., p. 392, note.

288 CHKOMATICS. No. XV.

Section Y.—Oft/ie Colours of Thin Plates.

The colours exhibited by Tery thin plates of transparent or
semitransparent substances have been well known to optical
philosophers, from the time that they were first noticed by
Boyle, and more particularly examined by Hooke and Newton.
They may be readily observed, by pressing together any two
clean pieces of common plate-glass, which have always sufficient
convexilies and concavities to exhibit them, touching each other
in some points, and leaving elsewhere a thin plate of air
between them ; or still more conveniently, by selecting from
the planoconvex lenses, kept by the opticians, such as have
their flatter ades very slightly convex, and are consequently
calculated to throw the spaces of equal thickness, and the
colours dependent on them, into the form of rings. The colours
are most distinct when they are formed in the light reflected
from the two surfaces in contact, especially when care is taken
to exclude the foreign light, reflected by the surfaces not con-
cerned in their production : and in this case they be^n from a
central dark spot, immediately surrounded by a bright light,
and then by rings more distinctly coloured ; while the colours,
exhibited in light transmitted through the glasses, begin from
a bright spot in the centre, surrounded by a dark ring ; being
always exactly complementary to the colours seen by reflection;
to which tiiey are also, as Mr. Arago has demonstrated, either
exactly or very nearly equal in intensity ; although they have
generally been supposed to be much less vivid, on account of
the diminution of their efiect on the eye by their mixture with
the whole of the beam of light which aflbrds them. But if we
employ, for the observation, two flattish pieces of glass, held in
such a position as to transmit the light received from one part,
and to reflect an image of another part, of an object equally
illuminated throughout its extent, the two series of colours will
destroy each other, and the whole appearance of rings will
vanish : when, on the contrary, the illumination of the object
varies materially, the rings will reappear in one or the other of
their forms, according to the diflerent intensities of the lights

No. XV. CHROMATICS. 289

received from its different parts : so that, as Mr. Arago has in-
geniously suggested, this test might he employed to answer the
purpose of a photometer, for ascertaining the equality of the
lights of two distant objects.

If any thin plate, affording colours, be inclined to the direc-
tion of the light passing through it, the appearance of the
colours will be changed either precisely or very nearly in the
same manner as if the thickness were reduced, in the ratio of
the radius to the cosine of the inclination within the plate ; at
least, if this proportion is not perfectly accurate, the deviations
from it, in the experiments of Newton, are manifestly within the
limits of the unavoidable errors of observation.

We are indebted to Mr. Arago for the important fact, that
the colours, observed in transmitted light, are distinguished by
a polarisation opposite or transverse to that which is appropriate
to transmitted light in general, and possessing the ordinary cha-
racter of the polarisation produced by partial reflection. It
is in light thus reflected that we must seek for one of the two
portions which are to be combined according to the laws of
interference, in the case of the colours seen by transmission,
and for both in the case of reflection. The light transmitted
simply through the plate will be followed by a portion which
has been reflected back from the second surface to the first, and
forwards again from the first to the second, and the difierence
of the times, occupied in these different paths, will obviously be
proportional to the thickness of the plate, and also, according
to the modification (B) of the law, to its refractive density : so
that the number of alternations of any given colour, between
the central spot of the rings and any given point, will be as the
thickness of the plate at that point ; and the numbers for dif-
ferent colours will be inversely as the magnitudes of the appro-
priate intervals ; the plate appearing light, when illuminated by
a homogeneous colour only, where the thickness corresponds to
any exact multiple of the interval, and dark at the intermediate
points; and this proportion is found to agree perfectly with
experiment. The two reflections within the plate, being always
of the same kind, will either not require any correction on
account of their nature (C), or will together add a whole inter-

VOL. I. u

290 CHROMATICS. No. XV.

val to the length of the path, an alteration which makes no
change in the appearances.

When the incidence is oblique, the actual length of the two
passages of the reflected ray across the plate, AB, 6C, is as
twice the secant of the angle of refraction ABD, and its
advance upon the surface, AC, as twice the tangent : and this
adyance, reduced to the direction of the transmitted ray AE
without the plate, must be subtracted from the retardation within
the plate ; the reduction being in the proportion of the radius to
the sine of the angle of incidence AC£, for which if we substitute
that of the radius to the sine of the angle of refraction ADF or
CDG, we shall have the deduction required to be made from

the length of the path within the plate, since the velocities vary
directly as these sines ; and by this deduction the secants, AB,
BC, will be reduced to the cosmos BF, BG : so that the true
retardation will always be proportional to the cosine of re-
fraction.

The same demonstration is applicable to the difierence of the
paths of the two portions of light reflected once only from the
upper and lower surfaces of the plates respectively ; supposing
A, the point of emergence of the transmitted ray, to become the
point of incidence of a new reflected ray HA. Hence it might
be expected that all the phenomena of colours should be ^e
same as in the case of transmitted light ; and this really appears
to happen when the observation is made on a plate of air con-
tained between a transparent substance and a polished surface
of gold or silver ; or on a plate of a refractive density inter-
mediate between the densities of the neighbourbg substances,

No. XV. CHBOMATICS. 291

as in the instance of a thin coat of smoke or of an oxide,
adhering to any polished metallic surface, which is at first of a
yellowish white, and, as it becomes thicker, changes to a yellow
and an orange colour ; but in more common cases there is a loss
of half an interval in one of the two reflections only, so that the
thicknesses affording a perfect coincidence, for any species of
colour, are always intermediate between the thicknesses affording^
the same colour by transmission ; and hence the tints of the two
series of rings are always complementary to each other, the
series seen by reflection always beginning from a dark central
spot, when they are exhibited by any detached transparent sub-
stance, as a soap bubble, a thin film of glass, or of talc, or by a
plate of air contained between two plates of glass, or between
a plate of glass and a piece of polished steel.

There is a peculiarity in the surface of silver and gold, and
perhaps of some other metals, that, besides the regular reflec-
tion at an angle equal to that of incidence, a considerable
quantity of light is dispersed irregularly ; and this light, as Mr.
Arago has observed, is polarised in a direction transverse to that
of the usual polarisation by reflection ; there is also in the
irregular reflection no loss of a half interval ; so that it exhibits,
with a piece of glass, a series of rings resembling those which
are produced by polished steel, except that their dimensions are
not varied exactly in the same proportion by the obliquity of the
incidence, because the light which forms them is not required
to pass towards the metal in an angle exactly equal to that
which it makes upon its return after reflection ; and there will
probably be considerable irregularities in the interval of retar-
dation, according to the mode of performing the experiment ;
although in general the irregular dispersion or diffiraction from
the glass is too weak te afibrd colours easily observable, when
the poation of the plate differs considerably from that in which
the light is regularly reflected. If a portion of polarised light
is incapable of interfering with another portion polarised in a
transverse direction, these rings ought to disappear when the
angle of incidence on the plate of glass is about 55"", since in
this case the light reflected by it is completely polarised in the
plane of incidence ; and this disappearance seems actually to

u 2

292 CHROMATICS. Na XV.

liave been observed in some of Mr. Arago's experiments, though
in others, where the metallic surface was less highly polished,
the polarisation of the dispersed light may have been less com-
plete, and the rings may still have been visible at this angle.
(MSmoirescTArcueilj Vol. III. p. 354, 859.)

Section VI. — Of the Colours of Double Plates.

When light is transmitted in succession through two plates,
differing but little in thickness, they exhibit an appearance of
colour similar to that which would be produced by a single
plate equal in thickness to their difference ; and this appear-
ance is wholly independent of the distance of the plates from
each other. It was first noticed by Mr. Nicholson, in the
glasses employed for the sights of sextants, and is attributed by
Dr, Young to " the rays twice reflected within the first glass
only, interfering with the rays twice reflected in the second
only:" in some circumstances, however, the light returning
from the second glass to the first, and again reflected by it,
may co-operate in the effect ; the interval of retardation being
the same in both cases. Mr. Knox has more lately described
some very striking appearances of colours obtained in this way,
by the combination of two pairs of lenses, each exhibiting their
appropriate rings when viewed separately, and affording together
a Uiird series of rings of larger dimensions, when the two former
are unequal in magnitude, and of straight bands when they are
eqnal. It is in fact easily demonstrable that in order that the
thicknesses of the plates of £dr, contained between two unequal
pairs of lenses, may be equal, the distances from the centres of
contact must be in a constant proportion ; and it is well known
that all the points, from which the lines drawn to two given
points are in a constant proportion, will be found in the circum-
ference of a circle, the diameter of which is a third proportional
to the difference and sum of the segments of the given distance
of the points ; so that the colours depending on this difference,
instead of beginning, as usual, from a white central spot, will
begin from a white ring, and will be arranged in concentric
rings on each side of it, precisely in the same order as when

No. XV. CHROMATICS. 293

they fonn concentric rings round an actual point of contact :
and when the curvatures of the two pairs of lenses are equal,
the diameter of the circle becoming infinite, it will obviously be
converted into a right line.

Dr. Brewster has observed a series of similar phenomena,
produced by two plates of equal thickness, but forming a small
angle with each other, so as to be differently inclined to the
li^t passing through them. The effect of the inclination
being to reduce the virtual thickness of the plate in the
ratio of the cosine, and the difference of the cosines of equi-
different arcs being simply as the sine of their half sum, it is
evident that the colours must correspond to a thickness which
varies nearly as the sine of the angle of incidence, considered
with regard to a plane bisecting the angle formed by the plates :
and this result agrees correctly with Dr. Brewster's experi-
ments.

Section VII, — Of the Colours of Supernumerary Rainbows
and Glories,

Within the common primary rainbow, and without the second-
ary, we sometimes observe a partial repetition of colours, more
or less distinctly marked, and extending occasionally to seve-
ral alternations ; the repetitions occupying somewhat narrower
spaces as they are more remote from the ordinary bows. These
appearances seem to have been first described by Mariotte ;
they have been since noticed by Langwith, Daval, and Dicque-
mare : and the term supernumerary rainbows has been very
properly applied to them. The coloured circles, called glories,
may generally be seen surrounding the shadows of our heads,
when we have an opportunity of standing on a high hill, and
observing them in a cloud below us : they are also sometimes
accompanied by a large white circle, which>' in an observation
of Ulloa, was 67*" in diameter; and such a circle may fre-
quently be distinguished when the sun shines on a mass of
vapour rising from a warm bath, of nearly the same dimensions,
or sometimes a little smaller. The whole of these phenomena
may be explained from the interference of some of the portions
of light regularly reflected within the minute drops of water

2d4f

CHROMATICS.

No. XV.

with other portions, incident at a different angle, but, after an
equal number of reflections, coinciding ultimately with them in
direction ; supposing only the clouds in question to afibrd a
number of these drops varying but little from each other in
diameter. We find by the well known mode of calculating the
greatest deviation that each order of reflections exhibits a zone
from 80*" to 10*" in breadth, through which a double light is
diffused by each drop ; and, besides this, when there have been
more than three reflections, the portions belonpng to the oppo-
site sides cross eadi other in one or more points, and surround
the drop ; or rather the observer, if we consider the effect of
the refraction of a multitude of drops situated in all directions.
Suppodng the index of refraction for the extreme rays 1.336,
and its logarithm .1258000, the results will be these—

After

Extreme Deviation.

Final Deviation.

1 reflection

41°

40'

13°

52'

2 „

8 „

4 „

5 „

6 „

34
ji*^*«

1 !•

We may obtain a more distmct idea of these duplicatures if
we represent them in a diagram, showing the angular extent of

■e

No. XV.

CHBOMATICa

295

the difiusion of light derived from each order of reflections, and
distinguishing by different kinds of lines the portions belonging
to the opposite halves of the drops : and it will be obvious, from
the inspection of this figure, that the appearances in question
have only been observed within some of the duplicatures of the
orders to which they belong, between the angles of extreme and
of final deviation. The tertiary and quaternary bows (III. IV.)
are evidently too near the luminary to be visible : the quinary
(V.) ought to be seen in the space between the primary and
secondary, but it is probably much too faint to be visible under
any circumstances. The duplicature belonging to the primary
rainbow exhibits two portions, for which we may calculate the
interval of retardation in parts of the radius of the drop, sup-
posing the velocity to be that which is appropriate to the air,
by taking twice the difference of the cosines of incidence on the
drop, and multiplying twice that of the cosine of refraction by
the index 1 .336 ; the difference of these differences giving the
interval for the two portions, of which the direction has been
found to coincide by a previous calculation.

Distance
from the

0°

Angle of
Reflection.

40° 2'
f42 591
136 231
144 2

134 32
44 45)

33 3
45 20
31 45

Difference

Distance

Angle of
Reflection.

of the
Paths.

from the
Edge.

.0000

5°

[45° 46

30 34

.0014

(46 9)

•

29 26

.0040

46 45

27 20

.0074

10
12

47 12
[25 24

.0113

47 321
23 33]

Difference
of the
Pathfl.

.0160
•0210
.0327
.0461
.0612

Hence it may be inferred that, supposing the extreme red to
re-appear at the distance of 2** from the primitive external
termination of the rainbow, the radius of the drop must be

•^^^^^^ = .00665, or yiir of an inch; the fourth alternation of

.004

the red being at the distance of 5% where the interval is .016.
The magnitude of the interval, at an equal distance from the
edge, varies but little with the refractive density : thus, for

296 CHROMATICS. No. XV-

yiolet light, the index of refraction being probably about
1.346, and its logarithm .1290000, the greatest deyiation will
be found 40° 14' ; and for a deyiation 2° less, the angles of
refraction must be 43° 30' and 33° 47', and the interval will be
little different from .00400.

The supernumerary bands of the secondary bow, formed by
the same drops, will be a little broader than these, ance it
appears, from a similar calculation, that the rays interfering
with each other, at the distance of a degree from the edge, will
exhibit an interval of .0011 of the radius only, instead of
.0014.

The supernumerary colours of the third and fourth bows will
be equally imperceptible with the bows themselves: but the
portions of light, four times reflected, will cross each other in
the point opposite to the sun, where their coincidence will be
perfect, and at other neighbouring points will afford an interval
nearly proportional to the distance from that point. We shall
find that the intervals for different deviations, supposed to be
measured in air, are these : —

DeviadoD.

Angle of Reflection.

inierrai ii
of the R

180°

24°

49'

.000

185
175

25
24

.096

190
170

26
23

in .

251

.195

Hence, supposing the first bright or greenish ring to appear
at the distance of 5° from the observer's head, the radius of

0000225

the drops must be about q^^ — = .000234, or -rijv of
an inch.

It might be questioned, whether the light, five times reflected,
could retain sufficient force to produce any sedsible effects by
these interferences, but since it exhibits no appearance of colour
between the primary and secondary rainbows, it must necessa-
rily be extremely faint. The interval which it affords, by the .
comparison of its two portions, agrees sufficiently well with that
which is derived from four reflections, to contribute in some
measure to the production of an alternation of light and shade ;

No. XV. cHROMATica 297

but the separate colours would be rather weakened than
strengthened by the mixture : dius, at the deviation of 5^, the
interyal is found to become .076 instead of .096 ; and at 10°^
.155 instead of .195 : and this difference is too considerable to
allow us to expect any material increase of brilliancy firom the
addition of the fifth reflection, however great its intensity
might be.

Supposing now a cloud to consist of spherules of which the
radius is .000234, we may inquire at wliat distance from the
outer edge of the primary rainbow the first additional red
of the supernumerary colours ought to be found : the interval

being in parts of the radius '^000234 ^ *^^^ ' *"*^ ^® ™*y
infer from the table, by taking the successive differences^ that
this distance will be about 18^; so that the semidiameter of
this red ring will be 42 — 18 = 24°: and the termination of
the primitive band of red, suppodng it to extend to one
fourth of a complete interval only, will be where the differ-
ence is .029, or at 7^° ; but for the violet the quarter of the in-

•0000042

terval will be, in parts of the radius, "^omT ^ -0183, which
answers to a distance from the edge of about 5j^° : and this
distance, measured from the edge of the violet, which is
somewhat less than 2° within that of the red, will extend
nearly to the same point as the red space : so that we shall
have a circle, about 70° in diameter, at the drcumference

of which all the colours will be united, and which will

consequently be white. This magnitude agrees tolerably well
with the direct observations of the phenomenon; and if we
wish to make the agreement more complete, we have only
to suppose the drops a little smaller, and the coloured glories,
which they are capable of aflbrding, a little larger. It has
already been remarked, that the non-appearance of the ordinary
rainbow, in this case, must be referred to the operation of some-
thing like diffiraction; although it is obvious that its form,
under such circumstances, would necessarily be somewhat
modified by the diffusion of the colours through a greater
space than that which they ordinarily occupy.

298 CHROMATICS. No. XV.

Section VIII — Of the Colours of Striated Substances.

It was obaerred by Boyle, that small scratches of any kind,
on the surfaces of polished substances, exhibited, when viewed
in the sunshine, a variety of changeable colours ; and the ob-
servation may easily be repeated with any piece of metal, not
too highly polished, and placed in a strong but limited light
Dr. Young ascertained by experiment that the colours afforded
by some regular lines, drawn on glass, always corresponded to
an interval, varying as the sine of the angle of deviation from
the position, in which an image of the luminous object was
exhibited by the regular reflection of the surface ; and it is
easily shown that, if we suppose two portions of light to be
reflected upon the opposite edges of the furrow, the difference
of their paths must vary in that proportion. Dr. Young had
conjectured that the colours of the integuments of some of the
coleopterous insects might be derived from furrows of this
nature ; but the conjecture has not been verified by observation.
Dr. Brewster has, however, very unexpectedly discovered that
some similar inequalities are the cause of the colours exhibited
by mother of pearl ; and he has confirmed the observation by
showing that impressions of the surface of this substance, taken
in black wax, in a hard cement, or in fusible metal, will often
exhibit a similar appearance. Where the form of the surface
of the mother of pearl is the most regular, it reflects, in an
oblique light, a white image of a luminous object, like that
which any other polished substance affords ; but on one side of
this image only, and at some little distance from it, we may
observe the first order of recurrent colours, beginning from
violet^ and occasioned in all probability by the reflections from
one side only of an infinite number of parallel strise, formed by
the terminations of a minute lamellated structure, nearly, but
not perfectly perpendicular to the general surfiice; one side
only of each of the little furrows being situated in such a
direction as to reflect an image of the luminous object to the
eye, and at such a distance that the whole may constitute a
regular series of equal intervals. By transmitted light this

No. XV. CHROMATICS. 299

substance generally appears of a red or a green coloar, chang-
ing more or less according to the obliquity, and apparently
belonging to some of the higher orders of recurrent colours.

Dr. Young has observed a series of these colours, produced
by the parallel lines of* some of Coventry's glass micrometers,
drawn at the distance of tIt of an inch from each other, in
which the first bright space, or. the confine between the green
and the red, corresponded to the interval of Tvi-sr of an inch,
or ,0000232;* and this result agrees very acciu'ately with
the general theory, the interval for the yellow, derived from
Newton's measurements, being .0000235 ; but in general these
lines exhibit colours much more widely extended, each separate
line consisting in reality of two or more scratches, at a minute
distance fix>m each other.

There is a remarkable peculiarity in the appearance both of
these colours, and of those which are exhibited by substances
naturally striated, as by mother of pearl, agate, and some other
semi-transparent stones; they lose the mixed character of
periodical colours, and resemble much more the ordinary pris-
matic spectrum, with intervals completely dark interposed.
This circumstance may be satisfactorily deduced from the
general law, if we consider that each interference depends not
only on two portions separated by a simple interval, but also on
a number of other neighbouring portions, separated by other
intervals which are its multiples ; so that unless the difference
of the two paths agrees very exactly with the interval appropri-
ate to each ray, the excels or defeat being multiplied in the
repetitions, the colour will disappear ; consequently, each of the
stripes which, in other cases, divide the space in which they ap-
pear almost equally between light and darkness, when homo-
geneous light is employed, becomes here a narrow line ; and
their succession affords a spectrum exhibiting very little mixture
of the neighbouring colours with each other, and nearly resem-
bling that which is afforded by the simple dispersion of the
prism ; except that, as in all other phenomena of periodical
colours, the blue and violet portions are much more contracted
than in the common spectrum.

♦ See p. 356 of this volume.

300 CHROMATICa No. XV.

Section IX. — Of the Colours of Mirrors and of thick

Flatea.

In all the species of periodical colours which hare been de-
scribed, the two portions of light conceraed have both been regu-
larly reflected from difierent surfaces. The methodical division
of the subject now leads us to the consideration of the colours
exhibited in light separately reflected from the same surface.
These may be denominated in general the colours of mirrors ;
and they will include, as a variety, those which are called by
Newton the colours of thick plates.

The general character of these colours is, that they are
observed in light reflected by small particles, or irregularly
dissipated by a single surface, first in the passage of the beam
of light towards the mirror, and then in its return : the difference
of the length of their paths affording, as usual, the interval of
retardation. Thus in Dr. Herschel's experiment of scattering
a fine powder in a beam of light reflected perpendicularly by a
concave mirror, and received on a screen in its return, it may
easily be shown that the colours will be precisely such as would
be exhibited by light transmitted through a tUn plate of air,
everywhere half as thick as the plate limited by two spherical
surfaces in contact ; the centre of the one surface being the
particle of powder, and that of the other its image formed by
the mirror. For in the direction of the principal ray, which is
perpendicular to the mirror, the paths of the light will be of
equal length, whether the dissipation takes place before or after
the reflection ; and in other parts, the whole length of the path
of the light passing fi:om any focal point to its conjugate focus
being the same, according to the definition of a conjugate focus
in the Huygenian theory, from whatever point of the mirror it
may be reflected, the light first dissipated will have advanced,
after its reflection, as fiair as the circumference of a circle, of
which the conjugate focus is the centre, at the same instant that
the portion coming directly from the powder, after a previous
reflection, will reach the circumference of the circle of which
the particle of powder is the centre ; so that the distance be-

No. XV. CHROMATICS. 301

tween these tvro circles must be the difference of the paths of
the two portions, and the colours the same as would be exhi-
bited by a plate of air of half the thickness, since such a plate is
twice traversed by the retarded light.

A similar appearance of colours had been obtained, by
earlier experimenters, from the interposition of a screen of
gauze, or of a semi-transparent substance, in the path of the
beam falling on the mirror. But the colours of Uiick plates,
obserred by Newton, are modified by the nature of the trans-
parent substance employed, and by the obliquity of the refracted
light The dissipation here takes place at the anterior surface
of a concave mirror of glass, and the reflection at the posterior,
which is coated with quicksilver : and if these two portions pro-
ceed, each with a slight divergence, from a perforation in a
screen situated near the centre of curvature of the mirror, they
will co-operate perfectly with each other in the circumference
of a circle described on the screen, of which the diameter is the
distance of the perforation frt>m its image ; since all the light
passing, in any given section of the mirror, with the same obli-
quity, through the glass as the beam itself passes in the prin-
cipal section, must be collected into a focal point situated in
some part of this circle, and will arrive at this point .at the
same time, whatever its situation in the section may have been :
the obliquity of the incident light being the same in every part
of the section, because the point of divergence is at the same
distance from Uie mirror as the centre of curvature. For the
other parts of the dissipated light, passing with different obli-
quities, the interval will be determined by tiie difference
between the lengths of the paths of the two portions of light
arriving at the given point, the one by regular refraction, after
being first dissipated and then reflected; the other by dis-
sipation, after being first regularly refracted and reflected.
And this interval agrees precisely with the law which Newton
has deduced from his experiments ; but the analogy which he
infers from it, between these colours and those of thin plates, is
in fiEu^t very far from amounting to identity : since, if they
belonged to the ordinary colours of thin plates, there is no
reason why the series should begin anew frx>m a certain arbitrary

302 CHBOiCATiGS. Na XV.

thickness, differing in every different experiment, which affords
a white of the first order.

Section X. — Of the Colours of ducted Light.

We are next to examine the case of light only once reflected,
and interfering with a portion of the same beam which has pur-
sued its course without interruption: a case which would
scarcely have required a separate consideration, but from the
difficulty of including it in a general definition with any others ;
although it is comprehended in the Newtonian description of
the colours of inflected light : but since the light is in this case
turned away from the substance near which it passes, it may
more properly be termed deflected, especially as the greater
number of the appearances* mentioned by Newton, as depending
on inflection, belong more properly to difiraction, and the term
inflection might consequently be misunderstood as relating
to them.

When a beam of light is received in a dark room, and
suffered to fall upon the edges of two extremely sharp knives
or razors, meeting each other in a very acute angle, the shadoifs
of the knives, received on a screen at some distance, will be
found to be bordered by several fringes of colours ; and the
angle will be bisected by a dark line. The distances from the
shadows, at which these fringes appear, agree in general with
the supposition of their depending on the interference of the
li^t, reflected from the edges of tlie req>ective knives, with the
uninterrupted light of the beam passing between them : but the
coincidence of these portions ought to be perfect in the imme-
diate neighbourhood of the pdnt in which tt^ shadows meet, and
the two last bright frii^es ought to unite there in an angle of
light. This, however, does not happen on account of the modi-
fication of the general law (C), which makes it necessary to allow
half an interval for the effect of a very oblique reflection : and
for the same reason, the space immediately next to the shadow
is always dark instead of being light If the knives are at all
blunt, the reflection from one to the other, where they meet,
causes the bisecting dark line to disappear ; but this source of

Na XV. CHROMATICS. 303

error may be avoided by causing one of them to advance a little
before the plane of the other.

Mr. Fresnel has repeated these experiments with all possible
care, and has ascertained that the points, in which the fringes
of any one colour are found, at different distances from their
origin, belong always to a hyperbola, as they ought to do
according to the calculation founded on the general law of
interference ; a fact which had before been inferred from other
measurements, but which had not been so distinctly proved by
direct experiments. Newton himself, indeed, was so fiir from
believing that these fringes are rectilinear^ as Mr. Fresnel
supposes^ that he expressly mentions their curvature, and infers
fit)m it that they are not derived from **the same light" in all
their parts ; imagining, perhaps, that each fringe was of the
nature of a caustic line, formed by reflection or refraction, in
which the light is everywhere more condensed than in the col-
lateral spaces, but which is by no means necessarily strai^t
Mr. Fresnel has also shown, that all the fringes are found
exactly at such distances from the true shadow, as would be
inferred from the supposition of the loss of half an interval by
reflection ; while some of the experiments of Newton appeared
to indicate a deviation from this law. It has been asserted, that
fringes of the same kind have been observed at the edges of a
detached beam of light, reflected into a dark space by a narrow
plane and polished surface ; and in this case it would be difficult
to point out in what manner the supposed oblique reflection
could be produced, or how a diffiraction of any kind could cause
the light to be redoubled back upon itself: but the experiment
does not appear to haye been hitherto performed with sufficient
attention to all possible sources of error.

Section XI. — Of the Colours of diffracted Light; including
those of Fibres^ and of Coronce.

The light reflected from each of the knife edges, in experi-
ments like those of Newton, not only produces colours by its
interference with the light proceeding uninterruptedly between
them, but also with another portion, diverging from the edge

304 CHROMATICS. No. XV.

of the opposite knife, and spreading into its shadow, lliis
tendency of light to diffuse itself was first described by
Grimaldi, under the appropriate name difiraction : but many
of the phenomena, in which it is concerned, having been attri-
buted by Newton to other causes, he appears almost to have
overlooked its existence.

The general law of interference is very directly applicable
to all phenomena of this kind: the fringes exhibited are
broader in tlie same proportion as the distance betweeti the
edges is narrower ; and they always depend on the difference
of the distance from the edges as the interval of retardation.
It is however necessary to suppose the same modification to
take place in difiraction as in oblique reflection, half an interval
being lost in both cases ; since the light which deviates the
least from a rectilinear direction, and which is derived from the
near approach of the two paths to equality, is always white.
But it is remarkable, that when the obliquity becomes a very
littie greater, the difiracted light seems to change its character
in this respect; for the colours occupy the same spaces as would
have belonged to them, if they had begun from a dark centre,
one of the portions only having lost a half interval in comparison
with the other : and of this circumstance no explanation has yet
been attempted.

The difiraction producing these fringes may easily be
detected within the eye itself, by holding any object near it,
in such a position as to intercept nearly all the light of a candle
except a narrow line at the edge ; this line will then appear to
be accompanied by other lines parallel to it, separated from it
by a dark space, and becoming wider when the object is
brought nearer to the eye. These fringes must be referred to
the light difiracted on one side round the object, so as to be
spread on the unenlightened part of the retina, and reflected on
the other from the margin of the pupil : for if we employ an
object narrower than the pupil, so as to observe them on both
sides of it, their magnitude will be altered by any change
in the aperture of the pupil, occasioned by admitting light to
the opposite eye, or otherwise. In such cases as this, where
one of the points of divergence is much nearer to the point of

No. XV. CHROMATICS. 305

interference than the other^ the interFal increases more rapidly
than the distance from the primitiye direction ; and the first
fnnges are much broader than those which succeed them ; the
mode of their formation approaching to that of the fringes seen
in deflected light, commonly called the exterior fringes of the
shadow ; wlnle the interior fringes belong more immediately to
the present subject, that of the colours of difiracted light.

When the distance of the points of divergence is more nearly
equal, the one being collateral to the other, the breadth of the
successive fnnges is also more uniform. Such is the appearance
of the colours exhibited by a number of equal fibres held
between the eye and a distant luminous object : their origin
being identical with those of the fringes produced in the
shadows of the kmves ; except that the difiracted rays come
from the remoter side of the fibres, and follow the refiected
rays, instead of preceding them. These colours may easily be
observed by looking at a candle through a lock of fine wool,
and still more distinctly by substituting for the wool some
of the seeds of the lycopodium, strewed on a piece of glass ; and
they become very large if we employ a few of the particles of
the blood, or the dust of the lycoperdon, or puff ball. Dr.
Young has made this appearance the foundation of a mode of
measuring the fineness of wool, which he has recommended for
agricultural purposes, though it seems hitherto to have been
found much too delicate to be employed by " the hard hands
of peasants/* with any advantage. The instrument, which be
has invented for this examination, is called the eriometer, and
its scale is calculated to express, in semidiameters of a circle,
formed round a central aperture in a card, or a plate of brass,
and marked by minute perforations, the distance at which the
lock of wool must be held, in order thiat the first bright ring of
colours, or the limit of the g^reen and the red surrounding it,
may coindde with the circle of points : and the actual measure,
expressed by a unit of this scale, is found to agree very nearly
with the thirty thousandth of an inch. Thus the particles of
water, which have been found capable of exhibitmg a glory
5^ from the shadow of the observer, being about ttVt of an
inch in diam^eter, they would correspond to number 14 of this

VOL. I. X

306 CHROMATICS. No. XV.

scale ; and the cotangent of the angle subtended by the aemi-
diameter of the bri^t drcle being 14, the angle itself will be
about 4^ ; consequently, if we looked at the sun through such
a cloud, he would appear to be surrounded by a Inright drde
of colours, 8^ in diameter, green within, and red without, and
attended by other colours, more or less distinctly marked, accord-
ing to. the degree of uniformity of the magnitude of the drops.
These drcles are called coronsB : their dimenrnMis vary con-
siderably : but they have seldom been observed quite so large as
these drops would make them ; and more commonly they seem
to depend on drops about a thousandth of an inch in diameter;
although it is not easy to ascertain the precise parts of the
rings, from which the measures have been taken by different
observers.

In the shadow of a larger substance, formed in a beam of
light admitted into a dark room, these colours are still per-
ceptible, be^nning from a white line in the middle ; but here
both the pcMtions, on which they depend, are diffracted into
the shadow ; and beyond its limits, th^ are lost in the stronger
light that passes on each side of it. Their appearance is
somewhat modified, when the shadow is formed by a body ter-
minating in an angle ; for the breadth of the fringes being
inversely as the breadth of tiie object which forms them, it is
obvious that this breadth must increase towards the point of the
shadow, like the distance of the fringes formed in tiie shadows
of Newton's knives : and the fringes seen within the angle must
necessarily assume the character of hyperbolas : nor. will this
form be materially altered, when the angle becomes a right
one, as in the crested fringes, noticed by Grimaldi ; although
the steps of the calculation, for determining their magnitude,
are in this case a little more complicated.

We find, in an elegant experiment of Mr. Biot, on the fringes
produced by diffraction, a singular confirmation of the truth of
the theory, which derives these colours from the diflerence of
the times occupied in the passage of the different portions of
light to the point of interference : although this eelebrated
author does not seem to have been aware of the nature of the
inference which may so naturally be drawn from it He found

No, XV. CHROMATIOB. 307

tbat the densities of the snbstaiiceS) from the margin of which
the diffracted light originated, bad no influence whatever on the
appearances produced by them : but when they were formed in
the light diffracted from substances placed at one end of a long
tube, and observed on a piece of glass fixed at the other end,
they became contracted, upon filling the tube with water, in
the proportion of 4 to 3; as was to be expected from the
diminished velocity which must be attributed, according to the
modification of the general law (B), to the passage of the light
through a denser medium.

SscTiON XIL— Of the Colours of Mixed Plates.

The colours of nuxed plates depend partly on diffitiction,
and partly either on reflection or on direct transmission : but
their essential character consists in the difierent nature of the
two mediums, through which the light passes after its sepa-
ration.

When a minute quantity of moisture is interposed between
two lenses, it readily divides itself into a great number of
smaller portions, scarcely distinguishable by the eye : and the
light transmitted through the lenses exhibits rings of colours
much larger than those which are ordinarily observed, aild
depending on the interval afforded by the difference of the
velocities in the different mediums, according to the inverse
proportion of the refractive densities. If they are viewed in a
direct and unconfined light, the rings belong to the series
commonly seen by transmission, beginning frt>m a light central
spot: both portions passing in this case simply through the
separate mediums, and arriving at the eye after some slight
diflfraction only, which affects both of them in an equal degree :
but if a distant dark object is situated immediately behind the
lenses, and they are illuminated by a light incident a little
obliquely, their character is changed, and they resemble the
colours commonly seen by reflection, one of the portions of
light being necessarily reflected, as in the case of the colours of
deflected light: so that, when the dark object is situated behind
one half of the glasses only, we observe the halves of two sets

x 2

308 CHROMATICS. Ko. XV.

of rings, of opposite characters, exhibiting everywhere tints
complementary to each other. The diameters of the rings vary
according to the refractive density of the liquid employed,
diminishing as that density increases, and becoming much
larger when two liquids, incapable of mixing with each other,
and diflfering but little in refractive density, as oil and water,
are employed instead of air and a single liquid.

The magnitude of the interval may also depend on that of a
minute transparent solid substance, immersed in a liquid,
instead of being limited by the distance of the two lenses : thus
the dust of the lycoperdon, mixed with water, gives it a purplish
hue, when seen by indirect, and a greenish by direct light : and
when salt is added to the water, or oil is substituted for it, the
difference of the velocities being lessened, the colours exhibited
rise in the series, as if the plate were made thinner.

Mr. Arago has very ingeniously applied the principle of the
production of these colours, to the construction of an instrument,
for measuring the refractive densities of different elastic fluids,
and of air in different states of humidity ; the fluids being con-
tained in two contiguous tubes of a given length through
which tiie two portions of light are made to pass, previously to
their reunion, and to the formation of the bands of colours ; and
it may easily be conceived, that the delicacy of such a test
must be great enough fcrr every determination that can be
required, either for the correction of astronomical observatkma,
or for the illustration of the optical properties of chemical com-
pounds.

Section XIIL — Of the Laws of the Polarisation of Light.

The colours first observed by Mr. Arago, in doubly re-
fracting crystals, and since more particularly analysed by Mr.
Biot, afford by far the most striking and interesting examples
of the colours of mixed plates. In order to understand the
laws of these phenomena, it is necessary to be previously
acquainted with the affections of polarised light, which were
first accurately investigated by Mains, and with the theory of
extraordinary refraction, derived by Huyghens, with equal ele-

No. XV. CHROMATICS. 309

gance and precision, from his peculiar hypothesis respecting the
nature of the transmission of li^t

1. Mr. Mains discovered, that at a certain angle of in-
cidence, the light partially reflected, by a transparent substance,
receives a peculiar modification with respect to the plane of
reflection, which is called polarisation in that plane.

2. Dr. Brewster observed, that the angle of complete polari-
sation is such, that the mean direction of the transmitted light
is perpendicular to that of the reflected portion ; the tangent of
the angle of incidence being equal to the index of the refractive
density of the medium.

3. A ray of polarised light is agun subdivided, in the usual
proportion, by a second refraction in the plane of polarisation :
but when it is refracted in a plane perpendicular to the plane
of polarisation, by a surface properly inclined, there is no partial
reflection: and in intermediate positions, the intensity of the
reflection is nearly as the square of the cosine of the angular
distance of the two planes.

4. A portion of the transmiJttedW^i is polarised in a direction
perpendicular to that of the plane of refraction, so that none of
this portion is reflected by a second surface parallel to the first ;
and when there are several parallel surfaces in succession, the
whole of the transmitted light becomes at last so polarised, that
none of it is partially reflected.

5. The same transverse polarisation will happen, in a greater
number of transmissions, when the angle differs from that of
complete polarisation : and in the same manner a second partial
reflection, by a surface parallel to the first, will produce a more
complete polarisation, when the first is imperfect.

6. A perfect polarisation in any new plane, by a partial
reflection at the appropriate angle, completely supersedes the
former polarisation ; but a reflection or refraction void of any
polarising eflect, which may be called a neutral reflection or
refraction, changes the direction of the plane of polarisation,
according to Mr. Biot*s experiments, into that of the image of
the former plane, supposed to be formed by the action of the
l^ven surface.

7. The light ordinarily refracted by a doubling crystal in

310 CHROMATICS. No. XV.

the plane of the principal section of the crystal, pasring through
its axis, is polarised in that direction : the light extraordinarily
refracted in the transverse direction.

8. Li^t previously polarised is transmitted by the ordinary
refraction when its plane of polarisation coincides with the
principal section, and by the extraordinary when it is perpen-
dicular to it. In intennediate directions, the quantity of light
transmitted by each refraction is, according to Mains, as the
square of the cosine and sine of the angle formed by the planes,
passing through the paths of the ray, and a line parallel to the
axis in each crystal, supposing the species of refraction to be
exchanged.

9. The rays of light ordinarily transmitted by doubling
crystals appear in general to retain their previous polarisation,
like rays transmitted through simple substances ; but the ex-*
traordinary refraction polarises them, according to Biot, like a
neutral reflection at a surface coinciding with the principal
section ; the new plane of polarisation taking the place of the
image of the former.

10. Reflections at metallic surfaces are generally neutral
with respect to polarisation : but in oblique planes they seem,
according to some experiments of Mains, to mix or depolarise
the light subjected to them.

Section XIV. — Qfthe Laws of extraordinary Befraction.

The extraordinary refraction of regular doubling crystals
may be correctly determined in all circumstances, by means -of
the Huyghenian supposition of an undulation direrging in the
form of a spheroid, from eyery point of the medium, the velocity
in any given direction being always proportional to the cor-
responding diameter, so that the successive spheroidal surfaces
remain always similar to each other. The relations of ib^
angles of incidence and refraction may be calculated by finding
the point, in which any of the spheroids, supposed to represent
the forms of the elementary undulations, at a given instant, is
touched by a plane pasang through that point of the surface,
at which the original beam of light would have arrived, at ilie
same instant, through the external medium; it may also be

Na XV. CHB0MATIC9S. 311

deduced, somewhat more simply, firom the determination of the
yelodty with which an expanding spheroidal undulation must
extend itself on any given sur&ce : a yelooity which imme-
diately gives us the direction of the ray in the surrounding
medium ; and the relation thus obtained will also obviously
hold good with respect to a ray returning in the opposite
direction.*

In common refractions, if we compare the space described
by an undulation on any given sur&ce with the radius, the
velocities appropriate to the different mediums will be repre-
sented by the sines of the respective angles. But the velocity,
with which a spheroidal undulation advances on any surface, is
evidently determined by the increment, or the fluxion, of the
perpendicular to the circumference of the section of the spheroid,
formed by that surfiu^ ; and calling this perpendicular y, the
velocity may be considered as proportional to its increment y' :
but the velocity in the surrounding medium is to that, with
which tiie axis x increases, as r to 1, r being the index of tiie
ordinary refractive density of the crystal, compared with that
of the surrounding medium, since the velocity in the direction
of the axis is the same as that which belongs to the ordinary
refractive denaty ; consequently, the increment of the path of
the undulation in the surrounding medium will be expressed
by rz^ and .v, the sine of refraction or inddence without the
crystal, will be to the radius as rx' to y\ and will be expressed

by -r , or by r-j-, the evanescent increments of any quantities

being always in the ratio of their fluxions : and the plane of
refraction or incidence, without the crystal, will always be per-
pendicular to the tangent of the section formed by the refracting
surface. The determination of the relation of the angles is
therefore reduced to the calculation of the value of y and of
its fluxion.

Supposing then the ratio of the greatest and least refractive
densities of the crystal, or of the equatorial diameter of the
spheroid 2A6 to tiie axis 2AC to be that of n to 1, n being
greater than unity, and the tangent of the angle ADE, formed

* Supra, No XIV., p. 263.

312

CHROMATICS.

No. XV.

by the axis with the refracting suriaoe D£, being called p;
the magnitude of the semidiameter AF, parallel to the surfiuse,
may be found by comparing the secants of the angles FAG,
HAG, subtended at the centre by the corresponding ordinates

of the ellipsis and the inscribed circle : for their tangents, FG,
HG, being represented by p and -£•, the secants wiU be V (1+

jn*), and V [ 1 + -^ j ; and the

semidiameter of the circle, AH,
being x, that of the ellipsis, AF,

will be n V^rxS 4?. But the tan-

gent of the angle GIF, made by
the tangent of the ellipsis with the
axis, is to that of the angle made
by the corresponding tangent of
the circle, GIH or GHA, that is,

J, as n to 1 ; consequently, - will be the tangent of the angle
made with the axis by the elliptic tangent, IF, or by the con-
jugate diameter AK; and if we substitute — for^i, we shall

find the length of this semidiameter AK = V ^ . ^jt, which is

mi -f pp

to that of the former AF in the ratio pfV (»^-H^) to 1.

« V (1 + W)
Hence, for the lesser semiaxis of the section formed by the
given surface, EL, calling AL the distance of its centre from
that of the spheroid, z, we have the mean proportional between

No. XV. CHROMATICS. 313

the segments of the . diameter *J ([AK + AL] - [AK —
AL]) = V(AKq-ALq)= V(^^a^ - ;r*), which must
be reduced in the ratio of the semi-conjugate diameters AK
and AF, so that it becomes n ^ i^rr^^ - I^TtL'^^^

s EL. But from the known similarity of the parallel sections
of a spheroid, the axes will be to each other as the semi-

diameter AF = n V ^ . ^ a is to iwj the equatorial semi-
diameter, a ratio which may be called that of 1 to m, m being
= V \ jf, pp\ so that the lesser axis EL being = n V \^ —

^^^\ the greater LP will be n V (a?- ^Jl^ nf ^).

Now, if 9 be the cotangent of the angle MNE, formed by
the plane of the ray's motion, in the external medium, with the
lesser axis of the section, or the tangent of the angle ELO
formed by the conjugate semidiameter LO with the same axis,
this semidiameter may be found by substituting q for p, m for
n, and the value of the semiaxis of the section for x^ in the ex-
pression for AF, the semidiameter parallel to the refracting

surface, and it becomes m tj ^ EL = n V ««. . ^(~ -

WOT ^ fy MOT ^ ifff^ \OTOT

\\^z\ = LO. Hence, smce all parallelograms described

about an ellipsis are equal, dividing the product of the semi-
axes EL.LP by this semidiameter, we shall have the required

J. , j^^ EL.LP LP ,mm + qq n .

perpendicular y = MQ = -^5- = — V -yq^-^ = -=• V

^TT?^ (**-l?T^ "«"^)- Now, in order to find the
fluxion of this quantity, increasing as the spheroid increases,
while the place of the centre of radiation remains unaltered,
we must make z constant while x varies, and we shall have

^y = ^^TT^*^-^(^--i^ nf^y which must
be equal to -y- ; consequently, V (a?*— ^qp^ ^ ^) " miT *
^ mm + qq . ^^ ^ iesser scmiaxb of the section, EL, which

314 GHBOMATIGS. No. XV,

was found = -^ V (a^ - SH^ '"'^) » be<»™e8 -^ x
fj ^^^^ . whence the semidiameter L M at the point of

incidencey which may be called tr, and which is analogous to
the conjugate diameter AK in the former section, will be

V zzTT^ ' l^ii^^Tir^ = ^^ ^ I . I, g. Hence it is

jnm + qq mmr » + W ■•■if * + W

obvious, that this semidiameter, in any one plane of incidence,
will be in a constant proportion to the ones, as Huyghens him-
self demonstrated : so that, supposing x to be constant, and z
to vary, the semidiameter to may be considered as an ordinate
in the elliptic section passing through the point of incidence M
and the diameter AK conjugate to the refracting surface,
which is also the path of a ray falling perpendicularly on that
surface from without : and the tangent of the angle ELM
formed by this semidiameter with the lesser axis of the given

section, will be — , which determines the intersection of this

oblique plane with the refracting surface.

But in order to find the angle made with the refracting
surface in a plane perpoidicular to it, we must compute LR,
the distance of the centre of the refracting section from the
point nearest to the centre of the spheroid : and the tangents
of the inclinations of the diameters to the axis being p and

^ , that of their mutual inclination will be ■"* "^^^, , since tan (a
+ A) = ^l^ ^ft> ^^^ ^® ^^°® ^^ ^^^ ^™® angle being ex-
pressed by ^S' 5^ ^^^^^ here ^^ + ^ Vi^*^ -H^^) " ^°
FAK = an ALR, which we may call t, and the cosine "**"^**°

^ •' 8eca B6c6

* Jl^-^^Tji^+p) * ^ • *°^ ^^ required distance LR will be
tZf and the distance of the centre of the spheroid from the
refracting surface AR = rz. But MS, the perpendicular
falling, from the pomt of incidence, on the lesser axis of the
section formed by the surface, being called u, the tangent of

the angle MLS, subtended by it at the centre being '^, and

No. XV. CHROMATICS. 315

its sine consequently ^^^^^), we have ti = ^(Srqr^ «
r jn + fl«) > ^^ ^® distance of this perpendicular from the
centre, LS = t? = V0«*"+^ ' or if we call the sine of ordinary
refraction — ^p and the sine of the inclination of the plane of
the ray's motion to the lesser axis, "TTT+^j K *od its cosine
ja^nq) = A» we have u :=: v? pkx^ and » = ~ />**• Hence
the cotangent of the angle ERM, formed by the line nearest
to the ray in the section with the lesser axis, will be ^ » if

the value of # be considered as positive, when the ray is inclined
on the refracting surface towards the axis of the crystal ; for in
this case the sign of t being negative, tz or LR will be sub-
tracted from V or LS ; and the reverse when s is negative.
We have also for the hypotenuse RM, or the distance of the
point of incidence from the point nearest to the centre of the
spheroid^ V (u^ + [v + tzj) ; consequently* the tangent of
RAM, the angle of incidence or refraction within the crystal,

will be —^ — ^ —. Now since it has been shown that

VC^^-iir+^iw"^) =7SrVTT^ ^,wehave2« ^^^—^—^

"" mmit' 1 + «/ ^ *°^ ^^ cotangent of the inclination of
the plane of refraction £RM, or Liif = J. + Jl , becomes

^ « MM «

^ + ^|r^VO-:^t'^** + *V).^; and since

T*j?* s= (m* - «*/)• [mW + A*] ar", the tangent of the angle of
incidence or refraction within the crystal, which is = V ( ^
+ -^+^.^ + ^)wUlbe represented by V

/«« M* y H-n* y , 2p (1 - wt) wwAg

^iii*(m« - i**^[iii»*» + A«]) '^ + (iw+/]p)mM^(m«-nV [»»•*•+*•])

+ fiiirr^T)' ''^^ ^*^"® ^' *^® perpendicular to the
surfEioe, AR or r2, is also of importance as immediately indi-

316 CHROMATIOB. No. XV.

eating, by its proportion to the axis ir, the velocity of the un-
dulation in the direction of the depth, which is therefore repre-
sented by V (»i" - ff^nflf + ^'l).

These expressions become somewhat simpler in many cases
of common occurrence ; thus, when the axis is parallel to the
8urfeoe,p = 0, m = n, and f = 0, consequently, the tangent of

refraction is -^ V ^^^^^^^, and the perpendicular velo-
dty n V (1 - [«*A* + **] f)- When the axis is perpendicular
to the surface, p is infinite, m = 1, and t is again = o ; and
the tangent of the angle of refraction is T/TY^r;^^)' the perpen-
dicular velocity being V (1 - vfp^).

The retardation, produced by the passage of light through
such a plate, being equal to the time occupied within the plate,
diminished by a time proportional to the product of the tangent
of the angle of refraction and the sine of the angle of inddence
(see Section V.) ; it will be expressed in the case of a plate

parallel to tiie axis, by ,^ (^^ [^ ^ ^j^^) - sp ^
1 - innkk + AA) > ^^^ yfhen the axis is perpendicular to the

P^ate, by ^(i^nuf,) - V a - nn,0 " Vd - nnfO = ^ V

(1 — nnpp). The efiect of any small change in the form of the
spheroid, on the retardation, may be found from the fluxions of
these quantities, supposing n to vary; which when properly

reduced, making n = 1, will be — r y (i"- ) ^ *°^

- 7(?^ d» respectively.

The values of r and n, for the principal substances, exhi-
biting the extraordinary refraction, which have been examined,
are these: —

Iceland crystal . .

r = 1.657 n:

= 1.1140 = 10:9

Arragcmite . .

1.693

1.1030 = 11 : 10

Ice ....

1.810

.9989 =890:891

Quartz . . .

1.558

.99444 = 179: 180

Sulfate of lime .

1.525

.99432 = 175: 176

Snl&teofbarita

1.635

.99295 = 142 : 148

No. XV. CHROMATICS. 317

In mica, aooording to Mr. Biot, and in arragonite, aco(»*ding
to Dr. Brewster, there are two axes of crystallization ; and the
refraction of such substances may probably be represented, by
suppomng all the circular sections of a spheroid to become
elUpses, so that the undulation may assume the shape of an
almond.

Section XV. — Of the Colours of doubly refracting Suiiiance$.

In the case of doubly refracting substances, the first difficulty
is, not to explun why the colours of double lights are some-
times produced, but why they are not more universally observ-
able ; since it might naturally be expected, as a consequence of
the general law of interference, that two portions of tiie same
beam, passing through a moderately thin plate of such a sub-
stance, in paths diflering but little from each other, and coin-
ciding again in direction, should, in all common cases, exhibit
colours nearly similar to those of ordinary thin plates. It
would, however, be difficult to conjecture, whether they ought
to resemble the colours seen by transmission or by reflection ;
and the fact is, that botii these series of colours are at once pro-
duced by the substances in question ; but they are so mixed,
that, without a particular arrangement, they always neutralise
each other ; and their formation appears to be also limited to
certain peculiar conditions of polarisation, consistent with Mr.
Arago's observation, on the non-interference of two portions of
light, polarised in transverse directions. Several of the cases,
indeed, in which they are exhibited, remain still involved in
some degree of obscurity ; but it is easy to analyse the most
important of the phenomena, and to reduce them, with great
predsion, to the general laws of periodical colours.

Mr. Mains has demonstrated, by satisSeK^tory experiments,
that a beam of light, admitted into a doubly refracting crystal,
is as much divided by partial reflection at the second surface,
as by transmission at the first : the directions and the relative
intensities of the two portions being precisely the same as those
of the two portions of ray similarly polarised, and returning
to the second surface from without in an equal angle ; so that
after a iarther transmission at the first surlkoe, all the portions

318 CHBOKATICS. No. XV.

become again parallel. When the ray ia in the direction of the
principal section, there b no separation^ each of the pencils pro-
ceedmg undiyided, as they would do if they passed through a
second crystal parallel to the first : and the separation becomes
the most complete when the plane of inddence makes an angle
of about 45"^ with the principal section ; each of the portions o
and ey into which the ray is divided upon its admission^ aflbrd-
ing them two reflections, oO and oE^ eO and eE^ of nearly equal
intensity. The times occupied by the portions oO, eE, will
differ most from each otheri while oE and eO will desoribe their
.paths in equal times of intermediate length: but of these eO
only win commonly interfere with oO, which has a similar
polarisation in the plane of incidence, and oE with eEj both be-
ing polarised in a transverse direction ; so that we have two
series of colours, depending on an equal interval, except so far
as they are distinguished by the inversion of one of the portions
belon^ng to the extraordinary reflection, which renders the
series of colours exhibited by them similar to that of the colours
of common thin plates seen by reflection, while the ordinary
reflection exhibits colours analogous to those of thin plates seen
by transmission.

Mr. Biot's usual mode of exhibiting these colours is to place
a thin plate of sulfate of lime, or of any other crystal, on a
black substance ; to allow it to reflect the white light of the
clouds at an angle of incidence of about 55"^ ; and to receive
this light on a black glass, at an equal angle of inddence, in a
plane transverse to the former, so that the plate may be viewed
by reflection in the black g^ass. In this arrangement the light
reflected from the upper surface of the plate, being polarised in
the first plane of reflection, is not reflected by the black glass,
and consequendy is incapable of rendering the colours less easily
perceptible by admixture with them : the beams oO and eOy
returning bytiie ordinary reflection, are also similarly polarised,
and will be transmitted or absorbed by the glass; but the
beams oE and eEy being polarised in a transverse direction, will
be partially reflected by it, aud will exhibit a very brilliant
colour, depending on their mutual interference. I^ on the con-
trary, the black glass be turned round the ray, so that the

No. XV. CHB01£ATIC8. 319

seoood plane of inddenoe may coincide with the first, the ordi-
nary rays only will be partially reflected by it» and the comple-
mentary colour will be exhibited by the union of the portions
oOf eO; but this colour will be less distinct, on account of its
mixture with the white light reflected by the first surface.

Appearances of a similar nature may also be observed in the
transmitted light ; each of the refiractions exhibiting the colour
complementary to that which it aflbrds by reflection, as happens
in the ordinary ooburs of thin plates ; and we must seek for the
portions of light which afibrd them, in the successiTe partial re-
flections at the two surfaces of the plate, as in the case of the
ordinary colours ; the light simply transmitted by the separate
refractions not exhibiting the ordinary efiects of interference, for
want of a similarity of polarisation. The obliquity of the ind-
dent light produces similar effects on both series.

Under some circxunstanoes of the reflection of rays near the
perpendicular, Mr. Biot obseryes that the plate assumes the
colour which is usually exhibited by a plate of twice the thick-
ness viewed a little more obliquely : and in such cases it is pro-
bable that tiie polarisation of the beams oO and eE has been so
modified hb to afford a partial interference ; and if this is not
the true explanation, it will not be diflbmlt to suppose the
interval to be doubled in some other manner by a repeated
reflection.

The effect of a plate of a double thickness is also produced
by two equal and parallel plates, through which the light passes
in succession, provided tiiat their axes of erystallixation be
parallel, and that they be of such a thickness, as to exhibit
in conjunction a colour more eadly observable than those which
they afford separately : a condition which is more generally
applicable to the case in which the axes are transverse to eadi
other, and one of tiie thicknesses is to be mbtracted from the
other ; since in tins situation the two portions of li^t must
always interchange their refractions, and that which has moved
the more slowly, in its passage through one of the plates, will
move the more rapidly in the other. This result is very accu-
rately confirmed by experiment, and certainly affords a very
striking illustration of the truth of the law of interference.

320 CHROMATICS. ITo. X V.

When we wish to examine the effects of the different obliqui-
ties of the inddent lights it is most convenient to employ a beam
previonsly polarised, which renders the separation of the difier-
ent portions by a subsequent reflection or refraction more easily
practicable : and for these purposes we may either make use of
plates of black glass, placed in proper situations, or polarising
piles, consisting of a number of oblique thin plates, which pro-
duce the efiect on the light transmitted through them, witii
less diminution of its intensity than would take place in a
single partial reflection. In some cases also, the light may be
analysed by causing it to pass tiut)ugh a piece of Iceland crys-
tal ; or through a thin plate of agate, which Dr. Brewster has
found to Iransndt only such light as is polarised in a particular
plane.

The measurements of the thickness of plates of doubly re-
fracting substances agree in general very accurately witii the
various tints exhibited by them in various situations with
respect to tiie axis, and with various obliquities of the inddent
light, according to the theory of periodical colours ; and the
agreement is always sufficiently perfect to convince us of the
dependence of the phenomena on the law of interference, even
if it should happen to require some unknown modification in
particular cases. In the first place, when the incidence is per-
pendicular, the thickness of the plates is precisely such as would
be inferred fit)m the theory, at least as nearly as the theory is
founded on observations sufficiently accurate, although this
thickness is often many hundred times as great as that of the
thin plates, with which it is to be compared : thus the greatest
disproportion of .the ordinary and exti*aordinary refitiction of
rock crystal, according to Malus's experiments, is that of 159
to 160 ; so tiiat the difference of the times, occupied by light in
passing through this substance, is to tiie interval, in virtue of
which a similar plate exhibits the common colours, as 1 to 380,
and to the interval in a plate of crown glass as 1 to 318 ; while
tiie experiments of Mr. Biot make the observed proportion that
of 1 to 360 ; the difference being no greater, than would arise
from an error of less tiian a thousandth part of the whole, in the
determination of one of the refractive densities.

No. XV, CHROMATICa 321

The effect of the obliquity of the incident light/ on the
colours exhibited by plates of rock crystal, agrees also perfectly
with the theory. The difference of the times required for the
ordinary and extraordinary refractions, which is always com-
paratively small, will vary as the fluxiqn of the retardation,
when the obliquity varies ; and the sine of ordinary refraction

being ^, the interval will be expressed by — r -jr^ - ^! dn,

when the axis is parallel to the surface of the .plate, and by — r

y^/^ X dw, when it is perpendicular. Taking, for example,

an experiment of Mr. Biot, on a plate in which the axis was
nearly perpendicular, the mean angle of refraction being
2P 38.5', the tint was a reddish white of the seventh order,
answering to the reflection from a plate of glass .0000496 of an
inch thick, in the experiments of Newton, while the colour ex-
hibited in a perpendicular light, by a plate of the same crystal,
in which the axis was parallel to the surface, would have been
expressed by the thickness .000332. In these two cases, the
values of the fluxion become — rdn and — .14633rd«; and re-
ducing the interval .000332 in this proportion, we find .00004^ 6
for the thickness of a plate of glass which ought to exhibit the
tint corresponding to the oblique incidence ; the difference from
the experiment being only one millionth of an inch, which would
scarcely make a sensible alteration in the colour observed.
When the thickness of such a plate is more considerable, or
when the excentricity of the extraordinary refraction is greater,
the colours differ, with the incidence, in different parts of the
plate ; and they are generally disposed in rings concentric with
the axis. These rings have been particularly described by Dr.
Brewster, as observed in the topaz : they are always interrupted
by a dark cross, occasioned by the want of light, properly polar-
ised to afford them, in the two transverse directions.

Mr. Biot has made a great number of experiments on the
colours of the plates of sulfate of lime, in the form denominated
Muscovy talc : they exhibit a general agreement with the re-
sults of the calculation, particularly with respect to the constancy
of the tint, in all moderate obliquities, when the inclination of
the axis to the plane of incidence is 45 ; but in other cases the

VOL. I. Y

322 CHROMATICS. No. XV.

agreement is somewhat less perfect, and the difference is too
great to be attributed altogether to accident. The most pro«
bable reason for this irregularity, under drcumstances so nearly
similar to those which accord with the theory in the case of
rock crystal, is the vr^xd of a perfect identity of the two refrac-
tions, in the direction of the supposed axis ; or, in the language
employed by Mr. Biot with respect to mica, the existence of a
double axis of extraordinary refraction : and it is the more
credible that such, a slight irregularity may have existed in the
sulfate of lime without having been observed, as Dr. Brewster
has lately detected a similar property in the arragonite, though
both Malus and Biot had examined this substance very care-
fully without being aware of it. The calculation of the extra-
ordinary refraction, in such a case, would afford but little ad-
ditional difficulty, if its characters were well determined : the
form, in which the undulations must be supposed to diverge,
might properly be termed an amygdaloid ; and the velocities
with which the sections, formed by the pven surface, would ex-
tend themselves, might be deduced from the properties of the
ellipsis, nearly in the same manner as they have been deter-
mined for the spheroid.* The difference of the results of the
calculation from the spheroid, and of Mr. Biotas experiments,
or rather of the empirical formula, derived from them, may be
seen in the subjoined table ; the first part of which, deduced
from the theory, is applicable to all substances affording a
regular extraordinary refraction, when the axis is either per-
pendicular or parallel to the surface of the plate. The first
column of decimals shows the equivalent variation of thickness
where the axis is perpendicular to the plate, being equal to

~jT\^z — ) » *^® product of the sine and tangent of refraction ; the

second represents the variation for an ordinary thin plate,
being proportional to the cosine V(l — §§) ; and the subsequent
columns are found by adding to the numbers of the second
column those of the first, multiplied by A', the square of the

* This was written before Fresnel's important inyestigation of the surface of
elasticity and more complete development of the theory of polarfeed light, which
was presenteKl to the AcwWmie des Sciences in 1821. The specolations in the text,
imperfect as they are, and in some important respects erroneous, are remarkable
anticipations of the true theory. — Note by the Editor.

No. XV. CHROMATICS. 323

sine of the inclination of the plane of incidence to the axis, since

i^hhee

V(i-ee)H

k **

Vd-ee) -

^ va-«)-

Angle of
Refraction.

Perpendicolar
Plate.

Parallel Plate. Inclination of the Plane of
Incidence to the principal Section.
- 22i^ 45° 77+° 90°

00^

.0000

1.0000 1.000

1.000

.1245

.9397 .958

1.002

1.046

1.064

.5394

.7660 .845

1.036

1.245

1.305

1.5000

.5000 .720

1.250

1.780

2.000

5.5851

.1736 .992

2.966

4.940

5.759

Biot—

' .969 .975

.995

1.023

1.038

.898 .920

1.000

1.112

1.175

.848 .882

1.097

1.396

1.588

80"

1.196 .921

1.440

2.338

3.562

There are also some circumstances in the experiments of
Mr. Biot on plates of rock crystal cut perpendicularly to the
axis, which cannot be sufficiently explained on any hypothesis,
without some further investigation. These plates seem to
transmit the beam of light subjected to the experiment, without
materially altering its polarisation, and then to produce differ-
ent colours, according to the situation of the substance subse-
quently employed for analysing the light : so that Mr. Biot
supposes the rays of light to be turned more or less by the
crystal, round an axis situated in the direction of their motion ;
and he has observed some similar efiects in oil of turpentine,
and in some other fluid& But it is highly probable, that all
these phenomena will ultimately be referred to some simpler
operation of the general law of interference.

Dr. Seebeck and Dr. Brewster have discovered appearances
of colours, like those of doubly refracting substances, in a
number of bodies which can scarcely be supposed to possess any
crystalline structure. They are particularly conspicuous in
large cubes of glass, which have been somewhat suddenly
cooled, so that their internal structure has been rendered un-
equal with regard to tension. The outside of a round mass,
thus suddenly cooled, being too large for the parts within it,

324 CHROMATICS. No. XV.

must necessarily be held by them in a state of compression
with respect to the direction of the circumference, while they
are extended in their turn by its resistance : although in the
direction of the diameter the whole will generally be in a state
of tension : so that the refractive density may naturally be
expected to be somewhat different in different directions, which
constitutes the essential character of oblique refraction : and
when the proportions of the external parts to the internal are
modified by the existence of angles, or other deviations from a
spherical form, the arrangement of the tensions must be altered
accordingly ; and there is no doubt that all the apparently
capricious variations of the rings and bands of colours which are
observed, might, by a careful and minute examination, be re-
duced to the natural consequences of these inequalities of den^ty,
so far at least as the laws of the extraordinary refraction alone
are concerned, although the separation of the light into two
portions might still remain unexplained. Effects of the same
kind are produced by the temporary operation of partial changes
of temperature, producing partial compression and extension
of the internal structure of the substance : and even a me-
chanical force, if sufficiently powerful, when applied externally
in a single direction, has been shown, by the same observers}
to produce a double refraction ; although the difference of the
densities, thus induced, is much too minute to be perceived in
any other way, than by means of these colours, which are, in
general, so mUch the more easily seen, as the cause which
excites them is the feebler.

Dr. Brewster has also shown, that the total reflection of light
within a denser medium, and the brilliant reflection at the sur-
faces of some of the metals, are capable of exhibiting some of
the appearances of colour; as if the light concerned were
divided into two portions, the one partially reflected in the first
instance, the other beginning to be refracted, and caused to
return by the continued operation of the same power. In the
case of silver and gold, it has already been observed, that there
appear to be two kinds of reflection, occa^oning opposite polari-
ties ; and these may possibly be concerned in the production of
this phenomenon. 1 he origindl interval appears to be extremely

No. XV. CHB0MATIC8. 325

minute, but it is capable of being increased by a repetition of
similar reflections, as well as by obliquity of inddence. Mr.
Biot has also found that such surfaces, combined with plates of
doubly refracting substances, either increase or diminish the
equivalent thickness, according to the direction of the polarisa-
tion which they occasion* In these and in a variety of similar
investigations, a rich harvest is opened, to be reaped by the
enlightened labours of future observers; and the more diffi-
culty we find in fully explaining the facts, upon the general
principles hitherto established, the more reason there is to hope
for an extension of the bounds of our knowledge of the optical
properties of matter, and of all the laws of nature connected
with them, when the examination of these apparent anomalies
shall have been still more diligently pursued.

Section XVL — Of the Nature of Light and Colours.

Notwithstanding the acknowledged impossibility of fully ex-
plaining all the phenomena of light and colours by any imagi-
nable hypothesis respecting their nature, it is yet practicable to
illustrate them very essentially, by a comparison with the known
effects of certain mechanical causes, which are observed to act
in circumstances somewhat analogous ; and as far as a theory
will enable us to connect with each other a variety of £Eu:ts, it
is perfectly justifiable to employ it hypothetically, as a tempo-
rary expedient for assisting the memory and the judgment,
until all doubts are removed respecting its actual foundation
in truth and nature. Whether, therefore, light may consist
merely in the projection of detached particles with a certain
velocity, as some of the most celebrated philosophers of modem
times assert, or whether in the undulations of a certain ethereal
medium, as Hooke and Huyghens maintained, or whether, as
Sir Isaac Newton believed, both of these causes are concerned
in the phenomena; without positively admitting or rejecting
any opinion as demonstrably true or false, it is our duty to
inquire what assistance can be given to our conception and
recollection, by the adoption of any comparison, which may be
pointedly applicable even to some insulated facts only. It has

326 CHROMATICS. No. XV.

however been thought desirable to separate this inyestigation,
as much as possible, from the relation of the iacts, in order to
avoid confounding the results of observation with the deductions
from mere hypothesis ; an error which has been committed by
some of the latest and most meritorious authors in this depart-
ment. It may be objected to some of the preceding sections,
that this forbearance has not been exercised with respect to the
general law of interference and its modifications : but it would
have been impossible to give any correct statement of the facts
in question, without determining whether the appearances de-
pend upon one or both of the portions of light supposed to be
concerned.

Art 1. (Sect. I.) The separation of colours is explained, in
the hypothesis of emission, by the supposition of an elective
attraction, different in intensity for the different rays of the
spectrum ; but for this difference no ulterior cause is assigned.
Any original difference of velocity is contrary to direct experi-
ment; and even the alterations of relative velocity, which must
inevitably be occasioned by a variety of astronomical causes,
have not been detected by the most accurate observations, in-
stituted for the express purpose of discovering them ; so that it
has been suggested, that there may possibly be a multitude of
rays of the same colour, moving with various velocities, and
only afiecting the sense when they have the velocity appropriate
to that colour in the eye. The name of elective attraction is
indeed little more than a mode of expresnng the fact, without
referring it to any simpler mechanical cause ; and in chemical
elective attractions the substances concerned are under very
different circumstances, with respect to contact, and with re-
spect to the probable influence of the form and bulk of their
integral particles ; at the same time it seems impossible to show
any absurdity in the suppontion of the existence of such an
elective attraction with regard to the different kinds of light
On the other hand, if we consider colours as depending on a
succession of equal undulations, of different magnitudes as the
colours are different, we may discover an analogy, somewhat
more approaching to a mechanical explanation, in the motions
of waves on the surface of a liquid ; the largest waves moving

No. XV. CHROMATICS. 327

with the greatest rapidity, although the approximate calculation,
derived from the moel approved theory, leads us to the same
expressions for the velocity, as are applicable to the transmisdon
of an impulse through an elastic fluid. The fact is, that a
larger wave moves more rapidly than a smaller, because the
pressure is not precisely limited to a perpendicular direction, as
the simplest calculation supposes, but operates also more or
less in an oblique direction, principally within a certain angular
limit; so that the utmost depth at which any difierence of
pressure can affect the liquid as a motive force, is that at which
this angle may be imagined to comprehend virtually the exact
breadth of a wave ; and since the velocity depends on the depth
of the fluid afiected at once by the pressure, the breadth becomes
in this manner an element of the determination. Thus also
the larger undulations, constituting red light, are found to move
more rapidly than those of the violet, which are supposed to be
smaller ; and there are many ways in which the difference may
be supposed to be occasioned, although not depending exactly
on the same cause as in the case of the waves on the surface of
a liquid. It is well known that sounds of all kinds move with
an equal velocity through the air; and all colours arrive through
the supposed elastic ether in the same time from the remotest
planets : but a refractive medium, however transparent, is not
to be considered as perfectly homogeneous : in many instances,
two medituns, of different qualities, seem to pervade every part
of a crystal, which is completely uniform in its appearance; and
it seems to be necessary, in every case, to suppose the particles
of material bodies scattered at considerable distances through a
medium which passes freely through their interstices ; so that
we may conceive the undulations of light to be transmitted
partly through the particles themselves, and partly through the
intervening spaces, the two portions meeting continually after a
certain very minute difference in the length of their paths : we
may then suppose the portion transmitted through the interstices
to be weakened by the irregularity of its passage, which will
aflect the smaller undulations more than the larger; and
when these portions are combined with the portions more
slowly transmitted through the particles themselves, these

328 CHROMATICS. No. XV.

last will bear a greater proportion to the former in the violet
than in the red light, and will have more influence on the
ultimate velocity, which will therefore be smaller for the
violet than for the red. This explanation may perhaps be
far from the best that the hypothesis in question might afford ;
but it will serve as an illustration of a possible mode, in which
the phenomenon may be referred to the established laws of
mechanios, without the continual introduction of new principles
and properties.

Art 2. (Sect. IV. A.) Most of the ordinary phenomena of
optics are capable of a sujBiciently satisfactory explanation, on
either of the hypotheses respecting the nature of light and
colours : but the laws of interference, which have been shown
to be so extensively applicable to the diversified appearances of
periodical colours, point very directly to the theory of undu-
lation ; so directly, indeed, that their establishment has been
considered, by many persons on the continent, as almost para-
mount to the establishment of that theory. It might not, how-
ever, be absolutely impossible, to invent some suppositions
respecting the effects of light, which might partially reconcile
these laws to the theory of emission. ThuS| if we suppose,
with Newton, the projected corpuscles of light to excite sensa-
tion by means of the vibrations of the fibres of the retina and
of the nerves, we may imagine that such vibrations must be
most easily produced by a series of particles following each
other at equal distances, each colour having its appropriate
distance in any given medium : it will then be demonstrable,
that any second series of similar particles, interfering with
them, in such a manner as to bisect their intervals, will destroy
their effect in exciting a vibratory motion ; each succeeding
particle meeting the fibre at the instant of its return from the
excursion occasioned by the stroke of the preceding, and thus
annihilating the motive effect of that stroke. But the illustra-
tion ends here : for it seems impossible to adapt it to the
greater number of the alternations which occur, during the
passage of a ray, through a given space, in a denser medium :
since it is an indispensable condition of the projectile theory,
that the velocity of light should be increased upon its entrance

No. XV. CHROMATICS. 329

into a medium of greater refractiye deosity. The Newtonian
theory, of fits of easy reflection and easy transmission, is still
more limited in its application ; since it attributes to one por-
tion of li^t those effects, which have been strictly demonstrated
to depend on the presence of two.

In the undulatory theory, the analogy between the laws of
interference, and the phenomena of the tides, and the effects of
the combination of musical sounds, is direct and striking. The
existence of an undulation of an elastic medium depends on
the recurrence of opposite motions, alternately direct and retro-
grade, at certain equal distances, in the same manner as a
series of waves consists in a number of alternate elevations and
depressions, and the succession of the tides in a number of
periods of high and low water. The spring and neap tides, de-
rived from the combination of the simple solar and lunar tides,
aflbrd a, magnificent example of the interference of two immense
waves with each other : the spring tide being the joint result of
the combination when they coincide in time and place, and the
neap when they succeed each other at the distance of half an
interval, so as to leave the effect of their difference only sen-
sible. The tides of the port of Batsha, described and explained
by Halley and Newton, exhibit a different modification of the
same opposition of undulations ; the ordinary periods of high
and low water being 'altogether superseded, on account of the
different lengths of the two channels by which the tides arrive,
affording exactly the half interval which causes the disappear-
ance of the alternation. It may also be very easily observed,
by merely throwing two equal stones into a piece of stagnant
water, that the circles of waves, which they occasion, obliterate
each other, and leave the surface of the water smooth, in cer-
tain lines of a hyperbolic form, while, in oUier neighbouring
parts, tiie surface exhibits the agitation belonging to both
series united. The beating of two musical sounds, nearly in
unison with each other, appears also to be an effect exactiy
resembling the succession of spring and neap tides, which may
be considered as the beatings of two undulations, related to
each other in frequency as 29 to 30 ; and the combination of
these sounds is still more identical with that which this theory

330 CHROMATICS. No. XV.

attributes to light ; since the elementary motions of the par-
ticles of the luminiferous medium are supposed to be principally
confined to the line of direction of the undulation, while the
most sensible effects of the waves depend immediately on their
ascent and descent, in a direction perpendicular to that of their
progresfflYe motion.

Art. 3. (Sect. IV. 6.) The diminution of the velocity of
light upon its entrance into a denser medium, in the direct pro-
portion of the refractive density, is one of the fundamental
principles of the undulatory theory, and is perfectly inadmisdble
on the supposition of projected corpuscles. But it must be
remembered, that the demonstration of the actual existence of
this proportion is somewhat indirect ; being only derived from
the necessity of admitting it, in the application of the laws of
interference to the observed phenomena : and we have no means
of obtaining an immediate measure of the velocity of light in
different mediums.

Art. 4. (Sect IV. C.) The loss of the half interval may be
explained in particular cases, without difficulty, although, in
other instances, the circumstances are too complicated to allow
us to appreciate theii' effects. In the direct transmission of a
ray of light through a plate of a transparent substance, we may
compare the denser medium to a series of elastic balls, larger
and heavier than another series in contact with them on each
side. Now, it is well known that a series of elastic balls trans-
mits any motion from one end to the other, while each ball
remains at rest, after having communicated the motion to the
next in order ; so that the last only ffies off, from having none
beyond it to impel ; and if the balls, instead of being only pos-
sessed of repulsive forces, were connected by elastic ligaments
of equal powers, a motion in a contrary direction would be trans-
mitted with equal ease ; the last ball, being retained by the
ligament, instead of flying off, would draw the last but one in
the same direction, itself remaining at rest after this negative
impulse ; and the motion would be communicated backwards in
the same manner throughout the series to the first ball : and
then, for want of further resistance, this ball would not remain
at rest after receiving the negative impulse, but would be

No. XV. CHROMATICS. 331

drawn forwards by it, so as to strike the second, precisely in
the same manner as at the beginning of the experiment ; and
this second positire impulse would proceed through the whole
series like the first. • Such is the nature of the longitudinal
vibrations of elastic rods, first observed by Chladni; the
cohesion of the substance supplying the place of the supposed
elastic ligaments: and in die case of an elastic fluid, the
pressure of the surrounding parts performs the same office ; a
negative impulse being always propagated tlirough it with the
same facility as a positive one. If, instead of a angle series of
balls, we now consider the efiect of two series, the second con-
sisting of larger balls than the first, the last ball of the smaller
series will not remain at rest after striking the first of the
larger, but will be reflected, so as to strike the last ball but one
in a retrograde direction ; and this retrograde impulse will be
continued to the first ball, constituting a positive impulse with
respect to the new direction, in which it is propagated. But if
the first series of balls be larger than the second, the last of
the larger balls will not be deprived of all its motion by
striking the first of the smaller, but will continue to move more
slowly in its first direction ; and the elastic ligaments will then
be called into action, so as to carry back, step by step, to the
first ball, this remaining impulse, which will become negative
with respect to the new direction of its transmission. And the
same must happen in the case of two elastic mediums in con-
tact, supposing them to be of equal elasticity, but of difierent
densities ; the direction of the elementary motions either coin-
ciding with that of the general impulse, or being opposite to it
in both mediums at once, when the reflection is produced by
the arrival of the undulation at the surface of a denser medium,
and being reversed when at the surfiice of a rarer : and it is
obvious that such an inversion of any regular undulation is pa-
ramount to its retardation or advancement, to the extent of half
of the interval which constitutes its whole breadth ; every afiec-
tion of such an undulation being precisely inverted, at the
distance of half the breadth of a complete alternation : and
these eflfects will not materially difier, whether the impulse be
supposed to arrive perpendicularly at the surface, or in an
oblique direction.

332 CHROMATICS. No. XV.

Art. 5. (Sect. IV. D. : Sect. XIII.) The experimenta of
Mr. Arago, not yet published in detail, which show that light
does not interfere with light polarised in a transTcrse direction,
lead US immediately to the consideration of the general pheno-
mena of polarisation, which cannot be said to have been by any
means explained on any hypothesis respecting the nature of
light It is certainly eader to conceiye a detached particle,
however minute, distinguished by its different sides, and having
a particular axis turned in a particular direction, than to ima-
gine how an undulation, resembling the motion of the air which
constitutes sound, can have any different properties, with respect
to the different planes which diverge from its path. But here
the advantage of the projectile theory ends ; for every attempt
to reduce the phenomena of polarisation to mechanical laws, by
the analogy of magnetism, has completely failed of enabling
us to calculate the results of the actions of the forces, supposed
to be concerned, in any correct manner ; to say nothing of the
extreme complication of the properties, which it would be ne-
cessary to attribute to the simplest and minutest substances, in
order to justify the original hypothesis of a polarity, existing
in all the particles of light, and a directive attraction, that is, a
combination of attraction and repulsion, in every reflecting or
refracting substance. In the undulatory theory, we may dis-
cover some distant analogies, su£Bcient to give us a conception
of the possibility of reconciling the facts with the theory, and
perhaps even of reducing those facts to some general laws
derived from it ; although it will be necessary, in this intricate
part of the inquiry, to proceed analytically rather than synthe-
tically, and to rest satisfied for the present, without bringing
the analysis to a termination by any means explanatory of all
the phenomena. Some of the supporters of this theory may
perhaps be of opinion, that its deficiencies are too strongly dis-
played by this attempt: but it is for them to find a more
complete solution of the diflSculties, if any such can be dis-
covered.

In the case of a wave, moving on the surface of a liquid,
considering the motion of the particles at some little distance
below the surface, as concerned in the propagation of an undu-
lation in a horizontal direction, wc may observe that there is

No. XV. CHROMATICS. 333

actually a lateral motion, throughout the liquid, in a plane of
which the direction is determined by that of gravitation : but
this happens because the liquor is more at liberty to extend
itself on this side than on any other, the force of gravitation
tending to bring it back, with a pressure of which the opera-
tion is analogous to that of elasticity ; and we cannot find a
parallel for this force in the motions of an elastic medium. It
is indeed very easy to deduce a motion, transverse to the gene-
ral direction, from the combination of two undulations pro-
ceeding from two neighbouring points, and interfering with
each other, when the difierence of their paths amounts to half
an interval ; for the result of this combination will be a regu-
lar though a very minute vibration in a transverse direction,
which will continue to take place throughout the line of the
propagation of the joint motions although certainly not wifJi
any force, that would naturally be supposed capable of produce
ing any perceptible efiects. There must even be a difference
in the motions of the particles in every simply diverging undu-
lation, in different parts of the spherical surface to which they
extend : for, supposing it to originate /rom a vibration in a
given plane, the velocity of the motion constituting the undu-
lation will be greatest in the direction of that plane, and will
disappear in a direction perpendicular to it, or rather will there
become transverse to the direction of the diverging radii : and
in all other parts there must be a very minute tendency to
a transverse motion, on account of the difference of the veloci-
ties of the collateral direct motions, and of the compressions
and dilatations which they ocdasion. When, also, a limited
undulation is admitted into a quiescent medium, it loses some
of its force by diflraction on each side, where it is unsupported
by the progress of the collateral parts : and if an undulation
were admitted by a number of minute parallel linear aper-
tures or slits, or reflected from an infinite number of small
wires, parallel to each other, it would still retain the impression
of the incipient tendency to diffraction in all its parts, producing
a modification of the motion, in a direction transverse to that
of the slits or wires. It is true that all these motions and
modifications of motion would be minute beyond the power of

334 CHROMATICS. N"o. XV.

imagination, eren when compared with other motions, them-
selves extending to a space far too minute to be immediately
perceived by the senses : and this consideration may perhaps
lessen the probability of the theory as a physical explanation
of the* £Eicts : but it would not destroy its utility as a mathe-
matical representation of them, provided that such a repre-
sentation could be rendered general, and reducible to calcula-
tion; and even in a physical sense, if the alternative were
unavoidable, it is easier to ima^ne the powers of perceiving
minute changes to be all but infinite, than to admit the por-
tentous complication of machinery, which must be heaped up,
in order to afford a solution of the difficulties, which beset the
application of the doctrine of simple projection to all the phe-
nomena of polarisation and of colours. It is not however
possible at present to complete such a mathematical theory,
even on imaginary grounds ; although a few further analogies
between polarisation and transverse motion force themselves on
our observation.

In the theory of emission, the resemblance of the phenomena
of polarisation to the selection of a certain number of particles,
having their axes turned in a particular direction, supposing
these axes, like those of the celestial bodies, to remain always
parallel, will carry us to a certain extent, in estimating the
quantity of light contained in each of the two pencils, into
which a beam is divided and subdivided: but it would soon
appear, that, after a few modifications, this parallelism could
no longer be supposed to be preserved : we should also find it
impossible to assign the nature and extent of any forces, which
might be capable of changing the former directions of the
axes, and fixing them permanently in new ones. The distinc-
tion of a fixed, a moveable, and a partial polarisation, which
has been imagined by Mr. Biot, must vanish altogether, upon
considering that all the etkcts, which he attributes to the
partial polarisation, are observable in experiments like those
of Mr. Knox, in which there is confessedly no polarisation
at all.

If we assume as a mathematical postulate, in the undulatory
theory, without attempting to demonstrate its physical founda-

No. XV. CHROMATICS. 335

tion, that a transverse motion may be propagated in a direct
line, we may derive from this assumption a tolerable illus-
tration of the subdivision of polarised light by reflection in an
oblique plane. Supposing polarisation to depend on a trans-
verse motion in the given plane, when a ray completely pola-
rised is subjected to simple reflection in a different plane, which
is destitute of any polarising action, and may therefore be
called a neutral reflection, the polar motion may be conceived
to be reflected, as any other motion would be reflected at a
perfectly smooth surface, the new plane of the motion being
always the image of the former plane : and the efiect of
refraction will be nearly of a similar nature. But when the
surface exhibits a new polarising influence, and the beams of
light are divided by it into two portions, the intensity of each
may be calculated, by supposing the polar motion to be re-
solved instead of being reflected, the simple velocities of the
two portions being as the cosines of the angles formed by the
new planes of motion with the old, and the energies, which are
the true measure of the intensity, as the squares of the sines.
We are thus insensibly led to confound the intensity of the
supposed polar motion, with that of the reflected light itself:
since it was observed by Mains, that the relative intensity of
the two portions, into which light is divided under such circum-
stances, is indicated by the proportion of the squares of the
cosine and sine of the inclination of the planes of polarisation.
The imaginary transverse motion might also necessarily be
alternate, partly from the nature of a continuous medium, and
partly fit>m the observed &ct, that there is no distinction
between the polarisations, produced by causes precisely op-
posed to each other, in the same plane.

Why light should or should not be reflected at certain
surfaces, when it has been previously polarised, cannot, even
with the greatest latitude of hypothecs, be very satisfactorily
explained, but it is remarkable that the transmission is never
wholly destroyed, or even weakened in any considerable pro-
portion. We might, indeed, assign a reason for the occur-
rence of a partial reflection or a total transmismon in tiie con-
stitution of the surface concerned : since every abrupt change

336 CHROMATICS. No. Xy.

of density miist necessarily produce a partial reflection, while a
gradual transition by insensible steps must transmit each im-
pulse with undiminished energy, and without any reflection of
finite intensity, as in the well known case of a collision, sup-
posed to be performed with the interposition of an infinite
number of balls of all possible intermediate magnitudes. If,
therefore, we could find any modification of light, which could
cause it to be transmitted from one medium to another iu a
more or less abrupt manner, we should thus be able to dis-
cover a cause of a variation of the intensity of the partial
reflection : and this seems to be the nearest approach that we
can at present make, to an explanation of the phenomenon,
according to the undulatory theory.

■"Art. 6. (Sect. V.) The equal intensity of the colours of
thin plates seen by reflection and by transmission, is a fact
which could not have been expected from the immediate ap-
plication of the law of interference, and which seems, therefore,
at first sight, to militate against its general adoption. But this
is only one of the many modifications of the law, which are the
immediate consequences of its connexion with the undulatory
theoi^ ; and it may be demonstrated, from the analogy of a
series of elastic bodies, that no material difierence in the in-
tensity of the two kinds of colours ought to be expected in such
circumstances. The intensity of a ray of light must always be
considered as proportional to the energy or impetus of the
elementary motions of the particles concerned, which varies as
the square of the velocity, and not simply as the velocity itself:
for if the velocity were made the measure of intensity, there
would be an actual gain of joint intensity, whenever a ray is
divided by partial reflection : since it follows from the laws
of the motion of the centre of inertia, that when a smaller
body strikes a larger, not the sum, but the difierence of the
separate momenta, will remain unchanged by the collision,
while the sum of the energies remains constant in all cir-
cumstances ; the square of a negative quantity being equal to
that of the same quantity taken positively. Thus, supposing
an elastic ball, 1, to strike another of which the mass is r, with
the velocity 1, the velocity of the transmitted impulse will be

No. XV. CHROMATICS. 33/

;rT-i> ^^^ ^^^^ ^f ^^^ reflected, ^-r^ — 1 = — , the

sum of the momeDta in the opposite directions being j-rri^
instead of 1, the original momentum; but the energies, ex-
pressed by the products of the masses into the squares of

the velocities, will be^^-^^, and f^^j respectively; and

the sum of these is (^^-^) = 1^- Now, when an impulse
arrives at the last of a series of larger particles, and is
reflected in an inverted form, if we substitute ~ for r, the

energies will be in the proportion of — , and I- 1 j,

or of 4 r and (I — r)*, which is the same as the former:
so that, according to this analogy, the subdivision of the
light at the second surface of a plate must be in the same
proportion as at the first We may call this proportion that of
nftovfy wf + rf being equal to 1 : we have then rf for the
energy of the first partial reflection, m^ff for the second ; and
ni?n^ for the third : for the first transmission, info the substance,
tn? ; for the second, out of it, m* ; for the third, after an inter-
mediate reflection, mV ; and for the foivth after two reflec-
tions, ifiV ; and the elementary velocities in either medium,
compared among themselves, will be as the square roots of the
respective energies. But it may be proved that, in all collisions
of two moving bodies, each of the motions produces its efiect
on the velodties after impulse, independently of the other : so
that the changes introduced, in consequence of the motion of
one of the bodies concerned, are the same as it would have
occasioned, if the other had been at rest ; and consequently,
if two undulations interfere in any manner, the joint velocities
of the particles must always be expressed by the addition
or subtraction of the separate velocities belonging to the
respective undulations. When, therefore, the beam first par-
tially reflected, of which the elementary velocity is expressed
by n, interferes with the beam transmitted back, afi»r reflection
at the second surface, with the velocity m"«, the joint velocity,
VOL. I. z

338 CHROMATICS. No. XV.

in the case of the perfect agreement of the motions, will be
n + m*n, and in case of their disagreement, n ^nfn ; the ener^es
being (» + wFnf and (n — nfhf ; the difference, which is the
true measure of the effect of the interference, being 4nM, that
is, four times the product of the respective velocities. But
when the light simply transmitted at the second sur&ce, with
the velocity m", interferes with the light transmitted after
two reflections, with the velocity m^n*, the quadruple product
becomes 4mV, only differing from the former in the ratio of
mf to 1, which is that of the intensity of the light transmitted
by the single surface to the intensity of the incident light, the
difference being much too slight to be directly perceived by
Uie eye : so that this result may be^ considered as agreeing
perfectly with Mr. Arago's observation.

We may also obtain, firom the analogy with tiie effects of
collision, an illustration of the intensity of the partial reflectiou
in different circumstances ; although it is not easy to say what
ought to be the precise value of r in the comparison. If we
imagined the two mediums to differ only in density, while their
elasticity remained equal, whidi is the simplest supposition, the
density must be conceived to vary as the square of the velocity
appropriate to the medium : but the value of r, thus deter-
mined, makes the partial reflection in general much too
intense, and it becomes necessary to suppose it weakened
by the intervention of a stratum of intermediate density,
such as there is every reason to attribute to the sur&ces
of material substances in general, from the considerations stated
in the article Cohesion.* However this may be, we shall
in general approach sufficiently near to a representation of
the phenomenon, by taking tlie mass r in the simple pro-
portion of the refractive density : thus, in the case of water,
making r = -1- , we have for the energy of the first partial re-
flection O^ririj "^ ^ * '^^"^ ^'"^^ ^ ^^'^^ ^' Bouguer's

experiments is .018; and the agreement is as accurate aa
could have been expected, even if the whole calculation had

* No. XX. in this rol.

No. XV. CHBOMATICS. 339

not been an imaginary structure. In the ease of glaaSy tlie
diflference is somewhat greater ; and it is natural to expect a
greater loss of light from a want of perfect polish in the

surface : for, taking r = -|- » we have if = .040, and Bouguer

found the reflection only .025. The surface of mercury re*
fleeted nearly .60 ; whence r should be about 8 : whether the
index of the refractive density can be so great as this, we have
no precise mode of determining ; but there seems to be some*-
thing in the nature of metallic reflection, not wholly dependent
on the density : thus it may be observed that potassium has a
very brilliant appearance, though its specific gravity is very
low ; at the same time, its great combu^tilHlity might ^ve it a
much higher rank among refractive substances than could
otherwise have been expected from its actual density.

Art 7. (Sect XIII.) Although the ingenuity of man has
not yet been able to devise anythmg like a satiafiictory reason
for the reflection of a polarised ray in one case, and its trans*
mission in another: yet several attempts have been made,
with various success, to reconcile the different hypotheses of
li^t with the other phenomena of oblique refraction. The
illustrious Mr. Laplace has undertaken to deduce the laws of
this refraction according to the projectile system, fiH>m the
general doctrines of motion; and he has si^ciently demon-
strated that the path followed by the lig^t is always such, as
to agree with the principle of the least action, supposing the
law of the velocities previously established ; or in other words,
that the sum of the products of the qpaces described, into the
respective velocities, is always the least posuble. To this
demonstration it has been objected, tiiat notwithstanding the
complication of its steps, it is in &et nothing more than the
simple translation of the ftmdaraental law of Huyghens into
anothar language ; for it is assumed in this theory, upon
obvious and intelligible grounds, that the path of light must
always be such, that the time may be equal with respect to
two neighbouring collateral parts of the undulation ; which is
the well known condition of a minimum of the whole time
employed ; and the time being always expressed by the space

z 2

342 CHROMATICS. No. XV.

to be inooDsiderable, the whole retardation will be as the square
of the cosine of ihe inclination to the axis, which is the well-
known proportion of the difference of the diameters of a circle,
and of an ellipua approaching near it We thus obtain a
general idea of the combination of two effects, whidi do not
appear to be related in anj other point of view, a regular
oblique refraction, and a distinct polarisation; further than
this, the comparison is by no means completely satisfactory :
and the great difficulty of all, which is to assign a sufficient
reason for the reflection or non-reflection of a polarked ray,
will probably long remain, to mortify the vanity of an ambitious
philosophy, completely unresolved by any theory.

No. XVI. REMA&KS ON BLOOD AND FUS« 343

No-XVL

REMARKS ON THE MEASUREMENT OF MINUTE PARTICI^ES,
ESPECIALLY THOSE OF THE

BLOOD MD OF PUS.

From Dr. Toong's Introdaciioii to Medical Literature, p. 545.

1. Oh the Form and Magnitude qfthe Particles of the Blood.

Thb fonn and magnitade of the coloured particles of the blood
IB a subject not only interesting and important in itself, but is
also capable of assisting, by means of comparatiye observations,
in the determination of the magnitade of the capillary arteries,
and the inyestigation of the resistance which tiiey exhibit; it
may also be of advantage to obtain some tests capable of ascer^
taining, whether these particles undergo any change in diseases
of various kinds, and what is their relation to the globules of
pus, and of other animal fluids : hitherto the measures of the
particles of blood, which have been considered by various
authors as the most accurate, have differed no less than in the
ratio of 2 to 5 ; and there is an equal degree of uncertainty re^
specting their form, some admitting the truth of Mr. Hewson's
opinions, and a greater number rejecting them without any satis-
fiictory evidence. In such examinations, it is only necessary to
employ a full and unlimited light, in order to obtain a very dis-
tinct outline of what appears manifestly to be a very simple
substance, and we thus seem to have the clear evidence of the
senses against Mr. Hewson : but we must remember, that where
the substances to be examined are perfectly transparent, it is
only in a confined and diversified light that we can gain a cor-
rect idea of their structare. The eye is best prepared lor the
investigation, by beginning with the blood of a skate, of which
the particles are so conspicuous, and of so unequivocal a form,
as to set aside at once the idea of a simple homogeneous sub^

544 REMARKS ON BLOOD AND PUS. No. XVI

Stance. They are oyal and depressed^ like an almond, but less
pointed, and a little flatter ; each of them contains a round
nucleus, which is wholly independent in its appearance of the
figure of the whole disc, being sometimes a little irregular in
ics form ; seldom deviating from its central situation, but often
remaining distinctly visible while the o\al part is scarcely per-
ceptible ; and as the portion of blood dries away, becoming
evidently prominent above the thinner portion. This nucleus is
about the size of a whole particle of Uie human blood, the
whole oval being about twice as wide, and not quite three times
as long ; the nucleus is very transparent, and forms a distinct
image of any large object which intercepts a part of the light
by which it is seen, but exhibits no inequalities of light and
shade, that could lead to any mistake respecting its form.
But if we place some particles of human blood under similar
circumstances, near the confine of light and shade, although
they -are little, if at all, less transparent, we immediately see
an annidar shade on the disc, which is most marked on the side
of the centre on which the marginal part appears the brightest,
and consequently indicates a depression in the centre, which
Delatorre mistook for a perforation. It is most observable
when the drop is drying away, so that the particles rest on the
glass : and when a smaller particle is viewed, it has merely
a dark central spot, without any lighter central space. Nor
have the particles ever appeared to me ^' as flat as a guinea,"
although their axis is sometimes not more than one third or
one fourth of their greatest diameter ; if they were much thin-
ner than this, their diameter would be more diminished than it
is when they become spherical, by the effect of an aqueous
fluid; while this form corresponds to a diminution to about
I of the original diameter. They may indeed possibly absorb
a part of the surrounding moisture in the change ; but they do
not seem to have their dimensions much affected by the fluid in
which they are suspended, since they may easily be spread thin
on glass, and dried, without much change of their magnitude,
at least in the direction of the surface to which they adhere ;
and they remain distinct as long as the access of moist air is
completely excluded. When they have been kept for some time

No. XVI. REMARKS ON BLOOD AND PUS. 345

in water, and a little solution of salt is added, their form and
structure, as Mr. Hewson has observed, are more easily ex-
amined, and appear to resemble those of a soft substance with
a denser nucleus, not altogether unlike the crystalline lens
together with the vitreous humour, as seen from behind : but
with respect to a central particle detached witlun a vesicle,
*^ like a pea in a bladder/' I cannot doubt that Mr. Hewson
was completely mistaken. I have never observed a prominence
in th6 outline of the particles of the human blood : and on the
other hand I am not perfectly confident that the apparent de-
pression, which is exhibited in some lights, may not depend on
some internal variaUon of the refractive density of the particle.
It has commonly been asserted, that these coloured particles
are readily soluble in water ; but this opinion appears to be
completely erroneous, and to depend partly on their passing
readily through filtering paper, a circumstance indeed already
observed by Berzelius, (Djurk, 11. p. iii.) and partly on Uie
extraction of a great part of their colouring matter, together
with which they lose much of their specific gravity., so that in-
stead of subsiding, they are generally suspended in the fluid ;
their presence may however still be detected by a careful ex-
amination, and they seem in this state to have recovered in
some measure their original form, which they had lost when first
immersed in the water. When the water is sufficiently diluted,
about three-fourths as much rectified spirits may be added to it
without destroying the appearance ; but after a few months it
becomes indistinct, although neither in this case nor in that of
complete putrefaction do the globules appear to become con-
stituent parts of a homogeneous fluid. The existence of solid
particles, in fluids which at first sight appear transparent, is the
most easily detected by looking through them at a small lumi-
nous object, either directly or by reflection, as, for example, at
die image of a candle seen at the edge of a portion of the fluid,
held in a teaspoon ; in this case, wherever Uiere are small par-
ticles in suspension, for instance, in milk diluted with water,
they will produce a minutely tremulous or sparkling appear-
ance, which is rendered still more distinct by the assistance of
a lens, and which depends on the divertiified interception of the

346 REMARKS ON BLOOD AND PUS. No. XVI.

light) while the particles are carried over each other by the in-
ternal motion of the fluid. Tliis test is applicable to all cases
of minute particles held in snspen^on; where, however, the j

greater nmnber of the particles are nearly equal in dimensions, J

the luminous object viewed through them exhibits a much more
striking appeaitooe, for it is surrounded by rings of colours,
somewhat resembling those of the rainbow, but diflerently
arranged, and often beautifully brilliant. The blood, a little
diluted, always exhibits them in great perfection, and they afibrd
a very accurate criterion for the distinction between pus and
mucus: mucus, containing no globules, affords no colours,
while those which are exhibited by pus exactly resemble the
appearance produced by the blood, the ringjB being usually of
the same dimensions : whence it follows tliat the globules are
also of the same size, for the dimensions of the rings vary with
those of the particles which produce them : and there can be
little doubt, from this circumstance, that the globules found in
pus are the identical globules of the blood, although probably
somewhat altered in the process of suppuration* A minute
quantity of the fluid to be examined in this manner may be put
between two small pieces of plate glass, and if we hold the
glass close to the eye, and look through it at a distant candle,
with a dark object behind it, the appearance, if the globules are
present, will be so conspicuous as to leave no doubt respecting
their existence.

2. Description of the Eriometer,

The rings of colours, which are here employed to discover
the existence of a number, of equal particles, may also be em-
ployed for measuring the oomparatiye and the real dimensions
of these particles, or of any pulverised or fibrous substances,
which are sufiSciently uniform in their diameters. Inunediately '^

about the luminous object, we see a light area, terminating in a
reddish dark margin, then a ring of bluish green, and without it
a ring of red : and the alternations of green and red are often
repeated several times, where the particles or fibres are suffi«-
ciently uniform. I observed some years ago that these rings
were the larger as the particles or fibres affording them were

No. XVI. REMARKS ON BLOOD AND PUS. 347

smaller, but that they were alwayB of the same magnitude for
the eame particles. It is therefore only necessary to measure
the angular magnitude of these rings, or of any one of them, in
order to identify the size of the particles which afibrd them ;
and having once established a scale, from an examination of a
sufficient number of substances of known dimensions, we may
thus determine the actual magnitude of any other substances
which exhibit the colours. The limit between the first green
ring, and the red which surrounds it, affords the best standard
of comparison, and its angular distance may be identified, by
projecting the rings on a dark surfSau^, pierced with a circle of
very minute holes, winch is made to coincide with the limit, by
properly adjusting the distance of the dark substance, and then
this distance, measured in semidiameters of the drcle of points,
gives the corresponding number of the comparative scale. Such
an instrument I have called an Eriometer, firom its utility in
measuring the fibres of wool, and I have given directions for
making it, to Mr. Iidler in Foley Street. The luminous point
is afforded by a perforation of a brass plate, which is surrounded
by the circle of minute holes ; the substance to be examined is
fixed on some wires, which are carried by a slider, the plate
being held before an Aigand lamp, or before two or three
candles placed ui a line ; the slider is drawn out to such a dis-
tance as to exhibit the required coincidence, and the index then
shows the number representing the magnitude of the substance
examined. The instrument may be rendered more portable,
though somewhat less accurate, by merely making the perfora-
tions in a blackened card, furnished with a graduated piece of
tape. An eye not shortsighted will generally require the assist-
ance of a lens, when the instrument is made of the most con-
venient dimenst<Mis, which I have found to be such as to have
two circles of points, one at i and the other i of an inch in
semidiameter, with their correeponding scales. The central
perforations are about -^ and Vt of an inch in diameter ; the
points 8 or 10 only in each circle, and as minute as possible.
The light of the sun might also be employed, by fixing the
circle of points at the end of the tube of a telescope : but it
rather adds glare dian distinctness to the colours : nor have I

348 REMARKS ON BLOOD AND PCS. No. XVI.

been able to gain anything by looking through coloiured glares,
or by using lights of different qualities. Where the object con-
sists of fibres which can be arranged in parallel directions,
a fine slit in the plate or card affords brighter colours than
a simple perforation, and the points must in this case be
arranged in lines parallel to the slit ; but if care is not taken
to stretch the fibres sufficiently, the employment of the slit in
this manner will make them appear coarser than they really are.
The colours will still appear, even if there be a considerable
difference in the dimensions of the fibres or particles, but they
will be so much the less distinct as the difference is greater. In
this case the measure indicated will be intermediate between the
extreme dimensions ; although most commonly it will be some-
what below the true mean, the colours exhibited by the finer
fibres prevailing in some degree over the rest. The latitude,
however, which the Eriometer affords in the regularity of the
substances measured by it, and its collecting into one result the
effect of many thousands of particles, or of an endless variety of
small differences in the diameters of fibres, give it an unques*
tionable preference over every kind of micrometer which mea-
sures a single interval only at once, with respect to all applica-
tions to agriculture or manufactures ; for in reality there is not
a single fibre of wool among the millions which constitute a
fleece, that preserves a uniform diameter throughout its length,
and the difference is still greater between the fibres which grow
on different parts of the animal ; so that to take a single mea-
surement, or even any practicable mumber of measurements, by
the most accurate micrometer, in the usual acceptation of the
term, for a criterion of the quality of a fleece, can tend only
to the propagation of error or conjecture in the semblance of
the minutest accuracy. Even with the Eriometer, the difficulty
of obtaining a fair average of the quality of a sample of wool is
extremely great ; it is absolutely necessary to preserve the fibres
as much as possible in their natural relative situation, and
to examine them near the middle of their length ; the ends
next the skin are almost always considerably finer, and the
outer ends generally coarser, than the rest; but this difference
is greater in some kinds of sheep than in others, and as far as I

No. XVL REMARKS ON BLOOD AND PUS. 349

have observed, it is less in the Merinos and their crosses than in
other sheep : there is also far less difference in the difierent
parts of the same fleece in these breeds than in others ; still
however this difference is very observable, although it is pro-
bable that some part of the sheep might be found, which in all
cases might fairly be considered as affording nearly the average
of the whole fleece ; and I imagine that the part of the back
about the loins is the most likely to be possessed of this property ;
so that the middle of the fibres of this part of the fleece might
be assumed, in the finer kinds of wool, as affording a fair
measure for the whole.

3. Scale of Eriometer.

The theory, which suggested to me the construction of the
eriometer, requires some corrections in its immediate applica-
tion, which depend on circumstances not completely understood:
at present therefore I shall only employ, for the determination
of the true value of the numbers of its scale, an experimental
comparison of its indications with some microscopical mea-
surements^ which Dr. Wo)laston has been so good as to perform
for me, with an admirably accurate micrometer of his own
invention.

The dust or seed of the lycoperdon bovista he finds to be
-rsV? of an inch in diameter : this substance gives very distinctly
3.5 on the scale of the Eriometer ; and 3.5 x 8500 = 29750.
The globules of the blood ^measured tAit ; and immediately
afterwards, when examined in the same state by the Eriometer,
indicated about 6^ ; and 6.5 x 4900 = 31850. A wire of pla-
tina, obtained by a very ingenious method, peculiar to Dr.
WoUaston, measured tiW ; and when coiled up gave n. 9 of
the Eriometer ; and 9 x 3200 » 28800. The mean of a consi-
derable number of comparative observations on fibres of wool,
between n. 20 and 30, afforded also 28800 for a product

A mean of these experiments pves very nearly TrJirT for the
unit of the scale of the Eriometer. Some former investigations
had led me to attribute to this unit a value somewhat smaller,
especially for the lowest numbers ; and I had obtained a for-
mula, and made a table, for ascertaining the true dimensions of

350

REMARKS ON BI,OOD AND PDS.

No. XVL

any substance measured by the instrument, according to the
result of these investigations ; but since my later experiments
seem to have superseded the mode of calculation which I had
adopted, I think it unnecessary to insert the table.

Having sufficiently ascertained the true value of the iudica-
tions of the eriometrical scale, I shall now enumerate the mea-
surements of the principal substances which I have examined
with the instrument.

4 Substances measured by the Eriometer.

Milk, diluted, very indistinct, about .

Dust of lycoperdon bovista, very distinct

Bullock*s blood, from beef

Smut of barley, called male ear

Blood of a mouse ....

Human blood diluted with water, 5 ; after standing

some days 6, or . . . ,
Blood recently diluted with serum only

Pus

Silk, very irregular, about %

Beaver wool, very even, (jointed)
Angola wool, about ....

Vigouia wool

Siberian hare's wool, Scotch hare's wool. Foreign

coney wool, Yellow rabbit's wool, about
Mole's fur, about

Skate's blood, very indistinct, about
American rabbit's wool, British coney wool,
Bufl&Jo's wool (B) .
Wool of the ovis montana (D) .
Finest seal wool, mixed, about
Shawl wool 18 or
Goat's wool ....
Cotton, very unequal, about
Peruvian wool, mixed, the finest locks
A small lock of Welsh wool (B)
Saxon wool, a few fibres 17, some 28, chiefly
An Escurial ram, at Ld. Somerville's show, 23 to

about

3
3i

6i
6i

7
8

74
12
13
14
15

154

164

184

No. xvr.

REMARKS ON BLOOD AND PUS.

Mr. Westeni's South Down, some specimens
Lioneza wool, 24 to 29, generally .
Paular wool, 24 to 29, generally
Alpacca wool, about
Farina of laurustinus
Ryeland Merino wool, Mr. Henty
Merino South Down wool, Mr. Henty
Seed of lycopodium, beautifully distinct
South Down ewe, Mr. W. B. .
Coarse wool, Sussex ....
Coarse wool, from some worsted

351

244

254

It would not be di£Scult to obtain from these measiu'es a
tolerable approximation to the value of wool at its usual prices.
If we square the number, and subtract 325, the remainder will
be about the number of pounds that are worth 100 guineas.
Thus, for good Lioneza, n. 25, 25x25-325 = 300, giving Is.
a pound ; for moderate South Down, n. 35, 35 x 35—300 = 900,
or 2«. 4J. a pound : which is probably about the proportional
value, though both the proportional and the real values must
fluctuate according to the demand of the manufacturer.

5, Microscopical fallacies,

I shall here take the liberty of inserting some remarks, which
I cannot attempt at present to render intelligible to any, who
have not entered into the minutest refinements of physical op-
tics : to such as are unacquainted with the latest investigations,
I fear they must appear involved in a degree of obscurity almost
enigmatical.

When a small object is viewed in a. microscope, especially
if the light is admitted by a limited aperture, it will often
appear to be surroimded by some lines of light and shade, or of
colours, which might be supposed to depend on its magnitude,
in the same way that the eriometrical colours are derived from
the magnitude of the objects examined. In reality, however,
their existence and their dimensions depend on the aperture of
the microscope, and not oa the magnitude of the particles in its
focus. To prove that this aperture may produce such an effect.

352 REMARKS ON BLOOD AND PUS. No. XVI.

hold any object, for instance, the finger or the nail, so as to
intercept all the light of a candle, except a narrow line, and
this line will seem to project other lines parallel to it into the
adjoining shade. Now these lines depend on the interposed
object on one side, and on the margin of the pupil on the other :
for if we take an object a little narrower than the pupil, we
may see them on both sides of it ; and causing the pupil to
contract by throwing more light on the opposite eye, they will
expand, as the space, through which they are admitted, is
diminished by the contraction. We may also very distinctly
observe, if we look in this manner at a narrow line of light
instead of a candle, that the dispersive powers of the eye mani-
festly convert its image on the retina into a spectrum of red,
green, and blue light; sufficiently confuting the conjectural
hypothesis of the achromatic property of its refractive sub-
stances. If again we substitute a minute hole or slit in a card
for the interposed object, the sides of this aperture will now
determine the magnitude of the fringes which are seen at the
edge of the candle, and their dimensions will be no longer
variable, whatever may be the state of the pupil. But the
candle must in this case either be placed at a distance or be
partly concealed from the eye, unless one edge of the aperture
project so far beyond the other, as to limit its visible extent
Now the substance, in which the lens of a microscope is con-
tained, presents a small aperture capable of exhibiting effects
of this kind, which however can only be expected to appear
when the light is peculiarly circumstanced. The aperture of
the highest magnifier that I have employed is A* of an inch,
which answers to about n. 330 of the scale of the Eriometer,
and would consequently exhibit a bright ring at -giv of the dis-
tance of a minute object viewed through it, while the darkest
part within this ring would be at about | of that distance: and
the focal distance of the lens being about -^V of an inch, the
diameter of the apparent dark circle would be ttItt of an inch,
and that of the bright one f^rr ; and the dimensions would be
nearly the same if any other small lens were employed, with
an aperture half as great as its focal distance ; so that the con-
stancy of such an appearance, notwithstanding a change of

No. XVI. REMARKS ON BLOOD AND PUS. 353

magnifiers, might inorease the probability of error. It is obvious
that a shade of this kind, surrounding the central parts of a
globule, if they happened to be much brighter than the rest,
might giye rise to a mistaken idea of inequalities in its form or
structure ; and it is possible that when a particle is darker than
the surrounding medium, some parts of its surface may have
lines of a similar nature projected on them in an inverse order.
The particles of the blood are about rrwv of an inch in dia-
meter, varying from tisW to .^^ ; and it is extremely possible
that an object of these dimensions may exhibit a light point
near its centre, which may be surrounded by a dark and then
by a light annular shade within the limits of its disc. There
are also several other sources of error in different lights, and in
a focus more or less imperfectly adjusted ; it is however suffi-
ciently evident that no fallacy of this kind can have given
rise to all the appearances, which have been already described,
as observable in the particles of the human blood, and still
less to those which are observable in the blood of some other
animals.

6. Changeable Colours,

In examining some of the dust of the lycoperdon, I had put
it with a drop of water on a glass, when I observed a purple
tinge in the water, which I thought at first was a stain extracted
from the powder; but the -water viewed separately was per-
fectly transparent, and the light transmitted directly through
tbe watei*, when the globules were present, was of a yellowish
green. After some consideration, I conjectured that this ap-
pearance of colour must be analogous to that of the mixed
plates which I had formerly observed, depending on the differ-
ence of refractive density of the water and the globules, (Nat.
Phil. I. 470. PL 30. F. 430,) and by substituting fluids of
different densities for water, I had the pleasure of finding my
conjecture confirmed ; for when the water was saturated witli
salt, the yellow green became nearly blue, and the purple redder
or browner ; and when olive oil was employed, the light directly
transmitted was purple, and the oblique light greenish ; in bal-
sam of Tolu again, this purple became red, and the indirect

VOL. L 2 a

354 REMARKS ON BLOOD AND PUS. No. XVI.

light afforded a faint blue. In air too, I found that the powder
appeared of a bright blue green by direct light, and of a pur-
plish hue with a light a little oblique ; but when the obliquity
became a little greater, the tint changed to a brownish yellow
green, which continued afterwards unchanged ; this alteration
may perhaps be derived from the admixture of a portion of
light coming round the particles by a more circuitous route.
By comparing the opposite effects of water and olive oil, of the
refractive densities 1.336 and 1.379, the refractive density of
the particles themselves may be calculated to be 1 .62, or some-
what less.

Grey beaver wool seems of a purplish hue in direct, and
greenish in oblique light, both in air and in olive oil ; its grey
colour seems to be derived from a mixture of these tints ; in
olive oil, the rings of colours which it affords are considerably
altered in their appearance, the reds becoming every where
very feint Lead precipitated from its acetate, or silver from its
nitrate, by common water, affords a reddish direct and a bluish
indirect light, and the same seems to be true of smoke, and of
other bodies consisting of very minute particles ; but when the
indirect light b very powerful, smoke sometimes appears reddish
in it, as might be expected from a collection of very small
opaque instead of transparent particles.

Mr. Delaval has observed that an infusion of sap green
appears of a bright red by transmitted light, and the case seems
perfectly analogous to that of the dust of the lycoperdon ; the
green becoming somewhat yellower, when the gum, with which
the colouring particles are mixed, is diluted with water. But
this is not the universal cause of a difference of colours exhibited
by pigments in different lights ; the carthamus, or pink dye com-
monly sold for domestic use, affords an unequivocal instance of
a substance exhibiting colours analogous to those of thin plates,
which have been adduced by Newton, in illustration of the
colours of natural bodies ; the reflected light being undeniably
of a yellow green, while the transmitted light is of a bright pink
colour. Here the light regularly reflected from the surface only,
especially when dry, gives the colour opposite to that of the
transmitted light ; all the light passing through the fluid, even

No. XVI. REMARKS ON BLOOD AND PUS. 355

indirectly, giving a pink colour. But the infusion of the lig-
num nephriticum seems to hold a middle place between this
substance and those which have been mentioned before; the
dry extract is of a brownish yellow only ; an infusion, not too
strong, ^ves the same colour, verging to orange, by direct trans-
mitted light, and a bright blue by light reflected, or obliquely
dispersed within the infusion, or at its surface. The solution of
the carthamus affords no green reflection from its surface, and
varies in its hue, in difierent lights, only from crimson to scarlet.
The tinging particles of the lignum nephriticum, like those of
the precipitated lead and silver, are probably extremely minute,
since the colour is but little changed by changing the density of
the fluid. It oflien happens that a blue colour, precisely like
that of this infusion, is' reflected by green glass bottles, which,
when seen by transmitted light, exhibit only a reddish brown
colour. The inner bark of the ash Lb also said to have a pro-
perty similar to that of the lignum nephriticum. (Murr. app*
med.) The particles of the blood do not derive their colour
from any of the causes which have been mentioned, since it may
be. extracted from them in a clear solution.

When I attempted to explain the colours of mixed plates*
which I had produced by partially moistening two lenses very
slightly convex, I observed that the reflection of the light from
the internal sur&ce of a denser medium must be supposed to
invert its properties with respect to the production of colours
by interference, as is naturally to be supposed on the prin-
ciples of the undulatory theory. But when the obliquity is so
considerable, it is not very easy to assign a reason for this in-
version; and the experiments, which I have now mentioned,
make it necessary to assume a law, which I cannot explain, that
every very oblique reflection inverts the properties of light with
respept to interference. This conclusion confirms the assertion
of Newton, that a dark space, bordered by light, will appear in
the centre of a portion of light transmitted between the edges
of two knives placed very near each other, and the opinion of
Mr. Jordan, that the augmentation of a shadow by difiraction
is to be considered as the first dark space belonging to the
coloured fringes. I had obtdncd a different result in an ex-

2 a2

356 REMARKS ON BLOOD AND PU». No. XVI.

periment similar to Newton's, because I was not aware of the
necessity of employing very sharp edges ; for when the edges
are blunt, the light is reflected from the one to the other in such
a manner, as wholly to destroy the appearance of a central dark
space ; but in any case this source of error may be avoided, by
causing one of the edges to advance a very little before the plane
of the other, so that half of the fringes may disappear. It is-
however necessary to suppose this inversion confined to cases
of extremely oblique reflection, for when the deviation of the
light from a rectilinear path becomes a little more considerable,
its effects are no longer perceptible; the second and third fringes
scarcely ever requiring any material corrections of the calcula-
tions from which it is excluded. The same inversion must also
be attributed to the light bent by di&action round the remoter
side of a fibre : for this light always co-operates in the first
instance with that which is reflected from the nearer side. The
extent of the central white light is indeed so great, that all the
coloured appearances may almost be considered as beginning
at such a distance, that the first dark space is exactly where
the simple calculation would lead us to expect the white ; since
the value of the unit of the Eriometer ought to be, according
to this calculation^ about tt^vt of an inch, instead of TTirr ;
and indeed this value agrees very accurately with experiment,
where the two portions of light concerned are exactly in similar
circumstances; as may be observed in some of the parallel
lines drawn on glass in Mr. Coventry's micrometers, probably
where they happen to be single, for in general they are double,
and exhibit colours corresponding to an interval much smaller
than their regular distance : but in some parts we may observe
colours exactly corresponding to their distance, for instance, to
fhr of an inch, according to the simple principle of considering
each unit as equal to about the 43000th of an inch. Hence
it seems that the necessity of a correction depends on the dif-
ferent state of the lights reflected from one side of a fibre, and
difiracted round its opposite side, and that when they proceed
in a similar manner from two neighbouring parallel lines, the
necessity no longer exists. What may be the cause of this
irregularity, will perhaps be understood when we understand the

No. XVI. REMARKS ON BLOOD AND PUS. 357

cause of the singular phenomena of oblique reflection discovered
by Mr. Malus, and we have no reason to expect to understand
it before.

7. Glories.

I have had an opportunity of ascertaining, that the clouds
which exhibit the white and coloured circles, sometimes deno-
minated glories, are certainly not composed of icy particles ;
and I have succeeded in deducing an explanation of these phe-
nomena from the same laws, which are capable of being applied
to so many other cases of physical optics. In the theory of
supernumerary rainbows, (Nat. Phil. I. 471. PI. 30. Fig. 451.
'^'^^^mpra^ p. 185), I have observed that the breadth of each bow
must be tlie greater as the drops which aflbrd it are smaller ;
and by considering the coloured figure, in which their production
is andysed, it will be obvious, that if we suppose the coloured
stripes extremely broad, they will coincide in such a manner in
one part as to form a white bow : the red, which projects beyond
the rest, being always broadest, so that if all the stripes be
supposed to expand, while they preserve their comparative
magnitude, the middle of the red may coincide with the middle
of the blue ; and it will appear on calculation that a white bow
will be formed, a few degrees within the usual place of the
coloured bow, when the drops are about lAv or ^ of an inch
in diameter. It is remarkable that in such cases the original
rsdnbow is altogether wanting, and probably for a similar rea-
son, we scarcely ever see a rainbow in a cloud which does not
consist of drops so large as to be actually falling, although I
have once seen such a rainbow ending abruptly at the bottom
of a cloud : it may be conjectured that the edge of the light is
in such cases so much weakened by diffitu!tion^ that it is too
faint to exhibit the effects occasioned by a larger drop. Dr.
Smith has made a remark somewhat similar, (Opt. r. 501.)
which, if not completely satisfactory upon the principles which
have been mentioned, is certainly altogether unintelligible upon
his own.

The coloured circles, immediately surrounding the shadows
of the observers, may be deduced from the effect of the same

358 REMARKS ON BLOOD AND PUS. No. XYI.

minute particles of water, upon the light which has been four,
and perhaps five, times reflected within the drops, which may,
after transmission, coincide in direction with another portion,
passing on the opposite side of the centre ; and the drops about
lAv or zi^ of an inch in diameter would in this manner produce
a fidnt corona, of such magnitude, that the limit of green and
red, employed in the use of the eriometer, should be at the
distance of about five degrees from the centre of the shadow ;
which, as nearly as I could estimate it, was its real distance in
the appearance that I observed.

No. XVII. CORRESPONDENICE ON OPTICAL SUBJECTS. 359

Na XVn.

SELECTIONS FROM CORRESPONDENCE RELATING TO

OPTICAL SUBJECTS.

1. — Ftom Dr., now Sir David Brewster to Dr. Young.

Dear Sir, Edinburgh, is, Hope-^twet, July 28, 1815.

I AVAIL myself of an opportunity which has just offered
itself, of sending you an account of an optical discorery in
which I have no doubt you will be much interested, as it
appears to give very great support to your opinion, that the
colours produced by the action of crystallised bodies upon
polarised light are referable to your theory of periodical
colours.

Haying some time ago discovered that all the metals acted
upon polarised light, like crystallised laminae, a single reflexion
polarising the same colour as a plate of sulphate of lime of a
given thickness, it occurred to me that a similar effect might
be produced by total reflexion from the second surfaces of all
transparent bodies. I had great difficulty, however, in making
this experiment, as almost every glass prism is more or less
crystallised ; but by enclosing water in hollow prisms of glass
that exercised no action upon light, I ascertained that one total
reflexion from the second surfaces of transparent bodies polar*
ises a tint of the first order of Newton*s table in exactly the
same manner as a plate of sulphate of lime. Two or more total
reflexions in the same plane polarise colours higher up the table
like two or more films of sulphate of lime, having their axes
coincident ; while two total reflexions at the same angle and in
planes of nght angles to eitch other, counteract each other like
two equally thick plates of sulphate of lime, having their axes
crossed at right angles.

360 CORRESPONDENCE RELATING TO No. XVII.

There is here no action of a doubly refracting force : the
totally reflected pencil consists of two oppositely polarised por-
tions, one of which, namely, the one which is polarised in a plane
at right angles to the plane of reflexion, has advanced farther
through the transparent body, and therefore suffers reflexion, or
changes its direction later than the other pencil. Hence
colours ought to be produced.

I have no doubt that the preceding phenomena will be con-
sidered as hostile to Biot's theory of oscillations, and as highly
favourable to your theory of recurrent colours.

I have lately found that muriate of soda and fluor spar,
which Mains, Biot, and Arago have always regarded as pos-
sessing simple refraction, do possess the properties of all doubly
refracting crystals : at a thickness of between 1 and 3 inches
they polarise the blue of the first order of Newton's table, and
exhibit regular optical axes. You would oblige me by not
mentioning any of the preceding properties, as I have not yet
finished the investigation of them.

I have the honour to be, dear Sir,

Ever most truly yours,

D. Brewster.

2. — Dr. Young to Sir D. Brewster.

Dear Sir, Worthing, ISth September, 1815.

I FEAR you have thought me remiss in not returning an
earlier answer to your kind communication : but I have been
waiting until I could find leisure to enter a little more fully
into the subject, than I had it in my power to do when I
received your letter. I am glad that a circumstance has
occurred, which has led you to think with some attention of my
theory of the interference of light, because I am quite certain
that it must become particularly interesting to you, who have
observed so many phenomena which appear to be most inti-
mately related to it, and which admit of a striking illustration
by its means. I cannot, however, see that it has much connexion
with the facts which you state respecting internal reflection : if
my memory does not deceive me, some former optician, perhaps

No. XVII. OPTICAL SUBJECTS. 361

Bouguer, has observed a little difference of colour in light
totally reflected, but I have no note of the obserration by me :
nor can I attempt to form anything like an opinion of the nature
of this appearance of colour, or of that which you mention as
produced by the reflection of metals, not tamUhed^ without a
more particular detul of the eiperiment

With respect to my own ftindamental hypotheses respectmg
the nature of light, I become less and less fond of dwelling on
them, as I learn more and more facts like those which Mr.
Mains discovered : because, although they may not be incom-
patible with these facts, they certainly give us no assistance
in explaining them. But this observation does not extend to
my laws of interference, as explanatory of the phenomena of
periodical colours : since almost every new case of the produc-
tion of colours, that has been lately discovered, ranges itself as
a simple consequence of these laws, and is as regularly dedu-
cible fi*om them by calculation, as the motions of the planets
are deducible from the laws of gravitation ; nor are the laws
any more invalidated by disproving the Huyghenian system of
undulations, than the Huyghenian law for the determination of
oblique refraction was rendered less accurate by Newton's
Bupposed confutation of the same hypothesis, or than the laws
of gravitation would be superseded if we allowed the validity
of poor Professor Vince's arguments against the Newtonian
ether.* You were apparently impressed with a very high
respect for the generalisation of the phenomena of the colours
of plates of doubly refracting crystals, which Mr. Biot had
obtained, although his rules are derived from direct observation
only, and merely enable us to ascertidn the colour of a given
plate of some few given substances, viewed in a given direction.

* Professor Vince's ' Memoir on the Cause of Grayitation ' was presented to the
Royal Society, and selected, in the first instance, as the Bakerian lecture, though it
was subsequently, upon a more deliberate examination, rejected as unlit for publica-
tion in the Transactions : it was atlerwards published as a pamphlet, with a preface,
in which the treatment it had experienced was in some degree attributed to the pre-
judice of the president. Sir J. Banks, against the party in the Society, headed by Dr.
Hutton, to which Uie author had attached himself. This pamphlet was made the
subject of some rery severe and somewhat unjust critiques by Dr. Young, under tlie
signature * Dytiscus,' in Nicholson's Journal for 1808 : neither the character nor the
works of Professor Vince were such as to justify the contemptuous reference to him
which is made in the text. — Note by the Editor,

362 CJORRESPONDENCE RELA'HNG TO No. XVII.

But you do not seem to have been at all struck by my deductions
of the same conclusions, or of conclusions agreeing equally well
with the experiments from the general principles of the laws of
interference only, combined with the known refractive powers
of the substances concerned: and I must confess that I was not
a little astonished to find that the deduction seemed to make
no impression whatever on Biot, when I sent it him : for in a
very civilly intended answer, he merely referred me to his own
book, which I have not read, and which I believe I shall not
read, because I find that he begins with suppositions respecting
the motions of the particles perfectly incompatible with the
general laws of mechanics. I can only suppose that I had not
made myself sufficientiy intelligible to him : otherwise the co-
incidence of my determination of the actual thickness affording
a ^ven colour and of the effects of obliquity and of inclination
in all directions with the diversified facts which he had ascer-
tained, must have been too striking to have escaped him.
Nothing can be simpler than the principle on which my
calculation proceeded, that there must be two sets of rays in
each direction, one transmitted by ordinary, the other by ex-
traordinary refraction, and that the difference of the velocities
of these rays must cause the phenomena of interference in the
same manner as if they had actually arrived by paths of different
lengths : and notlung can be more intricate than the residts, if
examined without a clue.

Had the consequences of this law been present to your mind,
you could scarcely have avoided observing how completely it
explained the phenomena that you have described in the printed
papers which you have been so good as to send me. The rays,
twice or thrice reflected, are transmitted through the two glasses
at an obliquity a littie different ; and this difference is sufficient
to produce the same phenomena as the passage of the rays
through a single plate of which the thickness corresponds to the
difference : just as in Mr. Knox's experiments the effects are
precisely the same as those of a plate of which the thickness is
equal to the difference of the thicknesses of the two plates of
glass ox of air concerned : a phenomenon which I had long ago
explained in my Lectures, in a remark on an observation of

No. XVII. OPTICAL SUBJECTS. 363

Mr. Nicholson : you will find it under the head Colours from
Interference, in the second volume. You have observed that
the Newtonian doctrine of fits of easy refraction and reflection
sufficiently explains the phenomena of thin and of thick plates ;
but you surely cannot have considered this subject with attention,
otherwise you would have been aware that when the path of the
ray is lengthened by its obliquity with respect to the surfaces,
the number of fits ought naturally to be increased, while in
reality it is diminished : and I must request you to tell me how
you explain the white circle from which the colours of thick
plates begin, by means of anything like the Newtonian doc-
trines ? You well know that whiteness never occurs in period-
ical colours, except at the be^nning of the series, where the
thickness is evanescent : and in order to obtmn this evanescent
measure, we must take the difierence between the lengths of
the paths of the two portions of light, passing with a ^ven
obliquity, and with an obliquity a little different from it ; an
explanation which is precisely the same with tliat of the colours
which you have lately observed by means of two separate
pieces, except that the Newtonian colours of thick plates are
formed in light irregularly dissipated, and yours in light regu-
larly reflected only.

You see that I leave polarisation completely out of the
question in all my calculations ; for, as far as I have been able
to discover, the phenomena of polarisation affect light of all
colours precisely in the same way, except so far as they happen
to separate portions of light from each other, which, in conse-
quence of otiber circumstances, are capable of exhibiting, when
so separated, particular colours ; and if the internal reflection
which you mention is found to affect some colours exclusively,
the phenomenon is altogether singular in this respect. But I
look forwards with great interest to the details of your experi-
ments, for at present I can form no very distinct idea of the
iacts. I must not omit to congratulate you on your happy
discovery of the determination of the angle of complete polari-
sation according to the difference of densities : * it appears to

* In the Philosophical TranMctions for 1813, for which the Copley medal wai
awarded.

364 CORRESPONDENCE RELATING TO No. XVII.

be the most important step that has been made in optics since
the first discoveries of Mr. Mains, although, like many other
great improvements, it seems sufficiently simple and obvious
when once known. I shall endeavour to find an opportunity of
forwarding to Mr. Knox the copy of your paper which you have
sent me for him ; it was too late to send it with the separate
copies of his paper in the * Philosophical Transactions.'
Believe me, dear Sir,

Very truly yours,

Thomas Young.

3. — Dr. Young to Sir David Brewster in Reply to Observa-
tions an the Priority of some of his Discoveries made likewise
by Dr. Seebeck, for which the French Institut had awarded
a joint Prize.

Dear Sir, London, 24th January, 1816.

I SHOULD have sent an earlier answer to your letter,
but I have been waiting to procure the information which you
require concerning Seebeck's papers. Unfortunately Dr. Wol-
laston has mislaid the copy from which my former abstract was
made, otherwise I should have begged him to let me send it to
you. The coloured plates bear a strong resemblance to your
figures, and there is no doubt of the identity of the appearances
.with those which you have described; there was, however,
certainly no mention of heat in that paper. The second
paper I have just procured from Dr. Thomson, who is in
possession of the whole journal, and in this he has most dis-
tinctly proved the dependence of the phenomena on the mode
of cooling the glass, so that there cannot be a doubt of bis
having a parallel claim with you, unless it can be proved that
he was previously acquainted with your discoveries, which is
indeed barely possible, but not very probable. The number in
which his paper is inserted contains an account of the contents
of Thomson's Annals for July and August, 1813, only ; and it
is very improbable that the editor would have inserted this
if any English publication of the kind a year later had been
current in Germany. Seebeck has also gone a little further in

No. XVII. opncAii SUBJECTS. 365

the inyestigation than you had done at the time, for he attempts
to show that the phenomena do not depend on anything like
crystallization, and this opinion is admirably confirmed by the
facts which you announce to me at the close of your letter.
Your researches are indeed so numerous, and you proceed
so rapidly in the career of discovery, that your subsequent
labours have often superseded the preceding, and for this
reason, as well as for many others, you might perhaps find
it more satisfactory in looking back on some of your papers, if
they had been a little more compressed in their bulk.* I am
perhaps too fond of concbeness myself, but in this instance
I am not the only one that regrets the unnecessary detail in
some of your papers. It is often true that '' half is more than
the whole," even of things which are all intrinsically valuable.
I must indeed congratulate you on the importance of the new
fact respecting the efiect of compression and extension in
causing a double refraction : perhaps I view it with a partial
eye as favourable to my own opinions ; but to me it appears to
exceed in value the determination of the angle of polarisation,
which the Council of the Royal Society fixed on as deserving .
to be distinguished from the rest of your communications. I
have demonstrated in an early number of the ' Quarterly
Review,' (though I must beg you not again to quote any
anonymous paper as mine,) that every undulation or other
impression must be propagated through a minutely stratified
substance in the form of a spheroid ; f and you have shown by
experiment that as soon as the density becomes greater in one
direction than in another, which is the condition on which my
demonstration proceeds, the propagation actually assumes a
spheroidal form : so that this single detached fact is completely
explained by my theory, while it has certainly no apparent con-
nexion with any other ; at the same time it gives me no assist-
ance with respect to the immediate phenomena of polarisation,

* Sir David Brewster, in his reply, gives very latufactory reasons for this want of
compression in some few of his memoirs. The progress of discovery in this rich £eld
of experimental research, in which many lahoureni were engaged, was so rapid, that
it was necessary, in order to secure priority of claim, to put them npon record with-
out allowing sufficient time for the suppression of much that was superfluous and
irrelevant. — Note by th^ Editor.

t See No. XII. p. 230.

366 CORRESPONDENCE RELATING TO No. XVII.

strictly so called. I must still wait for a more minute detail
of your experiments on internal reflection before I can admit
them as properly referable to my law of periodical colours ; it
is generally necessary, in the application of that law, to sup-
pose the reflection to take place exactly at the sur&ce, in a
sense almost strictly mathematical, except that in some cases
exactly half an interval appears to be lost, for every kind of
rays ; and I do not like the idea of supposing even a physical
surface to contain an appreciable space. You would, perhaps,
be able to throw some new light on this subject by examining
more particularly the reflection from the upper part of a soap
bubble. You know that a wine-glass dipped into soapy water,
and held against a window-shutter, affords the best sort of
bubble for the purpose : at the top is a black space almost in-
visible, but still exhibiting a faint reflection. What has always
struck me as the most remarkable about this space is its sharp
and very decided termination, whether viewed perpendicularly
or obliquely, while according to theory it ought to be gradually
shaded off. It might be imagined that the fluid is too thin
. to allow space for both the reflections which you suppose
to take place in some cases ; but there are many objections to
this explanation : possibly you may find something peculiar in
the polarisation of light by this reflection which may illustrate
the phenomenon. Biot highly approves the decision of the
Royal Society awarding you the Copleian medal, but he thinks
the paper particularly pointed out not the most unexcep-
tionable of your communications ; he does not, however, bring
any material objection against the universality of the law which
you have laid down in it, and which is certainly by far the
most important part of the paper. I hope you will be able to
add from time to time to our knowledge of these subjects, and
I could almost wish you could reserve something good for a
year or two, that you might have a fair chance for a more
valuable medal than the Copleian : unluckily your best dis-
coveries fall within a period when another invention has been
made public, which is so striking and so important to society
that I suppose it will scarcely fail of overcoming all competi-
tion. Have you seen the account of Biot's experiments on

No. XVII. OPTICAL SUBJECTS. 367

substances causing a polarised ray to revolye round its axis ?
He must have shown you some of them. I do not fully under-
stand them ; but if the fact is such as the simple statement
would induce one to suppose, it militates against the doctrine
of detached particles ; for if the particles had received a rota-
tory motion round an axis in their passage through the sub*
stance they would have retained it, and have turned further
round the further they travelled after their emersion. I do not
however profess to be sufficiently well acquainted with the
phenomena to appreciate the validity of the argument in its
whole extent. Pray have the kindness to give the enclosed
note to Dr. Duncan and the paper to Playfair, when you have
looked at it.

Believe me, dear Sir,

Your faithful, obedient servant,

Thomas Young.

Thursday. — The Council has adjudged the Rumfordian
Medal for the two years ending in November last to Dr.
Wells for bis Essay on Dew.

4,.— From Sir D. Brewster to Dr. Young.

Dear Sir VenUw by Peebles, October 4, 1817.

As you formerly requested me not to mention your
name in connexion with any of your anonymous works, I write
you at present chiefly to ask if you have any objections to have
your theory of the colours produced by the action of crystals
upon polarised light mentioned as your own. I have just come
to that part of my paper on the laws of polarisation and double
refraction, where I mean to introduce the subject* I have
little doubt that it will be found to represent all the phenomena ;
and my paper will furnish you with the means of putting it to
the most decisive test. I have succeeded in reducing the laws
of polarisation and double refraction to laws rigorously physical,

• ' Od the Laws of Polarisation and Doable Refraction in regularly Crystallised
Bodies,' read January 13th, 1818; in the Philosophical Transactions for that year,
p. 199.

368 CORRESPONDENCE RELATING TO No. XVII.

by which the tints and the aberration of the extraordinary ray
can, in every case, be computed, whatever be the number, the
position, and the character of the axes of extraordinary refrac-
tion. These laws are deduced from experiments made upon
60 crystals witii two axes, and 20 with one axis. The greater
number of the generalisations of Mr. Biot are completely
erroneous. His division of crystals into attractive and re-
pulsive is unfounded, and his results respecting the rotation of
the luminous particles are mere delusions.

• . • • t . *

I do not know if you have seen a notice which I sent
to Mr. Brande respecting tiie production of tlie complementary
colours at the separating surfaces of media : I wish very much
to have your opinion of this singular result, as it does not
appear to me to be explicable upon any theory but the one
which I formerly mentioned to you.

I am, dear Sir,

Ever most faithfully yours,

D. Brewster.

5. — Dr. Young to Sir David Brewster.

My dear Sir, Worthing, lOtb October, 1817.

I CAN have no objection to your quoting the article in
question, as one which you judge^ from internal evidence, to be
mine, if you feel no difficulty in hazarding such an opinion,
although, for many reasons, I do not wish to be considered as
avowing it publicly at present.

I shall be extremely interested in the details of the experi-
ments which you mention, though you do not say by what
channel you mean to make them public. I have long been
aware that Biotas oscillations were wretchedly visionary, and
that some of his experiments were inaccurate ; but others I
have found to agree extremely well with my calculations,
especially those on rock crystal. For the sulphate of lime I
have been obliged to suppose, unless his experiments are com-
pletely erroneous, that there are in your words " two axes of
crystallisation," or some similar irregularity. When I see your

No. XVII. OPTICAL SUBJECTS. 369

experiments I shall perhaps find in them sufficient data for
determining the proportions of the three axes of the distorted
spheroid which must, in such cases, be substituted for the sur-
face of revolution ; and there will then probably be no great
difficulty in calculating the refraction in the Huyghenian man-
ner, if you have not already done it Do you mean to deny
that the spheroid is sometimes oblate and sometimes oblong ?
which is what Biot's attraction and repulsion must be supposed
to mean. Surely this is proved by the interchange of the
refractions of two plates of the different substances, placed
with their principal sections parallel to each other, and showing
the colour dependent on this difference only. I have not seen
the note that you sent to Mr. Brande respecting the effects of
the common surfaces of different media ; and I fear that it will
be at least a month before I shall be able to ask him for it I
find that the surfaces of gold and silver exhibit the two species
of rings at once ; the one by regular reflection, the other by
irregular dispersion : and Mr. Arago has shown that the light
affording them is differently polarised. This has probably some
connexion with the effects that you mention.

I shall be most happy to receive fix>m you at all times any
accoimt of your interesting experiments, and of your investi-
gations respecting polarisation ; but do not send me any infor-
mation that you are not prepared to have mentioned agaif , for
I am always scribbling something anonymous ; and I am very
capable of introducing your experiments where perhaps you
would not wish them to appear : but I cannot help it — I can
only give you fiiir warning. I have, indeed, very lately been
entering into some optical subjects pretty much at large ; but
I do not tiiink that I shall resume the consideration of them for
a long time.

Believe me, dear Sir,

Yours very sincerely,

Thomas Young.

VOL. I. 2 b

370 CORRESEOKDBNCE RELATING TO No. XVII.

6. — Sir D. Brewster to Dr. Young.

My dear Sir VenUw by Peebles, Oetober 37th, 1817.

I WAS favoured with your letter of the 10th October,
and shall refer to the paper in the * Quarterly Review ' in the
way you mention. The manner, however, in which I would
have done it would not have been equivalent to your acknow-
ledging it, but would merely have shown my conviction that you
were the author. If your theory of the tints gives the same
values of them as those deduced from the law, which makes
them vary as the square of the sine of the angle which the
refracted ray forms with the axis of the crystal, it wiH suit all
crystals, whether they have one or more axes ; and if it does
not correspond with this law it cannot be correct. Biot's ex-
periments on rock crystal accord with the law, abstracting the
effect produced at oblique incidences^ which is, however, quite
a secondary one, and one which your theory ought not to em-
brace. In my paper which will be sent in the course of a
fortnight to the Royal Society, I have given general methods
by which the tint and the velocity of the extraordinary ray can
be computed in crystals with any number of axes of extraordi-
nary refraction, and hence not only the three axes, but all the
dimensions of what you very properly call the distorted spheroid^
can be easily ascertained : the ratio of the three axes is easily
found. ......

When I mentioDed to Mr. Biot, about a year ago, your de*
monstration, that an undulation propagated through a minutely
stratified substance, in which the density is greater in one
direction than in another, was spheroidical, he replied that both
Laplace and Poisson were of opinion that, in the present state
of mathematical analysis, the simplest case of undulation could
not be calculated ; and therefore that the above theorem was
not capable of demonstration. At such a distance from Edin-
burgh, I cannot command a sight of the volume of the ' Quar-
terly,' which contains your demonstration, otherwise I should
have studied it carefully. I recollect, however, not being able
to follow it thoroughly. I am exceedingly interested on this

No. XVII. OPTICAL 9UBJB0TB. 371

point, and am, therefore, anxious to know if yon are yourself
satisfied with the accuracy of the demonstration ; and if you
can assign any reason why the light which forms the ordinary
ray should not also be acted upon by the varying density of the
medium ?

I am, my dear Sir,

Ever most fisuthfnlly yours,

D. Brewster.

7. — Dr. Young to Sir David Brewster.

Dear Sir Worthing, 9th November, 1817.

I CANNOT help feeling a little disappointment in not
finding in your letter any such confirmation of my theory of
periodical colours as I had promised myself, from your having
mentioned your experiments as fully agreeing with it, or as
superseding those of Biot. You now tell me that my theory
ought not to embrace those very particulars to which I have
applied it in the paper that you intend to quote with approba-
tion. As to the variation proportional to the square of the
sine of the angle made with the axis, in perpendicular inci-
dences, it is so obvious a consequence of the properties of an
ellipsis difiering little from a circle, upon almost any supposi-
tion, that you could scarcely form a theory of double refraction
so erroneous as not to give you this law. What I profess to
have explained is that very efiTect of obliquity of incidence
which you seem now disposed to refer to another cause. But I
hope soon to have the pleasure of seeing the detailed account
of your experiments in the paper which you promise the Royal
Society.

I conclude that Mr. Biot had the candour to tell you that he
had read none of my papers whatever : he promised me that he
would attempt it in the course of the summer, but I dare say
he has not found leisure. Mr. Laplace has now arrived at so
happy a pre-eminence in science, that he thinks it sufficient to
assert where others would assign their reasons ; and having
once asserted, he is not very impatient to retract. He told mc
in July, as be had often declared before, that the Huyghenian

2 B 2

372 ' CORRESPONDENCE RELATINa TO No. XVII.

theory was incapable of determining the relation of the angles
of incidence and refraction; and when I could hardly help
smiling at the absurdity of the assertion, and endeavoured to
prove to him, in three words, how easily and necessarily the law
was deduced from the hypothesis, he be^ed me to send him a
short demonstration in writing, I did so, and instead of eiUier
admitting it, or endeavouring to point out its deficiency, he now
tells me that it is only an '* aper^u," a sketch, or a presumption.
(Infra, p. 374). This little occurrence is certainly of some value
to me, because it spares me a great deal of labour in entering
into any further controversy on such a subject with such a per-
son. With respect to Mr. Poisson, when we know how repeat-
edly and how deeply he has committed himself in praising and
in imitating some of Mr. Laplace's least successful speculations,
we cannot be surprised at his bearing him out on this point.
He praises, for instance, both the theory of capillary attraction
and tliat of oblique refraction as among the highest efforts of
human genius, while, to me, they both appear worse than
nugatory. Even within the last month I have received a paper
from Laplace, which is one of the most amusing instances of
waste of labour and calculation that can commonly be met
with. It is a determination of the correction required for Uie
length of a pendulum supported by a cylindrical axis, and the
ingenious author has exhausted all the powers of his analysis to
calculate tiie motions and rotatory powers of the different parts
of the pendulum in their various paths, when he might at once
have obtained the same result by simply calculating the curva-
ture of the padi of the centre of oscillation from the prpperties
of the epitrochoids.*

Without, however, entering into the discussion of the suf-
fideney or insufficiency of my demonstration of the elliptical form
of an undulation propagated through a stratified substance, I can
easily explain to you that it must, without all question, assume
an av€tl ferm, which is perhaps enough for your purpose. Mr.
Chladni has proved by experiment that the undulations consti-
tuting sound are propagated more slowly along a piece of wood

♦ See Dr. Toong's * Remarks on the Probabilities of Error iD Physical Obsenra-
tiona/ lie., Tol. U.,No. XXVIII.

N(K XVn. OPTICAL SUBJECTS. 373

cut across the grain, than in the direction of the fibre ; and you
may easily imagine that the velocity in an oblique direction
must be intermediate between the greatest and least velocities.
You must, therefore, either deny the accuracy of Chladni's
experiment, or admit the general fact of an oval undulation ;
and if you like Mr. Laplace's mode of reasoning, when he says
^' Nature t€Aes the form of the ellipse next to that of the
circle," you will have an apergu which he perhaps would like
better than my demonstration. I can, however, assign nothing
like a reason for the reflection or non-reflection of a ray accord-
ing to its polarisation.

I agree with you in thinking the experiments, showing the
eflect of doubly refracting substances on the polarisation of the
light by reflection, extremely important ; and I shall be glad to
know the details of your discoveries, when you have time to
prepare them for publication.

Believe me, my dear Sir,

Very truly yours,

Thomas Youno.*

• Sir D. Brewster's paper 'On the Laws of PoUrisation aod Double Refraction in
regularly Crystallised Bodies/ was read to the Royal Society in Janoary, 1818, and
was referred by the Council to Dr. Toong for examination. Whilst recognising to
their fullest extent the great value of the experimental and other results of this cele-
brated Memoir, he was not satisfied with some of the theoretical yiews which it con-
tained, and soj^ested to the author that they shoold either be withdrawn or modified.
A long and animated correspondence ensued between them, which ended with the
publication of the entire Memoir, with yery few alterations of any moment. The
following note was appended to i^ at the request of Dr. Toung :—

" My dear Sir,

** Tour experiments, on the colours afforded by crystals haying two optical
axei^ appear to establish a very important result in the theory of light; for supposing
them to be perfectly represented by your general law, it will follow that the tint
exhibited depends not on the difference of refractlye densities in the direction of the
ray transmitted, but on the greatest difference of rcfractiye densities in directions
perpendicular to that of the ray. These two conditions lead to the same result,
where the effect of one axis only is considered, but they yary materially where two
axes are supposed to be combined; and I do not immediately perceiye by what
modification it will be possible to aou>mmodate the laws of interference to these
experiments. There can be little doubt that the direction of the polarisation, in auch
cases, must be determined by that of the greatest and least of the refractiye densities
in question ; f and it seems to be yery possible to apply your mode of calculation to
many other phenomena, in which the polarising powers of different crystals are
combined.

** Belieye me, dear Sir, your yery £uthful seryant,

** Thos. Youno."

t Note added by Su" D. Brewster. — ^ This supposition of Dr. Young is perfectly
correct. In another paper, which will soon be submitted to the Royal Society,
I have giyen a general method of finding the direction of polarisation for any com-
bination of axes."

374 CORRBSPONDBafCB BELATING TO No. XVII.

8. — From M. Laplace to Dr. Young.
Monsieur, P»ri«» « 6 Oct. 1817.

J'ai regu la lettre que tous m^aves &it rhonneur de
m'ecrire, et dans laquelle vous cherehez a etablir que, suiyaiit
le systeme des ondulaticHis de la lumiere, lea onus d'ineideiioe
et de refraction sent en rapport constant, lonsqu'elle paase d'un
milieu dans un autre- Quelque ing^eux que aoit ce rai-
sonnement, je ne puis le regarder que oomme un ap^nQU,
et non comme une demonstration geinn^trique. Je persiste a
croire que le probleme de la propagation desondes, lorsquelles
traversent differens milienx, n'a jamais (^k rosdu, et qull sur-
passe peut-etre les forees actaelles de I'analyse. Descartes
expliqucHt oe rapport constant, an. moyen de deux suppositions ;
Tune, que la initesse des rayons lumineux parall^ement k la
surface du milieu refringent ne changeoit point par la rdfrao-
tion ; I'autre, que sa vltesse enti^re dans ce milieu ^toit la meme,
sous toutes les incidences ; mais comme il ne rattachoit aucune
de ces suppositions aux lois de la mecanique, son explication a
ete vivement combattue et rejett^e par le plus grand nombre
des physiciens jusqu'k ce que Newton ait fait voir que ces
suppositions resultoient de raction du milieu refringent sur la
lumiere ; alors on a eu une explication matbematique du ph&io-
mene dans le systeme de remission de la lumiere : systeme qui
donne encore Texplication la plus simple du phenomene de
I'aberration, que n'explique point le systeme des ondes lumi-
neuaes. Ainsi les suppositions de Descartes, comme plusienrs
aperqus de Kepler sur le systeme du monde, ont ete verifiees
par I'analyse : mais le merite de la decourerte d'une v^rit^ ap-
partient tout entier a celui qui la demontre. Je conyiens que
de Qouveaux phenomenes de la lumiere sontjusqu'apr^nttres
diffidles \ expliquer ; mais en les etudiant avec .un grand soin,
pour decouvrir les lois dont ils dependent, on parviendra
peut-etre un jour k reconnaftre dans les molecules lumineuses
des proprietes nouvelles qui donneront une explication matbe-
matique de ces phenomenes. Remonter des phenomenes aux
lois et des lois aux forces, est, comme vous le savez, ia vraie
niarche des sciences natiu'elles.

No. XVII. OPTICAL SUBJBOTS. 375

Monsieur Kater veut bien se charger de vous remettre deux
memoires qui paroltropt dans la prochaine Connaissance des
Terns. Uun est relatif a la longueur du pendule, 1 autre est
une application du calcul des probabilites a la geodesie. Je
regarde comme une chose importante, les applicatioos de ce
calcul aux sciences, et je desire beaucoup qu'elles se multipliait
Je joins a cet envoi, deux petites additions que j'ai faites a mop
* Essai Philosophique des Probabilites/ Je vous prie de vouloir
bien en rej^re une a M. Davy, avec miUe complimens de
ma part, et I'aBsurance de ma haute estime: ees additions
doivent etre substituees au lieu des ^lages 221 et suivantes de
I'ouvrage.

M. Kater a bien touJu me comvuniquer son resultat sur la
longueur du pendule a Londres. Ces experiances me paroisseut
etre d'uoe grande exactitude et faites par un procede ingl-
nieux. La seule chose qui en puisse laisser quelque incertitude,
est le changement de figure du systeme oacillant dans ses
deyx etats de suspension, en vertu de I'extension de ses parties
par les poids qu'elles supportent : il doit en resulter un de-
placement dans les centres d'oscillation, qui sans doute est tres
petit ; je suis meme porfce a croire qu*il est insensible ; mais
lorsqu'on yeut ai;teindre a la precision d'un cent-miUieme il est
laeoessaire d'appreeier toutes les causes d'erreur. J'ai ete
curieuK de comparer le r^liat de M. Eater a une formule
que M. Maihieu a conclue de Texperience de Borda, et puis de
la comparaisqn de toutes les experiences du pendule faites
jiifiqu'ici dans lets diyerses parties du monde, et que j'ai commu-
Qiquees k M. Kater. Cette &rmule donna 39^. 13842, pour
la l<wgueur du pendule au lieu ou M. Kater I'a trouvee
de 39Po. 1383

Yeiullez, Monsieur, agreer I'assur^nc^ de mia ioonsid^atioii

la plus distingqee.

Laplace.

La formule de la longueur dn pendule a secondes est 39^^
0819 + Op*. 212923 (sin^ lat.) : les pouces sont relatife a la
regie de la Societe Royale a la temperature 62° Fahrenheit.

376 CJOBRESPONDBNCE RELATING TO No. XVII.

9. — Frcm M. Frbsnel to Dr. Young.

Monsieur, p*^ i« 24 Mai, i8i«.

Je yous prie d*agr^r rhommage que je vous fais d'un
exemplaire de men m^moire sur la diffraction. Loraque je le
soumis a I'lnstitut, je ne connaissais pas vos experiences et la
consequence que yous en aYiez tiree, en sorte que je presentai
comme neuYes des explications que yous aYiez d^ja donn^
depuis longtemps. Je les ai retranchees dans le memoire im-
prime que j'ai Tbonneur de vous euYoyer, et je n'y ai laiss^
que celle des franges colorees des ombres, paroe que j'ai ajoute
quelque chose \ ce que yous aviez deja dit sur ce phenomene.

II m'a semble qu'il fallait supposer un changement d'une
demi-ondulation dans les rayons reflechis par les bords du
corps opaque pour que les formules s'accordassent avec les
obsenrations. Je n'ai pas pu jusquHci me rendre raison de ce
retard d^une demi-ondulation, mais la tache centrale des an-
neaux colores yus par reflexion presente un fait du meme
genre qui me parait tout aussi difficile a expliquer.

La theorie indique que la trajectoire des bandes interieures
sont des hyperboles, et cette consequence ne yous a point
echappe, comme M, Arago me I'a fait Yoir dans I'explication
d'une figure ou yous avez represente leur marche. Les franges
ext^rieures se propagent aussi suiYant des hyperboles,* comme
je I'ai reconnu, et la courbure de ces trajectoires, qui est nulle
pour les bandes interieures, devient sensible au contnure dans
les franges exterieures. C'est une remarque que j'ai eu le bon-
heur d'ajouter a la votre, et que j'ai Yerifiee par des obeerYations
plus exactes que celles qu'on avait pu £aire jusqu'a present
La demonstration exp^rimentale de ce fait surppenant, annonce
par la theorie des ondulations, a paru a M. Arago une des
preuYcs les plus frappantes de cette theorie, et une des plus
fortes objections centre le systeme de Newton.

Le moyen d'observation ou j'ai ete conduit, a de grands
aYantages sur ceux employ^ jusqu'a pr^nt, par sa commodiid,
sa precision et la fiicilit^ qu'il donne d'etudier les ph^nomenes

• See Letter 11, p. 381.

No. XVII. OPTICAL suBjEcrra, 377

dans des Giroonstances ou ils ^happent aux aiitres proc^6&
J'espere qu'il engagera les physiciens a s'ocouper davantage
de la difiraction, dont yous avez tire le premier des preuYes si
^Yidentes de la th^rie des ondulation&

En interceptant la lumibre d'un o6t^ du corps opaque, yous
aYez fait voir que les bandes int^rieures proYenaient de la ren-
contre des rayons infl^chis par ses deux bords. Vous aYex
encore d^montr^ Tinfluence des rayons lumineux les uns sur les
autres, en faisant passer la lumiere a traYcrs deux petits trous
tres Yoisins,et en formant de oette mani^re des bandes semblables
it celles qu'on obsenre dans Pint&ieur des ombres. II me sem-
ble qu'on ne pent faire aucune objection raisonnable aux con*
s^uences que yous aYez tir^ de cette belle experience.

N^anmoins, pour ^oigner toute id^ de Taction des bords
du corps, de I'^cran, ou des petits trous, dans la formation et
la disparition des franges interieures, j*ai cherch^ k en produire
de semblables au moyen du croisement des rayons r^flechis
par deux miroirs, et j'y suis panrenu apres quelques tatonne-
ments. J*ai remarqu^ que ces franges etaient toujours perpen-
diculaires a la ligne qui joignait les deux images du point
lumineux, et que leur direction etait independante de celle des
bords des miroirs. D*ailleur8 les rayons qui arnYaient a mon
oeil apres aYoir traYcrse la loupe, etaient partis de points iris
eloignes du bord commun des deux miroirs, et aYaient et^
reflechis r^gulierement. En mesurant la largeur de ces franges
nous aYons trouYe, M. Arago et moi, qu'elle s'accordait parfaite-
ment aYec celle deduite par la theorie de Tangle que fiusaient
entr eux les deux rayons Yisuels diriges sur les deux images du
point lumineux.

M. Arago a donne les details de cette experience dans le
tome 1^ des Annales de Physique et de Chimie, mois de Mars,
1816.

J'ai fait Yoir dans mon m^moire que, sur un meme point
d'une surface tr^ etroite ou d'une grande couYexite, les memes
rayons incidens peuYcnt etre refl^his dans des directions
differentes. Mais cela ne suffit pas pour expliquer les images
colorees reflechies par des cilindres metalliques d'un petit
diametre, parce qu'on pent en dire autant de tons les points de

378 CORRESPONDENOB RELATING TO No. XVII.

leur surface, en Borte que les diverses eouleuro resultant du
croiaement des ondulations ae superposeot et se confondent, a
moins que des asperiies du des rales n'interrompent la coDti-
Duite de la surface. En repetant demieremeDt TexperieDce de
Dntour, je Hie suis assure que les images colorees provenaient
de quelques raiai longitudiiialeSy oonuue le pensait M. Arago^
car en fiusant toumer le fil m^llique sur son axe, j*ai vu ces
images changer de plaoe. Je I'ai £Euft pofir ensnite an tour
aTee smn, de maniere a bit^ effacer les raies lon^tudinales, et
il n*a plus reflechi qu'une Imniere continne legerement irisee
dans le sens perpendkulaire a Faxe.* La grande eooTezite
de ces cilindres en isolant les raies fiEtvorise le developpement
des couleurs, et c'est la prohablement la principale caoae du
phenomene. Quand on cnut arcMr fait une deoou^erfte, on
n'appreod pas sans regret qu'ou a ete pr^enu, et je tous
avouerai frandiement, Monsieur, que c'est aussi le sentiment
que j'ai eprouT^, lorsque M. Arago m'a £ut voir qu'il n'y
a?ait qu'un petii aombre d'observations yeritablement neuves
dans le memoire que j'avais pr&ente a Tlnslitut. Mais si
quelque diose pou^ait me consoler de n'avoir pas I'avantage de
la priority, c'^tait de m'etre renoootr^ arec un savant qui a
enrichi la physique d'un si grand nombre de deoouvertes
importantes> et cela n'a pas peu coutribue en meme temps a
augmenter ma confiance dans la theorie que j'aTsis adoptee.
Je suis, arec la plus haute oonsideration, Monsieur,

Votre tresnhumble et tresK)beissant senriteur,

Freskel.

10. — From M. Abaoo to Dr. Young.

MoirSIEUR, P«n»» J« 13 JuUIet, 1816.

J'ai rhonneur de vous adresser quelques exemplaires
d'un memoire sur la dij&action de la lumiere, que j'ai fait
inserer dernierement dans le nouveau journal que nous redi-
geons, M. Gay-Lussac et moi, sous le titre 'cTArmales de Chimie

* ^ Letter 11, p. aS2.

No. XVIL OPTICAL SUBJBCTB. 379

et de Phfnque." L'autenr, M. Fresnel, ae ooD&oissait pas,
qaand il I'a compose, les exoellens ecrits que vous aves
publies, sur oette matiiere, dans les ^ Tranaacticmi Philosophi-
ques." Vous verrez que depuis que je lui en ai fait part il s'est
empresse de yous rendre justice et de recounaitre ranteriorite
de voa titres.

Le memoire de M. Fresnel me parait devoir etre coneid^re
comme la demonstratioii de Totre doctrine de$ inierffrenceg.
Je ne vols pas trop, ai eflet, comment les pardsans du systeme
de Remission pourront expliquer les trajectoires courbes des
bandes dif&act^ ; ou plutot, je devine deja, que pour ne jmls
abandonner la route quails out suivie jusqu^ present, ils reyo-
querent ce fait en doute, ou s'abstiendront d'en parler. Si le
▼olumineux ouyrage que M. Biot vient de publier sous le titre
de ' Tndte de Physique ezp^rimentale et mathematique' est
deja parvenu jusqu'en Angleterre, vous aurez eu Toocasion de
remarquer, par quels argumens pitoyables il pretend prouver,
oontre votre opnion, que deux faisoeaux lumineux, qui se
croisent, n*exercent jamais I'nn sur I'autre aucune influence
sensible. J'aurai, sous peu, rooeasion de m'occuper de cet
objet; en attendant j'ai ins^ daos nos 'Annales' deux
notes qui mettront le public au courant de la question, et qui
renferment un aper^ de vos ing^nieux travaux. L'nne
d'elles est relative a Texperience de la disparitton des bandes
int^eures que vous avez publiee dans les Transactions Flu-
loBophiques pour 1808, et a laquelle j'ai fait une modification
que me paratt importante par les consequences qui s'en de-
duisent. Cette modification consiste en ceci: que la disparition
de la totality des bandes difiractees, qui se forraent dans
I'lnt^rieur de Tombre d'un corps opaque, a ^alement lieu
lorsqu'on substitue un verre diaphane d'une certiune epaisseur
a r^cran opaque dont vous vous serviez. Ceci conduit k un
raoyen extremement precis, pour mesurer les plus petites
diffisrenees de refractk>n ; je le mettrai bient5t en pralique, et
j'ai toot lieu d'esperer qu'il reussira meme potnr les substances
gazeuses. Dans tous les cas, ces conrid^tions auront toujours
k mes yeux un grand prix, puisqu^elles ont 6te le pr^xte de
cette lettre et qu'elles m'auront fourne I'occaaon de vous

380 COBRESFOITDENCE BELATING TO No. XVIL

presenter les aasurances de la profomde estime que vos travaux
m'ont inspire depuis long temps.

Votre tres-humble et tres^obSssant serriteur,

F. Araoo.

Cette lettre Yous sera remise par M. Dupin, Tun de noe
ingenieurs les plus distingues. Mon excellent ami, M. de
Humbddt, qui a eu Tan dernier I'honneur de fidre Totre con-
naissance, s'est charge de yous le recommander.

♦1. — From Dr. Young to M. Arago.

Mt dear SfR London, 48, Welbeck-street, 12tli January, 1817.

I ha-^e long been intending to scold you for leaving
England without performing your promise of paying me another
visit with your friend Gay-Lussac at Worthing. I was the
more mortified at the circumstance, because J fear that you left
me under a mistaken apprehension that I had some engage-
ment for the day which made your company inconvenient to me ;
— this was very far from the truth; — and when I expressed
some regret that you had not written to give me notice of your
coming, it was more from feeling how easily it might have
happened that I might have been absent the whole of the
afternoon without seeing you, than from any partial engage-
ment which I had actually made. For the present, the obli-
gation is all on my side. I am sensible how great a compli-
ment you paid me in undertaking such a journey for such an
object; and I am conscious that I was unable to repay you
either by information or civilities of any kind. You were
already acquainted with everything that I meant to have told
you respecting my optical speculations ; and you did not give
me time to do the honours of the country by common hospitality.
I am, however, most happy to find that you are to return in the
spring, and then, I trust, that you will allow me to make up for
the deficiency.

No. XVII. OPTICAL suBjBcrrs. 381

I was reflecting, after you left me, on the very important
experiment which you made on the equality of the intensity of
colours formed in reflected and in transmitted light: you seemed
to regard it as forming a difficulty in my hypothesis ; but in
reality there is nothing in this fact at all unfavourable to that
theory, although it requires some modification of the general law
of interference, if we set out with considering the light as arriving
at any given point independently of the action of this law ; for
instance, in the present case of transmitted light, afUr two
internal reflections^ which woiild leave it less intense than you
actually found it But it is equally consistent with the theory
to consider the colour in question as being formed at the
instant of the second reflection ; and the analc^ with elastic
bodies fiilly justifies this mode of applying the law, so as to
consider the whole light once reflected, as interfering with an
equal portion of the transmitted light (Supra, p. 160.)

The same analogy is fully sufficient to explain the inversion
of the undulation, or the loss of half an interval, when a direct
partial reflection takes place from the surface of a rarer medium,
as, I believe, you are yourself aware. But Mr. Fresnel, in his
letter to me, mentions this fiict as equally inexplicable with the
inversion by extremely oblique reflection. I am sincerely de-
lighted with the success which has attended Mr. FresneFs
labours, as I beg you will tell him ; and I think some of his
proofe and illustrations very distinctly stated ; but I cannot fully
adopt your expression in the letter you wrote by Mr. Dupin,
that his memoir may be ** consid^re comme la demonstration
de la doctrine des interferences;" for neither I nor any of
those few who were acquainted with what I had written can
find a single new fact in it of the least importance : nothing
certainly half so important as your experiments on the colours
seen in transmitted light, or on the non-interference of light
polarised in opposite directions. Mr. Fresnel's words, in his
letter, are ^* les franges exterieures se propagent aussi suivant
des hyperboles comme je Tai reconnu, et la courbure de ces
trajectoires, qui est nulle pour les bandes interieures, devient
sensible au contraire dans les franges exterieures." Now you
are all well aware that tliis was known to Newton himself, and

382 CORRESPOKDENCE RELATINO TO No. XVII.

that he attempted to ehide the difficulty by saying that the
light was not the same ; and it was, therefore, unnecessary ibr
me to repeat it in the same form. And the precise hyperbolical
nature of the curves concerned is by no means a very strong
point in the chain of evidences, partly on account of the difficulty
of measuring the exact breadth of the fringes, and partly on
account of the loss of the half interval, not hitherto explained.
Mr. Fresnel has repeated some of Mr. Dutour^s experiments on
small cylinders, and has very truly observed that the spectra
move with the cylinders. This was the reason that I never
eonndered these expmments aa of any value, the circumstance
having been noticed by several authors, and, among the rest,
by Mr. Brou^m in 1796.

We have made but little progress in the measurement of the
pendulum, except that Major Katef's experiments are nearly
completed. Troughton is going on with his, but I am per-
suaded they can be of no use, from the nature of his suspension.
I have been calculating the effect of the flexure of a sprii^
in shortening the pendulum, and I find that it must be very
sensible in all imaginable cases, even when the elastic force of
the spring as an impelling power is wholly inconsiderable. I
hope in a few weeks to get a clockmaker to make a scapement
tor my pendulum, which shall not have any influence on its
rate ; or if otherwise, to make the experiments without a scape-
ment, as has been done in other instances ; but in this case it
would be necessary to fix the moveable weight at such points
as would afford coincidences at convenient intervals, and the
whole determination would be more laborious.

I have been reconsidering the theory of capillary attraction,
and have at last fully satisfied myself with respect to the iunda-
mental demonstration of the general law of superficial con-
traction, which I have deduced in a manner at once simple
and conclusive from the action of a cohesive force extending to
a considerable number of particles within a given insensible
distance. This solution has very unexpectedly led me to form
an estimate, something more than merely conjectural, though
not fully demonstrative, of the magnitude of the ultimate atoms
of bodies ; of water, for instance, about a million of which

No. XVII. opncAi scTBJEcrre. 383

wonld occupy a length equal to the drameter of one of the red
particles of blood. This, howeyer, yon may possibly regard as
a mere dream, and you are fully at liberty to do so.

I have also been reflecting on the possibility of giving an
imperfect explanation of the affection of light which constitutes
polarisation, without departing from the genuine doctrine of
undulations. It is a principle in this theory, that all undu-
lations are simply propagated through homogeneous mediums
in concentric spherical surfaces like tlie undulations of sound,
consisting simply in the direct and retrograde motions of the
particles in the direction of the radius, with their concomitant
condensation and rarefactions. And yet it is possible to ex-
plain in this theory a traDsv^*se yibration, propagated also in
the direction of the radius, and with equal velocity, the motions
of the particles being in a certain constant direction with
respect to that radius ; and this b a polarisation.* But its
inconceivable nrinuteness suggests a doubt as to the possibility
of its producing any sensible effects : in a physical sense, it is
almost an evanescent quantity, although not in a mathematical
one. Its foundation is this: suppose two particles to reflect
two portions of light, which interfere with each other, and form
a dark fringe, the one being situated at the distance of several
intervals from the other, in a direction transverse to that of the
fringe : it is obvious that their interference can never be so
completely effectual as not to leave some remains of the motions
combined with each other ; the direct motion of the one will
destroy the retrograde motion of the other : but the transverse
motions of each, with respect to the line bisecting their
directions, will conspire with each other and will produce a
single transverse vibratory motion. And who shall say that
this motion will be too minute to produce any efiect in any
ciroumstances ?

Pray give ray compliments to Mr. Gay-Lussac^ and tell him
that I was much disappointed in not having some farther con-
versation with him on elective attractions. Mrs. Y. begs to
unite with me in kind remembrances both to him and to your-

* This saggestion was a capital step in the undalatory theory of light. See Dr.
Whewell'H *HiRtorf of the Inductive tJciences/ rol. if. p. 417.

384 CJORRBSPONDENOE BELATINO TO No. XVII.

self. I am happy to hear that the work on Egypt is going on,
and that Mr. Jomard has married a pretty widow.

Belieye me, dear Sir,

Very sincerely yours,

Thomas Youno.

I know that I need not apolo^ze to you for writing in
English, as you read it with so much ease. I write it so much
faster than French, and of course better in a duplicate ratioy
although this is of less consequence.

12.— Dr. YouNO to M. Araqo.

Mt dear Sir, London, Welbeck-street, 22lnd April, 1817.

I HAVE been preparing for you a few memorandums on
the subject of the double internal reflection of light, which I
promised you at Paris, but I have not had time to put them
together in a very satisfactory form: having, however, an
opportunity of writing, I do not like to let it pass without
assuring you that I have not forgotten the many pleasant hours
that I spent at Paris, and the many kindnesses Uiat I received
there, for which I am indebted to no one so much as to your-
self. Pray tell me if I am not to expect the pleasure of seeing
you here shortly ; I understand that Mudge is ready to co-
operate in everything of every kind that you can desire of him.
With respect to the reflections in question, you were very
right in hesitating to admit the matter as self-evident, and I
was a littie precipitate in my way of stating it : for I find upon
calculation, that if we considered the simple velocities as the
measure of the intensity of light, the second reflection would
be no stronger in the case in question than in any other case,
and the .transmitted light ought to be much less strongly
coloured than the reflected, since the arrival of a new vibration
at the first sur&ce would not affect the velocity of the second
internal reflection, as added to or subtracted from the velocity
produced by its simple transmission. But the velocity alone is
not the measure of the intensity of light; and it must in

No. XVII. OPTICAL SUBJECTS. 385

general be considered as proportional to the square of the
velocity; consequently the degree in which the transmitted
light is strengthened or weakened by the interference with the
second reflection, must depend in some measure upon its own
intensity, and upon the density of the mediums concerned^ and
for a substance nearly of the refractive density of glass. The
intensity of the colours seen in the transmitted light will become
nearly or quite equal to that of the colours seen in the reflected
light, although the proportion will vary a little in difierent
cases. I do not know whether you will be quite satisfied with
this result, but it seems to me at least to assist us a little ; as
for the inversion, or the loss of half an interval between the
two reflections, there is no difliculty whatever, as you will see
when I send you the calculation which I have not had time
to copy out. Pray present my kind compliments to Madame
Arago, and believe me ever yours very sincerely,

Thobias Young.

13. — From Dr. Young to M. Arago.

My dear Sir, Worthing, 15th September, 1817.

I HAVE to thank you for the pleasure I have received
in looking over the numbers of your Annals, a work which
appears to me to be admirably conducted. I am in daily ex-
pectation of receiving the last cahier, which I believe is at my
house in London. I was rather surprised that you inserted
Mr. de Prony's paper on the metre, without adding the remark
"that the English standards being always employed at 62°,
the true length of the metre in English feet is not the number
which he has set down, but 39.3710 as I had made it,** or per-
haps .0008 more, according to some late experiments of which
Major Kater will tell you more. He will also show you my
letter to Mr. Laplace, on the subject which I mentioned to you,
if you think it worth reading, before he delivers it.

I have been amusing myself lately with revising some of my
investigations respecting light : but I do not know that I have
made out anything new that is very important : you will, how-

VOL. I. 2 c

386 CORRESPONDENCE RELATING TO No. XVII-

eyer, be interested in the result of a calculation which com-
pletely Bolyes your difficulty respecting the transmitted and
reflected rings. In the first place there is no doubt that the
intensity of light must be measured by the squares of the
velocities of the particles, and not by the simple momenta,
otherwise there would be an increase of the whole existing
quantity of light afker every partial reflection: and in the
second place you will find that the difierence in the squares of
the velocities of the compound transmitted undulations, at the
distance of half an interval, and a whole interval, is equal to the
difference of the squares in the case of reflection, except a slight
diminution exactly equal to that which would be produced by
viewing these last through the plate in question : and possibly
in the case of oblique incidences, even tliis difference would be
found to vanish.

I do not know whether it has occurred to you that the differ-
ence between the dimensions of the rings discoverable upon
silver as you first observed, from the light irregularly reflected,
and the ordinary rings, is perfectly intelligible from t£e circum-
stance of the difference of the interval of retardation in cases of
oblique incidence, the light not passing necessarily through the
plate in the same angle before and after its reflection. Have
you observed that steel reflects regularly a series of rings with
a black central spot, and gold ditto with a white one ?

I cannot yet satisfy myself respecting the true explanation
of Biot's experiments on oil of turpentine, and I shall be glad
to receive Mr. Fresners which you mentioned to me, as soon
as he is ready to make it .public. In short, the relation of
Biot's experiments is so mixed with his theory, that I am very
much at a loss to separate them.

Are we to entertain any hopes of seeing you this autumn ?
if not, you must come in the spring, and bring Madame Arago :
we will take care of her while you are engaged in Norfolk-
Pray give my compliments to Madame Biot when you see her.
I do not know what Sir C. Blagden has done respecting the
compass.

Ever yours,

Thomas Young.

No. XVII. OPTICAL SUBJECTS. 387

14.— i?V(wi M. Arago to Dr. Young. No date or place,

MON CHER ET ILLUSTRE CoNFRIIRE,

Je vous adresse, ci-joints, les remercimens de MM.
Haiiy, Prony et Poisson, au sujet de leur nominatioD comme
membres etrangers de la Societe Royale de Londres.* Pennet-
tez-moi, en men particidier, de vous prier de joindre aux bontes
sans nombre, que vous avez deja eues pour moi, celle d'etre
aupres de voire respectable Pr&ident et de vos confreres,
Tinterprete de ma profonde gratitude.

« ♦ * * «

J'ai un caique, de grandeur naturelle, que M. Jomard a fait
faire, a ma priere, de Tinscription hieroglyphique que vous
d&irez connaitre. J'attends une occasion favorable pour
vous le faire passer. Le rouleau est trop volumineux pour
que je puisse le confier a la poste.

Je vous remercie des details que vous avez bien voulu me
donner au sujet de la theorie des ondes ; vous savez que je suis
un de ses plus zel& proselytes : a ce titre je vois avec peine
qu'elle se complique un peu. II y a toujours la dispersion de la
lumiere et la polarisation qui attendent une explication plau-
sible. Cest a vous, dont la theorie des interferences portera
le nom dans la posterite, qu'il appartient de lever ces deux
grandes di£Scult&.

Permettez moi, avant de terminer cette lettre, de vous de-
mander des nouvelles de vos observations du pendule. M. de
Humboldt entre dans ce moment chez moi, et me charge de vous
faire ses complimens.

Adieu, Monsieur; presentez, je vous prie, mes respects a
Madame Young, et recevez les nouvelles assurances du bien
sincere attachement de votre tres humble serviteur,

P. Arago.

^ M«»ieurs Arago, Hatiy, Prony and Poisson were elected Foreign Members of the
Royal Society in 1818. This letter of acknowledgment was written therefore in that
year, —Not^ by the Editor,

2 C 2

388 CORRESPONDENCE RELATING TO No. XVII*

15. — From Dr. Young to M. Arago. For the * Armales ' if
you think proper.

My dear Sir, Worthing, 4th August, 1819.

You will imagine how greatly I have been interested
with the two principal papers in tlie * Annales de Chimie ' for
May. Perhaps, indeed, you will suspect that I am not a little
provoked to think that so immediate a consequeuce of the Huy-
ghenian system, as that which Mr. Fresnel has very ingeniously
deduced, should have escaped myself, when I was endeavouring
to apply it to the phenomena in question : but in fact, I am
still at a loss to understand the possibility of the thing ; for if
light has at all times so great a tendency to diverge into the
path of the neighbouring rays, and to interfere with them as
Huyghens supposes, I do not see how it escapes being totally
extinguished in a very short space, even in the most transparent
medium^ as I have observed in my first paper on the subject
(supra^ p. 149) : I cannot, however, deny the utility of Mr.
Fresnel's calculations. I hcive not yet seen his analysis ; but
the result may easily be exhibited in a very simple form, by
merely considering the eiSect of a pencil passing through a
small circular orifice, each point of which contributes equally to
furnish light to an object situated in the axis of the pencil : for,
supposing the area of the orifice to be x, the difierence of tlie
patiis of the rays passing through its centre and its circum-
ference will obviously vary as x, both these quantities being as
the square of the diameter ; we have also dx for the fluxion
of the area depending on its annular increment, and belong-
ing to the difference in the paths expressed by 4 ar, rf being the
distance of the object and a constant quantity; so that the
fluxion of the intensity of the light will be cos f ± x^dx^*

supposing the law of the undulations to be that of the cycloidal
pendulum, which is the simplest possible; consequently the
intensity for an orifice, of which the area is any finite quan-

♦ Note the correction of this in the next letter (16), p. 391, aa weU as Dr.
Young's reply, No. 17. --Note by Vie Editor,

No. XVII. OPTICAL SUBJECTS. 389

tity X will be c sin. ^ x which will vanish when ~ x becomes

equal to the breadth of a complete undulation ; a result equiva-
lent to the apparent inversion of the undulation by oblique
reflection, which I observed, but confessed myself " unable to
explain."

Believe me, my dear Sir,

Ever most truly yours,

Thomas Young.

Pray have the goodness to take charge of the enclosed for
Mrs. Kater, whose direction I do not know. I shall send Mr.
Jomard some hieroglyphics by Mr. Dupin : I have had some
late accounts from Egypt, confirming my interpretations.

16. — From M. Fresnel to Dr. Young.

Monsieur, Pa"«» le 19 Septembre, 1819.

J'ai I'honneur de vous adresser deux exemplaires de
mon memoire sur la difiraction, tel qu'il vient d'etre imprime
dans les ^Annalea de Physique et de Chimie.* II ne pouvait pas
y etre insere en totalite a cause de son etendue ; mais la partie
supprimee ne contenant guere que des objections contre le sys-
teme Newtonien, vous aurait presente peu d'interet. L'extrait
publie contient la partie essentielle de mon memdre : la theorie
de la difiraction et sa verification experimentale. Cette theorie,
comme vous Tavez tres bien dit, n'est autre chose que le principe
de Huyghens applique aux phenomenes en question. Sans
doute ce grand geometre en aurait aisement deduit les loix de
la diffraction, s'il avait songe a I'infiuence mutuelle que des
ondes produites par un mouvement oscillatoire doivent exercer
les unes sur les autres. Mais il vous etait reserve d'enrichir la
science du principe fecond des interferences, et de montrer par
une foule d'applications ingenieuses de quelle utilite il pouvait
etre en optique.

Le principe d'Huyghens me parait, aussi bien que celui des
interferences, une consequence rigoureuse de la coexistence des
petits mouvemens dans les vibrations des fluides. Une ondc

390 CORRESPONDENCE RELATING TO No. XVII.

derivee peut etre consideree comme Tassemblage d'une infioite
d'ebranlements simultanes ; on peut done dire, d'apres le principe
de la coexistence des petits mouvemens, que les vibrations
excitees par cette onde dans un point quelconque du fluide situe
au-dela, sont la somme de toutes les agitations qu*y aurait fait
naitre chacun de ces centres d'ebranlement en agissant isolement.
A la verite, d'apres la nature des ondes derivees, ces centres
d'ebranlementne peuvent pas produire de mouvement retrograde,
et les ondulations elementalres qui en emanent ne sauraient
avoir, dans des directions obliques a Timpulsion primitive, la
meme intensite que suivant la normale a Tonde generatrice.
Mais il est evident que le decroissement d'intensite doit suivre
une loi de continuite, et peut etre considere comme insensible
dans un intervalle angulaire tres petit : or, cette consideration
suffit pour la solution du probleme ; car des que I'inclinaison
des rayons est un peu prononcee, il est aise de voir qu'ils se
detruisent mutuellement.

Mais comment ces destructions mutuelles n'afiaiblissent-elles
pas considerablement la lumiere totale ? C'est ime consequence
gcnerale des vibrations des fluides elastiques que la somme des
forces vives reste toujours constante, de quelque maniere que
Ton subdivise et recompose le mouvement. On peut aisemout
verifier ce principe dans le cas tres simple des bandes obscures et
brillantes produites par I'interference de deux systemes d'ondes
lumineuses reflechies sur deux miroirs legerement inclines en-
tr'eux, qui sont d'une intensite sensiblement uniforme dans le
petit espace angulaire ou se forment les franges. On trouve
en integrant, que la somme des forces-vives d'une demi-frange,
dcpuis le point le plus sombre de la bande obscure, jusqa'au
point le plus eclatant de la bande brillante, est precisement la
nieme que dans les deux systemes d'ondes supposes independans
Tun de Tautre, malgre la destruction de mouvement qui r&ulte
de leur influence mutuelle dans les points de discordance ; parce-
qu'elle est exactement compensee par Taugraentation de mouve-
ment dans les points ou leurs vibrations s'accordent. En effet
81 Ton represente par a et a* les intensites des vitesses d'oscilla-
tion que les deux series d'ondes imprimeraicnt aux molecules
etherees, en agissant isolement, on a )K)ur Texpression de Tin-

No. XVII. OPTICAL SUBJECTS. 391

tensite d'oscillation du systeme d'ondes r&ultant du con-

cours des deux autres V <a* + a** + 2aa' cos (27r. -j^ J |, 25r

representant la circonference, dont le rayon est 1, X la longueur

d*ondulation, et - la difference des chemios parcourus dans le

point de la frange que Ton considere ; j'indique ici par x la
distance de ce point k celui d'accord parfait, c'est-a-dire au point

le plus eclaire de labande brillante : i estle rapport constant de

cette distance a Tintervalle correspondant entre les deux syst^mes
d'ondes. La force vive etant la masse multipliee par le carre de

la Vitesse sera proportionnelle a a* + a'* + 2aa' cos. f 2^. —\
et sa differentielle k jo* -h «" + 2aa' cos. \2w. ~ j| dx, dont
Tintegrale est {c? + a'*) a? — 5^ • 2a« sin. 1 2ir, — \ qui devient

(a*+ a!^) X, lorsque ~ egal a -j-, c*est-a-dire, lorsqu'on int^gre
depuis le point d*accord parfait jusqu'a celui ou les deux
systemes d'ondes diflFerent d'une demi-ondulation : or, d'apres
la meme notation {cf + a'*), x est precisement la somme des
forces vives que les deux systemes d'ondes apport«nt dans cet
intervalle d'une demi-frange, abstraction faite de leur inter-
ference ; rinfluence mutuelle qu'ils exercent Tun sur I'autre ne
diminue done pas la somme des forces vives.

Dans le calcul qui termine votre lettre a M. Arago, oil vous
appliquez le principe d'Huyghens au cas d'une ouverture cir-
culaire, il me semble, si je comprends bien votre notation, que
vous vous etes mepris sur la formule d'interfi^rence ; le fluxion
de I'intensite de la lumiere dans le point qui repond au centre

de Touverture n'est pas cos. f^ or Jir, mais cos. f i ^ x\dx,
dont I'integrale est 2c sin. (-5--^ ^n- Et en effet, cette ex-
pression, qui devient nulle quand -j x est egal a /, comme celle
que vous obtenez c sin. f ^ x\ s'accordc encore avec Texpe-

392 CX)RRESrONDEKCE RELATING TO No. XVII.

rience en ce qu'elle atteint son maximum lorsque -j x est la
moitie de / ; tandis que c sin. f -^ x\ devient alors une seconde

fois egal a zero, et ne pent pas en consequence representer
rintensit^ de la lumiere dans la projection du centre d'une
ouverture circulaire.

II est aise, sans le secours de Tanalyse et par une considera-
tion geometrique bien simple, de determiner les circonstances
de maximum ou de minimum de lumiere pour le point dont il
8*a^t. II suflBt de divisor par la pensee la surface de I'ouver-
ture circulaire en anneaux concentriques dont les circonferences
repondent a des differences d'une demi-ondulation dans les
chemins parcourus ; ces anneaux etant egaux ^ surface envoient^"'
chacun le memo nombre de rayons, et comme ces rayons sont
sensiblement egaux en intensite, d'apres mon hypothese, il est
clair qu'ils se detruisent tons mutuellement quand les anneaux
sont en nombre pair, et qu'ils doivent produire au contraire
par leur reunion la lumiere la plus vive possible, lorsque les
anneaux sont en nombre impair.

D*apres cette maniere d'en?isager les phenomenes de la
diffraction, il n'est plus necessaire de supposer ime inversion de
Tondulation dans les rayons reflechis sur le bord de I'ecran,
qui ne sont qu'une tres petite partie de ceux qui concourent a
la production des franges.* Mais je n'en crois pas moins a
cette inversion, du moins dans la reflexion produite par les corps
parfaitement transparens, tels que Teau, le verre, &c. Cette
opinion est fondee sur une hypothese a laquelle j'accordais la
preference depuis longtemps, et que je viens de verifier par des
experiences qui me paraissent decisives : je ne crois pas que la
reflexion soit occasionnee par une plus grande densite de Tether
dans le milieu refiingent, mais par des reflexions partielles sur
les particules propres du milieu, que je suppose dans une petite
epaisseur de la surface participcr a la fois aux vibrations des
rayons transmis et des rayons reflechis. II est aise de concevoir

• This was the correction of an important inaccaracy in Dr. Young's explanation
of the external fringe of shadows in diffraction, to which M. FrcsncPs fij-st explanation
(letter 9, p. 37«) was equally liable. See Dr. WhewelFs * History of the Inductive
Ssiienccs,' vol. il. p. 406. See also the next letter. No. 17. — Note by t/te Editor.

No. XVII, OPTICAL SUBJECTS^ 393

comment la reflexion devient insensible a une certaine dbtance
de la surface, lorsque les intervalles qui separent les particules
du milieu sont tres petits par rapport a la longueur d'une on-
dulation, puisqu'alors toutes les reflexions elementaires se
detruisent mutuellement dans I'interieur du corps.

Je vous prie d'avoir la bonte dofirir de ma part k la Societe
Royale de Londres un des exemplaires ci-joints de mon memoire
sur la diffraction.

J'ai riionneur d'etre ayec la plus haute consideration,

Monsieur,
Votre tres-humble et tres-obeissant serviteur,

A. Fresnel.

17. — From Dr. Young to M. Fresnel.

Worthing, 16th October, 1819.

Je vous remercie infiniment. Monsieur, pour le present que
vous m'avez fait de votre beau memoire, qui merite assurement
un rang distingue parmi les ecrits qui ont le plus contribue aux
progres de Toptique. Je n'ai pas la moindre idee d*insister
sur Toperation des rayons reflechis des bords d'un corps opaque ;
je savais meme tres bien, que quand on se sert de deux fentes
paralleles, il faut se rapporter au milieu de chacime pour
Tinterference, comme vous pouvez voir dans la figure 442 de
mes Lectures ; mais je n'avais jamais eu Theureuse idee
d'analyser les resultats de la combinaison des ondulations
particulieres, qui vous a si bien reussi, et ce qui m'en a
ompeche, c'est la di£Sculte que je sentais d'apprecier assez juste-
ment I'effet de Tobliquite, que vous n'avez pas trouve necessaire
de comprendre dans votre calcul. J*avoue que ma petite lettre
a M. Arago manque d'exactitude, et j'espere qu'l, ne Taura
pas publiee ; j'ai regarde la chose trop a la hate ; et comme
le seul resultat que je me sois donne la peine d'examiner etait
d'accord avec les votres, et avec Texperience, je m'en suis trop
aiscment satisfait Mais vous verrez par la petite table des
marges que je vous adresse avec cette lettre, que la vraie
maniere d'envisagcr la combinaison des ondulations m'etait
assez familicre ; en effet, nous n'avons qu'a divisor Ic trou circu-

394 CORRESPONDENCE RELATING TO No. XVII.

laire en petits anneaux ooncentriques d'une egale aire, qui
repondront a des differences egales dans les routes, et il s'ensuit
da principe connu de la combinaison des ondulations, dent je
me suis servi dans cette constroction, pour les marees^ que si
Ton represente les petites ondulations egales par les cotes d'un
polygone, inscrit dians un cercle, et formaut les angles exte-
rieurs egaux aux distances des ondulations sur le cercle qui les
mesure, les cordes de ces polygenes ou des arcs qui les repre-
sentent dans leur dernier etat, seront proportionnelles aux gran*
deurs des ondulations composees, et voila les doubles mma des
moities des angles, auxquels vous etes parvenu.

Huyghens aurait pu sans doute, comme vous remarquez,
soup(^onner cequi serait Teffet de Tinterference des ondulations,
mais il ne parait pas qu'il ait eu aucune idee de ce qui pouvait
constituer la difference des couleurs, quoiqu'il aurait pu adopter
la suggestion de Newton ou de Malebranche que j'ai citee, s*il
avait poursuivi plus loin ses recbercbes. J'avais remarque que
rinterference de deux systemes quelconques d'ondulations
n'alterait pas la somme des forces vives, et je vois que M .
Poisson a demontre quelques uns de mes resultats appartenant
a I'intensite de la lumiere d'une maniere plus directe, dans un
mmoire qu'il a eu la bonte de m'adresser. Si vous le voyez je
vous prie de Ten remercier de ma part, et de lui dire, qu' il
trouvera dans les Memoires de TAcademie de Berlin, pour
1775, des experiences de Lambert sur les flutes, comparees
avec la tbeorie de Bemouilli, qu'il ne parait pas connaitre.

La polarisation nous presente encore beaucoup de difficultes.
Je m'etais flatte, d apres ce que M. Biot venait d'annoncer dans
son memoire, que vous en aviez leve la plupart, et que vous
aviez explique la rotation apparente des rayons dans quelques
fluides que M. Biot a decouverte : si cela est vrai, je vous
serais extremement oblige si vous pouviez me donner quelque
idee de votre tbeorie. Vous trouverez quelques mots sur Toptique
dansl'extrait astronomique que j'ai I'honneur de vous adresser.

Je suis, Monsieur, avec les sentimens les plus distingues,
Votre tres-humble et tres-obeissant serviteur,

Thomas Young.

No. XVII. OPTICAL SUBJECTS. 395

18. — From M. Fresnel to Dr. Young.

Pazifl, 18 F^Trier, 1823.
• • • • •

J*AVAis remarque depuis six and que deux reflexions
totales, dans I'lnterieur du verre, ou d'un autre corps trans-
parent, peuventf sous une incidence convenable, imprimer a la
Aimiere polarisee dans I'azimut de 45^, la modification que
j'appelle maintenant polarisation circulaire^ et que je defi-
nissais par la reunion de deux systemes d'ondes egaux en
intensite, poIaris^ a angle droit, eft distans I'un de Tautre d'un
quart d'ondulation. J'avais observe aussi que cette difierence
de marche entre les deux systemes d'ondes, dont I'un est po-
larise suirant le plan de reflexion et I'autre suivant un plan
perpendiculaire, variait avec I'inclination des rayons et deve-
nait nuUe aux deux limites de la reflexion totale ; mais il me
semblait bien difficile de decouvrir suivant quelle loi, et je ne
Favais pas m£me essaye ; ce n'est que depuis tres pen de terns
que je me suis occupe de ce probleme ; j'en ai tronve la solution
beaucoup plus vite que je ne m'y attendais. J ai lu a ce sujet
un nouveau memoire a I'lnstitut, dans lequel je fais voir d'abord
comment on pent curriver aux formules d'intensite des rayons
reflechis sous des incidences obliques, en partant de la loi de
Descartes, et s'appuyant seulement sur le principe de la con-
servation des forces vivos et sur une bypothese mecanique tres
simple et tres admissible, dont je n'ai pas donne la demonstra-
tion, a la verite, mais qui me paratt facile a etablir. C'est a
Taide de ces formules que je suis parvenu a decouvrir la loi
dont je viens de parler : elles avaient ete publiees en 1 821,
dans les 'Annates de Chimie et de Physique,' a la fin de la der-
niere note ajoutee au rapport de M. Arago sur la coloration
des lames crystallisees : j'avais fait voir comment elles servent,
non seulement a calculer les proportions de lumiere directc
ou polarisee, reflechie sous toutes les incidences par les corps
transparens, mais encore a determiner d'avance la proportion
de lumiere polarisee, si ce sont des rayons directs qu'on
reqoit sur le corps transparent, ou la deviation du plan de po-
larisation, si Ton fait reiSechir de la lumiere polarisee dans un

396 CORRESPOin)ENCE RELATING TO No. XVII.

azimut oblique. Ainsi ces seules formules servcnt a calculer
les lois de tons les phenomenes que presentent la reflexion par-
tielle et totale a la premiere et a la seconde surface des corps
transparens.

On doit publier incessamment dans le * Bulletin des Sciences
de la Societe Philomatique,' un extrait de mon nouveau memoire.

Tons ces memoires, que demierement j*ai pr^ntes coup sur
coup a I'Academie des Sciences, ne m'en ont pas cependant
ouvert la porte. C'est M. Dulong qui a ete nomme pour
remplir la place vacante dans la section de physique par
Telection de M. Fourier a celle de secretaire perpetuel. Sans
doute M. Dulong est un physicien de beaucoup de merite ;
mais il est aussi habile chimiste, et pouvait entrer dans la
section de chimie, si' I'Academie avait eu grande envie de me
recevoir. Je n'ai pas meme ete presente sur le meme raug que
M. Dulong par les membres de la section de physique : ils Tout
mis le premier et moi le second. Vous Yoyez, Monsieur, que
la theorie des ondulations ne m'a point porte bonheur ; mais
cela ne m'en degoute pas ; et je me console de ce malheur en
m'occupant d'optique avec une nouvelle ardeur.

Agreez, Monsieur, I'expression de ma haute consideration, et
presentez, je vous prie, mes bommages a Madame Young.

A. Fresnel.

19. — From M. Fresnel to Dr. Young.
Monsieur, 27 Mars, I823.

J*Ai rhonneur de vous adresser sept cxemplaires d*un
extrait du memoire que je vous avais annonce dans ma der-
niere lettre, et qui a pour objet la recherche theorique et expe-
rimentale des lois suivant lesquelles la lumiere polarisee est
modifiee par sa reflexion totale dans Finterieur des corps trans-
parens; je vous prie de vouloir bien accepter un de ces
exemplaires, d'en ofllrir un de ma part k la Societe Royale, et
de remettre, ou faire parvenir, les autres a MM. WoUaston,
Dal ton, Herschel, Brewster et Leslie.

M. Brewster sera peut-etre surpris qu'en publiant cet extrait,
je n'y aie pas fait mention de ses recherches sur les eflTets de la

No. XVII. OPTICAL SUBJECTS. 397

reflexion totale, qui sont anterieures aux miennes : la raison
de men silence a cet egard tient d'abord an pen d'espacc dans
lequel j'etais forc^ de resserrer cet extrait ; et ensuite a la per-
suasion oil je suis que M. Brewster s'est complettement mepris
dans les lois qu il a donnees des phenomenes de coloration que
pr^nte la lumiere polarisee apres avoir eprouve la reflexion
totale. D'abord il n'a pas observe que ces couleurs ne sont
sensibles que dans les incidences voisines de la limite de la
reflexion partielle ; ce qui fait soup^onner que le verre dont il
se servait n'etait pas bien recuit ; en second lieu, il a avance
que ces couleurs, qu'il suppose pareilles a celles des lames
cristallisees, descendaient dans Tordre des anneaux par deux,
troisy quatre reflexions, &c., comme la teinte d'une lame
cristallisee dont on double, triple, quadruple Tepaisseur ; tan-
dis que des I'incidence de 48*^, par exemple, deux nouvelles
reflexions detruisent presque entierement I'efiet produit par les
deux premieres et ramenent sensiblement la lumiere a son
etat primitif de polarisation complette.

J'ignore au reste si le Dr. Brewster s'est occupe depuis des
memos phenomenes : je ne connais que le memdre qu'il a public
sur ce sujet dans les ^Transactions Philosophiques ' de 1816
ou 1817; memoire que M. Arago me montra lorsque je lui
communiquai, en 1817, mes premieres observations sur la de-
polarisation produite par la reflexion totale.

Agreez, Monsieur, Tassurance de mon devouement, et de la
haute consideration avec laquelle j'ai I'honneur d'etre
Voire tres-humble et obmssant serviteur,

A. Fresnel.

20. — From M. Fresnel to Dr. Young.

Monsieur, p»"*' ^« ^^ Septombre. 1823.

En vous ecrivant apr^ un silence aussi long j'aurais
desire pouvoir vous communiquer quelques nouvelles observa-
tions d'optique: malheureusement, depuis assez long temps
j'ai et^ constamment occupe d'affaires de service et de details
relati& k I'^clairage des phares. J'ai passe presque tout le
mois de Juillet dans la tour de Cordouan, a Tembouchure de la

398 CORRESPONDKNCE RELATING TO No. XVII.

Gironde, pour y installer un appareil lenticulaire ou dioptrique,
a feux toumaDS. N'ayant gueres ayec moi que de mauvais
ouvriers, j*ai ete oblige d'entrer dans les plus minutieux details
de cette installation, et de £sdre souvent moi-meme' rourrier.
La vivacite des eclats que presente le nouvel appareil a surpris
les marins. Quelques Anglais que la saison des bains avait
amenes k Royan, ont dit qu'ils n'avaient pas vu de phare aussi
brillant sur les cotes d' Angleterre. Je desirerais savoir ce qu'en
pensent vos marins, qui sont les plus experimentes de I'Europe ;
et s'ils trouvent que la duree de chaque apparition est suffisante
pour relever le phare a la mer, comme Testiment plusieurs
marins franqais que j'ai oonsultes sur ce sujet Get appareil
n'etant etabli dans le phare de Cordouan que depuis le 25
Juillet dernier, ce ne sera sans doute que dans un ou deux mois
d'ici que vous pourrez recueillir quelques observations de yos
marins sur le nouveau feu de Cordouan. Si vous avez la bonte
de me les communiquer, vous me rendrez un grand service.
Le phare de Cordouan devant servir k guider les batimens qui
entrent dans la Gironde, comme ceux qui passent au large,
j'ai tache de procurer aux premiers les avantages d'une lumiere
fixe, qui a quatre lieues marines de portee, et qui empeche de
perdre le phare de vue pendant les eclipses du feu toumant,
lorsqu'on approche des ecueils dont I'embouchure de la Gironde
est semee. Ce petit feu fixe est produit sans addition de lampes,
sans augmentation dans la depense d'huile, et seulement en
recueillant les rayons qui passent par dessous Tappareil toumant
et les reflechissant vers Thorison par de petites glaces etam^es
disposees comme les feuilles d'une jalousie. II n*est aucun
phare, je crois, dans lequel on tire autant parti de la quantite
d'huile employee. La consommation actuelle est d'une livre
et demie d'huile par heure, au plus ; tandis que celle de I'ancien
feu ^tait de trois livres ; en sorte qu'il resulte a la fois de ce
changement d'appareil, une economic annuelle de pres de six
mille francs et une grande augmentation de lumiere.

tTai I'honneur d'etre, avec la plus haute consideration,

Monsieur,
Votre tres-humble et tres-obfissant serviteur,

Fresnbl.

No. XVII. OPTICAL SUBJECTS. 399

21.'^From M. Fresnbl to Dr. Youno.

Monsieur, P«rw» !• is Octobre, 1824.

Je regrette beaucoup de ne pouvoir repondre en ce
moment a la demande obligeante que vous me fsdtes :* je suis
occupe du matin au soir par des examens que je fais k TEcoIe
Poljrtechnique, et je ne serai debarrasse que dans quinze jours
de cette penible occupation, qui m'a presque rendu malade.
II y a long-temps que je n'ai rien fait de neuf en optique.
Je crois vous avoir envoye, Monsieur, des extraits de mes
deux demiers memoires, sur la double refraction singuliere
que la lumiere subit en traversant le cristal de roche parallele-
ment a son axe, et sur la loi des modifications que la reflexion
totale, dans les corps diaphanes, imprime ala lumiere polarisee.
Mais je ne yous ai pas encore communique Textrait de mon
memoire sur la double refraction, publie dans le Bulletin des
Sciences de la Society Philomatique, livraison des mois
d'Arril et de Mai 1822^ parce que je n'en ai pas fait tirer
d'exemplaires. Si vous n'avez pas le Bulletin de la Societe
Philomatique, et que vous desiriez lire ce court extrait d'un
long memoire, j'en ferai faire une copie, que j'aurai Thon-
neur de voua envoyer par la voie que vous voudrez bien
m'indiquer.

J'ai maintenant, et depuis quelques annees, des idees theo-
riques assez arretees sur les principaux phenomenes de I'optique,
et je pourrais faire un article bien nourri en presentant ces vues
dans un cadre resserre ; mais ce n'est que dans quinze jours que
je pourrai» commencer k m'en occuper, et vous ne recevriez
mon article que dans un mois. Quant a Thistoire de la science,
personne n'est moins capable que moi de foumir des renseigne-
mens, n'ayajit pas I'avantage de pouvoir entendre les ouvrages
et les joumaux scientifiques ecrits en Anglais, et n'ayant meme
que le terns de lire tout ce qui se publie en France sur Toptique.

♦ Thii was in answer to an application, made through Dr. Young, by the pro-
prietors of the Encyclopaedia Dritanuica, for an article on Lights for their new
Sapplement. — Note by the Editor,

•too CORRESPONDENCE RELATING TO No. XVII.

Je vous prie de m'excuser men brouillon ; je suis accable par la
fatigue et le besoin de sommeil.

J ai rboimeur d'etre, avec la plus haute consideration,

Monsieur,
Votre tres-humble et tres-obeissant serviteur,

A. Fresnel.

22. — From M. Fresnel to Dr. Young.

Paris, le 26 Novembre, 1S24.

Monsieur, ^^ ^* F<m69 st, victor. No. 19.

Si j'ai tarde quelques jours a repondre a votre lettre du
17, c'est qu'une indisposition assez grave m'interdisait la plus
legfere occupation. Je n'ai encore en ce moment que le degre
de force qui suflSt pour ecrire une lettre. Cette indisposition
provient principalement de la fatigue de mes examens, et peut-
etre aussi du petit travail auquel je me suis livre en redigeant
un article pour la Revue Europeenne, comme je m'y etais
engage. Cette leqon severe m'avertit assez que je suis trop faible
pour multiplier mes engagemens, et que ma sant^ exige abso-
lument un repos de quelques mois. C'est avec regret que je me
vois dans I'impossibilite d'ecrire Texpose de mes idees theoriques
que vous me demandez.

£n y reflechissant bien, cependant, dois-je regretter de ne
pouvoir travailler pour un ouvrage anglais ? Avons-nous bien
de nous louer en France des jugemens qu'on porte en Angleterre
de nos travaux et de nos decouvertes ? Le Dr. Brewster pre-
tend que c'est d'apres ses idees qu'on a perfectionne Teclairage
du phare de Cordouan, quoique Tinvention et I'ex^ution des
lentilles k echelons soient toutes franQaises, du commencement
jusqu'a la fin. II reclame aussi la decouverte des modifica<-
tions imprimees par la reflexion totale k la lumiere polarisee ;
modifications dont il n'avait pas une idee bien juste, si j'en juge
par ce qu'il a public sur ce sujet. D'apres ce que m'a dit M.
Arago, il parait qu'on a fait tres pen d'attention en Angleterre
k la loi generale de la double refraction, ainsi qu'aux formules
que j ai donnees pour calculer les intensity de la lumiere
reflechie obliquement sur les corps transparens, et Ics deviations

No. XVII. OPTICAL SUBJECTS. 401

du plan de polarisation. Ces formules m'ont fait decouvrir la
loi assez compliquee des modifications singulieres que la reflexion
totale en dedans des milieux diaphanes imprime a la lumiere
polarisee ; mais il ne parait pas qu'on ait fait plus de cas chez
vous de cette decouverte que de celle de la double refraction
speciale des rayons qui traversent le cristal de roche parallelement
a son axe. Si je parvenais a demontrer a M. Herschel, a M.
WoUaston, et aux autres physiciens anglais, encore attaches au
systeme de Newton, que la theorie des ondes merite la prefer-
ence, ils ne manqueraient pas de dire que c'est uniquement a vos
travaux qu*on doit le rcnversement du systeme de remission et
les progres de la theorie des ondes. Si, desabusant vos savans
sur la polarisation mobile, je leur fesais adopter rexplication que
j'ai donn^ de la coloration des lames cristallisees et ces methodes
generales au moyen desquelles on pent calculer les teintes dans
tons les cristaux quand on connalt la double refraction de
chaque espece de rayon, ils diraient encore que Texplication de
ces phenomenes vous appartient : ils vous attribueraient egale-
ment celle des phenomenes compliques de la difiraction.

II me semble cependant (je ne sais si mon amour propre
m^ayeugle) que ce que vous m'aviez laisse k faire sur ces diverses
parties de I'optique etait aussi difficile que ce que vous aviez
fait. Vous aviez cueilli les fleurs, pourrais-je dire avec la
modestie anglaise, et j'ai creuse peniblcment pour decouvrir
les racines.

Je suis loin de pretendre a ce qui vous appartient, Monsieur,
comme vous I'avez vu dans le petit traite sur la lumiere insere
dans le supplement a la traduction fran^se de la Chimie de
Thomson, comme vous le verrez encore dans I'article que je
viens de rediger pour la * Revue Europeenne.' J'ai avou^ d'assez
bonne grace devant Ic public, en plusieurs occasions, Tanteriorite
de vos decouvertes, de vos observations, et meme de vos hypo-
theses. Cependant, entre nous, je ne suis pas persuade de la
justesse de ce mot spirituel par lequel vous vous compariez & un
arbre, et moi a une pomme que cet arbre aurait produite : j'ai la
conviction interieure que la pomme aurait pousse sans I'arbre,
car les premieres explications que je me suis donnees des phe-
nomenes de la difiraction et des anneaux color^, des lois de la

VOL. I. 2d

402 CORRESPONDENCE RELATING TO No. XVII.

reflexion et de la refraction, je les ai tirees de mon propre fonds,
sans avoir lu votre ouvrage ni celui de Huyghens. J'di remar-
que auBsi de moi-meme que la difierence de marche des rayons
ordinaires et extraordinaires au sortir d'une lame cristallisee
etdt egale a celle des rayons refl^cfais a la premiere et a la
seconde surface de la lame d'air qui donne la meme teinte dans
les anneaux colores. C*est lorsque je communiquai cette obser^
vation a M. Arago qu'il me parla pour la premiere fois de la
note que vous aviez publiee deux anB auparavant sur le meme
sujet, et a laquelle jusqu'alors il n'avoit pas compris grande chose.
Au reste, ceci ne me donne pas le droit de partager avec vous,
Monsieur, le merite de ces decouvertes, qui vous appartient
exclusivement par la priorfte : aussi, ai-je juge inutile d'informer
le public de tout ce que j'avais trouve de mon cote, mais apres
vous ; et si je vous en parle, c'est uniquement pour justifier ma
proposition paradoxale, que la pamme serait venue sans Farbre.
II y a longtems, Monsieur, que je desirais vous parler sur ces
sujets a ccBur ouvert, et vous montrer naivement toute Tetendue
de mes pretentions.

Admettons que mon amour propre soit trop exigeant, et qu'on
m ait assez rendu justice dans votre pays (car je suis peut-etre
effectivement un des Fran^ais qui out le moins a se plaindre de
vos compatriotes), je n'en serais pas moins etonne, je dirais
presque revolte, de ce qu'on me rapporte si souvent sur la
partialite choquante avec laquelle vos joumaux scientifiques ele-
vent tons les jours au-dessus des decouvertes fran^uses les plus
remarquables, ce qu'on a fait en Angleterre de plus insignifiant.
Certes, je suis loin de disconvenir que vous n*ayez sur nous,
surtout en politique, des superiorites incontestables ; mais vous
avouerez au moins que nous I'emportons de beaucoup en impar^
tialite et en amour de la justice.

Cette lettre vous paraitra peut-^tre, Monsieur, la boutade
d'un malade tourmente par la bile, et dont I'amour propre est
mecontent du peu d'attention qu'on a fait a ses travaux dans
votre pays. Je suis loin de nier le prix que j'attacherais aux
eioges des savans anglais, et de pretcudre qu*ils ne m'auraient
pas flatie agreablemeut. Mais, depuis longtems cette sensibilite,
ou cette vanite qu*on appelle amour de la gloire, s^est beaucoup

No. XVIT. OPTICAL SUBJECTS. 403

emoussee en moi : je trayaille bien moins pour capfer les suf-
frages du public que pour obtenir une approbation interieure
qui a toujours ete la plus douce recompense de mes efforts. Sans
doute }ai eu souvent besoin de Taiguillon de la vanite pour
m'exciter a poursuivre mes recherches dans les momens de
degout ou de decouragement ; mais tous les complimens que j'ai
pu recevoir de MM. Arago, de Laplace, ou Biot, ne m'ont
jamais fait autant de plaisir que la decouverte d'une verite
theorique et la confirmation de mes calcules par I'experience.
Le peu d'empressement que j'ai mis a publier mes memoires,
doDt il n'a gueres paru que des extraits, montre que je ne suis
pas tourment^ de la soif de la renommee, et que j'ai assez de
philosophic pour ne pas attacher trop d'importance aux jouis-
sances de la ?anite. Mais il est inutile de m'etendre davantage
BUT ce sujet en ecriyant a un homme trop superieur pour que
cette philosophic lui soit etrangere, et qui me comprendra et
me croira aisement.

Agrees, Monsieur, Tassurance de la haute con»deration ayee
laquelle j'ai Thonneur d'etre

Votre tr^s-humble et tres-obeissant senriteur,

Fresnel.

P.S. Je ne parlerai point a M. Arago de yotre seconde lettre.
Je lui ayais dit un mot de la premiere ; il a ete surpris que yous
me temoignassiez le desir d'ayoir un expose de mes idees
theoriques sur la lumiere pour un ouyrage ou yous lui ayez
recommande de ne rien mettre qui sentit Thypothese. II part
dans peu de jours pour Metz, ou il espere terminer dans ses
soirees son article sur la polarisation, par la description des
modifications que la reflexion totale imprime k la lumiere po-
larisee et des caracteres singuliers de la polarisation circulaire.

23. — From M. Frbsnel to Dr. Young.

Monsieur, ^"^ i« ^^ Janvier, I825.

LoRSQUE je yous ai ecrit ma demiere lettre, mon imagi-
nation etait fatiguee par des idees qui reyenaient sans cesse k

2 d2

404 CORRESPONDENCE RELATING TO Jfo. XVII.

ma pensee, comme cela arrive souvent aux maladesy et c etait
pour m'en debarrasser, que je les mettais sur le papier. Mais
j aurais du me bomer k cela, et ne pas vous enyoyer cette lettre
qui a dd vous paraltre assez ridicule et que je vous prie de jeter
au feu.

La peine que vous avez prise de transcrire les complimens
que vous m'avez adresses daus la preface de votre bel ouvrage
sur les hieroglyphes, me fait craindre que vous n'ayez pense que
mon amour-propre avait besoin de cette consolation. La verite
est que je n*eprouvais ni chagrin d'amour-propre ni sentiment
d'aigreur en ecrivant cette lettre, * qui, je Tavoue, n'a pas du
vous en paraltre exempte : je jetais sur le papier des idees
qui fatiguaient mon imagination.

J'ai beaucoup tarde a vous repondre, Monsieur, et vous avez
pu prendre mon silence pour un refus. J'ai toujours ete
languissant jusqu'a present, et je ne suis pas encore gueri. On
m'a recommande d'eviter soigneusement toute tension d'esprit.
II est resulte de ce long repos que je me trouve tres arriere dans
mes occupations obligees; en sorte que, lorsque je me sens
capable de travailler un peu, c'est a elles que je dois consacrer
de preference ces courts momens. J'ai cependant commence a
rediger une exposition de mes idees theoriques sur la polarisation
de la lumiere et les lois des interferences des rayons polarises :
j'espere que cette note sera terminee dans une dixaine de jours.
J'attendais toujours pour vous repondre que je I'eusse com-
mencee, esperant m'y mettre d'un jour a Tautre ; voila pour-
quoi, Monsieur, j'ai tant tarde a vous ecrire.

U est tres possible que vous n'ayez plus besoin maintenant
de ce petit memoire ; s'il vous etait inutile, je vous prie d'avoir
la bonte de m'en prevenir, afin que je ne vous fasse pas payer
mal a propos un assez fort port de lettre.

Vous avez pu trouver, Monsieur, dans le tome XVII. des
Annales de Chimie et de Physique, page 179 et suivantes, et
dans les divers extraits de mes memoires que j'ai eu Thonneur
de vous envoyer ou de vous indiquer, un apcrqu de mes travaux
et de mes idees theoriques sur la polarisation et la' double
refraction. La note que je me propose de vous envoyer con-
tiendra seulement la demonstration rigoureuse des vibrations

No. XVII. OPTICAL SUBJECTS. 405

transversales des rayons polarises, et Texplication theorique des
lois de rinterference de ces rayons, sur lesquelles reposent tons
mes calcules relatifs a la coloration des lames cristallisees ;
developpemens que je n'avais pu donncr, faute d'espace, dans
Tarticle des Annales que je viens de citer. Apropos de cette
theorie, il me semble que je puis en reclamer la seconde moitie.
Vous avez remarque et demontre le premier que les couleurs
des lames cristallisees provenaient de la difference de marche
des rayons ordinaires et extraordinaires ; mais il restait a ^tablir
le sens de polarisation de ces rayons dans les lames minces ; il
fallait expliquer pourquoi leurs interferences ne produLsaient
des couleurs que lorsqu'on analysait la lumiere emergente avec
un rhomboide de spath calcaire, ou par tout autre mode de
polarisation ; et pourquoi il etait encore necessaire que la lumi-
ere eiit rcQU une polarisation prealable avant de traverser la
lame cristallisee. * Je crois aussi etre le premier qui ait donne
des m^thodes siires et generales pour calculer les teintes que
la polarisation developpe dans les lames cristallisees.

Excusez, Monsieur, la brus({ue franchise de cette reclamation,
qui m'engage a vous faire un article de votre demiere lettre
ou vous me dites : ^^ Je crois avoir explique les phenomencs
desquels M. Biot avait tire cette notice imparfaite, avant que
vous eussiez publie la meme theorie, &c."

Agreez, Monsieur, I'assurance de la haute consideration
avec laquelle j'ai Thonneur d'etre

Votre tres-humble et tres-obeissant serviteur,

Frbsnel.

24. — Fr(ym Dr. Young to M. Araqo.

M V DEAR Sir, ^^''^^"' ^Sth Jaouary, 1825.

At last I have to congratulate you on the cessation of
your persecutions ; but I must tell you in the same breath,
that they are not quit^ at an end, and that you must positively
send me what you have written ; 1- on metallic mirrors ; 2. on
rings in plates perpendicular to the axis ; and 3. on the
absorption of light. I must protest against your having with-

406 CORRESPONDENCE RELATING TO No. XVII.

held theniy because ItJumght M. Blot's theory too foolish to
merit any serious attention : you may believe that I did not
mean to hint that your confutation of it was superfluous, so
far as it might have acquired any credit with the public : nor
did I say, or think, that the very refined and elegant experi-
ments of our friend Fresnel made the paper too long. I can
only say, that I hope to receive the three sections as they are
written by return of post : in that case only can I answer for
their being inserted, as I have told Mr. Napier that I have
received your la^t words, and I shall send him the translation in
a reasonable time. But I shall certainly not attempt to supply
the deficiency by anything of my own ; so that if you think the
article imperfect for want of them, you must be responsible
for your own omission. I shall only add a few theoretical
conjectures, quite independent of the series of your sections.
By his '* counbymeny** ^— - must certainly have meant Scotch-
men : he could not have meant to assert that the * Moni-
teur' had not been received in London at the time in question.
Apropos of priority of observations, pray have the goodness
to point out to Mr. Humboldt, in the * Bulletin Universel des
Sciences,' VII. 91, p. 177 aeq.^ this remarkable passage of Mr.
Champollion Figeac, *' Les signes 10 et 1 sont tels que M.
JoMARD les ei publics en 1819 ; les chiffi^s hieratiques ont ete
determine par mon frere."

Perhaps Mr. Ch. will say with that he had not seen the

^ Moniteur ' in which Mr. Humboldt stated the history of Mr.
Jomard's publication.* You will receive this, I hope, on
Thursday, so that I may get an answer by Friday's post if
possible ; if you cannot send your answer till Monday, still
send it and I will wait for it if I can. Adieu, mon cher
confrere.

Always yours,

Thomas Younq.

• See Vol. III. p. 208, of thw work.

No. XVII. OPTICAL SUBJECTS. 407

25. — Fr(m M. Fresnel to Dr. Young.

MONSIEUB, P*»^ 1« 4 Septembw, 1825.

LoBSQUE T0U8 me demandates, il y a environ un an, de
V0U8 faire part de mes vues theoriques sur la polarisation et la
double refraction, j'eus Thonneur de youb indiquer Vextrait de
mon memoire sur la double refraction qui avait et^ publie dans
les deux bulletins de la Societe Philomatique des mois d'Ayril
et Mai, 1822 : n'en ayant pas fiedt tirer d'exemplaires a part, je
ne pouvais pas vous en envoyer. Vous me r^pondites que
d'apr^ Tindication que je vous donnais, vous comptiez le
trouver ais^ment ; voil^ pourquoi je ne crus pas necessaire de
vous en faire une copie.

Je regrette de n'avoir point encore trouve le tems ni I'occa-
sion de fisdre imprimer le memoire en entier : ce que vous avez
la bonte de me dire sur Textrait me fait penser que la lecture
du memoire vous aurait offert quelque interet.

tTai eu I'honneur de vous envoyer cet hiver, avec un exem-
plaire de cet extrait, un petit memoire contenant des vues
theoriques sur la polarisation de la lumiere, dans lequel vous
avez pu remarquer uoe demonstration assez m^thodique de
I'existence ezclusive des vibrations transversales, A toutefois
vous avez eu le tems de le lire. C'^tait precis^ment ces
reflexions et ces developpemens que je me proposais de vous
communiquer pour I'Encyclopedie Britannique, mais ils sont
arrives trop tard : peut-etre meme ne les avez vous pas re^us.
Ce petit memoire avait ete insere dans le Bulletin des Sciences
de la Societe Philomatique du mois d'Octobre, 1824, publie,
je croia, vers le milieu Thiver.

Autant que je puis me rappeler, j'ai adress^ successivement
mes deux paquets d'exemplaires k Sir Humfrey Davy ; e'est M.
le Lieutenant-Colonel Wright qui a bien voulu se charger de les
lui faire passer par le courrier de I'Ambassade. Comme votre
lettre me fait supposer que vous ne les avez pas re9us, et que je
crains qu'il n'en soit de meme des autres savans auxquels je
desirais aussi les offirir, je joins a cette lettre sept exemplaires
de chaque espece, en vous priant de les donner aux physidens
qu'ils pourraient interesser.

408 CORRESPONDENCE RELATING TO No. XVIL

Je vous prie, Monsieur, d'avoir la bonte d'offrir mes remer-
cimens particuliers aux membres de la Societe Royale qui
ont bien voulu lui faire valoir mes travaux. Je n ai pas besoin
de dire que c'est a vous que je 6roi8 etre principalement
redevable de la faveor qu'elle vient de m'accorder.
Agreez, Monsieur,
Les sentiments de mon respectueux attachement,
Votre devoue serviteur,

Fresnel.

P.S. — Je me porte mieux depuis pliisieurs mois ; mais j'ai
ete toujours trop occupe pour me retablir entierement; je
compte faire incessamment un petit voyage, qui achevera,
j'espere, le retablissement de ma sante.

26. — From Dr. Young to M. Araoo.

My dear Sir, ^^^ Square, 29lii March, 1827.

In sending you my annual contribution to the improve-
ment of the ^ Connaissance des Tems,' I have also the pride
and pleasure to inform you that the Council of the Royal So*
ciety has done honour to us a//, by awarding to our friend
Fresnel the Kumford medal, which has been adjudged but
once since the death of Mains. In this determination the most
zealous supporter of the cause was Mr. Herschel : I was obliged
to be silent, from being too much interested in the subject, but
ill fact there was no opposition. The value of the medal is
GO/. ; there will be a sum of 50/. in money besides, which I shall
have to remit, arising from the accumulations from the value
of the medals not allotted. Thinking that this circumstance
would make our system a little more popular tiian hitherto, I
have determined to insert in my Astronomical and Nautical Col-
lections, a translation of M. Fresnel's Abstract, which is pub-
lished in 'Thomson's Chemistry,' and I trust he will not
dislike its appearance.

Pray remember me kindly to Humboldt, who, I find from
M. de Flavigny, is still at Paris, and tell him that I am still

No. XVII. OITICAL SUBJECTS. 409

going on, though slowly, with my Egyptian researches. It
would be very satisfactory to our Orientalists, if the exact his-
tory of the vase with the name of " Xerxes "* could be ascer-
tained, for some of them suspect it is a forgery, the writing
being in the wrong direction, that is, from right to left, in the
nail-headed character. Believe me always, my dear Sir, with
best compliments to Madame Arago,

Very truly yours,

Thomas Young.

il.^From Dr. Young to M. Fresnel.

Mt dear Sir London, 9, Park Square, 18th June, 1827.

I HAVE great pleasure in transmitting to you the Prize
Medal of Count Rumford, intended to be given biennially, to
the author of the most important discovery or improvement
relating to heat and light, which the Council of the Royal So-
ciety has thought it right to assign to your application of the
undulatory theory of light to the phenomenon of polarisation.
You will also have the goodness to call on Mr. Lafitte the
banker, whom I have ordered to pay you the sum of 55/. I65.
sterling, and who will return me your receipt for the amount
in French money : this sum being the accumulation derived from
the investment of the value of medals not adjudged. At last,
then, I trust you will no longer have to complain of the neglect
which your experiments have for a time undergone in this coun-
try. I should also claim some right to participate in the com-
pliment which is tacitly paid to myself in common with you by
this adjudication, but considering that more than a quarter of
a century is past since my principal experiments were made, I
can only feel it a sort of anticipation oi posthumous &me, which
I have never particularly coveted.

Believe me, dear Sir, with great respect,
Very truly yours,
Thomas Young, M.D., For. Sec. R.S.

• See Vol. JII p. 248, of this work.

410 CORRESPONDENCE RELATING TO No. XVII.

28. — From M. Araoo to Dr. Young.

MoN CHER Confrere, ^*™» ^ ^^^^ i^^t.

Je m'empresse de vous annoncer que rAcademie des
Sciences, sur la proposition d'une commission dont j'etais mem-
bre, et qui m'avait confie Thonneur de developper vos titres,
vient de vous nommer, a la place de Volta, Tun de ses huit
associes etrangers. Vos concurrens etaient MM. Olbers, Bessel,
Robert Brown, Blumenbach, Soemmerring, Leopold de Buch,
Dalton et Plana. Aussitot que le Roi aura confirme yotre
nomination, le secretaire de TAcademie yous la notifiera
officiellement

Vous avez sans doute app^ quelle perte cruelle les sciences
ont &ite le mois dernier. Le pauvre Fresnel etait deja a moiti^
eteint lorsque je lui remis vos medailles. Sa mort a plonge ici
dans la plus viye douleur tons ceux qui sout dignes d'apjMre-
cier Taccord d'un beau talent et d'un beau caractere.

Adieu I mon dier confrere. Pr^sentez, je vous prie, mes
hommages respectueux a Madame Young, et agreez la nouvelle
assiu'ance de mon attachement.

Votre tout d^voue,

F. Arago.

29. — From Dr. Young to M. Arago.

My DEAR Sir, London, Paric Square, 2nd September, 1827.

On my return from Liverpool a few days ago, I found
on my table your very obliging letter, annoimcing to me the
success of your kind exertions in my &vour, and my nomination
as one of die eight foreign associates of the Academy. If any
thing could add to the value of so distinguished a compliment,
it would be the consciousness of owing it chiefly to the good
opinion of so candid and so enlightened a judge as yourself.
I must however confess that I could not read, without some
confusion, my own name at the head of a list in which that of
Olbers was only the third : but I am so much the more obliged
to the Academy for its partiality to me.

No. XVII. OPnCAL SUBJECTS. 411

I do indeed deeply lament the fatality which has a second
time followed the adjudication of the Rumford medal. You do
not tell me how far our poor friend felt that gratification from
it» which it was our wish that he should receive, nor if he was
pleased with my having undertaken to translate his Abstract
into English. I have ordered the money which was due to him
to be piud to his brother, whom the bankers represent as his
natural heir : I suppose there can be no doubt that they are
right

Mrs. Young begs to unite with me in kind compliments and
thanks : we are all occupied at present with the marriage of
her sister, Miss Maxwell, which is to take place immediately,
to a Mr. Earle.

Believe me, cher confrhre^
Very truly yours,

Thomab Youkg.

412 POLARISATION OF LIGHT. No. XVIII.

No. XVIII.

THEOUETICAL INVESTIGATIONS INTENDED TO ILLUSTRATE THE;

PHENOMENA OF POLARISATION:

BEING AN ADDITION MADE BY DR. YOUNG TO M. ARAGO'S
'TREATISE ON THE POLARISATION OF LIGHT.'

From the Supplement to the Encyclopedia Britannica.

Written in January, 1823.

Wb are led from the facts which have been enumerated in the
11th section, respecting " uniaxal and biaxal crystals,"* to the
remarkable coincidence between the discoveries of Dr. Brewster
respecting crystals with two axes, and a theory which had been
published a few years earlier in order to illustrate the propaga-
tion of an undulation in a medium compressed or dilated in a
^ven direction only, and to prove that such an undulation must
necessarily assume a spheroidal form upon the mechanical
principles of the Huyghenian theory. As every contribution to the
investigation of so difficult a subject may chance to be of some
value, it will be worth while to copy this demonstration here,
from the Quarterly Review for Nov. 1809, Vol. II. p. 345.

[Here follows the investigation given above in No. XII. pp.
230, 231, and 232.]

* This refers to a former section, added also by Dr. Young (by whom M. Anigo's
memoir was translated and edited), containing an enmneration of the principal
minerals which possess one or two axes of double refraction, and referring more
especially to Sir David Brewster's remarkable observations on the effects of heat
and compression in producing or modifying double refraction. He found that com-
pression was capable of producing colour when a soft animal jelly was only touched
by the finger : and that when a negative crystal, like that of carbonate of lime,
where the Huyghenian undulations producing the extraordinary ray are propagated in
oblate spheroids, is compressed in the direction of the axis, the tints that it affords
"descend," and that they **rise" when they are dilated: whence it seems to follow,
that simple dilatation of a homogeneous substance in a given line will constitute that
line tiie axis of an oblate spheroid. M. Fresnel succeeded in exhibiting not only
colours by a strong pressure, but a very manifest reduplication of the image of a line,
seen through a glass strongly compiessed by screws. — Note by the Editor,

No. XVIII, POLARISATION OF LIGHT. 413

However satisfactorily such a mode of viewing the extraordi-
nary refraction may be applied to the subsequent discoveries
relating to the effects of heat and compression, there is another
train of ideas, which arises more immediately from the pheno-
mena of polarisation, and which might lead us to a more distinct
notion of the separation of the pencil into two or more portions,
though it does not seem to comprehend so entirely the pheno-
mena depending on spheroidal undulations.

We may begin this mode of considering the subject in the
words which have already been employed in the article Chro-
matics, «£pra, pp. 334, 335. " If we assume as a mathematical
postulate, in the imdulatory theory, without attempting to
demonstrate its physical foundation, that a transverse motion
may be propagated m a direct line, we may derive from this
assumption a tolerable illustration of the subdivision of polar-
ised light by reflection in an oblique plane. Supposing polar-
isation to depend on a transverse motion in the given plane ;
when a ray completely polarised is subjected to simple reflection
in a different plane, at a surface which is destitute of any
polarising action, and which may be said to afford a neutral
reflection, the polar motion may be conceived to be reflected,
as any other motion would be reflected at a perfectly smooth
surface, the new plane of the motion being always the image of
the former plane ; and the effect of refraction will be nearly of
a similar nature. But when the surface exhibits a new polar-
ising influence, and the beams of light are divided by it into
two portions, the intensity of each may be calculated, by
supposing the polar motion to be resolved instead of being
reflected, the simple velocities of the two portions being as the
cosines of the angle, formed by the new planes of motion with
the old, and the energies, which are the true measure of the
intensity, as the squares of the sines. We are thus insensibly
led to confound tiie intensity of the supposed polar motion with
that of the light itself; since it was observed by Mains, that the
relative intensity of the two portions, into which light is divided
under such circumstances, is indicated by the proportion of the
squares of the cosine and sine of the inclination of the planes of
polarisation. The imaginary transverse motion must also ne-

414 POLARISATION OF LIGHT. No. XVUI.

cessarily be alternate, partly from the nature of a continuous
medium, and partly from the observed fact, that there is no
distinction between the polarisations produced by causes pre-
cisely opposed to each other in the same plane." Another
analogous hint is found in the Philosophical Transactions for
1818, supra^ p. 373 " Supposing the experiments to be perfectiy
represented by [Dr. Brewster's] general law, it will follow that
the tint exhibited depends not on the difference of refracted
densities in the direction of the ray transmitted, but on the
greatest difference of refractive densities in directions perpen-
dicular to that of the ray. These two conditions lead to the
same result, where the effect of one axis only is considered, but
they vary materially where two axes are supposed to be
combined . • . There can be little doubt that the direction of the
polarisation, in such cases, must be determined by that of the
greatest and least of the refractive densities in question ;" a
•* supposition," which Dr. Brewster finds " quite correct."

We may add agsdn to these hints the consideration, that
when simple pressure or extension in the direction of any given
axis produces a spheroidal undulation in a medium before
homogeneous, this state is always accompanied by the con-
dition, that a ray describing the axis, while the densities in
all transverse directions remain equal, undergoes no subdivision,
but that a ray moving in the plane of the equator, to which the
perpendiculars are the axis and another equatorial diameter, un-
dergoes the greatest possible separation into parts that are respec-
tively polarised in the planes passing through these directions.

From these phenomena we are led to be strongly impressed
with the analogy of the properties of sound, as investigated
cursorily by Mr. Wheatstone, and in a more elaborate manner
by tiie multiplied experiments of Mr. Savart, which have shown
that, in many cases, the elementary motions of the substances
transmitting sound are transverse to the direction in which the
sound is propagated, and that they remain in general parallel
to the original impulse.

The next transition carries us from the mathematical postulate
here mentioned to the physical condition assumed by Mr. Fres-
nel, that the relative situation of the particles of the etherial

No. XVIII. POLARISATION OF LIGHT. 415

medium with respect to each other, is such as to produce an
elastic force tending to bring back a litie of particles, which has
been displaced, towards its original situation by the resistance
of the particles surrounding the liney and at the same time to
impel these particles in its own direction, and in that direction
only, or principally, while the aggregate effect is propagated in
concentric surfaces.

This hypothesis of Mr. Fresnel is at least very ingenious, and
may lead us to some satisfactory computations: but it is
attended by one circumstance which is perfectly appalling in its
consequences. The substances on which Mr. Savart made his
experiments were solids only ; and it is only to solids that such
a lateral resistance has ever been attributed : so that if we
adopted the distinctions laid down by the reviver of the imdula-
tory system himself, in his Lectures^ it might be inferred that
the luminiferous ether, pervading all space, and penetrating
almost all substances, is not only highly elastic, but absolutely
solid ! I ! The passage in question is this : (Vol. I. p. 627.)

*'The immediate cause of solidity, as distinguished from
liquidity, is the lateral adhesion of the particles to each other,
to which the degree of hardness or solidity is always proportional.
This adhesion prevents any change of the relative ntuation of
the particles, so that they cannot be withdrawn from their places
without experiencing a considerable resistance from the force of
cohesion, while those of liquids may remain equally in contact
with the neighbouring particles, notwitiistanding their change of
form. When a perfect solid is extended or compressed, the
particles, being retained in their situations by tiie force of
lateral adhesion, can only approach directly to each other, or be
withdrawn further from each other ; and the resistance is nearly
the same, as if the same substance, in a fluid state, were enclosed
in an unalterable vessel, and forcibly compressed or dilated.
Thus the resistance of ice to extension or compression is found
by experiment to differ very littie from that of water contained
in a vessel ; and the same effect may be produced even when
the solidity is not the most perfect that the substance admits ;
for the immediate resistance of iron or steel to flexure is the
samSf whether it may be harder or softer. It often happens.

416 POLARISATION OF LIGHT. No. XVIII.

however^ that the magnitude of the lateral adhesion is so much
limited, as to allow a capability of extension or compression,
and it may yet retain a power of restoring the bodies to their
original form by its reaction. This force may even be the
principal or the only source of the body's elasticity : thus when
a piece of elastic gum is extended, the mean distance of the
particles is not materially increased . . and the change of form
is rather to be attributed to a displacement of the particles than
to their separation to a greater distance from each other, and
the resistance must be derived fi*om the lateral adhesion only :
some other substances also, approaching more nearly to the
nature of liquids, may be extended to many times their original
length, with a resistance continually inci*easing; and in such cases
there can scarcely be any material changes of the specific gravity
of these substances. Professor Robison has mentioned the juice
of bryony as affording a remarkable instance of such viscidity.

**It is probable that the immediate cause of the lateral
adhesion of solids is a symmetrical arrangement of their consti-
tuent parts ; it is certain that almost all bodies are disposed, in
becoming solid, to assume the form of crystals, which evidently
indicates the existence of such an arrangement ; and all the
hardest bodies in nature are of a crystalline form. It appears,
therefore, consistent botli with reason and with experience to
suppose, that a crystallization more or less perfect is the uni-
versal cause of solidity. We may imagine, that when the
particles of matter are disposed without any order, they can
afford no strong resistance to a motion in any direction ; but
when they are regularly placed in certain situations with respect
to each other, any cliange of form must displace them in such a
manner, as to increase the distance of a whole rank at once ;
and hence they may be enabled to co-operate in resisting such
a change. Any inequality of tension in a particular part of a
solid, is also probably so far the cause of hardness, as it tends
to increase the strength of union of any part of a series of
particles which must be displaced by a change of form."

It must, however, be admitted, that this passage by no means
contains a demonstration of the total incapability of fluids to
transmit any impressions by lateral adhesion, and the hypothe-

No. XVIII. POLARISATION OF LIGHT. 417

sis remains completely open for discussion, notwithstanding the
apparent difficulties attending it ; which have appeared to bring
us very near to the case stated in the same lectures as a possible
one, that there may be independent worlds, some existing in
different parts of space, others pervading each other unseen and
unknaum in the same space.* We may perhaps accommodate
the hypothesis of Mr. Fresnel to tlie phenomena of the ordinary
and extraordinary refraction, by considering the undulations as
propagated through the given medium in two different ways ;
some by the divergence of the elementary motions in the direc-
tion of the ray, and others by their remaining parallel to the
direction of the impulse or of the polarisation : the former must
be supposed to furnish the spheroidal, the latter the spherical
refraction. It would indeed follow that the velocity of the
spherical undulation ought to vary by innumerable degrees,
within certain limits, according to the direction of the supposed
elementary motion : while in fact the actual velocity of the
spherical undulations seems to be uniformly equal to the velocity
in the direction of the axis : but this objection may be obviated
by supposing the surface so constituted, that for some unknown
reason the parallel elementary motion can only be propagated
in the regular manner when it takes place in the direction of the
axis, or when it is made to assume that direction : a condition
not very simple or natural, but by no means inconceivable ;
unless we saw any reason to consider the adhesion as a constant
force, independent of the direction, and equal to the least or
greatest elasticity, or unless it were possible to derive the phe-
nomena of two supposed axes of polarisation, which Mr. Fresnel
has explained on the hypothesis of two spheroids, from the sup-
position of two spherical undulations propagating oblique ele^
mentary motions in the direction of the actual polarisation as
already determined for these crystals.

If these conjectures should be found to afford a single step,
in an investigation so transcendently delicate, it will be best to
pause on them for a time, and to wait for further aid from a
new supply of experiments and observations.

* Vol. i., p. 610.
VOL. I. 2 E

418 ox THE COHESION OP FLUIDS. No. XIX

No. XIX.

AN ESSAY ON THE COHESION OF FLUIDS.

From the Philosophical Transactions for 1 805. .
Read December 20, 1804.

With some Alterations and Additions made bt the Author to

THE Reprint ok it in the Appendix to hib Lectures

ON Natural Philosophy in 1807.

I. — General Principles.

It has already been asserted, by Mr. Monge and others, that
the phenomena of capillary tubes are referable to the cohesive
attraction of the superficial particles only of the fluids employed ;
and that the surfaces must consequently be formed into curves
. of the nature of linteariae, which are supposed to be the results
of a uniform tension of a surface, resisting the pressure of a
fluid, either uniform, or varying according to a given law.
Segner, who appears to have been the first that maintained a
similar opinion, has shown in what manner the principle may be
deduced from the doctrine of attraction, but his demonstration
is complicated, and not perfectly satisfactory ; and in applying
the law to the forms of drops, he has neglected to consider the
very material efiects of the double curvature, which is evidently
the cause of the want of a perfect coincidence of some of his
experiments with his theory. Since the time of Segner, little
has been done in investigating accurately and in detail the
various consequences of the principle.

It will perhaps be most agreeable to the experimental philo-
sopher, although less consistent with the strict course of logical
argument, to proceed in the first place to the comparison of this
theory with the phenomena, and to inquire afterwards for its

No. XIX. ON THE COHESION OF FLUIDS. 419

foundation in the ultimate properties of matter. But it is
necessary to premise one observation, which appears to be new,
and which is equally consistent with theory and with experiment ;
that is, that for each combination of a solid and a fluid, there is
an appropriate angle of contact, between the surfaces of the
fluid, exposed to the air, and to the solid. This angle, for glass
and water, and in all cases where a solid is perfectly wetted by
a fluid, is evanescent: for glass and mercury, it is about 140'
in common temperatures, and when the mercury is moderately
clean.

11. — Form of the Surface of a Fluid.

It is well known, and it results immediately from the compo>
sition of forces, that where a line is equally distended, the force
that it exerts, in a direction perpendicular to its own, is directly
as its curvature ; and the same is true of a surface of simple
curvature ; but where the curvature is double, each curvature
has its appropriate eflect, and the joint force must be as the sum
of the curvatures in any two perpendicular directions. For this
sum is equal, whatever pair of perpendicular directions may be
employed, as is easily shown by calculating the versed sines of
two equal arcs taken at right angles in the surface. Now when
the surface of a fluid is convex externally, its tension is produced
by the pressure of the particles of tlie flxiid within it, arising
from their own weight, or from that of the surrounding fluid ;
but when the surface is concave, the tension is employed in
counteracting the pressure of the atmosphere, or where the
atmosphere is excluded, the equivalent pressure arising from
the weight of the particles suspended from it by means of their
cohesion, in the same manner as, when water is supported by
the atmospheric pressure in an inverted vessel, the outside of
the vessel sustains a hydrostatic pressure proportionate to the
height ; and this pressure must remain unaltered, when the
water, having been sufficiently boiled, is made to retain its
situation for a certain time by its cohesion only, in an exhausted
receiver. When, therefore, the surface of the fluid is terminated
by two right lines, and has only a simple curvature, the curva-
ture must be everywhere as the ordinate ; and where it has a

2 K 2

420 ON THE COHESION OF FLUIDS. No. XIX

double curvature, the sum of the curvatures in the difierent
directions must be as the ordinate. In the first case, the curve
may be constructed by approximation, if we set out from a point
at which it is either horizontal or vertical, and divide the height
into a number of small portions, and taking the radius of each
proportional to the reciprocal of the height of. its middle point,
above or below the general surface of the fluid, go on to add
portions of circles joining each other, until they have completed
as much of the curve as is required. In the second case it is
only necessary to consider the curve derived from a circular
basis, which is a solid of revolution ; and the centre of that
circle of curvature, which is perpendicular to the section formed
by a plane passing through the axis, is in the axis itself, conse-
quentiy in tiie point where tiie normal of the curve intersects tiie
axis : we must therefore here make the sum of this curvature,
and that of the generating curve, always proportional to the
ordinate. This may be done mechanically, by beginning at the
vertex, where the two curvatures are equal, then for each suc-
ceeding portion, finding the radius of curvature, by deducting
the proper reciprocal of the normal, at the beginning of the
portion, from the ordinate, and taking the reciprocal of the
remainder. In this case the analysis leads to fiuxional equations
of the second order, which appear to afibrd no solution by means
hitherto discovered ; but the cases of simple curvature may be
more easily subjected to calculation; the curvature varying
always as the ordinate, the curve belongs to the general descrip-
tion of an elastic curve.

III. — Analysis of the simplest Forms.*

Let the greatest ordinate of the curve (AB) be called a,
the arc of the circle of curvature at the vertex (AC) z, and
let us suppose, that while this circle is uniformly increased,
the curve (AD) flows with an equal angular velocity, then
the fluxion of the curve, being directly as the radius

* In the original Essay, the mathematical form of this investigation and the figures
were suppressed, the reasoning and the i-csults to wbicli it luads being express*^ in
ordinary language : even in its altered form the investigation is unduly concise and
obscure. — Note by the Editor,

No. XIX.

ON THE COHESION OF FLUIDS.

421

of curvature, will be inversely as the ordinate y, and

will be expressed by-—;

the fluxion of the absciss

will therefore be -j-, t

being the cosine of the
arc z, and r the radius,
and the fluxion of the

areawillbe— .

But?

is the fluxion of the sine
s of the arc z in the circle
to which it belongs ; con-
sequently, the area is ex-
pressed by CM, and is
equal to the rectangle con-
tained by the initial ordi-
note, and the sine cor-
responding to each point of the curve in the initial circle of curva-
ture. Hence it follows, that the whole area (ABEF or EFGH)
included by the ordinates where the curve is f^ertical and where it
is horizontal, is equal to the rectangle contained by the ordinate
and the radius of curvature.

In order to find the ordinate y, corresponding to a given

angular direction, and to a given arc z, we have + y =

or.

since — is the fluxion of the versed sine », + y = — , and ^

yy = at), whence yy = b"^ 2av. But at the summit of the
curve, when r = 0, y = a, therefore b = aa, and yy ^ aa —
2av ; and where the curve meets the absciss, y = 0 and a =
2v. If a = 4r, when y = 0, t; will be 2r, and the curve vrill
touch the horizontal line at an infinite distance, since its curva-
ture must be infinitely small ; if a be greater than 4r, the least
ordinate will be ^ {aa — 4ar). When the curve is vertical,
v = r, and yy =z aa — 2ar. The rectangle, contained by the
elevation above the general surface, and the diameter of the
circle of curvature, which is here 2ar, is constant in all circum-
stances for the same fluid, and may therefore be called the

422

ON THE COHESION OF FLUIDS.

No. XIX.

appropriate rectangle of the fluid ; and when the curve is in-
Cl^ \C finite, and a = 4^, this rectangle b equal to 8rr, or to ^aa^
so that r and a may be readily found from it : it is also equal
to the square of the ordinate at the vertical point, where yj/ =
aa — 2ar. K we describe a circle ABCD, of which the

AGE

diameter is a, the chord of the arc of this circle (AC, AB,)
corresponding in angular situation to the curve, will be equal
to the ordinate (EF, GH,) at the respective point ; for the
versed sine in this circle will be 2v, and the chord will be a
mean proportional between a and a—2v; in this case therefore,
where the curve is infinite, the ordinate varies as the sine of half
the angle of elevation.

For determining the absciss, it would be necessary to employ
an infinite series ; and the most convenient would perhaps be
that which is ^ven by Euler for the elastic curve, in the second
part of the third volume of the Acta Petropolitana.

IV. — Application to the Elevation of particular
Fluids.

The simplest phenomena, which afibrd us data for determining
the fundamental properties of the superficial cohesion of fluids,
are their elevation and depression between plates and in capil-
lary tubes, and their adhesion to the surface of solids, which are
raised, in a horizontal situation, to a certain height above the
general surface of the fluids. When the distance of a pair of

No. XIX. ON THE COHESION OF FLUIDS. 423

plates, or the diameter of a tube, is very minute, the curvature
may be considered as uniform, and the appropriate rectangle
may readily be deduced from the elevation, recollecting that
the curvature in a capillary tube is double, and the height
therefore twice as great as between two plates. In the case of
the elevation of a fluid, in contact with a horizontal surface, the
ordinate may be determined from the weight required to produce
a separation ; and the appropriate rectangle may be foilnd in
this manner also, the angle of contact being properly considered,
in this as well as in the former case. It will appear that these
experiments by no means exhibit an immediate measure of the
mutual attraction of the solid and fluid, as some authors have
supposed.

Sir Isaac Newton asserts, in his Queries, that water ascends
between two plates of glass at the distance of one hundredth of
an inch, to the height of about one inch, the product of the
distance and the height being about .01 ; but this appears to
be much too little. In the best experiment of Musschenbroek,
with a tube, half of the product was .0196; in several of
Weitbrecht, apparently very accurate, .0214. In Monge's
experiments on plates, the product was 2.6 or 2.7 lines, or
about .0210. Mr. Attwood says that, for tubes, the product is
.0530, half of which is .0265. Until more accurate experi-
ments shall have been made, we may be contented to assume
.02 for the rectangle appropriate to water, and .04 for the pro-
duct of the hefght in a tube by its bore. Hence, when the
curve becomes infinite, its greatest ordinate is .2, and the height
of the vertical portion, or the height of ascent against a single
vertical plane, .14, or nearly one-seventh of an inch.

Now when the horizontal surface of a solid is raised from a
vessel of water, the surface of the water is formed into a
lintearia, to which the solid is a tangent at its highest point, and
if the solid be still further raised, the water will separate ; the
surface of the water, being horizontal at the point of contact,
cannot add to the weight tending to depress the solid, which is
therefore simply the hydrostatic pressure of a column of water
equal in height to the elevation^ in this case one-fiftli of an
inch, and standing on the given surface. The weight of such a

424 ON THE COHESION OF FLUIDS. No. XIX.

column will be 504 grains for each square inch, and in Taylor's
well-known experiment the weight required was 50 grains.
But when the solid employed is small, the curvature of the ho-
rizontal section of the water, which is convex externally, will
tend to counteract the vertical curvature and to diminish the
height of separation ; thus, if a disc of an inch in diameter were
employed, the curvature in this direction would perhaps be
equivalent to the pressure of about one-hundredth of an inch,
and might reduce the height from 2 to about .19, and the
weight in the same proportion. There is, however, as great a
diversity in the results of different experiments on the force
required to elevate a solid from the surface of a fluid, as in
those of the experiments on capillary tubes ; and indeed the
sources of error appear to be here more numerous. Mr. Achard
found that a disc of glass, li inch French in diameter, required,
at 69^ Fahrenheit, a weight of 91 French grains to raise it
from the surfaice of water ; this is only 37 English grains
for each square inch ; at 44^° the force was tt greater, or 39i
grains, the difference being -rh* for each degree of Fahrenheit.
It might be inferred, from these experiments, that the height of
ascent in a tube of a ^ven bore, which varies in the duplicate
ratio of the height of adhesion, is diminished about y^^ for
every degree of Fahrenheit that the temperature is raised
above 50° ; there was, however, probably some considerable
source of error in Achard's experiments, for I find that this dimi-
nution does not exceed ttjVt. The experiments of Mr. Dutour
make the quantity of water raised equal to 44.1 grains for each
square inch. Mr. Achard found the force of adhesion of sul-
furic acid to glass, at 69° of Fahrenheit, 1.26, that of water
being I ; hence the height was as .69 to 1, and its square as
.47 to 1, which is the corresponding proportion for the ascent
of the acid in a capillary tube, and which does not very mate-
rially differ from the proportion of .395 to 1, assigned by Barruel
for this ascent. Musschenbroek found it .8 to 1, but bis acid
was probably weak. For alcohol the adhesion was as .593, the
height as .715, and its square as .510 : the observed proportion
in a tul)e, according to an experiment of Musschenbroek, was
about .550, according to Carre from .400 to .440. Tlie expo-

No. XIX. ON THE COHESION OF FLUIDS. 425

riments on sulfuric ether do not agree quite so well, but its
quality is liable to yery considerable variations. Dutour found
Uie adhesion of alcohol .58, that of water being, 1.

With respect to mercury, it has been shown by Professor
Casbois of Metz, and by others, that its depression in tubes of
glass depends on the imperfection of the contact, and that
when it has been boiled in the tube often enough to expel all
foreign particles, tlie surface may even become concave instead
of convex, and the depression be converted into an elevation.
Perhaps this change may be the effect of the commencement of
a chemical action between the mercury and the component
parts of the glass ; but in barometers constructed according to
the usual methods, the angle of the mercury will be found to
differ little from 140°; and in other experiments, when proper
precautions are taken, the inclination will be nearly the same.
The determination of this angle is necessary for finding the
appropriate rectangle for the curvature of the surface of
mercury, together with the observations of the quantity of
depression in tubes of a given diameter. The table published
by Mr. Cavendish, from the experiments of his father. Lord
Charles Cavendish, appears to be best suited for this purpose.
I have constructed a diagram, according to the principles
already laid down, for each case, and I find that the rectangle
which agrees best with the phenomena is .01. The mean
depression .is always .015, divided by the diameter of the tube:
in tubes less than half an inch in diameter, the curve is very
nearly elliptic ; and the central depression in the tube of a
barometer may also be found by deducting from the corre-
sponding mean depression the square root of one thousandth
part of its diameter. There is reason to suspect a slight inac-
curacy towards the middle of Lord Charles Cavendish's Table,
from a comparison with the calculated mean depression, as well
as from the results of the mechanical construction. The ellipsis
approaching nearest to the curve may be determined by the
solution of a biquadratic equation.

426 ON THE COHESION OF FLUIDS. No. XIX.

Diameter

Gnins in

Central de-

Cential de-

Central de-

Marginal de-

in inches.

an Inch.
C.

•ionbycalcu-
laUon. Y.

preMioit by
observation.

pression by preasion by pression bv
formula. Y. diagfam. Y. dia^^ram. V.

972

.025

.005

(.001)

.005

.066

675

.030

.007

.008

.007

.067

432

.037

.015

.017

.012

.069

.35

331

.043

.025

.024

.017

.072

.30

243

.050

.036

.033

.027

.079

.25

169

.060

.050

.044

.038

.086

.20

108

.075

.067

.061

.056

.096

.15

.100

.092

.088

.085

.116

.10

.150

.140

.101

The square root of the rectangle .01, or .1, is the ordinate
where the curve would become vertical if it were continued;
but in order to find the height at which the mercury adheres to
a vertical surface of glass, we must diminish this ordinate in the
proportion of the sine of 25° to the sine of 45°, and it will be-
come .06, for the actual depression in this case. The elevation
of the mercury that adheres to the lower horizontal surface of a
piece of glass, and the thickness at which a quantity of mer-
cury will stand when spread out on glass, supposing the angle
of contact still 140^, are found, by taking the proportion of the
sines of 20° and of 70*^ to the sine of 45°, and are therefore
.0484 and .1330 respectively. If, instead of glass, we em-
ployed any surface capable of being wetted by mercury, the
height of elevation would be .141, and this is the limit of the
thickness of a wide surface of mercury, supported by a sub-
stance wholly incapable of attracting it. Now the hydro-
static pressiu^e of a column of mercury .0484 in thickness, on a
disc of one inch diameter, would be 131 grains; to this the
surrounding elevation of the fluid will add about 11 grains for
each inch of the circumference, with some deduction for the
effectof the contrary curvature of the horizontal section, tending
to diminish the height ; and the apparent cohesion thus exhi-
bited will be about 160 grains, which is a little more than
four times as great as the apparent cohesion of glass and water.
With a disc 11 lines in diameter, Mr. Dutour found it 194
French grains, which is equivalent to 152 English grains,
instead of 160 for an inch; a result which is sufficient to con-
firm the principles of the calculation. The depth of a quantity
of mercury standing on glass I have found, by actual observa-
tion, to agree precisely with this calculation. Segner siiys that

No. XIX. ON THE COHESION OF FLUIDS. 427

the depth was .1358) both on glass and on paper ; the difference
is very trifling, but this measure is somewhat too great for glass
and too small for paper, since it appears from Dutour's experi-
ments that the attraction of paper to mercury is extremely weak.

If a disc of a substance capable of being wetted by mercury,
an inch in diameter, were raised from its surface in a position
perfectly horizontal, the apparent cohesion should be 381
grains, taking .141 as the height; and for a French circular
inch, 433 grains, or 528 French grains. Now in the experi-
ments of Morveau, the cohesion of a circular inch of gold to the
surface of mercury appeared to be 446 grains, of silver 429, of
tin 418, of lead 397, of bismuth 372, of zinc 204, of copper
142, of metallic antimony 126, of iron 115, of cobalt 8 ; and
this order is the same with that in which the metals are most
easily amalgamated with mercury. It is probable that such an
amalgamation actually took place in some of the experiments,
and affected their results ; for the process of amalgamation may
often be observed to begin almost at the instant of contact
of silver with mercury, and the want of perfect horizontality
appears in a slight degree to have affected them all. A
deviation of one-fiftieth of an inch would be sufficient to
have produced the difference between 446 grains and 528 ;
and it is not impossible that all the differences, as far down
as bismuth, may have been accidental. But if we suppose the
gold only to have been perfectly wetted by the mercury, and
all the other numbers to be -in due proportions, we may find
the appropriate angle for each substance by deducting from
180^, twice the angle, of which the sine is to the radius, as the
apparent cohesion of each to 446 grains ; that is, for gold 1,
for silver about .97, for tin .95, for lead .90, for bismuth .85,
for zinc .46, for copper .32, for antimony .29, for iron .26, and
for cobalt .02, neglecting the surrounding elevation, which has
less effect in proportion as the surface employed is larger.
Gellert found the depression of melted lead in a tube of glass
multiplied by the bore equal to about .054.

It would perhaps be possible to pursue these principles so far
as to determine in many cases the circumstances under which
a drop of any fluid would detach itself from a given surface.

428 ON THE COHESION OF FLUIDS. No. XIX.

But it is sufficient to infer, from the law of the superficial cohe-
sion of fluids, that the linear dimensions of similar drops,
depending from a horizontal surface, must vary precisely in the
same ratio as the heights of ascent of the respective fluids
against a vertical surface, or as the square roots of the heights
of ascent in a given tube ; hence the magnitudes of similar
drops of difierent fluids must vary as the cubes of the square
roots of the heights of ascent in a tube. I have mea-
sured the heights of ascent of water and of diluted spirit
of wine in the same tube, and I found them nearly as 100
to 64 : a drop of water falling from a large sphere of
glass weighed 1.8 grains, a drop of the spirit of wine about
.85, instead of .82, which is nearly the weight that would be
inferred from the consideration of the heights of ascent, com-
bined with that of the specific gravities. W'e may form a
conjecture respecting the probable magnitude of a drop> by in-
quiring what must be the circumference of the fluid that would
support by its cohesion the weight of a hemisphere depending
from it : this must be the same as that of a tube, in which the
fluid would rise to the height of one-third of its diameter ; and
the square of the diameter must be three times as great as the
appropriate product, or, for water, .12 ; whence the diameter
would be .35, or a little more than one-third of an inch, and
the weight of the hemisphere would be 2.8 grains. If more
water were added internally, the cohesion would be overcome,
and the drop would no longer be suspended ; but it is not easy
to calculate what precise quantity of water would be separated
with it. The form of a bubble of air rising in water is deter-
mined by the cohesion of the internal surface of the water,
exactly in the same manner as the form of a drop of water in
the air. The delay of a bubble of air at the bottom of a vessel
appears to be occasioned by a deficiency of the pressure of the
water between the air and the vessel ; it is nearly analogous to
the experiment of making a piece of wood remain immersed in
water, when perfectly in contact with the bottom of the vessel
containing it. This experiment succeeds, however, far more
readily with mercury, since the capillary cohesion of the
mercury prevents its insinuating itself under the wood.

No, XIX. ON THE COHESION OF FLUIDS. 429

V. — ^Of Apparent Attractions and Repulsions.

The apparent attraction of two floating bodies, round both of
which the fluid is raised by cohesive attraction, is produced by
the excess of the atmospheric pressure on the remote sides of
the solids, above its pressure on their neighbouring sides : or,
if the experiments are performed in a vacuum, by the equiva-
lent hydrostatic pressure or suction, derived from the weight
and the immediate cohesion of the intervening fluid. This
force varies ultimately in the inverse ratio of the square of the
distance ; for if two plates approach each other, the height of
the fluid, that rises between them, is increased in the simple
inverse ratio of the distance ; and the mean action, or negative
pressure of the fluid, on each particle of the surface, is also
increased in the same ratio. When the floating bodies are
both surrounded by a depression, the same law prevails, and its
demonstration is still more simple and obvious. The repulsion
of a wet and a dry body does not appear to follow the same
proportion : for it by no means approaches to infinity upon the
supposition of perfect contact; its maximum is measured by
half the sum of the elevation and depression on the remote
sides of the substances, and as the distance increases, this
maximum is only diminished by a quantity, which is initially
as the square of the distance. The figures of the solids con-
cerned modify also sometimes the law of attraction, so that, for
bodies surrounded by a depression, there is sometimes a maxi-
mum, beyond which the force again diminishes ; and it is hence
that a light body floating on mercury, in a vessel little larger
than itself, is held in a stable equilibrium without touching the
sides. The reason of this will become apparent, when we
examine the direction of the surface necessarily assumed, by
the mercury, in order to preserve the appropriate angle of
contact ; the tension acting with less force, when the surface at-
taches itself tu the angular termination of the float in a direction
less horizontal.

The apparent attraction produced between solids, by the
interposition of a fluid, does not depend on their being partially

430 ON THE COHESION OF FLUIDS. No. XIX.

immersed in it ; on the contrary, its effects are still more power-
fully exhibited in other situations; and, when the cohesion
between two solids is increased and extended by the intervention
of a drop of water or of oil, the superficial cohesion of these
fluids is AiUy sufficient to explain the additional effect. When
wholly immersed in water, the cohesion between two pieces of
glass is little or not at all greater than when dry : but if a
small portion only of a fluid be interposed, the curved sur&ce,
that it exposes to the air, will evidently be capable of resisting
as great a force, as it would support from the pressure of the
column of fluid, that it is capable of sustaining in a vertical
situation ; and in order to apply this force, we must employ, in
the separation of the plates, as great a force as is equivalent to
the pressure of a column, of the height appropriate to their
distance. Morveau found that two discs of glass, 3 inches
French in diameter, at the distance of one tenth of a line,
appeared to cohere with a force of 4719 grains, which is equi-
valent to the pressure of a column 23 lines in height : hence
the product of the height and the distance of the plates is 2.3
lines, instead of 2.65, which was the result* of Monge's ex-
periments on the actual ascent of water. The difference is
much smaller than the difference of the various experiments
on the ascent of fluids ; and it may easily have arisen from a
want of perfect parallelism in the plates ; for there is no force
tending to preserve this parallelism. The error, in the extreme
case of the plates coming into contact at one point, may reduce
the apparent cohesion to one half.

The same theory is suflicient to explain the law of the force,
by which a drop is attracted towards the junction of two plates,
inclined to each other, and which is found to vary in the inverse
ratio of the square of the distance ; whence it was inferred by
Newton that the primitive force of cohesion varies in the simple
inverse ratio of the distance, wliile other experiments lead us
to suppose that cohesive forces in general vary in the direct
ratio of the distance. But the difliculty is removed, and the
whole of the effects are satisfactorily explained, by considering
the state of the marginal surface of the drop. If the plates
were parallel, the capillary action would be equal on both sides

No. XIX. ON THE COHESION OF FLUIDS. 431

of the drop : but when they are inclined, the curvature of the
surface at the thinnest part requires a force proportioned to the
appropriate height to counteract it : and this force is greater
than that which acts on the opposite side. But if the two
plates are inclined to the horizon, the deficiency may be ma<ie
up by the hydrostatic weight of the drop itself ; and the same
inclination will serve for a larger or a smaller drop at the same
place. Now when the drop approaches to the line of contact,
the difierence of the appropriate heights for a small drop of a
given diameter will increase as the square of the distance de-
creases ; for the fluxion of the reciprocal of any quantity varies
inversely as the square of that quantity ; and, in order to
preserve the equilibrium, the sine of the angle of elevation of
the two plates must be nearly in the inverse ratio of the square
of the distance of the drop from the line of contact, as it actually
appears to have been in Hauksbee's experiments.

VI. — Physical Foundation of the Law of Superficial

Cohesion.

We have now examined the principal phenomena which are
reducible to the simple theory of the action of the superficial
particles of a fluid. We are next to investigate the natural
foundations upon which that theory appears ultimately to rest.
We may suppose the particles of liquids, and probably those of
solids also, to possess that power of repulsion, which has been
demonstratively shown by Newton to exist in aeriform fluids,
and which varies in the simple inverse ratio of the distance of
the particles from each other. In airs and vapours this force
appears to act uncontrolled ; but in liquids it is overcome by a
cohesive force, while the particles still retain a power of moving
freely in all directions ; and in solids the same cohesion is ac-
companied by a stronger or weaker resistance to all lateral
motion, which is perfectly independent of the cohesive force,
and which must be cautiously distinguished from it. It is
simplest to suppose the force of cohesion nearly or perfectly
constant in its magnitude, throughout the minute distance to
which it extends, and owing its apparent diversity to the

432 ON THE COHESION OF FLUIDS. No. XIX.

contrary action of the repulsive force, which varies with the
distance. Now in- the internal parts of a liquid these forces
hold each other in a perfect equilibrium, the particles being
brought so near, that the repulsion becomes precisely equal to
the cohesive force that urges them together ; but whenever
there is a curved or angular surface, it may be found, by col-
lecting the actions of the different particles, that the cohesion
must necessarily prevail over the repulsion, and must urge the
superficial parts inwards, with a force proportional to die cur-
vature, and thus produce the effect of a uniform tension of the
surface. For, if we consider the effect of any two particles in
a curved line on a third at an equal distance beyond them, we
shall find that the result of their equal attractive forces bisects
the whole angle formed by the lines of direction ; but that the
result of their repulsive forces, one of which is twice as great
as the other, divides it in the ratio of one to two, forming with
the former result an angle equal to one sixth of the whole ; so
that the addition of a third force is necessary, in order to
retain these two results in equilibrium ; and this force must be
in a constant ratio to the evanescent angle which is the measure
of the cui'vature, the distance of the particles being constant.
The same reasoning may be applied to all the particles which
are within the influence of the cohesive force ; and the con-
clusions are equally true if the coheaon is not precisely constant,
but varies less rapidly than the repulsion.

VII.— Cohesive Attraction of Solids and Fluids.

When the attraction of the particles of a fluid for a solid is
less than their attraction for each other, there will be an equi-
librium of the superficial forces, if the surface of the fluid make
with that of the solid a certain angle, the versed sine of which
is to the diameter, as the mutual attraction of the fluid and
solid particles is to the attraction of the particles of the fluid
among each other. For, when the fluid is surrounded by a
vacuum or by a gas, the cohesion of its superficial particles acts
with full force in producing a pressure ; but when it is any
where in contact with a solid substance of the same attractive

No. XIX. ON THE COHESION OF FLUIDS. 433

power with itself, the effects of this action must be as much
destroyed as if it were an internal portion of the fluid. Thus,
if we imagined a cube of water to have one of its halves con-
gealed, without any other alteration of its properties, ' it is
evident that its form and the equilibrium of the cohesive forces
would remain undisturbed : the tendency of the new angular
surface of the fluid water to contract would therefore be com-
pletely destroyed by the contact of a solid of equal attractive
force. If the solid were of smaller attractive force, the ten-
dency to contract would only be proportional to the difference
of the attractive forces or denmties, the effect of as many of the
attractive particles of the fluid being neutralised, as are equi-
valent to a solid of a like density or attractive power. For a
similar reason, the tendency of a given fluid, to contract the
sum of the surfaces of itself and a contiguous solid, will be
simply as the density of the solid, or as the mutual attractive
force of the solid and fluid. And it is indifferent whether we
consider the pressure produced by these supposed superficial
tensions, or the force acting in the direction of the surfaces to be
compared. We may therefore inquire into the conditions of
equilibrium of the three forces acting on the angular particles,
one in the direction of the surface of the fluid only, a second
in that of the common surface of the solid and fluid, and the
third in that of the exposed surface of the solid. Now, sup-
posing the angle of the fluid to be obtuse, the whole superficial
cohesion of the fluid being represented by the radius, the part
which acts in the direction of the surface of the solid will be
proportional to the cosine of the inclination ; and this force,
added to the force of the solid, will be equal to the force of the
common surfoce of the solid and fluid, or to the difference of
their forces ; consequently, the cosine added to t^ice the force
of the solid, will be equal to the whole force of the fluid, or to
the radius; hence the force of the solid is represented by
half the difference between the cosine and the radius, or by
half the versed sine ; or, if the force of the fluid be represented
by the diameter, the whole versed sine will indicate the force of
the solid. And the same result follows when the angle of the
fluid is acute. Hence we may infer, that if the solid have half
VOL. L 2 F

434 ON THE COHESION OF FLUIDS. No. XIX.

the attractiye force of the fluid, the surfaces will be perpen-
dicular; and this seems in itself reasonable, since two rect-
angular edges of the solid are equally near to the angular par-
ticles-with one of the fluid: and we may expect a fluid to rise
and adhere to the surface of every solid more than half as
attractive as itself; a conclusion which Clairaut has already
inferred, in a difierent manner, from prindples which he has
but cursorily investigated, in his treatise on the figure of the
earth.

The versed sine varies as the square of the sine of half the
angle : the force must therefore be as the square of the height
to which the fluid may be elevated in contact with a horizontal
surface, or nearly as the square of the number of grains ex-
pressing the apparent cohesion. Thus, according to the ex-
periments of Morveau, on the suppositions already premised,
we may infer that the mutual attraction of the particles of
mercury being unity, that of mercury for gold will be 1. or
more, lliat of silver about .94, of tin .90, of lead .81, of bis-
muth .72, of zinc .21, of copper .10, of antimony .08, of iron
.07, and of cobalt .0004 The attraction of glass for mercury
will be about one sixth of the mutual attraction of the particles
oi mercury : but when the contact is perfect, it appears to be
considerably greater.

Although the whole of this reasoning, on the attraction of
solids, is to be considered rather as an approximation than as a
strict demonstration, yet we are amply justified in concluding,
that all the phenomena of capillary action may be accurately
explamed and mathematically demonstrated from the general
law of the equable tension of the surface of a fluid, together
with the consideration of the angle of contact appropriate to
eveiy combination of a fluid with a solid. Some anomalies,
noticed by Musschenbroek and others, respecting in particular
the effects of tubes of considerable lengths, have not been
considered ; but there is great reason to suppose, that either
the want of uniformity in the bore, or some similar inaccuracy,
has been the cause of these irregularities, which have by no
means been sufficiently confirmed to afibrd an objection to
any theory. The principle which has been laid down respecting

No. XIX. ON THE C50HE8ION OF FLUIDS. 435

the contractile powers of the common sur&ce of a solid and
a fluid> is confirmed by an observation which I have made
on the small drops of oil which form themselves on water.
There is no doubt but that this cohesion is in some measure
independent of the chemical affinities of the substances con-
cerned : tallow, when solid, has a very evident attraction for
the water out of which it is raised ; and the same attraction
must operate upon an unctuous fluid to cause it to spread on
water, tHe fluidity of the water allowing this powerful agent to
exert itself with an unresisted velocity. An oil, which has
thus been spread, is afterwards collected, by some irregularity
of attraction, into thin drops, which the slightest agitation
again dissipates; their surface forms a very regular curve,
which terminates abruptly in a surface perfectly horizontal :
now it follows from the laws of hydrostatics, that Ae lower sur-
face of these drops must constitute a curve, of which the
extreme inclination to the horizon is to the inclination of the
upper surface, as the specific gravity of the oil to the difierence
between its specific gravity and that of water : consequently,
since the contractile forces are held in equilibrium by a force
which is perfectly horizontal, their magnitude must be in the
ratio that has been already assigned ; and it may be assumed
as consonant both to theory and to observation, that the con-
tractile force of the common surface of two substances is
proportional, other things being equal, to tjie difierence of their
densities. Hence, in order to explain the experiments of Boyle
on the efifects of a combination of fluids in capillary tubes, or
any other experiments of a similar nature, we have only to
apply the law of an equable tensiop, of which the magnitude is
determined by the difierence of the attractive powers of the
fluids.

I shall reserve some further illustrations of this subject for a
work which I have long been preparing for the press, and which
I flatter myself will contain a clear and simple explanation of
the most important parts of natural philosophy. I have only
thought it right, in the present Paper, to lay before the Royal
Society, in the shortest possible compass, the particulars of an
original investigation, tending to explain some facts, and

2 F 2

436 ON THE C!OHESION OF FLUIDS. No. XIX-

establish some analogies, which have hitherto been obscure and
unintelligible.

VIII. — Additional. Extracts from Laplace, with
Remarks.*

In an essay read to the Institute of France in December
1805, and published in 1806, as a supplement to the Meca-
nique Celeste, Mr. Laplace has advanced a theory of capillary
attraction, which has led him to results nearly similar to many
of those which are contidned in this paper. The coincidence is
indeed in some respects so striking, that it is natural, upon the
first impression, to inquire whether Mr. Laplace may not be
supposed either to have seen this essay, or to have read an
account of its contents in some periodical publication; but
upon further reflection, we cannot for a moment imagine a
person of so lugh and so deserved a reputation as Mr. Laplace,
to wish to appropriate to himself any part of the labours of others.
The path which he has followed is also extremely different from
that which I had taken ; several of the subjects which I had
considered as belonging to the discussion, have not occurred to
Mr. Laplace ; and it is much more flattering than surprising,
that, to an assembly of philosophers not extremely anxious to
attend to the pursuits of their contemporaries, investigations
should be communicated, by the most distinguished of their
members, as new and important, which had been presented, a
year before, to a similar society in this country. In order to
facilitate the comparison of the methods which have been
adopted, I shall insert here a translation of some parts of
Mr. Laplace's essay, which will also serve as an illustration of
the theory advanced in this paper; and I shall add some
remarks on the points in which those methods difier most.

«< I have considered," says Mr. Laplace, " in the tenth book of this
work, the phenomena derived from the refractive powers of transparent
bodies acting on light. This force is the result of the attraction of their
particles ; bat the law of this attraction cannot be detennined by the
phenomena, because they only require that it should be insensible at all

* The whole of thie article is an addition to the original memoir^— JStf.

No. XIX. ON THE COHESION OF FLUIDS. 437

sensible distances. Ail possible laws of attraction^ which fulfil this
condition, agree equally well with the difierent phenomena of refraction
indicated bj experience, the principal of which is the constant propor-
tion of the sine of refraction to that of incidence, in the passage of a ray
of light through a transparent body. It is only in this case, that this
kind of attraction has been subjected to an exact analysis. I shall now
submit to the consideration of mathematicians a second case, still more
remarkable than the first, on account of the variety and singularity of
the phenomena which depend on it, and which may be analysed with
equal accuracy ; this case is that of capillaiy action. The effects of
refiuctive powers belong to mechanics, and in particular to the theory
of projectiles ; those of capillary action relate to hydrostatics, or the
equilibrium of fluids, which are raised or depressed by its means, ac-
cording to certain laws, which I propose to explain."

I shall here take the liberty of observing, that the arguments
which I have formerly advanced in favour of the Huygenian
theory of light, would perhaps have occasioned some little hesi-
tation with respect to the action here supposed to be exerted
by transparent bodies on light, if they had ever been so fortu-
nate as to obtain Mr. Laplace's attention. Indeed an ^* attrac-
tion insensible at all sensible distances" would not explmn the
effects of what Newton calls inflection, which affects the rays
passing at a very considerable distance, at least as much as the
tenth or twentieth of an inch, on each side of an opaque sub-
stance, placed in a small pencil of light in a dark room.

** Clairaut is the fii-st, and has hitherto remained the only person,
that has subjected the phenomena of capillary tubes to a rigorous calcu-
lation, in his treatise on the figure of the earth. After having shown,
by arguments which are equally applicable to all the theories which have
been advanced, the inaccuracy and insufificiency of that of Jurin, he
enters into an exact analysis of all the forces which can contribute to
the elevation of a portion of water in a tube of glass. But his theory,
although explained with all the elegance peculiar to the excellent work
which contains it, leaves undetermined the law of the height of that
elevation, which is found from experiment to be inversely proportional
to the diameter of the tube. This great mathematician contents himself
with observing, that there must be an infinite variety of laws of attrac-
tion, which, if substituted in his formulas, would afibrd this conclusion,
'i'he knowledge of these laws is, however, the most delicate and the
most important part of the theory ; it b absolutely necessary for con-

438 ON THE COHESION OF FLUmS. No. XIX.

Decting together the different phenomena of capillary action ; and Clairaat
wonld himself have been aware of this necessity, if he had wished, for
example, to pass from capillary tabes to the sjiaces included between
two parallel planes, and to deduce from calculation the equality, which
is shown by experiment, between the height of ascent of a fluid in a
cylindrical tube, and its height between two parallel planes, of which
the distance is equal to the semidiameter of the tube ; a relation which
no one has yet attempted to explain. I endeavoured, long ago, to de-
termine the laws of attraction on which these phenomena depend ; some
later investigations have enabled me to demonstrate that they may all
be referred to the same laws which will account for the phenomena of
refraction, that is, to such as limit the sensible effect of the attraction to
an insensible distance ; and from these laws, a complete theory of capil-
lary action may be deduced."

It is time that Clairaut was the first that attempted to lay
the foundation of a theory of capillary action, but he is by no
means the only one that has made the attempt Segner
published, in the first volume of ihe Transactions of the Royal
Society of Gottingen, for 1751, an essay, in which he has gone
much further than Clairaut : it is true that he has made some
mistakes in particular cases ; but he begins, like Mr. Laplace,
from the eflects of an attraction insensible at all sensible
distances ; he ha^ demonstrated that the curvature of each point
of the surface of a fluid is always proportional to its distance
above or below the general level, and he has inferred, trom
earlier experiments, the true magnitude of this curvature at a
given height, both for water and for mercury, without material
error. We shall however find that the principles which Clairaut,
Segner, and Laplace have successively adopted, are insufficient
for explaining all the phenomena ; and that it is impossible to
account for them wiUiout introducing the consideration of a
repulsive force, which must indeed inevitably be supposed to
exist, even if its presence were not inferred from the efiects of
capillary action. "Attempts" have certainly been made to
explaiu the equality of the ascent of a fluid between the two
planes, and in a tube of which the radius is equal to their
distance : Mr. Leslie has made such an attempt, and with per-
fect snccess ; but, if I am not mistaken, the same explanation
had been given long before.

No. XIX. ON THE COHESION OF FLUIDS. 439

** Clairaut supposes, that a capillary tabe may exert a sensible action
on an infinitely narrow column of the fluid, situated in the axis of the
tube. In this respect, I am obliged to differ from him, and to agree
with Haoksbee, and with many other philosophers, in liking, that
capillary action, like refractive powers, and the forces of chemical affini-
ties, is only sensible at imperceptible distances. Hauksbee has observed
that when the internal diameters of several capillary tubes are equal, the
watec rises in them to the same height, whether they are very thin or very
thick. The cylindrical strata of glass, which are at a sensible distance
from the interior sur&ce, do not therefore contribute to the ascent of
the water, although each of them, taken sepcM^y, would cause it to
rise above its natural level. It is not the inl&position of the strata
which they surround, that prevents their action on the water ; for it is
natural lo suppose, that the force of capillary attraction is transmitted
through the substance of all material bodies, in the same manner as that
of gravitation ; this action is, therefore, only prevented by the distance
of the fluid from these strata ; whence it follows that the attraction of
glass for water is only sensible at insensible distances.

** Proceeding upon this principle, I have investigated the action of
a fluid mass, terminated by a portion of a concave or convex spherical
surface, upon a fluid colunm within it, contained in an infinitely narrow
cylindrical cavity or tube, directed towards the centre of the surface.
By this action I mean the pressure, which the fluid contained in the
tube would exert, in consequence of the attraction of the whole mass,
upon a flat basis, situated within the tube, perpendicular to its sides,
and at any sensible distance firom the external surface,^iaking this basis
for unity. I have shown that 'this action is either smaller or greater than
if the sur&ce were plane, accordingly as it is either concave or convex.
The algebraical formula, which expresses it, consists of two terms : the
first, which is much larger than the second, expresses the action of the
mass supposed to be terminated by a plane snr&ce ; and I conceive that
tills force is the cause of the suspension of mercuiy in the tube of a baro-
meter, at a height two or three times greater than that which is derived
from the pressure of the atmosphere, of the r^ractive powers of trans-
parent bodies, of cohesion, and of chemical affinities in general. The
second term expresses that part of the attraction, which is derived from
tlie curvature of the surface, that is, the attraction of the meniscus com-
prehended between that sur&oe and the plane which touches it This
action is either added to the former, or subtracted from it, accordingly
as the surfttce is convex or concave. It is inveisely proportional to
the radius of the spherical sur^u^e ; and it is indeed obvious, that, the
smaller the radius is, the greater is the meniscus near the point of
contact. Tliis second term expi esses the cause of capillar}* action, which

440 ON THE CSOHESION OF FLUIDS. No. XIX.

differs, in this respect, from the chemical affinities represented by the
first term."

It is indeed so " obvious," that the meniscus, which consti-
tutes the difference between a curved surface and a plane one,
is inversely proportional to the radius of curvature, that the
complicated calculations, which have led Mr. Laplace to this
conclusion, must be considered as wholly superfluous. The
attraction of the meniscus upon the evanescent column must be
confined to the edge which immediately touches the column,
extending only to an insensible distance on each side ; and the
situation of all the particles in this infinitely thin edge of the
meniscus, with respect to the column, being similar, whatever
the curvature may be, it b evident that their joint action must
be proportional to their number, that is, to the curvature of the
surface.

*' From these conclasions, relating to bodies which are terminated by
sensible portions of a spherical surface, I deduce this general theorem.
Whenever the attractive force becomes insensible at any sensible dis-
tance, the action of a body terminated by a curved surface, on an inter-
nal column, of infinitely small diameter, and perpendicular to the sur-
face at any point, is equal to the half sum of the actions, which would
be exerted on the same column by two spheres, having for their radii
the largest and the smallest of the radii of curvature at the given
point."

This theorem may be very simply inferred from the former,
by considering that, according to the principle laid down in the
second section of this essay, the sum of the thicknesses of the
evanescent meuiscoid, in any two planes passing through the
axis at right angles to each other, is equal to the sum of the
thicknesses of the two menisci formed by the largest and the
smallest radii of curvature ; consequently the sum of the whole
actions of these menisci must be twice as great as the action of
the roeniscoid.

*' By means of this theorem, and of the laws of the equilibrium of
fluids, we may determine the figure which must be assumed by a gra-
vitating fluid, inclosed in a vessel of any given form. We obtain from
these principles an equation of partial differences of the second order, the

No. XIX. ON THE COHESION OF FI.UI0S. • 441

int^pral of which cannot be found by any known method. If the figare
is such, as might be formed by the revolution of a curve jx>tind an axis,
the equation is reduced to common differences or fluxions, and its inte-
gral or fluent may be found very near the truth, when the sur&ce is very
small. I have shown in this manner, that, in very narrow tubes, the
sivface of the fluid approaches the nearer to that of a sphere, as the
diameter of the tube is smaller. If these segments are similar, in difler-
ent tubes of the same substance, the radii of their sur&ces will be
directly proportional to the diameters of the tubes. Now this simi-
larity of the spherical segments will easily appear, if we consider that
the distance, at which the action of the tube ceases to be sensible, is im-
perceptible ; so that if, by means of a very powerful microscope, it were
possible to make it appear equal to the thousandth part of a metre, it is
probable, that the same magnifying power would augment the appa-
rent diameter of the tube to several metres. The surface of the tube
may therefore be considered as nearly plane, within the limits of a circle
equal in radius to the distance at which its attraction becomes sensible ;
consequently the fluid within this distance will be elevated or depressed
with respect to the surface of the tube, almost precisely in the same
manner as if it were perfectly plane. Beyond this distance, the fluid
being subjected to no other sensible action than that of gravitation, and
that of its own attraction, the surface will be very nearly that of a
spherical segment, the marginal parts of which corresponding with
those of the surface of the fluid at the point which is the limit of the
sphere of the sensible activity of the tube, will be inclined very nearly
in the same angle to its surface, whatever its magnitude may be : hence
it follows, that all these segments will be similar.''

The ** near approadi " of the surface of a fluid in a very
small tube to a portion of a sphere is sufficiently obvious from
the fundamental principle, that the curvature is proportional to
the height above the general surface of the fluid ; for if the
diameter of the tube be small, this height will be so considerable,
that its variation at any part of the concave or convex surface
may be disregarded, and the curvature may consequently be
considered as uniform throughout the sur&ce. It is only upon
the supposition of a surface nearly approaching to a spherical
form, ^t Mr. Laplace has endeavoured to determine the
**' integral, very near the truth." He has deduced from the
expression, which indicates the curvature of the surface, another
which is simpler, and which might easily have' been inferred at
once from the uniform tension of the surface, as supporting at

442 ON THE COHESION OF FLUIDa No. XIX.

each point the weight of the portion of the fluid below it : he
has then supposed this weight to be the same as if the surface
were spherical, and has deduced from this supposition an ap-
proidmate expression, for the elevation corresponding to a given
angular position of the surface only. This formula is however
still only applicable to those cases, in which the surface may be
considered as nearly spherical ; and in these it is superfluous.
For example, if the surface of the mercury in a barometer be
depressed one twentieth of an inch, as it actually ijs in a tube
somewhat less than a quarter of an inch in diameter, Mr.
Laplace's formula fails so completely, as to indicate a concavity
instead of a convexity ; for a being the reciprocal of what I
have called the appropriate rectangle, and 9 being 50% the
term al^ becomes = 4, and makes the negative part of the
formula greater than the positive. When Mr. Laplace in-
vestigates the relation of the curvature and of the marginal
depression to the diameter of the tube, he simply considers the
whole surface as spherical ; but even on this supposition his
formula is by no means the most accurate that may be found,
and begins to be materially incorrect even when the diameter
of the tube amounts to one-fifth of an inch only. The formula
which I have already given in this paper is sufficiently accurate,
until the diameter becomes equal to half an inch ; but I shall
hereafter mention another, which comes much nearer to the
truth in all cases.

^' The comparison of these results shows the true cause of the ascent
or depression of fluids in capillary tabes, which is inversely proportional
to their diameters. If we imagine an infinitely narrow inverted siphon
to have one of its branches placed in the axis of the tube of glass, and
the other terminating in the general horizontal surface of the water in
the vessel, the action of the water in the tube on the first branch of the
siphon will be less, on account of the concavity of its surface, than the
action of the water of the vessel on the second ; the fluid must therefore
ascend in the tube, in order to compensate for this difference ; and, as it
has been shown, that the difference of the two actions is inversely pro-
portional to the diameter of the tube, the elevation of the fluid above
the general level must follow the same law.

*' If the sur&ce of the fluid within the tube is convex, as in the case
of mercury contained in a tube of glass, its action on the inverted siphon

No. XIX. ON THE COHESION OF FLUIDS. 443

will be greater than that of the fluid in the vessel ; the fluid must there-
fore be depressed in the tube, in proportion to ihe difierence, that is,
inversely in proportion to the diameter of the tube.

'* It appears, therefore, that the immediate attraction of a capillary
tube has no other eflect on the elevation or depression of the fluid con-
tained in it, than so far as it determines the inclination of the first por-
tion of the surface of the fluid, when it approaches the ^ides of the tube :
and that the concavity or convexity of the surface, as well as the mag^
nitude of its curvature, depends on this inclination. The friction of the
fluid, against the sides of the tube, may increase or diminish a little the
curvature of its surface, as we continually observe in the mercury of the
barometer; and in this case the capillary effects are increased or dimi-
nished in the same proportion. These eflects are also very sensibly
modified by the co-operation of the forces derived firom the concavity
and convexity of two difierent surfaces. It will appear hereafler, that
water may be raised, in a given capillary tube, to a greater height above
its natural level in this manner, than when the tube is immersed in a
vessel filled with that fluid."

It would perhaps be more correct to say in this case ^^ above
its apparent level :" for the real horizontal surface must here
be considered as situated above the lower orifice of the tube,
the weight of the portion of the fluid below it being as much
supported by the convexity of the surface of the drop, as if it
were contained in a vessel of any other kind.

<* The fluxional equation of the surface of a fluid, inclosed in a capil-
lar)" space of any kind, which may be referred to an axis of revolution,
leads to this general result, that if a cylinder be placed within a tube,
so that its axis may coincide with that of the tube, the fluid will rise in
this space to the same height, as in a tube of which the radius is equal
to this distance. If we suppose the radii of the tube and of the cylin-
der to become infinite, we obtain tlie case of a fluid contained between
two parallel vertical planes, placed near each other. The conclusion is
confirmed in this case by the experiments which were made long ago
in the presence of the Royal Society of London, under the inspection of
Newton, who has quoted them in his Optics ; that admirable work, in
which this profound genius, looking forwards beyond the state of science
in his own times, has suggested a variety of original ideas, which the
modem improvements of chemistry have confirmed. Mr. Hauy has
been so good as to make, at my request, some experiments on the case
which constitutes the opposite extreme, that is, with tubes and cylinders
of a very small diameter, and he has found the conclusion as correct in
this case, as in the former."

444 ON THE COHESION OF FLUIDS. No. XIX.

If indeed we may be allowed to place any confidence in the
fundamental principle of an equable tension of the surface of
the fluid, an equal length of the line of contact of the solid and
fluid supporting in all cases an equal weight, these results follow
of necessity, without any intricacies of calculation whatever.

'* The pheDomeDE exhibited by a drop of a fluid, moving, or sus-
pended iD eqailibrium, either in a conical capillary tabe, or between two
planes, inclined at a small angle to each other, are extremely proper to
confirm our theory. A small column of water, in a conical tube, open
at both ends, and held in a horizontal position, will move towards the
vertex of the cone ; and it is obvious that this must necessarily happen.
In fact, the snr&oe of the column is concave at both ends, but the radius
of this curvature is smaller at the end nearer the vertex than at the op-
posite end ; the action of the fluid upon itself is therefore less at the
narrower end, consequently the column must be drawn towards this
side. If the fluid employed be mercury, its surface will be convex, and
the radius of curvature will still be smaller towards the vertex than
towards the base of the cone ; but, on account of its convexity, the
action of the fluid upon itself will be greater at the narrower end, and
the column must therefore move towards the wider part of the tube.

" This actioif may be counterbalanced by the weight of the column,
60 as to be held in equilibrium by it, if we incline the axis of the tube
to the horizon. A very simple calculation is sufiicient to demonstrate
that if the length of the column is inconsiderable, the sine of the incli-
nation of the axis must be inversely proportional to the square of the
distance of the middle of the column flrom. the summit of the cone ; and
this law is equally applicable to the case of a drop of a fluid placed
between two planes, which form a very small angle with each other,
their horizontal mar^ns being in contact. These residts are perfectly
conformable to experiment, as may be seen in the 31st query of New-
ton's Optics. This great geometrician has endeavoured to explain them,
but his explanation, compared with that which has been here advanced,
serves only to show the advantages of a precise and mathematical inves-
tigation."

Mr. Laplace's superior skill in the most refined ^' mathema-
tical investigations '* might perhaps have enabled him to make
still more essential improvements, if it had been employed on
some other subjects of natural philosophy ; but his explanation
of these phenomena being exactly the same as that which I
had already published, in an essay not containing, in its original

No. XIX. ON THE COHESION OF FLUIDS. 445

8tate» any one mathematical symbol, it is obvious that the inac-
curacy of Newton's reasoning did not depend upon any defi-
ciency in his mathematical acquirements.

*• It may be sbown by calculation, that the sine of the inclination of
the axis of the cone to the horizon will be very nearly equal to the frac-
tioD of which the denominator is the distance of the middle of the drop
from the sammit of the cone, and the numerator the height to which
the fluid would rise in a cylindrical tube, of a diameter equal to that of
the cone at the middle of the column. If the two planes, inclosing a
drop of the same fluid, form with each other an angle, equal to that
which is formed by the axis of the cone and its sides, the inclination of
a plane, bisecting this angle, to the horizon, must be the same as that
of the axis of the cone, in order that the drop may remain in equilibrium.
Hauksbee has made, with very great care, an experiment of this kjnd,
which I have compared with the theorem here laid down ; and the near
agreement between the experiment and the theorem is amply suflicient
to confirm its truth."

If the height at which the fluid would stand, in a tube of the
diameter of the upper end of the column, be A ; the distance of
this end from the vertex of the cone being jr, and the length of
the column y, the height corresponding to the remoter end will

be — — , and the diflFerence of the heights h ^ = -^>

which must be the diflerence of the heights of the ends of the
drop, in order that it may remain in equilibrium ; but this
hdght istoyasAtox + y, consequently the axis of the tube
must be inclined to the horizon in an angle, of which the sine

is exactly — — ; the denominator being the distance of one end

of the vertex, and the numerator the height at which the fluid
would stand in a tube, of which the diameter is equal to that
of the colunm at the other end.

** This theoiy afibrds us also an explanation of another remarkable
phenomenon, which occurs in experiments of this nature. If a fluid be
either elevated or depressed between two vertical and parallel planes, of
which the lower ends are immersed in the fluid, the planes will tend to
approach each other. It is shown, by calculation, that if the fluid is
elevated between them, each plane is subjected to a pressure, urging it
towards the other plane, equal to that of a column of the same fluid, of
a height equal to the half sum of the elevations of the internal and ex-

446 ON THE COHESION OF FLUIDS. No. XIX.

ternal lines of contact, of the sur&ce of the fiaid with the plane, above
the general level, and standing on a base equal to a part of the plane
included between these lines. If the floid is depressed between the
planes, each of them will be forced inwards, by a pressure equal to that
of a column of the same fluid, of which the height is half the sum of
the depressions of the lines of contact of the external and internal sur-
fi^es of the fluid with the plane, and its base the part of the plane
comprehended between those lines."

In another part of his essay^ Mr. Laplace asserts that " this
force increases in the inverse ratio of the distance of the
planes." If this is not an error of the press, or of the pen, it can
only mean that the force increases as the distance diminishes ;
for the magnitude of the force is not simply in the inverse ratio
of the distances, but very nearly in the inverse ratio of their
squares, as I have already observed.

'' Since it has been hitherto usual with natural philosophers, to con-
sider the concavity and convexity of the surfaces of fluids in capillary
spaces, as a secondary efiect of capillary attraction only, and not as the
principal cause of phenomena of this kind, they have not attached much
importance to the determination of the curvature of these sorfeces. But
the theory, which has been here advanced, having shown that all these
phenomena depend principally on the curvature, it becomes of conse-
quence to examine it. Several experiments, which have been made
with great accuracy by Mr. Hauy, have shown, that in capillary tubes
of glass, of very small diameters, the concave surfaces of water and of
oils, and the convex surfaces of mercury, difler very little flrom the
form of a hemisphere."

Mr. Laplace informs us that MM. Haiiy and Tremery made
at his request several experiments, in which the mean ascent of
water, in a tube one thousandth part of a metre in diameter,
was 13.57 thousandths, and that of oil of oranges 6.74 The
product of the diameter and the height of ascent of water is
.039371 X .534 = .021 £. i., which is little more than half as
much as I have assigned for this product from the best experi-
ments of many other observers. Probably both these experi-
ments, and those of Newton or Hauksbee, were made with
tubes and plates either a little greasy or too dry ; and Mr.
Haiiy might be the more readily satisfied with the first results
that he obtained, from finding them agree nearly with those of

No. XIX. ON THE CX)HESION OF FLUIDS. 447

Newton, which Mr. Laplace wished to compare with them.
These gentlemen also found the depression of mercury in a
tube of the same diameter .2887 E. i., the product being
.011379 instead of .015, which is the ultimate product inferred
from Lord Charles Cavendish's experiments of a similar nature.
The observation of Mr. Haiiy, on the curvature of the surface
of mercury in a tube, is also fsx from being accurate ; Mr.
Laplace himself asserts that the angular extent of the surface
must £blI1 short of that of a heibisphere more or less, accord-
ingly as the tube has more or less attraction for the fluid ; and
it is easy to show that glass has a very considerable attraction
for mercury. The method that I took to ascertain the angle,
formed by the sur&ce of the mercury, with the side of the
tube, was to observe in what position the light reflected from it
began to reach the eye ; and I have every reason to think,
from the comparison of a great variety of experiments of dif-
ferent kinds, that the angle which I have assigned is very near
the truth.

I have lately repeated my calculations of the depression of
mercury, in barometer tubes of considerable diameter, with
great care, and by different methods. I had before formed a
table, by means of diagrams, which I had actually constructed
for each case, upon a Bu£Bciently accurate approximation : I
have now followed nearly the same steps in calculating, by
means of tables of sines and cosines, the precise form of the sur-
face in a variety of cases. Beginning from the vertex of the
curve, I have determined the mean curvature for every small
arc, from the approximate height of its middle point ; calcu-
lating with the assistance of a series of difierences, the normal
of the curve at each step for the same point, in order to find the
transverse curvature. I have also pursued, in some cases,
in order to confirm these calculations, a method totally difierent,
finding the mass of the quantity of fluid to be supported by the
tension of the surface at each concentric circle, and inferring
from its magnitude the inclination of the curve to the horizon ;
taking the height of the external circumference of each portion,
thus calculated, for the mean height; a supposition which
nearly compensates for the omission of the curvature of its sur-

448

ON THE OOHESIOX OF FLUIDS.

No. XIX.

&ce. But the accumulated effect of this curvature'becomes very
sensible in the vertical height of the surface, and I have there-
fore allowed for it, upon the supposition of a simple curvature
varying with the height; but this correction, for want of
including the effect of the variation of the transverse curvature,
is still a little too small ; the horizontal diameter of the surface^
however, agrees extremely well with the former mode of calcu-^
lation. In order that the results of these investigations may be
the more easily compared with each other and with experiment,
I shall insert some specimens, by means of which, if it be
required, the curves may be very correctly delineated.

Central Depression

.007.

FIB8T METHOD, BT THE CtTBVATUBE.

8E0OKD METHOD, BY THE TENSION.

Are.

Horixontal
ordinate.

Are.

Horiioiital
ordinate.

Deprenion.

.00000

.00700

.00°

.00000

.00700

.02444

.00721

.02

.02000

•00714

•04758

.00782

.04

.04000

.00757

•06651

.00865

.06

.05999

.00830

•08338

.00968

.08

.07997

.00939

.09791

.01082

.10

.09993

.01101

•11049

•01203

•12

.11985

.01302

.12153

•01329

.14

.13971

.01566

.13146

.01458

.16

.15948

.01909

•14022

.01589

.18

.17908

.02353

•14814

.01721

.20

.19842

.02922

.16177

.01986

.22

.21732

.03653

•17338

.02254

.24

.23550

.04530

•18344

.02524

.26

.25039

.05707

•19229

.02793

•2705

.25740

.06460

.20012

.03063

.21603

.03722

.22869

.04381

.23895

.05033

.24731

.05676

.25420

•06307

.25986

.06911

Central D

epression

.05.

FIBffT METHOD.

SEOOND METHOD.

Are.

ordiuftto.

Depnaion.

Are.

Horisontal
ordinate.

Depre^on.

0°

•00000

•05000

• 00°

.00000

•05000

.00349

.05003

.01

.01000

.05025

.00697

.05012

.02

•01999

.05101

.01044

.05027

.03

.02994

•05229

•01388

.05048

.04

.03982

•05409

.01729

.05075

.05

.04961

.05644

.02068

.05107

.06

•05926

.05938

No. XIX.

on THE COHESION OF FLUIDS.

449

FIRST XSTHOD— 00ta«IUA/.

Aic

9
10
12
14
16
18
20
25
30
35
40
45
50

Are.

5
10
15
20
25
30
35
40
45
50

Horizontal
ordinate.

.02402
.02731
.03056
.03375
.03995
.04589
.05157
.05697
.06209
.07363
.08365
.09224
.09958
.10581
.11106

DeproMion.

.05145
.05189
.05237
.05291
.05411
.05543
.05696
.05861
.06037
.06515
.07036
.07583
•08146
.08717
.09289

8E00ND MSTROD^wntmued.

Arc.

.07^

.08

.09

.10

.11

.12

.1214

Horiumtal
ordinate.

.06873
.07796
.08688
.09540
.10342
.11080
.11173

Depreaion.

•06294
.06718
.07212
.07783
.08436
.09170
.09280

3. Central Depression .14.

FIBST MKTHOD.

Horiiontal
ordinate.

.00000
.00623
.01234
.01832
.02405
.02950
.03459
.03931
.04361
.04749
.05091

OepreMioa.

.14000
.14027
.14108
.14240
.14421
.14646
•14911
.16211
.15541
.15897
.16270

Arc.

.00^

.01

.02

.03

.04

.05

.0655

SECOND METHOD.

Horiiontal
ordinate.

.00000
.01000
.01990
.02950
.03857
.04686
.05078

Depreation.

.1400
.1407
.1428
.1464
.1514
.1580
.1621

For representing the depression, thus determined, in a for-
mula capable of expressing it at once, in terms of the diameter
of the tube, I have deduced an approximate determination from
the supposition of a spherical sur&ce, and corrected it, by
comparison with the results of these calculations, so as to agree
with them all, without an error of one two-thousandth of an inch,
in the most unfavourable of the five cases compared. The

theorem is, first, e = ^'^ ^^ , which is nearly half the versed

sine of a spherical surface, and then/= ^-j- - |e — 14.56*,

which shows the central depression without any sensible error.

I have also found a formula, which expresses the difference
between the central and marginal depression, so that an ob-
servation on the height of the barometer may be corrected,
with equal accuracy, whether the elevation of the highest

VOL. I. . 2 G

450

ON THE COHESION OF FLUIDS.

No. XIX.

or lowest point of the surface has been measured, provided
that the tube be of moderate dimensions. This formula is^ =

15 (J^^^Z% + 18 • ^f ^ ^^^^ ^^^y '^^g^' ^* ^^^'^ ''^'''^^

some further correction, g being ultimately too great by .0069.
The results of these formulas are compared, in the first of the
following tables, with those of the calculations at large ; and in
the second, they are reduced into a form more immediately
applicable to practice, and are compared also with the table
published by Mr. Cavendish.

diameter.

True central
depression.

Fonnl.

True additional
depression at Form 9.
the margin.

.5197

.007

.0071

.0621

.0622

.3187

.025

.0250

.0535

.0534

.2221

.050

.0498

,0429

.0432

.1468

.090

.0905

.0313

.0311

.1018

.140
Obser

.1396

.0227

.0226

Diameter.

ved central

True central

T^e marginal

depr«wion.

1.00

.0022

.90

.0023

.80

.0026

.70

.0032

.60

.005

.0045

.0680

.50

.007

.0074

.0691

.45

.0100

.0703

.40

.015

.0139

.0722

.35

.025

.0196

.0753

.30

.036

.0280

.0798

.25

.050

.0404

.0872

.20

.067

.0589

.0989

.15

.092

.0880

.1196

.10

.140

.1422

.1646

.05

.2964

.3083

By continuing the calculations of the figure of some of these
curves to an arc of 90°, I have adapted them to the surface of
water contained in a cylindrical tube ; but in this case the scale
must be supposed to be augmented in the proportion of 1 to V 2*
The additional numbers stand thus in abstract.

1. Central Depression .025.

Are.

ordinate.

Depresrion.

Are.

Horiiontal
ordinate.

Deprassioi

0°

.00000

.02500

50°

.15934

.07847

.06214

.03023

.16768

.09039

.10280

.04097

.17296

.10169

.12969

.05340

.17580

•11228

.14793

.06606

.17665

•12203

No. XIX.

OS THE COHESION OF FLUIDS.

451

2. Central Depression .05.

Arc.

HorizonUl
oidinate. ^

Deprfflsion.

Alt.

ordinate.

DepitMioi

.00000

.05000

50^

.11105

.09289

.03375

.05291

.11911

.10414

.06209

.06037

.12494

.11492

.08365

.07036

.12769

.12518

.09958

.08146

.12853

.13470

3. Central Depression

.09.

Arc

HorisonUl
ordinate.

l)«pr«Mion.

Ak.

Horisontal
ordinate.

UepreMicn

0*"

.00000

.09000

50'

.07340

.12133

.01904

.09042

.08022

.13106

.03662

.09366

.08475

.14077

.05172

.10337

.08727

.15017

.06397

.11192

.08804

•15904

Hence, for water, we have the central elevation .035355,
.07071, and .12728, and the marginal elevation .17258, .19050,
and .22495, in tubes of which the diameters are .49964, .36354,
and .2490 respectively. The difference of the elevations is

expressed nearly by A = ^ sl\o1^d^T\Oi}d*J^ ^*^^^*^ ^ correct
in the extreme cases on both sides, and becomes, when d is .25,
and .5, .098, and .136 respectively, instead of .0977 and .137 ;
and when cf = 1, A = .141.

" Clairaut," says Mr. Laplace, ** has made this singular remark ;
that if the law of the attraction of the matter of the tube, for the fluid,
differs only in its intensity from that of the attraction of the particles of
the fluid among themselves, the fluid will be elevated above the level, as
long as the intensity of the first of these forces exceeds half that of the
second. If it be exactly half as great, it may easily be shown, that the
sur&ce of the fluid in the tube will be horizontal, and that it will not be
raised above the level. If the two forces be equal, the surface of the
fluid will be concave and hemispherical, and it will be elevated within
the tube. If the intensity of the attraction of the tube be wholly wantr
ing or insensible, the surface of the fluid will be hemispherical, but it
will be convex and depressed. Between these two limits, the surfiw^
will be that of a segment of a sphere, and it will be eitlier concave or
convex, accordingly as the intensity of the attraction of the matter ol*
the tube for the fluid is greater or less, than half of that of the mutual
attraction of the particles of the fluid."

These conclusions are in all probability nearly correct with
respect to very small tubes ; but it is remarkable that they
are not fairly deducible from Mr. Laplace's principles, nor

2 G 2

452 ON THE COHESION OF FLUIDS. No. XIX.

from those of Clairaut, whose steps he has followed ; and that
the expression, which he has derived from them, as indi-
cating the condition of equilibrium of the surface of a fluid
inclined to that of a solid, implies, by including an impossi-
bility, that such an equilibrium cannot subsist. This equation
requires that the attraction of the fluid, contained between the
surface and its extreme tangent, be more than equal to the
difference of the attraction of the two rectangular portions
composing the flat solid, and one similar portion of the fluid,
reduced only in the ratio of the sine of the angle occupied
by the termination of the fluid, to the radius : but it is very
evident that the action of the portion of the fluid, thus cut off
by the tangent, must be utterly evanescent, in comparison
with the other forces concerned, especially if we consider that
the surface of the fluid, as well as of that of the tube, within the
distance " of the sphere of activity of the attraction " is, to use
Mr. Laplace's terms, " almost absolutely plane." There can
therefore be no equilibrium upon these principles, when the
density of the solid is greater or less than half that of the fluid,
unless the surface of the fluid have a common tangent with that
of the solid : while, on the other hand, when the densities are
in this proportion, the surface will remain in equilibrium in any
position ; the action of the fluid being always proportional to
the chord of its angular extent, and composing, when combined
with that of the solid, a result perpendicular to the surface. If
Mr. Laplace had attempted to confirm or to confute my reasoning,
respecting the mutual attractions of solids and fluids, he would
probably have discovered the insufficiency of these principles,
and would perhaps have been induced to admit my explanation
of the foundation of the laws of superficial cohesion, as derived
from the combination of an attractive with a repulsive force^
varying according to a different law.

'' If the intensity of the attraction of the tube for the fluid exceeds
that of the attraction of the fluid for its own particles, I think it pro-
bable that, in this case, the fluid attaching itself firmly to the tnbe,
forms of itself an interior tube, which alone raises the fluid, so as to
make its sur&ce a concave hemisphere. It may reasonably be conjec-
tured, that this is the case with w^ater and with oils, in tubes of glass.

" The elevation of fluids between two vertical planes, which form

No. XIX. ON THE COHESION OF FLUIDS. 453

very small angles with each other, and their discharge through capillary
siphons, present a variety of phenomena, which are so many corollaries
from my theory. On the whole, if any person will take the trouble of
comparing it with the numerous experiments which have been made
on capillary action, he will see that the results of these experiments,
when made with proper precaution, may be deduced from it, not by
vague considerations, which always leave the subject in uncertainty, but
by a series of geometrical arguments, which appear to me to remove
every doubt respecting the truth of the theory. I wish that this appli-
cation of analytical reasoning, to one of the most curious departments of
natural philosophy, may be thought interesting by mathematicians, and
may induce them to make further attempts of a similar nature. Besides
the advantage of adding certainty to physical sciences, such investiga-
tions tend also to the improvement of the mathematics themselves, since
they frequently require the invention of new methods of calculation."

It mu6t be confessed that, in this country, the cultivation of
the higher branches of the mathematics, and the invention of
new methods of calculation, cannot be too much recommended
to the generality of those who apply themselves to natural phi-
losophy ; but it is equally true, on the other hand, that the
first matbematicians on the continent have exerted great inge-
nuity in involving tiie plainest truths of mechanics in the
intricacies of algebraical formulas, and in some instances have
even lost sight of the real state of an investigation, by attending
only to the symbols, which they have employed for expressing
its steps.*

* Laplace published, in 1807, a second * Supplement i la Throne de TAction
capiilaire,' in which he notices in the following terms the researches of Dr. Young: —
^ Lorsqne je m'occapais de oet objet, M. Thomas Young en faisait pareillemcnt le sujot
de recherches ingenieufles qu'il a ins^r^es dans les * Tranbactions Philosophiques'
pour Tannee 1805. En coroparant, avec Segner, la force capillaire, i la teiuiou d'unc
snr&ce, qui envelopperait les liquides, et en appliquant i cette force les resultats
connus sur la tension des siirfaceti, il a reconnu qu'il fallait avoir e'gaixt i la conrbure
des surfaces liquides, dans deux directions perpendiculaires entre elles ; il a de plus
suppose que ces surfaces, pour un mSme liquide, ooupent sous le m^me angle les parois
des tubes fonne's de la mJme mati^re, quelle que soit d'ailleurs leur figure ; ce qui,
comme on Ta vu, oesse d'etre exact auz eztremites de ces parois. Mais il n'a pas
tente, comme Segner, de deriver ces hypotheses de la loi de Pattraction des molecules,
de'croissante avec une extreme rapidite' ; ce qui e'tait indispensable pour les re'aliser.
EUes ne pouvaieot V6ixe que par une demonstration rigoureuse, pareille k celles que
nous avons donne'es dans la premiere methode i laquelle les explications de S^;ner et
de M. Thomas Young se rattachent, comme celle de Jurin se rattache i la seconde
methode."

These Supplements were made the subject of a Review by Dr. Young, in the fiist
number of the 'Quarterly Review,' for February, 1809. As tliis critique, however,
contains no observations of importance beyond those which are embodied in the
preceding additions to this Essay, it has not been thought necessar)- to reprint it.—
Note by the Editor,

454 COHESION. No. XX.

No. XX.

COHESION.*

From the Sapplement to the Encyclopsedia Britanuica.

The corpuscular forces, on which the mechanical properties of
the aggregates of matter depend, have been in some measure
considered, as far as they relate to solids, in the articles Bridge
and (Carpentry of this Supplement (Nos. LII. and LIII. in
Vol. II. of this work) : there are, however, other modifications of
these forces, which are principally exemplified in the Cohesion
OF Fluids (No. XIX.) ; and which afford us a series of pheno-
mena, highly interesting to the mathematician, on account of the
difficulty of investigating their laws, and of considerable
importance to the natural philosopher, from the variety of
forms in which they present themselves to his observation.

Section I. — Fundamental' Properties of the Cohesion of a
Single Fluid,

The three states of elastic fluidity, liquidity, and solidity, in
all of which the greater number of simple bodies are capable of
being exhibited at different temperatures, are not uncommonly
conceived to depend on the different actions of heat only, giving
a repulsive force to the particles of gases, and simply detaching
those of liquids, from that cohesion with the neighbouring
particles, which is supposed to constitute solidity. But these
ideas, however universal, may be easily shown to be totally
erroneous : and it will readily be found, that the immediate

* There appears in the same Supplement, under the title of * Elevation of Fluids,'
an article on the same subject by Mr. Ivory. It reproduces, under a more simple form,
the investigations of Laplace, and almost entirely ignores the previous as well as the
nearly contemporary researches of Dr. Young. " If the truth, '* says he, ** is to be
told, it may be affirmed, that reckoning back from the present time to the specu-
lations of the Florentine Academicians, the formula of Laplace and the remark of
Professor Leslie, relating to the lateral force, are the only approaches which have
been made to a sound physical account of the phenomena." The article in the text
was written in the summer of 1816.— Note by the Editor.

No. XX. COHESION. 455

effect of heat alone is by no means adequate to the explanation
of either of the changes of form in question.

There can never be rest without an equilibrium of force : and
if two particles of matter attract each other, and yet remain
without motion, it must be because there exists also a repulsive
force, equal, at the given distance, to the attractive force. If we
imagine the atoms of matter to be impenetrable spheres, only
resisting when their surfaces come into actual contact, it would
follow, that the degree of repulsive force exerted at the same
distance must be capable of infinite variation, so as to counter-
balance every possible modification of the attractive force, that
could operate between the particles. In this there would be no
mathematical absurdity, and it may sometimes even be conve-
nient to admit the hypothesis as an approximation: but we
know from physical considerations that the actual fact is other-
wise. The particles of matter are by no means incompressible :
the repulsion varies indeed very rapidly when they approach
near to each other ; but the distance of the particles, and the
density of the substance, must inevitably vary, in some finite
degree, from the effect of every force that tends to produce
either compression or expansion.

In elastic fluids, the law of the repulsive force of the particles
is perfectly ascertained ; and it has been shown to vary very
accurately in the inverse ratio of their mutual distances. It is
natural to inquire whether this repul^ve force, continued
according to the same law, would be capable of affording
the resistance exhibited by the same bodies in a liquid or solid
form, and holding the cohesive force in equilibrium : but in
order to answer this question, it would be necessary to determine
the law of the variation of the cohesive force with the variation
of the density. Now if this force extended to all particles
within a given distance of each other, whatever the density
might be, the number of particles similarly situated within tlie
sphere of action being as the density, and each one of this
number being attracted by an equal number, the whole cohesion
urging any two particles to approach each other would obviously,
as Laplace has observed, be as the square of the density : but
since this cohesive force would increase, with the increase of

456 COHESION. No. XX

density accompanying compresdon, more rapidly than any
repulsive force like that of elastic fluids, there could never be
an equilibrium between forces thus constituted : for, as Newton
has justly remarked, the force of repulsion must be supposed to
affect the particles immediately contiguous to each other only,
their number not increasing with the density. Nor is there
any reason to infer, from the phenomena of cohesion, that this
force extends to a given minute distance, rather than to a given
number of particles, as that of repulsion appears to do. It
would indeed be possible to assign a law for the variation of
cohesion, which would reduce the repulsion of liquids and of
elastic fluids to the action of the same force, without any other
modification than that which depends on the mutual distance of
the particles ; but this law is in itself so improbable, that it
cannot be considered as affording an admissible explanation of
the phenomena; for it would be required that the force of
cohesion should diminish, instead of increasing, with every
increase of density, and with a rapidity nineteen times as great
as the repulsion increased. For the height of the modulus of
elasticity of all kinds of gaseous substances remaining unaltered
by pressure, that of steam would still be only one twentieth as
high as the modulus of elasticity of water, even if the steam
were compressed by 1200 atmospheres; and the resistance to
any minute change of dimensions would be twenty times as
great in water as in steam of equal density, and the variation of
tiie repulsion would be in the same proportion. It is therefore
simplest to suppose the repulsion itself to be also twenty times
as great, and the cohesion little or not at all altered by the
effect of a slight compression or extension : and we shall have
no difficulty in imagining this abrupt change in the magnitude
of the repulsive force to depend on an increase of the number
of particles to which it extends ; supposing that when cohesion
be^ns to affect them, this number becomes four or five times as
great as before, and that it is not further increased by a greater
increase of density ; although, like the distance to which the
force of cohesion itself extends, it may be liable to some modi-
fication from the effects of a change of temperature. Thus it
is probable that the number of particles co-operating, both in

No. XX. COHESION. 457

repulsion and in cohesion, is diminished by the effect of heat;
for the diminution of the elasticity of a spring is much more
than proportional to the expansion of its substance, although
the primitive repulsive force of the single particles may very
possibly be as much augmented by an elevation of temperature
in this case as in that of an elastic fluid : the cohesive powers of
liquids are also diminished by heat, and indeed in a considerably
greater degree than the stiflbess of springs, although there can
be no doubt that there is a considerable analogy in these
changes. However this may be, it appears that the force of
cohesion cannot be supposed to vary much with the density,
and it is therefore allowable to consider it as constant, at all
distances, as far as its action extends ; while that of repulnon,
though it may operate in some degree at distances somewhat
greater, may still be considered, on account of its greater inten-
»ty at smaller distances, as equivalent to a resistance terminating
at a more minute interval than that to which the action of cohe-
sion extends.

The distance at which cohesion commences between the par-
ticles of gaseous fluids appears to depend entirely on the
temperature, and for any one fluid it is generally reduced
to one half by an elevation of about 100^ of Fahrenheit. In
whatever way the particles are caused to approach nearer than
this distance to each other, they become subject to the action of
this force, and rush together with violence, and with a great
extrication of heat, until the increased repulsion affords a
sufficient reostance to the cohesion, and the gas is converted
into a liquid. Superficial observers have sometimes imagined,
that liquids possessed little or no cohesion ; and it has generally
been supposed that their cohesive powers are far inferior to
those of solids. But that all liquids are more or less cohesive,
is sufficientiy shown by their remaining attached, in small
portions, to every substance capable of coming into intimate
contact with them, in opposition to the effect of gravitation, or
of any other force : and the cohesion of mercury is still more
fully exemplified by the well known experiment of a column,
standing at a height much exceeding that of the barometer,
when it has been brought, by strong agitation or otherwise, into

458 COHESION. No. XX.

perfect contact with the summit of the tube, and is then raised
into a vertical position ; the summit of the tube supporting, or
rather suspending the upper parts, and each stratum the stratum
immediately below it, with a force determined by the excess of
its height above that of the column equivalent to the atmosphe-
rical pressure. The perfect equality of the cohesion of a given
substance in the states of solidity and liquidity, appears, how-
ever, only to have been asserted in very modern times ; and
the assertion has only been confirmed by a single observation
of the sound produced by a piece of ice, compared with the
elasticity exhibited in Canton's experiments on the compres-
sibility of water ; the results demonstrating that the resistance
is either accurately or very nearly equal in both cases.

The real criterion of solidity is the lateral adhesion, which
prevents that change of internal arrangement, by which a fluid
can alter its external dimensions without any sensible difference
in the mutual distances of its particles taken collectively, and
consequently without any sensible resistance from the force of
cohesion. It is probable that this lateral adhesion depends
upon some symmetrical arrangement of the constituent parts of
the substance, while fluidity requires a total independence of
these particles, and an irregularity of situation affording a
facility of sliding over each other with little or no fnction.
The symmetry of arrangement, when continued uniformly to a
sensible extent, is readily discoverable by the appearance of
crystallization ; but there are several reasons for supposing it to
exist, though with perpetual interruptions, in more uniform
masses, or in amorphous solids. It is obvious that the lateral
adhesion, confining the particles so as to prevent their sliding
away, performs an office like that of the tube of a barometer to
which the mercury adheres, or like that of the vessels employed
by Canton and Zimmerman for confining water which is com-
pressed ; and enables the cohesive and repulsive powers of the
substances to be exhibited in their fiill extent Nor can we
obtain any direct estimate of these powers from the slight
cohesion exhibited, in some circumstances, by liquids in contact
with the surface of a solid which is gradually raised, and
carries witli it a certain portion of the liquid ; an experiment

No. XX. COHESION. 459

which had been often made, with a view of determining the
mutual attractions of solids and fluids, but which was first cor-
rectly explained, as Laplace observes, by our countryman Dr.
Thomas Young, from its analogy with the phenomena of capil-
lary tubes.

There are, however, still some difficulties in deducing these
phenomena from the elementary actions of the forces concerned,
whatever suppositions we may make respecting their primitive
nature. The intermediate general principle of a hydrostatic
force or pressure, proportional to the curvature of the surface,
had been employed long ago by Segner, and had been consi-
dered by him as the result of corpuscular powers extending to
an insensible distance only. But Segner's reasoning on this
point is by no means conclusive, and he has very unaccountably
committed a great error, in neglecting the consideration of the
effects of a double curvature. There is also an oversight in
sotne of the steps of the demonstration attempted by Dr. Young
in his Lectures^ which has been pointed out by an anonymous
writer in Nicholson's Journal: and Mr. Laplace's final equation
for determining the angle of contact of a solid and a liquid,
which Dr. Young had first shown to be constant, has been con-
sidered as completely inaccurate, and as involving an impossi-
bility so manifest, as to destroy all confidence in the theory
from which it was deduced. A demonstration, which appears
to be less exceptionable, was lately published in the Philo-
sophical Magazine ; and it may serve with some further illus-
trations for the present purpose.

It is only necessary to consider the actions of such of the
particles of the liquid, as are situated at a distance from the
surface shorter than that, to which the cohesive force extends ;
for all those which are more internal, must be urged equally in
all directions by the actions of the surrounding particles. Now
it will readily be perceived, that the first or outermost stratum
of particles will cohere very weakly with the stratum below it,
having only its own attraction to bind it down ; that the second
will be urged by a force nearly twice as great ; and that the
cohesion will gradually augment, by increments continually
diminishing, until we arrive at the depth of the whole interval

460 "" COHESION. No. XX.

to which the force extends: and below this it will remain
constant, the number of particles within the given distance not
undergoing any further change. It has been observed by Mr.
Laplace, that this partial diminution of ^e density of the
surface is likely to be concerned in facilitating the process of
evaporation ; and it has been cursorily suggested in another
quarter that the polarisation of light by oblique reflection may
be in some measure influenced by this gradation of density.
But its more immediate efiect must be to produce that uniform
tension of tlie surface, which constitutes so important a principle
in the phenomena of capillary action ; for since the cohesion in
the direction of the surface is the undiminished result of the
attractions of the whole number of particles constituting the
stratum, acting as they would do in any other part of the
substance, it follows that a small cubical portion of the liquid,
situated in any part of the space which we are considering, will
be pressed laterally by the whole force of cohesion, but aboVe
and below by that part only, which is derived from the action of
the strata above it ; so that this minute portion must necessarily
tend to extend itself upwards and downwards and to thicken the
superficial film; and at the same time to become thinner
in the direction of the surface, and to shorten it in all its
dimensions; unless this alteration be prevented by some
equivalent tension acting in a contrary direction: and this
tension must be always the same in the same liquid, whatever
its form may be, the thickness of the whole stratum being
always extremely minute in comparison with any sensible radius
of curvature.

Upon these grounds we may proceed to determine the actual
magnitude of the contractile force derived from a given cohe-
sion extending to a given distance. Supposing the corpuscular
attraction equable throughout the whole sphere of its action,
the aggregate cohesion of the successive parts of the stratum
will be represented by the ordinates of a parabolic curve ; for
at any distance x from the surface, the whole interval being a,
the fluxion of the force will be as Ax (a — x\ since a number
of particles proportional to Ax will be drawn downwards by a
number proportional to a, and upwards by a number propor-

No. XX. COHESION. 461

tional to x^ and the whole cohesion, at the gWen pointy will be
expressed by ax — ^jfi ; and this at last becomes ^a% which
must be equal to the undiminished cohesion in the direction of
the surface : consequently the difference of the forces acting
on the sides of the elementary cube will everywhere be as
io*— ax+4aj*, and the fluxion of the whole contractile force will
be dx (4a*— ax + ia*), the fluent of which, when a: = o, becomes
^', which is one-third of a X 4^*, the whole undiminished cohe*
sion of the stratum.

We may therefore conclude, in general, that the contractile
farce is one third of the whole cohesive force of a stratum of par-
ticlesy equal in thickness to the interval to which the primitive
equable cohesion extends; and if the cohesive force be not
equable, we may take the interval which represents its mean
extent, as affording a result almost equally accurate. In the
case of water, the tension of each inch of the surface is some-
what less than three gruns ; consequently we may consider the
whole cohesive and repulsive force of the superficial stratum as
equal to about nine grains. Now since there is reason to
suppose the corpuscular forces of a section of a square inch of
water to be equivalent to the weight of a column about 750,000
feet high, at least if we allow the cohesion to be independent
of the density, their magnitude will be expressed by 252.5 X
750,000 X 12 grains, which is to 9 as 252.5 X 1000,000 to 1 ;
consequently the extent of the cohesive force must be limited to
about the 250 millionth of an inch : nor is it very probable that
any error in the suppositions adopted can possibly have so far
invalidated this result as to have made it very many times
greater or less than the truth.

Within similar limits of uncertainty, we may obtain some-
thing like a conjectural estimate of the mutual distance of the
particles of vapours, and even of the actual magnitude of the
elementary atoms of liquids, as supposed to be nearly in
contact with each other ; for if the distance, at which the force
of cohesion begins, is constant at the same temperature, and if
the particles of steam are condensed when they approach within
this distance, it follows that at 6QP of Fahrenheit the distance
of the particles of pure aqueous vapour is about the 250

462 COHESION. No. XX.

millionth of an inch ; and since the density of this vapour is
about one-sixty thousandth of that of water, the distance of the
particles must be about forty times as great ; consequently the
mutual distance of the particles of water must be about the
ten thousandth millionth of an inch. It is true that the result
of this calculation will differ considerably according to the
temperature of the substiinces compared ; for the phenomena
of capillary action, which depend on the superficial tension,
vary much less with the temperature than the density of vapour
at the point of precipitation : thus an elevation of temperature,
amounting to a degree of Fahrenheit, lessens the force of elas-
ticity about one ten-thousandth, the superficial tension about
one thousandth, and the distance of the particles at the point of
deposition about a hundredth. This discordance does not,
however, wholly invalidate the general tenor of the conclusion ;
nor will the diversity resulting from it be greater than that of
the actual measurements of many minute objects, as reported
by different observers : for example those of the red particles of
blood, the diameter of wliich may be considered as about two
million times as great as that of the elementary particles of
water, so that each would contain eight or ten trillions of par^
tides of water, at the utmost. If we supposed the excess of
' the repulsive force of liquids above that of elastic fluids to
depend rather on a variation of the law of tlie force than of the
number of particles co-operating with each other, the extent of
the force of cohesion would only be reduced to about two-
thirds ; and on the whole it appears tolerably safe to conclude,
that, whatever errors may have affected the determination, the
diameter or distance of the particles of water is between the two
thousand and the ten thousand millionth of an inch.

Section II. — Relations of Heterogeneous Substances.

We must now return from this conjectural digression to the
regions of strict mathematical argument, and inquire into the
efilect of the contact of substances of different kinds on the ten-
sion of their common surfaces, and on the conditions required
for their equilibrium. Whatever doubts there may be respects
ing the variation of the number of particles co-operating when

No. XX. COHESION. 463

the actual density of the substance is changed, there can be
none respecting the consequence of the contact of two similar
substances of different densities ; for the less dense must neces-
sarily neutralize the effects of an equivalent portion of the
particles of the more dense, so as to prevent their being
concerned in producing any contractility in the common sur^
face, and the remainder, acting at the same interval as when
the substance remained single, must obviously produce an effect
proportional to the square of the number of particles concerned,
that is, of the difference of the densities of the substances.
This effect may be experimentally illustrated by introducing a
minute quantity of oil on the surface of the water contained in
a capillary tube, the joint elevation, instead of being increased
as it ought to be according to Mr. Laplace, is very conspicu-
ously diminished ; and it is obvious that since the capillary
powers are represented by the squares of the density of oil and
of its difference from that of water, their sum must be less than
the capillary power of water, which is proportional to the
square of the sum of the separate quantities.

Upon these principles we may determine the conditions of
equilibrium of several different substances meeting in the same
point, neglecting for a moment the consideration of solidity or
fluidity, as well as that of gravitation, in estimating the con-
tractile powers of the surfaces, and their angular situations.
We suppose then three liquids of which the densities are, A, B,
and C, to meet in a line situated in the plane termination of
the first, the contractile forces of the surfaces will then be ex-
pressed by (A— B)», (A— C)«, and (B— C)« ; and if these
liquids be so arranged as to hold each other in equilibrium,
whether with or without the assistance of any external force,
the equilibrium will not be destroyed by the congelation of the
first of the liquids, so that it may constitute a solid. Now,
unless the joint surface of the second and third coincides in
direction with that of the first, it cannot be held in equilibrium
by the contractility of this surface alone : but supposing these
two forces to be so combined as to produce a result perpendi-
cular to the surface of the first substance, this force may be
resisted by its direct attraction, the forces which tend to cause

464 COHESION. No. XX.

the oblique surface to move either way on it, balancing each
other, and the perpendicular attraction being counteracted
by some external force holding the solid in its situation:
consequently the force expressed by (B — C)*, reduced in the
proportion of the radius to the cosine of the angle, must become
equal to the difference of the forces (A — Bf and (A — C)*,
and if the radius be called unity, this cosine must be

(A - Cy - (A - B)2 _ 2AB ~ 2AC ~ (B> - C«) 2A-(B-fC) . . ,

(B - C)« - (B - c)8 = file » wmch

is the excess of twice the density of the solid above the sum
of the densities of the liquids, divided by the difference of
these densities ; and when there is only one liquid, and C = 0,

this cosine becomes ^ — 1, vanishing when 2 A = B, and the

density of the solid is half of that of the liquid, the angle then
becoming a right one, as Clairaut long ago inferred from other
considerations. Suppoang the attractive density of the solid
to be very small, the cosine will approach to — 1, and the
angle of the liquid to two right angles ; and on the other hand,
when A becomes equal to B, the cosine will be 1, and the angle
will be evanescent, the surface of the liquid coinciding in direc-
tion with that of the solid. If the density A be still further
increased, the angle cannot undergo any fiirther alteration, and
the excess of force will only tend to spread the liquid more
rapidly on the solid, so that a thin film would always be found
on its surface, unless it were removed by evaporation, or unless
its formation were prevented by some unknown circumstance
which seems to lessen the intimate nature of the contact of
liquids with dry solids. For the case of glass and mercury we

find — about J, and the cosine — f , which corresponds to an angle
of 139"^ ; and if we add a second liquid, the expression will be-

— 6 — C

come — gZTc* which will always indicate an angle less than ISO"',
as long as C remains less than 1, or as long as the liquid added
is less dense than glass. There must, therefore, have been a
slight inaccuracy in the observation mentioned by Mr. Laplace,
that the surface of mercury contained in a glass tube becomes
hemispherical under water : and if we could obtain an exact
measurement of the angle assumed by the mercury under

No. XX. COHESION. 465

these dreumstancesi we should at once be able to infer from it
the comparative attractive density of water and glass, which has
not yet been ascertained ; although it might be deduced vdth
equal ease from the comparative height of a portion of mercury,
contained in two imequal branches of the same tube, observed
in the air and under water. The cosine is more exactly —.735,

in the case of the contact of glass and mercury, and ^ = 265,
whence- = y-^, which is a disproportion somewhat greater

than that of the specific gravities, but it must probably vary
with the various kinds of glass employed.

There is also another mode of determining the angle of
contact of a solid with a single liquid, which has been ingeni-
ously suggested by Mr. Laplace ; it is derived from the prin-
ciple of the invariability of the curvature of the surface at a
^ven elevation ; and its results agree with those which we have
already obtained, except that it does not appear to be appli-
cable to the case of more than one liquid in contact with the
given solid. Supposing a capillary tube to be partially inserted
into a liquid, if we imagine it to be continued into a similar tube
of the liquid, leaving a cylinder or column of indefinite length
in the common cavity ; then the action of either tube, upon the
liquid immediately within it, will have no tendency either to
elevate or to depress the column : but the attraction of the
portion of the tube above the column will tend to raise it with
a certain force, and the lower end of the tube will exert an
equal force upon the portion of the column immediately below
it ; and this doul^ force will only be opposed by the single
attraction of the liquid continuation of the tube, drawing down
the column above it, so that the weight of the column suspended
will be as the excess of twice the attractive force of the solid
above that of the liquid. Now supposing two plates of the
solid in question to approach very near each other, so that the
elevation may be very great in comparison with the radius of
curvature of the surface, which in this case may be considered
as uniform ; the weight suspended will then be simply as the
elevation, which will be the measure of the efficient attractive
force, and will vary with it, if we suppose the nature of the
VOL, I. 2 H

466 COHESION. No. XX.

solid to vary, the radius of curvature varying in the inverse
ratio of the elevation : but the radius of curvature is to half
the distance of the plates, as unity to the numerical sine of
half the angular extent of the surface, or the cosine of the
angle of the liquid, so that this cosine will be inversely as the
radius, or directly as the elevation; that is, as the efficient
attractive force, which is expressed by 2 A — B, becoming
= — 1 when A vanishes, and consequently being always equal to

— ^^ — , as we have already found from other considerations. If

we wished to extend this mode of reasoning to the effect of a
repulsive force counteracting the cohesion, we should only have
to suppose the diameter of the tube diminished on each side
by the interval which is the limit of the repulsion, since beyond
this the repulsion could not interfere with the truth of the
conclusions, for want of any particles situated in the given
directions near enough to each other to exhibit it ; and within
the stratum more immediately in contact with the solid, the
forces may be supposed to balance each other by continuing
their action along its surface until they are opposed by similar
forces on the outside of the tube or elsewhere: and indeed
such a repulsive stratum seems in many cases to be required
for affording a support to the extended surface of the liquid
when the solid does not project beyond it. It may also be
shown, in a manner nearly similar, by supposing the column to
be divided into concentric cylinders, that the superficial curva-
ture of the liquid will not affect the truth of the conclusion.

Section III. — Forms of Surfaces of Simple Curvature.

We may now proceed upon the principle, admitted by all
parties, of a hydrostatic pressure proportional to the curvature
of the surface of the liquid, which is equivalent to a uniform
tension of that surface, and which either supports the weight
or pressure of the fluid within its concavity, or suspends an
equal column from its convexity, whether with the assistance
of the pressure of tho atmosphere, or more simply, by the
immediate effect of the same cohesion, that is capable of retain-
ing the mercury of the barometer in contact with the summit

No. XX. COHESION. 467

of the tube: and on this foundation, we may investigate the
properties of the forms assumed by the surface ; first consider*
ing the cases of simple currature, which are analogous to some
of the varieties of the elastic curve, and next those of the
surfisices having an axis of revolution, which will necessarily
involve us in still more complicated calculations.

A. Lei the height of the curve at its origin be a, the hori-
zontal absciss a?, the vertical ordinate y, the sine of the angular
elevation of the surface s, the versed sine t?, and the rectangle
contained by the ordinate and the radius of simple curvature r ;
then the area of the curve will be rs, and y = a/ (a* + 2 rv).

The fluxion of the curve z is jointly as the radius of curva-
ture -^ , and as the fluxion of the angle of elevation, which

we may call u?, or d? = — dtc, and dx = V (1 — «*)
d? = V (1 — «*) -— dttj ; but */ (1 — <•) du? = d*, consequently
Ax = --- d5, and ydi*, the fluxion of the area, becomes equal
to rd«, and the area itself to rs. In order to find y, we have
dy = sAz = .» — dtt? = — dt; ; whence ydy = 'rdt;, and y*
= 2ro -^ aay g becoming equal to a when v vanishes.

It may also be immediately inferred, that the area of the
curve must vary as the sine of the inclination of the surface,
from considering that, according to the principles of the resolu-
tion of forces, the tension being uniform, the weight which it
supports must be proportional to that sine.

Scholium. The value of r for water at common temperatures
is about one hundredth of a square inch, according to the
results of a variety of experiments compared by Dr. Young ;
or more correctly, if we adopt the more recent measurement
of Mr. Gay-Lussac, .0115 : for alcohol Mr. Gay-Lussac^s
experiments give r = .0047; and for mercury r = .0051.
Dr. Young had employed .005 for mercury, a number which
appears to be so near the truth, that it may still be retained for
the greater convenience of calculation. Hence in a very wide
vessel, the smallest ordinate a being supposed evanescent, and
g = J (2rr) = .1516 ^r, the height of the water rising against

2 H 2

468 COHESION. No. XX.

the side of the vessel, when t? = 1, will be .1516 ; and the utmost
height at which the water will adhere to a horizontal sarface,
raised above its general level, will be 2 ^ r = .2145. For mer-
cury, y becomes in these circumstances, »J (.0102t?) = .101 V ^>
and if « = .735, i? = .322, and the depression of the surface in
contact with a vertical surface of glass becomes .0573 ; and
again when v= 1.735, as in the case of a large portion of
mercury lying on a plate of glass, the height y is .133 : and
if the glass had no attraction at all for mercury, v would
become % and the height .1428. The actual tension of the
surface of mercury is to that of water as .0051 x 13.6 or
.06936 to .0115 ; that is a little more than six times as great ;
while the angle of contact of mercury with glass, which is more
attractive than water, would have led us to expect a dispro-
portion somewhat greater. If we take a mean of these results,
and estimate it at seven times, the value of ^r will be reduced,
by immersing mercury standing on glass into water, in the

ratio of Y X V ys^ » ™^ ^^ buoyant efiTect of the water
increases the value of r, so that V (2r) will be .09; and the
angle approaching to 180% the height will be about .127.

B. When the curve is infinite the absciss x becomes = ^ V r HL
2Vr + V Sr - yy) + V" (4r ~ y«), reckaninff from the greatest
ordinate y ss2 ^ r ; and the excess of the length of the curve
above the absciss is2 *J r ^ ij {At — y*).

In this case, a being = 0, y* = 2rv : but -j— = — -^ =

V (2» - w) " V (^ " 2n;)V(2ro " ^ (4r - yy)y > ^^ ''y tlie

common rules for finding fluents, x = j^ HL ^J^Zf. jilr^n\
+ V (4r — y*) ; which vanishes when y = 2 Vr : and for the
length of tiie curve, since -J^ = i = -^^^^ =
J ^ - wv) V ' subtracting the former fluxional coefficient from

this, we have ^ ^^^ j^ . for the fluxion of the difierence ; and
the fluent of this is — ./ (4r — y*).

No. XX. COHESION. 469

Corollary 1. Hence, where the curve is vertical, we find
X = .5328 Jr : and where the inclination amounts to a second,
X = 11.28 V r ; for example, in the case of water, ^r being .1072,
the latter value of x will become 1.21, and the former .056 :
so that the surface must be considered as sensibly inclined to
the horizon at the distance of more than an inch from the
vessel, but scarcely at an inch and a half: and for mercury,
these distances will be two-thirds as great. This circumstance
must not be forgotten when mercury is employed for an
artificial horizon; although, where the vessel is circular, the
surface becomes horizontal at its centre; and in other parts
the inclination is materially afiected by the double curvature.

Corollary 2. The form of the surface coincides in this
case with that of an elastic bar, or a slender spring, of infinite
length, supposed to be bent by a weight fixed to its extremity ;
since the curvature of such a spring must always be propor-
tional to its distance from the vertical line passing through
the weight We may therefore deduce from this proposition
the correction required for the length of a pendulum like
Mr. Whitehurst's, consisting of a heavy ball, suspended by a
very fine wire. Now the radius of curvature of the spring is

Y ^ (Art. Bridge, Prop. G.*) ; the modulus of elasticity, of

which M is the weight, being for iron or steel about
10,000,000 feet in height ; and since 80 inches of the wire
weighed 3 grains, the thickness a, supposing it to have been

1 or i of the breadth, as is usual in wire flattened for
hair springs, must have been about ttt of an inch : the
weight / was 12251 grains : and the weight M of ten mil-
lion feet must have been ~ X 12 X 10000000 grains; conse-

fl Mf!? 3 X 10000000 ^ 1000 _

quentiy ,^y = ^ ^^ ^g^Sl X 376 X 375y ^ 12251 X 375y —

■rrrr- , which is analogous to — in these propositions ; conse-

quentiy V r = tV ; and the whole value of ^ (4r-y*) from y =

2 ^ r to y = 0 is tV of an inch. Now supposing the spring to have

• In the Supplement to the Encyclopaedia BriUonica, reprinted in the lecond
volume of thii work, No. LIl.~>^otd hy the Editor,

470 COHESION. No. XX.

been firmly fixed at the axis of vibration, the excess of its length
above the ordinate will always be measured by 2 V " V
(4r-y*) ; but V(4r-y*) = V(4r-2n?) = Vr^(4-2t;), whichis
the chord of the supplement of the arc of vibration in the circle
of which the radius is ijr= -^'j and the ball will be drawn
above its path to a height equal to the distance between this
circle and another of twice the diameter, touching it- at its
lowest point : but a perpendicular falling fi'om this point on
the wire would always be found in a circle twice as much
curved as the first circle, and if it were made the centre of
vibration, the ball would always be raised twice as &r above
its original path as the distance between the first circle and
the second, which is the measure of the efiect of the curvature ;
so that the pendulum must be supposed to be shortened half
as much as this ; that is, in the present instance, ttt of an inch.
If however the spring remained, in Mr. Whitehurst's experi-
ments, at liberty to turn within the clip, and was firmly fixed
at a considerable distance above, the variation of the length
must have been only that which belongs to half of the arc of
vibration ; that is, one fourth as great as in the former case,
since the versed sine is initially as the square of the arc ; but
since it would aflect the spring both above and below the clip,
it would be doubled from this cause, and would amount to
Tf T of an inch : so that the true correction would be liable to
vary from .00735 to .00367, according to the mode of fixing
the wire. But since this error must have affected both Mr.
Whitehurst's pendulums in an equal degree, and the result was
deduced from the difierence, and not the proportion of the
lengths, it is free from any inaccuracy on this account. The
calculation however sufficiently proves the necessity of attending
to the efiect of different modes of fixing the spring, in order that
no variation may be made in the different experiments compared
without a proper correction. The elasticity of such a wire
as Mr. Whitehurst employed, could not have produced any
sensible error, by co-operating with the force of gravitation,
since it did not amount to one two-millionth part of the weight
of the ball.

Na XX COHESION. 471

C. The relation of the ordinate and absciss may be generally
expressed by means of an infinite series.

When the curve is concave towards the absciss throughout
its extent, the ordinate may be compared with the lengths of
hyperbolic and elliptic arcs, as Maclaurin has shown with
respect to the elastic curve {Fluxions, § 928) : but his solution
fails in the more ordinary cases of the problem ; and even
where it is applicable, the calculation is very little facilitated
by it Segner has made use of two different forms of infinite
series, each having its peculiar advantages with respect to
convergence in particular cases, and other forms may be
found, which will sometimes be more convenient than either

of these. The value of the cotangent j- being in general

1 - V 2r^- 2iv 2r - yt/ + aa 1

V(2r - vv) ~ V~(4r - 2fv) J^iiv) ~ V(4r -"y7+ a«)' V(yy - aa) '

we may retain either of these fractions, and expand the other
by means of the binomial theorem.

1. In the first place, making 4r + a' = c*, we have (c^— y*)-*

" c "*■ 2 ' c» + 4.2 ' c» "*■ 8.2.3 ' c7 +-' *°^ dy "" ^{yy - aa)

\c + 2 c» "♦" 4.2 c* '^"'J'^V(yy-aa) V c "'"2 cs "^4.2 tfi "^'"J
Now, in order to find the fluents of the separate terms, we
have first / ^ ^yy^l ^a) = hl (y + V Oy - oa]) ; and calling
this logarithm L,

f / V(i^) =(t-^+^V Cy» - ««) + S L ; .nd

^ o _dy /y^ 7^ 7.5aV 7.5.3a^\ . 7.5.3fle\T

^ ^Vto - «>) " \ 8 ■" S.6 "+■ 8.6.4 "" 8.6.4.2 J'^KJ^'^^J-f- 8.bA,iJ^'

whence by substitution we have

2r + oa^ , / 2r + gg 1 \ (t/ .r o ai <»* t \

^=4^+^^ + V20i^^T^)~r~4r+agr\2^Lr"^J+2"^;+"--

472 COHESION. No* XX.

2. If we reduce /t— ^ — ^. into a series, we have

d, ^ 2r + aa-yy .A . 1 . !!!. + ^ . ?L . \ Then,

for the fluents,

•' V(cc-yy) ^^"^ ^^' JyV(«^-yy) c+V(i?^)"-^'
i- dy _ _ V(££_:^y) L t ' .

J y^ jJicc-yy) " 2ccyy 2i« -^ '

J yft V(<»-yy) " \ 4c«y* + 8c*y»y ^ '^ ^'' ^ 8c*

and by combining these fluents we obtain a second series for x.

3. These series may be employed with advantage where the
initial ordinate is very small, the one being more convenient
for the upper, and the other for the lower part of the curve :
but where the elevation a is more considerable, the form of
the curve will be more readily determined by means of fluents
derived from circular arcs. Beginning with the expressions

-r- = f-77r:——->.9 wid v* = o* + 2rt>, WO may seek for a

dy V (2» - wr ^ ' ^

value of X in terms of v ; and since 2ydy - 2rdv, dy ss H

dv = ^(0^^+ 2rvy *^d ^ = Vcsi-L) • V(^^+i;S)' The
binomial {aa + 2rv) may then be expanded into a series of
integral powers of o, and the fluents may be found by means of
the equations / ^>2p - w) = / -7 ~ ^* *^® *"^ ®^ which t?
is the versed sine \ f -j- - s ^ w\ f -j^ = (| — 1 • 2 ^

- - W*d» /i>3 7 7,5 7.6.3 \

and/— = (- - - . 4,^ + j^ . 8« - jlj^jl . 16)
7.6.3 ,.

No. XX. COHESION. 473

4. Another series may be obtained by the expansion of
V55^ >°to ^j^ • (1 + i » + 5^5 1^ + j^ t^ + . . .>

whenee^g=(l-?SrOv^(l4-i^^+^

^ + ...): the fluxions belonging to the series (y* *a^)

dy» (y* — «*) dy> (y* ~ a*)*dy ; and the fluents of these are

HL (y + V |y-a»]) =L;4y V(y* - fl^)-*a'L;

(iV(y*-rf) + *a')yV(y*-<]?)+ ^L; which afford

a result somewhat resembling that which is deduced from the
first method.

5. We may also express ar in a series of integral powers of
y only, if we suppose it to begin at some point in which the
curve IS inclined to the horizon, where the height is/^ calling it

at other points /> + y ; and making ?- ^ r^ a + by + a^

+ . . . ; we have then x = /3 + ay + iJy* + \ci^ + . . .,
and the area /(p + y) dr = y + pay -h i pby^ + . . . + ^aj^

+ 2^Ay* + 3^^+ •••> ^^ch must be equal to rs (Prop,

A.) : but * = V(d;r« + dy«)= va + ^)> "^^^ "^^ ^ developed
by means of the Taylorian theorem ^ (A + H) » f A +

!i^H+ ^-^^- f + ..., taking A = a, and H = Jy

+ cy* + . . ., whence

H« = J^y« + 2icy + (2M + c»)y*+ ... .

H" = jy + 3y<?y* + . . . ; consequently rs = r^ . . .

y + jwy + (V + i«)»*+(ipc + ii)y»+ . . . ;

474 COHESION. No. XX.

tmd hence by eomparing the homologous terms, we find

y - VO + oa)'^ -^ ^a+aa) •*=/>« * OT^f ' *'

and 6 = ■""7(1 + <w)*; and in a similar manner we may
determine the subsequent coefficients; but the calculation is
somewhat laborious, and has no particular advantages.

6. We may still more readily obtain a similar series for y
in terms of the powers of x with constant coefficients ; calling

~, t^ and making i = &x + ca;^ + ir* -f . . ., whence y — a
^ * 6a:* + i ca^ + t ctr* + . . ., and the area yydx = ax

+ ^^-^/* + ...). But^ = y:p» + ^VcJ^ + ...
and ^ = VJ" + . . . : hence we have the equation

' 6ar + car* + dar* 4- . • •

- \V3? - *.3i»car»- ...

+ f Va? + . . . : consequently

6 =^,c-5^^J + 4ft»,andd =^c + ti»c-|y.

It is the less necessary to enter into any further detail of
these results, as we have a table, calculated by Segner, with
his son's assistance, which is sufficient to afford us a general
idea of the forms of the curve in different circumstances. The
unit of this table is the quantity ijr^ which Segner calls the
modulus of capillary attraction, and which for water is .1072
inch. The table begins with the extreme ordinate, where the
cinrve is vertical: we ha?e then the least ordinate, a\ the
greatest ordinate, where the curve again becomes horizontal,
and the absciss corresponding to the extreme ordinate and to
the greatest ordinate.

No. XX. COHBBION. 475

Bztram«

Lewt

OreatMt

Onatcst

Terminal

Ordinate. Oidinato.

Ordinate.

AbMiw.

AbRia.

lOO^r

99.99

100.01

.01

.000001

89.99

90.01

.01

.000002

79.99

80.01

.01

.000003

69.99

70.01

.01

.000004

59.99

60.02

,02

.000007

49.98

50.02

.02

.00001

44.98

45.02

.02

.00002

39.97

40.02

.02

.00003

34.97

35.03

.03

.00004

29.96

30.03

.03

.00006

24.96

25.04

.04

.0001

19.95

20.05

.05

.0002

14.93

15.07

.07

.0004

9.90

10.10

.10

.001

8.89

9.11

.11

.002

7.87

8.12

.13

.003

6.85

7.14

.14

.004

5.83

6.16

.17

.007

4.79

5.19

.21

.01

3.74

4.24

.26

.02

2.64

3.32

.37

.06

1.41

2.45

.65

.22

1.27

2.37

.71

.27

1.11

2.29

.79

.33

.94

2.21

.91

.47

.75

2.13

1.10

.65

.50

2.06

1.40

.96

,47

.40

2.04

1.64

1.18

1.445

.30

2.02

1.86

1.44

.428

.20

2.01

2.24

1.82

.418

.10

2.003

2.92

2.49

.4142

.01

2.000

5.22

4.80

.4142

.001

2.0000

7.52

7.09

4142

.0001

2.0000

9.82

9.39

.4142

.00001

2.0000

12.12

11.70

1.4142

.000001

2.0000

14.43

14.00

It may be observed that the last six values of the least
ordinate are in geometrical progression, while the absciss
increases in arithmetical progression; the difference of the
abscisses 2, 3, being the hyperbolical logarithm of 10, which is
the common multiplier of the ordinates. Although the table
appears to be generally accurate, yet we cannot always depend
on the last figures : thus the ultimate difference of the two last
columns Lb made .43, while it ought to be .53 (Prop. B. Cor. 1).
It is scarcely necessary to remark, that if we look, in the fourth
column, for half the distance between two parallel planes of
gkss, in a vertical position, the first and second columns will
give us the height to which water will rise between them, where
it touches the glass, and in the middle of the interval.

476 CJOHESiON. No. XX.

Section IV. — Surfaces of Double Curvature.

When the liquid is contained in a tube, or when it forms
itself spontaneously into a drop having an axis of revolution, it
becomes necessary to consider the effect of the tension in a
direction transverse to that of the principal section ; since the
curvature will cause it to exhibit an equal pressure, whatever
the direction of the section to which it belongs may be ; and the
curvatures of the sections perpendicular to each other will either
co-operate witii, or counteract each other, accordingly as the
convexities of both are on the same side, or on the opposite
sides, of the surface. But the simple consideration of the
tension, supporting the weight of the parts below, or the equi-
valent pressure in a contrary direction, will at once afford us
the equations necessary for the solution of the problem, without
any immediate reference to the curvature in question.

D. Hie form of a surface of revolution may he determined by
means of an infinite series.

The fluxion of the weight or mass of the parts, contained
within the cylindrical sur&ce, of which :i; is the radius or absciss,
and y the ordinate, being always proportional to y^rdx, and the
fluent tofyxdx ; and the extent of the circumference supporting
it varying also as a;, and the contractile force being diminished,
when reduced to the direction of gravitation, in the ratio of Uie
radius unity to the sine of the elevation «, it will always be pro-
portional to a» ; so that we have the general equation fyxdx >=
mx8. Now if we suppose y incomparably greater than a^ and
the surface extremely minute, the variation of y may be neg-
lected, and we have in this case fyxdx^iya?: and supposing
also « = I, and the curve vertical, iya? = 7aXf audi yx = ni;
X becoming also equal to the radius of curvature : but it is easy
to perceive that the height y must be twice as great, for any
value of or, as in the case of a simple curvature, since each
portion of the circumference has here only to support a wedge,
which is only half as heavy as a parallelepiped of the same

No, XX. COHESION. 477

height ; so that iyx will be equal to yx in Proportion A» and

In order to obtain a series for finding y, firom the equation
fyxdx = mxs^ we may put the tangent t = ^ - bx + ca? +
daf + ...y whence y = a + i&** + \ca^ -¥ \d3fi + ...,
and/y:rdar = J aa:« + ^ ia?* + j^j ca;« + g^ die* + . . . ; and

the value of * = ^ ^^ ^ ^^x being expanded into a series, as in
Propodtion C, n. 6, calling ^, or -, y, we find s = I fyxdx
=z bx + C3? + da? + ex'^ + ...

- ^ V x' ^ ...
= i ?«^ + 2H ?*^ + 4^6 ?«aJ^ + 6^ 9^'' + • • ' 5 consequently
ft = ^ ^a = — , and a = — = 2r&, and by continuing the cal-
culation and reducing the values, we find

/I ^A . 652 .M , g64S ^w ,

= 2.4i.6«.8*.10 5^^ + 2.4.6«.8«.10 ^^ + 2.4.6.8«.10 ^ ^

1260 „ • , 105 ^
2.4.6.8.10 S^'' + 2.4.6.8

1 ^, 5197 A^ 59855 •« ,

9 = 2.4«.6«.8M0M0 5^^ + 2.4,6«.8M0«.12 ^^ + 2.4.6.8«.10».12 ^ '^
70522.5 ,,, 17825 ,, .Jli_ jn
2.4.6.8.10«12 y ^^ -r 2.4.6.8.10.12 ^ ^ 2.4.6.8.10
» 1 <IJL 418 JM 1808084 ^jm ,
^ " 2.4«.6«..l2*.14g^+2.4.6«.8«..l2«.l4g^^+2.4.6.8«.10«.12«.14y^+-'

1 ^71 839412 -.8 _L

* = 2.4«.6«..14«.16 ? ^ + 2.4.6«.8«..14«.16 ?^ + •••

1 1 ^j^ 2779888 -^ ,

'^ ~ 2.4«.6«..16«.18 ^ + 2.4.6«.8«..16M8 V *^ ^ • • *

I 1 ^r . 22941328 .„ .

^ = 2.4«.6«..18«20. 5^^ + 2.4.6«.8«..18«.20 » '^ "*" ' ' "

478 OOHESIOK. No, XX.

We may here observe, that the numerical coeffidents of the

highest powers of b form the series i»^> o~~' o^ "'^ »&c-)

the ratio of the succesfflve terms of both continually approach-
ing to equality ; and those of the next in order, the series

S 2 8.5 4 8.6.7 6 3.5.7.9 8 . , . ^.

O • 6' 2X6 • 6' 2X6T • 6' 2.4.6.8.10 ' ?' ^= ''"* *«

laws of the numerical coefficients in general appear to be wholly
incapable of being reduced to any simple form. It will be con-
venient for calculation to form tables of the logarithmic values
of these coeffidents, which may be continued, by means of suc-
cessive differences, for as many terms as are requisite for any
practical purpose. The indices, with lines drawn over them,
are to be considered as negative numbers.

Logarithmic Coefficienia of the Value of the Sine.

» = ( 0. 0000000 -I- 13. 7811595 j»x"

+ 1. 0969100 jx» + 15. 9774. . . y"**"

+ "3. 7166987 j»jr* + 16. 1026. . . y"***

+ "4. 0354574 fu^ + 18. 160. . . . q'*x>*

+ "6. 1323674 q*a^ 4- 20. 16 q^x»

+ "8. 0531861 5»x'» +22. 10 j'V

+ li. 8278768 j«aJ« + 24. 0 j>»«»

+ 13. 4776288 q'x'* + ^.) J»«»

+ 15. 0182362 g»x" + (3. 8927900 qa*

+ 18. 4619336 gV» + ~8. 5337080 ^x*

+ 21. 8184809 q^ox^ + "4. 8558231 g»y

+ -.)** + ~5. 9851885 q*a^

+ (2. 3187587 qa? + ~6. 9727959 ^x^"

+ 3. 6375174 q'ai* + T 82 ^x"

+ J. 6482413 j»z« + "8. 57 yV«

+ _5. 4694937 ^;r« + "9. 27 y**"

+ 6. 1456895 ?»*'• +11. 97 j»ar"

+ "8, 7008651 5«a;» +11. 77 9»«a*»

+ "9. 1510234 q-'x'* • + . . .) i>a»

+ n. 5080209 j'x'o + (3. 5917600 yz»

No. XX. COHESION. 479

+ "3. 4368580 tfi* + . . .) i»a»

+ "4. 9595058 5»«* + (3. 1657913 jx*

+ ...>JV +...)i"x»>

+ "3. 3576767 jx* + . . .
+ "3. 3498514 ^a*

Logarithmic CoefficienU of the Value of the Ordinate y.

y = IT 1+ (0- 6989700 + 21. 60 y»a*

'+ J. 4948500,:^ +23.5 ,^*J^

+ 4. 9385474 j»** + 'J.-^ *"**

+ "6. 1323674 fz» + i?" ^958800

+ -1. 1323674 j'x' + 1- 5337080 ?.»

+ 10. 9740048 fx'o + 3 ^350043 <f>*

+ l2. 6817488 9«a:'* + ^ 1313165 j»««

+ 14. 2735087 gV* + i' 1769159 ^:f

+ 17. 7629636 /x" + _5. 08 j**"

+ 19. 1609036 g'a:" + _!* ^'^ 9**"

+ ...)*.» +J-51 '?'*"■

+ ■(1.0969100 +_9. 35 j-x'*

+ T 5406074 yx* +10.18 j»*"

+ "3. 6482413 g»x« + -•) *•«*

+ ^. 5486749 3»,« +(2.5917600

+ -3. 2918175 ^^ + J- ^^09*12 9^

+ T. 9049851 j'x'o + 2- 105«338 5«x^

+ 8. 4062959 <fx'* + ' *;„°^:„„

+ ro. 8090509, 'x" +(_2.43 8579

+ n. 1235822 A" + 2. 4959794 ,x»

+ 13. 3576... gV +_^..>»*

+ 15. 5176..., V +(2.3119193

+ 17. 607.... j"x» ■•■ •••)*"*"

+ I9. 637.... j-V* "*" ••

The calculation U somewhat facilitated by obtaining the value of xy Arom the
expansion of ^ V (1 — s*)~^' and deteimining the coefficients of the series for 9 at
once.— Jf9. note by Dr, Young, Oct, 8, 1818.

480 COHESION. Na XX.

E. Hie elewxHan or depresnon of a liquid contained ina given
tube may be found by reversing the series.

Having a given valae of x, the semidiameter of the tube, and
also of «, the elevation or depression of the sur&oe of the liquid
at the point of contact with the solid, we obtain an equation of
the form s=Kb + BV + Cft* + . - •, and from this we may
determine the central elevation or depression a-2rb by the
well known method of the reversion of series, which ^ves us the

- - 1 B . /C 8Bs\ . /D »BC . 12B8\ ,

value 6= ^*-i-/ - (jj-— ) ^ -(ij--X; + ^)*'-...
But it is more convenient to assume an approximate value of b,
a littie less than -^, and to find the corresponding value of $ ;
then once d« = Ad 6 + 3B J»di + 5CJ*dft + ..., if we
make Aft + SB^ + 50^+ ... = 2, we shall have ^f = ?•

do 6

consequently the small increments of e and b will be to each
other as 2 to ft, and we obtain the correction of ft from the
error of the calculated value of s : and if the calculation be
repeated with the corrected value of ft, the second result will
always be sufficiently near to the truth.

In order to judge of the accuracy of this mode of calculation,
which Mr. Laplace appears to have thought liable to some
undefined objection, it will be necessary to enter into the
details of its different elements, which will sufficiently show the
degree of convergence of the series, and the greatest posdble
amount of error.

Values of the Coefficients of s for Tubes of different Diameters,
r being .005, and » = .75.

D = 2x

t=bxx

+i»*»x +i»*»x

+6V5

1.0

47.176

7190

15.774

274

5.737

13.214 200

2.399

.8556 1.625

1.2717

.06311 .03693

.0311

1.0638

.01155 .00486

.00278

No. XX. COHESION. 481

Hence if

0.8073 .7248+.0252

.1147 .7237+.0265

.4160 .7160+.0254+.0060+[.0026]

1.503 .7211+.0240+.0041+[.0010J

5.776 .7345+.0122+.0024+ .0007 +[.0003]

14.004 .7449+.0040+.0008+ .0002 +[.0001]

It appears upon inspection of this table, that the coefficients
of bx alone always determine |{ of the value of the quantity
required, and these are easily calculated with perfect accuracy*
so that the error must always be far less than Vt* and in fact
the actual uncertainty never exceeds -nrif t of the whole, at least
in the last four examples. The differences of Mr. Laplace's
approximatory calculations from these results are incomparably
greater, so that we cannot hesitate to consider these differences
as errors. Indeed, when we recollect that in the method em-
ployed by Mr. Bouvard, under Mr. Laplace's directions, the
radius of curvature of each of the small portions, into which
the curve has been cut up, has been determined from the ordi-
nate at the beginning of the portion, it is obvious that the cur-
vature thus found must be less than the truth, and that in
order to obtain any required curvature of the whole surface the
depression must be increased in the same proportion : and there
is no ready way of appreciating the amount of this error. Dr.
Young had before attempted to avoid it, in making an esti-
mate of the same nature, by calculating for the middle of each
portion; but from some accident, the numbers of his table,
published in 1807, are generally a little too small, although
the method, which he then employed, is nearly the same as that
which Mr. Laplace afterwards adopted ; except that for the
lowest portion of the curve, Mr. Laplace had recourse to an
infinite series, applicable only to that part. The elements de-
duced in Nicholson's Journal for 1809,* from Mr. Gay Lussac's
experiments, which are r = .0051 and s = .7353, agree better
witli the numbers found in Mr. Laplace's table, than those from

♦ In an article contributed by Dr. Yooog.— i^Tote by the Editor,
VOL. I. 2 I

482 COHESION. No. XX.

which it was constructed, which were r = . 005038 and 5 = .729;
the depressions being always a little larger than the true results
from the elements assumed.

The value of the ordinate y depends also principally on the
first variable member of the series, although the subsequent
coefficients are not so inconsiderable as in the determination of
the sine. Thus taking ar = ,2, and b = 1.503, we have y =
a + .813Aa:«+.99 ^ar* + 2.97 ^o:* + ... = .01503 + .0489 +
,0054 + .0015 + [.0006] = .0714, which is the marginal de-
pression, leaving .0564 for the height of the convex portion
y — a. We may determine the effect of any small variations
in this height, in the same manner as that of the sine of the in-
clination : supposing them to depend on a change of the angle
of contact only, the quantity r remaining unaltered, it is obvious
that gand x must retain their value, while y and b only vary ;

and making Y = A* + 3By +. . . = A ^!^\ we have Y : * =

d (y — a) : dft. In the present instance, we find Y = .0489 -h
8 X .0054 + 5 X .0015 + ... = .070 ; and supposing, as in the
example suggested by Mr. Laplace, the variation of the height
y— a to be .00304, which is -sV of Y, that of b will be ^V of 6,
or .075, and the variation of the central depression a, .00075,
which is somewhat less than one fifth of the alteration in the
height of the convex portion ; but in smaller tabes it is obvious
that the variations of the depression a might much exceed that
of the height of the convex portion. Nothing can be easier or
more direct than this part of the calculation : and it is remark-
able that Mr. Laplace should have considered the awkward
contrivance of building up a curve, like the arch of a bridge,
with fourteen blocks on each side, as possessing anything like
an ^^ advantage '' over the series in the determination of this
variation.

If we wish to find the effect of a small variation of the dia-
meter of a tube, from D to D ± D', on the depression a of the
mercury contained in it, we may use for the interpolation the

formula- = 10 — 1, C being about 2.9 for tubes between

1 inch and ^^th of an inch in diameter, and being els^ewhere easily
deduced from the depressions already known. For variations of

No. XX. COHESION. 483

the cohesive power, and of its measure r we may suppose the
whole of the numbers of the table to be altered in the proportion
of the supposed alteration of Vr, and the change produced
by restoring the diameter to its former dimensions may then
be calculated like any other interpolation. There is also a
more comprehensive formula, which seems to express the
depresdon in tubes of all sizes with great accuracy: it is

this, a = D^4gD».4M4.«.aeD • *od it might even be possible to
shorten the original calculation by a comparison of the series
with the expansion of this empirical formula, if it were of any
further importance to facilitate the mode of computation. But
for all practical purposes, it will be sufficient to collect the
results already obtained into a comparative table, arranged in
chronological order : and it is renmrkable, that . they are all
comprehended, without any material exception, between the
two values assigned to each as near the truth in Dr. Young's
first table, the mean of those values never differing a thousandth
of an inch from the result of the more correct calculation ; while
the error of Lord Charles Cavendish's experiments, notwith-
standing their general accuracy, sometimes amounts to nearly
one hundredth.

2 I 2

484

COHESION.

No. XX.

s g

O O O

3 S

CP «S O e* «> P'

^ 3> 00 ^ to 31

Ml kO kO 1^1 Lfl T*-

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000000

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No. XXI. ON THE COHESION OF iXUIDS. 485

NaXXXL

ON THE COHESION OF FLUIDS,

BEING APPENDIX A. OF DR. YOUNG'S 'ELEMENTARY ILLUSTRATIONS
OF THE CELESTIAL MECHANICS OF LAPLACE/

PaSLXSHED IK 1821.

1. Theorbm. If there be a series of equal particles, arranged
at equal intervals in a right line, each attracting or repelling
its immediate neighbour only with a constant force /; the
force VM. acting on any obstacle M at one end of the whole
line Uj supposing the other to be fixed, will be equal to/1

The genend principle of virtual velocities is 2m Sis = 0 (see Ulustra-
tioDS, &C., p. 177), or, taking any one of the forces combined with each
other, as the result of the rest, and in an opposite direction, VJifiu =
2m Sis : and in applying this principle, the variations may be taken in
any manner capable of representing their relations to each other, without
confining them to such as are likely to occur in the natural phenomena
to be considered ; and the motive force VM may always be found, if we

can determine its equal ~c — . Now if the number of particles oon-

cemed be m, and their masses equal to unity, we shall have d« = - , since

we may suppose the particles to remain equally distributed throughout
the line after the variation of their distances, and S being ==/, we have
2mSis=/Zu; consequently VM=f.

2. Theorem. If an attractive or repulsive force extend to
a given distance c among a series of m particles situated at equal
distances in a right line, the mutual forces of any two particles
being /, and their masses each unity, the tension acting on an

obstacle at the end of the line u will be — —/
The number of particles in the line u being m, the number acting at

486 ON THE COHESION OF FLUIDS. No. XXI.

any one point will be 2»i- ; a»d when the length u is varied, the
variation of the distance of the remotest of these particles will be
hu-, while that of the particles at a smaller distance will be propor-
tionally smaller : and the mean variation of the distances of the particles
within the respective spheres of action will be half the extreme variatioD.

e
For each particle, therefore, the variation 2»»#53* will be ^ 5«- 2<?

m ' 'cc cc

-/= -wi/'^u, and for the whole line, consisting of m particles, m« ^

/5«, which, divided by 5tt, gives VM • /•

Corollary 1. Hence, if u be given, the tension will vary as the
square of the number of particles or density m, and as the square of tiie
extent of the sphere of action c, oonjointly.

Ck)ROLLART 2» If there be two forces, a cohesive force C, and a repul-
sive force 12, holding each other in equilibrium, but extending to the
different distances c and r, they will balance each other, in this hypo-
thetical case, if c* (7 = r* 22, that is, if the primitive forces of the single
pairs of particles be inversely as the squares of the minute distances to
which they extend.

SCHOUUM. It is obvious that the length u is indifferent to the force,

since m must vary as u, and —must remain constant, when the density
is given.

3. Theorem. If a flmd, composed of cohesive and repulsive
particles, holding each other in equilibrium, be contained be-
tween two parallel surfaces, of unlimited extent, the equal and

opposite forces, acting on either of the surfaces Jf, will be -d*

c^Mf\ cf being the density, « the circumference of a circle
divided by its diameter.

The number of particles in the space Mu being dMu^ the number of
those, which are within the limits of the sphere of action of each par-
ticle, will be ^irc'd. Supposing now the distance of the particles to
be varied by a slight change of the density, it is evident that the varia-
tion of the density will be in the duplicate proportion of that of the
distances, since if J = a;*, dJ = So?* dr ; and the variation of the whole

space Mu being 3/Sm, that of the density M = — 8w— , and that of any

No. XXI. ON THE COHESION OF FLUIDS. 487

linear distance c will be hc = — I M-, = i^M-, which will be the varia-

tion of the distance of the particles, at the surface of the sphere of action,
from its centre. But the mean distance of each elementary pyramid
from its vertex, or of the whole sphere firom the centre, is -}- of the height
or the radins, since the products of the elements of the content into the

distance added together and divided by the content, or ^^ = }. The

mean variation of distance for the whole fluid is therefore 4- c - ; and
this variation, multiplied by the number of particles within the sphere

d lie

of action, becomes -j^c*- ; which being again multiplied by the num-
ber of centres Mudj and by the force /, and divided by 5m, gives us
V=^d*e*Mf, for the whole force acting on the surface M.

Corollary. In this case if the two forces C and R hold each other
in equilibrium, we must have (?* C =^1* R^ and C must be to i?, for each
pair of particles, as f^ to cf* : each ibrce still varying as the square of the
density.

ScHOUUM 1. The determination of the attractive or repulsive force of
a sphere thus constituted may be illustrated and confirmed by a simpler
mode of considering the joint action of the particles of each hemisphere
which is easily shown to be half as great as if they were collected into
one line. For it is obvious that each particle in any spherical surface
must have its action on the central point reduced in the proportion that the
radius bears to its distance from the plane dividing the hemispheres,
consequently the whole force will be represented by the distance of the
centre of gravity of the surface, multiplied into the mass, or the number
of particles contained in it. Now the centre of gravity of a spherical
surface is situated in the middle of its absciss or versed sine, since the in-
crements of the surface are proportional to those of the versed sine.
Hence it follows, that the joint force of all the particles in each surface
is half what it would be, if they were all situated in the given direction :
and the proportion being the same for all the concentric sur&ces, it must
also remain the same for the whole hemisphere. If we had only to con-
sider the attractions of a series of particles, situated in a circular circum-
ference, upon a central particle, it might be shown, in a similar manner,
that they would be together equal to that of a number of particles re-
presented by the chord, supposed to be placed at the middle of the arc.

Scholium 2. If any of the elastic fluids, with which weare acquainted,
be considered as thus constituted, we must suppose the fourth power of

488 ON THE COHESION OF FLUIDS. No. XXL

the distance r to .vary inversely as the density rf, since the force Fis

foond to vary simply as the density, and--- =— ^« JUIf is constant It

would have been more natural to expect, that if c were not constant, its
cube c" would have varied inversely as the density, supposing the num-
ber of particles co-operating to be given. But in the Newtonian de-
monstration the elementary force/ is also supposed to vary inversely as
the distance, while the number of particles co-operating is invariable.
In this case the number of particles in the space Mu are as dMu^ and

the elementary forces as d^'^f the variations of the distances, for a given

value of ^«, being as cfiy so that the products of these quantities remain
constant, and the effective force is as the number of particles concemed,
or simply as d,

4. Lemma. If the height of a cone be a, the radius of the
base 6, and the obliqae side c, the mean distance of the base from

the vertex will be g* -^— •

For, if the fluxion of the radius of the base be do;, the product of the
elementary ring 2irxLXj into its distance V (fl"+a*")i will be 2irxAx

V(a*+«*); andsinced|(a«+a:«)"[s=f X2xda: V(«*+«')» wehave
Jivx V(a*+a:*)da:=-=-(a"+a:«)^, which becomes initially — a*,

2«'

and when a? = 6,-j-c', and the difference, divided by 7r6*, the area of

the base, that is, f -^ — , or | ,_ „ will be the mean distance of the

base from the vertex.

Corollary. For a solid cone, the mean distance becomes \ of that
of the base, as in the case of the sphere : and the expression becomes, in

this case, i-^rZ"«*

5. Theorem. The deficiency of the mutual actions of the
superficial particles of a fluid, of limited extent, deducts from
the tension i of the whole force of a stratum equal in thickness
to the radius of the sphere of equal action.

For the interior parts of the fluid, the actions of all the particles will
be the same as in a fluid of unlimited extent, that is, -zc*Mf, calling the
density unity, since its finite variations do not enter into the present

No. XXI. ON THE COHESION OF FLUIDS. 489

questioD, Bat for the paitides within the distance c of the surface, the
forces will be able to act on such a number of other particles only, as are
contained in a segment of the sphere, of which the versed sine is c+x^
the distance from the surface being x, which are not only fewer than in
the whole sphere, but are also at a smaller mean distance from the centre.
Each of these s^ments may be divided into two portions; that
which is contained between the centre and the spherical circum-
ference, and the cone, which lies between the centre and the plane
surface: the variation of the mean distance of the former will be
the same as for the whole sphere; but for the cone, instead of
the variation belonging to that of the corresponding portion of the
sphere, which will be expressed by the product of its contents into ■} of
the variation of the radius, we shall have the content of the cone into

the variation of its mean distance, or— (c* ^x*) x into 1 -r ^ • —

that is, -r (^* — *■) * «— » instead of 2wc (c^x) - into 4 c-- , or —

(c* — c*x) J , the difference being - (3c*— 40*0:+ **)i~ > ^^^ ^^h par-
ticle at the distance x from the surface ; and in order to find the total
difference for the whole stratum, we must multiply this by the fluxion of

«, and find the fluent, which will be j (3€*x^2c*x* +"i«»)— or, when

a=sc,J. J- *^* o"= {T ^*--» and for the length «» tt c* Ju, while the
force of the whole stratum of the thickness c, would have been

- c* c, substituting c for M in article 4, and the deficiency is to
3

the whole force as ^ to -^ or as 1 to 5.)

Ck)KOLLART. If the cohesive force C and the repulsive 22 be in equi-
librium for the whole fluid considered as incomparably -greater in
thickness than c or r, the diflerence of the forces with regard to the

superficial stratum on each side only, will ^^ « ' T (*^*^ ~ ^^^) • ^^^

it has been shown that c*C=r*B, consequently c*C - r^R^c^C

(c— r), and the joint deficiency in the cohesive force will be -• -^ c*(7

0-7>

Corollary 2. The deficiency being positive when c is greater than
r, it follows that if the superficial cohesion prevail in a fluid so con-
stituted, it must be because r is greater than c and the defect is greatest
with regard to the repulsive force. In such cases the fluid must be

jzd't

490 ON THE COHESION OF FLUIDS, No. XXL

slightly condensed in its interior parts, so as to prodaoe a resistance
equivalent to the excess of cohesion of the surface.

CoROLLABT 3. These conclusions are applicable, with slight modid-
cations only, to the case of a- repulsion like that of elastic fluids, as
assumed by Newton. For we have only to take r equal to the radius
of the actual mean sphere of action for the fluid in any given state of
compression, and the superficial deficiency of the force will be very nearly
as determined by this proposition, the distance r becoming in this case
somewhat smaller than the whole extent of the sphere of action. The
utmost possible cohesive force would be obtained fix>m ihe supposition

^iBt c is incomparably smaller than r, and this force would be-^ •^r'i^,

or j- of the repulsive force of a stratum of the interior part of the fluid

of the thickness r; but in every case that can actually occur, the
superficial force must probably be much less than tliis.

Scholium. On the whole we are fully justified in concluding that,
since the phenomena of capillary action necessarily lead us to infer the
existence of a superficial tension, and since without this supposition,
we should be obliged to admit the possibility of a perpetoal source
of motion, from an unequal hydrostatic pressure, upon any floating
body not homogeneous ; the existence of such a cohesive tension proves
that the mean sphere of action of the repulsive force is more extended
than that of the. cohesive : a conclusion, which, though contrary to the
tendency of some other modes of viewing the subject, shows the
absolute insufficiency of all theories built upon the examination of one
kind of corpuscular force alone. It must also be recollected that, as fiur
as our experiments enable us to observe, the repulsive force of solids
does actually extend farther than the cohesive, though, with respect
to its mean intensity, we have no direct method of ascertaining the
comparative extent of the spheres of action of the two forces.*

* Dr. Young refers, in the Preface of the work, from which the Article in the text
is taken, to the investigations which it contains as forming an important step in this
difficult theory, by aiding our conception of the existence of superficial cohesion,
consistently with the known laws of the pressure of fluids. — Note by the Editor,

No. XXU. HTDRATnJO 1NVE8TIGATI0N8. 491

No.xxn.
HYDRAULIC INVESTIGATIONS.

From the Philosophical Tnmnctions for 1808.
Read May 5, 1808.

I. — Of the Friction and Discharge of Fluids running in Pipesy
and of the Velocity of Rivers.

Having lately fixed on the discussion of the nature of inflam-
mation for the subject of an academical exercise,* I found it
necessary to examine attentively the mechanical principles of
the circulation of the blood, and to investigate mmutely and
comprehensively the motion of fluids in pipes, as afiected by,
friction, the resistance occaaoned by flexure, the laws of the
propagation of an impulse tlirough the fluid contained in an
elastic tube^ the magnitude of a pulsation in diflferent parts of
a conical vessel, and the effect of a contraction advancing pro-
gressively through the length of a given canal. The physio-
logical application of the results of these inquiries I shall have
the honour of laying before the Royal Society at a future time ;
but I have thought it advisable to communicate, in a separate
paper, such conclusions, as may be interesting to some persons,
who do not concern themselves with disquisitions of a physiolo-
gical nature; and I imagine it may be as agreeable to the
Society that they should be submitted at present to their con-
sideration, as that they should be withheld until the time
appointed for the delivery of the Croonian Lecture.

It has been observed by the late Professor Robison, that
the comparison of the Chevalier Dubuat's calculations with
his experiments is in all respects extremely satisfactory ; that
it exhibits a beautiful specimen of the means of expressing the

* On taking the degree of M.D. at Cambridge. This exercise has not been pre-
served.— Note by the Editor.

492 HYDRAULIC INVESTIGATIONS. No. XXII.

general result of an extensive series of observations in an
analytical formula, and that it does honour to tiie penetration,
skill, and address of Mr. Dubuat, and of Mr. De St. Honore,
who assisted him in the construction of his expressions.* I am
by no means disposed to dissent from this encomium ; and I
agree with Professor Bobison, and with all other late authors
on hydraulics, in applauding the unusually accurate coincidence
between these theorems and the experiments from which they
were deduced. But I have already taken the liberty of re-
markings in my lecture on the history of hydraulics,t that the
form of these expressions is by no means so convenient for
practice as it might have been rendered; and they are also
liable to still greater objections in particular cases, since, when
the pipe is either extremely narrow, or extremely long, they
become completely erroneous : for notwithstanding Mr. Dubuat
seems to be of opinion, that a canal may have a finite incli-
nation, and yet the water contained in it may remain perfectly
at rest, and that no force can be sufficient to make water flow
in any finite quantity through a tube less than one twenty-fifUi
of an inch in diameter, it can scarcely require an argument to
show that he is mistaken in both these respects. It was there-
fore necessary for my purpose to substitute, for the formulae of
Mr. Dubuat, others of a totally different nature ; and I could
follow Dubuat in nothing but in his general mode of con-
sidering a part of the pressure, or of the height of a ^ven
reservoir, as employed in overcoming the friction of the pipe
through which the water flows out of it ; a principle, which, if
not of his original invention, was certainly first reduced by him
into a practical form. By comparing the experiments, which
he has collected, with some of Gerstner, and some of my own,
I have ultimately discovered a formula, which appears to agree
fully as well as Dubuat*s, with the experiments from which his
rules were deduced, which accords better with Gerstner*s
experiments, which extends to all the extreme cases with equal
accuracy, which seems to represent more simply the actual
operation of the forces concerned, and which is directed in its

* See Sir David Brewster's edition of Dr. Kobison's works, vol. ii. p. 450,
* Theory of Rivers.* — Note by the Editor, \ Lecture xxx., p. 394,

No. XXn. HYDRAULIC INVESTIQATIONS. 493

application to practice, without the necessity of any successive
approximations.*

I began by examining the velocities of the water, discharged,
through pipes of a given diameter, with different degrees of
pressure ; and I found that the friction could not be represented
by any single power of the velocity, although it frequently
approached to the proportion of that power, of which the ex-
ponent is 1.8 ; but that it appeared to consbt of two parts, the
one varying simply as the velocity, the other as its square.
The proportion of these parts to each other must however be
considered as different, in pipes of different diameters, the first
part being less perceptible in very large pipes, or in rivers, but
becoming greater than the second in very minute tubes, while
the second also becomes greater, for each 'given portion of the
internal surface of the pipe, as the diameter is diminished.

If we express in the first place, all the measures in French
inches, calling the height employed in overcoming the friction
/, the velocity in a second v, the diameter of the pipe cf, and its

length /, we may make y*=a^^+ Sc^t;; for it b obvious

that the friction must be directiy as the length of the pipe ;
and since the pressure is proportional to the area of the

* In the article Hydraulics, which appeared in the Supplement of the Encyclo-
pfcdia Britannica, Dr. Young has not only recapitulated the principal results of this
Memoir, but has also notic^ the earlier attempts made by some other authors to
improve this department of science, and more especially those of Prony in his
' Rccherches Physico-Math^matiques sur la thebrie des eilux courantes,* which were
published in 1804. *' Ingenious and important," says he, **as the Chevalier Dubuat's
theory of the friction of fluids is admitted to hare been, it cannot be denied that it is
extremely deficient both in the distinct elucidation of the physical grounds of the
phenomena and with respect to the neatness and simplicity of the methods of calcula-
tion." M. Gerard, according to M. de Prony, was the first who entertained the fortu-
nate idea of applying the theory of M. Coulomb to the resifrtonre of water flowine in
pipes and canals : and in his two Memoira on the ' Theory of Running Waters, he
proposes a formula dependent on the sum of the velocity and ite square, for the
expression of the friction; and deducing a constant coefficient from twelve experi-
ments of Chezy and Dubuat, he obtains a formula equally correct with that of Dubuat
and far more simple.

*< M. Prony has obtained an expression founded on the same theoretical principles
that M. Gerard had adopted, but much more perfect and accurate and agreeing very
sufilciently with all those of Dubuat's experiments to which he has appUed it. But
M. de Prony*s formula, as well as Dubuat's, fiuls altogether when we attempt to
employ it for the computation of the discharge through very slender tubes; and
though his work was printed a year or two earlier than Dr. Young's investi^tions,
it would have been impossible for him to make any use of it in his physiological
inquiries, even if it had not been wholly unknown te him, since the resistences he had
to compute were principally such as occurred in tubes considerablj less than a
thousandth of an inch in diameter.** — NoU hy the Editor.

494 HYDRAULIC INVESTiaATIONS. No. XXII.

section, and the surface producing the friction to its circum-
ference or diameter, the relative magnitude of the friction
must also be inversely as the diameter, or nearly so, as
Dubuat has justly observed. We shall then find that

a must be .0000001 (430 + j-^T^'TTi)' ^^
c = .0000001 (^'^^ J, (1050 + ^% S))'* Hence it
is easy to calculate the velocity for any given pipe or
river, and with any given head of water. For the height
required for producing the velocity, independently of friction,
is, according to Dubuat,-^, or rather as it appears from

478

almost all the experiments which I have compared, ^l
and the whole height h is therefore equal ^ f + Tsq^ ^'

bcl

€ = -J , t^ + 2«? = M, whence t? = \/ (6A + «*) — «. In order

to adapt this formula to the case of rivers, we must make

/ infinite ; then b becomes ^, and *A = ~ • 7 = ~> * being

the sine of the inclination, and d four times the hydraulic
J . , J . . , c J (ads + a?) — c

mean depth; and smce e is here = -^ t? = »

and in most rivers, v becomes nearly ^ (20000 ds).

In order to show the agreement of these formulae with the
result of observation, I have extracted, as indiscriminately and
impartially as possible, forty of the experiments made and
collected by Dubuat ; I have added to these some of Grerstner's,.
with a few of my own ; and I have compared the results of
these experiments with Dubuat's calculations, and with my own
formulfiB, in separate columns. There are six of Dubuat's
experiments which he has rejected as irregular, apparently
without any very sufficient reason, since he has accidentally
mentioned that some of them were made with great care : I have
therefore calculated the velocities for these experiments in both
ways, and compared the results in the separate table on p. 495.

* It shonid always be kept in mind, that these and the subsequent formnUe
are almost entirely empirical. — Note by the Editor.

No. XXIL

HTDRAtTLIC mV^OlOATIONS.

495

Tabular Comparison of Hydraulic Experbmaai.

OcMervcT*

1
t

Dab.

mtio.

.^•..

e 1 /

.I^X'(20000A.)

Dubuat S

t$2.5

35723

15.96

12.56?

10.53

.0776 ]

LI. 10.

0537 424

952

11.1

558.5

6413

31.77

26.63?'

28.76

.0334 28.021.

0221 424

952

28.3

92.4

21827

.9.61

7.01?

8.38

.0775

8.14.

0649 415

914

9.3

75.6

27648

7.27

5.07?

6.55

.1112

6.27,

0923 413

887

7.5

17.6

9288

• .

6.70

5.86

.0120

5.97'.

0291 376

465

6.1

16.4

432

.. ,

82.52 i

U.61

.012430.67;.0255|374

451

27.6

11.7

1412

, ,

14.17 1

L3.59

.018214.05.0037 360

416

12.2

9.9

427

• . -

22.37 i

24.37

.0372 24.41 .0379'355 |414

21.7

5.8

212

-I'

27.51 27.19

.005127.34.0027 332 466 ' 23.5

1 1 1

ObMrrera.

Dob.

1^-

is-.

.KX

CoQplet

43200

145.08

39.16

40.51

.0148

38.49

.0075

376

469

84240

25.00

5.32

5.29

.0024

5.40

.0065

326

492

16.75

4.13

4.23

.0103

4.21

.0083

5.58

2.01

2.25

.0490

2.01

.0000

Bossut

2.0

2160

24.73

24.08

.0115

24 76

.0006

287

747

16.38

16.10

.0075

16.86

.0125

1080

35.77

35.10

.0082

35.05

,0089

360

58.90

58.80

.0007

56.85

.0154

1.3

2160

12.56

12.75

.0065

13.28

.0242

270

919

1080

28. 08

28.21

.0020

28.84

.0116

860

48.53

49.52

.0088

48.66

.0015

600

22.28

21.98

.0055

22.83

.0106

259

1063

12.22

11.76

.0167

11.92

.0108

Diibaat

737

23.7

28.67

29.41

.0111

30.11

.0213

12.2

19.99

19.95

.0009

20.67

.0145

4.2

10.56

10.66

.0041

10.90

.0137

117

84.95

85.52

.0029

83.12

.0069

58.31

58.47

.0014

58.41

.0012

.24

167

86.25

53.25

85.77

85.20

.0029

85.71

.0003

309

2268

41.25

73.81

73.90

.0005

74.67

.0050

20.17

51.96

50.14

.0155

50.87

.0093

5.00

23.40

23.19

.0039

23.09

.0058

.83

7.58

8.22

.0420

7.22

.0212

.16

36.25

51.25

64.37

64.95

.0031

64.08

.0021

402

2827

38.75

54.19

55.32

.0090

54.93

.0055

15.29

33.38

33.17

.0028

32.67

.0094

2.04

10.62

10.49

.0053

9.24

.0604

34.17

42.17

45.47

46.21

.0070

45.88

.0039

518

.3405

35.33

41.61

41.71

.0010

41.55

.0006

14.58

26.20

25.52

.0114

24.94

.0214

2.08

7.32

8.35

.0572

6.98

.0206

Mean

.0178

Menn

.0169

= L. 1.042

= L. 1.040

496

HYDBAULIC INVESTIGATIONS.

No. xxn.

ObMivets*

Dttb.

Log.

Log.
xat.

Gerstner

at55.5<^F.

10.7

24.2

23.9

.006

24.1

.002

349

2533

7.7

21.0

19.9

.023

19.1

.042

4.7

15.8

14.9

.026

13.9

.056

1.7

7.5

8.2

.039

6.9

• 036

2.5

5.0

.301

3.4

.133

10.7

27.1

23.4

.064

22.5

.081

488

3259

7.7

23.2

19.4

.077

18.5

.098

4.7

15.4

14.6

.024

13.5

.058

1.7

5.6

8.1

.160

6.7

.078

2.3

4.6

.301

3.4

.169

.0674

10.7

10.0

8.9

.051

10.1

.004

975

5700

7.7

7.2

7.4

.012

8.2

.057

4.7

4.5

5.6

.095

5.6

.095

1.7

1.5

3.1

.316

2.5

.222

1.8

.444

1.1

.342

Mean .129

Mean

.098

= L. 1.346

=L. 1.254

Y.ateo^

8.50

32.4

14.40

13.36

.032

2956

13882

Til

3.4-2
1.17

30.0
5.8

.53

.27

.52
.30

.008
.046

13404

452100

Mean .029

s:L.1.068

Dabuat .

255.25

36.35

86.31

84.2

.011

79.7

.035

287

747

36.25

122.59

117.8

.018

120.8

.007

259

1063

106.45

101.1

.022

104.1

.010

84.85

82.2

.013

.84.8

.000

59.25

57.5

.013

59.7

.004

27.08

118.67

111.5

.027

118.5

.000

MeaE

1.017

Mean .009

= L. 1.041

=L. 1.022

It appears from this comparison, that in the forty experiments
extracted from the collection, which served as a basis for
Dubuat's calculations, the mean error of his formula is ^^ of
the whole velocity, and that of mine tV only ; but if we omit
the four experiments, in which the superficial velocity only of a
river was observed, and in which I have calculated the mean
velocity by Dubuat's rules, the mean error of the remaining
36 is -bV) according to my mode of calculation, and tV according
to Mr. Dubuat's ; so that on the whole, the accuracy of the
two formulae may be considered as precisely equal with respect
to these experiments. In the six experiments which l^ubnat
has wholly rejected, the mean error of his formula is about ^,

No. XXII.

HYDRAULIC INVESTIGATIONS.

497

and that of mine ^. In fifteen of Gerstncr's experiments the
mean error of Dubuat's rule is one third, that of mine one
fourth ; and in the three experiments which I made with very
fine tubes, the error of my own rules is one fifteenth of the
whole, while in such cases Dubuat's formulae completely fail.

I have determined the mean error by adding together the
logarithmic ratios of all the results, and dividing the sum by
the number of experiments. It would be useless to seek for a
much greater degree of accuracy, unless it were probable that
the errors of the experiments themselves were less tfian those
of the calculations ; but if a sufficient number of extremely
accurate and frequently repeated experiments could be obtained,
it would be very possible to adapt my formula still more
correctly to their results.

In order to facilitate the computation, I have made a table
of the coefficients a and c for the different values of d^ all the
measures being still expressed in French inches.

Table of CoefficienJts for French Inches.

.Ux

a
A7x

c
.VX

a
A7x

c
A^X

a
A7x

c
A7X

430

900

400

719

4 i

319

540

257

1717

SOO

427

943

393

618

3 1

305

617

268

1895

400

426

946

387

560

2.5,

296

687

279

2008

800

423

950

380

492

288

751

803

2225

200

421

951

370

427

1.5'

275

866

349

2532

100

416

923

354

414

1 1

259

1063

402

2827

415

911

350

421

255

1123

.15

440

8026

413

896

345

433

.8'

252

1193

458

3116

410

872

340

440

.7i

249

1278

518

3405

408

840

335

462

248

1384

589

3693

406

792

'325

512

249

1524

646

3985

For example, in the last experiment, where d is 1, / 4, and
A 27.1, we have a = .0000259, b = ^^,^^,00182 = ^^^^ ^ =
.0001063, e = bcl:d== .22, and v = V(W + 6») - e = 118.46,
which agrees with the experiment within rfir of the whole. I
had at first employed for a the formula Y^HTd "*"rf "^ eS*

VOL. I.

2 K

498 HYDRAULIC INVESTIGATIONS. No. XXII.

but I found that the value thus determined, became too great
when d was about 20, and too small in some other cases.
Coulomb's experiments on the friction of fluids, made by
means of the torsion of wires, give about .00014 for the value
of e, which agrees as nearly with this table, as any constant
number could be expected to do. I have however reason
to think, from some experiments communicated to me by
Mr. Robertson Buchanan, that the value of a, for pipes above
half an inch in diameter, is somewhat too small ; my mode of
calculation, as well as Dubuat's, giving too great a velocity in
such cases.*

If any person should be desirous of making use of Du-
buat's formula, it would still be a great convenience to begin
by determining v according to this method; then, taking

b = ^ , p< . 478 > or rather, as Langsdorf makes it, ft = jn^liS*
to proceed in calculating v by the formula v = 148.5

(Vd - .2) • (^6,H.L.V(ft-H-6)"~ -^^^^ ^^ **^ <*®*«''-
mination of b will, in general, be far more accurate than the

* ** It must be confessed," says Dr. Young in the article Hjdnmlics referred to
before, *' that notwithstanding the convenience of this theory- ior calculation, with
the assistance of the tables of coefficients, their determination from the diameters of
the pipes is somewhat too complicated either for elegance or probability, if con-
sidered as representing the law of natnre. The formula of Prony, though it fails for
small pipes, has the advantage of great simplicity, and even of superior accuracy,
within certain limits of the magnitude of the pipes, although it seems to be indebted
for this accuracy to accidental causes only."

** If we take the equation v = ^^ + c ) -^ c ^^^^ .^ ^^^ ^^^ ^^ aasome

a
for a the constant mean value .0000377, and for c, .00003466, we shall have
V ^ fj (26520eb -f .845) - .919, which is equivalent to Prony's formola reduced

to French inches. For pipes, M. de Prony merely substitutes -r for s, neglecttog
entirely the height -rrr due to the velocity. He also gives a still simpler approxima-
tion for common purposes, v = fjl 26518^ ) ^' ^ ( ^^^^^^ ) which differs

very little from the rate given by Dr. Young in his Lectures, which is *' that the velo*
city is a mean proportional between the hyi&anlic mean depth and the fall in 2800
yards, for this, in French measures, would be ^ (27000cb). It is obvioui^ however*
that, in many cases, these formulie must require considerable modification, since, when
the velocity is great, the height due to it may become considerable, and since the
friction In small pipes is certainly increased beyond its mean value : nor can these
opposite causes of error be expected always to compensate each other even in pipes
of moderate dimensions." — Note by the Editor,

No. XXII.

HYDRAULIC INVESPTIOATIONS.

499

simple expression b

I + 45J

and the continued repetition of

the calculation, with approximate values of v^ may thus be
avoided. Sometimes, indeed, the values of v found by this
repetition, will constitute a diverging instead of a converging
series, and in such cases, we can only employ a conjectural
value of 17, intermediate between the two preceding ones.

Having sufficiently examined the accuracy of my formula,
I shall now reduce it into English inches, and shall add a
second table of the coefficients, for assisting the calculation. In

this case, a becomes .0000001 (413 + -^ -

1440

180

c = .0000001

(

900cM
rfJT 1136 ■*■ \fd
bcl

+ :;s(i085 + i^ +

d + 12.8 rf+.855
13_21 1.0563\

d "^ dd )

)

and

^ = fl/:rf-H.ooi7i> ^ ^ing — » and r = V (bh + 6*) - e?, or
= V( ^ + -- ) ^-j as before ; and in either case the superficial

velocity of a river may be found, very nearly, by adding to
the mean velocity v its square root, and the velocity at the
bottom by subtracting it.

Tabh of CoeffmenUfar

English Inches,

a
.KX

c
.irx

.ux

a
.KX

.ux

413

900

383

698

306

556

254

1779

500

410

944

377

597

292

635

268

1963

400

409

948

371

526

2.5

284

694

280

2082

300

406

951

364

482

277

774

305

2307

200

404

951

354

430

1.5

266

894

354

2631

100

399

918

339

413

251

1099

409

2943

398

903

336

421

248

1161

.15

447

3150

396

885

331

433

245

1234

466

3251

393

860

327

449

243

1322

528

3558

391

825

322

471

243

1433

599

3866

389

772

312

507

245

1578

657

4183

2k 2

500 HYDRAULIC INVESTIGATIONS. No. XXII.

II. — Oftlie Resistance occasioned by Flexure in Pipes or
Rivers,

Mr. Dubuat has made some experiments on the eflTect of
the flexure of a pipe in retarding the motion of the water flow-
ing through it ; but they do not appear to be by any means
suflScient to authorise the conclusions which he has drawn from
them. He directs the squares of the sines of the angles of
flexure to be collected into one sum, which, being multiplied
by a certain constant coefficient, and by the square of the
velocity, is to show the height required for overcoming the
resistance. It is, however, easy to see that such a rule must
be fundamentally erroneous, and its coincidence with some
experiments merely accidental, since the results afibrded by
it must vary according to the method of stating the problem,
which is entirely arbitrary. Thus it depended only on Mr.
Dubuat to consider a pipe bent to an angle of 144° as con-
sisting of a single flexure, as composed of two flexures of 72^
each, or of a much greater number of smaller flexures, although
the result of the experiment would only agree with the arbi-
trary division into two parts, which he has adopted. This diffi-
culty is attached to every mode of computing the eflfect either
from the squares of the sines or from the sines themselves ; and
the only way of avoiding it is to attend merely to the angles of
flexure as expressed in degrees. It is natural to suppose that
the efiect of the curvature must increase, as the curvature itself
increases, and that the retardation must be inversely pro-
portional to the radius of curvature, or very nearly so ; and
this supposition is sufficiently confirmed, by the experiments
which Mr. Dubuat has employed in support of a theory so
diflferent. It might be expected that an equal curvature would
create a greater resistance in a larger pipe than in a smaller,
since the inequality in the motions of the difierent parts of the
fluid is greater ; but this circumstance does not seem to have
influenced the results of the experiments made with- pipes of an
inch and of two inches diameter : there must also be some
deviation from the general law, in cases of very small pipes

No. XXII.

HYDRAULIC IJTV^ESTIGATIONS.

501

having a great curvature, but this deviation cannot be deter-
mined without further experiments. Of the 25 which Dubuat
has made, he has rejected 10 as irregular, because they do not
agree with his theory : indeed 4 of them, which were made
with a much shorter pipe than the rest, differ so manifestly
from them that they cannot be reconciled : but 5 others agree
sufficiently, as well as all the rest, with the theory which I have
here proposed, supposing the resistance to be as the angular
flexure, and to increase besides almost in the same proportion
as the radius of curvature diminishes, but more nearly as that
power of the radius of which the index is {. Thus if /? be
the number of degrees subtended at the centre of flexure,
and q the radius of curvature of the axis of the' pipe in French

inches, we shall have r = ^(mooQ °^*^'y> ^^» ^^""^ accurately,

r = * ^— . These calculations are compared with the

whole of Dubuat's experiments in the following table.

Table of Experiments

on the Resistance occasioned

by Flexure.

c»

Y. 1

Y. 2

»«

Y.l

Y.2

288

3.22

15030

4.75

6.71

6.98

288

3.22

3415

1.50

1.57

1.52

1.58

11330

3.50

5.06

5.26

144

.75

.78

.76

.79

7199

2.33

3.21

3.34

.37

.39

.38

.89

3510

1.08

1.56

1.62

196.5

6.12

.75

.78

.55

.62

ai6

7216

2.49

2.42

2.52

112.5

.53

1.50

3.63

3.00

144

1.50

1.66

1.61

1.67

720

3.22

5125

5.90

5.72

5.95

.75

.83

.80

.83

288

8458

1.64

1.59

1.54

1.60

196.5

6.12

1.50

1.66

1.16

1.31

860

.41

.40

.38

.40

147.4
98.3

1.12
.75

1.24
.83

.87
.58

.98
.65

821
3448

.39
1.33

.38

.37
1.21

.38
1.30

288

4.10

49.1

.37

.41

.29

.33

7449

2.90

2.59i2.78

112.5

.53

6.00

7.68

6.36

294.8

9.91

5.90

6.74

5.60

360

4.1

8.64

8.08 8.62

S88

3.22

3415

1.50

1.57

1.52

1.58

112.5

l.lj

In tlie last three experiments, the diameter of the pipe was
two inches. The radius of curvature is not ascertained within
the tenth of an inch, as Dubuat has not mentioned the thick-
ness of the pipes. The mean error of his formula in fifteen ex-
periments, and of mine in twenty, is i^y of the whole.

502 HYDRAULIC INVESTIGATIONa No. XXII.

III. — Of the Propagation of an Imptdse through an elastic

Tube,

The same reasoning that is employed for determining the
velocity of an impulse, transmitted through an elastic solid or
fluid body, is also applicable to the case of an incompressible
fluid contained in an elastic pipe ; the magnitude of the modulus
being properly determined according to the excess of pressure
which any additional tension of the pipe is capable of pro-
ducing ; its height being such as to produce a tension, which
is to any small increase of tension produced by the approadi of
two sections of the fluid in the pipe, as their distance to its
decrement : for in this case the forces concerned are precisely
similar to those which are employed in the transmission of an
impulse through a column of air enclosed in a tube, or through
an elastic solid. If the nature of the pipe be such, that its
elastic force varies as the excess of its circumference or dia-
meter above the natural extent, which is nearly the usual
constitution of elastic bodies, it may be shown that there is a
certain finite height which will cause an infinite extension, and
tliat the height of the modulus of elasticity, for each point, is
equal to half its height above the base of this imaginary
column ; which may therefore be called with propriety the
modular column of the pipe : consequently the velocity of an
impulse will be at every point equal to half of that which is
due to the height of the point above the base ; and the velocity
of an impulse ascending through the pipe being every where
half as great as that of a body falling through the correspond-
ing point in the modular column, the whole time of ascent
will be predsely twice as great as that of the descent of the
falling body ; and in the same manner, if the pipe be inclined,
the motion of the impulse may be compared with that of a body
descending or ascending freely along an inclined plane.

These propositions may be thus demonstrated : let a be the
diameter of the pipe in its most natural state, and let this dia-
meter be increased to b by the pressure of the column c, the
tube being so constituted that the tension may vary as the

No. XXII. HYDRAULIC INVESTIGATIONS. 503

force. Then the relative force of the column c is represented
by bcy since its efficacy increases, according to the law of hy«
drostatics, in the raUo of the diameter of the tube ; and this
force must be equal, in the state of equilibrium, to the tension
arising from the change from a to by that is, to b—a; conse-
quently the height c varies as — r^; &nd if the tube be enlarged

to any diameter ^, the corresponding pressure required to
distend it will be expressed by a height of the column equal to

(l - l) • i^a^ since*-^: c : :^ : (l - ^)^. Now

if the diameter be enlarged in such a degree, that the length
of a certain portion of its contents may be contracted in the
ratio 1 : 1 — r, r being very small, then the enlargement will

be in the ratio 1:1 + ^ » that is, x' will be ^ ; but the incre-
ment of the force, or of the height, is — • ^^ i which will

become ^ • j-r^* Now in a tube filled with an elastic fluid,
the height being A, the force in similar circumstances would be
rA, and if we make A = 2i ' S"T^> ^^ velocity of the pro-
pagation of an impulse will be the same in both cases, and
will be equal to the velocity of a body which has fallen through
the height i k. Supposing x infinite, the height capable of

producing the necessary pressure becomes r , which may

be called g, and for every other value of x this height is
( ^ ■" "ij^i ^^ ff'^ 7» or y - 2 A, since h becomes ^» so that
h is always equal to half the difierence between g and the
actual height of the column above the ^ven point, or to half
the height of the point above the base of the column.

If two values of x, with their con-esponding heights, are
given as b and z, corresponding to c and d^ and it is required

to find a ; we have — j— : c :: — j— : cf, dbx — dax = cbx — cAa,

and a = ^^^, or ^ = dT^' ^^ ^^ ^^^ ^^^f^^^ equiva-
lent to the tension vary in the ratio of any power m of the

504 HYDRAULIC INVESTIGATIONS. Ko. XXII.

diameter, so that, n being a small quantity, x = h (\ +n) and

, /I.N* 6c((l+n). (l + mii)-l) mn±n .
a =c (1 + fnn\ - = - .- ^ = , since

^ ^' « 6c ((1 +«).(! +«n)-(l+«)) «« '

the square of n is. evanescent, and - = - — . For example,
if m = 4, - = 4» and if m = 2, ft : a : : 3 : 2.

IV. — Of the Magnitude of a diverging Pulsation at different

Points.

The demonstrations of Euler, Lagrange, and Bemouilli,
respecting the propagation of sound, have determined that
the Telocity of the actual motion of the individual particles of
an elastic fluid, when an impulse is transmitted through a
conical pipe, or diverges spherically from a centre, varies in the
simple inverse ratio of the distance from the vertex or centre, or
in the inverse subduplicate ratio of the number of particles
affected, as might naturally be inferred from the general law of
the preservation of the ascending force or impetus, in all cases
of the communication of motion between elastic bodies, or the
particles of fluids of any kind. There is also another way of
considering the subject, by which a similar conclusion may be
formed respecting waves diverging from or converging to a
centre. Suppose a straight wave to be reflected backwards and
forwards in succession, by two vertical surfaces, perpendicular
to the direction of its motion ; it is evident that in this and
every other case of such reflections, the pressiure against the
opposite surfaces must be equal, otherwise the centre of inertia
of the whole system of bodies concerned would be displaced by
their mutual actions, which is contrary to the general laws of
the properties of the centre of inertia. Now if, instead of one
of the surfaces, we substitute two others, converging in a very
acute angle, the wave will be elevated higher and higher as it
approaches the angle : and if its height be supposed to be every
where in the inverse subduplicate ratio of the distance of the
converging surfaces, the magnitude of the pressure reduced to
the direction of the motion, will be precisely equal to that of
the pressure on the single opposite surface, which will not hap-
pen if the elevation vary inversely in the simple ratio of the

No. XXII. HYDRAULIC INVESTIGATIONS. 505

distance, or in that of any other power than its square root.
This mode of considering the subject afibrds us therefore an
additional reason for asserting, that in all transmissions of im-
pulses through elastic bodies, or through gravitating fluids, the
intensity of the impulse varies inversely in the subduplicate
ratio of the extent of the parts affected at the same time ; and
the same reasoning may without doubt be applied to the case of
an elastic tube.

There is however a very singular exception, in the case of
waves crossing each other, to the generaMaw of the preser-
vation of ascending force, which appears to be almost sufficient
to set aside the universal application of this law to the motions
of fluids. It is confessedly demonstrable that each of two
waves, crossing each other in any direction, will preserve its
motion and its elevation with respect to the surface of the fluid
affected by the other wave, in the same manner as if that
surface were plane: and when the waves cross each other
nearly in the same direction, both the height and the actual
velocity of the particles being doubled, it is obvious that the
ascending force or impetus is also doubled, since the bulk of
the matter concerned is only halved, while the square of the
velocity is quadrupled ; and supposing the double wave to be
stopped by an obstacle, its magnitude, at the moment of the
greatest elevation, will be twice as great as that of a single
wave in similar circumstances, and the height as well as the
quantity of matter, will be doubled, so that either the actual
or the potential height of the centre of gravity of the fluid
seems to be essentially altered, whenever such an interference
of waves takes place. This difficulty deserves the attentive
consideration of those who shall attempt to investigate either
the most refined parts of hydraulics, or the metaphysical prin-
ciples of the laws of motion.

V. — Of the Effect of a Contraction advancing through a
Canal.

If we suppose the end of a rectangular horizontal canal,
partly filled with water, to advance with a given velocity, less than

506 HYDRAULIC INVESTIGATIONS. No. XXII.

that with which a wave naturally moves on the surface of the
water, it may be shown that a certsun portion of the water
will be carried forwards, with a surface nearly horizontal, and
that the extent of this portion will be determined, very nearly,
by the difference of the spaces described, in any given time, by
a wave, moving on the surface thus elevated, and by the
moveable end of the canal. The form of the anterior termi-
nation of this elevated portion, or wave, may vary, according to
the degrees by which the motion may be supposed to have
commenced ; but whatever this form may be, it will cause an
accelerative force, which is sufficient to impart successively to
the portions of the fluid, along which it passes, a velocity equal
to that of the moveable end, so that the elevated surface of the
parts in motion may remain nearly horizontal : and this pro*
position will be the more accurately true, the smaller the
velocity of the moveable end may be. For calling this velocity
V, the original depth a, the increased depth :r, and the velocity
of the anterior part of the wave y, we have, on the supposition
that the extent of the wave is already become considerable,

X K -^r- , taking the negative or positive sign according to the

direction of the motion of the end ; since the quantity of
fluid, which before occupied a length expressed by ^, now

occupies the length y + r : and putting a u^x - z^z = -r^^'

The direction of the surface of the margin of the wave is in-
difierent to the calculation, and it is most convenient to suppose
its inclination equal to half a right angle, so that the ac-
celerating force, acting on any thin transverse vertical lamina,
may be equal to its weight : then the velocity 7/ must be such
that while the inclined margin of the wave passes by each
lamina, the lamina may acquire the velocity t? by a force equal
to its own weight ; consequentiy the time of its passage must
be equal to that in which a body acquires the velocity v, in
falling through a height by corresponding to that velocity : and .

26
this time is expressed by — ; but the space described by the

margin of the wave is not exactly z, because the lamina in
question has moved horizontally during its acceleration

No. XXIi HYDBAULIC INVESTIGATIONS. 507

through a space which must be equal to 6; the distance actu-

z±h 25

ally described will therefore be z + ft, and we have — — = — »
2r+J = ^,ar + fty-fei? = ?^ + 22y,y«Tity = ^ - 7'
(y + i ^y = II' "^ 16 5 ^^^ ^ ^^ ^® proper coeffi-
cient, V = m j/bf and t^ = nfbt JF "^ 15 ~ "** (I "^ lej*
y=:in V(| + ^j + f «, and y + t? = m V (5+^) + i ?.
But when v is small, we may take y + 1> nearly m V |'
and J? = J^^a) = V (2aA), and or = a + V (2ai), while the

height of a fluid, in which the velocity would be y, is nearly
a + I V (2aft) : consequently, when the velocity v is at all
considerable, y must be somewhat greater than the velocity
of a wave moving on the surface of the elevated fluid ; and
probably the surface of the elevated portion will not in this
case be perfectly horizontal ; but where v is small, y may

be taken, without material error, m^ ^, or even m V |*
which is the velocity of every small wave. The coefficient m
is here assumed the same for the motion of a wave, as for
the discharge through an aperture, and I have reason from
observation to think this estimation sufficiently correct.

Supposing now the moveable end of the canal to remain
open at the lower part as far as the height e, then the excess
of pressure occasioned by the elevation before it> and the
depression behind, will cause the fluid, immediately below the
moveable plane, to flow backwards, with the velocity deter*
mined by the height, which is the difierence between the levels ;
and the quantity thus flowing back, together with that which is
contained in the moveable elevation, must be equal to the
whole quantity displaced. But the depression behind the
moveable body must vary according to the circumstances of
the canal, whether it be supposed to end abruptly at the part
from which the motion begins, or to be continued backwards
without limit : in the first case, the elevation z will be to the

508 HYDRAULIC INVESTIGATIONS. No. XXII.

depression, as v to y— t?, the length of the same portion of
the fluid being varied inversely in that ratio ; in the second
case, the proportion will beasy-hwtoy — ©: and the

difierence of the levels will be first z + z ^-^^ = 5^^ or secondly
z + z ?-ZL? _ — f5L : and first, in V — c -f (v — t?) z = (a — c) t? ;
but, since y is here considered as equal to m V 2» P^^"
*^^8 Vi"'^*"^^'"*^" '"^' *°^' calling a — c, e,
m ij— c -{- mdz = me tj b^ ^ -^ c + dz-ejh^(?~^=^
^h + <Pz^ ^ 2.Z.W^^-(a + ^)^ = -i^and,

calling -^ + ?^^/, z =/- V (/^ '^) : and in the same

manner/ is found, for the second case, equal to ^/'^ , -4- ^-^^

For example, suppose the height a 2 feet, ft = i, c = 1, and
consequently « = 1, then d becomes i, t? = 4, and y = 8 ;
and in the first case z = .1, and in the second z = .14.

If V, the velocity of the obstacle, were great in comparison
with m V |> ^6 velocity of a wave, and the space c below the

obstacle were small, the anterior part of the elevation would
advance with a velocity considerably greater than the natural
velocity of the wave : but if the space below the obstacle bore
a considerable proportion to the whole height, the elevation z
would be very small, since a moderate pressure would cause
the fluid to flow back, with a sufficient velocity, to exhaust the
greatest part of the accumulation, which would otherwise take
place. Hence the elevation must always be less than tliat
which is determined by the equation m -^ zc ^ et?, and 2; is at

most equal to ( ^ j -\h\ but since the velocity of the ante-
rior margin of the wave can never materially exceed »» V |,
especially when z is small, and V | being in this case nearly
'^ i + 277fei ^»»V|-m^ft = m(vf + ^j— - ^ ft)

No. XXII. HYDRAULIC INVESTIGATIONS. 509

which, multiplied by z, shows the utmost quantity of the fluid
that can be supposed to be carried before the obstacle. Sup-
posing J = i a, this quantity becomes rnj^ • -^ • ^; and if -
be, for example, iV, it will be expressed by -nrkirv av,

A similar mode of reasoning may be applied to other cases
of the propagation of impulses, in particular to that of a con-
traction moving along an elastic pipe. In this case, an increase
of the diameter does not increase the velocity of the transmission
of an impulse ; and when the velocity of the contraction ap-
proaches to the natural velocity of an impulse, the quantity of
fluid protruded must, if possible, be still smaller than in an
open canal ; that is, it must be absolutely inconsiderable unless
the contraction be very great in comparison with the diameter
of the pipe, even if its extent be such as to occasion a friction
which may materially impede the retrograde motion of the
fluid. The application of this theory to the motion of the
blood in the arteries is very obvious, and I shall enlarge more
on the subject when I have the honour of laying before the
' Society the Croonian Lecture * for the present year.

The resistance opposed to the motion of a floating body,
might in some cases be calculated in a similar manner: but the
principal part of this resistance appears to be usually derived
from a cause which is here neglected ; that is, the force re-
quired to produce the ascending, descending, or lateral motions
of the particles, which are turned aside to make way for the
moving body : while in this calculation their direct and retro-
grade motions only are considered.

The same mode of considering the motion of a vertical
lamina may also be employed for determining the velocity of
a wave of finite magnitude. Let the depth of the fluid be a,
and suppose the section of the wave to be an isosceles triangle,
of which the height is ft, and half the breadth c : then the force
urging any thin vertical lamina in a horizontal direction will
be to its weight as b to c ; and the space </, through which it
moves horizontally, while half the waves pass it, will be such
that (c - d) .(a + 4 ft) = ac, whence d = ^^^. But the final

« No. XXIII., which follows.

510 HYDRAULIC INVESTIGATIONS. No. XXII.

velocity in this space is the same as is due to a height equal to
the space, reduced in the ratio of the force to the weight, that

is, to the height 2^73* and half this velocity is i m Vl 2a+6/
which is the mean velocity of the lamina. In the mean time
the wave describes the space c+rf, and its velocity is greater
than that of the lamina in the ratio of^ + 1 to 1, that is

?2±* .f 1 or '^+ 2 to 1, becoming m^^+ 1^ _A_ = ^
; which, when b vanishes, becomes m js/^j as in La-

grange's theorem, and, when b is small, m ^ .,^ ; but if a
were small, it would approach Ui m ^/ by the velodty due to
the whole height of the wave.

No XXIIL ON THE HEART AND ARTERIES. 511

No. xxin.

THE CROONIAN LECTURE.

ON THE PQNCnONS OP

THE HEART AND ARTERIES.

From the Philosophical Tnnaaotions for 1809, p. 1.
Read Nov. 10, 1808.

The mechanical motions, which take place in an animal body,
are regulated by the same general laws as the motions of
inanimate bodies. Tliufl the force of gravitation acts precisely
in the same manner, and in the same degree, on living as on
dead matter ; the laws of optics are most accurately observed
by all the refractive substances belonging to the eye ; and there
is no case in which it can be proved, that animated bodies are
exempted from any of the affections to which inanimate bodies
are liable, except when the powers of life are capable of insti-
tuting a process, calculated to overcome those affections, by
others, which are commensurate to them, and which are of a
contrary tendency. For example, animal bodies are incapable
of being frozen by a considerable degree of cold, because ani-
mals have the power of generating heat ; but the skin of an
animal has no power of generating an acid, or an alkali, to
neutralise the action of an alkaline or an acid caustic, and there-
fore its texture is destroyed by the chemical attraction of
such an agents when it comes into contact with it. As far,
therefore, as the functions of animal life depend on the locomo-
tions of the solids or fluids, those functions must be capable of
being illustrated by the conaderation of the mechanical laws of
moving bodies ; these laws being fully adequate to the expla-
nation of the connexion between the motive powers, which are
employed in the system, and the immediate effects, which they

512 ON THE FUNCTIONS OP THE No. XXIH.

are capable of producing, in the solids or fluids of the body :
and it is obvious that the inquiry, in what manner, and in what
degree, the circulation of the blood depends on the muscular
and elastic powers of the heart and of the arteries, supposing
the nature of those powers to be known, must become simply a
question belonging to the most refined departments of the
theory of hydraulics.

In examining the functions of the heart and arteries, I shall
inquire, in the fir^t place, upon the grounds of the hydraulic
investigations which I have already submitted to the Royal
Society, what would be the nature of the circulation of the
blood, if the whole of the veins and arteries were invariable in
their dimensions, like tubes of glass or of bone ; in the second
place, in what manner the pulse would be transmitted from the
heart through the arteries, if they were merely elastic tubes ;
and in the third place, what actions we can with propriety attri-
bute to the muscular coats of the arteries^ themselves. I shall
lastly add some observations on the disturbances of these motions
which may be supposed to bccur in different kinds of inflamma-
tions and of fevers.

When we consider the blood-vessels as tubes of invariable
dimensions, we may suppose, in order to determine the velocity
of the blood in their different parts, and the resistances opposed
to its motion, that this motion is nearly uniform, since the alter-
nations, arising from the pulsation of the heart, do not materially
affect the calculation, especially as they are much less sensible
in the smaller vessels than in the larger ones, and the principal
part of the resistance arises from these small vessels. We are
to consider the blood in the arteries as subjected to a certain
pressure, by means of which it is forced into the veins, where
the tension is much less considerable ; and this pressure, ori-
ginating from the contraction of the heart, and continued by
the tension of the arteries, is almost entirely employed in over-
coming the friction of the vessels : for the force required to
overcome the inertia of the blood is so inconsiderable, that
it may, without impropriety, be wholly neglected. We must
therefore inquire, what the magnitude of this pressure is, and
what degree of resistance we can suppose to arise from the

No. XXIII. HEART AND ARTERIES. 513

friction of the internal surface of the blood-vessels, or fix)m any
other causes of retardation. The magnitude of the pressure
has been ascertained by Hales's most interesting experiments
on a variety of animals, and may thence be estimated with suf-
ficient accuracy for the human body ; and for determining the
magnitude of Uie resistance, I shall employ the theorems which
I have deduced from my own experience on very minute
tubes, compared with those which had been made by former
observers under different circumstances; together with some
comparative experiments on the motion of water and of other
fluids in the same tubes.

Dr. Hales infers, from his experiments on quadrupeds of dif-
ferent sizes, that the blood in the human arteries is subjected to
a pressure, which is measured by a column of the height of
seven feet and a half: in the veins, on the contrary, the pressure
appears to amount to about six inches only ; so that the force
which urges the blood from the greater arteries through the
minuter vessels into the large veins, may be considered as equi-
valent to the pressure of a column of seven feet

In order to calculate the magnitude of the resistance, it is
necessary to determine the dimensions of the arterial system,
and the velocity of the blood which flows through it. Ac-
cording to the measurements of Keill and others, we may
take I of an inch for the usual diameter of the aorta, and
suppose each arterial trunk to be divided into two branches,
the diameter of each being about i of that of the trunk (or
more accurately 1 : 1.26 = 10 — .ioo567), and the joint areas
of the sections about a fourth part greater (or 1.2586 : 1 =
10.099896). This division must be continued twenty-nine times,
so that the diameter of the thirtieth s^ment may be only the
eleven hundredth part of an inch, that is nearly large enough
to admit two globules of the blood to pass at once. The
length of the first segment must be assumed about nine inches,
that of the last, the twentieth of an inch only ; and supposing
the lengths of the intermediate segments to be a series of
mean proportionals, each of them must be about one-sixth
part shorter than the preceding (or 1 : 1.961 = 10 — .07776),
the mean length of th^ whole forty-six inches, the capacity to

VOL. I. 2 L

514 ON THE FUNCTIONS OP THE No. XXIII.

that of the first segmeDt as 72.71 to 1, and coDsequently the
weight of the blood contained in the arterial system about 9.7
pounds. It is probable that this calculation approaches suffi-
ciently near to the truth ; for the whole quantity of blood in
the body being about 40 pounds, although some have sup-
posed it only 20, others no less than 100, there is reason to
believe that half of this quantity is contained in the veins of the
general circulation, and that the other half is divided, nearly
in equal proportions, between the pulmonary system and the
remaining arteries of the body, so that the arteries of the general
circulation may contain about 9 or 10 pounds. Haller allows
50 pounds of circulating fluid, partly serous, and partly red,
and supposes i of this to be contained in all the arteries taken
together: but in a determination which must be in great
measure conjectural we cannot expect perfect accuracy : and
according to Haller's own account of the proportions of the
sections of the arteries and veins, the large trunks of the veins
appear to be little more than twice as capacious as those of the
arteries, and the smaller branches much more nearly equal, so
that we cannot attribute to the arterial system less than ^ of the
whole blood.

It may be supposed that the heart throws out, at each pul-
sation, that is about seventy-five times in a minute, an ounce
and a half of blood : hence the mean velocity in the aorta be-
comes eight inches and a half in a second : and the velocity
in each of the succeeding segments must of course be smaller,
in proportion as the joint areas of all the corresponding sec-
tions are larger than the area of the aorta : for example, in
the last order of vessels, of which the diameter is the eleven
hundredth of an inch, the velocity will be one ninety-third of
an inch ; and this result agrees sufficiently well with Hales*s
observation of the velocity in the o4>illary arteries of a frog,
which was one-nmetieth part of an inch only. It is true, that
Haller is disposed to question the accuracy of this observa-
tion, and to attribute a much greater velocity to the blood
flowing through the capillary vessels, but he did not attempt
eitlier to measure the velocity, or to determine it by calcula-
tion : nor is this the onlv instance in which Haller has been

No. XXIII. HEART AND ARTERIE& 515

led to reason erroneously, from a want of mathematical know-
ledge : he may, however, have observed the particles of blood
moving in the axis of a vessel with a velocity much exceeding
the mean velocity of its whole contents. If we calculate upon
these foundations, from the formula which I have already
laid before the Society, it will appear that the resistance which
the friction of the arteries would occasion, if water drculated
in them instead of blood, with an equal velocity, must amount
to a force equivalent to the pressure of a column of fifteen
inches and a half: to this we may add about a fourth for the
resistance of the capillary veins, and we may estimate the
whole friction for water, at twenty inches. The only consi-
derable part of this force is derived from the term ^'^^^^ * in

the value of y*: this term increases for each successive seg-
ment in the ratio 1 : 1.49425 = 1 : n, and the sum of the

series is to the first term, as r- to 1. It appears also, that

a very small portion only of the resistance is created in the
larger vessels : thus, as frur as the twentieth division, at the
distance of an inch and a quarter only from the extreme capil-
lary arteries, the pressure of a column of one-twentieth of an
inch only is required for overcoming the whole friction, and at
the twenty-fifth division, where the artery does not much exceed
the diameter of a human hair, the height to which the water
would rise, in a tube fixed laterally into the artery, is only two
inches less than in the immediate neighbourhood of the heart.

In order to judge of the comparative resistance produced by
fluids of different degrees of viscidity, I employed the same
tubes, by means of which I had determined the fiiction of
water, in extreme cases, for ascertaining the efiect of different
substances held in solution in the water : since it is impossible
to make direct experiments on the blood in its natural state, on
account of its tendency to coagulate : and those substances
which have the power of preventing its coagulation, may na-
turally be supposed to produce a material change in its viscidity.
The diameter of one of the tubes, which was cylindrical, was

* Supra, p. 493. The expression for the friction referred to does not contain this
term.— Ab^tf by the Editor,

2 L 2

516 ON THE FUx^TCTIONS OF THE No. XXIIL

the fortieth part of an inch : the bore of the other was oval, as
is usual in tike finest tubes made for thermometers : the section
divided by one-fourth of the circumference, gave one hundred
and^ seventy seconds for the mean diameter. I caused some
milky and solutions of sugar of different strength, to pass
through these tubes: they were aU transmitted much more
sparingly than water, with an equal pressure, and the difference
was more considerable in the smaller than in the larger tube,
as might naturally be expected both from the nature of tbe
resistance, and from the result of Gerstner's experiments on
water at different temperatures. In the first tube the resistance
to the motion of milk was three times as great as that of
water, a solution of sugar in five times its weight of water pro-
duced twice as much resistance as water ; in twice its weight,
nearly four times as much as water : but in the narrower tube,
the weaker solution of sugar exhibited a resistance five times as
great as that of water, which is more than twice as much as
appeared in the larger tube. Hence there can be no doubt
that the resistance of the internal surface of the arteries to the
motion of the blood must be much greater than would be found
in the case of water : and supposing it about four times as
great, instead of 20 inches, we shall have 80, for the measure
of a column of which the pressure is capable of forcing the
blood, in its natural course, through the smaller arteries and
veins, which agrees very well with Hales's estimate.

This determination of the probable dimensions of the aiWial
system, and of the resistances occasioned by its different parts,
is in some few respects arbitrary, at the same time that it cannot
be materially altered without altering either the whole quantity
of blood contained in the body, the diameters of the smallest
capillary vessels, the mean number of bifurcations, or the mag-
nitude of the resistance, all of which are here assumed nearly as
they have been laid down by former observers : the estimation
of the length of the successive segments only is made in
such a maimer as to reconcile these data with eadb other, by
means of the experiments and calculations relating to the
friction of fluids in pipes. The effect of curvature in increasing
the resistance has been hitherto neglected ; it can be only

Na XXIII. HEART AND ABTERIES. 517

sensible in the lai^r vessels ; and supposing the flexures of
these to be equivalent to the circumferences of two circles,
each two inches in diameter, the radius q being 1, we have

r = '^^^^^^^P^'^^ = .0000045 x 720 x 64 = .207, or about one-

fifth of an inch, for the additional resistance arising from this
cause in the case of water, or four-fifths for blood, which is a
very inconsiderable part of the whole.

It nught be questioned whether the experiments which I have
made, with tubes -rhr of an inch in diameter, are sufficient for
determining with accuracy the degree in which the resistance
would be increased in tubes, of which the diameter is only one-
sixth part as great ; and it may be doubted whether the ana-
logy, derived from these experiments, can be safely employed
as a ground for asserting, that so large a portion of the arterial
pressure is employed in overcommg the resistance of the very
minute arteries But it must be remembered, that these expe-
riments are at least conclusive with respect to the arteries larger
than the tubes employed in them, and even those which are a
little smaller ; so that the remaining pressure, as observed in
experiments, can only be employed in overcoming the resist-
ance of the minuter arteries and veins, and these observations
tend therefore immediately to confirm the analogy drawn from
the experiments on the motion of water. It might indeed be
asserted that the viscidity of the blood exceeds that of water in
a much greater ratio than that which is here assigned ; but this
is rendered improbable by some experiments of Hales, in
which, when the intestines were laid open, on the side opposite
to the mesentery, so that many of the smaller arteries were
divided, the quantity of warm water which passed through
them with an equal pressure, was only about twelve times as
great as that of the blood which flows through them in their
natural state ; and it is probable that at least three or four times
as much of any fluid must have passed through them in their
divided, as in their entire state, unless we suppose that the coats
of the divided vessels, like many other muscular parts, are
capable of being contracted by the contact of water. In some
other experiments, it was found that a moderate degree of pres-

518 ON THE FUNCTIONS OF THE No. XXIIL

sure was capable of causing water to exude so copiously through
the exhalant vessels of the intestines, that it passed through Uie
aorta with a Telocity of about two inches in a second, although
these vessels do not naturally allow any passage to the blood :
on the other hand, it sometimes happened that very little water
would pass through such channels as naturally transmitted a
much larger quantity of blood : a circumstance which Dr. Hales
very judiciously attributes to the oozing of the water into tlie
cellular membrane surrounding the vessels, by means of which
they were compressed, and their diameters lessened. On the
whole, it is not improbable that, in some cases, the resistance,
opposed to the motion of the blood, may exceed that of water
in a ratio somewhat greater than I have assigned ; but this must
be in the minutest of the vessels, while in the larger arteries
the disproportion must be less : so that, however we may view
the subject, it appears to be established, that the only con^der-
able resistance which the blood experiences, occurs in the ex-
treme capillary arteries, of which the diameter scarcely exceeds
the hundredth part of an inch.

We cannot suppose that the dimensions of the sanguiferous
system agree uniformly, in all its parts, with the measmres
which I have laid down ; but the truth of the inference is not
affected by these variations. For example, there may perhaps
be some arteries communicating with veins, of which the dia-
meter exceeds the eleven-hundredth of an inch ; but there are
certainly many others which are much more minute ; and the
blood, or its more liquid parts, passing through these more
slowly, it must move more rapidly in the former, so that the
resistance may in all be equal to the pressure, and the mean
velocity may still remain such as is determined by the quan-
tity of blood passing through the aorta. There is indeed some
uncertainty in the measure of the globules of the blood, which
I have made the basis of the dimensions of the minute arteries :
and I have reason to think, that instead of Winr of an inch,
their greatest diameter does not exceed ttfW} or even tVt? :
the general results of the investigation are not however affected
by this difference : it will only require us to suppose the sub-
divisions somewhat more numerous, and the branches shorter.

No. XXIII. HEART AND ARTERIES. 519

These are the principal circumstances which require to be
conudered, with respect to the simple transmission of the blood
through the arteries into the yeins, without regard to the alter-
nate motions of the heart, and to the elastic and muscular
powers^ of the vessels. I shall next examine the nature and
velocity of the propagation of the pulse. The successiye trans^
mission of the pidsations of the heart, through the length of the
arteries, is so analogous to the motion of the waves on the sur-
face of water, or to that of a sound transmitted through the air,
that the same calculations will serve for determining the prin-
cipal afiections of all these kinds of motion ; and if the water,
which is agitated by waves, is supposed to flow at the same
time in a continued stream, and the air which conveys a sound
to be carried forwards also in the form of a wind, the similitude
will be still stronger. The coats of the arteries may perhaps
be considered, vrithout much inaccuracy, as perfectiy elastic,
that is, as producing a force proportional to the degree in which
they are extended beyond their natural dimensions; but it
is not impossible that there may be some bodies in nature,
which differ materially from this general law, especially where
the distension becomes considerable : thus there may be sub-
stances which exhibit a force of tension proportional to the
excess of the square, or the cube of their lengtib, beyond a cer-
tain given quantity. It is safest therefore to reason upon the
elasticity of any substance, from experiments made without any
great deviation from the circumstances to which the calculation
is to be applied.

For this purpose, we may again employ some of the many
excellent experiments contained in Hales's hsemastatics. It ap-
pears, that when any small alteration was made in the quantity
of blood contained in the arteries of an animal, the height of the
column, which measured the pressure, was altered nearly in
the same proportion, as far as we are capable of estimating the
quantity, which was probably contained in the larger vessels of
the animal. Hence it follows, that the velocity of the pulse
must be nearly the same as that of an impulse transmitted
through an elastic tube, under the pressure of a column of the
same height, as that which measures the actual arterial pres-

520 ON THE FUNCTIONS OF THE No. XXIII.

sure : that is, equal to that which is acquired by a heavy body
falling- freely through half this height. In mHn, this velocity
becomes about fifteen feet and a half in a second ; to which the
progressive motion of the blood itself adds about eight inches ;
and with this velocity, of at least sixteen feet in a second, it may
easily happen that the pulse may appear to arrive at the most
distant parts of the body without the intervention of any very
perceptible interval of time.

The velocity of the transmissicm of the pulse being known, it
is easy to determine the degree in which the arteries are dilated
during its passage through them. The mean velocity of the
blood in the aorta being eight inches and a half in a second, its
greatest velocity must be about three times as much, smce the
contraction of the heart is supposed to occupy only about one-
third part of the interval between two successive pulsations ; and
if the velocity of the pulse is sixteen feet in a second, that of
the blood itself must be about one-eighth part as great; so that
the column of blood occupying eight inches may occupy only
seven ; hence the diameter must increase in the ratio of about
fifteen to sixteen. The tension will also become one-eighth
greater, and the force of the heart must be capable of support-
ing a column of one hundred and one inches. This force
would, however, require to be somewhat increased, from the
consideration that the force required at the end of any canal
during the reflection of a pulsation or wave of any kind, is
twice as great as the force exerted during its transmission, and
the force employed in the origination of a wave or pulse in a
quiescent fluid, is the same as is required for its reflection ; on
the other hand, a weaker pulsation, proceeding into a narrower
channel, becomes more energetic, so that, from this considera-
tion, a force somewhat smaller would be required in the heart :
on the whole, however^ it appears probable, that the former of
these corrections must be the more considerable, and that the
force of the heart must be measured by the pressure of a
column, rather more than less than one hundred and one inches
high ; nor would this force by any means require a strong ex-
ertion of muscular power; for it only implies a 4;ension of
something less than three pounds for each inch of the circum-

Ko. XXIII. HEART AND ARTERIES. 521

ference of the greatest section of the heart ; and supposmg the
mean thickness half an inch, an equal number of the fibres of
some other muscles of the body would be capable of exerting a
force of more than two hundred pounds, in the state of the
greatest possible action.

The force, here assigned to each pulsation, agrees extremely
well with the inference that may be drawn from an experiment
of Hales on the ascent of the blood in a tube connected with an
artery of a horse. The whole height of the column being nine
feet, the blood rose about three inches higher during each pul-
sation, which was repeated fifty or sixty times in a minute : now
we may suppose the acceleration to have extended a little be-
yond the first half of the space thus described, so that two inches
were described in two-fifths of a second ; and if there had been
no friction, nor any other cause of retardation, there can be no
doubt that at least four inches would have been described in
the same time ; but the same column of nine feet, if it had
been actuated by its own weight, would have described thirty-
one inches in the same time; consequently the force with
which the blood was driven through the artery was nearly one-
eighth of the whole force of tension, as it appears in the former
calculation.

The magnitude of the pulse must diminish in the smaller
arteries in the subduplicate proportion of the increase of the
joint areas, in the same manner as the intensity of sound is
shown to decrease in diverging from a centre, in the subdupli-
cate ratio of the quantity of matter afiected by its motion at the
same time. For example, in the arteries of the tenth order, of
which the diameter is one-thirteenth of an inch, its magnitude
must be only one-third as great as in the aorta, that is, the
greatest progressive velodty of the blood must be eight inches
and a half iu a second only, and the dilatation one-fiftieth part
only of the diameter. In tiie vessels of the twentieth order, the
dilatation does not exceed riv of the diameter, which is itself
the 140th part only of an inch ; so that it is not surprising that
Haller should have been unable to discover any dilatation in
vessels of these dimensions, even with the assistance of a power-
ful microscope. If we estimated die magnitude of the pulse in

522 ON THE PUNCTIOTSS OF THE No. XXIIL

the aorta> from the excess of the temporary above the mean
Telocity, which would perhaps be justifiable, that magnitude
would become still less considerable.

These calculations agree extremely well with each other, and
with experiment, as far as they relate to the power of the heart,
and the affections of the smaller arteries. But there is reason
to think that the velocity of the pulse in the larger vessels is
much more considerable than has been here stated ; and their
dilatation is also less conspicuous, when they are exposed to
view, than it would probably be, if it were as great as is inferred
from the velocity here assigned. 1 have demonstrated, in the
hydraulic investigations which I lately laid before the Royal
Society, that the velodty of an impulse passing through a tube,
consisting of perfectly elastic materials, is half as great as that
of a body supposed to have fidlen from the given point to the
base of tiie modular column of the tube : and that the height
of this column is such that the tube would be extended without
limit by its pressure ; consequently it must be greater than the
height of a column equivalent to the pressure by which the tube
is burst. Now it has been ascertained by Dr. Hales, that the
pressxure, required for bursting one of the carotids of a dog, is
equal to that of a column of water one hundred and ninety feet
high ; nor does he remark that the artery was very materially
dilated; and deducting from this height the five feet whidi
express the actual pressure in the arteries of a dog, the remain-
ing one hundred and eighty-five feet will give a velocity of at
least fifty-four feet in a second, for the propagation of the pulse
in the dog. It is not however ascertained, that all the mem-
branes, which may have surrounded the artery in this experi-
ment, are called into' action in its ordinary pulsation, much
less that the force, developed by their tension, varies precisely
according to the general law of perfectiy elastic bodies : but
this mode of calculation is still amply sufficient to make it pro-
bable that the velocity of the pulsations, in the larger arteries,
must amount to at least forty feet in a second, although some
very considerable deductions must be made, on account of the
resistances of various kinds, which cannot be comprehended in
tlie calculation.

No. XXIII. HEART AND ABTERIE8. 523

The artery must not be suppoBed to subside, immediately
after each pulsation, precisely to its original dimensions, since
it must remain somewhat luller, in order to supply the capil-
lary arteries, and the veins, in the interyal between the two
succesfflve pulsations : and in this respect it difiers firom the
motions of a wave through a canal which is open on both
sides : but the difference may be understood, by supposing a
partial reflection of the pulse to take place at every point
where it meets with any resistance, which will leave a general
distension of the artery, without any appearance of a retrograde
pulsation.

I shall proceed to inquire, in the third place, into the nature
and extent of the functions which are to be attributed to the
muscular fibres of the coats of the arteries ; and I apprehend
that it will appear to be demonstrable, that they are much
less concerned in the progressive motion of the blood, than
is almost universally believed. The arguments which may
be employed to prove this, are nearly the same that I have
already stated, in examining the motion of a fluid, carried along
before a moving body in an open canal ; but in the case of an
elastic tube, the velocity of the transmission of an impulse
being rather diminished than increased by an increase of ten-
sion, the reasoning is still stonger and simpler ; for it may
here be safely asserted, that the anterior parts of tiie dilatation,
which must be forced along by any progressive contraction of
the tube, can only advance with the velocity appropriate to the
tube, and that its capacity must be proportionate to its length
and to the area of its section ; now the magnitude of its section
must be limited by that degree of tension which is sufficient to
force back through the contraction what remains of the dis-
placed fluid, and the length by the difference of the velocity
appropriate to the tube, and that with which the contraction
advances; consequently if tiie contraction advance with the
velocity of a pulsation, as any contractile action of the arte-
ries must be supposed to do, this length necessarily vanishes,
and with it the quantity of the fluid protruded ; the whole
being forced backwards, by the distending force which is ex-
erted by a very small dilated portion, immediately preceding

524 ON THE FUNCTIONS OF THE No. XXUL

the contraction. It might indeed be ima^ned, that the ooo-
traction follows the pulsation with a Telocity somewhat smaller
than its own ; but this opinion would stand on no other foun-
dation than mere conjecture, and it would follow, that the
pulse would always become more and more iiill, as it became
more dbtant fix>m the heart ; of which we have nothing like
eyidence: nor would a moderate contraction, even if this
supposition were granted, produce any material effect. For
example, if the velocity of the contraction were only half as
great as that of the pulsation, which is the most fayourable
proportion, it would be necessary, taking sixteen feet in a
second for the velocity of the pulsation, that the section of the
arteries should be contracted to about one-half, in order to
produce, by their progressive contraction only, the actual ve-
locity of the blood in the aorta ; one-sixteenth of the blood
being carried, in this case, before the contraction : but if the
contraction were only such, as to reduce the section of the ar-
tery to tV) which is probably more than ever actually happens,
the velocity produced would be only about -rV ^ much ; and
if the contraction were only to tVt) which is a sufficient allow-
ance for the smaller arteries, about rviw only of the actual
velocity in the aorta could be produced in this manner, even
upon a supposition much more favourable to the muscular
action of the arteries than the actual circumstances. A small
addition must be made to the force required for producing the
retrograde motion, on account of the friction to be overcome,
but the general reasoning is not affected by this correction.

The contraction of the artery might also be supposed to
remain after each pulsation, so that the vessel should not
be again dilated until the next pulsation, or, in other words,
a spontaneous dilatation might be supposed to accompany the
pulsation, instead of a contraction : but such a dilatation would
be useless in promoting the progressive motion of tbe blood,
since a larger quantity of blood, conveyed to the smaller ves-
sels, without an increased tension, would be ineffectual with
respect to the resistances which are to be overcome. It is
possible indeed that the muscular fibres of those arteries in
which the magnitude of the pulse is sensible, like the fibres of

No. XXni. HEART AND ARTEBIES. 525

the hearty may be inactive, or nearly so, during their dilatation,
and that they may contract after they have been once dis-
tended, with a force which is in a certain degree permanent ;
the greater momentum of the blood, which accompanies the
dilatation, enabling it to enter the minute arteries with equal
ease, although assisted by a tension somewhat smaller : so that
the same mean velocity may be sustained, as if the arteries
were simply elastic, and a little smaller in diameter, with a
very little less exertion of the heart. But the distribution of
the blood could never be materially diversified by any opera-
tion of this kind : for if any artery were for a moment dis-
tended by such a variation, so as to exceed its natural diameter
by one hundredth part only, a pressure would thence arise
equivalent to that of a column about two inches high, which
would, in spite of all resistances, immediately dissipate the
blood with a considerable velocity, and completely prevent any
local accumulation, unless the elastic powers of the vessel itself
were diminished ; and this is, perhaps, the most important, as
well as the best established inference from the doctrine that I
have advanced.

It appears that a mola has sometimes been found in the
uterus, totally destitute of a heart, in which the blood must
have circulated in its usual course through the veins and arte-
ries : in this case it cannot be ascertained whether there was
any alternate pulsation, or whether the blood was carried on
in a uniform current, in the same manner as the sap of a
vegetable probably circulates. If there was a pulsation, it may
have been maintained by a contraction of the artery, much
more considerable, and grower in its progress than usual ; and
with the assistance of a spontaneous dilatation ; the resistance
in the extreme vessels being also probably much smaller than
usual : if the motion was continued, it would lead us to ima-
gine that there may be some structure in the placenta capable
of assisting in the propulsion of the blood, as there may pos-
sibly be some arrangement in the roots of plants by which they
are calculated to promote the ascent of the sap. The circulation
in the vessels of the more imperfect animals, in which a great
artery supplies the place of a heart, is of a very different na-

526 ON THE FUNCTIONS OF THE Na XXIII.

ture from that of the more perfect animals : the great artery,
which performs the office of the heart, is here possessed of a
muscular power commensurate to its functions, and seems to
propel ilie blood, though much more slowly than in other cases,
by means of a true peristaltic motion. It appears also from
the observations of Spallanzani, that in many animals a portion
of the aorta, next the heart, is capable of exhibiting a conti-
nued pulsation, even when perfectly empty and separated from
tiie heart ; but this property is limited to a small part of the
artery only, which is obviously capable of being essentially
useful in propelling the blood when the valves of the aorta are
closed. The muscular power of the termination of the vena
cava is also capable of assisting the passage of tiie blood into the
auricle* It is not at all improbable that a muscle of involun-
tary motion, which had been affected throughout the whole
period of life by alternate contractions and relaxations, might
retain from habit the tendency to such contractions, even with-
out the neces^ty of supposing that the habit was originally
formed for a purpose* to be obtained by the immediate exertion
of the muscular power : but in fact the partial pulsation of the
vena cava isperfectiy well calculated to promote the temporary
repletion of the auricle, while it must retard, for a moment,
the column which is approaching, at a time that it could not be
received.

There is no difficulty in imagining what services the muscular
coats of the arteries may be capable of performing, without
attributing to them any immediate concern in supporting the
circulation. For since the quantity of blood in the system is on
many accounts perpetually varying, there must be some means
of accommodating the blood-vessels to their contents. This
circumstance was very evident in some of Hales's experiments,
when after a certain quantity of blood had been taken away,
the height of the column, which measured the tension of the
vessels, frequently varied in an irregular manner, before it
became stationary at a height proportional to the remaining
permanent tension. Haller also relates, that he has frequently
seen the arteries completely empty, although in some of his
observations there was probably only a want of red globules in

No. XXUI. HEART AND ABTERIES. 527

the blood which was flowing through them. Such alterations
in the capacity of the different parts of the body are almost
always to be attributed to the exertion of a muscular power.
A partial contraction of the coats of the smaller arteries may
also have an immediate effect on the quantity of blood con-
tained in any part, although very little variation could be pro-
ducexl in this manner by a change of the capacity of the larger
vessels.

According to tiiis statement of the powers which are con-
cerned in the circulation, it must be obvious that the nature
of the pulse, as perceptible to the touch, must depend almost
entirely on the action of the heart, since the state of the arte-
ries can produce very little alteration in its qualities. The
greater or less tension of the arterial system may indeed render
the artery itself, when at rest, somewhat harder or softer ; and,
if the longitudinal fibres give way to the tending force, it may
become also tortuous : possibly too a very delicate touch may
in some cases perceive a difference in the degree of dilatation^
although it is seldom practicable to distinguish the artery, in
its quiescent state, from the surrounding parts. But the sen-
sation, which is perceived when the artery is compressed, as
usual, by the finger, is by no means to be confounded with the
dilatation of the artery ; for in this case an obstacle is opposed
to the motion of the blood, against which it strikes witii the
momentum of a considerable column, almost in the same man-
ner as a stream of water strikes on the valve of the hydraulic
ram ; and in this manner, neglecting the diflerence of force
arising from the different magnitudes of the sections, the pres-
sure felt by the finger becomes nearly equal and similar to that
which is originally exerted by the heart : each pulsation pass^
ing under the finger, in the same time, as is required for the
contraction of the heart, although a very little later ; and more
or less so, in proportion as the artery is more or less distant ; the
artery remaining then at rest for a time equal to that in which
the heart is at rest. When therefore an artery appears to
throb, or to beat more strongly than usual, the circumstance is
only to be explained from its greater dilatation, which allows it
to receive a greater portion of the action of the heart, in the

528 ON THE FUNCnONS OF THE No, XXIII.

same manner as an aneurism exhibits a very strong pulsation,
without any increase of energy, either in itself, or in the neigh-
bouring vessels ; and on the other hand, when the pulsations of
the artery of a paralytic arm become feeble, we cannot hesitate
to attribute the change to its permanent contraction, since the
enlargement and contraction of the blood-vessels of a limb are
well known to attend the increase or diminution of its muscular
exertions. There is also another way, in which the diminution
of the strength of an artery may increase the apparent magnitude
of the pulse, that is, by diminishing the velocity with which the
pulsation is transmitted : for we have seen that the magnitude
of the pulse is in the inverse ratio of the length of the artery
distended at once ; and this length is proportional to the velo-
city of the transmission ; but it must be observed, that the
force of the pulse striking the finger would not be affected by
such a change, except that it might be rendered somewhat fuller
and softer, although a considerable throbbing might be felt in
the part, from the increased distension of the temporary diameter
of the artery. How little a muscular force is necessary far the
simple transmission of a pulsation, may easily be shown by
placing a finger on the vena saphena, and striking it with the
other hand at a distant part ; a sensation will then be felt pre-
cisely like that of a weak arterial pulsation.

The deviations from the natural state of the circulation,
which are now to be cursorily investigated, may be either gene-
ral or partial ; and the general deviations may consist either in
a change of the motion of the heart, or of the capacity of the
capillary arteries. When the motion of the heart is affected,
the quantity of blood transmitted by it may either remain the
same as in perfect health, or be diminished, or increased.
Supposing it to remain the same, the pulse, if more frequent,
must be weaker, and if slower, it mast be stronger ; but this
latter combination is scarcely ever observable; and in the
former case, the heart must either never be filled, perhaps on
account of too great irritability, or never be emptied, from the
weakness of its muscular powers. But the immediate effect of
such a change as this, in the functions depending on the circu-
lation, cannot be very material, and it can only be considered

No. XXIII. HEABT AND ARTERIES. 529

as an indication of a derangement in the nervous and muscular
system^ which is not likely to lead to any disease of the vital
fbnctions. When the quantity of the blood transmitted by the
heart is smaller than in health, the arteries must be contracted^
until their tension becomes only adequate to propel the blood,
through the capillary vessels, with a proportionally smaller
velocity, and the veins must of course become distended, unless
the muscular coats of the arteries can be sufficiently relaxed to
afford a diminished tendon, which is probably possible in a very
limited degree only. In this state the pulse must be small and
weak, and the arteries being partly exhausted, there will pro-
bably be a paleness and chilliness of the extremities : until the
blood, which is accumulated in the veins, has sufficient power
to urge the heart to a greater action, and perhaps from the
vigour which it may have acquired during the remission of its
exertions, even to a morbid excess of activity. Hence a con-
trary state may arise, in which the quantity of blood transmitted
by the heart is greater than in perfect health ; the pulse will
then be fiill and strong, the arteries being distended, so as to
be capable of exerting a pressure sufficient to maintain an
increased velocity, and to overcome the consequent increase of
redstance ; a state which perhaps constitutes the hot fit of fever ;
and which is probably sometimes removed in consequence of a
relaxation of the extreme arteries, which suffer the superfluous
blood to pass more easily into the veins. Such a relaxation,
when carried to a morbid extent, may also be a principal cause
of another general derangement of the curculation, the motion
of the blood being accelerated, and the arteries emptied, so that
the pulse may be small and weak, while the veins are over-
charged, and the heart exhausted by violent and fruitiess effiirts
to restore the equilibrium ;. and this state appears to resemble,
in many respects, the affections observed in typhus. When, on
the contrary, the capillary vessels are contracted, the arteries
are again distended, although without the excess of heat which
must attend their distension from an increased action of the
heart, and possibly without fever : an instance of this appears
to be exhibited in the shrinking of the skin, which is frequently
observable from the etkct of cold, and in the first impression
VOL. I. 2 m

530 ON TH£ FUNCnONS W THE No. XXIIL

prodooed by a cold bath : nor is it impoaaible, that sach a con-
traction may exiat in the cold fit of an intermittent, although it
seems more probable that a debility of the heart is the primary
canse of this afifoction.

Besides these general causes of derangemrat, which appear
to be more or less concerned in diferent kinds of fever, there
are other more partial ones, which seem to have a similar rela-
tion to local inflammations. The most obrious of these changes
are such as must be produced by partial dilatations or contrac-
tions of the capillary vessels: since, as I have endeavoured to
demonstrate, any supposed derailment in the actions of the
larger vessels must be excluded from the number of causes
which can materially aflect the circulation. It cannot be
denied, that a diminution of the elastic, or even of the muscular
force of the small arteries, must be immediately Avowed by
such a distension as will [wodnce a resistance equal to the
pressure : the distension will occasion an increase of redness,
and in most cases pain : the heat will also generally be in-
creased, on account of the increased quantity of blood which
will be allowed to pass through the part ; and since the hydro-
static pressure of the blood acquires greater force, as the artery
becomes more distended, it may be so weak as to continue to
gpve way, like a ligament which has been strained, until sup*
ported by the surrounding parts. In this state a larger supply
of blood will be ready for any purposes which require it, whe-
ther an iqjury is to be repaired, or a new substance fjurmed ;
and it is not imposaible, that this change in the state of the
minute vessels may ultimately produce some change in the pro-
perties of the blood itself.

The more the capillary arteries are debilitated and distended
the greater will be the mean velocity of the circulation ; but
whether or no the velodty will be increased in the vessels which
are thus distended, must depend on the extent of the aflected
part ; and it may frequently happen that the velocity may be
much more diminished on account of the dilatation of the space
which the blood is to occupy, than increased by the diminution
of the resistance. And on the other hand, the velocity may be
often increased, for a similar reason, at the place of a partial

No. XXIII. HEART AND ARTEBIES. 531

ooDtractiaii. Hence we may easily understand Boine of the
experiments which Dr. Wilson has related in his valuable
treatise on fevers ; the application of spirit of wine to a part of
the membrane of a firog's foot contracted the capillaiy arteries^
and at the same time accelerated the motion of the blood in
them, while in other parts, where inflammation was present, and
the vessels were distended, the motion of the blood was slower
than usual.

Another species of inflammation may jMPobably be occasioned
by a partial constriction or obstruction of the capillary arteries,
which must indeed be sujqioeed to exist where the blood has
become wholly stagnant, as Dr. Wilson in some instances found
it. This obstruction must however be extended to almost all
the branches, belonging to some small trunk, in which the
pressure remains nearly equal to the tensioo of the large arte-
ries : for in this case it will happen, that the whole pressure
will be continued throughout the obstructed branches, widiout
the subtraction of the most considerable part, whidi is usually
expended in overcoming the resbtances dependent on the velo-
pity ; so that the small brandies will be subjected to a pressure,
many times greater than that wfaicfa they are intended to with-
stand in the natural state of the circulation ; whence it may
eanly happen that they may be morbidly distended ; and this
distension may constitute an inflammation, attended by redness
and pain. Nor is it impossible tiiat obstructioos of this kind
may (Hriginate in a vitiated state of the blood itself, althou^ it
would be difficult to prove the truth of the conjecture ; it seems,
however, to be favoured by the observation of Haller, that little
clots of globules may often be observed in the arteries, when
the circulation is languid, and that they disappear when its
vigour is restored, especially after venesection. But if a very
small number only of capillary arteries be obstructed, other
minute branches will still be capable of receiring the blood,
which ought to pass throu^ them, without any great disten-
sion or increase of pressure : and this exception is sufficient to
exfdain another expmment of Dr. Wilson, in which a small
obstruction, caused by puncturing a membrane witli a hot
needle, failed to excite an inflammation. This species of

2 M 2

532 ON THE FUK0TION8 OF THE Na XXIII.

inflammatioD is probably attended by less heat than the for-
mer ; and where the obstniction is very great, it may perhaps
lead immediately to a mortification, which is called by the
Grermaos *^ a cold burning."

The most usual causes of inflammation appear to be easily
reconcileable with these conjectures. Suppose any considerable
part of the body to be afiected by cold ; the capillary vessels
will be contracted, and at the same time the temperature of
some parts of their contents will be lowered, firom both of which
causes the resistance will be increased, and the arteries in ge-
neral will be more or less overcharged : if then any other part
of the system be at the same time debilitated or overheated, its
arteries will be liable to be morbidly distended, and an inflam-
mation may thus arise, which may continue till the minute
vessels are supported and strengthened, by means of an efiusion
of coagulable lymph. The immediate efiect, either of cold or
of heat, may also sometimes produce such a degree of debility
in any part, as may lay the foundation of a subsequent inflam-
mation : but the first effect of heat in the blood-vessels i4>pears
to be the more ready transmission of the blood into the veins,
by means of which they become very observably prominent :
and cold, which checks the circulation in the cutaneous vessels,
probably occasions a livid hue, by retaining the blood stagnant
longer than usual in the capillary vessels of all kinds. It may
be objected, that an obstruction of the motion of the blood
through a great artery ought, upon these prindples, to produce
an inflammation in some distant part : but in this case, the
blood will still find its way very copiously into the parts sup-
plied by the artery, by means of some collateral branches,
which will always admit a much larger quantity of blood than
usually passes through them, whenever a very slight excess of
force can be found to carry it on, or when the blood which they
contain finds a readier passage than usual, by means of their
communication with such parts as are now deprived of their
natural supply.

[t is difficult to determine, whether blushing is more pro-
bably eflected by a constriction or by a relaxation of the
vessels concerned; it must, however, be chiefly an aflSsction

No. XXIII. HEABT AND ARTERIES. 533

of the smaller vessels, since the larger ones do not contain a
sufficient quantity of blood to produce so sudden an effect*
Perhaps the capillary vessels are dilated, while the arteries,
which are a little larger only, are contracted : possibly too an
obstruction may exist at the point of junction of the arteries
with the veins ; and where the blush is preceded by paleness,
such an obstruction is probably the principal cause of the whole
afiection.

Witii respect to the tendency of inflammation in general to
extend itself to tiie neighbouring parts, it is scarcely possible to
form any reasonable conjecture that can lead to its explanation:
this circumstance appears to be placed beyond the reach of any
mechanical theory, and to belong rather to some mutual com-
munication of the functions of the nervous system, since it is not
inflammation only, that is tiius promulgated, but a variety of
other local affections of a specific nature, which are usually com-
plicated with inflammation, although they may, perhaps, in some
cases, be independent of it. Inflammations, however, are cer^
tainly capable of great diversity in their nature, and it is not to
be expected, that any mechanical theory can do more than aflbrd
a probable explanation of the most material circumstances,
which are common to all the different species.

Besides these general illustrations of the nature of fevers and
inflammations, the theory which has been explained may some-
times be of use, in enabling us to understand the operation of
the remedies employed for relicTing them. Thus it may be
shown, that any diminution of the t^raion of the arterial system
must be propagated from the point at which it begins, as from
a centre, nearly in the same manner, and with the same velo-
city, as an increase of tension, or a pulsation of any kind would
be propagated. Hence the effect of venesection must be not
only more rapidly, but also more powerfully felt in a neigh-
bouring than in a distant part ; and although the mean or per-
manent tension of the vessels of any part must be the same,
from whatever vein the blood may have been drawn, provided
that they undergo no local alteration, yet the temporary change,
produced by opening a vein in their neighbourhood, may have
relieved them so effectually from an excess of pressure, as to

534 ON THE HEABT ASTD AJErTBRIE& Ko. XKIIL

allow them to recover thrir natural tone, which they could not
have done without such a partial exhaustion of the neighbonrii^
vessels. But ance it seems probable, that the minute arteries
are more affected by distension than the veins, there is reason
in general to expect a more speedy and efficadous relief in
inflammations, from opening an artery than a vein : this ope-
ration, however, can seldom be performed without material
inconvenience; but it is probably for a dmilar reason, that
greater benefit is often experienced from withdrawing a small
portion of blood by means of cupping or of leeches, than a much
larger quantity by venesection, since both the former modes of
bleeding tend to relieve the arteries, as immediately as the
veins, from that distension, which appears to constitute the
most essential charactmstic of inflammation. In a case of he-
morrhage from one of the sinuses of the brain, a very judicious
physician lately prescribed the digitalis : if the efiect of this
medicine tends principally to diminish the action of the heart,
^tB is commonly supposed, it was more likely to be injurious
than beneficial, since a venous plethora must be increased by
the inactivity of the heart ; but if the digitalis diminishes the
general tension of the arteries, in a greater {»H)portion than it
afiects the motion of the heart, it may possibly be advantageous
in venous hemorrhages. We have, however, no suflkient au-
thority for believing that it has any such efiect on the arterial
system in general.

' Although €be arguments, which J have advanced, appear to
me sufficient to prove that, in the ordinary state of the circo-
lation, the muscular powers of the arteries have very little
effect in propelling the blood, yet I neither expect nor derire
that the prevailing opinion Aould at once be universally aban-
doned. I wish, however, to protest once more against a hasty
rejection of my theory, from a su^)erfidal oonrideration <^
cases, like that which has been related by Dr. Clariie ; and to
observe again, that the ohjections which I have adduced against
the operation of the muscular powers of the arteries in the ordi-
nary circulation, not being applicable to these cases, they are by
BO means weakened by any inferences which can be drawn fixnti
them.

No. XXIV. ON THE EMFLOTMKMT OF OBUQUE BIDEBS. 535

No. XXIV.

REMARKS ON

THE EMPLOYMENT OF OBLIQUE RTOERS

AKD OH OTHEB

ALTERATIONS IN THE OONSTRUOTION OF SHIPS.

Beiro thb Subbtanob of ▲ Rbfort prbsbhtbd to thb Board of Admibaltt,
with additional demonstrations and illustrations.*

From the PhiloBophical TraniacUoiM for 1814.

Read March 24» 1814.

1. Intboductort Obsebvations.

The advantage derived from the employment of forces acting
obliquely with respect to each other, in a variety of cases which
occur in practical mechanics^ has been demonstratively esta-
blished by theoretical writers on the subject ; and attempts
have often been made to extend the application of the principle
very considerably in the art of ship-building ; but hitherto with
very little permanent success. Mr. Seppings's arrangements
are in many respects either new or newly modified ; and the
results of their actual employment, in the repair of the Tre-

* The Memoir which follows origiiiated in the drcnmetanoes detailed in the
IbHowiDg letter from Mr., afterwards Sir John Barrow, one of the eecretariei to the
Admiralty: —

'< Admiralty Offioe, 19th November, 1811.
** Sib,

** Mr. Seppfaigs, the master shipwri^t of Chatham Dockyard, having sab-
mitted to my Lords ONnmiBBionen of the Admiralty a model for the construction of
ships of war on a new principle, by which it wonld appear that an advantage is
obtained in point of strength and danbility, while at the same time^ a very consider-
able saving of timber is effected ; and my Lords having caosed a ship of seventy-fonr
guns to be fastened aooording to this new mode of construction, which, after a trial
of many months in the Noru Sea, has been found to answer every expectation that
the projector himself could have formed ; and being desirous of submitting this new

536 REHABKS ON THE Na XXIV.

mendous, appear to be sufficiently encouraging to entitle them
to a careful and impartial investigation, both with regard to
the theory on which they are supposed to be founded, and to
the facts which may be produced in their favour. The question,
respecting the best disposition of the timbers of a ship, is by
no means so easily discussed, as may be supposed by those,
who have considered the subject but superficially ; and if we
allowed ourselves to be influenced by a few hasty arguments or
experiments, we might be liable to die most dangerous errors :
on the other hand, it may easily happen that objections to the

principle to the connderatioo of such men of scienoo and practical experience as may
have turned their attention to mechanics in general, and more paiticolarly to the
conatmction and fastening of ships :

** I am therefore directed bj weir Lordships to acquaint yon that on Wednesday,
the 27th instant, at one o'clock, Mr. Seppings will be ordered to attend at this office,
for the purpose of exhibiting his model, and explaining the principle on which it is
constructed ; and to request the favour of your attendance on the occasion.

'* I am. Sir, your very humble servant,
" To Dr. Toungr " John Bjlbbow."

The following is Dr. Tonng*s reply :—

<' Welbeck-street, November 22, 1811.
<* Dear Sir,

** I ought perhaps to have returned an earlier answer to your official letter,
but I have made so many resolutions to forswear all further concern with the
mathematical sciences, that I could not at once determine again to deviate from them
by accepting their Lordships' invitation. Recollecting, liowever, that as 6u> as
I know, I am the only penion in this country that has communicated to the public
any attempts to improve the theory of carpentry (Lectures, Chap. XI V.)* and that
it would be scarcely decent to draw back on an occasion where I was called on to
aasist in a case of practiccU importance, I have overruled my hesitations, and shaU
attend on Wednesday with so much the more pleasure, as I csnnot help fancying,
from the little that I know of the question in agitation, I should he able, if I
had leisure to discuss it thoroughly, to reconcile the discordant opinions which seem
to prevail respecting it. « j ^^ ^^„ gj

'* Tour fidthful and obedient servant,

" T. TOUNO."

The proposition of the important changes which Sir Robert Seppings introduced
into our system of naval architecture occasioned no small amount of discussion.
They were advocated by Sir John Barrow, with mat vigour and ability, in several
articles of the ' Quarterly Review,' and were vehemently opposed by many of the
officials of the dockyards, as well as very generallv by the older officers of the navy.
Dr. Young's Report, and the Memoir which was founded upon it, though approving
generally of the proposed changes, was too reserved in its statements to satisfy either
party : whilst the mechanical and other problems which the investigation involved
wero of too high an order of difficultv to be easily understood by those persons for
whose especial benefit it was prepared. '* Though' science is much respected by their
Loi^ships," writes sn official of the Admfaralty, ** and your paper is much esteemed
by them, it is too learned."

The changes and improvements introduced by Sir R. Seppings, and their experi-
mental roMuts, form the subjects of several papers in the ' Philosophical Trans-
actions.'*

No. XXIV. EMPLOYMENT OF OBLIQUE RIDEBS. 587

application of those argaments or experiments, which may oocnr
at first flighty may be capable of being removed by a more
minute investigation: and the importance of the subject re-
quires that no assistance, which can be afforded by the abstract
sciences, should be withheld from the service of the public,
even by those who have no professional motives for devoting
themselves to it

2. Forces acting on a Ship.

Hie first consideration that is necessary, for enabling us to
judge of the propriety of any arrangement respecting the con-
struction of a ship, is to determine the nature and magnitude of
the forces which are to be resisted ; and the second, to inquire
in what manner the materials can be arranged, so as best to
sustain the strains which these forces occasion. The principal
forces, which act on a ship, are the weight of the whole fabric
with its contents, the pressure of the water, the impulse of the
wind, and the resistance of the ground or of a rock : and we
must endeavour to ascertain the degree in which any of them
have a tendency to bend the ship longitudinally or transversely,
or to break through any part of her. texture ; and to inquire
into those causes, which are likely to promote or to obviate the
decay of the substances employed.

3. Causes of abchino. Weight.

It is unnecessary to explain here the well known inequality
of the distribution of the weight and pressure, which causes
almost all ships to have a tendency to arch or hog, that is,
to become convex upwards, in the direction of their length. It
is possible that there may be cases in which a strain of a very
different nature is produced : but in ships of war, this tendency
appears to be universal. It is however very different in degree
in the different parts of a ship ; and of course, still more dif-
ferent according to the different modes of distribution of the
ballast and stores, which may occur in different ships : but in
ordinary cases, it will probably be found nearly such as is re-

5S6 HBHARKS OS THE No. XXIV.

presented in the ladcohtioiis sabjoined in the note,* deduced
from data whidi bave been obligii^ly furnished by an acute
and experienced member of the Navy Board.

* In 8 modern 74-gan ship, fitted for sea, the length being 176 feet, the bmdth
47^ the forces are tfaw distributed :—

Afteimort 40 f.

Weiffht 699

PMB68QK 627

DfiftreDoe72

Next 20

297

405

-108

1216

1098

118

290

409

-119

498

461

176 3000 3000 00

' Now the laws of eqailibriom will not aUow ns to suppoee these fones eonoeo-
trated in the middle of the respectiTe portions, or equally diBtribnted through them;
and it becomes necessary, that one of the weights should be situated further mrwaida ;
which must be that of the foremost portion, containing the bowsprit and its rigsin^
It is also natural to suppose the excesses of weight and pressure distributed wi^ as
, few abrupt chai^^ as possible, so as to neutralise each other at the oommon termi-
nation of the adjoining portions, and to become more unequal in parts more remote
from these neutral points. Thus the excess of weight in the first 49 feet being 72

144

tons, it may be supposed to begin at the rate of — tons per foot, and to dimmish

49 49

gradually and equably, so that its centre of action will be at the distance — from

the end : the excess of pressure must IncrMse in the nest place, until at the distanee

108
of 59 feet from the stem, it becomes per foot, and must then diminish untii it

vanishes at 69, wh«:>8 the excess of wei^t must beghi to prevail, becoming, at 94,

118

- — per foot, and vanishing at 119. The exceu of pvewnre might then be suppuieJ

to increase gradually through the next portion, in order to avoid an abrupt change at

its extremity; but this supposition would still be insufficient, and it becomes

necessary to imagine that for 6.6 feet the forces remain neutralised, and the

119
pressure then prevails, so that its excess becomes at last — = 17.7 per foot: it

must then decrease fbr 17.5 feet, and the eoBoess of weight at the extremity must
become 19,7 per foot, the neutral point being at 156.5. The equilibrium of the
forces will then be expressed by the equation 72 X 16.3-108 X 59 + US X 94 •-
119 X 134.5-155 X 144.8 + 192 x 169.5 = 0, which is sufficiently accurate fbr
every purpose.

From this distribution of the forces, we obtain a determination of the strain for
each point of the respective portions, which is in the joint ratio of the magnitudes
and distances of all the forces conoemed, on either side of the point, reduced into a

144 144 X

common lesult. For the first portion it is x Xix- i. — ._ .«ljr»

^ 49 ' 49 49 *

7H d^ - } . 1^ . ^, « being the distance from the stem: for the 2nd, 73<4P-16})
-J.!1«.£Z^: 8rf,«(*.161)-54(»-651)-^(.-«.).+ |,
j^. (« - 59)» : 4th, 73 (» - 16*) - 108 (. - 59) + 1 .2^ . ^^^s 5th.
78 (.-161) - 108 (. - 69) + 69 (« - 94) + ^ <* - 94)» - i . *^

No. XXiy. EMPLOYMENT OF OBCdtQUE RIDEBB. 539

Lanffitudinal Freuure.

To this strain another is added, from a cause, which, although
not very inconsiderable, appears hitherto to hare escaped notice;

(*_-94)» . g^ ^^ jjg ^ J25.6, 72 (a? - 16J) - 108 (a? -59) + 118 (a:- 94):

for the 7tli, we mutt «dd to thU esprasion ^^,— , (^ " 125. y^ ^ ^

lo*4 j9»«

the last 37 ftet, the stMia wUl be ezpnMsd hy (176 - «) 19.7 X i (176 - «)

-. } 19.7 ^}J^ ~ ^^*. Heoee we flad the sttvte, at eeTen pointi, 23 fcet diataat

19.6
fipoiD each other and from the ends, 605, 1993, 2815, 2244, 2655, 4610, and 1875 ;
and by taking the fluxion of « in the seTenth portioo, we dotennine the maximam at
141) feet, amounting to 5261 tons, suppoaed to act at the distance of one foot.

In order to form an idea of the conre which wonld be produced bj such a strain,
acting on a uniformly flexible substance, we may consider the cnrratnre as represented
by the second fluxion of the ordinate y, and by finding and correcting the fluent
separately for each portion, we may obtain the ordinate or foil at anjr given pofait
corrasponding to a giren extent of arching of the whole fabric. It will however be
sufficiently accurate for this purpose, to consider the forces as concentrated in a
limited number of points, diTiding those which act hi the extreme portions into two
parts, hi order tiut the curvature may be continued to the-ends ; so that the whole
of the forces may be thus distributed at 0, 86 ; at 821, ^^\ ^ ^d, —108 ; at 94,
118; at 134.5, - 119; at 144*8, «-155; at 168, 96; and at 176, 96. The
ttrain for each portion may then be represented by a » to, whence f s air —
&rJrir, jr s •xx ~ i&B*i + ci, andy = {<»* - )&v* -^ ex + d. It will be most
oonrenient in calculation, to make x begin anew with each portion, setting out from
the middle, and to divide the numbers by 100, in order to shorten the operations:
thus, for the middle p(Hrtion, from 88 to 59, the strain will be .2028 + .36^^ a
being .2028, and 6 s •. .86 ; and when x becomes .22, y is .00552, and when

X :» .29, f^ = .0740, and y = .0011 ; which ralues being substituted in the

i
equations for the next portion, we have o = .074, and d s .0011 : and by going
through the whole length in this manner, we find the foil at the extremes and at
seven equidistant intennediate points, .08697, .05325» .02514, .00552, 0, .00507,
.02591, .06705, and .12325. If we wish to find the point at which the curve is
parallel to the chord of the whole, we must inquire where e = (.12325 • .08697) :
1 .76, which will be at 98 feet, or 10 feet before the midships.

We must next determine the magnitude of the strain arising from the longitudinal
pressure acting on the lower part of the ship only. The resistance being supposed to
be proportional in the first instance to the degree of compression or extension, ac»
cording to the common and almost necessary li^ of the constitution of elastic bodies,
and vanring also in the direct ratio of the strength of the ftbric, which may be
assumed to be either equable, or, in the case of a ahip, proportional to the disCance
from a point more or less remote, we must fonn an equation of equilibrium for the
absolute equality of the forces in opposite directions, and another for their powers of
acting with respect to any given point as the fulcrum of a lever. Thus the fluxion
of the absolute resistance at the distance x from the upper surfoce, supposing the
strength to be as a+«, and the neutral point, at which the compression and exten-
sion cease, to be at the distance 6, will. be (b^x) c (a+«) jr s c (ab^ax-^hx^xx)
Xf which, when x k equal to the depth (2, must become equal to the force / pro-
ducing the strain, or /s c (abx — iaafi + ito* * l^*): and for the second
equation, referring the forces to tlie upper surfoce as a fulcrum, the fluent of o
(6 — jr) (a + x) xif must be equal to <>/, e being the distance at which the force e is
applied ; whence tf^c (ia6d>'- ^»eP + \f)d^ - ^d^)* Now if we make assdss x.

540 REMARKS ON THE No. XXIV.

that is, the partial pressure of the water in a longitadinal di-
rection, affecting the lower parts of the ship only, and tending
to compress and shorten the keel, while it has no immediate
action on the upper decks. The pressure, thus applied, must
obviously occasion a curvature, if the angles made with the
decks by the timbers are supposed to remain unaltered, while
the keel is shortened, in the same manner as any soft and thick
substance, pressed at one edge between the fingers, will become
concave jbA the part compressed (Lect. Nat PhiL I. PK 9. F.
117) ; and this strain, upon the most probable supposition re-
specting tile comparative streogth of the upper and lower parts
of the ship, must amount to more than one tiiird as much as
the mean value of the former, being equivalent to the effect of
a weight of about 1000 tons, acting on a lever of one foot in
lengtii, while the strain, arising from the unequal distribution of
the weight and the displacement, amounts where it is greatest,

the equations become c Q/nfl - Id*) = /, and e (fbifl - <j^) = tf^ and from Uie
former we hare e (^ - ^) = {d/; and, by subtraction, ^M^ = ({d - e)f:
conseqnently the force / may be considered as acting on a lever of the length c — Jd:
and if we take any other Talue for a, the fractional maltiplier of tf, instead of |, will

be ?iJL^: tiins if a - 4, we have « - jUf for tiie length of the lever. In

order to find the mean distance 0 at which tiie preBSore of the water acts, we maj
suppose the form of the mean transyerse section of the ship to be parsbolic, and the
area snch as to correspond to the bulk of 3000 tons of wttter, eaicfa containing 35
cubic feet, the length bemg 176 feet, and the breadth 47^, whence the depth must
be 18.84 feet : then the centre of gniyity of a parabola being at the distance of 1
of the depth from the vertex (Vince's Fluxions, p. 101), and the centre of oscillation
at 4, when the point of suspension u at the vertex (p. HI), the distance of these
points ^ will be increased to ^, when the point of suspension is removed to the
teimination of the absciss, and the distance of the centre of pressure from the
vertex will be{ — As.},and| X 18.84 s= 8.074, which, subtracted from
I X 40 => 17.777, leaves 9.703 for the length of the lever. Now the magnitude
of the pressure on this section must be to 3000 tons, as the depth of the centre of
gravity, 7536 feet, to 176, that is, 128.45 tons, which, acting at the distance
9.703, will produce a strain of 1247 tons, or, in the terms of l£e preceding calcu-
lation, .1247, which is the multiple of io;* indicating the fall. These different
causes of hrching being independent of each other in their operation, their effects
will be simply united into a common result : and the whole curvature of the ship,
supposing iU strength equable throughout its length, may be thus represented :

Distance

88 110 132 154 176

2224 2655 4610 1875 0

.00000 .00302 .01207 .02716 .04828

.00000 .00607 .02531 .06705 .12325

from the

stem . 0 22

Sti-ain . 1247+0 605

1993

2815

Fall . .04828 .02716

.01207

.00302

.08697 .05325

.02514

.00552

.13525 .08041 .03721 .00854 .00000 .00809 .03738 .09421 .17153
For 12 inches
of arching 10.58 6.29 2.91 .67 .00 .63 2.93 7.37 13.42.

r No. XXIV. EMPLOTMENT OF OBLIQUE RIDERS. 541

jj that is, about 37 feet from the head, to 5260, in a 74-gan ship

t of the usual dimensions ; and although the strain is conaderably

\ less thaD this exactly in the middle, and throughout the after-

I most half of the length, it is no where converted into a tendency

I to ^^sag," or to become concaye. It must, howeyer, be re-

j membered, that when arching actually takes place from the

operation of these forces, it depends upon the comparative
strength of the different parts of the ship and their fastenings,
whether the curvature shall vary more or less from the form,
which results from the- supposition of a uniform resistance
throughout the length. An apparent deviation may also arise
from the unequal distribution of the weight through the breadth
of the ship : thus the keel may actually sag, under the step of
the mainmast, even when' the strain, as here calculated, indi-
cates a contrary tendency with respect to the curvature of the
whole ship.

Force of the Waves.

The magnitude of the strain on the different parts of the ship
is subjected to very material alterations, when she is exposed
to the forces of the wind and waves. The effect of the wind
b generally compensated by a change of the situation of the
actual water-line, or line of fiuitation, so that its amount may
be estimated from the temporary or permanent inclination of
the ship; and the force of the waves may be more directly
calculated from their height and breadth. These two forces
can seldom be so applied, as to combine their effects in pro-
ducing a strain of the same kind in their foil extent ; it will
therefore be sufficient for our purpose to deternune the pro-
bable amount of the force of the waves, which is more materially
concerned in affecting the longitudinal curvature than that of
I the wind. As a fair specimen of the greatest strain that is

likely to arise from this cause in any common circumstances, we
may conmder the case of a series of waves 20 feet in height, and
70 in breadth ; the form being such, that the curvature of the
surface may be nearly proportional to the elevation or depres-
sion: a single wave might indeed act more powerfolly than
a continued series, but such a wave can scarcely ever occur

542 SJBICABBB ON THE No. XXIV.

singly. We shall tben find upon calculation,* lihat ihe greatest
strain takes place, in a 74-gun ship, at a distance of about 18
feet from the nudships, amounting to about 10,000 tons at die

• The stnin, produced by the pressure of waves of given magnitode, maj be
caknlated from the compuiaon of the duplaceBaent with respect to their anrlaioe^
compared with the displacement with respect to a level soHkce. It is true that the
pressare npon the ship's bottom is not immediately derived from the temporary
height of tiie nearest portion of water; bat the horiaontal motion of the water,
which is the proximate caase of the elevation, is equally capable of affecting the
flnid under a floating body, and of causing a pressure against it: the efloct bezn^
n«arly similar to the tiansmiasion of sound through an elastic medium. In other
cases, the actual height of the fluid, over every particle concerned in the transmission
of a wave, has been supposed, in calculations, to deteimine the pressue on it : bat,
whether from the necessary constitution of a fluid, or from the imperfect fluidity of
fluids actually existing, it appears that there is a lateral communication of pressuie'
within a certain angular extent, somewhat like the lateral friction attending the
motions of fluids; and this is the most probable cause of the deficiency in the
velocity of waves, when their breadth is very small in proportion to the depth of the
fluid. In the present calculation, however, the consideratioDS are more simple, and
we have only to determine the effect of the difference produced during the passage of
a wave in the quantity of water displaced by the ship, with respect to the genenl
surface. The total height of the waves being 20 feet, and the total breadth 70, the
section being supposed to constitute a figure of sines, the elevation or depression, at

the distance j: from the middle, will be 10 cos. jmt, p being Zlfzil s .08976, the

fluxion of the area lOi cos. px^ and the fluent _ sin. par ; at the constant ^'iirtanrf

P
t. from the middle, the fluxion of the p strain will he(x^z) lOi cos. par, in order to
find the fluent of which we must take the fluxions of x sin. px, and cos. |ur, which are

X sm,px+xpxco3.pXj and — px sin. px\ hence fxxcoB.px = -x sin. jpr+ — cos. px ;

P PP

and the fluent of the strain will be^ (x ^ z) mn.px -^ 1? cos. px + c, which

'^ 10 ^

must vanish when x = z, bo that c s= — _ cos. pz : now. when jr s a, the

corrected fluent becomes — (^^ -' O »^ pa -¥ — <km« pa — i- cos. pz ; and if

P PP PP

we take the fluxion of this, making z variable, we find, for the maximum,

— _ i sin. pa + — i shi. pz = 0, and sin. pz = sin. pci^ so that z must be

P P

a - 70 s 18, whence the greatest strain is found, — X 70 x .999 s 7791,

P
expressed in square foet of the longitudinal section, which, for a ship 47| Ibet

47 5
wide, may be reduced into tons, by multiplying it by — 1., and will become 10572.

35
It is true, that if the waves allowed time for the ascent or descent of the ship^
so that she might float in equilibrium, the greatest strain would be little more
than f of this weight; but the elevating force in the case here calculated being only
K of the whole weight, it would require almost a second to raise the ship 1.265
feet, and to restore the equilibrium ; so that notwithstanding its gradual application,
dependent on the progressive velocity of the waves, which varies with the depth of
the fluid, there must be an interval during whiph it operates veiy nearly in ils whole
extent, especially as the occurrence of a partial obstruction tends to increase the total
height of a wave at the point where it is situated.

•^;

Na XXIV. EMFLOTMENT OF OBLIQUE RIDERS. 543

inetant when the ship is in a horisontal position, while in more
oommon cases, when the waves are narrows, the strain will
be proportionally smaller and nearer to the extremity. Hence
it appears that the strain produced by the action of the waves
may very considerably exceed in magnitude the more per-
manent forces derived from the ordinary distribution of the
weight and pressure, being, according to this statement, nearly
three times as great ; so that when both strains co-operate, their
sum may be equivalent to about 15,000 tons, acting on a lever
of one foot, and their difference, in oppomte drcumstances,
to about 5000. There may possibly be cases in which the
pressure of the waves produces a still greater etkd than this;
it may also be observed, that the agitation accompanying it
tends to make the fiaLStenings give way much more readily, than
they would do if an equal force were applied less abruptly.
At the same time, it is not probable that this strain ever be*
comes so great, as to make the former perfectly inoonaderable
in comparison with it, espedally if we take into account the
uninterrupted continuance of its action : it appears therefore
to be highly proper that the provision made for counteracting
the causes of arching should be greater than for obviating the
strain in the contrary direction : for example, that if the
pieces of timber intended for opposing them were, on account
of the nature of their fastenings, or for any other reason, more
capable of resisting compression than extension, they should
be so placed as to act as shores rather than as ties : although
it by no means follows, from the form which the ship assumes
after oncebreaking, that the injury has been occauoned in the
first instance by the immediate causes of arching : since, when
the fastenings have been loosened by a force of any kind,
the ship will naturally give way to the more permanent pres-
sure, which continues to act on her in the state of weakness
thus superinduced.

4. Breaking Transversely.

The pressure of the water against the sides of a ship has
also a tendency to produce a curvature in a transverse direc-
tion, which is greatly increased by the distribution of the

544 REMARKS ON THE Ko. XXIV.

weight, the parts near the sides being the heaviest, while the
greatest vertical pressure of the water is in the neighbouriiood
of the keel. This pressure is often transmitted by the stan-
chions to the beams, so that they are forced upwards in the
middle : when they are unsupported, the beams are more ge-
nerally depressed in the middle by the weight of the load
which they sustain ; while the inequality of the pressure of the
water co-operates with other causes in promoting the separadon
of the sides of the ship from the beams of the upper decks. On
the other hand the weight of the mainmast often prevails
partially over that of the sides ; so that the keel is forced
rather downwards than upwards in the immediate neighbour-
hood of the midships. The tendency to a transverse curvature
is observable, when a ship rests on her side, in the opening of
the joints of the planks aloft, and in their becoming tighter
below: although this effect depends less immediately on the
absolute extension and compression of the neighbouring parts
than on the alteration of the curvature of the timbers in conse-
quence of the pressure.

5. Lateral Curvature.

In such a case there is aJso an obvious strun tending to pro-
duce a lateral curvature ; and shores are sometimes employed
to prevent its effects, when a ship is ^'hove down" on her aide.
This indeed is comparatively a rare occurrence ; but when a
series of large waves strikes a ship obliquely, they must often
act in a similar manner with immense force : the elevation
on one side may be precisely opposite to the depresaon on the
otiier ; and the strdn fi*om this cause can scarcely be less than
the vertical strain already calculated : but its effects are less
commonly observed, because we have not the same means of
ascertaining the weakness which results from it, by the operation
of a permanent cause. When a ship possesses a certain degree
of flexibility, she may in some measure elude the violence of this
force by giving way a little for the short interval occupied by
the passage of the wave ; but it may be suspected that her
sailing, in a rough sea, must be impaired by such a temporary
change of form.

J No. XXIV. EMPLOYMENT OF OBLIQUE RIDERS. 545

I 6. Grounding.

When a ship takes the ground, she may either give way at
^ once to the stroke of a rock, or rest on a bottom more or less

soft, until she is either wholly or partially abandoned by the
water. In the former case her resistance must depend in great
measure on the parts in the immediate neighbourhood of the
injury : in the latter it may happen, that she may be supported
by so large a surface, as to be more in danger of parting aloft
than of being crippled below. Commonly, however, die floor
timbers are forced in at one end, the first futtocks, which are
their immediate continuations, being broken off; and sometimes
the opposite ends of the floor timbers are forced out, especially
in large ships without riders, their attachment to the keel
remaining unimpaired.

7. Decay.

The causes which promote the decay of timber are only so
far understood, as we are acquainted by experience with their
effects. A partial exposure to moisture appears to be by far
the most general of these causes: it is well known that total
submersion does not accelerate decay ; a surface which is kept
moist by imperfect contact with another, so that a portion of
water is retained between them by capillary attraction, seems
always to be the part at which the timbers begin to rot : while
both the sur&ces completely exposed either to the drier air, or
to the water, and those which are wedged closely together,
and press strongly against each other, remain perfectly sound.

8. Means of Resistance.

We are next to inquire into the comparative advantages of
different angular positions of the timbers of' a ship for resistbg
the forces which have been described ; and in particular how
far the arrangements, which have been proposed by Mr. Sep-
pings, are better calculated for the purpose, than the common
modes of construction. Whatever opinion we may ultimately
form of these arrangements, they are by no means suflScienily

VOL. L 2 N

546 REMARKS OK THE No. XXIV.

justified by the experiments which have been exhibited in illus-
tration of them. These experiments show, that when two
parallel planks (fig. 132) have loose pieces interposed, ex-
tending perpendicularly from one to the other, they are in-
comparably weaker, with respect to any transverse force, than
when the intermediate pieces are in an oblique direction, so
as to constitute a frame, which can only be bent as a whole.
But it cannot for a moment be imagined, that the planks of a
ship are connected witli the timbers in as loose a manner as
these transverse braces, which will scarcely support their own
weight for the purpose of the experiment; and in fact the
comparison would have required, that the whole space included
by the parallelogram should be filled up in each case by dmilar
braces, or at least that the two planks should have been firmly
united at the loose end to the transverse braces (fig. 134) ; and
it ft demonstrable that in this case the same weight would have
broken the pins, as if one of the planks had been oblique, or as
if the planks had remained parallel, and had been connected by
oblique pieces.

Such a result would, however, be far from proving the in-
utility of the addition of oblique braces to a rectangular frame :
for the kind of strength required for any particular purpose
is not always determined by the magnitude of tlie force which
would be capable of breaking the substances concerned, although
the power of resisting such a force is properly called strength,
in the most limited sense of the term : but there are many
occasions on which stiflness or inflexibility b of still greater
consequence than strength, and others again on which flexi-
bility is of material advantage. A coach spring, consisting of
ten equal plates, would be rendered ten times as strong, if it
were united into one mass, and at the same time a hundred
times as stifle, bending only one hundredth of an inch with the
same weight that would bend it a whole inch in its usual state,
although nothing would be gained by the union with respect to
the power of resisting a very rapid motion, which I have, on
another occasion, ventured to call resilience. (Lect. Nat.
Phil. I. p. 143. II. p. 50.) Now it appears to be extremely
difficult to unite a number of parallel planks so firmly together,

No. XXIV. EMPLOYMENT OP OBLIQUE RIDERS. 547

by pieces croBsing them at right angles, as oomidietely to prevent
their sliding in any degree over each other : and a diagonal
brace of sufficient strength, even if it did not enable the planks
to bear a greater strain without giving way, might still be of
advantage, in many cases, by diminishing the degree in which
the whole structure would bend beiiH^ it broke.

The strength of a simple rectangular frame, firmly fixed at
one end, is rendered somewhat less than double by perfectly
fiutening the joints at the other (fig. 135,) and the sliffiiess is
nearly quadruple.*

llie comparative security, obtuned by the additiou of a dia-
gonal brace, is almost wiUiout limit Supposing any number
of planks of equal dimensions to lie amply on each other without
any adhesion, and to be firmly fixed at one end, their aggregate
strength will be very little greater than that of a single plank
of one sixth part of the common depth or thickness of each,
supported by a brace a little stronger, in the direction of the
diagonal of the whole, (fig. 136 ;) and the stifihess of the parallel
planks will be as many times less than that of such a frame, as
there are planks in one third of the whole series. Thus if we

* When two horixontal ban are finnly fixed at one end only, and simply onited
at the other end by a vertical piece, their inunediate joint force in resitting fleznre
mnains nnaitered : hnt if the rertical piece is finnly fixed to the ends of the bars, it
may be considered as a lever held in cquilibriom by four forces, arisinff from the
repolsiTe and cohesive powers of the sepazate bars; and the sum of Uieae forces
most vanish when reduced to the same direction, while the sum of their actions
refeired to any point as a fnlcmm, vanishes also : and it is obvious that tiie total
qompression of the one bar will initially be eqoal to the total extension of the other,
provided that their strength be equal. Hence, if the mean distance of the bars be a,
and the depth 6, reckomd between the centres of action of the respective forces,
which in perfectly elastic bodies will be | of the whole depth, the first force being «,
and the second — y, the third will be y, and the fourth .- x, and, from the
equilibrium with respect to the point of application of the first force as a fulcrum,

we have the equation — 6y + ay - (a + 6) x s 0, and x := " y, while the

a + 6

joint effect of all the forces in resisting the pressure of the weight is 2 (y + ' ) 6 : c,

0 bemg the length of the bars, or ^ • .^fL., while the resistance of the two

c a +6

single bars would be -Jf , the faidination of the elementary forces being here reduced
c

to -. : and since the magnitude of y at the instant of bceaking is given, the force

2c
will be augmented by the finnness of the connexion in the ratio of 2a to a + ft^
which is always less than that of 2 to 1. The stiffliess may be nearly quadrupled by
the fastenings, since the depression at the moment of breaking is reduced to little
more than one half.

2 N 2

548 REMARKS ON THE No. XXIV.

had twelve planks, dx inches deep, and one thick, with friction
rollers interposed, it is demonstrahly true, however surprising,
that they would be very little stronger in supporting a weight
at the end, than a single tie an inch square in its section, assisted
by a diagonal brace of equal relative strength : and also that
this apparently slight structure would be nearly four times as
stiff as the 12 planks, being depressed only one fourth as much,
with a given weight, as the planks with a similar force acting
on them.*

It is well knovm, that if the planks were firmly united into
one mass, their strength would be rendered 12 times as great
by the union, and their stiffiiess 144 times: but this is not the
greatest resistance of which the materials are capable, even
without any extension of their base of support : for if the
planks were connected in pairs at half the distance of the whole
depth, and allowed to move freely round fastenings perfectly
secure, their strength, speaking theoretically, would be greater
by nearly one half than if they formed a compact mass, while
their stiffiiess would be only about one fourth as great : and
an effect nearly similar might be produced if the respective
pairs were united by oblique braces, extending over half the
depth of the whole structure, although it would be very
difficult, in practice, to make the strength of an arrangement

• *< If one of the surfiuses of a beam were incompressible, and tlie oohesire force
of all its strata collected in the other, its strength would be six times as great as in
the nataral state." Lect. Nat. Ph. II. 50, Art. 835.— Hence a plank of | of the
actual depth, acting simply as a tie, supported by a brace fixed at the distance of the
depth, would be as strong as the original plank: and hj increasing the distance
of the point of support of the brace in the ratio of the number of planks, the
strength of the two arrangements will remain equal, without altering the dimensions
of the tie. The length of a plank being e (Lect. II. 48, Art. 326), the depth 6, the
height ^f the modulus of elasticity m, the depression d, and the foree applied at the

end equal to the weight of a shnilar plank of the length g, we hare m = — ^, and
rf = ^ g ; but, for a simple frame of two equal pieces, the force being g, the

longitudinal extending force will be the weight of j g, and the actual extension

^ g 2«3

—- . -jj^, and the depression — g^ half as great as that of a plank of the same

dimensions, when g is given, or supposing the weight on the frame sextuple, so as
to be equal to that supported by the plank of six times the depth, three times as
great; but^ by taking 12 planks together, we increase their stiffiiess only 12 tiroes,
while that of the frame is rendered 144 times as great by a simiUr extension of the
base, so that it becomes in tliis case 4 times as great as that of the 12 planks.

Ko. XXIV. EMPLOYMENT OF OBLIQUE RIDERS. 549

of this kind even equal to that of a compact mass, since the
fastenings could never be so perfect* as to bring every fibre of
each plank into its full action at once, as the theory supposes.
If the planks w.ere already united into a compact mass, so as
to be incapable of bending except as a whole, it is of impor-
tance to inquire whether any advantage would be gained by
the further addition of oblique braces : and it will appear
that if the braces were fixed to the outermost planks of the
series only, they would have no manner of effect either on
the strength or on the stiffness, whatever might be their
direction ; but if they were sufficiently fastened throughout
their extent to each plank with which they come into contact,
they would add both to tiie strength and to the stifihess, very
nearly in the same degree as if they were fixed in the direction
of the planks, at a distance from each other equal to their
shortest actual distance, so as to constitute as many ribs as
there are braces in a transverse line (fig. 137).* Hence,
although there is obviously no economy in such an employment
of oblique braces, yet it is by no means true that oblique
braces are incapable of adding to the strength of a structure
composed of pieces arranged at right angles; the assertion
might however be very nearly correct in circumstances ap-
proaching to those of one of the experiments which have been
exhibited for the purpose of illustrating the utility of such
braces. On the other hand, tiie advantage of employing ob-
lique braces must depend in great measure on the degree in
which the angular position of the structure would be suscep-
tible of variation without them ; since, when properly fastened,
they must universally t«nd to preserve the form unaltered,
although they are somewhat less calculated to add to the
ultimate strength of the principal tie or shore, than if tiieir
direction had been longitudinal. To take, for example, the case

* The extension and compression of the whole fabric being supposed equal, the
diagonal braces will undergo no change of length, and.therefore will not assist in the
resistance, if only attached at their extremities. But in reality, although the ex-
tension and compression may be very nearly equal in the first instant of the change
of form, the extension will always be much greater after a certain time, from the
imperfection of the fastenings, which will allow the parts to separate while their
own resistance prevents their compression in a material degree : so that oblique
braces, however fixed, must in this respect add considerably to the strength.

550 REMARKS ON THE No. XXIV.

of a ship's arching or bogging : if the strength were overcome
without any deficiency of stiffiiess, the upper decks and wales
would be elongated, and the butts of the {danks aloft parted,
while the keel would be somewhat shortened, and the planks
near it crippled, so that a ship 176 feet long and 40 feet deep,
arching one foot with a uniform curvature, would have the
length of the parts aloft, on the level of the quarter deck, 22
inches greater than that of the keel. If, on the contrary, the
strength were not overcome, but the stifihess only fiiiled, the
angular situation of the parts being altered, and the joints
simply becoming loose without parting, the planks would slide
on each other, and their square ends would no longer remain
in the same vertical line at the ports, while there would be no
material alteration in the comparative length of the decks and
keel, nor any permanent parting of the butts of the planks.

Grounds of Decision respecting Oblique Riders.

This comparison therefore brings the question, respecting
the general utility of oblique riders, into a very narrow com-
pass; and we have only to inquire in what way it is most
usual for ships to exhibit symptoms of weakness, in order to
decide it Now it will appear that, in cases of arching in
general, some of the butts of the planks are always found to
have parted aloft;, at the same time that the angular position of
some parts of the structure has as uniformly been more or less
altered; and very generally a certain degree of sliding is
observable in the planks at the sides of some of the ports.
This sliding is seen very distinctiy in the planks of the Albion
and of the Belliqueux, now at Chatham : at the same time there
are also obvious indications of a certain degree of extension
and compression : in the Albion, the butts of the planks have
parted so far, that in some instances pieces have been let in I

between them ; and in the Belliqueux, there is a space of about I

five inches between the middle of the deck transom and the I

carling which had originally been in contact with it In the
Asia, lately launched in the Medway, the arching amounted
to three inches and a quarter, and the comparative length of
the upper and lower parts was probably altered about two

No. XXIV. EMPLOYMENT OF OBLIQUE RIDEBS. 551

inches at most : the parting pf the butts amounted to iV of an
inch each '^ for upwards of fifty feet in length in the midships,
and for about eight feet from the top of the side/' making a
total exten^on of probably less than an inch : so that about
half the effect seems to have been produced in one way, and
half in the other : but apparently the greater half by the want
of stifihesB. It b also usually observable, that there has been
some degree of permanent compression or crippling below, the
butts of the planks opening when the cause of arching has been
remoyed, and the sheathing being more wrinkled tiian would
have happened from the simple bending of the planks. Where
it has been observed, that the fore part of all the treenails sup-
ported the pressure of the planks in the after part of the ship,
and the after part in the fore part of the ship, the observation
must probably have been made on the lower parts of the ship,
fitmi the eflect of a partial compression of this kind.

Authariiiet.

From this statement it appears that unless some very strong
fSsusts can be produced, to disprove the probability, that the
relative angular position of the parts constituting a ship may
always be materially altered, without an absolute failure of
strength, it cannot be denied that the principle of oblique
bracing offers a remedy for the tendency to' arch, whatever
doubts there may be of the efficacy of any particular mode of
applying it. And even if no observations could be produced in
confirmation of the frequent occurrence of such a change of the
angular situation of the timbers, the supposition that the stiff-
ness could be perfect in this respect, notwithstanding the unequal
shrinking of the timbers, and other similar circumstances, while
the ultimate strength gave way by the failure of the fastenings,
is in itself so highly improbable, that no pontive evidence would
be required for its complete rejection. We shall find, accord-
ingly, that Mr. Bouguer takes for granted the existence of a
partial flexure, as sufficiently admissible without direct proof,
and recommends the adoption of oblique planking as a remedy ;
and that other experienced authors have been equally favour-

552 REMARKS ON THE No. XXIV.

able to tfae employment of some similar arrangemeniB.* In
speaking of Mr. Gobert's mode of placing the ceiling of a ship
obliquely, Mr. Bouguer observes, that ^^ this method cannot fiiil
of producing the most desirable effects ; for when the planking
both within and without was arranged in the direction of the
keel, it happened, in case of the ship's arching, that the rect-
angles formed by the timbers and the planking, merely changed
their figure a little, so as to become rhomboids, two of the
angles opening a little, while the other two became more acute :
but when the planks of the ceiling are laid in an oblique direc-
tion, they serve as diagonals to the rectangles, so that a simple
change of the relative angular situations of the sides is not
sufficient to admit of the arching, without an alteration of tfae
length of the diagonals, which would afibrd a reastanoe incom-
parably greater, especially at the upright parts of the sides,
although at the floors it would have but littie effect" Traite
du Navire, 154. Mr. Groignard also, whose memoir, on the
improvement of ship-building, has been obligingly communicated
to me by an ingenious gentieman, formerly his pupil, although
he objects to Mr. Gobert's method, confesses that he " should
have very much approved this mode of disposing the ceiling, if
it had been possible to employ straight planks, having the same
obliquity without interruption, throughout the whole of the
ship's length ;'* but thinks, with Bouguer, that in carpentry,
" every interruption is to be avoided as dangerous ; " an objec-
tion so vague, as neither to require nor to admit a very distinct
reply. Don George Juan, too, after a calculation of the abso-
lute strength of the pieces of timber employed in the construc-
tion of a ship, very properly remarks, that the effect of arching
must be attributed not to their want of strength, but to *^ their
play on each other."

9. Mr. Seppinos's Braces.

It appears, therefore, to be sufficiently established, that the
principle of employing oblique timbers is a good one, provided
that it be so applied as to produce no practical inconvenience.

♦ See Sir John BaiTow*s obserrations on this statement. Quarterly Review^ vol.
xii., p. 457 — Note by the Editor.

No. XXIV. EMPLOYMENT OF OBUQUE RIDERS. 553

We must next inquire whether Mr. Seppings has introduced it
in a manner likely to be effectual, and not lifible to any material
objections. He places, on the sides of a 74-gun ship, several
series of oblique braces, principally between the ports, in the
place of the internal planking, making an angle of about 24^
with the decks ; consisting of planks 4 inches thick, and about
11 wide, coaked and bolted to the timbers, and abutting against
upright pieces similarly iastened. Now it follows, from what
has already been stated, that these pieces haye about four-fifths
as much effect in co-operating with the neighbouring parts,
which act horizontally, as if they had been placed in the same
situation with them, even on the supposition that the relative
angular situation of the pieces is unalterably fixed ; but for pre-
venting the alteration of this situation, there is no doubt of their
being very advantageously arranged, so &r as their strength is
sufficient ; and the existence of a tendency to such an alteration,
in a very material degree, appears to be altogether indisputable.
Below the gun deck, the oblique timbers are considerably
stronger, although they act under circumstances somewhat less
favourable.

If, however, the resistance of a part of a structure is very
immediately directed against a certain force, without au ade-
quate co-operation from other parts of that structure, and if
being abandoned by those parts, it is exposed to a strain which
it is too weak to withstand, it is obvious that it must inevitably
be the first to give way, and must leave the rest of the fabric
more exposed to be overpowered by such a force, than before
its introduction. We must therefore inquire, how far it is
possible that Mr. Seppings's braces should be so abandoned.
Now supposing a 74-gun ship to arch two feet, and one half of
the change to depend on the sliding of the planks over each
other, which will be allowed, by those who doubt the utility of
the arrangement, to be fully as much as can ever happen ; the
greatest fall of the surfiEU^ will be one foot in 44, and the length
of the brace will be diminished rhr or iV of an inch in the
length of six feet, which, with a moderate allowance for the
partial yielding of the fiistenings, it will be perfectly capable of
supporting without being crippled, although indeed it could

554 REMARKS ON THE No. XXIV.

scarcely support much more. It is obvious, however, that this
suppositioQ in many respects far exceeds the utmost that can
possibly happen : and it would even require a greater force to
produce such an effect on the braces, than any which the ship i

actually sustains. In order to calculate the magnitude of the \

greatest strain whidi these pieces could support, it will be
safest to proceed on the supposition, that each square inch of the
section of good oak timber is capable of resisting the pressure
of four tons on an average : it will then appear that a single
series of suctx braces, as Mr. Seppings employs, extending
throughout the length of each dde of the ship, would support
a weight of 143 tons, in whatever way the force counteracting
it might be applied.; and estimating the effect of all the braces
and riders as equivalent to about four such series, the whole
would resist a force of 570 tons ; while the greatest force de-
rived from the distribution of the weight together with the action
of such waves as we have considered, amounts to about 450
tons : so that the strength of these braces can scarcely be insuf- %

ficient to support the pressure, unless the ship should be left
dry, resting on the middle of her keel, and the braces should
be abandoned by all the other parts which usually co-operate
?rith them.* The fastenings must indeed be considerably
weaker than this, and the other parts of the ship considerably
stronger ; but since the fiistenings appear to possess sufficient
strength to resist any strain which is actually likely to afiect
them, there seems to be no inconvenience in their inferiority to
the other parts. In hct^ the Tremendous actually supported,
for three days, without any perceptible change of form, a strain

* If a jointed parallelogram, composed of pieces of inrariable length, having one
of its sides fixed in a Terti<»i position, be supported by a diagonal brace, the com-
pression or extension of the brace will be to the descent of a weight connected with
the moreable end of the parallelogtam, as the depth of the parallelogram to the
length of the brace, whatever the actual distance of the weight may be ; so that ,«

although the strain on the horizontal pieces increases with this distance, that which
affects the brace is independent of it ; the relative being to the absolute strength as
the depth of the frame is to the length of the brace. We most therefore inquire,
what is the greatest absolute force that can be supposed to urge a given portion of the
fabric in either direction : thus the excess of weight which has b^ attributed to the
bowsprit and the neighbouring parts being 192 tons at Id} feet from the head, this
force may be occasionally increased by a similar pressure derived from the effect of
the waves, which alone would amount to 302 tons at 35} feet from the head, and
which may sometimes co-operate with the former, so as to constitute a force of about
450 tons, about 25 feet from the head.

No. XXIV. EMPLOTHENT OF OBLIQUE RIDERS. 555

fully equal to that which is here calculated^ having been pur-
posely left on shores, which extended through 52 feet only of
her length. But it must be remembered, that such a force,
from its very gradual application, must be much less trying to
the ship's strength, than the more abrupt changes which occur
at sea, and it must on the whole be inferred, that it would be
unsafe to trust tg the braces alone, unsupported by the co-opera-
tion of the neighbouring parts. It would probably be easy to
add some further strength to these braces near the ends of the
ship, where the strain on them is the greatest, especially about
30 feet from the head, if it were found that tJiey gave way
before the rest of the timbers ; and it might also be posdUe to
replace them, if they had once fiiiled, with greater ease than
many other parts of the fiibric.

It may be questioned how far it is allowable to omit any part
of the inner planks between the ports, for which the braces are
a substitute, on account of their utility in securii^ the butts of
the planks, which are always made to shift where they are sup-
ported by this subsidiary tie : but with the outer planking which
remains, and with the partial assistance of the braces, to say
nothing of that of the shelf pieces, it can hardly be believed,
that the tie is more likely to part between two ports of the same
deck, than immediately over one of them.

It has been very ingeniously observed, that arching is not
only a part of the evil occasioned by a ship's weakness, but
that it has an immediate tendency to afford a partial remedy
for the cause which produces it, by makmg the displacement
greater at the extremities of the vessel, and smaller in the
middle ; but, in fact, this change appears to be too inconsi-
derable in its extent, to produce any material benefit: the
strain at the midships being diminished by each inch of arching
only 66 tons, supposed to act at one foot : so that very little
relief is obtained from the change, in comparison with the
whole strain.

Hie case of the Kent, which broke in a remarkable degree,
notwithstanding the employment of riders of large dimensions,
is perfectly recondleable with the principles which have been
laid down : indeed these riders, whic^ made an angle of a few

556 REMARKS ON THE No. XXIV.

degrees only with a vertical line, could have so little effect either
on the strength or on the stifihess of the structure, that there
was not the slightest reason to expect any material advantage
from their application.

The explanation which has been given of the imiversal
tendency of ships of war, in all common drcumstances, to arch
throughout their length, is sufficient to justify the different
directions in which Mr. Seppings now arranges his braces in the
different parts of the ship, since they must necessarily afford a
greater strength as shores than as ties, and since the most per-
manent and the greatest strain will generally be such as to call
them into action in this capacity. When, however, a ship is
compared to an inverted bridge, it must not be forgotten how
necessary it frequently becomes, to consider these braces in a
diiSerent capacity, and to provide for this contingency, as
indeed Mr. Seppings has not neglected to do, by employing
such fastenings, as are extremely well adapted to secure their
action as ties.

The shelf pieces, which Mr. Seppings employs, and the
superior strength of the fastenings of his decks to the breast
hooks and transoms, have so obvious a tendency to counteract
the causes of arching, that it is unnecessary to insist on their
utility ; the weight and expense of the shelf pieces are probably
the only drawbacks upon the advantages, which they, are so
manifestly calculated to afford, in resisting both a vertical and
a lateral strain ; and even in these respects, they appear to be
preferable to the wooden knees formerly employed.

The filling up the intervals of the timbers, throughout the
hold, with wedges of old stuff, is perhaps the most indisputably
beneficial of all the alterations which Mr. Seppings has either
introduced, or revived in an improved form. The strength,
which is thus obtained, acts immediately in the prevention of
arching, and is probably, in this respect, more thsoi an equiva-
lent to that of the internal planking, which has been omitted ;
while the cohesive strength of the external planking, considered
as a tie, is still probably more than sufficient for resbting the
smaller force, which occasionally operates in a contrary direction :
although the strength of the ship, for resisting such a force, is

No. XXIV. EMPLOYMENT OF OBLIQUE RmERS. 557

certainly much diminidhed by tbe change. From the manner in
which these wedges are driven by Mr. Seppings, it may easily
be understood, that they may tend to produce a convexity
below, without raising any part of the keel from the blocks,
merely causing it to press more strongly on them at the mid-
ships ; so that if this difference becomes equal to that of the
weight and pressure after launching or floating, there may be no
tendency to any further change whatever ; and hence it may hap-
pen, that without any other superiority of stiffiiess, or even of
workmanship, a ship may appear wholly exempt from arching,
as the Tremendous did, and some other ships are said to have
done. Without the operation of some such cause, even a hollow
cylinder of compact oak, 180 feet long, 50 feet in diameter, and
6 inches in thickness, if such a mass could be supposed to exist,
would exhibit, when immersed to the depth of its axis, a degree
of arching just perceptible, from the longitudinal pressure of
the water only, amounting to about iV of an inch ;* besides a
curvature proportionally greater from the other strains, which
have been already calculated. Mr. Seppings has also very pro-
perly introduced, in the Tremendous, an additional kelson on
each side of the step of the mainmast, in order to support its
weight, and to prevent the partial sagpng of the keel.

* *' The stiffneBS of a cylinder » to that of the circamscribing priitin, as three
times the bulk of the cylinder is to four times that of the prism** (Led, Nat.

Phil. II. 83, Art. 339. B.): but the radius of currature of a prismatic beam is ^H

12a/
(P. 46. Art. 321.), 6 being the depth, m the weight of the modolns, / the force, and
a the distance of itis application : and taking m for the weight of the modalns of the

cylinder, its radius of cunrature will be . But since the stifiness is as the

16a/
fourth power of the diameter (P. 49, Art. 333), that of the hollow cylinder in
question will be reduced in the ratio of 1 to 1 — .98^ = .0786. Now when a
cylinder is immersed to the depth of its axis, the calculation of the effect of the
longitudinal pressure exactly resembles that of the stiffness, the strain being to that
which would be the effect of the pressure on the ends of the circumscribing prism as
} X .7854 = .58905 to 1 ; but the strain on the prism would be = 50 x 25 x

50
12.5 X ~ : 35 = 7440 5, and for the cylinder, a/ = 4383 : and since the height

3
of the modulus of elasticity of oak is 5060000 feet (p. 509), and its specific gravity
nearly equal to that of water, or perhaps a little greater, we have m = 5060000 x

50 X 50 X .7854 : 35 tons, and Uie radius of curvature .0786 — = .0786

16a/
^ 50^ X 5060000 X .7854 : 35 _ ^ ^^^^ ^ 5060000 ^ ^^^3^

16 X I X .7854 X 50* • (24 X 35) 12 : 24 '

and dividing the square of 90 by twice this number, we have .0051, or one
sixteenth of an inch, for the vened sine or arching.

558 RBKAEKS OK THE No. XXIV.

10. Riders.

With respect to the transverse strain, or the tendency of the
sides to sink in comparison with the keel, some strength is pro-
bably gained by Mr. Seppings's mode of fixing the filling timbers
in the same manner as the frames : and some advantage must
be attributed to the co-operation of the wedges, or fillings in,
with the timbers, as £ur as their connexion is capable of bringing
them into action. . The common ceiling is by no means advan-
tageously placed for assisting in a resistance of this kind, rince
it can (mly act where the curvature would be increased by the
bending of the sides, and even there can only be compressed in
a transverse direction (fig. 138). The riders, commonly placed
upon it, on the contrary, are very favourably atuated for
assisting in this action ; but Mr. Seppings's riders are so mudi
more numerous, as to possess, notwithstanding their obliquity, a
still greater force. The iastenings of the beams to the sides are
also concerned in resisting a strain of this kind, as well as in
counteracting the tendency to an extension aloft, which is the
more immediate consequence of the unequal pressure of the
water against the ship's sides. Mr. Seppings's fastenings, so far
as they depend on the shelf pieces, have probably some advan-
tage over the more common ones ; but the iron knees which he
employs (fig. 139) do not appear to be quite so economically
arranged as the straps of a simpler form, which other builders
have used ; they aflbrd, indeed, a very direct connexion with
the timbers, and they save some valuable wood in the chocks
which support them : but still there appears to be some waste
of strength when they act as ties, from the great obliquity of the
shoulders, with respect to the direction of the force ; to say
nothing of the expense of the workmanship : and if, as Captain
Campbell seems to have suspected, there is any slight defidency
in the transverse strength of the Tremendous at the waterways,
the circumstance may afibrd a further reason for doubting of the
utility of these fastenings.

No. XXIV. EBffPLOYMENT OP OBLIQUE RIDERS. 559

11. Decks.

The least obvious advantage attributable to the obliquity
introduced by Mr. Seppings appears to be in his mode of laying
the planks of the decks ; parts which seem to be principally
required to co-operate with the sides of the ship as ties in a
longitudinal dinsction : for the slight curvature, which is ^ven
to tiiem, can no more render them incapable of such an action,
than the bending of a towing rope prevents its pulling along a
boat But in the first place, the lower decks can have little or
no action of this kind, from their near approach to the line, at
which extension ceases, and compression b^ns, at least until
some of the fastenings give way : and secondly, the upper decks
lose but one third of their strength in this capacity by having
their planks disposed at an angle of 45 degrees with the sides,
while the obliquity must be capable of affording some additional
power of resisting the violent action of the waves, which some-
times produces an immense strain in a transverse or lateral
direction, as well as of enabling the ship, in case of necessity, to
be more safely *^ hove down" on her side. There seems also
to be some convenience in havmg the ends of the planks covered
by the waterways, with respect to keeping the wings of the ship
dry, althou^ it has been suspected that the ends so covered
may be rendered somewhat more liable to decay. It may,
however, be apprehended, that any force, tending to shorten the
deck, will have some little efiect in forcing out the sides;
for instance, if the. whole deck became three inches shorter,
the length of the planks remaining the same, they must force
out each of the sides about a quarter of an inch, provided
that their connexion with the beams allowed such a change,
which appears indeed somewhat improbable. There may pos-
sibly be a slight difficulty in adjusting the planks to the curva-
ture of the beams; but this difficulty appears to be readily
overcome in other cases, as in that of the common ceiling. It
may hereafter deserve to be inquired, how far an oblique
direction of the carlings between the beams, which in their
present situation seem to contribute very litde to* the strength,

560 REMARKS ON THE ' No. XXIV.

might enable them to oo-operate in resisting a lateral force, if
the arrangement could be made without too much weakening
the beams, in procuring proper abutments for these piecea

12. FLooRa

It cannot easily be admitted, that Mr. Seppings's construction
aflbrds any additional strength to a ship's bottom in case of her
grounding. The fillings in between the timbers must indeed be
extremely useful in this respect, first by giving firmness in the
direction of the length, since even a straight plank is strength-
ened by having the incompressibility of its outside increased,
much more one that is curved, in however slight a d^ree ; and
secondly, by co-operating with the timbers, conadered as shores,
so far as the wedges are fixed in their places by their lateral
adhesion or otherwise.

The ceiling, which has been omitted, can have very little
efiect by its own strength in preventing the separation of the
timbers at the floor heads; but where there are transverse
riders, it must be of essential advantage in enabling these to
come into action, for the support of the neighbouring parts
exposed to pressure; somewhat more eflectually indeed, in
many cases, than Mr. Seppings's diagonal riders and their
trusses can do, notwithstanding the superiority of thdir number
and aggregate strength ; on account of the magnitude of the
intervening spaces, which might happen to receive the prindpal
stroke near their centrea This magnitude does not, however,
contribute by any means in the same proportion to the weakness
of the parts, as it would do if the surface were plane : and it is
not improbable, that for supporting the weight of the ship on a
very soft ground, Mr. Seppings's arrangement might affi)rd
equal strength with the common form, as seems to have been
exemplified by his experiment of leaving the Tremendous for
three days on fourteen shores only, without injury : but for
encountering the stroke of a rock, or of very hard ground, Mr.
Seppings's ship would probably be inferior, since in this case
greater 8ti£Biess, even with equal strength, would be detrimental
rather than beneficial ; while, on the other hand, she would on-

No. XXI V; AiMPLOYMENT OF OBLIQUE RIDERS. 561

deniably be less liable to suffer from any injury that might
happen to her outer planking only ; and, from her superiority
in this respect, might possibly sustain, without inconyenience, a
stroke, which would be ultimately fatal to a ship of a different
construction.

13. Durability.

There does not seem to be the slightest ground for the
apprehension, that the filling in should render the ship's timbers
liable to decay : on the contrary, the timbers of the Sandwich
were found perfectly sound in the lower half of their length,
opposite to the wedges which had been driyen in between them,
and completely decayed in the upper half, where they had
been exposed, in the usual manner, to the action of the confined
moist air and water ; and this result is perfectly conformable to
analogy with the few facts that haye been ascertained, respect-
ing the general causes of decay. The utility of the fiilling in, for
preyenting the accumulation of filth, and for keeping the ship
free from foul air, with respect to the comfort, and perhaps to
the health of the crew, is too obyious to require discussion.
How far the economy of timber may in all cases be so great as
Mr. Seppings is disposed to belieye, can best be ascertained by
those who are in the habit of estimating its yalue : but if the
durability of the yessel only were improyed, at an equal
expense, the adoption of his alterations would still be highly
adyisable.

14. Conclusion.

It is by no means impossible, that experience may suggest
some better substantiated objections to these innoyations, than
haye hitherto occurred : but none of those objections which have
yet been adyanced, appear to be sufiiciently valid to warrant a
discontinuance of the cautious and experimental introduction of
Mr. Seppings's arrangements, which has been commenced by
orders of the Board of Admiralty. The filling in seems to be
wholly unexceptionable ; the braces between the ports appear to
be decidedly more beneficial than the planks for which they are

VOL. L 2 0

562 ON THE EMPLOYMENT OF OBLIQinE RIDEBS. No. XXIV.

substituted ; and the coakiqgs seem to be very judiciously em-
ployed in yarious pajTts of the structure : but something more
may possibly be hereafter efiected for the further improyement
of the decks, and for the more complete provision of a substitute
for the thick stuff of the ceiling, in addition to the diagonal
riders, if experience should prove that there is any deficiency in
the resistance of these parts. But it must be remembered, in
forming conclusions from such experience, that when an arrange-
ment of any kind has nearly attained the maximum of its per-
fection, it may demonstrably be varied in a considerable degree,
without a proportional alteration of its effect; so that the
most correct knowledge of sdentific principles, and the minutest
accuracy in their application, must become indispensably neces-
sary, in order to secure us from the introduction of material
errors, derived from the latent operation of accidental causes,
foreign to the immediate subjects of investigation.

Welbeck Street, 80 Dec., 1811.

«xn^

OBLIQUE BIDERS.

Fi^.m-uo.

Iy.l32.

II II II

1^.13^

Jig.m

Iufl4'0.

Rgm.

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No. XXV. ELECTIVE ATTRACTIONS. 563

No. XXV.

A NUMERICAL TABLE OF

ELECnVE AHRACTIONS,

WITH RBMASKS ON THE SEQUBNCBB OF DOUBLE DBCOMPOBITIONa.

From the Philosophical Txaiuactioiis for 1809.

Read February 9, 1809.

Attemttb haye been made, by several chemists, to obtain a
series of numbers capable of representing the mutual attrac-
tive forces of the component parts of difierent salts ; but these
attempts have hitherto been confined within narrow limits, and
have indeed been so hastily abandoned, that some very im-
portant consequences, which necessarily follow from the general
principle of a numerical representation, appear to have been
entirely overlooked. It is not imposdble that there may be
some cases in which the presence of a fourth substance, besides
the two ingredients of the salt, and the medium in which
they are dissolved, may influence the precise force of their
mutual attraction, either by aflecting the solubility of the salt
or by some other unknown means, so that the number natu-
rally appropriate to the combination may no longer correspond
to its adSfections ; but there is reason to think tiiat such cases
are rare, and when they occur they may eaaly be noticed as
exceptions to the general rules. It appears, therefore, that
nearly all the phenomena of the mutual actions of a hundred
difierent salts may be correctly represented by a hundred num-
bers, while, in the usual manner of relating every case as a
different experiment, above two thousand separate articles would
be required.

Having been engaged in the collection of a few of the prin-

2o2

564 A NUMERICAL TABLE OF No. XXV.

cipal facts relating to chemistry and pharm£u;y, I was induced to
attempt the investigation of a series of these numbers ; and I
have succeeded, not without some difficulty, in obtaining such
as appear to agree sufficiently well with all the cases of double
decompositions which are fully established, the exceptions not
exceeding twenty out of about twelve hundred cases enu-
merated by Fourcroy. The same numbers agree in general
with the order of simple elective attractions as usually laid
down by chemical authors ; but it was of so much less impor-
tance to accommodate them to these that I have not been
very solicitous to avoid a few inconsistencies in this respect,
especially as many of the bases of the calculation remain uncer-
tain, and as the common tables of simple elective attractions
are certainly imperfect, if they are considered as indicating the
order of the independent attractive forces of the substances
concerned. Although it cannot be expected that these num-
bers should be accurate measures of the forces which they
represent, yet they may be supposed to be tolerable approxi-
mations to such measures; at least if any two of them are
nearly in the true proportion, it is probable that the rest
cannot deviate very far from it : thus, if the attractive force of
the phosphoric acid for potash is about eight-tenths of that
of the sulfuric acid for barita, that of the phosphoric acid for
barita must be about nine-tenths as great ; but they are calcu-
lated only to agree with a certain number of phenomena, and
will probably require many alterations as well as additions
when all other similar phenomena shall have been accurately
investigated.

There is, however, a method of representing the facts, which
have served as the bases of the determination, independently
of any hypothesis, and without being liable to the contingent
necessity of any future alteration, in order to make room for
the introduction of the affections of other substances ; and this
method enables us also to compare, upon general principles,
a multitude of scattered phenomena, and to reject many which
have been mentioned as probable, though doubtful, with the
omission of a very few only which have been stated as ascer-
tained. This arrangement simply depends on the supposition

No. XXV. ELECTIVE ATTRACrriOKS. 565

that the attractive force which tends to unite any two sub-
stances may always be represented by a certain constant
quantity.

From this principle it may be inferred, in the first place, that
there must be a sequence in the simple elective attractions. For
example, there must be an error in the common tables of elec-
tive attractions, in which magnesia stands above ammonia
under the sulfuric acid, and below it under the phosphoric, and
the phosphoric acid stands above the sulfuric under magnesia,
and below it under ammonia, since such an arrangement implies
that the order of the attractive forces is thb: phosphate of
magnesia, sulfate of magnesia, sulfate of ammonia, phosphate
of ammonia, and again phosphate of magnesia ; which forms a
circle, and not a sequence. We must therefore either place
magnesia above ammonia under the phosphoric acid, or the
phosphoric acid below the sulfuric under magnesia ; or we must
abandon the principle of a numerical representation in this
particular case.

In the second place there must be an agreement between the
simple and double elective attractions. Thus, if the fluoric
acid stands above the nitric under barita, and below it under
lime, the fluate of barita cannot decompose the nitrate of lime,
since the previous attractions of these two salts are respectively
greater than the divellent attractions of the nitrate of barita
and the fluate of lime. Probably, therefore, we ought to place
the fluoric acid below the nitric under barita; and we may
suppose that, when the fluoric add has appeared to form a pre-
cipitate with the nitrate of barita, there has been some Mlacy in
the experiment.

The third proposition is somewhat less obvious, but perhaps
of greater utility : there must be a continued sequence in the
order of double elective attractions ; that is, between any two
acids we may place the different bases in such an order that any
two salts, resulting from their union, shall always decompose
each other, unless each acid be united to the base nearest to
it : for example, sulfuric acid, barita, potass, soda» ammonia,
strontia, magnesia, glycina, alumina, zirconia, lime, phosphoric
acid. The sulfate of potass decomposes the phosphate of barita.

566 A NUMERICAL TABLE OF No. XXV.

because the difierence of the attractions of barita for the sulfuric
and phosphoric adds is greater than the difference of the simi-
lar attractions of potass ; and in the same manner the difference
of the attractions of potass is greater than that of the attrac-
tions ci soda ; consequentiy the difierence of the attractions of
barita must be much greater tiian that of the attractions of
soda, and the sulfate of soda must decompose the phosphate
of barita ; and in the same manner it may be shown tiiat each
base must preserve its relations of priority or posteriority to
every other in the series. It is also obvious that, for similar
reasons, the acids may be arranged in a continued sequence
between the different bases ; and wheH all the decompositions
of a certain number of salts have been investigated, we may
form two corresponding tables, one of the sequences of the bases
with the adds, and another of those of the adds with the dif-
ferent bases ; and if either or both of the tables are imperfect,
their defidencies may often be supplied, and their errors cor-
rected, by a repeated comparison with each other.

In forming tables of this kind from the cases collected by
Fourcroy, I have been obliged to reject some tacts whidi were
eridently contradictory to others, and these I have not thought
it necessary to mention ; a few, which are positively related,
and which are only inconsistent with the prindple of numerical
representation, I have mentioned in notes; but many others,
which have been stated as merely probable, I have omitted
without any notice. In the table of simple elective attractions
I have retained the usual order of the different substances ;
inserting again in parentheses such of them as require to be
transposed, in order to avoid inconsequences in the simple
attractions : I have attached to each combination marked with
an asterisc the number deduced from tiie double decompositions,
as expressive of its attractive force ; and where the number is
inconsistent with the corrected order of the simple elective
attractions, I have also inclosed it in a parenthesis. Such an
apparent inconsistency may perhaps in some cases be unavoid-
able, as it is possible that the different proportions of the masses
concerned, in the operations of simple and compound decompo-
sition, may sometimes cause a real difference in the comparative

I No, XXV. ELECTIVE ATTRACTIONS. 567

I magmtude of the attractive forces. Those numbers, to which

I no asterisc is affixed, are merely inserted by interpolation, and

I they can only be so far employed for determining the mutual

r actions of the salts to which they belong, as the results which

I they indicate would follow from the comparison of any other

[ numbers, intermediate to the nearest of those, which are more

correctly determined. I have not been able to obtain a sufficient

number of facts relating to the metallic salts, to enable me to

comprehend many of them in the tables.

It has been usual to distinguish che attractions, which produce
the double decompooltioDs of salts, into necessary and superflu-
ous attractions ; but the distinction is neither very accurate nor.
very important : tiiey might be still further divided, accordingly
as two, three, or the whole of the four ingredients concerned
are capable of simply decomposing the salt in which they are
not contained ; and if two, accordingly as they are previously
united or separate ; such divisions would however merely tend
to divert the attention from the natural operation of the joint
forces concerned.

It appears to be not improbable that the attractive force of
any two substances might, in many cases, be expressed by the
quotient of two numbers appropriate to the substances, or rather
by the excess of that quotient above imity ; thus the attractive
force of many of the acids for the three principal alkalies might
probably be correctly represented in this manner ; and where
the order of attractions is difierent, perhaps the addition of a
second, or of a second and third quotient, derived from a dif-
ferent series of numbers, would afford an accurate determination
of the relative force of attraction, which would always be the
weaker, as the two substances concerned stood nearer to each
other in these orders of numbers ; so that by affixing to each
simple substance two, three, or at most four numbers only, its
attractive powers might be expressed in the shortest and most
general manner.

I have thought it necessary to make some alterations in the
orthography generally adopted by chemists, not from a want of
deference to their individual authority, but because it appears
to me that there are certain rules of etymology which no modem

568 A NUMERICAL rABLE OF No. XXV.

author has a right to set aside. According to the orthography
unirersally established throughout the language, without any
material exceptions, our mode of writing Greek words is always
borrowed from the Romans, whose alphabet we hare adopted :
thus the Greek vowel Y, when alone, is always expressed in
Latin and in English by Y, and the Greek diphthong OT by U,
the Bomans having no such diphthong as OU or OY. The
French have sometimes deviated from this rule, and if it were
excusable for any, it would be for them, since their u and oic
are pronounced exactly as the T and OT of the Greeks pro-
bably were ; but we have no such excuse. Thus the Frendi
have used the term acaustique, which some English authors
have converted into ^^ acoustics ;" our anatomists, however,
speak, much more correctly, of the ^^ acustic " nerve. Instead
of glucine, we ought certainly, for a similar reason, to write
glydne ; or glycina, if the names of the earths are to end in a.
Barytes, as a single Greek word, means weight, and must be
pronounced b^y tes ; but as the name of a stone, accented on
the second syllable, it must be written barites ; and the pure
earth may properly be called barita. Yttri^ I have altered to
itria, because no Latin^word begins with a Y.

No. XXV.

ELECTIVE ATTRACTIONS.

569

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35s £

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570

▲ NUHEBICAIi TABLE OF

No. XXV.

NiTEIC

Acid.

NiTBio AND Muriatic Acids.

Barita

Potaas

Barita

Potass

Barita (10)

Potass

PotaM

Soda

Potass

Soda

Potass

Soda

Ammonia

Soda

Ammonia

Soda

Barita (10)

Strontia

Ammonia

MagnesU

Ammonia

Ammonia(7,ll)

Lime

Glycina

Magnesia

Glycina

Magnesia

Magnesia(7)

Magneoa(7)

1 Alumina

Glyoina

Alumina

Glydna

Strontia.

AmmomaC?) Zirconia(8)

Alumina

Ziroonia

i^lnn^in^

Lime

Glycina

Barita

Ziroonia

Barita

Zirconia

Glycina

Al"««i™»-

StTQDtia

Strontia (9)

Strontia

AlnwiiTxa

Zirconia

Lime

lime

Zirconia

MUBIATIC

Phosphoric

Flugbic

SULFUBOUS

BORACIC

Carbonic

(7) A triple salt is formed. (8) Fourcroy says, that the muriate of zirconia
decomposes the phosphates of barita and strontia. (9) According to Fourcro/s
account, the fluate of strontia decomposes the muriates of ammonia, and of all the
bases below it ; but he says in another part of the same volume, that the fluate of
strontia is an unknown salt. (10) According to Fourcroy's account of these com-
binations, barita should stand immediately below ammonia in both of these columns.
(11) With heat, the carbonate of lime decomposes the muriate of ammonia.

Fhosfhobic Acid.

Barita

Lime

Barita

Potass

Barita

Ume

Barita

Lime

Soda

Lime

Potass

Barita

Potass

Soda

Lime (13)

Soda

Strontia

Magnesia

Ammonia(12)

Ammonia

Magnesia

Ammonia

Magnesia

Magneda

Glycina?

Glycina

Alumina

Alnmifift.

Alumina

Zirconia

Ziroonia

Zirconia

Ziroonia

Fluoric

SULFDBOUB

BORAdC

Carbonic

(PH06PHO

(12) According to Fourcroy, the phosphate of ammonia decomposes the borate of
magnesia. (13) Fourcroy says, that the carbonate of lime decomposes the phosphates
of potass and of soda.

Fluoric Acid.

Lime lame

Potass

Potass Barita

Soda

Soda Strontia

Lime

Magnesia Potass

Barita

Ammonia Soda

Strontia

Glycina Ammonia

Ammonia(14)

Alumina Magnesia

Magnesia

Zirconia Glycina

Glycina

Strontia Alumina

Barita Zirconia

Ziroonia

SULFDBOUS BORACIC

Carbonic

(14) According to Fourcroy, the carbonate of ammonia decomposes the iluates of
barita and strontia.

No. XXV.

ELECTIVE ATTRACnOira.

571

SuurvBous Acid.

Bo&Actc Acid.

Bonta

Potass

Lime

Zirconia

Potass

Strontia

Soda

Strontia

Alnmina

Potass

Barita(15)

Barita

Glydna

Lime

Soda

Strontia

Zirconia

Ammonia

Barita

Anunoiiia

Ammonia

Alnmiim.

Magnesia

Strontia

Magnesia

Lime

Glyeina

Strontia

Magnesia

lime

Magnesia

Soda

Ammonia

Olyoioa

Glycioa

Amm^ynjii

Potass

Glyeina

Alumina

Aiftinimi

Soda

Barita

Alniwin^

Zirconia

Potass

Lime

Zirconia

BoBAdC

Cabbonic

(NlTBOUS)

(Phosfhoroob?)

Cabbokic

(15) Fourcroy says, that the snlfitte of barita decomposes the carbonate of
ammonia.

Table of the Sequences of the Adda toith different Bases.

Babita.

Stbontia.

Like.

Sulftirie

8 C

P P

Nitric

N SS

F F

Mnriatie

M F

B B

PhosphorfeSS

SS P

88 C

C B

S SS

Fluoric

B 8

C 8

Bonuslc

F M

N N

Cwbonie

P N

M M

Stboittia

LM FT

MO AM OL

•O

All

Potass
Soda

Mao-

NE8XA.

MAON.srAlOf.

/S B

Gltcina

N C

Alumina

M P

ZiBOONIA

P F

Each wHh every
■ubnqiient btM
In this order

F SS
SS S
B N

C M

\ AM*

* The acids in this Table are denoted by their initial letterB—Sulfiiric acid being
expressed by S, and Salfurons by SS: Lime is denoted by LM, Potass by FT,
Soda by SD, and so on. In Nicholson's Joomal for March, 1809, Dr. Yoang gave
the following technical hexameters, to aid the memory in retaining the order of these
sequences, the familiar knowledge of which \ar Important to chemists, and we may
add likewise to physicians in the composition of their medicines.

CoNTENTio aquatica; Victobia ; Requies.

ReBARisne modo posse ad/ore &elllca roSTRa ?
Des nautam satis apta ci6o re/overe aLiMenta ;
Cor superest sanam ; /laMtque oPTatos abaoDe
4%>iritns; has animi ira/eret ti6i acerriaiA OAZas.

AiST BsONTes animosns acer6o /cedere paLMas
Caesus /ert : ut pro refros monet aPTa sODales I
Si possit, /ato tu6icen meMor addat honores.
Postulat 08sa relate, hen 1 /le6ile condere mArMor.
^B est /ixa, &onum oceli OAZis fimitumm.

ALMa huic pax /iat or6i, lassis omniPOTens Des
O pater I Ut /le6o jossus canere armigeriiM yim I
Dire qpi/ex 6elli, ceag^a normAM abjicere omnem.
Pax /essos 6ona mulcet, OAZis lastior anri.

PnesuMAM OAZas nemp% ad/ore mrsos afr alto hue ;
Mira da6it Inorapax, /ortassis in ultioiA Mondi.

NoU by Otf Editor.

572

A NUMERICAL TABLE OF

No. XXV.

The oomparatiye use of this table may be understood firom
an example : if we suppose that the nitrate of barita decom-
poses the borate of ammonia, we must place the boradc add
above the nitric, between barita and ammonia in this table, and
consequently barita below ammonia, between the fluoric and
boracic in the former : hence the boracic and fluoric acids must
also be transposed between barita and strontia, and between
barita and potass ; or if we place the fluoric still higher than
the boracic in the first instauce, we must place barita below
ammonia between the mtric and fluoric adds, where indeed it
is not impossible that it ought to stand.

Numerical Table of Elective Attractions.

BABITA. SnONTIA. POTABB. SODA.

Solfbric add 1000* Solforic add 903* Solforie add 894* 885*
OxaHc 950 Phosphoric 827* Nitric 812* 804*

Sacdnic 990 Oxalic

Fluoric Tartaric

Phosphoric 906* Fluoric

825 Muriatic 804* 797*
757 Phosphoric 801* 795*
Saberic? 745 740

Mode

Nitric

Moriatic

Saberic

Citric

Tartaric

Arsemc

(Citric)

Lactic

(Fluoric)

Benzoic

Acetic

Boracic

Salfdrons

Nitrons

Oarbonic

Pnude

900 Nitric
849* Muriatic

754* Fluoric
748* Oxalic

840* (Succinic) 740 Tartaric

800 (Fluoric) 703* Arsenic

Succinic Succinic

760 Citric? 618 Citric

733^ Lactic 603 Lactic

730 Su^uroua 527* Bensoic

729 Acetic Sulfuxons

706* Araemc (733i) Acetic

597 Boradc
594 (Acetic)
(515)* Nitrous?
592* Carbonic
450
420*
400

513* Mucic
480 Boradc
430 Nitrous
419* Carbonic
Prusdc

671* 666*

650 645

616 611

614 609

612 607

610 605

609 604

608 603

488* 484*

486 482

484 480

482* 479*

440 437

306* 304*

300 298

LiMX.

OxaUc add 960
Sulfuric 668*
Tartaric 867
Succinic 866
Phosphoric 865*
Mudc 860

Nitric
Muriatic
Saberic

Fluoric

Areenic

£4ictic

(Stric

Malic

Benzoic

Acetic

Boradc

Snlfdroos 516*

(Acetic) 470

Nitrous 425

Carbomc 423*

Pruasic 290

741*

736*

735

734*

733f

732

731

700

590

537*

Maonbbia« Ammonia. Gltcina? Alumina. Zibooria?

Oxalic Add 820 Sulfhric acid 808* Solforic add 718* 709* 700*

Phosphoric Nitric 731* Nitric 642* 634* 626*

Sulfuric 810* Muriatic 729* Muriatic 639* 632* 625*

(Phosphoric) 736* Phosphoric 728* Oxalic 60o 594 588

Fluoric Suberic ? 720 Arsenic 580 575 570

Arsenic 733 Fluoric 613* Suberic? 535 530 525

Mudc 732^ Oxalic 611 Fluoric 534* 529* 524*

Succinic 732i Tartaric 609 Tartaric 520 515 510

Nitric 732* Arsenic 607 Succinic 510 505 500

Moriatic 728* Sucdni 605 Mudc 425 420 415

No. XXV.

ELECTIVE ATTRACTIONS.

573

BfAONESIA*

Saberic? 700

(Fluoric) 630*

Tartaric 618

Citric 615

Malic? 600?

Lactic 575

Benzoic 56b
Acetic

Boracic 459*

SalfViroiiB 439*

(Acetic) 430

Nitrons 410

Carhomc 366*

Pmsdc 280

SULFDBIC.

Barita 1000*

Strontia 908*

Potass

Soda

Lime

Magnesia

Ammonia 808*

Glydna 718*

Itria

Alnminii

Ziroonia

Ammonia. Gltcina? AunmiA. Ziboonia?

Citric 603 Citric 415 410 405

Lactic 601 Photphoric (648)* (642)* (636)*

Benioic 599 Lactic 410 405 400

Snlforoos 433* Benzoic 400 395 390

Acetic 432 Acetic 395 391 387

Mocic 431 Boracic 388* 385* 382*

Boracic 430* SoiaiTons 355* 351* 347*

Nitrons 400 Nitrons 340 336 332

Carbonic 339* Carbonic 325* 323* 321*

Pmssic 270 Pmssic 260 258 256

894*
885*
868*
810*

712

709*

700*.

Fluoric.

Lime

Barita

Strontia

734*
706*
703*
(620)*
Potass 671 *
Soda 666*

Ammonia 613 *
Glycina 534 *
Alumina 529 *
Ziroonia 524 *

Succinic.
Barita 930

lime 866

Strontia? 740
(Magnesia) 7d2i
Potass 612

Soda 607

Ammonia 605
Magtmia
Glycina? 510
Alumina 505
Zirconia? 500

Acids.

Nitric.

Barita 849*

Potass 812*

Soda 804*

Strontia 754*

Lime 741*

Magnesia 732*

Ammonia 731*

Glycina 642*

Alumina 634*

Ziroonia 626*

Muriatic.
Barita 840*

Potass
Soda
Strontia
Lime
Ammonia
Magnesia
Glycina
Alumina
Ziroonia

804*
797*
748*
736*
729*
728*
639*
632*
625*

Oxalic.

Lime 960

Barita 950
Strontia 825
Magnesia 820
Potass 650
Soda 645

Ammonia 611
Glycina? 600
Alumina 594
Zirconia? 588

Tartaric.
867
760
757
618
616
611
609
520
515
510

ARSBNia

Lime 733}

Barita 7d3|

Strontia 733i

Magnena 733

Potass 614

Soda 609

Ammonia 607

Glycina 580

Alumina 575

Ziroonia 570

Suberic.

Barita 800

Potass 745

Soda 740

Lime 735

Ammonia 720

Magnesia 700

Glycina? 535?

Alumina 530

Ziroonia? 525?

Camphoric.

Lime

Potass

Soda

Barita

Ammonia

Glycina?

Alumina

Ziroonia?

Magnesia

Phosphoric.

Barita 906*

Strontia 827 *

Lim€ (865)*

Potass 801 *

Soda 795*

Amm<mia (728)*

Magnesia 736 *

Glycina 648 *

Alumina 642 *

Ziroonia 636 *

TuNcwnc.
Lime
Barita
Strontia
Magnesia
Potass
Soda

Ammonia
Glycina
Alumina
Ziroonia

Citric.

Lime 781

Barita 730

Strontia 618

Magnesia 615

Potass 610

Soda 605

Ammonia 603

Glycina? 415?

Alumina 410

Zirconia 405

574

TABLE OP BLBCTIVK ATTRACTIONS.

Na XXV^.

Laohc.

Bemzoic.

SULVDROOB.

Acme

Barita

729

White ozid of

Barita

592*

Barita

594

PotaM

609

arsenic

Lime

516*

Potass

486

Soda

604

Potass

608

Potass

488*

Soda

482

Strontia

60S

Soda

608

Soda

484*

StrtMtui

480

Lime

(782)

Ammonia

599

Stnmtia

(527)*

Lime

476

Ammonia

601

Barita

597

Magnesia

439*

AmmooU

432

Magnesia

575

Lime

590

Ammonia

433*

Idagnesia

430

MetalUc ozida

Magnesia

560

Glycina

355*

Metallic ozids

Glycina

410

Glycina?

400?

Alnmifnt

851*

Glydna

395

Alumina

405

Alumina

395

Ziroonia

347*

Alumina

391

Ziioonia

400

Ziroonia?

890?

Ziroonia

387

M0CIC?

BOBACIC.

NimonsY

PHOePHDBOVS.

Barita

900

Lime

537*

Barita

450

Lime

Lima

860

Barita

515*

Potass

440

Barita

Potaai

484

Strontia

518*

Soda

437

Strontia

Soda

480

(459)*

Strontia

430

PMass

Amnwytift

481

Potass

482*

Lime

425

Soda

Qljcina

426

Soda

479*

Magnesia

410

Magnesia?

Alumina

420

Ammonia

430*

Ammonia

400

AT^>^1^|f»»^y

Ziroonia

415

Glycina

388*

Glycina

340

Glydna

Alumina

385*

Alumina

336

Alumina

Ziroonia

382*

Saroonia

332

ZinxNiia

CARBOMia

Pftusno.

Barita

420*

Barita

400

Strontia

419*

Strontia

Xmm

(423)*

Potass

300

Potass?

306*

Soda

298

Soda

304*

Lime

290

Magfima

(866)*

Magnesia

280

Ammonia

339*

Ammonia

270

Glycina

325*

Glycina?

260

Alumina

323*

Alumina?

258

Zirconia

321*

Ziroonia?

256

No. XXVI. ELEMENTS OP CHEMICAL PHILOBOPHY. 575

No. XXVI.

A REVIEW OF SIR HUMPHRY DAVY'S

ELEMENTS OF CHEMICAL PHILOSOPHY.

From the Qnarterlj Review for September, 1812.

In attempting a review of this work, we cannot avoid profess*
ing, that we are far from entertaining the impression of sitting
down as competent judges, to decide on the merits or demerits
of its author : on this point the public voice, not only within
our own islands, but wherever science is cultivated, has already
pronounced too definitive a sentence, to be weakened or
confirmed by any thing that we can suggest of exception or
approbation. Our humble labours, on such an occasion, must
be much more analytical and historical than critical ; at the
same time we are too well acquainted with the author's can-
dour, to suppress any remark which may occur to us, as
tending to correction or improvement. It has most assuredly
fallen to the lot of no one individual to contribute to the pro-
gress of chemical knowledge by discoveries so numerous and
important as those which have been made by Sir Humphry
Davy : and with regard to mere experimental investigation,
we do not hesitate to rank his researches as more splendidly
successful, than any which have ever before illustrated the
physical sciences in any of their departments. We are aware
that the Optics of Newton will immediately occur to our
readers as an exception ; but without attempting to convince
those who may difier from us on this point, we are disposed to
abide by the opinion, that for a series of well devised experi-
ments and brilliant discoveries, the contents of Davy's Bake-
rian Lectures are as much superior to those of Newton's
Optics, as the Principia are superior to these, or to any other

576 REVIEW OF SIR H. davy's No. XXVL

human work, for the accurate and refined application of a sub-
lime and simple theory to the most intricate and apparently
anomalous results, derived from previous observation.*

Discoveries so far outshining all that has been done in other
countries, and constituting so marked an era in the history of
chemistry, cannot be contemplated by any Englishman, who
possesses a taste for science, without some degree of national,
and even local exultation ; although it is true that other indi-
viduals and other coxmtries have contributed largely to the
success of the common cause ; some, by improving the prin-
ciples of other departments of physics which have been so
happily applied, or by furnishing the most powerful agents and
the most convenient instruments, which have been employed
with so much address ; and others by collateral or independent
speculations and researches, which have here been blended
together into one system.

From all these sources our author has derived the materials
of a volume, which, when compared even with the latest works
of a similar nature, exhibits a more rapid and triumphant pro-
gress of improvement than can be paralleled in the annals of
human invention. He has adverted, with a very laudable mo-
desty, to the favourable circumstances under which his re-
searches were conducted :

** Nothing tends so much," he observes, "to the advancement of
knowledge as the application of a new instrument. The native intel-
lectual powers of men in difierent times, are not so much the causes of
the different success of their labours, as the peculiar nature of the means
and artificial resources in their possession. Independent of vessels of
glass, there could have been no accurate manipulations in commcm
chemistry : the air-pump was necessary for the investigation of the pro-
perties of gaseous matter; and without the Voltaic apparatus, there

* Most persons will think this praise exaggerated. Dr. Young was not disposed
to do full justice to Newton's Optics, considering that it sanctioned some errors both
of experiment and reasoning which tended very seriously to retard the progress of
optical discovery : such were his statements of the law of double refraction through
Iceland crystal and of chromatic dispersion, aad his attempted proofof the necessary
diffusion of light transmitted through an aperture upon the nndulatory hypothesis.
It should not be forgotten, however, that it was in a great measare owing to the
very merits of this great work as a model of well-deri^ experiments and aocnraie
inductive reasoning that it acquired this somewhat fatal influence upon the opinions
of men of science. — Note by the Editor,

No. XXVI. ELEMENTS OF CHEMICAL PHILOSOPHY. 577

was no possibility of examining the relations of electricaT polarities to
chemical attractions.**

It must, however, be remembered, that almost every other
discovery of importance, which has been made in science, has
been facilitated by some previous steps, which have rendered
practicable what might otherwise have presented insuperable
obstacles to human ingenuity ; nor has such a preparation ever
been allowed to detract from the just applause, bestowed on
those who have been distinguished from their contemporaries
by a more successful exertion of talent.

Until the year 1806, Sir Humphry Davy had been remark-
able for the industrious and ingenious application of those
means of experiment only, which had been long known to
chemists ; he had acquired, at a very early period of his life,
a well established celebrity among* men of science throughout
Europe, by the originality and accuracy of his researches ; and
at the same time the fluent and impressive delivery of his
lectures had obtained him the most flattering marks of appro-
bation from the public of the metropolis. But it was in the
summer of that year, that in repeating some electro-chemical
experiments of very doubtful authority, he was led into a new
train of reasoning and investigation, which enabled him to
demonstrate the important laws of the connexion between the
electrical afiections of bodies and their chemical powers, lliis
was his first great discovery v and when he was complimented
on the occasion by the Institute of France with the prize
established by Buonaparte, it was only questioned, by those
who were capable of appreciating its importance, whether they
acted with strict impartiality in assigning to him tiie annual
interest only ; while he appeared to have a fair claim to the
principal, which was allotted, by the donor, 'to the author of a
discovery relating to electricity, paramount to that of Franklin
or of Volta. Our author's next great step was the decomposi-
tion of the alkalis, which he eflected the succeeding year : and
this, though less interesting and important with regard to the
fundamental theory of the science, was more brilliant and im-
posing, from its capability of being exhibited in a visible and
tangible form. The third striking feature, wliich distinguishes

VOL. I. 2 P

578 REVIEW OF SIB H. davy's No. XXVI.

the system adyaneed in the present work, is the assertion of
the existence of at least two empyreal principles ; oxygen, and
the elastic fluid called the oxymuriatic acid gas, being con-
sidered as possessing equal claims to the character of simple or
undecompounded substances. A fourth peculiarity, which, how-
ever, is less exclusively and ori^nally a doctrine of Sir Hum-
phry Davy, is the theory of the simplicity of the proportions in
which all bodies combine with each other ; a theory respecting
which hints may be found in the works of several chemists of
the last century, but for the explicit illustration, and general
and minute application of which, the science is principally in-
debted to our countryman Mr. Dalton ; although the work
before us tends much more to its confirmation than any other
mass of evidence which has yet been collected on the subject*
On each of these four principal novelties we shall make some
extracts and abstracts ; having first given a hasty outline of the
interesting sketch of the progress of chemistry which constitutes
the introduction.

We shall not attempt to follow our author in his inquiries
how far any of the Arabian physicians or ma^cians may be
said to have been the founders of the science of chemistry,
rather than the Greeks or Egyptians, or even to conjecture in
what sense Firmicus^ whom he has not mentioned, may have
intended to employ the term chymia, which he simply intro-
duces as a science or mystery : but contenting ourselves with
enumerating the names of Roger Bacon and Baal Valentine, as
the greatest chemists of the thirteenth and fifteenth centuries,
and Paracelsus, Agricola, and Libavius, of the sixteenth, we
shall hasten to the beginning of the seventeenth, as the true
period of the commencement of the pneumatic chemistry, under
the auspices of Van Helmont, who first distinctly observed the
properties of several elastic fluids, which he denominated gases ;
and more especially of Bey, who, in the year 1630, expressly
maintained the absorption of air by metals during their calci-
nation ; nor was it much later that Torricelli and Pascal began

* The account of the important labours of Berzelius upon the subject of definite
proportions was translated from the German by Dr. Young, and published in sac-
cessive numbers of tht Philosophical Magazine from January, 1813, to April, 1814.
— Note by the Editor,

No, XXVI. ELEMENTS OF CHEMICAL PHILOSOPHY. 579

to inyestigate the mechanical properties of the ur with mathe-
matical precbion. About the time of the foundation of the
Academy del Cimento, of the Royal Society, and the Pari^an
Academy of Sciences, which constitutes an era so important
in the progress of human knowledge^ the most distinguished
chemists in Germany were Glauber, Kunckel, Brandt, Hof-
mann, Beccher, and Stahl ; in France, Homberg, Geofiroy, and
the Lemerys; and in England Boyle, Hooke, Slare, and
Mayow ; but with regard to the philosophical theory, especially
of pneumatic chemistry, the English had advanced far beyond
their neighbours, even before the publication of the correct and
comprehensive speculations contained in the queries of Newton,
which marked the commencement of the eighteenth century,
and which may be considered as the basis of the more refined
and accurate cultivation of chemical science. In pursuit of
these suggestions, the order of chemical attractions appears to
have been first distinctly exhibited in a tabular form by Geof-
iroy, about the year 1718. The idea of a single combustible
principle, or phlogiston, is traced to Albertus Magnus, the
contemporary of Roger Bacon, and was received from Beccher
by Stahl, who advanced in support of it many ingenious experi-
ments; for example, the decomposition of Glauber's salt by
charcoal ; and this doctrine was almost universally adopted
throughout Europe, in preference to the more correct views of
Boyle, Hooke, and Mayow. The researches of these chemists
were, however, in some degree revived by the industrious Dr.
Hales, although he was unfortunately misled by the idea, that
all elastic fluids were essentially the same, and only distin-
guished by some accidental modifications, from the presence of
various effluvia. The error of this opinion was clearly and
elegantiy displayed by Dr. Black, who published, in 1756,
a littie essay on magnesia and fixed air, which may be con-
sidered as the true beginning of the pneumatic chemistry.
The earliest labours of Mr. Cavendish are dated in 1765, when
he invented the hydropneumatic apparatus^ discovered inflam-
mable air, and made many very important experiments on tiie
properties of gases of diflerent kinds. Dr. Priestley followed
the steps of Hales and Cavendish with the most distinguished

2p2

580 REVIEW OF SIR H. DAVY*S No. XXVI-

success, and discovered the existence of nitrous gas, nitrous
oxyd, and oxygen ; and exhibited, by means of the mercu-
rial apparatus, muriatic acid, sulphurous acid, and ammonia, in
a gaseous state. Macquer, Rouelle, Margra^ Pott, and above
all Bergman, were in the mean time diligently pursuing their
refined analyses on the continent : and Scheele was carrying
on a train of investigations much resembling those of Priestley,
ascertaining the composition of the atmosphere, and the pro*
perties of the fluoric and prussic acids, and the oxymunatic
acid gas. Of all these chemists. Black, Cavendish, Priestley,
and Scheele were unquestionably the greatest discoverers : the
facts, which they had brought forward, were in some measure
systematized by Lavoisier, to whom our author thinks that ^ no
other inquirer except Cavendish can be compared for precision
of logic, extent of view, and sagacity of induction.' Bayen^ in
1774, had shown that the calx of mercury was capable of being
rendered metallic, without the addition of any inflammable
substance, and hence had argued against the agency of phlo-
giston in the revival of metals in general. In the next year,
Lavoisier examined the air aflbrded by the calx during its
reduction, which was already known to Priestley and Scheele,
and called it oxygen : he demonstrated also the constitu-
tion of the carbonic acid gas, and showed that the nitrous and
sulphuric acids derive their properties from the combination of
their bases with oxygen: Mr. Cavendish soon after showed
the true nature of the basis of the nitric acid, and made a dis-
covery, which is perhaps of greater importance than any single
fact which human ingenuity has ascertained, either before or
since, that of the composition of water from oxygen and hydro-
gen. Soon after this, Mr. Berthollet proved that ammonia
consists of hydrogen and nitrogen ; and many other chemists
continued a series of researches, which appeared to illustrate
and confirm the doctrine of Lavoisier : the existence of phlo-
giston was, however, still very ably maintained by Mr. Caven-
dish in 1784, as the simpler of the two theories by which the
phenomena might be explained ; and other chemists rctained
the same opinion for a much longer period. In 1787, the
Frcnch chemists presented to the public their new system of

No. XXVI. ELEMENTS OF CHEMICAL PHILOSOPHY. 581

nomenclature, which certainly contributed in some degree to
the facility of acquiring the science, but still more to the
dissemination of the doctrines of the school from which it
proceeded.

"At the time (p. 63) when the antiphlogistic theory was established,
electricity had little or no relation to chemistry. The grand results of
Franklin, respecting the cause of lightning, had led many philosophers
to conjecture, that certain chemical changes in the atmosphere might be
connected with electrical phenomena ; and electrical discharges had been
employed by Cavendish, Priestley, and Vanmarum, for decomposing and
igniting bodies ; but it wa^ not till the era of the wonderful discovery
of Volta, in 1800, of a new electrical apparatus, that any great pro-
gress was made in chemical investigation by means of electrical com-
binations.

" By researches, the commencement of which is owing to Messrs.
Nicholson and Carlisle, in 1800, which were continued by Cruickshank,
Henry, Wollaston, Children, Pepys, Pfaff, Desormes, Biot, Thenard,
Hisinger, and Berzelius, it appeared that various compound bodies were
capable of decomix)8ition by electricity ; and experiments, which (says
our author) it was my good fortune to institute, proved that several
substances, which had never been sejMirated into any other forms of
matter in the common processes of experiment, were susceptible of ana-
lysis by electrical powers : in consequence of these circumstances, the
fixed alkalijs, and several of the earths have been shown to be metals
combined with oxygen ; various new agents have been furnished to che-
mistry, and many novel results obtained by their application, which, at
the same time that they have strengthened some of the doctrines of the
school of Lavoisier, have overturned others, and have proved that the
generalizations of the antiphlogistic philosophers were far from having
anticipated the whole progress of disco veiy.

** Certain bodies, which atti*act each other chemically, and combine
when their particles have freedom of motion, when brought into contact,
still preserving their aggregation, exhibit wliat may be called electrical
polarities ; and by certain combinations these polarities may be highly
exalted ; and in this case they become subservient to chemical decompo-
sitions ; and by means of electrical arrangements, the constituent parts of
bodies are separated in a uniform order, and in definite proportions.
Bodies combine with a force, which in many cases is corres|x>ndent to
their jwwer of exhibiting electrical polarity by contact ; and heat, or heat
and light, are produced in proportion to tlie energy of their combination.
Vivid inflammation occurs in a number of cases in which gaseous matter

582 REVIEW OF SIR H. davy's No. XXVI.

is not fixed ; and this phenomenon happens, in various instances, withoat
the interference of free or combined oxygen.

"Crystals of oxalic acid," for example, (p. 169,) "touched by dry
quicklime, exhibit electrical powers ; and the acid is negative, the lime
positive. All the acid crystals, upon which I have experimented, when
touched by a plate of metal, render it positive. And in Voltaic combi-
nations with single plates or arcs of metal, the metal is negative on the
side opposed to the acid, and positive on the side or pole opposed tjo the
alkali.

" Bodies that exhibit electrical effects previous to their chemical
action on each other, lose this power during combination. Thus, if
a polished plate of zinc is made to touch a surface of dry mercury, and
quickly separated, it is found positively electrical, and the effect is in-
creased by heat ; but if it be so heated as to amalgamate witli the surface
of the mercury, it no longer exhibits any marks of electricity. — ^When
any conducting substance, capable of combining with oxygen, has its
positive electricity increased, it will attract oxygen with more energy
from any imperfect conducting medium ; and metallic bodies, that in
their common state have no action upon water, such as silver, attract
oxygen from it easily, when connected with the positive pole in the
Voltaic circuit ; and bodies that act upon water, such as zinc and iron, so
as to decompose it slowly, refuse to attract oxygen from it, when they
are negatively electrified in the Voltaic circuit.

'* Acids, which are negative with respect to alkalis, metals, and earths,
are separated from these bodies in the Voltaic circuit at the positive sur-
face ; and alkalis, metals, and earths are separated firom acids at the
negative surface : and such are the attracting powers of these surfaces,
that acids are transferred through alkaline solutions, and alkalis through
acid solutions, to the surfaces where they have their points of rest It
is easy to show this by making a combination of three agate cups, one
containing sulphate of potassa, one weak nitric acid, and the third distilled
water, and connecting them by asbestus moistened in pure water, in such
a manner, that the surface of the acid is lower than the sur&ce of the
fluid in the other two cups. When two wires of platina, from a power-
ful Voltaic apparatus, are introduced into the two extreme cups, the
solution of the salt being positively electrified, a decomposition will take
place, and in a certain time a portion of potassa will be found dissolved
in the cup in contact with the negative wire, though the fluid in the
middle cup will still be sensibly acid.**

We must here take the liberty of remarking, that several of
these singular effects had been observed by Hisinger and Ber-
zelius in Sweden a year or two before the date of Sir Humphry

No. XXVL ELEMENTS OF CHEMICAL PHILOSOPHY. 583

Davy's discoveries ; but they had neither led those chemists to
entertain any suspicion of the true laws by which they are
governed, nor to apply them to tlie production of any unknown
substances. The first of the remarkable decompositions that
our author effected, by means of his newly established prin-
ciples, was that of potass^ or the vegetable fixed alkali, from
which he obtained the new metal potassium in October 1807.
When a thin piece of pure or caustic potass, in its usual state
of a dry hydret, or combination with water, '^ is placed between
two discs of platina connected with the extremities of a Voltaic
apparatus of 200 double plates, it will soon undergo fusion, oxy-
gen will separate at the positive surface, and small metallic
globules will appear at the negative surface, which consist of
potassium." It may also be procured by heating iron filings
to whiteness in a gunbarrel, and suffering melted potass to
come slowly into contact with them, as MM. Gay Lussac and
Thenard discovered ; and even by strongly igniting potass with
charcoal, as Mr. Curaudau has shown. Hiis metal is about
one seventh specifically lighter than water ; it fuses at about 1 50^
of Fahrenheit, and becomes gaseous below a red heat It
inflames violently when moistened, or when gently heated in
contact with the air, affording alkaline fiimes. Its powerful
attraction for oxygen renders it a very useful agent in chemical
analyses : naphtha seems to be almost the only substance in
whidi it can be kept with convenience.

Soda, the mineral alkali, affords, when similarly treated,
though not quite so easily, a metal much resembling potassium,
but a little heavier, though still lighter than water ; fusible at
about 200^, and evaporating at a strong red heat : our author
has very properly named it sodium ; it agrees with potassium
in most of its properties.

Barium was obtained in May 1808, in the form of a dark
grey mass, with little lustre, by means of a process suggested
by MM. Berzelius and Pontin. A portion of pure barita,
moistened with water, is placed oiwa plate of platina, which is
rendered positive by a Voltaic battery, while a globule of mer«
cury, placed in the paste, is made negative : an amalgam is
thus obtained, from which the mercury is expelled by distilla-

584 REVIEW OF SIR H. davy's No. XXVI.

tioQ in a tube of glass free from lead, filled with the vapour of
naphtha and hermetically sealed. Sir Humphry Davy had
before obtained it only in combination with iron.

Strontium and calcium are procured in the same manner
from strontia and lime ; strontium much resembles barium, cal-
cium is a little brighter and whiter. When the vapour of potas-
sium is made to pass through ignited barita or lime, or some of
their compounds, some potass is formed, and the earths are
either partially or completely reduced to a metallic state.

Magnesium may be obt^ed in either of these ways, though
more slowly by the electrochemical process : when the vapour
of potassium is employed in a thick tube of platina, a small
quantity of mercury may be afterwards introduced, which will
amalgamate with the metal, and when expelled, will leave it
in the form of a dark grey metallic film, not acting so rapidly
on water as any other of the alkaline metals.

Aluminium^ glycinium, zirconium, silicium, and itrium, have
been obtained less distinctly in separate forms. Aluminium,
for so we shall take the liberty of writing the word, in pre-
ference to aluminum, which has a less classical sound, withstands
all attempts to decompose the earth by electriiying mercury in
contact with it : but when a particle of iron is employed, with
an electrical power capable of fusing it, the iron is whitened,
and effervesces with water, affording a small portion of alumina.
By means of the vapour of potassium also, some gray metallic
particles may be obtained from ignited alumina : and glycina,
similarly treated, affords a dark coloured substance, which
regains the earthy appearance when heated in air, or moistened.
Similar particles obtained from zirconia are found to be partly
metallic, and partly of a chocolate brown colour. Silicium
seems to have an appearance somewhat resembling plumbago ;
its alloy with iron may be obtained like that of aluminium.
Itria also, treated with potassium, affords potassa, and acquires
a partial appearance of metallization.

Nor have the same powerful means of analysis been less suc-
cessful in demonstrating the composition of the boracic acid,
from which our author has obtained a substance too little re-
sembling a metal to be termed boracium, but which, from its

No. XXVI. ELEMENTS OF CHEMICAL PHILOSOPHY. 585

analogy to carbon, he had thought it right to distinguish by the
more appropriate than elegant name boron. It is procured
either by the electrical decomposition of the boracic acid, or by
igniting that acid with an equal weight of potassium in a tube
of iron. It is of a dark olive colour, neither fusible nor volatile
in any heat to which it has been exposed ; it sparkles very
brilliantly when thrown into oxygen gas, and a portion of it is
converted into boracic acid.

When the fluoboric add gas is decomposed by the combus-
tion of potassium, it affords fluate of potassa, and the boracic
acid only seems to be deprived of its oxygen ; but when potas-
sium is burned in the silicated fluoric acid, there is reason to
think that both the silica and the fluoric acid undergo a partial
decomposition, since the gas affords a smaller quantity of fluate
of lime than before the operation of the potassium : but the
base of the fluoric acid has never been separately exhibited, and
from the readiness with which the fluoboric gas enters into com-
binations, our author thinks it not altogether impossible that
the fluoric acid contained in it may be a simple empyreal prin-
ciple analogous to oxygen and to ' chlorine.' His opinions on
the nature of these substances, which constitute the third great
feature of the present work, require to be illustrated in his own
words ; p. 240.

** Scheele considered chlorine as an element of the mariatic acid, and
hence called it dephlogisticated marine acid. By that chemist it was
r^arded as aa undecompotmded body, Lavoisier and Bertbollet as-
serted that it was a compound of muriatic add gas and oxygen. This
idea is now universally given up ; but some chemists in France and
Scotland conceive that it is a compound of oxygen, and an unknown body,
which they call dry muriatic acid. The weight of chlorine, its absorb-
ability by water, its colour, and tlie analogy of some of its combinations
to bodies known to contain oxygen, are arguments in favour of its being
a compound ; and it is possible that oxygen may be one of its elements,
or that oxygen and chlorine are similarly constituted. 1 have made
a number of experiments with the hopes of detecting oxygen in it, but
without success; none of its compounds with inflammable bodies will
afford this principle ; charcoal, intensely ignited in it, undergoes no change,
nor is it altered by the strongest jwwers of electricity. Should oxygen
ever be procured from it, some other form of matter, possibly a new
one, will, at the same time, be discovered as entering into its consti-

586 REVIEW OF SIR H. davy's No. XXVL

tutioD, and till it is deoomponnded, it must be regarded, according to
the just logic of chemistry, as an elementary substance."

P. 237. '* Chlorine and oxygen are capable of existing in combinatioD,
and they form a peculiar gaseous matter. They do not unite, when
mixed together, but when existing in certain solids, they may be
detached in union. To make the compound of chlorine and oxygen,
hyperoxymuriate of potassa is introduced into a small retort of glass, and
twice as much muriatic acid as will cover it diluted with an equal vo-
lume of water. By the application of a gentle heat, the gas is evolved,
and it must be collected over mercury. I discovered this elastic sub-
stance in its pure form in January 1811, and gave to it the name of
euchlorine, from its bright yellow-green colour. Its smell is not unUke
that of burnt sugar. It must be collected and examined with great care,
and only in small quantities at a time ; a very gentle heat causes it to
explode, sometimes even the heat of the hand ; and its elements separate
from each other with great violence, producing light. None of the metals
that bum in chlorine act upon this gas at conamion temperatures ; but
• when the oxygen is separated they then inflame in the chlorine. The
proportion in which chlorine combines with bodies may be learned from
the decomposition of euchlorine ; the oxygen in wjiich is to the chlorine
as 15 to 67 in weight. If euchlorine be considered as consisting of one
proportion of oxygen to one of chlorine, then 67 will be the number
representing chlorine, which is most convenient as being a whole num-
ber. If euchlorine be supposed to contain two proportions of chlorine
and one of oxygen, then the number representing chlorine will
be 33.5. It will hereafter be shown that whichever of these data be
assumed, the relations of the number will haimonize with those g^uned
from various other combinations."

The doctrine of the simple proportions of combinations, ex-
emplified in these numbers, which was the fourth point that we
mentioned as particularly requiring to be noticed, is thus stated
in the introduction.

** Experiments made by Richter and Morveau had shown that, when
there is an interchange of elements between two neutral salts, there is
never an excess of acid or basis; and the same law seems to apply
generally to double decompositions. When one body combines with
another in more than one proportion, the second proportion appears to
be some multiple or divisor of the first ; and this circumstance, observed
and ingeniously illustrated by Mr. Dalton, led him to adopt the atomic
hypothesis of chemical changes, which had been ably defended by Mr.
Higgins in 1789 ; namely, that the chemical elements consist of certain

No. XXVI. ELEMENTS OF CHEMICAL PHILOSOPHY. 587

indestructible particles, which unite one and one, or one and two, or in
some definite numbers.** p. 56.

P. 117. *• Mr. Berthollet, to whom the first distinct views of the re-
lations of the force of attraction to quantity are owing, has endeavoured
to prove that these relations are universal, and that elective affinities
cannot strictly be said to exist He considers the powers of bodies to
combine as depending, in all cases, upon their relative attractions, and
upon their acting masses whatever these may be : and he conceives
that in all cases of decomposition, in which two bodies act upon a third,
that third is divided between them in proportion to their relative affini-
ties and their quantities of matter. Were this proposition strictly
correct, it is evident that there could be scarcely any definite proportions.
When an alkali precipitates an earth from its solution in an acid, the
earth, according to Mr. BerthoUef s ideas, ought to fall down in combi-
nation with a portion of acid. But if a solution of potassa be poured
into a sulphuric solution of magnesia, the precipitate produced, after
being well washed, affords no indication of the presence of acid ; and M.
PfafT has shown, by some very decisive experiments, that magnesia has
no action upon neutral combinations of alkalis and sulphuric acid ; and
likewise, that the tartarous acid is entirely separated from lime, and the
oxalic acid from oxyd of lead, by quantities of sulphuric acid merely
sufficient to saturate the two bases ; and these are distinct and simple
instances of elective attraction. Again, when one metal preci])itates
another from an acid solution, the body that falls down is usually free
from acid and oxygen : thus zinc precipitates lead and tin, and iron
copper ; and the whole of the oxygen and the acid is transferred from
one metal to the other."

Having exhibited this outline of the general doctrines which
Sir Humphry Davy has undertaken to maintain, we must pause
to consider how far he seems to have left any thing further to be
desired, with regard to the perfect establishment of either of
them. His electrochemical discoveries, and his decomposition
of the alkalis and earths, must ever remain incontestable
memorials of his ingenuity and success ; but on the subject of
the oxymuriatic acid gas, we cannot help thinking his tone
(p. 335) somewhat more decisive than the present state of the
iDvestigation altogether authorises. We do not see the abso-
lute necessity of considering every body as simple which has
never been decompounded, provided that there are strong ana-
logical reasons for suspecting that it is really a compound. In

588 REVIEW OF SIR H. davy's No. XXVL

the present instance, there are considerable difficulties on both
sides, and being much disposed to suspend our judgment until
further evidence can be obtained, we must confess that a new
nomenclature, founded on the adoption of the new opinion, and
tending to carry with it a tacit persuasion of its truth, appears
to us to be somewhat premature. Either hypothesis may be
employed for explaining the phenomena ; perhaps* the proba-
bility is in favour of Sir H. Davy's ; but the arguments, by which
it is supported, cannot yet be considered as finally conclusive.
We see ten or twelve different substances agreeing with the
muriatic acid in a very great majority of their properties, and
depending for these properties on the oxygen which they con-
ttdn, and one anomalous substance only, which possesses these
properties in a very slight degree, that is, sulphurated hydrogen,
and which is found to contain little or no oxygen : it does not
therefore appear to us to have been unphilosophical to suppose
that the muriatic acid also contained oxygen. It is true that
this presiunption is weakened by the failure of the newly
acquired powers of chemical electricity to obtain oxygen from
it ; but however great and wonderful those powers may be, they
are not altogether irresistible, since some of the metals of the
earth have been more easily exhibited by chemical than by
electrical means. The oxymuriatic acid gas approaches much
more nearly to the combinations of oxygen than to oxygen
itself, in the facility with which it unites with metals, and in
some other respects ; nor do the combinations of this gas appear
to resemble those of oxygen by any means so closely, as might
be expected from the analogy of two elementary principles
belonging to the same class. We are willing to allow, that the
necessity of supposing a portion of water inseparable from the
muriatic acid gas militates in some measure against the common
opinion ; but it must be remembered that neither the sulphuric
nor the nitric acid has ever been obtained without admixture,
either of water or of some other substance. On the other hand,
the theory of simple proportions affords an objection of con-
siderable weight to the doctrine advanced by our author ; for
the quantity of muriatic acid contained in some of the supersalts
and subsalts bears a regular relation to the oxygen of the

No. XXVI. ELEMENTS OP CHEMICAL PHILOSOPHY. 589

earths or oxyds on the common supposition, and not on that of
the elementary nature of chlorine; the patent yellow, for
example, if we mist&ke not^ is a substance which appears to be
produced by a regular process in a constant manner, and which
must, upon this hypothesis, be supposed to be a mixture of two
distinct combinations, governed by no intelligible law, while, if
we consider it as one of the submuriates of lead, it exhibits
a strict analogy with other substances. .

But even if we grant the existence of chlorine as an elemen-
tary principle, we cannot approve of distinguishing its combi-
nations by terminations only, much less by terminations so
simple as ane^ ana^ anee, which our author has proposed for the
different combinations of chlorine with any other simple sub-
stance. According to the Linnean precept, Phil. hot. § 287,
** a specific name must not be united to the generic as a termi-
nation," and Sir H. Davy has himself confessed that for calomel
and corrosive sublimate the terms mercurane and mercmrana
would be an insufficient distinction ; to say nothing of the inele-
gance of a French vowel in an English word, and of the difficulty
of preserving the terms distinct in translations into other
languages, which ought to induce us to refer all scientific
nomenclature to some common form, that of the Latin lan-
guage, for instance, whence they might be again derived for the
use of each modern language according to its characteristic
genius. We do not apprehend that the word ^* chlorid, follow-^
ing the analogy of oxyd," (p. vii,) would have been a more
^^ tlieoretical expression** than the termination ane, and we
might add to it, if necessary, dichlorid and trichlorid. In the case
of the earths and alkalis, there is a manifest reason for using
single words ; these substances, unlike the *' chlorids," occurring
continually in combination, it would require great circumlocu-
tion to express the most fiuniliar compounds, unless some such
abbreviation were permitted.

With regard to the fourth principal subject on which the
present work throws many new lights, that is, the simplicity of
the proportions of chemical combinations, the proofs are so
numerous and satisfactory, that there seems to be little room
left for argument. We must say that to us, the supposed

590 REVIEW OF SIB H. davy's No. XXVL

discovery of BerthoUet Bever carried with it any thing like con-
viction, and we always considered the praises and prizes which
were so liberally conferred on it, as so many instances only of
the facility with which the world is ready to bestow its appro-
bation on all the performances of a person once celebrated, and
frequently even the more enthusiastically the more paradoxical
they appear. At the same time we must observe, that the
objections of Pfaff are not so immediately applicable to Ber-
tbollet's doctrines as they appear at first sight to be; the
partition of one substance between two others being principally
asserted by BerthoUet, as existing in the state of solution, where
there is nothing to disturb it ; while he considers the crystal-
lization of one of the compounds as a new cause, perfectly
capable of modifying the previous arrangement of the sub-
stances. What Sir H. Davy attributes to the experiments of
Richter and Morveau was sufficiently understood by Bergman,
and still more explicitly demonstrated by the contemporary or
even earlier experiments of Wenzel. Kirwan's investigations
on this subject were well projected, but by no means happily
executed. Richter's first work on chemical combinations was
published in 1792 : his pompous and elaborate essays have all
ejaded in a short and imperfect table of proportions, which has
been, in a great measure, superseded by the more accurate re-
securches of Berzelius and other late chemists. Bergman had
also made experiments which prove that the oxygen, capable of
enabling one metal to form a salt, was sufficient to serve for the
oxydation of as much of another metal as precipitated it, and
entered into combination with the acid : but it was reserved
for Gay Lussac to place this law in a clear point of view, and
to establish and illustrate it by decisive experiments. The
principles of BerthoUet were strongly and suocessfully opposed
by Proust in 1804; he showed that in the combinations of
metals with oxygen and with sulphur, Certun fixed proportions
are always observed in preference to others ; his first experi-
ments on the sulphurets were made in 1801. The great
improvements in this doctrine, which are incoutestably of very
modem date, are the establishment of the simplicity of the num-
bers expressing the proportions of combinations, especially when

No. XXVI. ELEMENTS OF CHEMICAL FHILOSOPHT.

591

they relate to the volumes of elastic fluids, or to the compara-
tive relations of subsalts or supersalts, and of their identity in
compounds apparentiy of very different kinds ; for example, in
salts, sulphurets, and oxyds : and for these facts the science is
principally indebted, after Mr. Higgms, to Dalton, Gay Lussac,
Smithson, and WoUaston. The results of these principles may
be most conveniently compared by exhibiting them in a tabular
form ; and as no table of this kind is to be found in Sir H.
Davy's work, we shall here take the liberty of inserting such a
one, in which we have collected most of the numbers which he
has ascertained, together with some others which we have
deduced from the experiments of Berzelius and Richter.

Tabk of

the Proportional Weights of Che

mica

I Stibstances

entering into combination.

DiS00T6f9n«

W«fght.
combining.

Oxygen

—

Priestley

1774

—

•Chlorine*

Scheele

1774

—

Hydrogen

—

Cavendish

1766

—

Nitrogen

—

Rutherford

1772

—

Potassium

—

Davy

1807

—

Sodiom

—

Davy

1807

—

88 or 44

Barium

—

Davy

1808

—

180

Tellarinm

—

MiiUer

1782

—

*74' (60?)

Uranium

—

Klaproth

1789

—

77?

Chromium

—

Vanqnelin

1798

—

Antimony

—

165 (330?)

Manganesinm

—

Kaim

1770

—

103

Zinc

—

Tin

—

110

Molybdaenum

—

Hielm

1782

—

Iron

—

103

Cobalt

—

Brandt

1733

—

•166' (110)

Copper

—

*120' (128?)

Arsenic

—

Nickel

—

Cronstedt

1751

—

*55' (110?)

Bismuth

—

134

Silver

—

205'

Lead

—

398

Rhodium

—

WoUastcm

1804

—

Palhidinm

—

WoUaston

1803

—

'134' (106)

Mercury

—

380

Tungsteninm

—

Delhnyars

1781

—

Gold

♦

373 Ben.

Platina

—

Scheffer or Lewis

1750?

—

180 Ben.

Iridimn

—

Tennant

1803

—

592

REVIEW OF SIR H. DAVY^S

No. XXVI

SalMtenees.

Welghto
eomUnin«.

Osmiam

—

Tennant

1803

—

Titaninxn

—

Gregor?

1791

—

Columbiom

—

Hatchett?

1802

—

Tantaliimi

— -

Ekeberg

Cerinm

—

Hisinger and Bercelius 1804

—

86 (172?)

Strontiain

—

Davy

1808

—

Caldam

—

Davy

1808

—

Magnesiam

—

Davy

1808

—

38? (23?)

Gljcinium

—

Davy

—

39?

Itrium

—

Davy

—

111?

AlTiT«i"iinTTi

—

Davy

1808

—

S3?

Zirooniom

—

Davy

—

70?

Silidam

•—

Davy

—

31?

Carbon

—

11*4

Boron

—

Davy

1807

—

55?

PhosphoniB

—

Brandt

1669

—

20(25f)

Sulphur

—

Flaoric basis?

—

5-7?

Water

—

(I ox. II hydr. Cav.)

—

Ammonia

—

(I nitr. VI hydr.

Berth.)

—

Potass

—

(I ox. I pot)

—

Soda

—

•118' (59)

Banta

—

Scheele

1774

—

145

Strontia

—

Crawford

1790

—

105

Lime

—

Magnesia

—

Hofmann

—

•53* (38, B.)

Glycina

—

Vauquelin

1798

—

Itria

—

Gadolin

1794

—

126

Alamina

—

Margraff

—

Ziroonia

—

Klaproth

1788

—

Silica

—

Margraff

—

61 (30-5)

. Adds.

Weights.

Adds.

Weights.

Salpharic

Tungstic

128?

Salpharous

Columbic

Phosphoric

55 Berz.

Acetic

96 Ben.

Phosphorous

35 (105?)

Formic

64 Richt.

Carbonic

Oxalic

1,21 "~

Nitric

101

Nitrous

*8C' (71, B.)

Mellitic

Muriatic

(52, B.)

Tartaric

124 Ben.

Oxymuriatic

Citric

105 Bers.

Hjperoxymuriatic

Malic

Fluoric

21?

Mucic

Boracic

320?

Benzoic

Chromic

Succinic

79 Richt.

Molybdic

133

Moroxylic

110?

Molybdons

118

Camphoric

64?

Arsenic

135

Suberic

Arsenious

120

Lactic

No. XXVI. ELEMENTS OF CHEMICAL PHILOSOPHY. 593

By means of this table we can at once ascertain the propor*
tions of the component parts of any salt or other compound of
the substances contained in it : thus nitre consists of 90 potass
and 101 nitric acid, or of 47 per cent, alkali and 53 acid in its
dry state: or if we consider the white caustic potass, in the
driest stat« in which it is exhibited by any common means, when
it is still a hydret, and contains a portion of water, expressed
by 17, the number for potass will become 107 ; and the number
for the most concentrated liquid nitric acid, becoming in a simi-
lar manner 118, the proportion of alkali will be about 47) per
cent, instead of 47. And in a similar manner we find for the
sulphate of barita 145 and 75, or 66 per cent of earth, and 34
of dry acid, which is a result fully established by the meet
accurate analyses. It must, however, be observed, that the
number here assigned to the carbonic acid is that which belongs
to the alkaline subcarbonates, which are not, strictly speaking,
neutral salts ; and that there are some other apparent irregu-
larities of the same nature, in the operation of the laws of
ample proportions.

Besides the general doctrines which we have thus particu-
larly examined, there are many detached passages, which we
shall think it right to mention in the order of their occurrence ;
some on account of their novelty and interest, others because,
in a work so likely to be universally studied, we wish to leave
nothing unnoticed, which appears to require either correction
or explanation.

In speaking of Aristotle, (p. 5,) our author seems rather to
have been led away by a popular clamour, than to have studied
with attention the real tenor of that great observer's mode of
philosophizing. The *' practice of advancing general principles,
and applying them to particular instances," is so far from being
^^ fatal to truth m all sciences," that, when those principles are
advanced on sufficient grounds, it constitutes the essence of true
philosophy ; and Aristotle did not advance principles on physical
subjects without what he thought sufficient grounds. The
beauty of the theory of gravitation depends wholly on the
establishment of a general principle, and its application to par-
ticular instances : and even our author appears to have applied

VOL. I. 2 Q

594 REVIEW OF SIR H. davt's No. XXVI.

the general principle of simple proportions to particular in-
stances, almost in contradiction to his own earlier researches ;
where, for instance, he doubts the accuracy of his experiment
with diamond and potassium, because it " does not harmonise
with the doctrine of definite proportions." (p. 312.) In the case
of ammonia too, he has, perhaps, been partly induced by similar
considerations, to repeat his former analysis, in which he
" thought that a small quantity of water was found," and ^' very
delicate experiments " having convinced him (p. 269) that no
water is obtained, he has very candidly returned to Mr. Ber-
thollet's opinion respecting the constitution of this substance.

P. 69. " For any thing we know to the contrary, gravitation
and cohesion may be mere modifications of the same general
power of attraction." This is a mistake not altogether un-
common with those who have not sufficiently attended to the
mathematical characters of the forces concerned. Whether or
no these forces may be produced by any different modifications
of the game cause, we have no right even to conjecture ; but
their magnitude and the laws of their action are so totally dis-
similar, that they cannot possibly be considered as modifications
of the same power.

P. 70. There is an error in the comparative expansions of
solids and fluids as here related : ^^ 100,000 parts of glass,
raised from the degree of freezing to that of boiling water, be-
came 100,083; —the expansive power of liquids in general is
greater than that of solids, — 100,000 parts of mercury become
101,835," that is, in bulk ; but 100,000 parts of glass be-
come in bulk 100,250, not 100,083 only; and 100,000 of
zinc 100,910, its expansion being about half as much as that
of mercury, instead of one-sixth, as would be inferred from our
author's statement. P. 75. A " common thermometer " is not
" hermetically sealed " " at the moment of the ebullition of the
mercury ;" for, in this case, the fluid would sink within the
bulb at all common temperatures, unless the tube were much
longer than usual.

P. 76. Professor Leslie has complained, in the public papers,
that Sir H. Davy mentions a thermometer of Van Helmont, as
similar to his differential thermometer, while, in fact, Van

No. XXVI. ELEBIENTS OF CHEMICAL PHILOSOPHY. 595

Helmont's instrument was open at one end ; although his expla-
nations *^ incidentally involved the principle of the differential
thermometer, which the author never once dreamed of reducing
to use : " nor has the truth of this statement been disproved by
the person who has replied on bejialf of Sir H. Davy. The
*' principle of the differential thermometer *' is too simple to be
called an invention ; and it is only by its ingenious application
that Professor Leslie has made it an object of attention.

P. 79. A very amusing experiment, in which ether, floating
on water, is made to bum, without sensibly elevating the tem-
perature of the water one-eighth of an inch below the surface,
is adduced as a proof of the great diflSculty with which fluids
transmit heat downwards. But it must be remembered that
liquid ether is not susceptible of a temperature higher than 102%
and that a feverish hand, held at the surface of the water,
would heat it just as rapidly as the boiling ether ; and probably
much more so, since the capacity of ether for heat is less than
half of that of an aqueous fluid.

P. 80. ^' In solids the attractive force predominates over the
repulsive ; in fluids and in elastic fluids, they may be regarded
as in different states of equilibrium.*' It is difficult to conceive
how so much error and confusion could have been collected, by
such an author, into so short a sentence. When one of two
forces ^* predominates," there must be motion, and the parts of
a body cannot remain at rest : indeed so far is the attractive
fbrce from predominating in a solid rather than in a liquid, that
when water becomes solid, thb force gives way to the repulsive,
and the ice expands. Nor are the attractive and repulsive
forces in any ^^equilibrium" in elastic fluids; the repulsive
force exists here alone, and only compensated by external
pressure or gravitation. It is in liquids and in solids that the
attractive and repulsive forces exist in ^' different states of equi-
Ifbrium,*' and probably without differing materially in degree ;
for the compressibility of ice appears to differ very little from
that of water, and the immediate force of cohesion is intimately
connected with the compressibility: but the true distinction
between solids and liquids is the hardness or lateral adhesion of
the one, and the perfect freedom of lateral motion possessed by

2 Q 2

596 REVIEW OF SIR H. davy's No. XXVI.

the particles of the other ; and if it were necessary to assign
a cause for this distinction, there is none that we could point
out with greater probability, than a certain symmetry of arrange-
ment, or an approach to crystallization, in the particles of solids,
while those of fluids might be supposed to be collected together
without any uniform order, and so far to be perfectly indepen-
dent of each other.

P. 84. ^^ It appears from the researches of Professor Robi-
son, tliat in a vacuum all liquids boil about 145 "* lower than in
the open air.*' Such an observation as this could scarcely have
been made with any propriety, even before the speculations of
Mr. Dalton had assisted us in forming more correct ideas on
this subject. A liquid placed in a perfect vacuum might be
said to bdil at any temperature, however low : since the tem-
perature at which any liquid boils is wholly dependent on the
pressure to which it is subjected. We may however easily
understand Professor Robison's experiments, by interpreting
the term vacuum as relating to the receiyer of an ordinary air-
pump, not in the best repair, in which the mercurial gauge would
stand at about two-thirds of an inch ; for in such an atmosphere
as this, both water and alcohol would in reality have their usual
boiling points lowered about 145^.

P. 92. The ^^ inaccuracy" of the thermometer must be rather
increased than ^' counteracted," by the disparity of the expan-
sions of fluids and solids, if it is really such as oxvt author states it

P. 94. With respect to heat. Sir H. Davy still professes
himself an advocate of the opinion of Bacon and Newton, that
it depends on a vibratory motion of the particles of bodies : but,
however powerful we may allow some of his arguments to be,
we cannot agree with him in thinking, that the acknowledged
existence of a *^ motion " of expansion or contraction will go
very far to prove the intimate nature of the cause of that
expansion or contraction. *

P. 136. The capability of thin plates, to receive a much
stronger charge of electricity than thick, is here attributed to
'Hhe diflSculty with which non-conductors receive polarity."
Surely the explanations of Cavendish and Robison are much
more luminous.

No. XXVI. ELEMENTS OF CHEMICAL PfflLOSOPHY. 597

P. 141. The resemblance of the Aurora borealis to the dis-
charge of electricity through rare air was very naturally ad-
duced by Franklin in illustration of that phenomenon ; but it
cannot be admitted as a sufficient explanation, until it be shown
in what manner the magnetical effects of the Aurora borealis
are produced, or why its beams are always parallel to the
dipping needle.

P. 149. The different powers of Voltaic batteries consisting
of large and small plates, although perhaps somewhat too
strongly contrasted, are very happily exemplified by experiments
conducted in pursuit of Mr. Cavendish's idea of the different
effect of a great quantity of electric fluid, and a great intensity
of charge ; and very interesting accounts are given of the opera-
tion of Mr. Children's large plates, and of the gigantic apparatus
of 2000 double plates, procured by subscription for the use of
the Royal Institution : but we must be contented with merely
pointing out these experiments, without attempting to give a
particular abstract of them.

P. 168. Sir H. Davy confirms Mr. Ehrman's discovery of
unipolar bodies, which cUscharge the electricity of either end of
the Voltaic circuit taken separately, but when connected with
both, retain the character of one only ; soap, for instance, re-
mains positive, and the flame of a common taper negative.

P. 219. Note. The optical experiments here mentioned can-
not certainly be sufficiently ^* explained on the idea of attractive
poles on opposite sides of the particles of light" These experi-
ments prove, if they prove any thing, not only '* that homo-
geneous light, at certain equal distances in the direction of its
motion, is possessed of opposite qualities, capable of neutralising
each other;" but also that these qualities afiect the collateral
rays of any single beam in a manner precisely similar at equal
distances from the radiant point : so that it would be necessary
to suppose a continued stratum or film of particles to be thrown
off by every luminous pomt, many millions of millions of times
in a second, and to proceed in all directions, like an expanding
shell, with an inconceivable velocity, to immeasurable distances.
We do not state this as an impossibility, but as a condition
necessary to be taken into consideration, without which our

598 REVIEW OP SIR H, davy's No. XXVI.

author's conjecture would be wholly inapplicable to the pheno-
mena.

P. 233. It is very justly observed that atmospheric ak has
not been found to differ perceptibly in its composition in the
most dissimilar situations, containing always 21 of oxygen, and
79 of azote or nitrogen ; that is, as. it should have been added,
by measure.

P. 296. A peculiar hydrophosphoric gas is described, which
was discovered by the author in February, 1812; but which
appears to have been previously known to Bookman and others.
P. 320. Some very accurate and decisive experiments are men-
tioned, which seem to determine finally that the ^* carbureted "
or carboneted hydrogens contain no oxygen, and that they exist
exclusively in the forms of " carbureted hydrogen," and " super-
carbureted hydrogen," or defiant gas. P. 382. ^^ Stannane "
seems to have been known to Proust

P. 391. It is observed that the colours on a polished surface
of heated iron ^' cannot depend on oxydation, as they take place
under mercury." But they appear between the temperatures of
430** and 580°, when the mercury has not yet boiled, and when
we cannot be certain that all air has been excluded. In the case
of lead, there is positive evidence that these colours are derived
from the formation of litharge ; and in that of iron, it appears
almost impossible to doubt that they are the beginning of the
scales of oxyd, which are actually thrown off, when the heat
becomes more intense.

P. 435. Palladium is said not to have '^ been found in suffi-
cient quantities to be applied to the purposes of the artd." But,
if we are not misinformed, its ingenious discoverer, who seems
to set all quantity at defiance, has furnished an auropalladium,
or an alloy of this metal with gold, for the graduations of the
magnificent circular instrument, which has lately been con-
structed by Mr. Troughton for the Royal Observatory at Green-
wich; this alloy having the appearance and durability of platina,
and being of a hardness better adapted for receiving the
divisions.

P. 492. The " powder of Algarotti " seems to be a sub-
muriate, and not an ^^ oxyd " of antimony.

No. XXVI. ELEMENTS OF CHEMICAL PHILOSOPHY. 599

P. 498. The solution of potassium in hydrogen is made the
basis of the explanation of the ready production of potassium by
means of ignited iron filings : but we had before been told that
sodium may be obtained in the same manner, and that sodium
is not soluble in hydrogen ; pp. 331, 335. The attraction of
potass to the oxyd of iron is alleged by others, with more con-
sistency, as a predisposing or potential affinity.

The character of Sir Humphry Davy's researches has always
been that of the most interesting originality, and we have
certainly no reason to complain that he has in his experiments
very commonly forsaken the beaten path. But in a general
work like the present, it was impossible that everything which
was required should be supplied from what he had himself
discovered or confirmed, and in reporting the labours of others,
he has sometimes allowed inaccuracies to escape him, which
a little more plodding diligence might have avoided. The
processes for obtaining the metals in purity are often of this
description : they might perhaps very properly have been
omitted or deferred, as not sufficiently elementary to be read
with advantage by a student : but if they were to be inserted,
it would have been better to have rendered them a little more
intelligible : and the entering into such an explanation of each
process might often have led the author to have considered all
its steps with more attention, and to have inquired if they
afforded the best possible means of attaining the desired end.
He seems also in many instances to have trusted too much
to his memory in asserting the non-existence of certain com-
binations, especially those of several of the metals with carbon,
and of some with sulphur.

The present volume extends only to the general laws of
chemical changes, and the primary combinations of the unde-
compounded bodies : how many more such volumes will com-
plete the whole of the projected Elements, it has probably not
yet been possible to determine. With all its excellences, this
work must be allowed to bear no inconsiderable marks of
haste, and we could easily have conjectured, even if the author
had not expressly told us so in his dedication, that the period
employed on it " has been the happiest of his life." In that,

600 ELEMENTS OF CHEMICAL PHILOSOPHY. No. XXVI.

and in every other happiness which may have befallen him, we
shall ever most sincerely rejoice, nor shall we think the pnblic
will have any reason to reproach him with having done too little
for science, even if he should fail, at any fixture time, in his
avowed resolution of pursuing it ''with unabated ardour;" that
he has not yet so failed, is become, from a late accident, a
matter of public notoriety ; and if we may expect perseverance
to be at all commensurate to success, we have no reason to be
apprehensive of his passing any part of his life in inactivity.

The style and manner of this work are nearly the same with
those of the author's lectures delivered in the theatre of the
Royal Institution ; they have been much admired by some of
the most competent judges of good language and good taste ;
and it has been remarked that Davy was bom a poet, and has
only become a chemist by accident. Certainly the situation, in
which he was placed, induced him to cultivate an ornamented
and popular style of expression and embellishment ; and what
was encouraged by temporary motives has become natural to
him from habit Hence have arisen a multitude of sentimental
reflections, and appeals to the feelings, which many will think
beauties, and some only prettinesses ; nor is it necessary for us
to decide in which of the two classes of readers we wish our-
selves to be arranged, conceiving that in matters so indifferent
to the immediate object of a work, a great latitude may be
allowed to the diversity of taste and opinion.

Dr. Young also wrote notices, in the Quarterly Review, of Davy's Agricultanl
Chemistry and of Bancroft on Dyeing ; but as they contain few observations of much
originality or importance, it has not been thought necessary to reprint them. — Note
by the Editor,

END OP VOL. I.

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