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other in its reflected course, bj' the sJiortesl path and in the least time,
its velocity being uniform and equal before and after reflexion.

Fermat extended the same principle, called the " principle of least
time," to the case oi refraction according to the law of sines, provided
"we suppose the velocity diminished in the denser medium : that is, he
showed that the sum of the times, or of the spaces divided by the veloci-
ties, is a minimum.

Huyghens, adopting the theory of waves, deduced from it the law
of the sines; and as, in conformity with tliat theory, the velocity
must be diminished in the denser medium, on this theoiy the principle
of "least time " applies to the case of refraction, and that of reflexion
also easily follows as a pai-ticular case.

On the other hand, on the molecular theory, the law of refraction is
deduced on the principle of attraction, which the molecules undergo
in the medium, and it is a necessary consequence that the velocity
must be increased in the denser medium. Maupertuis, on these prin-
ciples, attempted an analogous investigation ; but here it was neces-
sary to adopt, not the principle of "least time," but that of "least
action," or that the sum of the products oJ' the spaces and velocities is
a minimum ; and, on this view, the law of the sines equally results as
that which fulfils the condition.

This refers to ordiuarj- refraction : when the same inquiry was ex-
tended to double refraction, or to the extraordinary' ray, more complex
considerations were introduced. This subject is fully discussed by
Dr. Young in his Life of Fermat. ( Wbi-ks, ed. Peacock, vol. ii. p.
584.) The same principle was the basis of Laplace's investigation of
double refraction, of which (" Sur la Loi de la Refraction Extraordi-
naire, &c.," Journal de Physique, 1809) Dr. Young produced his well-
known refutation in the Quarterly Review for the same year.

In the case of ordinary refraction, the investigation is very simple.
As it is not clearly stated, as far as we are aware, in any elementary
treatise, it may be satisfactory to some readers to have it briefly put
before them.

Let any lengths, respectively, of the incident and refracted rays be



190



FRESNEL.



rived at the result, setting out from the ideas he had
adopted of the nature of light. And, lastly, Newton de-

l V, described with the velocities vvf, which are in a constant ratio to
each other; and in times which will be — ^. Then, on the prin-
ciple of "least time," the condition is,

III . .
1 =mmimum;

V VI '

or, differentiating and multiplying by v v',

V dl + v (III = . . . . (1).
Then if X be the surface of the medium, taking equal increments
dxon each side of the point of incidence, and dropping perpendiculars




to give corresponding increments dldV, i and r being the angles of
incidence and refraction, we have geometrically -,



sm J

al^- dll=::

ax



dx



■ (2);



and substituting in (1) it becomes

d' sin i — i; sin r = 0,



But, as i is necessarily greater than ?', it follows that the v must be
greater than «': or the law of the sines fulfils the condition of "least
time" on the wave theory.

On the other hand, the principle of " least action" requires, instead
of equation (1), that we have

lv-\-l' vi = minimum,
or vdl+vldli=0:

whence, by precisely the same process, there results
. . vi .



PRINCIPLE OF LEAST ACTION. 191

duced it from the principle of attraction, because that law

■which can only agree with observation provided v' be greater than v,
or the velocity be increased in the refracting medium, which agrees
with the molecular theory.

On either supposition, if « = «/, and sin r positive, the case becomes
that of rejlexion, and we have i =: r, which is the law of reflexion,
whence Ptolemy's conclusion is manifest as a particular case of the
general theory. The case of reflexion is, in fact, nothing more than a
geometrical problem.

Let two points i e, be given without a given straight line x x/, and
let o be the point in that line at which straight lines drawn from i




and E make equal angles with x x'. Then taking any other pairs of
lines I L, L E, and i m, m k, terminating in the same points and meet-
ing X X' in L and in m, they will each form unequal angles with x x' ;
E L x' greater than i l x, and r m x' greater than i m x. Let i m and
L E intersect in k.

Then we have the angle k l m greater than i l x, which is
greater than the opposite and interior i M l; and therefore in the
triangle k l m, k m is greater than k l.

In the limit, when m approaches l, we have ultimately i k=i l,
and K R=M k; whence i l+l k+k r is less than i k+k m-|-m k, or
the pair of lines nearest to o are together less than the more remote.
The same reasoning will apply to all pairs of lines on either side of o;
therefore the lines meeting at o are a minimum.

It is an extension of this principle wliicli forms the basis of the in-
vestigations of Sir W. R. Hamilton. Observing that in some parallel
instances the action is, in fact, not a case of minimum, but of max-
imum, lie has adopted the more generic term, "stationary action; "
and upon this has based his fundamental idea of the " characteristic
function," by the aid of which his profound analytical system, ap-
plicable equally in questions of optics and dynamics, is constructed.
For an admirable exposition of the general principle tlie student
should consult Sir W. E. Hamilton's paper on " The Paths of Light
and of the Planets" in the Dublin University Review, Oct. 1833. —
Translaiar.



192



FRESNEL.



occupied the attention of the greatest geometers of the
seventeenth century.

Tlie question had arrived at this point, when a travel-
ler, returning from Iceland, brought to Copenhagen some
beautiful crystals from the Bay of Roerford. Their
great thickness and reaiarkable transparency rendered
them particularly proper for experiments on refraction.
Bartholinus (1669), to wliom they were sent, took care
to subject them to different trials ; but what was his
astonishment when he perceived that the light divided
itself into two distinct beams of precisely equal intensi-
ties, — when he recognized, in one word, that seen through
the Iceland spar (which has been since found in many
other localities, being nothing but carbonate of lime) all
objects appear double ! The theory of refraction, so
many times recast, had now need of a new examination.
At all events it was incomplete, for it spoke only of one
ray, and two were here seen. Besides, the direction and
the magnitude of the deviation of the two rays changed,
apparently in the most capricious manner, when we passed
from one face of a crystal to another, or when on one face
the dii-ection of the incident ray varied.*

Huyghens surmounted all these difficulties ; a general
law was found to comprehend in its announcement all the
lesser details of the phenomena ; but this law, in spite of
its simplicity and elegance, was misconstrued. Hypoth-
eses had been for so many ages useless or faitliless guides ;
they had been so long considered as constituting the whole
of physics, that, at the epoch of which I speak, experi-
menters had on this point arrived at a sort of reaction ;
and in such reactions, even in science, it is rare to be

* See above, note, p. 150.



LAW OF DOUBLE llEFRACTION. 193

able to keep a just mean. Huyghens had given his law
as the result of an hypothesis ; men rejected it therefore
■without examination. The measures on which it was
founded could not redeem it from what was thought
vicious in its origin. Newton himself took part among its
opponents ; and from this moment the progress of optics
■was arrested for more than a century. Since that period,
the numerous experiments and measures of two of the
most celebrated members of this Academy, WoUaston
and Mains, have replaced the law of Huyghens in the
rank to which it is entitled.*

* Newton had rejected Huyghens's law, and substituted one founded
on measures of his own. In 1788 Haiiy repeated the measurements,
and showed that Huj'ghens's rule was far more accurate than New-
ton's. In 1802 Wollaston repeated similar observations by his new
method, in ignorance of Huyghens's law; but found them well repre-
sented when that law was pointed oi;t to him — probably by Dr. Young,
as the circumstance is stated by him in an article in the Quarterly
Revieto, Nov. 1809, p. 338.

Some idea may be given of the simple geometrical construction de-
termining the direction of the extraordinary ray which results from
Huyghens's theory, as follows: Supposing portions of the concentric
sphere and spheroid within the crj-stal, whose axis a coincides with
the axis of revolution of the spheroid; and conceiving a second spher-
ical surface concentric, and of greater radius, as that which would
have been the wave surface if the velocity had remained undimin-




194 FRESNEL.

During the long discussions which took place among
physicists on the mathematical law according to which
double refraction is produced in Iceland spar, the exist-
ence of the second ray was generally considered as an
anomaly affecting half the incident light ; the other half,
it was said, obeyed the old law of refraction laid down by
Descartes : the carbonate of lime, in its crystallized state,
then, enjoys certain particular properties, but without
losing those which all ordinary transparent media i>os-
sess. All this was exact in the instance of tlie Iceland
spar, and it seemed as if it mi^lit witliont hazard be
asserted generally ; but in fact those who maintained
this deceived themselves. There are crystals in which
the principle of ordinary refraction is not veritied ; and
in which the two rays into which the incident light divides
itself both undergo anomalous refractions, where the law
of Descartes does not indicate the course of either ray.

When Fresnel for the first time published this unex-
pected result, he had as yet verified it only by the aid of
an indirect method, i-emarkable for the strange circum-
stance that the refraction of the rays was deduced from
experiments in which no refraction took place. Thus
our colleague found more than one incredulous reader.
The singularity of the discovery, perhaps, demanded
some hesitation : perhaps also in the eyes of some per-
sons, it had the fault, like the law of Huyghens, of being
the fruit of an hypothesis. However it may have been,
Fresnel met the difficulty boldly. By showing that in a
parallelopiped of topaz, ibrmed of two prisms of the same

ished; tlieu from the extremity t of the incident ray i as if produced
to meet this sphere, drawing tangent planes to the spliere and spheroid
respectively, the points of contact will give the position of the ordinary
and extraordinary rays o and e. See Peacoclv's Life of Young, p. 373. —
Translator.



BIAXIAL CRYSTALS. 195

angle, opposed, no ray passes between the opposite and
parallel faces without undergoing deviation, he rendered
all objections vain.*

* The paradoxical mention of proofs of refraction, ■where no refrac-
tion takes place, may need a brief explanation.

Fresnel's experiment, here referred to, was performed by means of
the simple interference of two rays produced by reflexion from plane
mirrors very little inclined from the same plane, or by transmission
through a very obtuse-angled prism. If, in the path (as explained in
a subsequent note) of each of the two interfering rays, plates of glass
of exactly the same thickness are interposed, the position of the stripes
remains unaltered; but if the plates be cut from a. biaxial crystal in
different directions with respect to its axis, but still of exactly the
same thickness, even if we employ those rays which correspond to the
ordinary rays in Iceland spar, there will be a displacement of the
stripes, showing a difference of velocity or refraction, in these rays,
on the principle hereafter explained, (see note infra.)

The more direct experiment alluded to consists in this: Fresnel cut
two prisms in different directions from the same crystal of topaz,
which, being cemented together with their axes in one line, were
ground together to exactly the same angle, and the whole achroma-
tized by another opposed prism. On looking through the two prisms
thus fixed side by side at a line of light, that line was seen to be bro-
ken at the junction, indicating different refractions in the two.

The law of Huyghens, or the construction of the sphere and sphe-
roid, was found to hold good not only in Iceland spar, but in many
Other doubly refracting crystals. But these were all characterized by
possessing only one axis or line along which there was no double re-
fraction, and which, by the aid of polarized light, is easily detected as
forming the centre of the rings.

Sir D. Brewster, in examining a vast varietj' of crystals, discovered
a class in which there was not one such axis, but tivo, and in which
the rings consequently assumed new and more complex forms, being
either arranged in two separate sets if the axes were distant, or in
coalescing curves if they were close.

For biaxial crystals Huyghens's law will not apply. The incident
ray is divided into two; but neither of them follows the law of the
sines represented by the sphere in his construction. One of the rays
is, indeed, usually less subject to deviation than the other, and thus,
for convenience, is still often called the ordinary ray; but both are, in
strictness, extraordinary rays.



196 FRESNEL.

Those physicists (I could here cite the names of some
of the most celebrated) who have sought to include in a

Hence the necessity for a more comprehensive theor}'. As Huy-
ghens had constructed sucli a theory by means of an independent
sphere and spheroid, Fresnel not only generalized the construction by
a method giving two curved surfaces of higher forms, but he did what
Huvghens's method did not effect, even in the simple case which he
considered — he showed also a necessary connection between tlie two
surfaces • they were in fact not two, but portions of one surface — parts
of the geometi-ical representation of the same algebraic equation, or,
in the h^nfuao'e of mathematicians, "a curve surface of two sheets."
Thus Frcsnel's theory showed not only the laws by which each ray
was refracted, but also why there must be two rays.

Of this more generalized mathematical investigation, the greater
part of the steps were omitted by Fresnel in his memoir, as being of
too complicated and tedious a nature for the patience of his readers;
he presents only the conclusions, which are derived from certain sup-
positions with respect to the elasticity of the ether, as being different
in different directions within the crystal, and ultimately lead to an
algebraic equation, representing a curved surface of the fourth order,
consisting of two sheets or portions, as the general form assumed by
the waves, but which in certain cases, as in calc spar, is reducible to
the simpler form of the sphere and spheroid of Huyghens.

For a connected view of these investigations the reader is referred
to Professor Powell's Treatise on the Undulatory Theory, fc. page 48.
London, 1841.

The mathematical investigation has since called forth much eluci-
dation, especially in supplying the suppressed processes of Fresnel, ia
which the analysis of Mr. A. Smith, as well as those of Sir J. Lub-
bock, Professor Sylvester, Sir W. K. Hamilton, and others, have been
eminently successful; while the last-named mathematician pointed
out the very curious consequence that this surface, mathematically
speaking, presents, at the extremities of the axis, conoidal cusps, — that
is depressions of a pointed funnel shape, — which, physically inter-
preted, would show that a ray passing along that direction ought to
emerge no longer a single ray, but spread out in a conical surface
whose surface would not be a point of light, but a ring with a dark
central space. This extraordinary prediction, so wholly unlike any
thing hitherto imagined, was, however, fully verified by the observa-
tions of Dr. Lloyd on a crystal of aragonite ; the phenomenon being
known by the name of '■^ conical reiva.ct\on."— Translator.



DOUBLE REFRACTION. 197

single rule all the possible cases of double refraction,
were thus misled, for they all admitted, as a fact of
which no one could doubt, that for half the light, for the
rays called "ordinary rays," the deviation ought to be
the same at the same incidence in whatever direction
the plane of incidence cut the crystal. The true law of
these complicated phenomena — the law which includes,
as particulai' cases, the laws of Descartes and of Huy-
ghens — is due to Fresnel. This discovery required in
an eminently high degree the union of a talent for exper-
iment with the genius of invention.

I freely admit that the phenomena of double refraction
recently analyzed by Fresnel, and tlie laws which con-
nect them, are not exempt from a certain complexity.
This is indeed a subject of regret — almost, I might say,
of lamentation — among some idle minds, who would wish
to reduce every science to those superficial notions of
which they might make themselves masters by a few
hours' work. But does not every one see that with such
ideas the sciences would not make any pi'ogress ; that to
neglect such phenomena because one feeble intellect may
experience some trouble in grasping them, would be to
be false to our vocation, and that thus we should often
allow the most important discoveries to pass by us.

Tims astronomy, while limited to a knowledge of the
constellations, and to some insignificant remarks on the
risings and settings of the stars, was within the capacity
of minds of any class : but could we then call it a science ?
From that time till after the most colossal labour which
one man ever went through, — Kepler had substituted
elliptic motions not uniform, for the circular and regular
motions which, according to the ancients, prevailed in the
planets, — his contemporaries might with equal right have



198 FRESNEL.

complained of complexity. But again, some time after,
in the hands of Newton, these motions, complex in ap-
pearance, became the basis of the greatest discoveries of
modern times, of a principle as simple as it is fertile ;
they served to prove that every planet is governed in its
elliptic course by a simple force, by an attraction emanat-
ing from the sun.

Those observers again, who, in their turn refining
upon Kepler, showed that simple elliptic motions would
not suffice to represent the true paths of the planets,
did not simplify the science. But besides that the
derangement (known under the name of perturbations)
would not the less have existed if, in the dislike of all
complexity, we had obstinately determined to shut our
eyes to them, I ought to "ay, that in studying them with
cai'e we have been conducted, among many other impor-
tant results, to the means of comparing the masses of the
different bodies of which our solar system is composed ;
and that if at the present day we know, for example, that
it requires not less than 350,000 times the globe of the
earth to form a weight equal to that of the sun, we owe
it to the observation of those very small inequalities,
which those would certainly have neglected, who at all
risks would admit nothing but simple phenomena.

Without extending these remarks farther, I may then
admit that optics would be a more easy science, more at
the command of the generality of men, more susceptible
of demonstration in public lectures, before the extension
of it which has been made in our times. But this exten-
sion is a real source of riches ; it has given occasion for
the most curious applications ; it has thence afforded
those indications of impossibilities in certain theories of
light, which may claim to rank among discoTeries ; for



DOUBLE KEFRACTION. 199

in the search after causes, we are often reduced to pro-
ceed by the method of exclusion, and in this point of view,
there is no experiment which is witliout use ; we can-
not muhiply them too much. That universal genius, Vol-
taire, who often took pleasure in concealing a profound
meaning under a burlesque form, compared a theory to a
mouse, which passes, he said, through nine holes, but is
caught in the tenth. It is in multiplying indefinitely the
number of these holes, or to speak in a manner less triv-
ial, the number of tests which a theory ought to satisfy,
that astronomy is placed in the rank which it occupies
in the estimation of men, and that it has become the first
of the sciences. It is in following the same route that
we sliall be able in like manner to give to other branches
of science the character of evidence which they yet want
in some respects. In every science of observation we
must distinguish the facts, the laws which connect them,
and the causes. Often the dilTiculties of the subject
arrest experimenters after the first step ; hardly ever do
they allow them to pass freely to the third. The pro-
gress which Fresnel made in the two former respects, in
the study of double refraction, by natural consequence,
led him to inquire on what so singular a phenomenon
depended. And here again he obtained striking suc-
cess. But pressed for time, I can only make known the
most prominent of his results.

When Huyghens published his Traitt de la Lumiere,
there were only known two crystals possessing double
refraction, — carbonate of lime and quartz. At present it
would be far shorter to enumerate those which have not
this property, than those which have it. Formerly, it
was necessary that a substance should distinctly show
double images, to allow us to assimilate it with Iceland



200 FRESNEL.

spar. Whenever the separation of the two rays was so
small as to escape detection by the eye, the observer
remained in doubt and did not venture to pronounce it
doubly refractive. Now, however, by the aid of a
method which a member of the Academy has pointed
out,* the existence of double refraction manifests itself
by characteristics quite independent of the separation of
the two images. No substance, however thin it may be,
possessed of this property, can escape this new mode of
examination. But, if it were certain that double refrac-
tion could not exist without our perceiving the very
manifest phenomena on which this method is founded, it
would not appear equally incontestable that it ought
necessarily to accompany them ; and a doubt in regard
to this might seem the more natural since the author of
this method has himself found certain jjlates of glass
which, without separating the images in a perceptible
degree, yet give birth to all the phenomena in question :
— since a distinguished philosopher of Berlin, M. See-
beck, afterwards proved that all glass rapidly cooled
enjoyed the same property ; — and since, lastly, a very
able experimenter of Edinburgh produced the same

* The author here alludes to his own discovery of the polarized
coloui\s, made also quite independently by Brewster about the same
time. These tints are now familiar to most persons by means of the
little instrument called'tlie polariscope. By placing a plate of sele-
nite, mica, &:c., far too thin to exhibit any separation of images, in
polarized light, and viewing it through an analyzer, these brilliant
tints convey distinct evidence of the existence of that propertj^, since
they are shown iheorclicaUy to depend solely upon its existence, how-
ever insensibly small its amount may be. It therefore seems impor-
tant for the verification of theory, to show independently its existence
in any substances which exhibit the tints. Glass ordinarily possesses
no such power; but plates of unannealed glass exhibit the tints.
Hence the importance of the experiments mentioned to show its
existence directlv. — Transhilor.



DOUBLE REFRACTION. 201

phenomena by compressing pieces of glass with groat
force in certain directions. To show that a piece of
ordinary glass, thus modified by cooling or compression,
always really separates the light into two rays, — and to
render this separation incontestably evident, was the
important problem which Fresnel proposed to himself,
and which he resolved in his usual happy manner.

By placing in the same line, and in a frame of iron
carrying powerful screws ingeniously arranged, a number
of prisms of glass, which by these screws were subjected
to very powerful pressure, Fresnel caused a manifest
double refraction to appear. In an optical point of view



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