EPOCH OF YOUNG AND FRESNEL.
_Sect._ 1.--_Introduction._
THE man whose name must occupy the most distinguished place in the
history of Physical Optics, in consequence of what he did in
reviving and establishing the undulatory theory of light, is Dr.
Thomas Young. He was born in 1773, at Milverton in Somersetshire, of
Quaker parents; and after distinguishing himself during youth by the
variety and accuracy of his attainments, he settled in London as a
physician in 1801; but continued to give much of his attention to
general science. His optical theory, for a long time, made few
proselytes; and several years afterwards, Auguste Fresnel, an
eminent French mathematician, an engineer officer, took up similar
views, proved their truth, and traced their consequences, by a
series of labors almost independent of those of Dr. Young. It was
not till the theory was thus re-echoed from another land, that it
was able to take any strong hold on the attention of the countrymen
of its earlier promulgator.
The theory of undulations, like that of universal gravitation, may
be divided into several successive steps of generalization. In both
cases, all these steps were made by the same persons; but there is
this difference:--all the parts of the law of universal gravitation
were worked out in one burst of inspiration by its author, and
published at one time;--in the doctrine of light, on the other hand,
the different steps of the advance were made and published at
separate times, with intervals between. We see the theory in a
narrower form, and in detached portions, before the widest
generalizations and principles of unity are reached; we see the
authors struggling with the difficulties before we see them
successful. They appear to us as men like ourselves, liable to
perplexity and failure, instead of coming before us, as Newton does
in the history of Physical Astronomy, as the irresistible and almost
supernatural hero of a philosophical romance. {93}
The main subdivisions of the great advance in physical optics, of
which we have now to give an account, are the following:--
1. The explanation of the _periodical colors_ of thin plates, thick
plates, fringed shadows, striated surfaces, and other phenomena of
the same kind, by means of the doctrine of the _interference_ of
undulations.
2. The explanation of the phenomena of _double refraction_ by the
propagation of undulations in a medium of which the optical
_elasticity_ is different in different directions.
3. The conception of _polarization_ as the result of the vibrations
being _transverse_; and the consequent explanation of the production
of polarization, and the necessary connexion between polarization
and double refraction, on mechanical principles.
4. The explanation of the phenomena of _dipolarization_, by means of
the interference of the _resolved parts_ of the vibrations after
double refraction.
The history of each of these discoveries will be given separately to
a certain extent; by which means the force of proof arising from
their combination will be more apparent.
_Sect._ 2.--_Explanation of the Periodical Colors of Thin Plates and
Shadows by the Undulatory Theory._
THE explanation of periodical colors by the principle of
interference of vibrations, was the first step which Young made in
his confirmation of the undulatory theory. In a paper on Sound and
Light, dated Emmanuel College, Cambridge, 9th July, 1799, and read
before the Royal Society in January following, he appears to incline
strongly to the Huyghenian theory; not however offering any new
facts or calculations in its favor, but pointing out the great
difficulties of the Newtonian hypothesis. But in a paper read before
the Royal Society, November 12, 1801, he says, "A further
consideration of the colors of thin plates has converted that
prepossession which I before entertained for the undulatory theory
of light, into a very strong conviction of its truth and efficiency;
a conviction which has since been most strikingly confirmed by an
analysis of the colors of _striated surfaces_." He here states the
general principle of interferences in the form of a proposition.
(Prop. viii.) "When two undulations from different origins coincide
either perfectly or very nearly in direction, their joint effect is
a combination of the motions belonging to them." He explains, by the
help of this proposition, the colors which were observed in
Coventry's {94} micrometers, in which instrument lines were drawn on
glass at a distance of 1⁄500th of an inch. The interference of the
undulations of the rays reflected from the two sides of these fine
lines, produced periodical colors. In the same manner, he accounts
for the colors of thin plates, by the interference of the light
partially reflected from the two surfaces of the plates. We have
already seen that Hooke had long before suggested the same
explanation; and Young says at the end of his paper, "It 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 opinion." He also quotes from Newton many
passages which assume the existence of an ether; of which, as we
have already seen, Newton suggests the necessity in these very
phenomena, though he would apply it in combination with the emission
of material light. In July, 1802, Young explained, on the same
principle, some facts in indistinct vision, and other similar
appearances. And in 1803,[75\9] he speaks more positively still. "In
making," he says, "some experiments on the fringes of colors
accompanying shadows, I have found so simple and so demonstrative a
proof of the general law of interference of two portions of light,
which I have already endeavored to establish, that I think it right
to lay before the Royal Society a short statement of the facts which
appear to me to be thus decisive." The two papers just mentioned
certainly ought to have convinced all scientific men of the truth of
the doctrine thus urged; for the number and exactness of the
explanations is very remarkable. They include the colored fringes
which are seen with the shadows of fibres; the colors produced by a
dew between two pieces of glass, which, according to the theory,
should appear when the thickness of the plate is _six_ times that of
thin plates, and which do so; the changes resulting from the
employment of other fluids than water; the effect of inclining the
plates; also the fringes and bands which accompany shadows, the
phenomena observed by Grimaldi, Newton, Maraldi, and others, and
hitherto never at all reduced to rule. Young observes, very justly,
"whatever may be thought of the theory, we have got a simple and
general law" of the phenomena. He moreover calculated the length of
an undulation from the measurements of fringes of shadows, as he had
done before from the colors of thin plates; and found a very close
accordance of the results of the various cases with one another. {95}
[Note 75\9: _Phil. Trans._ Memoir, read Nov. 24.]
There is one difficulty, and one inaccuracy, in Young's views at
this period, which it may be proper to note. The difficulty was,
that he found it necessary to suppose that light, when reflected at
a rarer medium, is retarded by half an undulation. This assumption,
though often urged at a later period as an argument against the
theory, was fully justified as the mechanical principles of the
subject were unfolded; and the necessity of it was clear to Young
from the first. On the strength of this, says he, "I ventured to
predict, that if the reflections were of the same kind, made at the
surfaces of a thin plate, of a density intermediate between the
densities of the mediums surrounding it, the central spot would be
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 inaccuracy of his calculations consisted in his considering the
external fringe of shadows to be produced by the interference of a
ray _reflected_ from the _edge_ of the object, with a ray which
passes clear of it; instead of supposing _all the parts_ of the wave
of light to corroborate or interfere with one another. The
mathematical treatment of the question on the latter hypothesis was
by no means easy. Young was a mathematician of considerable power in
the solution of the problems which came before him: though his
methods possessed none of the analytical elegance which, in his
time, had become general in France. But it does not appear that he
ever solved the problem of undulations as applied to fringes, with
its true conditions. He did, however, rectify his conceptions of the
nature of the interference; and we may add, that the numerical error
of the consequences of the defective hypothesis is not such as to
prevent their confirming the undulatory theory.[76\9]
[Note 76\9: I may mention, in addition to the applications which
Young made of the principle of interferences, his _Eriometer_, an
instrument invented for the purpose of measuring the thickness of
the fibres of wood; and the explanation of the supernumerary bands
of the rainbow. These explanations involve calculations founded on
the length of an undulation of light, and were confirmed by
experiment, as far as experiment went.]
But though this theory was thus so powerfully recommended by
experiment and calculation, it met with little favor in the
scientific world. Perhaps this will be in some measure accounted
for, when we come, in the next chapter, to speak of the mode of its
reception by {96} the supposed judges of science and letters. Its
author went on laboring at the completion and application of the
theory in other parts of the subject; but his extraordinary success
in unravelling the complex phenomena of which we have been speaking,
appears to have excited none of the notice and admiration which
properly belonged to it, till Fresnel's Memoir _On Diffraction_ was
delivered to the Institute, in October, 1815.
MM. Arago and Poinsot were commissioned to make a report upon this
Memoir; and the former of these philosophers threw himself upon the
subject with a zeal and intelligence which peculiarly belonged to
him. He verified the laws announced by Fresnel: "laws," he says,
"which appear to be destined to make an epoch in science." He then
cast a rapid glance at the history of the subject, and recognized,
at once, the place which Young occupied in it. Grimaldi, Newton,
Maraldi, he states, had observed the facts, and tried in vain to
reduce them to rule or cause. "Such[77\9] was the state of our
knowledge on this difficult question, when Dr. Thomas Young made the
very remarkable experiment which is described in the _Philosophical
Transactions_ for 1803;" namely, that to obliterate all the bands
within the shadow, we need only stop the ray which is going to
graze, or has grazed, one border of the object. To this, Arago added
the important observation, that the same obliteration takes place,
if we stop the ray, with a transparent plate; except the plate be
very thin, in which case the bands are displaced, and not
extinguished. "Fresnel," says he, "guessed the effect which a thin
plate would produce, when I had told him of the effect of a thick
glass." Fresnel himself declares[78\9] that he was not, at the time,
aware of Young's previous labors. After stating nearly the same
reasonings concerning fringes which Young had put forward in 1801,
he adds, "it is therefore the meeting, the actual crossing of the
rays, which produces the fringes. This consequence, which is only,
so to speak, the translation of the phenomena, seems to me entirely
opposed to the hypothesis of emission, and confirms the system which
makes light consist in the vibrations of a peculiar fluid." And thus
the Principle of Interferences, and the theory of undulations, so
far as that principle depends upon the theory, was a second time
established by Fresnel in France, fourteen years after it had been
discovered, fully proved, and repeatedly published by Young in
England. {97}
[Note 77\9: _An. Chim._ 1815, Febr.]
[Note 78\9: Ib. tom. xvii. p. 402.]
In this Memoir of Fresnel's, he takes very nearly the same course as
Young had done; considering the interference of the direct light
with that reflected at the edge, as the cause of the external
fringes; and he observes, that in this reflection it is necessary to
suppose half an undulation lost: but a few years later, he
considered the propagation of undulations in a more true and general
manner, and obtained the solution of this difficulty of the
half-undulation. His more complete Memoir on _Diffraction_ was
delivered to the Institute of France, July 29, 1818; and had the
prize awarded it in 1819:[79\9] but by the delays which at that
period occurred in the publication of the _Parisian Academical
Transactions_, it was not published[80\9] till 1826, when the theory
was no longer generally doubtful or unknown in the scientific world.
In this Memoir, Fresnel observes, that we must consider the effect
of _every portion_ of a wave of light upon a distant point, and
must, on this principle, find the illumination produced by any
number of such waves together. Hence, in general, the process of
integration is requisite; and though the integrals which here offer
themselves are of a new and difficult kind, he succeeded in making
the calculation for the cases in which he experimented. His _Table
of the Correspondences of Theory and Observation_,[81\9] is very
remarkable for the closeness of the agreement; the errors being
generally less than one hundredth of the whole, in the distances of
the black bands. He justly adds, "A more striking agreement could
not be expected between experiment and theory. If we compare the
smallness of the differences with the extent of the breadths
measured; and if we remark the great variations which _a_ and _b_
(the distance of the object from the luminous point and from the
screen) have received in the different observations, we shall find
it difficult not to regard the integral which has led us to these
results as the faithful expression of the law of the phenomena."
[Note 79\9: _Ann. Chim._ May, 1819.]
[Note 80\9: _Mém. Inst._ for 1821-2.]
[Note 81\9: _Mém. Inst._ p. 420-424.]
A mathematical theory, applied, with this success, to a variety of
cases of very different kinds, could not now fail to take strong
hold of the attention of mathematicians; and accordingly, from this
time, the undulatory doctrine of diffraction has been generally
assented to, and the mathematical difficulties which it involves,
have been duly studied and struggled with.
Among the remarkable applications of the undulatory doctrine to
diffraction, we may notice those of Joseph Fraunhofer, a {98}
mathematical optician of Munich. He made a great number of
experiments on the shadows produced by small holes, and groups of
small holes, very near each other. These were published[82\9] in his
_New Modifications of Light_, in 1823. The greater part of this
Memoir is employed in tracing the laws of phenomena of the extremely
complex and splendid appearances which he obtained; but at the
conclusion he observes, "It is remarkable that the laws of the
reciprocal influence and of the diffraction of the rays, can be
deduced from the principles of the undulatory theory: knowing the
conditions, we may, by means of an extremely simple equation,
determine the extent of a luminous wave for each of the different
colors; and in every case, the calculation corresponds with
observation." This mention of "an extremely simple equation,"
appears to imply that he employed only Young's and Fresnel's earlier
mode of calculating interferences, by considering two portions of
light, and not the method of integration. Both from the late period
at which they were published, and from the absence of mathematical
details, Fraunhofer's labors had not any strong influence on the
establishment of the undulatory theory; although they are excellent
verifications of it, both from the goodness of the observations, and
the complexity and beauty of the phenomena.
[Note 82\9: In Schumacher's _Astronomische Abhandlungen_, in French;
earlier in German.]
We have now to consider the progress of the undulatory theory in
another of its departments, according to the division already stated.
_Sect._ 3.--_Explanation of Double Refraction by the Undulatory
Theory._
WE have traced the history of the undulatory theory applied to
diffraction, into the period when Young came to have Fresnel for his
fellow-laborer. But in the mean time, Young had considered the
theory in its reference to other phenomena, and especially to those
of _double refraction_.
In this case, indeed, Huyghens's explanation of the facts of Iceland
spar, by means of spheroidal undulations, was so complete, and had
been so fully confirmed by the measurements of Haüy and Wollaston,
that little remained to be done, except to connect the Huyghenian
hypothesis with the mechanical views belonging to the theory, and to
extend his law to other cases. The former part of this task Young
executed, by remarking that we may conceive the _elasticity_ of the
{99} crystal, on which the velocity of propagation of the
luminiferous undulation depends, to be different, in the direction
of the crystallographic axis, and in the direction of the planes at
right angles to this axis; and from such a difference, he deduces
the existence of spheroidal undulations. This suggestion appeared in
the _Quarterly Review_ for November, 1809, in a critique upon an
attempt of Laplace to account for the same phenomena. Laplace had
proposed to reduce the double refraction of such crystals as Iceland
spar, to his favorite machinery of forces which are sensible at
small distances only. The peculiar forces which produce the effect
in this case, he conceives to emanate from the crystallographic
axis: so that the velocity of light within the crystal will depend
only on the situation of the ray with respect to this axis. But the
establishment of this condition is, as Young observes, the main
difficulty of the problem. How are we to conceive refracting forces,
independent of the surface of the refracting medium, and regulated
only by a certain internal line? Moreover, the law of force which
Laplace was obliged to assume, namely, that it varied as the square
of the sine of the angle which the ray made with the axis, could
hardly be reconciled with mechanical principles. In the critique
just mentioned, Young appears to feel that the undulatory theory,
and perhaps he himself, had not received justice at the hands of men
of science; he complains that a person so eminent in the world of
science as Laplace then was, should employ his influence in
propagating error, and should disregard the extraordinary
confirmations which the Huyghenian theory had recently received.
The extension of this view, of the different elasticity of crystals
in different directions, to other than uniaxal crystals, was a more
complex and difficult problem. The general notion was perhaps
obvious, after what Young had done; but its application and
verification involved mathematical calculations of great generality,
and required also very exact experiments. In fact, this application
was not made till Fresnel, a pupil of the Polytechnic School,
brought the resources of the modern analysis to bear upon the
problem;--till the phenomena of dipolarized light presented the
properties of biaxal crystals in a vast variety of forms;--and till
the theory received its grand impulse by the combination of the
explanation of polarization with the explanation of double
refraction. To the history of this last-mentioned great step we now
proceed. {100}
_Sect._ 4.--_Explanation of Polarization by the Undulatory Theory._
EVEN while the only phenomena of _polarization_ which were known were
those which affect the two images in Iceland spar, the difficulty
which these facts seemed at first to throw in the way of the
undulatory theory was felt and acknowledged by Young. Malus's
discovery of polarization by reflection increased the difficulty,
and this Young did not attempt to conceal. In his review of the
papers containing this discovery[83\9] he says, "The discovery
related in these papers appears to us to be by far the most
important and interesting which has been made in France concerning
the properties of light, at least since the time of Huyghens; and it
is so much the more deserving of notice, as it greatly influences
the general balance of evidence in the comparison of the undulatory
and projectile theories of the nature of light." He then proceeds to
point out the main features in this comparison, claiming justly a
great advantage for the theory of undulations on the two points we
have been considering, the phenomena of diffraction and of double
refraction. And he adds, with reference to the embarrassment
introduced by polarization, that we are not to expect the course of
scientific discovery to run smooth and uninterrupted; but that we
are to lay our account with partial obscurity and seeming
contradiction, which we may hope that time and enlarged research
will dissipate. And thus he steadfastly held, with no blind
prejudice, but with unshaken confidence, his great philosophical
trust, the fortunes of the undulatory theory. It is here, after the
difficulties of polarization had come into view, and before their
solution had been discovered, that we may place the darkest time of
the history of the theory; and at this period Young was alone in the
field.
[Note 83\9: _Quart. Rev._ May, 1810.]
It does not appear that the light dawned upon him for some years. In
the mean time, Young found that his theory would explain dipolarized
colors; and he had the satisfaction to see Fresnel re-discover, and
M. Arago adopt, his views on diffraction. He became engaged in
friendly intercourse with the latter philosopher, who visited him in
England in 1816. On January the 12th, 1817, in writing to this
gentleman, among other remarks on the subject of optics, he says, "I
have also been reflecting on the possibility of giving an imperfect
explanation of the affection of light which constitutes
polarization, {101} without departing from the genuine doctrine of
undulation." He then proceeds to suggest the possibility of "a
_transverse_ vibration, propagated in the direction of the radius,
the motions of the particles being in a certain constant direction
with respect to that radius; and this," he adds, "is
_polarization_." From his further explanation of his views, it
appears that he conceived the motions of the particles to be oblique
to the direction of the ray, and not perpendicular, as the theory
was afterwards framed; but still, here was the essential condition
for the explanation of the facts of polarization,--the transverse
nature of the vibrations. This idea at once made it possible to
conceive how the rays of light could have _sides_; for the direction
in which the vibration was transverse to the ray, might be marked by
peculiar properties. And after the idea was once started, it was
comparatively easy for men like Young and Fresnel to pursue and
modify it till it assumed its true and distinct form.
We may judge of the difficulty of taking firmly hold of the
conception of transverse vibrations of the ether, as those which
constitute light, by observing how long the great philosophers of
whom we are speaking lingered within reach of it, before they
ventured to grasp it. Fresnel says, in 1821, "When M. Arago and I
had remarked (in 1816) that two rays polarized at right angles
always give the same quantity of light by their union, I thought
this might be explained by supposing the vibrations to be
transverse, and to be at right angles when the rays are polarized at
right angles. But this supposition was so contrary to the received
ideas on the nature of the vibrations of elastic fluids," that
Fresnel hesitated to adopt it till he could reconcile it better to
his mechanical notions. "Mr. Young, more bold in his conjectures,
and less confiding in the views of geometers, published it before
me, though perhaps he thought it after me." And M. Arago was
afterwards wont to relate[84\9] that when he and Fresnel had
obtained their joint experimental results of the non-interference of
oppositely-polarized pencils, and when Fresnel pointed out that
transverse vibrations were the only possible translation of this
fact into the undulatory theory, he himself protested that he had
not courage to publish such a conception; and accordingly, the
second part of the Memoir was published in Fresnel's name alone.
What renders this more remarkable is, that it occurred when M. Arago
had in his possession the very letter of Young, in which he proposed
the same suggestion. {102}
[Note 84\9: I take the liberty of stating this from personal
knowledge.]
Young's first published statement of the doctrine of transverse
vibrations was given in the explanation of the phenomena of
dipolarization, of which we shall have to speak in the next Section.
But the primary and immense value of this conception, as a step in
the progress of the undulatory theory, was the connexion which it
established between polarization and double refraction; for it held
forth a promise of accounting for polarization, if any conditions
could be found which might determine what was the direction of the
transverse vibrations. The analysis of these conditions is, in a
great measure, the work of Fresnel; a task performed with profound
philosophical sagacity and great mathematical skill.
Since the double refraction of uniaxal crystals could be explained
by undulations of the form of a spheroid, it was perhaps not
difficult to conjecture that the undulations of biaxal crystals
would be accounted for by undulations of the form of an ellipsoid,
which differs from the spheroid in having its three axes unequal,
instead of two only; and consequently has that very relation to the
other, in respect of symmetry, which the crystalline and optical
phenomena have. Or, again, instead of supposing two different
degrees of elasticity in different directions, we may suppose three
such different degrees in directions at right angles to each other.
This kind of generalization was tolerably obvious to a practised
mathematician.
But what shall call into play all these elasticities at once, and
produce waves governed by each of them? And what shall explain the
different polarization of the rays which these separate waves carry
with them? These were difficult questions, to the solution of which
mathematical calculation had hitherto been unable to offer any aid.
It was here that the conception of transverse vibrations came in,
like a beam of sunlight, to disclose the possibility of a mechanical
connexion of all these facts. If transverse vibrations, travelling
through a uniform medium, come to a medium not uniform, but
constituted so that the elasticity shall be different in different
directions, in the manner we have described, what will be the course
and condition of the waves in the second medium? Will the effects of
such waves agree with the phenomena of doubly-refracted light in
biaxal crystals? Here was a problem, striking to the mathematician
for its generality and difficulty, and of deep interest to the
physical philosopher, because the fate of a great theory depended
upon its solution.
The solution, obtained by great mathematical skill, was laid before
the French Institute by Fresnel in November, 1821, and was carried
{103} further in two Memoirs presented in 1822. Its import is very
curious. The undulations which, coming from a distant centre, fall
upon such a medium as we have described, are, it appears from the
principles of mechanics, propagated in a manner quite different from
anything which had been anticipated. The "surface of the waves"
(that is, the surface which would bound undulations diverging from a
point), is a very complex, yet symmetrical curve surface; which, in
the case of uniaxal crystals, resolves itself into a sphere and a
spheroid; but which, in general, forms a continuous double envelope
of the central point to which it belongs, intersecting itself and
returning into itself. The directions of the rays are determined by
this curve surface in biaxal crystals, as in uniaxal crystals they
are determined by the sphere and the spheroid; and the result is,
that in biaxal crystals, _both_ rays suffer _extraordinary_
refraction according to determinate laws. And the positions of the
planes of polarization of the two rays follow from the same
investigation; the plane of polarization in every case being
supposed to be that which is perpendicular to the transverse
vibrations. Now it appeared that the polarization of the two rays,
as determined by Fresnel's theory, would be in directions, not
indeed exactly accordant with the law deduced by M. Biot from
experiment, but deviating so little from those directions, that
there could be small doubt that the empirical formula was wrong, and
the theoretical one right.
The theory was further confirmed by an experiment showing that, in a
biaxal crystal (topaz), neither of the rays was refracted according
to the ordinary law, though it had hitherto been supposed that one
of them was so; a natural inaccuracy, since the error was
small.[85\9] Thus this beautiful theory corrected, while it
explained, the best of the observations which had previously been
made; and offered itself to mathematicians with an almost
irresistible power of conviction. The explanation of laws so strange
and diverse as those of double refraction and polarization, by the
same general and symmetrical theory, could not result from anything
but the truth of the theory.
[Note 85\9: _An. Ch._ xxviii. p. 264.]
"Long," says Fresnel,[86\9] "before I had conceived this theory, I
had convinced myself by a pure contemplation of the facts, that it
was not possible to discover the true explanation of double
refraction, without explaining, at the same time, the phenomena of
polarization, which always goes along with it; and accordingly, it
was after having found {104} what mode of vibration constituted
polarization, that I caught sight of the mechanical causes of double
refraction."
[Note 86\9: _Sur la Double Réf., Mém. Inst._ 1826, p. 174.]
Having thus got possession of the principle of the mechanism of
polarization, Fresnel proceeded to apply it to the other cases of
polarized light, with a rapidity and sagacity which reminds us of
the spirit in which Newton traced out the consequences of the
principle of universal gravitation. In the execution of his task,
indeed, Fresnel was forced upon several precarious assumptions,
which make, even yet, a wide difference between the theory of
gravitation and that of light. But the mode in which these were
confirmed by experiment, compels us to admire the happy apparent
boldness of the calculator.
The subject of _polarization by reflection_ was one of those which
seemed most untractable; but, by means of various artifices and
conjectures, it was broken up and subdued. Fresnel began with the
simplest case, the reflection of light polarized in the plane of
reflection; which he solved by means of the laws of collision of
elastic bodies. He then took the reflection of light polarized
perpendicularly to this plane; and here, adding to the general
mechanical principles a hypothetical assumption, that the
communication of the resolved motion parallel to the refracting
surface, takes place according to the laws of elastic bodies, he
obtains his formula. These results were capable of comparison with
experiment; and the comparison, when made by M. Arago, confirmed the
formulæ. They accounted, too, for Sir D. Brewster's law concerning
the polarizing angle (see Chap. vi.); and this could not but be
looked upon as a striking evidence of their having some real
foundation. Another artifice which MM. Fresnel and Arago employed,
in order to trace the effect of reflection upon common light, was to
use a ray polarized in a plane making half a right angle with the
plane of reflection; for the quantities of the oppositely[87\9]
polarized light in such an incident ray are equal, as they are in
common light; but the relative quantities of the oppositely
polarized light in the reflected ray are indicated by the new plane
of polarization; and thus these relative quantities become known for
the case of common light. The results thus obtained were also
confirmed by facts; and in this manner, all that was doubtful in the
process of Fresnel's reasoning, seemed to be authorized by its
application to real cases. {105}
[Note 87\9: It will be recollected all along, that _oppositely_
polarized rays are those which are polarized in two planes
_perpendicular_ to each other. See above, chap. vi.]
These investigations were published[88\9] in 1821. In succeeding
years, Fresnel undertook to extend the application of his formulæ to
a case in which they ceased to have a meaning, or, in the language
of mathematicians, became _imaginary_; namely, to the case of
internal reflection at the surface of a transparent body. It may
seem strange to those who are not mathematicians, but it is
undoubtedly true, that in many cases in which the solution of a
problem directs impossible arithmetical or algebraical operations to
be performed, these directions may be so interpreted as to point out
a true solution of the question. Such an interpretation Fresnel
attempted[89\9] in the case of which we now speak; and the result at
which he arrived was, that the reflection of light through a rhomb
of glass of a certain form (since called _Fresnel's rhomb_, would
produce a polarization of a kind altogether different from those
which his theory had previously considered, namely, that kind which
we have spoken of as _circular polarization_. The complete
confirmation of this curious and unexpected result by trial, is
another of the extraordinary triumphs which have distinguished the
history of the theory at every step since the commencement of
Fresnel's labors.
[Note 88\9: _An. Chim._ t. xvii.]
[Note 89\9: _Bullet. des Sc._ Feb. 1823.]
But anything further which has been done in this way, may be treated
of more properly in relating the verification of the theory. And we
have still to speak of the most numerous and varied class of facts
to which rival theories of light were applied, and of the
establishment of the undulatory doctrine in reference to that
department; I mean the phenomena of depolarized, or rather, as I
have already said, _di_polarized light.
_Sect._ 5.--_Explanation of Dipolarization by the Undulatory Theory._
WHEN Arago, in 1811, had discovered the colors produced by polarized
light passing through certain crystals,[90\9] it was natural that
attempts should be made to reduce them to theory. M. Biot, animated
by the success of Malus in detecting the laws of double refraction,
and Young, knowing the resources of his own theory, were the first
persons to enter upon this undertaking. M. Biot's theory, though in
the end displaced by its rival, is well worth notice in the history
of the subject. It was what he called the doctrine of _moveable
polarization_. He conceived that when the molecules of light pass
through {106} thin crystalline plates, the plane of polarization
undergoes an oscillation which carries it backwards and forwards
through a certain angle, namely, twice the angle contained between
the original plane of polarization and the principal section of the
crystal. The intervals which this oscillation occupies are lengths
of the path of the ray, very minute, and different for different
colors, like Newton's fits of easy transmission; on which model,
indeed, the new theory was evidently framed.[91\9] The colors
produced in the phenomena of dipolarization really do depend, in a
periodical manner, on the length of the path of the light through
the crystal, and a theory such as M. Biot's was capable of being
modified, and was modified, so as to include the leading features of
the facts as then known; but many of its conditions being founded on
special circumstances in the experiments, and not on the real
conditions of nature, there were in it several incongruities, as
well as the general defect of its being an arbitrary and unconnected
hypothesis.
[Note 90\9: See chap. ix.]
[Note 91\9: See MM. Arago and Biot's Memoirs, _Mém. Inst._ for 1811;
the whole volume for 1812 is a Memoir of M. Biot's (published 1814);
also _Mém. Inst._ for 1817; M. Biot's Mem. read in 1818, published
in 1819 and for 1818.]
Young's mode of accounting for the brilliant phenomena of
dipolarization appeared in the _Quarterly Review_ for 1814. After
noticing the discoveries of MM. Arago, Brewster, and Biot, he adds,
"We have no doubt that the surprise of these gentlemen will be as
great as our own satisfaction in finding that they are perfectly
reducible, like other causes of recurrent colors, to the general
laws of the interference of light which have been established in
this country;" giving a reference to his former statements. The
results are then explained by the interference of the ordinary and
extraordinary ray. But, as M. Arago properly observes, in his
account of this matter,[92\9] "It must, however, be added that Dr.
Young had not explained either in what circumstances the
interference of the rays can take place, nor why we see no colors
unless the crystallized plates are exposed to light previously
polarized." The explanation of these circumstances depends on the
laws of interference of polarized light which MM. Arago and Fresnel
established in 1816. They then proved, by direct experiment, that
when polarized light was treated so as to bring into view the most
marked phenomena of interference, namely, the bands of shadows;
pencils of light which have a common origin, and which are polarized
in the parallel planes, interfere completely, while those which are
{107} polarized in _opposite_ (that is, perpendicular,) planes do
not interfere at all.[93\9] Taking these principles into the
account, Fresnel explained very completely, by means of the
interference of undulations, all the circumstances of colors
produced by crystallized plates; showing the necessity of the
_polarization_ in the first instance; the _dipolarizing_ effect of
the crystal; and the office of the _analysing plate_, by which
certain portions of each of the two rays in the crystal are made to
interfere and produce color. This he did, as he says,[94\9] without
being aware, till Arago told him, that Young had, to some extent,
anticipated him.
[Note 92\9: _Enc. Brit._ Supp. art. _Polarization._]
[Note 93\9: _Ann. Chim._ tom. x.]
[Note 94\9: Ib. tom. xvii. p. 402.]
When we look at the history of the emission-theory of light, we see
exactly what we may consider as the natural course of things in the
career of a false theory. Such a theory may, to a certain extent,
explain the phenomena which it was at first contrived to meet; but
every new class of facts requires a new supposition,--an addition to
the machinery; and as observation goes on, these incoherent
appendages accumulate, till they overwhelm and upset the original
frame-work. Such was the history of the hypothesis of solid
epicycles; such has been the history of the hypothesis of the
material emission of light. In its simple form, it explained
reflection and refraction; but the colors of thin plates added to it
the hypothesis of fits of easy transmission and reflection; the
phenomena of diffraction further invested the particles with complex
hypothetical laws of attraction and repulsion; polarization gave
them sides; double refraction subjected them to peculiar forces
emanating from the axes of crystals; finally, dipolarization loaded
them with the complex and unconnected contrivance of moveable
polarization; and even when all this had been assumed, additional
mechanism was wanting. There is here no unexpected success, no happy
coincidence, no convergence of principles from remote quarters; the
philosopher builds the machine, but its parts do not fit; they hold
together only while he presses them: this is not the character of
truth.
In the undulatory theory, on the other hand, all tends to unity and
simplicity. We explain reflection and refraction by undulations;
when we come to thin plates, the requisite "fits" are already
involved in our fundamental hypothesis, for they are the length of
an undulation; the phenomena of diffraction also require such
intervals; and the intervals thus required agree exactly with the
others in magnitude, {108} so that no new property is needed.
Polarization for a moment checks us; but not long; for the direction
of our vibrations is hitherto arbitrary;--we allow polarization to
decide it. Having done this for the sake of polarization, we find
that it also answers an entirely different purpose, that of giving
the law of double refraction. Truth may give rise to such a
coincidence; falsehood cannot. But the phenomena become more
numerous, more various, more strange; no matter: the Theory is equal
to them all. It makes not a single new physical hypothesis; but out
of its original stock of principles it educes the counterpart of all
that observation shows. It accounts for, explains, simplifies, the
most entangled cases; corrects known laws and facts; predicts and
discloses unknown ones; becomes the guide of its former teacher,
Observation; and, enlightened by mechanical conceptions, acquires an
insight which pierces through shape and color to force and cause.
We thus reach the philosophical _moral_ of this history, so
important in reference to our purpose; and here we shall close the
account of the discovery and promulgation of the undulatory theory.
Any further steps in its development and extension, may with
propriety be noticed in the ensuing chapters, respecting its
reception and verification.
[2nd Ed.] [In the _Philosophy of the Inductive Sciences_, B. xi. ch.
iii. Sect. 11, I have spoken of the _Consilience of Inductions_ as
one of the characters of scientific truth. We have several striking
instances of such consilience in the history of the undulatory
theory. The phenomena of fringes of shadows and colored bands in
crystals _jump together_ in the Theory of Vibrations. The phenomena
of polarization and double refraction _jump together_ in the Theory
of Crystalline Vibrations. The phenomena of polarization and of the
interference of polarized rays _jump together_ in the Theory of
Transverse Vibrations.
The proof of what is above said of the undulatory theory is
contained in the previous history. This theory has "accounted for,
explained, and simplified the most entangled cases;" as the cases of
fringes of shadows; shadows of gratings; colored bands in biaxal
crystals, and in quartz. There are no optical phenomena more
entangled than these. It has "corrected experimental laws," as in
the case of M. Biot's law of the direction of polarization in biaxal
crystals. It has done this, "without making any new physical
hypothesis;" for the transverse direction of vibrations, the
different optical elasticities of crystals in different directions,
and (if it be adopted) the hypothesis of finite {109} intervals of
the particles (see chap. x. and hereafter, chap. xiii.), are only
limitations of what was indefinite in the earlier form of the
hypothesis. And so far as the properties of visible radiant light
are concerned, I do not think it at all too much to say, as M.
Schwerd has said, that "the undulation theory accounts for the
phenomena as completely as the theory of gravitation does for the
facts of the solar system."
This we might say, even if some facts were not yet fully explained;
for there were till very lately, if there are not still, such
unexplained facts with regard to the theory of gravitation,
presented to us by the solar system. With regard to the undulatory
theory, these exceptions are, I think, disappearing quite as rapidly
and as completely as in the case of gravitation. It is to be
observed that no presumption against the theory can with any show of
reason be collected from the cases in which classes of phenomena
remain unexplained, the theory having never been applied to them by
any mathematician capable of tracing its results correctly. The
history of the theory of gravitation may show us abundantly how
necessary it is to bear in mind this caution; and the results of the
undulatory theory cannot be traced without great mathematical skill
and great labor, any more than those of gravitation.
This remark applies to such cases as that of the _transverse fringes
of grooved surfaces_. The general phenomena of these cases are
perfectly explained by the theory. But there is an interruption in
the light in an oblique direction, which has not yet been explained;
but looking at what has been done in other cases, it is impossible
to doubt that this phenomenon depends upon the results of certain
integrations, and would be explained if these were rightly performed.
The phenomena of _crystallized surfaces_, and especially their
effects upon the plane of polarization, were examined by Sir D.
Brewster, and laws of the phenomena made out by him with his usual
skill and sagacity. For a time these were unexplained by the theory.
But recently Mr. Mac Cullagh has traced the consequences of the
theory in this case,[95\9] and obtained a law which represents with
much exactness, Sir D. Brewster's observation.
[Note 95\9: Prof. Lloyd's _Report, Brit. Assoc._ 1834, p. 374.]
The phenomena which Sir D. Brewster, in 1837, called a _new property
of light_, (certain appearances of the spectrum when the pupil of
the eye is half covered with a thin glass or crystal,) have been
explained by Mr. Airy in the _Phil. Trans._ for 1840.
Mr. Airy's explanation of the phenomena termed by Sir D. {110}
Brewster a _new property of light_, is completed in the
_Philosophical Magazine_ for November, 1846. It is there shown that
a dependence of the breadth of the bands upon the aperture of the
pupil, which had been supposed to result from the theory, and which
does not appear in the experiment, did really result from certain
limited conditions of the hypothesis, which conditions do not belong
to the experiment; and that when the problem is solved without those
limitations, the discrepance of theory and observation vanishes; so
that, as Mr. Airy says, "this very remarkable experiment, which long
appeared inexplicable, seems destined to give one of the strongest
confirmations to the Undulatory Theory."
I may remark also that there is no force in the objection which has
been urged against the admirers of the undulatory theory, that by
the fulness of their assent to it, they discourage further
researches which may contradict or confirm it. We must, in this
point of view also, look at the course of the theory of gravitation
and its results. The acceptance of that theory did not prevent
mathematicians and observers from attending to the apparent
exceptions, but on the contrary, stimulated them to calculate and to
observe with additional zeal, and still does so. The acceleration of
the Moon, the mutual disturbances of Jupiter and Saturn, the motions
of Jupiter's Satellites, the effect of the Earth's oblateness on the
Moon's motion, the motions of the Moon about her own centre, and
many other phenomena, were studied with the greater attention,
_because_ the general theory was deemed so convincing: and the same
cause makes the remaining exceptions objects of intense interest to
astronomers and mathematicians. The mathematicians and optical
experimenters who accept the undulatory theory, will of course
follow out their conviction in the same manner. Accordingly, this
has been done and is still doing, as in Mr. Airy's mathematical
investigation of the effect of an annular aperture; Mr. Earnshaw's,
of the effect of a triangular aperture; Mr. Talbot's explanation of
the effect of interposing a film of mica between a part of the pupil
and the pure spectrum, so nearly approaching to the phenomena which
have been spoken of as a new Polarity of Light; besides other labors
of eminent mathematicians, elsewhere mentioned in these pages.
The phenomena of the _absorption_ of light have no especial bearing
upon the undulatory theory. There is not much difficulty in
explaining the _possibility_ of absorption upon the theory. When the
light is absorbed, it ceases to belong to the theory. {111}
For, as I have said, the theory professes only to explain the
phenomena of _radiant visible_ light. We know very well that light
has other bearings and properties. It produces chemical effects. The
optical polarity of crystals is connected with the chemical polarity
of their constitution. The natural colors of bodies, too, are
connected with their chemical constitution. Light is also connected
with heat. The undulatory theory does not undertake to explain these
properties and their connexion. If it did, it would be a Theory of
Heat and of Chemical Composition, as well as a Theory of Light.
Dr. Faraday's recent experiments have shown that the magnetic
polarity is directly connected with that optical polarity by which
the plane of polarization is affected. When the lines of magnetic
force pass through certain transparent bodies, they communicate to
them a certain kind of circular polarizing power; yet different from
the circular polarizing power of quartz, and certain fluids
mentioned in chapter ix.
Perhaps I may be allowed to refer to this discovery as a further
illustration of the views I have offered in the _Philosophy of the
Inductive Sciences_ respecting the _Connexion of Co-existent
Polarities_. (B. v. Chap. ii.)]