F. (François) Arago.

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not only in the general principles, but even in the
details of this science. It must be observed, however,
that all the deductions of Malus, even the most plausible
at the present day, can be subverted by a single word ;
it sutfices to cite, in contradiction to all the phenomena

* The masonry encasing and supporting the earthworks in a for-

136 MALUS.

which our friend alleges, the instance of the light which
is engendered in a vacuum, by the aid of the voltaic
current, passed through simple substances, such as car-
bon, platinum, &c.

In the second part of the memoir Malus seeks to
establish that the different natures of various lights only
differ from each other in the greater or less proportion
of caloric which they contain. The red light would
thus be the most heating, the violet the least so, which
agrees with experiment. According to a singular opinion
professed by the author, all rays, if possessing a certain
high intensity, ought to produce the sensation of white-

The third part of the work is devoted to mechanical
consequences which result by analysis from the supposi-
tions explained in the first two sections. It may suffice
to say, that the author finds, like all the partisans of the
system of emission, that the velocity of light ought to be
greater in water than in air : every one therefore will see
how superfluous it would be now to go into a discussion
of the details of such a subject.

* The "singular" opinion here ascribed to Mains is perhaps not
altogether without foundation, at least in some cases. It is certain
that while the prismatic spectrum of the white light of the clouds
present a clear yellow and green portion, that same portion, when the
direct rays of the sun are substituted, appears to the eye intensely
brilliant and white. And it is far from certain that in some other ex-
periments, which have been the occasion of some little controversy
and where the colour of certain parts of the spectrum has appeared
to undergo a change, the intensity of the light reaching the eye may
not be concerned. In fact, the sensation of colour is one so entirely de-
pendent on unknown physiological causes, that we can hardly venture
to predict what the result may be on different individual eyes, though
all the optical conditions may be precisely the same. It may not be
altogether without a bearing on this subject, to remark the extremely
contradictory statements made by different observers as to the colour
of intenselv brilliant meteors. — Trunslntor.


The memoir of which I am speaking was destined for
the Institute of Egypt. I find in fact, in a letter from
Malus to Lancret, the following passage : —

" I send you, my dear Lancret, the work of which I
have already spoken : mark out for me those things in it
which any one might call repetitions of what has been
already said, or which are useless ; if, after this expurga-
tion, it sliould be reduced to zero, we will put it aside,
and there will be no more question about it."

It is just to remark, after the critique from which I
could not abstain when I considered that my task was
not that of a panegyrist but of a biographer aiming at the
truth, that the third part of the memoir was written be-
fore the publication of the fourth volume of the Mecan-
ique Celeste, in which the same subject is treated with
the greatest care. I would add that no army in the
world ever before counted in its ranks an officer who
occupied himself in the spare hours of advanced_ posts
with researches so complete and so profound. The truth
of this remark is not affiscted by the recollection which it
brings up of the expedition of Alexander. It is true,
men of science, at the recommendation of Aristotle, then
accompanied the great general : but their mission was
solely to collect the scientific achievements of the con-
quered nations, and not to make advances in the sciences
by their own labours. This difference, altogether in
favour of the French army, deserves, I think, to be here

I see by a letter of Lancret, of the 14th Vendemiaire,t

* If this comparison were worth carrying out. the author might
have added that the men of science in Alexander's expedition were
not officers of (he army charged at the same time with onerous and
hazardous duties, but leisurely investigators, having no other occupa-
tion. — Translator.

t October 5, 1800.

138 MALUS.

an. IX., that Malus was occupied theoi*eticallj with that
most important meteorological question, the distribution
of heat in different climates. I have never been able to
find what has become of this work.


On the 20th April, 1807, Malus presented to the first
class of the Institute, a treatise on analytical optics, in
which he treats of rays of light by geometry of three

The choice of academicians to whose examination the
work was entrusted, sufficiently indicates the reputation
which the author had already acquired. These commis-
sioners were Lagrange, Laplace, Monge, and Lacroix.
The report of this distinguished commission was pre-
sented by Lacroix, and bears date the 19 th October,
1807. ■

The author of the memoir examines the nature and
relative position of the surfaces formed by straight lines
successively intersecting one another according to given
laws. After having deduced from his researches some
general theorems, of a very remarkable kind, he pro-
ceeds to make an application of them to the case of rays
of light proceeding in similar directions, either by reflex-
ion or by refraction. He thus generalizes the theory
of plane caustics, formerly broached by Tschirnhausen.
Among the curious results which he deduces from his
formulas, we Avill merely quote the following : —

" Reflexion and refraction furnish sometimes optical
images which are erect in one of their dimensions and
inverted in the other."

The report, for which I will not presume to substitute
my personal opinion, concludes in these terms : —


"To apply thus, without any limitation on its general-
ity, calculation to phenomena ; — to deduce, from a single
consideration of a very general kind, all the solutions
■which before were only obtained from particular consid-
erations, — is truly to write a treatise on analytical optics,
which, concentrating the whole science in a single point
of view, cannot but contribute to the extension of its do-

The Academy decided (which is the highest degree of
approbation it can bestow) that the memoir of Malus
should be printed in the liecueil des Savants Etrangers*


On the 16th November, 1807, Malus presented to the
Academy a memoir in which he treats a point of optics
of great importance, a question, in fact, involving no less

* Malus's analytical theory contained in bis Traite c? Opiique, is pre-
fixed to his prize memoir on Double Refraction, Paris, 1810.

The ordinary deviations by reflexion or refraction which rays un-
dergo on impinging on given surfaces, may be investigated in all the
simpler cases by means of elementary geometrical constructions, lead-
ing to the theory of foci, caustics, &c. But more general investiga-
tions of the same kind have been pursued by considering the algebraic
equations of rays undergoing such deviations. This higher generaliza-
tion leads to, and includes, the same results. An excellent discussion
of the subject treated in this point of view will be found in Dr. Lloyd's
Treatise on Light and Vision. It is a still higher generalization of this
kind which was followed out by Malus. The reader who is desirous
of seeing a condensed abstract of the leading mathematical principles
involved, is referred to a brief but luminous summary drawn up by
the Eev. A. Neate, M. A., and inserted in Professor Powell's Elemen-
tary Treatise on Optics, p. 71, Oxford, 1833. But the entire subject
has been treated by a far higher analysis with extreme generality, and
by a new and powerful principle of his own, by Sir W. E. Hamilton,
in his essay on the Theory of Systems of Rays. Mem. of R. Irish Acad-
emy, vols. XV. and xvi., and Supplement, vol. xviii. — Translator.

140 MALUS.

than the grounds for a decision between the claims of the
two rival theories of light.

The celebrated physicist Wollaston, some years before,
had proposed a method by means of which to deduce the
refractive power of all substances whether transparent or
opaque. This method rests on the determination of the
angle under which these substances applied immediately
in contact with one of the surfaces of a prism of glass,
through which we look at them, begin to cease to be

Now according to the theory of reflexion* expounded

* To render what follows intelligible, many readers may find it per-
haps desirable if we here exi^lain, very briefly, the view of ordinary
reflexion and refraction of light as explained respectively by the emis-
sion and the loave theories.

On the former a molecule of light resembles an elastic body, which
if projected obliquely against a hard plane surface, by the principles
of mechanics rebounds at ore anale equal to that at which it im-

In refraction the investigation is more difficult: a molecule of light
is here supposed to enter, projected with great velocity, among the
molecules of the refracting transparent medium which are at such
relative distances as to allow it freely to pass among them; but at its
first entry among them it is of course aitracled by them ; it then be-
comes a problem of dynamics, requiring the aid of the higher mathe-
matics, to determine what will be the path which it will pursue under
their influence. In general it is clear, that under these united attrac-
tions urging it on, its velocity will be accelerated: but to go into the
complete solution, would be beyond the limits of a note. It was fully
investigated by Newton {Principia, lib. i. sect. xiv. prop. 94), where
he demonstrates that on these principles the deviation of the refracted
ray will follow the law that the sines of the angles of incidence and re-
fraction are in a constant ratio.

Similar investigations have been pursued by Laplace, more espec-
ially with regard to atmospherical refraction, the atmosphere being
supposed to consist of strata of different densities. {Mec. Celeste, vol.
iv. liv. X. ch. i. 2, 3.)

On the wave hypothesis, the explanation admits of a very simple
kind of illustration.



in the^^lOth hook of the Mecaniqne Celeste, and founded
on the corpupfidar hypothesis, tlie formulas would be dif-
ferent foi" opaque and for transparent bodies. It is on

A set of waves propagnted circularly from any source, when they
get to a considerable distance, may be regarded as proceeding in par-
allel planes. In all cases, the portions of circles or spheres which are
their true form have a common tangent which marks what is called
the "front" of the wave.

But whenever waves encounter any kind of obstacle, or enier any
new medium, then, from and round each point of such encounter, a new
set of spherical waves begins to spread. In denser media these new
waves spread more slowly than in rarer, but when the obstacle is still
surrounded by the same meiium, then the velocity is unaltered.

On these principles the ordinary laws of reflexion and refraction are
proved on the theory of waves.

In reflexion, if parallel waves u u' follow at equal intervals A, u im-

pinging on the surface at o, will cause a new circular wave to spread
backwards from that point as a centre; when the next wave «' im-
pinges at o', it will do the same, and so on in succession. But when
the wave from o' has spread to a radius =2., that from o will have
spread to a radius =2/1, and so on. Hence to these contemporaneous
circular waves drawing a common tangent v v' t this will be the front
of the reflected waves, and the radii to the points of contact o v, oi vi,
will give the inclination of the refected rays, which is easily seen to be
equal to that of the incident, since ol v' =oi u=2., and a v—2ol v, whence
o'—oi t, and the triangles upon these equal bases being right-angled,
the angle v t o=u o c', or the angle of incidence, is equal to that of



this point then, they would say, that WoUaston was de-
ceived. The object which Malus proposed in his memoir
was to submit this point to a decisive experimental test.
He chose a substance, beeswax, whose refractive power
could be measured in the transparent state, and in the
opaque state by the method of Wollaston. He applied
to the angles of disappearance corresponding to these two
conditions, and sufficiently different one from the other,
the formulas of the Mecanique Celeste, and he found there
would result refractive powers perfectly identical. This

For refraction ; by an analogous construction, tlie circles which

spi'ead in the denser medium are smaller than those in the first, the
radii being diminished in the ratio of the velocities or inversely as the
densities. Thus when the new wave originating at o' has spread to »/,
that from o will have spread to doiible the same radius at v. The com-
mon tangent or front of the refracted waves will be inclined at an angle
t V, which is easilj^ determined by drawing the pai-allel through i of
the incident light, whence we have {i and r being the angles of inci-
dence and :-efraction) m t=o t sin. i, and o v=o t sin. r ; but o v and u t
being the radii of waves in the two media, are in the constant ratio of
the densities =fJ.; hence sm. i =(" sin. »•, which is the experimental
law of refraction.



identity of the refractive powers of wax, wben transparent
and when opaque, which seemed to be a necessary result,
appeared both to tlie author, to Laplace, and to all the

The law of refraction may also be more briefly deduced thus : tak-
ing the fronts of the incident and refracted rays perpendicular to their

directions, their inclinations will be determined by the relative veloci-
ties with which those fronts advance; and while the incident front has
advanced through a space d, that of the refracted will have advanced
through cZj proportional to their velocities; or,
d V

But geometrically for any breadth,

d = b sin. i

dT = b sin. r.

fi = ■ ,

sin. »•
which is the law of refraction.

This method, though in a less concise form, is given by Mr. Power
( On Absorption of Rays, ^-c, Pldlos. Trans. 1854, part i.,) who never-
theless calls in question the principle of the assumption that the front
of the rays is stricth'- perpendicular to their direction, and proposes a
more general view: from which, without any assumption as to the
nature or law of refraction, he shows that the formula of the sines is
directly deducible from his analysis. Objections, however, have been
raised against his reasoning. — Translator.

144 MALUS.

mathematicians and physicists of the Emission School in
Europe, to afford a mathematical proof of the truth of
the emission theory. It is assuredly a singular thing that
there should be this perfect identity of refractive powers
calculated from angles of disappearance differing from
each other, and according to formulas very dissimilar be-
tween themselves.

But what proof was there that the refractive powers
ought to be identical? Ought we to suppose that the
change from the solid to the fluid state in any substance
would be without influence on its refractive power?
Might we not cite cases in which heat modifies the re-
fractive power of bodies independently of their density ?
Again, were the temperature of the wax and its density
well ascertained at the moment of the experiment such
as Malus was obliged to make it ? Besides, would it be
strange to suppose that within those limits where the
action of bodies on light opei'ates, there are no sub-
stances truly opaque !

Now that the system of emission is overthrown without
hope of restoration, I endeavour to recall all the circum-
stances by which Malus might possibly have been misled.
But, for my own part, I feel sure that I do not deceive
myself in affirming that the memoir of which we are
speaking offers a new proof of the mathematical spirit
and experimental talent which Malus possessed in so
high a degree. We ought only to regret that the
conclusions in the report were so explicit that they
represented the atomic theory of light as completely
established ; and that such a decision, emanating from
individuals so competent as Laplace, Haiiy, and Gay-
Lussac, may perhaps have contributed to alienate our
illustrious associate from that experimental path which



Fresnel a few years afterwards showed to be so astonish-
ingly fruitful in results.*

* In the remarks here made by Arago on Malus's investic;ation of
the refractive powers of solid and liquid wax, there appears some lit-
tle obscurit}' of statement, and a degree of importance attached to the
result as decisive between the rival theories, which it does not appear
to deserve.

Perhaps for the general reader a few words explanatory of the
method may be necessary, in order to see the general bearing of the

When a ray passes out of a denser medium m into a rarer n, the
angle of refraction r will be greater than that of incidence /, according
to the well-known law of the sines, which liere becomes sin r=u
sin i. But /i being constant for the same two substances, there is a

certain limit to i when sin r=l or r=90° or sin i=- that is, the

refracted ray coincides with the bounding surface of the media, or it
ceases to be refracted: and if i exceed tiiis value, sin r would be
greater than unity, which is impossible, or the ray cannot emerge from
the denser medium, but must remain wholly within it. This alone,
however, does not prove that it will be reflected. Experiment, how-

ever, shows that it is, and the precise angle i at whicli this beo-Ins to

1 °

take place, or when sin i= - for any_;joJ;' of media, can bo easily and

accurately determined; thus ji is found for that pair of substances
but It is the compound ratio of the separate refractive powers of each
out of vacuum or air; if, therefore, one of these is known, the other
is deduced.

On this principle Dr. Wollaston's method was founded (P/«7. Trans.


146 MALUS.

1802). Any substance n, of less refractive power than glass in opti-
cal contact with the base of a glass prism m, can be seen by an ej'e

at e at any incidence within the limit just mentioned, or while the ray
i entering the other side of the prism and impinging on its base, is in-
capable of being refracted out at the base, and therefore reflected from
within; but as soon as this limit is exceeded, or the ray is refracted
out at the base, then n ceases to be visible at e. The exact incidence
or " critical angle " at ■which this takes place, is measured by an
appropriate apparatus, and the refractive index for n deduced, that of
the prism being known, a series of substances being applied in suc-
cession, whether transparent or opaque. Dr. Wollaston in this way
determined their refractive indices. As the different primary rays
have indices a little differing, and which are greatest for red light, Dr.
Young remarked that the limit thus found applies in strictness to the
extreme red ray.

In this way Dr. Wollaston found the refractive indices as follows: —
White wax, boiling - - . 1.542

Ditto cold - - - - 1.535

In the same way ]\Ialus found

Waxat 14° Reaum. (=63^ Fahr.) - - 1.5123

Ditto melting ... 1.4503

Ditto boiling - - - 1.4416

(These numbers are all lower than the former, probably from a dif-
ferent sort of wax being used.)

Dr. Wollaston, in applying the simple calculation above indicated
to the observed angles, did not question the vei-y natural assumption,
that the same Ibrmulas would apply to the observed angles equally,
whether the substance was opaque or transparent, solid or fluid.

Laplace, in a theoretical investigation founded on certain consider-
ations derived from the molecular theory, framed his formulas on the
assumption that the conditions were different for opaque and for
transparent bodies, and even for the same substances in the two states
respectively. The que;.tion at issue was the truth of this assump- -
tion, though it must be confessed that little appears in the tenth book



On the 4tli January, 1808, the Academy proposed, as
the subject for a prize in physical science to be decided
in 1810, the following question : —

" To give a mathematical theory, confirmed by experi-
ment, of the double refraction which light undergoes in
passing through diiferent ci'ystallized bodies."

The memoir of Malus received the prize. Doubtless
fearing lest he should be forestalled by some of the com-
petitors, in the discovery of the singular properties of
light which he had observed, this eminent physicist com-
municated the most essential parts of his researches to
the Academy on the 12th December, 1808, without
waiting for the period at which, according to the pro-
gramme, the competition was to be closed. It is then to
the end of the year 1808 that the immortal discoveries
belong of which I proceed immediately to give you an
analysis. The commission appointed to judge of the com-
petitors was composed of Lagrange, Haiiy, Gay-Lussac,

of the Mec. Celeste by which this conclusion can be considered as

Malus observed by Wollaston's method the angles at which the dis-
appearance took place in wax, solid and in fusion. Tliese angles were
different; and calculated in the usual way, the indices of refraction
resulted different also (as seen in the above tabular view).

The same observed angles, however, calculated b}' Laplace's for-
mula gave the resulting index the same in both cases.

Now Laplace, Malus, and the emissiouists, considered the identity
of refractive power thus resulting to be a necessary truth — why so, we
do not see; it is obviously, at best, a mere consequence of the assump-
tion made at the first. The result is no proof of its truth, and decides
nothing either way. Arago's laboured remarks therefore seem super-
fluous. — Translator.

148 MALUS.

and Biot. The report was presented by Lagrange, and
thus nothing was wanting duly to signalize the important
discovery of Malus.


. We must go back to Erasmus Bartholimus to find the
first observations relative to the existence of double
refraction in Iceland spar, also called calc spar, or rhom-
boidal carbonate of lime. Huyghens had occupied him-
self with the study of these phenomena, and pointed out
a geometrical construction of a very simple and elegant
kind by which we can determine, in all directions and at
all incidences, the position of the extraordinary ray rela-
tive to the ray properly called the ordinary ray, whose
position is determined by the well-known law of the
sines, made known by Descartes. Huyghens arrived at
the discovery of this construction by means of an ellip-
soid, which, as he tells his readers, he derived from con-
siderations borrowed from the theory of waves.

The reporter of the Academy on Malus's memoir of
the 12th December, 1808, entitled Memoir on a Prop-
erty of Light reflected hy transparent Bodies, who was
no other than Laplace, wished that Huyghens had been
contented to have given his law as the result of experi-
ence only. But I may be permitted to ask. Is not the
hatred of theory carried too far when it leads to the sug-
gestion of dissimulation or the want of sincerity ?

Newton contended for substituting other rules instead
of that of Huyghens ; but these have not been found
conformable to facts.

Among modern observers, "Wollaston was the first who
established the truth of the principles laid down by the
Dutch philosopher. To make this verification he availed


himself of the ingenious method by which he found the
index of refraction by means of total reflexion. It ap-
pears that in 1808 these verifications had not appeared
sufficient to the physicists of the Academy of Sciences,
since they proposed the question as the subject of a prize
for experimenters. However this may have been, Malus
translated the construction of Huyghens into analytical

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