the velocity of the rays of light amounts to 77,000 leagues (192,000
English miles) per second, the philosophers who profess to explain the
force of attraction by the impulsive energy of a fluid, will see what
prodigious velocities they must satisfy.
The reader cannot fail again to remark the sagacity with which Laplace
singled out the phenomena which were best adapted for throwing light
upon the most obscure points of celestial physics; nor the success with
which he explored their various parts, and deduced from them numerical
conclusions in presence of which the mind remains confounded.
The author of the _Mécanique Céleste_ supposed, like Newton, that light
consists of material molecules of excessive tenuity and endued in empty
space with a velocity of 77,000 leagues in a second. However, it is
right to warn those who would be inclined to avail themselves of this
imposing authority, that the principal argument of Laplace, in favour of
the system of emission, consisted in the advantage which it afforded of
submitting every question to a process of simple and rigorous
calculation; whereas, on the other hand, the theory of undulations has
always offered immense difficulties to analysts. It was natural that a
geometer who had so elegantly connected the laws of simple refraction
which light undergoes in its passage through the atmosphere, and the
laws of double refraction which it is subject to in the course of its
passage through certain crystals, with the action of attractive and
repulsive forces, should not have abandoned this route, before he
recognized the impossibility of arriving by the same path, at plausible
explanations of the phenomena of diffraction and polarization. In other
respects, the care which Laplace always employed, in pursuing his
researches, as far as possible, to their numerical results, will enable
those who are disposed to institute a complete comparison between the
two rival theories of light, to derive from the _Mécanique Céleste_ the
materials of several interesting relations.
Is light an emanation from the sun? Does this body launch out
incessantly in every direction a part of its own substance? Is it
gradually diminishing in volume and mass? The attraction exercised by
the sun upon the earth will, in that case, gradually become less and
less considerable. The radius of the terrestrial orbit, on the other
hand, cannot fail to increase, and a corresponding effect will be
produced on the length of the year.
This is the conclusion which suggests itself to every person upon a
first glance at the subject. By applying analysis to the question, and
then proceeding to numerical computations, founded upon the most
trustworthy results of observation relative to the length of the year in
different ages, Laplace has proved that an incessant emission of light,
going on for a period of two thousand years, has not diminished the mass
of the sun by the two-millionth part of its original value.
Our illustrious countryman never proposed to himself any thing vague or
indefinite. His constant object was the explanation of the great
phenomena of nature, according to the inflexible principles of
mathematical analysis. No philosopher, no mathematician, could have
maintained himself more cautiously on his guard against a propensity to
hasty speculation. No person dreaded more the scientific errors which
the imagination gives birth to, when it ceases to remain within the
limits of facts, of calculation, and of analogy. Once, and once only,
did Laplace launch forward, like Kepler, like Descartes, like Leibnitz,
like Buffon, into the region of conjectures. His conception was not then
less than a cosmogony.
All the planets revolve around the sun, from west to east, and in planes
which include angles of inconsiderable magnitude.
The satellites revolve around their respective primaries in the same
direction as that in which the planets revolve around the sun, that is
to say, from west to east.
The planets and satellites which have been found to have a rotatory
motion, turn also upon their axes from west to east. Finally, the
rotation of the sun is directed from west to east. We have here then an
assemblage of forty-three movements, all operating in the same
direction. By the calculus of probabilities, the odds are four thousand
millions to one, that this coincidence in the direction of so many
movements is not the effect of accident.
It was Buffon, I think, who first attempted to explain this singular
feature of our solar system. Having wished, in the explanation of
phenomena, to avoid all recourse to causes which were not warranted by
nature, the celebrated academician investigated a physical origin of the
system in what was common to the movements of so many bodies differing
in magnitude, in form, and in distance from the principal centre of
attraction. He imagined that he discovered such an origin by making
this triple supposition: a comet fell obliquely upon the sun; it pushed
before it a torrent of fluid matter; this substance transported to a
greater or less distance from the sun according to its mass formed by
concentration all the known planets.
The bold hypothesis of Buffon is liable to insurmountable difficulties.
I proceed to indicate, in a few words, the cosmogonic system which
Laplace substituted for that of the illustrious author of the _Histoire
Naturelle_.
According to Laplace, the sun was at a remote epoch the central nucleus
of an immense nebula, which possessed a very high temperature, and
extended far beyond the region in which Uranus revolves in the present
day. No planet was then in existence.
The solar nebula was endued with a general movement of revolution
directed from west to east. As it cooled it could not fail to experience
a gradual condensation, and, in consequence, to rotate with greater and
greater rapidity. If the nebulous matter extended originally in the
plane of the equator as far as the limit at which the centrifugal force
exactly counterbalanced the attraction of the nucleus, the molecules
situate at this limit ought, during the process of condensation, to
separate from the rest of the atmospheric matter and form an equatorial
zone, a ring revolving separately and with its primitive velocity. We
may conceive that analogous separations were effected in the higher
strata of the nebula at different epochs, that is to say, at different
distances from the nucleus, and that they give rise to a succession of
distinct rings, included almost in the same plane and endued with
different velocities.
This being once admitted, it is easy to see that the indefinite
stability of the rings would have required a regularity of structure
throughout their whole contour, which is very improbable. Each of them
accordingly broke in its turn into several masses, which were plainly
endued with a movement of rotation, coinciding in direction with the
common movement of revolution, and which in consequence of their
fluidity assumed spheroidal forms.
In order, then, that one of those spheroids might absorb all the others
belonging to the same ring, it will be sufficient to assign to it a mass
greater than that of any other spheroid.
Each of the planets, while in the vaporous condition to which we have
just alluded, would manifestly have a central nucleus gradually
increasing in magnitude and mass, and an atmosphere offering, at its
successive limits, phenomena entirely similar to those which the solar
atmosphere, properly so called, had exhibited. We here witness the birth
of satellites, and that of the ring of Saturn.
The system, of which I have just given an imperfect sketch, has for its
object to show how a nebula endued with a general movement of rotation
must eventually transform itself into a very luminous central nucleus (a
sun) and into a series of distinct spheroidal planets, situate at
considerable distances from each other, revolving all around the central
sun in the direction of the original movement of the nebula; how these
planets ought also to have movements of rotation operating in similar
directions; how, finally, the satellites, when any of such are formed,
cannot fail to revolve upon their axes and around their respective
primaries, in the direction of rotation of the planets and of their
movement of revolution around the sun.
We have just found, conformably to the principles of mechanics, the
forces with which the particles of the nebula were originally endued, in
the movements of rotation and revolution of the compact and distinct
masses which these particles have brought into existence by their
condensation. But we have thereby achieved only a single step. The
primitive movement of rotation of the nebula is not connected with the
simple attraction of the particles. This movement seems to imply the
action of a primordial impulsive force.
Laplace is far from adopting, in this respect, the almost universal
opinion of philosophers and mathematicians. He does not suppose that the
mutual attractions of originally immovable bodies must ultimately reduce
all the bodies to a state of rest around their common centre of gravity.
He maintains, on the contrary, that three bodies, in a state of rest,
two of which have a much greater mass than the third, would concentrate
into a single mass only in certain exceptional cases. In general, the
two most considerable bodies would unite together, while the third would
revolve around their common centre of gravity. Attraction would thus
become the cause of a sort of movement which would seem to be explicable
solely by an impulsive force.
It might be supposed, indeed, that in explaining this part of his system
Laplace had before his eyes the words which Rousseau has placed in the
mouth of the vicar of Savoy, and that he wished to refute them: "Newton
has discovered the law of attraction," says the author of _Emile_, "but
attraction alone would soon reduce the universe to an immovable mass:
with this law we must combine a projectile force in order to make the
celestial bodies describe curve lines. Let Descartes reveal to us the
physical law which causes his vortices to revolve; and let Newton show
us the hand which launched the planets along the tangents of their
orbits."
According to the cosmogonic ideas of Laplace, comets did not originally
form part of the solar system; they are not formed at the expense of the
matter of the immense solar nebula; we must consider them as small
wandering nebulæ which the attractive force of the sun has caused to
deviate from their original route. Such of those comets as penetrated
into the great nebula at the epoch of condensation and of the formation
of planets fell into the sun, describing spiral curves, and must by
their action have caused the planetary orbits to deviate more or less
from the plane of the solar equator, with which they would otherwise
have exactly coincided.
With respect to the zodiacal light, that rock against which so many
reveries have been wrecked, it consists of the most volatile parts of
the primitive nebula. These molecules not having united with the
equatorial zones successively abandoned in the plane of the solar
equator, continued to revolve at their original distances, and with
their original velocities. The circumstance of this extremely rare
substance being included wholly within the earth's orbit, and even
within that of Venus, seemed irreconcilable with the principles of
mechanics; but this difficulty occurred only when the zodiacal substance
being conceived to be in a state of direct and intimate dependence on
the solar photosphere properly so called, an angular movement of
rotation was impressed on it equal to that of the photosphere, a
movement in virtue of which it effected an entire revolution in
twenty-five days and a half. Laplace presented his conjectures on the
formation of the solar system with the diffidence inspired by a result
which was not founded upon calculation and observation.[39] Perhaps it
is to be regretted that they did not receive a more complete
development, especially in so far as concerns the division of the matter
into distinct rings; perhaps it would have been desirable if the
illustrious author had expressed himself more fully respecting the
primitive physical condition, the molecular condition of the nebula at
the expense of which the sun, planets, and satellites, of our system
were formed. It is perhaps especially to be regretted that Laplace
should have only briefly alluded to what he considered the obvious
possibility of movements of revolution having their origin in the action
of simple attractive forces, and to other questions of a similar nature.
Notwithstanding these defects, the ideas of the author of the _Mécanique
Céleste_ are still the only speculations of the kind which, by their
magnitude, their coherence, and their mathematical character, may be
justly considered as forming a physical cosmogony; those alone which in
the present day derive a powerful support from the results of the recent
researches of astronomers on the nebulæ of every form and magnitude,
which are scattered throughout the celestial vault.
In this analysis, we have deemed it right to concentrate all our
attention upon the _Mécanique Céleste_. The _Système du Monde_ and the
_Théorie Analytique des Probabilités_ would also require detailed
notices.
The _Exposition du Système du Monde_ is the _Mécanique Céleste_ divested
of the great apparatus of analytical formulæ which ought to be
attentively perused by every astronomer who, to use an expression of
Plato, is desirous of knowing the numbers which govern the physical
universe. It is in the _Exposition du Systéme du Monde_ that persons
unacquainted with mathematical studies will obtain an exact and
competent knowledge of the methods to which physical astronomy is
indebted for its astonishing progress. This work, written with a noble
simplicity of style, an exquisite propriety of expression, and a
scrupulous accuracy, is terminated by a sketch of the history of
astronomy, universally ranked in the present day among the finest
monuments of the French language.
A regret has been often expressed, that Cæsar, in his immortal
_Commentaries_, should have confined himself to a narration of his own
campaigns: the astronomical commentaries of Laplace ascend to the origin
of communities. The labours undertaken in all ages for the purpose of
extracting new truths from the heavens, are there justly, clearly, and
profoundly analyzed; it is genius presiding as the impartial judge of
genius. Laplace has always remained at the height of his great mission;
his work will be read with respect so long as the torch of science shall
continue to throw any light.
The calculus of probabilities, when confined within just limits, ought
to interest, in an equal degree, the mathematician, the experimentalist,
and the statesman. From the time when Pascal and Fermat established its
first principles, it has rendered and continues daily to render services
of the most eminent kind. It is the calculus of probabilities, which,
after having suggested the best arrangements of the tables of population
and mortality, teaches us to deduce from those numbers, in general so
erroneously interpreted, conclusions of a precise and useful character:
it is the calculus of probabilities which alone can regulate justly the
premiums to be paid for assurances; the reserve funds for the
disbursement of pensions, annuities, discounts, &c.: it is under its
influence that lotteries, and other shameful snares cunningly laid for
avarice and ignorance, have definitively disappeared. Laplace has
treated these questions, and others of a much more complicated nature,
with his accustomed superiority. In short, the _Théorie Analytique des
Probabilités_ is worthy of the author of the _Mécanique Céleste_.
A philosopher, whose name is associated with immortal discoveries, said
to his audience who had allowed themselves to be influenced by ancient
and consecrated authorities, "Bear in mind, Gentlemen, that in questions
of science the authority of a thousand is not worth the humble reasoning
of a single individual." Two centuries have passed over these words of
Galileo without depreciating their value, or obliterating their truthful
character. Thus, instead of displaying a long list of illustrious
admirers of the three beautiful works of Laplace, we have preferred
glancing briefly at some of the sublime truths which geometry has there
deposited. Let us not, however, apply this principle in its utmost
rigour, and since chance has put into our hands some unpublished letters
of one of those men of genius, whom nature has endowed with the rare
faculty of seizing at a glance the salient points of an object, we may
be permitted to extract from them two or three brief and characteristic
appreciations of the _Mécanique Céleste_ and the _Traité des
Probabilités_.
On the 27th Vendemiaire in the year X., General Bonaparte, after having
received a volume of the _Mécanique Céleste_, wrote to Laplace in the
following terms:--"The first _six months_ which I shall have at my
disposal will be employed in reading your beautiful work." It would
appear that the words, the first _six months_, deprive the phrase of the
character of a common-place expression of thanks, and convey a just
appreciation of the importance and difficulty of the subject-matter.
On the 5th Frimaire in the year XI., the reading of some chapters of the
volume, which Laplace had dedicated to him, was to the general "a new
occasion for regretting, that the force of circumstances had directed
him into a career which removed him from the pursuit of science."
"At all events," added he, "I have a strong desire that future
generations, upon reading the _Mécanique Céleste_, shall not forget the
esteem and friendship which I have entertained towards its author."
On the 17th Prairial in the year XIII., the general, now become emperor,
wrote from Milan: "The _Mécanique Céleste_ appears to me destined to
shed new lustre on the age in which we live."
Finally, on the 12th of August, 1812, Napoleon, who had just received
the _Traité du Calcul des Probabilités_, wrote from Witepsk the letter
which we transcribe textually:--
"There was a time when I would have read with interest your _Traité du
Calcul des Probabilités_. For the present I must confine myself to
expressing to you the satisfaction which I experience every time that I
see you give to the world new works which serve to improve and extend
the most important of the sciences, and contribute to the glory of the
nation. The advancement and the improvement of mathematical science are
connected with the prosperity of the state."
I have now arrived at the conclusion of the task which I had imposed
upon myself. I shall be pardoned for having given so detailed an
exposition of the principal discoveries for which philosophy, astronomy,
and navigation are indebted to our geometers.
It has appeared to me that in thus tracing the glorious past I have
shown our contemporaries the full extent of their duty towards the
country. In fact, it is for nations especially to bear in remembrance
the ancient adage: _noblesse obligé_!
FOOTNOTES:
[22] The author here refers to the series of biographies contained in
tome III. of the _Notices Biographiques_.--_Translator_.
[23] These celebrated laws, known in astronomy as the laws of Kepler,
are three in number. The first law is, that the planets describe
ellipses around the sun in their common focus; the second, that a line
joining the planet and the sun sweeps over equal areas in equal times;
the third, that the squares of the periodic times of the planets are
proportional to the cubes of their mean distances from the sun. The
first two laws were discovered by Kepler in the course of a laborious
examination of the theory of the planet Mars; a full account of this
inquiry is contained in his famous work _De Stella Martis_, published in
1609. The discovery of the third law was not effected until, several
years afterwards, Kepler announced it to the world in his treatise on
Harmonics (1628). The passage quoted below is extracted from that
work.--_Translator_.
[24] The spheroidal figure of the earth was established by the
comparison of an arc of the meridian that had been measured in France,
with a similar arc measured in Lapland, from which it appeared that the
length of a degree of the meridian increases from the equator towards
the poles, conformably to what ought to result upon the supposition of
the earth having the figure of an oblate spheroid. The length of the
Lapland arc was determined by means of an expedition which the French
Government had despatched to the North of Europe for that purpose. A
similar expedition had been despatched from France about the same time
to Peru in South America, for the purpose of measuring an arc of the
meridian under the equator, but the results had not been ascertained at
the time to which the author alludes in the text. The variation of
gravity at the surface of the earth was established by Richer's
experiments with the pendulum at Cayenne, in South America (1673-4),
from which it appeared that the pendulum oscillates more slowly--and
consequently the force of gravity is less intense--under the equator
than in the latitude of Paris.--_Translator_.
[25] It may perhaps be asked why we place Lagrange among the French
geometers? This is our reply: It appears to us that the individual who
was named Lagrange Tournier, two of the most characteristic French names
which it is possible to imagine, whose maternal grandfather was M. Gros,
whose paternal great-grandfather was a French officer, a native of
Paris, who never wrote except in French, and who was invested in our
country with high honours during a period of nearly thirty years;--ought
to be regarded as a Frenchman although born at Turin.--_Author_.
[26] The problem of three bodies was solved independently about the same
time by Euler, D'Alembert, and Clairaut. The two last-mentioned
geometers communicated their solutions to the Academy of Sciences on the
same day, November 15, 1747. Euler had already in 1746 published tables
of the moon, founded on his solution of the same problem, the details of
which he subsequently published in 1753.--_Translator_.
[27] It must be admitted that M. Arago has here imperfectly represented
Newton's labours on the great problem of the precession of the
equinoxes. The immortal author of the Principia did not merely
_conjecture_ that the conical motion of the earth's axis is due to the
disturbing action of the sun and moon upon the matter accumulated around
the earth's equator: he _demonstrated_ by a very beautiful and
satisfactory process that the movement must necessarily arise from that
cause; and although the means of investigation, in his time, were
inadequate to a rigorous computation of the quantitative effect, still,
his researches on the subject have been always regarded as affording one
of the most striking proofs of sagacity which is to be found in all his
works.--_Translator_.