in 1799; the third volume appeared in 1802, the fourth volume in 1805;
as regards the fifth volume, Books XI. and XII. were published in 1823,
Books XIII. XIV. and XV. in 1824, and Book XVI. in 1825. The _Théorie
des Probabilités_ was published in 1812. We shall now present the reader
with the history of the principal astronomical discoveries contained hi
these immortal works.
Astronomy is the science of which the human mind may most justly boast.
It owes this indisputable preëminence to the elevated nature of its
object, to the grandeur of its means of investigation, to the certainty,
the utility, and the unparalleled magnificence of its results.
From the earliest period of the social existence of mankind, the study
of the movements of the heavenly bodies has attracted the attention of
governments and peoples. To several great captains, illustrious
statesmen, philosophers, and eminent orators of Greece and Rome it
formed a subject of delight. Yet, let us be permitted to state,
astronomy truly worthy of the name is quite a modern science. It dates
only from the sixteenth century.
Three great, three brilliant phases, have marked its progress.
In 1543 Copernicus overthrew with a firm and bold hand, the greater part
of the antique and venerable scaffolding with which the illusions of the
senses and the pride of successive generations had filled the universe.
The earth ceased to be the centre, the pivot of the celestial movements;
it henceforward modestly ranged itself among the planets; its material
importance, amid the totality of the bodies of which our solar system is
composed, found itself reduced almost to that of a grain of sand.
Twenty-eight years had elapsed from the day when the Canon of Thorn
expired while holding in his faltering hands the first copy of the work
which was to diffuse so bright and pure a flood of glory upon Poland,
when Würtemberg witnessed the birth of a man who was destined to achieve
a revolution in science not less fertile in consequences, and still more
difficult of execution. This man was Kepler. Endowed with two qualities
which seemed incompatible with each other, a volcanic imagination, and a
pertinacity of intellect which the most tedious numerical calculations
could not daunt, Kepler conjectured that the movements of the celestial
bodies must be connected together by simple laws, or, to use his own
expressions, by _harmonic_ laws. These laws he undertook to discover. A
thousand fruitless attempts, errors of calculation inseparable from a
colossal undertaking, did not prevent him a single instant from
advancing resolutely towards the goal of which he imagined he had
obtained a glimpse. Twenty-two years were employed by him in this
investigation, and still he was not weary of it! What, in reality, are
twenty-two years of labour to him who is about to become the legislator
of worlds; who shall inscribe his name in ineffaceable characters upon
the frontispiece of an immortal code; who shall be able to exclaim in
dithyrambic language, and without incurring the reproach of any one,
"The die is cast; I have written my book; it will be read either in the
present age or by posterity, it matters not which; it may well await a
reader, since God has waited six thousand years for an interpreter of
his works?"[23]
To investigate a physical cause capable of making the planets revolve in
closed curves; to place the principle of the stability of the universe
in mechanical forces and not in solid supports such as the spheres of
crystal which our ancestors had dreamed of; to extend to the revolutions
of the heavenly bodies the general principles of the mechanics of
terrestrial bodies,--such were the questions which remained to be solved
after Kepler had announced his discoveries to the world.
Very distinct traces of these great problems are perceived here and
there among the ancients as well as the moderns, from Lucretius and
Plutarch down to Kepler, Bouillaud, and Borelli. It is to Newton,
however, that we must award the merit of their solution. This great man,
like several of his predecessors, conceived the celestial bodies to have
a tendency to approach towards each other in virtue of an attractive
force, deduced the mathematical characteristics of this force from the
laws of Kepler, extended it to all the material molecules of the solar
system, and developed his brilliant discovery in a work which, even in
the present day, is regarded as the most eminent production of the human
intellect.
The heart aches when, upon studying the history of the sciences, we
perceive so magnificent an intellectual movement effected without the
coöperation of France. Practical astronomy increased our inferiority.
The means of investigation were at first inconsiderately entrusted to
foreigners, to the prejudice of Frenchmen abounding in intelligence and
zeal. Subsequently, intellects of a superior order struggled with
courage, but in vain, against the unskilfulness of our artists. During
this period, Bradley, more fortunate on the other side of the Channel,
immortalized himself by the discovery of aberration and nutation.
The contribution of France to these admirable revolutions in
astronomical science, consisted, in 1740, of the experimental
determination of the spheroidal figure of the earth, and of the
discovery of the variation of gravity upon the surface of our planet.
These were two great results; our country, however, had a right to
demand more: when France is not in the first rank she has lost her
place.[24]
This rank, which was lost for a moment, was brilliantly regained, an
achievement for which we are indebted to four geometers.
When Newton, giving to his discoveries a generality which the laws of
Kepler did not imply, imagined that the different planets were not only
attracted by the sun, but that they also attract each other, he
introduced into the heavens a cause of universal disturbance.
Astronomers could then see at the first glance that in no part of the
universe whether near or distant would the Keplerian laws suffice for
the exact representation of the phenomena; that the simple, regular
movements with which the imaginations of the ancients were pleased to
endue the heavenly bodies would experience numerous, considerable,
perpetually changing perturbations.
To discover several of these perturbations, to assign their nature, and
in a few rare cases their numerical values, such was the object which
Newton proposed to himself in writing the _Principia Mathematica
Philosophiæ Naturalis_.
Notwithstanding the incomparable sagacity of its author the Principia
contained merely a rough outline of the planetary perturbations. If this
sublime sketch did not become a complete portrait we must not attribute
the circumstance to any want of ardour or perseverance; the efforts of
the great philosopher were always superhuman, the questions which he did
not solve were incapable of solution in his time. When the
mathematicians of the continent entered upon the same career, when they
wished to establish the Newtonian system upon an incontrovertible basis,
and to improve the tables of astronomy, they actually found in their way
difficulties which the genius of Newton had failed to surmount.
Five geometers, Clairaut, Euler, D'Alembert, Lagrange, and Laplace,
shared between them the world of which Newton had disclosed the
existence. They explored it in all directions, penetrated into regions
which had been supposed inaccessible, pointed out there a multitude of
phenomena which observation had not yet detected; finally, and it is
this which constitutes their imperishable glory, they reduced under the
domain of a single principle, a single law, every thing that was most
refined and mysterious in the celestial movements. Geometry had thus the
boldness to dispose of the future; the evolutions of ages are
scrupulously ratifying the decisions of science.
We shall not occupy our attention with the magnificent labours of Euler,
we shall, on the contrary, present the reader with a rapid analysis of
the discoveries of his four rivals, our countrymen.[25]
If a celestial body, the moon, for example, gravitated solely towards
the centre of the earth, it would describe a mathematical ellipse; it
would strictly obey the laws of Kepler, or, which is the same thing, the
principles of mechanics expounded by Newton in the first sections of his
immortal work.
Let us now consider the action of a second force. Let us take into
account the attraction which the sun exercises upon the moon, in other
words, instead of two bodies, let us suppose three to operate on each
other, the Keplerian ellipse will now furnish merely a rough indication
of the motion of our satellite. In some parts the attraction of the sun
will tend to enlarge the orbit, and will in reality do so; in other
parts the effect will be the reverse of this. In a word, by the
introduction of a third attractive body, the greatest complication will
succeed to a simple regular movement upon which the mind reposed with
complacency.
If Newton gave a complete solution of the question of the celestial
movements in the case wherein two bodies attract each other, he did not
even attempt an analytical investigation of the infinitely more
difficult problem of three bodies. The problem of three bodies (this is
the name by which it has become celebrated), the problem for determining
the movement of a body subjected to the attractive influence of two
other bodies, was solved for the first time, by our countryman
Clairaut.[26] From this solution we may date the important improvements
of the lunar tables effected in the last century.
The most beautiful astronomical discovery of antiquity, is that of the
precession of the equinoxes. Hipparchus, to whom the honour of it is
due, gave a complete and precise statement of all the consequences which
flow from this movement. Two of these have more especially attracted
attention.
By reason of the precession of the equinoxes, it is not always the same
groups of stars, the same constellations, which are perceived in the
heavens at the same season of the year. In the lapse of ages the
constellations of winter will become those of summer and reciprocally.
By reason of the precession of the equinoxes, the pole does not always
occupy the same place in the starry vault. The moderately bright star
which is very justly named in the present day, the pole star, was far
removed from the pole in the time of Hipparchus; in the course of a few
centuries it will again appear removed from it. The designation of pole
star has been, and will be, applied to stars very distant from each
other.
When the inquirer in attempting to explain natural phenomena has the
misfortune to enter upon a wrong path, each precise observation throws
him into new complications. Seven spheres of crystal did not suffice for
representing the phenomena as soon as the illustrious astronomer of
Rhodes discovered precession. An eighth sphere was then wanted to
account for a movement in which all the stars participated at the same
time.
Copernicus having deprived the earth of its alleged immobility, gave a
very simple explanation of the most minute circumstances of precession.
He supposed that the axis of rotation does not remain exactly parallel
to itself; that in the course of each complete revolution of the earth
around the sun, the axis deviates from its position by a small quantity;
in a word, instead of supposing the circumpolar stars to advance in a
certain way towards the pole, he makes the pole advance towards the
stars. This hypothesis divested the mechanism of the universe of the
greatest complication which the love of theorizing had introduced into
it. A new Alphonse would have then wanted a pretext to address to his
astronomical synod the profound remark, so erroneously interpreted,
which history ascribes to the king of Castile.
If the conception of Copernicus improved by Kepler had, as we have just
seen, introduced a striking improvement into the mechanism of the
heavens, it still remained to discover the motive force which, by
altering the position of the terrestrial axis during each successive
year, would cause it to describe an entire circle of nearly 50° in
diameter, in a period of about 26,000 years.
Newton conjectured that this force arose from the action of the sun and
moon upon the redundant matter accumulated in the equatorial regions of
the earth: thus he made the precession of the equinoxes depend upon the
spheroidal figure of the earth; he declared that upon a round planet no
precession would exist.
All this was quite true, but Newton did not succeed in establishing it
by a mathematical process. Now this great man had introduced into
philosophy the severe and just rule: Consider as certain only what has
been demonstrated. The demonstration of the Newtonian conception of the
precession of the equinoxes was, then, a great discovery, and it is to
D'Alembert that the glory of it is due.[27] The illustrious geometer
gave a complete explanation of the general movement, in virtue of which
the terrestrial axis returns to the same stars in a period of about
26,000 years. He also connected with the theory of gravitation the
perturbation of precession discovered by Bradley, that remarkable
oscillation which the earth's axis experiences continually during its
movement of progression, and the period of which, amounting to about
eighteen years, is exactly equal to the time which the intersection of
the moon's orbit with the ecliptic employs in describing the 360° of the
entire circumference.
Geometers and astronomers are justly occupied as much with the figure
and physical constitution which the earth might have had in remote ages
as with its present figure and constitution.
As soon as our countryman Richer discovered that a body, whatever be its
nature, weighs less when it is transported nearer the equatorial
regions, everybody perceived that the earth, if it was originally
fluid, ought to bulge out at the equator. Huyghens and Newton did more;
they calculated the difference between the greatest and least axes, the
excess of the equatorial diameter over the line of the poles.[28]
The calculation of Huyghens was founded upon hypothetic properties of
the attractive force which were wholly inadmissible; that of Newton upon
a theorem which he ought to have demonstrated; the theory of the latter
was characterized by a defect of a still more serious nature: it
supposed the density of the earth during the original state of fluidity,
to be homogeneous.[29] When in attempting the solution of great problems
we have recourse to such simplifications; when, in order to elude
difficulties of calculation, we depart so widely from natural and
physical conditions, the results relate to an ideal world, they are in
reality nothing more than flights of the imagination.
In order to apply mathematical analysis usefully to the determination of
the figure of the earth it was necessary to abandon all idea of
homogeneity, all constrained resemblance between the forms of the
superposed and unequally dense strata; it was necessary also to examine
the case of a central solid nucleus. This generality increased tenfold
the difficulties of the problem; neither Clairaut nor D'Alembert was,
however, arrested by them. Thanks to the efforts of these two eminent
geometers, thanks to some essential developments due to their immediate
successors, and especially to the illustrious Legendre, the theoretical
determination of the figure of the earth has attained all desirable
perfection. There now reigns the most satisfactory accordance between
the results of calculation and those of direct measurement. The earth,
then, was originally fluid: analysis has enabled us to ascend to the
earliest ages of our planet.[30]
In the time of Alexander comets were supposed by the majority of the
Greek philosophers to be merely meteors generated in our atmosphere.
During the middle ages, persons, without giving themselves much concern
about the nature of those bodies, supposed them to prognosticate
sinister events. Regiomontanus and Tycho Brahé proved by their
observations that they are situate beyond the moon; Hevelius, Dörfel,
&c., made them revolve around the sun; Newton established that they move
under the immediate influence of the attractive force of that body, that
they do not describe right lines, that, in fact, they obey the laws of
Kepler. It was necessary, then, to prove that the orbits of comets are
curves which return into themselves, or that the same comet has been
seen on several distinct occasions. This discovery was reserved for
Halley. By a minute investigation of the circumstances connected with
the apparitions of all the comets to be met with in the records of
history, in ancient chronicles, and in astronomical annals, this eminent
philosopher was enabled to prove that the comets of 1682, of 1607, and
of 1531, were in reality so many successive apparitions of one and the
same body.
This identity involved a conclusion before which more than one
astronomer shrunk. It was necessary to admit that the time of a complete
revolution of the comet was subject to a great variation, amounting to
as much as two years in seventy-six.
Were such great discordances due to the disturbing action of the
planets?
The answer to this question would introduce comets into the category of
ordinary planets or would exclude them for ever. The calculation was
difficult: Clairaut discovered the means of effecting it. While success
was still uncertain, the illustrious geometer gave proof of the greatest
boldness, for in the course of the year 1758 he undertook to determine
the time of the following year when the comet of 1682 would reappear. He
designated the constellations, nay the stars, which it would encounter
in its progress.
This was not one of those remote predictions which astrologers and
others formerly combined very skilfully with the tables of mortality, so
that they might not be falsified during their lifetime: the event was
close at hand. The question at issue was nothing less than the creation
of a new era in cometary astronomy, or the casting of a reproach upon
science, the consequences of which it would long continue to feel.
Clairaut found by a long process of calculation, conducted with great
skill, that the action of Jupiter and Saturn ought to have retarded the
movement of the comet; that the time of revolution compared with that
immediately preceding, would be increased 518 days by the disturbing
action of Jupiter, and 100 days by the action of Saturn, forming a
total of 618 days, or more than a year and eight months.
Never did a question of astronomy excite a more intense, a more
legitimate curiosity. All classes of society awaited with equal interest
the announced apparition. A Saxon peasant, Palitzch, first perceived the
comet. Henceforward, from one extremity of Europe to the other, a
thousand telescopes traced each night the path of the body through the
constellations. The route was always, within the limits of precision of
the calculations, that which Clairaut had indicated beforehand. The
prediction of the illustrious geometer was verified in regard both to
time and space: astronomy had just achieved a great and important
triumph, and, as usual, had destroyed at one blow a disgraceful and
inveterate prejudice. As soon as it was established that the returns of
comets might be calculated beforehand, those bodies lost for ever their
ancient prestige. The most timid minds troubled themselves quite as
little about them as about eclipses of the sun and moon, which are
equally subject to calculation. In fine, the labours of Clairaut had
produced a deeper impression on the public mind than the learned,
ingenious, and acute reasoning of Bayle.
The heavens offer to reflecting minds nothing more curious or more
strange than the equality which subsists between the movements of
rotation and revolution of our satellite. By reason of this perfect
equality the moon always presents the same side to the earth. The
hemisphere which we see in the present day is precisely that which our
ancestors saw in the most remote ages; it is exactly the hemisphere
which future generations will perceive.
The doctrine of final causes which certain philosophers have so
abundantly made use of in endeavouring to account for a great number of
natural phenomena was in this particular case totally inapplicable. In
fact, how could it be pretended that mankind could have any interest in
perceiving incessantly the same hemisphere of the moon, in never
obtaining a glimpse of the opposite hemisphere? On the other hand, the
existence of a perfect, mathematical equality between elements having no
necessary connection--such as the movements of translation and rotation
of a given celestial body--was not less repugnant to all ideas of
probability. There were besides two other numerical coincidences quite
as extraordinary; an identity of direction, relative to the stars, of
the equator and orbit of the moon; exactly the same precessional
movements of these two planes. This group of singular phenomena,
discovered by J.D. Cassini, constituted the mathematical code of what is
called the _Libration of the Moon_.
The libration of the moon formed a very imperfect part of physical
astronomy when Lagrange made it depend on a circumstance connected with
the figure of our satellite which was not observable from the earth, and
thereby connected it completely with the principles of universal
gravitation.
At the time when the moon was converted into a solid body, the action of
the earth compelled it to assume a less regular figure than if no
attracting body had been situate in its vicinity. The action of our
globe rendered elliptical an equator which otherwise would have been
circular. This disturbing action did not prevent the lunar equator from
bulging out in every direction, but the prominence of the equatorial
diameter directed towards the earth became four times greater than that
of the diameter which we see perpendicularly.
The moon would appear then, to an observer situate in space and
examining it transversely, to be elongated towards the earth, to be a
sort of pendulum without a point of suspension. When a pendulum deviates
from the vertical, the action of gravity brings it back; when the
principal axis of the moon recedes from its usual direction, the earth
in like manner compels it to return.
We have here, then, a complete explanation of a singular phenomenon,
without the necessity of having recourse to the existence of an almost
miraculous equality between two movements of translation and rotation,
entirely independent of each other. Mankind will never see but one face
of the moon. Observation had informed us of this fact; now we know
further that this is due to a physical cause which may be calculated,
and which is visible only to the mind's eye,--that it is attributable to
the elongation which the diameter of the moon experienced when it passed
from the liquid to the solid state under the attractive influence of the
earth.
If there had existed originally a slight difference between the
movements of rotation and revolution of the moon, the attraction of the
earth would have reduced these movements to a rigorous equality. This
attraction would have even sufficed to cause the disappearance of a
slight want of coincidence in the intersections of the equator and orbit
of the moon with the plane of the ecliptic.
The memoir in which Lagrange has so successfully connected the laws of
libration with the principles of gravitation, is no less remarkable for
intrinsic excellence than style of execution. After having perused this
production, the reader will have no difficulty in admitting that the
word _elegance_ may be appropriately applied to mathematical researches.