The Marquis de Laplace, peer of France, one of the forty of the French Academy, member of the Academy of Sciences and of the Bureau des Longitudes, an associate of all the great Academies or Scientific Societies of Europe, was born at Beaumont-en-Auge of parents belonging to the class of small farmers, on the 28th of March, 1749; he died on the 5th of March, 1827.

The first and second volumes of the Mecanique Celeste were published 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 Theorie des Probabilites 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 preeminence 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 Wuertemberg 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 cooeperation 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 Philosophiae 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 deg. 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 deg. 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 Brahe proved by their observations that they are situate beyond the moon; Hevelius, Doerfel, &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.

In this analysis we have merely glanced at the astronomical discoveries of Clairaut, D'Alembert, and Lagrange. We shall be somewhat less concise in noticing the labours of Laplace.

After having enumerated the various forces which must result from the mutual action of the planets and satellites of our system, even the great Newton did not venture to investigate the general nature of the effects produced by them. In the midst of the labyrinth formed by increases and diminutions of velocity, variations in the forms of the orbits, changes of distances and inclinations, which these forces must evidently produce, the most learned geometer would fail to discover a trustworthy guide. This extreme complication gave birth to a discouraging reflection. Forces so numerous, so variable in position, so different in intensity, seemed to be incapable of maintaining a condition of equilibrium except by a sort of miracle. Newton even went so far as to suppose that the planetary system did not contain within itself the elements of indefinite stability; he was of opinion that a powerful hand must intervene from time to time, to repair the derangements occasioned by the mutual action of the various bodies. Euler, although farther advanced than Newton in a knowledge of the planetary pertubations, refused also to admit that the solar system was constituted so as to endure for ever.

Never did a greater philosophical question offer itself to the inquiries of mankind. Laplace attacked it with boldness, perseverance, and success. The profound and long-continued researches of the illustrious geometer established with complete evidence that the planetary ellipses are perpetually variable; that the extremities of their major axes make the tour of the heavens; that, independently of an oscillatory motion, the planes of their orbits experienced a displacement in virtue of which their intersections with the plane of the terrestrial orbit are each year directed towards different stars. In the midst of this apparent chaos there is one element which remains constant or is merely subject to small periodic changes; namely, the major axis of each orbit, and consequently the time of revolution of each planet. This is the element which ought to have chiefly varied, according to the learned speculations of Newton and Euler.

The principle of universal gravitation suffices for preserving the stability of the solar system. It maintains the forms and inclinations of the orbits in a mean condition which is subject to slight oscillations; variety does not entail disorder; the universe offers the example of harmonious relations, of a state of perfection which Newton himself doubted. This depends on circumstances which calculation disclosed to Laplace, and which, upon a superficial view of the subject, would not seem to be capable of exercising so great an influence. Instead of planets revolving all in the same direction in slightly eccentric orbits, and in planes inclined at small angles towards each other, substitute different conditions and the stability of the universe will again be put in jeopardy, and according to all probability there will result a frightful chaos.[31]

Although the invariability of the mean distances of the planetary orbits has been more completely demonstrated since the appearance of the memoir above referred to, that is to say by pushing the analytical approximations to a greater extent, it will, notwithstanding, always constitute one of the admirable discoveries of the author of the Mecanique Celeste. Dates, in the case of such subjects, are no luxury of erudition. The memoir in which Laplace communicated his results on the invariability of the mean motions or mean distances, is dated 1773.[32] It was in 1784 only, that he established the stability of the other elements of the system from the smallness of the planetary masses, the inconsiderable eccentricity of the orbits, and the revolution of the planets in one common direction around the sun.

The discovery of which I have just given an account to the reader excluded at least from the solar system the idea of the Newtonian attraction being a cause of disorder. But might not other forces, by combining with attraction, produce gradually increasing perturbations as Newton and Euler dreaded? Facts of a positive nature seemed to justify these fears.

A comparison of ancient with modern observations revealed the existence of a continual acceleration of the mean motions of the moon and the planet Jupiter, and an equally striking diminution of the mean motion of Saturn. These variations led to conclusions of the most singular nature.

In accordance with the presumed cause of these perturbations, to say that the velocity of a body increased from century to century was equivalent to asserting that the body continually approached the centre of motion; on the other hand, when the velocity diminished, the body must be receding from the centre.

Thus, by a strange arrangement of nature, our planetary system seemed destined to lose Saturn, its most mysterious ornament,—to see the planet accompanied by its ring and seven satellites, plunge gradually into unknown regions, whither the eye armed with the most powerful telescopes has never penetrated. Jupiter, on the other hand, the planet compared with which the earth is so insignificant, appeared to be moving in the opposite direction, so as to be ultimately absorbed in the incandescent matter of the sun. Finally, the moon seemed as if it would one day precipitate itself upon the earth.

There was nothing doubtful or speculative in these sinister forebodings. The precise dates of the approaching catastrophes were alone uncertain. It was known, however, that they were very distant. Accordingly, neither the learned dissertations of men of science nor the animated descriptions of certain poets produced any impression upon the public mind.

It was not so with our scientific societies, the members of which regarded with regret the approaching destruction of our planetary system. The Academy of Sciences called the attention of geometers of all countries to these menacing perturbations. Euler and Lagrange descended into the arena. Never did their mathematical genius shine with a brighter lustre. Still, the question remained undecided. The inutility of such efforts seemed to suggest only a feeling of resignation on the subject, when from two disdained corners of the theories of analysis, the author of the Mecanique Celeste caused the laws of these great phenomena clearly to emerge. The variations of velocity of Jupiter, Saturn, and the Moon flowed then from evident physical causes, and entered into the category of ordinary periodic perturbations depending upon the principle of attraction. The variations in the dimensions of the orbits which were so much dreaded resolved themselves into simple oscillations included within narrow limits. Finally, by the powerful instrumentality of mathematical analysis, the physical universe was again established on a firm foundation.

I cannot quit this subject without at least alluding to the circumstances in the solar system upon which depend the so long unexplained variations of velocity of the Moon, Jupiter, and Saturn.

The motion of the earth around the sun is mainly effected in an ellipse, the form of which is liable to vary from the effects of planetary perturbation. These alterations of form are periodic; sometimes the curve, without ceasing to be elliptic, approaches the form of a circle, while at other times it deviates more and more from that form. From the epoch of the earliest recorded observations, the eccentricity of the terrestrial orbit has been diminishing from year to year; at some future epoch the orbit, on the contrary, will begin to deviate from the form of a circle, and the eccentricity will increase to the same extent as it previously diminished, and according to the same laws.

Now, Laplace has shown that the mean motion of the moon around the earth is connected with the form of the ellipse which the earth describes around the sun; that a diminution of the eccentricity of the ellipse inevitably induces an increase in the velocity of our satellite, and vice versa; finally, that this cause suffices to explain the numerical value of the acceleration which the mean motion of the moon has experienced from the earliest ages down to the present time.[33]

The origin of the inequalities in the mean motions of Jupiter and Saturn will be, I hope, as easy to conceive.

Mathematical analysis has not served to represent in finite terms the values of the derangements which each planet experiences in its movement from the action of all the other planets. In the present state of science, this value is exhibited in the form of an indefinite series of terms diminishing rapidly in magnitude. In calculation, it is usual to neglect such of those terms as correspond in the order of magnitude to quantities beneath the errors of observation. But there are cases in which the order of the term in the series does not decide whether it be small or great. Certain numerical relations between the primitive elements of the disturbing and disturbed planets may impart sensible values to terms which usually admit of being neglected. This case occurs in the perturbations of Saturn produced by Jupiter, and in those of Jupiter produced by Saturn. There exists between the mean motions of these two great planets a simple relation of commensurability, five times the mean motion of Saturn, being, in fact, very nearly equal to twice the mean motion of Jupiter. It happens, in consequence, that certain terms, which would otherwise be very small, acquire from this circumstance considerable values. Hence arise in the movements of these two planets, inequalities of long duration which require more than 900 years for their complete development, and which represent with marvellous accuracy all the irregularities disclosed by observation.

Is it not astonishing to find in the commensurability of the mean motions of two planets, a cause of perturbation of so influential a nature; to discover that the definitive solution of an immense difficulty—which baffled the genius of Euler, and which even led persons to doubt whether the theory of gravitation was capable of accounting for all the phenomena of the heavens—should depend upon the fortuitous circumstance of five times the mean motion of Saturn being equal to twice the mean motion of Jupiter? The beauty of the conception and the ultimate result are here equally worthy of admiration.[34]

We have just explained how Laplace demonstrated that the solar system can experience only small periodic oscillations around a certain mean state. Let us now see in what way he succeeded in determining the absolute dimensions of the orbits.

What is the distance of the sun from the earth? No scientific question has occupied in a greater degree the attention of mankind; mathematically speaking, nothing is more simple. It suffices, as in common operations of surveying, to draw visual lines from the two extremities of a known base to an inaccessible object. The remainder is a process of elementary calculation. Unfortunately, in the case of the sun, the distance is great and the bases which can be measured upon the earth are comparatively very small. In such a case the slightest errors in the direction of the visual lines exercise an enormous influence upon the results.

In the beginning of the last century Halley remarked that certain interpositions of Venus between the earth and the sun, or, to use an expression applied to such conjunctions, that the transits of the planet across the sun's disk, would furnish at each observatory an indirect means of fixing the position of the visual ray very superior in accuracy to the most perfect direct methods.[35]

Such was the object of the scientific expeditions undertaken in 1761 and 1769, on which occasions France, not to speak of stations in Europe, was represented at the Isle of Rodrigo by Pingre, at the Isle of St. Domingo by Fleurin, at California by the Abbe Chappe, at Pondicherry by Legentil. At the same epochs England sent Maskelyne to St. Helena, Wales to Hudson's Bay, Mason to the Cape of Good Hope, Captain Cooke to Otaheite, &c. The observations of the southern hemisphere compared with those of Europe, and especially with the observations made by an Austrian astronomer Father Hell at Wardhus in Lapland, gave for the distance of the sun the result which has since figured in all treatises on astronomy and navigation.

No government hesitated in furnishing Academies with the means, however expensive they might be, of conveniently establishing their observers in the most distant regions. We have already remarked that the determination of the contemplated distance appeared to demand imperiously an extensive base, for small bases would have been totally inadequate to the purpose. Well, Laplace has solved the problem numerically without a base of any kind whatever; he has deduced the distance of the sun from observations of the moon made in one and the same place!

The sun is, with respect to our satellite, the cause of perturbations which evidently depend on the distance of the immense luminous globe from the earth. Who does not see that these perturbations would diminish if the distance increased; that they would increase on the contrary, if the distance diminished; that the distance finally determines the magnitude of the perturbations?

Observation assigns the numerical value of these perturbations; theory, on the other hand, unfolds the general mathematical relation which connects them with the solar parallax, and with other known elements. The determination of the mean radius of the terrestrial orbit then becomes one of the most simple operations of algebra. Such is the happy combination by the aid of which Laplace has solved the great, the celebrated problem of parallax. It is thus that the illustrious geometer found for the mean distance of the sun from the earth, expressed in radii of the terrestrial orbit, a value differing only in a slight degree from that which was the fruit of so many troublesome and expensive voyages. According to the opinion of very competent judges the result of the indirect method might not impossibly merit the preference.[36]

The movements of the moon proved a fertile mine of research to our great geometer. His penetrating intellect discovered in them unknown treasures. He disentangled them from every thing which concealed them from vulgar eyes with an ability and a perseverance equally worthy of admiration. The reader will excuse me for citing another of such examples.

The earth governs the movements of the moon. The earth is flattened, in other words its figure is spheroidal. A spheroidal body does not attract like a sphere. There ought then to exist in the movement, I had almost said in the countenance of the moon, a sort of impression of the spheroidal figure of the earth. Such was the idea as it originally occurred to Laplace.

It still remained to ascertain (and here consisted the chief difficulty), whether the effects attributable to the spheroidal figure of the earth were sufficiently sensible not to be confounded with the errors of observation. It was accordingly necessary to find the general formula of perturbations of this nature, in order to be able, as in the case of the solar parallax, to eliminate the unknown quantity.

The ardour of Laplace, combined with his power of analytical research, surmounted all obstacles. By means of an investigation which demanded the most minute attention, the great geometer discovered in the theory of the moon's movements, two well-defined perturbations depending on the spheroidal figure of the earth. The first affected the resolved element of the motion of our satellite which is chiefly measured with the instrument known in observatories by the name of the transit instrument; the second, which operated in the direction north and south, could only be effected by observations with a second instrument termed the mural circle. These two inequalities of very different magnitudes connected with the cause which produces them by analytical combinations of totally different kinds have, however, both conducted to the same value of the ellipticity. It must be borne in mind, however, that the ellipticity thus deduced from the movements of the moon, is not the ellipticity corresponding to such or such a country, the ellipticity observed in France, in England, in Italy, in Lapland, in North America, in India, or in the region of the Cape of Good Hope, for the earth's materials having undergone considerable upheavings at different times and in different places, the primitive regularity of its curvature has been sensibly disturbed by this cause. The moon, and it is this circumstance which renders the result of such inestimable value, ought to assign, and has in reality assigned the general ellipticity of the earth; in other words, it has indicated a sort of mean value of the various determinations obtained at enormous expense, and with infinite labour, as the result of long voyages undertaken by astronomers of all the countries of Europe.

I shall add a few brief remarks, for which I am mainly indebted to the author of the Mecanique Celeste. They seem to be eminently adapted for illustrating the profound, the unexpected, and almost paradoxical character of the methods which I have just attempted to sketch.

What are the elements which it has been found necessary to confront with each other in order to arrive at results expressed even to the precision of the smallest decimals?

On the one hand, mathematical formulae, deduced from the principle of universal attraction; on the other hand, certain irregularities observed in the returns of the moon to the meridian.

An observing geometer who, from his infancy, had never quitted his chamber of study, and who had never viewed the heavens except through a narrow aperture directed north and south, in the vertical plane in which the principal astronomical instruments are made to move,—to whom nothing had ever been revealed respecting the bodies revolving above his head, except that they attract each other according to the Newtonian law of gravitation,—would, however, be enabled to ascertain that his narrow abode was situated upon the surface of a spheroidal body, the equatorial axis of which surpassed the polar axis by a three hundred and sixth part; he would have also found, in his isolated immovable position, his true distance from the sun.

I have stated at the commencement of this Notice, that it is to D'Alembert we owe the first satisfactory mathematical explanation of the phenomenon of the precession of the equinoxes. But our illustrious countryman, as well as Euler, whose solution appeared subsequently to that of D'Alembert, omitted all consideration of certain physical circumstances, which, however, did not seem to be of a nature to be neglected without examination. Laplace has supplied this deficiency. He has shown that the sea, notwithstanding its fluidity, and that the atmosphere, notwithstanding its currents, exercise the same influence on the movements of the terrestrial axis as if they formed solid masses adhering to the terrestrial spheroid.

Do the extremities of the axis around which the earth performs an entire revolution once in every twenty-four hours, correspond always to the same material points of the terrestrial spheroid? In other words, do the poles of rotation, which from year to year correspond to different stars, undergo also a displacement at the surface of the earth?

In the case of the affirmative, the equator is movable as well as the poles; the terrestrial latitudes are variable; no country during the lapse of ages will enjoy, even on an average, a constant climate; regions the most different will, in their turn, become circumpolar. Adopt the contrary supposition, and every thing assumes the character of an admirable permanence.

The question which I have just suggested, one of the most important in Astronomy, cannot be solved by the aid of mere observation on account of the uncertainty of the early determinations of terrestrial latitude. Laplace has supplied this defect by analysis. The great geometer has demonstrated that no circumstance depending on universal gravitation can sensibly displace the poles of the earth's axis relatively to the surface of the terrestrial spheroid. The sea, far from being an obstacle to the invariable rotation of the earth upon its axis, would, on the contrary, reduce the axis to a permanent condition in consequence of the mobility of the waters and the resistance which their oscillations experience.

The remarks which I have just made with respect to the position of the terrestrial axis are equally applicable to the time of the earth's rotation which is the unit, the true standard of time. The importance of this element induced Laplace to examine whether its numerical value might not be liable to vary from internal causes such as earthquakes and volcanoes. It is hardly necessary for me to state that the result obtained was negative.

The admirable memoir of Lagrange upon the libration of the moon seemed to have exhausted the subject. This, however, was not the case.

The motion of revolution of our satellite around the earth is subject to perturbations, technically termed secular, which were either unknown to Lagrange or which he neglected. These inequalities eventually place the body, not to speak of entire circumferences, at angular distances of a semi-circle, a circle and a half, &c., from the position which it would otherwise occupy. If the movement of rotation did not participate in such perturbations, the moon in the lapse of ages would present in succession all the parts of its surface to the earth.

This event will not occur. The hemisphere of the moon which is actually invisible, will remain invisible for ever. Laplace, in fact, has shown that the attraction of the earth introduces into the rotatory motion of the lunar spheroid the secular inequalities which exist in the movement of revolution.

Researches of this nature exhibit in full relief the power of mathematical analysis. It would have been very difficult to have discovered by synthesis truths so profoundly enveloped in the complex action of a multitude of forces.

We should be inexcusable if we omitted to notice the high importance of the labours of Laplace on the improvement of the lunar tables. The immediate object of this improvement was, in effect, the promotion of maritime intercourse between distant countries, and, what was indeed far superior to all considerations of mercantile interest, the preservation of the lives of mariners.

Thanks to a sagacity without parallel, to a perseverance which knew no limits, to an ardour always youthful and which communicated itself to able coadjutors, Laplace solved the celebrated problem of the longitude more completely than could have been hoped for in a scientific point of view, with greater precision than the art of navigation in its utmost refinement demanded. The ship, the sport of the winds and tempests, has no occasion, in the present day, to be afraid of losing itself in the immensity of the ocean. An intelligent glance at the starry vault indicates to the pilot, in every place and at every time, his distance from the meridian of Paris. The extreme perfection of the existing tables of the moon entitles Laplace to be ranked among the benefactors of humanity.[37]

In the beginning of the year 1611, Galileo supposed that he found in the eclipses of Jupiter's satellites a simple and rigorous solution of the famous problem of the longitude, and active negotiations were immediately commenced with the view of introducing the new method on board the numerous vessels of Spain and Holland. These negotiations failed. From the discussion it plainly appeared that the accurate observation of the eclipses of the satellites would require powerful telescopes; but such telescopes could not be employed on board a ship tossed about by the waves.

The method of Galileo seemed, at any rate, to retain all its advantages when applied on land, and to promise immense improvements to geography. These expectations were found to be premature. The movements of the satellites of Jupiter are not by any means so simple as the immortal inventor of the method of longitudes supposed them to be. It was necessary that three generations of astronomers and mathematicians should labour with perseverance in unfolding their most considerable perturbations. It was necessary, in fine, that the tables of those bodies should acquire all desirable and necessary precision, that Laplace should introduce into the midst of them the torch of mathematical analysis.

In the present day, the nautical ephemerides contain, several years in advance, the indication of the times of the eclipses and reappearances of Jupiter's satellites. Calculation does not yield in precision to direct observation. In this group of satellites, considered as an independent system of bodies, Laplace found a series of perturbations analogous to those which the planets experience. The rapidity of the revolutions unfolds, in a sufficiently short space of time, changes in this system which require centuries for their complete development in the solar system.

Although the satellites exhibit hardly an appreciable diameter even when viewed in the best telescopes, our illustrious countryman was enabled to determine their masses. Finally, he discovered certain simple relations of an extremely remarkable character between the movements of those bodies, which have been called the laws of Laplace. Posterity will not obliterate this designation; it will acknowledge the propriety of inscribing in the heavens the name of so great an astronomer beside that of Kepler.

Let us cite two or three of the laws of Laplace:—

If we add to the mean longitude of the first satellite twice that of the third, and subtract from the sum three times the mean longitude of the second, the result will be exactly equal to 180 deg..

Would it not be very extraordinary if the three satellites had been placed originally at the distances from Jupiter, and in the positions, with respect to each other, adapted for constantly and rigorously maintaining the foregoing relation? Laplace has replied to this question by showing that it is not necessary that this relation should have been rigorously true at the origin. The mutual action of the satellites would necessarily have reduced it to its present mathematical condition, if once the distances and the positions satisfied the law approximately.

This first law is equally true when we employ the synodical elements. It hence plainly results, that the first three satellites of Jupiter can never be all eclipsed at the same time. Bearing this in mind, we shall have no difficulty in apprehending the import of a celebrated observation of recent times, during which certain astronomers perceived the planet for a short time without any of his four satellites. This would not by any means authorize us in supposing the satellites to be eclipsed. A satellite disappears when it is projected upon the central part of the luminous disk of Jupiter, and also when it passes behind the opaque body of the planet.

The following is another very simple law to which the mean motions of the same satellites of Jupiter are subject:

If we add to the mean motion of the first satellite twice the mean motion of the third, the sum is exactly equal to three times the mean motion of the second.[38]

This numerical coincidence, which is perfectly accurate, would be one of the most mysterious phenomena in the system of the universe if Laplace had not proved that the law need only have been approximate at the origin, and that the mutual action of the satellites has sufficed to render it rigorous.

The illustrious geometer, who always pursued his researches to their most remote ramifications, arrived at the following result: The action of Jupiter regulates the movements of rotation of the satellites so that, without taking into account the secular perturbations, the time of rotation of the first satellite plus twice the time of rotation of the third, forms a sum which is constantly equal to three times the time of rotation of the second.

Influenced by a deference, a modesty, a timidity, without any plausible motive, our artists in the last century surrendered to the English the exclusive privilege of constructing instruments of astronomy. Thus, let us frankly acknowledge the fact, at the time when Herschel was prosecuting his beautiful observations on the other side of the Channel, there existed in France no instruments adapted for developing them; we had not even the means of verifying them. Fortunately for the scientific honour of our country, mathematical analysis is also a powerful instrument. Laplace gave ample proof of this on a memorable occasion when from the retirement of his chamber he predicted, he minutely announced, what the excellent astronomer of Windsor would see with the largest telescopes which were ever constructed by the hand of man.

When Galileo, in the beginning of the year 1610, directed towards Saturn a telescope of very low power which he had just executed with his own hands, he perceived that the planet was not an ordinary globe, without however being able to ascertain its real form. The expression tri-corporate, by which the illustrious Florentine designated the appearance of the planet, implied even a totally erroneous idea of its structure. Our countryman Roberval entertained much sounder views on the subject, but from not having instituted a detailed comparison between his hypothesis and the results of observation, he abandoned to Huyghens the honour of being regarded as the author of the true theory of the phenomena presented by the wonderful planet.

Every person knows, in the present day, that Saturn consists of a globe about 900 times greater than the earth, and a ring. This ring does not touch the ball of the planet, being everywhere removed from it at a distance of 20,000 (English) miles. Observation indicates the breadth of the ring to be 54,000 miles. The thickness certainly does not exceed 250 miles. With the exception of a black streak which divides the ring throughout its whole contour into two parts of unequal breadth and of different brightness, this strange colossal bridge without piles had never offered to the most experienced or skilful observers either spot or protuberance adapted for deciding whether it was immovable or endued with a movement of rotation.

Laplace considered it to be very improbable, if the ring was immovable, that its constituent parts should be capable of resisting by their mere cohesion the continual attraction of the planet. A movement of rotation occurred to his mind as constituting the principle of stability, and he hence deduced the necessary velocity. The velocity thus found was exactly equal to that which Herschel subsequently deduced from a course of extremely delicate observations.

The two parts of the ring being placed at different distances from the planet, could not fail to experience from the action of the sun, different movements of rotation. It would hence seem that the planes of both rings ought to be generally inclined towards each other, whereas they appear from observation always to coincide. It was necessary then that some physical cause should exist which would be capable of neutralizing the action of the sun. In a memoir published in February, 1789, Laplace found that this cause must reside in the ellipticity of Saturn produced by a rapid movement of rotation of the planet, a movement the existence of which Herschel announced in November, 1789.

The reader cannot fail to remark how, on certain occasions, the eyes of the mind can supply the want of the most powerful telescopes, and lead to astronomical discoveries of the highest importance.

Let us descend from the heavens upon the earth. The discoveries of Laplace will appear not less important, not less worthy of his genius.

The phenomena of the tides, which an ancient philosopher designated in despair as the tomb of human curiosity, were connected by Laplace with an analytical theory in which the physical conditions of the question figure for the first time. Accordingly calculators, to the immense advantage of the navigation of our maritime coasts, venture in the present day to predict several years in advance the details of the time and height of the full tides without more anxiety respecting the result than if the question related to the phases of an eclipse.

There exists between the different phenomena of the ebb and flow of the tides and the attractive forces which the sun and moon exercise upon the fluid sheet which covers three fourths of the globe, an intimate and necessary connection from which Laplace, by the aid of a series of twenty years of observations executed at Brest, deduced the value of the mass of our satellite. Science knows in the present day that seventy-five moons would be necessary to form a weight equivalent to that of the terrestrial globe, and it is indebted for this result to an attentive and minute study of the oscillations of the ocean. We know only one means of enhancing the admiration which every thoughtful mind will entertain for theories capable of leading to such conclusions. An historical statement will supply it. In the year 1631, the illustrious Galileo, as appears from his Dialogues, was so far from perceiving the mathematical relations from which Laplace deduced results so beautiful, so unequivocal, and so useful, that he taxed with frivolousness the vague idea which Kepler entertained of attributing to the moon's attraction a certain share in the production of the diurnal and periodical movements of the waters of the ocean.

Laplace did not confine himself to extending so considerably, and improving so essentially, the mathematical theory of the tides; he considered the phenomenon from an entirely new point of view; it was he who first treated of the stability of the ocean. Systems of bodies, whether solid or fluid, are subject to two kinds of equilibrium, which we must carefully distinguish from each other. In the case of stable equilibrium the system, when slightly disturbed, tends always to return to its original condition. On the other hand, when the system is in unstable equilibrium, a very insignificant derangement might occasion an enormous dislocation in the relative positions of its constituent parts.

If the equilibrium of waves is of the latter kind, the waves engendered by the action of winds, by earthquakes, and by sudden movements from the bottom of the ocean, have perhaps risen in past times and may rise in the future to the height of the highest mountains. The geologist will have the satisfaction of deducing from these prodigious oscillations a rational explanation of a great multitude of phenomena, but the public will thereby be exposed to new and terrible catastrophes.

Mankind may rest assured: Laplace has proved that the equilibrium of the ocean is stable, but upon the express condition (which, however, has been amply verified by established facts), that the mean density of the fluid mass is less than the mean density of the earth. Every thing else remaining the same, let us substitute an ocean of mercury for the actual ocean, and the stability will disappear, and the fluid will frequently surpass its boundaries, to ravage continents even to the height of the snowy regions which lose themselves in the clouds.

Does not the reader remark how each of the analytical investigations of Laplace serves to disclose the harmony and duration of the universe and of our globe!

It was impossible that the great geometer, who had succeeded so well in the study of the tides of the ocean, should not have occupied his attention with the tides of the atmosphere; that he should not have submitted to the delicate and definitive tests of a rigorous calculus, the generally diffused opinions respecting the influence of the moon upon the height of the barometer and other meteorological phenomena.

Laplace, in effect, has devoted a chapter of his splendid work to an examination of the oscillations which the attractive force of the moon is capable of producing in our atmosphere. It results from these researches, that, at Paris, the lunar tide produces no sensible effect upon the barometer. The height of the tide, obtained by the discussion of a long series of observations, has not exceeded two-hundredths of a millimetre, a quantity which, in the present state of meteorological science, is less than the probable error of observation.

The calculation to which I have just alluded, may be cited in support of considerations to which I had recourse when I wished to establish, that if the moon alters more or less the height of the barometer, according to its different phases, the effect is not attributable to attraction.

No person was more sagacious than Laplace in discovering intimate relations between phenomena apparently very dissimilar; no person showed himself more skilful in deducing important conclusions from those unexpected affinities.

Towards the close of his days, for example, he overthrew with a stroke of the pen, by the aid of certain observations of the moon, the cosmogonic theories of Buffon and Bailly, which were so long in favour.

According to these theories, the earth was inevitably advancing to a state of congelation which was close at hand. Laplace, who never contented himself with a vague statement, sought to determine in numbers the rapid cooling of our globe which Buffon had so eloquently but so gratuitously announced. Nothing could be more simple, better connected, or more demonstrative, than the chain of deductions of the celebrated geometer.

A body diminishes in volume when it cools. According to the most elementary principles of mechanics, a rotating body which contracts in dimensions ought inevitably to turn upon its axis with greater and greater rapidity. The length of the day has been determined in all ages by the time of the earth's rotation; if the earth is cooling, the length of the day must be continually shortening. Now there exists a means of ascertaining whether the length of the day has undergone any variation; this consists in examining, for each century, the arc of the celestial sphere described by the moon during the interval of time which the astronomers of the existing epoch called a day,—in other words, the time required by the earth to effect a complete rotation on its axis, the velocity of the moon being in fact independent of the time of the earth's rotation.

Let us now, after the example of Laplace, take from the standard tables the least considerable values, if you choose, of the expansions or contractions which solid bodies experience from changes of temperature; search then the annals of Grecian, Arabian, and modern astronomy for the purpose of finding in them the angular velocity of the moon, and the great geometer will prove, by incontrovertible evidence founded upon these data, that during a period of two thousand years the mean temperature of the earth has not varied to the extent of the hundredth part of a degree of the centigrade thermometer. No eloquent declamation is capable of resisting such a process of reasoning, or withstanding the force of such numbers. The mathematics have been in all ages the implacable adversaries of scientific romances.

The fall of bodies, if it was not a phenomenon of perpetual occurrence, would justly excite in the highest degree the astonishment of mankind. What, in effect, is more extraordinary than to see an inert mass, that is to say, a mass deprived of will, a mass which ought not to have any propensity to advance in one direction more than in another, precipitate itself towards the earth as soon as it ceased to be supported!

Nature engenders the gravity of bodies by a process so recondite, so completely beyond the reach of our senses and the ordinary resources of human intelligence, that the philosophers of antiquity, who supposed that they could explain every thing mechanically according to the simple evolutions of atoms, excepted gravity from their speculations.

Descartes attempted what Leucippus, Democritus, Epicurus, and their followers thought to be impossible.

He made the fall of terrestrial bodies depend upon the action of a vortex of very subtle matter circulating around the earth. The real improvements which the illustrious Huyghens applied to the ingenious conception of our countryman were far, however, from imparting to it clearness and precision, those characteristic attributes of truth.

Those persons form a very imperfect estimate of the meaning of one of the greatest questions which has occupied the attention of modern inquirers, who regard Newton as having issued victorious from a struggle in which his two immortal predecessors had failed. Newton did not discover the cause of gravity any more than Galileo did. Two bodies placed in juxtaposition approach each other. Newton does not inquire into the nature of the force which produces this effect. The force exists, he designates it by the term attraction; but, at the same time, he warns the reader that the term as thus used by him does not imply any definite idea of the physical process by which gravity is brought into existence and operates.

The force of attraction being once admitted as a fact, Newton studies it in all terrestrial phenomena, in the revolutions of the moon, the planets, satellites, and comets; and, as we have already stated, he deduced from this incomparable study the simple, universal, mathematical characteristics of the forces which preside over the movements of all the bodies of which our solar system is composed.

The applause of the scientific world did not prevent the immortal author of the Principia from hearing some persons refer the principle of gravitation to the class of occult qualities. This circumstance induced Newton and his most devoted followers to abandon the reserve which they had hitherto considered it their duty to maintain. Those persons were then charged with ignorance who regarded attraction as an essential property of matter, as the mysterious indication of a sort of charm; who supposed that two bodies may act upon each other without the intervention of a third body. This force was then either the result of the tendency of an ethereal fluid to move from the free regions of space, where its density is a maximum, towards the planetary bodies around which there exists a greater degree of rarefaction, or the consequence of the impulsive force of some fluid medium.

Newton never expressed a definitive opinion respecting the origin of the impulse which occasioned the attractive force of matter, at least in our solar system. But we have strong reasons for supposing, in the present day, that in using the word impulse, the great geometer was thinking of the systematic ideas of Varignon and Fatio de Duillier, subsequently reinvented and perfected by Lesage: these ideas, in effect, had been communicated to him before they were published to the world.

According to Lesage, there are, in the regions of space, bodies moving in every possible direction, and with excessive rapidity. The author applied to these the name of ultra-mundane corpuscles. Their totality constituted the gravitative fluid, if indeed, the designation of a fluid be applicable to an assemblage of particles having no mutual connexion.

A single body placed in the midst of such an ocean of movable particles, would remain at rest although it were impelled equally in every direction. On the other hand, two bodies ought to advance towards each other, since they would serve the purpose of mutual screens, since the surfaces facing each other would no longer be hit in the direction of their line of junction by the ultra-mundane particles, since there would then exist currents, the effect of which would no longer be neutralized by opposite currents. It will be easily seen, besides, that two bodies plunged into the gravitative fluid, would tend to approach each other with an intensity which would vary in the inverse proportion of the square of the distance.

If attraction is the result of the impulse of a fluid, its action ought to employ a finite time in traversing the immense spaces which separate the celestial bodies. If the sun, then, were suddenly extinguished, the earth after the catastrophe would, mathematically speaking, still continue for some time to experience its attractive influence. The contrary would happen on the occasion of the sudden birth of a planet; a certain time would elapse before the attractive force of the new body would make itself felt on the earth.

Several geometers of the last century were of opinion that the force of attraction is not transmitted instantaneously from one body to another; they even assigned to it a comparatively inconsiderable velocity of propagation. Daniel Bernoulli, for example, in attempting to explain how the spring tide arrives upon our coasts a day and a half after the sizygees, that is to say, a day and a half after the epochs when the sun and moon are most favourably situated for the production of this magnificent phenomenon, assumed that the disturbing force required all this time (a day and a half) for its propagation from the moon to the ocean. So feeble a velocity was inconsistent with the mechanical explanation of attraction of which we have just spoken. The explanation, in effect, necessarily supposes that the proper motions of the celestial bodies are insensible compared with the motion of the gravitative fluid.

After having discovered that the diminution of the eccentricity of the terrestrial orbit is the real cause of the observed acceleration of the motion of the moon, Laplace, on his part, endeavoured to ascertain whether this mysterious acceleration did not depend on the gradual propagation of attraction.

The result of calculation was at first favourable to the plausibility of the hypothesis. It showed that the gradual propagation of the attractive force would introduce into the movement of our satellite a perturbation proportional to the square of the time which elapsed from the commencement of any epoch; that in order to represent numerically the results of astronomical observations it would not be necessary to assign a feeble velocity to attraction; that a propagation eight millions of times more rapid than that of light would satisfy all the phenomena.

Although the true cause of the acceleration of the moon is now well known, the ingenious calculation of which I have just spoken does not the less on that account maintain its place in science. In a mathematical point of view, the perturbation depending on the gradual propagation of the attractive force which this calculation indicates has a certain existence. The connexion between the velocity of perturbation and the resulting inequality is such that one of the two quantities leads to a knowledge of the numerical value of the other. Now, upon assigning to the inequality the greatest value which is consistent with the observations after they have been corrected for the effect due to the variation of the eccentricity of the terrestrial orbit, we find the velocity of the attractive force to be fifty millions of times the velocity of light!

If it be borne in mind, that this number is an inferior limit, and that 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 Mecanique Celeste 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 Mecanique Celeste 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 nebulae 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 Mecanique Celeste 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 nebulae 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 Mecanique Celeste. The Systeme du Monde and the Theorie Analytique des Probabilites would also require detailed notices.

The Exposition du Systeme du Monde is the Mecanique Celeste divested of the great apparatus of analytical formulae 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 Systeme 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 Caesar, 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 Theorie Analytique des Probabilites is worthy of the author of the Mecanique Celeste.

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 Mecanique Celeste and the Traite des Probabilites.

On the 27th Vendemiaire in the year X., General Bonaparte, after having received a volume of the Mecanique Celeste, 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 Mecanique Celeste, 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 Mecanique Celeste 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 Traite du Calcul des Probabilites, wrote from Witepsk the letter which we transcribe textually:—

"There was a time when I would have read with interest your Traite du Calcul des Probabilites. 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 oblige!

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.