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. * 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 aban. don 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 in. creased 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 reigps the most satisfac. tory 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 plauet.f In the time of Alexander, comets were supposed by the majority of the Greek philosophers to be merely meteors generated in our atmos. pbere. Daring the Middle Ages, persons, without giving themselves much concern about the nature of those bodies, supposed them to prognosti. in the year 1674, but the account of his observations with the pendulum during his residence there was not published until 1679, nor is there to be found any allusion to them during the intermediate interval, either in the volumes of the Academy of Sciences or any other publication. We have no means of ascertaining how Newton was first induced to suppose that the figure of the earth is spheroidal, but we know, upon his own authority, that as early as the year 1667 or 1663, he was led to consider the effects of the centrifugal force in diminishing the weight of bodies at the equator. With respect to Huyghens, he appears to have formed a conjecture respecting the spheroidal figure of the earth independently of Newton; bat his method for computing the ellipticity is founded upon that given in the Principia.TRAXSLATOR. * Newton assumed that a homogeneous fluid mass of a spheroidal form would be in equilibrium if it were endued with an adequate rotatory motion and its constituent particles attracted each other in the inverse proportion of the square of the distance. Maclaurin first demonstrated the truth of this theorem by a rigorous application of the ancient geometry.-TRANSLATOR. + The results of Clairaut's researches on the figure of the earth are mainly embodied in a remarkable theorem discovered by that geometer, and which may be enunciated thus: The sam of the fractions expressing the ellipticity and the increase of gravity at the pole is equal to two and a half times the fraction expressing the centrifugal force at the equator, the unit of force being represented by the force of gravity at the equator. This theorem is independent of any hypothesis with respect to the law of the densities of the successive strata of the earth. Now the increase of gravity at the pole may be ascertained by means of observations with the pendulum iu different latitudes. Hence it is plain that Clairaut's theorem furnishes a practical method for determining the valuo of the earth's ellipticity.TRANSLATOR. cate sinister events. Regiomontanus and Tycho Brahé proved by their observations that they are situate beyond the moon; Hevelius, Dörfel, &c., made them revolve around the sun; Newton established that they move under the immediate influence of the attractire 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 philoso. pher 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 as. tronomer shrank. 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 plan. ets ? The answer to this question would introduce comets into the category of ordinary planets, or would exclude them forever. 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 re-appear. He designated the constellations, nay, the stars, which it would encoun. ter in its progress. This was not one of those remote predictions which astrologers and others formerly combined very skillfully with the tables of mortality, so that they might not be falsified during their life-time: 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 inore 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 forever 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 labors 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 endeavoring 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, matbematical equality between elements having do 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 situated 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 toward 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 toward 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 princi. pal 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 attrib. utable 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 origiually 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 discor. eries of Clairaut, D'Alembert, and Lagrange. We shall be somewhat less concise in noticing the labors 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 tbe 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 ; be 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 perturbations, refused also to admit that the solar system was constituted so as to endure forever. Never did a greater philosophical question offer itself to the inquiries of mankind. Laplace attacked it with boldness, perse Terance, and success. The profound and loug.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 experience a displacement in virtue of which their intersections with the plane of the terrestrial orbit are each year directed toward 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; rariety 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 toward 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.* * The researches on the secular variations of the eccentricities and inclinations of the planetary orbits depend upon the solution of an algebraic equation equal in degree to the number of planets whose mutual action is considered, and the co-efficients of wbich involve the values of the masses of those bodies. . It may be shown that if the roots of this equation be equal or imaginary, the corresponding element, whether the eccentricity or the inclination, will increase indefinitely with the time in the case of each planet; but that if the roots, on the other hand, be real and unequal, the value of the element will oscillate in every instance within fixed limits. Laplace proved by a general analysis that the roots of the equation are real and unequal; whence it followed that neither the eccentricity nor the inclination will vary in any case to an indefinite extent. But it still remained uncertain whether the limits of oscillation were not in any instance so far apart that the variation of the element (whether the eccentricity or the inclination) might lead to a complete destruction of the existing physical condition of the planet. Laplace, indeed, attempted to prove, by means of two well-known theorems relative to the eccentricities and inclinations of the planetary orbits, that if those elements were once small they would always remain so, provided the planets all revolved around the sun in one common direction, and their masses were inconsiderable. It is to these theorems that M. Arago manifestly alludes in the text. Le Verrier and others have, however, remarked that they are inadequate to assure the permanence of the existing physical condition of several of the planets. In order to arrive at a definite conclusion on this subject, it is indispensable to have recourse to the actual solution of the algebraic question above referred to. This was the course adopted by the illus. trious Lagrange in his researches on the secular variations of the planetary orbits. (Mem. Acad. Berlin, 1783-'84.) Having investigated the values of the masses of the planets, he then determined, by an approximate solution, the values of the several roots of the algebraic equation upon which the variations of the eccentricities and inclinations of the orbits de |