[25] It may perhaps be asked why we place Lagrange among the French geometers? This is our reply: It appears to us that the individual who was named Lagrange Tournier, two of the most characteristic French names which it is possible to imagine, whose maternal grandfather was M. Gros, whose paternal great-grandfather was a French officer, a native of Paris, who never wrote except in French, and who was invested in our country with high honours during a period of nearly thirty years;—ought to be regarded as a Frenchman although born at Turin.—Author.
[26] The problem of three bodies was solved independently about the same time by Euler, D'Alembert, and Clairaut. The two last-mentioned geometers communicated their solutions to the Academy of Sciences on the same day, November 15, 1747. Euler had already in 1746 published tables of the moon, founded on his solution of the same problem, the details of which he subsequently published in 1753.—Translator.
[27] It must be admitted that M. Arago has here imperfectly represented Newton's labours on the great problem of the precession of the equinoxes. The immortal author of the Principia did not merely conjecture that the conical motion of the earth's axis is due to the disturbing action of the sun and moon upon the matter accumulated around the earth's equator: he demonstrated by a very beautiful and satisfactory process that the movement must necessarily arise from that cause; and although the means of investigation, in his time, were inadequate to a rigorous computation of the quantitative effect, still, his researches on the subject have been always regarded as affording one of the most striking proofs of sagacity which is to be found in all his works.—Translator.
[28] It would appear that Hooke had conjectured that the figure of the earth might be spheroidal before Newton or Huyghens turned their attention to the subject. At a meeting of the Royal Society on the 28th of February, 1678, a discussion arose respecting the figure of Mercury which M. Gallet of Avignon had remarked to be oval on the occasion of the planet's transit across the sun's disk on the 7th of November, 1677. Hooke was inclined to suppose that the phenomenon was real, and that it was due to the whirling of the planet on an axis "which made it somewhat of the shape of a turnip, or of a solid made by an ellipsis turned round upon its shorter diameter." At the meeting of the Society on the 7th of March, the subject was again discussed. In reply to the objection offered to his hypothesis on the ground of the planet being a solid body, Hooke remarked that "although it might now be solid, yet that at the beginning it might have been fluid enough to receive that shape; and that although this supposition should not be granted, it would be probable enough that it would really run into that shape and make the same appearance; and that it is not improbable but that the water here upon the earth might do it in some measure by the influence of the diurnal motion, which, compounded with that of the moon, he conceived to be the cause of the Tides." (Journal Book of the Royal Society, vol. vi. p. 60.) Richer returned from Cayenne 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 1668, 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; but his method for computing the ellipticity is founded upon that given in the Principia.—Translator.
[29] 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.
[30] 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 sum 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 in different latitudes. Hence it is plain that Clairaut's theorem furnishes a practical method for determining the value of the earth's ellipticity.—Translator.
[31] 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 coefficients of which 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 definitive conclusion on this subject, it is indispensable to have recourse to the actual solution of the algebraic equation above referred to. This was the course adopted by the illustrious Lagrange in his researches on the secular variations of the planetary orbits. (Mem. Acad. Berlin, 1783-4.) 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 depended. In this way, he found the limiting values of the eccentricity and inclination for the orbit of each of the principal planets of the system. The results obtained by that great geometer have been mainly confirmed by the recent researches of Le Verrier on the same subject. (Connaissance des Temps, 1843.)—Translator.
[32] Laplace was originally led to consider the subject of the perturbations of the mean motions of the planets by his researches on the theory of Jupiter and Saturn. Having computed the numerical value of the secular inequality affecting the mean motion of each of those planets, neglecting the terms of the fourth and higher orders relative to the eccentricities and inclinations, he found it to be so small that it might be regarded as totally insensible. Justly suspecting that this circumstance was not attributable to the particular values of the elements of Jupiter and Saturn, he investigated the expression for the secular perturbation of the mean motion by a general analysis, neglecting, as before, the fourth and higher powers of the eccentricities and inclinations, and he found in this case, that the terms which were retained in the investigation absolutely destroyed each other, so that the expression was reduced to zero. In a memoir which he communicated to the Berlin Academy of Sciences, in 1776, Lagrange first showed that the mean distance (and consequently the mean motion) was not affected by any secular inequalities, no matter what were the eccentricities or inclinations of the disturbing and disturbed planets.—Translator.
[33] Mr. Adams has recently detected a remarkable oversight committed by Laplace and his successors in the analytical investigation of the expression for this inequality. The effect of the rectification rendered necessary by the researches of Mr. Adams will be to diminish by about one sixth the coefficient of the principal term of the secular inequality. This coefficient has for its multiplier the square of the number of centuries which have elapsed from a given epoch; its value was found by Laplace to be 10".18. Mr. Adams has ascertained that it must be diminished by 1".66. This result has recently been verified by the researches of M. Plana. Its effect will be to alter in some degree the calculations of ancient eclipses. The Astronomer Royal has stated in his last Annual Report, to the Board of Visitors of the Royal Observatory, (June 7, 1856,) that steps have recently been taken at the Observatory, for calculating the various circumstances of those phenomena, upon the basis of the more correct data furnished by the researches of Mr. Adams.—Translator.