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Jupiter and Saturn were subject, and which seemed to increase continually without' any appearance of returning into themselves, were not accounted for; so that the problem of their disturbance was either imperfectly resolved, or they must be supposed to be subject to the action of a force different from their mutual attraction. In the course of about twenty centuries to which astronomical observation had extended, it appeared that the motion of Jupiter had been accelerated by 3° 23', and that of Saturn retarded by 5° 13'. This had been first remarked by Dr Halley, and had been confirmed by the calculations of all the astronomers who came after him.
With a view to explain these appearances, Euler, resuming the subject, found two inequalities of long periods that belonged to Jupiter and Saturn; but they were not such as, either in their quantity or in their relation to one another, agreed with the facts observed. Lagrange afterwards undertook the same investigation; but his results were unsatisfactory; and Laplace himself, on pushing his approximation farther than either of the other geometers had done, found that no alteration in the mean motion could be produced by the mutual action of these two planets. Physical astronomy was now embarrassed with a great difficulty, and at the same time was on the eve of one of the noblest discoveries it has ever made. The same Lagrange, struck with this circumstance, that the calculus had never yet given
any inequalities but such as were periodical, applied himself to the study of this general question, whether, in our planetary system, such inequalities as continually increase, or continually diminish, and by that means affect the mean motion of the planets, can ever be produced by their mutual gravitation. He found, by a method peculiar to himself, and independent of any approximation, that the inequalities produced by the mutual action of the planets, must, in effect, be all periodical: that amid all the changes which arise from their mutual action, two things remain perpetually the same; viz. the length of the greater axis of the ellipse which the planet describes, and its periodical time round the sun, or, which is the same thing, the mean distance of each planet from the sun, and its mean motion remain constant. The plane of the orbit varies, the species of the ellipse and its eccentricity change; but never, by any means whatever, the greater axis of the ellipse, or the time of the entire revolution of the planet. The discovery of this great principle, which we may consider as the bulwark that secures the stability of our system, and excludes all access to confusion and disorder, must render the name of Lagrange for ever memorable in science, and ever revered by those who delight in the contemplation of whatever is excellent and sublime. After Newton's discovery of the elliptic orbits of the planets, Lagrange's discovery of their
periodical inequalities is, without doubt, the noblest truth in physical astronomy; and, in respect of the doctrine of final causes, it may truly be regarded as the greatest of all.
The discovery of this great truth, however, on the present occasion, did but augment the difficulty with respect to those inequalities of Jupiter and Saturn, that seemed so uniform in their rate; and it became now more than ever probable, that some extraneous cause, different from gravitation, must necessarily be recognized.
It was here that Laplace stepped in again to extricate philosophers from their dilemma. On subjecting the problem of the disturbances of the two planets above mentioned, to a new examination, he found that some of the terms expressing the inequalities of these planets, which seemed small, as they involved the third power of the eccentricities, had very long periods, depending on five times the mean motion of Saturn minus twice the mean motion of Jupiter, which is an extremely small quantity, the mean motion of Jupiter being to the mean motion of Saturn in a ratio not far from that of five to two. Hence it appeared, that each of these planets was subject to an inequality, having a period of nine hundred and seventeen years, amounting in the case of the former, when a maximum, to 48' 44", and in that of the other to 20′ 49′′, with opposite signs.
These two results, therefore, are deduced from the theory of gravitation, and, when applied to the comparison of the ancient and modern observations, are found to reconcile them precisely with one ano. ther.
The two equations had reached their maximum in 1560: from that time, the apparent mean motions of the planets have been approaching to the true, and became equal to them in 1790. Laplace has further observed, that the mean motions which any system of astronomy assigns to Jupiter and Saturn, give us some information concerning the time when that system was formed. Thus, the Hindoos seem to have formed their system when the mean motion of Jupiter was the slowest, and that of Saturn the most rapid; and the two periods which fulfil these conditions, come very near to the year 3102 before the Christian era, and the year 1491 after it, both of them remarkable epochs in the astronomy of Hindostan.
Thus, a perfect conformity is established between theory and observation, in all that respects the disturbances of the primary planets and of the moon; there does not remain a single inequality unexplained; and a knowledge is obtained of several, of which the existence was indicated, though the law could not have been discovered by observation alone.
The discoveries of Laplace had first been communicated in the Memoirs of the Academy of
Sciences; as those of the other mathematicians above mentioned had been, either in these same Memoirs, or in those of Petersburgh and Berlin. An important service is rendered to science, by bringing all these investigations into one view, as is done in the Mécanique Céleste, and deducing them from the same principles in one and the same method. Laplace, though far from the only one who had signalized himself in this great road of discovery, being the person who had put the last hand to every part, and had overcome the difficulties which had resisted the efforts of all the rest, was the man best qualified for this work, and best entitled to the honour that was to result from it. Indeed, of all the great co-operators in this unexampled career of discovery, Lagrange and Laplace himself were the only survivors when this work was published.
We cannot dismiss the general consideration of the problem of the Three Bodies, and of the Second Book of the Mécanique Céleste, without taking notice of another conclusion that relates particularly to the stability of the planetary system. The orbits of the planets are all ellipses, as is well known, having the sun in their common focus; and the distance of the focus from the centre of the ellipsis, is what astronomers call the eccentricity of the orbit. In all the planetary orbits, this eccentricity is small, and the ellipse approaches nearly to a circle.