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mean quantity, by which it alternately increases and diminishes in the course of periods, which are not all of the same length, but by which, in the course of many ages, a compensation ultimately takes place.
Still, however, the secular inequalities to which 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. La Grange afterwards undertook the same investigation; but his results were unsatisfactory; and La Place himself, on pushing his approximation further 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 La Grange, 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 inequa lities 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 priniple, which we may consider as the bulwark that secures the sta
bility of our system, and excludes all access to confusion and disorder, must render the name of La Grange 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, La Grange'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 La Place stepped in again to extricate philoso phers 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 signes.
These two results, therefore, are deduced from the theory of gravitation, and, when applied to the comparison of the antient and modern observations, are found to reconcile them precisely with one another. 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. La Place 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 fulfill 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 La Place 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 Mechanique Céleste, and deducing them from the same principles in one and the same method. La Place, 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 cooperators in this unexampled career of discovery, La Grange and La Place 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 Mechanique 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. These eccentricities, however, continually change, though very slowly, in the progress of time, but in such a manner, that none of them can ever become very great. They may vanish, or become nothing, when the orbit will be exactly circular; in which state, however, it will not continue, but change in the course of time, into an ellipsis, of an eccentricity that will vary as before, so as never to exceed a certain limit. What this limit is for each individual planet, would be difficult to determine, the expression of the variable eccentricities being necessarily very complex. But, notwithstanding of this, a general theorem, which shows that none of them can ever become great, is the result of one of La Place's investigations. It is this: If the mass of each planet be multiplied into the square of the eccentricity of its orbit, and this product into the square root of the axis of the same orbit, the sum of all these quantities, when they are added together, will reS
VOL. XI. NO. 22.
main for ever the same. This sum is a constant magnitude, which the mutual action of the planets cannot change, and which nature preserves free from alteration. Hence no one of the eccentricities can ever increase to a great magnitude; for as the mass of each planet is given, and also its axis, the square of the eccentricity in cach, is multiplied into a given coefficient, and the sum of all the products so formed, is incapable of change. Here, therefore, we have again another general property, by which the stability of our system is maintained; by which every great alteration is excluded, and the whole made to oscillate, as it were, about a certain mean quantity, from which it can never greatly depart.
If it be asked, is this quantity necessarily and unavoidably permanent in all systems that can be imagined, or under every possible constitution of the planetary orbits? We answer, by no means: if the planets did not all move one way,-if their orbits were not all nearly circular, and if their eccentricities were not small, the permanence of the preceding quantity would not take place. It is a permanence, then, which depends on conditions that are not necessary in themselves; and therefore we are authorized to consider such permanence as an argument of design in the construction of the universe.
When we thus obtain a limit, beyond which all the changes that can ever happen in our system shall never pass, we may be said to penetrate, not merely into the remotest ages of futurity, but to look beyond them, and to perceive an object, situated, if we may use the expression, on the other side of infinite dura
Though in the detail into which we have now entered, we have anticipated many things that may be thought to belong to another place, we think that the leading facts are in this way least separated from one another. La Place, after treating of the problemst of the Three Bodies generally in the Second book, to which the observations made above chiefly refer, resumes the consideration of the same problem, and the application of it to the tables of the planets in the Sixth and Seventh. These we shall be able to pass over slightly, as much of what might be said concerning them, is contained in the preceding remarks. We go on now to the Second volume, which treats of the figure of the planets, and of the tides.
In the First book, a foundation was laid for this research by the general theorems that were investigated concerning the equilibrium of fluids and the rotation of bodies. These are applied here; first, to the figure of the planets in general; and afterwards particularly to the figure of the earth.
The first inquiry into the physical causes which determine the
figure of the earth and of the other planets, was the work of Newton, who showed, that a fluid mass revolving on its axis, and its particles gravitating to one another with forces inversely as the squares of their distances, must assume the figure of an oblate spheroid; and that, in the case of a homogeneous body, where the centrifugal force bore the same ratio to the force of gravity that obtains at the surface of the earth, the equatorial diameter of the spheroid must be to the polar axis as 231 to 230. The method by which this conclusion was deduced, was however by no means unexceptionable, as it took for granted, that the spheroid must be elliptical. The defects of the investigation were first supplied by Maclaurin, who treated the subject of the figure of the earth in a manner alike estimable for its accuracy and its elegance. His demonstration had the imperfection, at least in a certain degree, of being synthetical; and this was remedied by Clairaut; who, in a book on the figure of the earth, treated the subject still more fully; simplified the view of the equilibrium that determines the figure; and showed the true connexion between the compression at the poles and the diminution of gravity on going from the poles to the equator, whatever be the internal structure of the spheroid. Several mathematicians considered the same subject afterwards; and, in particular, Le Gendre proved, that, for every fluid mass given in magnitude, and revolving on its axis in a given time, there are two elliptic spheroids that answer the conditions of equilibrium; in the instance of the earth, one of these has its eccentricity in the ratio of 231 to 230; the other, in the ratio of 680 to 1.
The results of those investigations, with the addition of several quite new, are brought together in the work before us, and deduced according to the peculiar methods of the author. These theoretical conclusions are next applied to the experiments and observations that have been actually made, whether by determining the length of the second's pendulum in different latitudes, or by the measurement of degrees. After a very full discussion, and a comparison of several different arches, on each of which an error is allowed, and this condition superadded, that the sum of the positive and negative errors shall be equal, and, at the same time, the sum of all the errors, supposing them positive, shall be a minimum, La Place finds that the result is not reconcileable with the hypothesis of an elliptic spheroid, unless a greater error be admitted in some of the degrees than is consistent with probability. In this determination, however, the Lapland degree is taken as measured by Maupertuis, and the other academicians who assisted him. The cor rection by the Swedish mathematicians was not made when this part of La Place's work was published. If that correction is attend