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It is the glory of the Newtonian philosophy, not to have been limited to the precise point of perfection to which it was carried by its author; nor, like all the systems which the world had yet seen, from the age of Aristotle to that of Descartes, either to continue stationary, or to decline gradually from the moment of its publication. Three geometers, who had studied in the schools of Newton and of Leibnitz, and had greatly improved the methods of their masters, ventured, nearly about the same time, each unknown to the other two, to propose to himself the problem which has since been so well known under the name of the Problem of Three Bodies. Clairaut, D'Alembert and Euler, are the three illustrious men, who, as by a common impulse, undertook this investigation in the year 1747; the priority, if any could be claimed, being on the side of Clairaut. The object of those geometers was not merely to explain the lunar inequalities that had been observed; they aimed at something higher; viz. from theory to investigate all the inequalities that could arise as the effects of gravitation, and so to give an accuracy to the tables of the moon, that they could not derive from observation alone. Thus, after having ascended with Newton from phenomena to the principle of gravitation, they were to descend from that principle to the discovery of new facts; and thus, by the twofold method of analysis and composition, to apply to their theory the severest test, the only infallible criterion that at all times distinguishes truth from falsehood. Clairaut was the first who deduced, from his solution of the problem, a complete set of lunar tables, of an accuracy far superior to any thing that had yet appeared, and which, when compared with observation, gave the moon's place, in all situations, very near the truth.
Their accuracy, however, was exceeded, or at least supposed to be exceeded, by another set produced by Tobias Mayer of Gottingen, and grounded on Euler's solution, compared very diligently with observation. The expression of the lunar irregularities, as deduced from theory, is represented by the terms of a series, in each of which there are two parts carefully to be distinguished; one, which is the sine or cosine of a variable angle determined at every instant by the time counted from a certain epocha; another, which is a coefficient or multiplier, in itself constant, and remaining always the same. The determination of this constant part may be derived from two different sources; either from our knowledge of the masses of the sun and moon, and their mean distances from the earth; or from a comparison of the series above mentioned, with the observed places of the moon, whence the values of the coeffici
ents are found, which makes the series agree most accurately with observation. Mayer, who was himself a very skilful astro nomer, had been very careful in making these comparisons; and thence arose the greater accuracy of his tables. The problem of finding the longitude at sea, which was now understood to de pend so much on the exactness with which the moon's place could be computed, gave vast additional value to these researches, and established a very close connexion between the conclusions of theory, and one of the most important of the arts, Mayer's tables were rewarded by the Board of Longitude in England; and Eu ler's, at the suggestion of Turgot, by the Board of Longitude in France.
It may be remarked here, as a curious fact in the history of science, that the accurate solution of the problem of the Three Bodies, which has in the end established the system of gravitation on so solid a basis, seemed, on its first appearance, to threaten the total overthrow of that system. Clairaut found, on determining, from his solution, the motion of the longer axis of the moon's orbit, that it came out only the half of what it was known to be from astronomical observation. In consequence of this, he was persuaded, that the force with which the earth attracts the moon, does not decrease exactly as the squares of the distances increase, but that a part of it only follows that law, while another follows the inverse of the biquadrate or fourth power of the distances, The existence of such a law of attraction was violently opposed by Buffon, who objected to it the want of simplicity, and argued that there was no sufficient reason for determining what part of the attraction should be subject to the one of these laws, and what part to the other. Clairaut, and the other two mathematicians, (who had come to the same result), were not much influenced by this metaphysical argument; and the former proceeded to inquire what the proportion was between the two parts of the attraction that followed laws so different.
He was thus forced to carry his approximation further than he had done, and to include some quantities that had before been rejected as too small to affect the result, When he had done this, he found the numerator of the fraction that denoted the part of gravity which followed the new law, equal to no thing; or, in other words, that there was no such part. The candour of Clairaut did not suffer him to delay, a moment, the acknowledgement of this result; and also, that when his calculus was rectified, and the approximation carried to the full length, the motion of the moon's apsides as deduced from theory, coincided exactly with observation,
Thus, the lunar theory was brought to a very high degree of perfection; and the tables constructed by means of it, were found to give the moon's place true to 30". Still, however, there was one inequality in the moon's motion, for which the principle of gravitation afforded no account whatever. This was what is known by the name of the moon's acceleration. Dr Halley had observed, on comparing the ancient with modern observations, that the moon's motion round the earth appeared to be now performed in a shorter time than formerly; and this inequality appeared to have been regularly, though slowly, increasing; so that, on computing backward from the present time, it was necessary to suppose the moon to be uniformly retarded, (as in the case of a body ascending against gravity), the effect of this retardation increasing as the squares of the time. All astronomers admitted the existence of this inequality in the moon's motion; but no one saw any means of reconciling it with the principle of gravitation. All the irregularities of the moon arising from that cause had been found to be periodical; they were expressed in terms of the sines and cosines of arches; and though these arches depend on the time, and might increase with it continually, their sines and cosines had limits which they never could exceed, and from which they returned perpetually the same order. Here, therefore, was one of the greatest anomalies yet discovered in the heavens-an inequality that increased continually, and altered the mean rate of the moon's motion. Various attempts were made to explain this phenomenon, and those too attended with much intricate and laborious investigation.
To some it appeared, that this perpetual decrease in the time of the moon's revolution, must arise from the resistance of the medium in which she moves, which, by lessening her absolute velocity, would give gravity more power over her; so that she would come nearer to the earth, would revolve in less time, and therefore with a greater angular velocity. This hypothesis, though so unlike what we are led to believe from all other appearances, must have been admitted, if, upon applying mathematical reasoning, it had been found to afford a good explanation of the appearances. It was found, however, on trial, that it did not; and that the moon's acceleration could not be explained by the supposed resistance of the ether.
Another hypothesis occurred, from which an explanation was attempted of this and of some great inequalities in the motions of Jupiter and Saturn, that seemed not to return periodically, and were therefore nearly in the same circumstances with the moon's accele ration. It was observed, that most of the agents we are acquainted
with take time to pass from one point of space to another; that the force of gravity may be of this sort, and may not, any more than light, be instantaneously transmitted from the sun to the planets, or from the planets to one another. The effect that would arise from the time thus taken up by gravity, in its transmission from one point of space to another, was therefore investigated by the strictest laws of geometry; but it was found, that this hypothesis did not, any more than the preceding, afford an explanation of the moon's acceleration.
By this time also, it was demonstrated, that there was not, and could not be in our system, any inequality whatever produced by the mutual gravitation of the planets, that was not periodical, and that did not, after reaching a certain extent, go on to diminish by the same law that it had increased.
An entire suspense of opinion concerning the moon's acceleration therefore took place, till La Place found out a truth that had eluded the search of every other mathematician. It was known to him, both from the investigation of La Grange, and from his own, that there are changes in the eccentricities of the planetary orbits, extremely slow, and of which the full series is not accomplished but in a very long period. The eccentricity of the earth's orbit is subject to this sort of change; and as some of the lunar inequalities are known to depend on that eccentricity, they. must vary slowly along with it; and hence an irregularity of a very long period in the moon's motion. On examining further, and the examination was a matter of great difficulty, La Place found this inequality to answer very exactly to what we have called the acceleration of the moon; for though, in strictness, it is not uniform, it varies so slowly, that it may be accounted uniform for all the time that astronomical observation has yet existed. It is a quantity of such a kind, and its period of change is so long, that for an interval of two thousand years, it may be considered as varying uniformly. Two thousand years are little more than an infinitesimal in this reckoning; and as an astronomer thinks he commits no error when he considers the rate of the sun's motion as uniform for twenty-four hours, so he commits none when he regards the rate of this equation as continuing the same for twenty centuries. That man, whose life, nay, the history of whose species, occupies such a mere point in the duration of the world, should come to the knowledge of laws that embrace myriads of ages in their revolution, is perhaps the most astonishing fact that the history of science exhibits.
Thus La Place put the last hand to the theory of the moon, nearly one hundred years after that theory had been propounded in the first edition of the Principia.
The branch of the theory of disturbing forces that relates to the action of the primary planets on one another, was cultivated during the same period, with equal diligence, and with equal success. In the years 1748 and 1752, the academy of sciences proposed for prize questions the inequalities of Jupiter and Saturn; both the prizes were gained by Euler, whose researches have thrown so much light on all the more difficult questions, both of the pure and the mixt mathematics. There was a particular difficulty that attended this inquiry, and distinguished it greatly from the case of the moon disturbed in its motion by so distant a body as the sun. In the case of Jupiter and Saturn, the disturbing body may be as near to the one disturbed as this last is to the body about which it revolves; for the distance of Saturn from Jupiter may sometimes be nearly the same with that of Jupiter from the sun. In such cases, the means of obtaining a series expressing the force of the one planet on the other, and converging quickly, was quite different from any thing required in the case of the moon, and was a matter of extreme difficulty. No man was more fit than Euler to contend with such a difficulty; he accordingly overcame it; and his mode of doing so has served as the model for all the similar It resulted from his inresearches that have since been made. vestigation, that both the planets were subject to considerable inequalities, depending on the action of one another, but all of them periodical, and returning after certain stated intervals, not exceeding twenty or thirty years, nearly in the same order.
Though this agreed well with astronomical observations so far as it went, yet it afforded no account of two inequalities of very long periods, or perhaps of indefinite extent, which, by the comparison of ancient and modern observations, seemed to affect the motions of these two planets in opposite directions.
This was a subject, therefore, that remained for further discussion. In the mean time, it was considered that the other planets must no doubt be affected in the same way; and both Euler and Clairaut gave computations of the disturbance which the earth suffers from Jupiter, Venus, and the Moon. The same was extended to the other planets; and a great additional degree of accuracy was thus given to all the tables of the planetary mo
In the course of these researches, the change in the obliquity cf the ecliptic came first to be perfectly recognized, and ascribed to the action of the planets above named on the earth. It was area Euler, that the change in this obliquity is periodical, it the others we have already seen; that it is not a constant dimu, but a small and slow oscillation on each side of a