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 coefficients are found, which make the series agree most accurately with observation. Mayer, who was himself a very skilful astronomer, 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 depend 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 connection 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 Euler'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 farther 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 nothing; or, in other words, that there was no such part. The candour of Clairaut did not suffer him to delay, a moment, the acknowledgment 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 motion'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 ap. peared 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 those 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 in 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 |