In the greater part of this investigation, the theorems are such as have been long since deduced by more ordinary methods : the deduction of them here was however essential, in order to preserve the unity of the work, and to show that the simpler truths, as well as the more difficult, make parts of the same system, and emanate from the same principle. These more elementary investigations have this further advantage, that the knowledge of the calculus, and of the methods peculiar to this work, is thus gra. dually acquired, by beginning from the more simple cases; and we are prepared, by that means, for the inore difficult problems that are to follow, The general methods of integrating the differential equations above mentioned, are laid down in the Fifth Chapter, which deserves to be studied with particular attention, whether we would improve in the knowledge of the pure or the mixt mathematics, The calculus of variations is introduced with great effect in the last article of this chapter, A very curious subject of investigation, and one that we believe to be altogether new, follows in the next chapter, In the general movement of a system of bodies, such as is here supposed, and such, too, as is actually exemplified in nature, every thing is in moțion; not only every body, but the plane of every orbit, The mutual action of the planets changes the positions of the planes in which they revolve; and they are perpetually made to depart, by a small quantity, on one side or another, each from that planė in which it would go on continually, if their mụtual action were to cease. The calculus makes it appear, that the in, clinations of these orbits in the planetary system is stable, or that the planes of the orbits oscillåte a little, bạckwards and for, wards, on each side of a fixt and immoveable plane. This plane is shown to be one, on which, if every one of the bodies of the system be projected by a perpendicular let fall from it, and if the mass of each body be multiplied into the area described in a given time by its projection on the said plane, the sum of all these products shall be a maximum. Froin this condition, the method of determining the immoveable plane is deduced ; and in the progress of science, when observations made at a great distance of time shall be compared together, the reference of them to an immoveable plane must become a înatter of great importance to astronomers. As the great problem resolved in this first book is that which is called the problem of the three bodies, it may be proper to give some account of the steps by which mathematicians have been gradually conducted to a solution of it so perfect as that which is given by La Place. The problem is,--Having given the masses given ; e lines, and the one anoisy and masses of three bodies projected from three points given in posi. tion with velocities given in their quantity and direction, and supposing the bodies to gravitate to one another with forces that áre as their masses directly and the squares of their distances inversely, to find the lines described by these bodies, and their position, at any given instant. . The problem may be rendered still more general, by supposing the number of bodies to be greater than three.' To resolve the problem in the general form contained in either of these enunciations, very far exceeds the powers even of the nost improved analysis. ' In the cases, however, where it applies to the heavens, that is, when one of the bodies is very great and powerful in respect of the other two, a solution by approximation, and having any required degree of accuracy, may be obtained. · When the number of bodies is only two, the problem admits of a complete solution. Newton had accordingly resolved the problem of two bodies gravitating to one another, in the most perfect manner; and had shown, that when their mutual gravitation is as their masses divided by the squares of their distances, the orbits they describe are conic sections. The application of this theorem and its corollaries to the motions of the planets round the sun, furnished the most beautiful explanation of natural phenomena that had yet been exhibited to the world ; and however excellent, or in some respects superior, the analytical methods may be that have since been applied to this problem, we hope that the original demonstrations will never be overlooked. When Newton, however, endeavoured to apply the same methods to the case of a planet disturbed in its motion round its primary by the action of a third body, the difficulties were too great to be completely overcome. The efforts, nevertheless, which he made with instruments that, though powerful, were still inadequate to the work in which they were employed, displayed, in a striking manner, the resources of his genius, and conducted him to many valuable discoveries. Five of the most considerable of the inequalities in the moon's motion were explained in a satisfactory manner, and referred to the sun's action ; but beyond this, though there is some reason to think that Newton ato tempted to proceed, he has not made us acquainted with the route which he pursued. It was evident, however, that beside these five inequalities, there were many more, of less magnitude indeed, but of an amount that was often considerable, though the laws which they were subject to were unknown, and were never likely to be discovered by observation alone. 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 con. tinue 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 ef. fects 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 theoTy 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 their meowledge of from two ents ents are found, which makes the series agree most accurately with observation. Mayer, who was himself a very skilful astrom 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 inyerse 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 now 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 obseryation, Thus, · 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 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 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 acceleyation. It was observed, that most of the agents we are acquainted |