The solution of every mechanical problem, in which the acting forces were known, as in the motions of the solar system, had been reduced by D’Alembert and Lagrange to such a state that the difficulties were only mathematical; that is, no farther advances could be made, except in pure analysis. We cannot expect the general reader to know what is meant by the words, _solution of a Differential Equation_; but he may be made aware that there is a process so called, which, if it could be successfully and exactly performed in all cases, would give the key to every motion of the solar system, and render the determination of its present, and the prediction of its future state, a matter of mathematical certainty. Unfortunately, in the present state of analysis, such precision is unattainable; and its place is supplied by slow and tedious approximations. These were begun by Newton, whose object being to establish the existence of universal gravitation, he was content to show that all the phenomena which might be expected to result, if that theory were true, did actually take place in the solar system. But here, owing to the comparatively imperfect state of mathematical analysis, he could do little more than indicate the cause of some of the principal irregularities of that system. His successors added considerably to the number of phenomena which were capable of explanation, and thereby increased the probability of the hypothesis. Lagrange, the great rival of Laplace, if we consider his discoveries, and his superior in the originality of his views, and the beauty of his analysis, added greatly to the fund; but it was reserved for the latter to complete the system, and, extending his views beyond the point to which Newton directed his attention, to show that there is no marked phenomenon yet observed by astronomers, regarding the relative motions of the planets or their satellites, but what must necessarily follow, if the law of gravitation be true. We shall select a few instances of the success of his analysis. The average motions of Jupiter and Saturn had been observed to vary; that of the former being accelerated, and of the latter retarded. This fact, which Euler had attempted in vain to explain, was linked by Laplace to the general law, and shown to follow from it. A somewhat similar acceleration in the moon’s mean motion was demonstrated, as we have observed more fully in the life of Halley, to arise from a small alteration in the form of the earth’s orbit, caused by the attraction of the planets. A remarkable law attending the motions of the satellites of Jupiter, viz.—that the mean motion of the first satellite, together with double that of the second, is always very nearly equal to three times that of the third—was so far connected with the general law, that if, in the original formation of the system, that relation had been nearly kept, the mutual attractions, instead of altering it, would tend to bring it nearer the truth.