(304.) The legacy of research which Newton may be said to have left to his successors was truly immense. To pursue, through all its intricacies, the consequences of the law of gravitation; to account for all the inequalities of the planetary movements, and the infinitely more complicated, and to us more important ones, of the moon; and to give, what Newton himself certainly never entertained a conception of, a demonstration of the stability and permanence of the system, under all the accumulating influence of its internal perturbations; this labour, and this triumph, were reserved for the succeeding age, and have been shared in succession by Clairaut, D’Alembert, Euler, Lagrange and Laplace. Yet so extensive is the subject, and so difficult and intricate the purely mathematical enquiries to which it leads, that another century may yet be required to go through with the task. The recent discoveries of astronomers have supplied matter for investigation, to the geometers of this and the next generation, of a difficulty far surpassing any thing that had before occurred. Five primary planets have been added to our system; four of them since the commencement of the present century, and these, singularly deviating from the general analogy of the others, and offering _cases of difficulty_ in theory, which no one had before contemplated. Yet even the intricate questions to which these bodies have given rise seem likely to be surpassed by those which have come into view, with the discovery of several comets revolving in elliptic orbits, like the planets, round the sun, in very moderate periods. But the resources of modern geometry seem, so far from being exhausted, to increase with the difficulties they have to encounter, and already, among the successors of Lagrange and Laplace, the present generation has to enumerate a powerful array of names, which promise to render it not less celebrated in the annals of physico-mathematical research than that which has just passed away.