FORMERLY FELLOW, NOW HONORARY FELLOW OF TRINITY COLLEGE; SECOND EDITION. LONDON: MACMILLAN AND CO., 1884. PREFACE TO THE FIRST EDITION. IN laying this work before the public, I think it right to state the object for which it was originally composed, and the circumstances which have in some degree changed its destination. The treatise was originally designed for a class of readers who might be supposed to possess a moderate acquaintance with the phænomena and the terms of astronomy; geometrical notions sufficient to enable them to understand simple inferences from diagrams ; two or three terms of algebra as applied to numbers; but none of that elevated science which has always been used in the investigation of these subjects, and without which scarcely an attempt has been made to explain them. I proposed to myself, therefore, this general design: to explain the perturbations of the solar system, as far as I was able, without introducing an algebraic symbol. It will readily be believed that, after thus denying myself the use of the most powerful engine of mathematics, I did not expect to proceed very far. In my progress, however, I was surprised to find that a general explanation, perfectly satisfactory, might be offered for almost every inequality recognised as sensible in works on Physical Astronomy. I now began to conceive it possible that the work, without in the smallest degree departing from the original plan, or giving up the original object, might also be found useful to a body of students, furnished with considerable mathematical powers, and in the habit of applying them to the explanation of difficult physical problems. With this idea, the treatise is now printed in a separate form. The utility of a popular explanation of profound physical investigations is not, in my opinion, to be restricted to the instruction of readers who are unable to pursue them with the powers of modern analysis. Much is done when the interest of a good mathematician is excited by seeing, in a form that can be easily understood, results which are important for the comprehension of the system of the universe, and which can be made complete only by the application of a higher calculus. That such an interest has operated powerfully in our Universities, I have no doubt. How many of our students would have known anything of the Lunar Theory, if they had not been enjoined to read Newton's eleventh section? And how many at this time possess the least acquaintance with the curious and complicated, but beautiful, theory of Jupiter's satellites, of which no elementary explanation is laid before them? But this is not all. The exercise of the mind in understanding a series of propositions, where the last conclusion is geometrically in close connexion with the first cause, is very different from that which it receives from putting in play the long train of machinery in a profound analytical process. The degrees of conviction in the two cases are very different. It is known to every one who has been engaged in the instruction of students at our Universities, that the results of the differential calculus are received by many, rather with the doubts of imperfect faith than with the confidence of rational conviction. Nor is this to be wondered at; a clear understanding of many difficult steps, a distinct perception that every connexion of these steps is correct, and a general comprehension of the relations of the whole series of steps, are necessary for complete confidence. An unusual combination of talents, attainments, and labour, must be required, to appreciate clearly the evidence for a result of deep analysis. I am not unwilling to avow that the simple considerations which have been forced upon me in the composition of this treatise, have, in several instances, contributed much to clear up my view of points, which before were obscure, and almost doubtful. To the greater number of students, therefore, I conceive a popular geometrical explanation is more useful than an algebraic investigation. But even to those who are able to pursue the investigations with a skilful use of the most powerful methods, I imagine that a popular explanation is not unserviceable. The insight which it gives into the relation of some mechanical causes and geometrical effects, may powerfully, yet imperceptibly, |