satisfactory examination in the mere book learning of the subject. The most valuable effect however of the study of science is to awaken originality or at least activity of thought, and to prepare the student for a state of intellect beyond mere logical assent to a number of tolerably easy propositions. In order that his mind may be aroused from the unproductive torpidity of mere acquiescense in prescribed demonstrations of standard theorems, it is necessary that he should exercise his invention in the solution of problems by the application of his primary principles and theorems. As the best method of acquiring a power of independent thought, I would strongly urge him to be in the habit of concentrating his attention on the kindred properties of each fundamental idea, and to abandon the too common practice of the indiscriminate solution of problems connected together by no principle of affinity. In order to facilitate this process of study, I have written this and other similar works. By the classification of numerous problems in distinct groups, I have endeavoured to induce the student to expend an adequate amount of thought at one time in the exclusive consideration of the phenomena of each important theorem, regarded as a characteristic type of its corresponding group, hoping to convey to his mind, through the medium of an enlarged comparison of associated properties, an abiding and energetic conception of each grand principle of the Science. For far the greater number of the problems, which are contained in this Volume, I am indebted to the printed Examination papers, proposed by the University Examiners for the Mathematical Tripos and by the Examiners in the various Colleges. My reason for so largely availing myself of these sources is twofold; first, because it is impossible for any one man to invent a large number of problems equal in value to the best of those contributed by many; and, secondly, because, by bringing together, as I have done, so many of the problems which have of late years been actually proposed to Candidates for the various University and College distinctions, I have prepared a collection of problems which may certainly be regarded as fair specimens of the class of problems to which, in this branch of Mechanics, the University of Cambridge directs the attention of its younger members. As the doctrine of Cycloidal Oscillations is especially mentioned in the Schedule of the subjects prescribed for the First Three Days of the Examination for the Mathematical Tripos, I have thought it desirable to publish, in an Appendix to this Volume, a new demonstration, by Mr R. L. Ellis of Trinity College, of the tautochronism of the Cycloid, which I think cannot fail to be interesting to many of those who are engaged in the study of Elementary Mechanics. The utility of a work of this kind to a beginner in mathematical studies would be much impaired, were the answers to the various problems, the solutions of which are not given in full, very frequently erroneous. I have accordingly taken much pains to render the results accurate: I cannot however hope to have escaped the commission of sundry errors, and should feel much indebted to any one who would be so kind as to direct my attention to any mistakes, the rectification of which might be effected either in a new table of errata or ultimately in a second edition of the work, if called for by the public. CAMBRIDGE, |