ILLUSTRATIVE OF THE THEORY OF CURVES BY JAMES THOMSON, LL.D. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF GLASGOW. SECOND EDITION, WITH MANY CHANGES AND IMPROVEMENTS. LONDON: SIMMS AND MINTYRE, 13, PATERNOSTER ROW; AND 26, DONEGALL STREET, BELFAST. PREFACE. THE following publication is intended for the use of persons commencing the study of the Differential and Integral Calculus. It has, therefore, been the aim of the author to render the investigations as simple and easy as possible; and he has illustrated the subjects of inquiry by numerous Examples. Most of the Sections, also, contain a number of questions without solutions, for exercising the student on the principles established in the work, and for training his mind to habits of thought, and cultivating his powers of investigation. For reasons assigned in the APPENDIX, the Method of Limits has been adopted as the basis of the work, instead of Lagrange's Method, which was employed in the former edition. Besides this fundamental change, a great portion of the volume has been re-written; and the whole work, it is hoped, has been very materially improved. The APPENDIX exhibits the elements and some of the leading properties of the Conic Sections, and of many other Curves, several of which are referred to in the preceding parts of the work. It contains, also, short articles on the Convergence of Series, on Definite Integrals, and on the principal methods of investigation that have been employed by mathematicians in the higher branches of science. These will be useful to persons who have but limited access to books, and they may occasionally save time and trouble to others. As the cost of the former edition was felt by many individuals to be an objection to the use of the book, the present edition is published in a different form, and at a reduced price. The amount of matter which the work contains is much larger than would the size of the volume would seem to indicate; and it will be found to be sufficient for the majority of students. Persons, however, who shall have made themselves acquainted with what is here given, will find it easy to read more extensively on subjects which the limits of the present work have made it necessary to treat briefly, and will be able to acquire an acquaintance with others which have been altogether omitted; and, for this purpose, they may have recourse to one or more of the works mentioned in the APPENDIX, such as the treatises of Cournot; Duhamel, and De Morgan. It may be remarked also, that the student who wishes to prosecute the study of mathematical science extensively, will find the most interesting and valuable applications of the Differential and Integral Calculus in the modern works on Mechanics, Physical Astronomy, and other branches of Natural Philosophy. The study of these subjects, indeed,-supplying, as it does, a large amount of valuable knowledge, and displaying in the most striking manner the wonderful triumphs of modern science,ought to be kept in view as a most important sequel to the study of pure mathematics. Glasgow College, Oct. 2, 1848. |