ELEMENTS OF GEOMETRY AND TRIGONOMETRY FROM THE WORKS OF A. M. LEGENDRE ADAPTED TO THE COURSE OF MATHEMATICAL INSTRUCTION J. HOWARD VAN AMRINGE, A.M., PH.D. PROFESSOR OF MATHEMATICS IN COLUMBIA COLLEGE NEW YORK .:. CINCINNATI .:. CHICAGO AMERICAN BOOK COMPANY Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39.48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1992. PREFACE. O F the various treatises on Elementary Geometry which have appeared during the present century, that of M. Legendre stands pre-eminent. Its peculiar merits have won for it not only a European reputation, but have also caused it to be selected as the basis of many of the best works on the subject that have been published in this country. In the original treatise of Legendre, the propositions are not enunciated in general terms, but by means of the diagrams employed in their demonstration. This departure from the method of Euclid is much to be regretted. The propositions of Geometry are general truths, and ought to be stated in general terms, without reference to particular diagrams. In the following work, each proposition is first enunciated in general terms, and afterward with reference to a particular figure, that figure being taken to represent any one of the class to which it belongs. By this arrangement. the difficulty experienced by beginners in comprehending abstract truths is lessened, without in any manner impairing the generality of the truths evolved. The term solid, used not only by Legendre, but by many other authors, to denote a limited portion of space, seems calculated to introduce the foreign idea of matter into a science which deals only with the abstract properties and relations of figured space. The term volume has been introduced in its place, under the belief that it corresponds more exactly to the idea intended. Many other departures have been made from the original text, the value and utility of which have been made manifest in the practical tests to which the work has been subjected. In the present edition, numerous changes have been made, both in the Geometry and in the Trigonometry. The definitions have been carefully revised - the demonstrations have been harmonized, and, in many instances, abbreviated—the principal object being to simplify the subject as much as possible, without departing from the general plan. These changes are due to Professor Peck, of the Department of Pure Mathematics and Astronomy in Columbia College. For his aid, in giving to the work its present permanent form, I tender him my grateful acknowledgments. The edition of Legendre, referred to in the last paragraph, will not be altered in form or substance; and yet, Geometry must be made a more practical science. To attain this object, without deranging a system so long used, and so generally approved, an Appendix has been prepared and added to Legendre, embracing many Problems of Geometrical construction, and many applications of Algebra to Geometry. It would be unjust to those giving instruction, to add to their daily labors, the additional one, of finding appropriate solutions to so many difficult problems: hence, a Key has been made for the use of Teachers, in which the best methods of construction and solution are fully given. FISHKILL-ON-HUDSON, June, 1875. CHARLES DAVIES. NOTE. The edition of Legendre referred to in the foregoing preface was prepared by the late Professor Davies the year before his lamented death. The present edition is the result of a careful re-examination of the work, into which have been incorporated such emendations, in the way of greater clearness of expression or of proof, as could be made without altering it in form or substance. Practical exercises have been placed at the end of the several books, and comprise additional theorems, problems, and numerical exercises upon the principles of the Book or Books preceding. They will, it is hoped, be found of service in accustoming students, early in and throughout their course, to make for themselves practical application of geometric principles, and constitute, in addition, a large body of review and test questions for the convenience of teachers. The Trigonometry has been carefully revised throughout, to simplify the discussions and to make the treatment conform in every particular to the latest and best methods. It is believed that in clearness and precision of definition, in general simplicity and rigor of demonstration, in orderly and logical development of the subject, and in compactness of form, Davies' Legendre is superior to any work of its grade for the general training of the logical powers of pupils, and for their instruction in the great body of elementary geometric truth. COLUMBIA COLLEGE, N. Y., June, 1885. J. H. VAN AMRINGE, Editor of Davies' Course of Mathematies. |