OF GEOMETRY AND TRIGONOMETRY; WITH PRACTICAL APPLICATIONS. BY BENJAMIN GREENLEAF, A. M., AUTHOR OF A MATHEMATICAL SERIES. IMPROVED ELECTROTYPE EDITION. BOSTON: PUBLISHED BY ROBERT S. DAVIS & CO. NEW YORK: D. APPLETON & CO., AND PHINNEY, BLAKEMAN, & MASON. PHILADELPHIA: J. B. LIPPINCOTT AND COMPANY. COLUMBUS, OHIO: RILEY AND BOWLES. QA Han 529. Math 5081;1,5 6793 ; GREENLEAF'S MATHEMATICAL SERIES. Greenleaf's New Primary Arithmetic, Greenleaf's Intellectual Arithmetic, Upon the Inductive Plan. Being an Advanced Intellectual Course, Greenleaf's Common School Arithmetic, Greenleaf's National Arithmetic, Being a Complete Course of Higher Arithmetic, for High Schools, Greenleaf's Treatise on Algebra, For Academies and High Schools, and for advanced Students in Com- Greenleaf's Geometry and Trigonometry, With Practical Applications. Designed for High Schools, Academies, Complete Keys to the Intellectual, Common School, and National Entered according to Act of Congress, in the year 1858, by in the Clerk's Office of the District Court of the District of Massachusetts. Entered according to Act of Congress, in the year 1861, by in the Clerk's Office of the District Court of the District of Massachusetts. Cambridge: Electrotyped and Printed by Welch, Bigelow, & Co. PREFACE. THE preparation of this treatise was undertaken at the earnest solicitation of many teachers, who, having used the author's Arithmetics and Algebra with satisfaction, were desirous of seeing his series rendered more nearly complete by the addition of the Elements of Geometry and Trigonometry. That there are peculiar advantages in a graded series of textbooks on the same subject, few, if any, properly qualified to judge, will doubt. The author, therefore, feels justified in introducing this volume to the attention of the public. In the Elements of Geometry, he has followed, in the main, the simple and elegant order of arrangement adopted by Legendre; but in the methods of demonstration no particular authority has been closely followed, the aim having been to adapt the work fully to the latest and most approved modes of instruction. In this respect, there will be found incorporated a considerable number of important improvements. More attention than is usual in elementary works of this kind has been given to the converse of propositions. In almost all cases where it was possible, the converse of a proposition has been demonstrated. The demonstration of Proposition XX. of the first book is essentially the one given by M. da Cunha in the Principes Mathématiques, which has justly been pronounced by the highest mathematical authorities to be a very important improvement in elementary geometry. It has, however, never before been introduced into a text-book by an American author. The Applications of Geometry to Mensuration, given in the eleventh and twelfth books, are designed to show how the theoretical principles of the science are connected with manifold practical results. The Miscellaneous Exercises, which follow, are calculated to test the thoroughness of the scholar's geometrical knowledge; and sufficient Applications of Algebra to Geometry are given to show the relation existing between these two branches of the mathematics. The Elements of Plane and Spherical Trigonometry present a complete system, theoretical and practical, fully adapted to the wants of advanced classes. The trigonometric functions, in this treatise, have been regarded as ratios, since this improved method has not only now superseded the ancient method in English and French works, but has been approved and adopted generally by the best Ameriican mathematicians. Reference, however, is made to whatever is especially valuable in the old method. In the preparation of this work the author has received valuable suggestions from many eminent teachers, to whom he would here express his sincere thanks. Especially would he acknowledge his great obligations to H. B. Maglathlin, A. M., who for many months has been associated with him in his labors, and to whose experience as a teacher, skill as a mathematician, and ability as a writer, the value of this treatise is largely due. BENJAMIN GREENLEAF. NOTICE. A KEY, comprising the Solutions of the Problems contained in this work, is published, for Teachers only; and the same will be mailed, postpaid, to the address of any Teacher who will forward thirty-six cents in stamps to the Publishers. |