BY BENJAMIN GREENLEAF, A. M., LEACH, SHEWELL AND SANBORN, BOSTON AND NEW YORK. NEW COMPREHENSIVE SERIES. An ENTIRELY NEW MATHEMATICAL COURSE, fully adapted to the best methods of Modern Instruction. * Other Books of a Complete Series, in preparation. **KEYS to the PRACTICAL ARITHMETIC, ALGEBRAS, GEOMETRY and TRIGONOMETRY, in separate volumes. Entered according to Act of Congress, in the year 1863, by in the Clerk's Office of the District Court for the District of Massachusetts. RIVERSIDE PRESS: PRINTED BY H. O. HOUGHTON AND COMPANY. PREFACE. THE preparation of this treatise has been undertaken at the earnest solicitation of many teachers, who, having used the author's Arithmetics and Algebra with satisfaction, have been desirous of seeing his series rendered more complete by the addition of the Elements of Geometry. 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 common with most compilers of the present day, 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, it is believed, 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 Mathé matiques, 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 Application 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 Geometrical Exercises, which follow, are calculated to test the thoroughness of the scholar's geometrical knowledge, besides being especially adapted to develop skill and discrimination in the demonstration of theorems and the solution of problems unaided except by principles. Sufficient Applications of Algebra to Geometry are given to show the relation existing between these two branches of the mathematics. The problems introduced in connection therewith will be found to be, not only of a highly interesting character, but well calculated to secure valuable mental discipline. 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. BRADFORD, Mass., June 25, 1858. NOTICE. A KEY, comprising the Solutions of the Problems contained in the last four Books of this Geometry, has been published, for Teachers mly; and the same will be mailed, post-paid, to the address of any Teacher who will forward sixty cents in stamps to the Publishers. |