or GEOMETRY; WITH PRACTICAL APPLICATIONS ΤΟ MENSURATION. BY BENJAMIN GREENLEAF, A. M., Permanent Electrotype Edition. BOSTON: PUBLISHED BY ROBERT S. DAVIS & CO. NEW YORK: MASON, BAKER, & PRATT, 142 GRAND STREET. PHILADELPHIA: J. A. BANCROFT & COMPANY. ST. LOUIS: HENDRICKS & CHITTENDEN. CHICAGO: S. C. GRIGGS & CO. 1874. GREENLEAF'S NEW COMPREHENSIVE SERIES. An ENTIRELY NEW MATHEMATICAL COURSE, fully adapted to the best methods of Modern Instruction. GREENLEAF'S NEW PRIMARY ARITHMETIC. GREENLEAF'S NEW INTELLECTUAL ARITHMETIC. GREENLEAF'S NEW ELEMENTARY ARITHMETIC. GREENLEAF'S NEW PRACTICAL ARITHMETIC. GREENLEAF'S NEW ELEMENTARY ALGEBRA. GREENLEAF'S NEW HIGHER ALGEBRA. GREENLEAF'S NEW ELEMENTARY GEOMETRY. GREENLEAF'S ELEMENTS OF TRIGONOMETRY. GREENLEAF'S GEOMETRY AND TRIGONOMETRY. 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 1858 by in the Clerk's Office of the District Court for the District of Massachusetts. RIVERSIDE PRESS: PRINTED BY H. O. HOUGHTON AND COMPANY. HARVARD LIBRARY 347X175 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 |