TREATISE ON GEOMETRY AND TRIGONOMETRY: FOR COLLEGES, SCHOOLS AND PRIVATE STUDENTS. WRITTEN FOR THE MATHEMATICAL COURSE OF JOSEPH RAY, M. D., BY ELI T. TAPPAN, M. A., PROFESSOR OF MATHEMATICS, OHIO UNIVERSITY. CINCINNATI: WILSON, HINKLE & CO. THE BEST AN CHEAPEST. RAY'S MATHEMATICAL SERIES. Ray's Primary Arithmetic: Simple Mental Lessons and Tables. For little Learners. Ray's Intellectual Arithmetic: the most interesting and valuable Arithmetic extant. Ray's Rudiments of Arithmetic: combining mental and practical exercises. For beginners. 1 Ray's Practical Arithmetic: a full and practical treatise on the inductive and analytic methods of instruction. Ray's Higher Arithmetic: the principles of Arithmetic analyzed and practically applied. Ray's Test Examples: three thousand practical problems for the slate or blackboard. For drill exercises and review. Ray's New Elementary Algebra: a simple, thorough, and progressive elementary treatise. For Schools and Academies. Ray's New Higher Algebra: a progressive, lucid, and comprehensive work. For advanced Students and for Colleges. Ray's Elements of Geometry: a comprehensive work on Plane and Solid Geometry, with numerous practical exercises. Ray's Geometry and Trigonometry: Plane and Spherical Trigonometry, with their applications; also a complete set of Logarithmic tables, carefully corrected. Ray's Differential and Integral Calculus: in course of preparation, and to be published during the present year. To be followed, at an early day, by other works, forming a complete Mathematical Course for Schools and Colleges. Entered according to Act of Congress, in the year 1868, by SARGENT, WILSON & HINKLE, In the Clerk's Office of the District Court of the United States, for the ELECTROTYPED AT THE FRANKLIN TYPE FOUNDRY, CINCINNATI. PREFACE. THE science of Elementary Geometry, after remaining nearly stationary for two thousand years, has, for a century past, been making decided progress. This is owing, mainly, to two causes: discoveries in the higher mathematics have thrown new light upon the elements of the science; and the demands of schools, in all enlightened nations, have called out many works by able mathematicians and skillful teachers. Professor Hayward, of Harvard University, as early as 1825, defined parallel lines as lines having the same direction. Euclid's definitions of a straight line, of an angle, and of a plane, were based on the idea of direction, which is, indeed, the essence of form. This thought, employed in all these leading definitions, adds clearness to the science and simplicity to the study. In the present work, it is sought to combine these ideas with the best methods and latest discoveries in the science. By careful arrangement of topics, the theory of each class of figures is given in uninterrupted connection. No attempt is made to exclude any method of demonstration, but rather to present examples of all. The books most freely used are, "Cours de géométrie élémentaire, par A. J. H. Vincent et M. Bourdon;" "Géométrie théorique et pratique, etc., par H. Sonnet;" "Die |