A TREATISE ON SPECIAL OR ELEMENTARY GEOMETRY. SCHOOL EDITION. INCLUDING PLANE, SOLID, AND SPHERICAL GEOMETRY, AND PLANE AND BY EDWARD OLNEY, PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN. NEW YORK: SHELDON & COMPANY, 677 BROADWAY. NARTAND COLLEGE LIVRARY EY EXCHANGE FROM Stoddard's Mathematical Series. STODDARD'S JUVENILE MENTAL ARITHMETIC STODDARD'S INTELLECTUAL ARITHMETIC SHORT AND FULL COURSE FOR GRADED SCHOOLS. STODDARD'S PICTORIAL PRIMARY ARITHMETIC STODDARD'S COMPLETE ARITHMETIC 165 The Combination School Arithmetic being Mental and Written Arithmetic in one book, will alone serve for District Schools. For Academies a full high course is obtained by the Complete Arithmetic and Intellectual Arithmetic. STODDARD'S HIGHER MATHEMATICS. By Prof. A COMPLETE SCHOOL ALGEBRA in one vol., 390 pages, $1.50. Designed fer Elementary and higher classes in Schools and Academies. EDWARD OLNEY, University of Michigan. A GEOMETRY AND TRIGONOMETRY in one vol. By Prof. EDWARD OLNEY. One vol. 8vo, A GENERAL GEOMETRY AND CALCULUS in one vol. Price $2.25. 2.50. The other books of Stoddard's Series will be published as rapidly as possible. Entered according to Act of Congress, in the year 1871, BY SHELDON & COMPANY, In the Office of the Librarian of Congress at Washington. PREFACE. THIS treatise on the Special or Elementary Geometry consists of four parts. PART I. is designed as an introduction. In it the student is made familiar with the geometrical concepts, and with the fundamental definitions and facts of the science. The definitions here given, are given once for all. It is thought that the pupil can obtain his first conception of a geometrical fact, as well, at least, from a correct, scientific statement of it, as from some crude, colloquial form, the language of which he will be obliged to replace by better, after the former shall have become so firmly fixed in his mind, as not to be easily eradicated. No attempt at demonstration is made in this part, although most of the fundamental facts of Elementary Plane Geometry are here presented, and amply and familiarly illustrated. This course has been taken in obedience to the canon of the teacher's art, which prescribes "facts before theories." Moreover, such has been the historic order of development of this, and most other sciences; viz., the facts have been known, or conjectured, long before men have been able to give any logical account of them. And does not this indicate what may be the natural order in which the individual mind will receive science? When the student has become familiar with the things (concepts) about which his mind is to be occupied, and knows some of the more important of their properties and relations, he is better prepared to reason upon them. PART II. contains all the essential propositions in Plane, Solid, and Spherical Geometry, which are found in our common text-books, with their demonstrations. The subject of triedrals and the doctrine of the sphere are treated with more than the ordinary fullness. The earlier sections of this part are made short, each treating of a single subject, and the propositions are made to stand out prominently. At the close of each section are Exercises designed to illustrate and apply the principles contained in the section, rather than to extend the pupil's knowledge of geometrical facts. These features, together with the synopses at the close of the sections, practical teachers cannot fail to appreciate. PART III., which is contained only in the University Edition, has |