A TREATISE ON SPECIAL OR ELEMENTARY GEOMETRY. IN FOUR PARTS. UNIVERSITY EDITION. INCLUDING PLANE, SOLID, AND SPHERICAL GEOMETRY, AND PLANE AND PART III. WHICH DISTINGUISHES THE UNIVERSITY FROM THE SCHOOL EDITION, 160 PROPOSITIONS AND PROBLEMS FOR EXERCISE IN GEOMETRICAL AN INTRODUCTION TO THE MODERN GEOMETRY. A453 052 Stoddard's Mathematical Series. STODDARD'S JUVENILE MENTAL ARITHMETIC SHORT AND FULL COURSE FOR GRADED SCHOOLS. STODDARD'S PICTORIAL PRIMARY ARITHMETIC 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. 74271 HIGHER MATHEMATICS, BY EDWARD OLNEY, PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN. A COMPLETE SCHOOL ALGEBRA, in one vol, 410 pages, $1.50. Designed for Elementary and higher classes in Schools and Academies. A GEOMETRY AND TRIGONOMETRY, in one vol., 8vo, Price $2.50 EDITION, in Price $3.00 Price 2.50 The other books of this Series will be published as rapidly as possible. Entered according to Act of Congress, in the year 1872, by SHELDON & COMPANY, In the Office of the Librarian of Congress, at Washington. OF C 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 been written with special reference to the needs of students in the University of Michigan. Our admirable system of public HighSchools, of which schools there is now one in almost every considerable village, promises ere long to become to us something near what the German Gymnasia are to their Universities. In order to promote the legitimate development of these schools, it is necessary that the University resign to them the work of instruction in the elements of the various branches, as fast and as far as they are prepared in sufficient numbers to undertake it. It is thought that these schools should now give the instruction in Elementary Geometry, which has hitherto been given in our ordinary college course. The first two parts of this volume furnish this amount of instruction, and students are expected to pass examination upon it on their entrance into the University. This amount of preparation enables students to extend their knowledge of Geometry, during the Freshman year in the University, considerably beyond what has hitherto been practicable. As a text-book for such students, Part III. has been written. At this stage of his progress, the student is prepared to learn to investigate for himself. Hence he is here furnished with a large collection of well classified theorems and problems, which afford a review of all that has gone before, extend his knowledge of geometrical truth, and give him the needed discipline in original demonstration. To develop the power of independent thought, is the most difficult, while it is the most important part of the teacher's work. Great pains have therefore been taken, in this part of the work, to render such aid, and only such, as a student ought to require in advancing from the stage in which he has been following the processes of others, to that of independent reasoning. In the second place, this part contains what is usually styled Applications of Algebra to Geometry, with an extended and carefully selected range of examples in this important subject. A third purpose has been to present in this part an introduction to what is often spoken. of as the Modern Geometry, by which is meant the results of modern thought in developing geometrical truth upon the direct method. While, as a system of geometrical reasoning, this Geometry is not philosophically different from that with which the student of Euclid is familiar, and which is properly distinguished as the special or direct method, the character of the facts developed is quite novel. So much so, indeed, that the student who has no knowledge of Geometry but that which our common text-books furnish, knows absolutely nothing of the domain into which most of the brilliant advances of |