ELEMENTS OF GEOMETRY PLANE AND SOLID BY JOHN MACNIE, A.M. !་ AUTHOR OF "THEORY OF EQUATIONS" EDITED BY EMERSON E. WHITE, A.M., LL.D. AUTHOR OF WHITE'S SERIES OF MATHEMATICS NEW YORK: CINCINNATI .:. CHICAGO AMERICAN BOOK COMPANY White's Series of Mathematics. QA453 ORAL LESSONS IN NUMBER. (For Teachers.) FIRST BOOK OF ARITHMETIC. NEW COMPLETE ARITHMETIC. SCHOOL ALGEBRA. (In Preparation.) ELEMENTS OF GEOMETRY. ELEMENTS OF TRIGONOMETRY. (In Preparation.) COPYRIGHT, 1895, BY AMERICAN BOOK COMPANY, Printed at In this treatise, an endeavor is made to present the elements of geometry with a logical strictness approaching that of Euclid, while taking advantage of such improvements in arrangement and notation as are suggested by modern experience. It has been carefully kept in mind that the purpose of such a work is only in a secondary degree the presentation of a system of useful knowledge. A much more important purpose is to afford those who study this subject the only course of strict reasoning with which the great majority of them will ever become closely acquainted. A mind that, by exercise in following and weighing examples of strict logical deduction, has learned to appreciate sound reasoning, and, by practice on suitable exercises, has been trained to reason out a sound logical deduction for itself, has gained what is of far greater importance than mere knowledge; it has gained power. A treatise on rational geometry ought, accordingly, to have for guiding principles those laid down by Pascal as the chief laws of demonstration, substantially as follows: to leave no obscure terms undefined; to assume nothing not perfectly evident; to prove everything at all doubtful, by reference to admitted principles. In accordance with the first principle, great care has been taken in the wording of the definitions. In the case of some terms, such as straight line and angle, for which no definitions quite free from objection have as yet been proposed, those adopted have been chosen, not as theoretically perfect, but as best suited to the comprehension of the beginner, and most available in deducing the properties of the things defined. The use of hypothetical constructions has been abandoned for several reasons. To assume them silently, as is now usually done, is unwarrantable in a treatise upon a science supposed, above all others, to consist of a series of rigorous deductions from admitted truths. Why state so carefully that we must assume the possibility 3 183969 |