5 gist PREFACE THIS text-book aims to carry out the spirit of t rable suggestions made by the Committee on S School studies, appointed by the National Educatio ciation. While the book speaks for itself, some of ing features may here be pointed out. (1) It aims at a combination of Euclidean ri modern methods of presentation suitable for beg the study of demonstrative geometry; but the rig regarded as consisting so much in excessive for expression as in soundness of structural developme (2) It regards the postulates as a body of fun conventions that constitute à definition of Euclide from which (with the definitions of particular figur properties of such space are to be unfolded by a logical steps. (3) It regards the postulates of construction as ing or defining the province of elementary as disti from higher geometry. Accordingly no hypotheti is made the basis of an argument until its constru been proved to be reducible to the construction p and thus problems, no less than theorems, have th in the logical development of the subject. (4) The theorems and problems are arranged in groups and subgroups with reference to their u principle, thus exhibiting the gradual unfoldin 6 gist 0 PREFACE THIS text-book aims to carry out the spirit of t rable suggestions made by the Committee on S School studies, appointed by the National Educatio ciation. While the book speaks for itself, some of ing features may here be pointed out. (1) It aims at a combination of Euclidean ri modern methods of presentation suitable for beg the study of demonstrative geometry; but the rig regarded as consisting so much in excessive for expression as in soundness of structural developme (2) It regards the postulates as a body of fun conventions that constitute à definition of Euclide from which (with the definitions of particular figu properties of such space are to be unfolded by a logical steps. (3) It regards the postulates of construction as ing or defining the province of elementary as disti from higher geometry. Accordingly no hypotheti is made the basis of an argument until its constru been proved to be reducible to the construction po and thus problems, no less than theorems, have th in the logical development of the subject. (4) The theorems and problems are arranged in groups and subgroups with reference to their u lence of statements that differ only in form, and also distinguish between different statements that may seem to be alike. (6) The mode of treating ordinary size-relations is purely geometrical. "This method being pure and thoroughly elementary, and involving no abstraction, is surely better suited to the beginner. Indeed, the student is most likely to become a sound geometer who is not introduced to the notion of numerical measures until he has learned that geometry can be developed independently of it altogether. For this notion is subtle, and highly artificial from a purely geometrical point of view, and its rigorous treatment is difficult. The student generally only half comprehends it, so that for him demonstrations lose more in rigor as well as in vividness and objectivity by its use than they gain in apparent simplicity. Moreover, the constant association of number with the geometric magnitudes as one of their properties, tends to obscure the fundamental characteristic of these magnitudes their continuity."* Words suggestive of measurement, such as length, area, distance, etc., are accordingly not used in the purely geometrical chapters. (7) The Euclidean doctrine of ratio and proportion is presented in a modernized form, which shows its naturalness and generality, and renders it easier of application than the unsatisfactory numerical theory which is so often allowed to usurp its place, although it is generally conceded by mathematicians that Euclid's treatment of proportion is one of the most admirable and beautiful of his contributions to geometry. (8) There is a chapter on mensuration, in which measurenumbers are introduced as a natural outgrowth from the general notion of ratio, and the irrational numbers that cor *See Report of Conference of School and College Teachers embodied in the Report of the Committee of Ten, p. 113. (Published for the National Educational Association by American Book Company, 1894.) |