M3 PREFACE This text-book aims to carry out the spirit of the admi. rable suggestions made by the Committee on Secondary School studies, appointed by the National Educational Association. While the book speaks for itself, some of its leading features may here be pointed out. (1) It aims at a combination of Euclidean rigor with modern methods of presentation suitable for beginners in the study of demonstrative geometry; but the rigor is not regarded as consisting so much in excessive formality of expression as in soundness of structural development. (2) It regards the postulates as a body of fundamental conventions that constitute a definition of Euclidean space, from which (with the definitions of particular figures) other properties of such space are to be unfolded by a series of logical steps. (3) It regards the postulates of construction as determining or defining the province of elementary as distinguished from higher geometry. Accordingly no hypothetical figure is made the basis of an argument until its construction has been proved to be reducible to the construction postulates; and thus problems, no less than theorems, have their place in the logical development of the subject. (4) The theorems and problems are arranged in natural groups and subgroups with reference to their underlying principle, thus exhibiting the gradual unfolding of the space relations. (5) Elementary ideas of logic are introduced comparatively early, so that the student may easily recognize the equiva V M306204 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.) respond to the ratios of incommensurable magnitudes are given simple logical treatment based on the general theory of ratio, without resorting to the notion of a limit, which has no natural connection with the subject. (9) The measurement of the circle is based on the correct definition of the length of a curved line (in terms of a straight measuring-unit) given by the best continental writers. Here the idea of a limit is imbedded in the definition; but the existence and uniqueness of the limit must be proyed before we can speak of the “length of an arc” so as to make it the subject of our discourse; otherwise we are using a word that has not been completely defined. Similar statements may be made with regard to the area of the circle. As far as the author is aware this plan has not hitherto been followed in any text-book in the English language. It is hoped that this important topic has been presented in a rigorous and simple manner. (10) Throughout the book there is an endeavor to develop the student's power of invention and generalization, without encouraging looseness, or introducing discouraging difficulties. These features have received the approval of several experienced educators. Special acknowledgments are due to Professors Wait, Jones, Tanner, and Stecker for assistance and advice. SUGGESTIONS TO TEACHERS It is suggested to teachers that the introductory articles be read and discussed in class in an informal way, with the aim of drawing out and clarifying those ideas of spacerelations which the students may already possess. Some of the introductory matter can be passed over lightly on first reading, and returned to when necessary. Teachers may exercise their discretion with regard to articles in small print throughout the book. For a shorter course, any of the following groups of articles may be omitted without breaking the continuity of the subject: Book I. 180-186, 195–213, 232–247. Book IV. 10. Most of the exercises that are given in immediate connection with the propositions should be solved by the student; but only a few of those placed at the end of sections need be taken on a first reading. They are all carefully graded, and many suggestions are given. The author will be glad to hear from any person who may meet with any error or difficulty. As some teachers may wish to use the Socratic or heuristic methods of instruction in certain parts of the work, the arrangement and development of the topics are such as to lend themselves easily to these valuable pedagogical methods, without interfering with the more formal presentation that is appropriate to a course in demonstrative geometry. The actual details of any such method are, however, left to individual discretion, as the skillful teacher has usually no difficulty in reconciling the claims of pedagogy and sound reasoning. viii |