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THIS book is designed for the use of beginners in geometry in secondary schools, and has been used for several years in a simpler form by the teachers of mathematics in the Somerville (Mass.) English High School. No attempt has been made to depart in the least from the subject-matter given in the regular Euclidian geometries; the variation from the well-known texts has been more in regard to the presentation of the theorems and the method of development. To those teachers who feel the need of a book intermediate in treatment between a syllabus and a geometry of completely demonstrated theorems, this work is confidently submitted.

The most prominent characteristic of this book is that it is a working book, -a book in the use of which the pupil is obliged to show the result of individual effort. The distinctive features that bring about this result and that also differentiate this book from any yet published are the following:

(1) The development of definitions. To prevent the perfunctory memorizing of definitions without a thorough understanding of their meanings, the various geometrical terms and concepts are to be developed in the class as they are needed. This may be accomplished by encouraging the pupils not only to present to the class their own ideas regarding the various terms, but also to bring to the recitation definitions which they have obtained from various sources. Under the careful guidance of the teacher, satisfactory definitions may thus be decided upon and then recorded by each pupil in a note-book kept for the


(2) Suggested demonstrations. Demonstrations are not given in full, but are outlined by means of hints or suggestions. Although consider


able assistance must necessarily be given in the development of the first few theorems, it is expected that the suggestions for the remaining theorems will be, in general, sufficient guides to enable the pupil to work out his own proofs. Constant watchfulness will be necessary on the part of the teacher in order to secure logical demonstrations. The pupil must keep constantly in mind that a geometrical demonstration is a series of steps or statements leading in logical order from the hypothesis to the required conclusion, each step being dependent on some geometrical fact already known, that is, a definition, an axiom, or some theorem previously proved. That reasons may be readily and accurately given, definitions and statements of theorems should be committed to memory.

(3) The construction of the figures by the pupil. - Space for the figure is provided under each theorem. The appropriate figure is to be constructed and lettered by the pupil in accordance with the directions in the hypothesis and the accompanying suggestions. Especial emphasis is laid on the correct construction of figures in order to fix firmly in the mind of the pupil the exact conditions under which the theorems are to be proved. To prevent the incorrect drawing of figures, the class and the teacher should determine exactly what each figure is to be before it is constructed in the book.

Problems in construc

(4) The use of the problems of construction. tion are given whenever needed as an assistance in the proper construction of the figures for the theorems. It follows that the method of proof for a problem cannot always be explained when the problem is given. In the review, however, the explanation of all problems should be thoroughly understood.

(5) The choice and the arrangement of the theorems. — In regard to the choice of theorems, effort has been made to select the theorems that will give a clear and comprehensive view of the subject. The Syllabus of Propositions prepared by the Mathematical Department of Harvard University has been used as a basis of selection, but has not been followed exactly. The arrangement of theorems, although strictly logical, has been purposely varied from the arrangements in the best known

text-books that the pupil may be as independent of outside assistance as possible.

(6) The use of exercises. - Över one hundred and fifty exercises are given. These furnish a minor feature only, but are of special value in the review.

This book differs radically from the geometries used in the majority of secondary schools. It is the conviction, however, of those who know its methods and have seen its practical working that the use of the book will infuse new life in the study of geometry, awaken greater interest in both teacher and pupil, secure a great gain in the amount of individual effort, increase the spirit of self-reliance, and greatly strengthen the power of deductive reasoning.

Grateful acknowledgments are due to associate teachers and others for aid and suggestions in the preparation of this book.


February, 1903.

J. A. A.

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