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Educ T 149,15.895




JUN 12 1916

Ar 16 1921




Norwood Press

J. S. Cushing Co. - - Berwick & Smith Co.
Norwood, Mass., U.S.A.


As a result of the widespread discussion during recent years on the improvement of our courses in elementary geometry, the majority of thoughtful teachers appear to have reached substantial agreement on at least one point: To begin the course in plane geometry in the traditional formal manner is pedagogically irrational and scientifically unnecessary. There has accordingly arisen an increasing demand for a textbook which will supply a pedagogically rational approach to the study of Plane Geometry, without sacrificing the logical structure of the subject.

The present text aims to supply this demand. The beginning should be thoroughly concrete, informal, and to the pupil natural and interesting. More formal methods should be introduced gradually at a time when the pupil can understand their significance and value. These ends we have sought to attain by making a systematic study of geometric drawing the basis of the first chapter. Throughout this chapter points, lines, etc., are concrete things to be drawn with a pencil or a piece of chalk; the reasoning involved is couched in easy, natural language, without any of the stiffness of a formal arrangement. While the development of the subject matter of this chapter is wholly systematic, the emphasis is intended to be placed on the exercises which have been carefully selected both with respect to interest and for drill in developing the power of making inferences. The study of this first chapter will, we confidently believe, accomplish two ends: (1) It will lead the pupil to gain a thorough understanding of the fundamental notions of geometry; (2) it will develop in him the power of attack and insight into the spirit of a geometric problem. Beginning with more formal methods in the second chapter,

he will be in a position to pursue the remainder of the course with understanding, pleasure, and profit. Our confidence in this respect is not based merely on theoretical considerations, but rather on the fact that this method of approach has been used for several years with most satisfactory results in actual class work with large numbers of pupils.

Among other more or less characteristic features of our text we would mention briefly the following:

(1) The systematic use of symmetry as a method of proof. That this can be made a useful and active means of inference may be seen, for example, by reference to pages 10 to 17. Frequently where a rather long process of reasoning by congruence yields but a single detail of a figure, the method of symmetry enables one to grasp the whole relationship by one mental act. Above all other advantages, however, this method of inference has shown itself by actual classroom experience to appeal strongly to the pupil as in his eyes a natural method of procedure.

(2) The disposal of the so-called incommensurable cases on purely practical grounds, but in a way which is entirely in the spirit of the rigorous logical development.

(3) The introduction of the definitions and elementary applications of the trigonometric ratios, without lengthening the course. This has been made possible by the omission (or insertion in the exercises) of a number of theorems hitherto retained which are not necessary links in the logical development or do not serve a useful purpose as basal theorems. The introduction of this material seems to us highly desirable from whatever point of view the study of geometry in secondary schools be advocated, whether cultural, disciplinary, or utilitarian.

(4) The two-color printing of most of the figures. The device of printing the auxiliary lines and construction lines in a subdued color we hope will prove a welcome innovation, tending to emphasize the distinction and improving the appearance of the page.

We have taken great care in the selection and grading of the numerous exercises, and have made a consistent effort throughout to develop independence and resourcefulness on the part of the

pupils. Analysis is continually resorted to or suggested in the solution of problems, and genetic methods are given the preference over didactic in the proof of theorems wherever possible.

In conclusion, it is a pleasure to acknowledge our indebtedness to such excellent works as Borel's "Géométrie," the "Traité de Géométrie," by Rouche et Comberousse, "Ebene Geometrie" by Mahler, "Congruent Figures" by Henrici, and many other texts printed in this country and abroad. Our cordial thanks are due also to several friends who have taken a live interest in our work, and especially to Dr. F. M. Morgan of Dartmouth College, who, in reading portions of the manuscript and in rendering valuable assistance in seeing the book through the press, has found occasion to make a number of helpful suggestions and criticisms.

J. W. Y.,

A. J. S.

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