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SOLID GEOMETRY

BY

FLETCHER DURELL, PH.D.

HEAD OF THE MATHEMATICAL DEPARTMENT, THE LAWRENCEVII LE SCHOOL

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PREFACE

ONE of the main purposes in writing this book has been to try to present the subject of Geometry so that the pupil shall understand it not merely as a series of correct deductions, but shall realize the value and meaning of its principles as well. This aspect of the subject has been directly presented in some places, and it is hoped that it pervades and shapes the presentation in all places.

Again, teachers of Geometry generally agree that the most difficult part of their work lies in developing in pupils the power to work original exercises. The second main purpose of the book is to aid in the solution of this difficulty by arranging original exercises in groups, each of the earlier groups to be worked by a distinct method. The pupil is to be kept working at each of these groups till he masters the method involved in it. Later, groups of mixed exercises to be worked by various methods are given.

In the current exercises at the bottom of the page, only such exercises are used as can readily be solved in connection with the daily work. All difficult originals are included in the groups of exercises as indicated above.

Similarly, in the writer's opinion, many of the numeri

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cal applications of geometry call for special methods of solution, and the thorough treatment of such exercises should be taken up separately and systematically. [See pp. 304-318, etc.] In the daily extempore work only such numerical problems are included as are needed to make clear and definite the meaning and value of the geometric principles considered.

Every attempt has been made to create and cultivate the heuristic attitude on the part of the pupil. This has been done by the method of initiating the pupil into original work described above, by queries in the course of proofs, and also at the bottom of different pages, and also by occasional queries in the course of the text where definitions and discussions are presented. In the writer's opinion, the time has not yet come for the purely heuristic study of Geometry in most schools, but it is all-important to use every means to arouse in the pupil the attitude and energy of original investigation in the study of the subject.

In other respects, the aim has been to depart as little as possible from the methods most generally used at present in teaching geometry.

The Practical Applications (Groups 88-91) have been drawn from many sources, but the author wishes to express his especial indebtedness to the Committee which has collected the Real Applied Problems published from time to time in School Science and Mathematics, and of which Professor J. F. Millis of the Francis W. Parker School of Chicago is the chairman. Page 360 is due almost entirely to Professor William Betz of the East High School of Rochester, N. Y.

LAWRENCEVILLE, N. J., Sept. 1, 1904.

FLETCHER DURELL.

TO THE TEACHER

1. IN working original exercises, one of the chief dif ficulties of pupils lies in their inability to construct the figure required and to make the particular enunciation from it. Many pupils, who are quite unable to do this preliminary work, after it is done can readily discover a proof or a solution. In many exercises in this book the figure is drawn and the particular enunciation made. It is left to the discretion of the teacher to determine for what other exercises it is best to do this for pupils.

2. It is frequently important to give partial aid to the pupil by eliciting the outline of a proof by questions such as the following: "On this figure (or, in these two triangles) what angles are equal, and why ?" "What lines are equal, and why?" etc.

3. In many cases it is also helpful to mark in colored. crayon pairs of equal lines, or of equal angles. Thus, in the figure on p. 37 lines AB and DE may be drawn with red crayon, AC and DF with blue, and the angles A and D marked by small arcs drawn with green crayon. If colored crayons are not at hand, the homologous equal parts may be denoted by like symbols placed on them, thus:

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In solving theorems concerning proportional lines, it is occasionally helpful to denote the lines in a proportion

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