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book in a natural and attractive pedagogical order. After the introductory chapter there are presented a few easy problems of construction in which the pupil will make free use of his ruler and compasses, and be introduced to the idea of a formal proof in connection with matters which he clearly sees need proving; other similar problems are inserted where they come in most naturally and are of most immediate interest.

Abstract discussions, such as the theory of limits, have been postponed as far as possible in order to secure for their comprehension greater mathematical maturity. Young pupils do not readily assimilate theoretical or abstract principles, and it is best that these should be brought into but little prominence in the early parts of the subject.

The exercises have been carefully selected and are intended to form an integral part of the work. They bear a direct relation, in most cases, to the propositions with which they are associated; those inserted at the ends of sections or chapters are for the most part of a more difficult character and may be omitted on a first reading.

It is hoped that the classified summary at the end of each chapter will prove serviceable for purposes of review and for reference, and will also aid the pupil in systematizing his knowledge.

Care has been taken to state the fundamental assumptions the postulates-upon which the science rests, in as clear a form as possible, and to distinguish between assumption and axiomatic truth. Some things have been assumed which are often made matters of demonstration; for example, the fact that the perimeter of a regular polygon inscribed in a circle approaches a limit as the number of its sides is indefinitely increased. This admits of a rigorous proof, to be sure, but the proofs given in text-books on elementary geometry are as a rule either unsatisfactory or beyond the appreciation of the pupil. I have preferred openly to assume the property, -an assumption at which the pupil does not hesitate.

A circle has been defined as a particular kind of line, and a polygon as a figure made up of points and lines. The area of such a figure is defined to be the surface enclosed by it. This accords, I believe, with the best usage, though perhaps not with the common usage. Everywhere outside of a class in elementary geometry a circle is so understood. Neither in more. advanced mathematics nor in everyday life is it thought of as a portion of a plane, in accordance with the common definition, and I see no reason why the pupil should be obliged to change his idea of such a figure upon entering the geometry class, and change back again immediately upon leaving that class.

The Appendix contains a short chapter on Plane Trigonometry, which is intended to serve as a brief introduction to the subject, to meet the needs of those preparatory pupils who take up the study of physics or mechanics, not as a substitute for a complete course. It may be found sufficient also to meet the practical needs of those who do not continue their studies beyond the high school.

The teacher will do well to consider the following suggestions:

1. Read the introductory chapter carefully, then talk it over with the class in an informal way, calling attention to the geometrical principles involved, most of which will be readily accepted by your pupils. Do not assign this chapter as a lesson.

2. Proceed very slowly at first. Remember that your pupils already have some geometrical ideas. Draw these out, clarify and fix them. Do not break down, but build on what your pupils already have.

3. Do not be too strenuous at first about a formal demonstration. Emphasize the geometric truth presented. Fix as your ideal an elegant, faultless proof, and gradually work up to it.

4. Remember that in this subject the primary object should be the acquisition of geometric knowledge and the development of the geometric sense. Logical reasoning and rhetorical demonstration are secondary aims, to be sure, but the first object should be Geometry.

Since my chief desire has been to produce a text-book adapted to the needs of the class-room, I have not hesitated to make free use of many existing texts, both old and new, and from them have derived much valuable help and many suggestions. The exercises in particular have been gathered from a variety of sources; only a few of them are new. To many friends, and in particular to my colleague, Professor Henry S. White, to Mr. B. Annis, of the Hartford, Connecticut, High School, and to Mr. J. F. Petrie, of North-western University Academy, I am greatly indebted for valuable suggestions and criticisms. Also to my pupil, Miss Elda L. Smith of Springfield, Illinois, my sincerest thanks are due for patient and careful work in testing all exercises and in the tedious task of proof-reading.

I shall be glad to be informed of any errors that may have been overlooked, and to receive suggestions for improvement either in matter or arrangement.



May 30, 1901.

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