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DEVELOPED BY THE

SYLLABUS METHOD

BY

EUGENE RANDOLPH SMITH, A.M.

HEAD OF THE DEPARTMENT OF MATHEMATICS, POLYTECHNIC PREPARATORY
SCHOOL, BROOKLYN, NEW YORK

(FORMERLY HEAD OF THE DEPARTMENT OF MATHEMATICS,
MONTCLAIR HIGH SCHOOL)

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NEW YORK .:. CINCINNATI .:. CHICAGO
AMERICAN BOOK COMPANY

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PREFACE

THE belief that the proofs of Geometry should be, as far as possible, worked out by the pupils, either in class discussions or individually, is becoming more widespread every year. The day of memorizing proofs will soon be past, and the most efficient method for mental training along logical lines will be the one generally adopted. This syllabus is written with the hope of encouraging teachers to undertake Geometry by the "no text" method. The author believes very decidedly that this method gives a maximum of mental training with a minimum waste of energy

The list of theorems is based on the latest reports of the Mathematical Associations, and, while much shorter than that in many of the text-books, it will be found sufficient to prepare the pupils for any of the colleges. It contains all the theorems of the “New England List” with a few additions that simplify proofs.

The order is the development of ten years' class use, and will be found different from that of any text. Whenever a theorem has seemed to be simplified, either in content or in proof, by making a change in its place in the order of theorems, that change has been tried in class, and has been made permanent if it proved of advantage. Any teacher using this book should feel equally free to make changes in the order if he is convinced that there is a decided advantage in the change.

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The chapter on Logic has been found of great assistance in helping the pupils to think accurately, and it is certain to save more time for a class than its discussion requires.

The definitions and axioms are given in quite complete form, not for assignment to the class, as this part of the work should be developed before the text is given to a class for study; but as a guide for the teacher, both in order and in subject-matter, and as a reference book for the pupil. Good results can be secured by withholding the book from the pupils until part, at least, of the preliminary matter has been discussed.

The subjects of “existence" and "betweenness" have not been considered to any great extent, as they do not seem worth the time and effort required, except to a student of the more advanced pure mathematics. “Location" and "intersection," on the other hand, are of such vital importance in considering the correctness of proofs that they have received some attention. The aim throughout has been to arrange a system of Geometry that should be natural, reasonably complete, and suitable to afford as much mental training as the maturity of the pupils would allow. The author has not hesitated to assume any axiom that would help more than its presence would complicate; on the other hand, he has left out things that seemed to require more than they gave.

Geometry itself has no concern with measurement by means of a unit. The applications of Geometry to such measurement are, however, very frequent and very important, and while this book presupposes geometrical proofs to as great a degree as seems possible without unnecessarily complicating the subject, there has been no attempt to draw a hard and fast line of demarcation between Geometry and its applications. If a teacher believes in distinguishing sharply between the different branches of mathematics, the study of the lengths of line sects and the calculation of areas can be put under the head of MENSURATION.

The exercises are in two divisions, those under the theorems and those in general lists. The exercises under the theorems have been chosen to illustrate the uses of the various theorems, and they should therefore be of great help to the teacher. The general lists give the pupils practice in finding for themselves what principles underlie the proofs. Probably no class could finish all the exercises in the book in one year, but the teacher can easily choose those best suited to his purposes. There are several pages of college examination questions. Some of these are duplicates of exercises scattered through the book, but the differences in wording, as well as the desire to let students know what type of questions examiners ask, has prompted leaving them in the book.

This book has been written with little reference to the order and methods of other texts, for it is a compilation that has grown naturally from class work. The author is, of course, indebted for many of the ideas used to numerous works on mathematics and its pedagogy, but in many cases it is now impossible to tell from what source the suggestion first arose. He wishes, however, to acknowledge his special indebtedness to Dr. William H. Metzler of Syracuse University, for assistance and encouragement in the writing of this book.

EUGENE R. SMITH.

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