Plato (429-348 B.C.) was an Athenian philosopher, who laid great stress on the remarkable relations between mathematics and philosophy. He made the study of geometry indispensable to that of philosophy (see page 2), and taught that the secret of the universe is to be found in number and form.

His greatest contribution to geometry was in emphasizing the need of formal definitions and axioms, and in developing the analytic method of discovering proofs. See page 107.

Above the entrance to his school of philosophy was the inscription:

"He who knows not geometry may not enter here."

WITH

PROBLEMS AND APPLICATIONS

REVISED EDITION

BY

H. E. SLAUGHT, PH.D., Sc.D.

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO

PREFACE

THE movement for reform in the teaching of geometry now seems to have crystallized into certain rather definite conclusions. In rewriting this book the authors have sought to include the improvements resulting from this movement and at the same time to keep what was best in the older texts.

First among the improvements may be mentioned practical applications of geometry to everyday life. These are a feature of the new book. For example, see exercise 7 on page 161, a practical problem which should be brought to the notice of all interested in the game of football.

Among other improvements may be mentioned informal proofs of certain propositions which, though obvious even to the beginner, usage forbids stating as axioms; also, the relegating of difficult theoretical topics like limits to the end of the book, where the incommensurable cases are all treated together. In many respects the older treatment of limits is obscure and sometimes misleading. It is believed that the method here adopted is more defensible from every point of view.

The extreme inductive method of twenty years ago is introduced here only in a modified form. Theorems are proved completely, especially in the early part of the course. Propositions quoted in the proofs are at first given in full, later in part, and still later only by reference. After theorems have thus been quoted several times the pupil is left to his own devices in supplying the reasons.

Among the features of the older books which have been retained are the orderly and clear arrangement of the page and the simplicity of language in the definitions and proofs. The new book avoids technical and unusual language; the

982097