Klein - Elementary mathematics from an advanced standpoint. Geometry

ELEMENTARY MATHEMATICS

FROM AN ADVANCED STANDPOINT

GEOMETRY

FELIX KLEIN

Translated from the Third German Edition by

E. R. Hedrick

Vice President and Provost

The University of California

and

C. A. Noble

Professor of Mathematics, Emeritus

The University of California

DOVER PUBLICATIONS, INC.
MINEOLA, NEW YORK

Bibliographical Note

This Dover edition, first published in 2004, is an unabridged republication of an earlier Dover reprint (1949) of the translation first published by The Macmillan Company, New York, in 1939. The translation follows volume 2 (Geometrie) of the three-volume third German edition of Elementarmathematik vom hheren Standpunkte aus, published by J. Springer, Berlin, in 19241928.

Library of Congress Cataloging-in-Publication Data

Klein, Felix, 1849-1925.

[Elementarmathematik vom hheren Standpunkte aus. English]

Elementary mathematics from an advanced standpoint. Geometry / Felix Klein; translated from the third German edition by E.R. Hedrick and C.A. Noble.

p. cm.

Originally published: New York : Macmillan, 1939.

A translation of v. 2 of the authors three-volume work entitled Elementarmathematik vom hoheren Standpunkte aus.

Includes index.

ISBN 0-486-43481-8 (pbk.)

1. GeometryStudy and teaching. I. Title: Geometry. II. Title.

QA461.K4513 2004
372.7dc22

2004045586

Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

In the preface to , devoted to geometry, could appear very soon. Nevertheless it has been possible to complete it, thanks to the diligence of Mr. Hellinger.

Concerning the origin and purpose of this series of lectures I have nothing especial to add to what was said in the foreword to . However, a word seems necessary concerning the new form which this second part has assumed.

This form is, in fact, quite unlike that of . I made up my mind to give, above all, a comprehensive view of the field of geometry, of such a range as I should wish every teacher in a higher school to have; the discussions about geometric instruction were pushed into the background and were placed in connected form at the end, insofar as there was room.

The choice of this new order was motivated partly by the desire to avoid a stereotyped form. There were, however, weightier and deeper reasons. In geometry we possess no unified textbooks corresponding to the general level of the science, such as exist in algebra and analysis, thanks to the model French Cours. We find, rather, a single page here, another there, of an extensive subject, just as it has been developed by one or another group of investigators. In contrast to this, it seemed to be demanded by the pedagogic and the general scientific purpose which I am pursuing that I attempt a more unified presentation.

I close with the wish that the two complementary parts of my Elementary Mathematics from an Advanced Standpoint which are herewith completed may find the same friendly reception in the teaching world as the lectures on the organization of mathematical instruction by Mr. Schimmack and myself, which appeared last year.

Gttingen, Christmas, 1908

Klein

In accord with the comprehensive plan for the new edition of my mimeographed lectures, which I developed in the preface to the third edition of the first volume, the text and presentation of this, the second volume, have remained unaltered, except for small changes in detail and a few insertions.

The two supplements, which concern literature of a scientific and pedagogic character which was not considered in the original text, were prepared by Mr. Seyfarth after repeated conferences with me. He assumed again the major portion of the burden entailed by the publication. Messrs. Hellinger, Vermeil, and Walther assisted him in the proof reading. Mr. Vermeil undertook the preparation of the two indexes. I feel under obligation to these gentlemen, and also to the firm of Julius Springer, which showed on all occasions a willing spirit of accommodation.

Gttingen, May, 1925

Klein

The favorable reception given to the English translation of volume one of Kleins three-volume work entitled Elementary Mathematics from an Advanced Standpoint seems to justify the appearance of the present book, which is a translation of volume two. Professor Courant of New York University, who, while he was a professor at Gttingen, suggested the English translation of Kleins books, has been generously helpful in smoothing the way for the printing of volume two in the United States.

The Translators

Newly added remarks are indicated by square brackets.

Gentlemen! The course of lectures which I now begin will be an immediate continuation of, and a supplement to, my course of last Winter. My purpose now, as it was then, is to gather together all the mathematics that you studied during your student years, insofar as this could be of interest for the future teacher, and, in particular, to show its bearing in the business of school instruction. I carried out this plan, during the winter semester, for Arithmetic, Algebra, and Analysis. During the current semester, attention will be given to geometry, which was then left to one side. In this course, comprehension of our considerations will be independent of a knowledge of the preceding course of lectures. Moreover, I shall give the whole a somewhat different tone: In the foreground I shall place, let me say, the encyclopedic idealyou will be offered a survey of the entire field of geometry into which you can arrange, as into a rigid frame, all the separate items of knowledge which you have acquired in the course of your study, in order to have them at hand when occasion to use them arises. Only afterward shall I emphasize that interest in mathematical instruction which was always my starting point last winter.

I am glad to refer to a vacation course for teachers of mathematics and physics which was given here in Gttingen during the Easter vacation in 1908. In it I gave an account of my winter lectures. In connection with this, and also with the address of Professor Behrendsen of the local gymnasium, there arose an interesting and stimulating discussion concerning the reorganization of school instruction in arithmetic, algebra, and analysis, and more particularly about the introduction of differential and integral calculus into the schools.toward the elimination of the old complaint which we have had to hear continuallyand often justlyfrom the schools: University instruction provides, indeed, much of a special nature, but it leaves the beginning teacher entirely without orientation as to many important general things which he could really use later.

Concerning now the material of these lectures, let me say that, as in the preceding course, I shall now and then have to presuppose knowledge of important theorems from all of the fields of mathematics which you have studied, in order to lay emphasis upon a general survey of the whole. To be sure, I shall always try to assist your memory by brief statements, so that you can easily orient yourself in the literature. On the other hand, I shall draw attention, more than is usually done, as I did in , to the historical development of the science, to the accomplishments of its great pioneers. I hope, by discussions of this sort, to further, as I like to say, your