THEIR HISTORY AND SOLUTION
BY HEINRICH DORRIE
TRANSLATED BY DAVID ANTIN
NEW YORK DOVER PUBLICATIONS, INC.
Copyright © 1965 by Dover Publications. Inc.: originally published in German under the title of Triumph der Mathematik, © 1958 by PhysicaYerlag. WUrzburg. All rights reserved under Pan American and International CopyrightConventions.
Published in Canada by General Publishing Company. Ltd., 30 Lesmill Road. Don Mills. Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd .• 10 Orange Street, London WC 2.
This Dover edition. first published in 1965. is a new translation of the unabridged text of the fifth edition of the work published by the Physica-Yerlag. Wiirzburg, Germany. in 1958 under thetitle Triumph der Mathematik: Hundert beruhmte Probleme aw %wei Jahrtawenden mathematischer Kultur. This authorized translation is published by special arrangement with the German-language publishers. Physica-Yerlag. Wiirzburg.
Standard Book Number: 486-61348-8 Library of Congress Catalog Card Number:65-140JO
Manufactured in the United States of America Dover Publications, Inc. 180 YarickStreet New York. N.Y. 10014
A book collecting the celebrated problems of elementary mathematics that would commemorate their origin and, above all, present their solutions briefly, clearly, and comprehensibly has long seemed a necessary and attractive task to the author. The restriction to problems of elementary mathematics was considered advisable in view of those readers who haveneither the time nor the opportunity to acquaint themselves in any detail with higher mathematics. Nevertheless, in spite of this limitation a colorful and compelling picture has emerged, one that gives an idea of the amazing variety of mathematical methods and one that will-I hope-enchant many who are interested in mathematics and who take pleasure in characteristic mathematical thought processes. Inthe present work there are to be found many pearls of mathematical art, problems the solutions of which represent, in the achievements ofa Gauss, an Euler, Steiner, and others, incredible triumphs of the mathematical mind. Because the difficult economic situation at the present time barred the publication of a larger work, a limit had to be set to the scope and number of the problems treated.Thus, I decided on a round number of one hundred problems. Moreover, since many of the problems and solutions require considerable space despite the greatest concision, this had to be compensated for by the inclusion of a number of mathematical miniatures. Possibly, however, it may be just these little problems, which are, in their way, true jewels of mathematical miniature work, that will find thereadiest readers and win new admirers for the queen of the sciences. As we have indicated already, a knowledge of higher analysis is not assumed. Consequently, the Taylor expansion could not be used for the treatment of the important infinite series. I hope nonetheless that the derivations we have given, particularly the striking derivation of the sine and cosine series, will please and will not befound unattractive even by mathematically sophisticated readers.
On the other hand, in some of the problems, e.g., the Euler tetrahedron problem and the problem of skew lines, the author believed it necessary not to dispense with the simplest concepts of vector analysis. The characteristic advantages of brevity and elegance of the vector method are so obvious, and the time and effortrequired for mastering it so slight, that the vectorial methods presented here will undoubtedly ~pur many readers on to look into this attractive area. For the rest, only the theorems of elementary mathematics are assumed to be known, so that the reading of the book will not entail significant difficulties. In this connection the inclusion of the little problems may in fact increase the...