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Elementary Mathematics from a Higher Standpoint

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Elementary Mathematics from a Higher Standpoint Felix Klein Elementary Mathematics from a Higher Standpoint Volume III: Precision Mathematics and Approximation Mathematics Elementary Mathematics from a Higher Standpoint Felix Klein Elementary Mathematics from a Higher Standpoint Volume III: Precision Mathematics and Approximation Mathematics Translated by Marta Menghini in collaboration with Anna Baccaglini-Frank Mathematical advisor for the English translation: Gert Schubring Felix Klein Translated by: Marta Menghini Mathematical advisor for the English translation: Gert Schubring ISBN 978-3-662-49437-0 ISBN 978-3-662-49439-4 (eBook) DOI 10.1007/978-3-662-49439-4 Library of Congress Control Number: 2016943431 Translation of the 3rd German edition „Elementarmathematik vom höheren Standpunkte aus“, vol. 3 by Felix Klein, Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band 16, Verlag von Julius Springer, Berlin 1928. © Springer-Verlag Berlin Heidelberg 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publica- tion does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Cover Illustration: The publisher could not determine a copyright holder and believes this picture to be in the public domain. If the copyright holder can prove ownership, the publisher will pay the customary fee. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg Preface to the 2016 Edition The present volume contains the translation of Felix Klein’s “Präzisions- und Approximationsmathematik”, which appeared in 1928 as volume III of the series Elementarmathematik von einem höheren Standpunkte aus1 (Elementary mathe- matics from a higher standpoint). In turn, the 1928 edition was a re-edition of the lecture course delivered by Felix Klein in 1901 with the title “Anwendung der Differential- und Integralrechnung auf die Geometrie: eine Revision der Prinzi- pien”, published in 1902 in lithographic form2 (a reprint of 1908, edited by Conrad Heinrich Müller, left the text essentially unchanged). Klein participated, together with Fritz Seyfarth, in the whole project of re-editing the three volumes on Elementary mathematics from a higher standpoint, but he died in 1925, after the first two volumes had appeared. The third volume was, therefore, edited only by Seyfarth; changes and insertions had been nevertheless discussed with Klein, as Seyfarth writes in his preface. The notes added are probably only by Seyfarth. We left, as in the 1928 edition, the notes by Seyfarth in square brackets, and labelled “translator’s note” the notes added in the present edition. Of course, the notes by Klein himself are included without any additional marks. In the text the reader will find, again in square brackets, the page numbering of the original edition. Cross references in notes and in the text refer to this numbering, as well as the name index and the subject index (that is, the original text has not been changed to this respect). Similarly for the references to Volume 1 and 2 of 1924/25, whose original layout is marked in the current new edition. Moreover, we left, as in the German edition, the comma to separate the integer part from the fractional part of a decimal number. 1 Felix Klein, 1928. Elementarmathematik vom höheren Standpunkte aus, 3: Präzisions- und Approximationsmathematik; ausgearbeitet von C. H. Müller; für den Druck fertig gemacht und mit Zusätzen versehen von Fr. Seyfarth. 3. Aufl., Berlin: J. Springer, 1928 Series: Die Grundlehren der mathematischen Wissenschaften, 16. Reprint in 1968. 2 Felix Klein, 1902. Anwendung der Differential- und Integralrechnung auf die Geometrie: eine Revision der Prinzipien: Vorlesung gehalten während des Sommersemesters 1901; ausgearbeitet von C. Müller, Leipzig: Teubner. v vi Preface to the 2016 Edition In the present translation we have added, when possible, the first names of the persons mentioned. In the German edition, as it was customary at that time, the first names were indicated only with the initials. The bibliographic references in the notes have also been completed, when needed. “Präzisions- und Approximationsmathematik” was never translated into English, unlike the first two volumes; although the translators of the first two volumes were aware of the existence of volume III, they gave no reason for not having translated it. While the Spanish, the Russian and the Japanese translations also did not include volume III3, there is a complete translation of all the three volumes into Chinese – first published in 1989 by Hubei Educational Press, Beijing, and published again by Chiu Chan Publishing (Republic of China, Taipei) in 1996. On the occasion of the 13th International Congress on Mathematics Education (ICME-13, Hamburg 2016), the will of the International Commission on Mathe- matical Instruction (ICMI), together with Springer, is of closing this gap in the English version. We also proudly note that Felix Klein was the first president of ICMI, which was founded in Rome in 1908. A translation after nearly a century, joined with a new edition of the two first volumes, might seem strange. Undoubtedly, this translation has an historical value, since Klein is one of the greatest mathematicians of history4. It also has a math- ematical value – because of the interesting approaches and the links with the ap- plications. But, above all, it has a didactical value, concerning the training of mathematics teachers. The volumes on Elementary Mathematics from a higher standpoint are still apt to this aim, for the simple reason that today’s school math- ematics is not very different from the school mathematics proposed at the time of Klein. Even if today we can find some snapshots of the mathematics of the 20th cen- tury (which is indeed too specialised) and also contents of probability and statistics, the remaining curricular contents have been developed well before the 20th cen- tury. The higher mathematics for these contents is mainly found in the period of the systematization of the mathematical theories, which corresponds to the period of Klein’s writings. And, indeed, most of the contents of Klein’s volumes are still offered in the many courses (which often bear the same name of Elementary math- ematics from a higher standpoint) that all over the world are devoted to pre-service training of mathematics teachers. For this third volume in particular, the motivations for the kind of study it pro- poses are given by Klein himself during his lifelong career. The third volume focuses on those properties that applied mathematicians take for granted when studying certain phenomena from a mathematical point of view. These properties must be seen as supplementary conditions (and constraints) to be required for the ideal objects of pure mathematics. However, in the meantime, these very proper- 3 There is a recent translation of Volumes I and II into Portuguese and it might be extended to volume III. 4 Books by Felix Klein are still in use today: Klein’s Nicht-euklidische Geometrie has been re- edited in 2006, the English version of the Lectures on the Ikosahedron re-edited in 2007, and his Development of mathematics in the nineteenth century was translated in 1979. Preface to the 2016 Edition vii ties prove to be the more intuitive ones. Therefore the comparison moves towards another field: it is a comparison between properties that can be considered only in the theoretical field of abstract mathematics and properties that can be grasped by intuition. Here the problem proves to become pertinent for mathematics teaching. Klein had always been convinced of the importance of maintaining the link of mathematics with its applications and had manifested this interest ever since his initial work at the university of Erlangen, in 1872. The necessary link between pure and applied mathematics has been aptly characterised by Klein, in the 6th of his American Conferences (The Evanston Colloquium, 1894, p. 46): I should lay particular stress on the heuristic value of the applied sciences as an aid to discovering new truths in mathematics. Thus I have shown (in my little book on Riemann’s theories) that the Abelian integrals can best be un- derstood and illustrated by considering electric currents on closed surfaces. In an analogous way, theorems concerning differential equations can be de- rived from the consideration of sound-vibrations; and so on. In this way Klein stresses the importance of the connection of mathematics with other disciplines that can foster intuition; and, in particular, the importance of main- taining
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