E-Book Overview
These three volumes constitute the first complete English translation of Felix Klein’s seminal series “Elementarmathematik vom höheren Standpunkte aus”. “Complete” has a twofold meaning here: First, there now exists a translation of volume III into English, while until today the only translation had been into Chinese. Second, the English versions of volume I and II had omitted several, even extendedparts of the original, while we now present a complete revised translation into modern English.
The volumes, first published between 1902 and 1908, are lecture notes of courses that Klein offered to future mathematics teachers, realizing a new form of teacher training that remained valid and effective until today: Klein leads the students to gain a more comprehensive and methodological point of view on school mathematics. The volumes enable us to understand Klein’s far-reaching conception of
elementarisation, of the “elementary from a higher standpoint”, in its implementation for school mathematics.
In Volume III, Klein explores the relationship between precision and approximation mathematics. He crosses the various fields of mathematics – from functions in one and two variables to practical geometry to space curves and surfaces – underlining the relation between the exactness of the idealised concepts and the approximations to be considered in applications. Logical procedures are confronted with the way in which concepts arise starting from observations. It is a comparison between properties pertaining only to the theoretical field of abstract mathematics and properties that can be grasped by intuition. The final part, which concerns gestalt relations of curves and surfaces, shows Klein to be the master of the art of description of geometric forms.
E-Book Content
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 DOI 10.1007/978-3-662-49439-4
ISBN 978-3-662-49439-4 (eBook)
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 publication 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