
06592123
b
2016e.00798
Klein, Felix
Elementary mathematics from a higher standpoint. Volume III: precision mathematics and approximation mathematics. Translated from the 3rd German edition by Marta Menghini and Anna BaccagliniFrank. Mathematical advisor for the English translation: Gerd Schubring.
Berlin: Springer (ISBN 9783662494370/pbk; 9783662494394/ebook; 9783662495155/set). xviii, 259~p. (2016).
2016
Berlin: Springer
EN
I10
G10
Zbl 1341.11001
Zbl 1341.51001
JFM 54.0068.03
doi:10.1007/9783662494394
Publisher's description: These three volumes constitute the first complete English translation of Felix Klein's seminal series ``Elementarmathematik vom h\"oheren 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 extended parts 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 farreaching 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. For Volumes I and II see [Zbl 1341.11001; Zbl 1341.51001]. For the original 3rd German edition see [JFM 54.0068.03]