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Klein, Felix
Elementary mathematics from a higher standpoint. Volume II: Geometry. Translated from the 4th German edition by Gert Schubring.
Berlin: Springer (ISBN 9783662494431/pbk; 9783662494455/ebook; 9783662495155/set). xvi, 315~p. (2016).
2016
Berlin: Springer
EN
G10
Zbl 1341.11001
Zbl 1339.26002
JFM 51.0055.13
doi:10.1007/9783662494455
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. This volume II presents a paradigmatic realisation of Klein's approach of elementarisation for teacher education. It is shown how the various geometries, elaborated particularly since the beginning of the 19th century, are revealed as becoming unified in a new restructured geometry. As Klein liked to stress: ``Projective geometry is all geometry". NonEuclidean geometry proves to constitute a part of this unifying process. The teaching of geometry is discussed in a separate chapter, which provides moreover important information on the history of geometry teaching and an international comparison. For Volumes I and III see [Zbl 1341.11001; Zbl 1339.26002]. See the review of the 3rd German edition in [JFM 51.0055.13].