These three volumes constitute the first complete English translation of Felix Kleinâs seminal series âElementarmathematik vom höheren Standpunkte ausâ. âCompleteâ 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 Kleinâs far-reaching conception of elementarisation, of the âelementary from a higher standpointâ, in its implementation for school mathematics. This volume II presents a paradigmatic realisation of Kleinâs 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: âProjective geometry is all geometryâ. Non-Euclidean 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.
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