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 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 far-reaching 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". 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.
