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.
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.
