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.
