By Howard Eves
From the book's preface:
Since writing the preface of the 1st variation of this paintings, the gloomy plight there defined of starting collegiate geometry has brightened significantly. The pendulum turns out certainly to be swinging again and a goodly volume of good textual fabric is showing.
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Additional resources for A Survey of Geometry (Revised Edition)
Riemann does not seem to be aware that there is no explicit concept of space in Greek mathematics; see Jammer 1966, 15-24, and Huggett 1999 (a useful anthology of writings about space). By “multiply extended quantity” he means the generalization of a single Euclidean magnitude to a magnitude characterized by many numbers. The particular case that engages him is that of ordinary space, which could be considered to be a three-fold extended magni tude, since any point can be characterized by three numbers (say, the Cartesian coordinates of that point x }y,z).
5 On this assumption, all effects proceeding from ponderable bodies and im pacting ponderable bodies via empty space must be propagated through this substance. Therefore, the forms of motion of light and heat the heavenly bodies send forth to each other must also be forms of motion of this substance. 6 I now assume that the real motion of the substance in empty space is com pounded of the motion that must be assumed for the explanation of gravitation and the motion that must be assumed for the explanation of light.
But Riemann is aware that it is possible that such a homogenous sense of distance might not apply to all possible spaces; in some spaces there might be certain special points or regions near which the distance function is changed. ] 6 . [As he indicated in his introduction, Riemann considers that the mere ex istence of such manifolds does not settle the question of their metric (distance). That is, given a (possibly curved) surface in a space of arbitrary dimension, what sort of distance-functions (“metric relations”) can describe that surface?
A Survey of Geometry (Revised Edition) by Howard Eves