TABLE OF CONTENTS As far as we know, geometry originated from land measurements and area and volume computations made by the early Babylonians and Egyptians in the years approximately from 4000 to 1500 B.C. The word geometry itself is derived from the Greek word for "earth measure. " The Greeks (600 to 300 B.C.) developed the empirical geometry of the Babylonians and Egyptians into a more systematic science which ultimately prepared the ground for Euclid's outstanding Elements. Euclid, who lived in Alexandria in about 300 B.C., gave a systematic logical organization of what is now called Euclidean geometry. Euclid's work was so solid that it took 2000 years before geometries other than Euclidean geometry were discovered.
The structure of an affine space does not encompass distances and angles. If distances and angles are defined in an affine space, one refers to it as a Euclidean space. In general, affine maps do not preserve distances and angles. Those that do are the Euclidean motions.
The Euclidean space is an affine space where, in addition, the distance between two points is defined in the usual way. This distance between points induces a compatible length of vectors in the underlying linear space and facilitates the introduction of angles. However, a Euclidean space can be introduced more geometrically by means of a gauge ellipsoid.
Literature: Berger, Perdoe, Coxeter
