Analytic geometry (Plane and solid)

Geometry, at its start, was largely arithmetical in nature. It dealt with lengths from the surveyor's point of view, according to which every length is expressible as an exact multiple of some one length selected as the unit, or of certain subdivisions of this selected length such that it contains one subdivision as exact integral number of times. The Greeks, however, soon saw that there existed lengths which could not be expressed as exact integral or fractiona1 multiples of the unit of 1ength. Pythagorus, for example, showed that if the side of a square was one unit of length, then no subdivision of that unit could be found which would enable the diagonal to be expressed in terms of the side by means of a finite ratio.