An introduction to mathematics, by A. N. Whitehead.

COORDINATE GEOMETRY 119
of first thoughts on the subject-which was
the cause of the comparative sterility of the
study of the science during so many centuries.
Coordinate geometry, and Descartes its inventor, must have the credit of disclosing the
true simple objects for geometrical thought.
In the place of a bit of a straight line, let
us think of the whole of a straight line
throughout its unending length in both directions. This is the sort of general idea from
which to start our geometrical investigations.
The Greeks never seem to have found any
use for this conception which is now fundamental in all modern geometrical thought.
Euclid always contemplates a straight line as
drawn between two definite points, and is
very careful to mention when it is to be produced beyond this segment. He never thinks
of the line as an entity given once for all as a
whole. This careful definition and limitation,
so as to exclude an infinity not immediately apparent to the senses, was very characteristic of the Greeks in all their many
activities. It is enshrined in the difference
between Greek architecture and Gothic architecture, and between the Greek religion and
the modern religion. The spire on a Gothic
cathedral and the importance of the unbounded straight line in modern geometry
are both emblematic of the transformation of
the modern world.