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

182 INTRODUCTION TO MATHEMATICS
an angle; for it is dependent on no arbitrary
unit of length, and on no arbitrary way of
dividing up any arbitrarily assumed angle,
AP
such as a right-angle. Thus the fraction AP
represents the magnitude of the angle AOP.
Now draw PM perpendicularly to OA. Then
the Greek mathematicians called the line PM
the sine of the arc AP, and the line 031 the
cosine of the arc AP. They were well aware
that the importance of the relations of these
various lines to each other was dependent on
the theory of similarity which we have just
expounded. But they did not make their
definitions express the properties which arise
from this theory. Also they had not in their
heads the modern general ideas respecting
functions as correlating pairs of variable numbers, nor in fact were they aware of any
modern conception of algebra and algebraic
analysis. Accordingly, it was natural to
them to think merely of the relations between
certain lines in a diagram. For us the case
is different: we wish to embody our more
powerful ideas.
Hence, in modern mathematics, instead
of considering the arc AP, we consider
AP
the fraction -, which is a number the
same for all lengths of OP; and, instead of
considering the lines PM and OM, we con