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

72 INTRODUCTION TO MATHEMATICS
thirds, and with allied notions. Thus as the
first generalization of number we place the
concept of fractions. The Greeks thought of
this subject rather in the form of ratio, so
that a Greek would naturally say that a
line of two feet in length bears to a line of
three feet in length the ratio of 2 to 3.
Under the influence of our algebraic notation we would more often say that one line
was two-thirds of the other in length, and
would think of two-thirds as a numerical
multiplier.
In connection with the theory of ratio, or
fractions, the Greeks made a great discovery,
which has been the occasion of a large amount
of philosophical as well as mathematical
thought. They found out the existence of
"incommensurable" ratios. They proved,
in fact, during the course of their geometrical
investigations that, starting with a line of any
length, other lines must exist whose lengths
do not bear to the original length the ratio
of any pair of integers-or, in other words,
that lengths exist which are not any exact
fraction of the original length.
For example, the diagonal of a square cannot be expressed as any fraction of the side of
the same square; in our modern notation the
length of the diagonal is 9/2 times the length
of the side. But there is no fraction which
exactly represents V2. We can approximate