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

78 INTRODUCTION TO MATHEMATICS
index. Thus 7 is the index of 4, and also of
-, and of -. Let the fractions in each class
be all fractions which have some specified
index, which may therefore also be called
the class index. Now arrange these classes
in the order of magnitude of their indices.
The first class has the index 2, and its only
member is 1; the second class has the index
3, and its members are 1 and 2; the third
class has the index 4, and its members are -,
-, -; the fourth class has the index 5, and its
members are -~,, 3, I; and so on. It is easy
to see that the number of members (still including fractions not in their lowest terms)
belonging to any class is one less than its
index. Also the members of any one class
can be arranged in order by taking the first
member to be the fraction with numerator 1,
the second member to have the numerator 2,
and so on, up to (n - 1) where n is the index.
Thus for the class of index n, the members
appear in the order.
1 2 3 n-l
1,2,- 3 n.i.., n —. The members
n- 1 n- ' n- 3' 1
of the first four classes have in fact been
mentioned in this order. Thus the whole set
of fractions have now been arranged in an
order like that of the integers. It runs thus:
1121 r21 3 1 2 3 4
1 2' 15 3' ' l_ 1' 4' 3' 2' 1'"'