The Principles of Mathematics (1903)

§ 144

The present chapter, in so far as it deals with relations of integers, is essentially confined to finite integers: those that are infinite have no relations strictly analogous to what are usually called ratios. But I shall distinguish ratios, as relations between integers, from fractions, which are relations between aggregates, or rather between their magnitudes of divisibility; and fractions, we shall find, may express relations which hold where both aggregates are infinite. It will be necessary to begin with the mathematical definition of ratio, before proceeding to more general considerations.(§ 144 ¶ 1)

Ratio is commonly associated with multiplication and division, and in this way becomes indistinguishable from fractions. But multiplication and division are equally applicable to finite and infinite numbers, though in the case of infinite numbers they do not have the properties which connect them with ratio in the finite case. Hence it becomes desirable to develop a theory of ratio which shall be independent of multiplication and division.(§ 144 ¶ 2)

Two finite numbers are said to be consecutive when, if u be a class having one of the numbers, and one term be added to u, the resulting class has the other number. To be consecutive is thus a relation which is one-one and asymmetrical. If now a number a has to a number b the nth power of this relation of consecutiveness (powers of relations being defined by relative multiplication), then we have a+n=b. This equation expresses, between a and b, a one-one relation which is determinate when n is given. If now the mth power of this relation holds between a′ and b′, we shall have a′+mn=b′. Also we may define mn as 0+mn. If now we have three numbers a, b, c such that ab=c, this equation expresses between a and c a one-one relation which is determinate when b is given. Let us call this relation B. Suppose we have also a′b′=c. Then a has to a′ a relation which is the relative product of B and the converse of B′, where B′ is derived from b′ as B was derived from b. This relation we define as the ratio of a′ to a. This theory has the advantage that it applies not only to finite integers, but to all other series of the same type, i.e. all series of the type which I call progressions.(§ 144 ¶ 3)