The Principles of Mathematics (1903)

§ 146

The above theory of ratio has, it must be confessed, a highly artificial appearance, and one which makes it seem extraordinary that ratios should occur in daily life. The fact is, it is not ratios, but fractions, that occur, and fractions are not purely arithmetical, but are really concerned with relations of whole and part.(§ 146 ¶ 1)

Propositions asserting fractions show an important difference from those asserting integers. We can say A is one, A and B are two, and so on; but we cannot say A is one-third, or A and B are two-thirds. There is always need of some second entity, to which our first has some fractional relation. We say A is one-third of C, A and B together are two-thirds of C, and so on. Fractions, in short, are either relations of a simple part to a whole, or of two wholes to one another. But it is not necessary that the one whole, or the simple part, should be part of the other whole. In the case of finite wholes, the matter seems simple: the fraction expresses the ratio of the number of parts in the one to the number in the other. But the consideration of infinite wholes will show us that this simple theory is inadequate to the facts.(§ 146 ¶ 2)