The Principles of Mathematics (1903)

§ 151

In fixing the meaning of such a term as quantity or magnitude, one is faced with the difficulty that, however one may define the word, one must appear to depart from usage. This difficulty arises wherever two characteristics have been commonly supposed inseparable which, upon closer examination, are discovered to be capable of existing apart. In the case of magnitude, the usual meaning appears to imply (1) a capacity for the relations of greater and less, (2) divisibility. Of these characteristics, the first is supposed to imply the second. But as I propose to deny the implication, I must either admit that some things which are indivisible are magnitudes, or that some things which are greater or less than others are not magnitudes. As one of these departures from usagei s unavoidable, I shall choose the former, which I believe to be the less serious. A magnitude, then, is to be defined as anything which is greater or less than something else.(§ 151 ¶ 1)

It might be thought that equality should be mentioned, along with greater and less, in the definition of magnitude. We shall see reason to think, however—paradoxical as such a view may appear—that what can be greater or less than some term, can never be equal to any term whatever, and vice versâ. This will require a distinction, whose necessity will become more and more evident as we proceed, between the kind of terms that can be equal, and the kind that can be greater or less. The former I shall call quantities, the latter magnitudes. An actual foot-rule is a quantity: its length is a magnitude. Magnitudes are more abstract than quantities: when two quantities are equal, they have the same magnitude. The necessity of this abstraction is the first point to be established.(§ 151 ¶ 2)