ABSTRACT
Among the traditional problems of mathematical philosophy, few are more important than the relation of quantity to number. Opinion as to this relation has undergone many revolutions. Euclid, as is evident from his definitions of ratio and proportion, and indeed from his whole procedure, was not persuaded of the applicability of numbers to spatial magnitudes. 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. 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.
