ABSTRACT
There is no doubt that the notion of half a league, or half a day, is a legitimate notion. It is therefore necessary to find some sense for fractions in which they do not essentially depend upon number. Fractions have the advantage that they introduce a discrimination of greater and smaller among infinite aggregates having the same number of terms. There is no difficulty in connecting the theory of ratio with the usual theory derived from multiplication and division. But the usual theory does not show, as the present theory does, why the infinite integers do not have ratios strictly analogous to those of finite integers. The fact is, that ratio depends upon consecutiveness. And consecutiveness as defined does not exist among infinite integers, since these are unchanged by the addition of 1.
