Numerical abstraction is thus seen to be a different process from logical abstraction (p. 27), for in the latter process we drop out of notice the very existence of difference and plurality. In forming the abstract notion *hardness*, we ignore entirely the diverse circumstances in which the quality may appear. It is the concrete notion *three hard objects*, which asserts the existence of hardness along with sufficient other undefined qualities, to mark out *three* such objects. Numerical thought is indeed closely interwoven with logical thought. We cannot use a concrete term in the plural, as *men*, without implying that there are marks of difference. But when we use an abstract term, we deal with unity. The origin of the great generality of number is now apparent. Three sounds differ from three colours, or three riders from three horses; but they agree in respect of the variety of marks by which they can be discriminated. The symbols 1 + 1 + 1 are thus the empty marks asserting the existence of discrimination. But in dropping out of sight the character of the differences we give rise to new agreements on which mathematical reasoning is founded. Numerical abstraction is so far from being incompatible with logical abstraction that it is the origin of our widest acts of generalization. *Concrete and Abstract Number.* The common distinction between concrete and abstract number can now be easily stated. In proportion as we specify the logical characters of the things numbered, we render them concrete. In the abstract number three there is no statement of the points in which the *three* objects agree; but in *three coins*, *three men*, or *three horses*, not only are the objects numbered but their nature is restricted. Concrete number thus implies the same consciousness of difference as abstract number, but it is mingled with a groundwork of similarity expressed in the logical terms. There is identity so far as logical terms enter; difference so far as the terms are merely numerical. The reason of the important Law of Homogeneity will now be apparent. This law asserts that in every arithmetical calculation the logical nature of the things numbered must remain unaltered. The specified logical agreement of the things must not be affected by the unspecified numerical differences. A calculation would be palpably absurd which, after commencing with length, gave a result in hours. It is equally absurd, in a purely arithmetical point of view, to deduce areas from the calculation of lengths, masses from the combination of volume and density, or momenta from mass and velocity. It must remain for subsequent consideration to decide in what sense we may truly say that two linear feet multiplied by two linear feet give four superficial feet; arithmetically it is absurd, because there is a change of unit.