As a general rule we treat in each calculation only objects of one nature. We do not, and cannot properly add, in the same sum yards of cloth and pounds of sugar. We cannot even conceive the result of adding area to velocity, or length to density, or weight to value. The units added must have a basis of homogeneity, or must be reducible to some common denominator. Nevertheless it is possible, and in fact common, to treat in one complex calculation the most heterogeneous quantities, on the condition that each kind of object is kept distinct, and treated numerically only in conjunction with its own kind. Different units, so far as their logical differences are specified, must never be substituted one for the other. Chemists continually use equations which assert the equivalence of groups of atoms. Ordinary fermentation is represented by the formula C^{6} H^{12} O^{6} = 2C^{2} H^{6} O + 2CO^{2}. Three kinds of units, the atoms respectively of carbon, hydrogen, and oxygen, are here intermingled, but there is really a separate equation in regard to each kind. Mathematicians also employ compound equations of the same kind; for in, *a* + *b* √ - 1 = *c* + *d* √ - 1, it is impossible by ordinary addition to add *a* to *b* √ - 1. Hence we really have the separate equations *a* = *b*, and *c* √ - 1 = *d* √ - 1. Similarly an equation between two quaternions is equivalent to four equations between ordinary quantities, whence indeed the name *quaternion*. *Analogy of Logical and Numerical Terms.* If my assertion is correct that number arises out of logical conditions, we ought to find number obeying all the laws of logic. It is almost superfluous to point out that this is the case with the fundamental laws of identity and difference, and it only remains to show that mathematical symbols do really obey the special conditions of logical symbols which were formerly pointed out (p. 32). Thus the Law of Commutativeness, is equally true of quality and quantity. As in logic we have AB = BA, so in mathematics it is familiarly known that 2 × 3 = 3 × 2, or *x* × *y* = *y* × *x*. The properties of space are as indifferent in multiplication as we found them in pure logical thought. Similarly, as in logic triangle or square = square or triangle, or generally A ꖌ B = B ꖌ A, so in quantity 2  +  3  =  3  +  2, or generally *x* + *y* = *y* + *x*. The symbol ꖌ is not identical with +, but it is thus far analogous. How far, now, is it true that mathematical symbols obey the Law of Simplicity expressed in the form AA = A, or the example Round round = round? Apparently there are but two numbers which obey this law; for it is certain that *x* × *x* = *x* is true only in the two cases when *x* = 1, or *x* = 0.