160 THE PRINCIPLES OF SCIENCE. [CHAP 



plied by two linear feet give four superficial feet ; arith- 

 metically it is absurd, because there is a change of unit. 



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 speci- 

 fied, 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 H 12 0' = 2C 2 H 6 + 2001 



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. Mathe- 

 maticians also employ compound equations of the same 

 kind ; for in, + & Y/ i = c + d ^/ I, it is impossible 

 by ordinary addition to add a to b ^/ I. Hence we 

 really have the separate equations a = b, and c V I = d 

 s/ I. 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 



