PRINCIPLES OF NUMBER. 179 



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 similarity or 

 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 numbered 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 in reality 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 in what sense 

 we may truly say that two linear feet multiplied by two 

 linear feet give four superficial feet, but arithmetically 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 velo 

 city, or length to density, or weight to value. The unit 

 numbered and added must have a basis of homogeneity, 

 or must be reducible to some common denominator. 

 Nevertheless it is quite 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 



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