PRINCIPLES OF NUMBER. 183 



symbols, but two of these laws seem to be inapplicable 

 simply because they are presupposed in the definition of 

 the mathematical unit. Logic thus lays down the con- 

 ditions of number, and the science of arithmetic developed 

 as it is into all the wondrous branches of mathematical 

 calculus is but an outgrowth of logical discrimination. 



Principle of Mathematical Inference. 



As I have asserted, the universal principle of all 

 reasoning is that which allows us to substitute like for 

 like. I have now to point out that in the mathema- 

 tical sciences this principle is involved in each step of 

 reasoning. It is in these sciences indeed that we meet 

 with the clearest cases of substitution, and it is the 

 simplicity with which the principle can be applied which 

 probably led to the comparatively early perfection of the 

 sciences of geometry and arithmetic. Euclid, and the 

 Greek mathematicians from the first, recognised equality 

 as the fundamental relation of quantitative thought, but 

 Aristotle rejected the exactly analogous, but far more 

 general relation of identity, and thus crippled the formal 

 science of logic as it has descended to the present day. 



Geometrical reasoning starts from the Axiom that 

 ' things equal to the same thing are equal to each other/ 

 Two equalities enable us to infer a third equality ; and this 

 is true not only of lines and angles, but of areas, volumes, 

 numbers, intervals of time, forces, velocities, degrees of 

 intensity, or, in short, anything which is capable of being 

 equal or unequal. Two stars equally bright with the 

 same star must be equally bright with each other, and two 

 forces equally intense with a third force are equally 

 intense with each other. It is remarkable that Euclid 

 has not expressly stated two other axioms, the truth of 

 which is necessarily implied. The second axiom should 



