viii.] PRINCIPLES OF NUMBER. 163 



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 

 arid 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 explicitly 

 stated two other axioms, the truth of which is necessarily 

 implied. The second axiom should be that " Two things of 

 which one is equal and the other unequal to a third com- 

 mon thing, are unequal to each other." An equality and 

 inequality, in short, give an inequality, and this is equally 

 true with the first axiom of all kinds of quantity. If 

 Venus, for instance, agrees with Mars in density, but Mars 

 differs from Jupiter, then Venus differs from Jupiter. A 

 third axiom must exist to the effect that " Things unequal 

 to the same thing may or may not be equal to each 

 other." Two inequalities give no ground of inference what- 

 ever. If we only know, for instance, that Mercury and 

 Jupiter differ in density from Mars, we cannot say whether 

 or not they agree between themselves. As a fact they do 

 not agree ; but Venus and Mars on the other hand both 

 differ from Jupiter and yet closely agree with each other. 

 The force of the axioms can be most clearly illustrated by 

 drawing equal and unequal lines. 1 



1 Elementary Lessons in Logic (Macmillan), p. 123. It is pointed 

 out in the preface to this Second Edition, that the views here given 

 were partially stated by Leibnitz. 



M 2 



