70 LECTURES AND ESSAYS [1869-1873] 



of such consequences as that the diagonal of a parallelo 

 gram divides it into two triangles, Euclid offers no other 

 proof than an appeal to the eye. 1 In fact, no other proof 

 can be offered. Yet surely it will not be asserted that this 

 too is an induction. In one word, if no proposition is fairly 

 demonstrated where it is essential to look at the figure, 

 there is no sound demonstration in synthetic geometry. 



Finally, Professor Bain himself seems not quite satisfied 

 as to the inductive nature of Euc. I. 4. The proof,&quot; 

 he says, &quot; rests solely on definitions,&quot; and hence &quot; the 

 proposition cannot be real the subject and predicate 

 must be identical.&quot; Surely an identical proposition is 

 not an induction ! And surely, too, the proof rests not 

 on definitions merely, but on definitions and the use of the 

 figure ! But I do not think that Professor Bain means 

 to speak here in strict logical terms, for he straightway 

 adds in explanation, &quot; The proposition must, in fact, be 

 a mere equivalent of the notions of line, angle, surface, 

 equality a fact apparent in the operation of understand 

 ing these notions. It is implicated in the experience 

 requisite for mastering the indefinable elements of 

 geometry, and should be rested purely on the basis of 

 experience.&quot; We should have known better what this 

 sentence means, if the author had adopted here the 

 distinction between synthetic and analytic judgments. 

 He cannot mean that a truth that is an induction, and 

 rests on experience, is an analytic judgment, that it can 

 be reached by a purely formal dividing and compounding 

 of the definitions of terms. Such a proposition could be 

 shown to be true without any figure or any experiment. 

 Yet the proposition is, we are told, involved in the 

 notions ; we cannot know what lines, angles, etc., are 

 without knowing this too. If this means anything, it 

 means that Euc. I. 4 is a synthetic judgment a priori ; 

 and that, after all, Kant and the mathematicians are 

 right, and Mr. Mill and the empirical logicians wrong. 



1 Logic, vol. ii. p. 218. 



