68 LECTURES AND ESSAYS [1869- 



syllogistic inference from Euclid s axioms that he errs. 

 Professor Bain does not attempt to defend the blunder 

 of his predecessor. He admits that Euclid s proof cannot 

 be reduced to a chain of syllogisms. But, instead of 

 surrendering Mr. Mill s theory of mathematical reasoning, 

 he concludes that Euclid has not demonstrated his 

 proposition that the superposition which he enjoins is 

 only an experiment, and that &quot; if his readers had not made 

 actual experiments of the kind indicated, they could not 

 be convinced by the reasoning in the demonstration.&quot; l 



Now I believe, and in my former paper expressly 

 pointed out, that the position that Euc. I. 4 is really an 

 inductive truth, and that the usual demonstration is not 

 in itself convincing, is the only ground that remains to 

 Mr. Mill and his adherents. So far, then, I am confirmed 

 by Professor Bain : it remains only to show that this 

 new position is mathematically as untenable as that from 

 which Mr. Mill has been dislodged. If Professor Bain 

 grants that the proof of Euc. I. 4 is not by syllogism from 

 axioms if, again, mathematically it is plain that there 

 is none the less a real proof, not merely an induction we 

 shall have gone far to establish the validity of proof by 



intuition. 



Professor Bain teUs us that Euclid, while professedly 

 going through a process of pure deduction, requires us 

 to conceive an experimental proof. There is surely an 

 ambiguity here. Does Mr. Bain mean that Euclid merely 

 calls to our mind former concrete experiments with 

 triangles of card-board or paper, for these alone are actual 

 and concrete to our author? Does Euclid s &quot;experi 

 ment &quot; agree with the descriptions of experiments in 

 books of Physics, save only in this, that we have all made 

 Euclid s experiment before ? Clearly not. In picturing 

 to myself an experimental proof in the usual sense, I 

 imagine mentally, or with the help of a diagram, certain 

 arrangements, and then I am told to imagine a certain 



1 Logic, vol. ii. p. 217. 



