177 
of Edinburgh, Session 1869 - 70 . 
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 tells 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 “ experi- 
ment ” agree with the descriptions of experiments in books of 
Physics, save only in this, that we have all njade Euclid’s experi- 
ment before ? Clearly not. In picturing to myself an experi- 
mental 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 result following — or rather, I am told to believe 
this result, for to picture it is quite superflous and often impossible. 
Euclid, on the other hand, tells me to superpose ideally the point 
A on C, the line AB on CD, and so forth, and then I do not require 
to be told that the coincidence of the whole triangles follows. I 
have no choice to imagine coincidence or non-coincidence. I see 
that it follows, and that quite apart from previous experiment. 
Professor Bain allows the possibility of ideal experiments on 
mathematical forms.* I presume, therefore, that he will not deny 
that the intelligent reader of our proposition does, as he reads, 
make a valid experiment in favour of the proposition. But if this 
be so, where is the deception in Euclid’s proof, and what is the 
necessity of supplementing that proof by further “ideal” or 
“actual experiments”? The course of Euclid’s argument shows 
that the two triangles are not only equal, but equal in virtue of the 
way in which they have been constructed, viz., the equality of the 
two sides and the included angle. The fact that the. proof is not 
syllogistic does not make it any the less a case of that parity of 
* Logic, vol. i. p. 225. 
