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reasoning which Professor Bain, in another connection, admits to 
be not induction but demonstration.* 
Our author draws a broad line between the fourth proposition, 
with its “ appeal to experiment or trial in the concrete,” and the 
mass of geometrical proofs in which the figure is referred to for 
verification only, “the effect of every construction and every step 
of reasoning being judged of by actual inspection.” But if the in¬ 
spection follows the construction, what is the construction itself? 
A construction is not proved by syllogism from axioms. It is 
necessarily drawn, and in the drawing (mental or other) looked at. 
Every construction involves a figure and an intuition, which, while 
it looks at the individual figure, sees in it the general truth.f Mr 
Bain grants that of such consequences as that the diagonal of a 
parallelogram divides it into two triangles, Euclid offers no other 
proof than an appeal to the eye.J 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 de¬ 
monstration in synthetic geometry. 
Finally, Professor Bain himself seems not quite satisfied as to 
the inductive nature of Euc. I. 4. “The proof,” he says, “rests 
solely on definitions,” and hence “ the proposition cannot be real— 
the subject and predicate must be identical.” 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 explana¬ 
tion, “ The proposition must, in fact, be a mere equivalent of the 
notions of line, angle, surface, equality—a fact apparent in the 
operation of understanding 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.” 
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 in- 
* Logic, vol. ii. p. 5. 
t Cf. Kant, Krit. d. r. Vern. p. 478. Ed. Hartenstein, 1867. 
t Logic, vol. ii. p. 218. 
