1873] BAIN ON GEOMETRY 69 



result following or rather, I am told to believe this 

 result, for to picture it is quite superfluous and often 

 impossible. Euclid, on the other hand, tells me to super 

 pose 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 coin 

 cidence 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 experi 

 ments on mathematical forms. 1 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 &quot; ideal &quot; or &quot; actual 

 experiments &quot; ? 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 reasoning which 

 Professor Bain, in another connection, admits to be not 

 induction but demonstration. 2 



Our author draws a broad line between the fourth 

 proposition, with its &quot; appeal to experiment or trial in 

 the concrete,&quot; and the mass of geometrical proofs in which 

 the figure is referred to for verification only, &quot; the effect 

 of every construction and every step of reasoning being 

 judged of by actual inspection.&quot; But if the inspection 

 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. 8 Mr. Bain grants that 



J Logic, vol. i. p. 225. - Ibid. vol. ii. p. 5. 



8 Cf. Kant, Kritik der reinen Vernunjt, p. 478, Ed. Hartenstein, 1867. 



