176 
Proceedings of the Royal Society 
3. Note on Professor Bain’s Theory of Euclid 1. 4. By Wm. 
Robertson Smith, M.A., Assistant to the Professor of 
Natural Philosophy. Communicated by Professor Tait. 
In a paper communicated to this Society last session, I pointed 
out that the proof of Euc. I. 5, given by Mr Mill, is unsound; 
endeavouring, at the same time, to show that this is no mere 
accident, but that it is impossible to give a mathematically correct 
analysis of the processes of Synthetic Geometry on any theory 
that holds figures to be merely illustrative, and does not admit 
that intuition in the Kantian sense — i.e., actual looking at a single 
engraved or imaginary figure — may he a necessary and sufficient 
step in a demonstration perfectly general. I now venture to draw 
the attention of the Society to the confirmation which I conceive 
that this argument derives from the way in which Euc. I. 4 is 
treated by Professor Bain in his recent “ Logic ” — a book which, 
on the whole, is based on Mr Mill’s principles, and which is mainly 
original in an attempt, which I cannot regard as felicitous, to 
bring these principles into closer contact with the special sciences, 
especially with Physics and Mathematics. 
It will he remembered that Mr Mill, undertaking to demonstrate 
Euc. I. 5 from first principles, has to supply, in the course of his 
proof, a demonstration of Euc. I. 4, and it is in the attempt to give 
to this process the form of 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 sur- 
rendering Mr Mill’s theory of mathematical reasoning, he concludes 
that Euclid has not demonstrated his proposition — that the super- 
position which he enjoins is only an experiment, and that “if his 
readers had not made actual experiments of the kind indicated, 
they could not be convinced by the reasoning in the demonstra- 
tion.” * 
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 
* Logic, vol. ii. p. 217= 
