414 Mr. J. P. Nichol on Parallel Lines. 



" The truth is, there is an unfortunate and illogical inversion 

 in the first book of the ' Elements.' The proposition required 

 by Euclid is the thirty-second, viz. that the sum of the three angles 

 of a triangle is equal to two right angles ; given that proposition, 

 the difficulty about parallels disappears, inasmuch as their pro- 

 perties may be deduced from it by the negative process, although 

 with a certain difficulty : but Euclid deduces this proposition 

 from the subject of parallels, having first assumed their theory 

 under guise of what he most unjustifiably terms an axiom. If 

 we are correct then, the question turns on this, — Can we logically 

 establish the thirty-second proposition without appealing to the 

 doctrine of parallels ? Assuredly there is no reason, on the face 

 of the subject, that should cause geometers to shrink from this 

 attempt ; but it is just as certain that the apparent difficulty of 

 succeeding in it — attested by innumerable failures — indicates 

 a defect in the statement of the usual axioms or fundamental 

 propositions regarding our primary discernments concerning 

 Space. The defect is a very important one : were it supplied, a 

 great change would pass over all arrangements and methods 

 of development in our elementary geometry. The defect is in 

 Euclid's inadequate conception of the necessarily distinctive nature 

 of two definite attributes of geometrical quantity —/orma«t??H«^- 

 nitude. The Greek geometer did not trace out the manner in 

 which we acquire our notions of these attributes ; and he did not 

 therefore recognize it as an axiom, that the attribute oi/orm has 

 no dependence on the attribute of magnitude. The phsenomena 

 of universal belief indeed amply sustain the proposition, — 'If any 

 figure exists or is conceivable, it must exist or be conceivable with 

 the same fwm, whatever its magnitude ; ' or any other statement 

 involving the truth, that in our perception of the geometrical qua- 

 lities of an object, form alone is definite, magnitude being indefinite. 

 The physical process of perception reveals the root of that belief; 

 the notion of magnitude involving an estimate of the distance of 

 the object, while the notion of form is, at its source, independent 

 of every variable quantity. These views may be sustained by 

 the high authority of Laplace. The following is a note attached 

 to his Systeme du Monde : — ' Les tentatives des geometres, pour 

 demontrer le postulatuji d'Euclide sur les paralleles, ont ete 

 jusqu'a present inutiles. Cependant jjersonne ne revoque en doute 

 ce posTULATUM ct les theoremes qu'Euclide en a deduit. La per- 

 ception de I'etendue renfenne done une propriete speciale evidente 

 par elle-meme, et sans laquelle on ne peut rigoureusement etablir 

 les proprietes des paralleles. L'idee d'une etendue limitee, par 

 exemple du cercle, ne contient rien qui depende de sa grandeur 

 absolue. Mais si nous diminuons, par la pensee, son rayon, nous 

 sommes partes invinciblement a diminuer, dans le meme rapport, sa 



