DEMONSTRATION, AND NECESSARY TRUTHS. 261 



These are reducible to two, which I shall endeavour to state as 

 clearly and as forcibly as possible. 



5. In the first place it is said that if our assent to the 

 proposition that two straight lines cannot inclose a space, 

 were derived from the senses, we could only be convinced of 

 its truth by actual trial, that is, by seeing or feeling the 

 straight lines; whereas in fact it is seen to be true by merely 

 thinking of them. That a stone thrown into water goes to the 

 bottom, may be perceived by our senses, but mere thinking 

 of a stone thrown into the water would never have led us to 

 that conclusion : not so, however, with the axioms relating to 

 straight lines: if I could be made to conceive what a straight 

 line is, without having seen one, I should at once recognise 

 that two such lines cannot inclose a space. Intuition is " ima- 

 ginary looking ;"* but experience must be real looking : if we 

 see a property of straight lines to be true by merely fancying 

 ourselves to be looking at them, the ground of our belief cannot 

 be the senses, or experience ; it must be something mental. 



To this argument it might be added in the case of this 

 particular axiom, (for the assertion would not be true of all 

 axioms,) that the evidence of it from actual ocular inspection 

 is not only unnecessary, but unattainable. What says the 

 axiom ? That two straight lines cannot inclose a space ; that 

 after having once intersected, if they are prolonged to infinity 

 they do not meet, but continue to diverge from one another. 



valid mode of inductive proof ; proof by approximation. Though experience 

 furnishes us with no lines so unimpeachably straight that two of them are inca- 

 pable of inclosing the smallest space, it presents us with gradations of lines 

 possessing less and less either of breadth or of flexure, of which series the 

 straight line of the definition is the ideal limit. And observation shows that 

 just as much, and as nearly, as the straight lines of experience approximate to 

 having no breadth or flexure, so much and so nearly does the space-inclosing 

 power of any two of them approach to zero. The inference that if they had 

 no breadth or flexure at all, they would inclose no space at all, is a correct in- 

 ductive inference from these facts, conformable to one of the four Inductive 

 Methods hereinafter characterized, the Method of Concomitant Variations ; of 

 which the mathematical Doctrine of Limits presents the extreme case, 

 * Whewell's History of Scientific Ideas, i. 140. 



