174 REASONING. 



This, foi* instance, they would be able to do, if they could prove chrono- 

 logically that we had the conviction (at least practically) so early in infan- 

 cy as to be anterior to those impressions on the senses, upon which, on the 

 other theory, the conviction is founded. This, however, can not be proved: 

 the point being too far back to be within the reach of memory, and too ob- 

 scure for external observation. The advocates of the a priori theory are 

 obliged to have recourse to other arguments. These ai'e reducible to two, 

 which I shall endeavor 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 can not 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 recognize 

 that two such lines can not inclose a space. Intuition is " imaginary look- 

 ing ;"* 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 can not be the senses, or experience ; it 

 must be something mental. 



To tliis 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 can not inclose a space ; 

 that after having once intersected, if they are prolonged to infinity they do 

 not moot, but continue to diverge from one another. How can this, in any 

 single case, be proved by actual observation ? We may follow the lines to 

 any distance we please ; but we can not follow them to infinity : for aught 

 our senses can testify, they may, immediately beyond the farthest point to 

 which we have traced them, begin to approach, and at last meet. Unless, 

 therefore, we had some other proof of the impossibility than observation 

 affords us, we should have no ground for believing the axiom at all. 



To these arguments, which 1 trust I can not be accused of understating, 

 a satisfactory answer will, I conceive, be found, if we advert to one of the 

 characteristic properties of geometrical forms — their capacity of being 

 painted in the imagination with a distinctness equal to reality : in other 

 words, the exact resemblance of our ideas of form to the sensations which 



true, for two of them may, and sometimes do, inclose a small portion of space. In neither 

 case, therefore, does experience prove the axiom. 



Those who employ this argument to show that geometrical axioms can not be proved by 

 induction, show themselves unfamiliar with a common and perfectly valid mode of inductive 

 proof; proof by approximation. Though experience furnishes us with no lines so unira- 

 peachably straight that two of them are incapable 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 tiie 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- in closing 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 inductive 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 Scientijic Ideas, \., 140. 



