DEMONSTRATION AND NECESSARY TRUTHS. 



153 



18 derived from the same sources as 

 every other part. * 



This, for instance, they would be 

 able to do, if they could prove chrono- 

 logically that we had the conviction 

 (at least practically) so early in infancy 

 as to be anterior to those impressions 

 on the senses, upon which, on the other 

 theory, the conviction is founded. This 

 however, cannot be proved : the point 

 being too far back to be within the 

 reach of memory, and too obscure for 

 external observation. The advocates 

 of the d priori theory are obliged to 

 have recourse to other arguments. 

 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 enclose 

 a space, were derived from the senses, 

 we could only be convinced of its 

 truth by actual trial, that is, by see- 

 ing or feeling the straight lines; 



* Some persons find themselves pre- 

 vented from believing that the axiom, Two 

 straight lines cannot enclose a space, could 

 ever become known to us through experi- 

 ence, by a diflSculty which may be stated 

 as follows. If the straight lines spoken of 

 are those contemplated in the definition — 

 lines absolutely without breadth and abso- 

 lutely straight ; — that such are incapable 

 of enclosing a space is not proved by ex- 

 perience, for lines such as these do not pre- 

 sent themselves in our experience. If, on 

 the other hand, the lines meant are such 

 straight lines as we do meet with in ex- 

 perience, lines straight enough for practical 

 purposes, but in reality slightly zig-zag, 

 and with some, however trifling, breadth ; 

 as applied to these lines the axiom is not 

 true, for two of them may, and sometimes 

 do, enclose a small portion of space. In 

 neither case, therefore, does experience 

 prove the axiom. 



Those who employ this argument to 

 show that geometrical axioms cannot be 

 proved by induction, show themselves 

 unfamiliar with a common and perfectly 

 valid mode of inductive proof — proof by 

 approximation. Though experience fur- 

 nishes us with no lines so uuimpeachably 

 straight that two of them are incapable of 

 enclosing 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 defini- 

 tion is the ideal limit. And observation 



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 merely 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 enclose a space. Intuition is 

 " imaginary looking ; " * but experi- 

 ence must be real looking : if we see a 

 property of straight lines to be true by 

 merely fancjnng ourselves to be look- 

 ing 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 imnecessary but unattainable. 

 What says the axiom? That two 

 straight lines cannot enclose a space ; 

 that after having once intersected, if 

 they are prolonged to infinity they do 

 not meet, but continue to diverge from 

 one another. How can this, in any 

 single case, be proved by actual ob- 

 servation ? We may follow the lines 

 to any distance we please ; but we 

 cannot 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 



shows that just as much, and as nearly, as 

 the straight lines of experience approxi- 

 mate to having no breadth or flexure, so 

 much and so nearly does the space-enclos- 

 ing power of any two of them approach to 

 zero. The inference that if they had no 

 breadth or flexure at all, they would enclose 

 no space at all, is a correct inductive in- 

 ference from these facts, conformable to 

 one of the four Inductive Methods herein- 

 after characterised — the Method of Conco- 

 mitant Variations, of which the Mathe- 

 matical Doctrine of Limits presents the 

 extreme case. 



* Whewell's History of Scientific Ideas, I. 

 140. 



