DEMONSTKATIOV, AXD NECCSSAUY TRUTHS. 155 



the present case the conditions which qualify a real, object to be the 

 representative of its clas*, arc completely fulfilled by an object existing 

 only in our fancy. "Without denying, therefore, the possibility ot 

 satisfying ourselves that two sti-aight lines cantiot inclose a s])ace, by 

 merely thinking of straight lines without actually looking at them ; I 

 contend, tluit we do not beUeve this truth on the ground of the imagi- 

 nary intuition simply, but because we know that the imaginary lines 

 exactly resemble real ones, and that we may conclude from them to 

 real ones with quite as much certainty as we could conclude from one 

 real line to another. The conclusion, therefore, is still an induction 

 from obsei-Vation. And we should not be authorized to substitute 

 obsenation of the image in our mind, for obsei-vation of the reality, if 

 we had not learnt by long continued experience that all the properties 

 of the reality are faithfully represented in the image ; just as we 

 should be scientifically warranted, in describing the sliape and color of 

 an animal Nvhich we had never seen, fi-om a plaotogenic picture made 

 of it with a daguerreotype ;• but not until we had leanit by ample 

 exj>erience, that observation of such a picture is precisely equivalent 

 to observation of the original. 



These considerations also remove the objection aiising from the 

 impossibity of ocularly following the lines in. their prolongation to 

 infinity. For though, in order actually to see that two given lines 

 never meet, it would be necessary to follow them to infinity : yet 

 A\'ithout doing so we may know that if they ever do meet, or indeed if, 

 after diverging fi-om one another, they begin again to approach, this 

 must take place not at an infinite, but at a finite distance. Supposing, 

 therefore, such to be the case, we can transport ourselves thither in 

 imagination, and can frame a mental image of the appearance which 

 one or both of the lines must present at that point, which we may rely 

 upon as being precisely similar to the reality. Now, -^vhether we fix 

 our contemplation upon this imaginary picture, or call to mind the 

 generalizations we have had occasion to make fi'om fonner ocular 

 obser\-atic»n, we shall either way be equally satisfied, that a line which, 

 after diverging fi-om another straight line, begins to approach to it, 

 produces the impression on our senses which we describe by the 

 expression, " a bent line," not by the expression, " a straight line." 



§ 6. The first of the two great arguments in support of the theory 

 that axioms are a priori truths, having, I tliink, been sufficiently an- 

 swered ; I proceed..to the second, on which most stress is usually laid, 

 and which is chiefly insisted upon by Mr. ^Vliewell. Axioms (it is 

 asserted) are conceived by iis not only as tnie, but as universally and 

 necessarily true. Now, experience cannot possibly give to any propo- 

 sition this character. I may have seen snow a hundred times, and 

 may have seen that it was white, but this cannot give me entire assur- 

 ance even that all snow is white ; much less that snow must be white. 

 "However many instances we may have observed of the truth of a 

 proposition, there is nothing to assure us that the next case shall not be 

 an exception to the rule. If it be strictly true that every ruminant 

 animal yet known has cloven hoofs, we still cannot be sure that some 

 creature will not hereafter be discovered which has the first of these 



attributes, without having the other Experience must always consist 



of a limited number of observations : and, however numerous these 



