t54 



REASONING. 



proof of the impossibility than obser- 

 vation affords us, we should have no 

 ground for believing the axiom at 

 all. 



To these arguments, which I trust 

 I cannot be accused of understating, 

 a satisfactory answer will, I conceive, 

 be found, if we advert to one of the 

 characteristic properties of geometri- 

 cal forms — their capacity of being 

 painted in the imagination with a dis- 

 tinctness equal to reality : in other 

 words, the exact resemblance of our 

 ideas of form to the sensations which 

 suggest them. This, in the first place, 

 enables us to make (at least with a 

 little practice) mental pictures of all 

 possible combinations of lines and 

 angles, which resemble the realities 

 quite as well as any which we could 

 make on paper ; and in the next place, 

 make those pictures just as fit sub- 

 jects of geometrical experimentation as 

 the realities themselves ; inasmuch as 

 pictures, if sufficiently accurate, ex- 

 hibit of course all the properties which 

 would be manifested by the realities 

 at one given instant, and on simple 

 inspection : and in geometry we are 

 concerned only with such properties, 

 and not with that which pictures 

 could not exhibit, the mutual action 

 of bodies one upon another. The 

 foundations of geometry would there- 

 fore be laid in direct experience, even 

 if the experiments (which in this case 

 consist merely in attentive contem- 

 plation) were practised solely upon 

 what we call our ideas, that is, upon 

 the diagrams in our minds, and not 

 upon outward objects. For in all 

 systems of experimentation we take 

 some objects to serve as representa- 

 tives of all which resemble them ; and 

 in the present case the conditions 

 which qualify a real object to be the 

 representative of its class are com- 

 pletely fulfilled by an object existing 

 only in our fancy. Without denying, 

 therefore, the possibility of satisfying 

 ourselves that two straight lines can- 

 not enclose a space, by merely think- 

 ing of straight lines without actually 

 looking at them ; I contend that we 



do not believe this truth on the ground 

 of the imaginary 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, there- 

 fore, is still an induction from obser- 

 vation. And we should not be autho- 

 rised to substitute observation of the 

 image in our mind for observation of 

 the reality if we had not learnt by long- 

 continued experience that the proper- 

 ties of the reality are faithfully repre- 

 sented in the image ; just as we should 

 be scientifically warranted in describ- 

 ing an animal which we have never 

 seen from a picture made of it with 

 a daguerreotype ; but not until we 

 had learnt by ample experience that 

 observation of such a picture is pre- 

 cisely equivalent to observation of the 

 original 



These considerations also remove 

 the objection arising from the impos- 

 sibility 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 without doing so we may know 

 that if they ever do meet, or if, after 

 diverging from one another, they be- 

 gin 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 our- 

 selves thither in imagination, and 

 can frame a mental image of the ap- 

 pearance which one or both of the 

 lines must present at that point, 

 which we may rely on as being pre- 

 cisely similar to the reality. Now, 

 whether we fix our contemplation 

 upon this imaginary picture, or call 

 to mind the generalisations we have 

 had occasion to make from former 

 ocular observation, we learn by the 

 evidence of experience, that a line 

 which, after diverging from another 

 straight line, begins to approach to 

 it, produces the impression on our 

 which we describe by the ex- 



