DEMONSTRATION, AND NECESSARY TRUTHS. 175 



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 an- 

 gles, 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 subjects of 

 geometrical experimentation as the realities themselves ; inasmuch as pic- 

 tures, if sufliciently accurate, exhibit 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 therefore 

 be laid in direct experience, even if the experiments (which in this case 

 consist merely in attentive contemplation) were practiced solel}^ upon 

 what Ave 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 representatives of all which resemble them ; and 

 in the present case the conditions which qualify a real object to be the rep- 

 resentative of its class, are completely fulfilled by an object existing only 

 in our fancy. Without denying, therefore, the possibility of satisfying our- 

 selves that two straight lines can not inclose a space, by merely thinking 

 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 observation. And we should not be author- 

 ized to substitute observation of the image in our mind, for observation 

 of the I'eality, if we had not learned by long-continued experience that the 

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

 we should be scientifically warranted in describing an animal which we 

 have never seen, from a picture made of it with a daguerreotype ; but not 

 until we had learned by ample experience, that observation of such a pic- 

 ture is precisely equivalent to observation of the original. 



These considerations also remove the objection arising from the im- 

 possibility 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 anoth- 

 er, they begin again to approach, this must take place not at an infinite, 

 but at a finite distance. Supposing, therefoi'e, 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 on as being precisely similar to the reality. Now, 

 whether we fix our contemplation upon this imaginary picture, or call to 

 mind the generalizations 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 senses which we describe by the expression, " a bent 

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



* Dr. Whewell {Philosophy of Discovery, p. 289) thinks it unreasonable to contend that we 

 know by experience, that our idea of a line exactly resembles a real line. "It does not ap- 

 pear," he says, "how we can compare our ideas with the realities, since we know the realities 

 only by our ideas." We know the realities by our sensations. Dr. Whewell surel}' does not 

 hold the "doctrine of perception by means of ideas," which Reid gave himself so much trou- 

 ble to refute. ' 



