DEMONSTRATION, AND NECESSARY TRUTHS. 265 



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, therefore, is 

 still an induction from observation. And we should not be 

 authorized 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 properties of the reality are faith- 

 fully represented in the image ; just as we should be scienti- 

 fically warranted in describing 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 precisely equivalent to observation of the 

 original. 



These considerations also remove the objection arising from 

 the impossibility of ocularly following the lines in their pro- 

 longation 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 begin again to approach, this must take place 

 not at an infinite, but at a finite distance. Supposing, there- 

 fore, 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 occa- 

 sion 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 ex- 

 pression, " 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 appear," he says, "how we can compare our ideas 



