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THE POPULAR SCIENCE MONTHLY.— SUPPLEMENT. 



inductions from the evidence of our senses ; now, 

 we must have " some other proof of the impossi- 

 bility than observation affords us." 



This further proof, it appears, consists in the 

 attentive contemplatation of mental pictures of 

 straight lines and other geometrical figures. Such 

 pictures, if sufficiently accurate, exhibit, of course, 

 all the properties of the real objects, and in the 

 present case the conditions which qualify a real 

 object to be the representative of its class are 

 completely fulfilled. Such pictures, Mill admits, 

 must be sufficiently accurate ; but what, in geom- 

 etry, is sufficient accuracy ? The expression is, 

 to my mind, a new and puzzling one. Imagine, 

 since Mill allows us to do so, two parallel straight 

 lines. What is the sufficient accuracy with which 

 we must frame our mental pictures of such lines, in 

 order that they shall not meet ? If one of the 

 lines, instead of being really straight, is a portion 

 of a circle having a radius of a hundred miles, then 

 the divergence from perfect straightness within 

 the length of one foot would be of an order of 

 magnitude altogether imperceptible to our senses. 

 Can we, then, detect in the mental picture that 

 which cannot be detected in the sensible object? 

 This can hardly be held by Mill, because he says, 

 further on, that we are only warranted in substi- 

 tuting observation of the image in our mind for 

 observation of the reality by long-continued ex- 

 perience that the properties of the reality are 

 faithfully represented in the image. 



Now, since we may (at least with a little prac- 

 tice) form mental pictures of all possible combina- 

 tions of lines and angles, we may, I presume, 

 form a picture of lines which are so nearly paral- 

 lel that they will only meet at a distance of 100,- 

 000 miles. If we cannot do so, how can we de- 

 tect the difference between such lines and those 

 that are actually parallel ? Mill meets this diffi- 

 culty. If two lines meet at a great distance, 



"we can transport ourselves thither in imagina- 

 tion, and can frame a mental image of the appear- 

 ance 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 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 wc describe by the expression, 

 ' a bent line,' not by the expression, ' a straight 

 line.' " i 



1 Book II., chapter v., section 5, end of fourth para- 

 graph. 



In this passage we have somewhat unexpect- 

 edly got back to the senses. We may call to mind 

 the generalizations from former ocular observa- 

 tion, and we have the evidence of experience to 

 distinguish between the impressions made on our 

 senses by a bent line and a straight line. But 

 what will happen if the bent line be a circle w ith 

 a radius of a million miles ? Have we the evi- 

 dence of experience that two such lines, which 

 seem to be parallel for the first hundred miles, 

 afterward begin to approach, and finally intersect ? 

 If so, our senses must enable us to see clearly 

 and to exactly measure quantities a hundred 

 miles away. Or again, if there be two lines which 

 close in front of me are one foot apart, but which 

 a hundred miles away are one foot plus the thou- 

 sandth of an inch apart, they are not parallel 

 Will my senses enable me to perceive the magni- 

 tude of the thousandth part of an inch placed a 

 hundred miles off? 



But we have had enough of this trifling. Any 

 one who has the least knowledge of geometry 

 must know that a straight line means a, perfectly 

 straight line : the slightest curvature renders it 

 not straight. Parallel straight lines mean per- 

 fectly parallel straight lines ; if they be in the 

 least degree not parallel, they will, of course, 

 meet sooner or later, provided that they be in the 

 same plane. Now, Mill said that we get an im- 

 pression on our senses of a straight line ; it is 

 through this impression that we are enabled to 

 form images of straight lines in the mind. We 

 are told, 1 moreover, that the imaginary lines ex- 

 actly resemble real ones, and that it is long-con- 

 tinued observation which teaches us this. It fol- 

 lows most plainly, then, that the impressions on 

 our senses must have been derived from really 

 straight lines. Mill's philosophy is essentially 

 and directly empirical ; he holds that we learn 

 the principles of geometry by direct ocular per- 

 ception, either of lines in Nature, or their images 

 in the mind. Now, if our observations had been 

 confined to lines which are not parallel, we could 

 by no possibility have perceived, directly and 

 ocularly, the character of lines which are paral- 

 lel. It follows, that ice must have perceived per- 

 fectly parallel lines and perfectly straight lines, 

 although Mill previously told us that he considered 

 the existence of such things to be " inconsistent with 

 the physical constitution of our planet, at least, if 

 not of the universe." 



Perhaps it may be replied that Mill simply 

 made a mistake in saying that no really straight 



1 Same section, about thirteen lines from the end 

 of the third paragraph. 



