NECESSARY TRUTHS— EFFECTS OF EXPERIENCE. 657 



mean two things of an identical kind. The axiom of con- 

 stant results (see above, p. 645) holds in geometry. Tlie 

 same forms, treated in the same way (added, subtracted, 

 or compared), give the same results — how shouldn't they ? 

 The axioms of mediate comparison (p. 645), of logic (p. 648), 

 and of number (p. 654) all apply to the forms which we 

 imagine in space, inasmuch as these resemble or differ 

 from each other, form kinds, and are numerable things. 

 But in addition to these general principles, which are true 

 of space-forms only as they are of other mental conceptions, 

 there are certain axioms relative to space-forms exclusively, 

 which we must briefly consider. 



Three of them give marks of identity among straight 

 lines, planes, and parallels. Straight lines which have two 

 points, planes which have three points, parallels to a given 

 line which have one point, in common, coalesce throughout. 

 Some say that the certainty of our belief in these axioms 

 is due to repeated experiences of their truth ; others that 

 it is due to an intuitive acquaintance with the jjroperties 

 of space. It is neither. We exj)erience lines enough which 

 pass through two points only to separate again, only we 

 won't call them straight. Similarly of planes and parallels. 

 We have a definite idea of what we mean by each of these 

 words ; and when something different is offered us, we see 

 the difference. Straight lines, planes, and parallels, as they 

 figure in geometry, are mere inventions of our faculty for 

 apprehending serial increase. The farther continuations 

 of these forms, we say, shall bear the same relation to their 

 last visible parts which these did to still earlier parts. It 

 thus follows (from that axiom of skipped intermediaries 

 which obtains in all regular series) that parts of these 

 figures separated by other parts must agree in direction, 

 just as contiguous parts do. This uniformity of direction 

 throughout is, in fact, all that makes us care for these 

 forms, gives them their beauty, and stamps them into fixed 

 conceptions in our mind. But obviously if two lines, or 

 two planes, with a common segment, were to part company 

 beyond the segment, it could only be because the direction 

 of at least one of them had changed. Parting company in 

 lines and planes means changing direction, means assuming 



