﻿Elements of Geometry. 25 



9. Euclid's definition of parallel lines must be rejected 

 entirely, for, since all plane surfaces are limited, the 

 criterion suggested is inapplicable. Since the crude de- 

 finition of parallelism is similarity of direction, we may 

 try to define parallel lines as those which being in the 

 same plane make the same angle with any third line. 

 We thereby imply the axiom of parallels in the form 

 (Prop. I. 29) that such lines which make the same angle 

 with one straight line make the same angle with any other ; 

 we imply also that the angles which are to be equal are the 

 "exterior" and "interior" opposites or the "alternate" 

 angles, since if the interior angles are compared the 

 proposition is not true. But the definition is not very 

 satisfactory; for, when the lines are edges, there is not 

 always an exterior or an alternate angle. It is better 

 to adopt the substance of Axiom I. 12 as a definition, 

 and to say that lines in one plane are parallel when the 

 sum of the interior angles is equal to two right angles. 

 This much abused axiom seems to me a very ingenious 

 way out of a real difficulty. We then assert the axiom of 

 parallels in the form (implied by I. 32) that if any two 

 straight lines in a plane are cut by any third line, the 

 sum of the interior angles is the same for all third lines. 

 The merit of this axiom is that it indicates clearly that the 

 " axiom of parallels " is really something concerning all 

 straight lines in a plane and not only parallel lines, and 

 that parallel lines are merely a particular case of other 

 pairs of lines. The propositions that parallel lines never 

 do intersect and that the angle between them is zero follow 

 immediately. 



The axiom of parallels is a law of measurement because 

 it is involved in the measurement of the angle between lines 

 w r hich do not intersect. Its use for this purpose requires 

 that at some point of a straight line it should always be 

 possible to place a straight line parallel to a given straight 

 line. This proposition is not true for concave surfaces, but 

 the complexities arising from this failure and the means of 

 avoiding them may be left for the present ; they are dealt 

 with more naturally in connexion with "space." If the 

 axiom were not used, we could not by our present methods 

 measure the angle between non-intersecting straight lines : 

 first, because the definition of equality given above, though 

 sufficient for such lines, is not necessary : second, because 

 the definition of addition is wholly unsatisfactory. 



There has been so much discussion of the necessity of the 



