356 PROCEEDINGS OP THE AMERICAN ACADEMY. 



It is one proposition to say, as the deviation-postulate does, that 

 straight line L 3I 2 cannot take the course LM % L X as described, thereby 

 excluding M 2 L x from being the continuation of L M 2 , and another to 

 say, with the fifth postulate, that the same line must meet P P s , for in 

 this case both M 1 L 1 and M 2 H.bxq excluded from being the continuation 

 of L M,. 



The two postulates exclude the line L M. z L x (see Figure) from being 

 straight, but respectively on different grounds. 



The deviation-postulate, considering also the ideal property referred 

 to above as belonging to a straight line, excludes L M. 2 Ij x from being 

 straight on the ground of its being a course inconsistent with a generic 

 property of the straight line. 



The fifth postulate makes the exclusion, because L M» L x being so con- 

 ditioned at its intersection with L P that L M., and P P 3 are subject to 

 the fifth postulate, does not take the course which, according to this 

 postulate, it should take, or because it does not take the course of one 

 of two straight lines related to each other in a certain way. 



Now the line L 3L L x , considered as a part of diagram (see Figure), 

 embodies both the two sets of conditions which the two postulates 

 respectively present for the line not being straight. It is submitted 

 that either of the two sets may be used for arguing that the line is not 

 straight. 



The deviation-postulate may be stated as follows : 



Given in a plane two straight-line segments, having a common point 

 outside of a given third straight line in the same plane, and situated on 

 different sides of the perpendicular from the common point to the third 

 straight line ; let each of the two straight-line segments, considered as pro- 

 ceeding each from the common point, diverge from the third straight line, 

 then postulate: The two straight lines of the segments do not coincide. 



The above postulate is in one respect interesting in comparison with 

 the fifth postulate, in that the former concerns special conditions which 

 prevent a si?igle line in a plane from being straight; whereas the latter 

 tells us that two straight lines in a plane, and related to each other in a 

 certain way, intersect. 



Ax Additional Difference Between the Two Postulates. 



The deviation-postulate is a postulate about two straight lines, and 

 states a condition which prevents a certain line from being straight (see 

 the Figure). Given two adjoining straight-line segments L J\L and M, L^ 



