HAY. — DEVIATION FROM THE STRAIGHT LINE IN A PLANE. 357 



having a given point in common and related in a certain way, as indi- 

 cated in description of the Figure, to a given third straight line P P 3 , 

 which the' segments do not meet. The point in common excludes 

 parallels. 



With the given relation to each other that the two segments have 

 a given point in common, a possible case for the two straight lines of 

 these segments would be that they coincide. But, according to the 

 postulate, the relations of the two straight lines to the given third straight 

 Hue are such that they do not coincide. Then the condition of these 

 two lines is that they intersect at one point. The part of this condition 

 which the postulate of the two lines states, is 



The exclusion of coinciding. 



Fifth postulate : With the relation to each other that the two straight 

 lines of the postulate do not coincide, and are related in a certain way 

 to a third straight line which they both intersect ; then postulate. The 

 two straight lines of the postulate intersect. This, the intersection, is 

 equivalent to 



Excluding parallels. 



Therefore, the postulate in this case virtually states the exclusion of 

 parallels, whereas, in the case of the deviation-postulate, it states exclu- 

 sion of coinciding. 



Two uses may be made of the above deviation-postulate : one, to show 

 that the sum of the angles of the rectilinear triangle is equal to 180°; 

 and the other, to show that if that sum is supposed less than 180° then 

 the axiom of the plane does not hold good. 



