354 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



a = ang. M, Q P 



/3 = aug. QM 2 P 2 



y = ang. M 2 L Q 



Supposing we are dealing with a plane, and, the sura of the angles of 

 the rectilinear triangle not being greater than 180°, we have 



a = j3 

 y < 90° 



Thus in the Figure, the straight lines L M 2 and P P z are subject to 

 the fifth postulate, as the Figure represents straight lines in a plane. 

 As this postulate concerns straight lines, it cannot, if true, be inconsistent 

 with any other true proposition about such lines. 



L L x 



Mo 



Figure. 



As to the question of line L M 2 L x in the Figure being straight or not, 

 besides what may be deduced from the fifth postulate, use may also be 

 made of the property mentioned above. In accordance with this, the 

 straight line of L M 2 should coincide with that of M 2 Z b if L M. 2 X/ 1 is 

 straight. 



But in the Figure, straight-line portion L M 2 may be considered as 

 converging towards PP S , and straight-line portion M. 2 L x as diverging 

 from PP 3 , the two lines having point il/ 2 in common, and being on differ- 

 ent sides of the perpendicular M 2 P 2 , and on the same side of P P 8 . 



Now the straight-line portions L M., and M 2 Z x being conditioned as 

 above described, the postulate may be stated : Straight line passing 

 through points L and M 2 cannot coincide with straight line passing 

 through points M 2 and L x . The condition stated by the postulate ex- 

 cludes the line L M 2 L x from being straight, and the postulate may be 

 designated as postulate for a certain form of deviation from the straight 

 line in a plane, or briefly as deviation -postulate. 



