ON A POSTULATE RESPECTING A CERTAIN FORM OF 



DEVIATION FROM THE STRAIGHT LINE 



IN A PLANE. 



By G. Hay. 



Presented Oct. 11, 1905. Received Oct. 16, 1905. 



In studying the relation of the axiom of the plane to the supposition of 

 the sum of the angles of the rectilinear triangle being less than 180°, it 

 becomes important to decide whether a certain line which presents itself 

 in our supposed plane is to be considered a straight line or not. 



" Der Begriff der Geraden ist vermoge seiner Einfachheit nicht definir- 

 bar, Richtung ohne die Gerade unverstandlich." Baltzer, Elemente 

 der Mathematik, zweiter Band, Leipzig, 1874. Legendre alludes to the 

 difficulty of defining the straight line. See Erdmanu, Axiome der 

 Geometrie, Leipzig, 1877, p. 19. 



Suppose in a plane a given straight line ; and in the same plane a line 

 defined by its relations to the given straight line. It may be important 

 to decide whether the line so defined is to be considered straight or not. 



In the absence of a satisfactory definition of the straight line it is pro- 

 posed to exclude directly from the category of straight lines by a postu- 

 late certain lines in a plane, as being, according to the postulate, forms of 

 deviation from the ideal straight line. 



A necessary property of the latter shall be the id^al property of it, 

 that the straight line of any two of its points coincides with that of any 

 other two. This property however leaves the straight line undefined, 

 for a similar relation of any two of the points to their line would hold of 

 the points of a given circular arc, with fixed centre, in a plane. 



In the Figure which represents straight lines in a plane, LP, M * P% 

 and L X P Z are each a straight line perpendicular to straight line PP S . 



LP > M,P 2 



L l P 3 >M 2 P, 



QP=M 2 P 2 = HP Z 



