HAY. — DEVIATION FROM THE STRAIGHT LINE IN A PLANE. 355 



The ground of this postulate may be shown by means of the following 

 subsidiary postulate. The plane is here regarded as a perfectly flat 

 surface. Consider the straight line in which the limited straight line 

 L M 2 lies ; and in this straight line, outside the portion L M 2 and adjoin- 

 ing M 2 an additional small portion of it, small enough to lie on the same 

 side of P P 3 as J/ 2 does. Now the subsidiary postulate : The additional 

 small straight-line portion commencing at M 2 will converge towards P P 3 

 as L M 2 so converges. Consequently M 2 L x , diverging from PP 3 , must 

 be in a different straight line from that in which L M % lies, as they both 

 pass through point M 2 , and in the respective portions near to M 2 the one 

 converging to, and the other diverging from, P P 3 , and therefore by the 

 ideal property mentioned above, line L M 2 L x is not straight. 



It may be noted that the convergence of L M 2 towards P P 3 depends on 

 the given conditions of the construction of the diagram; and in using this 

 convergence in the reasoning it is not necessary to consider whether the 

 straight line of L M 2 , when prolonged, meets P P 3 or not. That is, the 

 reasoning of the subsidiary postulate does not make use of the relation of 

 the fifth postulate which holds valid between the lines L M 2 and P P s . 



Comparison of the Deviation-Postulate avith tiie Fifth 

 Postulate as They are Indicated on the Figure. 



In using the former to show that iif 2 A is n «t straight, it is not 

 necessary to consider whether the course of the straight line L M 2 meets 

 PP 3 or not; but only that the two limited straight lines LM 2 and 

 M 2 Li being differently inclined to PP 3 , the one converging towards and 

 the other diverging from it, then, according to the postulate, the straight 

 lines of L M 2 and M 2 L l cannot coincide. This reasoning, it is submitted, 

 does not make use of the fifth postulate. 



That L M 2 L x is not straight may also be considered as a consequence 

 of the fifth postulate, because by that postulate the straight line LM t 

 would have to meet PP 3 ; but the line LM 2 L X not doing so, cannot be 

 straight. Here we take into account whether the line L M 2 L x meets 

 PP 3 or not, which was not the case in the preceding reasoning. 



That the two postulates which we are comparing are not identical 

 will appear from the following : 



The fifth postulate excludes L M 2 from taking the course M 2 H, which, ' 

 as Jf.,P 2 equals HP 3 , can never meet PP 3 ; but the other postulate 

 considers only those continuations of L M 2 which, like M 2 L U diverge 

 from P P 3 . and disregards course of M, H. 



