80 THE FOURTH DIMENSION. 



in demonstration of these axioms, especially of the eleventh, all 

 turn out on close examination to be pseudo-proofs. Legendre drew 

 attention to the fact that either of the following axioms might be 

 substituted for the eleventh : 



a) Given a straight line, there can be drawn through a point 

 in the same plane with that line, one and one line only which shall 

 not intersect the first (parallels) however far the two lines may be 

 produced ; 



b) If two parallel lines are cut by a third straight line, the in- 

 terior alternate angles will be equal. 



c) The sum of the angles of a triangle is equal to two right 

 angles, that is, to the angle of a straight line or to 180. 



By the aid of any one of these three assertions, the eleventh 

 axiom of Euclid, may be proved, and, vice versa, by the aid of the 

 latter each of the three assertions may be proved, of course with 

 the help of the other two axioms, eight and twelve. The percep- 

 tion that the eleventh axiom does not admit of demonstration with- 

 out the employment of one of the foregoing substitutes may best 

 be gained from the consideration of- congruent figures. Every 

 reader will remember from his first instruction in geometry that the 

 congruence of two triangles is demonstrated by the superposition 

 of one triangle on the other and by then ascertaining whether the 

 two completely coincide, no assumptions being made in the deter- 

 mination except those above mentioned. 



In the case of triangles which are congruent, as are I and II in 

 the preceding cut, this coincidence may be effected by the simple 

 displacement of one of the triangles ; so that even a two-dimensional 

 being, supposed to be endowed with powers of reasoning, but only 

 capable of picturing to itself motions within a plane, also might 

 convince itself that the two triangles I and II could be made to 

 coincide. But a being of this description could not convince itself 



