64 SCIENCE ABSOLUTE OF SPACE. 



Millan, 1895. It is of the greatest importance 

 for every teacher to know and connect the 

 commonest forms of assumption equivalent to 

 Euclid's Axiom XL If in a plane two straight 

 lines perpendicular to a third nowhere meet, 

 are there others, not both perpendicular to 

 any third, which nowhere meet? Euclid's 

 Axiom XI is the assumption No. Playf air's 

 answers no more simply. But the very same 

 answer is given by the common assumption of 

 our geometries, usually unnoticed, that a circle 

 may be passed through any three points not 

 costraight. 



This equivalence was pointed out by Bolyai 

 Farkas, who looks upon this as the simplest 

 form of the assumption. Other equivalents 

 are, the existence of any finite triangle whose 

 angle-sum is a straight angle; or the existence 

 of a plane rectangle; or that, in triangles, the 

 angle-sum is constant. 



One of Legendre's forms was that through 

 every point within an angle a straight line 

 may be drawn which cuts both arms. 



But Legendre never saw through this mat- 

 ter because he had not, as we have, the eyes 

 of Bolyai and Lobachevski to see with. The 

 same lack of their eyes has caused the author 

 of the charming book " Euclid and His Modern 



