CONCRETE REPRESENTATIONS OF NON- 

 EUCLIDEAN GEOMETRY 



INTRODUCTORY NOTE 



WHEN Euclid composed his logical system of the Elements 

 of Geometry he was no doubt aware that it was based upon 

 many unproved assumptions. Some of these assumptions 

 are explicitly stated, either as postulates or as axioms (or 

 common notions). The fifth postulate, often given as the 

 eleventh or the twelfth axiom, is a lengthy statement relating 

 to parallel straight lines, and is conspicuous by its want of 

 any intuitive character : ' If a straight line falling on two 

 straight lines make the interior angles on the same side less 

 than two right angles, the two straight lines, if produced 

 indefinitely, meet on that side on which are the angles less 

 than two right angles.' The universal converse of this state- 

 ment is proved (with the help of another assumption, that the 

 straight line is of unlimited extent) in Prop. 17, while its 

 contrapositive is proved (again with the same assumption) 

 in Prop. 28 of the First Book. Such considerations induced 

 geometers and others to attempt its demonstration. Hundreds 

 of such attempts have been made, with a display of great 

 ingenuity. All these attempts, however, if they do not 

 actually involve fallacious reasoning, are based upon some 

 equivalent assumption either tacit or expressed. 



An entirely different mode of attack was devised by a 

 Jesuit, Gerolamo Saccheri. 1 He attempted to discover con- 



1 G. Saccheri, Eudides db omni naevo vindicatus, Milan, 1733. This work was for a 

 long time forgotten. It was brought to the notice of Beltrami in 1889, who published 

 an account of it in the Rendiconti of the Lincei Academy. It has been translated into 

 English by G. B. Halsted, Amer. Math. Mon., 1-5 (1894-98), German by Stackel and 

 Engel in Theorie der Paralkllinien, 1895, and Italian (Manuali Hoepli, 1904). 



