NON-EUCLIDEAN GEOMETRY. 643 



What are the fundamental principles of geometry? What is its origin; 

 its nature; its scope? 



These are questions which have at all times engaged the attention of 

 mathematicians and thinkers, but which took on an entirely new aspect, thanks 

 to the ideas of Lobachevski and of Bolyai. 



For a long time we attempted to demonstrate the proposition known as the 

 postulate of Euclid; we constantly failed; we know now the reason for these 

 failures. 



Lobachevski succeeded in building a logical edifice as coherent as the 

 geometry of Euclid, but in which the famous postulate is assumed false, and 

 in which the sum of the angles of a triangle is always less than two right 

 angles. Riemann devised another logical system, equally free from contra- 

 diction, in which the sum is, on the other hand, always greater than two right 

 angles. These two geometries, that of Lobachevski and that of Riemann, are 

 what are called the non-Euclidean geometries. The postulate of Euclid then 

 can not be demonstrated; and this impossibility is as absolutely certain as any 

 mathematical truth whatsoever. 



It was the attainment of this very perception which in fact led to 

 the creation of the non-Euclidean geometry. Says Lobachevski in the 

 introduction to his ' New Elements of Geometry ' : 



The futility of the efforts which have been made since Euclid's time during 

 the lapse of two thousand years awoke in me the suspicion that the ideas em- 

 ployed might not contain the truth sought to be demonstrated. When finally 

 I had convinced myself of the correctness of my supposition I wrote a paper 

 on it [assuming the infinity of the straight]. 



It is easy to show that two straights making equal angles with a third 

 never meet. 



Euclid assumed inversely, that two straights unequally inclined to a third 

 always meet. 



To demonstrate this latter assumption, recourse has been had to many 

 different procedures. 



All these demonstrations, some ingenious, are without exception false, 

 defective in their foundations and without the necessary rigor of deduction. 



John Bolyai calls his immortal two dozen pages (the most extraor- 

 dinary two dozen pages in the whole history of thought), ' The Science 

 Absolute of Space, independent of the truth or falsity of Euclid's 

 Axiom XI. (which can never be decided a priori).' 



Later we read on the title page of W. Bolyai's ' Kurzer Grundriss ' : 

 'the question, whether two straights cut by a third, if the sum of the 

 interior angles does not equal two right angles, intersect or not? no 

 one on the earth can answer without assuming an axiom (as Euclid 

 the eleventh)' [the parallel postulate]. 



With the ordinary continuity assumptions or the Archimedes pos- 

 tulate, it suffices to know the angle-sum in a single rectilineal triangle 

 in order to determine whether space be Euclidean or non-Euclidean. 



How peculiarly prophetic or mystic then that the clairvoyant in- 

 spiration of the genius of Dante, the voice of ten silent centuries, 

 should have connected with the wisdom of Solomon and the special 

 opportunity vouchsafed him by God a question whose answer would 

 have established the case of Euclidean geometry seven hundred years 

 before it was born, or that of non-Euclidean geometry three thousand 

 years before its creation. 



I. Kings 3 : 5 is : 



