MISCELLANEOUS PAPERS. 375 



that a set of axioms may be taken contradicting in whole or part 

 those of Euclid, and a geometry be built thereon as consistent and as 

 logical as his. Take, for example, one axiom of Euclid, the compli- 

 cated and unwieldly one whose statement is : "If two lines are cut by 

 a third, and the sum of the interior angles on the same side of the 

 cutting line is less than two right angles, the lines will meet on that 

 side when sufficiently produced." Euclid proved that lines making 

 with a transversal equal alternate angles are parallel; and then, in or- 

 der to prove that parallels cut by a transversal make alternate angles 

 he used the complicated axiom previously given. Many were the at- 

 tempts to accomplish this by reasoning about the nature of the straight 

 line and plane angle. Legendre, a celebrated French geometer, at- 

 tempted it, and in the course of his reasoning proved that the sum of 

 the angles of a triangle can never exceed two right angles, but could 

 not prove that there exists a triangle the sum of whose angles is two 

 right angles. 



The time came when some mathematicians began to believe that 

 Euclid's axiom was not capable of proof, and that a geometry could 

 be constructed on the supposition that the axiom is not always true. 



As early as 17()6 a paper was written by Lambert, in which he main- 

 tains that the parallel axiom needs proof, and gives some of the char- 

 acteristics of geometries in which this axiom does not hold. The 

 greatest mathematician of the nineteenth century, Gauss, sought to 

 prove the axiom of parallels for years, but he never published any- 

 thing on the subject. 



About the year 1830, two men — one a Russian, Lobachevsky, the 

 other a Hungarian, Johann Bolyai — first asserted and then proved 

 that the axiom of parallels is not necessarily true. Thirty years 

 passed before any attention was given to the work of the Russian and 

 the Hungarian. Clitford, an Englishman, one of the most brilliant 

 of mathematicians, in 1870, wrote that "several new ideas have come 

 to me lately: First, I have procured Lobachevsky's 'Geometrical 

 Studies upon the Theory of Parallels,' a small tract, of which Gauss 

 says: 'The author has treated the matter with the hand of a master, 

 and with the true geometrical spirit. I must call your attention to 

 this book, the reading of which will not fail to cause you the most 

 lively pleasure.' " 



The "Theory of Parallels" is not a very high-sounding title, but it 

 reveals to us a "new kind of universal space." Gauss, in 1846, says : 

 "The non-Euclidean geometry contains nothing in it that is contra- 

 dictory, although at first view very many of the results have the air 

 of paradoxes. These apparent contradictions must be regarded as 

 the effect of an illusion, due to the habit we have of considering the 

 Euclidean geometry as rigorous." 



