March 29, 1895.] 



SCIENCE. 



357 



done to the memory of a man who dui-ing his 

 lifetime never received any public recogni- 

 tion for his scientific work. At the present 

 time no competent mathematician doubts 

 the value of Lobachevskj''s investigations 

 in non-Euclidean geometry. For those not 

 familiar with modern mathematical thought 

 it is, however, difticult, if not impossible, to 

 fully appreciate the true value of this sub- 

 ject; they are inclined to attribute undue 

 importance to its possible bearings on non- 

 mathematical questions and to neglect and 

 underrate what is most valuable. 



The starting point for Lobachevsky's re- 

 searches, as for those of all the earlier 

 writers on non-Eviclidean geometry (Sac- 

 cheri, Lambert, the two Bolyais), is given 

 by the theory of parallels in elementary 

 plane geometiy which is based by Euclid 

 on his fifth postulate (usually called his 

 " eleventh axiom "). This postulate I'efers 

 to two lines cut by a transversal, and states 

 that if the sum of the interior angles on 

 one side of the transversal be less than two 

 right angles the lines will meet on this 

 side if sufficiently produced. The numer- 

 ous attempts that have been made to make 

 a theorem of this propo.sition, and to prove 

 it, have always remained as futile as the 

 attempts to square the circle. They have 

 only shown that it can be replaced by other 

 postulates, such as that only one parallel 

 can be drawn to a given line through a 

 given point, or that the sum of the angles 

 of a triangle is equal to two right angles, etc. 



Does it follow that these postulates ex- 

 press an absolute necessary truth ? Cer- 

 tainly not. For it can be shown — and 

 this is just what Lobachevsky did — that a 

 perfectly consistent system of geometry can 

 be constructed bj' rejecting Euclid's postu- 

 late and its equivalents, and assuming, say, 

 that more than one parallel can be drawn 

 to a given line through a given point, or 

 that the sum of the angles of a triangle is 

 less than two right angles. 



The question of the character of the .so- 

 called geometrical axioms thus assumes an 

 aspect very difl'erent from the one it had at 

 the beginning of the present century, when 

 they were comnnmly regarded as necessary 

 logical truths. It is, however, not for the 

 mathematician to decide whether ultimately 

 these axioms express facts of obsei-vation 

 unconsciously acquired and made familiar 

 through the constant perception of an actu- 

 ally existing space. For him they represent 

 mere assumptions selected for the purpose 

 of defining his space or his methods of 

 measuring this space. 



It would, of course, be very important to 

 know which of the different spaces that the 

 mathematician can thus define corresponds 

 most closely to the facts of observation. 

 But this question is difficult to decide ; for 

 while the ordinary Euclidean space appears 

 in this respect to satisfy all demands, the 

 non-EucUdean spaces do the same, at least, 

 approximatelj' within certain limits ; and 

 all our observations give only approximate 

 results and are confined within a narrow 

 range of space. 



What the mathematician has gained 

 through the generalization of non-Euclidean 

 geometrj' is a broader horizon and a vastly 

 extended field of research. The multifarious 

 relations by which this new science is con- 

 nected with the various banches of geometry 

 are admirably set forth by Professor F. 

 Klein, of Gottingen, in his Vorlemngen iiber 

 HicM-EiikUdkche Geometrie (1889-90). These 

 lectures also trace the historical development 

 of the subject since the times of Gauss. A 

 few more recent investigations were dis- 

 cussed by him in the Evaiuton CoUoqumm 

 (New York, Macniillan, 1894), in the 6th 

 and 11th lectures. 



What Professor Vasiliev tells us about 

 Bartels, who in his earlier j^ears had inti- 

 mately associated with Gauss, and later, as 

 the first professor of mathematics at the 

 University of Kazan, became the teacher 



