706 



SCIENCE. 



[N. S. Vol. XIV. No. 358. 



that it was reassuring when on October 22 

 (old style), 1900, the Commission of the 

 Phy si CO- Mathematical Society of Kazan 

 found the scientific merits of the works of 

 these authors, A. N. Whitehead and Wm. 

 Killing equal for the great Lobachevski 

 prize and had to decide between them by 

 the drawing of lots. 



In his report on the work of Whitehead, 

 Sir Robert Ball says of the ' Universal 

 Algebra ' : 



" Several other writers, to whom of 

 course Mr. Whitehead makes due acknowl- 

 edgment, have approached the study of 

 non- Euclidean geometry by the aid of 

 Grassmann's methods, but the systematic 

 and most instructive development of the 

 subject in book VI. is, I believe, new, as 

 are also many of the results obtained. 



'* The superiority of Whitehead's methods 

 appears to lie in the two following features : 



" 1"^. That he can treat n dimensions by 

 practically the same formulae as those used 

 for two or three dimensions. 



'' In this I think he has made a consider- 

 able advance upon the methods, ingenious 

 and beautiful as some of them no doubt 

 are, which have been used by previous in- 

 vestigators. 



"2°. The various kinds of space, para- 

 bolic, hyperbolic and elliptic (of two kinds), 

 present themselves in Whitehead's methods 

 quite naturally in the course of the work, 

 where they appear as the only alternatives 

 when certain assumptions have been made. 



' ' Moreover the results have been ob- 

 tained in such a way that it is easy for the 

 reader to develop for one of the other 

 spaces properties treated out in full for one 

 space only. 



" The book deserves in the highest de- 

 gree the attention of the student of modern 

 mathematical methods, and it marks so 

 great an advance that it is, in my judg- 

 ment, well worthy of the important prize in 

 view of which this report is prepared. 



" Mr. Whitehead's memoir on geodesies 

 in elliptic space appears to me to indicate 

 great power in dealing with a very difficult 

 problem. I believe it to be of much im- 

 portance, as the geodesies in the generalized 

 space conceptions had been but little 

 studied." 



In the corresponding report on the work 

 of Killing, Professor Engel, of Leipsic, says 

 of the ' Grundlagen der Geometrie ' : 



" This work is, from the first to the last 

 page, a justification and detailed develop- 

 ment of the circle of ideas which we are 

 accustomed to understand under the ex- 

 pression ' non-Euclidean geometry.' " 



'' Already so many preliminary questions 

 have been settled," said Killing in the 

 preface to his first volume, " that the final 

 solution can be hoped for at a not too dis- 

 tant time." 



" These words written in 1893," says 

 Engel, " have meanwhile most recently 

 (1899) found a highly striking confirmation 

 in many directions through Hilbert's inves- 

 tigations. 



" The geometries possible with the Eu- 

 clidean, namely the Lobachevski-Bolyaian, 

 the Riemannian and the elliptic, Killing 

 develops, each for itself, in Euclidean way 

 up to a certain grade. 



''Also it should not be forgotten that 

 Killing was the first^ who (1879, Crelles 

 Journal, Bd. 83) made clear the difference 

 between the Biemannian and the elliptic 

 space (or as he calls it, the Polar form of 

 the Riemannian). 



" The fourth section treats the Clifibrd- 

 Klein space-forms, in whose investigation 

 Killing himself has taken a conspicuous 

 part (by a work in Bd. 39 of the Mathe- 

 matische Annalen, 1891). The great impor- 

 tance of these space-forms rests upon this, 

 that they show with especial clearness, 

 what a mighty difierence it makes whether 

 we, from the beginning, assume the geo- 

 metric axioms as valid for space as a whole 



