Febbuary 18, 1898.] 



SCIENCE. 



231 



THE FIRST AWARD OF THE LOBACSEVSEI 

 PRIZE. 



The Jjobach^vski prize is adjudged every 

 three years. Its value is five hundred 

 roubles. It is given for work in geometry, 

 preferably non-Euclidean geometry. All 

 works published within the six years pre- 

 ceding the award of the prize, and sent by 

 their authors to the Physico-Mathematical 

 Society of Kazan, are allowed to compete 

 if published in Russian, French, German, 

 iEnglish, Italian or Latin. 



The Society has now in formal session 

 :awarded the prize to Sophus Lie, professor 

 of mathematics at the University of Leip- 

 zig, for his work ' Theorie der Transforma- 

 tionsgruppen, Band III., Leipzig, 1893.' In 

 this work the theory of non-Euclidean 

 geometry has been exhaustively re- stated 

 and re-established in a profound investiga- 

 tion of the work of Helmholtz on the space- 

 problem. 



To the genius of Helmholtz is due the 

 ■conception of studying the essential charac- 

 teristics of a space by a consideration of the 

 movements possible therein. 



But since the time when Helmholtz did 

 his work on this subject the greatest of 

 living mathematicians, Sophus Lie, formerly 

 of Christiania, has enriched mathematics 

 with a new instrument, the Theory of 

 Oroups, which its creator has applied with 

 tremendous power to the Helmholtz treat- 

 ment. Lie finds, as was almost inevitable, 

 that certain details had escaped the great 

 physicist, but that, with the tact of true 

 genius, he had kept his main results free 

 from error, though there comes to light a 

 superfluity in his explicit assumptions, an 

 unconscious assumption now seen to be 

 mathematically important for the rigor of 

 the demonstration, and at least one definite 

 error in minor results. 



Lie's method is in general the following. 

 Consider a tri-dimensional space, in which 

 a point is defined by three quantities, x, y, z. 



A movement is defined by three equa- 

 tions : 



^' = / (^> 2/) 2) ; 2/' = V (», y, h) ] «' = '^ (», 

 y, 0- 



By this transformation an assemblage. A, 

 of points (a;, y, z) becomes an assemblage, 

 A', of points (a', y', s'). 



This represents a movement which 

 changes A to A'. 



Now make, in regard to the space to be 

 studied, the following assumptions : 



1st. Assume : In reference to any pair of 

 points which are moved, there is something 

 which is left unchanged by the motion. 



That is, after an assemblage of points, A, 

 has been turned by a single motion into an 

 assemblage of points, A', there is a certain 

 function, F, of the coordinates of any pair of 

 the old points {x^, y^, z^), (x^, y^, z^) which 

 equals that same function, F, of the corre- 

 sponding new coordinates {x\, y\, z'^, 

 (^'2- y'v ^'2) ; tliat iS; F (x^, 2/1, z^, a;,, y^, s,) = 

 F(a;'i, 2/'i, s'l, a;'^, 2/2, z'j). 



This something corresponds to the Cayley 

 definition of the distance of two points 

 when interpreted as completely independ- 

 ent of ordinary measurement by superpo- 

 sition of ^n unchanging sect as unit for 

 length. 



This independence, involving the deter- 

 mination of cross-ratio without any use of 

 ordinary ratio, without using congruence, 

 without using motion, Cayley never clearly 

 saw. It follows from the profound pure 

 projective geometry of von Staudt. 



2d. Assume : If one point of an assem- 

 blage is fixed, every other point of this as- 

 semblage, iiyithout any exception, describes a 

 surface (a two-dimensional aggregate). 



When two points are fixed a point in 

 general (exceptions being possible) de- 

 scribes a curve (a one-dimensional aggre- 

 gate). Finally, if three arbitrary points 

 are fixed, all are fixed (exceptions being 

 possible). With these assumptions Lie 

 proves exhaustively that the general results 



