May 19, 1911] 



SCIENCE 



765 



bert's which is entitled "Foundations of 

 Geometry, ' ' which bears then the same title 

 as his "Festschrift," but where he takes, 

 however, a wholly different point of view. 

 In his "Festschrift," in fact, as we have 

 seen by the preceding analysis, the rela- 

 tions of the notion of space and the notion 

 of group resulting from the works of Lie 

 are laid aside or relegated to an inferior 

 place. The general properties of groups 

 do not appear in the list of fundamental 

 assumptions. Not so in the memoir of 

 which we are to speak. 



As regards the ideas of Lie, the progress 

 made is considerable. Lie supposed his 

 groups defined by analytic equations. Hil- 

 bert's hypotheses are far more general. 

 Without doubt this is still not entirely 

 satisfactory, since though the form of the 

 group is supposed any whatever, its matter, 

 that is to say, the plane which undergoes 

 the transformations, is still subjected to 

 being a numher-manifold in Lie's sense. 

 Nevertheless, this is a step in advance, and 

 besides Hilbert analyzes better than any 

 one before him the idea of numher-mani- 

 fold and gives outlines which may become 

 the germ of an assumptional theory of an- 

 alysis situs. 



It is impossible not to be struck by the 

 contrast between the point of view here 

 taken by Hilbert and that adopted in his 

 "Festschrift." In this "Festschrift" the 

 continuity assumptions took lowest rank 

 and the great thing was to know what 

 geometry became when they were put aside. 

 Here, on the contrary, continuity is the 

 point of departure and Hilbert is above all 

 anxious to see what we get from continuity 

 alone, joined to the notion of group. 



It remains for us to speak of a memoir 

 entitled "Surfaces of Constant Curva- 

 ture. ' ' 



We know that Beltrami has shown that 

 there are in ordinary space surfaces which 



image the non-euelidean plane; these are 

 the surfaces of constant negative curva- 

 ture; we know what an impulse this dis- 

 covery gave to the non-euclidean geometry. 

 But is it possible to represent the non- 

 euclidean plane entire on a Beltrami sur- 

 face without singular point? Hilbert has 

 proved that it is not. 



As to the surfaces of constant positive 

 curvature, to which Riemann's geometry 

 corresponds, Hilbert proves that besides 

 the sphere there is no other closed surface 

 of this sort. 



{To ie concluded) 



SCIENTIFIC NOTES AND NEWS 



Dr. David Starr Jordan has tendered to 

 President Taft his resignation as interna- 

 tional commissioner of fisheries, this position 

 having been created three years ago under the 

 treaty of April 11, 1908, with Great Britain. 

 Under the terms of the appointment, the work 

 of the commissioner ceases on the completion 

 of the series of fishery regulations of the 

 boundary waters, and the technical investiga- 

 tions necessary for their completion. This 

 work being finished, the administration of the 

 treaty passes to the Bureau of Fisheries. 



Dr. Willy Kukenthal, professor of zool- 

 ogy at Breslau, has been appointed exchange 

 professor at Harvard University during the 

 academic year of 1911-12. 



Dr. Edward Mlner Gallaudet has retired 

 from the presidency of Gallaudet College, 

 which he has held for fifty-four years. 



Dr. Oscar Riddle, of the University of 

 Chicago, has returned from a year of study 

 and travel in Europe. He spent the past six 

 months at the Zoological Station at Naples, 

 whence he now returns to Chicago to take 

 charge of the preparation for publication of 

 the manuscripts left by the late Professor C. 

 O. Whitman. He will also continue certain 

 features of Professor Whitman's investiga- 

 tions. 



Professor George E. Sever has been 

 elected president of the Columbia Chapter of 



