Februakv 27, 1903.] 



SCIENCE. 



347 



didates for the degree of Ph.D., and the cata- 

 logue of mathematical apparatxis, hy Professor 

 Walter Dyck. Its principal publication has 

 been eleven volumes of 'Annual Reports,' con- 

 taining elaborate reviews of progress in the 

 various branches of mathematics and its ap- 

 plications. In 1894, at the Vienna meeting, 

 the publication of a 'mathematical lexicon' 

 was decided upon, but at the Frankfort meet- 

 ing of 1896 it was concluded best to combine 

 this idea with the ' EncyclopEedia of Mathe- 

 matical Sciences,' undertaken by Professor 

 W^ F. Meyer, of Clausthal, and Professor H. 

 Burkhardt, of. Zurich. The mathematical 

 union has, therefore, united with the Scientific 

 Association of GfJttingen and the academies 

 of science at Munich and Vienna in becoming 

 responsible for this latter work. It was ori- 

 ginally estimated that the encyclopedia would 

 consist of seven volumes in ten distinct parts 

 besides the general index, but the portions 

 already printed show that the whole work will 

 be larger than was expected. The publication 

 has proceeded by parts as follows: 1898, one; 

 1899, four; 1900, three; 1901, three; 1902, 

 four. These parts are scattered through the 

 encyclopedia as follows : Volume I., seven 

 parts and completed ; II., four parts ; IH., 

 one part ; IV., three parts. It may, therefore, 

 be expected that five or six years will still 

 elapse before we shall approach the end of this 

 great work. 



Mr. Abbe exhibited the fifteen parts already 

 received, and they were examined in detail 

 by the audience. He remarked upon the 

 chapters treating of the theory of numbers ; 

 that on mathematical economics; the memoirs 

 on differential equations; the chapter on 

 mathematical apparatus and machinery, and 

 especiallj' the two memoirs by Professor A. 

 E. H. Love, of Oxford, on the physical basis 

 and the theoretical development of hydro- 

 dynamics. 



Professor Frank H. Bigelow, also of the 

 Weather Bureau, sununarized the 'Applica- 

 tions of Mathematics in Meteorology.' Me- 

 teorology has suffered in the past by the mis- 

 application of mathematical theories to the 

 explanation of cyclones and anticyclones, and 

 also of the general circulation. It has been 



shown that Ferrel's vortex and Oberbeck's vor- 

 tex do not agree with the modern observations 

 of the local circulation of the air; also, that 

 the theories of these authors must be greatly 

 modified to fit the facts of the movements of 

 the atmosphere in general. Similarly, there 

 has been a tendency to misapply the theory 

 of least squares, and the probability curves, 

 in discussing the periodic cycles observed in 

 the solar and terrestrial atmospheres. These 

 theorems require that the events shall be inde- 

 pendent of one another, but in such thermo- 

 dynamic circulations this is not the case. 



The next paper was ' On the Foundations oi 

 Geometry and on Possible Systems of Geom- 

 etry,' by Dr. Henry Freeman Stecker, of Cor- 

 nell University. In the absence of Dr. Stecker 

 his paper was presented by Mr. Radelfinger. 



After an introduction on the assumption 

 which must be made in constructing a geom- 

 etry. Dr. Stecker reviewed the criticisms of 

 Moore and Schur of Hilbert's classic paper 

 of 1899, recently translated,, and announced 

 the conclusions that in spite of all criticisms 

 and attempted improvements Hilbert's system 

 has ' withstood all attacks and remains not 

 only apparently sound in logic, but the sim- 

 plest of such systems as have thus far been 

 constructed.' 



An account was next given of Hilbert's 

 second, 'and recent, great memoir. Math. An- 

 nalen, Bd. 56, which has for its object to es- 

 tablish Lie's well-known and indispensable 

 results, without the assumption, made by Lie, 

 that the functions defining the displacements 

 are differentiable. In solving the problem 

 Hilbert makes use of Cantor's theory of point- 

 assemblages and Jordan's theory of a closed 

 curve free from double points. Hilbert's re- 

 sults, so far as they go, establish the independ- 

 ence of Lie's results of the assumption stated 

 above, but they have yet to be extended to 

 elliptic geometry and also to space. 



In conclusion, a thesis by Hamel, a pupil of 

 Hilbert's, was discussed, which leads to the 

 conclusion that ' from the standpoint of the 

 calculus of variations the Euclidean geometry 

 is the simplest possible.' 



A fourth paper, by Mr. F. G. Kadclfinger, 



