588 



SCIENCE. 



[N. S. Vol. XVII. No. 432 



various functions of an angle or of half the 

 angle, together with the usual T-square and 

 straight edge. By these instruments a plane 

 triangle is constructed having sides and 

 angles proportional to the proper trigonomet- 

 rical functions of three knov^n parts of the 

 spherical triangle to be solved; then the re- 

 maining parts may be scaled off. 



The next paper was a report by Mr. F. G. 

 Radelfinger ' On the Analytic Representation 

 of Function.' 



The general problem of obtaining an an- 

 alytic development coextensive with the do- 

 main of existence of a function was stated 

 by way of introduction; then the author re- 

 viewed the most important recent . researches 

 bearing on this general problem. He con- 

 fined himself in the main to a synopsis of the 

 results obtained by Mittag-Leffler in his four 

 ' notes,' which have appeared at intervals dur- 

 ing the last three years in his journal, the 

 Acta Mathematica. 



The analytic developments constructed by 

 Mittag-LefBer are in the form of series n 

 times infinite, which can be transformed into 

 singly infinite series of rational polynomials; 

 these developments converge within an ex- 

 tended region, which for n= co coincides 

 with a star introduced by him; the star em- 

 braces the whole plane excepting radial cross- 

 cuts extending from each singular point to 

 infinity. Several methods of constructing 

 these expressions have been made use of and 

 expounded in the several notes. Borel's work 

 was also reviewed, and the extension thereof 

 by Mittag-Leffler. 



The next paper was ' On the Foundations 

 of Geometry and on Possible Systems of 

 Geometry,' by Dr. Henry Freeman Stecker, 

 of Cornell University. In the absence of Dr. 

 Stecker his paper was presented by Mr. Rad- 

 elfinger. 



After an introduction on the assumptions 

 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 concluson that in spite of all criticisms 

 and attempted improvements, Hilbert's sys- 

 tem has ' withstood all attacks, and remains 



not only apparently sound in logic, but the 

 simplest of such systems as have thus far 

 been constructed.' 



An account was next given of Hilbert's 

 second, and recent, great memoir, Mathe- 

 matische Annalen, Bd. 56, which has for its 

 object to establish Lie's well-known and in- 

 dispensable 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 results, so far as they go, 

 establish the independence 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.' 



Ohaeles K. Wead, 



Secretary. 



NEW YORK ACADEMY OP SCIENCES. 

 SECTION OF ANTHEOPOLOGY AND PSYCHOLOGY. 



The regular meeting of the section was 

 held February 23, in conjimction with the 

 New York branch of the American Psycholog- 

 ical Association, Professor Thorndike presid- 

 ing. Afternoon and evening sessions were 

 held, the members dining together at the close 

 of the afternoon session. The following 

 papers were presented: 



' Phonetic Surveys,' Professor E. W. Scrip- 

 ture. After brief mention of the phonetic 

 surveys being carried on by Grierson in India 

 and Guilleron in France, a description was 

 given of the chief talking machine methods 

 that may be used for this purpose. It was 

 pointed out that the advances in the construc- 

 tion of phonographs, graphophones and 

 gramophones during the last couple of years 

 have been so great as to revolutionize these 

 methods. It was also explained that making 

 a speech record was like taking a photograph; 

 everybody can take a picture, but a good pic- 

 ture requires skill. By use of the grapho- 



