August 14, 1896.] 



SCIENCE. 



201 



The Congress was decidedly cosmopolitan in 

 the authorship of the papers presented and the 

 subjects treated. Of the forty-five papers, 

 fourteen were from America, two from Austria, 

 four from France, twenty from Germany, three 

 from Italy, one from Russia and one from 

 Switzei'land. Germany was oflicially repre- 

 sented at the Congress by an Imperial Commis- 

 sioner, Prof. Felix Klein, of Gottingen, who 

 brought nearly all the papers contributed by 

 his countrymen. It is a singular circumstance 

 that the British Empire did not furnish a single 

 contribution nor a single representative. 

 Eoughly classified, seven of the papers'deal with 

 geometry, ten with theory of functions, eight 

 with the theory of groups, seven with the theory 

 of numbers, two each with differential equa- 

 tions, invariants and mechanics, and seven 

 with miscellaneous subjects. 



In his opening address before the combined 

 Congresses on Mathematics and Astronomy, 

 Prof. Klein drew attention to a matter of great 

 interest to all scientists, viz. , a present marked 

 revulsion from the tendency of mathematics to 

 run into isolated specialities, which has been so 

 pronounced for a century. At present the 

 movement is decidedly toward unification and 

 breadth, not only in mathematics itself, but in 

 its relation to other sciences. The general 

 conceptions of ' function ' and of ' group ' are 

 powerful coordinating elements. Two of the 

 Chicago papers are especially mentioned by 

 Klein as representing the new tendency. One 

 of these, by Schonflies, deals with the connec- 

 tion between the theory of groups and crystal- 

 lography. The other, by Burkhardt, discusses 

 the relations between astronomical problems 

 and the theory of linear differential equations. 

 A third paper, by Fricke, on the automorphic 

 functions and arithmetic {i. e., theory of num- 

 bers), illustrates the work of Klein's own school 

 in the unification of strictly mathematical 

 branches. 



Of the other papers on the theory of functions, 

 one isby Bolza, on Weierstrass' system of hyper- 

 elliptic integrals ; one by Hermite, on certain 

 fundamental propositions in the theory of ellip- 

 tic functions; one by Krause, on the transforma- 

 tion of the fifth degree of the hyperelliptic 

 functions of the first order ; two by Macfarlane, 



on the definition of the trigonometric functions, 

 and on the principles of elliptic and hyperbolic 

 analysis; one by Pincherle, in summary of cer- 

 tain results relative to the theoiy of I'ecurrent 

 systems of functions; two by Pringsheim, on the 

 expansion of functions in series and conver- 

 gency and divergency; and one by Stringham, 

 on a formulary introductory to elliptic func- 

 tions. Maschke, Moore, Meyer, de Perott, 

 Taber and Cole have articles on special topics 

 in the theory of groups. Halsted gives an ac- 

 count of some salient points in the history of 

 non-Euclidean and hyper-geometries. It is, of 

 course, impossible here to give any detailed ac- 

 count of the contents of these or the other 

 valuable papers in the list. The moral of the 

 present publication, as representing the Inter- 

 national Congress, lies not so much in the 

 specific contents of the book, as in the fact that 

 America's workers in mathematics are con- 

 stantly coming into closer affiliation with 

 those of other lands and strengthening their 

 scientific position by this connection. The 

 other sciences have long had this advantage, 

 but mathematics has received it only within a 

 few years. Our position in this country has been 

 so isolated, and our science so backward here in 

 development, that it will probably be news to 

 most of our colaborers in other fields that the 

 development of mathematics in the present 

 century has probably been as great in actual 

 permanent substance as that of any other 

 science now existing. And it is the function of 

 such meetings as that at Chicago to stimulate 

 the growth of the science and to secure it the 

 recognition to which it is entitled. 



F. N. Cole. 

 Columbia University. 



recent text-books of geometry. 



Elements of Geometry. By George C. Edwards, 

 Associate Professor of Mathematics in the 

 University of California. New York, The 

 Macmillan Company. 1895. 8°. Pp. xvi+ 

 293. 



Plane and Solid Geometry. Suggestive Method. 

 By C. A. Van Velzer, Professor of Mathe- 

 matics in the University of Wisconsin, and 

 Geo. C. Shutts, Professor of Mathematics in 

 the State Normal School, Whitewater, Wis. 



