i 9 2 SCIENCE PROGRESS 



sary and sufficient conditions that P Q should be Dedekindian 

 or semi-Dedekindian when P and Q are well-ordered series. 

 In this notation the field of P Q is the class of Cantor's " Bele- 

 gungen." The greater part of the paper makes use of the 

 symbols introduced by Whitehead and Russell in their Prin- 

 cipia Mathematica. 



D. Hilbert (17-18 ; two papers) gives a research, based on 

 the axiomatic method, on the foundations of physics, and 

 F. Klein and Hilbert (21) contribute a note on the first of 

 these papers. On the theory of relativity we may notice 

 papers by A. Einstein (14 ; two papers), G. Nordstrom (48), 

 A. D. Fokker (48), Sir Oliver Lodge (42), G. W. Walker (43), 

 J. P. Kuenen (46-7), and L. de la Rive (50). 



On the calculus of probabilities, cf. O. Knopf (31), J. Haag 

 (34), and F. Schuh (48). 



Arithmetic and Theory of Numbers. — R. D. Carmichael 

 (Amer. Math. Monthly, 19 19, 26, 137-46) gives an account of 



Fermat's numbers — = 2 2 -f 1. In view of the known facts 



n 



F 



about the factors of — , Fermat's question whether (2&) 2 " -f 1 



n 



F . 

 is always a prime except when divisible by an — is without 



further particular interest, and in the present paper all the 

 essential facts known about the factorisation of the numbers 



F 



— are gathered together and proved in the simplest ways pos- 



sible. Also a few minor results are given which appear to be 

 novel. N. M. Shah and B. M. Wilson (Proc. Cambridge Phil. 

 Soc, 1 919, 19, 238-44) give some calculations which originated 

 in a request made by G. H. Hardy and J. E. Littlewood that 

 they should check a suggested asymptotic formula for the 

 number of ways of expressing a given even number n as the 

 sum of two primes — Goldbach's empirical theorem being 

 that any even number is the sum of two primes. Other allied 

 formulae are also discussed. Hardy and Littlewood (ibid. 

 245-54) give some indication of the genesis of their particular 

 formulae and others of the same character. S. Ramanujan 

 (ibid. 207-10) guesses by induction a general theorem — as 

 yet unproved — on congruence properties of p(n), the number 

 of partitions of n, and gives proofs of two particular cases of 



