178 SCIENCE PROGRESS 



unlikely therefore," says Dresden, " that we shall have to 

 place Barrow at last on a par with, if not above, Newton and 

 Leibniz among the inventors of the calculus." 



G. H. Hardy (P.C.P.S. 191 8, 19, 148-56) gives a very 

 penetrating analysis of the paper of 1847 m which Stokes ad- 

 vanced his discovery of uniform convergence. The thorough- 

 ness of this examination makes it very important. 



Logic, Principles, and Aggregates. — In Hartog's work of 

 191 5 (Science Progress, 191 7, 11, 453, and 12, 7), it was 

 proved, though not in all respects satisfactorily, that, for any 

 given aggregate M, there is always one and only one ordinal 

 number (a) such that : (1) For every number 7 less than a 

 there is a part of M which can be arranged in type 7 ; (2) No 

 part of M can be put in type a. Philip E. B. Jourdain (C.R. 

 191 8, 166, 520-3, 984-6 ; Nt. 191 8, 101, 84, 304; Mind, 191 8, 

 27, 386-8) defines completely a method of removing the mem- 

 bers of those parts of M which can be well ordered (not neces- 

 sarily in the order, if there is one, in which M is given) and 

 forming with them well-ordered parts of M which actually 

 exhaust M, though none of the parts of M first mentioned 

 need be presupposed to exhaust M. Thus any aggregate 

 can be well ordered, Zermelo's " axiom " can be proved quite 

 generally, and Hartog's work is completed. As this result 

 gives the solution of a problem in pure mathematics which 

 has been much discussed, especially during the past fourteen 

 or fifteen years, and has many far-reaching consequences, a 

 detailed account of it is given elsewhere in the present number 

 of Science Progress. 



C. D. Broad (Mind, 191 8, 27, 284-303) describes a nota- 

 tion for the logic of relations which presents certain advan- 

 tages — especially in the case of relations of a high degree of 

 polyadicity — over that used by Whitehead and Russell in 

 their Principia Mathematica. 



An English translation of Hermann Minkowski's famous 

 lecture on space and time, which was first published in 1909, 

 is given in the Monist for April, 191 8 (28, 288-302). 



W. M. Thorburn (Mind, 191 8, 27, 345-53) continues his 

 elaborate researches to prove that the principle of methodology 

 known as " Occam's Razor," which has come to be of such 

 great importance in logic and the principles of mathematics, 

 was not stated by Ockham in the form with which he is usually 



