RECENT ADVANCES IN SCIENCE 365 



author says that we cannot urge logical objections against this 

 theory of classes, " but only practical ones, because the language 

 in use is modified. Many mathematicians will find it a novelty 

 to write ' 1 = proper fraction ' to say that unity is the upper 

 limit of proper fractions " (p. 7 ; cf. p. 13). 



A manuscript of the late Louis Couturat on symbolic logic 

 and the calculus of probabilities is printed in the Rev. de Meta- 

 phys. et de Morale for May, 1917 (24, 291-313). This manu- 

 script dates from before 1902 and will not be reprinted in the 

 forthcoming Manuel of his (cf. Science Progress, 191 7, 12, 6). 



A. N. Whitehead (Proc. Aristot. Soc. 1916, 16, 104-29, 

 and pp. 191-228 of The Organisation of Thought reviewed else- 

 where in the present number) discusses the theories of space, 

 time, and relativity, and brings into relation with one another 

 the standpoints of mathematical physics, experimental psy- 

 chology, metaphysics, and mathematics. It will be seen that 

 this paper continues on the same lines Whitehead's important 

 work. Sir Joseph Larmor (Proc. Aristot. Soc. 1916, 16, 130-32) 

 has a short article entitled, " Relativity : A New Year Tale." 



B. Petronievics (Rend, delta R. Accad. dei Lincei, 191 7, 26, 

 309-16) tries to show that Fontenelle's " attempt at a rational 

 theory of infinite numbers " had an " historical value " in that it 

 may have affected the ideas of Cantor and Veronese. There do 

 not, however, seem to the reviewer to be any grounds whatever 

 for this supposition, and it is and has always been recognised 

 that Fontenelle's theory is fundamentally unsound and quite 

 different in its nature from, at least, Cantor's theory. 



Arithmetic, Theory of Numbers, and Algebra. — G. Peano 

 (Rend, delta R. Accad. dei Lincei, Rome, 1916, 25, 8-14) gives 

 the rules for numerical approximations under an elementary 

 form without presupposing the differential calculus ; and then 

 gives from these rules the form of the rules of derivation. 



G. H. Hardy and S. Ramanujan (Proc. Lond. Math. Soc. 19 16, 

 16, 112-32) prove a theorem on the distribution of integers of 

 various types by an interesting method. The theorem includes 

 the " highly composite " numbers studied by Ramanujan 

 (ibid. 1915, 14, 347-409 ; see Science Progress, 191 6, 10, 434). 

 . C. Burali-Forti (II Pitagora, 191 7, 1, 1-6) gives a new defini- 

 tion of the complex numbers of algebra. 



Sir Thomas Muir read to the Royal Society of South Africa 

 on April 18, 191 7 (Nature, 191 7, 99, 360), a note on the expan- 





