RECENT ADVANCES IN SCIENCE 267 



We know, one of Weierstrass's pupils, and has always regarded, 

 not without reason, his old master's theory of real numbers as 

 presenting certain advantages over the theories of Meray, 

 Cantor, Heine, and Dedekind. The exposition which Mittag- 

 Leffler {Rev. gen. des Sci. 191 5, 28, 504) gives of the foundations 

 of the theory of numbers is thus very interesting, especially in 

 connection with a paper on Weierstrass's theory which the 

 same author published in 1909. 



In a very able article on the calculus of probabilities and 

 intuition, S. Pincherle (Scientia, 1916, 19, 417) points out that 

 the causes which we consider in such schematic representations 

 of reality as mechanics and mathematical physics are reduced 

 to a small number of predominating ones while the others are 

 deliberately left on one side ; but in sciences such as political 

 economy, medicine, meteorology, and biology, the selection of 

 dominant causes is not imposed by the question in itself, and a 

 subjective element comes into play. Thus, in the latter cases, 

 the statistical method, which is closely connected with the 

 calculus of probabilities, must supplement deduction. The 

 very definition of probability is founded on ignorance of which, 

 if any, causes are dominant, and this ignorance is translated 

 into a principle of equivalence between the various causes 

 possible. Pincherle goes on to refer to modern researches into 

 probabilities in which there are an infinite number (in the 

 Cantorian sense) of possibilities, and the part played by the 

 intuition in the calculus of probabilities ;' but what seems most 

 important to the present writer is that some light may be 

 thrown on the question as to whether, as Russell maintains, 

 probability is fundamental in the principle of induction. If, 

 in fact, probability depends on the notion of cause, it would 

 seem that the law of causality, which depends on the principle 

 of induction, cannot depend on probability. 



In connection with the theory of relativity and an optical 

 geometry of space and time, we may refer to two papers by Dr. 

 H. Bateman which are difficult to abstract on account of their 

 technical character : " Time and Electromagnetism " {Mess. 

 of Math. 191 5, 45, 97) an d "The Structure of the ^Ether " 

 {Bull. Amer. Math. Soc. 191 5, 21, 299). 



Theory of Numbers. — In a paper read to the Royal Society 

 on May 11, 1916, Major P. A. Macmahon gave a detailed study 

 of the enumeration of the partitions of multipartite numbers. 



