RECENT ADVANCES IN SCIENCE 9 3 



axiom plays a part in geometry which is to a certain extent 

 analogous to that played by the metrical axiom of Archimedes. 

 In the second place, Dr. B. A. Bernstein (Trans. Amer. Math. 

 Soc. 1916, 17, 50) improved on Sheffer's (1913) set of five 

 postulates for Boole's logic by choosing the same primitive 

 ideas (class and operation) as Sheffer, and deducing Sheffer's 

 five postulates from four independent ones. This is the last 

 chapter so far in the history of reduction in the number of 

 postulates for the algebra of logic : in 1904 Huntington had 

 got down as far as nine. 



Theory of Numbers and Algebra. — G. H. Hardy (Proc. Lond. 

 Math. Soc. 1916, 15, 1) attacks a new aspect of the problem 

 he calls " Dirichlet's divisor problem." Dirichlet gave a formula 

 for the sum of terms d (n) (the number of divisors of n, unity 

 and n itself included) when n varies from 1 to x, and the problem 

 in question is to determine as precisely as possible the maximum 

 order of the error term in this formula. 



G. A. Miller (Proc. Nat. Acad. Set. Washington, D.C., 1916, 2, 

 No. 1 ) finds that the degree of transitivity of a substitution-group 

 of degree n which does not include the alternating group of 

 this degree is always less than 5/2 \/n — 1 . 



In continuation of some former researches of his, O. C. 

 Hazlett (Amer. Journ. Math. 1916, 38, 109) deals with the 

 classification and invariantive characterisation of nilpotent 

 algebras, and completely solves his problem for algebras in a 

 small number of units, if the commutative and associative laws 

 are assumed. 



Here we may mention two papers dealing with vital sta- 

 tistics and the application of the theory of probability to 

 pathometry. Prof. Karl Pearson, in a paper read to the Royal 

 Society on February 24, gave the nineteenth of his mathematical 

 contributions to the theory of evolution. Of the other paper 

 we will attempt a somewhat fuller account. In investigating 

 the mathematical theory of epidemics, we may begin with 

 observed statistics, try to fit analytical laws to them, and so 

 work backwards to the underlying cause, as in the work of 

 Farr, Evans, and Brownlee ; or we may follow an a priori 

 method and assume a knowledge of the causes, construct 

 our different equations on that supposition, follow up the 

 logical consequences, and finally test the calculated results 

 by comparing them with the observed statistics. This latter 



