RECENT ADVANCES IN SCIENCE 5 



head and Russell, for example, fail to give a correct theory of 

 mathematical deduction. 



Arithmetic, Theory of Numbers, and Algebra. — L. R. Ford 

 (Proc. Edinburgh Math. Soc. 191 6-1 7, 35; Research Paper, 

 191 7, No. 5) proves Hurwitz's (1891) theorem on rational ap- 

 proximations to irrational numbers by considering the geometry 

 of the classic modular division of the half-plane, and thus ex- 

 hibits anew the remarkable connection between this geometry 

 and the theory of numbers. J. H. Grace (Proc. Lond. Math. 

 Soc. 19 1 9, 17, 247-58) proves a theorem, allied with those of 

 Hermite and Minkowski, on rational approximations, and 

 gives a classification of such approximations. Grace (ibid. 

 316-9) gives a result on Diophantine approximation. 



L. R. Ford (Proc. Edinburgh Math. Soc. 191 6-1 7, 35 ; Re- 

 search Paper, 191 7, No. 2) finds some striking properties of an 

 interesting class of continued fractions in which any real 

 number can be developed in an infinity of ways, and contrasts 

 these fractions with the continued fractions usually employed. 



H. Hallberg (Journ. Indian Math. Soc. 191 8, 10, 454-72) 

 concludes his paper (cf. Science Progress, 191 8, 12, 543) on 

 infinite series and arithmetical functions, in which he tries to 

 show how asymptotic formulae in the theory of numbers may 

 easily be obtained in a manner somewhat different from the 

 usual one, which rests chiefly on the theory of Dirichlet's series. 



G. N. Bauer and H. L. Slobin (Amer. Math. Monthly, 191 8, 

 25, 435-40) give several theorems on a certain system of alge- 

 braic and transcendental equations. H. T. Burgess (ibid. 441- 

 4) explains a scheme which works with ease and simplicity in 

 practice for finding a fundamental system of solutions of a 

 system of linear equations. E. T. Whittaker (Proc. Edinburgh 

 Math. Soc. 191 7-1 8, 36, 103-6 ; Research Paper, 191 8, No. 2) 

 gives a formula for the solution of algebraic or transcendental 

 equations. 



Whittaker (ibid. 107-15 ; and No. 3) gives some theorems 

 on determinants whose elements are determinants. Whittaker 

 also (ibid. 1916-17, 35; Research Paper, 1917, No. 1) finds 

 what Sylvester called " the latent roots " of all compound 

 determinants and Brill's (1870) determinants in terms of the 

 latest roots of the determinant of which they are formed. 

 Haripada Datta (ibid. 191 5-16, 34; Research Paper, 191 6, 

 No. 5) obtains theorems on zero-axial skew Pfaffians and sym- 



