8 SCIENCE PROGRESS 



the common rule that " the approximation tt = %\ will generally 

 get the first three figures of your answer right " is a sound 

 working rule, because it will succeed in 843 cases out of 1 ,000, 

 chosen at random, supposing the answer to contain ir as a factor. 



L. E. Dickson (Bull. Amer. Math. Soc. 191 6, 23, 1 09-11) 

 gives an extension of the theory of numbers by means of cor- 

 respondences between fields, and H. S. Vandiver (ibid, n 1-4) 

 has a note on the distribution of quadratic residues for a rational 

 prime modulus. 



The analytic theory of numbers might be classed under 

 " analysis," but the subject will be dealt with under the present 

 heading. We have thus one more example of how the nature of 

 a branch of science cannot be described by means of a finite 

 number of classifications. 



G. H. Hardy has written (Cambridge, 19 16) a report to the 

 University of Madras on S. Ramanujan's mathematical work in 

 England since 1914. Ramanujan's work is distinguished by 

 strikingly original and unusual methods, and is on extremely im- 

 portant subjects in definite integrals, series, the Zeta function, 

 the analytic theory of numbers, and so on. One of Rama- 

 nujan's last papers, of which some account is given in this 

 report, but which was published since the report was printed, 

 is a paper on the expression of a number in the form ax 2 + by 2 + 

 cz 2 + du 2 (Proc. Camb. Phil. Soc. 191 7, 19, 11 -21), where he 

 shows that there are exactly 55 sets of values of a, b, c, d, for 

 which this is true. 



G. H. Hardy (Mess, of Math. 19 16, 46, 104-7) considers 

 certain multiple integrals and series which occur in the analytic 

 theory of numbers and in particular in his paper in the Proc. 

 Lond. Math. Soc. for 191 6 (15, 192-213 ; see Science Progress, 

 1916, 11, 268). 



A. A. Bennett (Proc. Nat. Acad. Sci., Washington, D.C. 

 191 6, 2, No. 10) extends the special algebraic work of H. B. 

 Fine to general analysis. 



Analysis. — W. A. Hurwitz and L. L. Silverman (Trans. 

 Amer. Math. Soc. 191 7, 18, 1-20) consider the consistency and 

 equivalence of certain definitions of the summability of diver- 

 gent series. It is not by any means necessary that two different 

 generalisations of the conception of convergence should be 

 consistent, but the authors find that all definitions of sum- 

 mability of a certain class are so in fact. 



