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Descriptio, he did not do very much more than mention the 

 Constructio, which was written long before the Descriptio, 

 though it was only published posthumously in 1619. However, 

 it seems that the Constructio gave an account from which the 

 mode of genesis of Napier's great invention may be gathered, 

 whereas the kinematical and almost fluxional considerations 

 in the Descriptio seem to be of origin later than the invention 

 of the logarithms. For this reason, the long paper by Prof. 

 H. S. Carslaw (Math. Gaz. 191 5, 8, jj, 115), which gives, in 

 modern notation, Napier's method as explained in his Con- 

 structio, is very welcome. An account of the contents of the 

 Memorial Volume published to commemorate the Napier Ter- 

 centenary is given by Dr. C. G. Knott in the number of this 

 Quarterly for October 191 5 (10, 189). 



W. D. MacMillan (Proc. Nat. Acad. Sci. Washington, D.C., 

 191 5, 1, No. 7) gives some theorems connected with irrational 

 numbers, as such numbers play an important part in the 

 convergence of some series occurring in celestial mechanics. 



The number d(N) of divisors of N varies with extreme 

 irregularity as N tends to infinity, and a theorem of Dirichlet 

 on the sum d(\) + d(2) + ^(3) + . . . + d(N) has aroused a good 

 deal of interest in recent years with Voronoi, Landau, and 

 Hardy. S. Ramanujan (Proc. Lond. Math. Soc. 191 5, 14, 347), 

 in a paper on highly composite numbers, proves a large number 

 of results which add a good deal to our knowledge of the be- 

 haviour of d(N). The same author (Journ. Indian Math. Soc. 

 I 9 I 5> 7, 131) gives an investigation on the best possible upper 

 limit for d(N), using only purely elementary reasoning. 



G. Humbert (Compt. Rend. 191 5, August 30) writes on the 

 reduction of Hermite forms in an imaginary quadratic corpus. 



During 191 5 papers have been read to the Paris Academy 

 of Science by G. Mittag-Leffler on a new theorem in the theory 

 of Dirichlet's series, and by G. H. Hardy on the problem of 

 divisors of Dirichlet. In the same year, G. H. Hardy and Dr. 

 Marcel Riesz published a small work on the general theory of 

 Dirichlet's series, which is reviewed elsewhere in the present 

 number, and which is so up-to-date that it includes even a 

 reference to the above paper by Mittag-LefTler (see pp. 8, 75). 



Analysis. — Arnaud Denjoy, in a paper read to the Paris 

 Academy in 191 5, has studied the theory of the four deriva- 

 tives of a function — a theory which, principally owing to the 



