160 SCIENCE PROGRESS 



Machines of Direct Multiplication '' ; Mrs. E. Gifford, " New Table of Natural 

 Sines"; Dr. J. R. Milne, "The Arrangement of Mathematical Tables"; Mr. 

 T. C. Hudson, "Note on 'Critical' Tables"; Prof. J. E. A. Steggall, "On a 

 Possible Economy of Entries in Tables of Logarithmic and other Functions " ; 

 Dr. A. Hutchinson, "The Graphical Treatment of some Crystallographic 

 Problems " ; Mr. William Schooling, " A Method of Computing Logarithms by 

 Simple Addition" ; Mr. A. K. Erlang, " How to Reduce to a Minimum the Mean 

 Error of Tables " ; Dr. W. F. Sheppard, " Extension of Accuracy of Mathematical 

 Tables by Improvement of Differences " ; Dr. Artemas Martin, " A Method of 

 Finding without the Use of Tables the Number Corresponding to a given Natural 

 Logarithm"; Mr. H. S. Gay, " Approximate Determination of the Functions of 

 an Angle, and the Converse"; M. Albert Quiquet, "Life Probabilities: on a 

 Logarithmic Criterion of Dr. Goldziher, and on its Extension." 



The following remarks may be useful to a mathematician who reads through 

 this volume with the intention of forming a sound opinion on the various historical 

 questions that arise about the discovery of logarithms. On arithmetic and algebra 

 in Napier's time, see p. 69 (Glaisher) ; on the method of " Prosthaphaeresis " or 

 the method of replacing the product of trigonometrical functions by a sum of other 

 trigonometrical functions, referred to on p. 7 of Lord Moulton's Address, cf. p. 73 

 (Glaisher), pp. 99-100 (Cajori), and pp. 219-21 (Sampson); on the comparison 

 of an arithmetical progression with a geometrical progression, cf. p. 7 (Moulton) 

 and pp. 8 [-2 (Smith); and on methods of calculation with Napier and later 

 writers, see pp. 70-71 (Glaisher). With Lord Moulton's theory should also be 

 compared Prof. Carslaw's theory mentioned in this quarterly for January 1916 

 (P- 434). 



The only mistake which there appears to be in this volume is that curious 

 mistake of spelling " Tabullen " in the name of Burgi's Progress Tabulen, which 

 Prof. Knott has made both in his article in this quarterly referred to above and 

 on p. vi of the volume under review. This is all the more strange as there is a 

 facsimile of Burgi's title-page facing p. 210. 



Philip E. B. Jourdain. 



A Course of Modern Analysis. An Introduction to the General Theory of 

 Infinite Processes and of Analytical Functions, with an Account of the 

 Principal Transcendental Functions. Second Edition, completely Revised. 

 By E. T. Whittaklr, D.Sc, F.R.S., Professor of Mathematics in the 

 University of Edinburgh, and G. N. Watson, M.A., Fellow of Trinity 

 College, Cambridge, and Assistant Professor of Pure Mathematics at 

 University College, London. [Pp. 560.] (Cambridge : at the University 

 Press, 191 5. Price i8j. net.) 

 It is scarcely necessary to give a formal review merely on the occasion of a 

 second edition of so admirable and well known a book as this one ; but, as we 

 happen to know, many have been long awaiting the second edition, and our 

 readers may therefore be glad to be furnished with some idea as to the changes 

 and additions made in it. In the first place, for the second edition, Prof. 

 Whittaker's name is associated with that of Mr. G. N. Watson, who, Prof. Whit- 

 taker tells us in his Preface, is responsible for the new chapters on Riemann 

 Integration, on Integral Equations, and on the Riemann-Zeta Function. The 

 second edition adopts the Peano decimal system of paragraphing, much to the 

 advantage of the work. The two Parts of the original work still stand ; but 

 the original Chapter 4 on Uniform Convergents is suppressed and partly replaced 



