518 SCIENCE PROGRESS 



have been mentioned, and a full list given of the historical reports of Meyer, 

 Hilbert, Schoenflies, Burkhardt, and others (cf. p. 65). In the second Part we miss 

 references to the Portsmouth Catalogue, Rigaud's Historical Essay, and Rouse 

 Ball's Essay {on some of Newton's early manuscripts), Mrs. De Morgan's Memoir 

 of her husband, Babbage's Passages from the Life of a Philosopher, and Campbell 

 and Garnett's Life of Maxwell. 



In the next edition we hope that the usefulness of Prof. Loria's book will be 

 increased by the addition of a detailed analysis of those works, such as magazines, 

 which deal with many questions of science or history. What most of us want to 

 know is where exactly to find reliable information about a particular thing, theory, 

 or person. 



(2) The idea of an international catalogue of recent books is a good one ; but 

 this catalogue is by no means complete, especially as regards English books, and 

 the prices are always given in Italian money, so that it is of not so much use to, 

 say, an English buyer in England as it might be. Still, it will be found useful, 

 especially to those who are interested in the technical arts. 



Philip E. B. Jourdain. 



The Integration of Functions of a Single Variable. By G. H. Hardy, M.A., 

 F.R.S., Fellow and Lecturer of Trinity College, and Cayley Lecturer in 

 Mathematics in the University of Cambridge. No. 2 of the Cambridge 

 Tracts in Mathematics and Mathematical Physics. [Second Edition. 

 Pp. viii + 67.] (Cambridge : University Press, 1916. Price is. 6d. net.) 



In the first edition (1905) of this useful tract the subject of indefinite integration 

 of what the author called and calls " elementary functions " was given in a very 

 thorough and systematic form. An " elementary function " is a member of the 

 class of functions which comprises rational functions, algebraic functions — explicit 

 or implicit, the exponential function, the logarithmic function, and all functions 

 which can be defined by means of any finite combination of the symbols proper to 

 the preceding four classes of functions. The integration of all such functions was 

 treated with particular reference to the work of Abel, Liouville, and Tschebyschef ; 

 and an admirable Bibliography of the papers by these and other mathematicians 

 was given. 



The new edition differs from the first only in one very important point. Mr. 

 J. E. Littlewood discovered that a proof of Abel's given in the first edition was 

 invalid ; a new proof due to Mr. H. T. J. Norton is given in this edition, and a 

 few sections on pp. 36-41 (cf. pp. 66-67) are consequently completely rewritten. 

 There are many other minor alterations in this edition, and the Bibliography has 

 been expanded. 



This is probably the most useful Tract in the whole series for one who means 

 to study mathematics systematically. 



Philip E. B. Jourdain. 



The Elements of Non-Euclidean Plane Geometry and Trigonometry. By 

 H. S. Carslaw, Sc.D., D.Sc, Professor of Mathematics in the University 

 of Sydney. [Pp. xii + 179, with 116 figures.] (London: Longmans 

 Green & Co., 1916. Price $s, net.) 



This little book will be welcomed by all who read it ; and those especially who 

 have to spend much time in teaching elementary geometry will be refreshed to 



