34 o SCIENCE PROGRESS 



and functions of Legendre and Jacobi ; and Chapter XII to the Theta functions. 

 To return for a moment to Chapter VII, the definition of residues (p. 256) is the 

 now usual one as the coefficient of the first term in the expansion in negative 

 powers. This definition is rather an anachronism, for Cauchy defined residues 

 and extensively used them long before Laurent's theorem was discovered. It 

 would be a good thing if more use were made of the original definition and the 

 many applications of the calculus of residues. In quite modern times the import- 

 ance of these applications has been shown in the masterly work of Lindelof. 



The last three chapters of the book are both interesting and important, 

 especially from the point of view of the students for whom the book was 

 originally written. In these chapters Prof. Pierpont, having in mind the needs 

 of students of applied mathematics, dwells at some length on the theory of 

 linear differential equations, especially as regards the functions of Legendre, 

 Laplace, Bessel, and Lame. It is new and refreshing to find, in a text-book of 

 function-theory, a good discussion of the modern point of view in the theory 

 of differential equations. It is quite true that Cauchy ought to be regarded as 

 the founder of the modern theory of these equations, but the remarks on p. 2 are 

 misleading in so far as they mention Cauchy's work on the theory of functions of 

 a complex variable only in connection with his theory of differential equations. 

 Cauchy's theory of functions of a complex variable first grew up as a result of his 

 work on the evaluation of definite integrals. 



The only mistake which the reviewer has hitherto detected is a misspelling of 



Rodrigues's name (p. 502). The volume is well written and well printed, and is 



uniform with Prof. Pierpont's two volumes on the theory of functions of real 



variables. 



Philip E. B. Jourdain. 



Dialogues concerning Two New Sciences. By Galileo Galilei. Translated 

 from the Italian and Latin into English by Henry Crew and Alfonso de 

 Salvio of Northwestern University. With an Introduction by Antonio 

 Favaro, of the University of Padua. [Pp. xxvi + 300.] (New York : The 

 Macmillan Company, 1914. Price Ss. bd. net.) 



There have been two previous English translations of Galileo's Duscorsi e 

 Dhnostrazioni matematiche of 1638 : one by Thomas Salusbury published in 1665, 

 and one by Thomas Weston published in 1730. Both are now very scarce, 

 especially that of Salusbury ; and the present handsome translation will be 

 welcome to all English-speaking students. As a frontispiece there is a very good 

 reproduction of Subterman's painting of Galileo, and the translation is made with 

 great care from Favaro's national edition of the works of Galileo. In the national 

 edition, the Leyden text of 1638 has been followed faithfully but not slavishly 

 (p. xii), and the manuscript corrections and additions of Galileo himself have been 

 used in this translation. However, all the other comments and annotations in the 

 national edition have been omitted in this translation save here and there a foot- 

 note intended to economise the reader's time. To each of these footnotes has 

 been attached the signature " [Trans.]" in order to preserve the original as nearly 

 intact as possible (p. vi). The numbers of pages inserted in the text refer to the 

 national edition. 



The " two new sciences " created by Galileo are the theory of the strength of 

 materials described in the first two days of the dialogue, and the theory of the 

 uniform acceleration shown by falling bodies described in the third and fourth 

 days. The dialogue is held between Salviati, who represents the opinions of 



