160 SCIENCE PROGRESS 



them as means for developing a given function in certain forms. The rest of the 

 book is devoted to the treatment of such functions as the Gamma function and 

 the elliptic functions of Weierstrass and Jacobi, and a slightly new point of view in 

 British and American treatises on the theory of functions is introduced by a treat- 

 ment of linear differential equations of the second order from a modern point of 

 view, including equations of Fuchsian type and the solution of differential equations 

 by definite integrals. The book has certain advantages as a text-book. 



Philip E. B. Jourdain. 



Sur les Problemes celebres de la Geometrie elementaire 11011 resolubles avec 

 la regie et le compas. By F. Gomes Teixeira. [Pp.132.] (Coi'mbre : 

 Imprimerie de PUniversite, 191 5.) 



This exceedingly useful compilation is a very complete summary of the various 

 solutions that have been given of the three celebrated problems : the duplication 

 of the cube, the trisection or multisection of the angle, and the quadrature of the 

 circle. The contributions of each mathematician are dealt with in separate 

 sections and in an analytical manner, so that modern methods and notations are 

 used throughout. Such a method certainly conveys a clear idea of the point at 

 issue to modern students, and perhaps this indicates the object of the book. The 

 problem and solutions of the duplication of the cube are traced from Hippocrates 

 to Montucci (1869), those of the division of the angle are traced from Hippias to 

 Kempe, and those of the quadrature of the circle from the approximations given in 

 the Rhind papyrus to the expressions of Mansion (1910). A fourth chapter is on 

 the impossibility of the resolution of the above problems by the ruler and compass, 

 and is a very useful analytical treatment from a modern point of view. 



Philip E. B. Jourdain. 



Elementi della Teoria delle Equazioni Integrali Lineari. By Giulio 

 Vivanti, Ordinary Professor at the Royal University of Pavia. [Pp. xvi + 

 398.] (Milano : Ulrico Hoepli, 1916. Price 4.50 lire.) 



This is a triple volume in the well-known series of " Manuali Hoepli," and is as 

 well printed and pleasant to handle as all the volumes of this series. It might 

 well be questioned whether the pages are not rather too small considering the size 

 of type and thickness of the book. It is not that the type is too large : it is of a 

 very convenient size ; but, especially in mathematical works, it helps our faculty 

 of understanding to be able to take in a good deal with one glance of the eye, and 

 not to have to turn over pages in the middle of an argument. As for the matter 

 of the book, it forms a most valuable and complete summary of our present know- 

 ledge of the theory of integral equations. There is at the end (pp. 367-398) a 

 bibliographical list of books and memoirs on the subject which seems to be very 

 thoroughly done ; a great deal of space is saved by omitting the full titles of the 

 works listed, and giving instead a short indication of what the work is about. 

 After some preliminary considerations on analytic functions, linear differential 

 equations, and properties of determinants, the second chapter deals with the 

 equations of Volterra and Fredholm. The third chapter is on relations between 

 integral equations and self-adjoint linear differential equations of the second order, 

 and the fourth chapter is on some applications to mathematical physics— theory 

 of potential, vibrations of a cord and membrane, and the movement of heat. 



Philip E. B. Jourdain. 



