

RECENT ADVANCES IN SCIENCE 363 



more or less conscious intellectual need is creditable to Des- 

 cartes, though he was mistaken in thinking that Fermat's 

 method was not general. Milhaud also (ibid. 464-9) gives 

 several examples from the correspondence of Descartes to show 

 how naturally and easily he worked with infinitesimal con- 

 siderations, to show that, in opposition to the opinions of some, 

 Descartes was by instinct accessible to all new methods and 

 ideas in mathematics. 



L. E. Dickson (Annals of Math. 191 7, 18, 161-87) gives a 



i most detailed and important paper on Fermat's theorem that 

 x» 4- y n = z n has no integral solutions different from zero 

 when n is greater than 2, and the origin and nature of the 

 theory of algebraic numbers. The first part of the paper con- 



i tains the early history of the theorem mentioned and a sum- 

 mary of the more important results concerning it which have 

 been proved without the theory of algebraic numbers ; the 

 second part enables the reader to obtain a clear insight into the 



U origin, nature and use of ideals ; and the third part is on those 



; papers which treat Fermat's theorem by means of algebraic 

 numbers. 



As a sequel to his study of the enigma of imaginary numbers 

 (cf. Science Progress, 191 7, 12, 6), Gino Loria (Scientia, 191 7, 

 22, 1-1 3) shows that geometry in the course of its evolution was 

 continually meeting a quite analogous conception which passed 

 through several stages of development strikingly like those 

 which Loria brought out in the case of the evolution of the 

 treatment of imaginary numbers. In this article most atten- 

 tion is paid to the work of Carnot, Monge, Poncelet, Chasles, 

 Steiner, and von Staudt. 



The most interesting of the instalment of letters between 

 Schlafli and Casorati published in the first number of the Boll, 

 di Bibl. e St. delle Sci. Mat. for 191 7 (19, 9-14) is an animated 

 letter in which Schlafli gives an extension of a method of his 

 to prove the existence of Abel 's integrals . Some correspondence 

 between Schlafli and Cremona from 1867 to 1887 is also given 



{ibid. 43-9)- 



Sir Thomas Muir (Quart. Journ. of Math. 191 7, 47, 344-84) 



gives a sixth list of writings on determinants in continuation 



' of his former lists published in the same Journal. The present 



list is concerned mainly with the five years following 1910, but 



gives also additional titles for all the periods specially covered 



