352 Notices respecting New Books. 



"Whittaker's own mathematical institute in the University of 

 Edinburgh is explicitly recognized as adding to the sum of mathe- 

 matical knowledge. Through his inspiration the ' Proceedings of 

 the Edinburgh Mathematical Society,' which for many years were 

 mainly devoted to geometrical developments, now contain research 

 papers on transcendental functions, certain results of which find 

 their place in this valuable treatise on Modern Analysis. The 

 centre of gravity of mathematical development in the United 

 Kingdom has appreciably shifted since Professor Whittaker 

 flitted from Cambridge to Edinburgh by way of Dublin. 



It should be said in conclusion that the book is beautifully 

 printed, and that its usefulness is greatly enhanced by a very 

 full general subject index as well as a complete list of authors 

 quoted. 



Diophantine Analysis. By Robert D. Carmichael. New York : 



John Eiiey & Sons. London : Chapman & Hall. 

 This is No. 16 of the mathematical monographs edited by 

 Mansfield Merriman and Robert S. Woodward. The most 

 familiar Diophantine problem is that connected with the property 

 of the right-angled or Pythagorean triangle, and may be stated 

 in these words : to find two integers the sum of whose squares is a 

 square. Theorems of this nature have had, and no doubt will 

 always have, a great fascination for all interested in the theory of 

 numbers ; and the fascination is probably the greater on account 

 of the lack of a general method of investigation. Eermat is 

 universally recognized as the great master o£ Diophantine analysis. 

 He left behind him many theorems without proof ; and some of 

 these still await a demonstration. Eermat's Last Theorem that, 

 if n is an integer greater than 2, there do not exist integers x, y, z, 

 all different from zero, such that x n -\-y n =z n , is one of those for 

 which no general solution has yet been found. The Academy of 

 Sciences of Grottingen holds a sum of a hundred thousand marks 

 to be presented as a prize to" the person who first gives a rigorous 

 proof of the theorem. Professor Carmichael devotes the fifth 

 chapter of his book to a discussion of this equation, the earlier 

 chapters being taken up w T ith the consideration of equations of the 

 second, third, and fourth degrees. The sixth and last chapter 

 deals with what is called the method of functional equations, a 

 method of considerable use in the investigation of Diophantine 

 problems. In particular, it is applied to the solution of another of 

 Eermat's problems, namely : To find three squares such that the 

 product of any two of them, added to the sum of those two, gives 

 a square. The book is clearly written and fully carries out the 

 aim of the author, which is " to systematize, as far as possible, a 

 large number of isolated investigations and to organize the frag- 

 mentary results into a connected body of doctrine. The principal 

 single organizing idea here used and not previously developed 

 systematically in the literature is that connected with the notion 

 of a multiplicative domain." 



