xiv PREFACE. 



and those of Chapter XVII relate to special systems of two quadratic 

 functions or equations and do not possess sufficient general interest to 

 warrant mention here. The last remark applies also to Chapter XVIII, 

 which treats of three or more quadratic functions. 



Chapter XIX begins with the history of the problem of finding three 

 integers x, y, z such that 2 +?/ 2 , z 2 +2 2 , y 2 +z 2 are all perfect squares. Solu- 

 tions involving arbitrary parameters, but obtained under special assump- 

 tions, were found by Saunderson (who was blind from infancy) and Euler 

 in their Algebras of 1740 and 1770. The problem is equivalent to that of 

 finding a rectangular parallelepiped having rational values for the edges 

 and the diagonals of the faces. If we impose the further restriction that 

 also a diagonal of the solid shall be rational, we have a difficult problem 

 which has been recently attacked but not solved. 



The problem of finding n squares the sum of any n I of which is a square 

 was treated at length by Euler for n = 4, and for any n by Gill by use of 

 trigonometric functions. The problem of finding three squares the sum of 

 any two of which exceeds the third by a square was treated by four special 

 methods by Euler in a posthumous paper, as well as by Legendre and others. 

 The problem of making a quadratic form in x and y, one in x and z, and one 

 in y and z simultaneously equal to squares has received much attention during 

 the past hundred years. Beginning with Diophantus, there is an extensive 

 early literature on the problem of finding n numbers such that the product 

 of any two of them increased by a given number shall be a square. 



Euler developed an interesting method (p. 522) to make several func- 

 tions simultaneously equal to squares. He selected a suitable auxiliary 

 function / such that solutions of /=0 can be readily found. For any set 

 of solutions, P 2 / is evidently a square, whatever be the function P. 

 Many further problems occur in this long chapter, which closes with an 

 account of rational orthogonal substitutions. 



The nature of Chapter XX will be illustrated by means of an example 

 of considerable interest for the history of algebraic numbers. Feraiat 

 stated that he had a proof that 25 is the only integral square which if 

 increased by 2 becomes a cube. Euler, in attempting a proof in his Algebra 

 of 1770, assumed that 2 +2 = 3 implies that each factor #V^2 is the 

 cube of a number p+q V 2, where p and q are integers, although he knew 

 that a like assumption is not valid when 2 is replaced by other numbers. 

 The justification of his assumption in the first example is due to the fact 

 that for these numbers p+gV-2 factorization into primes is unique and 

 to the further fact that 1 are the only ones of these numbers which 

 divide unity. Instead of this explanation by means of algebraic numbers, 

 we may employ the theory of classes of binary quadratic forms, as was done 

 by Pepin (p. 541). 



In the 69 pages of Chapter XXI report is made on about 500 papers on 

 Diophantine equations of degree 3. The method by which Diophantus 



