xvi PREFACE. 



equivalents of these facts were obtained by Cauchy in 1826 without their 

 geometrical setting. Levi in 1906-9 defined a configuration of rational 

 points on a cubic curve without double points to be the set of all rational 

 points which can be derived from one or more rational points by the opera- 

 tions of finding the tangential to a point of the set and of finding the third 

 intersection of the curve and the secant joining two points of the set. In 

 1917 A. Hurwitz called such a set of points a complete set and obtained 

 theorems on the number of rational points on the cubic curve. Mordell 

 made use of the invariants of F. 



The problem of finding n rational numbers the cube of whose sum in- 

 creased (or decreased) by any one of the numbers gives a cube was treated 

 for n = 3 by Diophantus and his commentators, by Ludolph van Ceulen in his 

 Dutch work on the circle, by van Schooten, J. Pell and others the simplest 

 answer being that by Hart (p. 611). 



Chapter XXII devotes 57 pages to reports on 400 papers on Diophantine 

 equations of degree 4. Fermat's proof of his challenge theorem that no 

 rational right triangle has an area which is a rational square is of special 

 interest, as it illustrates in detail his method of indefinite descent; his proof 

 also shows that the difference of two biquadrates is never a square. Leibniz 

 left a manuscript giving a proof. 



Fermat affirmed that the smallest rational right triangle whose hypote- 

 nuse and the sum of whose legs are squares has its sides expressed by 

 numbers of thirteen digits. The problem is equivalent to that of finding two 

 numbers (for n numbers, pp. 665-7) whose sum is a square and whose sum 

 of squares is a biquadrate, and was proposed in this form by Leibniz and 

 treated several times by Euler, and at great length by Lagrange in 1777, who 

 found it necessary to solve several equations of the form az 4 +fa/ 4 = c2 2 . 

 The extensive literature on the latter equation is reviewed on pp. 627-634; 

 some of the methods employed apply also when there occurs a term dx z y 2 

 in the equation (pp. 634-9). 



Just as in algebra no general equation of degree exceeding 4 can be 

 solved by radicals, so in Diophantine analysis nearly all the problems for 

 which solutions have been found are those which reduce finally to the 

 question of making a given binary form /of degree ^4 equal to a square or 

 higher power. Among the methods (pp. 639-644) of making a quartic 

 function f(x) of special type equal to a square are the rather obvious 

 methods of Fermat; the method of Euler of reducing/ to the form P 2 +QR, 

 where P, Q, R are quadratic functions of x, so that/= (P+Qy)* becomes an 

 equation quadratic in x and in y ; and the invariantive methods of Mordell 

 and Haentzschel. Euler 's method is similar to that employed by him in 

 the problem of the multiplication of an elliptic integral; Jacobi noted a 

 generalization by use of Abel's theorem (p. 641). 



Euler, after solving A 4 +# 4 = C 4 +D 4 by several methods, stated (p. 648) 

 that it is impossible to find three biquadrates whose sum is a biquadrate, 

 and that he believed it possible to assign four biquadrates whose sum is a 

 biquadrate. But his investigation was incomplete and led to no example. 



