PREFACE. xiii 



where / is any quadratic form in four variables, with simple criteria in the 

 case of a 2 +fa/ 2 -f c2 2 -|-dw 2 = 0; and noted that, when there is a fifth term 

 ev 2 , the equation is solvable in integers not all zero if the coefficients are odd 

 and not all of the same sign. Minkowski (p. 433) proved the generalization 

 that zero can be represented rationally by every indefinite quadratic form 

 in five or more variables, and gave invariantive criteria for four or fewer 

 variables. 



Chapter XIV reports on many elementary papers on squares in arith- 

 metical or geometrical progression. While there is a simple, general, 

 formula for three squares in arithmetical progression, known by Vieta, 

 Fermat, and Frenicle, there do not exist four distinct squares in arithmetical 

 progression. 



Chapter XV opens with a collection of the problems from Diophantus, 

 in which it is a question of finding values of the unknowns for which several 

 linear functions of them become equal to squares. Such problems were 

 treated by Brahmegupta in the seventh century, by Vieta in 1591, and by 

 Bachet, Fermat, Prestet, Ozanam, and others, in the seventeenth century. 

 One of the problems studied most frequently is that of finding three numbers 

 such that the sum and difference of any two of them are squares; it was 

 treated by Petrus in 1674, Leibniz in 1676, Rolle in 1682, Landen in 1775, 

 by Euler in his Algebra and elsewhere, as well as by various later writers 

 (all cited in note 28, p. 448). 



The story of congruent numbers, given in Chapter XVI, is a long one, 

 beginning with Diophantus. If x and k are rational numbers such that 

 x z +k and x 2 k are both rational squares, k is called a congruent number. 

 Diophantus knew that x 2 +y 2 = z* implies z 2 2xy=(xy} 2 , so that 2xy is a 

 congruent number. This topic was the chief subject of two Arabic manu- 

 scripts of the tenth century. Leonardo Pisano, in his Liber Quadratorum 

 of 1225, treated the subject at length and with skill, making repeated use 

 of the fact that any integral square is a sum of consecutive odd numbers 

 beginning with unity. In particular, he stated, but did not completely 

 prove, that no congruent number is a square, which implies that the area 

 of a rational right triangle is never a square and that the difference of two 

 biquadrates is not a square, results of special importance historically. 

 Although part of Leonardo's work was incorporated in the arithmetics of 

 Luca Paciuolo, Ghaligai, Feliciano, and Tartaglia, the original seemed to 

 be lost and Cossali made a laborious, but unsuccessful, attempt to recon- 

 struct it. The original was found and published by Prince Boncompagni 

 in 1854 and in the Scritti di Leonardo Pisano, II, 1862. The most important 

 later papers on congruent numbers are those by Euler, Genocchi, Woepcke, 

 Collins, and Lucas. 



The related problem of concordant forms is to make x^+my 2 and 

 x 2 -\-ny z both squares and was studied by the same writers, especially by 

 Euler in several of his memoirs. The remaining problems of this chapter 



