xiv PEEFACE.

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 x2+t/2, x2+z*, yz+z2 are all perfect squares. Solutions involving arbitrary parameters, but obtained under special assumptions, 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—1 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 functions simultaneously equal to squares. He selected a suitable auxiliary function / such that solutions of /=0 can be readily found. For any set of solutions, P2— / 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. Fermat 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 xz+2=fi implies that each factor z±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+q^^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