PREFACE. xv 



expressed the difference of two given rational cubes as a sum of two positive 

 rational cubes was given in his Porisms, a work which has not been pre- 

 served. The formula (p. 550) which Vieta used in 1591 for this purpose 

 is valid only when the greater of the given cubes exceeds the double of the 

 smaller. While also Bachet could solve only this case, Girard and Fermat 

 showed how, by employing Vieta's three formulas in turn, to solve the 

 remaining case as well as the problem to express a sum of two given rational 

 cubes as another such sum. The last problem had been proposed by Fermat 

 to the English mathematicians Brouncker and Wallis, who gave merely solu- 

 tions derived from known solutions by multiplication by a constant. The 

 general solution in integers of this problem was first given by Euler in 

 1756-57. His solution was expressed in a simpler form by Binet in 1841 

 and deduced elegantly by Hermite in 1872 by means of the ruled lines on the 

 corresponding cubic surface (a method extended to a certain equation of 

 degree n by Brunei, p. 556). Report is made on pp. 560-1 on Japanese 

 writings during 1826-45 on this subject. The related problem of rinding 

 three equal sums of two cubes arose in the question of finding four integers 

 the sum of any two of which is a cube. 



There are many minor papers of recent decades which give relations 

 between five or more cubes, or express a sum of three cubes as a square. 

 The problem of making a binary cubic form equal to a cube was treated by 

 obvious elementary methods by Fermat and Euler, and recently by bi- 

 rational transformation by von Sz. Nagy, and by covariants by Haentzschel. 

 To make a binary cubic form equal to a square, Fermat and Euler equated 

 it to the square of a linear or quadratic function, and Lagrange used the 

 norm of an algebraic number (p. 570), while Mordell in 1913 employed the 

 theory of invariants. 



Since every rational number is a sum of three rational cubes (p. 726), 

 it is an interesting question to determine the rational numbers which are 

 sums of two rational cubes, or, if we prefer, the integers A for which 

 x z +y z = Az* is solvable in integers. Reports on fifty papers on this subject 

 are given on pp. 572-8. Euler proved that the problem is impossible if 

 A = 1 and A = 4, and that x= y if A = 2. Legendre erred in his statement 

 that it is impossible if A = 6. In 1856 Sylvester stated that it is impossible 

 if A = p, 2p, 4p 2 , 4g, <? 2 , 2q 2 , where p and q are primes of the respective forms 

 181 + 5 and 18Z + 11. In 1870 Pepin proved these and similar results. 

 Using also analogous facts proved by Sylvester in 1879, we can state whether 

 or not any proposed number, not exceeding 100, is a sum of two rational 

 cubes. 



There are 42 papers (pp. 582-8) on the problems of rinding numbers 

 in arithmetical progression the sum of whose cubes is a cube or a square. 



If F(x, y, z) = is a homogeneous cubic equation with rational coefficients 

 and if P is a rational point (i.e., having rational coordinates) on the curve 

 F = 0, the tangent at P cuts the curve in a new rational point, called the 

 tangential to P. Similarly, the secant through two rational points on the 

 curve cuts it in a third rational point. Curiously enough, the analytic 



