552 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



He would be content if Brouncker would divide 8+1 into two other rational 

 cubes. 



Without indicating his method, Frenicle 43 gave the solutions 



9 3 +10 3 = l 3 4-12 3 , 9 3 +15 3 = 2 3 +16 3 , 15 3 +33 3 = 2 3 +34 3 , 

 16 3 +33 3 = 9 3 +34 3 , 19 3 +24 3 = 10 3 +27 3 . 



J. Wallis 44 gave 22 additional solutions 



27 3 +30 3 = 3 3 +36 3 , (4i) 3 +(7^) 3 = l 3 +8 3 , . 



" If these do not suffice, I will furnish as many as he wishes; and so easily 

 that in an hour I would promise a hundred . . . ." Letter XXVI contains 

 Frenicle's reply; he points out that all of Wallis' solutions were obtained 

 from the known solutions by simple multiplication or division. " You 

 should therefore not be astonished that he agrees so readily to furnish a 

 hundred such combinations in an hour; what is easier than to multiply or 

 divide small numbers? Indeed, it would be still easier to indicate the 

 divisions, not making the reductions, unless he wished to disguise more 

 his artificial solutions." Frenicle added that it would have been easy to 

 give essentially new solutions and then cited 13 such (Oeuvres de Fermat, 

 III, 535). Wallis (p. 538, letter XXVIII) claimed that Frenicle had 

 been guilty of the same fault. 



Wallis (p. 599, letter XLIV, June 30, 1658) was not more fortunate 45 

 in regard to Fermat 's problem to express 9 as the sum of two positive cubes; 

 he expressed 9 as the difference of the cubes of 20/7 and 17/7, and said that 

 the method to employ to express 9 as the sum of two cubes would be to 

 find in a table of cubes two whose sum is 9 times a cube ! Vieta and Bachet 

 had found no difficulty in expressing jB 3 +Z) 3 as a difference of two cubes, 

 but had not attacked the more difficult problem x 3 +y 2 = B 3 -\-D 3 . 



J. Prestet 46 treated the problem to find two cubes whose sum equals the 

 difference of two given cubes (even when the smaller exceeds one-half the 

 greater), using first (3) and then (1). To find two cubes whose difference 

 is the sum B 3 -\-D 3 of two given cubes, solve (2), then z 3 -\-vP = x 3 y 3 , and 

 then t 3 f 3 = z 3 +v 3 . To find two cubes whose difference is B 3 D 3 , solve 

 (1) and then z 3 v 3 = x 3 +y 3 . 



L. Euler 47 noted that there exist integral solutions of 



(4) A 3 +B 3 +C 3 = D 3 . 



Euler 48 derived Vieta's formula (2) and noted that it does not give all 

 the solutions. For J5 = 4, D = 3, we have 37?/ = 465, 37z = 472, whereas 



43 Commercium Epistolicum de Wallis, letter X, Brouncker to Wallie, Oct. 13, 1657; French 



transl. in Oeuvres de Fermat, 111, 419-420. 



44 Commercium, letter XVI, Wallis to Digby, Nov., 1657. Oeuvres de Fermat, III, 436. 

 48 Cf. Frenicle, letter to Digby, Oeuvres de Fermat, III, 605, 609. 



48 Nouveaux elemens des math., Paris, 2, 1689, 260-1. 

 47 Corresp. Math. Phys. (ed., Fuss), 1, 1843, 618, Aug. 4, 1753. 



" Novi Comm. Acad. Petrop., 6, 1756-7, 155; Comm. Arith., I, 193; Op. Om., (1), II, 428. 

 Reproduced without reference by E. Waring, Meditationes Algebr., ed. 3, 1782, 325. 



