CHAP, xxi] CUBE OF SUM PLUS ANY ONE A CUBE. 609 



bers are the ratios of 2837107, 29G6301, 2981888 to 150G9223. Or we may 

 take a 1 = 103/(9-23), a 2 = 12/(9-23), a a = l/9, whence s = 9-23/1053, giving 

 xi = 7777016/43243551, etc. Or, ai = 41/64, a 2 = 39/64, a 3 = 3/63, 



s = 8- 16/185, x l = 1545784/6331625. 



Or, ! = 67/88, a 2 = 87/88, a 3 = 22/88, s = 176/221, x, = 3045672/10793861. 

 For four numbers, take on, -, or 4 to be the ratios of 4684, 4836, 3485, 

 3315 to 1360. Then 2(64-a*)= 2 , = 16027/1360; the desired numbers 

 are x i =(Wal')s 3 , s = '2Xi=l/t. His 411 solution of V, 18 is the same as in 

 Diophantus. 



The answer of van Schooten 408 was given without details in the Ladies' 

 Diary, 1717, Question 51. J. Hampson 412 gave without details the smaller 

 answer to Diophantus, V, 19: 13851/D, 19467/Z), 18954/D, where D = 85 184. 

 He 413 also stated two answers to Diophantus V, 18: ratios of 23625, 1538 

 and 18577 to 157464; ratios of 18954, 4184 and 271 to 132651. 



J. Landen 414 took zy, zx, zv as the numbers in Diophantus V, 20, and 

 p 2 z as their sum, and zs, zr, zq as the roots of the cubes, finding the answer 

 341/D, 854/D, 250/D, where D = 4913; no details were given. 



The " Repository solution " 415 is a repetition of that by Diophantus as 

 completed by Bachet, 404 leading to 162707336/d, 134953209/d, 68574961/d, 

 where d = 549353259. It is also noted that 37 is the sum of the cubes of 

 18/7 and 19/7, whence s = 2/3 and a new answer is 6S256/&, 67229/fc, 31213/&, 

 where & = 250047. 



For Diophantus V, 18, J. Bennett 416 took nx as one number and s as 



the sum of the three. Let s 3 +no: = (s+rc) 3 , whence x = ^4n 3s 2 3s/2. 

 Taking n/s 2 = 21, 31, 57, we get nz = 63s 3 , 124s 3 , 342s 3 , which will be the 

 desired numbers if their sum is s, i. e., if s = l/23. J. Ryley 417 used the 

 numbers x, y, axy, then a 3 +z = a 3 s 3 , a 3 +y = a?n 3 give x, y. Let 

 a 3 +a x 2/ = m 3 a 3 . Then a 2 /=l, /=m 3 +n 3 +s 3 3. Take n = 2 r, 

 s = 1-f-r, /= (2vm 3r) 2 , which gives r in terms of m, v. 



T. Leybourn 418 noted that Diophantus V, 18 is satisfied by taking 

 (a 3 w 6 )y 3 , (6 3 w 6 )y 3 , (c 3 u^v* as the numbers, if u 2 v is their sum. The 

 latter requires F = a 3 +& 3 +c 3 3u 6 = D. Take a = p-f<?, b = r p, c = s. 

 Then F = 3(g+r)p 2 +3(^ 2 -r 2 )p+g 3 +r 3 +s 3 -3w 6 . Take 3(?+r)=n 2 ; then 

 F=(mnp} 2 determines p rationally. By trial, he found that 7^=(23) 2 if 

 a = 4, 6 = 5, c = 7, u = l. For Diophantus V, 20, he 419 took (a?+u*)v 3 , -, 

 as the numbers and u-v as their sum. Then 



411 The Elements of Algebra, London, Book III, 1674, 101. 



412 Ladies' Diary, 1747, 27, Quest. 275. 



413 Ibid., 1748, 27, Quest. 288. 



414 Ladies' Diary, 1749, 26, Quest. 304; C. Button's Diarian Miscellany, 2, 1775, 270; Ley- 



bourn's Math. Quest, proposed in Ladies' Diary, 2, 1817, 7-9. 



416 The Diarian Repository; or, Math. Register . . . Collection of Math. Quest, from 



Ladies' Diary, by a Society of Mathematicians, London, 1774, 81-2. 

 418 Ladies' Diary, 1805, 43-4, Quest. 1132; Leybourn's M. Quest. L. D., 4, 1817, 46-7. 



417 The Diary Companion, Supplement to Ladies' Diary, London, 1805, 46-7. 



418 Leybourn's Math. Quest, proposed in Ladies' Diary, 1, 1817, 405-7. 



419 Ibid., 2, 1817, 7-9. 

 40 



