344 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xn 



The least t is 990, the side of A being then 27180. He considered also 

 higher values of t, but gave no final answer to (l)-(9). 



G. H. F. Nesselmann 11 argued that the final part of the epigram leading 

 to conditions (8) and (9) was a later addition, partly since he believed that 

 triangular numbers were not employed in Archimedes' time (a view already 

 expressed by G. S. Kliigel 12 ). 



O. Terquem 13 stated that the tenth condition added by Hermann is 

 incompatible with the earlier conditions. 



A. J. H. Vincent 14 regarded as spurious the conditions relating to the 

 cows. By the first three conditions, we have (10). Then Y + Z = 2471m 

 is to be a A and this is the case if m = 99 122314, the side of the A being 



244628. Then 4^W + X is approximately 861182, which is very nearly 

 the area of Sicily in square stades, in accord with Vincent's interpretation 

 of the condition to replace (8). 



C. F. Meyer 15 duplicated the paper by Lessing and discussion by Leiste, 

 adding merely that, in attempting to make em 2 + 1 a square by the con- 

 venient method of Kawsler, he had carried the development of Va into a 

 continued fraction to the 240th quotient without finding the period. 



A. Amthor 16 showed that Wurm's problem (l)-(7), (9) is satisfied by 

 taking u = v/20, v = 117423 in Leiste's values of W, - , z, since then 



Y + Z = 1643921-1643922/2, W + X = 1485583-1409076. 



For the main problem (l)-(9), he satisfied (8) by taking v = /-4657n 2 , 

 / = 3-11-29 = 957, as in Leiste. Then in (9), viz., Y + Z = q(q +.l)/2, 

 set t = 2q + 1, u = 2 -4657?i. We obtain the Pell equation 



2 - Du* = 1, D = 2-7-/-35S = 4729494. 



He found that the continued fraction for ^D has a period of 91 terms 

 and obtained as the least solutions 



T = 109 931 986 732 829 734 979 866 232 821 433 543 901 088 049, 

 U = 50 549 485 234 315 033 074 477 819 735 540 408 986 340. 



It remains to derive the least solutions t, u in which u is divisible by 

 2-4657, so that n shall be integral. By proving and applying general 

 lemmas concerning t k + u k VZ> = (T + U VZJ) fc , he found that, for k = 2329, 

 tk, Uk is the desired pair. He verified that W has 206545 digits. 



B. Krumbiegel 17 made a historical and philological discussion of the 

 problem and concluded that, while the epigram itself is probably subsequent 

 to Archimedes, the problem itself is due to him. This accords with the 



11 Die Algebra der Griechen, Berlin, 1842, 488. On p. 485, his g = 57- should be 54- -. 



12 Math. Worterbuch, 1, 1803, 184. Cf. M. Cantor, Geschichte Math., ed. 2, 1, 297; ed. 3, I, 



312. 



13 Nouv. Ann. Math. 14, 1855, Bull. Bibl., 113-124, 130-1. He at first attributed incorrectly 



Hermann's paper to F. E. Theime. 



"Nouv. Ann. Math., 14, 1855, Bull. Bibl., 165-173; 15, 1856, Bull. Bibl., 39-42 (restored 

 Greek text and French transl.). 



15 Ein diophantisches Problem, Progr., Potsdam, 1867, 14 pp. 



16 Zeitschrift Math. Phys., 25, 1880, Hist.-Lit. Abt., 153-171. 



17 Zeitschrift Math. Phys., 25, 1880, Hist.-Lit. Abt., 121-136. 



