CHAP. II] SYSTEM OF LINEAR EQUATIONS. 79 



Luca Paciuolo 183 treated the solution of 



p + c + 7r + a = 100, \p + \c + TT + 3a = 100, 



giving the single solution p = 8, c = 51, TT = 22, a = 19. Many solutions 

 were found by P. A. Cataldi. 184 



Christoff Rudolff 185 stated the following problem. To find the number 

 of men, women and maidens in a company of 20 persons if together they 

 pay 20 pfennige, each man paying 3, each woman 2 and each maiden \ . 

 The answer is given to be 1 man, 5 women, 14 maidens. [The only solu- 

 tion of x + y + z = 20, 3x + 2y + \z = 20 in positive integers is x = 1, 

 y = 5, z = 14.] The solution is said to be found by the rule called Cecis 

 or Virginum. 



C. G. Bachet de Meziriac 186 solved in integers the system of equations 



x + y + z = 41, 4x + 3y + \z = 40. 



Multiplying the second by 3 and subtracting the first, he obtained 

 llx + Sy = 79. Since y = 9| lf:r, x must have one of the values 1, 

 , 7. By the value of Sz in terms of x, I + 3x must be divisible by 8. 

 Hence x = 5, so that y = 3, z = 33. He treated RudolffV 85 and a similar 

 system and found 81 sets of positive integral solutions of 



x + y + z + w = 100, 3z + y + \z + \w = 100. 



J. W. Lauremberg 187 described and illustrated by examples the rule called 

 Cecis [Coeci] or Virginum 188 for solving indeterminate linear equations, 

 referring to the Arabs [although known to the Indians]. 



Rene" Francois de Sluse 189 (1622-1685) treated the problem to divide a 

 given number 6 into three parts the sum of whose products by given 

 numbers z, g, n shall be p. Call the first and second parts a and e. Then 



. p rib + ne ge 



za + ge + n(b a e) p. a = - - . 



z n 



Take p = b = 20, z = 4, g = J, n = J. Then a = (60 - e)/15. 



Johann Pratorius 190 solved the following problem: Anna took to market 

 10 eggs, Barbara 30, Christina 50. Each sold a part of her eggs at the 

 same price per egg and later sold the remainder at another price. Each 



183 Summa de Arithmetica, 1523, fol. 105; [Suma . . ., Venice, 1494]; same solution by 

 N. Tartaglia, General Trattato di Nvmeri . . . , I, 1556. 



184 Regola della Quantita o Cosa di Casa, Bologna, 1618, 16-28. 



185 Kiinstliche Rechnung, 1526; Niirnberg, 1534, f. nvij a and b; Niirnberg, 1553 and Vienna, 



1557, f. Rvii a and b. 



186 Diophantus Alex. Arith., 1621, 261-6; comment on Dioph., IV, 41. 



187 Arithmetica, Sorae, Denmark, 1643, 132-3. Cf. H. G. Zeuthen, 1'interme'diaire des math., 



3, 1896, 152-3. 



188 According to O. Terquem, Nouv. Ann. Math., 18, 1859, Bull. Bibl., 1-2, the term problem 



of the virgins arose from the 45 arithmetical Greek epigrams, Bachet, 186 pp. 349-370, 

 and J. C. Heilbronner, Historia Math. Universae, 1742, 845. Cf. Sylvester 54 of Ch. III. 



189 MS. No. 10248 du fonds latin, Bibliotheque Nationale de Paris, f. 194, "De problematibus 



arith. indefinites," Prob. 2. 



190 Abentheuerlicher Gliickstopf, 1669, 440. Cf . Kastner. 197 



