78 HISTORY OF THE THEORY OF NUMBERS. [CHAP, n 



These determine x, y, z, t in terms of u. Since It = 47w, he took u = 7, 

 whence t = 47, x = 33, y = 13, z = 1. His next indeterminate problem 179 is 



Since the problem is impossible if x : and z 2 are positive, change x\ to XL 

 Now 2 = 4#i. Take x 2 = 4, whence Xi = xa = 1, x* = 4, t = 11. 



For 180 z + T/ + 2 = 30, %x + %y + 2z = 30, we have y + lOz = 120, 

 ?/ -f z < 30, ^ ^ 9. The case 2 = 10 is impossible. For z = 11, we get 

 y = 10, z = 9. The same problem with the constant term 30 replaced 

 by 29 or 15 is treated similarly. 



Finally, 181 consider the system 



x + y + z + t = 24, f + | + 2z + 3t = 24. 



5 o 



Hence 2y + 27z + 42t = 288, y + z + t < 20. Thus z is even and < 10. 

 The cases z = 6, z = 8 are impossible. Thus there are only two solutions : 



2 = 2, t = 5, y = 12, x = 5; z = 4, t = 4, ?/ = 6, x = 10. 



Regiomontanus (1436-1476) proposed in a letter (cf. de Murr, 86 p. 144) 

 the problem to solve in integers 



x + y +z = 240, 97z + 5Gy + 3z = 16047. 



J. von Speyer gave the solution 114, 87, 39 (de Murr, p. 167). 

 Estienne de la Roche 182 treated the solution in integers of 



x -j- y + z a, mx + ny + pz = b. 



His rule [applied to the case a = b = 60, m = 3, n = 2, p = |] is as follows. 

 Let p be the least of m, n, p. From the second equation subtract the 

 product of the first by p; we get 



(m - p)x + (n - p)y = b - ap [fz + % = 30]. 



To avoid fractions, multiply by 2. Thus 5x + 3y = 60. Although 

 x = 60/5 gives an integral solution, the corresponding y is zero and is 

 excluded. The next smaller values 11 and 10 for x lead to fractions for y, 

 while x = 9 gives y = 5 [whence z = 46]. For x = 1, 2, , the least x 

 yielding an integer for y is x = 3, whence y = 15, z = 42. The problem 

 may be impossible, as shown by the case a = b = 20, m = 5, n = 2, p = %, 

 whence 9z + 3y = 20. 



179 Scritti, II, 238-9 (De quatuorhominibusetbursa). Genocchi, 178 172-4. Three misprints 



in the account by Terquem. 



180 Scritti, II, 247-8 (De auibus emendis). Genocchi, 218-22. For analogous problems, 



see Liber Abbaci, Scritti, 1, 1857, 165-6. 



181 Scritti, II, 249 (Item passeres). Genocchi, 222-4. 



182 Larismetique & Geometric, Lyon, 1520, fol. 28; 1538. Cf. L. Rodet, Bull. Math. Soc. 



France, 7, 1879, 171 [162]. 



