444 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xv 



Brahmegupta 2 (born 598 A.D.) made ax-\-\ and bx-\- 1 both squares, 

 viz., of (3a+6)/(o-6) and (a+36)/(a-6), by taking x = 8(a+6)/(a-6) 2 . 

 He made ( 80-81, p. 369) x+y, x y, xy+l all squares by taking 



whence 



/2a 2 V 



He made ( 82-85, pp. 370-1) x+a and x+b squares by taking 



*-(! 



whence 



To make ax+6 a square ( 86-87, pp. 371-2), put it equal to an arbi- 

 trarily assumed square and solve the equation for x. 



Bhascara 3 (born 1114 A.D.) made 3y+l and 5y+l squares by equating 

 the first to (3n+l) 2 , whence 5y+l = 15n?+Wn+l = D for n = 2 or 18. 



Alkarkhi 4 (beginning of eleventh century) solved x + W = y 2 , rr+15 = 2 2 

 by setting z = y+2/3 in y~+5 = z 2 ', also, #+3 = y 2 , x+5 =z 2 by taking z-\-y = 4, 



G. Gosselin 5 found three numbers (13/9, 133/9, 253/9) in A. P. which 

 become squares when increased by 4; three numbers (1/9, 15/9, 48/9) 

 whose sum is a square, the first a square, and the sum of the first and either 

 of the other two is a square; four numbers (25, 16, 12, 11) whose sum is a 

 square, while the excess of the first over the second, second over third, 

 third over fourth are squares. 



Rafael Bombelli 6 required three numbers, the sum of any two of which 

 increased by 6 and the sum of all three increased by 6 are squares. He 

 gave 38 4 /5, 55Vs, 14 49 /ioo- He found (p. 458) a number which added to 

 4 and to 6 makes two squares. 



F. Vieta 7 generalized the method of Diophantus III, 10 [11]. If the 

 numbers are x, y, z, let 



x+y = (A+B)*-a, y+z=(A+DY-a, x+y+z=(A+G)--a. 

 Then 



x = 2AG+G 2 -2AD-D*, z = 2AG+G 2 -2AB-B*, 



x+z+a = 4AG+2G*-2AB-B 2 -2AD-D*= D, 

 say F 2 , by choice of a rational A. 



2 Brahme-sphut'a-sidd'hanta, Ch. 18 (Algebra), 78-79. Algebra, with arith. and mensura- 



tion, from the Sanscrit of Brahmegupta and Bhdscara, transl. by Colebrooke, 1817, 

 pp. 368-9. 



3 Vija-gan'ita, 197; Colebrooke, 2 p. 259. 



4 Extrait du Fakhri, French transl. by F. Woepcke, Paris, 1853, 86, 101. 

 6 De Arte magna, seu de occulta parte numerorum, Paris, 1577, 74-5. 



6 L'algebra opera, Bologna, 1579, 496. 



7 Zetetica, 1591, V, 4[5], Francisci Vietae opera mathematica, ed. Francisci a Schooten, 



Lugd. Bat., 1646, p. 77. 



