444 HISTORY OF THE THEORY OF NUMBERS. [CHAP. XV

Brahmegupta2 (born 598 A.D.) made ax+I and bx +1 both squares, viz., of (3a+&)/(a-5) and (a+36)/(a-b), by taking x = 8(a+fyf(a-ty2. He made (§§ 80-81, p. 369) x+y, x—y, xy+1 all squares by taking

whence

2a2

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

To make ax+b a square (§§ 86-87, pp. 371-2), put it equal to an arbitrarily assumed square and solve the equation for x.

Bhascara3 (born 1114 A.D ) made 3y+l and 5y+l squares by equating the first to (3n+l)2, whence 52/+l = 15n2+10n+l= D for n = 2 or 18.

Alkarkhi4 (beginning of eleventh century) solved z+10 = ?/2, z+15=z2

G. Gosselin5 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 Bombelli6 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 384/s, SS'/s, 1449/ioo- He found (p. 458) a number which added to 4 and to 6 makes two squares.

F. Vieta7 generalized the method of Diophantus III, 10 [11]. If the numbers are x} y, z, let

Then

x = 2AG+G2-2AD-D*, z

*-2AB-B2-2AD~D2= D,

say F~, by choice of a rational A.

2 Brahme-sphut'a-sidd'hanta, Ch. 18 (Algebra), §§78-79. Algebra, with arith. and mensuration, from the Sanscrit of Brahmegupta and Bhascara, transl. by Colebrooke, 1817, pp. 368-9.

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

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

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

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

7Zetetica, 1591, V, 4[5], Francisci Vietae opera mathematica, ed. Francisci a Schooten, Lugd. Bat., 1646, p. 77.