162

HISTORY OF ALGEBRA.

TRACT 33.

or £x - v'-lx — 2. But, instead of this way, the Hinduassumes 2x' 1 for the number of bees, which at once givesthis simpler form 2x' l — x 2 — x — 2, or ~x^ — x — 2, thesolution of which is much easier.

On Indeterminate Problems of the Second Degree , Diophan-tus, and the Beej Gunnit, or rather Bija Ganita.

The 16th question of the 6th book of Diophantus, is asfollows:—

“ Having two numbers given, if one of these drawn intoa certain square, and the other subtracted from the product,make a square; it is required to find another square, greaterthan the former, which shall do the same thing. For in-stance, given the two numbers 3 and 11, and a certainsquare 25, which drawn into 3, and 11 taken from the pro-duct, leaves the square of 8 ; to find another square greaterthan 25 having the same property. Put its side In+ 5,then its square is la + 10n + 25 ; triple of this diminishedby 11 leaves 3 q + 30n + 64 equal a square; let its side be2n — 8, which gives n = 62; then 62 + 5 = 67 is the side,and 4489 the square sought.”

In the Bija Ganita this problem is solved very generallyand scientifically, by the assistance of another, which wasunknown in Europe till the middle of the 17th century;and first applied to questions of this nature by Euler, iuthe middle of the 18th century.—With the affirmative sign,the Bija Ganita rule for finding new values of ax' l + b—y %is this: Suppose ag 1 + b = h? a particular case: find m andn such that an 1 + 1 = m z ; then is x = mg + nh, andy = mh + ang.

General methods, according to the Hindus, for the so-lution of indeterminate problems of the first and seconddegrees, are found in the 4th and 5th chapters of the BijaGanita, which differ much from Diophantus’s work. Itcontains, in Mr. Strachey’s opinion, which is highly proba-ble, a great deal of knowledge and skill, which the Greeks