- HINDU ALGEBRA 421
4.2. Bhaskara II. In the five centuries between Brahmagupta and Bhaskara II
(who will henceforth be referred to simply as Bhaskara), the idea of using symbols
for the unknown in an equation seems to have taken hold in Hindu mathematics. In
Section 4 of his Vija Ganita (Algebra) Bhaskara reports that the initial syllables of
the names for colors "have been selected by venerable teachers for names of values
of unknown quantities, for the purpose of reckoning therewith" (Colebrooke, 1817,
p. 139). He proceeds to give the rules for manipulating expressions involving such
quantities; for example, the rule that we would write as (—x — 1) + (2x — 8) = ÷ — 9
is written
ya 1 ru 1
ya2 ru8
Sum ya 1 ru 9,where the dots indicate negative quantities. The syllable ya is the first syllable of
the word for black, and ru is the first syllable of the word for species.^11
By the time of Bhaskara, the distinction between a rational and an irrational
square root was well known. The Sanskrit word is carani, according to the com-
mentator Krishna (Colebrooke, 1817, p. 145), who defines it as a number "the root
of which is required but cannot be found without residue." Bhaskara gives rules
such as V8 + V2 = y/l8 and V8 - %/2 = s/2.
Bhaskara's algebraic rules go beyond what is taught even today as standard
algebra. He says that a nonzero number divided by zero gives an infinite quotient.
This fraction [3/0], of which the denominator is cipher, is termed
an infinite quantity.
In this quantity consisting of that which has cipher for its di-
visor, there is no alteration, though many be inserted or extracted;
as no change takes place in the infinite and immutable GOD, at the
period of the destruction or creation of worlds, though numerous
orders of beings are absorbed or put forth. [Colebrooke, 1817, pp.
137-138]Both the Vija Ganita and the Lilavati contain problems on simple interest in
which an unknown principal is to be found given the rate of simple interest and the
amount to which it accrues after a given time. These equations are linear equations
in one unknown.
The Lilavati contains a collection of problems in algebra, which are sometimes
stated as though they were intended purely for amusement. For example, the rule
for solving quadratic equations is applied in the Vija Ganita (Colebrooke, 1871,
p. 212) to find the number of arrows ÷ that Arjuna (hero of the Bhagavad Gita)
had in his quiver, given that he shot them all, using |r to deflect the arrows of
his antagonist, 4^/x to kill his antagonist's horse, six to kill the antagonist himself,
three to demolish his antagonist's weapons and shield, and one to decapitate him.
In other words, ÷ = \x + 4>/r + 10.
Bhaskara gives a criterion for a quadratic equation to have two (positive) roots.
He also says that "if the solution cannot be found in this way, as in the case of cubic(^11) There is no evidence that Bhaskara knew of Diophantus; the fact that both describe a power of
the unknown using a word whose meaning is approximated by the English word species is simply
a coincidence.