The History of Mathematics: A Brief Course

420 14. EQUATIONS AND ALGORITHMS

Europe around the same time (1693, by Leibniz), but in a comparatively limited
context. As Smith and Mikami say {1914, p. 125),

It is evident that Seki was not only the discoverer but that he had a

much broader idea than that of his great German contemporary.

4. Hindu algebra

The promising symbolic notation of the Bakshali Manuscript was not adopted im-
mediately throughout the world of Hindu mathematics. In particular, Aryabhata I
tended to work in prose sentences. He considered the problem of finding two num-
bers given their product and their difference and gave the standard recipe for solving
it.

4.1. Brahmagupta. The techniques involved with the kuttaka (pulverizer) belong
to algebra, but since they are applied in number theory, we discussed them in that
connection in Section 4 of Chapter 7. Brahmagupta also considered many problems
that require finding the lengths of lines partitioning a polygon into triangles and
quadrilaterals.
Brahmagupta's algebra is done entirely in words; for example (Colebrooke,
1817, p. 279), his recipe for the cube of a binomial is:

The cube of the last term is to be set down; and, at the first remove

from it, thrice the square of the last multiplied by the preceding;

then thrice the square of the preceding term taken into that last

one; and finally the cube of the preceding term. The sum is the

cube.

In short, (a + b)^3 = a^3 +3a^2 b+3ab^2 +b^3. This rule is used for finding successive
approximations to the cube root, just as in China and Japan. Similarly, in Section 4
(Colebrooke, 1817, p. 346), he tells how to solve a quadratic equation:

Take the absolute number from the side opposite to that from

which the square and simple unknown are to be subtracted. To

the absolute number multiplied by four times the [coefficient of the]

square, add the square of the [coefficient of the] middle term; the

square root of the same, less the [coefficient of the] middle term,

being divided by twice the [coefficient of the] square is the [value

of the] middle term.

Here the "middle term" is the unknown, and this statement is a very involved
description of what we write as the quadratic formula:

\j4oc + b^2 -b 2 é

÷ = when ax + bx = c.

2a

Brahmagupta does not consider equations of degree higher than 2.