- INDIA 175
4. India
The Sulva Sutras contain rules for finding Pythagorean triples of integers, such as
(3,4,5), (5,12,13), (8,15,17), and (12,35,37). It is not certain what practical use
these arithmetic rules had. They may have been motivated by religious ritual. A
Hindu home was required to have three fires burning at three different altars. The
three altars were to be of different shapes, but all three were to have the same area.
These conditions led to certain "Diophantine" problems, a particular case of which
is the generation of Pythagorean triples, so as to make one square integer equal to
the sum of two others.
One class of mathematical problems associated with altar building involves an
altar of prescribed area in layers. In one problem from the Bodhayana Sutra the
altar is to have five layers of bricks, each layer containing 21 bricks. Now one cannot
simply divide a pile of 105 identical bricks into five layers and pile them up. Such
a structure would not be stable. It is necessary to stagger the edges of the bricks.
So that the outside of the altar will not be jagged, it is necessary to have at least
two different sizes of bricks. The problem is to decide how many different sizes of
bricks will be needed and how to arrange them. Assuming an area of one square
unit (actually, the unit is 1 square vyayam, about 7 square meters), the author
suggests using three kinds of square bricks, of areas ^, ^, and § square unit. The
first, third, and fifth layers are to have nine of the first kind and 12 of the second.
The second and fourth layers get 16 of the first kind and five of the third.
4.1. Varahamihira's mystical square. According to Hayashi (1987), around
the year 550 the mathematician Varahamihira wrote the Brhatsamhita, a large book
devoted mainly to divination. However, Chapter 76 also discusses the mixing of
perfumes from 16 substances, grouped in fours and mixed according to proportions
given by the rows of the following square array:2 3 5^8
5 8 2 3
4 1 7 6
7 6 4 1
Thus, the mysticism surrounding these squares penetrated even practical aspects of
life. Hayashi notes that the Sanskrit word for the square itself, kacchaputa, means
a box with compartments, but originally meant a tortoise shell. The resemblance
to the Luo Shu is probably a coincidence.4.2. Aryabhata I. In verses 32 and 33 of the Aryabhatiya we find a method
of solving problems related to the problem of Sun Zi that leads to the Chinese
remainder theorem. However, the context of the method and the description leave
much to be desired in terms of clarity. It would have helped if Aryabhata had
included specific examples. Such examples were provided by later commentators,
and the process was described more clearly by Brahmagupta.4.3. Brahmagupta. A century after Aryabhata, Brahmagupta called the method
the kuttaka (pulverizer). We shall exclude certain complications in Brahmagupta's
presentation and present the method as simply as possible. The kuttaka provides the
following visual implementation of an algorithm for solving the equation ax = by+c,
with b > a > 0 and a and b relatively prime. As an example, we shall find all