212 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS1 1
12 1(^1 3 3 1)
(^1 4 6 4 1)
(^1 5 10 10 5 1)
(^1 6 15 20 15 6 1)
(^1 7 21 35 35 21 7 1)
FIGURE 3. The Meru Prastara.
Permutations and combinations. The metaphysics of the Jainas, based on a clas-
sification of sentient beings according to the number of senses possessed, led them
to a mathematical topic related to number theory. They called it vikalpa, and we
know it as the basic part of combinatorics.
A typical question might be: "How many groups of three can be formed from
a collection of five objects?" We know the answer, as did the early Jaina mathe-
maticians. In the Bhagabati Sutra, written about 300 BCE, the author asks how
many philosophical systems can be formed by taking a certain number of doctrines
from a given list of basic doctrines. After giving the answers for 2, 3, 4, and so on,
the author says that enumerable, unenumerable, and infinite numbers of things can
be discussed, and "as the number of combinations are formed, all of them must be
worked out."
The general process for computing combinatorial coefficients was known to the
Hindus at an early date. Combinatorial questions seemed to arise everywhere for
the Hindus, not only in the examples just given but also in a work on medicine
dating from the sixth century BCE (Biggs, 1979, p. 114) that poses the problem of
the number of different flavors that can be made by choosing subsets of six basic
flavors (bitter, sour, salty, astringent, sweet, hot). The author gives the answer as
6+15 + 20 + 15 + 6+1, that is, 63. We recognize here the combinatorial coefficients
that give the subsets of various sizes that can be formed from six elements. The
author did not count the possibility of no flavor at all.
Combinatorics also arose in the study of Sanskrit in the third century BCE
when the writer Pingala gave a rule for finding the number of different words
that could be formed from a given number of letters. This rule was written very
obscurely, but a commentator named Halayudha, who is believed to have lived in
the tenth century CE (Needham, 1959, p. 37), explained it as follows. First draw a
square. Below it and starting from the middle of the lower side, draw two squares.
Then draw three squares below these, and so on. Write the number 1 in the middle
of the top square and inside the first and last squares of each row. Inside every
other square the number to be written is the sum of the numbers in the two squares
above it and overlapping it.