- COMBINATORICS 213
This array of numbers, which is known as Pascal's triangle because of a treatise
on it written by Pascal in the seventeenth century, was studied four centuries before
Pascal by Jordanus Nemorarius, who developed many of its properties (Hughes,
1989). Pascal's triangle also occurs in Chinese manuscripts some four centuries
before Pascal's treatise. In China the inspiration for the study of this diagram
arose in connection with the extraction of cube roots and higher roots. The diagram
appears in the Xiangjie Jiuzhang Suan Fa (Detailed Analysis of the Mathematical
Methods in the Nine Chapters) of Yang Hui, written in 1261 (Li and Du, 1987,
p. 122). But in India we find it 300 years before it was published in China and
700 years before Pascal. Moreover it purports to be only a clarification of a rule
invented 1200 years earlier!^16 Its Sanskrit name is Meru Prastara (see Fig. 3),
which means the staircase of Mount Meru.^17
According to Singh (1985), Pingala's work on poetry also leads to another
interesting combinatorial topic, recognized as such by the Hindu mathematicians.
We treated this topic above as number theory, but it will bear repeating as com-
binatorics. To simplify the explanation as much as possible, suppose that a line of
poetry is to be written using short beats and long ones, a long one being equivalent
to two short ones. If a line contains ç beats, how many arrangements are possible.
Just to get started, we see that there is obviously one line of one beat (short), two
lines of two beats (two short or one long), three lines of three beats (short-long,
long-short, short-short-short), and five lines of four beats (long-long, short-short-
long, short-long-short, long-short-short, short-short-short-short). Since a line with
ç + 1 beats must begin with either a short or a long beat, we observe that those
beginning with a short beat are in one-to-one correspondence with the lines of ç
beats, all of which can be obtained by removing the initial short beat, while those
beginning with a long beat are in a similar correspondence with lines of ç — 1 beats.
It follows that the number of lines with n + l beats is the sum of the numbers with
ç — 1 and n. Once again we generate the Fibonacci sequence.
Bhaskara II knew the rules for combinatorial coefficients very well. In Chapter 4
of the Lilavati (Colebrooke, 1817, pp. 49-50), he gives an example of hexameter
and asks how many possible combinations of long and short syllables are possible.
He prescribes setting the numbers from 1 to 6 down "in direct and inverse order,"
that is, setting down the 2x6 matrix
6 5 4 3 2 1
1 2 3 4 5 6'From this array, by forming the products from the left and dividing, he finds the
number of verses with different numbers of short syllables from 1 to 5 as6 „ 6-5 6-5-4 nn 6-5-4-3 ir 6-5-4-3-2 „- = 6, = 15, = 20, = 15, = 6.
1 '1-2 ' 1-2-3 1-2-3-4 1-2-3-4-5
(^16) That claim cannot be verified, however. Evidence indicates that knowledge of the combinatorial
coefficients arose in India around the time of Aryabhata I, in the sixth century (Biggs, 1979, p.
115).
(^17) In Hindu mythology Mount Meru plays a role similar to that of Mount Olympus in Greek
mythology. One Sanskrit dictionary gives this mathematical meaning of Meru Prastara as a
separate entry. The word prastara apparently has some relation to the notion of expansion as
used in connection with the binomial theorem.