unknown integer, N, is divided by a set of integers, {a 1 ,a 2 ,..., an}, one by one,
the remainders are {r 1 ,r 2 ,..., rr}. What is that number, N?”
Va r a ̄hamihira is one of the most famous authorities on astrology in India. He
flourished in Avantı ̄ (modern Ujjain) in the sixth century ad.
He divided the “astral science” (jyotih.s ́a ̄stra) into three major “branches”
(skandhas), namely, (1) mathematics including mathematical astronomy (gan.ita
ortantra), (2) horoscopic astrology (hora ̄), and (3) natural astrology or divina-
tion in general (sam.hita ̄)(Br.hatsam.hita ̄1.8–9, 2.2, 2.19), and wrote several
books each in the second and third branches. His only work in the first branch,
Pañcasiddha ̄ntika ̄, is a compendium of the texts of five earlier astronomical
schools, and no work on mathematics proper is known to have been written by
him, but his works are important from the view point of the history of mathe-
matics as well.
In the Pañcasiddha ̄ntika ̄, zero occurs as a number, that is, the object of mathe-
matical operations like addition, subtraction, etc. For example, he states the
mean daily velocity of the sun in each of the 12 zodiacal signs beginning with
Aries as follows: “The daily velocity of the sun is in order 60 ·minutesÒminus 3,
3, 3, 3, 2, 1; plus 1, 1, 1, 1; and minus 0, 1” (Pañcasiddha ̄ntika ̄3.17).
Presumably, the existence of both the zero symbol for vacant places in the
place-value notation and calculations by this notation brought about the
concept of zero as a number, because we cannot calculate, for instance, 15 + 20
=35 without the rule, 5 + 0 =5. This is not the case with an abacus where no
symbol exists for vacant places.
He expressly stated the “graphic procedure” for constructing a sine table, a
method which had been only alluded to by A ̄ryabhat.a I, and gave a sine table for
the radius, R=120 (ibid. 4.1–15).
In a chapter on the combination of perfume of his work on divination,
Br.hatsam.hita ̄(76.22), he provided a rule for calculating the number of combi-
nations, nCr, when rthings are taken at a time from nthings, and a method called
“spread by token” (los.t.aka-prasta ̄ra) for enumerating all the possible combina-
tions correctly. In the same chapter he gave the correct numbers, 84 and 1820,
of 9 C 3 and 16 C 4 respectively, but he mistakenly regarded 4! ¥ 4 ¥ 16 C 4 = 174720
as the number of possible combinations when 4 are taken at a time out of 16
ingredients for perfumes, in the ratio, 1:2:3:4 (ibid. 76.13–21); the correct
number should be 16 P 4 = 16 ¥ 15 ¥ 14 ¥ 13 =43680.
In the same chapter he utilized a magic square of order four in order to pre-
scribe the perfumes called “good for all purposes.” It consists of the four rows:
2, 3, 5, 8; 5, 8, 2, 3; 4, 1, 7, 6; and 7, 6, 4, 1 (ibid. 76.23–6). This is the oldest
magic square in India. It is irregular because it is made not of the numbers 1 to
16 but of two sets of the numbers 1 to 8, but it is “pan-diagonal” in the sense
that not only the two main diagonals but also all the “broken” diagonals each
amounts to the magic constant, 18 in this case. Presumably, Vara ̄hamihira first
constructed a regular magic square with the numbers 1 to 16, and then modi-
fied it by subtracting 8 each from the numbers greater than 8 to arrive at his
irregular square.
368 takao hayashi