II.3 The seventh to eighth centuries – restructuring
The seventh century saw a restructuring of Indian mathematics. It was divided
into two major fields, that is, arithmetic with mensuration and algebra, which
were later called pa ̄t.ı ̄-gan.ita (“mathematics of algorithms”) and bı ̄ja-gan.ita
(“mathematics of seeds”), respectively. The “seeds” here means algebraic equa-
tions (samı ̄karan.a, lit. “to make ·both sidesÒequal”), which, just like seeds of
plants, have “the potentiality to generate” the solutions of mathematical prob-
lems. A work for pa ̄t.ı ̄-gan.itausually consists of two categories of rules often
accompanied by examples, that is, parikarman (basic operations) and vyavaha ̄ra
(practical or applied mathematics).
Brahmagupta included two chapters corresponding to these two fields in his
astronomical work, Bra ̄hmasphut.asiddha ̄nta(ad628), and began to use the word
gan.aka(“calculator”) in the sense of one who knows mathematical astronomy.
According to his own words, Brahmagupta, son of Jis.n.u, wrote the Bra ̄hmas-
phut.asiddha ̄ntain 25 chapters at the age of 30 in S ́aka 550 =ad628. This means
that he was born in ad598. He was still active at the age of 67 (ad665), when
he composed another work on astronomy, the Khan.d.akha ̄dyaka.
In the Bra ̄hmasphut.asiddha ̄nta, five chapters are particularly concerned with
mathematics. They are: chap. 12, “Mathematics,” on arithmetic and mensura-
tion, chap. 18, “pulverizer,” on algebra (the title has simply been taken from the
name of the first topic in the chapter), chap. 19, “Knowledge about Gnomon and
Shadow,” on measurements of shadows and lights, chap. 20, Answer to <the
Problems of> Piling of Meters” (chandas ́cityuttara, on combinatorics concerning
prosody, and a small section called “sine-production” (jya ̄-utpatti) in chap. 21
(“Spherics”) on trigonometry.
It is in chap. 12 that he gives his famous theorem on the diagonals of a cyclic
quadrilateral. In chap. 18, he prescribes rules for surds, negative quantities,
zero, and unknown quantities, and provides rules called varga-prakr.ti(“square
nature”) for quadratic indeterminate equations of the type, Px^2 +t=y^2.
Bha ̄skara I flourished in the first half of the seventh century in Saura ̄s.t.ra,
perhaps in Valabhı ̄ near modern Bha ̄vnagar, and composed three works as the
expositions of the teachings of A ̄ryabhat.a I, “based on the continuity of tradi-
tion” (samprada ̄ya-aviccheda ̄t).They are, in chronological order, Karmanibandha
(“Treatise on ·AstronomicalÒ Computation”) alias Maha ̄bha ̄skarı ̄ya (“Large
·BookÒof Bha ̄skara”), a prose commentary on the A ̄ryabhat.ı ̄ya(written in S ́aka
551 =ad629), and an abridged version of the first work also called Karmani-
bandhaaliasLaghubha ̄skarı ̄ya(“Small·BookÒof Bha ̄skara”).
Particularly important for the history of Indian mathematics is the second
work, that is, the commentary on the A ̄ryabhat. ̄yaı , which provides valuable
information on, among other things, mathematical procedures and expressions
of his time.
A vacant place (kha) in the decimal place-value notation was indicated by a
small circle (bindu, lit. “a dot”), which was also put on the right shoulder of a
indian mathematics 369