(747), consists of ten topics, which presumably cover the entire mathematics
known to the Indians of those days. They are: basic computation (parikamma),
procedure or applied mathematics (vavaha ̄ra), rope (rajju), quantity (ra ̄si), reduc-
tion of fractions (kala ̄savanna), “as many as” (ja ̄vam.ta ̄vati), square (vagga), cube
(ghana), square of square (vaggavagga), and choice (vikappa, combinatorics). The
Jainas played an important role in the making of Indian mathematics.
It was, however, neither gan.ana ̄norsam.khya ̄nabutgan.itathat was used later
as the most general term for mathematical science. A ̄ryabhat.a I, in the first verse
of his A ̄ryabhat.ı ̄ya(ad499 or a little later), enumerates the three subjects to be
dealt with in its subsequent chapters, namely, gan.ita,ka ̄la-kriya ̄(time-reckoning),
andgola(spherics).
The Vedic numerals continued to be used in later Hindu society as well, while
the Buddhists and the Jainas each developed their own systems of numerals for
numbers greater than a thousand.
Apart from the Indus script, the earliest extant script in India is the one called
Karos.t.hı ̄ of, probably, Aramaic origin. Its use was restricted to north-western
India and central Asia from the fourth century bcto the fourth century ad. At
nearly the same time, in the As ́okan edicts, appeared another script called
Bra ̄hmı ̄ which was to become the origin of many varieties of south and south-
east Asian scripts. Its relationship to the former is not certain. These scripts had
their own numerical symbols, but in both systems particular symbols were used
not only for units but also for tens, hundreds, thousands, etc., and some of them
were made by the principle of addition, and others by the principle of multipli-
cation. The numeral systems in both scripts were therefore not based on a place-
value system, and the Bra ̄hmı ̄, non-place-value, numerals continued to be used
in epigraphy even after the place-value system was introduced in daily calcula-
tions and in mathematical literature in the early centuries of the Christian era.
The oldest datable evidence of the decimal place-value system in India is
found in the Yavanaja ̄taka(ad269/270) of Sphujidhvaja, a book on astrology. It
is not certain if the decimal place-value notation in India had a symbol for zero
from the beginning. We have to keep it in mind that, historically, not all place-
value notations were accompanied by a zero symbol as the sexagesimal notation
of the Old Babylonians proves. There is no reference to zero in the Yavanaja ̄taka.
In Pin.gala’s Chandah.su ̄tra (8.28–31), a work on Sanskrit prosody, zero (s ́u ̄nya)
and two (dvi) are used as signs for the multiplication by two and for the squar-
ing, respectively, in the computation of powers of two, which occurs in the right-
hand side of the equation, 2 + 22 + 23 +...+ 2 n= 2 ¥ 2 n-2. The date of the
work is controversial: some ascribe it to ca. 200 bcbut others to the third century
ador later. The oldest datable evidence of zero as a symbol as well as of that as
a number are found in Vara ̄hamihira’s Pañcasiddha ̄ntika ̄(ca.ad550).
II.2 The fifth to sixth centuries – beginning
A ̄ryabhat.a I is so called to distinguish him from another astronomer of the same
name, the author of the Maha ̄siddha ̄nta, who is called A ̄ryabhat.a II.
366 takao hayashi