angled triangles and a central square, while discussing the validity of
A ̄ryabhat.a’s statement, “A square (varga) is an equi-quadri-lateral ·figureÒ”
(A ̄ryabhat. ̄yaı 2.3). Bha ̄skara’s square demonstrates that 4 ¥(ab/2)+(a-b)^2 =c^2 ,
from which he would easily obtain the equation, a^2 +b^2 =c^2 , numerically.
TheBakhsha ̄lı ̄ Manuscript, whose original title is not known, is the oldest
extant manuscript in Indian mathematics. It was unearthed in a deteriorated
condition at Bakhsha ̄lı ̄ near Peshawar (now in Pakistan) in 1881 and is now
preserved in the Bodleian Library at Oxford University. The extant portion of the
manuscript consists of 70 fragmentary leaves. It is written on birchbark with
the earlier type of the S ́a ̄rada ̄ script, which was used in the northwestern part of
India from the eighth to the twelfth centuries ad. The language is Sanskrit but
it has largely been influenced by the vernacular(s) of those regions.
The work is a loose compilation of mathematical rules and examples collected
from different works. They are written in verse, and the examples are solved in
prose commentaries on them and are often given verifications of the answers.
The dates so far proposed for the Bakhsha ̄lı ̄ work vary from the third to the
twelfth centuries ad, but a recently made comparative study has shown many
similarities, particularly in the style of exposition and terminology, between the
Bakhsha ̄lı ̄ work and Bha ̄skara I’s commentary on the A ̄ryabhat.ı ̄ya.This seems
to indicate that both works belong to the nearly same period, although this does
not deny the possibility that some of the rules and examples in the Bakhsha ̄lı ̄
work date from anterior periods.
The rules that occur in the extant portion of the Bakhsha ̄lı ̄ work are: (1) arith-
metical operations such as addition, etc.; (2) general rules applicable to different
kinds of problems such as the rule of three, regula falsi, etc.; (3) rules for purely
numerical problems such as algebraic equations and arithmetical progressions;
(4) rules for problems of money such as buying and selling, etc.; (5) rules for
problems of travelers such as equations of journeys, etc.; and (6) rules for geo-
metric problems such as the volume of an irregular solid. The Bakhsha ̄lı ̄ work
employs a decimal place value notation with a dot for zero.
S ́rı ̄dhara, who flourished between Brahmagupta and Govindasva ̄min, is one
of the earliest mathematicians who wrote separate treatises for the two major
fields,pa ̄t.ı ̄-gan.itaandbı ̄ja-gan.tia, although his work on the latter is known only
from a quotation. He included many new topics such as combinations of the six
tastes, the hundred fowls problem, the cistern problem, etc., in his Pa ̄t.ı ̄gan.ita.
II.4 The ninth to fourteenth centuries – later developments
A follower of the A ̄ryabhat.a school of mathematics and astronomy, Govin-
dasva ̄min flourished in the first half of the ninth century in Kerala. His com-
mentaries on Bha ̄skara I’sMaha ̄bha ̄skarı ̄yaand on the latter half of Para ̄s ́ara’s
Hora ̄s ́a ̄stra (between 600 and 750) are extant, but three works of his, Govindakr.ti
on astronomy, Govindapaddhati on astrology, and Gan.itamukha on mathematics,
are known only from references and quotations by later writers.
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