Govindasva ̄min shows a keen interest in the logical foundations of the rule of
three. He provides a definition, with a detailed explanation, of the four terms
(prama ̄n.a,prama ̄n.a-phala,iccha ̄, and ichha ̄-phala) of the rule of three in his com-
mentary on the Maha ̄bha ̄skarı ̄ya(1.7), and, in three verses cited by S ́an.kara,
compares these four terms to the constituent parts of the inference (anuma ̄na)of
Indian logicians. A typical inference according to them is as follows: “That
mountain has fire because of its having smoke. That which has smoke has fire,
like a kitchen.” According to Govinda, the four terms of the rule of three corre-
spond in order to the smoke and fire in the kitchen and the same two in the
mountain, and thus the rule of three can be regarded as an inference.
A Jaina mathematician, Maha ̄vı ̄ra wrote a book for pa ̄t.ı ̄-gan.ita entitled
Gan.itasa ̄rasam.graha during the reign of King Amoghavars.a (ca. 814/815–80).
The work is quite voluminous and comprises more than 1130 verses for rules
and examples. He seems to be the first in India who admitted two solutions of a
quadratic equation.
Bha ̄skara II was born in ad1114 to a family which produced a number of
scholars and literary men before and after him. He lived in Vijjad.avid.a at
the foot of the Sahya mountain situated at the northern end of the Western
Ghats, and completed his main work, Siddha ̄ntas ́iroman.i, when he was 36
years old (ad1150). He also wrote an astronomical manual, Karan.akutu ̄hara,in
ad1183, and a commentary (date unknown) on Lalla’sS ́is.yadhı ̄vr.ddhidatantra.
The Siddha ̄ntas ́iroman.i consists of four parts. Two of them, Lı ̄la ̄vatı ̄ and
Bı ̄jagan.ita, are on mathematics, and the other two, Grahagan.ita ̄dhya ̄yaand
Gola ̄dhya ̄ya, on astronomy. These four parts were often regarded as independent
works.
The most popular among them was the Lı ̄la ̄vatı ̄, which is a well organized text-
book ofpa ̄t. ̄-ganı .itawritten in a plain and elegant Sanskrit. It circulated all over
India and was commented upon in Sanskrit and in north Indian languages (such
as Mara ̄t.hı ̄ and Gujara ̄tı ̄) by a number of persons and translated not only into
Indian languages of the north and of the south (such as Kan.n.ad.a and Telugu)
but also several times into Persian.
Bha ̄skara II included a whole theory of the pulverizer in the Lı ̄la ̄vatı ̄, a book
ofpa ̄t.ı ̄. This was possible because it required neither algebraic symbolism nor
“intelligence” (mati) which were essential for bı ̄ja-gan.itaor algebra according to
Bha ̄skara II.
The last chapter, “the nets of digits,” deals with permutations of numerical
figures. The last problem, for example, reads as follows: “How many varieties of
numbers are there with digits placed in five places when their sum is thirteen?
It should be told, if you know.”
TheBı ̄jagan.itais the culmination of Indian algebra. Bha ̄skara II’s main con-
tribution to algebra is his treatment of various types of equations of order two
or more (up to six) with more than one unknown. The equations of higher
orders are solved by reducing them to quadratic equations. In his solutions, the
square nature and pulverizer, as well as the “elimination of the middle term”
(which is the name given to the solution procedure of quadratic equations), play
important roles.
372 takao hayashi