Sridhara 

Born: about 870 in possibly Bengal, India
Died: about 930 in India Sridhara is now believed
to have lived in the ninth and tenth centuries. However, there has
been much dispute over his date and in different works the dates of
the life of Sridhara have been placed from the seventh century to the
eleventh century. The best present estimate is that he wrote around
900 AD, a date which is deduced from seeing which other pieces of
mathematics he was familiar with and also seeing which later
mathematicians were familiar with his work. We do know that Sridhara
was a Hindu but little else is known. Two theories exist concerning
his birthplace which are far apart. Some historians give Bengal as
the place of his birth while other historians believe that Sridhara
was born in southern India. Sridhara is
known as the author of two mathematical treatises, namely the
Trisatika (sometimes called the Patiganitasara ) and the Patiganita.
However at least three other works have been attributed to him,
namely the Bijaganita, Navasati, and Brhatpati. Information about
these books was given the works of Bhaskara II (writing around 1150),
Makkibhatta (writing in 1377), and Raghavabhatta (writing in 1493).
We give details below of Sridhara's rule for solving quadratic
equations as given by Bhaskara II. There is
another mathematical treatise Ganitapancavimsi which some historians
believe was written by Sridhara. Hayashi in [7], however, argues that
Sridhara is unlikely to have been the author of this work in its
present form. The Patiganita is written in
verse form. The book begins by giving tables of monetary and
metrological units. Following this algorithms are given for carrying
out the elementary arithmetical operations, squaring, cubing, and
square and cube root extraction, carried out with natural numbers.
Through the whole book Sridhara gives methods to solve problems in
terse rules in verse form which was the typical style of Indian texts
at this time. All the algorithms to carry out arithmetical operations
are presented in this way and no proofs are given. Indeed there is no
suggestion that Sridhara realised that proofs are in any way
necessary. Often after stating a rule Sridhara gives one or more
numerical examples, but he does not give solutions to these example
nor does he even give answers in this work.
After giving the rules for computing with natural numbers, Sridhara
gives rules for operating with rational fractions. He gives a wide
variety of applications including problems involving ratios, barter,
simple interest, mixtures, purchase and sale, rates of travel, wages,
and filling of cisterns. Some of the examples are decidedly
nontrivial and one has to consider this as a really advanced work.
Other topics covered by the author include the rule for calculating
the number of combinations of n things taken m at a time. There are
sections of the book devoted to arithmetic and geometric
progressions, including progressions with a fractional numbers of
terms, and formulae for the sum of certain finite series are
given. The book ends by giving rules, some
of which are only approximate, for the areas of a some plane
polygons. In fact the text breaks off at this point but it certainly
was not the end of the book which is missing in the only copy of the
work which has survived. We do know something of the missing part,
however, for the Patiganitasara is a summary of the Patiganita
including the missing portion. In [7] Shukla
examines Sridhara's method for finding rational solutions of Nx2 1 =
y2, 1  Nx2 = y2, Nx2 C = y2, and C  Nx2 = y2 which Sridhara gives
in the Patiganita. Shukla states that the rules given there are
different from those given by other Hindu mathematicians.
Sridhara was one of the first mathematicians to give a rule
to solve a quadratic equation. Unfortunately, as we indicated above,
the original is lost and we have to rely on a quotation of Sridhara's
rule from Bhaskara II: Multiply both sides
of the equation by a known quantity equal to four times the
coefficient of the square of the unknown; add to both sides a known
quantity equal to the square of the coefficient of the unknown; then
take the square root. To see what this means
take ax2 + bx = c.
Multiply both sides by 4a to get 4a2x2 +
4abx = 4ac then add b2 to both sides to
get 4a2x2 + 4abx + b2= 4ac + b2
and, taking the square root 2ax +
b = v(4ac + b2). There is no suggestion that
Sridhara took two values when he took the square root.
Article by: J J O'Connor and E F Robertson Source:
www.history.mcs.standrews.ac.uk/Mathematicians



