Sridhara

Quick Info

Born
possibly Bengal, India
Died
India

Summary
Sridhara was an Indian mathematician who wrote on practical applications of algebra and was one of the first to give a formula for solving quadratic equations.

Biography

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 non-trivial 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 $Nx^{2} ± 1 = y^{2}, 1 - Nx^{2} = y^{2}, Nx^{2} ± C = y^{2}$, and $C - Nx^{2} = y^{2}$ 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
$ax^{2} + bx = c$.
Multiply both sides by $4a$ to get
$4a^{2}x^{2} + 4abx = 4ac$
then add $b^{2}$ to both sides to get
$4a^{2}x^{2} + 4abx + b^{2}= 4ac + b^{2}$
and, taking the square root
$2ax + b = √(4ac + b^{2})$.
There is no suggestion that Sridhara took two values when he took the square root.

References (show)

1. D Pingree, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. G G Joseph, The crest of the peacock (London, 1991).
3. K Shankar Shukla, The Patiganita of Sridharacarya (Lucknow, 1959).
4. B Datta, On the relation of Mahavira to Sridhara, Isis 17 (1932), 25-33.
5. Ganitanand, On the date of Sridhara, Ganita Bharati 9 (1-4) (1987), 54-56.
6. T Hayashi, Sridhara's authorship of the mathematical treatise Ganitapancavimsi, Historia Sci. (2) 4 (3) (1995), 233-250.
7. K Shankar Shukla, On Sridhara's rational solution of Nx2+1=y^2, Ganita 1 (1950), 1-12.
8. A I Volodarskii, Mathematical treatise Patiganita by Sridhara (Russian), in 1966 Phys. Math. Sci. in the East (Russian) 'Nauka' (Moscow, 1966), 141-159
9. A I Volodarskii, Notes on the treatise Patiganita by Sridhara (Russian), in 1966 Phys. Math. Sci. in the East (Russian) 'Nauka' (Moscow, 1966), 182-246.