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# hindu mathematicians

Hindu Mathematicians

The earliest written documents about Mathematics are Sulvasutra. Their date is uncertain but is placed in eighth century B. C. The rules relate to the proportions of alters in the temples. the numbers which satisfy the equation x2 + y2 = z2 are known though whether Pythagoras Theorem is proved is not clear. There is evidence of knowledge of irrational numbers and the use of gnomon (observation of sun's meridian by seeing the shadow of a pillar). The Sulvasutra states that the diagonal of a unit square is 1 + 1/3 + 1/3.4 + 1/3.4.34 (i. e. 1.4142156 or √2). The area of circle is asserted to be (1/8 + 1/8.29 - 1/8.29.6 + 1/8.29.6.8)2 d2 where d is the diameter of the circle.

Sulvasutra was followed by commentaries on it by Apastamba, Baudhyana and Katyayana. the statements in Baudhyana commentaries include the following -

"The chord stretched across a square produces an area twice the size."

"The diagonal of an oblong produces by itself both the areas which the two sides of the oblong produce separately." (if x and y are the sides of the rectangle and z is the diagonal then x2 + y2 = z2 )

In Lalitavistara, the text speaks of the prowess of Budhha in arithmetic. There is nothing definite but it is clear that the arithmetic was not limited to the meager needs of trade.

A noteworthy contribution of India was the introduction of zero and of place value of digits. But this is well known and need not be written about.

Suryasidhhanta was an important work produced probably written at the beginning of fifth century A. D. This deals with the mathematics of stars. Varaharamitra speaks of five Sidhhantas of this period. amongst them is Paulisa Sidhhanta of about the same period as Surya Sidhhanta. This speaks of the sines of angles. In modern terminology, it may be expressed as

Sin 300 = ½ ; π = √10 ; Sin 600 = √3/4 ;

Sin2 θ = (1/2 sin2θ)2 + [{1 - sin (900 - 2θ)}/2]2

There was also tables for sines of various angles.

Aryabhatta was born at Kusumpura, a small town on the banks of Ganga near Patna in Bihar. His work is called Aryabhattiyam and it consists of Gitika, a collection of astronomical tables. Another work is Aryastatata which includes Ganita, Kalakriya and Gola treatise on arithmetic, time and circle. Arithmetic carries numeration, planes, solid numbers, square root, sum of arithmetical series, roots of quadratic equation. It also contains discussion about indeterminate linear equations. Area of a triangle is given as product of the perpendicular from the vertex to the base and half the base. Volume of sphere is given by πr2 √πr2. The value of π is given as 3.1416. Aryabata also gives a rule for finding sines and Gitika has a brief table of these functions. The solution of the indeterminate linear equations is given by means of continued fractions.

Varaharamitra is known for his work on Astronomy. He wrote several works of which Pancha Sidhhantika is most famous. It treats of astrology and astronomy. It includes computation for calculating the position of the planets. He taught about the sphericity of earth.

Brahamagupta lived in seventh century A. D. His observatory was at Ujjain. At the age of 30, he wrote Brahanasidhhanta consisting of 21 chapters. He defines ganaca - mathematician - as one who knows twenty one logistics and eight determinations. these included such routine things as addition and also out of way things like measuring by means of a shadow. He dealt with integers, fractions, progressions, barter, rule of three, simple interest, mensuration of plane figures, problems on volumes and shadows. His formula for area of quadrilateral is √(s-a)(s-b)(s-c)(s-d) where s = ½ (a + b+ c + d). [It is true for cyclical quadrilaterals only]. he uses 3 as approximate value of π and √10 as its 'neat' value. He was the first to apply algebra to astronomy. Aryabhata had considered the question of the integral solutions of equation ax ± by = c. Brhamagupta gave the actual solutions as x = ±cq -bt and y = ±cp + at where t is zero or an integer and p/q is the penultimate convergent of a/b. He also considered the equation of the form Du2 + 1 = t2 . (Its solution was given by Bhaskara later on)

For the right angled triangle, he gave the figures of 2mn, m2 - n2, m2 + n2 and alternatively √m, ½ (m/n -n) and ½ (m/n + n).

In South India, Mahavirachrya wrote Ganita Sara Sangraha in the days of Rashtrakutas, in the eighth century. His work consists of nine volumes and deals with mathematical operations (though excluding addition and subtraction being considered too simple. Besides square roots, he also writes about cubes and cube roots, summation of series, arithmetic and geometric series, and also the truncated series (some initial members have been left out). He does not give square roots of negative numbers as , according to him, they are not squares. His division is by means of multiplication with inverted fraction. (Division by p/q is equivalent to multiplication by q/p). He also talks of quadratic equations and of the area of trapeziums He also gives to formula for its diagonals as

√(ac+bd)(ab+cd) /(ad + bc) or √(ac+ bd)( ad + bc)/(ab + cd). He tries to give treatment of the ellipse. He gives the volume of sphere as approximately 9/2(1/2 d)3 and more accurately as 9/10. 9/2 (1/2 d)3. π is thus taken as 3.0375.

Sridhara

Probably born in 991 A. D. His best work is Ganita Sara also known as Trisatika, as it contained 300 couplets. The subjects considered are numeration, measures. These included series of natural numbers, multiplication, division, zero, squares, cubes, fractions, rule of three, interest, alloys, partnerships, mensuration and shadow reckoning.

Bhaskara

Bhaskara lived in the 12th century. A native of Bidar but worked in Ujjain. His best work is called Lilavati, probably named after his daughter. The book follows the arrangements adopted in Trishatika of Sridhara. He talked, for the first time of division by zero giving infinity as the answer. Other work of Bhaskar is Beej Ganit. It deals, inter alia, with negative numbers. The unknown quantities are called 'colours'. Surds are treated extensively. Simple and quadratic equations are dealt with. A third work of importance is Sidhhanta Shiromani. In this book is included Goladhia which deals with astronomy and spherecity of earth.

It is mentioned in an inscript that Changadwa, grandson of Bhaskara was court astrologer to King Singhana and that he founded a college to teach the doctrines of Bhaskara.

Bakhshali Manuscript

It is of uncertain origin and date. (Probably of 10th century A. D. ) It contains problems relating to both arithmetic and algebra. The writer and his location are not known.

There is nothing worth mentioning about development of Mathematics after the invasion of the Muslims. In fact this led to deterioration in all avenues of human knowledge being the result, probably, of the deliberate policy of the Government of the day. Mention can be made only of two persons who wrote commentaries on Bhaskara works viz Suryadas (1535 A. D.) and Ganesha about the same time). In the 17th century, the only name is Ranganatha (1621 A. D.) but he did not introduce anything new.

Comments on Hindu mathematics

Alberuni says, "What we have of sciences is nothing but the scanty remains of the bygone better days," He prepared the summary for Hindu Mathematics in the 11th century.

Summary of Arab achievements - We are struck by the interest of the Arabs in science but by their lack of originality. They received their astronomy, first from the Hindus and than from the Greek, their geometry solely and their algebra chiefly from the Greeks and their trigonometry largely from the HIndus. They originated nothing of importance, either in arithmetic or in geometry; but they systematized algebra to some extent; improved on astronomy of their predecessors and made some contributions to trigonometry.

(From History of Mathematics Vol. I David Eugene Smith (Dover Publications New York - 1958)