Ancient Indian mathematicians-Bhaskara I

Bhaskara I

Born: about 600 in (possibly) Saurastra (modern Gujarat state), IndiaDied: about 680 in (possibly) Asmaka, India

We have very little information about Bhaskara I's life except what can be deduced from his writings. Shukla deduces from the fact that Bhaskara I often refers to the Asmakatantra instead of the Aryabhatiya Ⓣ that he must have been working in a school of mathematicians in Asmaka which
was probably in the Nizamabad District of Andhra Pradesh. If this is correct, and it does seem quite likely, then the school in Asmaka would have been a collection of scholars who were followers of Aryabhata I and of course this fits in well with the fact that Bhaskara I himself was certainly a follower of Aryabhata I.

There are other references to places in India in Bhaskara's writings. For example he mentions Valabhi (today Vala), the capital of the Maitraka dynasty in the 7^th century, and Sivarajapura, which were both in Saurastra which today is the Gujarat state of India on the west coast of the continent. Also mentioned are Bharuch (or Broach) in southern Gujarat and Thanesar in the eastern Punjab which was ruled by Harsa for 41 years from 606. Harsa was the preeminent ruler in north India through the first half of Bhaskara I's life. A reasonable guess would be that Bhaskara was born in Saurastra and later moved to Asmaka.

Bhaskara I was an author of two treatises and commentaries to the work of Aryabhata I. His works are the Mahabhaskariya Ⓣ, the Laghubhaskariya Ⓣand the Aryabhatiyabhasya Ⓣ. The Mahabhaskariya Ⓣ is an eight chapter work on Indian mathematical astronomy and includes topics which were fairly standard for such works at this time. It discusses topics such as: the longitudes of the planets; conjunctions of the planets with each other and with bright stars; eclipses of the sun and the moon; risings and settings; and the lunar crescent.

Bhaskara I included in his treatise the Mahabhaskariya Ⓣ three verses which give an approximation to the trigonometric sine function by means of a rational fraction. These occur in Chapter 7 of the work. The formula which Bhaskara gives is amazingly accurate and use of the formula leads to a maximum error of less than one percent. The formula is

sin x = 16x (π - x)/[5π² - 4x (π - x)]

and Bhaskara attributes the work as that of Aryabhata I. We have computed the values given by the formula and compared it with the correct value for sin x for x from 0 to π/₂ in steps of π/₂₀.

x = 0	formula = 0.00000	sin x = 0.00000	error = 0.00000
x = π/20	formula = 0.15800	sin x = 0.15643	error = 0.00157
x = π/10	formula = 0.31034	sin x = 0.30903	error = 0.00131
x = 3π/20	formula = 0.45434	sin x = 0.45399	error = 0.00035
x = π/5	formula = 0.58716	sin x = 0.58778	error = -0.00062
x = π/4	formula = 0.70588	sin x = 0.70710	error = -0.00122
x = π/10	formula = 0.80769	sin x = 0.80903	error = -0.00134
x = 7π/20	formula = 0.88998	sin x = 0.89103	error = -0.00105
x = 2π/5	formula = 0.95050	sin x = 0.95105	error = -0.00055
x = 9π/20	formula = 0.98753	sin x = 0.98769	error = -0.00016
x = π/2	formula = 1.00000	sin x = 1.00000	error = 0.00000

In 629 Bhaskara I wrote a commentary, the Aryabhatiyabhasya Ⓣ, on the Aryabhatiya Ⓣ by Aryabhata I. The Aryabhatiya Ⓣ contains 33 verses dealing with mathematics, the remainder of the work being concerned with mathematical astronomy. The commentary by Bhaskara I is only on the 33 verses of mathematics. He considers problems of indeterminate equations of the first degree and trigonometric formulae. In the course of discussions of the Aryabhatiya Ⓣ, Bhaskara I expressed his idea on how one particular rectangle can be treated as a cyclic quadrilateral. He was the first to open discussion on quadrilaterals with all the four sides unequal and none of the opposite sides parallel.

One of the approximations used for π for many centuries was √10. Bhaskara I criticised this approximation. He regretted that an exact measure of the circumference of a circle in terms of diameter was not available and he clearly believed that π was not rational.

In [11], [12], [13] and [14] Shukla discusses some features of Bhaskara's mathematics such as: numbers and symbolism, the classification of mathematics, the names and solution methods of equations of the first degree, quadratic equations, cubic equations and equations with more than one unknown, symbolic algebra, unusual and special terms in Bhaskara's work, weights and measures, the Euclidean algorithm method of solving linear indeterminate equations, examples given by Bhaskara I illustrating Aryabhata I's rules, certain tables for solving an equation occurring in astronomy, and reference made by Bhaskara I to the works of earlier Indian mathematicians.