Bhaskara I

Quick Info

about 600
(possibly) Saurastra (modern Gujarat state), India
about 680
(possibly) Asmaka, India

Bhaskara I was an Indian mathematicians who wrote commentaries on the work of Aryabhata I.


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 7th 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 pre-eminent 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
sinx=16x(πx)/[5π24x(πx)]\sin x = 16x (\pi - x)/[5\pi^{2} - 4x (\pi - 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 sinx\sin x for xx from 0 to π/2\pi/_{2} in steps of π/20\pi/_{20}.
x=0x = 0formula = 0.00000sinx\sin x = 0.00000error = 0.00000
x=π/20x = \pi/_{20} formula = 0.15800sinx\sin x = 0.15643error = 0.00157
x=π/10x = \pi/_{10} formula = 0.31034sinx\sin x = 0.30903error = 0.00131
x=3π/20x = 3\pi/_{20}formula = 0.45434sinx\sin x = 0.45399error = 0.00035
x=π/5x = \pi/_{5}formula = 0.58716sinx\sin x = 0.58778error = -0.00062
x=π/4x = \pi/_{4}formula = 0.70588sinx\sin x = 0.70710error = -0.00122
x=π/10x = \pi/_{10}formula = 0.80769sinx\sin x = 0.80903error = -0.00134
x=7π/20x = 7\pi/_{20}formula = 0.88998 sinx\sin x = 0.89103error = -0.00105
x=2π/5x = 2\pi/_{5}formula = 0.95050sinx\sin x = 0.95105error = -0.00055
x=9π/20x = 9\pi/_{20}formula = 0.98753sinx\sin x = 0.98769error = -0.00016
x=π/2x = \pi/_{2}formula = 1.00000sinx\sin x = 1.00000error = 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.

References (show)

  1. D Pingree, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
  2. K Shankar Shukla, Bhaskara I, Bhaskara I and his works II. Maha-Bhaskariya (Sanskrit) (Lucknow, 1960).
  3. K Shankar Shukla, Bhaskara I, Bhaskara I and his works III. Laghu-Bhaskariya (Sanskrit) (Lucknow, 1963).
  4. R C Gupta, Bhaskara I's approximation to sine, Indian J. History Sci. 2 (1967), 121-136.
  5. R C Gupta, On derivation of Bhaskara I's formula for the sine, Ganita Bharati 8 (1-4) (1986), 39-41.
  6. T Hayashi, A note on Bhaskara I's rational approximation to sine, Historia Sci. No. 42 (1991), 45-48.
  7. P K Majumdar, A rationale of Bhaskara I's method for solving ax ± c = by, Indian J. Hist. Sci. 13 (1) (1978), 11-17.
  8. P K Majumdar, A rationale of Bhatta Govinda's method for solving the equation ax - c = by and a comparative study of the determination of 'Mati' as given by Bhaskara I and Bhatta Govinda, Indian J. Hist. Sci. 18 (2) (1983), 200-205.
  9. A Mukhopadhyay and M R Adhikari, A step towards incommensurability of π and Bhaskara I : An episode of the sixth century AD, Indian J. Hist. Sci. 33 (2) (1998), 119-129.
  10. A Mukhopadhyay and M R Adhikari, The concept of cyclic quadrilaterals: its origin and development in India (from the age of Sulba Sutras to Bhaskara I, Indian J. Hist. Sci. 32 (1) (1997), 53-68.
  11. K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya, Ganita 22 (1) (1971), 115-130.
  12. K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya II, Ganita 22 (2) (1971), 61-78.
  13. K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya III, Ganita 23 (1) (1972), 57-79
  14. K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya IV, Ganita 23 (2) (1972), 41-50.
  15. I I Zaidullina, Bhaskara I and his work (Russian), Istor. Metodol. Estestv. Nauk No. 36 (1989), 45-49.

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Written by J J O'Connor and E F Robertson
Last Update November 2000