Bhaskara I 

Born: about 600 in (possibly) Saurastra (modern
Gujarat state), India Died: 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 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 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 (p  x)/[5p2  4x (p 
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 p/2 in
steps of p/20. x = 0 formula = 0.00000 sin x =
0.00000 error = 0.00000 x = p/20 formula = 0.15800
sin x = 0.15643 error = 0.00157 x = p/10 formula =
0.31034 sin x = 0.30903 error = 0.00131 x = 3p/20
formula = 0.45434 sin x = 0.45399 error = 0.00035
x = p/5 formula = 0.58716 sin x = 0.58778 error = 0.00062
x = p/4 formula = 0.70588 sin x = 0.70710 error = 0.00122
x = p/10 formula = 0.80769 sin x = 0.80903 error = 0.00134
x = 7p/20 formula = 0.88998 sin x = 0.89103 error =
0.00105 x = 2p/5 formula = 0.95050 sin x =
0.95105 error = 0.00055 x = 9p/20 formula =
0.98753 sin x = 0.98769 error = 0.00016 x = p/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 p for many centuries was v10. 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 p 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. Article by: J J O'Connor
and E F Robertson Source:
wwwhistory.mcs.standrews.ac.uk/Mathematicians



