| Paramesvara |
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Born: about 1370 in Alattur, Kerala, India
Died: about 1460 in India Paramesvara was an Indian astronomer and
mathematician who wrote many commentaries on earlier works as well as
making many observations. Although his father has not been
identified, we know that Paramesvara was born into a Namputiri
Brahmana family who were astrologers and astronomers. The family home
was Vatasseri (sometimes called Vatasreni) in the village of Alattur.
This village was in Kerala and Paramesvara himself gives its
coordinates with respect to Ujjain. This puts it at latitude 10 51'
north. It is on the north bank of the river Nila at its mouth. From
Paramesvara's writing we know that Rudra was his teacher, and
Nilakantha, who knew Paramesvara personally, tells us that
Paramesvara's teachers included Madhava and Narayana. We can be
fairly confident that the dates we have given for Paramesvara are
roughly correct since he made eclipe observations over a period of 55
years. We will say a little more about these observations below. He
played an important part in the remarkable developments in
mathematics which took place in Kerala in the late 14th and early
part of the 15th century. The commentaries by Paramesvara on a number
of works have been published. For example the Karmadipika is a
commentary on the Mahabhaskariyam, an astronomical and mathematical
work by Bhaskara I, and its text is given in [3]. In [2] the text of
Paramesvara's commentary on the Laghubhaskariyam of Bhaskara I is
given. Munjala wrote the astronomical work Laghumanasam in the year
932 and Paramesvara wrote a commentary (see [4]). It is a work
containing typical topics for Indian mathematical astronomy works of
this period: the mean motions of the heavenly bodies; the true
motions of the heavenly bodies; miscellaneous mathematical rules; the
systems of coordinates, direction, place and time; eclipses of the
sun and the moon; and the operation for apparent longitude. Aryabhata
gave a rule for determining the height of a pole from the lengths of
its shadows in the Aryabhatiya. Paramesvara gave several illustrative
examples of the method in his commentary on the Aryabhatiya. Like
many mathematicians from Kerala, Madhava clearly had a very strong
influence on Paramesvara. One can see throughout his work that it is
teachings by Madhava which direct much of Paramesvara's mathematical
ideas. One of Paramesvara's most remarkable mathematical discoveries,
no doubt influenced by Madhava, was a version of the mean value
theorem. He states the theorem in his commentary Lilavati Bhasya on
Bhaskara II's Lilavati. There are other examples of versions of the
mean value theorem in Paramesvara's work which we now consider. The
Siddhantadipika by Paramesvara is a commentary on the commentary of
Govindasvami on Bhaskara I's Mahabhaskariya. Paramesvara gives some
of his eclipse observations in this work including one made at
Navaksetra in 1422 and two made at Gokarna in 1425 and 1430. This
work also contains a mean value type formula for inverse
interpolation of the sine. It presents a one-point iterative
technique for calculating the sine of a given angle. In the
Siddhantadipika Paramesvara also gives a more efficient approximation
that works using a two-point iterative algorithm which turns out to
be essentially the same as the modern secant method. See [8] and [9]
for further details. The expression for the radius of the circle in
which a cyclic quadrilateral is inscribed, given in terms of the
sides of the quadrilateral, is usually attributed to Lhuilier in
1782. However Paramesvara described the rule 350 years earlier. If
the sides of the cyclic quadrilateral are a, b, c and d then the
radius r of the circumscribed circle was given by Paramesvara as: r2
= x/y where x = (ab + cd) (ac + bd) (ad + bc) and y = (a + b + c - d)
(b + c + d - a) (c + d + a - b) (d + a + b - c). The original text by
Paramesvara describing the rule is given in [7]. Paramesvara made a
series of eclipse observations between 1393 and 1432 which we have
referred to above. The last observation which we know he made was in
1445 but Nilakantha quotes a verse by Paramesvara in which he claims
to have made observations spanning 55 years. The known observatons by
Paramesvara do not quite square with this statement, there being a
discrepancy of three years. Although we do not know when Paramesvara
died we do know, again from Nilakantha, that the two knew each other
personally. Since we have a definite date for Nilakantha's birth of
1444 it is hard to believe that Paramesvara died before 1460. Using
his observations, Paramesvara made revisions of the planetary
parameters and, like many other Indian astronomers, he constantly
attempted to compare the theoretically computed positions of the
planets with those which he actually observed. He was keen to improve
the theoretical model to bring it into as close an agreement with
observations as possible. Article by: J J O'Connor and E F Robertson
Source: www.history.mcs.st-andrews.ac.uk/Mathematicians
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