|Born: 505 in Kapitthaka, India
Died: 587 in India Our knowledge of Varahamihira is very limited
indeed. According to one of his works, he was educated in Kapitthaka.
However, far from settling the question this only gives rise to
discussions of possible interpretations of where this place was.
Dhavale in  discusses this problem. We do not know whether he was
born in Kapitthaka, wherever that may be, although we have given this
as the most likely guess. We do know, however, that he worked at
Ujjain which had been an important centre for mathematics since
around 400 AD. The school of mathematics at Ujjain was increased in
importance due to Varahamihira working there and it continued for a
long period to be one of the two leading mathematical centres in
India, in particular having Brahmagupta as its next major figure.
The most famous work by Varahamihira is the Pancasiddhantika
(The Five Astronomical Canons) dated 575 AD. This work is important
in itself and also in giving us information about older Indian texts
which are now lost. The work is a treatise on mathematical astronomy
and it summarises five earlier astronomical treatises, namely the
Surya, Romaka, Paulisa, Vasistha and Paitamaha siddhantas. Shukla
states in :-
The Pancasiddhantika of Varahamihira is one of the most
important sources for the history of Hindu astronomy before the time
of Aryabhata I I.
One treatise which Varahamihira summarises was the
Romaka-Siddhanta which itself was based on the epicycle theory of the
motions of the Sun and the Moon given by the Greeks in the 1st
century AD. The Romaka-Siddhanta was based on the tropical year of
Hipparchus and on the Metonic cycle of 19 years. Other works which
Varahamihira summarises are also based on the Greek epicycle theory
of the motions of the heavenly bodies. He revised the calendar by
updating these earlier works to take into account precession since
they were written. The Pancasiddhantika also contains many examples
of the use of a place-value number system.
There is, however, quite a debate about interpreting data
from Varahamihira's astronomical texts and from other similar works.
Some believe that the astronomical theories are Babylonian in origin,
while others argue that the Indians refined the Babylonian models by
making observations of their own. Much needs to be done in this area
to clarify some of these interesting theories.
In  Ifrah notes that Varahamihira was one of the most
famous astrologers in Indian history. His work Brihatsamhita (The
Great Compilation) discusses topics such as :-
... descriptions of heavenly bodies, their movements and
conjunctions, meteorological phenomena, indications of the omens
these movements, conjunctions and phenomena represent, what action to
take and operations to accomplish, sign to look for in humans,
animals, precious stones, etc.
Varahamihira made some important mathematical discoveries.
Among these are certain trigonometric formulae which translated into
our present day notation correspond to
sin x = cos(p/2 - x),
sin2x + cos2x = 1, and
(1 - cos 2x)/2 = sin2x.
Another important contribution to trigonometry was his sine
tables where he improved those of Aryabhata I giving more accurate
values. It should be emphasised that accuracy was very important for
these Indian mathematicians since they were computing sine tables for
applications to astronomy and astrology. This motivated much of the
improved accuracy they achieved by developing new interpolation
The Jaina school of mathematics investigated rules for
computing the number of ways in which r objects can be selected from
n objects over the course of many hundreds of years. They gave rules
to compute the binomial coefficients nCr which amount to
nCr = n(n-1)(n-2)...(n-r+1)/r!
However, Varahamihira attacked the problem of computing nCr
in a rather different way. He wrote the numbers n in a column with n
= 1 at the bottom. He then put the numbers r in rows with r = 1 at
the left-hand side. Starting at the bottom left side of the array
which corresponds to the values n = 1, r = 1, the values of nCr are
found by summing two entries, namely the one directly below the (n,
r) position and the one immediately to the left of it. Of course this
table is none other than Pascal's triangle for finding the binomial
coefficients despite being viewed from a different angle from the way
we build it up today. Full details of this work by Varahamihira is
given in .
Hayashi, in , examines Varahamihira's work on magic
squares. In particular he examines a pandiagonal magic square of
order four which occurs in Varahamihira's work.
Article by: J J O'Connor and E F Robertson