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Varahamihira

Article by: J J O’Connor and E F Robertson

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 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. Varahamihira was one of the most famous astrologers in Indian history. His work Brihatsamhita (The Great Compilation) discusses topics such as [1]:- 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 formulas 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 methods. 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 [5]. Hayashi, in [6], examines Varahamihira’s work on magic squares. In particular he examines a pandiagonal magic square of order four
which occurs in Varahamihira’s work.

References:

Books:

G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
Articles:
G Abraham, Theories for the motion of the Sun in the Pancasiddhantika, Arch. Hist. Exact Sci. 34 (3) (1985), 221-230.
D G Dhavale, The Kapitthaka of Varahamihira, in Proceedings of the Symposium on Copernicus and Astronomy, New Delhi, 1973,
Indian J. History Sci. 9 (1) (1974), 77-78; 141.
Ghayasuddin, Varahamihira, the best Sanskrit source of al-Biruni on Indian Jyotisa, in Proceedings of the Symposium on al-Biruni
and Indian Sciences, New Delhi, 1971, Indian J. History Sci. 10 (2) (1975), 139-152;159-160.
R C Gupta, Varahamihira’s calculation of öC¨ and the discovery of Pascal’s triangle, Ganita Bharati 14 (1-4) (1992), 45-49.
T Hayashi, Varahamihira’s pandiagonal magic square of the order four, Historia Math. 14 (2) (1987), 159-166.
T S Kuppanna Sastry, The Vasistha-Paulisa Venus in the Pancasiddhantika of Varahamihira, Indian J. Hist. Sci. 14 (2) (1979),
150-154.
O Neugebauer and D Pingree, The Pancasiddhantika of Varahamihira. I, Hist.-Filos. Skr. Danske Vid. Selsk. 6 (1) (1970).
O Neugebauer and D Pingree, The Pancasiddhantika of Varahamihira. II, Hist.-Filos. Skr. Danske Vid. Selsk. 6 (2) (1971).
K S Shukla, The Panca-siddhantika of Varahamihira. II, Ganita 28 (1977), 99-116.
K S Shukla, The Pancasiddhantika of Varahamihira. I, in Proceedings of the Symposium on Copernicus and Astronomy, New Delhi,
1973, Indian J. History Sci. 9 (1) (1974) 62-76; 141.
K S Shukla, The Pancasiddhantika of Varahamihira. I (errata insert), Ganita 24 (1) (1973), 59-73.

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