# Kamalakara

### Quick Info

Born
Benares (now Varanasi), India
Died
India

Summary
Kamalakara was an Indian astronomer and mathematician who combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists.

### Biography

Kamalakara was an Indian astronomer and mathematician who came from a family of famous astronomers. Kamalakara's father was Nrsimha who was born in 1586. Two of Kamalakara's three brothers were also famous astronomer/ mathematicians, these being Divakara, who was the eldest of the brothers born in 1606, and Ranganatha who was younger than Kamalakara.

As was common throughout the classical period of Indian mathematics, members of the family acted as teachers to other family members. In particular Kamalakara was taught by his elder brother Divakara while Divakara himself had been taught by their uncle Siva. Pingree writes in [1]:-
[Kamalakara] combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists (especially Ulugh Beg). Following his family's tradition he wrote a commentary, Manorama, on Ganesa's Grahalaghava and, like his father, Nrsimha, another commentary on the Suryasiddhanta, called the Vasanabhasya ...
Kamalakara's most famous work, the Siddhanta-tattva-viveka, was commented on by Kamalakara himself. The work was completed in 1658. It is a work of fifteen chapters covering standard topics for Indian astronomy texts at this time. It deals with the topics of: units of time measurement; mean motions of the planets; true longitudes of the planets; the three problems of diurnal rotation; diameters and distances of the planets; the earth's shadow; the moon's crescent; risings and settings; syzygies; lunar eclipses, solar eclipses; planetary transits across the sun's disk; the patas of the moon and sun; the "great problems"; and a final chapter which forms a conclusion.

The third chapter of the Siddhanta-tattva-viveka contains some of the most interesting mathematical results. In that chapter Kamalakara used the addition and subtraction theorems for the sine and the cosine to give trigonometric formulae for the sines and cosines of double, triple, quadruple and quintuple angles. In particular he gives formulae for $\sin(A/2)$ and $\sin(A/4)$ in terms of $\sin(A)$ and iterative formulae for $\sin(A/3)$ and $\sin(A/5)$. See for example [7] and [8] for a discussion of the details of Kamalakara's work in this area.

The Siddhanta-tattva-viveka is a Sanskrit text and in it Kamalakara makes frequent use of the place-value number system with Sanskrit numerals. This and many other aspects of the work are discussed in [3].

### References (show)

1. D Pingree, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. G G Joseph, The crest of the peacock (London, 1991).
3. S Dvivedi, The Siddhantatattvaviveka of Kamalakara (Benares, 1935).
4. A K Bag, Indian literature on mathematics during 1400-1800 A.D., Indian J. Hist. Sci. 15 (1) (1980), 79-93.
5. R C Gupta, Kamalakara's mathematics and construction of Kundas, Ganita Bharati 20 (1-4) (1998), 8-24.
6. R C Gupta, Addition and subtraction theorems for the sine and the cosine in medieval India, Indian J. History Sci. 9 (2) (1974), 164-177.
7. R C Gupta, Sines and cosines of multiple arcs as given by Kamalakara, Indian J. History Sci. 9 (2) (1974), 143-150.
8. R C Gupta, Sines of sub-multiple arcs as found in the Siddhanta-tattva-viveka, Ranchi Univ. Math. J. 5 (1974), 21-27.
9. D Pingree, Islamic astronomy in Sanskrit, J. Hist. Arabic Sci. 2 (2) (1978), 315-330; 425.
10. A N Singh, Hindu trigonometry, Proc. Benares Math. Soc. 1 (1939), 77-92.