c. 1500

Early Calculus

Mādhavan of Sangamagrāmam (c. 1350–c.1425), Nīlakantha Somāyaji (1444–1544)

Astronomical research in India through the Middle Ages was initially based on the early findings and writings of Aryabhata and other mathematicians and astronomers; it was ultimately expanded by the creation of dedicated research and teaching groups like the Kerala school of astronomy and mathematics, founded in the fourteenth century by the mathematician Mādhavan of Sangamagrāmam.

Mādhavan and subsequent Kerala mathematicians like Nīlakantha Somāyaji developed mathematical methods of estimating the motions of the planets based initially on geometry and trigonometry and later on newly developed techniques for modeling complex curves and mathematical shapes using combinations of functions. Among these shapes were parabolas, hyperbolas, and ellipses; their work on ellipses proved especially applicable to astronomy because they were able to show that Aryabhata’s earlier conjecture was correct: the paths of the planets could be described by elliptical orbits. The new mathematical methods developed at Kerala that focused on series of functions were early versions of calculus, predating the European development of calculus some 200 years later by scientists like Isaac Newton.

Nīlakantha’s work Aryabhatiyabhasya (a commentary on Aryabhata’s Aryabhatiya), published around 1500, further demonstrated that a rotating Earth and a partially heliocentric solar system provided a more accurate way of fitting the planetary orbits. In his model, Mercury, Venus, Mars, Jupiter, and Saturn all orbited the Sun, but the Sun orbited Earth. A similar model was adopted by the sixteenth-century Danish astronomer Tycho Brahe, and some aspects of Nīlakantha’s model are also consistent with the fully heliocentric cosmology proposed in 1543 by Polish astronomer Nicolaus Copernicus.

The contributions of the Kerala school, and perhaps of Indian mathematics and astronomy in general, may have previously been underappreciated in the West. It seems clear now that they should be counted among the “shoulders of giants” that supported the later discoveries of Copernicus, Newton, and others.

Planetary orbital calculations by mathematicians from the Kerala school in southern India, active between the fourteenth and sixteenth centuries, fit a heliocentric model for the solar system. This figure shows some examples from modern Indian physicists reconstructing the geometry used by Kerala school astronomers.