Alathiyur, on the northern bank of the Bharathappuzha near Tirur in what is now Malappuram district, was the home of an astronomer whose careful, unglamorous work reset the calendar of an entire region. Parameshvara belonged to the Vatasseri illam there, and Sanskrit texts render the village name as Asvatthagrama, the village of the peepal tree, translating the Malayalam directly. He was born around 1380, though one respected reckoning puts it a decade earlier, and he seems to have lived to about eighty, dying around 1460. In the political geography of his day the village lay within the small kingdom of Tanur; today it is a modest place on the road between Tirur and Kuttippuram, and the river he used as a horizon for his measurements still runs past it.

An inherited profession

Parameshvara's family were Namboothiri Brahmins who practised astronomy and astrology as a hereditary calling, a common arrangement in medieval Kerala where the accuracy of temple calendars depended on a handful of such households. He was, by his own tradition, descended through a line of pupils from Govinda Bhattathiri, the thirteenth-century author of a widely used astrological work. What lifted him out of routine practice was his teachers. He studied under a scholar named Rudra, another named Narayana, and, most importantly, under Madhava of Sangamagrama, the mathematician whose infinite series for sine, cosine and the arctangent anticipated European results by roughly three centuries. Parameshvara is one of the few people we can name with confidence as Madhava's direct student, and he is also one of the main reasons Madhava's results survived at all, since he quoted them in his own writings.

Fifty-five years at the eyepiece

The quality that distinguishes Parameshvara from most Indian astronomers before him is patience with observation. Indian astronomy was rich in computation but comparatively thin in sustained record-keeping; Parameshvara spent more than half a century closing that gap. From the 1390s to the 1430s, and by some accounts into the 1440s, he watched eclipses and planetary positions and compared what he saw against what the standard tables predicted. His texts mention specific occasions and places, including observations made at Navakshetra in 1422 and at Gokarna in 1425 and again in 1430, which is the sort of detail that makes the record credible rather than rhetorical.

What he found was drift. The parameters in circulation in Kerala descended ultimately from Aryabhata and had been fixed locally in the seventh century as the Parahita system; by Parameshvara's lifetime they no longer matched the sky. Rather than patch them, he derived a corrected set from his own observations and published it in 1431 as the Drgganita, a title that can be read as computation that agrees with sight. The name was the argument: a system should be judged against what the eye records. The Drig system was adopted across much of Kerala and remained the working basis of astronomical almanacs there long after his death, a rare case of an individual's measurements changing regional practice.

Mathematics well ahead of its time

Parameshvara's mathematics is scattered through astronomical works rather than gathered into a treatise, which has meant it took a long while to be noticed outside India. Two results stand out. The first is a formula for the radius of the circle passing through the four corners of a cyclic quadrilateral, expressed in terms of the four sides. It is a genuine extension of the classical formulae for triangles, and it was not obtained again in Europe until Simon Lhuilier published it in 1782, roughly three and a half centuries later.

The second is more subtle. In working out how to invert his sine tables, Parameshvara wrote down a rule that amounts to a mean value form of the relationship between a function's change and its rate of change, essentially a special case of what later became the mean value theorem of differential calculus. He also set out iterative procedures for computing sines, including a two-point scheme that behaves like the modern secant method. None of this was framed as calculus in the European sense, and it would be an overstatement to say he invented the discipline; but the underlying insight, that a curve can be approximated locally by tracking rates of change, is unmistakably present.

A prolific commentator

Around twenty-five manuscripts have been identified as his. Many are commentaries, which in the Indian tradition is not a lesser form of authorship: a commentary is where a scholar tests, corrects and extends an inherited text. He wrote on Bhaskara I's Mahabhaskariya and Laghubhaskariya, on Aryabhata's Aryabhatiya, on Munjala's Laghumanasa, on the Suryasiddhanta, and on the Lilavati of Bhaskara II. His independent works include the Goladipika on spherical astronomy, of which two rather different versions survive, the second dated 1443, and short specialised manuals on eclipse computation, one of which is anchored to an epoch in July 1411.

The chain of teachers

His influence ran forward through his own family. His son Damodara was also his pupil, and Damodara in turn taught Nilakantha Somayaji, the astronomer who in the following century proposed a partially heliocentric planetary model in which Mercury, Venus, Mars, Jupiter and Saturn orbit the Sun. Parameshvara is thus a central link in an unbroken sequence of teachers running from Madhava in the late fourteenth century to Jyeshthadeva and the Malayalam mathematical prose of the sixteenth, one of the longest continuous research traditions anywhere in the medieval world. That it was sustained from a cluster of river villages in Malappuram, by families who taught their own children, is the part most often left out of the story.