Anantashreni: Infinite Series Two Centuries Before Europe

The foundational concept of calculus, worked out in a Kerala village

Discover how the Kerala school of Madhava developed the concept of anantashreni, the infinite series, which is the foundational idea of modern calculus, in a small Malayalam speaking village in the fourteenth century, more than two hundred years before Newton or Leibniz.

The Problem of the Endless Sum

Around 1400 CE, in a thatched-roof pāṭhaśālā in the village of Saṅgamagrāma on the Periyar river in central Kerala, a Nambudiri Brahmin in his late fifties named Mādhava sat cross-legged with a palm-leaf folio on his lap. On the leaf, with an iron stylus, he was working the oldest unsolved problem in geometry: how to write the length of a curved arc, or the value of π, or the sine of an arbitrary angle, as an exact number rather than a shrinking approximation. The tools around him were simple. A bundle of palm-leaf folios. A stylus. A small brass bowl of turmeric powder to rub into the scored letters so they darkened and stayed readable. The problem in front of him was older than any of them, and it had no finite answer. He knew it. And he had decided to give the answer as an infinite one.

Before there was calculus, there was a problem that made calculus inevitable. Ordinary algebra handles finite sums with ease. Five plus three is eight. A quadratic has two roots and you can write them down. But many of the most useful quantities in nature, the length of a curved arc, the area under a parabola, the exact value of pi, the sine of an angle that is not a round fraction of the circle, are not finite sums at all. They are the limits of processes that require you to add up infinitely many terms, each one smaller than the last, and trust that the total will settle down to a definite value. This is the idea of the anantashreni, the infinite series. It is the doorway through which all of calculus eventually walks. And the doorway was built, opened, and walked through first not in seventeenth century England or Germany but in a small village on the Nila river in fourteenth century Kerala, by a mathematician named Madhava of Sangamagrama and his students.