The Kerala school, and calculus before Newton, the gist

The Kerala school, and calculus before Newton

Between about 1350 and 1600, a chain of teachers and pupils living a short walk apart on one river in Kerala worked out endless sums that give you pi and the sine, roughly two hundred years before Europe did, and the honest answer to "so did they invent calculus?" is neither yes nor no.

This is the short way in. The essay has the working, the sources and the sky.

Around 1530 one of them, Jyeshthadeva, wrote a book that does not just give the rules but proves them, and he wrote it in Malayalam, the language spoken around him, when serious science in India was written in Sanskrit verse. A palm-leaf copy of that book sits in London, with a note in the front, in an Englishman's hand, saying that this is the one with the proofs in it.

From about 1350 to about 1600, a line of teachers and their pupils in Kerala did some of the best mathematics being done anywhere in the world, and nearly all of them lived within a short walk of one another on the bank of a single river. It runs west across the middle of the state. Malayalam calls it the Nila; maps call it the Bharathappuzha. Near its mouth, in what is now the town of Tirur, three small places stand close together.

Start with pi. Pi is what you get when you compare the distance round a circle with the distance straight across it. It is about 3.14159, and the digits never stop.

These men found a way to reach it by adding fractions forever. Take one. Subtract a third. Add a fifth. Subtract a seventh. Carry on with odd numbers underneath. Multiply your running total by four, and it creeps towards pi. An endless sum like that is called a series. Europe found the same series in the 1670s.

Creeping is the problem. A hundred steps of that sum only get you to 3.13, which is not yet 3.14. The Kerala texts fix this. They give a small extra fraction to add at the point where you stop, worked out so that your answer jumps much closer. With the best of the three fixes they record, thirty-one steps are enough for nine correct digits. Knowing how wrong you are is often harder than being roughly right, and they knew.

The results are credited to Madhava, who worked around 1400. Not one page of his mathematics survives in his own arrangement. We have it because his pupils' pupils wrote it down and said whose it was.

So was it calculus? Partly. They could do what we call integration, for the particular problems they needed it for, and they proved their steps. They did not have the general machinery that came later in Europe: no idea of a function, no method of differentiation you can apply to anything you like, no symbols. And there is no evidence any of it reached Europe. Both halves of that are worth saying out loud.

Words decoded

the Nila, or Bharathappuzha: the river in Kerala on whose lower bank almost all of these men lived
pi: the number relating the distance round a circle to the distance across it, about 3.14159
series: an endless sum, where you keep adding terms and the running total closes in on some number
Madhava of Sangamagrama: the mathematician at the head of this line, working roughly 1340 to 1425; nobody has settled where his village was
Yuktibhasha (युक्तिभाषा): "the speech of reasoning", the book of proofs written around 1530 by Jyeshthadeva
Malayalam: the everyday language of Kerala, as against Sanskrit, the learned language almost all Indian science was written in
integration: the calculus method for adding up an endless number of tiny pieces to get a total, such as an area
differentiation: the calculus method for measuring how fast something is changing at one instant
palm-leaf manuscript: a book written with a stylus on dried palm leaves, cut long and narrow, stacked and tied with a cord through a hole
jyotisha (ज्योतिष): the Indian tradition of sky-reading, covering both the astronomy and the astrology

Why anyone should careThese were working astronomers, and the principle one of them insisted on in 1431, that a computed sky must be checked against the sky you can see, still separates one kind of Indian almanac from another. Their story is also the cleanest available test of two bad habits: claiming everything for the old tradition, and granting it nothing. Neither habit survives contact with what is actually written on the leaves.

Read it properly, 8 min The Kerala school, and calculus before Newton