Madhava's Pi: The 'Gregory-Leibniz' Series
The infinite series for pi that should bear Madhava's name
Discover the infinite series for pi developed by Madhava of Sangamagrama around 1400 CE, later independently rediscovered by Gregory in 1671 and Leibniz in 1674, who got the European credit that should have been Madhava's.
The Verse of the Odd Numbers
Sometime between 1380 and 1400 CE, in the village of Saṅgamagrāma on the Periyar river in central Kerala, Mādhava of Saṅgamagrāma scored a single Sanskrit couplet onto a palm-leaf folio with an iron stylus. The verse begins vyāse vāridhi-nihate. Translated, it tells a student to take the diameter of a circle, multiply it by four, and then subtract one-third of that, add one-fifth of that, subtract one-seventh, and keep going through the odd numbers forever, alternating minus and plus. The running total, in the limit, is the circumference of the circle. Rearrange, and you get the value of π.
It was the earliest known written statement of what modern textbooks call the Gregory-Leibniz series. Gregory would post it in Scotland in 1671. Leibniz would publish it in Germany in 1674. Neither man would know that a Nambudiri Brahmin in a Kerala village had beaten them to it by almost three centuries, and in a form sharper than either of theirs.
Write down the number one. Subtract one third. Add one fifth. Subtract one seventh. Keep going forever, alternating plus and minus, with each successive odd number as the next denominator. The running total, in the limit, is exactly one quarter of pi. Multiply by four and you have pi itself. This is one of the most beautiful identities in mathematics. It ties together the ratio of a circle's circumference to its diameter, the odd numbers, and the operation of infinite addition. It appears in every modern calculus textbook under the names of two European mathematicians, James Gregory of Scotland and Gottfried Wilhelm Leibniz of Germany, who independently wrote it down in 1671 and 1674 respectively. Neither man was the first. The identity was first written down in Sanskrit verse, with a full derivation and a set of correction terms Gregory and Leibniz never knew, by a Kerala astronomer named Madhava of Sangamagrama, working on the Malabar coast of south India, between roughly 1380 and 1420 CE. Gregory and Leibniz were almost three centuries late.