Journey
Mathematics and astronomy — Kusumapura to Sangamagrama
A thousand years of mathematical astronomy in one line of descent: Āryabhata at Pātaliputra about 499, Brahmagupta in Rajasthan, Varāhamihira at Ujjain, Bhāskara II in the Deccan, and then the Kerala school, which arrived at infinite series for sine, cosine and pi two centuries before Newton and Leibniz. Madhava's Kerala school wrote its results in Sanskrit verse, which is part of why Europe did not notice.
Stops (7)
- Āryabhata · Āryabhata — Kusumapura, 499
He wrote the Āryabhatīya at twenty-three: 121 verses covering arithmetic, algebra, trigonometry and planetary motion, with a sine table, a value of pi accurate to four decimals described as approximate, and the statement that the earth rotates on its axis and that eclipses are shadows rather than demons. Kusumapura is generally identified with Pātaliputra.
- Varāhamihira · Varāhamihira — Ujjain, sixth century
Author of the Pañchasiddhāntikā, which summarises five earlier astronomical systems including one derived from Greek sources, and the Brihat Samhita, an encyclopaedia covering everything from planetary motion to architecture, gemstones and the behaviour of clouds. Ujjain sat on the meridian Indian astronomy used as its prime, which is why the tradition keeps returning to it.
- Brahmagupta · Brahmagupta — Bhillamāla, 628
The Brāhmasphutasiddhānta is the first text anywhere to treat zero as a number with rules of its own and to work systematically with negative quantities, which he described as debts. He also gave the formula for the area of a cyclic quadrilateral. The book reached Baghdad within a century and a half and is how the decimal system entered Arabic mathematics.
- Bhāskara II · Bhāskara II — Ujjain, 1150
The Siddhānta Shiromani, whose Līlāvatī section is arithmetic set as problems addressed to a girl of that name and was used as a textbook for six centuries. He worked with the Pell equation by the chakravāla method, and his treatment of instantaneous motion contains the differential principle in a recognisable form five hundred years before it was named.
- Sangamagrama · Sangamagrāma — Mādhava, about 1400
Mādhava of Sangamagrāma, near Irinjalakuda in Kerala, arrived at infinite series for sine, cosine and the arctangent, and a series for pi with error terms to speed convergence — the results Europe reached with Newton and Leibniz two hundred and fifty years later. Almost none of his own writing survives; he is known through his successors' citations.
- Nīlakantha Somayāji · Nīlakantha Somayāji — Tirur, 1500
He worked out a planetary model in which Mercury and Venus orbit the sun while the sun orbits the earth — the same geometry Tycho Brahe proposed a century later — and wrote the Tantrasangraha and a commentary on Āryabhata that preserves much of Mādhava. He also argued explicitly that astronomical parameters must be revised against observation.
- Jyeshthadeva · Jyeshthadeva — the Yuktibhāshā, about 1530
He wrote the Yuktibhāshā in Malayalam rather than Sanskrit, and it is the first mathematical text anywhere to give systematic proofs rather than results — derivations of the series, of integration for the area of a sphere, worked out and justified step by step. A vernacular textbook of calculus a century before calculus.