Bhaskara's Siddhanta: Mathematical Elegance

The 12th-century masterwork that refined Indian astronomy

Explore Bhaskara II's Siddhanta Shiromani, the crown jewel of Indian astronomical mathematics.

The Crown at Ujjain

Around 1150 CE, at the astronomical observatory on the dusty plain of Ujjain, a thirty-six-year-old mathematician named Bhaskara II sat down to write what he would call Siddhanta Shiromani, 'the crown jewel of treatises.' Palm-leaf manuscripts lay around him, the Indian astronomical inheritance of the previous seven centuries, from Aryabhata's terse verses of 499 CE through Brahmagupta's Brahmasphutasiddhanta of 628 CE. His first book, the arithmetic textbook Lilavati, named for his daughter, was already in use across the subcontinent. The treatise he was now beginning was different in ambition. He intended it to be the final word on the mathematics of the heavens. He was not boasting. The book he produced would stand for six centuries as the most advanced work on astronomy and mathematics anywhere in the world, and it would do so while reading, in places, like love poetry.

A Crown in Four Parts

Siddhanta Shiromani is structured like a spiral staircase. The reader enters through Lilavati, the arithmetic section, which teaches fractions, interest, combinatorics, and the rule of three through riddles addressed to a young girl. Climbing higher, Bijaganita introduces algebra: quadratic equations, the treatment of negative numbers as debt, and the cyclic chakravala method for solving indeterminate equations that Europeans would later call Pell's equation. The third section, Grahaganita, turns the tools of arithmetic and algebra onto the heavens, computing mean and true positions of the planets. The summit is Goladhyaya, the 'chapter on the sphere,' where Bhaskara treats the Earth as a ball hanging in space, describes the instruments needed to measure the heavens, and slips in a casual remark that would have astonished Galileo.

The famous verse sits in Goladhyaya's Bhuvanakosha section. The Earth, Bhaskara writes, possesses an attractive force, and by that force heavy bodies in space are drawn toward her by her own power. So drawn, they appear to fall. In uniform space, where would the Earth itself fall? That is not proto-physics. That is gravity, written five hundred years before Newton, embedded in a verse that also answers the naive objection that a spherical Earth should tumble through the sky.