Indian mathematician and astronomer (1114 – c. 1185 CE), head of the astronomical observatory at Ujjain — the same post Brahmagupta had held five centuries earlier. He completed the chakravāla method for solving Pell's equation, worked with concepts that closely anticipate differential calculus, and wrote the celebrated textbook Līlāvatī, reportedly named for his daughter.
Bhāskara II (also called Bhāskarācārya, "Bhāskara the Teacher") lived in 12th-century India and became head of the same astronomical observatory in Ujjain that Brahmagupta had led five hundred years before him. He is best known today for the Līlāvatī, a beautifully written mathematics textbook that, according to a popular but historically unverifiable legend, he named after his daughter to console her after an astrological mishap prevented her wedding.
The Līlāvatī is famous for posing mathematical problems as charming stories and riddles rather than dry exercises — a pedagogical style still admired today. One famous problem asks the reader to find how many bees are in a swarm, given a series of poetic clues about fractions of the swarm flying to different flowers.
| Born | 1114 CE, Vijjadavida (possibly modern Bijapur/Jalgaon region), India |
| Died | c. 1185 CE |
| Position | Head of astronomical observatory, Ujjain |
| Major work | Siddhānta Śiromaṇi (1150 CE), comprising Līlāvatī and Bījagaṇita |
The chakravāla ("cyclic") method solves Pell's equation Nx² + 1 = y² for integer solutions. It was initiated by Brahmagupta (628 CE) and Jayadeva, and completed by Bhāskara II in the Bījagaṇita. The method uses a clever cyclic algorithm to generate successively better approximate solutions until an exact one is found — an approach that predates European work on the same equation (misleadingly called "Pell's equation" after John Pell, who had little to do with it) by roughly 500 years. Lagrange gave a complete European theory only in 1766.
In the Siddhānta Śiromaṇi, Bhāskara showed that when a variable quantity reaches a maximum value, its "differential" vanishes — essentially the calculus fact that a function's derivative is zero at a maximum. He applied a form of this idea to planetary motion, computing what is effectively an instantaneous rate of change. This represents a genuine anticipation of a core calculus concept, though it was not developed into the full, general differential and integral calculus that emerged in Europe five centuries later.
Bhāskara proposed that a number divided by zero yields infinity (khahara): a/0 = ∞. This is closer to the modern limiting perspective than earlier Indian attempts (Brahmagupta had said 0/0 = 0), though it is still not the modern treatment, which regards division by zero as simply undefined within ordinary arithmetic.
The Bījagaṇita contains a proof of the Pythagorean theorem using a rearrangement (dissection) argument, accompanied only by a diagram and the single word Paśya ("Behold!") — trusting the visual proof to speak for itself without further verbal elaboration.
The popular story that Bhāskara named his textbook after his daughter Līlāvatī, whose wedding was ruined by an astrological miscalculation, appears in later commentarial traditions but is not verified in Bhāskara's own writing or in contemporary sources. Historians of Indian mathematics (Plofker, Hayashi) treat the story as legendary rather than established biographical fact, though it remains a beloved part of the popular tradition surrounding the text.
Claims that Bhāskara II "discovered calculus" or "invented differential calculus" circulate widely in popular and some nationalist historical writing, but professional historians of mathematics (Plofker 2009; Katz 1995) urge caution: Bhāskara's results on rates of change and maxima are genuine, specific, and impressive applications to particular astronomical problems, but they do not constitute the general, systematic theory of differentiation and integration — with its unifying Fundamental Theorem connecting the two — that Newton and Leibniz developed. The precise degree of continuity or independent parallel development between such Indian results and later European calculus remains a subject of ongoing, careful scholarly investigation rather than a settled question in either direction.
The standard critical edition and translation of the Līlāvatī and Bījagaṇita is Colebrooke (1817), supplemented by more recent scholarly analysis in Plofker (2009) placing the works in fuller historical and mathematical context, including their relationship to earlier Indian texts (Āryabhaṭīya, Brāhmasphuṭasiddhānta) and later commentarial tradition (particularly the influential 16th-century commentary by Gaṇeśa Daivajña).