Bhaskara II (Bhaskaracharya): Life, Lilavati and His Mathematics

Bhaskara II, known as Bhaskaracharya, was a 12th-century Indian mathematician and astronomer, born in 1114, whose Siddhanta Shiromani includes the Lilavati on arithmetic and the Bijaganita on algebra. His solutions to hard equations in whole numbers, using the chakravala ("cyclic") method, are among the high points of medieval mathematics anywhere. He is called "II" to distinguish him from the 7th-century astronomer Bhaskara I.

Life

What we know of Bhaskara's life comes mostly from his own verses. At the end of the Goladhyaya he says he was born in Shaka 1036 (1114 CE) and wrote the Siddhanta Shiromani at the age of thirty-six, in 1150. He names his home as Vijjadavida near the Sahyadri mountains and his father Maheshvara as his teacher. A stone inscription of 1207 at Patan, near Chalisgaon, in Maharashtra records that his grandson Changadeva founded a school there to teach Bhaskara's works.

The Siddhanta Shiromani

The "Crown of Treatises" has four parts.

• Lilavati (arithmetic): operations with numbers, fractions, interest, series, mensuration of figures and solids, and combinations, set as verse problems often addressed to a girl, Lilavati. A legend first written down centuries later, in Faizi's Persian translation of 1587, says she was his daughter; the story cannot be confirmed.
• Bijaganita (algebra): rules for positive and negative numbers and zero, equations of one and several unknowns, quadratic equations, and indeterminate equations.
• Grahaganita (mathematics of the planets): mean and true positions of the planets, eclipses and related calculations.
• Goladhyaya (the sphere): the shape of the Earth and the celestial sphere, astronomical instruments, and his own commentary.

He also wrote a prose commentary, the Vasanabhashya, explaining his rules.

The chakravala method

The Bijaganita solves equations of the form Nx² + 1 = y² in whole numbers. Building on the earlier work of Brahmagupta and of Jayadeva, Bhaskara used the chakravala, a cyclic method that reaches a solution step by step. His worked example for N = 61 gives the smallest solution, x = 226,153,980 and y = 1,766,319,049. European mathematicians took up the same problem in the 17th century, and in Europe it later came to be called Pell's equation.

Zero, motion and astronomy

Bhaskara wrote that a number divided by zero gives a quantity he called khahara, which stays unchanged whatever is added to or taken from it; historians compare this with the modern idea of infinity, while noting that his treatment was not fully consistent. In computing the true motion of planets he used the idea of a planet's instantaneous motion (tatkalika gati), which historians compare with ideas later found in the differential calculus. These are comparisons made by historians, not claims that he discovered calculus.

Legacy

The Lilavati was used as a school textbook across India for centuries, translated into Persian and later into English, and commentaries on Bhaskara's works were written well into the colonial period. ISRO's experimental remote-sensing satellite of 1979 carries the name Bhaskara.

Frequently asked questions

Who was Bhaskaracharya?

Bhaskara II, a 12th-century Indian mathematician and astronomer, author of the Siddhanta Shiromani, including the Lilavati and the Bijaganita.

When was Bhaskara II born?

In 1114 CE, by his own statement in the Goladhyaya.

What is the Lilavati?

The arithmetic part of the Siddhanta Shiromani, written as verse problems.

What is the chakravala method?

Bhaskara's cyclic method for solving equations of the form Nx² + 1 = y² in whole numbers.

Was Lilavati Bhaskara's daughter?

A legend recorded in 1587 says so, but there is no earlier evidence.

How is Bhaskara II different from Bhaskara I?

Bhaskara I was a 7th-century astronomer of Aryabhata's school; Bhaskara II lived in the 12th century.

Sources

• Kim Plofker, Mathematics in India (Princeton University Press, 2009).
• J. J. O'Connor and E. F. Robertson, "Bhaskara II", MacTutor History of Mathematics, University of St Andrews.
• Henry Thomas Colebrooke, Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhascara (John Murray, 1817).

Last reviewed: 12 November 2026.

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