Bhāskara II

Not to be confused with Bhāskara I.
Bhāskara II

Bhāskara[1] (also known as Bhāskarāchārya ("Bhāskara the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I) (1114–1185), was an Indian mathematician and astronomer. He was born in Bijapur in Karnataka.[2]

Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India.[3] His main work Siddhānta Shiromani, (Sanskrit for "Crown of Treatises")[4] is divided into four parts called Lilāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya,[5] which are also sometimes considered four independent works.[6] These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karaṇa Kautūhala.[6]

Bhāskara's work on calculus predates Newton and Leibniz by over half a millennium.[7][8] He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.[9]

Date, place, and family

Bhāskara gives his date of birth, and date of composition of his major work, in a verse in the Āryā metre:[6]

śhaka-nṛpa samaye 'bhavat mamotpattiḥ /
rasa-guṇa-varṣeṇa mayā
siddhānta-śiromaṇī racitaḥ //

This reveals that he was born in 1036 of the Śhaka era (1114 CE), and that he composed the Siddhānta Śiromaṇī when he was 36 years old.[6] He also wrote another work called the Karaṇa-kutūhala when he was 69 (in 1183).[6] His works show the influence of Brahmagupta, Sridhara, Mahāvīra, Padmanābha and other predecessors.[6]

He was born near Vijjadavida (believed to be Bijjaragi of Vijayapur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of medieval India. He lived in the Sahyadri region (Patnadevi, in Jalgaon district, Maharashtra).[1]

History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father Mahesvara[1] (Maheśvaropādhyāya[6]) was a mathematician, astronomer[6] and astrologer, who taught him mathematics, which he later passed on to his son Loksamudra. Loksamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings.

The Siddhanta-Shiromani


The first section Līlāvatī (also known as pāṭīgaṇita or aṅkagaṇita) it is the name of his daughter. consists of 277 verses.[6] It covers calculations, progressions, measurement, permutations, and other topics.[6]


The second section Bījagaṇita has 213 verses.[6] It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) Pell's equation, solving it using a kuṭṭaka method.[6] In particular, he also solved the case that was to elude Fermat and his European contemporaries centuries later.[6]


In the third section Grahagaṇita, while treating the motion of planets, he considered their instantaneous speeds.[6] He arrived at the approximation:[10]

for close to , or in modern notation:[10]

In his words:[10]

bimbārdhasya koṭijyā guṇastrijyāhāraḥ phalaṃ dorjyāyorantaram

This result had also been observed earlier by Muñjalācārya (or Mañjulācārya) in 932, in his astronomical work 'Laghu-mānasam, in the context of a table of sines.[10]

Bhāskara also stated that at its highest point a planet's instantaneous speed is zero.[10]


Bhaskara's Method for Pythagorean Theorem.

Some of Bhaskara's contributions to mathematics include the following:


Bhaskara's arithmetic text Leelavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.

Lilavati is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and measurement. More specifically the contents include:

His work is outstanding for its systemisation, improved methods and the new topics that he has introduced. Furthermore, the Lilavati contained excellent recreative problems and it is thought that Bhaskara's intention may have be.


His Bijaganita ("Algebra") was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root).[15] His work Bijaganita is effectively a treatise on algebra and contains the following topics:

Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax2 + bx + c = y.[15] Bhaskara's method for finding the solutions of the problem Nx2 + 1 = y2 (the so-called "Pell's equation") is of considerable importance.[13]


The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discovered spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, discoveries first found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for and .


His work, the Siddhānta Shiromani, is an astronomical treatise and contains many theories not found in earlier works. Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry, differential calculus and integral calculus that are found in the work are of particular interest.

Evidence suggests Bhaskara was acquainted with some ideas of differential calculus.[15] Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'.[16]

Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work and further advanced the development of calculus in India.


Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Earth to orbit the Sun, as 365.2588 days which is the same as in Suryasiddhanta. The modern accepted measurement is 365.2563 days, a difference of just 3.5 minutes.

His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere.

The twelve chapters of the first part cover topics such as:

The second part contains thirteen chapters on the sphere. It covers topics such as:


The earliest reference to a perpetual motion machine date back to 1150, when Bhāskara II described a wheel that he claimed would run forever.[18]

Bhāskara II used a measuring device known as Yaṣṭi-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.[19]


In his book Lilavati, he reasons: "In this quantity also which has zero as its divisor there is no change even when many quantities have entered into it or come out [of it], just as at the time of destruction and creation when throngs of creatures enter into and come out of [him, there is no change in] the infinite and unchanging [Vishnu]".[20]


It has been stated, by several authors, that Bhaskara II proved the Pythagorean theorem by drawing a diagram and providing the single word "Behold!".[21][22] Sometimes Bhaskara's name is omitted and this is referred to as the Hindu proof, well known by schoolchildren.[23]

However, as mathematics historian Kim Plofker points out, after presenting a worked out example, Bhaskara II states the Pythagorean theorem:

Hence, for the sake of brevity, the square-root of the sum of the squares of the arm and upright is the hypotenuse: thus it is demonstrated.[24]

This is followed by:

And otherwise, when one has set down those parts of the figure there [merely] seeing [it is sufficient].[24]

Plofker suggests that this additional statement may be the ultimate source of the widespread "Behold!" legend.

See also


  1. 1 2 3 Pingree 1970, p. 299.
  2. Mathematical Achievements of Pre-modern Indian Mathematicians by T.K Puttaswamy p.331
  3. Chopra 1982, pp. 52–54.
  4. Plofker 2009, p. 71.
  5. Poulose 1991, p. 79.
  6. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 S. Balachandra Rao (July 13, 2014), "ನವ ಜನ್ಮಶತಾಬ್ದಿಯ ಗಣಿತರ್ಷಿ ಭಾಸ್ಕರಾಚಾರ್ಯ", Vijayavani, p. 17
  7. Seal 1915, p. 80.
  8. Sarkar 1918, p. 23.
  9. Goonatilake 1999, p. 134.
  10. 1 2 3 4 5 S. Balachandra Rao (July 13, 2014), "ನವ ಜನ್ಮಶತಾಬ್ದಿಯ ಗಣಿತರ್ಷಿ ಭಾಸ್ಕರಾಚಾರ್ಯ", Vijayavani, p. 21
  11. Verses 128, 129 in Bijaganita Plofker 2007, pp. 476-477
  12. 1 2 Mathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy
  13. 1 2 Stillwell1999, p. 74.
  14. Students& Britannica India. 1. A to C by Indu Ramchandani
  15. 1 2 3 50 Timeless Scientists von K.Krishna Murty
  16. Shukla 1984, pp. 95–104.
  17. Cooke 1997, pp. 213–215.
  18. White 1978, pp. 52–53.
  19. Selin 2008, pp. 269–273.
  20. Colebrooke 1817.
  21. Eves 1990, p. 228
  22. Burton 2011, p. 106
  23. Mazur 2005, pp. 19-20
  24. 1 2 Plofker 2007, p. 477


Further reading

External links

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