Bhāskara II

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Bhāskara II
Bornc.1114 AD
Diedc.1185 AD
Other namesBhāskarācārya
Academic background
Academic work
Era Shaka era
Main interests Algebra, Calculus, Arithmetic, Trigonometry
Notable works Siddhānta Shiromani (Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya), Karaṇa-Kautūhala
Bhaskara's proof of the Pythagorean Theorem. Teorema de Pitagoras.Bhaskara.svg
Bhaskara's proof of the Pythagorean Theorem.

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


Born in a Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara was the leader of a cosmic observatory at Ujjain, the main mathematical centre of ancient India. [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-Śiromani, (Sanskrit for "Crown of Treatises") [4] is divided into four parts called Līlāvatī , 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 Karaṇā 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 Shaka 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, Śrīdhara, Mahāvīra, Padmanābha and other predecessors. [6]

He was born in a Deśastha Rigvedi Brahmin family [10] 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 centre of medieval India. He lived in the Sahyadri region (Patnadevi, in Jalgaon district, Maharashtra). [11]

History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father Maheśvara [11] (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. He died in 1185 CE.

The Siddhānta-Śiromani


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


The second section Bījagaṇita(Algebra) 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: [12] It consists of 451 verses

for close to , or in modern notation: [12]

In his words: [12]

bimbārdhasya koṭijyā guṇastrijyāhāraḥ phalaṃ dorjyāyorantaram

This result had also been observed earlier by Muñjalācārya (or Mañjulācārya) mānasam, in the context of a table of sines. [12]

Bhāskara also stated that at its highest point a planet's instantaneous speed is zero. [12]


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


Bhaskara's arithmetic text Līlāvatī 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.

Līlāvatī 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 systematisation, improved methods and the new topics that he introduced. Furthermore, the Lilavati contained excellent problems and it is thought that Bhaskara's intention may have been that a student of 'Lilavati' should concern himself with the mechanical application of the method.[ citation needed ]


His Bījaganita (" 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). [17] His work Bījaganita 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. [17] Bhaskara's method for finding the solutions of the problem Nx2 + 1 = y2 (the so-called "Pell's equation") is of considerable importance. [15]


The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed 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, results 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.[ citation needed ] 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. [17] 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'. [18]

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, Bhāskara 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 approximately 365.2588 days which is the same as in Suryasiddhanta. [ citation needed ] The modern accepted measurement is 365.25636 days, a difference of just 3.5 minutes. [20]

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. [21]

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. [22]


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]". [23]


It has been stated, by several authors, that Bhaskara II proved the Pythagorean theorem by drawing a diagram and providing the single word "Behold!". [24] [25] Sometimes Bhaskara's name is omitted and this is referred to as the Hindu proof, well known by schoolchildren. [26]

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. [27]

This is followed by:

And otherwise, when one has set down those parts of the figure there [merely] seeing [it is sufficient]. [27]

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


A number of institutes and colleges in India are named after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College of Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications and Geo-Informatics in Gandhinagar.

On 20 November 1981 the Indian Space Research Organisation (ISRO) launched the Bhaskara II satellite honouring the mathematician and astronomer. [28]

Invis Multimedia released Bhaskaracharya, an Indian documentary short on the mathematician in 2015. [29] [30]

See also

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Narayaṇa Paṇḍita (1325–1400) was a major mathematician of India. Plofker writes that his texts were the most significant Sanskrit mathematics treatises after those of Bhaskara II, other than the Kerala school. He wrote the Ganita Kaumudi in 1356 about mathematical operations. The work anticipated many developments in combinatorics. About his life, the most that is known is that:

His father’s name was Nṛsiṃha or Narasiṃha, and the distribution of the manuscripts of his works suggests that he may have lived and worked in the northern half of India.

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<i>Līlāvatī</i> Mathematical treatise by Bhāskara II

Līlāvatī is Indian mathematician Bhāskara II's treatise on mathematics, written in 1150. It is the first volume of his main work, the Siddhānta Shiromani, alongside the Bijaganita, the Grahaganita and the Golādhyāya.

Indian mathematics emerged in the Indian subcontinent from 1200 BC until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, and Varāhamihira. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.

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The chakravala method is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, although some attribute it to Jayadeva. Jayadeva pointed out that Brahmagupta's approach to solving equations of this type could be generalized, and he then described this general method, which was later refined by Bhāskara II in his Bijaganita treatise. He called it the Chakravala method: chakra meaning "wheel" in Sanskrit, a reference to the cyclic nature of the algorithm. C.-O. Selenius held that no European performances at the time of Bhāskara, nor much later, exceeded its marvellous height of mathematical complexity.

<i>Aryabhatiya</i> Sanskrit astronomical treatise by the 5th century Indian mathematician Aryabhata

Aryabhatiya or Aryabhatiyam, a Sanskrit astronomical treatise, is the magnum opus and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimated scripture of book around 510 CE based on speculative parameters in text.

History of trigonometry

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).

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A timeline of algebra and geometry

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In mathematics, Bhaskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhaskara I, a seventh-century Indian mathematician. This formula is given in his treatise titled Mahabhaskariya. It is not known how Bhaskara I arrived at his approximation formula. However, several historians of mathematics have put forward different hypotheses as to the method Bhaskara might have used to arrive at his formula. The formula is elegant, simple and enables one to compute reasonably accurate values of trigonometric sines without using any geometry whatsoever.

Sudhakara Dvivedi (1855–1910) was an Indian scholar in Sanskrit and mathematics.

Bijaganita was Indian mathematician Bhāskara II's treatise on algebra. It is the second volume of his main work Siddhānta Shiromani, Sanskrit for "Crown of treatises," alongside Lilāvati, Grahaganita and Golādhyāya.

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.


  1. Mathematical Achievements of Pre-modern Indian Mathematicians by T.K Puttaswamy p.331
  2. Sahni 2019, p. 50.
  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 (13 July 2014), ನವ ಜನ್ಮಶತಾಬ್ದಿಯ ಗಣಿತರ್ಷಿ ಭಾಸ್ಕರಾಚಾರ್ಯ, Vijayavani, p. 17
  7. Seal 1915, p. 80.
  8. Sarkar 1918, p. 23.
  9. Goonatilake 1999, p. 134.
  10. The Illustrated Weekly of India, Volume 95. Bennett, Coleman & Company, Limited, at the Times of India Press. 1974. p. 30. Deshasthas have contributed to mathematics and literature as well as to the cultural and religious heritage of India. Bhaskaracharaya was one of the greatest mathematicians of ancient India.
  11. 1 2 Pingree 1970, p. 299.
  12. 1 2 3 4 5 Scientist (13 July 2014), ನವ ಜನ್ಮಶತಾಬ್ದಿಯ ಗಣಿತರ್ಷಿ ಭಾಸ್ಕರಾಚಾರ್ಯ, Vijayavani, p. 21
  13. Verses 128, 129 in Bijaganita Plofker 2007 , pp. 476–477
  14. 1 2 Mathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy
  15. 1 2 Stillwell1999, p. 74.
  16. Students& Britannica India. 1. A to C by Indu Ramchandani
  17. 1 2 3 50 Timeless Scientists von K.Krishna Murty
  18. Shukla 1984, pp. 95–104.
  19. Cooke 1997, pp. 213–215.
  20. IERS EOP PC Useful constants. An SI day or mean solar day equals 86400 SI seconds. From the mean longitude referred to the mean ecliptic and the equinox J2000 given in Simon, J. L., et al., "Numerical Expressions for Precession Formulae and Mean Elements for the Moon and the Planets" Astronomy and Astrophysics 282 (1994), 663683.
  21. White 1978, pp. 52–53.
  22. Selin 2008, pp. 269–273.
  23. Colebrooke 1817.
  24. Eves 1990 , p. 228
  25. Burton 2011 , p. 106
  26. Mazur 2005 , pp. 19–20
  27. 1 2 Plofker 2007 , p. 477
  28. Bhaskara NASA 16 September 2017
  29. "Anand Narayanan". IIST .
  30. "Great Indian Mathematician - Bhaskaracharya". indiavideodotorg. 22 September 2015.


Further reading