Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with
Bhāskara I, was an
astronomer. From verses, in his main work, Siddhānta Shiromani (सिद्धांतशिरोमणी), it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Sahyadri mountain ranges of
Western Ghats, believed to be the town of Patan in Chalisgaon, located in present-day Khandesh region of
Maharashtra by scholars. He is the only ancient mathematician who has been immortalized on a monument. In a temple in Maharashtra, an inscription supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him. Colebrooke who was the first European to translate (1817) Bhaskaracharya II's mathematical classics refers to the family as
Maharashtrian Brahmins residing on the banks of the
Born in a Hindu
Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of a cosmic observatory at
Ujjain, the main mathematical centre of ancient India. 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. His main work Siddhānta-Śiromaṇi, (
Sanskrit for "Crown of Treatises") is divided into four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which are also sometimes considered four independent works. These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karaṇā Kautūhala.
Date, place and family
Bhāskara gives his date of birth, and date of composition of his major work, in a verse in the
This reveals that he was born in 1036 of the
Shaka era (1114
CE), and that he composed the Siddhānta Shiromani when he was 36 years old. Siddhānta Shiromani was completed during 1150 CE. He also wrote another work called the Karaṇa-kutūhala when he was 69 (in 1183). His works show the influence of
Mahāvīra, Padmanābha and other predecessors. Bhaskara lived in
Patnadevi located near Patan (Chalisgaon) in the vicinity of Sahyadri.
He was born in a
Deśastha Rigvedi Brahmin family near Vijjadavida (Vijjalavida).
Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has given the information about the location of Vijjadavida in his work Marīci Tīkā as follows:
This description locates Vijjalavida in Maharashtra, near the
Vidarbha region and close to the banks of
Godavari river. However scholars differ about the exact location. Many scholars have placed the place near Patan in (Chalisgaon Taluka of
Jalgaon district) whereas a section of scholars identified it with the modern day Beed city. Some sources identified Vijjalavida as
Karnataka. Identification of Vijjalavida with
Telangana has also been suggested.
Bhāskara is said to have been the head of an
astronomical observatory at
Ujjain, the leading mathematical centre of medieval India. 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 (Maheśvaropādhyāya) was a mathematician, astronomer and astrologer, who taught him mathematics, which he later passed on to his son Lokasamudra. Lokasamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings. He died in 1185 CE.
The first section Līlāvatī (also known as pāṭīgaṇita or aṅkagaṇita), named after his daughter, consists of 277 verses. It covers calculations, progressions,
measurement, permutations, and other topics.
The second section Bījagaṇita(Algebra) has 213 verses. It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called)
Pell's equation, solving it using a kuṭṭaka method. In particular, he also solved the case that was to elude
Fermat and his European contemporaries centuries later.
In the third section Grahagaṇita, while treating the motion of planets, he considered their instantaneous speeds. He arrived at the approximation: It consists of 451 verses
Solutions of indeterminate quadratic equations (of the type ax2 + b = y2).
Integer solutions of linear and quadratic indeterminate equations (Kuṭṭaka). The rules he gives are (in effect) the same as those given by the
Renaissance European mathematicians of the 17th century.
Chakravala method for solving indeterminate equations of the form ax2 + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.
The first general method for finding the solutions of the problem x2 − ny2 = 1 (so-called "
Pell's equation") was given by Bhaskara II.
Diophantine equations of the second order, such as 61x2 + 1 = y2. This very equation was posed as a problem in 1657 by the French mathematician
Pierre de Fermat, but its solution was unknown in Europe until the time of
Euler in the 18th century.
interest and interest computation.
Indeterminate equations (
Kuṭṭaka), integer solutions (first and second order). His contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the
renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of
Aryabhata and subsequent mathematicians.
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.
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). His work Bījaganita is effectively a treatise on algebra and contains the following topics:
Solutions of indeterminate equations of the second, third and fourth degree.
Quadratic equations with more than one unknown.
Operations with products of several unknowns.
Bhaskara derived a cyclic,
chakravala method for solving indeterminate quadratic equations of the form ax2 + bx + c = y. Bhaskara's method for finding the solutions of the problem Nx2 + 1 = y2 (the so-called "
Pell's equation") is of considerable importance.
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 .
Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. 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 '
There is evidence of an early form of
Rolle's theorem in his work. The modern formulation of Rolle's theorem states that if , then for some with .
He gave the result that if then , thereby finding the derivative of sine, although he never developed the notion of derivatives.
Bhaskara uses this result to work out the position angle of the
ecliptic, a quantity required for accurately predicting the time of an eclipse.
In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a 1⁄33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time.
He was aware that when a variable attains the maximum value, its
He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero. In this result, there are traces of the general
mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value theorem was later found by
Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.
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. The modern accepted measurement is 365.25636
days, a difference of 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
The twelve chapters of the first part cover topics such as:
diurnal rotation.(Diurnal motion is an astronomical term referring to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle, that is called the diurnal circle.)
The earliest reference to a
perpetual motion machine date back to 1150, when Bhāskara II described a
wheel that he claimed would run forever.
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.
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]".
It has been stated, by several authors, that Bhaskara II proved the Pythagorean theorem by drawing a diagram and providing the single word "Behold!". Sometimes Bhaskara's name is omitted and this is referred to as the Hindu proof, well known by schoolchildren.
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.
This is followed by:
And otherwise, when one has set down those parts of the figure there [merely] seeing [it is sufficient].
Plofker suggests that this additional statement may be the ultimate source of the widespread "Behold!" legend.
^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.
^ "1.Ignited minds page 39 by APJ Abdul Kalam,2 Prof Sudakara Divedi (1855-1910),3. Dr B A Salethor (Indian Culture),4. Govt of Karnataka Publications,5.Dr Nararajan (Lilavati 1989),6.Prof Sinivas details(Ganitashatra Chrithra by1955,7.Aalur Venkarayaru (Karnataka Gathvibaya 1917,8 Prime Minister Press Statement at sarawad in 2018,9. Vasudev Herkal (Syukatha Karnataka articles),10. Manjunath sulali (Deccan Herald 19/04/2010,11 Indian Archaeology 1994-96 A Review page 32,Dr R K Kulkarni (Articles)"
^IERS EOP PC Useful constants.
An SI day or mean solar day equals 86400
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), 663–683.
Selin, Helaine, ed. (2008), "Astronomical Instruments in India", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition), Springer Verlag Ny,
Shukla, Kripa Shankar (1984), "Use of Calculus in Hindu Mathematics", Indian Journal of History of Science, 19: 95–104