| Bhāskara II | |
| Birth Place: | Vijjadavida, Maharashtra (probably Patan[1] [2] in Khandesh or Beed[3] [4] [5] in Marathwada) |
| Death Place: | Ujjain, Madhya Pradesh |
| Other Names: | Bhāskarācārya |
| Occupation: | Astronomer, mathematician |
| Era: | Shaka era |
| Discipline: | Mathematician, astronomer, geometer |
| Main Interests: | Algebra, arithmetic, trigonometry |
| Notable Works: |
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Bhāskara II (in Sanskrit pronounced as /bʰɑːskərə/; 1114–1185), also known as Bhāskarāchārya, was an Indian polymath, mathematician, astronomer and engineer. From verses in his main work, Siddhāṁta Śiromaṇī, it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Satpuda mountain ranges of Western Ghats, believed to be the town of Patana in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars.[6] 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.[7] [8] Henry 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 Godavari.[9]
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.
Bhāskara gives his date of birth, and date of composition of his major work, in a verse in the Āryā metre:
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 Brahmagupta, Śrīdhara, 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[10] 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:[3]
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[11] whereas a section of scholars identified it with the modern day Beed city.[1] Some sources identified Vijjalavida as Bijapur or Bidar in Karnataka.[12] Identification of Vijjalavida with Basar in Telangana has also been suggested.[13]
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
61x2+1=y2
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
\siny'-\siny ≈ (y'-y)\cosy
y'
y
| d | |
| dy |
\siny=\cosy
This result had also been observed earlier by Muñjalācārya (or Mañjulācārya) mānasam, in the context of a table of sines.
Bhāskara also stated that at its highest point a planet's instantaneous speed is zero.
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.
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. 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
\sin\left(a+b\right)
\sin\left(a-b\right)
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. 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'.
f\left(a\right)=f\left(b\right)=0
f'\left(x\right)=0
x
a<x<b
x ≈ y
\sin(y)-\sin(x) ≈ (y-x)\cos(y)
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.[18] The modern accepted measurement is 365.25636 days, a difference of 3.5 minutes.[19]
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.
Bhāskara II invented a variety of instruments one of which is 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:
This is followed by:
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.[20]
Invis Multimedia released Bhaskaracharya, an Indian documentary short on the mathematician in 2015.[21] [22]