Bhāskara II (; 1114–1185), commonly known as Bhāskarāchārya (), was an Indian polymath, mathematician, and astronomer whose works represent the zenith of the classical Siddhānta school of Jyotiṣa (astronomy and applied mathematics—one of the six Vedāngas or auxiliary disciplines of Vedic scholarship). From verses in his main work, Siddhānta Śiromaṇi, it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Satamala mountain ranges of Western Ghats, believed to be the town of Patana in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars. 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.गणिती (Marathi term meaning Mathematicians) by Achyut Godbole and Dr. Thakurdesai, Manovikas, First Edition 23, December 2013. p. 34.Henry Colebrooke who was the first European to translate (1817) Bhaskaracharya's mathematical classics refers to the family as Maharashtrian Brahmins residing on the banks of the Godavari.Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhascara by Henry Colebrooke, Scholiasts of Bhascara p., xxvii
Born in a Hindu Deshastha Brahmin family lineage of scholars, mathematicians and astronomers, Bhāskara II was the head of the astronomical observatory at Ujjain, the historic reference meridian (madhyarekhā) and premier institutional center of mathematical astronomy in 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 He belongs from a long line of court scholars, mostly astronomer-astrologers in western India. His grandson Caṅgadeva founded a school in 1207 for the study of Bhāskara’s works, helping to ensure their continued dissemination. His works were also notable for their careful organization, clear exposition, and inclusion of worked examples and solutions.
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 Bijapur or Bidar in Karnataka."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)" Identification of Vijjalavida with Basar in Telangana has also been suggested.B.I.S.M. quarterly, Poona, Vol. 63, No. 1, 1984, pp 14-22 However, the identifications suggested by these sources remain untenable.
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 Siddhānta-Śiromaṇi He wrote the Siddhānta Śiromaṇi (which may be translated as "Crest-jewel of Siddhāntas") in 1150 at the age of 36. It became an important astronomical treatise, even though, since it was an orthodox Brāhmapakṣa work, it did not supplant the canonical texts of the other astronomical traditions.The Siddhānta Śiromaṇi is sometimes said to include the Līlāvatī and Bījagaṇita together with its two astronomical sections, and Bhāskara considered the subjects of these works to be closely related. Yet as the Līlāvatī and Bījagaṇita each have their own titles and were usually transmitted as separate manuscripts, the Siddhānta Śiromaṇi is normally regarded as a separate work concerned with astronomy.The Siddhānta Śiromaṇi consists of two main parts. The first of these, called Grahagaṇita ("planetary calculations"), sets out the methods for calculating the mean and true motions of the planets, the Three Questions, lunar and solar eclipses, and other astronomical phenomena. The second part, known as Golādhyāya, is concerned with spherical astronomy and contains treatments of the celestial sphere, the shape of the sphere, the Earth, topics connected with the gaṇita section, astronomical instruments, the seasons, and questions meant to test the student's knowledge.
Līlāvatī
The Līlāvatī, composed by Bhāskara II, is one of the most famous Sanskrit mathematical works, covering a wide range of topics in arithmetic, progressions, geometric measurement, and related mathematical techniques. The title, meaning “Beautiful” or “Playful". While a popular legend—first recorded in 1587 by the Mughal poet Faizi in his persian translation—claims the work was named after Bhāskara's widowed daughter, no contemporary evidence confirms that it was named after her.
Like classical Indian mathematical texts, the work was composed in highly compressed metered Sanskrit verse (padya), designed for memorization, rhythmic recitation, and oral transmission, and were often accompanied by prose commentaries. In Līlāvatī, mathematical problems are addressed playfully to an inquisitive young woman or student using imagery from nature—such as flights of swans, swarms of bees, peacock, and blooming lotuses this also demonstrates the integration of aesthetics and the functional role of mathematics in classical Indian education systems.
Bījagaṇita The work known as the Bījagaṇita ("Seed of Computation") was presumably meant to be a more advanced treatise on mathematics than the Līlāvatī. As a result, it has survived in a smaller number of manuscripts and commentaries and seems to have had a narrower readership. It is also distinguished by the small number of works that compete with it, and is considered to be the earliest surviving independent treatise in the Indian mathematical tradition that is entirely devoted to algebra. Similar to the Līlāvatī, it is made up of verses that are continuously numbered and are grouped into sections covering various mathematical subjects; these sections form the basis for the arrangement of the work's mathematical content.
Bījagaṇita contains systematic rules concerning arithmetic operations with positive and negative numbers, zero, unknowns indicated by colours, and surds; it provides methods for solving linear and quadratic equations, indeterminate equations, and equations with more than one unknown, including the pulverizer and the cyclic method for Pell-type equations; the work also examines both negative and positive roots and offers procedures for solving algebraic problems which extend and systematise earlier Indian methods.
Grahagaṇita In the third section, Grahagaṇita (451 verses), Bhāskara II treated the instantaneous motion (tātkālika gati) of the planets and distinguished it from their varying apparent motion from their mean motion(madhyama gati). In his astronomical work, Bhāskara gives a result that looks like a precursor to infinitesimal methods:
if then .
This can be interpreted as the discovery that cosine is the derivative of sine,although he did not develop the notion of a derivative.
He observed that when a planet's apparent motion reaches an extremum("parama phala"), its instantaneous variation becomes zero, an early astronomical recognition of the principle that motion is stationary at an extremum.
In the Grahagaṇita, Bhāskara describes this rule in verse:'bimbārdhasya koṭijyā guṇastrijyāhāraḥ phalaṃ dorjyāyorantaram"The product of the cosine of the semi-diameter multiplied by the variation in the anomaly and divided by the radius gives the difference between the two sines."
A related use of Sine differences occurs in Mañjula's astronomical work from the tenth century, the Laghu-mānasa, this text including a highly condensed sine table derived from three "sine-difference" values.
Mathematics Some of Bhaskara's contributions to mathematics include the following:
Proofs of the Pythagorean Theorem. In his algebraic treatise Bijaganita, two methods for calculating the hypotenuse of a right triangle are described. These are equivalent to proofs of the Pythagorean Theorem.Verses 128, 129 in Bijaganita , and the prose commentaries following them. Also, see Hayashi 2009, pp.67-68 that includes a figure not found in . The first method involves dividing the right triangle using a perpendicular line drawn from the right angle to the hypotenuse, and then applying the principle of similar figures and proportions.(See the figure.) The second method involves calculation of the area of the square on the hypotenuse. First, the square is divided into smaller pieces, one square with side length a-b and four triangles, illustrating the area is . (see the figure on the left.) Here, the length of the hypotenuse is denoted by c, and the lengths of the other two sides are denoted by a and b. Then he computes the hypotenuse of a right angled triangle, where a=15 and b=20. Next, the author states a fact equivalent to, and presents a figure (the figure on the right) as a demonstration. (The figure accompanies no further explanation.) Finally, he presents as a concise method. In Bījagaṇita and Līlāvatī, Bhāskara II provides geometric and algebraic demonstrations for the relationship between the sides of a right-angled triangle (known in Indian geometry as bhuja (base), koti (altitude) and karna (hypotenuse) from the ancient Śulba Sūtra tradition of Baudhāyana. Rather than relying on deductive axiomatic proofs, Bhāskara offers a visual dissection-and-rearrangement demonstration (kṣetragata-upapatti) for what is known in the West as the Pythagorean property: four congruent right triangles with sides and are arranged within a square on the hypotenuse , leaving the central square of side . This demonstrates the geometric equality: accompanied with his famous prompt Paśya ('Behold!'). (Chapter 4 Mathematics in India, Kim Plofker) Solutions of indeterminate quadratic equations (of the type ax2 + b = y2). Integer solutions of linear and quadratic indeterminate equations (Kuṭṭaka). A cyclic 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.