# Bhāskara II

**Bhāskara**^{[1]} (1114–1185) also known as **Bhāskarāchārya** ("Bhāskara, the teacher"), and as **Bhaskara II** to avoid confusion with Bhāskara I, 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 Karaṇaa 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]}

On 20 November 1981 the Indian Space Research Organisation (ISRO) launched the Bhaskara II satellite honouring the mathematician and astronomer.^{[10]}

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

rasa-guṇa-porṇa-mahīsama

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, 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 centre 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. He died in 1185 CE.

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]}

In the third section *Grahagaṇita*, while treating the motion of planets, he considered their instantaneous speeds.^{[6]} He arrived at the approximation:^{[11]}

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.^{[11]}

Bhāskara also stated that at its highest point a planet's instantaneous speed is zero.^{[11]}

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 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 *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).^{[16]} 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 ax^{2} + bx + c = y.^{[16]} Bhaskara's method for finding the solutions of the problem Nx^{2} + 1 = y^{2} (the so-called "Pell's equation") is of considerable importance.^{[14]}

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.^{[16]} 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'.^{[17]}

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.^{[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 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.^{[20]}

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.^{[21]}

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]".^{[22]}

It has been stated, by several authors, that Bhaskara II proved the Pythagorean theorem by drawing a diagram and providing the single word "Behold!".^{[23]}^{[24]} Sometimes Bhaskara's name is omitted and this is referred to as the *Hindu proof*, well known by schoolchildren.^{[25]}

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.^{[26]}

And otherwise, when one has set down those parts of the figure there [merely] seeing [it is sufficient].^{[26]}

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