Bhāskara II

The first section Līlāvatī (also known as pāṭīgaṇita or aṅkagaṇita), named after his daughter, consists of 277 verses.[5] It covers calculations, progressions, measurement, permutations, and other topics.[5]

The second section Bījagaṇita(Algebra) has 213 verses.[5] It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) Pell's equation, solving it using a kuṭṭaka method.[5] In particular, he also solved the ${\displaystyle 61x^{2}+1=y^{2}}$ case that was to elude Fermat and his European contemporaries centuries later.[5]

In the third section Grahagaṇita, while treating the motion of planets, he considered their instantaneous speeds.[5] He arrived at the approximation:[12]

${\displaystyle \sin y'-\sin y\approx (y'-y)\cos y}$ for ${\displaystyle y'}$ close to ${\displaystyle y}$, or in modern notation:[12]

${\displaystyle {\frac {d}{dy}}\sin y=\cos y}$.

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]

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:

• Definitions.
• Properties of zero (including division, and rules of operations with zero).
• Further extensive numerical work, including use of negative numbers and surds.
• Estimation of π.
• Arithmetical terms, methods of multiplication, and squaring.
• Inverse rule of three, and rules of 3, 5, 7, 9, and 11.
• Problems involving 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).[17] His work Bījaganita is effectively a treatise on algebra and contains the following topics:

• Positive and negative numbers.
• The 'unknown' (includes determining unknown quantities).
• Determining unknown quantities.
• Surds (includes evaluating surds).
• Kuṭṭaka (for solving indeterminate equations and Diophantine equations).
• Simple equations (indeterminate of second, third and fourth degree).
• Simple equations with more than one unknown.
• Indeterminate quadratic equations (of the type ax² + b = y²).
• Solutions of indeterminate equations of the second, third and fourth degree.
• Quadratic equations.
• 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.[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 ${\displaystyle \sin \left(a+b\right)}$ and ${\displaystyle \sin \left(a-b\right)}$.

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.[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]

• There is evidence of an early form of Rolle's theorem in his work
  • If ${\displaystyle f\left(a\right)=f\left(b\right)=0}$ then ${\displaystyle f'\left(x\right)=0}$ for some ${\displaystyle \ x}$ with ${\displaystyle \ a<x<b}$
• He gave the result that if ${\displaystyle x\approx y}$ then ${\displaystyle \sin(y)-\sin(x)\approx (y-x)\cos(y)}$, thereby finding the derivative of sine, although he never developed the notion of derivatives.[19]
  • 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 differential vanishes.
• 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.

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.