Bhāskara II

Bhāskara^[1] (also known as Bhāskara II and Bhāskarāchārya ("Bhāskara the teacher"), (1114–1185), was an Indian mathematician and astronomer. He was born near Vijjadavida (Bijāpur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of ancient India. He lived in the Sahyadri region.^[1]

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.^[2] His main work Siddhānta Shiromani, (Sanskrit for "Crown of treatises,"^[3]) is divided into four parts called Lilāvati, Bijaganita, Grahaganita and Golādhyāya.^[4] These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karan Kautoohal.

Bhāskara's work on calculus predates Newton and Leibniz by half a millennium.^[5]^[6] 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.^[7]

Family

Bhaskaracharya was born into a family belonging to the Deshastha Brahmin community.^[8] 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] was as an 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.

Mathematics

Some of Bhaskara's contributions to mathematics include the following:

A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a² + b² = c².^{[citation needed]}

In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained.

Solutions of indeterminate quadratic equations (of the type ax² + b = y²).^{[citation needed]}

Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century^{[citation needed]}

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.^{[citation needed]}

The first general method for finding the solutions of the problem x² − ny² = 1 (so-called "Pell's equation") was given by Bhaskara II.^[9]

Solutions of 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.^{[citation needed]}

Solved quadratic equations with more than one unknown, and found negative and irrational solutions.^{[citation needed]}

Preliminary concept of mathematical analysis.^{[citation needed]}

Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.^{[citation needed]}

Conceived differential calculus, after discovering the derivative and differential coefficient.^{[citation needed]}

Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works.^{[citation needed]}

Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)^{[citation needed]}

In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.)

Arithmetic

Bhaskara's arithmetic text Lilavati 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 mensuration. 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 (Kuttaka), integer solutions (first and second order)^{[citation needed]}. His contributions to this topic are particularly important^{[citation needed]}, since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century^{[citation needed]}, 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.^{[citation needed]}

His work is outstanding for its systemisation, improved methods and the new topics that he has introduced.^{[citation needed]} Furthermore the Lilavati contained excellent recreative 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]}

Algebra

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)^{[citation needed]}. His work Bijaganita is effectively a treatise on algebra and contains the following topics:

Positive and negative numbers.
Zero.
The 'unknown' (includes determining unknown quantities).
Determining unknown quantities.
Surds (includes evaluating surds).
Kuttaka (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^{[citation needed]}. Bhaskara's method for finding the solutions of the problem Nx² + 1 = y² (the so-called "Pell's equation") is of considerable importance.^[9]

Trigonometry

The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions^{[citation needed]}. He also discovered spherical trigonometry, along with other interesting trigonometrical results^{[citation needed]}. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation^{[citation needed]}. Among the many interesting results given by Bhaskara, discoveries first found in his works include the now well known results for $\sin \left(a+b\right)$ and $\sin \left(a-b\right)$ :^{[citation needed]}

Calculus

His work, the Siddhanta 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.^{[citation needed]}

Evidence^{[citation needed]} suggests Bhaskara was acquainted with some ideas of differential calculus. It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement^{[citation needed]}. 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'.^[10]

There is evidence of an early form of Rolle's theorem in his work^{[citation needed]}:
- If $f\left(a\right)=f\left(b\right)=0$ then $f'\left(x\right)=0$ for some $\ x$ with $\ a<x<b$

He gave the result that if $x\approx y$ $x\approx y$ then $\sin(y)-\sin(x)\approx (y-x)\cos(y)$ $\sin(y)-\sin(x)\approx (y-x)\cos(y)$ , thereby finding the derivative of sine, although he never developed the notion of derivatives.^[11]
- 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.^{[citation needed]}

He was aware that when a variable attains the maximum value, its differential vanishes.^{[citation needed]}

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.^{[citation needed]} He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.^{[citation needed]} In this result, there are traces of the general mean value theorem^{[citation needed]}, one of the most important theorems in analysis^{[citation needed]}, which today is usually derived from Rolle's theorem^{[citation needed]}. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.^{[citation needed]}

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.^{[citation needed]}

Astronomy

Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara 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 365.2588 days ^{[citation needed]} which is same as in Suryasiddhanta. The modern accepted measurement is 365.2563 days, a difference of just 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 sphere.

The twelve chapters of the first part cover topics such as:

Mean longitudes of the planets.
True longitudes of the planets.
The three problems of diurnal rotation.
Syzygies.
Lunar eclipses.
Solar eclipses.
Latitudes of the planets.
Sunrise equation
The Moon's crescent.
Conjunctions of the planets with each other.
Conjunctions of the planets with the fixed stars.
The patas of the Sun and Moon.

The second part contains thirteen chapters on the sphere. It covers topics such as:

Praise of study of the sphere.
Nature of the sphere.
Cosmography and geography.
Planetary mean motion.
Eccentric epicyclic model of the planets.
The armillary sphere.
Spherical trigonometry.
Ellipse calculations.^{[citation needed]}
First visibilities of the planets.
Calculating the lunar crescent.
Astronomical instruments.
The seasons.
Problems of astronomical calculations.

Engineering

The earliest reference to a perpetual motion machine date back to 1150, when Bhāskara II described a wheel that he claimed would run forever.^[12]

Bhāskara II used a measuring device known as Yasti-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.^[13]

Legends

His book on arithmetic is the source of interesting legends that assert that it was written for his daughter, Lilavati. In one of these stories, which is found in a Persian translation of Lilavati, Bhaskara II studied Lilavati's horoscope and predicted that her husband would die soon after the marriage if the marriage did not take place at a particular time. To alert his daughter at the correct time, he placed a cup with a small hole at the bottom of a vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. He put the device in a room with a warning to Lilavati to not go near it. In her curiosity though, she went to look at the device and a pearl from her nose ring accidentally dropped into it, thus upsetting it. The marriage took place at the wrong time and she was soon widowed.

Bhaskara II conceived the modern mathematical convention that when a finite number is divided by zero, the result is infinity. 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]".^[14]

Notes

^ ^a ^b ^c Pingree 1970, p. 299.
^ Chopra 1982, pp. 52–54.
^ Plofker 2009, p. 71.
^ Poulose 1991, p. 79.
^ Seal 1915, p. 80.
^ Sarkar 1918, p. 23.
^ Goonatilake 1999, p. 134.
^ Chopra 1982, p. 52.
^ ^a ^b Stillwell1999, p. 74.
^ Shukla 1984, pp. 95–104.
^ Cooke 1997, pp. 213–215.
^ White 1978, pp. 52–53.
^ Selin 2008, pp. 269–273.
^ Colebrooke 1817.

References

Sarkār, Benoy Kumar (1918), Hindu achievements in exact science: a study in the history of scientific development, Longmans, Green and co.
Seal, Sir Brajendranath (1915), The positive sciences of the ancient Hindus, Longmans, Green and co.
Pingree, David Edwin (1970), Census of the Exact Sciences in Sanskrit, vol. Volume 146, American Philosophical Society, ISBN 978-0-87169-146-0 {{citation}}: |volume= has extra text (help)
Plofker, Kim (2009), Mathematics in India, Princeton University Press, ISBN 978-0-691-12067-6
Colebrooke, Henry T. (1817), Arithmetic and mensuration of Brahmegupta and Bhaskara
White (1978), "Tibet, India, and Malaya as Sources of Western Medieval Technology", Medieval religion and technology: collected essays, University of California Press, ISBN 978-0-520-03566-9 {{citation}}: Text "Lynn Townsend" ignored (help)
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, ISBN 978-1-4020-4559-2
Shukla, Kripa Shankar (1984), "Use of Calculus in Hindu Mathematics", Indian Journal of History of Science, 19: 95–104 {{citation}}: Cite has empty unknown parameter: |coauthors= (help)
Cooke, Roger (1997), "The Mathematics of the Hindus", The History of Mathematics: A Brief Course, Wiley-Interscience, pp. 213–215, ISBN 0-471-18082-3
Poulose, K. G. (1991), K. G. Poulose (ed.), Scientific heritage of India, mathematics, vol. Volume 22 of Ravivarma Samskr̥ta granthāvali, Govt. Sanskrit College (Tripunithura, India) {{citation}}: |volume= has extra text (help)
Chopra, Pran Nath (1982), Religions and communities of India, Vision Books, ISBN 978-0-85692-081-3
Goonatilake, Susantha (1999), Toward a global science: mining civilizational knowledge, Indiana University Press, ISBN 978-0-253-21182-8
Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2001), Mathematics across cultures: the history of non-western mathematics, vol. Volume 2 of Science across cultures, Springer, ISBN 978-1-4020-0260-1 {{citation}}: |volume= has extra text (help)
Stillwell, John (2002), Mathematics and its history, Undergraduate texts in mathematics, Springer, ISBN 978-0-387-95336-6

External links

Template:Persondata

[FOOTNOTEPingree1970299-1] Pingree 1970, p. 299.

[FOOTNOTEChopra198252–54-2] Chopra 1982, pp. 52–54.

[FOOTNOTEPlofker200971-3] Plofker 2009, p. 71.

[FOOTNOTEPoulose199179-4] Poulose 1991, p. 79.

[FOOTNOTESeal191580-5] Seal 1915, p. 80.

[FOOTNOTESarkar191823-6] Sarkar 1918, p. 23.

[FOOTNOTEGoonatilake1999134-7] Goonatilake 1999, p. 134.

[FOOTNOTEChopra198252-8] Chopra 1982, p. 52.

[FOOTNOTEStillwell199974-9] Stillwell1999, p. 74.

[FOOTNOTEShukla198495–104-10] Shukla 1984, pp. 95–104.

[FOOTNOTECooke1997213–215-11] Cooke 1997, pp. 213–215.

[FOOTNOTEWhite197852–53-12] White 1978, pp. 52–53.

[FOOTNOTESelin2008269–273-13] Selin 2008, pp. 269–273.

[FOOTNOTEColebrooke1817-14] Colebrooke 1817.

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