Bhaskara (1114 – 1185) (Kannada Kannada is one of the major Dravidian languages of India, spoken predominantly in the state of Karnataka. Kannada, whose native speakers are called Kannadigas (ಕನ್ನಡಿಗರು Kannadigaru), number roughly 38 million, making it the 27th most spoken language in the world. It is one of the scheduled languages of India and the official and ಭಾಸ್ಕರಾಚಾರ್ಯ) (also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher")) was an Indian India, officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the south, the Arabian Sea on the west, and the Bay of Bengal on the east, India has a coastline of 7,517 mathematician Indian mathematics is the mathematics that emerged in South Asia from ancient times until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first recorded in Indian mathematics. Indian and astronomer Indian astronomy—the earliest textual mention of which is given in the religious literature of India —became an established tradition by the 1st millennium BCE, when Jyotiṣa Vedānga and other ancillary branches of learning called Vedangas began to take shape. During the following centuries a number of Indian astronomers studied various. He was born near Bijjada Bida (in present day Bijapur district Bijapur is a district in the state of Karnataka in southern India. The city of Bijapur is the headquarters of the district, and is located 530 km northwest of Bangalore. Bijapur is well known for the great monuments of historical importance built during the Adil Shahi dynasty, Karnataka Karnataka (Kannada: ಕರ್ನಾಟಕ, pronounced [kəɾˈnɑːʈəkɑː] ) is a state in the southern part of India. It was created on November 1, 1956, with the passing of the States Reorganisation Act. Originally known as the State of Mysore, it was renamed Karnataka in 1973 state, South India Southern India, also known as the Dravida in the Indian anthem, is the area encompassing India's states of Andhra Pradesh, Karnataka, Kerala and Tamil Nadu as well as the union territories of Lakshadweep and Pondicherry, occupying 19.31% of India's area. South India lies in the peninsular Deccan Plateau and is bounded by the Arabian Sea in the) into the Deshastha Brahmin Majority speak Marathi. Also a small number of Deshastha resident outside Maharashtra speak local languages such as Gujarati, Hindi, Kannada and Konkani family. Bhaskara was head of an astronomical Astronomy is the scientific study of celestial objects and phenomena that originate outside the Earth's atmosphere (such as the cosmic background radiation). It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the formation and development of the universe observatory at Ujjain Ujjain pronunciation (help·info) (also known as Ujain, Ujjayini, Avanti, Avantikapuri), is an ancient city of Malwa region in central India, on the eastern bank of the Kshipra River Hindi: क्षिप्रा (today part of the state of Madhya Pradesh). It is the administrative centre of Ujjain District and Ujjain Division, the leading mathematical centre of ancient India. His predecessors in this post had included both the noted Indian mathematician Brahmagupta Brahmagupta ( listen ) (598–668) was a great Indian mathematician and astronomer. Brahmagupta wrote important works on mathematics and astronomy. In particular he wrote Brahmasphutasiddhanta (Correctly Established Doctrine of Brahma), in 628. The work was written in 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamala (598–c. 665) and Varahamihira Daivajna Varāhamihira , also called Varaha, or Mihira was an Indian astronomer, mathematician, and astrologer who lived in Ujjain. He is considered to be one of the nine jewels (Navaratnas) of the court of legendary king Vikramaditya (thought to be the Gupta emperor Chandragupta II Vikramaditya). Though little is known about his life, he. He lived in the Sahyadri The Western Ghats also sometimes known as the Sahyadri Mountains, is a mountain range along the western side of India. It runs north to south along the western edge of the Deccan Plateau, and separates the plateau from a narrow coastal plain along the Arabian Sea. The range starts near the border of Gujarat and Maharashtra, south of the River region.

It has been recorded that his great-great-great-grandfather held a hereditary post as a court scholar, as did his son and other descendants. His father Mahesvara 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.^[1]

Bhaskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. His main works were the Lilavati Lilavati was Indian mathematician Bhāskara II's treatise on mathematics in the twelfth century (dealing with arithmetic Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers. In common usage, it refers to the simpler properties when), Bijaganita (Algebra Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure) and Siddhanta Shiromani (written in 1150) which consists of two parts: Goladhyaya (sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in three dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The) and Grahaganita (mathematics of the planets A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.[a]).

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 Persian is an Iranian language within the Indo-Iranian branch of the Indo-European languages. It is widely spoken in Iran, Afghanistan, Tajikistan, Uzbekistan and to some extent in Iraq, Bahrain, and Oman. New Persian, which usually is called also by the names of Farsi, Parsi, Dari or Parsi-ye-Dari (Dari Persian), can be classified linguistically 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 Lilavati was Indian mathematician Bhāskara II's treatise on mathematics in the twelfth century, 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]".^[2]

Mathematics

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

• A proof of the Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem (in British English) is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle in British English). It states: by calculating the same area Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential in two different ways and then canceling out terms to get a² + b² = c².

• In Lilavati, solutions of quadratic In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is, cubic where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function. The integral of a cubic function is a quartic function and quartic where a is nonzero; or in other words, a polynomial of degree of four. Such a function is sometimes called a biquadratic function, but the latter term can occasionally also refer to a quadratic function of a square, having the form indeterminate equations An indeterminate equation, in mathematics, is an equation for which there is an infinite set of solutions; for example, 2x = y is a simple indeterminate equation. Indeterminate equations cannot be directly solved from the given information. For example, the equations.

• Solutions of indeterminate quadratic equations In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is (of the type ax² + b = y²).

• Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Florence in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historic era, but since the changes of the Renaissance were not uniform across Europe, this is a general use of the European mathematicians of the 17th century

• A cyclic Chakravala method The chakravala method is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, although some attribute it to Jayadeva (c. 950 ~ 1000 CE). Jayadeva pointed out that Brahmagupta's approach to solving equations of this type could be generalized, and he then described this 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.

• His method for finding the solutions of the problem x² − ny² = 1 (so-called "Pell's equation") is of considerable interest and importance.

• Solutions of Diophantine equations In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface or more of the second order, such as 61x² + 1 = y². This very equation was posed as a problem in 1657 by the French France (pronounced /ˈfræns/ franss or /ˈfrɑːns/ frahns; French pronunciation (help·info): [fʁɑ̃s]), officially the French Republic (French: République française, pronounced: [ʁepyblik fʁɑ̃sɛz]), is a member state of the European Union located in its western region, with several overseas territories and islands located on other mathematician Pierre de Fermat Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to modern calculus. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the, but its solution was unknown in Europe until the time of Euler Leonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is pronounced /ˈɔɪlər/ OY-lər in English and [ˈɔʏlɐ] in German; the pronunciation /ˈjuːlər/ EW-lər is incorrect in the 18th century.

• Solved quadratic equations In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is with more than one unknown, and found negative Being negative or non-negative is a property of a number which is real, or a member of a subset of real numbers such as rational and integer numbers. A negative number is one that is less than zero, such as −, −1.414, −1. A positive number is one that is greater than zero, such as , 1.414, 1. Zero itself is neither positive nor negative. The and irrational In mathematics, an irrational number is any real number that is not a rational number—that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Informally, this means numbers that cannot be represented as simple fractions. It can be proved that irrational numbers are precisely those real solutions.

• Preliminary concept of mathematical analysis Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function. It also includes the theories of differentiation, integration and.

• Preliminary concept of infinitesimal Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite calculus Calculus is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in, along with notable contributions towards integral calculus Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral.

• Conceived differential calculus In mathematics, differential calculus is a subfield of calculus concerned with the study of how functions change when their inputs change. The primary object of study in differential calculus is the derivative. A closely related notion is the differential. The derivative of a function at a chosen input value describes the behavior of the function, after discovering the derivative In calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity at which the vehicle is traveling. Conversely, the and differential In calculus, the differential represents the principal part of the change in a function y = ƒ with respect to changes in the independent variable. The differential itself is defined by an expression of the form coefficient.

• Stated Rolle's theorem In calculus, a branch of mathematics, Rolle's theorem essentially states that a differentiable function, which attains equal values at two points, must have a point somewhere between them where the slope of the tangent line to the graph of the function is zero, 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.

• Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)

• In Siddhanta Shiromani, Bhaskara developed spherical trigonometry Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles. This is of great importance for calculations in astronomy and earth-surface, orbital and space navigation along with a number of other trigonometric Trigonometry is a branch of mathematics that studies triangles, particularly right triangles. Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships, as well as describing angles in general and the motion of waves such as sound and light waves results. (See Trigonometry section below.)

Arithmetic

Bhaskara's arithmetic Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers. In common usage, it refers to the simpler properties when text Lilavati Lilavati was Indian mathematician Bhāskara II's treatise on mathematics in the twelfth century covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry In mathematics, a plane is a flat surface Chyea. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry, solid geometry In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. It was developed following the development of plane geometry. Stereometry deals with the measurements of volumes of various solid figures: cylinder, circular cone, truncated cone,, 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.
• Arithmetical and geometrical progressions.
• Plane (geometry).
• Solid geometry.
• Permutations and combinations.
• Indeterminate equations (Kuttaka), 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 systemisation, improved methods and the new topics that he has introduced. 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.

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). 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. Bhaskara's method for finding the solutions of the problem Nx² + 1 = y² (the so-called "Pell's equation") is of considerable importance.

He gave the general solutions of:

• Pell's equation using the chakravala method.
• The indeterminate quadratic equation using the chakravala method.

He also solved^{[citation needed]}:

• Cubic equations.
• Quartic equations.
• Indeterminate cubic equations.
• Indeterminate quartic equations.
• Indeterminate higher-order polynomial equations.

Trigonometry

The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discovered 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, discoveries first found in his works include the now well known results for and :

Calculus

His work, the Siddhanta 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. It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement. 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'.^[3]

• There is evidence of an early form of Rolle's theorem in his work:
  • If then for some with

• He gave the result that if then , thereby finding the derivative of sine, although he never developed the general concept of differentiation.^[4]
  • 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.

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.^[5]

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.^[6]

References

1. ^ Plofker, Kim (2007). pp. 447. "Bhāskara, who lived in the Sahyadri region in western Maharashtra, was born in 1114 into a family whose members may have filled hereditary posts as court scholars (at least, it is recorded that his great-great-great-grandfather held such a position under a noble patron, as did Bhaskara's son and some other descendants). Hardly anything is known about the other events of Bhāskara's life; it is speculated that he may have had a daughter named Lilavati because of his allusions to a girl so addressed in his book on arithmetic, and his son's son helped to set up a school in 1207 for the study of Bhaskara's writings."
2. ^ Arithmetic and mensuration of Brahmegupta and Bhaskara, H.T Colebrooke, 1817
3. ^ Shukla, Kripa Shankar (1984). "Use of Calculus in Hindu Mathematics". Indian Journal of History of Science 19: 95–104.
4. ^ Cooke, Roger (1997). "The Mathematics of the Hindus". The History of Mathematics: A Brief Course. Wiley-Interscience. pp. 213–214. ISBN 0471180823.
5. ^ Lynn Townsend White, Jr. (April 1960), "Tibet, India, and Malaya as Sources of Western Medieval Technology", The American Historical Review 65 (3): 522-6
6. ^ Ōhashi, Yukio (2008), "Astronomical Instruments in India", in Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition) edited by Helaine Selin, Springer, pp. 269-273, ISBN 978-1-4020-4559-2

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