History of science
Background
Theories/sociology
Historiography
Pseudoscience
By era
In early cultures
in Classical Antiquity
In the Middle Ages
In the Renaissance
Scientific Revolution
By topic
Natural sciences
Astronomy
Biology
Chemistry
Ecology
Geography
Geology
Physics
Social sciences
Economics
Linguistics
Political science
Psychology
Sociology
Technology
Agricultural science
Computer science
Materials science
Medicine
Navigational pages
Timelines
Portal
Categories

Bhaskara (1114-1185), also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher"), was an Indian mathematician-astronomer. He was born near Bijjada Bida (in present day Bijapur district, Karnataka state, South India) in Deshastha Brahmin family and became head of the astronomical observatory at Ujjain, continuing the mathematical tradition of Varahamihira and Brahmagupta. Science is a body of empirical and theoretical knowledge, produced by a global community of researchers, making use of specific techniques for the observation and explanation of real phenomena, this techne summed up under the banner of scientific method. ... Image File history File links Download high resolution version (1020x1508, 359 KB) Book cover Frontispiece of : Tabulae Rudolphinae : quibus astronomicae . ... The sociology and philosophy of science, as well as the entire field of science studies, have in the 20th century been preoccupied with the question of large-scale patterns and trends in the development of science, and asking questions about how science works both in a philosophical and practical sense. ... The historiography of science is the historical study of the history of science (which often overlaps the history of technology, the history of medicine, and the history of mathematics). ... A pseudoscience is any body of knowledge purported to be scientific or supported by science but which fails to comply with the scientific method. ... In prehistoric times, advice and knowledge was passed from generation to generation in an oral tradition. ... The Ptolemaic system of celestial motion, from Harmonia Macrocosmica, 1661. ... The history of science in the Middle Ages refers to the discoveries in the field of natural philosophy throughout the Middle Ages - the middle period in a traditional schematic division of European history. ... Leonardo da Vincis Vitruvian Man, an example of the blend of art and science during the Renaissance. ... // The event which most historians of science call the scientific revolution can be dated roughly as having begun in 1543, the year in which Nicolaus Copernicus published his De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) and Andreas Vesalius published his De humani corporis fabrica (On the... Natural philosophy or the philosophy of nature, known in Latin as philosophia naturalis, is a term applied to the objective study of nature and the physical universe before the development of modern science. ... Astronomy is probably the oldest of the natural sciences, dating back to antiquity, with its origins in the religious practices of pre-history: vestiges of these are still found in astrology, a discipline long interwoven with astronomy, and not completely different from it until about 1750‑1800 in the Western... The history of biology dates as far back as the rise of various civilization as classic philosophers did their own ways of biology as a system of understanding life. ... Portrait of Monsieur Lavoisier and his Wife, by Jacques-Louis David The history of chemistry may be said to begin with the distinction of chemistry from alchemy by Robert Boyle in his work The Skeptical Chymist, which was written after a long and tearfilled talk with his father, and alchymist... ÛEcology is generally spoken of as a new science, having only become prominent in the second half of the 20th Century. ... Wikipedia does not yet have an article with this exact name. ... The growth of physics has brought not only fundamental changes in ideas about the material world, mathematics and philosophy, but also, through technology, a transformation of society. ... For more, see: Social science#History In ancient philosophy, there was no difference between the liberal arts of mathematics and the study of history, poetry or politics—only with the development of mathematical proof did there gradually arise a perceived difference between scientific disciplines and others, the humanities or liberal... The term economics was coined around 1870 and popularized by Alfred Marshall, as a substitute for the earlier term political economy which has been used through the 18th-19th centuries, with Adam Smith, David Ricardo and Karl Marx as its main thinkers and which today is frequently referred to as... Efforts to describe and explain the human language faculty have been undertaken throughout recorded history. ... Antecedents of political science While the study of politics is first found in the Western tradition in Ancient Greece, political science is a late arrival in terms of social sciences. ... The history of psychology as a scholarly study of the mind and behavior dates, in Europe, back to the Late Middle Ages. ... Sociology is a relatively new academic discipline among other social sciences including economics, political science, anthropology, and psychology. ... The wheel was invented circa 4000 BC, and has become one of the worlds most famous, and most useful technologies. ... Agronomy today is very different from what it was before about 1950. ... The history of computer science began long before the modern discipline of computer science that emerged in the 20th century. ... The History of materials science is rooted in the history of the Earth and the culture of the peoples of the Earth. ... This article does not cite its references or sources. ... Alternative meanings: Timeline is a 1999 science fiction novel by Michael Crichton Timeline is a 2003 film based on the novel. ... Events January 7 - Matilda, daughter of Henry I of England, marries Henry IV, Holy Roman Emperor Births Deaths Categories: 1114 ... Events April 25 - Genpei War - Naval battle of Dan-no-ura leads to Minamoto victory in Japan Templars settle in London and begin the building of New Temple Church End of the Heian Period and beginning of the Kamakura period in Japan. ... The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BCE) and Vedic civilization (1500-500 BCE) to modern India (21st century CE). ... An astronomer or astrophysicist is a person whose area of interest is astronomy or astrophysics. ... Bijapur (Kannada: ವಿಜಾಪುರ) is a district in the state of Karnataka. ... Karnātakā   (Kannada: ಕನಾ೯ಟಕ) (IPA: ) is one of the four southern states of India. ... South India is a linguistic-cultural region of India that comprises the four Indian states of Andhra Pradesh, Karnataka, Kerala and Tamil Nadu and the Union Territory of Pondicherry, whose inhabitants are collectively referred to as South Indians. ... Deshastha Brahmins (Marathi: देशस्थ ब्राह्मण) are a Hindu Brahmin sub-caste primarily from the Indian state of Maharashtra and Northern Karnataka. ... A giant Hubble mosaic of the Crab Nebula, a supernova remnant. ... Ujjain   (Hindi:उज्जैन) (also known as Ujain, Ujjayini, Avanti) is an ancient city of central India, in the Malwa region of Madhya Pradesh, on the eastern bank of the Kshipra River. ... Varahamihira (505 – 587) was an Indian astronomer, mathematician, and astrologer born in Ujjain. ... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ...

In many ways, Bhaskara represents the peak of mathematical and astronomical knowledge in the 12th century. He reached an understanding of calculus, astronomy, the number systems, and solving equations, which were not to be achieved anywhere else in the world for several centuries or more. His main works were the Lilavati (dealing with arithmetic), Bijaganita (Algebra) and Siddhanta Shiromani (written in 1150) which consists of two parts: Goladhyaya (sphere) and Grahaganita (mathematics of the planets). (11th century - 12th century - 13th century - other centuries) As a means of recording the passage of time, the 12th century was that century which lasted from 1101 to 1200. ... Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ... A giant Hubble mosaic of the Crab Nebula, a supernova remnant. ... In mathematics, a number system is a set of numbers, or number-like objects, together with one or more operations, such as addition or multiplication. ... Wikipedia does not yet have an article with this exact name. ... Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Events Åhus, Sweden gains city privileges City of Airdrie, Scotland founded King Sverker I of Sweden is deposed and succeeded by Eric IX of Sweden. ... A sphere is a perfectly symmetrical geometrical object. ... The eight planets and three dwarf planets of the Solar System. ...

Legends

Lilavati, 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, found in a Persian translation of Lilavati, Bhaskara 2 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 prevent that, he placed a cup with a small hole at the bottom of the 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 wrong time and she was widowed soon. Bhaskara is said to have taught her mathematics to console her in her grief and to have written the book for her. Wikipedia does not yet have an article with this exact name. ... Persian (local name: Fārsī or Pārsī ) is an Indo-European language spoken in Iran, Afghanistan, Tajikistan and by minorities in Uzbekistan, Turkmenistan, Pakistan, India, Azerbaijan, Armenia, Georgia, Southern Russia, neighboring countries, and elsewhere. ...

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.

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

• Solutions of indeterminate quadratic equations (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 European mathematicians of the 17th Century

• 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.

• 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 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.

• Solved quadratic equations with more than one unknown, and found negative and irrational solutions.

• Preliminary concept of mathematical analysis.

• Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.

• He conceived differential calculus, after discovering the derivative and differential coefficient.

• 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.

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

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

In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ... Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ... An indeterminate equation is an equation for which there is an infinite set of solutions – for example, 2x = y. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Raphael was famous for depicting illustrious figures of the Classical past with the features of his Renaissance contemporaries. ... (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... The Chakravala method is a cyclic algorithm to solve quadratic integer equations. ... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ... In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ... Events January 8 - Miles Sindercombe, would-be-assassin of Oliver Cromwell, and his group are captured in London February - Admiral Robert Blake defeats the Spanish West Indian Fleet in a battle over the seizure of Jamaica. ... Pierre de Fermat Pierre de Fermat (August 17, 1604 – January 12, 1668) is aFrench lawyer at the Parlement of Toulouse, southern France, and a mathematician who is given credit for the development of modern calculus. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Negative has meaning in several contexts: Look up negative in Wiktionary, the free dictionary. ... In philosophy: Irrationality In music: Irrational rhythm In economics: Irrational exuberance In mathematics: Irrational number Proof that e is irrational Quadratic irrational List of integrals of irrational functions See also: rational This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ... This article deals with the concept of an integral in calculus. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... In mathematics, a derivative is the rate of change of a quantity. ... A differential can mean one of several things: Differential (mathematics) Differential (mechanics) Differential signaling is used to carry high speed digital signals. ... In calculus, Rolles theorem states that if a function f is continuous on a closed interval and differentiable on the open interval , and then there is some number c in the open interval such that . Intuitively, this means that if a smooth curve is equal at two points then... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine and cosine. ...

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. Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... Wikipedia does not yet have an article with this exact name. ... In mathematics, plane geometry may mean: geometry of the Euclidean plane; or sometimes geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others; or geometry of the hyperbolic plane or two-dimensional spherical 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. ... The gnomon is the part of a sundial which casts the shadow. ... In mathematics, more precisely in algebra, an indeterminate is a quantity that is not known, and cannot be solved for. ... In combinatorial mathematics, a combination of members of a set is a subset. ...

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. Archimedes' 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. 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... A negative number is a number that is less than zero, such as −3. ... In phonetics, surd is an older (and now rarely-used) alternate name for a voiceless consonant. ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... In mathematics, multiplication is an elementary arithmetic operation. ... In algebra, the square of x is written x2 and is defined as the product of x with itself: x × x. ... The Rule of three is the method of finding the fourth term of a mathematical proportion when three terms are given, given that the products of the first and fourth terms are equal to the product of the second and third. ... Interest is the rent paid to borrow money. ... In mathematics, plane geometry may mean: geometry of the Euclidean plane; or sometimes geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others; or geometry of the hyperbolic plane or two-dimensional spherical 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 has been suggested that this article or section be merged into Combination. ... In mathematics, more precisely in algebra, an indeterminate is a quantity that is not known, and cannot be solved for. ... Raphael was famous for depicting illustrious figures of the Classical past with the features of his Renaissance contemporaries. ... (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... (11th century - 12th century - 13th century - other centuries) As a means of recording the passage of time, the 12th century was that century which lasted from 1101 to 1200. ... Statue of Aryabhata on the grounds of IUCAA, Pune. ...

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: Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ...

• 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. A negative number is a number that is less than zero, such as −3. ... 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... In phonetics, surd is an older (and now rarely-used) alternate name for a voiceless consonant. ... An indeterminate equation is an equation for which there is an infinite set of solutions – for example, 2x = y. ... In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... The Chakravala method is a cyclic algorithm to solve quadratic integer equations. ... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

He gave the general solutions of:

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

He also solved: Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

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

Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...

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 : Events Åhus, Sweden gains city privileges City of Airdrie, Scotland founded King Sverker I of Sweden is deposed and succeeded by Eric IX of Sweden. ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from the Greek Trigona = three angles and metron = measure [1]) is a branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ...

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. Infinitesimal calculus is an area of mathematics pioneered by Gottfried Leibniz based on the concept of infinitesimals, as opposed to the calculus of Isaac Newton, which is based upon the concept of the limit. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from the Greek Trigona = three angles and metron = measure [1]) is a branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... This article deals with the concept of an integral in calculus. ...

Evidence suggests Bhaskara was fully acquainted with the principle of differential calculus, and that his researches were in no way inferior to Newton's work five centuries later, aside from the fact that it seems he did not understand the utility of his researches, and thus historians of mathematics generally neglect his outstanding 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'. Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...

• He also gives the now well known results for and

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

• He was the first to calculate the differential of as
  • 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. In calculus, Rolles theorem states that if a function f is continuous on a closed interval and differentiable on the open interval , and then there is some number c in the open interval such that . Intuitively, this means that if a smooth curve is equal at two points then... A differential can mean one of several things: Differential (mathematics) Differential (mechanics) Differential signaling is used to carry high speed digital signals. ... The plane of the ecliptic is well seen in this picture from the 1994 lunar prospecting Clementine spacecraft. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... A differential can mean one of several things: Differential (mathematics) Differential (mechanics) Differential signaling is used to carry high speed digital signals. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... In calculus, Rolles theorem states that if a function f is continuous on a closed interval and differentiable on the open interval , and then there is some number c in the open interval such that . Intuitively, this means that if a smooth curve is equal at two points then... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ... Parameshvara (परमेश्वर) (1360-1425) was a major mathematician of the Kerala school. ... (14th century - 15th century - 16th century - other centuries) As a means of recording the passage of time, the 15th century was that century which lasted from 1401 to 1500. ... Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician from Kerala, South India. ... Events Europe has about 74 million inhabitants. ... Events Foundation of the Katholieke Universiteit Leuven, Belgium Births John II, Duke of Lorraine (died 1470) Edmund Sutton, English nobleman (died 1483) Deaths January 18 - Edmund Mortimer, 5th Earl of March, English politician (born 1391) March 17 - Ashikaga Yoshikazu, Japanese shogun (born 1407) May 24 - Murdoch Stewart, 2nd Duke of... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... This 14th-century statue from south India depicts the gods Shiva (on the left) and Uma (on the right). ... (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ...

Astronomy

The study of astronomy in Bhaskara's works is based on the heliocentric solar system of gravitation earlier propunded by Aryabhata in 499, where the planets follow an elliptical orbit around the Sun, and the law of gravity described by Brahmagupta in the 7th century. Bhaskara's contributions to astronomy include accurate calculations of many astronomical results based on this heliocentric solar system of gravitation. One of these contributions is his accurate calculation of the sidereal year, the time taken for the Earth to orbit the Sun, as 365.2588 days. The modern accepted measurement is 365.2596 days, a difference of just one minute. Heliocentric Solar System Heliocentrism (lower panel) in comparsion to the geocentric model (upper panel) In astronomy, heliocentrism is the belief that the Sun is at the center of the Universe and/or the Solar System. ... Major features of the Solar System (not to scale, from left to right): Pluto, Neptune, Uranus, Saturn, a comet, Jupiter, the asteroid belt, the Sun, Mercury, Venus, Earth & Moon, and Mars. ... Gravity redirects here. ... Statue of Aryabhata on the grounds of IUCAA, Pune. ... Events March 1 - Pope Symmachus makes Antipope Laurentius bishop of Nocera in Campania. ... The ellipse and some of its mathematical properties. ... The Sun is the star at the center of the Solar System. ... It has been suggested that gravitation be merged into this article or section. ... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... The 7th century is the period from 601 - 700 in accordance with the Julian calendar in the Christian Era. ... In astronomy, heliocentrism is the theory that the Sun is at the center of the Universe and/or the Solar System. ... Major features of the Solar System (not to scale, from left to right): Pluto, Neptune, Uranus, Saturn, a comet, Jupiter, the asteroid belt, the Sun, Mercury, Venus, Earth & Moon, and Mars. ... Gravity redirects here. ... The sidereal year is the time for the Sun to return to the same position in respect to the stars of the celestial sphere. ...

His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere. A sphere is a perfectly symmetrical geometrical object. ...

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.
• Risings and settings.
• 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: Longitude, sometimes denoted by the Greek letter λ (lambda),[1][2] describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ... A planet (from the Greek πλανήτης, planetes or wanderers) is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. ... Diurnal motion is an astronomical term referring to the apparent daily motion of stars in orbit around the Earth, caused by the Earths rotation around its axis. ... Look up Syzygy in Wiktionary, the free dictionary. ... An eclipse refers to the phenomenon of one body passing into the shadow cast by another body. ... Photo taken during the 1999 eclipse. ... Latitude, usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ... Rising may refer to: Rising, an album by Rainbow, released in 1976. ... Apparent magnitude: up to -12. ... This article or section does not cite its references or sources. ... For alternate meanings see star (disambiguation) Hundreds of stars are visible in this image taken by the Hubble Space Telescope of the Sagittarius Star Cloud in the Milky Way Galaxy. ... Pata can refer to: an Indian weapon, see Pata (weapon). ...

• 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.
• First visibilities of the planets.
• Calculating the lunar crescent.
• Astronomical instruments.
• The seasons.
• Problems of astronomical calculations.

He also showed that when a planet is at its furthest from the Earth, or at its closest, the equation of the centre (measure of how far a planet is from the position it is to be predicted to be in by assuming it to movie uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero. Cosmography is the science that maps the general features of the universe; describes both heaven and earth (but without encroaching on geography or astronomy) A representation of the earth or the heavens. ... Mean Motion, , is a measure of how far a satellite has progressed around its orbit, from perigee. ... (This page refers to eccentricity in mathematics. ... In the Ptolemaic system of astronomy, the epicycle (literally: on the cycle in Greek) was a geometric model to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets. ... Armillary sphere An armillary sphere (also known as spherical astrolabe) is a model of the celestial sphere, invented by Eratosthenes in 255 BC. Its name comes from the Latin armilla (circle, bracelet), since it has a skeleton made of graduated metal circles linking the poles and representing the equator, the... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... The ellipse and some of its mathematical properties. ... Lunar may refer to: an adjective that means having to do with or pertaining to the Moon, or to moons in general. ... This article is about divisions of a year. ...

Influence

Some scholars have suggested that Bhaskara's work influenced later developments in the Middle East and Europe. His work was perhaps known to Islamic mathematicians as soon as it was written, and influenced their subsequent writings. The results thus became indirectly known in Europe by the end of the 12th century, but the text itself was not introduced until much later. (Ball, 1960) (See Possible transmission of Kerala mathematics to Europe for other evidence.) A map showing countries commonly considered to be part of the Middle East The Middle East is a region comprising the lands around the southern and eastern parts of the Mediterranean Sea, a territory that extends from the eastern Mediterranean Sea to the Persian Gulf. ... World map exhibiting the location of Europe. ... Islamic mathematics is the profession of Muslim Mathematicians. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ...

There have also been several allegedly unscrupulous attempts to argue that there are traces of Diophantine influence in Bhaskara's work, but this is seen as an attempt by Eurocentric scholars to claim European influence on many great non-European works of mathematics. Particularly in the field of algebra, Diophantus only looked at specific cases and did not achieve the general methods of the Indians. The study of Diophantine equations in India can also be traced back to the Sulba Sutras written from 800 BC to 500 BC, which pre-date Diophantus' work by many centuries. Diophantine means pertaining to the ancient Greek mathematician Diophantus. ... Eurocentrism is the practice, conscious or otherwise, of placing emphasis on European (and, generally, Western) concerns, culture and values at the expense of those of other cultures. ... Cover of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. ... In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Centuries: 10th century BC - 9th century BC - 8th century BC Decades: 850s BC 840s BC 830s BC 820s BC 810s BC - 800s BC - 790s BC 780s BC 770s BC 760s BC 750s BC Events and Trends 804 BC - Hadad-nirari IV of Assyria conquers Damascus. ... Centuries: 7th century BC - 6th century BC - 5th century BC Decades: 550s BC - 540s BC - 530s BC - 520s BC - 510s BC - 500s BC - 490s BC - 480s BC - 470s BC - 460s BC - 450s BC Events and Trends 509 BC - Foundation of the Roman Republic 508 BC - Office of pontifex maximus created...

References

• W. W. Rouse Ball. A Short Account of the History of Mathematics, 4th Edition. Dover Publications, 1960.
• George Gheverghese Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition. Penguin Books, 2000.
• O'Connor, John J., and Edmund F. Robertson. "Bhaskara". MacTutor History of Mathematics archive. St Andrews University, 2000.
• Ian Pearce. Bhaskaracharya II at the MacTutor archive. St Andrews University, 2002.

Penguin Books is a British publisher founded in 1935 by Allen Lane. ... University of St Andrews The University of St Andrews was founded between 1410-1413 and is the oldest university in Scotland and the third oldest in the United Kingdom. ...

See also

• Bhaskara I
• Indian mathematics
• List of Indian mathematicians

Bhāskara, or Bhāskara I, (c. ... The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BCE) and Vedic civilization (1500-500 BCE) to modern India (21st century CE). ... The chronology of Indian mathematics spans from the Indus valley civilization and the Vedas to Modern times. ...

External links

• Bhaskara
• Calculus in Kerala