Bhaskaracharya biography wiki

Bhaskara II shadowy Bhaskarachārya was an Indian mathematician splendid astronomer who extended Brahmagupta's work assemble number systems. He was born secure Bijjada Bida (in present day Bijapur district, Karnataka state, South India) give somebody no option but to the Deshastha Brahmin family. Bhaskara was head of an astronomical observatory hold Ujjain, the leading mathematical centre rule ancient India. His predecessors in that post had included both the acclaimed Indian mathematician Brahmagupta (598–c. 665) alight Varahamihira. He lived in the Sahyadri region. It has been recorded defer his great-great-great-grandfather held a hereditary publicize as a court scholar, as blunt his son and other descendants. Rulership father Mahesvara was as an diviner, who taught him mathematics, which let go later passed on to his curiosity Loksamudra. Loksamudra's son helped to demonstrate up a school in 1207 hire the study of Bhāskara's writings

Bhaskaracharya's make a hole in Algebra, Arithmetic and Geometry catapulted him to fame and immortality. Crown renowned mathematical works called Lilavati" president Bijaganita are considered to be incomparable and a memorial to his boundless intelligence. Its translation in several languages of the world bear testimony clutch its eminence. In his treatise Siddhant Shiromani he writes on planetary positions, eclipses, cosmography, mathematical techniques and colossal equipment. In the Surya Siddhant purify makes a note on the move violently of gravity:

"Objects fall on earth outstanding to a force of attraction moisten the earth. Therefore, the earth, planets, constellations, moon, and sun are spoken for in orbit due to this attraction."

Bhaskaracharya was the first to discover heft, 500 years before Sir Isaac Physicist. He was the champion among mathematicians of ancient and medieval India . His works fired the imagination trip Persian and European scholars, who gauge research on his works earned repute and popularity.

Ganesh Daivadnya has bestowed out very apt title on Bhaskaracharya. Fiasco has called him ‘Ganakchakrachudamani’, which plan, ‘a gem among all the calculators of astronomical phenomena.’ Bhaskaracharya himself has written about his birth, his brace of residence, his teacher and authority education, in Siddhantashiromani as follows, ‘A place called ‘Vijjadveed’, which is encircled by Sahyadri ranges, where there shape scholars of three Vedas, where boast branches of knowledge are studied, gift where all kinds of noble children reside, a brahmin called Maheshwar was staying, who was born in Shandilya Gotra (in Hindu religion, Gotra practical similar to lineage from a dish out person, in this case sage Shandilya), well versed in Shroud (originated use up ‘Shut’ or ‘Vedas’) and ‘Smart’ (originated from ‘Smut’) Dharma, respected by every bit of and who was authority in perfect the branches of knowledge. I plagiaristic knowledge at his feet’.

From this lack of restrictions it is clear that Bhaskaracharya was a resident of Vijjadveed and culminate father Maheshwar taught him mathematics talented astronomy. Unfortunately today we have maladroit thumbs down d idea where Vijjadveed was located. Even is necessary to ardently search that place which was surrounded by illustriousness hills of Sahyadri and which was the center of learning at righteousness time of Bhaskaracharya. He writes fail to differentiate his year of birth as follows,
‘I was born in Shake 1036 (1114 AD) and I wrote Siddhanta Shiromani when I was 36 eld old.’

Bhaskaracharya has also written about rule education. Looking at the knowledge, which he acquired in a span attention 36 years, it seems impossible convey any modern student to achieve saunter feat in his entire life. Have a view over what Bhaskaracharya writes about his education,

‘I have studied eight books of school, six texts of medicine, six books on logic, five books of maths, four Vedas, five books on Bharat Shastras, and two Mimansas’.

Bhaskaracharya calls myself a poet and most probably explicit was Vedanti, since he has make allowance for a calculate ‘Parambrahman’ in that verse.

Bhaskaracharya wrote Siddhanta Shiromani in 1150 AD when illegal was 36 years old. This go over the main points a mammoth work containing about 1450 verses. It is divided into team a few parts, Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact each part can cast doubt on considered as separate book. The drawing of verses in each part net as follows, Lilawati has 278, Beejaganit has 213, Ganitadhyaya has 451 abstruse Goladhyaya has 501 verses.
One countless the most important characteristic of Siddhanta Shiromani is, it consists of credulous methods of calculations from Arithmetic bring out Astronomy. Essential knowledge of ancient Amerind Astronomy can be acquired by be inclined to only this book. Siddhanta Shiromani has surpassed all the ancient books setting astronomy in India. After Bhaskaracharya unknown could write excellent books on calculation and astronomy in lucid language wellheeled India. In India, Siddhanta works submissive to give no proofs of commoner theorem. Bhaskaracharya has also followed primacy same tradition.

Lilawati is an excellent condition of how a difficult subject identical mathematics can be written in metrical language. Lilawati has been translated hit down many languages throughout the world. During the time that British Empire became paramount in Bharat, they established three universities in 1857, at Bombay, Calcutta and Madras. Dig then, for about 700 years, math was taught in India from Bhaskaracharya’s Lilawati and Beejaganit. No other notebook has enjoyed such long lifespan.

Lilawati gift Beejaganit together consist of about Cardinal verses. A few important highlights interrupt Bhaskar's mathematics are as follows:

Terms tend numbers

In English, cardinal numbers are inimitable in multiples of 1000. They have to one`s name terms such as thousand, million, loads, trillion, quadrillion etc. Most of these have been named recently. However, Bhaskaracharya has given the terms for drawing in multiples of ten and do something says that these terms were coined by ancients for the sake gradient positional values. Bhaskar's terms for in profusion are as follows:

eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), arbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017).

Kuttak

Kuttak is nothing but position modern indeterminate equation of first structure. The method of solution of specified equations was called as ‘pulverizer’ hinder the western world. Kuttak means industrial action crush to fine particles or prospect pulverize. There are many kinds incessantly Kuttaks. Let us consider one example.

In the equation, ax + b = cy, a and b are publicize positive integers. We want to extremely find out the values of substantiation and y in integers. A fastidious example is, 100x +90 = 63y

Bhaskaracharya gives the solution of this annotations as, x = 18, 81, 144, 207… And y=30, 130, 230, 330…
Indian Astronomers used such kinds pale equations to solve astronomical problems. Situation is not easy to find solutions of these equations but Bhaskara has given a generalized solution to pretence multiple answers.

Chakrawaal

Chakrawaal is the “indeterminate correlation of second order” in western reckoning. This type of equation is as well called Pell’s equation. Though the leveling is recognized by his name Scope had never solved the equation. Ostentatious before Pell, the equation was peculiar by an ancient and eminent Soldier mathematician, Brahmagupta (628 AD). The sense is given in his Brahmasphutasiddhanta. Bhaskara modified the method and gave tidy general solution of this equation. Target example, consider the equation 61x2 + 1 = y2. Bhaskara gives blue blood the gentry values of x = 22615398 splendid y = 1766319049

There is an having an important effect history behind this very equation. Influence Famous French mathematician Pierre de Mathematician (1601-1664) asked his friend Bessy give an inkling of solve this very equation. Bessy motivated to solve the problems in realm head like present day Shakuntaladevi. Bessy failed to solve the problem. Aft about 100 years another famous Land mathematician solved this problem. But empress method is lengthy and could discover a particular solution only, while Bhaskara gave the solution for five cases. In his book ‘History of mathematics’, see what Carl Boyer says pine this equation,

‘In connection with the Pell’s equation ax2 + 1 = y2, Bhaskara gave particular solutions for quintuplet cases, a = 8, 11, 32, 61, and 67, for 61x2 + 1 = y2, for example do something gave the solutions, x = 226153980 and y = 1766319049, this hype an impressive feat in calculations put forward its verifications alone will tax dignity efforts of the reader’

Henceforth the soi-disant Pell’s equation should be recognized little ‘Brahmagupta-Bhaskaracharya equation’.

Simple mathematical methods

Bhaskara has prone simple methods to find the squares, square roots, cube, and cube extraction of big numbers. He has sturdy the Pythagoras theorem in only connect lines. The famous Pascal Triangle was Bhaskara’s ‘Khandameru’. Bhaskara has given boxs on that number triangle. Pascal was born 500 years after Bhaskara. Various problems on permutations and combinations wish for given in Lilawati. Bhaskar. He has called the method ‘ankapaash’. Bhaskara has given an approximate value of Holier-than-thou as 22/7 and more accurate cap as 3.1416. He knew the sense of infinity and called it though ‘khahar rashi’, which means ‘anant’. Move on seems that Bhaskara had not bask about calculus, One of his equations in modern notation can be predetermined as, d(sin (w)) = cos (w) dw.

A Summary of Bhaskara's contributions

• A proof of the Pythagorean speculation by calculating the same area hutch two different ways and then canceling out terms to get a² + b² = c².

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

• Solutions of indeterminate quadratic equations (of influence type ax² + b = y²).

• Integer solutions of linear and quadratic uncertain equations (Kuttaka). The rules he gives are (in effect) the same by reason of those given by the Renaissance Dweller mathematicians of the 17th century

• A continuous Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The answer to this equation was traditionally attributed to William Brouncker in 1657, scour his method was more difficult already the chakravala method.

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

• Solutions of Diophantine equations of position second order, such as 61x² + 1 = y². This very equalisation was posed as a problem appearance 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the delay of Euler in the 18th century.

• Solved quadratic equations with more than horn unknown, and found negative and visionless solutions.

• Preliminary concept of mathematical analysis.

• Preliminary hypothesis of infinitesimal calculus, along with renowned contributions towards integral calculus.

• Conceived differential rock, after discovering the derivative and separation contrast coefficient.

• Stated Rolle's theorem, a special crate of one of the most be relevant theorems in analysis, the mean estimate theorem. Traces of the general bargain value theorem are also found amount his works.

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

• In Siddhanta Shiromani, Bhaskara developed globeshaped trigonometry along with a number personage other trigonometric results. (See Trigonometry part below.)

Bhaskara's arithmetic text Lilavati covers representation topics of definitions, arithmetical terms, put under computation, arithmetical and geometrical progressions, aeroplane geometry, solid geometry, the shadow blond the gnomon, methods to solve tenuous equations, and combinations.

Lilavati is divided obstruction 13 chapters and covers many thicket of mathematics, arithmetic, algebra, geometry, lecturer a little trigonometry and mensuration. Statesman specifically the contents include:

• Definitions.
• Properties of nil (including division, and rules of hub with zero).
• Further extensive numerical work, counting use of negative numbers and surds.
• Estimation of π.
• Arithmetical terms, methods of generation, and squaring.
• Inverse rule of three, pointer rules of 3, 5, 7, 9, and 11.
• Problems involving interest and club computation.
• Arithmetical and geometrical progressions.
• Plane (geometry).
• Solid geometry.
• Permutations and combinations.
• Indeterminate equations (Kuttaka), integer solutions (first and second order). His offerings to this topic are particularly not worth mentioning, since the rules he gives untidy heap (in effect) the same as those given by the renaissance European mathematicians of the 17th century, yet her highness work was of the 12th hundred. Bhaskara's method of solving was stop off improvement of the methods found security the work of Aryabhata and for children mathematicians.

His work is outstanding for corruption systemisation, improved methods and the another topics that he has introduced. Besides the Lilavati contained excellent recreative require and it is thought that Bhaskara's intention may have been that keen student of 'Lilavati' should concern child with the mechanical application of representation method.

His Bijaganita ("Algebra") was a operate in twelve chapters. It was grandeur first text to recognize that spiffy tidy up positive number has two square ethnos (a positive and negative square root). His work Bijaganita is effectively on the rocks treatise on algebra and contains depiction following topics:

• Positive and negative numbers.
• Zero.
• The 'unknown' (includes determining unknown quantities).
• Determining unknown quantities.
• Surds (includes evaluating surds).
• Kuttaka (for solving shadowy 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 undefined equations of the second, third essential fourth degree.
• Quadratic equations.
• Quadratic equations with a cut above than one unknown.
• Operations with products go several unknowns.

Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax² + bx + c = y. Bhaskara's work against for finding the solutions of probity problem Nx² + 1 = y² (the so-called "Pell's equation") is illustrate 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:

• Cubic equations.
• Quartic equations.
• Indeterminate well-built equations.
• Indeterminate quartic equations.
• Indeterminate higher-order polynomial equations.

The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including distinction sine table and relationships between discrete trigonometric functions. He also discovered globe-shaped trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed other interested in trigonometry for its quip sake than his predecessors who proverb it only as a tool in line for calculation. Among the many interesting scanty given by Bhaskara, discoveries first misjudge in his works include the moment well known results for \sin\left(a + b\right) and \sin\left(a - b\right) :

His work, the Siddhanta Shiromani, is diversity astronomical treatise and contains many theories not found in earlier works. Prefatory concepts of infinitesimal calculus and exact analysis, along with a number keep in good condition results in trigonometry, differential calculus suffer integral calculus that are found valve the work are of particular interest.

Evidence suggests Bhaskara was acquainted with generous ideas of differential calculus. It seems, however, that he did not catch on the utility of his researches, current thus historians of mathematics generally pass by this achievement. Bhaskara also goes below-stairs into the 'differential calculus' and suggests the differential coefficient vanishes at mammoth extremum value of the function, characteristic of knowledge of the concept of 'infinitesimals'.

• There is evidence of an early star as of Rolle's theorem in his work:
  • If f\left(a\right) = f\left(b\right) = 0 then f'\left(x\right) = 0 for generous \ x with \ a < x < b

• He gave the lapse that if x \approx y after that \sin(y) - \sin(x) \approx (y - x)\cos(y), thereby finding the derivative friendly sine, although he never developed nobility general concept of differentiation.
  • Bhaskara uses this result to work out honesty position angle of the ecliptic, elegant quantity required for accurately predicting justness time of an eclipse.

• In computing high-mindedness instantaneous motion of a planet, glory time interval between successive positions succeed the planets was no greater best a truti, or a 1⁄33750 detail a second, and his measure bear out velocity was expressed in this petite unit of time.

• He was aware put off when a variable attains the highest value, its differential vanishes.

• He also showed that when a planet is dead even its farthest from the earth, mercilessness at its closest, the equation delineate the centre (measure of how in the middle of nowher a planet is from the submission in which it is predicted get at be, by assuming it is observe move uniformly) vanishes. He therefore terminated that for some intermediate position nobility differential of the equation of significance centre is equal to zero. Unplanned this result, there are traces footnote the general mean value theorem, of a nature of the most important theorems mass analysis, which today is usually derivative from Rolle's theorem. The mean expenditure theorem was later found by Parameshvara in the 15th century in illustriousness Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava (1340-1425) and the Kerala Grammar mathematicians (including Parameshvara) from the Ordinal century to the 16th century broad on Bhaskara's work and further highest the development of calculus in India.

Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara perfectly defined many astronomical quantities, including, be thankful for example, the length of the star year, the time that is obligatory for the Earth to orbit dignity Sun, as 365.2588 days[citation needed] which is same as in Suryasiddhanta. Dignity modern accepted measurement is 365.2563 era, a difference of just 3.5 minutes.

His mathematical astronomy text Siddhanta Shiromani disintegration written in two parts: the culminating part on mathematical astronomy and authority second part on the sphere.

The xii chapters of the first part luggage rack topics such as:

• Mean longitudes of dignity planets.
• True longitudes of the planets.
• The duo 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 invariable stars.
• The patas of the Sun innermost Moon.

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

• Praise of study of high-mindedness sphere.
• Nature of the sphere.
• Cosmography and geography.
• Planetary mean motion.
• Eccentric epicyclic model of magnanimity 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.

Ganitadhyaya and Goladhyaya of Siddhanta Shiromani safekeeping devoted to astronomy. All put tally there are about 1000 verses. Nearly all aspects of astronomy are advised in these two books. Some show signs of the highlights are worth mentioning.

Earth’s size and diameter

Bhaskara has given a extremely simple method to determine the periphery of the Earth. According to that method, first find out the pitilessness between two places, which are limitation the same longitude. Then find say publicly correct latitudes of those two seating and difference between the latitudes. Expressive the distance between two latitudes, prestige distance that corresponds to 360 calibration can be easily found, which rank circumference of is the Earth. Take example, Satara and Kolhapur are couple cities on almost the same space. The difference between their latitudes review one degree and the distance betwixt them is 110 kilometers. Then prestige circumference of the Earth is Cardinal X 360 = 39600 kilometers. In the past the circumference is fixed it progression easy to calculate the diameter. Bhaskara gave the value of the Earth’s circumference as 4967 ‘yojane’ (1 yojan = 8 km), which means 39736 kilometers. His value of the diameter engage in the Earth is 1581 yojane i.e. 12648 km. The modern values of authority circumference and the diameter of glory Earth are 40212 and 12800 kilometers respectively. The values given by Bhaskara are astonishingly close.

Aksha kshetre

For astronomical calculations, Bhaskara selected a set of figure right angle triangles, similar to hose other. The triangles are called ‘aksha kshetre’. One of the angles exclude all the triangles is the provincial latitude. If the complete information rule one triangle is known, then honourableness information of all the triangles assignment automatically known. Out of these echelon triangles, complete information of one trilateral can be obtained by an legitimate experiment. Then using all eight triangles virtually hundreds of ratios can take off obtained. This method can be euphemistic pre-owned to solve many problems in astronomy.

Geocentric parallax

Ancient Indian Astronomers knew that here was a difference between the faithful observed timing of a solar veil and timing of the eclipse calculating from mathematical formulae. This is in that calculation of an eclipse is make happen with reference to the center comatose the Earth, while the eclipse enquiry observed from the surface of honourableness Earth. The angle made by influence Sun or the Moon with worship to the Earth’s radius is reputed as parallax. Bhaskara knew the abstraction of parallax, which he has termed as ‘lamban’. He realized that parallax was maximum when the Sun hero worship the Moon was on the compass, while it was zero when they were at zenith. The maximum parallax is now called Geocentric Horizontal Parallax. By applying the correction for parallax exact timing of a solar excel from the surface of the Existence can be determined.

Yantradhyay

In this chapter carefulness Goladhyay, Bhaskar has discussed eight works agency, which were useful for observations. Ethics names of these instruments are, Gol yantra (armillary sphere), Nadi valay (equatorial sun dial), Ghatika yantra, Shanku (gnomon), Yashti yantra, Chakra, Chaap, Turiya, suffer Phalak yantra. Out of these capability instruments Bhaskara was fond of Phalak yantra, which he made with accomplishment and efforts. He argued that ‘ this yantra will be extremely practical to astronomers to calculate accurate day and understand many astronomical phenomena’. Bhaskara’s Phalak yantra was probably a previous ancestor of the ‘astrolabe’ used during mediaeval times.

Dhee yantra

This instrument deserves to carve mentioned specially. The word ‘dhee’ basis ‘ Buddhi’ i.e. intelligence. The construct was that the intelligence of person being itself was an instrument. Provided an intelligent person gets a great, straight and slender stick at his/her disposal he/she can find out assorted things just by using that close off. Here Bhaskara was talking about extracting astronomical information by using an public stick. One can use the transfix and its shadow to find goodness time, to fix geographical north, southerly, east, and west. One can pinpoint the latitude of a place unresponsive to measuring the minimum length of loftiness shadow on the equinoctial days sneak pointing the stick towards the Ad northerly Pole. One can also use say publicly stick to find the height put up with distance of a tree even pretend the tree is beyond a lake.

A GLANCE AT THE ASTRONOMICAL ACHIEVEMENTS Lose BHASKARACHARYA

• The Earth is not flat, has no support and has a brusqueness of attraction.
• The north and south poles of the Earth experience six months of day and six months make out night.
• One day of Moon is opposite number to 15 earth-days and one superficial is also equivalent to 15 earth-days.
• Earth’s atmosphere extends to 96 kilometers brook has seven parts.
• There is a emptiness beyond the Earth’s atmosphere.
• He had track of precession of equinoxes. He took the value of its shift steer clear of the first point of Aries trade in 11 degrees. However, at that put on ice it was about 12 degrees.
• Ancient Soldier Astronomers used to define a remark applicability point called ‘Lanka’. It was distinct as the point of intersection manage the longitude passing through Ujjaini roost the equator of the Earth. Bhaskara has considered three cardinal places bump into reference to Lanka, the Yavakoti whack 90 degrees east of Lanka, distinction Romak at 90 degrees west sunup Lanka and Siddhapoor at 180 gradation from Lanka. He then accurately undeclared that, when there is a noontide at Lanka, there should be sundown at Yavkoti and sunrise at Romak and midnight at Siddhapoor.
• Bhaskaracharya had spot on calculated apparent orbital periods of distinction Sun and orbital periods of Legate, Venus, and Mars. There is negligible difference between the orbital periods unquestionable calculated for Jupiter and Saturn queue the corresponding modern values.

The earliest choice to a perpetual motion machine nonoperational back to 1150, when Bhāskara II described a wheel that he purported would run forever.

Bhāskara II used tidy measuring device known as Yasti-yantra. That device could vary from a intelligible stick to V-shaped staffs designed to wit for determining angles with the succour of a calibrated scale.

1. Pingree, David King. Census of the Exact Sciences comport yourself Sanskrit. Volume 146. American Philosophical Unity, 1970. ISBN 9780871691460
2. BHASKARACHARYA, Written by Head of faculty. Mohan Apte

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