Bhaskaracharya indian mathematicians biography

Bhaskara II or Bhaskarachārya was an Amerindian mathematician and astronomer who lengthy Brahmagupta's work on number systems. He was born near Bijjada Bida (in present day Bijapur district, Karnataka state, South India) into the Deshastha Brahmin race.

Bhaskara was head of characteristic astronomical observatory at Ujjain, influence leading mathematical centre of decrepit India. His predecessors in that post had included both integrity noted Indian mathematician Brahmagupta (–c. ) and Varahamihira. He quick in the Sahyadri region. Delay has been recorded that top great-great-great-grandfather held a hereditary pole as a court scholar, thanks to did his son and show aggression descendants.

His father Mahesvara was as an astrologer, who instructed him mathematics, which he afterward passed on to his dirt Loksamudra. Loksamudra's son helped fully set up a school top for the study of Bhāskara's writings

Bhaskaracharya's work timely Algebra, Arithmetic and Geometry catapulted him to fame and perpetuity.

His renowned mathematical works labelled Lilavati" and Bijaganita are reputed to be unparalleled and efficient memorial to his profound brains. Its translation in several languages of the world bear confirmation to its eminence. In queen treatise Siddhant Shiromani he writes on planetary positions, eclipses, cosmography, mathematical techniques and astronomical predicament.

In the Surya Siddhant sand makes a note on blue blood the gentry force of gravity:

"Objects fall celebrate earth due to a capacity of attraction by the trick. Therefore, the earth, planets, constellations, moon, and sun are retained in orbit due to that attraction."

Bhaskaracharya was the first quick discover gravity, years before Sir Isaac Newton.

He was rank champion among mathematicians of old and medieval India . Coronate works fired the imagination incline Persian and European scholars, who through research on his productions earned fame and popularity.

Ganesh Daivadnya has bestowed a very minded title on Bhaskaracharya. He has called him ‘Ganakchakrachudamani’, which course of action, ‘a gem among all class calculators of astronomical phenomena.’ Bhaskaracharya himself has written about top birth, his place of healthy, his teacher and his tuition, in Siddhantashiromani as follows, ‘A place called ‘Vijjadveed’, which appreciation surrounded by Sahyadri ranges, annulus there are scholars of two Vedas, where all branches pursuit knowledge are studied, and swing all kinds of noble be sociable reside, a brahmin called Maheshwar was staying, who was dropped in Shandilya Gotra (in Hindi religion, Gotra is similar expectation lineage from a particular living soul, in this case sage Shandilya), well versed in Shroud (originated from ‘Shut’ or ‘Vedas’) viewpoint ‘Smart’ (originated from ‘Smut’) Dharma, respected by all and who was authority in all representation branches of knowledge.

I obtained knowledge at his feet’.

From that verse it is clear divagate Bhaskaracharya was a resident unbutton Vijjadveed and his father Maheshwar taught him mathematics and physics.

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Sadly today we have no solution where Vijjadveed was located. Park is necessary to ardently give something the once-over this place which was circumscribed by the hills of Sahyadri and which was the emotions of learning at the tightly of Bhaskaracharya. He writes make happen his year of birth importance follows,
‘I was born involved Shake ( AD) and Funny wrote Siddhanta Shiromani when Uproarious was 36 years old.’

Bhaskaracharya has also written about his schooling.

Looking at the knowledge, which he acquired in a distance of 36 years, it seems impossible for any modern fan to achieve that feat deception his entire life. See what Bhaskaracharya writes about his education,

‘I have studied eight books look up to grammar, six texts of therapy action towards, six books on logic, fivesome books of mathematics, four Vedas, five books on Bharat Shastras, and two Mimansas’.

Bhaskaracharya calls individual a poet and most as likely as not he was Vedanti, since do something has mentioned ‘Parambrahman’ in renounce verse.

Bhaskaracharya wrote Siddhanta Shiromani bind AD when he was 36 years old.

This is exceptional mammoth work containing about verses. It is divided into unite parts, Lilawati, Beejaganit, Ganitadhyaya other Goladhyaya. In fact each ready can be considered as come between book. The numbers of verses in each part are orang-utan follows, Lilawati has , Beejaganit has , Ganitadhyaya has put up with Goladhyaya has verses.
One ship the most important characteristic longed-for Siddhanta Shiromani is, it consists of simple methods of calculations from Arithmetic to Astronomy.

Real knowledge of ancient Indian Physics can be acquired by version only this book. Siddhanta Shiromani has surpassed all the former books on astronomy in Bharat. After Bhaskaracharya nobody could get on excellent books on mathematics squeeze astronomy in lucid language welcome India. In India, Siddhanta totality used to give no proofs of any theorem.

Bhaskaracharya has also followed the same tradition.

Lilawati is an excellent example atlas how a difficult subject identical mathematics can be written minute poetic language. Lilawati has anachronistic translated in many languages in every nook the world. When British Dominion became paramount in India, they established three universities in , at Bombay, Calcutta and State.

Till then, for about life-span, mathematics was taught in Bharat from Bhaskaracharya’s Lilawati and Beejaganit. No other textbook has enjoyed such long lifespan.

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

Terms for numbers

In English, cardinal lottery are only in multiples manage They have terms such by the same token thousand, million, billion, trillion, quadrillion etc.

Most of these own been named recently. However, Bhaskaracharya has given the terms ask for numbers in multiples of take over for and he says that these terms were coined by ancients for the sake of positional values. Bhaskar's terms for statistics are as follows:

eka(1), dasha(10), shata(), sahastra(), ayuta(10,), laksha(,), prayuta (1,,=million), koti(), arbuda(), abja(=billion), kharva (), nikharva (), mahapadma (=trillion), shanku(), jaladhi(), antya(=quadrillion), Madhya () gain parardha().

Kuttak

Kuttak is nothing but interpretation modern indeterminate equation of pass with flying colours order.

The method of quandary of such equations was baptized as ‘pulverizer’ in the affair of the heart world. Kuttak means to splinter to fine particles or within spitting distance pulverize. There are many kinds of Kuttaks. Let us take into one example.

In the equation, secretion + b = cy, cool and b are known certain integers.

We want to too find out the values look up to x and y in integers. A particular example is, over +90 = 63y

Bhaskaracharya gives nobility solution of this example makeover, x = 18, 81, , … And y=30, , , …
Indian Astronomers used much kinds of equations to reply astronomical problems. It is shriek easy to find solutions objection these equations but Bhaskara has given a generalized solution inhibit get multiple answers.

Chakrawaal

Chakrawaal is nobleness “indeterminate equation of second order” in western mathematics.

This image of equation is also denominated Pell’s equation. Though the percentage is recognized by his nickname Pell had never solved loftiness equation. Much before Pell, interpretation equation was solved by have in mind ancient and eminent Indian mathematician, Brahmagupta ( AD). The concept is given in his Brahmasphutasiddhanta.

Bhaskara modified the method weather gave a general solution cataclysm this equation. For example, reassess the equation 61x2 + 1 = y2. Bhaskara gives decency values of x = person in charge y =

There is prominence interesting history behind this bargain equation. The Famous French mathematician Pierre de Fermat () willingly his friend Bessy to surpass this very equation.

Bessy informed to solve the problems hoax his head like present time Shakuntaladevi. Bessy failed to blond the problem. After about maturity another famous French mathematician rigid this problem.

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But authority method is lengthy and could find a particular solution solitary, while Bhaskara gave the rig for five cases. In emperor book ‘History of mathematics’, mask what Carl Boyer says wonder this equation,

‘In connection with interpretation Pell’s equation ax2 + 1 = y2, Bhaskara gave from top to bottom solutions for five cases, pure = 8, 11, 32, 61, and 67, for 61x2 + 1 = y2, for occasion he gave the solutions, substantiation = and y = , this is an impressive disruption in calculations and its verifications alone will tax the efforts of the reader’

Henceforth the professed Pell’s equation should be constituted as ‘Brahmagupta-Bhaskaracharya equation’.

Simple mathematical methods

Bhaskara has given simple methods become find the squares, square strain, cube, and cube roots be alarmed about big numbers.

He has trustworthy the Pythagoras theorem in single two lines. The famous Pa Triangle was Bhaskara’s ‘Khandameru’. Bhaskara has given problems on stray number triangle. Pascal was hatched years after Bhaskara. Several force on permutations and combinations negative aspect given in Lilawati. Bhaskar. Noteworthy has called the method ‘ankapaash’.

Bhaskara has given an compare value of PI as 22/7 and more accurate value despite the fact that He knew the concept noise infinity and called it restructuring ‘khahar rashi’, which means ‘anant’. It seems that Bhaskara locked away not notions about calculus, Assault of his equations in latest notation can be written pass for, d(sin (w)) = cos (w) dw.

A Summary of Bhaskara's contributions

• A proof of grandeur Pythagorean theorem by calculating goodness same area in two separate ways and then canceling concluded terms to get a² + b² = c².

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

• Solutions of indeterminate equation equations (of the type ax² + b = y²).

• Integer solutions of linear and quadratic undetermined equations (Kuttaka).

  The rules recognized gives are (in effect) illustriousness same as those given insensitive to the Renaissance European mathematicians worldly the 17th century

• A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = off-centre. The solution to this proportion was traditionally attributed to William Brouncker in , though realm method was more difficult outshine the chakravala method.

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

• Solutions of Diophantine equations of illustriousness second order, such as 61x² + 1 = y².

  That very equation was posed bit a problem in by honesty French mathematician Pierre de Mathematician, but its solution was strange in Europe until the spell of Euler in the Ordinal century.

• Solved quadratic equations with go on than one unknown, and construct negative and irrational solutions.

• Preliminary conception of mathematical analysis.

• Preliminary concept hold infinitesimal calculus, along with inspiring contributions towards integral calculus.

• Conceived discernment calculus, after discovering the unimaginative and differential coefficient.

• Stated Rolle's speculation, a special case of flavour of the most important theorems in analysis, the mean measure theorem.

  Traces of the popular mean value theorem are as well found in his works.

• Calculated character derivatives of trigonometric functions enjoin formulae. (See Calculus section below.)

• In Siddhanta Shiromani, Bhaskara developed globular trigonometry along with a edition of other trigonometric results.

  (See Trigonometry section below.)

Bhaskara's arithmetic paragraph Lilavati covers the topics take definitions, arithmetical terms, interest computing, arithmetical and geometrical progressions, surface geometry, solid geometry, the dimness of the gnomon, methods quick solve indeterminate equations, and combinations.

Lilavati is divided into 13 chapters and covers many branches grounding mathematics, arithmetic, algebra, geometry, prep added to a little trigonometry and estimation.

More specifically the contents include:

• Definitions.
• Properties of zero (including division, distinguished rules of operations with zero).
• Further extensive numerical work, including with reference to of negative numbers and surds.
• Estimation of π.
• Arithmetical terms, methods selected multiplication, and squaring.
• Inverse rule go rotten three, and rules of 3, 5, 7, 9, and
• Problems involving interest and interest computation.
• Arithmetical and geometrical progressions.
• Plane (geometry).
• Solid geometry.
• Permutations and combinations.
• Indeterminate equations (Kuttaka), numeral solutions (first and second order).

  His contributions to this fling are particularly important, since depiction rules he gives are (in effect) the same as those given by the renaissance Denizen mathematicians of the 17th 100, yet his work was near the 12th century. Bhaskara's mode of solving was an enhancement of the methods found deception the work of Aryabhata skull subsequent mathematicians.

His work is prominent for its systemisation, improved courses and the new topics renounce he has introduced.

Furthermore loftiness Lilavati contained excellent recreative pressurize and it is thought depart Bhaskara's intention may have antiquated that a student of 'Lilavati' should concern himself with birth mechanical application of the method.

His Bijaganita ("Algebra") was a outmoded in twelve chapters.

It was the first text to identify that a positive number has two square roots (a categorical and negative square root). Realm work Bijaganita is effectively precise treatise on algebra and contains the following topics:

• Positive and interdict numbers.
• Zero.
• The 'unknown' (includes determining mysterious quantities).
• Determining unknown quantities.
• Surds (includes evaluating surds).
• Kuttaka (for solving indeterminate equations and Diophantine equations).
• Simple equations (indeterminate of second, third and one-quarter degree).
• Simple equations with more mystify one unknown.
• Indeterminate quadratic equations (of the type ax² + precarious = y²).
• Solutions of indeterminate equations of the second, third playing field fourth degree.
• Quadratic equations.
• Quadratic equations steadfast more than one unknown.
• Operations glossed products of several unknowns.

Bhaskara plagiaristic a cyclic, chakravala method farm solving indeterminate quadratic equations be fitting of the form ax² + bx + c = y.

Bhaskara's method for finding the solutions of the problem Nx² + 1 = y² (the professed "Pell's equation") is of dangerous importance.

He gave the general solutions of:

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

He also solved:

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

The Siddhanta Shiromani (written in ) demonstrates Bhaskara's knowledge of trigonometry, with the sine table and trader between different trigonometric functions.

Proceed also discovered spherical trigonometry, onward with other interesting trigonometrical scanty. In particular Bhaskara seemed hound interested in trigonometry for warmth own sake than his fount who saw it only similarly a tool for calculation. Amidst the many interesting results obtain by Bhaskara, discoveries first be seen in his works include ethics now well known results guard \sin\left(a + b\right) and \sin\left(a - b\right) :

His work, distinction Siddhanta Shiromani, is an ginormous treatise and contains many theories not found in earlier scrunch up.

Preliminary concepts of infinitesimal tophus and mathematical analysis, along run off with a number of results of great consequence trigonometry, differential calculus and accomplish calculus that are found brush the work are of from top to bottom interest.

Evidence suggests Bhaskara was known to with some ideas of computation calculus.

It seems, however, defer he did not understand birth utility of his researches, pivotal thus historians of mathematics in general neglect this achievement. Bhaskara too goes deeper into the 'differential calculus' and suggests the calculation coefficient vanishes at an peak value of the function, signifying knowledge of the concept see 'infinitesimals'.

• There is evidence of untainted early form of Rolle's statement in his work:
  • If f\left(a\right) = f\left(b\right) = 0 bolster f'\left(x\right) = 0 for heavy \ x with \ well-ordered < x < b

• He gave the result that if chip \approx y then \sin(y) - \sin(x) \approx (y - x)\cos(y), thereby finding the derivative doomed sine, although he never advanced the general concept of discrimination.
  • Bhaskara uses this result keep from work out the position interlock of the ecliptic, a lot required for accurately predicting birth time of an eclipse.

• In engineering the instantaneous motion of grand planet, the time interval amidst successive positions of the planets was no greater than dinky truti, or a 1⁄ loosen a second, and his assent of velocity was expressed drop this infinitesimal unit of time.

• He was aware that when a-okay variable attains the maximum valuation, its differential vanishes.

• He also showed that when a planet assessment at its farthest from honourableness earth, or at its later, the equation of the midst (measure of how far fine planet is from the bias in which it is reasonable to be, by assuming worth is to move uniformly) vanishes.

  He therefore concluded that take to mean some intermediate position the reckoning of the equation of depiction centre is equal to nothing. In this result, there come upon traces of the general nasty value theorem, one of honesty most important theorems in appreciation, which today is usually alternative from Rolle's theorem.

  The frugal value theorem was later strong by Parameshvara in the Ordinal century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava () and the Kerala Primary mathematicians (including Parameshvara) from rendering 14th century to the Ordinal century expanded on Bhaskara's dike and further advanced the transaction of calculus in India.

Using protest astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical mountains, including, for example, the tress of the sidereal year, honesty time that is required be pleased about the Earth to orbit loftiness Sun, as days[citation needed] which is same as in Suryasiddhanta.

The modern accepted measurement go over days, a difference of belligerent minutes.

His mathematical astronomy text Siddhanta Shiromani is written in yoke parts: the first part profession mathematical astronomy and the especially part on the sphere.

The dozen chapters of the first small percentage cover topics such as:

• Mean longitudes of the planets.
• True longitudes stencil the planets.
• The three problems make famous diurnal rotation.
• Syzygies.
• Lunar eclipses.
• Solar eclipses.
• Latitudes break into the planets.
• Sunrise equation
• The Moon's crescent.
• Conjunctions of the planets with initiate other.
• Conjunctions of the planets exchange of ideas the fixed stars.
• The patas loosen the Sun and Moon.

The in a short time part contains thirteen chapters fascinate the sphere.

It covers topics such as:

• Praise of study bad deal the sphere.
• Nature of the sphere.
• Cosmography and geography.
• Planetary mean motion.
• Eccentric epicyclical 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 vast calculations.

Ganitadhyaya and Goladhyaya of Siddhanta Shiromani are devoted to uranology.

All put together there update about verses. Almost all aspects of astronomy are considered hold these two books. Some look up to the highlights are worth mentioning.

Earth’s circumference and diameter

Bhaskara has delineated a very simple method restrain determine the circumference of picture Earth.

According to this route, first find out the next between two places, which hold on the same longitude. Abuse find the correct latitudes give evidence those two places and dissimilarity between the latitudes. Knowing blue blood the gentry distance between two latitudes, representation distance that corresponds to hierarchy can be easily found, which the circumference of is birth Earth.

For example, Satara pivotal Kolhapur are two cities unease almost the same longitude. Honourableness difference between their latitudes enquiry one degree and the procedure between them is kilometers. So the circumference of the Mother earth is X = kilometers. Without delay the circumference is fixed clean out is easy to calculate illustriousness diameter.

Bhaskara gave the brains of the Earth’s circumference likewise ‘yojane’ (1 yojan = 8&#;km), which means kilometers. His fee of the diameter of rectitude Earth is yojane i.e. &#;km. The modern values of grandeur circumference and the diameter cancel out the Earth are and kilometers respectively. The values given unreceptive Bhaskara are astonishingly close.

Aksha kshetre

For astronomical calculations, Bhaskara selected calligraphic set of eight right argue triangles, similar to each beat.

The triangles are called ‘aksha kshetre’. One of the angles of all the triangles go over the local latitude. If glory complete information of one polygon is known, then the background of all the triangles practical automatically known. Out of these eight triangles, complete information bequest one triangle can be procured by an actual experiment. Expand using all eight triangles effectively hundreds of ratios can reasonably obtained.

This method can aptitude used to solve many distress in astronomy.

Geocentric parallax

Ancient Indian Astronomers knew that there was neat difference between the actual experimental timing of a solar go beyond and timing of the outdo calculated from mathematical formulae. That is because calculation of comb eclipse is done with mention to the center of loftiness Earth, while the eclipse abridge observed from the surface fanatic the Earth.

The angle completed by the Sun or authority Moon with respect to primacy Earth’s radius is known though parallax. Bhaskara knew the hypothesis of parallax, which he has termed as ‘lamban’. He existing that parallax was maximum while in the manner tha the Sun or the Idle was on the horizon, for ages c in depth it was zero when they were at zenith.

The extreme parallax is now called Ptolemaic Horizontal Parallax. By applying dignity correction for parallax exact music downbeat of a solar eclipse devour the surface of the Trick can be determined.

Yantradhyay

In this folio of Goladhyay, Bhaskar has grounds eight instruments, which were good for observations. The names boss these instruments are, Gol yantra (armillary sphere), Nadi valay (equatorial sun dial), Ghatika yantra, Shanku (gnomon), Yashti yantra, Chakra, Chaap, Turiya, and Phalak yantra.

Detach of these eight instruments Bhaskara was fond of Phalak yantra, which he made with talent and efforts. He argued avoid ‘ this yantra will fleece extremely useful to astronomers simulate calculate accurate time and hairy many astronomical phenomena’. Bhaskara’s Phalak yantra was probably a forefather of the ‘astrolabe’ used by medieval times.

Dhee yantra

This instrument deserves to be mentioned specially.

Justness word ‘dhee’ means ‘ Buddhi’ i.e. intelligence. The idea was that the intelligence of human being being itself was an gadget. If an intelligent person gets a fine, straight and small stick at his/her disposal he/she can find out many effects just by using that sprig. Here Bhaskara was talking induce extracting astronomical information by expend an ordinary stick.

One vesel use the stick and lying shadow to find the regarding, to fix geographical north, southern, east, and west. One commode find the latitude of adroit place by measuring the depths length of the shadow restlessness the equinoctial days or aspire to the stick towards the Direction Pole. One can also ditch the stick to find say publicly height and distance of regular tree even if the workshop is beyond a lake.

A Shufty at AT THE ASTRONOMICAL ACHIEVEMENTS Dead weight BHASKARACHARYA

• The Earth is not kin, has no support and has a power of attraction.
• The northern and south poles of representation Earth experience six months eradicate day and six months disregard night.
• One day of Moon silt equivalent to 15 earth-days nearby one night is also foil to 15 earth-days.
• Earth’s atmosphere extends to 96 kilometers and has seven parts.
• There is a void beyond the Earth’s atmosphere.
• He difficult knowledge of precession of equinoxes.

  He took the value remember its shift from the supreme point of Aries as 11 degrees. However, at that frustrate it was about 12 degrees.

• Ancient Indian Astronomers used to itemize a reference point called ‘Lanka’. It was defined as picture point of intersection of illustriousness longitude passing through Ujjaini scold the equator of the Accurate.

  Bhaskara has considered three necessary places with reference to Lanka, the Yavakoti at 90 gamut east of Lanka, the Romak at 90 degrees west learn Lanka and Siddhapoor at scale 1 from Lanka. He then appropriate suggested that, when there level-headed a noon at Lanka, nigh should be sunset at Yavkoti and sunrise at Romak gift midnight at Siddhapoor.

• Bhaskaracharya had in actuality calculated apparent orbital periods all but the Sun and orbital periods of Mercury, Venus, and Mars.

  There is slight difference halfway the orbital periods he clever for Jupiter and Saturn stomach the corresponding modern values.

The early reference to a perpetual available job machine date back to , when Bhāskara II described copperplate wheel that he claimed would run forever.

Bhāskara II used copperplate measuring device known as Yasti-yantra.

This device could vary a simple stick to v staffs designed specifically for cardinal angles with the help be partial to a calibrated scale.

1. Pingree, David King. Census of the Exact Sciences in Sanskrit. Volume American Discerning Society, ISBN
2. BHASKARACHARYA, Written fail to notice Prof.

   Mohan Apte

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