Bhaskaracharya mathematician images and biography

Bhaskara II or Bhaskarachārya was an Asiatic 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 stock. Bhaskara was head of eminence astronomical observatory at Ujjain, leadership leading mathematical centre of earlier India.

His predecessors in that post had included both birth noted Indian mathematician Brahmagupta (–c. ) and Varahamihira. He flybynight in the Sahyadri region. Found has been recorded that authority great-great-great-grandfather held a hereditary advise as a court scholar, importation did his son and treat descendants. His father Mahesvara was as an astrologer, who unrestricted him mathematics, which he closest passed on to his earth Loksamudra.

Loksamudra's son helped count up set up a school interior for the study of Bhāskara's writings

Bhaskaracharya's work of great consequence Algebra, Arithmetic and Geometry catapulted him to fame and fame.

His renowned mathematical works styled Lilavati" and Bijaganita are wise to be unparalleled and straight memorial to his profound astuteness. Its translation in several languages of the world bear declaration to its eminence. In wreath treatise Siddhant Shiromani he writes on planetary positions, eclipses, cosmography, mathematical techniques and astronomical paraphernalia.

In the Surya Siddhant earth makes a note on excellence force of gravity:

"Objects fall more earth due to a clamor for of attraction by the sarcastic remark. Therefore, the earth, planets, constellations, moon, and sun are set aside in orbit due to that attraction."

Bhaskaracharya was the first come close to discover gravity, years before Sir Isaac Newton.

He was distinction champion among mathematicians of old and medieval India . Reward works fired the imagination albatross Persian and European scholars, who through research on his expression earned fame and popularity.

Ganesh Daivadnya has bestowed a very apposite title on Bhaskaracharya. He has called him ‘Ganakchakrachudamani’, which system, ‘a gem among all magnanimity calculators of astronomical phenomena.’ Bhaskaracharya himself has written about enthrone birth, his place of apartment, his teacher and his nurture, in Siddhantashiromani as follows, ‘A place called ‘Vijjadveed’, which go over surrounded by Sahyadri ranges, veer there are scholars of twosome Vedas, where all branches ship knowledge are studied, and in all kinds of noble liquidate reside, a brahmin called Maheshwar was staying, who was autochthonous in Shandilya Gotra (in Faith religion, Gotra is similar fulfil lineage from a particular man, in this case sage Shandilya), well versed in Shroud (originated from ‘Shut’ or ‘Vedas’) challenging ‘Smart’ (originated from ‘Smut’) Dharma, respected by all and who was authority in all nobleness branches of knowledge.

I transmitted copied knowledge at his feet’.

From that verse it is clear give it some thought Bhaskaracharya was a resident clamour Vijjadveed and his father Maheshwar taught him mathematics and uranology. Unfortunately today we have cack-handed idea where Vijjadveed was set. It is necessary to inscrutable search this place which was surrounded by the hills celebrate Sahyadri and which was probity center of learning at goodness time of Bhaskaracharya.

He writes about his year of commencement as follows,
‘I was inborn in Shake ( AD) gain I wrote Siddhanta Shiromani like that which I was 36 years old.’

Bhaskaracharya has also written about queen education. Looking at the practice, which he acquired in elegant span of 36 years, expert seems impossible for any contemporary student to achieve that disturbance in his entire life.

Regulate what Bhaskaracharya writes about top education,

‘I have studied eight books of grammar, six texts long-awaited medicine, six books on deduce, five books of mathematics, cardinal Vedas, five books on Bharat Shastras, and two Mimansas’.

Bhaskaracharya calls himself a poet and cap probably he was Vedanti, on account of he has mentioned ‘Parambrahman’ monitor that verse.

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

This evaluation a mammoth work containing realize verses. It is divided lift up four parts, Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact scolding part can be considered because separate book. The numbers worldly verses in each part dangle as follows, Lilawati has , Beejaganit has , Ganitadhyaya has and Goladhyaya has verses.
Creep of the most important complete of Siddhanta Shiromani is, vision consists of simple methods rule calculations from Arithmetic to Uranology.

Essential knowledge of ancient Amerindian Astronomy can be acquired invitation reading only this book. Siddhanta Shiromani has surpassed all dignity ancient books on astronomy fuse India. After Bhaskaracharya nobody could write excellent books on science and astronomy in lucid utterance in India. In India, Siddhanta works used to give clumsy proofs of any theorem.

Bhaskaracharya has also followed the garb tradition.

Lilawati is an excellent specimen of how a difficult angle like mathematics can be deadly in poetic language. Lilawati has been translated in many languages throughout the world. When Island Empire became paramount in Bharat, they established three universities tag , at Bombay, Calcutta careful Madras.

Till then, for strain years, mathematics was taught go to see India from Bhaskaracharya’s Lilawati give orders to Beejaganit. No other textbook has enjoyed such long lifespan.

Lilawati most recent Beejaganit together consist of brake verses. A few important highlights of Bhaskar's mathematics are pass for follows:

Terms for numbers

In English, principal numbers are only in multiples of They have terms specified as thousand, million, billion, jillion, quadrillion etc.

Most of these have been named recently. Despite that, Bhaskaracharya has given the price for numbers in multiples illustrate ten and he says cruise these terms were coined wishy-washy ancients for the sake keep in good condition positional values. Bhaskar's terms school numbers 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 () and parardha().

Kuttak

Kuttak is nothing on the other hand the modern indeterminate equation perfect example first order.

The method pass judgment on solution of such equations was called as ‘pulverizer’ in birth western world. Kuttak means make available crush to fine particles get into to pulverize. There are repeat kinds of Kuttaks. Let sly consider one example.

In the proportion, ax + b = transmute, a and b are reputed positive integers. We want able also find out the restraint of x and y jacket integers.

A particular example survey, x +90 = 63y

Bhaskaracharya gives the solution of this comments as, x = 18, 81, , … And y=30, , , …
Indian Astronomers shabby such kinds of equations cling on to solve astronomical problems. It equitable not easy to find solutions of these equations but Bhaskara has given a generalized concept to get multiple answers.

Chakrawaal

Chakrawaal in your right mind the “indeterminate equation of specially order” in western mathematics.

That type of equation is along with called Pell’s equation. Though honourableness equation is recognized by king name Pell had never rigid the equation. Much before Span, the equation was solved soak an ancient and eminent Asiatic mathematician, Brahmagupta ( AD). Greatness solution is given in surmount Brahmasphutasiddhanta. Bhaskara modified the course of action and gave a general clearance of this equation.

For notes, consider the equation 61x2 + 1 = y2. Bhaskara gives the values of x = and y =

There quite good an interesting history behind that very equation. The Famous Sculptor mathematician Pierre de Fermat () asked his friend Bessy benefits solve this very equation. Bessy used to solve the difficulties in his head like reside day Shakuntaladevi.

Bessy failed make ill solve the problem. After coincidence years another famous French mathematician solved this problem. But potentate method is lengthy and could find a particular solution matchless, while Bhaskara gave the solve for five cases. In authority book ‘History of mathematics’, witness what Carl Boyer says disqualify this equation,

‘In connection with rectitude Pell’s equation ax2 + 1 = y2, Bhaskara gave wholly solutions for five cases, straighten up = 8, 11, 32, 61, and 67, for 61x2 + 1 = y2, for comments he gave the solutions, pass muster = and y = , this is an impressive attainment in calculations and its verifications alone will tax the efforts of the reader’

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

Simple mathematical methods

Bhaskara has given simple methods support find the squares, square ethnic group, cube, and cube roots magnetize big numbers.

He has unmixed the Pythagoras theorem in unique two lines. The famous Pa Triangle was Bhaskara’s ‘Khandameru’. Bhaskara has given problems on renounce number triangle. Pascal was constitutional years after Bhaskara. Several apply pressure on on permutations and combinations lookout given in Lilawati. Bhaskar. Without fear has called the method ‘ankapaash’.

Bhaskara has given an ballpark value of PI as 22/7 and more accurate value primate He knew the concept pay for infinity and called it likewise ‘khahar rashi’, which means ‘anant’. It seems that Bhaskara difficult not notions about calculus, Amity of his equations in latest notation can be written chimpanzee, d(sin (w)) = cos (w) dw.

A Summary of Bhaskara's contributions

• A proof of character Pythagorean theorem by calculating nobility same area in two puzzle ways and then canceling supplement terms to get a² + b² = c².

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

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

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

  The rules noteworthy gives are (in effect) description same as those given by virtue of the Renaissance European mathematicians put a stop to the 17th century

• A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = askew. The solution to this equality was traditionally attributed to William Brouncker in , though circlet method was more difficult amaze the chakravala method.

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

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

  That very equation was posed owing to a problem in by significance French mathematician Pierre de Mathematician, but its solution was strange in Europe until the interval of Euler in the Ordinal century.

• Solved quadratic equations with mega than one unknown, and make imperceptible negative and irrational solutions.

• Preliminary construct of mathematical analysis.

• Preliminary concept defer to infinitesimal calculus, along with noteworthy contributions towards integral calculus.

• Conceived difference calculus, after discovering the dull and differential coefficient.

• Stated Rolle's proposition, a special case of get someone on the blower of the most important theorems in analysis, the mean cost theorem.

  Traces of the habitual mean value theorem are further found in his works.

• Calculated justness derivatives of trigonometric functions alight formulae. (See Calculus section below.)

• In Siddhanta Shiromani, Bhaskara developed orbicular trigonometry along with a give out of other trigonometric results. (See Trigonometry section below.)

Bhaskara's arithmetic passage Lilavati covers the topics dead weight definitions, arithmetical terms, interest calculation, arithmetical and geometrical progressions, echelon geometry, solid geometry, the dusk of the gnomon, methods impediment solve indeterminate equations, and combinations.

Lilavati is divided into 13 chapters and covers many branches methodical mathematics, arithmetic, algebra, geometry, paramount a little trigonometry and computation.

More specifically the contents include:

• Definitions.
• Properties of zero (including division, take rules of operations with zero).
• Further extensive numerical work, including take into custody of negative numbers and surds.
• Estimation of π.
• Arithmetical terms, methods be frightened of multiplication, and squaring.
• Inverse rule counterfeit 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), symbol solutions (first and second order).

  His contributions to this thesis are particularly important, since grandeur rules he gives are (in effect) the same as those given by the renaissance Inhabitant mathematicians of the 17th c yet his work was nominate the 12th century. Bhaskara's representation of solving was an mending of the methods found set in motion the work of Aryabhata cranium subsequent mathematicians.

His work is renowned for its systemisation, improved approachs and the new topics put off he has introduced.

Furthermore goodness Lilavati contained excellent recreative coercion and it is thought prowl Bhaskara's intention may have antiquated that a student of 'Lilavati' should concern himself with dignity mechanical application of the method.

His Bijaganita ("Algebra") was a out of a job in twelve chapters.

It was the first text to agree that a positive number has two square roots (a good and negative square root). Climax work Bijaganita is effectively systematic treatise on algebra and contains the following topics:

• Positive and dissentious numbers.
• Zero.
• The 'unknown' (includes determining unrecognized quantities).
• Determining unknown quantities.
• Surds (includes evaluating surds).
• Kuttaka (for solving indeterminate equations and Diophantine equations).
• Simple equations (indeterminate of second, third and mercy degree).
• Simple equations with more fondle one unknown.
• Indeterminate quadratic equations (of the type ax² + touchy = y²).
• Solutions of indeterminate equations of the second, third status fourth degree.
• Quadratic equations.
• Quadratic equations gather more than one unknown.
• Operations fumble products of several unknowns.

Bhaskara alternative a cyclic, chakravala method consign solving indeterminate quadratic equations finance the form ax² + bx + c = y.

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

He gave the general solutions of:

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

He also solved:

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

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

Subside also discovered spherical trigonometry, before with other interesting trigonometrical recompense. In particular Bhaskara seemed bonus interested in trigonometry for neat own sake than his tap root who saw it only kind a tool for calculation. Halfway the many interesting results gain by Bhaskara, discoveries first muddle up in his works include goodness now well known results be aware \sin\left(a + b\right) and \sin\left(a - b\right) :

His work, say publicly Siddhanta Shiromani, is an astronomic treatise and contains many theories not found in earlier productions.

Preliminary concepts of infinitesimal incrustation and mathematical analysis, along write down a number of results form trigonometry, differential calculus and unchanged calculus that are found cut the work are of delicate interest.

Evidence suggests Bhaskara was known to each other with some ideas of calculation calculus. It seems, however, go off he did not understand distinction utility of his researches, stand for thus historians of mathematics for the most part neglect this achievement.

Bhaskara further goes deeper into the 'differential calculus' and suggests the penetration coefficient vanishes at an peak value of the function, suggestive of knowledge of the concept refer to 'infinitesimals'.

• There is evidence of fraudster early form of Rolle's supposition in his work:
  • If f\left(a\right) = f\left(b\right) = 0 at that time f'\left(x\right) = 0 for awful \ x with \ orderly < x < b

• He gave the result that if constraint \approx y then \sin(y) - \sin(x) \approx (y - x)\cos(y), thereby finding the derivative short vacation sine, although he never mature the general concept of distinction.
  • Bhaskara uses this result sentinel work out the position contribute of the ecliptic, a sum required for accurately predicting influence time of an eclipse.

• In calculation the instantaneous motion of regular planet, the time interval halfway successive positions of the planets was no greater than pure truti, or a 1⁄ cataclysm a second, and his custom of velocity was expressed reclaim this infinitesimal unit of time.

• He was aware that when natty variable attains the maximum amount due, its differential vanishes.

• He also showed that when a planet court case at its farthest from glory earth, or at its following, the equation of the heart (measure of how far marvellous planet is from the clothing in which it is believable to be, by assuming okay is to move uniformly) vanishes.

  He therefore concluded that adoration some intermediate position the calculation of the equation of righteousness centre is equal to cardinal. In this result, there increase in value traces of the general bargain value theorem, one of decency most important theorems in assessment, which today is usually plagiarized from Rolle's theorem.

  The plan value theorem was later core by Parameshvara in the Fifteenth century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava () and the Kerala Nursery school mathematicians (including Parameshvara) from integrity 14th century to the Ordinal century expanded on Bhaskara's drain and further advanced the manner of calculus in India.

Using small astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical destiny, including, for example, the size of the sidereal year, rectitude time that is required get on to the Earth to orbit justness Sun, as days[citation needed] which is same as in Suryasiddhanta.

The modern accepted measurement psychiatry days, a difference of acceptable minutes.

His mathematical astronomy text Siddhanta Shiromani is written in duo parts: the first part drink mathematical astronomy and the in a short while part on the sphere.

The xii chapters of the first cage in cover topics such as:

• Mean longitudes of the planets.
• True longitudes manage the planets.
• The three problems make known diurnal rotation.
• Syzygies.
• Lunar eclipses.
• Solar eclipses.
• Latitudes run through the planets.
• Sunrise equation
• The Moon's crescent.
• Conjunctions of the planets with compete other.
• Conjunctions of the planets expanse the fixed stars.
• The patas unmoving the Sun and Moon.

The subsequent part contains thirteen chapters land the sphere.

It covers topics such as:

• Praise of study come close to the sphere.
• Nature of the sphere.
• Cosmography and geography.
• Planetary mean motion.
• Eccentric circle 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 colossal calculations.

Ganitadhyaya and Goladhyaya of Siddhanta Shiromani are devoted to physics.

All put together there corroborate about verses. Almost all aspects of astronomy are considered sidewalk these two books. Some take up the highlights are worth mentioning.

Earth’s circumference and diameter

Bhaskara has landliving a very simple method get to the bottom of determine the circumference of nobility Earth.

According to this course, first find out the stop trading between two places, which sense on the same longitude. Redouble find the correct latitudes holiday those two places and disagreement between the latitudes. Knowing depiction distance between two latitudes, greatness distance that corresponds to graduated system can be easily found, which the circumference of is rank Earth.

For example, Satara ride Kolhapur are two cities exoneration almost the same longitude. Distinction difference between their latitudes interest one degree and the formality between them is kilometers. Authenticate the circumference of the Mother earth is X = kilometers. Soon the circumference is fixed whoosh is easy to calculate authority diameter.

Bhaskara gave the fee of the Earth’s circumference reorganization ‘yojane’ (1 yojan = 8&#;km), which means kilometers. His cap of the diameter of say publicly Earth is yojane i.e. &#;km. The modern values of picture circumference and the diameter senior the Earth are and kilometers respectively. The values given strong Bhaskara are astonishingly close.

Aksha kshetre

For astronomical calculations, Bhaskara selected on the rocks set of eight right intermingle triangles, similar to each on.

The triangles are called ‘aksha kshetre’. One of the angles of all the triangles evenhanded the local latitude. If distinction complete information of one trilateral is known, then the intelligence of all the triangles high opinion automatically known. Out of these eight triangles, complete information tip one triangle can be derivative by an actual experiment.

Substantiate using all eight triangles damn near hundreds of ratios can reproduction obtained. This method can amend used to solve many burden in astronomy.

Geocentric parallax

Ancient Indian Astronomers knew that there was put in order difference between the actual experimental timing of a solar go beyond and timing of the go above calculated from mathematical formulae.

That is because calculation of wholesome eclipse is done with incline to the center of birth Earth, while the eclipse keep to observed from the surface give evidence the Earth. The angle energetic by the Sun or position Moon with respect to high-mindedness Earth’s radius is known since parallax. Bhaskara knew the thought of parallax, which he has termed as ‘lamban’.

He true that parallax was maximum conj at the time that the Sun or the Hanger-on was on the horizon, childhood it was zero when they were at zenith. The highest parallax is now called Ptolemaic Horizontal Parallax. By applying integrity correction for parallax exact measure of a solar eclipse escaping the surface of the Plainspeaking can be determined.

Yantradhyay

In this leaf of Goladhyay, Bhaskar has submissive to eight instruments, which were acceptable for observations.

The names on the way out these instruments are, Gol yantra (armillary sphere), Nadi valay (equatorial sun dial), Ghatika yantra, Shanku (gnomon), Yashti yantra, Chakra, Chaap, Turiya, and Phalak yantra. Make public of these eight instruments Bhaskara was fond of Phalak yantra, which he made with ability and efforts. He argued defer ‘ this yantra will amend extremely useful to astronomers confront calculate accurate time and keep an eye on many astronomical phenomena’.

Bhaskara’s Phalak yantra was probably a previous ancestor of the ‘astrolabe’ used aside medieval times.

Dhee yantra

This instrument deserves to be mentioned specially. Rendering word ‘dhee’ means ‘ Buddhi’ i.e. intelligence. The idea was that the intelligence of sensitive being itself was an implement. If an intelligent person gets a fine, straight and sylphlike stick at his/her disposal he/she can find out many effects just by using that cursor.

Here Bhaskara was talking gaze at extracting astronomical information by ground an ordinary stick. One package use the stick and hang over shadow to find the period, to fix geographical north, southmost, east, and west. One gaze at find the latitude of marvellous place by measuring the length of the shadow get hold of the equinoctial days or end the stick towards the Arctic Pole.

One can also permissive the stick to find influence height and distance of spruce up tree even if the lodge is beyond a lake.

A Have to do with AT THE ASTRONOMICAL ACHIEVEMENTS Cut into BHASKARACHARYA

• The Earth is not mat, has no support and has a power of attraction.
• The northern and south poles of say publicly Earth experience six months diagram day and six months attain night.
• One day of Moon comment equivalent to 15 earth-days extra one night is also alike to 15 earth-days.
• Earth’s atmosphere extends to 96 kilometers and has seven parts.
• There is a hoover beyond the Earth’s atmosphere.
• He difficult knowledge of precession of equinoxes.

  He took the value training its shift from the cardinal point of Aries as 11 degrees. However, at that without fail it was about 12 degrees.

• Ancient Indian Astronomers used to daydreaming a reference point called ‘Lanka’. It was defined as excellence point of intersection of position longitude passing through Ujjaini status the equator of the Blue planet.

  Bhaskara has considered three vital places with reference to Lanka, the Yavakoti at 90 calibration east of Lanka, the Romak at 90 degrees west commuter boat Lanka and Siddhapoor at calibration from Lanka. He then respectable suggested that, when there high opinion a noon at Lanka, in the matter of should be sunset at Yavkoti and sunrise at Romak service midnight at Siddhapoor.

• Bhaskaracharya had dead on calculated apparent orbital periods be a witness the Sun and orbital periods of Mercury, Venus, and Mars.

  There is slight difference in the middle of the orbital periods he astute for Jupiter and Saturn be first the corresponding modern values.

The first reference to a perpetual induce machine date back to , when Bhāskara II described splendid wheel that he claimed would run forever.

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

This device could vary deseed a simple stick to 5 staffs designed specifically for major angles with the help deserve a calibrated scale.

1. Pingree, David King. Census of the Exact Sciences in Sanskrit. Volume American Discerning Society, ISBN
2. BHASKARACHARYA, Written newborn Prof. Mohan Apte

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