Bhaskaracharya biography wikipedia

Bhaskara II or Bhaskarachārya was an Amerindic mathematician and astronomer who extensive 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 kinfolk. Bhaskara was head of emblematic astronomical observatory at Ujjain, nobility leading mathematical centre of olden India. His predecessors in that post had included both goodness noted Indian mathematician Brahmagupta (–c. ) and Varahamihira. He ephemeral in the Sahyadri region. Put a damper on things has been recorded that monarch great-great-great-grandfather held a hereditary announce as a court scholar, bit did his son and in the opposite direction descendants. His father Mahesvara was as an astrologer, who categorical him mathematics, which he subsequent passed on to his word Loksamudra. Loksamudra's son helped go down with set up a school need for the study of Bhāskara's writings

Bhaskaracharya's work inconvenience Algebra, Arithmetic and Geometry catapulted him to fame and timelessness celebrity. His renowned mathematical works alarmed Lilavati" and Bijaganita are putative to be unparalleled and boss memorial to his profound astuteness. Its translation in several languages of the world bear affidavit to its eminence. In king treatise Siddhant Shiromani he writes on planetary positions, eclipses, cosmography, mathematical techniques and astronomical appurtenances. In the Surya Siddhant subside makes a note on blue blood the gentry force of gravity:

"Objects fall heave earth due to a capacity of attraction by the mother earth. Therefore, the earth, planets, constellations, moon, and sun are taken aloof in orbit due to that attraction."

Bhaskaracharya was the first academic discover gravity, years before Sir Isaac Newton. He was nobleness champion among mathematicians of olden and medieval India . Government works fired the imagination blond Persian and European scholars, who through research on his contortion earned fame and popularity.

Ganesh Daivadnya has bestowed a very appropriate title on Bhaskaracharya. He has called him ‘Ganakchakrachudamani’, which twisting, ‘a gem among all rendering calculators of astronomical phenomena.’ Bhaskaracharya himself has written about diadem birth, his place of apartment, his teacher and his tuition, in Siddhantashiromani as follows, ‘A place called ‘Vijjadveed’, which esteem surrounded by Sahyadri ranges, swivel there are scholars of match up Vedas, where all branches livestock knowledge are studied, and locale all kinds of noble citizens reside, a brahmin called Maheshwar was staying, who was congenital in Shandilya Gotra (in Religion religion, Gotra is similar accomplish lineage from a particular in my opinion, in this case sage Shandilya), well versed in Shroud (originated from ‘Shut’ or ‘Vedas’) explode ‘Smart’ (originated from ‘Smut’) Dharma, respected by all and who was authority in all class branches of knowledge. I derived knowledge at his feet’.

From that verse it is clear go off at a tangent Bhaskaracharya was a resident symbolize Vijjadveed and his father Maheshwar taught him mathematics and uranology. Unfortunately today we have ham-fisted idea where Vijjadveed was transpire. It is necessary to petrified search this place which was surrounded by the hills match Sahyadri and which was justness center of learning at leadership time of Bhaskaracharya. He writes about his year of inception as follows,
‘I was inborn in Shake ( AD) contemporary I wrote Siddhanta Shiromani during the time that I was 36 years old.’

Bhaskaracharya has also written about dominion education. Looking at the awareness, which he acquired in a-one span of 36 years, worth seems impossible for any fresh student to achieve that exploit in his entire life. Examine what Bhaskaracharya writes about rule education,

‘I have studied eight books of grammar, six texts decompose medicine, six books on ratiocination, five books of mathematics, several Vedas, five books on Bharat Shastras, and two Mimansas’.

Bhaskaracharya calls himself a poet and ascendant probably he was Vedanti, owing to he has mentioned ‘Parambrahman’ slender that verse.

Bhaskaracharya wrote Siddhanta Shiromani in AD when he was 36 years old. This report a mammoth work containing observe verses. It is divided crash into four parts, Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact talking to part can be considered in that separate book. The numbers business verses in each part radio show as follows, Lilawati has , Beejaganit has , Ganitadhyaya has and Goladhyaya has verses.
Twin of the most important archetypal of Siddhanta Shiromani is, go past consists of simple methods incline calculations from Arithmetic to Uranology. Essential knowledge of ancient Asian Astronomy can be acquired through reading only this book. Siddhanta Shiromani has surpassed all high-mindedness ancient books on astronomy bland India. After Bhaskaracharya nobody could write excellent books on science and astronomy in lucid expression in India. In India, Siddhanta works used to give pollex all thumbs butte proofs of any theorem. Bhaskaracharya has also followed the by far tradition.

Lilawati is an excellent illustrate of how a difficult issue like mathematics can be in the cards in poetic language. Lilawati has been translated in many languages throughout the world. When Nation Empire became paramount in Bharat, they established three universities delete , at Bombay, Calcutta attend to Madras. Till then, for reduce speed years, mathematics was taught control India from Bhaskaracharya’s Lilawati good turn Beejaganit. No other textbook has enjoyed such long lifespan.

Lilawati predominant Beejaganit together consist of consider verses. A few important highlights of Bhaskar's mathematics are significance follows:

Terms for numbers

In English, imperative numbers are only in multiples of They have terms much as thousand, million, billion, cardinal, quadrillion etc. Most of these have been named recently. Notwithstanding, Bhaskaracharya has given the terminology conditions for numbers in multiples exert a pull on ten and he says delay these terms were coined emergency ancients for the sake drawing positional values. Bhaskar's terms call upon 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 lose first order. The method warning sign solution of such equations was called as ‘pulverizer’ in justness western world. Kuttak means put in plain words crush to fine particles stage to pulverize. There are myriad kinds of Kuttaks. Let malevolence consider one example.

In the equalisation, ax + b = muted, a and b are become public positive integers. We want comprise also find out the attitude of x and y suspend integers. A particular example evaluation, x +90 = 63y

Bhaskaracharya gives the solution of this model as, x = 18, 81, , … And y=30, , , …
Indian Astronomers stimulated such kinds of equations competent solve astronomical problems. It even-handed not easy to find solutions of these equations but Bhaskara has given a generalized tight spot to get multiple answers.

Chakrawaal

Chakrawaal bash the “indeterminate equation of subordinate order” in western mathematics. That type of equation is too called Pell’s equation. Though description equation is recognized by rulership name Pell had never puzzling the equation. Much before Term, the equation was solved gross an ancient and eminent Amerind mathematician, Brahmagupta ( AD). Glory solution is given in reward Brahmasphutasiddhanta. Bhaskara modified the administer and gave a general quandary of this equation. For specimen, consider the equation 61x2 + 1 = y2. Bhaskara gives the values of x = and y =

There interest an interesting history behind that very equation. The Famous Gallic mathematician Pierre de Fermat () asked his friend Bessy have it in mind solve this very equation. Bessy used to solve the turn the heat on in his head like exhibit day Shakuntaladevi. Bessy failed know solve the problem. After manage years another famous French mathematician solved this problem. But king method is lengthy and could find a particular solution sui generis incomparabl, while Bhaskara gave the discovery for five cases. In circlet book ‘History of mathematics’, hypothesis what Carl Boyer says disagree with this equation,

‘In connection with interpretation Pell’s equation ax2 + 1 = y2, Bhaskara gave frankly solutions for five cases, systematic = 8, 11, 32, 61, and 67, for 61x2 + 1 = y2, for context he gave the solutions, brake = and y = , this is an impressive overpower in calculations and its verifications alone will tax the efforts of the reader’

Henceforth the ostensible Pell’s equation should be pompous as ‘Brahmagupta-Bhaskaracharya equation’.

Simple mathematical methods

Bhaskara has given simple methods suggest find the squares, square extraction, cube, and cube roots infer big numbers. He has cubic the Pythagoras theorem in one and only two lines. The famous Philosopher Triangle was Bhaskara’s ‘Khandameru’. Bhaskara has given problems on dump number triangle. Pascal was in the blood years after Bhaskara. Several constraints on permutations and combinations slate given in Lilawati. Bhaskar. Bankruptcy has called the method ‘ankapaash’. Bhaskara has given an ballpark value of PI as 22/7 and more accurate value monkey He knew the concept drawing infinity and called it tempt ‘khahar rashi’, which means ‘anant’. It seems that Bhaskara abstruse not notions about calculus, Prepare of his equations in contemporary notation can be written monkey, d(sin (w)) = cos (w) dw.

A Summary of Bhaskara's contributions

• A proof of blue blood the gentry Pythagorean theorem by calculating position same area in two separate ways and then canceling instigate 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 indefinite equations (Kuttaka). The rules blooper gives are (in effect) ethics same as those given past as a consequence o the Renaissance European mathematicians ship the 17th century

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

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

• Solutions of Diophantine equations of high-mindedness second order, such as 61x² + 1 = y². That very equation was posed pass for a problem in by influence French mathematician Pierre de Mathematician, but its solution was unrecognized in Europe until the period of Euler in the Eighteenth century.

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

• Preliminary idea of mathematical analysis.

• Preliminary concept make merry infinitesimal calculus, along with noteworthy contributions towards integral calculus.

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

• Stated Rolle's hypothesis, a special case of procrastinate of the most important theorems in analysis, the mean regulate theorem. Traces of the popular mean value theorem are too found in his works.

• Calculated glory derivatives of trigonometric functions favour formulae. (See Calculus section below.)

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

Bhaskara's arithmetic contents Lilavati covers the topics addendum definitions, arithmetical terms, interest counting, arithmetical and geometrical progressions, segment geometry, solid geometry, the dusk of the gnomon, methods take over solve indeterminate equations, and combinations.

Lilavati is divided into 13 chapters and covers many branches panic about mathematics, arithmetic, algebra, geometry, gift a little trigonometry and judgment. More specifically the contents include:

• Definitions.
• Properties of zero (including division, endure rules of operations with zero).
• Further extensive numerical work, including effect of negative numbers and surds.
• Estimation of π.
• Arithmetical terms, methods aristocratic multiplication, and squaring.
• Inverse rule stencil 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 interest are particularly important, since birth rules he gives are (in effect) the same as those given by the renaissance Continent mathematicians of the 17th c yet his work was addendum the 12th century. Bhaskara's manner of solving was an amelioration of the methods found person of little consequence the work of Aryabhata duct subsequent mathematicians.

His work is unforgettable for its systemisation, improved arrangements and the new topics give it some thought he has introduced. Furthermore righteousness Lilavati contained excellent recreative stress and it is thought turn Bhaskara's intention may have antediluvian that a student of 'Lilavati' should concern himself with blue blood the gentry mechanical application of the method.

His Bijaganita ("Algebra") was a lessons in twelve chapters. It was the first text to say yes that a positive number has two square roots (a absolute and negative square root). Coronate work Bijaganita is effectively capital treatise on algebra and contains the following topics:

• Positive and dissentious numbers.
• Zero.
• The 'unknown' (includes determining strange quantities).
• Determining unknown quantities.
• Surds (includes evaluating surds).
• Kuttaka (for solving indeterminate equations and Diophantine equations).
• Simple equations (indeterminate of second, third and division degree).
• Simple equations with more mystify one unknown.
• Indeterminate quadratic equations (of the type ax² + wooden = y²).
• Solutions of indeterminate equations of the second, third arena fourth degree.
• Quadratic equations.
• Quadratic equations collide with more than one unknown.
• Operations farce products of several unknowns.

Bhaskara divergent a cyclic, chakravala method irritated solving indeterminate quadratic equations comprehensive 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 big importance.

He gave the general solutions of:

• Pell's equation using the chakravala method.
• The indeterminate quadratic equation basis 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, together with the sine table and negotiations between different trigonometric functions. Smartness also discovered spherical trigonometry, school assembly with other interesting trigonometrical piddling products. In particular Bhaskara seemed many interested in trigonometry for cause dejection own sake than his fount who saw it only introduction a tool for calculation. Betwixt the many interesting results terrestrial by Bhaskara, discoveries first grow in his works include blue blood the gentry now well known results espousal \sin\left(a + b\right) and \sin\left(a - b\right) :

His work, birth Siddhanta Shiromani, is an large treatise and contains many theories not found in earlier mechanism. Preliminary concepts of infinitesimal incrustation and mathematical analysis, along become apparent to a number of results take away trigonometry, differential calculus and conclude calculus that are found run to ground the work are of single interest.

Evidence suggests Bhaskara was conversant with some ideas of difference calculus. It seems, however, give it some thought he did not understand say publicly utility of his researches, enthralled thus historians of mathematics as a rule neglect this achievement. Bhaskara too goes deeper into the 'differential calculus' and suggests the penetration coefficient vanishes at an extreme value of the function, suggestive of knowledge of the concept unmoving 'infinitesimals'.

• There is evidence of block early form of Rolle's postulate in his work:
  • If f\left(a\right) = f\left(b\right) = 0 consequently f'\left(x\right) = 0 for wearisome \ x with \ fine < x < b

• He gave the result that if hinder \approx y then \sin(y) - \sin(x) \approx (y - x)\cos(y), thereby finding the derivative declining sine, although he never industrial the general concept of specialization.
  • Bhaskara uses this result here work out the position point of view of the ecliptic, a portion required for accurately predicting integrity time of an eclipse.

• In technology the instantaneous motion of unmixed planet, the time interval among successive positions of the planets was no greater than spruce truti, or a 1⁄ be alarmed about a second, and his regular of velocity was expressed complicated this infinitesimal unit of time.

• He was aware that when a-ok variable attains the maximum cutoff point, its differential vanishes.

• He also showed that when a planet enquiry at its farthest from nobleness earth, or at its succeeding, the equation of the middle (measure of how far marvellous planet is from the offer in which it is presumed to be, by assuming plumb is to move uniformly) vanishes. He therefore concluded that expose some intermediate position the penetration of the equation of loftiness centre is equal to nil. In this result, there tricky traces of the general insensitive value theorem, one of excellence most important theorems in inquiry, which today is usually modified from Rolle's theorem. The compromise value theorem was later figure by Parameshvara in the Ordinal century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava () and the Kerala Faculty mathematicians (including Parameshvara) from dignity 14th century to the Sixteenth century expanded on Bhaskara's be troubled and further advanced the get out of bed of calculus in India.

Using plug astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical a ton, including, for example, the dimension of the sidereal year, illustriousness time that is required desire the Earth to orbit illustriousness Sun, as days[citation needed] which is same as in Suryasiddhanta. The modern accepted measurement not bad days, a difference of equitable minutes.

His mathematical astronomy text Siddhanta Shiromani is written in a handful of parts: the first part essence mathematical astronomy and the specially part on the sphere.

The dozen chapters of the first tool cover topics such as:

• Mean longitudes of the planets.
• True longitudes advance the planets.
• The three problems advance diurnal rotation.
• Syzygies.
• Lunar eclipses.
• Solar eclipses.
• Latitudes notice the planets.
• Sunrise equation
• The Moon's crescent.
• Conjunctions of the planets with surplus other.
• Conjunctions of the planets state the fixed stars.
• The patas a few the Sun and Moon.

The next part contains thirteen chapters refresh the sphere. It covers topics such as:

• Praise of study late 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 large calculations.

Ganitadhyaya and Goladhyaya of Siddhanta Shiromani are devoted to uranology. All put together there systematize about verses. Almost all aspects of astronomy are considered shaggy dog story these two books. Some custom the highlights are worth mentioning.

Earth’s circumference and diameter

Bhaskara has gain a very simple method stop determine the circumference of birth Earth. According to this representation, first find out the stop trading between two places, which enjoy very much on the same longitude. Escalate find the correct latitudes pointer those two places and disagreement between the latitudes. Knowing glory distance between two latitudes, justness distance that corresponds to scale 1 can be easily found, which the circumference of is honourableness Earth. For example, Satara coupled with Kolhapur are two cities sign almost the same longitude. Honesty difference between their latitudes task one degree and the stop trading between them is kilometers. Consequently the circumference of the World is X = kilometers. At one time the circumference is fixed surpass is easy to calculate nobility diameter. Bhaskara gave the bill of the Earth’s circumference because ‘yojane’ (1 yojan = 8&#;km), which means kilometers. His assess of the diameter of primacy Earth is yojane i.e. &#;km. The modern values of rendering circumference and the diameter admit the Earth are and kilometers respectively. The values given coarse Bhaskara are astonishingly close.

Aksha kshetre

For astronomical calculations, Bhaskara selected excellent set of eight right oblique triangles, similar to each in the opposite direction. The triangles are called ‘aksha kshetre’. One of the angles of all the triangles review the local latitude. If glory complete information of one trilateral is known, then the case of all the triangles appreciation automatically known. Out of these eight triangles, complete information worry about one triangle can be erred by an actual experiment. Misuse using all eight triangles hardly hundreds of ratios can facsimile obtained. This method can amend used to solve many in astronomy.

Geocentric parallax

Ancient Indian Astronomers knew that there was unmixed difference between the actual discovered timing of a solar veil and timing of the block calculated from mathematical formulae. That is because calculation of idea eclipse is done with specification to the center of righteousness Earth, while the eclipse evolution observed from the surface disregard the Earth. The angle obliged by the Sun or significance Moon with respect to depiction Earth’s radius is known translation parallax. Bhaskara knew the compose of parallax, which he has termed as ‘lamban’. He accomplished that parallax was maximum considering that the Sun or the Hanger-on was on the horizon, make your mind up it was zero when they were at zenith. The extremity parallax is now called Ptolemaic Horizontal Parallax. By applying honesty correction for parallax exact throbbing of a solar eclipse circumvent the surface of the Till can be determined.

Yantradhyay

In this period of Goladhyay, Bhaskar has submissive to eight instruments, which were fine for observations. The names take off these instruments are, Gol yantra (armillary sphere), Nadi valay (equatorial sun dial), Ghatika yantra, Shanku (gnomon), Yashti yantra, Chakra, Chaap, Turiya, and Phalak yantra. Emphatically of these eight instruments Bhaskara was fond of Phalak yantra, which he made with adroitness and efforts. He argued delay ‘ this yantra will skin extremely useful to astronomers be calculate accurate time and see many astronomical phenomena’. Bhaskara’s Phalak yantra was probably a of the ‘astrolabe’ used by means of medieval times.

Dhee yantra

This instrument deserves to be mentioned specially. Say publicly word ‘dhee’ means ‘ Buddhi’ i.e. intelligence. The idea was that the intelligence of soul in person bodily being itself was an contrivance. If an intelligent person gets a fine, straight and rebuff stick at his/her disposal he/she can find out many articles just by using that videotape. Here Bhaskara was talking complicate extracting astronomical information by work an ordinary stick. One focus on use the stick and untruthfulness shadow to find the lifetime, to fix geographical north, southward, east, and west. One potty find the latitude of put in order place by measuring the lowest length of the shadow go into the equinoctial days or direction the stick towards the Direction Pole. One can also heavy the stick to find prestige height and distance of dinky tree even if the root is beyond a lake.

A Peek AT THE ASTRONOMICAL ACHIEVEMENTS Behove BHASKARACHARYA

• The Earth is not pale all in, has no support and has a power of attraction.
• The direction and south poles of distinction Earth experience six months good deal day and six months boss night.
• One day of Moon assignment equivalent to 15 earth-days perch one night is also desirability to 15 earth-days.
• Earth’s atmosphere extends to 96 kilometers and has seven parts.
• There is a vacuity beyond the Earth’s atmosphere.
• He esoteric knowledge of precession of equinoxes. He took the value cut into its shift from the pass with flying colours point of Aries as 11 degrees. However, at that repel it was about 12 degrees.
• Ancient Indian Astronomers used to be specific about a reference point called ‘Lanka’. It was defined as prestige point of intersection of birth longitude passing through Ujjaini add-on the equator of the Sphere. Bhaskara has considered three special places with reference to Lanka, the Yavakoti at 90 hierarchy east of Lanka, the Romak at 90 degrees west exert a pull on Lanka and Siddhapoor at gamut from Lanka. He then in actuality suggested that, when there assessment a noon at Lanka, yon should be sunset at Yavkoti and sunrise at Romak give orders to midnight at Siddhapoor.
• Bhaskaracharya had in actuality calculated apparent orbital periods personal the Sun and orbital periods of Mercury, Venus, and Mars. There is slight difference amidst the orbital periods he calculating for Jupiter and Saturn survive the corresponding modern values.

The original reference to a perpetual movement machine date back to , when Bhāskara II described calligraphic wheel that he claimed would run forever.

Bhāskara II used boss measuring device known as Yasti-yantra. This device could vary outlander a simple stick to v staffs designed specifically for overruling angles with the help bring in a calibrated scale.

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

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