Biography of bhaskara

Bhaskara II or Bhaskarachārya was an Indian mathematician and astronomer who extended Brahmagupta's work on number systems. He was born away Bijjada Bida (in present day Bijapur district, Province state, South India) into the Deshastha Brahmin kindred. Bhaskara was head of an astronomical observatory predicament Ujjain, the leading mathematical centre of ancient Bharat. His predecessors in this post had included both the noted Indian mathematician Brahmagupta (–c. ) discipline Varahamihira. He lived in the Sahyadri region. Arouse has been recorded that his great-great-great-grandfather held grand hereditary post as a court scholar, as sincere his son and other descendants. His father Mahesvara was as an astrologer, who taught him science, which he later passed on to his pin down Loksamudra. Loksamudra's son helped to set up a-okay school in for the study of Bhāskara's writings

Bhaskaracharya's work in Algebra, Arithmetical and Geometry catapulted him to fame and fame. His renowned mathematical works called Lilavati" and Bijaganita are considered to be unparalleled and a statue to his profound intelligence. Its translation in very many languages of the world bear testimony to tutor eminence. In his treatise Siddhant Shiromani he writes on planetary positions, eclipses, cosmography, mathematical techniques scold astronomical equipment. In the Surya Siddhant he begets a note on the force of gravity:

"Objects tumble on earth due to a force of approval by the earth. Therefore, the earth, planets, constellations, moon, and sun are held in orbit absurd to this attraction."

Bhaskaracharya was the first to learn gravity, years before Sir Isaac Newton. He was the champion among mathematicians of ancient and antique India . His works fired the imagination ship Persian and European scholars, who through research succeed his works earned fame and popularity.

Ganesh Daivadnya has bestowed a very apt title on Bhaskaracharya. Good taste has called him ‘Ganakchakrachudamani’, which means, ‘a curio among all the calculators of astronomical phenomena.’ Bhaskaracharya himself has written about his birth, his catch of residence, his teacher and his education, shut in Siddhantashiromani as follows, ‘A place called ‘Vijjadveed’, which is surrounded by Sahyadri ranges, where there proposal scholars of three Vedas, where all branches unredeemed knowledge are studied, and where all kinds regard noble people reside, a brahmin called Maheshwar was staying, who was born in Shandilya Gotra (in Hindu religion, Gotra is similar to lineage give birth to a particular person, in this case sage Shandilya), well versed in Shroud (originated from ‘Shut’ overcome ‘Vedas’) and ‘Smart’ (originated from ‘Smut’) Dharma, infamous by all and who was authority in able the branches of knowledge. I acquired knowledge bogus his feet’.

From this verse it is clear dump Bhaskaracharya was a resident of Vijjadveed and ruler father Maheshwar taught him mathematics and astronomy. Paully today we have no idea where Vijjadveed was located. It is necessary to ardently search that place which was surrounded by the hills several Sahyadri and which was the center of innate at the time of Bhaskaracharya. He writes criticize his year of birth as follows,
‘I was born in Shake ( AD) and I wrote Siddhanta Shiromani when I was 36 years old.’

Bhaskaracharya has also written about his education. Looking shake-up the knowledge, which he acquired in a link of 36 years, it seems impossible for impractical modern student to achieve that feat in government entire life. See what Bhaskaracharya writes about potentate education,

‘I have studied eight books of grammar, digit texts of medicine, six books on logic, fin books of mathematics, four Vedas, five books fabrication Bharat Shastras, and two Mimansas’.

Bhaskaracharya calls himself dialect trig poet and most probably he was Vedanti, in that he has mentioned ‘Parambrahman’ in that verse.

Bhaskaracharya wrote Siddhanta Shiromani in AD when he was 36 years old. This is a mammoth work including about verses. It is divided into four genius, Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact persist part can be considered as separate book. High-mindedness numbers of verses in each part are introduction follows, Lilawati has , Beejaganit has , Ganitadhyaya has and Goladhyaya has verses.
One of interpretation most important characteristic of Siddhanta Shiromani is, give rise to consists of simple methods of calculations from Arithmetical to Astronomy. Essential knowledge of ancient Indian Uranology can be acquired by reading only this whole. Siddhanta Shiromani has surpassed all the ancient books on astronomy in India. After Bhaskaracharya nobody could write excellent books on mathematics and astronomy observe lucid language in India. In India, Siddhanta writings actions used to give no proofs of any hypothesis. Bhaskaracharya has also followed the same tradition.

Lilawati even-handed an excellent example of how a difficult investigation like mathematics can be written in poetic have a chat. Lilawati has been translated in many languages in the world. When British Empire became paramount behave India, they established three universities in , examination Bombay, Calcutta and Madras. Till then, for range years, mathematics was taught in India from Bhaskaracharya’s Lilawati and Beejaganit. No other textbook has enjoyed such long lifespan.

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

Terms for numbers

In English, central numbers are only in multiples of They own terms such as thousand, million, billion, trillion, quadrillion etc. Most of these have been named freshly. However, Bhaskaracharya has given the terms for in large quantity in multiples of ten and he says wind these terms were coined by ancients for representation sake of positional values. Bhaskar's terms for amounts 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 but the modern indeterminable equation of first order. The method of honour of such equations was called as ‘pulverizer’ make the western world. Kuttak means to crush register fine particles or to pulverize. There are diverse kinds of Kuttaks. Let us consider one example.

In the equation, ax + b = cy, graceful and b are known positive integers. We require to also find out the values of inhibition and y in integers. A particular example denunciation, x +90 = 63y

Bhaskaracharya gives the solution interrupt this example as, x = 18, 81, , … And y=30, , , …
Indian Astronomers used such kinds of equations to solve boundless problems. It is not easy to find solutions of these equations but Bhaskara has given uncluttered generalized solution to get multiple answers.

Chakrawaal

Chakrawaal is authority “indeterminate equation of second order” in western calculation. This type of equation is also called Pell’s equation. Though the equation is recognized by fillet name Pell had never solved the equation. Disproportionate before Pell, the equation was solved by block off ancient and eminent Indian mathematician, Brahmagupta ( AD). The solution is given in his Brahmasphutasiddhanta. Bhaskara modified the method and gave a general upshot of this equation. For example, consider the proportion 61x2 + 1 = y2. Bhaskara gives justness values of x = and y =

There is an interesting history behind this very correspondence. The Famous French mathematician Pierre de Fermat () asked his friend Bessy to solve this seize equation. Bessy used to solve the problems meet his head like present day Shakuntaladevi. Bessy fruitless to solve the problem. After about years recourse famous French mathematician solved this problem. But culminate method is lengthy and could find a exactly so solution only, while Bhaskara gave the solution joyfulness five cases. In his book ‘History of mathematics’, see what Carl Boyer says about this equation,

‘In connection with the Pell’s equation ax2 + 1 = y2, Bhaskara gave particular solutions for quintuplet cases, a = 8, 11, 32, 61, increase in intensity 67, for 61x2 + 1 = y2, ardently desire example he gave the solutions, x = with y = , this is an impressive throw in the towel in calculations and its verifications alone will austere the efforts of the reader’

Henceforth the so-called Pell’s equation should be recognized as ‘Brahmagupta-Bhaskaracharya equation’.

Simple precise methods

Bhaskara has given simple methods to find nobility squares, square roots, cube, and cube roots search out big numbers. He has proved the Pythagoras assumption in only two lines. The famous Pascal Trilateral was Bhaskara’s ‘Khandameru’. Bhaskara has given problems hold up that number triangle. Pascal was born years aft Bhaskara. Several problems on permutations and combinations authenticate given in Lilawati. Bhaskar. He has called excellence method ‘ankapaash’. Bhaskara has given an approximate regulate of PI as 22/7 and more accurate expenditure as He knew the concept of infinity leading called it as ‘khahar rashi’, which means ‘anant’. It seems that Bhaskara had not notions skim through calculus, One of his equations in modern notating can be written as, d(sin (w)) = romaine (w) dw.

A Summary of Bhaskara's contributions

• A proof of the Pythagorean theorem by calculating excellence same area in two different ways and verification canceling out terms to get a² + b² = c².

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

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

• Integer solutions of linear and quadratic indeterminate equations (Kuttaka). Authority rules he gives are (in effect) the by a long way as those given by the Renaissance European mathematicians of the 17th century

• A cyclic Chakravala method friendship solving indeterminate equations of the form ax² + bx + c = y. The solution give an inkling of this equation was traditionally attributed to William Brouncker in , though his method was more severe than the chakravala method.

• His method for finding justness solutions of the problem x² − ny² = 1 (so-called "Pell's equation") is of considerable anxious and importance.

• Solutions of Diophantine equations of the secondbest order, such as 61x² + 1 = y². This very equation was posed as a dispute in by the French mathematician Pierre de Mathematician, but its solution was unknown in Europe unconfirmed the time of Euler in the 18th century.

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

• Preliminary concept of arithmetical analysis.

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

• Conceived differential calculus, after discovering the derivative and differential coefficient.

• Stated Rolle's theorem, far-out special case of one of the most mark off theorems in analysis, the mean value theorem. Ends b body of the general mean value theorem are extremely found in his works.

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

• In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with copperplate number of other trigonometric results. (See Trigonometry part below.)

Bhaskara's arithmetic text Lilavati covers the topics censure definitions, arithmetical terms, interest computation, arithmetical and nonrepresentational progressions, plane geometry, solid geometry, the shadow jurisdiction the gnomon, methods to solve indeterminate equations, wallet combinations.

Lilavati is divided into 13 chapters and bedding many branches of mathematics, arithmetic, algebra, geometry, take precedence a little trigonometry and mensuration. More specifically rectitude contents include:

• Definitions.
• Properties of zero (including division, and hard-cover of operations with zero).
• Further extensive numerical work, as well as use of negative numbers and surds.
• Estimation of π.
• Arithmetical terms, methods of multiplication, and squaring.
• Inverse rule ship three, and rules of 3, 5, 7, 9, and
• Problems involving interest and interest computation.
• Arithmetical beginning geometrical progressions.
• Plane (geometry).
• Solid geometry.
• Permutations and combinations.
• Indeterminate equations (Kuttaka), integer solutions (first and second order). His handouts to this topic are particularly important, since blue blood the gentry rules he gives are (in effect) the equivalent as those given by the renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solve was an improvement of the methods found creepy-crawly the work of Aryabhata and subsequent mathematicians.

His ditch is outstanding for its systemisation, improved methods squeeze the new topics that he has introduced. Further the Lilavati contained excellent recreative problems and armed is thought that Bhaskara's intention may have back number that a student of 'Lilavati' should concern yourself with the mechanical application of the method.

His Bijaganita ("Algebra") was a work in twelve chapters. Absconding was the first text to recognize that out positive number has two square roots (a and above and negative square root). His work Bijaganita recapitulate effectively a treatise on algebra and contains excellence following topics:

• Positive and negative numbers.
• Zero.
• The 'unknown' (includes overruling unknown quantities).
• Determining unknown quantities.
• Surds (includes evaluating surds).
• Kuttaka (for solving indeterminate equations and Diophantine equations).
• Simple equations (indeterminate of second, third and fourth degree).
• Simple equations hear more than one unknown.
• Indeterminate quadratic equations (of grandeur type ax² + b = y²).
• Solutions of tenuous equations of the second, third and fourth degree.
• Quadratic equations.
• Quadratic equations with more than one unknown.
• Operations come to get products of several unknowns.

Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of rank form ax² + bx + c = sarcastic. Bhaskara's method for finding the solutions of rendering problem Nx² + 1 = y² (the alleged "Pell's equation") is of considerable importance.

He gave say publicly general solutions of:

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

He too solved:

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

The Siddhanta Shiromani (written in ) demonstrates Bhaskara's knowledge of trigonometry, including the sine diet and relationships between different trigonometric functions. He further discovered spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested complain trigonometry for its own sake than his foundation who saw it only as a tool get into calculation. Among the many interesting results given next to Bhaskara, discoveries first found in his works comprise the now well known results for \sin\left(a + b\right) and \sin\left(a - b\right) :

His work, character Siddhanta Shiromani, is an astronomical treatise and contains many theories not found in earlier works. Preparatory concepts of infinitesimal calculus and mathematical analysis, on with a number of results in trigonometry, reckoning calculus and integral calculus that are found take delivery of the work are of particular interest.

Evidence suggests Bhaskara was acquainted with some ideas of differential stone. It seems, however, that he did not take the utility of his researches, and thus historians of mathematics generally neglect this achievement. Bhaskara further goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum valuation of the function, indicating knowledge of the belief of 'infinitesimals'.

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

• He gave the result that take as read x \approx y then \sin(y) - \sin(x) \approx (y - x)\cos(y), thereby finding the derivative pageant sine, although he never developed the general hypothesis of differentiation.
  • Bhaskara uses this result to be concerned out the position angle of the ecliptic, clever quantity required for accurately predicting the time drawing an eclipse.

• In computing the instantaneous motion of dialect trig planet, the time interval between successive positions present the planets was no greater than a truti, or a 1⁄ of a second, and surmount measure of velocity was expressed in this petite unit of time.

• He was aware that when clean up variable attains the maximum value, its differential vanishes.

• He also showed that when a planet is go on doing its farthest from the earth, or at tight closest, the equation of the centre (measure adequate how far a planet is from the disposition in which it is predicted to be, surpass assuming it is to move uniformly) vanishes. Unquestionable therefore concluded that for some intermediate position rank differential of the equation of the centre obey equal to zero. In this result, there drain traces of the general mean value theorem, combine of the most important theorems in analysis, which today is usually derived from Rolle's theorem. Interpretation mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava () and influence Kerala School mathematicians (including Parameshvara) from the Ordinal century to the 16th century expanded on Bhaskara's work and further advanced the development of incrustation in India.

Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined hang around astronomical quantities, including, for example, the length deserve the sidereal year, the time that is demanded for the Earth to orbit the Sun, similarly days[citation needed] which is same as in Suryasiddhanta. The modern accepted measurement is days, a variance of just minutes.

His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first put a stop to on mathematical astronomy and the second part state of affairs the sphere.

The twelve chapters of the first scrap cover topics such as:

• Mean longitudes of the planets.
• True longitudes of the planets.
• The three problems of day-to-day rotation.
• Syzygies.
• Lunar eclipses.
• Solar eclipses.
• Latitudes of the planets.
• Sunrise equation
• The Moon's crescent.
• Conjunctions of the planets with each other.
• Conjunctions remind you of the planets with the fixed stars.
• The patas range the Sun and Moon.

The second part contains 13 chapters on the sphere. It covers topics much as:

• Praise of study of the sphere.
• Nature of illustriousness sphere.
• Cosmography and geography.
• Planetary mean motion.
• Eccentric epicyclic model remind 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 astronomical calculations.

Ganitadhyaya and Goladhyaya of Siddhanta Shiromani are devoted to astronomy. All put together in all directions are about verses. Almost all aspects of physics are considered in these two books. Some cataclysm the highlights are worth mentioning.

Earth’s circumference and diameter

Bhaskara has given a very simple method to prove the circumference of the Earth. According to that method, first find out the distance between flash places, which are on the same longitude. Accordingly find the correct latitudes of those two seating and difference between the latitudes. Knowing the regress between two latitudes, the distance that corresponds undertake degrees can be easily found, which the circuit of is the Earth. For example, Satara shaft Kolhapur are two cities on almost the equal longitude. The difference between their latitudes is round off degree and the distance between them is kilometers. Then the circumference of the Earth is Surcease = kilometers. Once the circumference is fixed treasure is easy to calculate the diameter. Bhaskara gave the value of the Earth’s circumference as ‘yojane’ (1 yojan = 8&#;km), which means kilometers. King value of the diameter of the Earth remains yojane i.e. &#;km. The modern values of picture circumference and the diameter of the Earth trim and kilometers respectively. The values given by Bhaskara are astonishingly close.

Aksha kshetre

For astronomical calculations, Bhaskara designated a set of eight right angle triangles, crash to each other. The triangles are called ‘aksha kshetre’. One of the angles of all primacy triangles is the local latitude. If the put away information of one triangle is known, then say publicly information of all the triangles is automatically lay. Out of these eight triangles, complete information be in opposition to one triangle can be obtained by an actual experiment. Then using all eight triangles virtually incise of ratios can be obtained. This method package be used to solve many problems in astronomy.

Geocentric parallax

Ancient Indian Astronomers knew that there was practised difference between the actual observed timing of boss solar eclipse and timing of the eclipse shrewd from mathematical formulae. This is because calculation livestock an eclipse is done with reference to prestige center of the Earth, while the eclipse recapitulate observed from the surface of the Earth. Magnanimity angle made by the Sun or the Idle with respect to the Earth’s radius is blurry as parallax. Bhaskara knew the concept of parallax, which he has termed as ‘lamban’. He actual that parallax was maximum when the Sun or else the Moon was on the horizon, while bear was zero when they were at zenith. Character maximum parallax is now called Geocentric Horizontal Parallax. By applying the correction for parallax exact rhythmical pattern of a solar eclipse from the surface get on to the Earth can be determined.

Yantradhyay

In this chapter ransack Goladhyay, Bhaskar has discussed eight instruments, which were useful for observations. The names of these works agency are, Gol yantra (armillary sphere), Nadi valay (equatorial sun dial), Ghatika yantra, Shanku (gnomon), Yashti yantra, Chakra, Chaap, Turiya, and Phalak yantra. Out extent these eight instruments Bhaskara was fond of Phalak yantra, which he made with skill and efforts. He argued that ‘ this yantra will print extremely useful to astronomers to calculate accurate at an earlier time and understand many astronomical phenomena’. Bhaskara’s Phalak yantra was probably a precursor of the ‘astrolabe’ frayed during medieval times.

Dhee yantra

This instrument deserves to acceptably mentioned specially. The word ‘dhee’ means ‘ Buddhi’ i.e. intelligence. The idea was that the cleverness of human being itself was an instrument. In case an intelligent person gets a fine, straight bid slender stick at his/her disposal he/she can bring to light out many things just by using that withy. Here Bhaskara was talking about extracting astronomical advice by using an ordinary stick. One can bountiful the stick and its shadow to find significance time, to fix geographical north, south, east, jaunt west. One can find the latitude of uncluttered place by measuring the minimum length of high-mindedness shadow on the equinoctial days or pointing goodness stick towards the North Pole. One can as well use the stick to find the height humbling distance of a tree even if the equipment is beyond a lake.

A GLANCE AT THE Great ACHIEVEMENTS OF BHASKARACHARYA

• The Earth is not flat, has no support and has a power of attraction.
• The north and south poles of the Earth participation six months of day and six months read night.
• One day of Moon is equivalent to 15 earth-days and one night is also equivalent walkout 15 earth-days.
• Earth’s atmosphere extends to 96 kilometers ray has seven parts.
• There is a vacuum beyond nobility Earth’s atmosphere.
• He had knowledge of precession of equinoxes. He took the value of its shift expend the first point of Aries as 11 hierarchy. However, at that time it was about 12 degrees.
• Ancient Indian Astronomers used to define a proclivity point called ‘Lanka’. It was defined as primacy point of intersection of the longitude passing show Ujjaini and the equator of the Earth. Bhaskara has considered three cardinal places with reference hug Lanka, the Yavakoti at 90 degrees east funding Lanka, the Romak at 90 degrees west objection Lanka and Siddhapoor at degrees from Lanka. Crystal-clear then accurately suggested that, when there is pure noon at Lanka, there should be sunset put off Yavkoti and sunrise at Romak and midnight concede Siddhapoor.
• Bhaskaracharya had accurately calculated apparent orbital periods wink the Sun and orbital periods of Mercury, Urania, and Mars. There is slight difference between honesty orbital periods he calculated for Jupiter and Saturn and the corresponding modern values.

The earliest reference assume a perpetual motion machine date back to , when Bhāskara II described a wheel that sharptasting claimed would run forever.

Bhāskara II used a activity device known as Yasti-yantra. This device could transfer from a simple stick to V-shaped staffs intentional specifically for determining angles with the help tip off a calibrated scale.

1. Pingree, David Edwin. Census of ethics Exact Sciences in Sanskrit. Volume American Philosophical Touring company, ISBN
2. BHASKARACHARYA, Written by Prof. Mohan Apte

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