bhaskara 2 biography in english

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Bhaskara II – History, Biography, and Accomplishments

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Who was Bhaskara II?

General history of bhaskara ii’s life, the lilavati, the bijaganita, the ganitadhyaya, the goladhyaya, the bhaskara equation, spherical trigonometry, rolle’s theorem, foundations of calculus, bhaskara ii’s legacy.

Bhaskaracharya is just one of the many historical figures that you’ll rarely hear about, unless you happen to be from his home country of India. Also known as Bhaskara II, or Bhaskara the Teacher, Bhaskaracharya formulated what would go on to become the foundations of the mathematics that we use today.

While he came to the wrong conclusions often, Bhaskara’s work served to set the stage for those who would later come along and repurpose his work within their correct frames of reference. Much of his work still continues to be used as-is, without modification, because of how well-thought-out it was. In fact, today, most people still use Bhaskara’s work without even realizing it!

Labeled as one the “greatest mathematicians of medieval India” , the 12 th -Century mathematician Bhaskara II wrote many books containing mathematical and astronomical feats which would not be discovered elsewhere for another 500 years.

While very intelligent in his own right, Bhaskara did not develop mathematics himself. He had access to the writing of the Ancient Greek scholars, as well as that of the many Indian mathematicians who had come before him, especially Bhaskara I, a famous Indian mathematician from the 7th Century.

It was Bhaskara I’s influence that inspired Bhaskara II to join Ujjain, India’s “most prestigious mathematical center” at the time. It was at Ujjain where Bhaskara II would formulate the ideas which would go on to become his legacy.

However, he is not well known outside of India. Sure, Bhaskaracharya received high acclaim at home, but he never received much acknowledgment in Europe or the Middle East. Over the years, most of his work went uncredited, or was otherwise given to others.

Bhaskara developed solutions and foundations for what would eventually become calculus, as well as the second proof of the Pythagorean Theorem – yet, no one talks about him outside of his home country.

Still, Bhaskara II’s legacy lives on through other Ujjain mathematicians , and through his son, who established a school dedicated to teaching his work.

While we may know his work, the man behind it is a mystery. Most of what we now know about Bhaskara comes from his son and the records kept by the Ujjain, where Bhaskara served as head of the astronomical observatory .

From these records, we know that Bhaskara was born in 1114 A.D., near Bijjada Bida (present-day Bijapur), but not much else, especially about his early life. We do know that he was the son of an equally-famous Indian mathematician and astrologer, Mahesvara . Today, many experts agree that it was Mahesvara who taught Bhaskara everything he knew and developed throughout the course of his life.

While at Ujjain, he wrote most of his books. Writing in verse (as was Indian custom at the time), Bhaskara wrote on several topics of mathematics and astronomy , such as trigonometry, algebra, and calculus.

Upon his death in 1185, Bhaskara’s work was picked up by Madhava of Sangamagrama, among other Indian mathematicians at Kerala School . These mathematicians would expand upon Bhaskara’s work, to establish the foundations for calculus development in India.

Bhaskara II’s Writings and Legacy

While he published many books, Bhaskara is mostly remembered for his 1150 A. D. masterpiece, the Siddhanta Siromani (Crown of Treatises). Released as a 4-part series rather than as a single book, Bhaskara wrote the 1450-verse tome when he was 36 years old.

Each part of the book focused on a separate field of astronomy and mathematics . Often referred to as separate books themselves, these parts were called the Lilavati, Bijaganita, Grahagaṇita, and Goladhyaya .

Named after his daughter Lilavati, Bhaskara wrote his first Siddhanta Siromani volume as condolence for her husband’s death, because of a contraption he had made to ensure that the marriage happened at a certain time had failed.

The Lilavati contains thirteen chapters on several mathematical topics, including mensuration and trigonometry , although it mostly worked to define terms and the properties of the number zero. Because the book was a textbook, Bhaskara included several problems for readers to work through.

At its heart, the book is mostly about indeterminate equations and obtaining integer solutions for them . Bhaskara delved into several ways to solve the equations, from multiplication to squaring . He even discussed the Rule of Three , something that Europe would not discover for themselves for another 500 years.

Some of the other important subjects were:

• Zero and dividing by zero
• Rules of Operations
• Negative numbers
• Surds or non-whole number square roots
• Integer solutions to Indeterminant equations

The 12-chapter Bjiganita is all about algebra, including the first written record of the positive and negative square roots of numbers . The book also continued Bhaskara’s work on indeterminant equations and solutions to unknown quantities.

Expanding upon previous works by Aryabhata and Brahmagupta, Bhaskara used several pages of the Bjiaganita to improve the Kuttaka methods for solving equations, including both indeterminant and Diophantine varieties .

The book also furthered Bhaskara’s work on the Number Zero, leading to one of his few failures. The book details his conclusions that dividing by zero would produce infinity. And while there are certainly situations in which this works, it is a flawed solution that would serve to break mathematics in the long run, if true.  It would take European mathematicians to eventually come along and realize this, and then declare that dividing by zero was impossible .

Still, it bears saying that the Bjiaganita contains many other conclusions that panned out successfully, including the chakravala method solutions to Pell’s equation.

Some of the other topics in the book include quadratic and simple equations, along with methods for determining surds.

The third book deals with mathematical astronomy . Derived from concepts first envisioned by Aryabhata, Bhaskara uses his third publication to describe the heliocentric view of the solar system and the elliptical orbits of planets, based on Brahmagupta’s law of gravity .

Throughout the twelve chapters, Bhaskara discusses such topics as the mean and true longitudes and latitudes of the planets, as well as the nature of lunar and solar eclipses . He also examines planetary conjunctions, the orbits of the sun and moon, as well as issues arising from diurnal rotations.

He also wrote estimates for values such as the length of the year , which was so accurate that we were only off their actual value by a minute!

Bhaskara’s final, thirteen-chapter publication is all about spheres and similar shapes , such as armillary spheres, rings, and hoops . Most of the book deals with spherical trigonometry, in which Bhaskara found the sine of many angles, from 18 to 36 degrees. The book even includes a sine table, along with the many relationships between trigonometric functions.

Other topics in the Goladhyaya include:

• Cosmography
• Geography and the seasons
• Planetary mean motion
• Ellipses and lunar crescents

Bhaskara also discussed an epicycle model for planetary orbits. This model proposed that some planets, such as the sun and moon , move in small circles as they move in their orbits. While the model does not line up with reality, Bhaskara notes how difficult it is to calculate astronomical phenomena using the astronomical instruments that were available at the time.

Throughout the entire Siddhanta Siromani series, Bhaskara reached many mathematical achievements that rivaled the works of both Europe and China.

However, none of these would come close to the equation that now bears his name – despite most people having never heard of it, even though they know what it is and how to use it.

This is because most of the world simply refers to Bhaskara’s discovery as the quadratic formula for the solutions of quadratic equations:

X = -b/a ± sqrt(b 2 -4ac)/a

Originally used just for real roots , others would later expand this solution to include complex number solutions as well, making the quadratic formula one of Bhaskara’s most important discoveries.

Along with the quadratic formula, Bhaskara also created other insights into quadratic equations.

For instance, he also gave us the formula for finding the square root of a sum of a number and a square root:

Sqrt(a ± sqrt(b)) = sqrt(½(a + sqrt(a 2 – b)) ± sqrt(½(a – sqrt(a 2 – b))

Other Achievements in Mathematics and Astronomy

While the quadratic formula was big, it was not Bhaskara’s only achievement that he wrote about in his treatises.

Bhaskara developed many of the trigonometric identities and formulas used throughout mathematics, navigation, geodesy, and astronomy.

For instance, the Goladhyaya is the first publication to report the now-famous formulas:

• sin(a + b) = sin(a) cos(b) + cos(a) sin(b)
• sin(a – b). sin(a) cos(b) – cos(a) sin(b)

Aside from that, Bhaskara also made advancements in other areas of math and astronomy, including (but not limited to) his work on what would become Rolle’s Theorem.

In particular, Bhaskara noted that the distance between a planet’s real and predicated locations become zero when the planet is at its farthest, or nearest, points to Earth. This solution to what was the center equation hinted towards the general mean value theorem which would replace it.

Bhaskara’s other major contributions are the foundations of what would later become calculus. In particular, he also dabbled in differential calculus, integral calculus, and mathematical analysis – even though he never fully recognized their usefulness. He just developed the principles to solve problems in astronomy.

While Isaac Newton and Gottfried Leibniz would later serve to bring everything together, Bhaskara was the one who formulated the foundations long before they were born. He even coined the concepts of the derivative and differential coefficients.

While the man himself is not well-known outside of India, Bhaskara’s work traveled the globe . Other Indian mathematicians picked up his work immediately upon his death, with the establishing of his school. Outside of India, his work got compiled into the works of Islamic scholars , only to be reassigned back to him recently.

Still, Bhaskara’s influence on mathematics cannot be denied or understated.

In fact, there is strong evidence that Islamic mathematicians knew about Bhaskara and his ideas as he published them. Many reported Islamic mathematic discoveries to show direct influence, starting from the end of the 12th Century, ultimately culminating in a famous 1587 Persian translation of the Lilavati.

In Europe, Bhaskara’s influence came through those Islamic channels . While no one in Europe would know his name until the 20th Century, European scholars were using Bhaskara’s work all throughout the Renaissance .

Some scholars do debate whether Bhaskara II was influenced by Diophantine, although these claims were later disputed, as similar Indian developments on Diophantine equations going as far back as the Sulba Sutras were written from 800 to 500 BCE.

Nevertheless, Bhaskara II’s work will continue to inspire other Indians to take up mathematics for centuries to come.

Bhāskara II Biography

Born In: Bijjaragi, Vijayapur, Karnataka

Bhaskara II, also known as Bhaskara or as Bhaskaracharya, was a 12th century Indian mathematician. He was also a renowned astronomer who accurately defined many astronomical quantities, including the length of the sidereal year. A brilliant mathematician, he made the significant discovery of the principles of differential calculus and its application to astronomical problems and computations centuries before European mathematicians like Newton and Leibniz made similar discoveries. It is believed that Bhaskara II was the first to conceive the differential coefficient and differential calculus. The son of a mathematician and astronomer, he was trained by his father in the subjects. Following in his father’s footsteps the young man too became a renowned mathematician and astronomer and was considered the lineal successor of the noted Indian mathematician Brahmagupta as head of an astronomical observatory at Ujjain. Bhaskara II wrote the first work with full and systematic use of the decimal number system and also wrote extensively on other mathematical techniques and on his astronomical observations of planetary positions, conjunctions, eclipses, cosmography, and geography. In addition, he also filled many of the gaps in his predecessor Brahmagupta’s work. In recognition of his invaluable contributions to mathematics and astronomy, he has been called the greatest mathematician of medieval India.

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Also Known As: Bhaskara the teacher, Bhaskara Achārya, Bhaskara II, Bhāskarācārya

Died At Age: 71

Born Country: India

Astronomers Mathematicians

Died on: 1185

place of death: Ujjain, Madhya Pradesh, India

You wanted to know

What are bhāskara ii's notable contributions to mathematics.

Bhāskara II made significant contributions to algebra, arithmetic, geometry, and trigonometry. He is known for his work on indeterminate equations, quadratic equations, and the concept of zero.

What is Bhāskara II's famous work in the field of astronomy?

Bhāskara II's most famous work in astronomy is the Siddhānta Shiromani, where he discusses planetary positions, eclipses, and the cosmology of his time.

How did Bhāskara II contribute to the development of trigonometry?

Bhāskara II provided important advancements in trigonometry by introducing the concept of sine and exploring trigonometric ratios in his mathematical works.

What is the significance of Bhāskara II's Lilavati?

Lilavati, a mathematical treatise by Bhāskara II, covers various topics such as arithmetic, geometry, and algebra. It is known for its innovative problem-solving techniques and mathematical principles.

How did Bhāskara II impact the understanding of calculus?

Bhāskara II's works laid the foundation for calculus concepts, such as differential calculus and infinite series. His contributions to mathematical analysis had a lasting impact on the development of calculus.

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Bhāskara II, a renowned Indian mathematician and astronomer, is credited with developing the concept of positive and negative numbers in the 12th century, a groundbreaking achievement that significantly advanced mathematics.

In his mathematical treatise, Lilavati, Bhāskara II included fascinating puzzles and riddles to engage and challenge his readers, showcasing his playful and creative approach to teaching.

Bhāskara II's work also delved into the field of calculus, with his contributions foreshadowing some of the principles later formalized by European mathematicians like Newton and Leibniz.

Beyond his mathematical prowess, Bhāskara II was also an accomplished poet, weaving his love for literature and mathematics together in his writings, showcasing his multifaceted talents.

See the events in life of Bhāskara II in Chronological Order

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Bhaskara II - The Great Indian Mathematician

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30 September 2020                

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Introduction

Bhaskara ii (1114 – 1185), also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher"), was an Indian mathematician and astronomer. The ii has been appended to his name to distinguish him from the 7th-century astronomer Bhaskara i.

Who is Bhāskara ii?

Bhaskara was born to Mahesvara, a famous astrologer, near Bijjada Bida (present-day Bijapur district) in the state of Karnataka. Bhaskaracharya was the first mathematician to write a work with full and systematic use of the decimal number system.

It is believed his father taught him mathematics, which he later passed on to his son Loksamudra.

Bhaskara II became the head of the astronomical observatory at Ujjain to join Ujjain, India’s “most prestigious mathematical centre” at the time. It was at Ujjain where Bhaskara II would formulate the ideas which would go on to become his legacy. Mathematicians of renown such as Varahamihira and Brahmagupta had worked there and built up a strong school of mathematical astronomy. He can be called a pioneer in understanding the number systems and solving equations.

Works of Bhaskara ii

Bhaskara developed an understanding of calculus, the number systems, and solving equations, which were not to be achieved anywhere else in the world for several centuries.

Bhaskara is mainly remembered for his 1150 A. D. masterpiece, the Siddhanta Siromani (Crown of Treatises) which he wrote at the age of 36. The treatise comprises 1450 verses which have four segments. Each segment of the book focuses on a separate field of astronomy and mathematics.

• Lilavati: A treatise on arithmetic, geometry and the solution of indeterminate equations
• Bijaganita: ( A treatise on Algebra), 
• Goladhyaya: (Mathematics of Spheres),
• Grahaganita: (Mathematics of the Planets).

He also wrote another treatise named Karaṇā Kautūhala.

Lilavati 

Lilavati is composed in verse form so that pupils could memorise the rules without the need to refer to written text. Some of the problems in  Leelavati  are addressed to a young maiden of that same name. There are several stories around Lilavati being his daughter Lilavati has thirteen chapters which include several methods of computing numbers such as multiplications, squares, and progressions, with examples using kings and elephants, objects which a common man could easily associate with.

Here is one poem from Lilavati:

A fifth part of a swarm of bees came to rest

 on the flower of Kadamba,

 a third on the flower of Silinda

 Three times the difference between these two numbers

 flew over a flower of Krutaja,

 and one bee alone remained in the air,

attracted by the perfume of a jasmine in bloom

 Tell me, beautiful girl, how many bees were in the swarm ?

Step-by-step explanation:

Number of bees - x

A fifth part of a swarm of bees came to rest on the flower of Kadamba- \(1/5x\)

A third on the flower of Silinda- \(1/3x\)

Three times the difference between these two numbers flew over a flower of Krutaja- \(3 \times (1/3-1/5)x\)

The sum of all bees:

\[\begin{align}&x=1/5x+1/3x+3 \times (1/3-1/5)x+1\\&x=8/15x+6/15x+1\\&1/15x=1\\&x=15\end{align}\]

\[3+5+6+1=15\]

The  Bijaganita  is a work in twelve chapters. In  Bījagaṇita  (“Seed Counting”), he not only used the decimal system but also compiled problems from Brahmagupta and others. Bjiganita is all about algebra, including the first written record of the positive and negative square roots of numbers. He expanded the previous works by Aryabhata and Brahmagupta, Also to improve the Kuttaka methods for solving equations. Kuttak means to crush fine particles or to pulverize. Kuttak is nothing but the modern indeterminate equation of first order. There are many kinds of Kuttaks. For example- In the equation, \(ax + b = cy\) , a and b are known positive integers, and the values of x and y are to be found in integers. As a particular example, he considered \(100x + 90 = 63y\)

 Bhaskaracharya gives the solution of this example as, \(x = 18, 81, 144, 207...\) and \(y = 30, 130, 230, 330...\) It is not easy to find solutions to these equations. He filled many of the gaps in Brahmagupta’s works.

 Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form \(ax^2 + bx + c = y.\) Bhaskara’s method for finding the solutions of the problem \(Nx^2 + 1 = y^2\) (the so-called “Pell’s equation”) is of considerable importance.

The book also detailed Bhaskara’s work on the Number Zero, leading to one of his few failures. He concluded that dividing by zero would produce an infinity. This is considered a flawed solution and it would take European mathematicians to eventually realise that dividing by zero was impossible.

Some of the other topics in the book include quadratic and simple equations, along with methods for determining surds.

Touches of mythological allegories enhance Bhaskasa ii’s Bījagaṇita. While discussing properties of the mathematical infinity, Bhaskaracharya draws a parallel with Lord Vishnu who is referred to as Ananta (endless, boundless, eternal, infinite) and Acyuta (firm, solid, imperishable, permanent): During pralay (Cosmic Dissolution), beings merge in the Lord and during sṛiṣhti (Creation), beings emerge out of Him; but the Lord Himself — the Ananta, the Acyuta — remains unaffected. Likewise, nothing happens to the number infinity when any (other) number enters (i.e., is added to) or leaves (i.e., is subtracted from) the infinity. It remains unchanged.

Grahaganita

The third book or the Grahaganita deals with mathematical astronomy . The concepts are derived from the earlier works Aryabhata. Bhaskara describes the heliocentric view of the solar systemand the elliptical orbits of planets, based on Brahmagupta’s law of gravity .

Throughout the twelve chapters, Bhaskara discusses topics related to mean and true longitudes and latitudes of the planets, as well as the nature of lunar and solar eclipses . He also examines planetary conjunctions, the orbits of the sun and moon, as well as issues arising from diurnal rotations.

He also wrote estimates for values such as the  length of the year , which was so accurate that we were only of their actual value by a minute!

Bhaskara’s final, thirteen-chapter publication, the Goladhyaya is all about spheres and similar shapes. Some of the topics in the Goladhyaya include Cosmography, geography and the seasons, planetary movements, eclipses and lunar crescents.

The book also deals with spherical trigonometry, in which Bhaskara found the sine of many angles, from 18 to 36 degrees. The book even includes a sine table, along with the many relationships between trigonometric functions.

 In one of the chapters of Goladhyay, Bhaskara ii has discussed eight instruments, which were useful for observations. The names of these instruments are Gol yantra (armillary sphere), Nadi valay (equatorial sundial), Ghatika yantra, Shanku (gnomon), Yashti yantra, Chakra, Chaap, Turiya, and Phalak yantra. Out of these eight instruments, Bhaskara was fond of Phalak yantra, which he made with skill and efforts. He argued that „ this yantra will be extremely useful to astronomers to calculate accurate time and understand many astronomical phenomena‟.

Interestingly, Bhaskara ii also talks about astronomical information by using an ordinary stick. One can use the stick and its shadow to find the time to fix geographical north, south, east, and west. One can find the latitude of a place by measuring the minimum length of the shadow on the equinoctial days or pointing the stick towards the North Pole

Bhaskaracharya had calculated the apparent orbital periods of the Sun and orbital periods of Mercury, Venus, and Mars though there is a slight difference between the orbital periods he calculated for Jupiter and Saturn and the corresponding modern values.

A medieval inscription in an Indian temple reads:-

Triumphant is the illustrious Bhaskaracharya whose feats are revered by both the wise and the learned. A poet endowed with fame and religious merit, he is like the crest on a peacock.

Bhaskara ii’s work was so well thought out that a lot of it being used today as well without modifications. On 20 November 1981, the Indian Space Research Organisation (ISRO) launched the Bhaskara II satellite in honour of the great mathematician and astronomer.

It is a matter of great pride and honour that his works have received recognition across the globe.

Frequently Asked Questions (FAQs)

When was bhaskara ii born.

Bhaskar ii was born in Circa 1114.

Where was Bhaskara ii born?

He was born in Bijapur, Karnataka.

When did Bhaskara ii die?

Bhaskara ii died in Circa 1185.

Where did Bhaskara ii die?

Probably Ujjain

Bhāskara II

Bhaskara (1114 – 1185), also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher"), was an Indian mathematician and astronomer. He became head of the astronomical observatory at Ujjain , continuing the mathematical tradition of Varahamihira and Brahmagupta.

• 3.1 Arithmetic
• 3.2 Algebra
• 3.3 Trigonometry
• 3.4 Calculus
• 4 Astronomy
• 6 Final Days
• 8 References

In many ways, Bhaskara represents the peak of mathematical and astronomical knowledge in the twelfth century. He reached an understanding of calculus, astronomy , the number systems, and solving equations, which were not to be achieved anywhere else in the world for several centuries. His main works were the Lilavati (dealing with arithmetic ), Bijaganita (Algebra) and Siddhanta Shiromani (written in 1150) which consists of two parts: Goladhyaya ( sphere ) and Grahaganita (mathematics of the planets ).

Bhaskara, born in 1114 C.E. (1114 – 1185), also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher"), was an Indian mathematician and astronomer. He was born near Bijjada Bida (in present day Bijapur district, Karnataka state, South India) into the Deshastha Brahmin family. Bhaskaracharya, as he was known in India, was born to Mahesvara, another famous astrologer. He became head of the astronomical observatory at Ujjain , the leading mathematical center in India at the time, thus continuing the mathematical tradition of previously renowned mathematicians, Varahamihira and Brahmagupta.

Bhaskaracharya learnt mathematics from his father. After being introduced to the works of a previous famous mathematician, Brahmagupta, Bhaskaracharya was so inspired that he devoted himself to mathematics for the rest of his life. After his daughter, Lilavati, was widowed at the age of six, he even influenced her to study mathematics—it is not known, however, how great of a mathematician she became. When it came to algebra, Bhaskaracharya followed Brahmagupta’s work closely as his guru, and went about extending Brahmagupta’s works.

As a mathematician, Bhaskara represents the peak of mathematical and astronomical knowledge in the twelfth century. As J. J. O’Connor and E. F. Robertson stated in their article for the School of Mathematics and Statistics, “[Bhaskaracharya] reached an understanding of the number systems and solving equations which was not to be achieved in Europe for several centuries.” Bhaskaracharya was the first mathematician to write a work with full and systematic use of the decimal number system. His main work, written when he was only 30, includes the Siddhanta Siroman (written in 1150), which was segmented into four—these were the Lilavati (dealing with arithmetic ), Bijaganita (Algebra), “Goladhyaya” ( sphere ), and “Grahaganita” (mathematics of the planets ). There are also “Karanakutuhala” or “Brahmatulya,” a simplified version of the “Siddhanta Shiromani” and the “Vivarana,” a commentary on the “Shishyadhividdhidatantra.”

The books written by Bhaskaracharya were essentially textbooks, and had been simplified to help and stimulate student’s interests. The book became so well-known that even four or five centuries after it was written, it was translated into Persian.

Bhaskaracharya was known as an original thinker. Dilip M. Salwi of Our Scientists stated, “He was the first mathematician to declare confidently that any term divided by zero is infinity and the sum of any term and infinity is infinity” (though we now know that any term divided by zero is undefined). Still, he has made several major contributions to the world of mathematics. He is the first to introduce Chakrawal, a cyclic method of solving algebraic equations. In fact, it took six centuries after his finding for European mathematicians such as Galois, Euler, and Lagrange to rediscover this method and call it “inverse cyclic” method of solution.

Any trace of calculus first appeared in Bhaskaracharya’s works—Salwi stated, “Determination of the area and volume of a sphere in a rough integral calculus manner was also mentioned for the first time in his book. It contained important formulas and theorems in trigonometry and permutation and combination.” Though unknown by most, Bhaskaracharya can be considered the founder of differential calculus, for it was he who founded such methods centuries before Isaac Newton and Gottfried Leibniz came about it. At his time, no one took note of his great achievements. In astronomy, Bhaskaracharya is renowned for his concept of Tatkalikagati, instantaneous motion in physical terms. This advance allows astronomers to accurately measure planetary movements.

He wrote his second book, “Karanakutuhala,” at the age of 69. This consists of various astronomical calculations and remains, to this day, a reference book in the making the calendars based on astronomical happenings (analogical to horoscopes and numerology based systems).

Lilavati , his book on arithmetic, is the source of interesting legends that assert that it was written for his daughter, Lilavati. In one of these stories, found in a Persian translation of Lilavati , Bhaskaracharya studied Lilavati's horoscope and predicted that her husband would die soon after the marriage if the marriage did not take place at a particular time. To prevent that, he placed a cup with a small hole at the bottom of a vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. He put the device in a room with a warning to Lilavati to not go near it. In her curiosity though, she went to look at the device and a pearl from her nose ring accidentally dropped into it, thus upsetting it. The marriage took place at wrong time and she was soon widowed.

Mathematics

Some of Bhaskara's contributions to mathematics include the following:

• A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a 2  +  b 2  =  c 2 .
• In Lilavati , solutions of quadratic, cubic and quartic indeterminate equations.
• Solutions of indeterminate quadratic equations (of the type ax 2 + b = y 2 ).
• Integer solutions of linear and quadratic indeterminate equations (Kuttaka) . The rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the seventeenth century.
• A cyclic, Chakravala method for solving indeterminate equations of the form ax 2 + bx + c = y . The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.
• His method for finding the solutions of the problem x 2  −  ny 2 = 1 (so-called "Pell's equation") is of considerable interest and importance.
• Solutions of Diophantine equations of the second order, such as 61 x 2 + 1 = y 2 . This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the eighteenth century.
• Solved quadratic equations with more than one unknown, and found negative and irrational solutions.
• Preliminary concept of mathematical analysis.
• Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.
• He conceived differential calculus, after discovering the derivative and differential coefficient.
• Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also 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 a number of other trigonometrical results. (See Trigonometry section below.)

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

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

• Definitions.
• Properties of zero (including division, and rules of operations with zero).
• Further extensive numerical work, including use of negative numbers and surds.
• Estimation of π.
• Arithmetical terms, methods of multiplication, and squaring.
• Inverse rule of three, and rules of 3, 5, 7, 9, and 11.
• Problems involving interest and interest computation.
• Arithmetical and geometrical progressions.
• Plane geometry.
• Solid geometry.
• Permutations and combinations.
• Indeterminate equations (Kuttaka), integer solutions (first and second order). His contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the seventeenth century, yet his work was of the twelfth century. Bhaskara's method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.

His work is outstanding for its systemization, improved methods and the new topics that he has introduced. Furthermore the Lilavati contained excellent recreative problems and it is thought that Bhaskara's intention may have been that a student of 'Lilavati' should concern himself with the mechanical application of the method.

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

• Positive and negative numbers.
• The 'unknown' (includes determining 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 with more than one unknown.
• Indeterminate quadratic equations (of the type ax 2 + b = y 2 ).
• Solutions of indeterminate equations of the second, third and fourth degree.
• Quadratic equations.
• Quadratic equations with more than one unknown.
• Operations with products of several unknowns.

Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax 2 + bx + c = y. Bhaskara's method for finding the solutions of the problem Nx 2 + 1 = y 2 (the so-called "Pell's equation") is of considerable importance.

He gave the general solutions of:

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

He also solved:

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

Trigonometry

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

Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect his outstanding achievement. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of ' infinitesimals '. [1]

• Bhaskara uses this result to work out the position angle of the ecliptic , a quantity required for accurately predicting the time of an eclipse.
• In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti , or a fraction of a second, and his measure of velocity was expressed in this infinitesimal unit of time.
• He was aware that when a variable attains the maximum value, its differential vanishes.
• He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the center (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the center is equal to zero . In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value theorem was later found by Parameshvara in the fifteenth century in the Lilavati Bhasya , a commentary on Bhaskara's Lilavati .

Madhava (1340-1425) and the Kerala School mathematicians (including Parameshvara) from the fourteenth century to the sixteenth century expanded on Bhaskara's work and further advanced the development of calculus in India.

The study of astronomy in Bhaskara's works is based on the heliocentric solar system of gravitation earlier propunded by Aryabhata in 499, where the planets follow an elliptical orbit around the Sun , and the law of gravity described by Brahmagupta in the seventh century. Bhaskara's contributions to astronomy include accurate calculations of many astronomical results based on this heliocentric solar system of gravitation. One of these contributions is his accurate calculation of the sidereal year, the time taken for the Earth to orbit the Sun, as 365.2588 days. The modern accepted measurement is 365.2596 days, a difference of just one minute (analyzed by naked eyes and this accuracy is achieved in the absence of any sophisticated instrument).

His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere .

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

• Mean longitudes of the planets.
• True longitudes of the planets.
• The three problems of diurnal rotation.
• Lunar eclipses .
• Solar eclipses .
• Latitudes of the planets.
• Risings and settings.
• The Moon 's crescent.
• Conjunctions of the planets with each other.
• Conjunctions of the planets with the fixed stars.
• The patas of the Sun and Moon.

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

• Praise of study of the sphere.
• Nature of the sphere.
• Cosmography and geography.
• Planetary mean motion.
• Eccentric epicyclic model of the planets.
• The armillary sphere.
• Spherical trigonometry.
• Ellipse calculations.
• First visibilities of the planets.
• Calculating the lunar crescent.
• Astronomical instruments.
• The seasons.
• Problems of astronomical calculations.

He also showed that when a planet is at its furthest from the Earth, or at its closest, the equation of the centre (measure of how far a planet is from the position it is to be predicted to be in by assuming it to movie uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.

Some scholars have suggested that Bhaskara's work influenced later developments in the Middle East and Europe. His work was perhaps known to Islamic mathematicians as soon as it was written, and influenced their subsequent writings. The results thus became indirectly known in Europe by the end of the twelfth century, but the text itself was not introduced until much later (Ball, 1960). (See Possible transmission of Kerala mathematics to Europe for other evidence.) The Mughal emperor Akbar commissioned a famous Persian translation of the Lilavati in 1587.

There have also been several allegedly unscrupulous attempts to argue that there are traces of Diophantine influence in Bhaskara's work, but this is seen as an attempt by Eurocentric scholars to claim European influence on many great non-European works of mathematics. Particularly in the field of algebra, Diophantus only looked at specific cases and did not achieve the general methods of the Indians. The study of Diophantine equations in India can also be traced back to the Sulba Sutras written from 800 B.C.E. to 500 B.C.E. , which pre-date Diophantus' work by many centuries.

Bhaskara was a natural born teacher and mathematician. As is common at his time, generations of a family would be mathematicians, with each father passing on knowledge to their sons. Bhaskara himself passed on his knowledge to his son, Loksamudra. Bhaskaracharya passed away in 1185 C.E. at Ujjain.

• ↑ Kripa Shankar Shukla, 1984, Use of Calculus in Hindu Mathematics, Indian Journal of History of Science 19: 95-104.
• ↑ Roger Cooke, "The Mathematics of the Hindus", The History of Mathematics: A Brief Course (Wiley-Interscience, 1997, ISBN 0471180823 ), 213-214.

References ISBN links support NWE through referral fees

• Joseph, George Gheverghese. The Crest of the Peacock: Non-European Roots of Mathematics , 2nd ed. Penguin Books, 2000. ISBN 978-0140125290
• Rouse Ball, W. W. A Short Account of the History of Mathematics , 4th ed. Dover Publications, 1960. ISBN 0486206300
• Salwi, Dilip M. Our Scientists . New Delhi: Children's Book Trust, 1986. ISBN 8170113180

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(1114–85?). Indian mathematician Bhaskara II was the leading mathematician of the 12th century. He wrote the first work with full and systematic use of the decimal number system.

Bhaskara II was born in 1114, in Biddur, India. He was the lineal successor of the noted Indian mathematician Brahmagupta (598–c. 665) as head of an astronomical observatory at Ujjain, India, the leading mathematical center of ancient India.

In his mathematical works, particularly Lilavati (“The Beautiful”) and Bijaganita (“Seed Counting”), he not only used the decimal system but also compiled problems from Brahmagupta and others. He filled many of the gaps in Brahmagupta’s work, especially in obtaining a general solution to the Pell equation ( x 2 = 1 + py 2 ) and in giving many particular solutions. Bhaskara II anticipated the modern convention of signs (minus by minus makes plus, minus by plus makes minus) and evidently was the first to gain some understanding of the meaning of division by zero, for he specifically stated that the value of 3 / 0 is an infinite quantity, though his understanding seems to have been limited, for he also stated wrongly that a / 0 × 0 = a . Bhaskara II used letters to represent unknown quantities, much as in modern algebra, and solved indeterminate equations of 1st and 2nd degrees. He reduced quadratic equations to a single type and solved them and investigated regular polygons up to those having 384 sides, thus obtaining a good approximate value of pi = 3.141666.

In other of his works, notably Siddhantasiromani (“Head Jewel of Accuracy”) and Karanakutuhala (“Calculation of Astronomical Wonders”), he wrote on his astronomical observations of planetary positions, conjunctions, eclipses, cosmography, geography, and the mathematical techniques and astronomical equipment used in these studies. Bhaskara II was also a noted astrologer, and tradition has it that he named his first work, Lilavati , after his daughter in order to console her. His astrological meddling coupled with an unfortunate twist of fate is said to have deprived her of her only chance for marriage and happiness. He died around 1185, probably in Ujjain.

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Indian mathematics - redressing the balance, ian g pearce, the classical period: v. bhaskaracharya ii.

... Because of his work India gave a definite 'quota' to the forward world march of the science. [ LG, P 104]

Mathematician : Bhaskara II Acharya

• 1 Mathematician
• 2 Nationality
• 4 Theorems and Definitions
• 5 Publications
• 6 Notable Quotes
• 7 Also known as

Mathematician

Indian mathematician and astronomer.

One of the first to identify zero as a number in its own right.

He was so influential that his works were still being copied by as late as $1800$.

Nationality

• Born: 1114 in Vijayapura, near Bijjada Bida (in present day Bijapur district, Karnataka state), South India
• Died: 1185 in Ujjain, India

Theorems and Definitions

• Bhaskara's Lemma
• A proof of Pythagoras' Theorem by dropping a perpendicular and investigating the relations between the ratios of the sides of the resulting similar triangles.
• A proof of Pythagoras' Theorem by calculating the same area in two different ways and then canceling out terms to get $a^2 + b^2 = c^2$.
• Solutions of quadratic , cubic and quartic indeterminate equations .

Results named for Bhaskara II Acharya can be found here .

Publications

All written in ca. $1150$:

• Lilavati ("The Beautiful", about arithmetic , in particular rules for divisibility by $9$, $3$, $5$, $7$ and $11$)
• Bijaganita ("Seed Arithmetic", about algebra and the solution of equations)
• Goladhyaya (sphere)
• Grahaganita (mathematics of the planets).
• Vasanabhasya of Mitaksara ( Bhaskara 's own commentary on the Siddhanta Shiromani )
• Karanakutuhala (Calculation of Astronomical Wonders) or Brahmatulya (a simplified version of the Siddhanta Shiromani )
• Vivarana (a commentary on the Shishyadhividdhidatantra of Lalla .

Notable Quotes

Also known as.

In Kannada: ಭಾಸ್ಕರಾಚಾರ್ಯ

The name can also be rendered Bhāskara .

Known as Bhāskara II , Bhāskara Āchārya ("Bhāskara the teacher"), or Bhāskarāchārya , to distinguish him from Bhaskara I ).

• John J. O'Connor and Edmund F. Robertson : "Bhaskara II Acharya" : MacTutor History of Mathematics archive
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• 1998:  David Nelson : The Penguin Dictionary of Mathematics  (2nd ed.)  ... (previous)  ... (next) : Bhaskara  (1114 - c . 1185)
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Bhaskaracharya – The Crown Jewel of Mathematics and Astronomy

Some of Bhaskara’s contributions to mathematics include the following:

-Bhaskara is the first to give the general solution to the quadratic equation ax2 + bx + c = 0, the answer being x = (-b ± (b2 – 4ac)1/2)/2a.

-He also gives the (now) well known results for sin (a + b) and sin (a – b).

-Stated Rolle’s Theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works.

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

-Bhaskara also goes deeper into the ‘differential calculus’ and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of ‘infinitesimals’

Bhaskaracharya wrote Siddhanta Shiromani in 1150 AD when he was 36 years old. This is a mammoth work containing about 1450 verses. It is divided into four parts and each part can be considered as separate book. The numbers of verses in each part are as follows, 1. Lilawati has 278 2. Beejaganit has 213 3. Ganitadhyaya has 451 4. Goladhyaya has 501 verses.

One of the most important characteristic of Siddhanta Shiromani is, it consists of simple methods of calculations from Arithmetic to Astronomy. Essential knowledge of ancient Indian Astronomy can be acquired by reading only this book. Siddhanta Shiromani has surpassed all the ancient books on astronomy in India. After Bhaskaracharya nobody could write excellent books on mathematics and astronomy in lucid language in India.

Lilawati is an excellent example of how a difficult subject like mathematics can be written in poetic language. Lilawati has been translated in many languages throughout the world. Lilavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry. Bhaskara’s method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.

His work Bijaganita is effectively a treatise on algebra and contains the following topics: – Positive and negative numbers. – Zero. – The ‘unknown’. – Surds. – Kuttaka (modern indeterminate equation of first order) – Simple equations (indeterminate of second, third and fourth degree). – Simple equations with more than one unknown. – Indeterminate quadratic equations (of the type ax2 + b = y2). – Quadratic equations. – Quadratic equations with more than one unknown. – Operations with products of several unknowns.

Ganitadhyaya and Goladhyaya of Siddhanta Shiromani are devoted to astronomy. All put together there are about 1000 verses. Almost all aspects of astronomy are considered in these two books.

Bhaskara has given a very simple method to determine the circumference of the Earth. According to this method, first find out the distance between two places, which are on the same longitude. Then find the correct latitudes of those two places and difference between the latitudes. Knowing the distance between two latitudes, the distance that corresponds to 360 degrees can be easily found, which the circumference of the Earth.

He also showed that when a planet is farthest from, or closest to, the Sun, the difference between a planet’s actual position and its position according to “the equation of the centre” (which predicts planets’ positions on the assumption that planets move uniformly around the Sun) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.

A Glance at the Astronomical Achievements of Bhaskaracharya

1. The Earth is not flat, has no support and has a power of attraction.

2. The north and south poles of the Earth experience six months of day and six months of night.

3. One day of Moon is equivalent to 15 earth-days and one night is also equivalent to 15 earth-days.

4. Bhaskaracharya had accurately calculated apparent orbital periods of the Sun and orbital periods of Mercury, Venus, and Mars. There is slight difference between the orbital periods he calculated for Jupiter and Saturn and the corresponding modern values.

5. Earth’s atmosphere extends to 96 kilometers and has seven parts.

6. There is a vacuum beyond the Earth’s atmosphere.

There have been several unscrupulous attempts to argue that there are traces of Diophantine influence in Bhaskara’s work, but this once again seems like an attempt by European scholars to claim European influence on (all) the great works of mathematics. These claims should be ignored. Particularly in the field of algebra, Diophantus only looked at specific cases and did not achieve the general methods of the Indians.

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Bhaskara II Biography

Bhaskara II was the head of the astronomical observatory at Ujjain, the chief mathematical center of ancient India. It goes to the credit of Varahamihira and Brahmagupta, the leading mathematicians who worked there and built up this school of mathematical astronomy. He wrote six books and the seventh book, which is attributed to him, is considered to be a forgery. The subjects of his six works are arithmetic, algebra, trigonometry, calculus, geometry, and astronomy. The six works are: Lilavati on mathematics; Bijaganita on algebra; the Siddhantasiromani which is divided into two parts: mathematical astronomy and sphere; the Vasanabhasya of Mitaksara which is Bhaskaracharya’s views on the Siddhantasiromani ; the Karanakutuhala or Brahmatulya in which he simplified the concepts of Siddhantasiromani ; and the Vivarana which comments on the Shishyadhividdhidatantra of Lalla. From the mathematical point of view the first three of these works are the most interesting.

Bhaskara II wrote Siddhanta Shiromani at the age of 36 in 1150 AD. This colossal work is divided into four parts Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya and consists of about 1450 verses. Each part of the book consists of huge number of verses and can be considered as a separate book: Lilawati has 278, Beejaganit has 213, Ganitadhyaya has 451 and Goladhyaya has 501 verses. He formulated simple ways of calculations from Arithmetic to Astronomy in this book. He wrote Lilawati is an excellent lucid and poetic language. It has been translated in various languages throughout the world.

Few important contributions of BhaskarII to mathematics are as follows:

Terms for numbers In English, the multiples of 1000 are termed as thousand, million, billion, trillion, quadrillion etc. These terms were named recently in English, but Bhaskaracharya gave the terms for numbers in multiples of ten which are as follows: eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), arbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017).

Kuttak Kuttak according to modern mathematics is ‘indeterminate equation of first order’. In the western world, the method of solving such equations was called as ‘pulverizer’. Bhaskara suggested a generalized solution to get multiple answers for these equations.

Chakrawaal Chakrawaal, according to western mathematics is the “indeterminate equation of second order”. It is also called Pell’s equation. The equation was first solved by an ancient Indian mathematician, Brahmagupta (628 AD) and given in his Brahmasphutasiddhanta. Bhaskara II converted the method and gave a generalized solution.

Simple mathematical methods Bhaskara II suggested simple methods to calculate the squares, square roots, cube, and cube roots of big numbers. The Pythagoras theorem was proved by him in only two lines. Bhaskara’s ‘Khandameru’is the famous Pascal Triangle. Pascal, the European mathematician was born 500 years after Bhaskara. In Lilawati, he solved several problems on permutations and combinations and called the method as ‘ankapaash’. He even gave an approximate value of PI as 22/7, which is 3.1416. He was even familiar with the concept of infinity and called it as ‘khahar rashi’, which means ‘anant’.

He died in 1185 in Ujjain, India

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Bhāskara II

• Reference work entry
• First Online: 01 January 2016
• Cite this reference work entry

• R. C. Gupta  

27 Accesses

Undoubtedly, the greatest name in the history of ancient and medieval Indian astronomy and mathematics is that of Bhāskarācārya (b. AD 1114). His Līlāvatī is the most popular book of traditional Indian mathematics. He is usually designated as Bhāskara II in order to differentiate him from his earlier namesake who flourished in the early part of the seventh century.

According to Bhāskara’s own statement towards the end of his Golādhyāya , he was born in Śaka AD 1036 or AD 1114. He also adds that he came from Vijjaḍaviḍa near the Sahya mountain. This place is usually identified with the modern Bijapur in Mysore. S.B. Dikshit is of the opinion that Bhāskara’s original home was Pāṭaṇa (in Khandesh), where a relevant inscription was discovered by Bhau Daji in 1865. According to the inscription, Manoratha, Maheśvara, Lakṣmīdhara, and Caṇgadeva were the names of the grandfather, father, son, and grandson, respectively, of Bhāskara. Caṇgadeva was the chief astronomer in the court of the king...

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Colebrooke, H. T. (1817). Algebra with arithmetic and mensuration from the sanscrit of Brahmegupta and Bháscara . London: Murray. Reprint. Wiesbaden: Martin Sandig. 1973.

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Helaine Selin

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Gupta, R.C. (2016). Bhāskara II. In: Selin, H. (eds) Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7747-7_9219

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Published : 27 March 2016

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Discussed in biography.

…all Indian mathematical classics), particularly Līlāvatī (“The Beautiful”) and Bījagaṇita (“Seed Counting”), he not only used the decimal system but also compiled problems from Brahmagupta and others. He filled many of the gaps in Brahmagupta’s work, especially in obtaining a general solution to the Pell equation ( x 2 = 1 +…

Indian mathematics

…arithmetic and algebra, the 12th-century Lilavati (“The Beautiful”) and the more advanced Bijaganita , by Bhaskara II, followed the conventional definition of the eight operations. Bhaskara asserted, however, that the “Rule of Three” (of proportionality) is the truly fundamental concept underlying both arithmetic and algebra: