bhaskara 2 biography in english

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Bhāskara II

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Bhāskara II (born 1114, Biddur, India—died c. 1185, probably Ujjain) was the leading mathematician of the 12th century, who wrote the first work with full and systematic use of the decimal number system .

Bhāskara II was the lineal successor of the noted Indian mathematician Brahmagupta (598– c. 665) as head of an astronomical observatory at Ujjain , the leading mathematical centre of ancient India . The II has been attached to his name to distinguish him from the 7th-century astronomer of the same name .

In Bhāskara II’s mathematical works (written in verse like nearly 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 + py 2 ) and in giving many particular solutions (e.g., x 2 = 1 + 61 y 2 , which has the solution x = 1,766,319,049 and y = 226,153,980; French mathematician Pierre de Fermat proposed this same problem as a challenge to his friend Frenicle de Bessy five centuries later in 1657). Bhāskara 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 . Bhāskara 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 π = 3.141666.

In other of his works, notably Siddhāntaśiromaṇi (“Head Jewel of Accuracy”) and Karaṇakutūhala (“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. Bhāskara II was also a noted astrologer , and, according to a legend first recorded in a 16th-century Persian translation, he named his first work, Līlāvatī , after his daughter in order to console her. He tried to determine the best time for Līlāvatī’s marriage by using a water clock consisting of a cup with a small hole in the bottom floating in a larger vessel. The cup would sink at the beginning of the correct hour. Līlāvatī looked into the water clock, and a pearl fell off of her clothing, plugging up the hole. The cup never sank, depriving her of her only chance for marriage and happiness. It is unknown how true this legend is, but some problems in Līlāvatī are addressed to women, using such feminine vocatives as “dear one” or “beautiful one.”

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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 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

How To Cite

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Lilavati was the name of Bhaskaracharya's daughter. From casting her horoscope, he discovered that the auspicious time for her wedding would be a particular hour on a certain day. He placed a cup with a small hole at the bottom of the vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. When everything was ready and the cup was placed in the vessel, Lilavati suddenly out of curiosity bent over the vessel and a pearl from her dress fell into the cup and blocked the hole in it. The lucky hour passed without the cup sinking. Bhaskaracharya believed that the way to console his dejected daughter, who now would never get married, was to write her a manual of mathematics!

In the inverse method, the operation is reversed. That is the fruit to be multiplied by the augment and divided by the demand. When fruit increases or decreases, as the demand is augmented or diminished, the direct rule is used. Else the inverse. Rule of three inverse: If the fruit diminish as the requisition increases, or augment as that decreases, they, who are skilled in accounts, consider the rule of three to be inverted. When there is a diminution of fruit, if there be increase of requisition, and increase of fruit if there be diminution of requisition, then the inverse rule of three is employed.

Example: On an expedition to seize his enemy's elephants, a king marched two yojanas the first day. Say, intelligent calculator, with what increasing rate of daily march did he proceed, since he reached his foe's city, a distance of eighty yojanas, in a week?

Example: Say quickly, mathematician, what is that multiplier, by which two hundred and twenty-one being multiplied, and sixty-five added to the product, the sum divided by a hundred and ninety-five becomes exhausted.

Joy and happiness is indeed ever increasing in this world for those who have Lilavati clasped to their throats, decorated as the members are with neat reduction of fractions, multiplication and involution, pure and perfect as are the solutions, and tasteful as is the speech which is exemplified.

Example: Tell quickly the result of the numbers three and four, negative or affirmative, taken together; that is, affirmative and negative, or both negative or both affirmative, as separate instances; if thou know the addition of affirmative and negative quantities.

The characters, denoting the quantities known and unknown, should be first written to indicate them generally; and those, which become negative should be then marked with a dot over them. Example: Subtracting two from three, affirmative from affirmative, and negative from negative, or the contrary, tell me quickly the result ...

A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

Example: Inside a forest, a number of apes equal to the square of one-eighth of the total apes in the pack are playing noisy games. The remaining twelve apes, who are of a more serious disposition, are on a nearby hill and irritated by the shrieks coming from the forest. What is the total number of apes in the pack?

Example: The horses belonging to four men are 5 , 3 , 6 and 8 . The camels belonging to the same men are 2 , 7 , 4 and 1 . The mules belonging to them are 8 , 2 , 1 and 3 and the oxen are 7 , 1 , 2 and 1 . all four men have equal fortunes. Tell me quickly the price of each horse, camel, mule and ox.

A morsel of tuition conveys knowledge to a comprehensive mind; and having reached it, expands of its own impulse, as oil poured upon water, as a secret entrusted to the vile, as alms bestowed upon the worthy, however little, so does knowledge infused into a wise mind spread by intrinsic force. It is apparent to men of clear understanding, that the rule of three terms constitutes arithmetic and sagacity constitutes algebra. Accordingly I have said ... The rule of three terms is arithmetic; spotless understanding is algebra. What is there unknown to the intelligent? Therefore for the dull alone it is set forth.

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.

References ( show )

D Pingree, Biography in Dictionary of Scientific Biography ( New York 1970 - 1990) . See THIS LINK .
Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Bhaskara-II
R Calinger ( ed. ) , Classics of Mathematics ( New Jersey, 1995) .
G Ifrah, A universal history of numbers : From prehistory to the invention of the computer ( London, 1998) .
G G Joseph, The crest of the peacock ( London, 1991) .
K S Patwardhan, S A Naimpally and S L Singh, Lilavati of Bhaskaracarya ( Delhi 2001) .
S P Arya, On the Brahmagupta- Bhaskara equation, Math. Ed. 8 (1) (1991) , 23 - 27 .
B Chaudhary and P Jha, Studies of Bhaskara's works in Mithila, Ganita Bharati 12 (1 - 2) (1990) , 27 - 32 .
B Datta, The two Bhaskaras, Indian Historical Quarterly 6 (1930) , 727 - 736 .
R C Gupta, Bhaskara II's derivation for the surface of a sphere, Math. Education 7 (1973) , A 49 -A 52 .
R C Gupta, The last combinatorial problem in Bhaskara's Lilavati, Ganita Bharati 18 (1 - 4) (1996) , 14 - 20 .
M G Inamdar, A formula of Bhaskara for the chord of a circle leading to a formula for evaluating sin a∞, Math. Student 18 (1950) , 9 - 11
A A Krishnaswami Ayyangar, Remarks on Bhaskara's approximation to the sine of an angle, Math. Student 18 (1950) , 12 .
V Madhukar Mallayya, Arithmetic operation of division with special reference to Bhaskara II's Lilavati and its commentaries, Indian J. Hist. Sci. 32 (4) (1997) , 315 - 324 .
V Madhukar Mallayya, Pancarasikadau - the Indian golden rule compound ( with special reference to Bhaskara's Lilavati and its commentaries, Gurukula Kangri Vijnana Patrika Aryabhata 1 (1998) , 29 - 50 .
V Madhukar Mallayya, Various methods of squaring with special reference to the Lilavati of Bhaskara II and the commentary Kriyakramakari of Sankara and Narayana, Ganita Sandesh 11 (1) (1997) , 31 - 36 .
V Madhukar Mallayya and K Jha, Bhaskara's concept of numeration in decuple proportions - earliest reference in Vedas with Yaska's 'Nirukta' throwing light on the notion of succession in enumeration : an anticipation of Peano's axioms, Ganita-Bharati 17 (1 - 4) (1995) , 85 - 91 .
S A Naimpally, Approximate Formula For The Length of a Chord, Ganita Bharati 9 (1987) 57 - 58 .
C-O Selenius, Rationale of the chakravala process of Jayadeva and Bhaskara II, Historia Math. 2 (1975) , 167 - 184 .
C-O Selenius, Kettenbruch theoretische Erklarung der zyklischen Methode zur Losung der Bhaskara-Pell-Gleichung, Acta Acad. Abo. Math. Phys. 23 (10) (1963) .
S R Sinha, Bhaskara's Lilavati, Bull. Allahabad Univ. Math. Assoc. 15 (1951) , 9 - 16 .
D A Somayaji, Bhaskara's calculations of the gnomon's shadow, Math. Student 18 (1950) , 1 - 8 .

Additional Resources ( show )

Other pages about Bhaskara II:

See Bhaskara II on a timeline
Pell's equation
Heinz Klaus Strick biography

Other websites about Bhaskara II:

Dictionary of Scientific Biography
MathSciNet Author profile

Honours ( show )

Honours awarded to Bhaskara II

Popular biographies list Number 5

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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

$\sin \left(a+b\right)$

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]

$f\left(a\right)=f\left(b\right)=0$

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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Bhaskara II and the Marvels of Siddhanta Shiromani: Unveiling Ancient India’s Mathematical Legacy

Post author By Mala Chandrashekhar
Post date May 12, 2023
No Comments on Bhaskara II and the Marvels of Siddhanta Shiromani: Unveiling Ancient India’s Mathematical Legacy

Introduction

In the rich tapestry of India’s intellectual history, the name Bhaskara II shines as a brilliant mathematician and astronomer who left an indelible mark on the world of science. Born in the 12th century, Bhaskara II, also known as Bhaskaracharya, is celebrated for his groundbreaking work, “Siddhānta Shiromani.” This monumental treatise, divided into four distinct parts, laid the foundation for numerous mathematical and astronomical discoveries that continue to inspire and amaze scholars and enthusiasts today. In this blog post, we delve into the life and achievements of Bhaskara II and explore the enduring legacy of “Siddhānta Shiromani.”

Bhaskara II: The Mathematical Genius

Bhaskara II was born in 1114 CE in Bijapur, a town in present-day Karnataka, India. From a young age, he displayed an exceptional aptitude for mathematics and astronomy. Under the guidance of his father, Maheswara, Bhaskara II honed his skills and embarked on a journey of exploration and discovery that would change the course of mathematics and astronomy.

“Siddhānta Shiromani”: The Crown Jewel

“Siddhānta Shiromani” (translated as “Crown of Treatises”) is Bhaskara II’s magnum opus. It comprises four major sections:

Lilavati: This section focuses on arithmetic and geometry. It is named after Bhaskara II’s daughter, Lilavati, and presents mathematical concepts and problems in a poetic and engaging manner. The Lilavati covers topics like algebra, number theory, and combinatorics, making it accessible and enjoyable for students of all ages.
Bijaganita: In this part, Bhaskara II delves into algebra and provides a comprehensive exploration of equations, including quadratic and indeterminate equations. His work in this area predates similar European developments by centuries.
Golādhyāya: Bhaskara II’s Golādhyāya is dedicated to the study of spheres and planetary motion. It contains precise calculations of the positions of celestial bodies and their orbits, demonstrating his profound understanding of astronomy.
Grahaganita: This section is a comprehensive treatise on mathematical astronomy. Bhaskara II presents detailed methods for calculating planetary positions, eclipses, and other celestial phenomena. His work in this domain had a significant influence on later astronomers.

Legacy and Impact

Bhaskara II’s “Siddhānta Shiromani” stands as a testament to his unparalleled genius and deep insights into mathematics and astronomy. His contributions laid the foundation for future developments in these fields, not only in India but also in other parts of the world. Some of the notable aspects of his legacy include:

Influence on European Mathematics: Bhaskara II’s work, especially in algebra and number theory, found its way to Europe through Arabic translations. European mathematicians like Fibonacci were influenced by his ideas, which played a crucial role in the development of modern mathematics.
Astronomical Accuracy: Bhaskara II’s precise astronomical calculations provided valuable data for future astronomers, aiding in the refinement of planetary models and predictions of celestial events.
Educational Legacy: The poetic and engaging style of “Lilavati” made mathematics more accessible and appealing, contributing to the spread of mathematical knowledge among students in India and beyond.

Bhaskara II’s “Siddhānta Shiromani” is a jewel in the crown of India’s mathematical and astronomical heritage. His profound contributions to algebra, number theory, and astronomy have left an indelible mark on the world of science and continue to inspire mathematicians and astronomers to this day. Bhaskara II’s work is a testament to the enduring power of human curiosity and the boundless possibilities of the human mind, and it reminds us of the rich intellectual heritage of ancient India.

By Mala Chandrashekhar

Introducing Blogger Mala Chandrashekhar - A specialist academically trained in modern Western sciences, yet deeply enamored with India's timeless ethnic arts, crafts, and textiles. Her heart beats for the rich and glorious cultural and spiritual heritage of India, and she has dedicated her entire blog to spreading the immortal glories of ancient India worldwide. Through her simple yet impactful blog posts, Mala aims to reach every nook and corner of the globe, sharing India's beauty and wisdom with the world.

But Mala doesn't stop at just sharing her own thoughts and ideas. She welcomes constructive criticisms and suggestions to improve her blog and make it even more impactful. And if you share her passion for India's culture and heritage, she extends a warm invitation for high-quality guest blog posts.

Ready to dive into the world of India's ageless beauty? Follow Mala on LinkedIn, Twitter & Facebook and join her in spreading the magic of ancient India to the world.

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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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MATHEMATICIAN

Bhāskara II

1114 - 1185

Bhāskara II ([bʰɑːskərə]; c.1114–1185), also known as Bhāskarāchārya (lit. 'Bhāskara the teacher'), was an Indian polymath, mathematician, astronomer and engineer. From verses in his main work, Siddhāṁta Śiromaṇī, it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Satpuda mountain ranges of Western Ghats, believed to be the town of Patana in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars. In a temple in Maharashtra, an inscription supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him . Read more on Wikipedia

Since 2007, the English Wikipedia page of Bhāskara II has received more than 1,827,094 page views. His biography is available in 54 different languages on Wikipedia (up from 52 in 2019) . Bhāskara II is the 63rd most popular mathematician (up from 69th in 2019) , the 43rd most popular biography from India (up from 44th in 2019) and the 3rd most popular Indian Mathematician .

Bhāskara II is most famous for his text, Siddhānta Shiromani. It is a commentary on the Brahma Sutras, one of the most important texts in Vedanta philosophy.

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Among MATHEMATICIANS

Among mathematicians , Bhāskara II ranks 63 out of 1,004 . Before him are Gaspard Monge , Al-Battani , Émilie du Châtelet , August Ferdinand Möbius , Abu al-Wafa' Buzjani , and Hermann Minkowski . After him are Urbain Le Verrier , Simon Stevin , Friedrich Bessel , Abraham de Moivre , Ludolph van Ceulen , and Giuseppe Peano .

Most Popular Mathematicians in Wikipedia

Gaspard Monge

1746 - 1818

Émilie du Châtelet

1706 - 1749

August Ferdinand Möbius

1790 - 1868

Abu al-Wafa' Buzjani

Hermann Minkowski

1864 - 1909

Urbain Le Verrier

1811 - 1877

Simon Stevin

1548 - 1620

Friedrich Bessel

1784 - 1846

Abraham de Moivre

1667 - 1754

Ludolph van Ceulen

1540 - 1610

Giuseppe Peano

1858 - 1932

Contemporaries

Among people born in 1114 , Bhāskara II ranks 1 . After him are Otto of Freising , Ramon Berenguer IV, Count of Barcelona , Gerard of Cremona , Berenguer Ramon, Count of Provence , Dirk VI, Count of Holland , Rainald of Dassel , and Gertrude of Sulzbach . Among people deceased in 1185 , Bhāskara II ranks 6 . Before him are Baldwin IV of Jerusalem , Pope Lucius III , Afonso I of Portugal , Chrétien de Troyes , and Ibn Tufail . After him are Andronikos I Komnenos , Agnes of Courtenay , Emperor Antoku , Richeza of Poland, Queen of Castile , John Kinnamos , and Beatrice of Rethel .

Others Born in 1114

Otto of Freising

1114 - 1158

Ramon Berenguer IV, Count of Barcelona

1114 - 1162

Gerard of Cremona

1114 - 1187

Berenguer Ramon, Count of Provence

1114 - 1144

Dirk VI, Count of Holland

1114 - 1157

Rainald of Dassel

1114 - 1167

Gertrude of Sulzbach

1114 - 1146

Others Deceased in 1185

Baldwin IV of Jerusalem

1161 - 1185

Pope Lucius III

RELIGIOUS FIGURE

1110 - 1185

Afonso I of Portugal

1109 - 1185

Chrétien de Troyes

1135 - 1185

PHILOSOPHER

Andronikos I Komnenos

1118 - 1185

Agnes of Courtenay

1133 - 1185

Emperor Antoku

1178 - 1185

Richeza of Poland, Queen of Castile

1140 - 1185

John Kinnamos

1143 - 1185

Beatrice of Rethel

1130 - 1185

Among people born in India , Bhāskara II ranks 43 out of 1,861 . Before him are Sathya Sai Baba (1926) , Jiddu Krishnamurti (1895) , Rani of Jhansi (1828) , Amir Khusrow (1253) , Sita (null) , and Alauddin Khalji (1266) . After him are Tulsidas (1532) , Amartya Sen (1933) , Ghalib (1797) , Tipu Sultan (1750) , Savitribai Phule (1831) , and Meera (1498) .

Others born in India

Sathya Sai Baba

1926 - 2011

Jiddu Krishnamurti

1895 - 1986

Rani of Jhansi

1828 - 1858

Amir Khusrow

1253 - 1325

Alauddin Khalji

1266 - 1316

1532 - 1623

Amartya Sen

1933 - Present

1797 - 1869

Tipu Sultan

1750 - 1799

Savitribai Phule

1831 - 1897

1498 - 1546

Among MATHEMATICIANS In India

Among mathematicians born in India , Bhāskara II ranks 3 . Before him are Srinivasa Ramanujan (1887) , and Brahmagupta (598) . After him are Augustus De Morgan (1806) , S. R. Ranganathan (1892) , Madhava of Sangamagrama (1350) , Bhāskara I (600) , Jai Singh II (1688) , D. R. Kaprekar (1905) , Nilakantha Somayaji (1444) , Harish-Chandra (1923) , and Kātyāyana (-200) .

Indian born Mathematicians

Srinivasa Ramanujan

1887 - 1920

Brahmagupta

Augustus De Morgan

1806 - 1871

S. R. Ranganathan

1892 - 1972

Madhava of Sangamagrama

1350 - 1425

Jai Singh II

1688 - 1743

D. R. Kaprekar

1905 - 1986

Nilakantha Somayaji

1444 - 1544

Harish-Chandra

1923 - 1983

200 BC - 159 BC

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The Mysterious India

Explore the wonders of mysterious India

Bhaskaracharya – The great Astronomer and Mathmatician

T he period between 500 and 1200 AD was the golden age of Indian Astronomy. During this golden period an Indian wizard was born who contributed greatly to the conception of Astronomy and Mathematics. He was none other than Bhaskaracharya.

Bhaskaracharya was the leading mathematician and Astronomer of the 12th century, who wrote the first work with full and systematic use of the decimal number system. He was born near Vijjadavida (Bijapur in modern Karnataka). Bhaskaracharya’s name was actually ‘Bhaskara’ only but the title ‘Acharya’ was added and conferred to mean “Bhaskara the Teacher”. He is also known as Bhaskaracharya II.

Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He became head of the astronomical observatory at Ujjain, the leading mathematical centre in India at that time. Outstanding mathematicians such as Varahamihira and Brahmagupta had worked there and built up a strong school of mathematical astronomy.

Buy The Surya Siddhanta and Siddhanta Siromani

There are six well known works of Bhaskaracharya. They are :- Lilavathi – Mathematics, Bijaganita – Algebra, Ganitadhyaya – mathematical astronomy, Goladhyaya – sphere, Karanakutuhala – Calculation of Astronomical Wonders, Vasanabhasya – Bhaskara’s own commentary on the Siddhanta Shiromani, and Vivarana which is a commentary on the Shishyadhividdhidatantra of Mathematician and Astronomer Lalla .

Bhaskara was known not only for his mathematical scholarship, but also for his poetic inclinations. He wrote Lilawati in an excellent lucid and poetic language. It has been translated in various languages throughout the world. It was written for his daughter, Lilavati. The Lilavati deals with arithmetic and geometry; it is said that the name is after his daughter Lilavati, who was according to her horoscope to remain unmarried.

Buy Lilavati of Bhaskracarya : A Treatise of Mathematics of Vedic Tradition

In his mathematical works, particularly Lilavati and Bijaganita, 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 (x2 = 1 + py2) and in giving many particular solutions.

Bhaskara 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. Bhaskara used letters to represent unknown quantities, much as in modern algebra, and solved indeterminate equations of 1st and 2nd degrees.

Brahmagupta was Bhaskara’s role model. To Brahmagupta he pays homage at the beginning of his Siddhanta Siromani. Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately calculated the time that earth took to revolve around the Sun as 365.2588 days that is a difference of 3 minutes of modern acceptance of 365.2563 days.

Bhaskaracharya was the first to discover gravity , 500 years before Sir Isaac Newton. He is also known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. Bhaskara’s work on calculus predates Newton and Leibniz by over half a millennium.

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.

Some other achievements of Bhaskaracharya were:

The Earth is not flat, has no support and has a power of attraction.
The north and south poles of the Earth experience six months of day and six months of night.
One day of Moon is equivalent to 15 earth-days and one night is also equivalent to 15 earth-days.
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.
Earth’s atmosphere extends to 96 kilometers and has seven parts.
There is a vacuum beyond the Earth’s atmosphere.

Bhaskaracharya, or Bhaskara II (1114 – 1185) is regarded almost without question as the greatest mathematician of all time and his contribution to not just Indian, but world mathematics is undeniable. He was perhaps the last and the greatest astronomer that India ever produced.

Source: Free Press Journal & Veda Wikidot

Posted in: Famous People
Tagged in: Aryabhata , Astronomy , Bhaskaracharya , Brahmagupta , mathematicians , Mathematics , Surya Siddhant

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12 Facts About Bhaskaracharya: An Extra-Ordinary Mathematician

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Bhaskaracharya, also known as Bhaskara II, was born circa 1114 in Bijapur, Karnataka to a Brahmin family of scholars. He is considered to be the greatest mathematician from medieval India. As Indians, it is important for us to know the history of India and the story of her greatest children. Bhaskara was one such child who went on to touch great heights in the field of mathematics. Here are 12 Facts about Bhaskaracharya, the mathematician from medieval India.

Also read: 15 Books On Indian History To Understand Our Country Better

Now, let’s begin our list of 12 Facts about Bhaskaracharya:

1. A man of many talents.

He is believed to have been the head of the astronomical observatory in Ujjain.

2. Well ahead of his time.

Bhaskara’s work on calculus was written over half a millennium before Newton and Leibniz’s work on the same subject was published.

3. ISRO’s tribute.

As a tribute to the mathematician, ISRO in 1981 launched a satellite named Bhaskara II in his honor.

4. Siddhanta Siromani, his most important work.

His main work, Siddhanta Siromani is divided into four parts: Lilavati, Bijaganita, Grahaganita, and Goladhyaya.

5. The first section: Lilavati

Named after Bhaskara’s daughter, Lilavati contains 277 verses.

6. The second section: Bijaganita.

The second section, Bijaganita is a section of Algebra, and it has 213 verses. The section discusses subjects like zero, infinity, and integers. The work includes a discussion on what is now called Pell’s equation, which he seems to have solved using a Kuttaka method.

7. The Third Section: Grahaganita

The third section, Grahaganita, discusses the motion of planets in which he takes in regard to their instantaneous speeds.

In it, Bhaskara mentions that at its highest, a planet’s instantaneous speed is zero.

9. Proving the Pythagorean theorem.

Bhaskara also proves the Pythagorean theorem by calculating the same area in two distinct ways to get a²+b²=c².

11. The Chakravala method.

He used the Chakravala method, a cyclic algorithm, for solving indeterminate equations of the form ax2 + bx + c = y. However, historically, the attribution of this equation has been given to William Brouncker.

12. The properties of zero.

He discussed the properties of zero, addressing the various rules of operations with zero.

I'm a writer, poet, and I sometimes make art and sing for fun. I cook when I'm sad. I also like watching stuff and obsessing over details. I do not read as much as I'd wish I did.

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Bhāskara II

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First Online: 01 January 2016
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R. C. Gupta

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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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Gupta, R. C. (1974). Addition and subtraction theorems for the sine and their use in computing tabular sines. Mathematics Education, 8 (3B), 43–46.

Gupta, R. C. (1975). The Līlāvatī rule for computing sides of regular polygons. Mathematics Education, 9 (2B), 25–29.

Hayashi, T. (2002). Notes on the differences between the two recensions of the Lilavati of Bhaskara II. Sciamvs, 3 , 193–230.

Jha, A. (Ed.). (1949). Bīja Gaṇita of Bhāskarācārya . Banaras, India: Chowkhamba.

Kunoff, S. (1991). A curious counting/summation formula from the ancient Hindus. In F. Abeles et al. (Eds.), Proceedings of the sixteenth annual meeting of the CSPM (pp. 101–107). Toronto, Canada: CSHPM.

Mallayya, V. M. (1997). Arithmetic operation of division with special reference to Bhaskara I’s “Lilavati” and its commentaries. Indian Journal of History of Science, 32 , 315–324.

Sastri, B. D. (Ed.). (1929). The Siddhānta-śiromaṇi by Bhāskarācārya with his own Vasanābhāṣya . Benares, India: Chowkhamba.

Selenius, C. O. (1975). Rationale of the Chakravāla process of Jayadeva and Bhāskara II. Historia Mathematica, 2.2 , 167–184.

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Srinivasiengar, C. N. (1967). The history of ancient Indian mathematics . Calcutta, India: World Press.

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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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Bhāskara II
Bhaskara's proof of the Pythagorean Theorem. Bhāskara II [a] ([bʰɑːskərə]; c.1114-1185), also known as Bhāskarāchārya (lit. ' Bhāskara the teacher '), was an Indian polymath, mathematician, astronomer and engineer. From verses in his main work, Siddhāṁta Śiromaṇī, it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Satpuda mountain ...
Bhāskara II
Bhāskara II (born 1114, Biddur, India—died c. 1185, probably Ujjain) was the leading mathematician of the 12th century, who wrote the first work with full and systematic use of the decimal number system. Bhāskara II was the lineal successor of the noted Indian mathematician Brahmagupta (598- c. 665) as head of an astronomical observatory ...
Bhaskara II
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 ...
Bhāskara II Biography
Bhāskara II. 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 ...
Bhaskara II (1114
R C Gupta, Bhaskara II's derivation for the surface of a sphere, Math. Education 7 (1973), A 49-A 52. R C Gupta, The last combinatorial problem in Bhaskara's Lilavati, Ganita Bharati 18 (1-4) (1996), 14-20. M G Inamdar, A formula of Bhaskara for the chord of a circle leading to a formula for evaluating sin a∞, Math. Student 18 (1950), 9-11
Bhaskara 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.
Bhāskara II
General. 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.
Bhaskara II
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 ...
Bhāskara II
Bhāskara II, also known as Bhāskarāchārya, was an Indian polymath, mathematician, astronomer and engineer. From verses in his main work, Siddhāṁta Śiromaṇī, it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Satpuda mountain ranges of Western Ghats, believed to be the town of Patana in Chalisgaon, located in present-day Khandesh region of ...
Bhāskara II
Bhāskara II. ( b. 1115) astronomy, mathematics. Bhāskara II has been one of the most impressive Indian astronomers and mathematicians, not only to modern students of the history of science but also to his, contemporaries and immediate successors. An important inscription discovered at Pātnā, near Chalisgaon in East Khandesh, Mahārāstra ...
Bhāskara II
Bhāskara II was an Indian (Hindu) astronomer of the twelfth century. The number "II" is added by modern historians to differentiate him from his namesake ( Bhāskara I ) of the seventh century. Bhāskara II is frequently called Bhāskarācārya (Master Bhāskara). He probably lived in Vijjayapura; his father was Maheśvara who was also an ...
Bhāskara II
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 ...
Bhaskara II and the Marvels of Siddhanta Shiromani: Unveiling Ancient
In this blog post, we delve into the life and achievements of Bhaskara II and explore the enduring legacy of "Siddhānta Shiromani." Bhaskara II: The Mathematical Genius. Bhaskara II was born in 1114 CE in Bijapur, a town in present-day Karnataka, India. From a young age, he displayed an exceptional aptitude for mathematics and astronomy.
Līlāvatī
Līlāvatī is a treatise by Indian mathematician Bhāskara II on mathematics, written in 1150 AD. It is the first volume of his main work, the Siddhānta Shiromani, [1] alongside the Bijaganita, the Grahaganita and the Golādhyāya. [2]A problem from the Lilavati by Bhaskaracharya. Written in the 12th century. This appeared on page 18 of The Mathematical Mystery Tour by UNESCO in 1989.
Bhaskara II Biography
Bhaskara II Biography. Bhaskara II was the head of the astronomical observatory at Ujjain, the chief mathematical center of ancient India. ... Terms for numbers In English, the multiples of 1000 ...
Bijaganita
128710435. This is the PDF of the book Bijaganita with English Translation by Edward Strachey. It is the earliest record of the algebra and its rules as formulated by Hindu mathematicians of India. Bhaskara was a mathematician of great renown. I have no monetary benefits from this work *& it is intended for non-commercial purposes only.
Siddhānta Shiromani
Use of the Pythagorean theorem in the corner. 1650 edition. Siddhānta Śiromaṇi (Sanskrit: सिद्धान्त शिरोमणि for "Crown of treatises") [1] is the major treatise of Indian mathematician Bhāskara II. [2] He wrote the Siddhānta Śiromaṇi in 1150 when he was 36 years old. The work is composed in Sanskrit ...
Bhāskara II Biography
Since 2007, the English Wikipedia page of Bhāskara II has received more than 1,812,669 page views. His biography is available in 54 different languages on Wikipedia (up from 52 in 2019) . Bhāskara II is the 63rd most popular mathematician (up from 69th in 2019) , the 43rd most popular biography from India (up from 44th in 2019) and the 3rd ...
Bhaskaracharya
He was none other than Bhaskaracharya. Bhaskaracharya was the leading mathematician and Astronomer of the 12th century, who wrote the first work with full and systematic use of the decimal number system. He was born near Vijjadavida (Bijapur in modern Karnataka). Bhaskaracharya's name was actually 'Bhaskara' only but the title 'Acharya ...
Bhaskara
Bhaskara. 1114-1185. Indian Mathematician and Astronomer. Bhaskara, one of the greatest medieval Indian scholars, pioneered learning in a number of areas, most notably in his approximations of π.Director of the astronomical observatory at Ujjain, he was at the center of scientific activities in the India of his time, and his work in number systems and equations represented a level of ...
12 Facts About Bhaskaracharya: An Extra-Ordinary Mathematician
Here are 12 Facts about Bhaskaracharya, the mathematician from medieval India. Also read: 15 Books On Indian History To Understand Our Country Better. Now, let's begin our list of 12 Facts about Bhaskaracharya: 1. A man of many talents. He is believed to have been the head of the astronomical observatory in Ujjain. 2.
Bhāskara II
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 ...
Bhāskara I
Bhāskara (c. 600 - c. 680) (commonly called Bhāskara I to avoid confusion with the 12th-century mathematician Bhāskara II) was a 7th-century Indian mathematician and astronomer who was the first to write numbers in the Hindu-Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's ...

Bhāskara II

Bhaskara II – History, Biography, and Accomplishments

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Bhaskara II - The Great Indian Mathematician

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Bhaskara II and the Marvels of Siddhanta Shiromani: Unveiling Ancient India’s Mathematical Legacy

By Mala Chandrashekhar

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