Bhaskaracharya II was an Indian mathematician and astronomer in the 12th century. He was born in Bijapur in modern Karnataka.He was the greatest mathematician of medieval India.Bhaskaracharya wrote Siddhanta Shiromani in 1150 AD. This is a mammoth work containing about 1450 verses. It is divided into four parts, Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact each part can be considered as separate book. The numbers of verses in each part are as follows, Lilawati has 278, Beejaganit has 213,Ganitadhyaya has 451 and Goladhyaya has 501 verses.These four sections deal with arithmetic, algebra, mathematics of the planets,and spheres respectively.
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 anywhere in the world !
Bhaskaracharya's work on calculus predates Newton and Leibniz by over half a millennium. He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. Bhaskaracharya was a pioneer in some of the principles of differential calculus. He was the first to conceive the differential coefficient and differential calculus.
Bhaskaracharya gives his date of birth, and date of composition of his major work, in a verse in the Arya metre:
rasa-guṇa-pūrṇa-mahīsama
śaka-nṛpa samaye 'bhavat mamotpattiḥ /
rasa-guṇa-varṣeṇa mayā
siddhānta-śiromaṇī racitaḥ //
This reveals that he was born in 1036 of the Śaka era (1114 CE), and that he composed the Siddhānta Śiromanī when he was 36 years old.He wrote Karana-kutūhala when he was 69 (in 1183)
Bhaskaracharya's work in Algebra, Arithmetic and Geometry catapulted him to fame and immortality. His renowned mathematical works called Lilavati and Bijaganita are considered to be unparalleled and a memorial to his profound intelligence. Its translation in several languages of the world bear testimony to its eminence. In his treatise Siddhant Shiromani he writes on planetary positions, eclipses, cosmography, mathematical techniques and astronomical equipment. In the Surya Siddhant he makes a note on the force of gravity:
"Objects fall on earth due to a force of attraction by the earth. Therefore, the earth, planets, constellations, moon, and sun are held in orbit due to this attraction."
Bhaskaracharya was the first to discover gravity, 500 years before Isaac Newton. His contribution to mathematics is unmatched and incomparable. Here is a summary of a few of his contributions :
- A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a² + b² = c².
- In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations.
- Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
- 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 17th century
- A cyclic Chakravala method for solving indeterminate equations of the form ax² + 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 x² − ny² = 1 (so-called "Pell's equation") is of considerable interest and importance.
- Solutions of Diophantine equations of the second order, such as 61x² + 1 = y².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 the18th 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.
- 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.
- In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results.
Astronomical achievements of Bhaskaracharya :
- The Earth is not flat, has no support and has a power of attraction.
- The north and south poles of the Earth 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.
- Earth’s atmosphere extends to 96 kilometers and has seven parts.
- There is a vacuum beyond the Earth’s atmosphere.
- He had knowledge of precession of equinoxes. He took the value of its shift from the first point of Aries as 11 degrees. However, at that time it was about 12 degrees.
- Ancient Indian Astronomers used to define a reference point called ‘Lanka’. It was defined as the point of intersection of the longitude passing through Ujjaini and the equator of the Earth. Bhaskara has considered three cardinal places with reference to Lanka, the Yavakoti at 90 degrees east of Lanka, the Romak at 90 degrees west of Lanka and Siddhapoor at 180 degrees from Lanka. He then accurately suggested that, when there is a noon at Lanka, there should be sunset at Yavkoti and sunrise at Romak and midnight at Siddhapoor.
- Bhaskaracharya had accurately calculated apparent orbital periods of the Sun and orbital periods of Mercury, Venus, and Mars.
Bhaskaracharya is one of the greatest mathematician ever to have lived, and yet he is forgotten. Lets bring him back by rediscovering all the knowledge that is already there, hidden, in our ancient texts.
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