भास्कराचार्य BHASKARACHARYA - The Real Discoverer of Gravitational Force

 

भास्कराचार्य

 Bhaskaracharya 

Bhaskaracharya  was Great Bharatiya Mathematician and Astronomer. He was the leader of a cosmic observatory at Ujjain, the main mathematical center of ancient Bharat. He was the one who gave us the concept of Infinity. He has given his birth and other information as below;

                           रसगुणपूर्णमही समशकनृपसमयेऽभवन्मोत्पत्तिः।

                            रसगुणवर्षेण मया सिद्धान्तशिरोमणि रचितः॥

translation to Hindi : शक संवत १०३६ में मेरा जन्म हुआ और छत्तीस वर्ष की आयु में मैंने सिद्धान्तशिरोमणि की रचना   की।

Translation to English: I was born in 1036 of the Shaka era (1114 CE), and I composed the Siddhānta-Śiromaṇī(सिद्धान्तशिरोमणि)  when I was 36 years old. 

He was born in a Deśastha Rigvedi Brahmin family near Vijjadavida (believed to be Bijjaragi of Vijayapur in modern Karnataka). He lived in the Sahyadri region (Patnadevi, in Jalgaon district, Maharashtra). He is the only ancient mathematician who has been immortalized on a monument. In a temple in Maharashtra, an inscription supposedly created by his grandson Cangadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him
Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India.
His main work Siddhānta-Śiromaniसिद्धान्तशिरोमणि (Translation: "Crown of Treatises") is divided into four parts called Līlāvatī लीलावती, Bījagaṇitaबीजगणित, Grahagaṇita ग्रहगणित and Golādhyāya  गोलाध्याय which are also sometimes considered four independent works.These four sections ,as the names suggests, deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karaṇā Kautūhala करणकौतूहल
He , not Newton, was the first to discover Gravitational Forceगुरुत्वाकर्षण शक्ति Bhaskaracharya has written about the gravity of the earth in his 'Siddhānta-Śiromaniसिद्धान्तशिरोमणि' book that 'the earth pulls the celestial objects towards itself with a special force. Because of this the celestial bodiesआकाशीय पिण्ड  fall on the earth'.

Bhaskaracharya accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Earth to orbit the Sun, as approximately 365.2588 days which is the same as in Suryasiddhanta. The modern accepted measurement is 365.25636 earth days, a difference of just 3.5 minutes.

Grahaganita ग्रहगणित Mathematical Astronomy

In the third section Grahagaṇita, while treating the motion of planets, he considered their instantaneous speeds. He arrived at the approximation:It consists of 451 verses

${\displaystyle \sin y'-\sin y\approx (y'-y)\cos y}$ for ${\displaystyle y'}$ close to ${\displaystyle y}$, or in modern notation:

${\displaystyle {\frac {d}{dy}}\sin y=\cos y}$.

In his words

बिम्बार्धस्य कोटिज्या गुणस्त्रिज्याहारः फलं दोर्ज्यायोरान्तरम्

He also stated that at its highest point a planet's instantaneous speed is zero

 part of his book was divided into following parts;

• Mean longitudes of the planets.
• True longitudes of the planets.
• The three problems of diurnal rotation.(astronomical term refers to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle, that is called the diurnal circle.)
• Syzygies(roughly straight-line configuration of three or more celestial bodies in a gravitational system.)
• Lunar eclipses.
• Solar eclipses.
• Latitudes of the planets.
• Sunrise equation
• The Moon's crescent.
• Conjunctions of the planets with each other.
• Conjunctions of the planets with the fixed stars.
• The paths of the Sun and Moon.

 गोलाध्यायGolādhyāya

this Part of his book on Sphere was divided into following parts;

• 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.

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. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhaskaracharya  was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus. There is evidence of an early form of Rolle's theorem in his work. The modern formulation of Rolle's theorem states that if , then  for some  with . He gave the result that if  then , thereby finding the derivative of sine, although he never developed the notion of derivatives . He 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 trutiत्रुटि, or a 1⁄33750 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 centre vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre 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 15th century in the Lilavati Bhasya, a commentary on Bhaskaracharya 's Lilavati.

Engineering

The earliest reference to a अमरगति perpetual motion machine date back to 1150, when Bhaskaracharya described a wheel that he claimed would run forever.

Bhaskaracharya used a measuring device known as  यस्तियंत्रYaṣṭi-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.

Arithmetic

His arithmetic text Līlāvatī 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.

Līlāvatī is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and measurement. 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.
• Quadratic , Cubic and Quartic Indeterminate equations (Kuṭṭaka कूटटक), 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 17th century, yet his work was of the 12th 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 systematisation, improved methods and the new topics that he introduced. Furthermore, the Lilavati contained excellent 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.

Algebra

His Bījagaṇitaबीजगणित ("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 Bījaganita 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).
• Kuṭṭaka कूटटक (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 ax² + b = y²).
• 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.

He derived a cyclic, chakravala methodचक्रवाला विधि for solving indeterminate quadratic equations of the form ax² + bx + c = y^] his method for finding the solutions of the problem Nx² + 1 = y² (the so-called "Pell's equation") is of considerable importance

Trigonometry

The Siddhānta Shiromani (written in 1150) demonstrates his knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed spherical trigonometry, along with other interesting trigonometrical results. In particular He seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by him, results found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for sin(a + b) and sin(a - b)