Masters of Math - Bhāskara I & II

There were two great Mathematician in ancient India with name Bhāskara(भास्कर).

• Bhāskara(भास्कर-1)  (c. 600 – c 680) was a 7th-century Indian mathematician      
• Bhāskara II (भास्कर-2) also known as Bhāskarāchārya (AD 1114–1185) was a 12th century mathematician

Bhāskara I

He was born at Bori, in Parbhani district of Maharashtra state in India. He was scholar of Aryabhata's astronomical school. Also know for his contributions to the study of fractions.

Bhāskara is the author of three works: 

• the Mahābhāskarīya, 
• the Laghubhāskarīya, and 
• the Āryabhatīyabhāṣya.

 The Laghubhāskarīya also contains eight chapters:

1. On the mean longitudes of the planets.
2. On the true longitudes of the planets.
3. On the three problems relating to diurnal motion.
4. On lunar eclipses.
5. On solar eclipses.
6. On the visibility of the moon and on its cresent.
7. On the heliacal risings and settings of the planets and on their conjunctions.
8. On the conjunctions of the planets with the stars.

Bhāskara-1's sine approximation formula : 

This formula is given in his treatise titled Mahabhaskariya. 

The above state the rule (for finding the bhujaphala and the kotiphala, etc.) without making use of the Rsine-differences 225, etc. Subtract the degrees of a bhuja (or koti) from the degrees of a half circle (that is, 180 degrees). Then multiply the remainder by the degrees of the bhuja or koti and put down the result at two places. At one place subtract the result from 40500. By one-fourth of the remainder (thus obtained), divide the result at the other place as multiplied by the 'anthyaphala (that is, the epicyclic radius). Thus is obtained the entire bahuphala (or, kotiphala) for the sun, moon or the star-planets. So also are obtained the direct and inverse Rsines.

In modern mathematical notations, for an angle x in degrees, this formula given

                           

Bhaskara I's sine approximation formula can be expressed using the radian measure of angles as follows.

               

This is indeed remarkable approximation formula for sin x. Parts of Mahabhaskariya were later translated into Arabic.   

Fibonacci Series Number   

Bhaskara already dealt with the assertion that if p is a prime number, then 1 + (p–1)! is divisible by p. This is now known as Wilson's theorem.

Pell equations

First let us say what Pell's equation is. We are talking about the indeterminate quadratic equation

nx² + 1 = y²

the equation is an important one for several reasons – only some of which will be touched upon here – and its solution furnishes an ideal introduction to a whole branch of number theory, Diophantine Approximation.


Bhaskara stated theorems about the solutions of today so called Pell equations. For instance, he posed the problem: "Tell me, O mathematician, what is that square which multiplied by 8 becomes - together with unity - a square?" In modern notation, he asked for the solutions of the Pell equation . It has the simple solution x = 1, y = 3, or shortly (x,y) = (1,3), from which further solutions can be constructed, e.g., (x,y) = (6,17).

Bhāskara II

Bhāskara-II(also known as Bhāskarāchārya)

 

Thomas Colebrooke (1765-1837) was an English Orientalist and a Sanskrit scholar, had translated a work by the Indian Mathematician Bhaskara II also known as Bhaskarachaya (Bhaskara the teacher).

 Bhaskarachaya’s works represents a significant contribution to mathematical and astronomical knowledge in that period.  

Also Bhaskara’s work on calculus predates Newton and Leibniz by 500 years.

His book on arithmetic was written for his daughter Lilavati (meaning the one possessing beauty in Sanskrit) After studying his daughter’s horoscope, Baskara learned that she would not marry and remain childless unless she married at a certain time.  To avoid this fate he invented an ‘alarm  clock’;  a complicated device with a cup and a container of water  ensuring the girl would be married at the auspicious hour.He warned Lilavati not to go near; intrigued by the contraption

Lilavati had a look, a pearl from her wedding gown dropped in, upsetting the mechanism.  The moment passed and the wedding did not take place. Devastated by the tragic turn of events her father promised to write a book in her name that would remain until the end of time; ‘akin to a second life’.

Whilst making love a necklace broke. A row of pearls mislaid. One sixth fell to the floor. One fifth upon the bed. The young woman saved one third of them. One tenth were caught by her lover. If six pearls remained upon the string How many pearls were there altogether?

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 a2 + b2 = c2.
• ·         In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained.
• ·         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.
• ·         Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.

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.(Diurnal motion is an astronomical term referring 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.
• ·         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.