When Bhaskara tried to cheat destiny

Bhaskaracharya or Bhaskara , the great mathematician and astronomer of the 12^th Century was also a well-known astrologer. When his daughter, Leelawathi, was born, he decided to cast her horoscope.

When he was calculating the planetary positions in the horoscope of his daughter, he found to his horror that Leelawathi would become a widow within an year of her marriage.

Bhaskara then decided to groom his daughter in the best possible way. He also wanted to educate her and her brother Lopamudra. Both seem to have taken after Bhaskara as they loved mathematics.

When Leelawathi reached marriageable age, the search for a suitable companion started. Bhaskara had not forgotten the horoscope he had cast of his daughter  and he decided to cheat fate by fixing an auspicious time for the marriage.

Bhaskara, by his precise mathematical calculations, concluded that if the mangalasutra was tied at a particular time, both his favourite child and his son-in-law would live a long and eventful life. He then determined to endure that nothing went wrong and himself determined the exact time when the mangalasutra has to be tied.

All arrangements for the marriage were being made and the marriage party had come to Bhaskara’s house in Ujjaian. Bhaskara was the head of the Ujjain Observatory and he had invited several people to the wedding.

Bhaskara had decided to rely on science to ensure that the mangalasutra was tied at the most auspicious time he had predicted. In this way, he thought he could change the destiny of his daughter.

Bhaskara had kept a simple device in the house to give him the exact mathematical time when the mangalasutra would have to be tied. He took two vessels and placed them one above the other. He filled the upper vessel with water and made a tiny hole in the centre. Water would trickle down to the lower vessel. The time was calculated by estimating the depth of the water that had collected in the lower vessel.

The unit of time was a Nadika. One Nadika was equal to 24 minutes. Thus, this was the water clock that was used in India. It was very accurate. 

Bhaskara had warned Leelawathi not to go near the clock. Curiosity, thy name is woman, says an adage. Leelawathi went near the clock as she was fascinated by the simple mechanism. She bent to look into the vessel. A small pearl set in her nose ring fell into the upper vessel. It sank down and it covered the tiny hole, thus blocking the free floor of water into the lower vessel.

Leelawathi was not aware of the nose ring having fallen into the vessel. Bhaskara too did not know what happened. When he saw that the water flow had stopped, he thought that the auspicious time had arrived.

He got the marriage ceremony performed. This was the very time he had wanted to avoid. Thus all the minute calculations  that Bhaskara had come up with went awry. His daughter was married at the most inappropriate time. Bhaskara realised that something had gone wrong when he went to check the time and found that the water flow had stopped.

However, the marriage had already taken place and nothing could be done to undo the prediction. Leelawathi’s husband died soon after marriage and she came back home.

Feeling cheated by destiny, Bhaskara consoled his daughter saying that he will make her immortal. “I will see you’re your name will live on forever”, he said.

He then began writing his magnum opus, on Mathematics and Astronomy and called it Sidhanta Shiromani.

The first portion of the book was called Leelawathi and it dealt with mathematics. He named it after his daughter.

Leelawathi was good at mathematics and in the book, Bhaskara addresses mathematical and algebraic problems to her. Leelawathi and other portions of Sidhanta Shiromani are in verse form.

Leelawathi has 278 verses and it deals with arithematic, algebra and geometry.

He poses the question to Leelawathi as follows:   

“Partha was determined to slay Karna. Posing his mathematical query, he asks, With half the arrows in his quiver Partha (Arjuna) destroyed those of Karna; with four times the square root of the total, he killed Karna’s horses. Six arrows were needed to kill Karna. With three arrows, Partha cut down Karna’s royal standard, his umbrella and bow, With the last arrow Arjuna killed Karna. Tell me O Leelawathi, how many arrows did Aruna have in his quiver?         
In another verse, he says a peacock which was sitting on a nine foot high pillar saw a snake going towards its anthill at the foot of the pillar. The snake was at a distance of three times the pillar’s height when the peacock swooped down. If both the peacock and snake travelled at the same speed, tell me O Leelawathi how far from the pillar did the peacock peck at the snake?

However, Bhaskara could never forget the incident of the pearl from the nose ring falling into the vessel and ruining Leelawathi’s marriage.

In perhaps the most poignant verse, he says

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 (total)  were there in the necklace.
   

Leelawathi was first translated into Persian by Abdul Faizi on orders of the Mughal Emperor Akbar. The other portions of Sidhanta Shiromani were translated into Persian during the reign of Akbar’s grandson, Emperor Shahajan.

The story of Leelawathi is narrated by Faizi in his Persian narration and other sources. Leelawathi remained the standard mathematical text for centuries till the advent of the British and the adoption of the English mathematical text.

If you have the time and interest, please go through Bhaskara’s Sidhantha. I can assure you that it is not boring as European mathematical texts. Nor is it uninteresting. It is in Sanskrit and it has been translated into almost all languages, including English.

  

The Indian mathematician who predated Newton and Liebniz

The city of Bijapur is not only famous for its link with the Adil Shahis but it is also the birthplace of several eminent personalities in the field of literaure and science.

If Narayana of Bijapur, better known as Kumar Valmiki, wrote Torvi Ramayana and Naga Chandra, a devout Jain penned the Jain version of the Ramayana,  there was another writer who became a famous mathematician and astronomer. He is Bhaskaracharya.
Bhaskaracharya or Bhaskara is believed to be born near Bijapur.

Though there are no precise records to tell us where or when he was born, it is believed that he was born in 1114 A D in a Deshastha Brahmin family somewhere near Bijjada Bida, which was the earlier name of Bijapur.

His father, Mahadeva, was an astrologer. It was his father who taught the young Bhaskara the rudiments of mathematics( Bhaskara passed on this knowledge to his son Lokasamudra). In one of his books, Bhaskara himself speaks about how and where he studied. He said he had studied grammer, vedas, Bharata shastras, Mimasa, mathematics, medicine and logic by the time he reached 36 years of age. 

He calls himself a poet. He is the author of  Siddhanta Shiromani (Crown of Treatises), a book of 1450 verses written in 1150 AD. The book can be broadly classified into four parts - Lilawathi, Beejaganita, Ganitadhyaya and Goladhyaya. There are 278 verses in Lilawathi, 213 in  Beejaganita, 451 verses in Ganitadhyaya and 501 verses in Goladhyaya.

This is a book on calculations in arthematic and astronomy. Even today, it is regarded by experts as being one of the nest books in ancient Indian astronomy.

Lilawathi mainly deals with mathematics. Pleas read it to know how mathematics can be written in poetic form. The language is simple and lucid. It was only after the British set up the universities in Bombay, Calcutta and Madrtas did that Indian come to know about present day mathematics. Till then, Indians relied on this magnificent work to unbdersytanbd the nuances of mathematics.

The Lilawathi and Beejaganita is a mathematician’s delight. It gives us easy methods to find the squares, square roots, cube, and cube roots of  numbers. Bhaskara was such a genius that he has proved the Pythagoras theorem in two lines.

Pascal’s Triangle was Bhaskara’s Khandameru. Bhaskara has worked out this number triangle 500 years before Pascal.

Lilawathi has also details on  permutations and combinations in a unique method called Ankapaash. Bhaskara has given us the  approximate value of PI as 22/7 and its more accurate value as 3.1416. He knew infinity and called it khahar rashi.

The two sections has problems on spherical trignometry, interest computation, plane and solid geometry.

Lilawathi is sub-divided into 13 chapters and each covers different branches of arithematic, algebra, geometry and mathematics..

Beejaganita  deals with Algebra  in twelve chapters. It was the first book to recognize that a positive number has two square roots (a positive and negative square root).

It deals with positive and negative numbers, Zero, simple equations, Diophantine and indeterminate quadratic equations.

Chakravala was a cyclic format devised by Bhaskara to solve indeterminate quadratic equations of the form ax² + bx + c = y.

His method also helps us solve other similar problems such as quartic equations, cubic equations, indeterminate cubic equations, indeterminate quartic equations and higher-order polynomial equations.

Check out his poetic forms on Trignomotry and the Sine functions and calculus (integral and differential calculus). His work on calculus predates Newton and Leibnez.

He has also accurately measures the distance between the planets and their farthest and nearest positions with earth as the base.

The two sections has problems on spherical trignometry, interest computation, plane and solid geometry.

Lilawathi is sub-divided into 13 chapters and each covers different branches of arithematic, algebra, geometry and mathematics..

Beejaganita  deals with Algebra  in twelve chapters. It was the first book to recognize that a positive number has two square roots (a positive and negative square root).

It deals with positive and negative numbers, Zero, simple equations, Diophantine and indeterminate quadratic equations.

Chakravala was a cyclic format devised by Bhaskara to solve indeterminate quadratic equations of the form ax² + bx + c = y.

His method also helps us solve other similar problems such as quartic equations, cubic equations, indeterminate cubic equations, indeterminate quartic equations and higher-order polynomial equations.

Bhaskara’s contribution to astronomy is too immense to be written in this part of the article. He calcluated the orbital periods of several planets. He also wrote about several astronomical instruments that he used for his calculations.
He died in 1185. His son Lopamudra built school in 1206 to teach his father's works.
A persian translation of  Siddanta Shiromani says Bhaskara wrote it for his daughter Lilawathi. It also has an interesting anecdote on his daughter's marriage. I will write about this in another article.