BHASKARA IICHRISTIN SAM

LIFE SKETCH OF BHASKARA ll

• Bhaskara II is a famous mathematician of ancient India. He was born in 1114 A.D. in the city of Bijapur, Karnataka state, India. Peoples also know him as Bhaskaracharya, which means “Bhaskara the Teacher”.

• Bhaskara II became head of the astronomical observatory at Ujjain, which was the leading mathematical centre in India at that time He wrote six books and but a seventh work, which is claimed to be by him, is thought by many historian to be a late forgery

• The three most important books he published were Lilavati(The Beautiful), which is about mathematics; Bijaganita (Seed Counting), which is about algebra; and an astronomical work, Karanakutuhala (The Calculation of Astronomical Wonders). Lilavati is the first known published work that uses the decimal position system.

• His father name was Mahesvara. By profession he was an astrologer, who taught him mathematics, which he later passed on to his son Loksamudra. In many ways, Bhaskaracharya represents the peak of mathematical knowledge in the 12th century

Bhaskaracharya's significant contribution to mathematics

• 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 seventeenth 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.

LILAVATHI

• Lilavati is the first part of Bhaskaracharya's work Siddhantashiromani which he wrote atthe age of 36.

• Lilavati mainly deals with what we call as `Arithmetic' in today's mathematical parlance.

• It consists of 279 verses written in Sanskrit in poetic form (terse verses).

• There are certainverses which deal with Mensuration (measurement of various geometrical objects), Volumeof pyramid, cylinders, heaps of grains etc., wood cutting, shadows, trigonometric relations

• Bhaskaracharya wrote this work by selecting good parts from Sridharacharya'sTrishatika and Mahaviracharya's Ganitasarasamgraha and adding material of his own.

THE NAME ‘Lilavati’

• Lilavati has an interesting story associated with how it got its name.

• Bhaskaracharya created a horoscope for his daughter Lilavati, stating exactly when she needed to get married.

• He placed a cup with a small hole in it in a tub of water, and the time at

• which the cup sank was the optimum time Lilavati was to get married. • Unfortunately, a pearl fell into the cup, blocking the hole and keeping

it from sinking.• Lilavati was then doomed never to wed, and her father Bhaskara

wrote her a manual on mathematics in order to console her, and named it Lilavati.

• This appears to be a myth associated with this classical work.

PUZZLES• Out of a group of swans, 7/2 times the square root of

the number are playing on the shore of a tank. The two remaining ones are swimming in the water. What is the total number of swans?

SOLUTION:let the total no. of swans = x2

No. of swans playing on the shore of a tank. =7/2x

No. of swans swimming in the water =2

Total no. of swans = No. of swans playing on the shore of a tank + No. of swans swimming in the water

X2=7/2X+2

2 X2 = 7X+42 X2-7X-4 = 0

2 X2-8X+X-4 = 0(2X+1) (X-4) =0X-4 = 0X = 4

Total no. of swans = X2 = 16

• Bhaskaracharya gives a very interesting puzzle from the epic Mahabharata where Arjuna uses a certain number of arrows (say x) to destroy the horses of Karna, a certain number to destroy his chariot, flag, bow and to cut o his head. The solution of the puzzle is the root of a quadratic equation.

1/4th of a herd of camels were seen in a forest . Twice the square root of the herd had gone to the mountain remaining 15 were seen at the bank of river. find the no. of camels.

SOLUTION:let the total no. of camels= x2

No. of camels gone to the mountain = 2x

No. of camels seen in a forest = 1/4 x2

No. of remaining camels = 15

Total no. of camels = No. of camels gone to the mountain + No. of camels seen in a forest + No. of remaining camels

X2 = 1/4 x2 + 2x + 15

4 x2 = x2 + 8x + 60

3 x2 = 8x + 60

3 x2 (a) - 8x(b) – 60 (c)= 0

X = (8 +28)/6 = 6

or

x = (8- 28)/6 = -3.33

Total no. of camels = x2 = 36

A snakes hole is at the foot of a pillar , and a peacock is perched on its summit . Seeing the snake At the distance of thrice the pillar, gliding towards his hole,he pounce obliquely upon him. Say at how many cubits from the snakes hole do they meet, both proceeding on equal distance?

SOLUTION: HN = XHS = 27PH = 9PN = SN = 27 – X

In right triangle PHN.By Pythagoras theoremPH2 + HN2 = PN2

92+X2 = (27-X)2

81 + X2 = 729 + 54X + X2

(729 – 81)/54 = XX = 12

SIDDANTA SHIROMANI

• Siddhānta Shiromani (Sanskrit for "Crown of treatises") is the major treatise of Indian mathematician Bhāskarāchārya. He wrote Siddhanta Sherman in 1150 AD when he was 36 years old. The work is composed in Sanskrit Language in 1450 verses

• Lilavati

The name of the book comes from his daughter Līlāvatī. It is the first volume of Siddhānta Shiromani. The book contains thirteen chapters, 278 verses, mainly arithmetic and mensuration.

• Bijaganita

It is the second volume of Siddhānta Shiromani. It is divided into six parts, contains 213 verses and is devoted to algebra.

• Ganitadhyaya and Goladhyaya

Ganitadhyaya and Goladhyaya of Siddhanta Shiromani are devoted to astronomy.).