BHASKARACHARYA-II

( on the occasion of 900th Birth Anniversary )

Dr.S.Balachandra RaoHon Director,

Gandhi centre for science & human values,

Bharatiya Vidya Bhavan, Bangalore

FROM LILAVATI :

*

BHASKARA’S TIME

According to his own statement “ he belonged to Vijjada vida( Bijjada bida) near the line of Sahyadri mountains”.

The place Vijjada vida is identified as present Bijapura of Karnataka, but some scholars have identified the place with a other place in Maharashtra.

Born in 1114 CE (This year 900th Birth Anniversary)

Bhaskara’s father was Maheshwara,a scholarly person belonging to the Shandilya gotra.

BHASKARA’S WORKS

Siddantha Siromani ( Grahaganitam and Goladhyaya) This text was composed by Bhaskara ,when he was 36 years old( in 1150CE)

Lilavati

Bijaganitam

Karana kutuhalam This text was composed by Bhaskara ,when he was 69 years old ( in1183CE )

Vasana Bhasya

ABOUT HIS WORKS According to some sources Siddantha Siromani consists

of four parts namely , Lilavati, Bijaganitam, Grahaganitam and Goladhyaya.

The first two are independent texts deal exclusively with Mathematics and the last two with Astronomy. This text was composed by Bhaskara ,when he was 36 years old

Lilavati is an extremely popular text dealing arithmetic , elementary algebra, permutations of digits, progressions, geometry and mensuration,etc.

Bijaganitam is a treatise on advanced algebra. Grahaganitam and Goladhyaya are completely devoted

to computations of planetary motions , eclipses and rationales of spherical astronomy.

Karana kutuhalam is another smaller astronomical karana text with ready-to-use tables.

Vasana Bhasya is a detailed commentary on his works with very interesting and illustrative examples.

BHASKARA’S CONTRIBUTION TO ASTRONOMY Bhaskara’s Siddantha siromani merits as the best

and exhaustive text for understanding Indian Astronomy.

He gets the credit of being the first among Hindu astronomers in introducing the moon’s equation which is now called “evection’’ into siddhantic text. It is remarkable discovery by Bhaskara which preceded even the western countries by four centuries.

The chapter on spherical astronomy, Goladhyaya, is very important from the point of theoretical astronomy. Rationale for the formulae used are provided.

Large number of astronomical instruments are given in “yantradhyaya”.

He has improved the formulae and methods adopted by earlier Indian Astronomers.

BHASKARA II ON DIFFERENTIALS

He introduces the concept of instantaneous motion (tatkalika gati) of a planet .

He clearly distinguishes between sthula gati (average velocity) and sukshma gati ( accurate velocity) in terms of differentials. The concepts are basic to differential calculus.

If y and y’ are the mean anomalies of a planet at the end of consecutive intervals, according to Bhaskara sin y’ – sin y = (y ’ - y) cos y

The above result equivalent to d (sin y)=cos y dy in our modern notation.

BHASKARA II ON CALCULUS

Bhaskara further state that the derivative vanishes at the maxima.

“Where the planet’s motion is maximum, there the fruit of the motion is absent”

KUTTAKA , BHAVANA AND CHAKRAVALA Ancient Indian mathematical treatises contain

ingenious methods for finding integer solutions of indeterminate( or Diophantine) equations.

The three landmarks in this area are the Kuttaka method of Aryabhata-I for solving the linear indeterminate equation ay- bx = c ,

the Bhavana method of Brahmagupta (628 CE) and Chakravala algorithm by Jayadeva( who lived prior to 1073 CE) and Bhaskara II for solving the quadratic indeterminate equation

Nx2 + 1 = y2

CHAKRAVALA METHOD Bhaskara II illustrated the Chakravala with difficult

numerals N= 61 and N=67.

For 61x2 + 1 = y2 ,the smallest solution in positive integers is x = 226153980 , y = 1766319049.

( Contrast it with the minimum solution of 60x2 + 1 = y2:

it is x=4 and y=31)

Narayana pandita ( 1350 C E) too discussed solutions of the equation Nx2 + 1 = y2 and illustrated the

method with N=97 and N=103 .

INDETERMINATE ANALYSIS

The equations ay-bx =c and Nx2 + 1 = y2 important equations in modern mathematics .But the Indian works on such indeterminate equations during 5th - 12th centuries were too advanced to be appreciated or noticed by Arabs and Persian scholars and did not get transmitted to Europe during the medieval period

Fyzi translator of Bhaskara’s Lilavati into Persian also omitted the portion on indeterminate equations.

Pierre de Fermat(1601-65), a French mathematician Challenges his fellow European mathematicians to solve the equation 61x2 + 1= y2

INDETERMINATE ANALYSIS

Fermat had asserted in his correspondence of 1659 that he had proved by his own method of “descent” that the equation Nx2 + 1 = y2 has infinitely many integer solutions( when N is a positive integer which is not a perfect square). The proof has not been found in any of his writings.

This problem was again taken up by Euler(1707-83) and initial discoveries were made. Later Lagrange(1736-1813) published the formal proofs of all these in his book “ Additions to Euler’s elements of Algebra”.

THE LABEL PELL’S EQUATION

The equation Nx2 + 1 = y2 was attributed to the English mathematician John Pell(1611-85) by Euler although there is no evidence that Pell had investigated the equation.

Because of Euler’s mistaken attribution it remained as Pell’s equation, even though it is historically wrong.

As suggested by R.Sridharan the equation should be called “Brahmagupta’s equation” as attribute to the genius who contributed the equation thousands of years before the time Fermat and Pell.

COMPLIMENTS AND REMARKS

“Bhaskara's Chakravala method is beyond all praise : It is certainly the finest thing achieved in the theory of numbers before Lagrange”

- Hankel, the famous German mathematician.

Regarding Fermat’s challenge, Andre Weil remarks“What would have been Fermat’s astonishment if

some missionary , just back from India had told him that his problem had been successfully tackled there by native mathematicians almost five centuries earlier”

CYCLIC QUADRILATERALS WITH RATIONAL SIDES

The credit of Constructing a Cyclic - Quadrilateral with rational sides goes to Brahmagupta (628 CE)

The only cyclic -quadrilateral that was known to western countries till 18th century was with sides 39,52,60,25and it was referred to as Brahmagupta’s quadrilateral.

The German Mathematician, Kummer (1810-1893) in one of his papers shows that Brahmagupta’s simple method enables us to construct any number of such quadrilaterals and expresses his great admiration for Brahmagupta.

CYCLIC – QUADRILATERAL WITH RATIONAL SIDE:

FROM LILAVATI, A MANUSCRIPT LEAF SHOWING THE CONSTRUCTION OF QUADRILATERAL

BHASKARA’S WORK ON CYCLIC- QUADRILATERAL WITH INTEGER SIDES

Bhaskara had explained a construction of cyclic –quadrilateral by considering two right angled triangle with integer sides ( the palm leaf of the manuscript is shown).

In the above manuscript the two triangles are of the sides (3,4,5 ) and (5,12,13) resulting in a cyclic quadrilateral with sides 52,39,25 and 60. same as Brahmagupta’s quadrilateral ! !

SOME INTERESTING EXAMPLES FROM LILAVATI

“A beautiful pearl necklace of a young lady was torn and were all scattered on the floor. 1/3rd of the pearls was on the floor and 1/5th on the bed , 1/6th was found by the pretty lady , 1/10th was collected by the lover and six pearls were seen hanging in the necklace” (Li.54)

Solution : If the number of pearls in the necklace is ‘ x ’ then the problem yields the equation ,

on solving it ,We get x = 30

PROBLEM FROM LILAVATI “Partha ,with rage ,shot a round of arrows to

Karna in the war. With half of those arrows he destroyed Karna’s arrows, then killed his horses with four times the square-root , hit shalya with six arrows, destroyed umbrella , flag and bow with three arrows and finally beheaded Karna with one arrow . How many arrows did Arjuna shoot? (Li 71)

Solution: Let the number of arrows used by Arjuna be ‘x’ then the equation is

CONTINUED

therefore x=100 or x = 4 ,Here 4 is not admissible, hence x=100 is the valid answer

NUMBER THEORY PROBLEM IN LILAVATI

Generating ‘a’ and ‘b’ such that

are both perfect squares. Bhaskara’s work on finding such numbers is

really a wonderful part in Lilavati. Bhaskara gives a= 8x4+1 and b=8 x3

11 2222 baandba

x a= 8x4+1 b=8 x3 a2+b2-1 a2-b2-1

1 9 8 144=122 16=42

2 129 64 20736=1442 12544=1122

3 649 216 467856=6842 374544=6122

4 2049 512 4460544=21122

3936256=19842

PROBLEM ON PERMUTATION & COMBINATIONS How many variations of form of god, lord

Shiva are possible by arrangement in different ways of ten items , held in his several hands, namely pasha , ankusha , sarpa , damaru ,kapala,

shula , khatvanga ,shakti , shara and chapa? Also those of Lord Vishnu by the exchange of gada , chakra , saroja (lotus) and shanka(conch)?

SOLUTION

Lord Shiva has 10 items held in his hands .These can exchanged among themselves in 10! Ways

Answer =10! =36,28,800 ways Lord Vishnu has 4 items held in his four hands and

those can be exchanged among themselves in 4! Ways.

Answer= 4! =24 waysRemark :There is an idol of Lord Shiva with 10 hands

in the outer courtyard of Sri Chennakeshava temple at Belur in Karnataka. Bhaskara may had got the inspiration for this problem from this unusual idol of Lord Shiva with 10 hands and 24 forms of Lord Vishnu, which is located in the same temple.

AN IDOL OF LORD SHIVA IN BELUR

AN INTERESTING PROBLEM ON PERMUTATIONS

If any two or more numbers are taken then how many two or more (respective) digit numbers can be formed and what is their sum?

Ex: If 3 and 5 are considered then the two digit numbers that are possible to form are 2, they are 35 and 53 and their sum is 88.

The same can be calculated by Bhaskara’s method

Sum= 88101532

!2

EXAMPLE 2: If 3,5,8 are taken then three digit numbers that are

possible to form are 3!=6 The numbers are 358,385,538,583,835,853 The sum can be obtained by adding them. By Bhaskara’s formula

Sum =

Where as the sum of 358 + 385 + 538 + 583 + 835 + 853 = 3552

This method can be extended to any numbers to form any digit numbers and general formula is

3552)111)(16(2101018533

!3 2

numbersnsumofn

nsum n ''1010101

! 2

PEACOCK-SNAKE PROBLEM “A snake ‘s hole is at the foot of a pillar 9 ft

high and a Peacock is perched on its summit. Seeing at a distance of thrice the height of the pillar moving (crawling) towards its hole,the peacock pounces obliquely upon the snake . Say quickly at what distance from the snake’s hole they meet? if both move at same speed?

AC2 = AB2 + BC2

CE = AC = (27 – x). (27 – x)2 = 92 + x2

729 – 54x + x2 = 81 + x2

54x = 729 – 81 = 648 x = 12 ft.

PEACOCK-SNAKE PROBLEM

A PAGE FROM LILAVATI

CUBIC AND BIQUADRATIC EQUATIONS

Bhaskara gives the solutions of cubic and bi-quadratic equations in his Bijaganitam

Solve the cubic equation

FOURTH DEGREE EQUATION

Solve the bi-quadratic equation

Therefore x=11

PROBLEM FROM LILAVATI ON SURFACE AREA AND VOLUME OF A SPHERE Bhaskara has given the correct relation

between the Diameter, the surface area and the volume of a Sphere in his Lilavati.

In a circle the circumference multiplied by one-fourth the diameter is the area. Which,

multiplied by four is its surface area going around like a net around a ball .This surface area multiplied by the diameter and divided by Six is the volume of the Sphere.

SURFACE AREA AND VOLUME OF A SPHERE

If the diameter of a circle is ‘2r’ and its circumference is then

The area of a circle =

Surface area of a Sphere = 4( area of a circle)=

Volume of a sphere = (surface area of a sphere)2r/6

=

24 r

rr 224

1

3

3

4r

COMMENTARIES ON BHASKARA’S WORKS

Krishna Daivajna( c.16th century) is known for his commentary on the Bijaganitam of

Bhaskara II, known as Bija pallavam.o Ganesha Daivajna (c.16th century)has written

the commentary on Lilavati called Buddi vilasini.

o Suryadasa(early 16th century) has written commentary on both Bijaganitam and Lilavati.

o Sumati Harsha (c.1621) has written commentary on Karanakutuhalam called Ganaka kumuda kaumudi.

BIBLIOGRAPHY Indian Mathematics and Astronomy some Landmarks ,

Dr.S.Balachandra Rao, revised 3rd edition , Bhavan’s Gandhi centre, Bangalore.

Mathematics in India, Culture and History of mathematics-7,Kim Plofker , Hindustan Book Agency , New Delhi.

Studies in the History of Indian Mathematics, Culture and History of mathematics-5 , C.S.Seshadri , Hindustan Book Agency , New Delhi.

Lilavati of Bhaskaracarya with Kriya- kramakari , K.V.Sarma , VVBIS, panjab University.

Lilavati ,2 vols, V.G.Apte ,Anandashrama Press , Pune. Sisya –dhi- Vrddidha-tantra of Lalla, 2 vols, Dr.Bina Chatterjee, Indian

National Science Academy, New Delhi. Sri Bhaskaracharya virachita “Lilavati” in Kannada, K S Nagarajan,

SSVM, Bangalore. Lilavti-108 selected Problems in Kannada, Dr.S.Balachandra Rao,

Navakarnataka Publications, Bangalore (In Press)

Thank

you