Bhaskara II was an Indian mathematician and astronomer born in 1114 CE in Bijjada vida, present day Bijapur, Karnataka. Some of his major works include Siddhanta Siromani, Lilavati, Bijaganitam, and Karanakutuhalam. In his works, he made important contributions to calculus, arithmetic, algebra, trigonometry, and astronomy. He introduced concepts related to differential calculus and was the first to use the moon's equation in astronomical calculations. Bhaskara II is also known for his work on indeterminate equations such as Pell's equation and methods to find integer solutions.

1. BHASKARACHARYA-II
( on the occasion of 900th Birth Anniversary )
Dr.S.Balachandra Rao
Hon Director,
Gandhi centre for science & human values,
Bharatiya Vidya Bhavan, Bangalore

2. FROM LILAVATI :
*

3. 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.

4. 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

5. 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.

6. 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.

7. 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.

8. 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”

9. 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

10. 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 .

11. 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

12. 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”.

13. 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.

14. 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”

15. 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.

16. CYCLIC – QUADRILATERAL WITH RATIONAL
SIDE:

17. FROM LILAVATI, A MANUSCRIPT LEAF SHOWING
THE CONSTRUCTION OF QUADRILATERAL

18. 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 ! !

19. 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

20. 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

21. CONTINUED
 therefore x=100 or x = 4 ,Here 4 is not admissible,
hence x=100 is the valid answer

22. 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
   
1
1 2
2
2
2



 b
a
and
b
a
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

23. 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)?

24. 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 ways
Remark :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.

25. AN IDOL OF LORD SHIVA IN BELUR

26. 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=
   88
10
1
5
3
2
!
2




27. 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
(
2
10
10
1
8
5
3
3
!
3 2






  
numbers
n
sumof
n
n
sum n
'
'
10
10
10
1
! 2










28. 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.

29. PEACOCK-SNAKE PROBLEM

30. A PAGE FROM LILAVATI

31. CUBIC AND BIQUADRATIC EQUATIONS
 Bhaskara gives the solutions of cubic and bi-
quadratic equations in his Bijaganitam
 Solve the cubic equation

32. FOURTH DEGREE EQUATION
 Solve the bi-quadratic equation
 Therefore x=11

33. 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.

34. 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 =
2
4 r

 
r
r 
2
2
4
1
3
3
4
r


35. 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.

36. 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)

37. Thank you