Bhaskara 2

bhabhaajajajha

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. FROMLILAVATI:
*

3. BHASKARA’STIME
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 1114CE (This year 900
th
Birth Anniversary)
Bhaskara’s father was Maheshwara,a scholarly person
belonging to the Shandilya gotra.

4. BHASKARA’SWORKS
SiddanthaSiromani ( 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. ABOUTHISWORKS
According to some sources SiddanthaSiromaniconsists 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
Lilavatiis an extremely popular text dealing arithmetic ,
elementary algebra, permutations of digits, progressions,
geometry and mensuration,etc.
Bijaganitamis a treatise on advanced algebra.
GrahaganitamandGoladhyayaare completely devoted to
computations of planetary motions , eclipses and rationales of
spherical astronomy.
Karanakutuhalamis another smaller astronomical karana
text with ready-to-use tables.
VasanaBhasyais a detailed commentary on his works with
very interesting and illustrative examples.

6. BHASKARA’SCONTRIBUTIONTOASTRONOMY
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. BHASKARAII ONDIFFERENTIALS
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. BHASKARAII ONCALCULUS
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, BHAVANAANDCHAKRAVALA
Ancient Indian mathematical treatises contain
ingenious methods for finding integer solutions of
indeterminate( or Diophantine) equations.
The three landmarks in this area are theKuttaka
methodof Aryabhata-I for solving the linear
indeterminate equation ay-bx = c ,
the Bhavana method of Brahmagupta (628 CE)
and Chakravalaalgorithmby Jayadeva( who
lived prior to 1073 CE) and Bhaskara II for solving
the quadratic indeterminate equation
Nx
2
+ 1 = y
2

10. CHAKRAVALAMETHOD
Bhaskara II illustrated the Chakravala with difficult numerals
N= 61 and N=67.
For 61x
2
+ 1 = y
2
,the smallest solution in positive integers
is x = 226153980 , y = 1766319049.
( Contrast it with the minimum solution of 60x
2
+ 1 = y
2
:
it is x=4 and y=31)

Narayana pandita ( 1350 C E) too discussed
solutionsof the equation
Nx
2
+ 1 = y
2and illustrated
the method with N=97 and N=103 .

11. INDETERMINATE ANALYSIS
The equations ay-bx =cand Nx
2
+ 1 = y
2
important
equations in modern mathematics .But the Indian works
on such indeterminate equations during 5
th
-12
th
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 Lilavatiinto 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 61x
2
+ 1= y
2

12. INDETERMINATE ANALYSIS
Fermat had asserted in his correspondence of 1659
that he had proved by his own method of “descent”
that the equation Nx
2
+ 1 = y
2
has infinitely many
integer solutions( when Nis 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. THELABELPELL’SEQUATION
The equationNx
2
+ 1 = y
2
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 ANDREMARKS
“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
centuriesearlier”

15. CYCLICQUADRILATERALS WITHRATIONALSIDES
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 18
th
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 WITHRATIONAL
SIDE:

17. FROMLILAVATI, AMANUSCRIPTLEAFSHOWING
THECONSTRUCTION OFQUADRILATERAL

18. BHASKARA’SWORKONCYCLIC-
QUADRILATERAL WITHINTEGERSIDES
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. SOMEINTERESTINGEXAMPLESFROMLILAVATI
“A beautiful pearl necklace of a young lady was torn
and were all scattered on the floor. 1/3
rd
of the
pearls was on the floor and 1/5
th
on the bed , 1/6
th
was found by the pretty lady , 1/10
th
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. PROBLEMFROMLILAVATI
“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=100is the valid answer

22. NUMBERTHEORYPROBLEMINLILAVATI
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= 8x
4
+1 and b=8 x
3  11
2222
 baandba
x a= 8x
4
+1 b=8 x
3
a
2
+b
2
-1 a
2
-b
2
-1
1 9 8 144=12
2
16=4
2
2 129 64 20736=144
2
12544=112
2
3 649 216 467856=684
2
374544=612
2
4 2049 512 4460544=2112
2
3936256=1984
2

23. PROBLEMONPERMUTATION& 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. ANIDOLOFLORDSHIVAINBELUR

26. ANINTERESTINGPROBLEM 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=8810153
2
!2


27. EXAMPLE2:
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(210101853
3
!3
2
   numbersnsumof
n
n
sum
n
''1010101
!
2


28. PEACOCK-SNAKEPROBLEM
“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?
AC
2
= AB
2
+ BC
2
CE = AC = (27 –x).
(27 –x)
2
= 9
2
+ x
2
729 –54x+ x
2
= 81 + x
2
54x= 729 –81 = 648
x= 12 ft.

29. PEACOCK-SNAKEPROBLEM

30. A PAGEFROMLILAVATI

31. CUBICANDBIQUADRATICEQUATIONS
Bhaskara gives the solutions of cubic and bi-
quadratic equations in his Bijaganitam
Solve the cubic equation

32. FOURTHDEGREEEQUATION
Solve the bi-quadratic equation
Therefore x=11

33. PROBLEMFROMLILAVATION
SURFACEAREAANDVOLUMEOFASPHERE
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. SURFACEAREAANDVOLUMEOFASPHERE
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
4r rr22
4
1 3
3
4
r

35. COMMENTARIES ONBHASKARA’SWORKS
Krishna Daivajna( c.16
th
century) is known for his
commentary on the Bijaganitam of
Bhaskara II, known as Bijapallavam.
oGanesha Daivajna (c.16
th
century)has written the
commentary on Lilavaticalled Buddi vilasini.
oSuryadasa(early 16
th
century) has written
commentary on both Bijaganitamand Lilavati.
oSumatiHarsha(c.1621) has written commentary on
KaranakutuhalamcalledGanakakumuda
kaumudi.

36. BIBLIOGRAPHY
Indian Mathematics and Astronomy some Landmarks ,
Dr.S.Balachandra Rao, revised 3
rd
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, 2vols, 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