Mahavira was an influential 9th century Indian mathematician from southern India. He wrote the Ganita-Sara Samagraha, one of the earliest and most important texts entirely devoted to mathematics in India. The text consisted of 9 chapters covering all mathematical knowledge of the mid-9th century, including arithmetic operations, fractions, equations, areas, and volumes. It built upon the work of earlier mathematicians like Brahmagupta but also advanced mathematical understanding with original contributions such as techniques for solving various types of equations and approximate formulas for the areas of ellipses. The text provides a valuable record of the state of Indian mathematics in the 9th century.

1. By
Mrs. Sarita D. Mirzapure.
N.M.Model Jr. College
Nagpur.

2.  Mahavira
 Born: about 800 ad. in Mysore, India
 Died: about 870 in India

3.  Mahavira was of the Jain religion and was
familiar with jain mathematics. He worked in
Mysore in southern India where he was a
member of a School of mathematics.
 This great mathematician monk wrote “Ganita-
Sara Samagraha” in 850 AD during the regime of
the great Rashtrakuta king Amoghavarsha .

4. This book was designed as an updating of
Bramhagupta’s book.
Filliozat writes :This book deals with the
teaching of Bramha-gupta but contains both
simplifications and additional information.....
Although like all Indian versified texts,it is
extremely condensed, this work ,from a
educational point of view, has a significant
advantage over earlier texts.

5. This book consisted of nine chapters and
included all mathematical knowledge of mid-
ninth century India.
There were many Indian mathematian during
the time of Mahavira ,but their work on mathe-
matics is always contained in text which discuss
other topics such as Astronomy .
The Ganita Sara Samgraha is the only early
Indian text which is devoted entirely to
mathematics .

6.  In the introduction to the work Mahavira paid
tribute to the mathematicians whose work
formed the basis of his book.
 Mahavira writes:-
With the help of accomplished holy sages, who
are worthy to be worshipped by the lords of the
world....I glean from the great ocean of the
knowledge of numbers a little of its essence,

7.  in the manner in which gems are picked
from the sea , gold from the stony rock and
the pearl from the oyster shell;
and I give out according to power of my
intelligence, the Sara Samgraha, a small work
on arithmetic, which is however not small in
importance.

8.  The nine chapters of the Ganita Sara Samgraha
are:
1. Terminology
2. Arithmetical operations
3. Operations involving fractions
4. Miscellaneous operations
5. Operations involving the rule of three
6. Mixed operations
7. Operations relating to the calculations of area
8. Operations relating to the excavations
9. Operations relating to the shadows.

9.  Ex. on Indeterminate equation.
 Three merchants find a purse lying in the road.
One merchant says, If I keep the purse,I shall have
twice as much money as the two of you together.
Give me the purse and I shall have three times as
much, said the second. The third said ,I shall be
much better off than either of you if I keep the
purse,I shall have five times as much as the two of
you together.
 How much money is in the purse?
 How much money does each have?

10.  If the first merchant has x,
the second y,
the third z,
the amount in the purse is p,
Then
p+x = 2(y+z); p+y = 3(x+z); p+z = 5(x+y)
Solution: p=15, x=1, y=3, z=5.

11.  Some of the interesting things in Ganita-sara-
samgraha are:
 A naming shame for numbers from 10^24,which are
eka,dasha,.......mahakshobha.
 Formulas for obtaining cubes of sums.
 No real square root of the negative numbers can exists.
 Techniques for least common denominators.
 Techniques for combinations.
 Techniques for solving linear,quadratic,higher order
equations.
 Several arithmetic and geometric series.
 Techniques for calculating areas and volumes.

12.  “Construction of a Rhombus of a given area "is a
topic from the 7th chapter “Operations relating to
the calculations of area ”from Ganita Sara
Samgraha .

13. To construct a Rhombus of a given
area.
 To construct a Rhombus of an area =2mn
 Consider a Rectangle ABCD with sides 2m
and 2n . 2m
D C
2n
A B

14.  Area of Rectangle ABCD =2m×2n
=4mn.
 Let E,F,G,H be the midpoints of the sides
of a Rectangle . D G C
 Join POINTS :E,F,G,H .
H F
A E B

15. Now to prove that Quadrilateral EFGH is a
Rhombus, and its area=2mn.
D G C
n n
H F
n n
B
A m E m B

16.  From the fig.
 In .EFGH
 lt.(EF)=lt.(FG)=lt.(GH)=lt.(EH)= √m
2
+n
2
 EFGH is a Rhombus.
 Now Area of EFG=1/2(mn)

17.  Similarly
Area of GFC=Area of HDG
=Area of AEH= ½(mn)
 Sum of area of all four s = 4 (1/2 mn )
= 2 mn .

18.  Area of Rhombus = Area of Rectangle ABCD
- Sum of area of all four triangles.
= 4 mn -2 mn
=2 mn
 Hence proved : Area of Rhombus = 2 mn
 So this is the simplest method for the construction of
a Rhombus of given area .

19.  Mahavira also attempts to solve certain mathematical
problems which had not been studied by other Indian
mathematicians .
 For ex. he gave an approximate formula for the area
and the perimeter of an Ellipse.
 Hayashi writes : the formulas for conch like fig. have
so far been found only in the works of Mahavira and
Narayana.

20. References
 An article recently published in Vishva Viveka(a Hindi
quarterly published from New Orleans,USA)by
Prof.S.C.Agrawal and Dr. Anupama Jain-via web sights.
 An article by J J O’Corner and E F Robertson- via web-
sights.
 Hindi Translation on Ganita Sar Samgraha by Prof.
L.C. Jain. Via-www.jaingranths.com