India's narayan-pandit[1]

for student

1. 05/08/15 1India's Contribution to Geometry
India's Contribution to Geometry
Narayan Pandit
Presented by:-
Mrs . Geeta Ghormade
Innovation & Research Cell , MGS
Nagpur

2. 05/08/15 2India's Contribution to Geometry
Scripts:
1) An arithmetical treatise - Ganit Kaumudi,
2) An algebraic treatise – Bijaganita Vatamsa
•Lived in 14
th
century AD in the period of (1340 - 1400)
•Mathematician of medieval period .
•Kerala School of Mathematics
Narayan Pandit

3. Ganit Kaumudi
05/08/15 India's Contribution to Geometry 3
Chapter 4 - Triangles,
quadrilaterals, circle, their areas,
formation of integral triangle and
quadrilaterals, cyclic quadrilaterals

4. 05/08/15 4India's Contribution to Geometry
Formulae for Triangle
•Area of triangle =
•If a, b, c are sides of the triangle and s is
semi perimeter i.e. 2 s = a + b + c then
Area of triangle = [s (s-a) (s-b) (s-
c)]
1/2
•Circum radius =
•Radius of inscribed circle =
2
HeightBase´
altitude
sidesofproduct
´2
Perimetrer
Area´2

5. 05/08/15 5India's Contribution to Geometry
Narayana’s Results for Circum radius
1) R = [ BC
2
+ {(AD
2
- BD × DC)/AD}
2
]
1/2

2) R =

A
B C
D
2
1
2
1
altitudesofproduct
flanksofproductdiagonalsofoduct ´Pr

6. 05/08/15 India's Contribution to Geometry 6
Narayana’s Results for Circumradius
R =
2
1
altitudesofproduct
flanksofproductdiagonalsofoduct ´Pr
From ADB
R = =
D
altitude
sidesofoduct
2
Pr
1
2
.
P
BDAD
From ACB
R = =
D
altitude
sidesofoduct
2
Pr
2
2
.
P
BCAC
21
.
...
2
1
pp
BCADBDAC
R=

7. 05/08/15 India's Contribution to Geometry 7
Area of Triangle
The area of triangle is the product of sides divided
by 4 times the circum radius
R
cba
A
4
=
BC = a
CA = b
AB = c
‘O’ is the centre of circum - circle
R = Circum radius
A
B
C
O

8. 05/08/15 India's Contribution to Geometry 8
A
B
C
E
D
O
R
cba
APROOF
4
:=-
)1(
2
.
2
1
.
2
1
--------=
=
=D
a
A
AD
ADaA
ADBCABCofArea

9. 05/08/15 India's Contribution to Geometry 9
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ë
é
DD
=Ð@Ð
---Ð@Ð
DD
similarareACEandADB
CD
arcsametheininscribedAnglesEB
ACEandADBIn
2
p
A
B
C
E
D
O

10. 05/08/15 India's Contribution to Geometry 10
A
B
C
E
D
O
R
abc
A
From
a
A
bR
c
b
AD
R
c
csst
AC
AD
AE
AB
similarareACEandADB
4
)1(
2
.
1
2
2
)(
=
-----=
=
------=
DD

11. 05/08/15 India's Contribution to Geometry 11
If the altitude is produced to meet the circum-circle , the
portion beneath the base can be calculated using the sutra
Meaning:- The lower part of the
altitude which touches the circum-
circle is product of the parts of the
base divided by the altitude
DE = (BD × DC) / AD.
BE = (BD × AC ) / AD.
CE = (CD × AB) / AD
A
B
C
E
D
O

12. Third Diagonal of a quadrilateral
12
Definition:- When the top side and the flank side of a
quadrilateral are interchanged a third diagonal is generated called
as a ‘para’
In a quadrilateral ABCD interchange the
sides CD & CB.
Select a point P on the circum-circle such that
BP = CD and DP = BC
Then AP is third diagonal

13. 05/08/15
13
India's Contribution to Geometry
Area of a Cyclic Quadrilateral
The area of cyclic quadrilateral is given by the product
of three diagonals divided by twice the circum -diameter
In quad .ABCD
AC and BD are original diagonals .
AP is third diagonal.
D
APBDAC
ABCDA
2
)(
´´
=

14. 05/08/15 India's Contribution to Geometry 14
)..(
4
)..(
4
4
..
4
..
)()(
ABDPBPAD
R
AC
ABBCCDAD
R
AC
R
ABCBAC
R
ADCDAC
ACBAACDAralquadrilateofArea
+=
+=
+=
D+D=
Area of a Cyclic Quadrilateral

15. 05/08/15 India's Contribution to Geometry 15
R
BDAPAC
BDAP
R
AC
ABDPBPAD
R
AC
4
..
).(
4
)..(
4
=
=
+=
Ptolemy’s Theorem
In a cyclic quadrilateral ,
sum of product of opposite sides = product of diagonals
AD . BP + DP . AB =AP . BD

16. 05/08/15 India's Contribution to Geometry 16
Area of a cyclic Quadrilateral
When the diagonal is multiplied by the sum of products of the
sides about the other diagonal and divided by four times the
circum –radius , that will be the area of isosceles trapezia and
other cyclic quadrilateral

17. 05/08/15 India's Contribution to Geometry
17
When the diagonal is multiplied by the sum of products
of the sides about the other diagonal and divided by four
times the circum –radius , that will be the area of
isosceles trapezia and other cyclic quadrilateral
A B
C
D
L
K
R
DCBCABADBD
R
ABBCDCADAC
ABCDA
4
)..(
4
)..(
)(
+
=
+
=
Area of a cyclic Quadrilateral

18. 05/08/15 India's Contribution to Geometry 18
Area of a cyclic Quadrilateral
A B
C
D
L
K
R
BCABDCADAC
R
BCABAC
R
DCADAC
BKACDLAC
ABCAADCAABCDA
4
)..(
2
..
22
.
.
2
2
.
2
.
)()()(
+
=
+=
+=
+D=

19. 05/08/15 India's Contribution to Geometry 19
Diagonals of Cyclic Quadrilateral
•AB = a, BC = b, CD = c,
DA = d, DB = x, AC = y,
AP = z (Third Diagonal)
•Diagonal AC =
[(ac+bd)(ad+bc)/(ab+cd)]
1/2
•Diagonal BD =
[(ad+bc)(ac+bd)/(ab+cd)]
1/2
•Diagonal AP =
[(ab+cd)(ad+bc)/ ac+bd)]
1/2

20. 05/08/15 India's Contribution to Geometry 20
Formation of Integral Triangle
(Rational Triangle)
Rational Triangle :- A triangle in the Euclidean plane such that
all three sides measured relative to each other are integer .
He gave a rule of finding rational triangles
whose sides differ by one unit of length
( 3 , 4 , 5 )
( 13 ,14 ,15 )
(51 , 52 , 53 )
(193 , 194 , 195 )
(2701 ,2702 , 2703 )

21. 05/08/15 India's Contribution to Geometry 21
Ganita Kaumudi
Ganit Kaumudi is for removing the
darkness and increasing the
knowledge of mathematics which is
like a sea and which is the life of
many people.

22. Ganita Kaumudi
May this Ganita Kaumudi with
its pleasant light puff up our
pride in Bharatiya Ganita and
empower us to touch new
horizons in the subject.

23. References
05/08/15 India's Contribution to Geometry 23
Sr noPublication Name of the bookAuthor
1 . Motilal
Banarasidass
Pvt.Ltd
Geometry in Ancient
and Medieval India
T.A.Sarasvati
Amma
2. ------ The Ganit Kaumudi of
Narayana Pandit
Paramananda
Singh

24. 24
Thank You
Any Questions???
05/08/15