india's narayan-pandit[1]

05/03/23 1India's Contribution to Geometry

India's Contribution to Geometry

Narayan PanditPresented by:- Mrs . Geeta Ghormade Innovation & Research Cell , MGS Nagpur

05/03/23 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

Ganit Kaumudi

05/03/23 India's Contribution to Geometry 3

Chapter 4 - Triangles, quadrilaterals, circle, their areas, formation of integral triangle and quadrilaterals, cyclic quadrilaterals

05/03/23 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 =

2HeightBase

altitudesidesofproduct

2

PerimetrerArea2

05/03/23 5India's Contribution to Geometry

Narayana’s Results for Circum radius

1) R = [ BC2+ {(AD2 - BD × DC)/AD}2 ]1/2

2) R =

A

B CD

21

21

altitudesofproductflanksofproductdiagonalsofoduct Pr

05/03/23 India's Contribution to Geometry 6

Narayana’s Results for Circumradius

R = 21

altitudesofproductflanksofproductdiagonalsofoduct Pr

From ADB

R = =

altitudesidesofoduct

2Pr

12.PBDAD

From ACB

R = =

altitudesidesofoduct

2Pr

22.PBCAC

21....

21

ppBCADBDACR

05/03/23 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

RcbaA

4

BC = a CA = bAB = c

‘O’ is the centre of circum - circle R = Circum radius

A

B C

O

05/03/23 India's Contribution to Geometry 8

A

B C

E

D

ORcbaAPROOF

4:

)1(2

.21

.21

aAAD

ADaA

ADBCABCofArea

05/03/23 India's Contribution to Geometry 9

similarareACEandADB

CD

arcsametheininscribedAnglesEBACEandADBIn

2

A

B C

E

D

O

05/03/23 India's Contribution to Geometry 10

A

B C

E

D

O

RabcA

FromaA

bRc

bAD

Rc

csstACAD

AEAB

similarareACEandADB

4

)1(2.12

2

)(

05/03/23 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

DO

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

05/03/2313

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.

DAPBDACABCDA

2)(

05/03/23 India's Contribution to Geometry 14

)..(4

)..(4

4..

4..

)()(

ABDPBPADRAC

ABBCCDADRAC

RABCBAC

RADCDAC

ACBAACDAralquadrilateofArea

Area of a Cyclic Quadrilateral

05/03/23 India's Contribution to Geometry 15

RBDAPAC

BDAPRAC

ABDPBPADRAC

4..

).(4

)..(4

Ptolemy’s TheoremIn a cyclic quadrilateral , sum of product of opposite sides = product of diagonalsAD . BP + DP . AB =AP . BD

05/03/23 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

05/03/23 India's Contribution to Geometry17

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

CD

L

K

RDCBCABADBD

RABBCDCADAC

ABCDA

4)..(

4)..()(

Area of a cyclic Quadrilateral

05/03/23 India's Contribution to Geometry 18

Area of a cyclic Quadrilateral

A B

CD

L

K

RBCABDCADAC

RBCABAC

RDCADAC

BKACDLACABCAADCAABCDA

4)..(2

..22

..2

2.

2.

)()()(

05/03/23 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

05/03/23 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 )

05/03/23 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.

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.

References

05/03/23 India's Contribution to Geometry 23

Sr no Publication Name of the book Author

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

Thank You

Any Questions???

05/03/23