india's narayan-pandit[1]

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06/07/22 1 India's Contribution to Geometry India's Contribution to Geometry Narayan Pandit Presented by:- Mrs . Geeta Ghormade Innovation & Research Cell , MGS Nagpur

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Page 1: 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

Page 2: India's  narayan-pandit[1]

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

Page 3: India's  narayan-pandit[1]

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

Page 4: India's  narayan-pandit[1]

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

Page 5: India's  narayan-pandit[1]

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

Page 6: India's  narayan-pandit[1]

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

Page 7: India's  narayan-pandit[1]

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

Page 8: India's  narayan-pandit[1]

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

A

B C

E

D

ORcbaAPROOF

4:

)1(2

.21

.21

aAAD

ADaA

ADBCABCofArea

Page 9: India's  narayan-pandit[1]

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

similarareACEandADB

CD

arcsametheininscribedAnglesEBACEandADBIn

2

A

B C

E

D

O

Page 10: India's  narayan-pandit[1]

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

)(

Page 11: India's  narayan-pandit[1]

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

Page 12: India's  narayan-pandit[1]

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

Page 13: India's  narayan-pandit[1]

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)(

Page 14: India's  narayan-pandit[1]

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

)..(4

)..(4

4..

4..

)()(

ABDPBPADRAC

ABBCCDADRAC

RABCBAC

RADCDAC

ACBAACDAralquadrilateofArea

Area of a Cyclic Quadrilateral

Page 15: India's  narayan-pandit[1]

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

Page 16: India's  narayan-pandit[1]

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

Page 17: India's  narayan-pandit[1]

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

Page 18: India's  narayan-pandit[1]

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.

)()()(

Page 19: India's  narayan-pandit[1]

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

Page 20: India's  narayan-pandit[1]

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 )

Page 21: India's  narayan-pandit[1]

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.

Page 22: India's  narayan-pandit[1]

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.

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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

Page 24: India's  narayan-pandit[1]

24

Thank You

Any Questions???

05/03/23