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Presented by:-Mrs .Swati Fating Mathematics DepartmentV.T. Convent, Atrey LayoutNagpur

India’s Contribution to Geometry

01-12-2012 MGS

Statue of Aryabhata

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Aryabhata ( 476 – 550 CE )

Birth place : Kusumpur means Patliputra (modern Patna, Bihar ) Era : Gupta era

Texts of Aryabhata

1) Aryabhatta-Siddhanta

2) Aryabhatiya

3) Suryasiddhanta 4) Brahmasiddhanta

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Major Contributions

•Aryabhatiya covers Arithmetic, Algebra, Plane Trigonometry, and spherical trigonometry. Mensuration Geometric progressions, gnomon / shadows and indeterminate equations (kuTTaka)

• It also contains continued Fraction, Quadratic equations sums-of-power series, and a table of sines

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•Knowledge of zero was implicit in Aryabhata's place-value system ,as a place holder for the powers of ten with null coefficients.

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 π is an Irrational number

Add 4 to 100, multiply by 8, then add 62000, then divide by 20000. The result is "approximately" circumference of a circle of diameter 20000.Π =62832/20000 = 3.1416.

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A

C

ZY

XB

AC = light houseXY = obstacleXB = Shadow of obstacleYZ = perpendicular drawn from tip of obstacle to light house

ZCZY

XYXB

YZCBXY

ZCZYXYXB

XYACZYXY

Shadow of obstacle =

But ZC = AC –AZ = AC-XY (Reason AZ=XY)

Shadow of obstacle =

heightofdifferncethembetweencedisobstacleofHeight tan Shadow of obstacle =

Problems from Aryabhatiya

Sholka

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Obstacle Height = Radius of Earth Height of light source = radius of sun Distance between obstacle&light source=distance between sun & earth

Distance between sun & Earth = 1.5 x 108 Km (Approximate)Radius of sun = 696000 KmRadius of sun = 6400 Km

Shadow of Earth =

= 1392000 Km

6400696000105.16400 8

689600109600 8

=

Shadow of Earth

Meaning of Shloka :The distance between the tips of the shadows & multiplied by the length of the shadow & divided by the difference between the two is the koti . This koti is multiplied by the length of the shadow & divide bythe lngth of shadowsgives the length of bhuja.

Shloka

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BA P S X

YQ

C AC = light housePQ,XY = obstacle of same lengthAS = length of shadow of obstacle PQ from light houseAB = length of shadow of obstacle PQ from light house

XBAB

XYAC

similarBXYareBAC

,

PSAS

PQAC

SPQSACalso

PSXBSB

XBAB

PSXBASAB

XBAB

PSAS

XBAB

PQXYPQAC

XYACBut

)(------1

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Problems

PSXBSBXBAB

Koti =Shadow of XY = Shadow x difference difference between Shadows between tip

of shadows

----------2

From the equation 1 & 2

PSASPQ

XBABXYAC

But XY or PQ = height of obstacle AB or AS = Distance from light house to tip of the shadow XB or PS = length of shadow AC = height of light house (Bhuja) Bhuja = AC = obstacle x Distance from Shadow

light house to tip of the shadow

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A

A

C

DB E

The distance between the house and the man is divided by the sum of the heights of the house and the man’s eyes and multiplied by the height of the eyes. The tip of the image of the house will be seen when the reflecting water is at a distance equal to the above product.

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Problems

AB = height of house (object)CD = height of man’s eyeE = Reflecting point

DECDBEhouseABofheight

BDDE

DEBEDE

CDABCD

similarareCDEABE

&

Also from the same pair of similar triangles the height of the house

DECDBEAB

“The young learners must study the contribution of ‘Aryabhata’ to attain knowledge . The tradition of excelling in the field of Astronomy 7 mathematics by Indian scientists is continuing till date. Without learning the contributions by Indian Astronomers like Aryabhata one can not become a true astronomer”. Newton

“ In the whole history of mathematics , there has been no more revolutionary step than the one which Hindus made when invented the sign 0 for the empty column of the counting frame” Lancelet Hogben in “mathematics for the Million’’

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Thoughts

References

Sr.no

Author Name of the book

Publication

1 Mohan Apte Aryabhatiya Rajhans Publication

2 Murty, Rao, Kannan

I-SERVE Institute of scientific Research on vedas

3 Saraswati Amma

4 Internet websites

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Thank You