1. 1
Presented by:-
Mrs .Swati Fating
Mathematics Department
V.T. Convent, Atrey Layout
Nagpur
India’s Contribution to Geometry
01-12-2012 MGS

2. Statue of Aryabhata
2
Aryabhata ( 476 – 550 CE )
Birth place : Kusumpur means Patliputra (modern Patna, Bihar )
Era : Gupta era

3. Texts of Aryabhata
1) Aryabhatta-Siddhanta
2) Aryabhatiya
3) Suryasiddhanta
4) Brahmasiddhanta
3

4. 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
4

5. 5
•Knowledge of zero was implicit in
Aryabhata's place-value system ,as a place
holder for the powers of ten
with null coefficients.

6. 6
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.

7. 7
A
C
Z
Y
X
B
AC = light house
XY = obstacle
XB = Shadow of obstacle
YZ = perpendicular drawn
from tip of obstacle to light
house
ZC
ZY
XY
XB
YZCBXY
=∴
∆≈∆
ZC
ZYXY
XB
×
=∴∴
XYAC
ZYXY
−
×
Shadow of obstacle =
But ZC = AC –AZ = AC-XY (Reason AZ=XY)
Shadow of
obstacle =∴
heightofdiffernce
thembetweencedisobstacleofHeight tan×
Shadow of obstacle =
Problems from Aryabhatiya
Sholka

8. 9
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 Km
Radius of sun = 6400 Km
Shadow of Earth =
= 1392000 Km
6400696000
105.16400 8
−
××
689600
109600 8
×
=
Shadow of Earth

9. 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
10

10. BA P S X
YQ
C AC = light house
PQ,XY = obstacle of same length
AS = length of shadow of
obstacle PQ from light house
AB = length of shadow of
obstacle PQ from light house
XB
AB
XY
AC
similarBXYareBAC
=∴
∆∆ ,
PS
AS
PQ
AC
SPQSACalso
=∴
≈∆
PSXB
SB
XB
AB
PSXB
ASAB
XB
AB
PS
AS
XB
AB
PQXY
PQ
AC
XY
AC
But
−
=∴
−
−
=∴
=∴
== )(
------1
11
Problems

11. PSXB
SBXB
AB
−
×
=∴
∴
Koti =Shadow of XY = Shadow x difference difference between Shadows
between tip
of shadows
÷
----------2
From the equation 1 & 2
PS
ASPQ
XB
ABXY
AC
×
=
×
=
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
÷
12

12. 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.
13
Problems
AB = height of house (object)
CD = height of man’s eye
E = Reflecting point
DE
CDBE
houseABofheight
BD
DE
DEBE
DE
CDAB
CD
similarareCDEABE
×
=∴
=
+
=
+
∆∆ &
Also from the same pair of similar triangles the height of the
house
DE
CDBE
AB
×
=

13. “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’’
14
Thoughts

14. References
Sr.n
o
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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15. 1/12/2012 16
Thank You