1. Shadow Problems

2. Overview
Indian Mathematicians who have worked in this field
Shadow Phenomenon and Geometry
The Great Bhahmagupta and his contribution to shadow problems
Shadow Problems by Bhahmagupta
References
Contribution of other Indian Mathematicians to shadow problems

3. Shadow Phenomenon and Geometry:
 A shadow is an area where direct light from a light
source cannot reach due to obstruction by an object.
 Shadow measurements and calculation based on them
formed an important part of astronomy and therefore of
mathematics from very early time.
Using shadows is a quick way to estimate the heights
of trees, flagpoles, buildings, and other tall
objects.
 Shadow phenomenon also plays an important part in :
– Photography
– Astronomy
– Convolution applications in mathematics

4. Prominent Mathematicians who have contributed
in this field:
Brahmagupta
 Aryabhatta
 Sridhara
 Bhaskara
 Narayana
 Mahavira

5. Brahmagupta:
Brahmagupta was one of the great Indian
mathematician and astronomer who wrote
many important works on mathematics and
astronomy.
His best known work is
the Brahmasphutasiddhanta written in AD
628.
Brahmagupta was the first to use zero as a
number. He gave rules to compute with zero.
Brahmagupta used negative numbers and
zero for computing. The modern rule that two
negative numbers multiplied together equals a
positive number first appears in
Brahmasphutasiddhanta.

6. Brahmagupta’s contribution:
Brahmagupta has formulated different rules for
calculating
time of the day from shadow measurement
length of the shadow from the known height of gnomon,
the light and the horizontal distance between the two
for finding the height and distance of the light by
measuring the shadow lengths of the gnomons at two
distances from the light
height and distance of objects by observing their reflections
in water

7. What is Gnomon?
- also called as "shanku-yantra"
The gnomon is the part of a sundial that
casts the shadow.
Gnomon is an ancient Greek word
meaning "indicator“.
 It is used for a variety of purposes in
mathematics and other fields.
The term was used for an L-shaped instrument like a steel square
used to draw right angles.
This shape may explain its use to describe a shape formed by
cutting a smaller square from a larger one

8. Shadow Problems
by Brahmagupta

9. 1.Problem combining shadow and reflection to
find the height at which the light from a source is
projected:
The problem is defined as
To calculate the ascent of the sun’s rays on a
wall from the known ratio of the shadow to the object
and the distance between the water and the wall.
Solution is given as
The distance between the water and the wall
divided by the ratio of the shadow to the object is the
height of ascent.

10. C
D
S
A
B E
Where,
SE=incident ray striking the
reflecting surface at E
EA=reflected ray striking
the wall at A
CD= gnomon in the path of
the incident ray
DE=shadow of gnomon

11. 2.Determining the height and distance of the object
by observing the reflection from two different
distances:
AB= object whose height
is to be determined
C1D1 and C2D2= two
positions of the observer
E1,E2= two points of
reflection
A
B E1
C1
D1 E2
C2
D2
The distance between the first and second
positions of the water divided by the difference between
the distance of the man from the water, when multiplied
by the height of the eyes, is the height, and the same,
when multiplied by the distance between the water and
the man, is the difference between the water and the
house.

12. Consider ABE1 , C1E1D1 and, ABE2, C2E2D2
A
B E1
C1
D1 E2
C2
D2
==
=

13. 3.Determining height and distance of the object by
observing their reflections in water:
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 then the
reflecting water is at a distance equal to above product.

14. A
A
B
C
D
E
Where,
AB=object (house)
CD=height of the man’s eyes
E=reflecting point
The man will be able to see the
tip of the image when
Also from the same pair of similar
triangles, the height of the object (house) can
be given by

15. Contribution by other Indian mathematicians:
Aryabhatta:
Calculating the height of the source of light and its
horizontal distance from the observer with the help of two
shadow- throwing gnomons .
Where ,
S=source of light
AB and A1B1= two equal
gnomons
BC and B1C1=shadow
gnomons
B
D
A A1
B1
S
C C1

16. The shadow problem in Lilavati are purely
geometrical and evidently modeled on Aryabhatta
treatment.
Sridhara has rules for calculating the time of the
day from the length of the shadow and vice-versa.
Mahavira gives the time honoured method of
fixing the cardinal directions explained in Sulabh-
Sutra.

17. References:
1) “Geometry in Ancient and Medieval India”, Dr. T. A. Sarsvati
Amma, Motilal Banarsidass Publishers Private Limited, page
251-260.
2) http://en.wikipedia.org/wiki/Brahmagupta
3) http://www.storyofmathematics.com/indian_brahmagupta.ht
ml
4) http://www.encyclopedia.com/topic/Brahmagupta.aspx

18. THANK YOU