Shadow Problems

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

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– Projection applications in mathematics

Prominent Mathematicians who have contributed in this field:

Brahmagupta

Aryabhatta

Sridhara

Bhaskara

Narayana

Mahavira

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.

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

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

Shadow Problems by Brahmagupta

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.

C

D

S

A

B E

Where,SE=incident ray striking the reflecting surface at EEA=reflected ray striking the wall at ACD= gnomon in the path of the incident ray DE=shadow of gnomon

2.Determining the height and distance of the object by observing the reflection from two different distances:

AB= object whose height is to be determinedC1D1 and C2D2= two positions of the observerE1,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.

Consider

ABE1 , C1E1D1 and, ABE2, C2E2D2

A

B E1

C1

D1 E2

C2

D2

= =

=

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.

A

A

B

C

DE

Where,AB=object (house)CD=height of the man’s eyesE=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

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 lightAB and A1B1= two equal gnomons BC and B1C1=shadow gnomons

BD

A A1

B1

S

C C1

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.

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/Brahmagupta3)http://www.storyofmathematics.com/

indian_brahmagupta.html4)

http://www.encyclopedia.com/topic/Brahmagupta.aspx

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