If we place a gnomon at different locations on earth and closely watch the shadow during daytime, we will observe that the angle between the vertical and the mid day sun will be different at different places (or latitudes). Can we use this to find the latitude of a place? Or can we use this method to find out the difference in latitude for two different places? Should we walk all around the earth and find the circumference and then the radius of the earth to do so? Well we don’t have to go to so much trouble. Should we use a map with a scale that was made over hundreds of years by other people walking long distances and surveying the land? No! There is no need to do it. Should we copy the radius of the earth from a handbook and use it? No! No!
This is getting a bit irritating! Lets go to the heart of the matter right away. As before A and B are two locations on the surface of the earth of radius R. Two sticks (Gnomon) of equal length (AP=BQ) are placed in the two locations. The shadow length AM is shorter than shadow length BN. The direction of the mid -day sun at A and B are MP and NQ. The shorter shadow implies that the sun has a higher altitude at A than at B. The angle APM between the vertical and the direction of the sun at A can be measured by constructing similar triangles on paper or by using trigonometry. The angle BQN between the vertical and the direction of the sun at B, can be similarly measured
Look at the figure below. Let the circle below represent the earth. The vertical at A and B is defined as the line joining the centre of the earth to the points A and B respectively. The verticals through them are OAP and OBQ respectively. The shortest shadow is cast by the mid day sun. The rays of the sun falling on the earth may be considered parallel, as the sun is very far compared to the size of the earth. From measurement we know the angles APM and BQN and the difference between them. It is clear from the geometry that this is equal to the angle AOB subtended by the arc AB at the centre of the earth.
Suppose the direction of the noon day sun at station A is as shown in the diagram below
ie the sun is exactly overhead at noon at A and casts no shadow! The angle POQ is still the difference in the angle between the vertical angle of the noon sun at the two places. From simple geometry angle POQ = angle BQN, the vertical angle of the noon sun at B. So you can easily measure it at B. If station A is at the equator, it directly measures the LATITUDE at B! Now March 20 is the equinox day, On this the sun shines directly over the equator. If we perform the experiment by measuring the vertical angle of the noon day sun on that day everywhere in the world we can measure our own latitude!
WHAT are we assuming here?
The Earth is spherical.
The Sun is very very far as compared to the size of the earth
The sun shines exactly overhead on the equator on March 20, 2010, the equinox day. This we took from an Astronomy Handbook as the day when sun’s declination is 0 degrees.
We can also measure latitude on any day of the year. But that is another measurement and another adventure in science!
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