When we want to know the time, we consult a watch. Time the watch gives us is called the Indian Standard Time . The life of our country is regulated by this time. We all know that this time we use in our day to day life is the local time of a place which is on a longitude which divides India in to two halves in the east-west direction. That place is at 82.5 degrees E away from the Greenwich which is the zero longitude. 'E' shows that it is east of Greenwich. It falls in between two big cities, Allahabad and Varanasi. So whenever someone asks you the time, you look at you watch and tell him/her the time of the place at 82.5 degrees and not the time at your place. That's why when it is noon or rather the LOCAL noon at your place your watch does not show 12 o'clock in the afternoon! If you are east of the 82.5 degrees then your local noon will happen before the IST noon and viceversa for the person west of 82.5 degrees.
This difference can be used to get your longitude. Lets see it this way. The whole earth is 360 degrees and it takes 24 hours by your watch to rotate back to same point after one rotation.
 

Now if two places are separated by 15 degrees in longitude on the earth, they will have a difference of 1 hour in their time. You already know that at the local noon the shadow of your gnomon is the shortest. So by measuring the difference (in time) in the local noon at two difference places, we can get the longitude difference.

In our experiment on 20th of March this year, we are going to measure the time of shortest shadow at our place and at a place near the meridian at IST (82.5 degrees E). Let this be T.
The time difference (T- 12) expressed in minutes and divided by 4 should give us gives us the difference it degrees in Longitude.

But there is a complication! The standard time refers to the local noon at 82.5 for a FICTITIOUS SUN! The earth DOES NOT go around the sun at a uniform rate! It moves faster in winter and slower in summer. Again the axis of rotation of the earth is tilted its orbit around the sun. Both these make the sun act like a clock that runs sometimes slow and some times fast! This is very inconvenient to say the least. So our day to day watch pretends that there is a fictious sun that keeps uniform time through out the year. The time it takes for the earth around the sun once is divided by 365 and then by 24 to define a standard hour. Standard times of all countries use this method.
So the local noon at 82.5E occurs LATER than 12 o'clock on the watch on some days and EARLIER on other days! This correction, which changes through out the year, is called the equation of time. Fortunately it is the SAME at ALL longitudes and latitudes and is known for every date.

Now we have two options

Option 1
Get the local noon time in IST by looking at our watch (the time at which we get the shortest shadow in our place) t
Add or subtract the equation of time correction to 12 o'clock to get the local noon time at 82.5 E T
Find T-t in minutes and divide by 4 to get the longitude difference.
Add or subtract to 82.5E and we find OUR longitude!

Option 2
we have another bunch of kids at 82.5 degrees and they measure their local noon t
we measure ours T
Find the the difference in minutes and divide by 4 to find the longitude difference.
Add or subtract from 82.5 and we find our longitude. Notice that the equation of time correction cancels out and so we don't need to know it!

The second option is ALWAYS better as TWO actual readings are taken rather than one reading and we DON'T need any one to tell us about the correction due to equation of time.
In understanding how science works it is always much much better to assume concepts and the minimum number of known quantities. Then the measurement gives support to the CONCEPT assumed rather than claiming accuracy of a number which experts have already measured in any case! More than that it teaches us that cooperation is as IMPORTANT in science as it is in other aspects of life, if not more so.
Now getting a person at exactly 82.5 is also a task. We have a few kids in Varanasi(82 degrees) who are doing this experiment, so a time difference in our time and varanasi time will give us the difference in our longitude within required error.

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