SHADOWS
R. Shankar, The Institute of Mathematical Sciences, Chennai

Vikram, a young elephant in the Mudumalai camp, was resting near the camp. He saw his friend Lakshmi, a mahout's daughter, come running home from school.
"Why are you running in this heat?" he asked.
"To keep cool" came the reply.
"Are you feeling OK?" Vikram was genuinely concerned that the heat had affected her mind.
"Of course I am OK. I was running to keep up with the shadow of the cloud".
This answer did not reassure Vikram, "But chasing clouds to keep cool is crazy!"
Lakshmi laughed "Yes it is. But it is a lot of fun! Also, it has got me here to you quicker and I can rest in your shadow".
She lay down on the grass using her school bag as a pillow. It was not yet noon so Vikram was a bit puzzled. "How come you have come back so early?"
"They wanted to do some repairs to the school building so they let us off early".
There was a pleasant breeze and soon she dozed off.
A little later she woke up with the sun on her face. "Why did you move?"
Vikram, who had remained where he was despite spotting some delicious leaves nearby, protested strongly, "What do mean? I did not budge an inch!"
Lakshmi got up and picked up her bag, "Then you should have told your shadow also not to move".
"OK, I'll do that next time. But now maybe both of us can move. Come, I'll give you a lift home".
He lifted his foreleg to form a step. Lakshmi climbed on it, clambered on to his back and they set off towards the elephant camp.
"Actually, my shadow did not move. It only became smaller" said Vikram, who had been giving the matter some thought.
"That is true", replied Lakshmi, "Your shadow will only move when you move but it is long in the morning, shortest at noon and again becomes long in the evening".
"Yes, that is true. But why does that happen?" Vikram had really never thought about shadows earlier.
"The sun moves. Your shadow is just the part of the ground where the direct sunlight has been cut off by your body. So when the sun is low, your shadow is long and when it is high in the sky it is short". 
Vikram thought about it a bit more. "So it is nice when it is cloudy, because we are in the shadow of the clouds?"
"Yes, but the clouds are only water vapour so they do not cut off the sunlight completely."
"Then it must have been a really thick cloud passing by on 22nd July morning that made it quite dark".
"Ah no! That was an eclipse. It was actually the shadow of the moon".
"What nonsense! The moon is there only at night. This happened in the morning".
"No, no! The moon is there all the time. We cannot see it in the day because the sun is very bright. Actually, the moon is circling the earth, the earth is circling the sun and the earth is also spinning. We see the sun and the moon moving across the sky because the earth is spinning. Every day it spins around once."
"I am not sure I understand, my head is spinning with all this circling, but then why this eclipse?"
"Thats because the moon is circling the earth. It completes one circle in around 28 days or just about a month. When it is between the sun and the earth, its shadow falls on the earth and we have an eclipse". 
"Then we should have an eclipse every month", reasoned Vikram.
This was very logical and Lakshmi was confused. "That should be true but it does not happen, I really don't know why. They never told us this in school".
This was a puzzle. Why do we not have an eclipse every month? As they walked back, both of them thought about it but could not figure it out.
"We must try it out" said Lakshmi as they reached the camp.
"How?" asked Vikram. "Our teacher did tell us how to make a model to understand
this. Come, I'll show you".
She went to her father. "Appa, can I borrow your volleyball and torch?"
"There is a match tomorrow. If it gets punctured there will be a lot of trouble".
"But it won't get punctured. We will only keep it on a table and shine light on it".
"OK, but be careful. Also don't shine too much light and finish the battery".
"Actually, we should do this at night" said Lakshmi. "But I will not be allowed out of my bedroom at night."
"I'll stand near the window and my shadow will block the light", replied Vikram.
So he stood near the window, Lakshmi closed all the doors of the room and it was reasonably dark. She put the volleyball on the table, and placed the torch on a shelf about two meters away and adjusted its position so that the light beam fell on the volley ball.
"See! The volleyball is like the earth. Half the earth is lit up and half of it has no light. As it spins, the part that had light will become dark and the part that was dark will get the light". 
"Yes, I now understand. If I was on that volleyball, I would see the sun move across the sky. Now what about the moon?"
So Lakshmi made a small ball out of plastacine, about the size of a table tennis ball, stuck it on a stick and held it about half a meter away from the volley ball. The shadow of the plastacine ball (and the stick) could clearly be seen on the volleyball.
"Oh! Only a small portion of the volley ball has the shadow" exclaimed Vikram.
"Yes. In an actual eclipse, people in that shadow region will have total darkness. It did not happen in Mudumalai, but I read that it happened in other places in India like Surat, Banaras, Patna, Assam and Arunachal Pradesh". As she was speaking, Lakshmi's hand shook a little bit and the shadow disappeared.
"Oops, its gone. I have to adjust it so that the plastacine is almost directly in the line between the torch and the centre of the volley ball for the shadow to fall on it", muttered Lakshmi as she again adjusted the stick to get the shadow back on the ball.
"AHA! Maybe that is it!" exclaimed Vikram excitedly. "Maybe the moon is not directly in line each time it circles the earth. Maybe that happens only once in many years!"
"You could be right, but how can we be sure that this is the reason that we do not have an eclipse every month?"
Both of them thought hard. Lakhmi thought aloud. "It should be somehow possible to do this by looking at the positions of the moon and the sun".
"Yes, come to think of it, if I were on that volleyball, I would see both the sun and the moon move across the sky", Vikram continued on that line of thought.
Lakshmi considered this. "You are absolutely right. What you are saying is that from the earth, it looks as if the moon and sun are circling the earth once every day. But these two circles may not be the same. They could be on different planes. In that case, the moon may not be directly in line every time it circles around the earth, sometimes it could be a little above and sometimes a little below. Then an eclipse will not occur every month".
"We must check this out" said Vikram.
"But how do we do this? The sun can be seen only in the day and the moon only at night".
"Let us think about it. We must have some way to record their paths and then compare them. But enough for today. We will do it some other day".