Michael Fowler

Thales' tricks: 2500 years ago, Thales brought back to Greece from Egypt some practical ways to use geometrical ideas. In particular, he showed how to find the height of something by measuring its shadow, and how to find the distance of a boat at sea by measuring line of sight angles from two places on the shore a known distance apart. To see exactly what he did, it's simplest just to do it.

1. Find the height of a lamp post in the parking lot behind the Physics Building by measuring the length of its shadow, and measuring the length of your own shadow. Explain clearly how knowing your own height means you can figure the height of the lamp post.

2. Since we don't have a handy boat at sea, we settle for measuring the distance of the Rotunda from the next path down across the Lawn (not the path right by the Rotunda). Go to one end of that next path down, and measure the angle between the line of the path and a pencil, say, pointing straight to the middle of the Rotunda. Now do the same from the other end of the path. Next, pace off the path, after measuring the length of your pace. Now draw a triangle with baseline to represent the path, in some units you decide, such as 1 inch = 100 feet, (state what unit you decide to use!) draw the other two sides to represent lines from the ends of the path to the middle of the Rotunda. Use a protractor to get the angle equal to what you measured at the Lawn, then use your drawing to measure the distance of the Rotunda from the path.

3. We can now measure the height of the Rotunda without getting close to it. The triangle you drew for question 2 above was to be used to find the distance of the Rotunda from the midpoint of the path. For this question, you must actually go to the midpoint of the path and there measure the angle between a pencil pointing at the topmost point of the Rotunda and the horizontal. Now, back to the drawing board, draw a triangle having as baseline the line from the midpoint of the path to the middle of the Rotunda, and having the angle you just measured (so the other point in the triangle is the top of the Rotunda). If you drew this triangle accurately, the short side of the triangle is the height of the Rotunda -- but, in writing this up, you should make clear why this is so.

4. The point of the above exercise is to show that the size of an inaccessible object can be determined by observing it from more than one place. Can this trick be used to find the distance and size of the moon?

5. By measuring some shadow, find how high in the sky the sun gets in the middle of the day, that is, what maximum angle does a pencil pointing directly at the sun make with the horizontal? What direction is the sun when this angle is maximum? Would this angle be measurably different in New York?