*Due 2:00p.m., Tuesday 10 September.*

*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 a pyramid 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.*

*These simple ideas are closely related to astronomical methods,
for example finding the distance to the moon and the sun.*

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. To test your understanding of the Pythagoreans' argument that
the square root of 2 is not a ratio of two whole numbers, see
if you can prove the square root of 3 isn't either, that is, there
are no two whole numbers *m*, *n* such that *m*^{2}
= 3*n*^{2}. HINT: the argument is a lot like that
for 2, but this time is *not* in terms of odd numbers and
even numbers. Can you see what the corresponding sets of numbers
are this time? What does *m*^{2} = 3*n*^{2
} tell us about possible factors of *m*?

Read the lectures on Babylon and Early Greek Science I've put on the Web,
and read *Uncommon Sense*, pages 81 to 91.