In the last post, I presented some practical puzzles. That is, practical a few thousand years ago — mostly for fun now.

The first was to determine the cardinal compass points to within a degree or so with no tool more complicated than a stick. Actually things are easier if you have two sticks, or one stick and some small pebbles. The solution is simple. It only takes time.

Place the stick in the ground — somewhere reasonably flat like a sandy beach. Do this in the morning and place a pebble or make a mark where the shadow of the stick ends. Repeat this several times during the day and the line you trace out will run East and West almost exactly, certainly more accurately than a typical compass. From that line, you can easily construct North and South.

The line you traced out is almost East and West, but it slants slightly. If you do the same thing on the next day, and if your stick is tall enough, and your pebbles small enough, you will trace out a similar line offset a bit. This is because or the Earth’s rotation around the sun. You could use multiple measurements, and taller sticks, to improve you accuracy. However, in the process you are well on the way to solving the next puzzle which is to determine the length of a year in terms of days. The ancients in the Northern hemisphere thought of the seasons as the sun moving south in the winter, stopping, and returning in the summer etc. This means that the noon shadow of a stationary stick will grow longer as winter approaches and shorter in summer. At locations south of the northernmost latitude of the sun’s voyage, the noon shadow will actually change sides (opposite for the southern hemisphere). Careful study of the shadow lengths over a long period can allow an accurate measurement of the length of a year expressed in normal days.

Obviously if a stick can serve such a lofty purpose, then it should be more than a simply piece of wood. Think of Egypt and all those obelisks. Other cultures used structures built on top of high isolated hills to get the advantage of longer shadows. There are limits to the accuracy due to the finite size of the sun and the properties of light. This limits the gains to be had by going even higher. Over years, these proto-scientific laboratories became venerated and often the center of cult activity.

To estimate the relative distance of the sun and moon, Greek observers knew the moon is a sphere lit on one side by the sun. They waited until the moon was as nearly half full as they could estimate. That is, the border between light and dark on the moon was a straight line. At that moment, the sun was at 90 degrees from the observer to the moon (moon at the apex of the angle). The observer could quickly measure the angle between the sun and moon (observer at the apex of the angle). This gives two angles of a triangle (try drawing it), which also determines the third angle. The three sides are unknown, and would remain unknown for some time, but simple geometry allows one to compute the ration of the legs on the triangle knowing all the angles. I have seen different numbers reported for what the ancients found, but in all cases, they certainly knew the sun was at least 90 times further away than the moon, and since both appear to be the same size (think eclipses), the sun must be much bigger.

I will discuss answers to the other puzzles in another post.