■ How can an analog watch tell compass directions?

It's spring equinox today, so we celebrate the onset of spring tackling a simple trick of nature. If you spent time in scouts, you will know that on a fine sunny day, while lost in a forest you can use your hand watch as a compass to find North (or South) and orientate yourself. The idea is you point the hour hand into the sun's direction, and bisect (cut in half) the angle between said hand and your watch dial's 12 o'clock marker, and that's the south direction if you are in northern hemisphere (north for Australians). See Figure 1 below.

Figure 1. Heuristic turns watch into compass

Figure 2. Trivial proof of bisection heuristic for sunrise

Many websites talk of this neat scout trick, but I haven't found any explanation for the bisection rule. We don't like things unexplained round here, and as it turns out the "proof" of this rule of thumb is pleasingly simple and easy to grasp. For example at sunrise (see figure 2), and assuming for simplicity that the day lasts 12 hours (from 6am to 6pm, it is equinox time), pointing the hour hand to the sun aligns the watch east/west, so the angle from 6 to 12 o'clock is 180°, and half the angle is 180/2 = 90 degrees, straight due south. Likewise it is easy to show that at 12 noon, the angle is 0° so the hour hand points directly south.

The semi-circle in figure 2 represents the path of sun in the heavens, from dawn to dusk; the watch is drawn sideways

Why is the sun south of us? In northern hemisphere, above the tropic of cancer, the sun always stays below the zenith, or straight above our heads, because of Earth's inclined axis of daily rotation. This is especially so during winter, when the sun stays low all day, even below the tropic of cancer. But what about the angle bisection rule? Are we sure that it holds for all times, not just for sunrise and noon?

The elder scouts are trully wise, and here's why. Half a turn round a watch, it is 6 hours, whereas the "heavenly watch" that represents the sun's path is 12 hours, so the hour hand travels twice as fast as the sun, that's why the angle must be cut in half. See the little animation below.

Say the time is 8 in the morning. If the watch is aligned east-west as in figure 2 further up, the hour hand is pointing at 10 am "sun time". If we turn our clock to point into the sun, then half the angle till 12 o'clock is due south. QED.

Now we understand why the bisection rule works, we also understand the limitations of the procedure (other than the obvious necessity of a sunny day and an old school non-digital watch, showing the correct time)

Ancient people knew when 12 noon was exactly: when the sun is at its highest point and shadows are shortest. With our modern watches and time zones, an indicated 12 o'clock isn't necessarily noon (for example in Cyprus I measured noon to be at 11.30 am). So the technique won't give you the true north direction.
If we are in summer time (daylight saving), 12 noon is really showing as 1pm, so we need to calculate watch angles to 1pm, not 12.
If you live close to the equator, within the tropic zone, sun can be north or south, so you need your calendar to use the bisection rule — which could make escapes from the jungle a little trickier <g>