"In case one desires to locate north and has no compass, a watch may be used. Point the hour hand to the sun. In the morning, halfway between the outside end of the hour hand and noon is due south; in the afternoon, one must reckon halfway backward; for instance, at 8 A.M., point the hour hand to the sun and find the place halfway to noon. It will be at 10, which is due south. At 4 o'clock, point the hour hand at the sun and reckon halfway, and the south will be found at 2 o'clock," wrote Ernest Thompson Seton, artist, writer, and founder of the Woodcraft Indians youth program (which inspired the Boy Scouts) in his book The Birch Bark Rule, which I found online at http://www.inquiry.net/outdoor/skills/seton/watch_compass.htm.
This rule for finding south using a watch is commonly reproduced and can be found on many websites, for example http://www.ehow.com/how_2043433_use-watch-as-compass.html or http://www.youtube.com/watch?v=-UdurZmsLPo. (In honor of Seton, I'll call it Seton's rule.) However, in general, it doesn't give the correct direction of south. If an observer's watch isn't set to local apparent time, Seton's rule obviously doesn't give the right results: see chapter 2 of "Sundials: Their Theory and Construction" by Albert Waugh for an explanation. This isn't the only problem: suppose an observer's latitude is equal to the declination of the sun, so that in the morning the sun rises straight up from the east. Then the Sun's azimuth, measured east of north, takes only the value of 90 degrees; but since the hour hand is constantly changing position, south from Seton's rule must too be constantly changing position (never even getting close to the correct direction). In this brief page, I'm going to explain why Seton's rule fails and calculate the magnitude of its failure at two latitudes.
What exactly does Seton's rule predict? Let's work the rule in reverse: suppose we know the direction of south and the hour (assume local apparent time), and we want to find the azimuth of the sun, measured east of north. We point the line bisecting the angle between the hour hand and the 12 due south. The hour hand now effectively gives the azimuth of the sun. Starting from midnight, each hour the angle between the hour hand and the 12 increases by 30 degrees, or the angle between true south and the hour hand increases by 15 degrees, so each hour the azimuth of the sun increases 15 degrees, southing at noon (12 hours past midnight). The awkward bisection comes from our twelve-hour watches, as Seton rightly points out, "If our timepieces were rational and had a face showing 24 hours, the hour hand pointed to the sun would make 12 o'clock, noon, always south."
Seton's rule predicts that the sun's azimuth increases linearly over the whole day. But this is usually not how the sun behaves. Consider the sun at middle latitudes: when it rises, it moves only a little in azimuth, but a lot in altitude; when it souths, it moves a lot in azimuth, but very little in altitude. This is why hour lines in horizontal or vertical sundials designed for middle latitudes don't subtend equal angles from the dial center. Spherical trigonometry gives an equation for the azimuth of the sun as a function of hour angle, latitude, and solar declination: azimuth, z = arctan[sin(hour angle)/(sin(latitude)cos(hour angle) - cos(latitude)tan(solar declination))]. Intuitively, the difference between real solar azimuth and azimuth as predicted by Seton's rule is greatest when the sun rises highest in the sky, since the sun must waste a lot of time rising and so vary in azimuth extremely quickly (like the latitude = declination example I gave where the azimuth changes between the values 90 and 270 degrees instantly).
I calculated the difference between real solar azimuth and azimuth as predicted by Seton's rule for a latitude of 40 degrees north and a solar declination of -23.5 degrees (northern hemisphere summer solstice). You should try this for yourself: the difference is very large, amounting to a maximum of 36.4 degrees and generally around 15-25 degrees. This means that Seton's rule may give dangerously inaccurate results. However, for a latitude of 90 degrees, the equation for solar azimuth reduces to arctan(tan(hour angle) or just hour angle; therefore, the sun appears to move only in azimuth and not at all in altitude, assuming constant declination. Here the sun's azimuth really does increase linearly over the whole day.
So for an observer at the North Pole, Seton's rule works perfectly. But the rest of us had better be careful.