Holy moly, if you want to see a great post you should read Ethan’s post on the solar analemma. If you photograph the sun in the sky at the same time each day, it won’t be in the same spot. The orbital motion of the earth, your location on the curve of the earth, and the tilt of the earth’s axis causes the sun to appear in slightly different locations each day and the path it traces out is sort of a figure-8 called an analemma.
Is it just a mathematical curiosity? Well, mostly yes other than to astronomers. But this was not always so. If you were an 17th century sailor plying your trade across the Atlantic ocean it’s pretty important to know where the heck you are. GPS wasn’t even a gleam in an engineer’s eye, so if you’re a pirate in the Sargasso…
…you can use this predictable solar pattern to find out where you are. Sort of. We can try a simplified version itself. Grab a dowel rod or other straight stick of known length and stick it in the ground. As the sun moves across the sky, the end of the shadow will move across the ground. The end of the shadow will trace out a vaguely parabolic shape, and at some point the length of the shadow will reach a minimum. This will happen when the sun has reached its highest point in that day’s circle. This will be sometime around noon, though there’s a huge amount of variance due both to the whole analemma thing and to civil corrections like daylight savings time.
Let’s say you find that minimum shadow length and measure it. The angle that describes the altitude of the sun will be:
The angle of the sun at that moment in comparison to your latitude is relatively easy to calculate or look up in a table. It’s called the solar declination, and in degrees it’s 0 at the equinoxes and either 26.433 or -26.433 at the solstices. The solar declination is really a measure of the seasonal tilt of the earth with respect to the sun, and so the angle of the sun in the sky when it’s at its highest is 90 degrees plus the solar declination minus your latitude. A little mind-bending, but you can see how it works by thinking about a picture of the scenario, here from Wikipedia:
Rearrange the relationship we talked about and you’ll see that your latitude = 90 degrees plus the solar declination minus the sun’s altitude (theta). With some accuracy and care, it’s relatively easy to get your latitude to within a fraction of a degree, which is the same as determining your north-south position to a few miles.
Your longitude is much more difficult. If you know the exact time of day you can use a method similar to this one, but before accurate clocks this was out of the question. Navigators at sea thus had a very difficult time finding east-west position with any precision. John Harrison was the man who more or less put this problem to bed, and in fact it’s a distant descendant of his method of extremely accurate timekeeping that GPS uses today.