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Calculating the Height of the Sun from the Length of a Shadow
The shadow boxes on the back of the astrolabe are nothing more than simple trigonometric calculators. Notice that the one on the left is labeled with the number 12 in the corner while the one on the right is labeled with the number 10. The 10 shadow box produces nice round numbers for the purpose of computation (e.g. it's relatively easy to divide by ten.) The twelve shadow box is useful for two reasons: 1.) twelve has a large number of integer factors that simplifies calculations, and 2.) twelve is twice the height of a human measured with his or her own feet. As it turns out, most people are approximately six feet tall when they pace off their own height using their feet (try it!) Smaller people have smaller feet, so the ratio of height to foot size is usually pretty close to 6 to 1.
This suggests an easy way to determine the sun's height. Simply observe where the shadow of the top of your head falls. Now pace off the distance to your shadow. The inverse tangent of your height divided by the distance to the tip of your shadow is the height of the sun. Fortunately, the shadow boxes make the computation of an arctangent relatively easy.
Example: If your shadow is four feet long measured with your own feet, what is its height? The left shadow box is actually designed for a 12 foot gnomon, so you need to multiply the length of your shadow by two. Now place the edge of the alilade against eight on the left shadow box. The alilade points to ~56 degrees on the altitude scale, which is the height of the sun.
But wait, what if your shadow is longer than you are tall? There are no markings on the left shadow box for shadows longer than 12 feet. If this is the case (a common occurrence in winter at northern latitudes), then we need to turn the problem around. We need to ask the question: How tall would I have to be to cast a 12 foot shadow? By a simple ratio of triangles, one can solve this problem. The only thing left to do is to find your reduced height on the vertical part of the left shadow box.
Example: Your shadow is twelve feet long measured with your own feet. What is the height of the sun? This example is particularly easy. The answer to the question, How tall would I have to be to cast a 12 foot shadow? is just six, since that is indeed the length of the shadow my six foot body is casting. By aligning the alilade with the vertical scale of the left shadow box, one can see that it point to ~26 degrees. This is the height of the sun.
Comments: Most cases are not so straightforward. If I had cast an eight foot shadow, for example, I would have had to solve the ratio 6/8 = x/12 which gives me x = 9. This equation derives from the similarity of triangles and can be stated in plain English as "a six foot gnomon casts an eight foot shadow, so an x foot gnomon would cast a twelve foot shadow." In this case, I would set the alilade against 9 on the vertical part of the left shadow box. If you prefer just to memorize a general formula, remember the following rules:
If your shadow is shorter than six feet, multiply it by two and find it on the horizontal part of the shadow box.
If your shadow is longer than six feet, find 72 divided by the length of your shadow on the vertical part of the shadow box.
If you use a gnomon other than your own body, you might want to consider using the right shadow box with ten tick marks per side. It's often easier to scale your gnomon/shadow to ten rather than to twelve. The right hand shadow box can be used in exactly the same way as the left hand box.