The Gnomon can be used to study the path of the sun
The gnomon is a scientific instrument that can be used for finding the declination of the sun through the year, among other things. It is one of the first scientific instruments ever made, originating with the Chaldean astronomers of Babylon and from there brought to the Greek world. It is still very instructive, an excellent example of random and systematic errors of measurement, vividly demonstrating astronomical principles. The Greek gnwmwn, originally meaning one who knows, or judge, was also used for the index of a sundial, as well as for the carpenter's square. It is easy to make a good portable gnomon, and easy to use it.
To make a gnomon, you will need a straight piece of wood for the base (I used a redwood 1 x 3 by 22 in. long), a length of metal rod (mine was 1/8 in. diameter brass rod about 10 in. long), a piece of millimeter paper, a small spirit level, and some household cement. The rod should be straight. Finish the wood as nicely as you want, then drill a 1/8 in. hole about 1 in. from one end on the center line. You must use a drill press so that the hole is perpendicular, to hold the gnomon at right angles to the base. Great accuracy in perpendicularity is not required. Saw off the piece of rod with a hacksaw, and finish the ends with a file. Then cement the rod in the hole, getting it as closely as possible to a right angle from the board. Prepare a strip of millimeter paper about 4 cm wide long enough to go from a couple of inches from the rod to near the end of the board. This may have to be in two pieces. With a pencil and ruler, draw a line down the middle of the strip. Cement the strip to the board so that the extended line passes through the base of the rod, and that some centimeter line is exactly a certain number of centimeters from the center of the rod. When this is done, mark every 50 or 100 mm on the line to make readings easy. Make sure that the base and the scale are long enough to measure the lowest elevation of the sun at noon (about 66° minus your latitude). Measure the height of the top of the rod from the board, and note the value in mm on the paper strip. The rod is the actual gnomon, while the base used for making a measurement. A small pocket spirit level can be obtained at low cost, and should be cemented to the board alongside the scale. All this can be done in fifteen or twenty minutes!
To use the gnomon, set it in the sun and rotate the base so that the shadow of the rod falls on the center line. The scale for reading the length of the shadow must be accurately horizontal, so use the spirit level to make sure of this, shimming the base as necessary with pieces of cardboard. Now, read the length of the shadow on the scale to the nearest millimeter. Note that the end of the shadow is not sharp - why not? - so measure to the center of the penumbra. The altitude of the sun is the arctangent of the ratio of the length of the gnomon to the length of the shadow, easily computed on a scientific pocket calculator. My gnomon can measure altitudes from 28° to 78° , or just about enough to measure the noon elevation of the sun for any date at my latitude of 40° . I should have made the board a little longer, since the minimum solar altitude at noon is 26° . If the board is set accurately north-south on the meridian, local noon occurs when the shadow is on the center line. This is the time to measure the altitude of the sun to determine its declination. When daylight savings time is in effect, noon is really at 1 pm. The sun is not exactly on the meridian at standard time noon, but the difference is not great.
The declination of the sun d (its angle north or south of the celestial equator) is the sum of the solar altitude a at noon and your latitude f , less 90° , as you can easily deduce from the Figure. Z is the zenith, P the pole, E the equator, and N,S are the north and south points. On a piece of graph paper, plot the declination along the y-axis, and the day of the year along the x-axis. Number the days beginning with March 21, which is near the vernal equinox, and allow for declinations in the range ± 30° . Plot the declination for every day that is sunny at noon - in Denver this can be done with few gaps - and missing a few days will not matter in any case. After you have done this for a full year, you will appreciate the sun's motion much more clearly.
Today, 5 September 1998, at apparent noon, the length of the shadow was 158 mm, giving an altitude of 56.7° , and a solar declination of 56.7° + 39.7° - 90° = +6.6° . The Astronomical Almanac 1998, page C14, gives the declination of the sun at this time as +6.7° , so my result is very good. I am pretty sure I can read the shadow to about 1 mm more or less. This is an estimate of a random error of measurement. Taking shadow lengths of 157 and 159 mm, I find that the corresponding difference in elevation is only 0.4° , so the accuracy of the result is consistent with this. If you are fortunate, your first attempt will be a degree or so off, and you can have the satisfaction of tracking down the reason for the discrepancy. I had to do this before I obtained a result free of systematic error. Don't give up until you can get results accurate to within half a degree, which this instrument is capable of.
A systematic error is a bias that is not a result of the ordinary uncertainty in reading a scale, or of other random fluctuations. It cannot be removed by averaging repeated measurements. The systematic errors of the gnomon are mainly constant ones. For example, an inaccurate level, wrong height of the gnomon, and incorrect latitude all will cause the measured value to deviate from the correct value. If you place the gnomon in the same place for all measurements, without leveling, there will be a constant systematic error in elevation. Each measurement can be corrected for this error once it has been determined, saving the bother of leveling each time. The level and latitude should be known to 0.1° (1/32" in 18" for level, and within 6 miles for latitude). Your latitude can be obtained from a USGS topographic map.
If the gnomon points to the celestial pole instead of being vertical, the shadow moves uniformly with the hour angle of the sun, as in the equatorial dial shown in the picture, which is at Cranmer Park in Denver, Colorado. In the winter, the lower face of the dial must be used instead. The dial shows apparent solar time, of course, and must be adjusted by the equation of time and for the longitude to show standard time.
Use the gnomon to measure the height of a tree. This can be done at any time, not just at noon, and does not require any trigonometry. If you know the declination of the sun at the particular time, the gnomon can be used to determine the latitude. This information is given very accurately in the Astronomical Almanac for the year, but an approximate value can be determined from the date in any year, for example from your annual plot. The gnomon can also be used to determine the time, when you know the sun's declination and the meridian, and the direction of the meridian, when you know the time. This does require some trigonometry, spherical trigonometry in fact. The gnomon can easily be used to determine the approximate date - figure out how to do this. This operation has been carried out to determine the date of the vernal equinox since very ancient times. What are some other ways of finding the vernal equinox? The full moon, as well as the sun, casts a measurable shadow. Mark the end of the shadow in the dark, then read the length of the shadow with a light.
I used my gnomon to measure and plot the declination of the sun throughout a year. This gave a vivid appreciation for the movement of the sun. It was very pleasing to see the plot develop, and showed how to find the latitude to a degree or better with such a simple instrument. With a permanent gnomon, properly fixed, the instant of apparent noon could be obtained and compared with the standard time, thereby finding the equation of time and the variations in the sun's motion through the year.
Composed by J. B. Calvert
Created 3 September 1999
Last revised 25 November 2000