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Latitude and the sundial
This is one of a series of articles written for "Clocks" magazine by the late Noel Ta'bois, and reproduced with permission here as a memorial to him.
This article originally appeared in Clocks in November 1985
Figure 1 shows the angulations of horizontal sundials on the earth at varying latitudes. N is at the north pole, B, C, and D are at latitudes 52, 38, and 16 degrees north, suitable for London, Washington DC, and Rangoon respectively. As the speed of rotation of the earth is virtually constant the gnomon of N which is a vertical rod in line with the earth's axis casts 24 shadows spaced 15 degrees apart at hourly intervals, as shown in plan view at N in Figure 2. The 24 radial lines of the circles B, C, and D are a projection of the equally spaced lines of N on to the respective planes, whose angles relative to N increase as they approach the equator E.
One can study the effect of projection on to variously angled planes by displaying a transparency of, say, a spoked wheel (or a clock face) on to a large piece of white card and then tilting the card to varying degrees. When the card is held at right angles to the beam from the projector the spokes (or the chapters) will be equally spaced as in N. As the card is tilted away from the vertical the spokes at six and 12 o'clock appear closer while those at three and nine become more widely spaced. Try it. The effect is clearly illustrated in B, C, and D in figure 2. The extent of the shift of the lines can be judged by comparing them with the dotted lines drawn at 45 degrees on both sides of each noon line.
Inside each circle is a smaller circle which shows a plan of the dial plate for the horizontal sundial at the appropriate latitude. The half hours are marked, and the gnomons have been 'rabatted' to the plane of the dial plates. To rabat is to revolve a plane about its intersection with another plane until the two coincide. This is a useful procedure much employed in sundial drawings to reduce three dimensions to two. Here the effect is to hinge the gnomons along their bases and fold them down on to the dial plates. B, C, and D also illustrate how the shapes of the gnomons change with latitude. Because the styles - the shadow casting edges of the gnomons - are all parallel to the earth's axis each makes an angle with the local horizontal plane equal to the latitude of that location, as shown in figure 1.
At the equator all the hour lines will have converged to lie along the noon line, while the style will lie along the noon line, while the style will lie on the dial plate. The result is a useless sundial! To make it functional, the style has to be lifted away from the dial late as illustrated by the dotted line at E in figure 1, producing what is known as a polar dial.
In northern mid-summer N will receive the sun for 24 hours a day, while at the equator, where night and day are always equal in length, E can never have more than 12 hours' sunshine. At intermediate latitudes the maximum number of hours decreases as one proceeds from the pole to the equator. There is no point in marking a sundial to show more than the maximum possible hours of sunshine. In B of figure 2 the hours run from 4am to 8pm, in C from 4.30am to 6.30pm.
Thus latitude affects the design of a dial in three ways: the spacing of the hour lines; the shape of the gnomon; and the time period covered. I have discussed only horizontal dials in the northern hemisphere but the principles are the same anywhere in the world for any dial working on the sun's hour angle: a propjection of N or S on to the dial plate, and the style set parallel to the earth's axis. For the southern hemisphere, rotate dials N, B, C and D of figure 1 through 180 degrees so that the styles point to the south celestial pole, number the hour lines anticlockwise, and they become S, H, G, and F, suitable for the South Pole, the Falkland Islands, Melbourne in Australia, and Brasilia respectively.
For convenience figure 1 shows all the dials on the same line of longitude. This does not matter because a sundial can be moved along a line of latitude without affecting its design. But this movement will alter the time shown by the dial by four minutes for each degree of longitude as I explained last month. Latitude interests the diallist when he is designing a sundial, longitude when reading the time by one.
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