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Calculating the hour lines
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 April 1986

Geometrical methods of drawing the hour lines of a sundial are well covered in textbooks and although the theory is straightforward the practice can be tedious, particularly for declining dials where there are masses of muddling construction lines. Trigonometrical methods are simpler, quicker and more accurate.

Now that inexpensive scientific pocket calculators are readily available, even those of you with little mathematical knowledge, for whom this article is written, can easily use spherical trigonometry to calculate the hour lines of a sundial as I will show by considering horizontal and direct south-facing vertical dials. These will establish the principles, and once understood there should be little difficulty in progressing to the more complicated procedures.

Study the instruction book for your own calculator because its method of use may differ slightly from mine which I give here. Note that the calculator will show results to many more places of decimals than is normally required, and that the last figure(s) of a long number may vary in individual instruments. Three keys on the calculator are marked 'sin' 'cos' and 'tan' which stand for sine, cosine and tangent.

Though an advantage, it is not essential to know the meanings of these terms to be able to obtain, for example, the tangent of an angle by entering that angle on the keyboard and pressing the 'tan' key.

The calculator has made the use of logarithms unnecessary, but formulae are frequently given in logarithmic form only and it is necessary to convert these for the calculator. This is simplicity itself: omit 'log' wherever it appears and substitute 'multiply' or 'divide' for 'plus' or 'minus' respectively. Thus:

log tan A = log sin L + log tan HA
becomes
tan A = sin L X tan HA...(1)
and
log cos A = log sin D - log cos L becomes
cos A - sin D/cos L.

I will now describe in details how to use equation (1) to calculate the hour lines for a horizontal dial for, say, latitude 52 degrees. A is the angle between the hour line and the noon line, and L is the latitude of the place where the dial is to be used, both in degrees. Minutes and seconds of arc must be expressed as decimals of a degree. HA is the sun's hour angle, 15 degrees an hour, starting from noon. Thus, the hour angles for 11am and 1pm are both 15, for 10am and 2pm both 30 and so on.

The lines for subdivisions of the hours are calculated in exactly the same way as for the whole hours, bearing in mind that the hour angle for minutes is M x 15/60 where M is the number of minutes required. This is equivalent to four minutes a degree. Thus the hour angle for half an hour is 7.5 and for five minutes 1.25 degrees. From this it follows that the hour angle for, say, 2.35 is 30 + 7.5 + 1.25 = 38.75 degrees.

To calculate the angle which the 1pm line makes with the noon line, first enter 52 on the calculator keyboard and press the key marked 'sin'. The calculator display shows 0.7880107, which is sin 52, the sine of the latitude. This figure should be put in the memory as it will be required for all the other hour line calculations.

Next press the following keys in turn: X, 1, 5 tan, to instruct the calculator to multiply 0.7880107 by the tangent of 15 which is 0.2679491 and which is shown in the display. Press the = key and that instruction is carried out to display 0.2111468, the tangent of the required angle. The key which reveals this angle is marked either 'arc tan' or 'tan-1, and pressing it shows that the angle is 11.922699, or 11.9 degrees corrected to one decimal place.

To find the angle for the 2pm line, first press the 'RM' (recall memory) key, which again enters the value of sin 52 in the display, followed by X, 3, 0, tan, =, arc tan, to obtain 24.463552, or 24.5 degrees which is the required angle correct to one decimal place.

Following the same procedure for the 3pm, 4pm and 5pm lines to obtain 38.2, 53.8 and 71.2 degrees. At 3pm the hour angle is 45 degrees and you may have noticed that its tangent is 1. This is because the 'opposite' and 'adjacent' (referred to last month) are the equal sides of an isosceles triangle. The procedure can therefore be shortened to: RM, arc tan. If this short cut confuses you than go through the full procedure as for the other lines; the result will be the same.

The hour angle for 6pm is 90 degrees and its tangent is infinity, which is outside the limits of the calculator (and incidentally, the computer also). If entered it will prevent further use of the calculator until cancelled. But it does not have to be entered because on any horizontal or direct north- or south-facing vertical sundial the six o'clock hour lines always make a right angle with the noon line.

No further calculations are needed because the morning hour lines and a mirror image of those for the afternoon, the angle for 11am being the same as that for 1pm, 10am the same as 2pm and so on, and the lines before 6am and after 6pm are a continuation of those 12 hours away. On ANY sundial with a flat dial plate, lines twelve hours apart always lie in the same straight line; they must do, because the earth rotates 180 degrees in 12 hours.

The angles for the vertical direct south dial may be calculated form equation (1) in exactly the same way as for the horizontal dial, but using the co-latitude for L, which is the latitude subtracted from 90. Thus the co-latitude of 52 is 90 - 52 = 38 degrees. Alternatively, one may use the formula!

tan A = cos L X tan HA...(2)

where L is the latitude of the dial's location. Cos L is, of course, entered in the memory. You may check for yourself that these two methods of approach give the same result by using the calculator to show that the sine of any latitude is the same as the cosine of the co-latitude.

I use equation (1) for horizontal dials and (2) for vertical dials because it avoids the arithmetic needed to determine the co-latitude and hence eliminates a possible, though unlikely, error.

Having calculated the angles of the hour lines they may be drawn either with the aid of a protractor or by making direct use of the tangent values of the angles. I prefer the latter method because it is easy and accurate, and I will explain it next month.