Adjusting for longitude

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 June 1986

A Sundial east of a time meridian reads fast, and a dial west slow. To illustrate the effects to best advantage I have selected (by means of a map and a pin) a location well removed from the Greenwich meridian: Tralee, near the west coast of Ireland, with a latitude of 52 degrees 16 minutes (52.267) north and longitude of 9 degrees 41 minutes (9.683) west.

At four minutes for each degree of longitude, a sundial at Tralee reads nearly 39 minutes slower than one at Greenwich. This difference can be permanently eliminated by marking the positions of the shadows at 39 minutes before each hour with lines which are then labelled as hours. Just how this is done is not quite so straightforward as the few books which briefly mention it would have us believe!

First calculate the hour angles for the wanted hour lines in the normal way, 15 degrees an hour, working backwards and forwards from noon, as shown in columns 1 and 2 of the table on this page. I have included a few subdivision of hours (indented in column 1) to show how easily they can be incorporated. The half of a quarter was usual on sundials made in the days of one-handed clocks.

From the figures in column 2, the shadow angles (the angles of the hour lines) for a conventional dial are calculated and these angles are measured backwards and forwards from the substyle, the line on which the gnomon stands. One becomes accustomed to measuring angles from the noon line because in unmodified horizontal and direct north and south facing dials this line and the substyle coincide. As can be seen in the modified dial for Tralee shown on this page the noon line is not now vertical but the gnomon and substyle remain in their original vertical positions. It would be instructive for you to draw the unmodified dial for comparison. Thinking of shadow angles being measured from the substyle will help you when you come to draw hour lines for declining dials.

Next fill in column 3, which shows the hour angles from which the amended shadow angles will be calculated. Opposite zero HA in column 2, enter the longitude of Tralee expressed in decimals. This is the new HA for noon. With this figure as the starting point, add to it the HA shown in column 2 when moving away from the substyle and subtract from it the HA when moving towards the substyle.

In the finished dial the new noon line must lie in the old am area for a dial west of the meridian and in the pm area for one east. If the noon line is on the wrong side the addition and subtraction were started in the wrong directions.

Working downwards from noon, subtracting 10 from 9.683 to get the HA for 12.40 gives -0.317. The minus indicates that the substyle has been passed and a line is drawn across the table as shown to indicate its position. The rest of column 3 can be completed either by subtracting the value in column 2 from 9.683 and ignoring the minus sign, or by adding progressively to 0.317 the differences between adjacent figures in column 2.

Figures greater than 90 degrees (shown in brackets) are ignored, because their tangents are negative. While these negative numbers give the correct final results when properly applied, the beginner would be wise to avoid them. The am times in column 1 which are thereby eliminated must be included at the pm end, or vice versa.

By applying column 3 figures to the formulae H = cos L X tan HA and V = 1/(2cos L X tan HA), the hour lines are calculated and drawn by the tangent method explained last month. as this is a vertical dial north of the equator, figures above the substyle line in the table are measured clockwise on the drawing, and those below anticlockwise.

The angular values in column 4 are included for reference; they need not be calculated because the dial plate can be drawn using only the values of H and V.

Notice that the substyle, which is vertical, falls at about 12.39. This is correct, because when the style casts its shadow in this position the sun is due south, which is noon local time and 12.39 Greenwich time.

The use of amended hour angles, as in column 3, for longitude adjustment may be applied to any sundial.

1	2	3*	4	5	6
TIME	LOCAL
	HA°	HA°	A°	H	V
...6.00am	....90	....(99.683)
........6.30	....82.5	....(92.183)
...7.00	....75	.....84.683	.......81.4		.......0.076
...8.00	....60	.....69.683	.......58.8		.......0.303
...9.00	....45	.....54.683	.......40.8		.......0.579
..10.00	....30	.....39.683	.......26.9		.......0.985
..11.00	....15	.....24.683	.......15.7	........0.281
..12.00	......0	.......9.683	.........6.0	........0.104
.......12.30	....7.5	.......2.183	.........1.3	........0.023
.......12.35	.....8.75	.......0.933	.........0.6	........0.010
.......12.40	....10	.......0.317	.........0.2	........0.003
.......12.45	....11.25	.......1.567	.........1.0	........0.017
.......12.52h	....13.125	.......3.442	.........2.1	........0.037
....1.00	....15	.......5.317	.........3.3	........0.057
....2.00	....30	.....20.317	.......12.8	........0.227
....3.00	....45	.....35.317	.......23.4	........0.434
....4.00	....60	.....50.317	.......36.4		.......0.678
....5.00	....75	.....65.317	.......53.1		.......0.375
....6.00	....90	.....80.317	.......74.1		.......0.139
.........6.30	....97.5	....87.817	.......86.4		.......0.031
....7.00pm	...105	....(95.317)

*Columns 3-6 with longitude adjustment