Sundial Designer

A sundial looks like the simplest clock there is. A stick, a plate, some lines. Most of the cheap ones sold for gardens do not keep time, and the reason is worth understanding before you cut metal or stone.

The part that matters is one straight edge. Dial makers call the shadow-casting edge the style, and the piece that carries it the gnomon. On every dial described here the style is set parallel to the Earth’s axis, so that it points at the celestial pole. That single requirement decides the angle of the gnomon, the direction it must face, the spacing of every line on the plate, and the reason a dial made for one town reads wrong in another.

One edge, aimed at the pole

Seen from the ground, the sun appears to circle the Earth’s axis once a day, at a steady 15 degrees an hour. Make the style a short copy of that axis and the plane holding the style and the sun swings around the style at the same steady rate. The shadow is simply where that plane meets the dial.

The most direct dial uses this without any arithmetic. Put a plate parallel to the equator, stand a rod square to it, and the hour lines are 15 degrees apart all the way round. That is the equatorial dial. It needs no calculation and works at any latitude, provided the rod points at the pole, which means the plate leans from the level by the co-latitude, 90 degrees minus the latitude.

It has a cost. From the autumn equinox to the spring equinox the sun is below the plane of the plate, so the light falls on the underside and the lines have to be marked on both faces. Near the equinoxes the sun runs almost along the plate and no clear shadow forms on either face.

Why the lines on a flat dial crowd toward noon

Lay the plate level instead and the even spacing is lost. The shadow plane still turns 15 degrees an hour around the style, but a level plate cuts that plane at a slant, and the slant stretches some hours more than others. The rule for a horizontal dial is short: the tangent of the line’s angle from the noon line equals the sine of the latitude times the tangent of the hour angle.

Take a place at 40.71 degrees north. One hour from noon the hour angle is 15 degrees and the line sits 9.91 degrees from the noon line. Two hours gives 20.63 degrees, three hours gives 33.11 degrees, four gives 48.48, five gives 67.67. The six o’clock lines are always square to the noon line. So the first hour after noon takes under 10 degrees of the plate and the fifth takes more than 19.

Now see what latitude does. At 50 degrees north the same five lines fall at 11.60, 23.86, 37.45, 53.00 and 70.72 degrees. At the pole the sine is one, the lines return to an even 15 degrees, and the level dial has become an equatorial one. At the equator the sine is zero and the dial fails: the style would have to lie flat along the plate.

This is the plain reason a dial must be made for its latitude. The gnomon of a horizontal dial rises at an angle equal to the latitude, and the lines are spaced for that same number. A dial drawn for 45 degrees and set up at 30 is wrong twice over. It can be rescued by tilting the whole plate until the style points at the pole again, but not by turning it.

Walls, and walls that face the wrong way

A dial on a wall follows the same logic with the roles reversed. On a wall facing due south in the northern hemisphere, the tangent of each line’s angle equals the cosine of the latitude times the tangent of the hour angle, and the style stands out from the wall at the co-latitude. Such a wall can only ever see the sun between six in the morning and six in the evening.

Few walls face due south. A wall turned toward the west sees less of the morning and more of the evening, and its dial is lopsided. The gnomon no longer stands on the noon line: the line under the style, called the sub-style, swings to the afternoon side on a west-facing wall and to the morning side on an east-facing one. The noon line itself still hangs straight down from the dial center.

Three formulas handle this, and all three need the wall’s declination, the angle between the wall’s outward direction and due south. One gives the hour lines, one the sub-style angle, one the height of the style above the wall. At zero declination they collapse into the direct south dial. Measuring the wall’s angle is the hard part, and an error there spoils the whole layout.

Turn the wall all the way to face east or west and the style becomes parallel to it. The lines stop meeting at a point and run parallel instead. That is the polar dial, more often built as a plate tilted up to the latitude under a raised style. Each line sits at the style’s height times the tangent of the hour angle, so the three o’clock line is exactly one style height from the center. A style 2.5 inches high needs a plate 37.98 inches wide to show from 6:30 to 5:30.

South of the equator everything is mirrored. The style points at the south celestial pole, a wall dial faces north, and the hours run the other way round the plate. A dial for 50 degrees south is the mirror image of one for 50 degrees north.

The three reasons a good dial disagrees with your watch

A dial that is built and set perfectly shows local apparent solar time, and a watch does not. The gaps are known and can be allowed for.

The first is longitude. Civil time is the same across a whole zone, but the sun crosses each meridian at its own moment. The offset is 4 minutes for every degree between the dial and the meridian that defines its zone, and it never changes. Omaha, at 96 degrees west in a zone defined by 90, has mean solar time 24 minutes later than its clocks all year. A maker can build this in by shifting every hour angle before drawing the lines, and the dial then shows zone time rather than local time.

The second is the equation of time, the difference between apparent solar time and mean solar time. The Earth’s orbit is not a circle and its axis is tilted, so the sun crosses the sky at slightly different rates through the year. The sum of these small daily differences reaches about 16 minutes. The U.S. Naval Observatory gives the value at the end of July as about 7 minutes behind. No fixed set of straight hour lines can absorb this. It is handled with a table or a curve beside the dial.

The third is daylight saving. In the United States the clocks move forward on the second Sunday of March and back on the first Sunday of November, and other countries keep other rules or none. A dial cannot know them. The usual answer is a second ring of numerals one hour later.

Put the three together for a level dial at 40.71 north, 74.01 west, on a clock set five hours behind Greenwich. The zone meridian is 75 west, so the dial stands 0.99 degrees east of it and its sun runs about 4 minutes early. On 3 November the equation of time is about plus 16 minutes. A dial with plain solar lines there reads close to 12:20 when the watch says noon. With lines drawn for zone time it reads about 12:16, and the rest has to come from the table.

Setting it down, and what it will never do

The lines assume the style points at the pole, so the dial must be aimed at true north or true south. A compass points at magnetic north, and the difference varies from place to place and drifts with time; the model used to predict it is reissued every five years. The Sundial Primer does not recommend the compass as a method at all, and an uncorrected one is worse. A plumb line’s shadow at local apparent noon gives the meridian directly.

A marker on the style, called a nodus, adds a calendar. Its shadow traces a curve across the plate each day: a straight line at the equinoxes and a bowed curve at each solstice, with every other day falling between the two. On a horizontal dial the summer solstice curve is the one nearest the gnomon. The sun’s declination swings between about 23.4 degrees north and south, and that sets where the outer curves fall.

There are limits no layout removes. The sun is not a point. It is about half a degree wide, so the edge of a shadow is a soft band rather than a line. The sun takes two minutes to move its own width across the sky, and the band takes that long to pass a mark. Near the horizon the air bends the light by about 34 arcminutes, lifting the sun above where the geometry puts it. Dial lines are drawn for the geometric sun, so the first and last minutes of the day are the least trustworthy.

What the method can do is still remarkable. A triangle of metal and a few straight lines, all derived from one latitude and one direction, will hold local solar time for as long as the plate stays level and the Earth keeps its tilt. The corrections are not flaws in the dial. They are the difference between the sun’s time and the time people agreed to keep.