procedure

This page describes the construction of an equatorial, a horizontal, and a vertical sundial
The description is limited to sundials in the northern hemisphere.

Skipping a large part of history, we will begin in the modern sundial era.
For our purposes, that is when in Western Europe the pole style shadow casting device came into general use.
While we do not have an exact date, such an old pole style sundial, on Jacobi Church, Utrecht, is dated 1463.

What is a pole style? It is a facsimile of the Earth's axis, or celestial axis, around which we see the Sun revolve daily.
We will realize this axis by placing a rod parallel to the Earth's or celestial axis.
We call this rod a pole style.

The pole style is always directed north-south.
It intersects the horizontal plane at an angle equal to the latitude of its location.
In Utrecht, this would be 52 degrees; in Cape Town, -35.
The figure above shows this for both locations.

We call the angle between the pole style and the dial plane the style height.
The line in the dial plane right under it we call the sub-style.

Equatorial sundial

We now place a sundial plane perpendicularly to the pole style. This plane will be parallel to the plane of the equator, and we therefore call the resulting sundial an equatorial sundial.
In fact, there are two sundials. The sundial plane has a top as well as a bottom face, and each is a sundial face.

In our summer, the sun runs its daily orbit to the north of the equatorial plane, and therefore shines on this equatorial plane from sunrise until sunset.
The pole style will produce on this plane a shadow line that begins from the style foot, and neatly turns with the sun.
The sun advances fifteen degrees through its orbit every hour (360/24=15), and so after every hour the shadow line will have turned a further fifteen degrees.
It is therefore easy to draw hour lines on this sundial: there is one hour line every fifteen degrees.

(Greek letter phi = latitude)

Which numerals shall we mark them with? We will establish a convention:
When the sun reaches the highest point of its daily arc - in our case in the south - and we observe true noon, we call that XII (twelve) o'clock APPARANT SOLAR TIME.
We mark the hour line corresponding to the shadow at that time with XII. The next hour lines we mark I, II, et cetera.
Roman numerals are preferred for apparent solar time, but not obligatory.
In The Netherlands, for a latitude of about 52 degrees, we should draw hour lines from IV (4 a.m.) to VIII (8 p.m.).

On the other side we also apply hour lines every fifteen degrees, taking care to have them run counter-clockwise this time.
The hours from VI (a.m.) to VI (p.m.) should be drawn here.

The construction of this simple equatorial sundial is the basis for the design of the other dials.
In the following constructions this is the pattern with the white background color.

Horizontal sundial

Draw a right-angled triangle ABC as shown in the figure below.
The angle C is equal to the latitude for the intended location of the sundial.
As an example, we will draw a sundial for Utrecht, latitude 52 degrees north.

Beside the triangle, draw a larger rectangle with a diameter DAC, using measure AC from the auxiliary figure.
Circle AB to the extension of AC to find AD, using that for measure AD in the rectangle.
Draw a semicircle around D, and draw lines to divide it into twelve sectors of fifteen degrees each.
This is in fact the equatorial sundial used as an auxiliary figure for the construction.
Connect the intersections of these lines and the rectangle side through A, with point C, so creating the hour lines on the horizontal plane.
To find the early hours before VI (a.m.) and the late hours after VI (p.m.), we extend the hour lines for VII and VIII and for IV and V respectively.
Add the proper numerals and our horizontal sundial is finished.

Longitude correction

The horizontal sundial just constructed reads apparent solar time.
However, that is not the time that our wrist watch or clock reads.
For The Netherlands and surrounding countries that would be Central European Time, which is based on the apparent solar time for a longitude of 15 degrees east.
When it is XII o'clock apparent solar time there, we call it 12 o'clock CET.
For Utrecht, with longitude 5 degrees east, it takes another forty minutes before it is XII o'clock.
Every degree more to the west from Utrecht adds another four minutes.
To the east of Utrecht, every degree subtracts four minutes.
Our sundial can, however, show CET if we shift the hour lines by forty minutes. This is called longitude correction.
For daylight saving time, CEST, we simply add one hour to the reading, or we can add numerals showing one hour later.
In the vertical direct south dial about to be constructed, we will add this longitude correction.

Vertical direct south sundial

This sundial is to go on a wall facing exactly south.
A moment's consideration will show us that the angle between the pole style and the wall is 90 - 52 = 38 degrees.
Again, draw a right-angled triangle ABC, but now with an angle of 38 degrees at C.
Again circle AB, and transfer the necessary measures to the rectangle beside it.
Draw the auxiliary equatorial sundial, but also rotate it over 40/4 = 10 degrees to the left to apply the longitude correction.
Draw the hour lines from C and add the correct numerals.
Notice that now the hour lines run counter-clockwise, and that you need less.
A vertical direct south dial can never show more than twelve hours.

Equation of Time

Does this sundial read the time such as we use in our daily life?
That is, apart from the hour to be added during daylight saving time?
No.
We should apply one more correction: the equation of time.
We may think that the sun always takes exactly 24 hours to complete one revolution about us.
In reality, things are different.
Sometimes the sun revolves more quickly, sometimes more slowly.
This is due to the elliptical orbit of the earth about the sun, and the slanted attitude of the earth's axis.
The difference for any one day is not large, about thirty seconds at most, but the cumulative effect is appreciable.
For completeness' sake, we present a graph showing the value for the equation of time over the course of the year.
Before you set your wristwatch to the sundial, the equation of time should be subtracted from the dial reading.

graph showing equation of time and solar declination.
red: equation of Ttime
blue: solar declination

Computerprogram

The computer program ZW2000, which can be downloaded from this site, will calculate and draw not only the sundials just described, but also sundials for any other - arbitrarily oriented - plane.
It can also add other kind of lines, but a description of those is outside the scope of this text.