Capricorn: Dec.22-Jan.29 The Sundial Primer
created by Carl Sabanski
Capricorn: Dec.22-Jan.29

The Sundial Primer Index

Vertical Declining Sundial

Declining Sundial: a vertical dial which does not face any of the cardinal points of the compass. The sub-style will be displaced from the noon line, although the latter will still lie vertically below the dial centre.

Vertical Sundial: any dial in which the dial plate is vertical.

Declination (of a wall) (sometimes called the declining angle or the deviation, to avoid confusion with the sun's declination) (d, DEC): the angle, measured in a horizontal plane, that a wall's perpendicular makes with due south (i.e. a wall facing S has d=0). Walls declining westward have positive declinations, those eastward, negative.

Latitude: is the angular position of a place north or south of the equator. Positive values in the Northern hemisphere, negative in the Southern hemisphere.

Style Height / Style Angle (SH): of a polar style is the angle that the style makes with the sub-style line.

Sub-Style (line): the line lying in the dial plane which is perpendicularly below (or behind for a vertical dial) the style.

Sub-Style Angle (SD): the angle that the sub-style makes with the noon line, measured positively clockwise (towards the p.m. hours for a south-facing vertical dial).

Hour Line: the line on a dial plate indicating the shadow position at a particular time (includes fractional as well as whole hours).

Hour Angle (h, HA): the angle corresponding to the sun's position around its daily (apparent) orbit. Measured westward from local noon, it increases at a rate of 15 per hour. Thus 3 pm (Local Apparent Time) is 45 and 9 am is -45

Hour Line Angle (X, HLA): the angle that an hour line on a dial plate makes with the noon line. For a vertical dial, the angle increases counter-clockwise.

Horizontal Line: a line drawn on a dial plate at its intersection with a plane which passes through the nodus and is parallel to the plane of the horizon.

As the vertical declining sundial is designed for a wall that does not face directly north, south, east or west, the first task is to determine the wall's declination (d). Please read the "Declination of a Wall" page to find out how to determine the declination of your wall. As illustrated in Figure 1 there are four types of vertical declining dials:

  • SWD - southwest decliners

  • SED - southeast decliners

  • NWD - northwest decliners

  • NED - northeast decliners


Figure 1: Types of Vertical Declining Sundials

Figure 1: Types of Vertical Declining Sundials (CAD)

The vertical declining sundial must be designed for the particular latitude () where it is to be used. The style height and sub-style angle must be determined.

The style height (SH) of the gnomon is calculated as follows:

SH = arcsin (cos d * cos )

The sub-style angle is calculated as follows:

SD = arctan (sin d / tan )

On east decliners the sub-style falls before noon and on west decliners after noon.

The hour line angles (X, HLA) can be calculated as follows:

X = arctan { [cos * tan (h)] / [cos d + sin d * sin * tan (h)] }

where h is the hour angle, in degrees, given by:

h = (T24 - 12) * 15

and T24 is the time in 24-hour clock notation (hours after midnight) in decimal hours.

Click here to download a spreadsheet that will perform these calculations for you.

Figure 2 illustrates the layout for a southwest declining dial. It is located at a latitude of 44.23 N and longitude of 9.14 E. The declination of the wall is 42.2 W. The figure shows all the components mentioned above.

Figure 2: Verical Declining Sundial

Figure 2: Vertical Declining Sundial (ZW200/CAD)

The sun can only shine on a vertical direct south sundial in the Northern Hemisphere and a vertical direct north sundial in the Southern Hemisphere between 6 A.M. and 6 P.M. If the dial is rotated to the east the dial will see the sun earlier in the morning but the sun will move off the dial earlier in the evening. If the dial is rotated to the west the opposite is true and it will not see the sun until later in the morning but it will stay on the dial later in the evening. Only a maximum of 12 hours of sun can be seen by these  sundials. Throughout most of the year the sun will illuminate the sundial for less than 12 hours. The twelve hours that can be included on the sundial are those that fall below the horizontal line shown in Figure 2. 

Visit the "Sun Charts" page for some interesting information. Note the sun chart that is provided for latitude 50 N. The only days that the sun will shine on a sundial for a period of 12 hours is the equinoxes. And this is only for a vertical direct south sundial. On all other days the sun will illuminate the dial for less than 12 hours. Create a sun chart for your latitude and you can find out the maximum period that your sundial will be illuminated by the sun. This will help to determine the number of hours to include on the dial plate.

It is interesting to note that once the calculations have been completed for one vertical declining dial the information is available to design the other three dials. For example, having designed the dial in Figure 2 for a vertical south dial located at a latitude of 44.23 N and declining 42.2 W the information is available for:

  • A vertical south dial declining 42.2 east.

  • A vertical north dial declining 42.2 west.

  • A vertical north dial declining 42.2 east.

This can be seen by comparing the two sundials in Figures 3 and 4.

Figure 3: Vertical Declining Southeast Sundial

Figure 3: Vertical Declining Southeast Sundial (SHADOWS)

Figure 4: Vertical Declining Southwest Sundial

Figure 4: Vertical Declining Southwest Sundial (SHADOWS)

Vertical Declining Sundial

For an image complete with shadow click here.