Article of the month April

Horizontal sundial construction using four circles

The equatorial sundial, with evenly spaced hour lines every 15 degrees, plays a large role in sundial construction.
The figure below shows this for a horizontal sundial.
See also Article of the month September 2005.

CDE, right, is a style triangle for a horizontal dial for 52 degrees latitude. DE is the radius of the equatorial sundial in the construction above left.
First, we should construct the fan of hour lines every fifteen degrees.
This is not difficult, but old literature mentions an interesting construction using only four circles.

Draw a coordinate system with X- and Y-axes, and in the centre a circle of radius R taken from the style triangle as shown in the first figure. The circle intersects the Y-axis in A and B.
Using a pair of compasses in B, draw a circle of radius 2R through A.
From A, draw lines to the intersections of both circles with the X-axis.
Draw also a line through A and parallel to the X-axis.
Our fan now contains lines meeting the Y-axis at angles of 45, 60 and 90 degrees.

The large circle intersects the X-axis in the P and Q.
With the compass in P and Q respectively, two circles of radius 2R through A are added to the figure.
From A, draw lines to the intersections of these circles with the X-axis.
We have now added the angles of 15 and 75 degrees to our fan.

The three large circles intersect in S and T.
The lines AS and AT add the angles of 30 degrees.

Of course, the entire construction could have been done in a single figure. For clarity, the figures show it in three steps.
It is now a simple matter to construct the horizontal sundial, using the principles illustrated in the first figure.