The dial needs to be properly oriented so you have to find which way is north. The easiest way I know of is to go at night to the spot you've chosen for your dial and find the north star, (apologies to people living in the southern hemisphere) and, with help if you can get it, lay out stakes or markers in a straight north/south line.
Then in the daylight mark your north/south line with something like a chalk line if you are going to be painting on a paved surface or stakes with string stretched out between them if you are working in the garden. You'll also need mark a second line perpendicular to the first (running east/west--you can use a framing square to lay it out) and crossing it at the center of your dial. It will need to extend the full width you have chosen for your dial. Remember, these lines will be only used to measure from so you want something temporary.
Now you should have two lines crossing at a right angle. These lines are the x and y axes of a cartesian coordinate system. The rest of the markings are found by measuring from these lines.
As you know, you stand on the central scale of dates and tell the time from your shadow. On the equinoxes, around March 21 and September 21, you will stand at the center of your dial, where the two lines cross. On those days the sun rises due east and sets due west and the length of night and day are equal. This means that the markers for 6 am and 6 pm will lie on the east/west line, that is, their y coordinate will be 0. Since the width of the dial is determined by the distance between the two 6:00 markers we can see that their x coordinates will be width/2 and -width/2. In fact, we can see that the hour markers are all symetrically arranged around the north/south line. This saves us a considerable amount of work because we don't have to calculate each half separatly. If you have understood everything up to this point then you can mark 6:00 am at the coordinates (-width/2,0) and 6:00 pm at (width/2,0). Use a tape measure and measure out width/2 from the north/south line both directions along the east/west line.
The rest of the coordinates will be found in the table below--at least they will after you fill them in! Eventually I hope to write a Java applet or something that will fill the table out for you. In the meantime, get out your calculator (or spreadsheet) and fill out the table using your latitude and chosen dial width.
|Noon||0||0||Width/2 * sin(Lat)|
|11 am & 1 pm||Width/2 * .259||Width/2 * .996 * sin(Lat)|
|10 am & 2 pm||Width/2 * .5||Width/2 * .866 * sin(Lat)|
|9 am & 3 pm||Width/2 * .707||Width/2 * .707 * sin(Lat)|
|8 am & 4 pm||Width/2 * .866||Width/2 * .5 * sin(Lat)|
|7 am & 5 pm||Width/2 * .966||Width/2 * .259 * sin(Lat)|
|6 am & 6 pm||Width/2||0||0|
Perhaps you've noticed that the hours before 6:00 am and after 6:00 pm are not listed in the table. This is because the ellipse is also symmetrical around the x axis (the line between the 6:00 marks.) The 5:00 am mark is exactly below (I call the north side 'the top') the 7:00 am mark, just measure down from the axis instead of up. To determine the earliest and latest times the sun is up at your latitude ...
you can consult a chart such as the one in Waugh's Sundials: Their Theory and Construction or make a paper analemmatic dial at my You Can Make a Sundial! website. I'll have to make a chart or find a link or a Java applet for earliest sunrise/latest sunset, won't I?
That is all there is to laying out the hour markers. Calculate the distances in the table above and then carefully measure along the X axis, then carefully place your square at that point and measure the Y distance up or down. Next we turn to the central scale of dates.
Note: In case you are interested we figure x is the sine of the sun's hour angle multiplied by the length of the semi-major axis (or sMa) of the ellipse and y=cos(hour angle)*sin(latitude)*sMa. The sun's hour angle is 0 at local apparent noon and increases 15 degrees per hour--that's 360 degrees per day!
It's time to use your calculator again but this time you won't have to figure out any x values. The date scale is entirely on the north/south line in the center of the dial. Fill out the table below using your latitude.
|Jan. 1||cos(Lat) * Width/2 * -.4272||July 1||cos(Lat) * Width/2 * .4245|
|Feb. 1||cos(Lat) * Width/2 * -.3115||Aug. 1||cos(Lat) * Width/2 * .3249|
|March 1||cos(Lat) * Width/2 * -.1405||Sept. 1||cos(Lat) * Width/2 * .1495|
|April 1||cos(Lat) * Width/2 * .0743||Oct. 1||cos(Lat) * Width/2 * -.0507|
|May 1||cos(Lat) * Width/2 * .2679||Nov. 1||cos(Lat) * Width/2 * -.2493|
|June 1||cos(Lat) * Width/2 * .4040||Dec. 1||cos(Lat) * Width/2 * -.3973|
|June 21||cos(Lat) * Width/2 * .4336||Dec. 21||cos(Lat) * Width/2 * -.4336|
If you've gotten this far you shouldn't have much trouble laying out the date scale. Just measure along the north/south line. Measure up from the east/west line for positive values and down for negative values.
email me with your comments, criticisms, corrections or suggestions.