The following instructions for laying out the hour lines of a horizontal sundial use the centuries old method of concentric circles. The problem with the original method was that it made no provision for the physical width of the gnomon, thereby causing the shadow of the style to be displaced west of the morning hour lines and east of the afternoon hour lines. The amount of dispacement was equal to half the width of the gnomon. Since a gnomon does have width, its inclined surface has two styles. The eastern edge is one style and the western edge is the other. These two styles come into play in the following order as the day progresses:
Time frame Style Shadow falls in the Before 6am LAT Eastern Southwest quadrant of the dial plate 6am - noon LAT Western Northwest quadrant of the dial plate Noon - 6pm LAT Eastern Northeast quadrant of the dial plate After 6pm LAT Western Southeast quadrant of the dial plate Note: LAT = Local Apparent Time
The concentric circles method given below has been modified to yield hour line placement such that each hour line receives the shadow from the correct style at the correct time of day. I figured out the modification myself and was rather pleased to have done so, but cannot claim originality as I have since discovered the modification in an old sundial book.
To correctly lay out the hour lines of a sundial, the latitude and longitude of the dial's location must be known. The instructions that follow are for a dial located at 38 deg north latitude, 81 deg west longitude. When designing your dial, just substitute the coordinates for your location.
Figure 1: Begin by drawing the right triangle ABC such that angle BAC is equal to the latitude (38 deg in this case). The size of the triangle will determine the size of the concentric circles later on. The larger the circles, the more accurately you will be able to lay your hour lines. I would suggest a triangle size such that side AB, the hypotenuse, is about four to six inces long. This will yield a construction diagram that is reasonably accurate without being unduly large.
Figure 2: The length of the following three lines should be an inch or so longer than twice the length of line AB in the triangle in figure 1.
Draw vertical lines AB and CD parallel to each other and separated by the width of the gnomon. Line AB is the substyle of the western style and line CD is the substyle of the eastern style. Points A and C are at the north end while B and D are at the south end.
Draw EF, the east-west transverse line, across and normal to (at right angles to) AB and CD, intersecting AB and CD at G and H. Points G and H are where the western and eastern styles intersect the dial plate.
Figure 3: Draw the following eight arcs using the radii and center given:
Using AB from figure 1 as the radius and G as the center: In quadrant II from AG to EG In quadrant IV from BG to FG Using AB from figure 1 as the radius and H as the center: In quadrant I from FH to CH In quadrant III from EH to DH Using BC from figure 1 as the radius and G as the center: In quadrant II from AG to EG In quadrant IV from BG to FG Using BC from figure 1 as the radius and H as the center: In quadrant I from FH to CH In quadrant III from EH to DH
Figure 4: Before drawing in the guide lines in this figure, you must first determine the angle of each guide line.
Since the sun travels across 15 degrees of arc each hour, the guide lines, within each quadrant, will have 15 degrees spacing between them but their positions must first be adjusted for longitude.
Each time zone around the world takes its time from a standard meridian within the zone (thus the term "Standard Time"). The standard meridians are spaced 15 degrees apart beginning at longitude 0 degrees in Greenwich England and proceeding both east and west from that point. For example, the standard meridian in the United States for Eastern Standard Time is longitude 75 degrees West. The standard meridian for Malta is longitude 15 degrees east.
The guide lines need to be adjusted by the angular difference between the standard meridian for the zone in which the dial is located and the local meridian of the dial. In this example the dial is located at longitude 81 degrees West. This is 6 degrees west of the standard meridian for Eastern Standard Time.
Since noon Local Apparent Time (LAT) is defined as the point when the sun transits the local meridian, the angle of the quide line, before correcting for longitude, would be 0 degrees. Each hour before and after noon makes an angle with the local meridian that is a multiple of 15 degrees. This is shown in the following table:
LAT Guide line angle 4 a.m. -120 degrees 5 a.m. -105 degrees 6 a.m. -90 degrees 7 a.m. -75 degrees 8 a.m. -60 degrees 9 a.m. -45 degrees 10 a.m. -30 degrees 11 a.m. -15 degrees Noon 0 degrees 1 p.m. +15 degrees 2 p.m. +30 degrees 3 p.m. +45 degrees 4 p.m. +60 degrees 5 p.m. +75 degrees 6 p.m. +90 degrees 7 p.m. +105 degrees 8 p.m. +120 degrees
The angles are shown with a minus sign for the morning hour lines and a plus sign for the afternoon hour lines. If the dial is west of the standard meridian then subtract the difference in longitude (6 degrees in this case) from the angles in the table above. Conversely, if the dial is east of the standard meridian then add the difference in longitude to the angles in the table above. After correction, we have the following guide line angles:
LAT Guide line angle Guide line angle (before correction) (after correction) 4 a.m. -120 degrees -126 degrees 5 a.m. -105 degrees -111 degrees 6 a.m. -90 degrees -96 degrees 7 a.m. -75 degrees -81 degrees 8 a.m. -60 degrees -66 degrees 9 a.m. -45 degrees -51 degrees 10 a.m. -30 degrees -36 degrees 11 a.m. -15 degrees -21 degrees Noon 0 degrees -6 degrees 1 p.m. +15 degrees +9 degrees 2 p.m. +30 degrees +24 degrees 3 p.m. +45 degrees +39 degrees 4 p.m. +60 degrees +54 degrees 5 p.m. +75 degrees +69 degrees 6 p.m. +90 degrees +84 degrees 7 p.m. +105 degrees +99 degrees 8 p.m. +120 degrees +114 degrees
The guide lines will be drawn in four groups: one group per quadrant using the pivot point given, Each guide line should be long enough to cut both the inner and outer arc of the quadrant. (Actually, only two short line segments need be drawn where the guide line intersects the two arcs.)
In quadrant I using pivot point H, draw those guide lines having positive values between 0 and +90 degrees. Measure the angles clockwise from line CH.
In quadrant III using pivot point H, draw those guide lines having negative values greater than -90 degrees. Measure the angles counter-clockwise from line CH.
In quadrant II using pivot point G, draw those guide lines having negative values between 0 and -90 degrees. Measure the angles counter-clockwise from line AG.
In quadrant IV using pivot point G, draw those guide lines having positive values greater than +90 degrees. Measure the angles clockwise from line AG.
Figure 5: This figure is a close-up of the pivot points G and H to illustrate how the four groups of guide lines radiate into their respective quadrants.
Figure 6: Each of the guide lines intersects both the inner and outer arcs in its quadrant. From the intersection point on the inner arc draw a vertical line. From the intersection point on the outer arc draw a horizontal line. These lines will intersect at some point between the two arcs. The intersection points will be above the guide lines in the upper two quadrants and below the guide lines in the lower two quadrants.
Figure 7: This figure shows the intersection points more clearly be having the bulk of the guide lines removed, leaving only the intersection points illustrated.
Figure 8: It is now time to draw the hour lines that will appear on the finished dial.
From pivot point G, draw lines through the intersection points in quadrants II and IV.
From pivot point H, draw lines through the intersection points in quadrants I and III.
Figure 9: This figure shows only the hour lines and the position of the gnomon. All construction lines have been removed.
Figure 10: This figure shows a simple example of how the hour lines have been applied to a finished dial.
When trasfering the hour lines to the dial plate, great care must be taken to ensure the positional relationship between the hour lines and the two substyles is maintained.
The gnomon may be constructd in any shape or design you desire; from a simple triangle to one that is pierced and intricately carved. The gnomon must however meet the following requirements:
1. The two styles must be straight and parellel to each other.
2. The substyles and styles must both lie on the meridian of the dial plate. Since your dial has two styles and two substyles, they must be placed parallel to the plane of the meridian and positioned equidistantly on the east and west sides of the meridian of the dial plate.
3. The gnomon must be perpendicular to the dial plate.
4. The angle between the styles and the dial plate must be equal to the latitude of the dial's location.
5. The point where the styles intersect the dial plate must be on the east-west transverse line as drawn in figure 2. On a dial with no correction for longitude (one that shows Local Apparent Time), the line between 6am and 6pm is the transverse line.
It is left to you, the diallist, to transfer the finished hour lines to an appropriate medium. Almost any material can be used to construct the dial: wood; plastic; ceramic; stone; metal (brass, bronze, copper etc.). Use that material with which you have skill in working. Apply your artistic talents! Be creative! You may wish to enlist the aid of an artistic friend to embellish your finished dial. If you should choose to use wood, be sure to apply some type of transparent protective agent such spar varnish to the finished dial to protect it from the elements.