A Universal Sundial Ruler for Shadow-Following Clocks

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1 A Universal Sundial Ruler for Shadow-Following Clocks by Heinz Sigmund (Heidelberg, Germany) Preliminary Remarks My interest in the combination of sundials and mechanical clocks was confirmed by a fine handcrafted work by David Buschmann dating from 1660, which I admired on a visit to the Deutsches Museum in Munich, Germany. From there I developed some geometrical possibilities, taking into consideration especially the elliptical distortion of the equatorial circle for horizontal and vertical sundials. Then, several years later, after having studied the history of gnomonic schemas and rulers for designing flat sundials, once more I was fascinated, thinking that there must be other simple possibilities to combine sundials with modern timekeepers. and mathematicians, which led to a large variety of graphical layouts. I would like to mention only some well-known authors, such as Dürer, Benedetti, Clavius, Foster, Serle, Leybourn and Ozanam. To describe all these interesting geometrically similar solutions goes beyond the scope of this article. It would be like bringing brass to gold to do so, because we have plenty of well-researched and detailed information on this subject in excellent articles published during the last 20 years, most of them in The Compendium and especially by Fred Sawyer. A Simple Sundial Ruler For the geometrical construction, we refer to the wellknown fundamental diagram, which Albrecht Dürer published in During the last centuries, schemas or dialing scales have been described by many famous astronomers Figure 1 shows that the radius of the circles for different geographical latitudes depends on the basic Figure 1. Fundamental diagram according to Albrecht Dürer. 264 May June 2017 NAWCC Watch & Clock Bulletin

2 Figure 2. Equatorial dial. triangle (upper right-hand side) which represents the needed planes for drawing flat sundials. The diagonal line marks the direction of the inclined gnomon positioned toward geographical North/South. Figure 2 explains the design in detail for a geographical latitude of 50 (Horizontal Sundial) or 40 (Vertical Sundial). We now have two centers: Point B for the Equatorial Dial, and Point D for Horizontal and Vertical Dials. The hour lines starting from B include 15 for each hour and may simply be drawn by a compass (that means: division of the circumference of the circle in 24 equal parts) or by using a protractor. The meeting points of the two hour line systems form the Equinoctial Line. To complete the design, the missing hour lines must be added symmetrically on the right side. It is important to put the numbers in the right order: for the horizontal dial they go from right to left (i.e., clockwise) and for the vertical dial from left to right. Figure 3 shows the three parts of our sundial ruler. (Print and cut them out to make your own instrument.) The shown latitude scale (blue ground) has been obtained by continuing the procedure (Figure 2) for different latitudes. In our example the geographical distances are drawn mostly for every 2nd degree, starting at a position on the Pole (90 ) for the Figure 3. Parts of the ruler. NAWCC Watch & Clock Bulletin May June

3 horizontal dial and at the Equator for the vertical dial and ending (for practical reasons) at 18 or 72. Both scales turn to infinity! The other two scales mark the distance of the hour points from the center (A = Noon). To finish the instrument, the two hour rulers have to be fixed at the points C1 and C2, so that they may be turned around these centers. If we lay them on the larger ruler, we have a short and easily transportable instrument (Photo 1c). As a designing tool, the two yellow arms must be stretched so that they form an exact right angle (Photo 1b). Instead of using it for layouts (especially for large sundials), we can use the backside directly as a universal sundial. Therefore, we only add (in the middle of the blue ruler) a runner, where you can move a gnomon (e.g., in the form of a wooden stylus), which has to be inclined corresponding to the geographical latitude (Photo 1a). Then the complete instrument should be oriented exactly to the geographical South Pole as a horizontal dial and on a wall facing direct south as a vertical dial. The shadow cast by the inclined gnomon then marks (on the central line) True Local Time (TLT) for the chosen geographical Latitude. Remarks: The scales also may be calculated by the following formulas (see Figure 2): AD = R/sin ø (for the distance AD of the gnomon s center) L = R tan t (for the distance L from noon on the equinoctial line) Clockwork as Shadow Follower Remember that the equatorial dial has equidistant hour lines (see Figure 2); we can then install a 24-hour clock in the center (B). So if it is hand-driven by a clock s movement, it can now follow the shadow cast by an inclined gnomon fixed in the center D. If we wish the hand to touch the equinoctial line exactly as the meeting line of the two faces, the clock`s hand should be changed to the suitable length. We have solved this problem in different ways and describe these for each presented model (Photos 2-4). The dial in Photo 3/Figure 5 has been constructed as a vertical dial consisting of two parts in black and white. The upper half representing the sundial is translucent, so that the shadow cast by a gnomon outside can be seen from behind, looking from north to south. The other part has a 12-hour clock movement modified by cogs to 24-hours indication because of Photo 1a. Ruler as universal sundial. 266 May June 2017 NAWCC Watch & Clock Bulletin

the required direction of turning (i.e., counterclockwise). To use this dial, it has to be turned in so that the (white) hand and the (dark) shadow line cross themselves over the equinoctial line.

In Photo 2/Figure 4 the different hours have been marked by the colors of the spectrum and the vertical sundial (upper side) is posed partially over the face of the 24-hour clock; thus, we avoid the

4 the required direction of turning (i.e., counterclockwise). To use this dial, it has to be turned in so that the (white) hand and the (dark) shadow line cross themselves over the equinoctial line. Photo 1b, above. Ruler as drawing instrument. Photo 1c, left. Folded ruler. In Photo 2/Figure 4 the different hours have been marked by the colors of the spectrum and the vertical sundial (upper side) is posed partially over the face of the 24-hour clock; thus, we avoid the above-mentioned problem. In this model, we added two other hands (for minutes and seconds) for better accuracy. It would also be possible to adjust the dial for legal time by superimposing a turnable disk on the 24-hour clock. The two described models show that the equinoctial line as boundary between the dial faces is only reached by the hour line of 8 a.m. or 4 p.m., because of the quickly increasing values of the tangents. The disadvantage of this geometrical layout is well known and has been described by Sawyer in Horizontal Layouts. To avoid this inconvenience, I refer to the other method proposed there also. Figure 6 demonstrates the procedure: The hour lines before 9 a.m. and after 3 p.m. are drawn from new centers, which can be obtained by extending the six hour lines horizontally from the center for the chosen geographical latitude, until they cross the diagonal line (i.e., the prolonged hour line for 3 p.m./9 a.m. on the equatorial dial). NAWCC Watch & Clock Bulletin May June

5 Photo 2, above. Colored vertical dial. Photo 3, left. Translucent vertical dial. 268 May June 2017 NAWCC Watch & Clock Bulletin

6 Now we must deal with three centers for our clock movement and two other tangential lines forming right angles to the equinoctial line. In Photo 4b we give the solution to this problem: a 12-hour clockwork modified to a 24-hour indication (by a supplementary cog) is joined by a chain to the two other centers. Because our dial (Photo 4a) is conceived for a horizontal plane, we must observe that the gnomon s shadow is moving clockwise (in the northern hemisphere) so the three hands (all driven by the same clock movement) must turn counterclockwise. Figure 6 also shows that the shadow cast by the gnomon can sometimes touch the end of two hands on their tangential lines. As we mentioned above, the hour hand cannot always touch exactly the equinoctial line, so we propose a kind of limitation mechanism (Figure 7) bringing the hand`s point in the suitable position by pushing the point or the end of the hand that, however, is still turning around its center. Our last combination of sundials with clocks can be regarded as a vertical Shadow Simulation Dial indicating (without sunshine) the position of the shadow by a clock s hour hand (Photo 5a). There also may be an added inclined gnomon, allowing then to orient the face of the dial toward south, when its shadow falls straight on the black hour hand of the clock (Photo 5c). The inside mechanism for this simulation is shown in Photo 5b: first, there are three hands in the form of gliders/runners of which two are fixed in their centers, where they can be turned around. One of them (on the button) is driven by the movement of a 24-hour clock; the other sticks in the center of a vertical sundial. Its runner/glider is joined to the hour hand on the face outside and also to the clockwork runner taking it with himself while gliding it in the equinoctial line. I refer the reader to Sigmund s Combinations of Vertical and Horizontal Sundials with Mechanical Clocks. In this context, here is a convenient method to determine the declination of a vertical wall from direct south by one of the described Sun s shadow Followers. The procedure is as follows: Choose one of the vertical dials (I recommend the dial from Photo 2), and adjust the clock movement Figure 4. Vertical dial (spectral colors). Figure 5. Black-and-white face for a vertical dial. exactly to True Local Time, considering the geographical longitude distance to the corresponding time meridian and respecting also the amount of the equation of time for the day. Then turn the instrument until sundial and clock show the same time and are then also positioned to direct south (as described above). Immediately measure (by a simple protractor) the declination of the wall from south. NAWCC Watch & Clock Bulletin May June

7 Figure 6. Face of the Sun s shadow Follower. 270 May June 2017 NAWCC Watch & Clock Bulletin

Photo 4a. Horizontal dial with shadow Follower clock. Bibliography Gunella, Alessandro. The Pons Asinorum of Clavius (1586). The Compendium Vol. 10, No. 2 (June 2003): Rohr, Rene R. J.

1 (March 2012): Towards a General Theory of Dialing Scales, The Compendium No. 4 (pp. 14-20) Sigmund, Heinz. Azimut-und Höhenanzeigen auf ebenen Sonnenuhren.

8 Photo 4a. Horizontal dial with shadow Follower clock. Bibliography Gunella, Alessandro. The Pons Asinorum of Clavius (1586). The Compendium Vol. 10, No. 2 (June 2003): Rohr, Rene R. J. Sundials: History, Theory and Practice. New York, NY: Dover Publications, Sawyer, Fred. Horizontal Layouts 1-4. The Compendium Vol. 19, No. 1 (March 2012): Towards a General Theory of Dialing Scales, The Compendium No. 4 (pp ) Sigmund, Heinz. Azimut-und Höhenanzeigen auf ebenen Sonnenuhren. Jahresschrift der Deutschen Gesellschaft für Chronometrie No. 46 (2007) Bestimmung der Nord-Süd-Richtung durch Entwurf einer deklinierenden Vertikalsonnenuhr. Jahresschrift der Deutschen Gesellschaft für Chronometrie No. 48 (2009). Photo 4b. Mechanism (backside). NAWCC Watch & Clock Bulletin May June

9 Photo 5a, above. Shadow simulation dial. Photo 5b, left. Mechanism (backside). 272 May June 2017 NAWCC Watch & Clock Bulletin

10 Photo 5c, above. Shadow simulation dial with inclined gnomon. Figure 7, left. Limitation for the hour hand. NAWCC Watch & Clock Bulletin May June

Combinations of Vertical and Horizontal Sundials With Mechanical Clocks. The Compendium Vol. 12, No.3 (September 2005). Orientierung und Navigation mit [Sonnen-] Uhren.

11 Combinations of Vertical and Horizontal Sundials With Mechanical Clocks. The Compendium Vol. 12, No.3 (September 2005). Orientierung und Navigation mit [Sonnen-] Uhren. Jahresschrift der Deutschen Gesellschaft für Chronometrie No. 49 (2010). Verbindung von Vertikal- und Horizontalsonnenuhren mit Räderuhren. Jahresschrift der Deutschen Gesellschaft für Chronometrie No. 43 (2004). Zeichengeräte zur Markierung von Zeitpunkten auf ebenen Sonnenuhren. Jahresschrift der Deutschen Gesellschaft für Chronometri No. 44 (2005). Sonderegger, Helmut. A Universal Sundial presented by Johannes Gaupp. The Compendium Vol. 20, No. 2 (June 2013) Syndram, Dirk. Wissenschaftliche Instrumente und Sonnenuhren. Callwey: München, Editor s Note: The original version of this article was published in the December 2016 issue of The Compendum (Vol. 23, No. 4), the journal of the North American Sundial Society. It is reprinted here with permission. The Linnaeus Flower Dial By Stephen Luecking A garden dial offers a dialist, who is also an avid gardener, the opportunity to integrate the dial with another form of timekeeping proffered by Carl Linnaeus ( ), the founder of biologic taxonomy. Linnaeus noted that certain species of flowers regularly bloom during specific periods of daylight hours. A garden, he surmised, could comprise a series of beds arranged according to the hours of a clock face. The gardener could simply plant those flowers coming into bloom in the bed corresponding to the prescribed hour on the clock. Substitute the face of a sundial for that of the clock and, as the day progresses, the flowers will bloom in step with the movement of the gnomon s shadow. Editor s Note: The original version of this article was published in the December 2015 issue of The Compendum (Vol. 22, No. 4), the journal of the North American Sundial Society. It is reprinted here with permission. 274 May June 2017 NAWCC Watch & Clock Bulletin

Compendium* A Mean Time Sundial by Hendrik Hollander

Volume 3 Number ISSN 074-397 (printed) ISSN 074-8059 (digital) June 006 The Journal of the North American Sundial Society Compendium* A Mean Time Sundial by Hendrik Hollander Time and reflection change