Bannekar - Drawing a Dial |

Posted: Friday, 15 April 2011 21:45 |

## Sundials for StartersThis article appeared in
The Compendium in March 2007By Robert L. Kellogg, Ph. D. Benjamin Banneker, 1731-1806 , is one of the nation's best-known African American inventors. He was born in Maryland and in 1791 played an important part in surveying the newly designed Federal Territory, now called the District of Columbia. In his youth, Banneker was inspired to build his own clock after an acquaintance gave him a watch. He took the watch apart to find out how it worked and made drawings of each component, and based on his drawings, he carved larger versions of the components out of wood and constructed a clock that kept accurate time for more than 50 years. As mathematician, he designed an Almanac that was a rival of Benjamin Franklin’s famous publication. As astronomer, clockmaker, and mathematician, he was expected to know how to design sundials, although none exist bearing his mark. In an age before pocket calculators, how would Banneker design a sundial? The graphical method is available in modern texts such as Waugh’s 1973 classic “Sundials: Theory and Construction”. Want to lay out a horizontal sundial without sines, cosines, and tangents? Then this “Sundials for Starters” is for you. Step 1. On a large drawing paper draw two perpendicular lines. Where they intersect is point “O”, the origin. Somewhere “south” on the vertical line mark a point that we’ll call “C”. From C, draw a slant line that is angled away from vertical by an angle L corresponding to your latitude. To do this, you’ll need a protractor. We’re going to lay out a triangle with this new line such that C,P,O form a right triangle with OC as the hypotenuse and sides CP and PO that meet at right angles. The result should look like Figure 1. Step 2. Use a compass to draw an arc of radius OP centered at O. This is “Circle Arc #1” in Figure 2. Where this circle crosses the vertical line, mark the point as “N” and draw another circle of the same radius around N. This is “Circle Arc #2”. Where the two circles intersect, mark a point “A”. The result should now look like Figure 2. Step 3: Using point A as the center, draw another circle arc of the same radius. It should pass through point O and through point N. Use the intersection of this circle and Circle Arc #1 to form a bisecting line NB, as shown in Figure 3. The bisecting line does not necessarily pass through point P, and in fact for clarity the lines CP and PO have been erased. We now have points O, B, and A on the circumference of Circle Arc #2. Step 4: The circumference arcs OB and BA can be bisected again, producing two more intermediary points on the circumference of Circle Arc #2. All of these distances are equally spaced at 15 degree intervals. Using a divider or compass with pencil, copy that distance and “mirror” the points to the left of O along the Circle Arc #2 circumference. In fact, one additional point can be put beyond A and its mirror image counterpart. The last part of Step 4 is to draw lines from the center O through each circumference point until it touches the initial horizontal line HH’, resulting in Figure 4. Step 5: The lines drawn from O through the circumference points to the horizontal line HH’create the auxiliary hour lines starting at noon (the original vertical line) and proceed on the right to 1, 2, 3, 4, and 5 (the rightmost line) and from noon backward counting leftward 11, 10, 9, 8, and 7 (the leftmost line). Now, remember the point C at the bottom of the drawing paper? From the horizontal line HH’connect the intersecting points of the 7 to noon and noon to 5 lines to C, forming the hour lines as shown in Figure 5. The 6am and 6pm hour lines radiate from C and are parallel to HH’. These are added at the bottom of Figure 5. Step 6: We now have the sundial plate in Step 5, its just that we might not recognize it. Centering our attention at point C, with a little bit of clean-up, the radiating hour lines can be trimmed (in this case in a circle). I’ve done the dial plate as part of a circle (traditional), but it could be designed using an octagonal design or a square design. And that initial right triangle that was formed in Step 1? It can now be used to form the gnomon, the dial’s shadow caster. The triangle stands upright, perpendicular to the dial plate that we’ve just designed. Its pointy end C’ is placed on the dial plate at C, with the rest of the gnomon following the noon line. There you have it: dial construction in six steps. It can be done with only compass and straight edge except for setting the latitude angle in Step 1. And once you have a drawing template for the sundial, you can make not only a dial for yourself, but one for your neighbor. In fact, it is most probable that Bannekar made templates for others to use. It would be wonderful if anyone finds one of these 250 year old drawings. |