The Graduation of a Circle or Arc Using a "Dividing Engine"
In the late 1990's I determined to make myself a "dividing engine", seen in the photo above, to aid in dividing the various instruments I make. I did not make a true automated engine, rather I made a device where the angles are set manually then a line is scribed manually with a scriber or knife. This makes for a much easier project, but still greatly enhances my ability to turn out graduated arcs up to 5 1/4" radii in a reasonable time.
The engine is based on two surplus components I had previously obtained, a c. 1900 small bench top drill press, and a discarded precision rotary stage made from the main scale of a dismantled David White Pilot Balloon Theodolite* (a machinists rotary table would actually be preferred and what I would use if I were making another engine). The drill press frame and table was then used as a frame to align all of the parts and the rotary stage was centered onto the drill press table. The resulting device is seen in the image to the left with the parts labeled. I discuss the use of the device below with the division of my Mariner's Astrolabe (2014). If you wish additional information, please contact me by email.
The dividing engine is shown with the centering tool (the original tight-fit drill press quill with a tapered center on a press fit aluminum shaft replacing the original drill shaft). I replaced the drill press handle with a clamping system to lock the centering and ruling tools in place and to ensure their precise location.
The astrolabe is next placed on the table and the centering post is used to center it based on a small centered hole in the astrolabe (the hole will later be carefully enlarged to its final diameter to accommodate the pin). At this point the astrolabe may be fixed in place with a low m.p. wax or clamps (the minimum two clamps were left off for visual clarity for this and the next two images).
The centering clamp is now replaced with the post for the dividing rule. In my example the post is a carefully machined aluminum rod that just fits in the quill hole and that has had a square face machined into one end that is parallel to the rod center is extends to the exact center of the rod. This facet accommodates the 1"x 1/2" x 5 3/4" precision ground steel stock used a a rule.
The rule, which was removed to allow it to go through the quill hole, is now attached with two bolts and tightened with the rule parallel to the item to be scribed.
The engine is now shown, with the astrolabe clamped in place, ready for a scribe line.
I now manually scribe a single line one or more times depending on how deep I want the graduation and the resistance of the material. Depending on the project and the material I have used a variety of knives (sharpend on a single side as chisel or both) as well as scribes in this process.
One turn of the micrometer on my rotary stage moves the astrolabe by one degree and the next graduation can now be scribed. (My machine is capable, in careful hands, of graduations to better than 0.1 degree, but such precision has not been needed on any of my projects thus far.)
This image shows the setting for a graduation at 351.0°.
For a full circle, such as on the scales of my torquetum or planisphaeric astrolabe, this process of setting, scribing and setting must be repeated 360 times. Periodically the clamps also get in the way, so a third clamp is put into an appropriate spot, and the obstructing clamp then removed. I always keep at least two clamps in place to keep the object from moving. Because of the design of my engine with the micrometer and readout moving with table, I actually graduate two sets of 180° with requisite two centerings etc.
Finally, I always use a divides or trammel points to set out at least the 60° arcs on my circles with fine pin pricks at these points to help me confirm my graduations and to allow resetting my engine if necessary.
* A description of these instruments and their use etc. is available at Martin Brenner's pilot weather balloon (Pibal) theodolite site.