There are a number of different methods available for graduating a circle. I describe a few older, traditional methods I am familiar with.
- Geometrical construction: This is the oldest method, based on Euclidian geometry. The basic tool is a pair of dividers or compass. You may recall from geometry that the circumference of a circle may be readily divided into six equal segments using a dividers (or compass) set to the circles radius. Each of these segments then encompasses 60°. These segments may in turn be bisected to give segments including 30° arcs. The 30° arcs are bisected again to give 15° arcs. Here a problem is encountered - bisecting 15° will give 7.5° - we cannot create a scale with degree graduations by bisection alone. Thus the 15° arc is trisected. But, as the Greeks realized, one cannot trisect perfectly with compasses. So at this point one determines the trisection by trial and error - one attempts the trisection making very light marks, readjusts the compass etc. until the arc is trisected as perfectly as possible. Then the 15° arc is properly trisected with the adjusted compass or divider. The resulting 5° arcs are similarly divided into five 1° segments. Of course once the dividers is set, one can work around the entire circle, filling in the 1° graduations in the gaps. This basic method, using beam compasses for their extra rigidity, a few tricks and extraordinary craftsmanship produced the finest instruments made until the development of high quality dividing engines by Jesse Ramsden in the late 18th century. In fact, large orservatory instruments continued to be divieded "by hand" into the nineteenth century. The best practitioners were able to graduate arcs to tolerances better than a thousandth of an inch. John Bird, graduated instruments by feel to closer tolerances than can be percieved by eye. (Some of the greatest practitioners are discussed in Chapter 4, "Dividing as a high art," in Chapman in the References section.)
In a similar manner, a 90° quadrant may be divided. As an example, you may follow the step-by-step graduation of a bronze quadrant by following this link.
- Copying or transfer: In this method one uses a model to strike off the divisions in a second device. As an example, one can use a protractor to mark off the angles on a quadrant or astrolabe, or one can use one instrument to set one's dividers for transferring dimensions etc. to a blank for making a second instrument. Some of the earliest European Astrolabes were probably made in this manner. In this case the method was particularly advantageous since the maker didn't have to understand the mathematical model underlying the astrolabe! Many lower quality instruments have probably been made by copying over the ages.
- Dividing Engine: The modern dividing engine was invented by Jesse Ramsden and contributed to the solution of the longitude problem in the 18th century. You can see an image and description of his original engine now held at the Smithsonian Institution in Washington D.C. I built a manually operated graduation device modelled on but less automated then standard engines. You can view a photo essay showing the graduation of a mariner's astrolabe.