Figuring a mirror refers to the act of changing the mirror's surface from a spherical cross section to a parabolic cross section.   This is accomplished by deepening the curvature of the mirror in the central region only.  A parabolic reflector will bring all parallel rays of light entering the telescope to a common focus. (A spherical mirror cannot do this.)

A common example of a parabolic reflector is a telescope in reverse, and is the reflector in your automobile's sealed beam headlights. The bulb is placed at the parabolic reflector's focal point, and the light rays eminating from the bulb are directed out the front in a tight narrow parallel beam of light.

The idea here is to gradually shorten the radius of curvature from 2743.2000 mm at a radius of 2.5" at right angles to the mirror's central axis, to a radius of curvature of 2737.3201 mm on the mirror's central axis. This is a difference on only 5.8801 mm out of a distance of 2.7 meters.

The foucult test will be used in this proceedure with two separate mappings at each test. The first mapping will measure the mirror's surface from a radius of 2.5" out to the edge of the mirror, while the second mapping will map the mirror's surface from its central axis to the 2.5" radius.

The method is to polish the mirror more severley at the center than at the edges. If the radius of curvature of the perfectly spherical mirror was +2.59 mm larger than the 108" radius of curvature desired, then we want to polish 2.59 mm off the outside edge of the mirror while polishing 7.76 mm off the center of the mirror, making the polishing three times more active in the central region.  This would leave us with a radius of curvature of exactly 2743.2 mm in the outer region of the mirror, and a radius of curvature of 2737.32 mm along the mirror's central axis, a difference of 5.88 mm.

By doctoring the grinding tool, and lengthening my strokes I knew I could polish out the central axis of the mirror 3 times faster than I polished the mirror's outer regions.

Here I made three trips around the barrel with 16 long strokes for each trip, and a fourth trip around the barrel with 16 short strokes. The idea was not to remove burden from the central region alone, which would result in two joining curves that would function as a parabola alright, but would give my mirror a seam in the form of a ridge causing Coma error.

That fourth trip around the barrel deepened the outer region once for every 3 times I deepened of the mirror's inner region. This "Flaring" of the central region gave me one smooth parabolic curve instead of the usual two joining circular curves. This was an inovation I worked out for myself before I started the polishing procedure. Thus I was able to leave enough burden on the mirrors outer regions to make this possible.

I was prompted to do this because I knew what a parabolic curve should look like from my studies of the quadratic equation, and though the instructions I was following assured me two joining circular segments will produce a parabolic curve, I wasn't convinced. I drew a pair of circular segments on a piece of graph paper, and superimposed the drawing of a true parabolic curve with the same end points and depth over the first drawing. Surprisingly, there was little difference. I could describe the two joined circular segments as a parabola with a small ridge in it. Understanding the problem was to grind away this ridge while deepening the central region of my mirror lead to the above solution.

The tool is thoroughly cleaned off, and a new coating of pitch is applied so that the top surface of the pitch is flat, and just barely covers the central region of the tool. The grooves that are cut into the pitch are not evenly spaced, being cut more and more wider as you move towards the outside edge of the tool.  The polishing motion is the same as the grinding and polishing motions before, only now some of the strokes are shorter.  Frequent mapping is the key once more, even though the testing is more complex.

I stopped doing this when my outside radius of curvature was exactly 2743.200 mm, and found my radius of curvature along the central axis was in deed 2737.32 mm confirming the accuracy of the 3:1 grinding ratio from center to edge.   I was now finished. and the last polishing grade of Rouge was so fine, my mirror had a beautiful lustered finish.

Instead of using a pair of masks to separate the outer region and inner region as directed, I relied on frequent mapping of the whole surface. I was looking for a smooth and symmetrical transition in my data,- from the Outer Radius of Curvature to the Inner Radius of Curvature.   I gave precedence to grinding adjustments indicated by the data over blindly following even my own stroke pattern.  Following directions would have given me two separate spherical regions and their coma producing seam.   A vernier caliper with a micrometer readout carefully strapped onto the side of the main yard stick close to the desired Radius of Curvature being measured, enabled acurate fractional millimeter readings to be made.