1)Take any two arbitrarily sized glass blanks, where the diameter of the
smaller blank is less than the radius of the larger, but more than a quarter
of the diameter of the larger.
(M/4 < T < M/2). For our purposes here, let
us take, say, a situation where M is a blank of 6 inches in diameter and T
is one of 2 inches in diameter.
2) Center M on a slowly rotating turntable
3) MAKING THE ORBIT: On thin wood (I find 0.25-inch plywood to be best) draw a straight line whose length is (M + T) / 2. In this case (6 + 2) / 2 = 4. This is the major axis of the ellipse. The distance of the foci from the ends of this major axis is equal to T/2, or the radius of T. Mark these positions off carefully and hammer two nails into them. Next hammer a third nail at one of the ends of the major axis just drawn and tightly encircle a non-stretchable string, such as twine, around all three and tie it tightly. Remove this third nail, while keeping the tied loop of string intact. Thus, trace out the ellipse carefully onto the wood with a pen. Cut this ellipse out with a jigsaw, but cut a couple millimeters inside the line, since the saw blade tends to take off an extra millimeter or two when it is cutting. I find it best to leave a bit of extra space uncut inside the ellipse, so that I can sandpaper down the ellipse to the exact line. I generally can get an ellipse which is 99 per cent or better of the specifications defined in this way.
4) Now, fix this ellipse, this "orbit," above M so that its downward face is above the face of M which is facing up. You don't want M rubbing against the wood. Fix the "orbit" in position above M such that M is centered at one focus of this ellipse. Since the focus will always be the distance of the radius of T from the end of the ellipse, an easy way to position the ellipse is to simply center T on M at the focus: the edge of T will exactly hit the inner edge of the ellipse at the point of the ellipse which is closest to the center of M (which I often call "perihelion"). Make sure the ellipse is firmly held down in position and cannot move at all. It is best to screw it down, so that it can be removed and replaced in the exact same position at will. This way you can remove M for cleaning, and so on, between stages.
5) Pitch a holder to the top of T, and connect this holder to a handle, which allows T to rotate on its own axis, while the handle is being held motionless. (There are different ways to do this, of course.)
6) Place required grade of abrasives onto the upward-facing surface of M, with liquid. Hold T by handle, such that its downward face is pressed against the upward face of M. Place the edge of T against the inside of the ellipse, at the same time, and using your hand, revolve T around inside the ellipse, such that, at all times, the edge of T is touching the inside of the ellipse. As T rubs against the ellipse, this rubbing will naturally cause T to rotate on its own axis, while you revolve T around its "orbit."
So we get three motions, but only one need be actually produced by a machine:
1. M rotates on its own axis. This is done by the machine, the turntable.
2. T revolves around its orbit. This motion is carried out by hand, in this case.
3. The rubbing of T against the orbit, causes T to rotate on its own axis.
7) Polishing: Because it is impossible for pitch to uniformly conform to the shape of T, we must depart from the method, and use the standard method of polishing.