Circular slide rules are virtually the same as a Mannheim, but bent around in a circle. The cursor is a pair of radial arms that move around the origin. The advantage of the circular slide rule is that you can fit a longer scale, and many more tick marks, within a certain area. Cylindrical Slide rules are by far my most favorite, probably because I've never actually see one up close, and can fantasize a bit about them. They are also similar to the Mannheims, but the scales are wrapped around a cylinder.
Slide rules work through the magic of logarithms.
If you express the horizontal location of a number using its logarithm, you can find their product merely by adding the logs, and seeing what number has its log at the sum of the distances. Exponents are merely an extension of this. Trig and other mathematical functions are carefully calculated and graphically drawn on the slide rule to work as addition or direct comparison. Slide Rules have etched on them a variety of scales, depending on what one you have or need. These are the basic multiplication scales.