Slide rules were analog devices in which the number of significant digits were proportional to the size of the rule (and the smallest graduations that a typical person could resolve.) By wrapping the scales around a circle, the manufacturer could increase the scale length while keeping the overall length modest. (But with an increase in width which made many of them too large for pockets.)
The Loga slide rule shown below was all metal except for the cursors. It was 5 inches in diameter and had a pivoting stand on the bottom so the cursors and slide could easily be moved while it was sitting on a desk. The black protrusion on the left controlled the rotating portion (all but the two outermost scales) and the bottom cursor. The top cursor was positioned directly. On this slide rule, the multiplication scales were labeled A and B and the squaring scale was labeled n2.
Picture of a Loga circular slide rule (~70K)
The Atlas slide rule shown below was also all metal, but instead of having a rotating scale like the Loga, it used two pointers and a solid disk. The top pointer moved freely by itself but moving the bottom pointer also caused the top pointer to move. Another important difference is that while the loga had multiple circular scales on the front, the Atlas had a single spiral scale allowing much greater accuracy. (Trig. scales were on the back.) The rule was 8.3" in diameter and was copyrighted in 1931.
Picture of an Atlas circular slide rule (~138K)
The Fowler's Calculator shown below was a variation on the circular slide rule. The entire scale disk was rotated by turning the top knob. No part of the scale rotated with respect to another part. Instead, the second knob, rotated a second hairline (in black) while the other (red) hairline was fixed relative to the metal case. (This made it more like the Gunter's line with the hairlines serving as dividers.) To multiply 25x14, the user rotated the scale to align 25 with the fixed red index. Next the user moved black index to 1. The user then moved the scale to align 14 with the black index. The result of 35 could be read from the fixed red index. Once the decimal was positioned the result was 350. This could be chained as needed and because it was circular, there was no need for folded scales. This calculator was 3.4" in diameter and ruggedly made for traveling. (Though its thick metal case will get your carry-on luggage inspected these days.) It came in a heavy duty felt lined leather case.
The model displayed is the "Fowler's Twelve-Ten" and was made around 1950. Besides the typical multiply and reciprocal scales, it included scales for converting twelfths to/from tenths, meters to yards, Kg to lbs, in to cm, etc. Many other models with different scales were available and they came in sizes both larger and smaller than the one pictured. Scales on the Fowler's were obviously labeled like "recips" rather than the more common A, B...
Picture of the Fowler Calculator (~72K)
The Lietz circular slide rule shown below was smaller overall and simpler in construction than the two above. It was 4.2" in diameter and .05" thick over most of its surface (.4" thick at the center.) It was just a single piece of metal with a single pointer on the back and two pointers on the front. The back pointer aided reading the scales C, S, ST and T. The front pointers were used alone or together for the C, CI, A (decimal squares), Af (fractional squares), L, Fa (fraction addition/subtraction), LL1, LL2, DS (drill size), DT (double depth of threads) and M (metric conversion) scales.
Multiplication was performed with both pointers using the C scale. To multiply X by Y, the user set the lower pointer to X and held it there while moving the upper pointer to 1. Then the user released the lower pointer and moved the upper pointer to Y (which moved the lower pointer to the result.) As always, division was simply the reverse.
Picture of the Lietz Slide Rule (~58K)