# Usage of reverse side of circular slide rule

I have a circular slide rule, but I can't figure out what the markings on the back are used for.

The front face has two logarithmic scales, the outer one runs from $$1$$ to $$100$$ (exclusive), and the inner scale runs from $$1$$ to $$10$$, so it's easy to take square roots or square numbers using the front face.

Here's a demonstration of multiplying $$4$$ by $$2$$ . The slide rule doesn't have instructions, so I'm just guessing on what the proper procedure is for multiplying.

First, rotate the multiplicand so it's right under the marker under the top button, and rotate the needle so it points to $$1$$, the initial multiplier.

Next, rotate so that the needle points to the desired multiplier (in this case $$2$$). The top marker now points to $$8$$ as desired.

So, that all makes sense, but the back of the slide rule is completely inscrutable to me.

It seems to have multiple scales, with the innermost scale arranged in some kind of spiral. The scales appear to logarithmically spaced, similar to the scales on the front. However, I can't figure out what computation the scale on the back is supposed to be used for.

• How much do you want for that? I would love to own such a wonderful artifact – Carl Witthoft May 17 at 11:35
• never mind - found a bunch on ebay :-) – Carl Witthoft May 17 at 12:01
• @Carl Witthoft: There are many interesting old toys on e-Bay:-) – Alexandre Eremenko May 17 at 22:00
• This one is called a KL-1 circular slide rule. They seem plentiful. Mine is a little under 2in in diameter. – Gregory Nisbet May 17 at 22:04

The scales on the back seem to me to be easier to understand than on the front, but without having it in my hand that may be just an illusion.

The spiral scale is, like the circular one, one decade per revolution. So stretching it out into a line, you have the ex-circular scale being one decade long, at so many inches per decade, while the spiral scale is two decades long at the same number of inches per decade.

This could be useful (for example) if you want to get 3 x 4 = 12 rather than 3 x 4 = 1.2 “and then to position the decimal point, count on your fingers how many times you have got past the 1 on the scale”.

I hope this helps, but it is all a total guess.

## jumping in here...(CWitthoft)

this is exactly correct. See this explanation page that has a representative device you can play with.

The outermost scale for the sine, the scale adjacent to it is degrees, it is marked by S and has angles from 0 to 90. For example, sin 30=0.5. However sin 8=0.139. The part of the outer scale from 1 to 2 has a finer division.

The inner spiral scale marked T is also degrees from 0 to 45, and it is used to read the tangent on the outermost scale. You put the arrow against degree on the inner (spiral) scale and read tangent on the outer scale. For example sin 45=0.707, tan 45=1 tan 12=0.21.

• This is what I said isn’t it? – Michael E2 May 18 at 2:14
• @Michael: Yes. We entered our answers simultaneously. – Alexandre Eremenko May 18 at 11:24
• I see. Since yours was posted two hours after mine, I was didn’t realize it but that can happen – Michael E2 May 18 at 20:00

The S scale maps degrees to sines on the outer scale. The T scale maps degrees to tangents.