A few months ago (woah, so long already!) I had an impulse buy: I purchased a circular slide rule from Etsy. It was cheap, and I had always wanted one, so... I just bought it (a neat addition to my Addiator.)
I guess if you are geeky enough to read mostlymaths.net, you know how a slide rule works. Although I knew how to use it, getting to grips with it took a little while. Just to make sure you follow along here's a brief explanation.
In a normal rule (as pictured below) you can measure a distance, add the distance to another one and get a new distance. 4 and 4 are 8, see?
A slide rule works exactly the same, but the scale on the rule is logarithmic. This means that if you measure the distance between 1 and 3, and add it to the point 3, you get 9 (which is 3 times 3.) This is because in a linear scale distance measures the difference (b-a) and in a logarithmic scale it measure ratios (b/a)
Usually you will do this using a piece labeled the same as the slide rule:
In a normal slide rule (the long wooden or aluminium thing) you have several scales, all placed together with an indicator to know where to measure. In the simplest setting a slide rule is basically what is shown above: a way to fix a distance and displace it along. You could do it with your fingers, even!
In a basic model of circular slide rule (like the KL-1) you have something that works much like this. See the picture below:
If you look carefully you'll see a marker (which is place just below a knob, and fixed in the glass cover) and a red marking needle (it's hard to see, but it's red.) In this picture you'll also see the logarithmic scale (below, from 1 to 9) and a square scale just on top (from 1 to 90.)
The black knob (on top of the fixed marker) displaces the scale, leaving the red marking needle wherever it is (it moves the paper where the scale is written!) and the red knob (the other one) moves just the needle, hovering above the paper.
If you set the black marker on 1 by moving the black knob and then the red marker on (for example) sqrt(2) (which is the number just below 2 in the square scale!) you are fixing the ratio 1-sqrt(2) (roughly 1-1.4)
If you now twist the black knob, you are moving the scale while the distance black-red remains perfectly fixed. So if you place the black marker above 2, you are "adding" the ratio of sqrt(2) and 2, which means 2 times sqrt(2) (roughly 2.8):
So, no mysteries in how this works! In the back there are a few trigonometric scales (known in the slide rule lingo as the S and T scales, for sin and tan):
For the sin and tan scales, you place the cursor (the red marker) in the degrees scale (the one from 10 to 90) and you read the sin of it on top. For the tan, you need to check the degrees in the spiraling scale in the center, and read the tan of the angle also on top.
What's it good for?
Such a small circular slide rule has a very low precision (2 digits more or less.) This essentially makes it pretty much useless for me, since I have more or less the same precision in mental division or multiplication.
Anyway, I have used it several times when checking traffic numbers for websites, when estimating daily visits from monthly or bimonthly numbers. Once you set the number of days as ratio, it's pretty fast.
An interesting use would be for quick currency conversions while abroad (you fix the ratio and can easily convert from currencies) but since I travel mostly in Europe, I can't use it for this (Norway, Iceland, Great Britain and Denmark are good targets still.)
People still use slide rules though: nomographs (what the sin and tan scales are, actually) are a quick way to compute things, and are used in aviation, electric engineering and other fields where speed is interesting and 5 digit precision is not as important.