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Years ago, before everyone (or anyone) had electronic calculators, I had a pocket slide rule which I used in secondary school until the first TI-30 cane out.

Recently I dug it out. Here's a photo of one end of it.

As you can see, there's a number $C$ marked at about $1.128$ (times some power of $10$; with a slide rule you supply that yourself) on the C and D scales. Reading across to the A scale, its square is about $1.27$. By the C1 scale (which reads reciprocals of the C scale) its reciprocal is about $0.886$ (times some power of $10$).

The only two special numbers marked are $C$ and $pi$.

I'm not sure whether it's some frequently used constant that's used (eg) in some branch of engineering, or a number which is useful for some trick for using the slide rule.

Unlike $pi$, which is marked on most of the scales, this mysterious $C$ only appears on the C and D scales, which are the main ones used for multiplication and division.

If you need me to, I can give more explanation of the various scales on the rule and how calculations are done. That might give some clues as to what $C$ is for.

I'm sure the instructions explained what $C$ was, but I last saw those in the 1970s.

Has anyone any idea what $C$ is and why it would be useful on a slide rule?

I found the answer by googling "slide rule markings"! It took me straight to the Glossary of the International Slide Rule Museum, which gives C its own entry:

C - Gauge mark found on the C and D scales denoting $sqrt{4/pi} = 1.128$ for calculating the area of a circle and the volume of a cylinder. Place the C mark on the C scale over the diameter of a circle on the D scale. The area of the circle is found above the index on the A scale. If this is the base of a cylinder, without moving the slide, move the cursor to the height of the cylinder on the B scale. The volume is read on the A scale. This gauge mark was rendered obsolete with the advent of multi-lined cursors.

And there is, of course, so much more at that site.