We're used to the idea of using logarithms to convert multiplication problems into addition problems, because of the relation:

log(xy) = log(x) + log(y)

So if you have a table of logarithms, to multiply x and y you can look up the logarithms of each, add them, then find the number in your tables whose logarithm is that sum. Three table-lookups and an addition. [Well, in practice you do the last lookup in a table of antilogarithms, but that's only a minor benefit, it's not actually necessary to the calculation.]

However, there are other ways to make multiplication "simpler" than trying to directly multiply, such as this identity:

cos x cos y = [cos (x + y) + cos (x - y)]/2

This sort of relationship was actually used to do multiplication; it's a bit more complicated than logarithms, but it works well enough if you haven't invented logs yet, but

*do* have cosine tables. It works like this.

You want to multiply two numbers, X and Y (and let's assume that they have already been scaled to lie in a range where this will work).

You find the numbers that they are the cosines of in your tables, X = cos x, Y = cos y. Add and difference those two numbers, giving (x+y) and (x-y). Look those numbers up in your cosine tables. Average them. That's the product. Four table lookups, an addition and a halving (and some rescaling back to the original problem, presumably, which probably just involves moving a decimal point), all much easier than general multiplication.

So far so good.

But here's what I am wondering. Humans came up with a nifty device to automate that multiplication via addition of logs, the slide rule. Is it possible to build a device that does mutliplication using a trig identity like the one above, perhaps some sort of "swivel rod"?

If we do it in two parts, I think maybe we can do it with a slide rule; one side does the additions and subtraction (and halving; halving is inverse doubling, which is just addition, so it can also be done there). The other does the conversions to and from normal scale to cosine scale - but you'd be forever transferring numbers across the different scales.

So I wonder if it can all be combined in a single, simpler system, whether it's a kind of slide rule or something more complex involving actual rotations to do the cosines (accuracy may be an issue though).

I don't think a device was ever made that did this, but I think that it might be possible.

If it were, I'd love to have one.