I explained the basics of how a slide rule works in the previous post. But how does a **circular** slide rule work?

Apparently the prop Mr. Spock is holding is an E6B aircraft slide rule. It includes a circular slide rule and more functionality.

Start with an ordinary straight slide rule, with each bar labeled 1 on the left and 10 on the right end. Now imagine bending this into a circle so that you now have two concentric circles. For *x* between 1 and 10, the label for *x* is located at

log_{10 }*x* × 360°.

Suppose you want to multiply *x* and *y*, and *xy* is less than 10, i.e. log_{10} *x* + log_{10} *y* < 1. Then the multiplication procedure works exactly analogously to how it would have with a straight slide rule. Line up the 1 of the inner ring with the position of *x* on the outer ring. The position of *y* on the inner ring lines up with *xy* on the outer ring because it is at position

log_{10 }*x* × 360° + log_{10 }*y* × 360° = log_{10 }*xy* × 360°.

But what if *xy* > 10, i.e. log_{10} *x* + log_{10} *y* > 1? In that case the *y* on the inner ring is at position

(log_{10 }*x* + log_{10} *y*) × 360°− 360° = (log_{10 }*x* + log_{10} *y* − 1) × 360°

which is equal to

(log_{10 }*x* + log_{10} *y* − log_{10 }10) × 360° = log_{10 }(*xy*/10) × 360°

and so the *y* on the inner ring is aligned with the position labeled *xy*/10. As we noted in the previous post, slide rules only give you the significand (mantissa) of the result, so a slide rule doesn’t know the difference between *xy* and *x*y/10. That’s up to you to keep up with.

## E6B Flight Computer

**Update**: After first posting this article, I bought an E6B flight computer from Amazon for about $14. I bought the paper version because I’m certainly not going to wear it out. But if you’d like a nicer one, they make the same product in metal.

Here’s a photo of mine:

You can slide the horizontal piece out and essentially have a circular slide rule with some extra markings.

The mathologer just had a great video on this

https://m.youtube.com/watch?v=ZIQQvxSXLhI

The writers obviously thought it would make Spock look intelligent to TV viewers of the era, but how strangely anachronistic it seems now for a 23rd-century science officer to be using a slide rule.