Why are there A and B scales on a slide rule?

Question

Why do they usually put A and B scales next to each other on a slide rule?

It's an almost universal construction but I can't think of a single calculation that would need sliding A and B scales next to each other. To me it makes more sense to put a single A scale at the top and put something useful in the sliding part (eg. CF/DF).

Question: What sort of calculations would require (or greatly benefit from) having A+B scales together? Can anybody come up with an example calculation?

Or: Is it simply that A+B next to each other looks nice and symmetrical and they didn't know what else to put there? (which is what I suspect)

FWIW: I own some slide rules that don't do this, they have CF/DF there and an A scale at the top.

I also know of slide rules where they put two K scales next to each other (on the back, with A/B on the front), so somebody must have thought it was useful, I just can't think of what the use might be.

Answer 1

After some research and thinking about this, I think can answer my own question: The reason is purely historical.

The biggest clue was in the fact that the A+B scales are called 'A' and 'B'. This implies that they came before 'C' and 'D'.

If we start with the precursor to the slide rule, the Gunter Scale, it only has what we'd call an 'A' scale. This makes perfect sense when you think that multiplication on these rules was done with a single scale and a pair of dividers. A useful scale has to repeat itself for this method to work.

When William Oughtred made the first slide rule in 1622 he did it by putting two Gunter Scales next to each other. This naturally creates a rule with what we'd call 'A' and 'B' scales.

Gunter scales and Oughtred-style slide rules were the norm and lived side by side until Amédée Mannheim created what we'd recognize as a modern slide rule in 1859 (and 'Gunters' were used in marine navigation long afterwards because they had other scales designed to help with that).

Bottom line: In 1859 it made perfect sense to place A+B scales on the rule, it was simply what people expected to see.

Mannheim's 1859 design continued to be the basis of slide rule designs right up until slide rules disappeared in the 1970s so it was natural to keep the traditional A+B scales alive.

Footnote: There are slide rules without A+B scales, I own a couple of them myself, but they're in the minority.

Answer 2

You do all the same sorts of calculations on A and B as on C and D. But the A and B scales are less compressed than C/D, so the calculations on them are more accurate.

Answer 3

I cannot think of a single calculation that would need both A and B scales.

But suppose you have a long list of numbers that all need to be multiplied by the same amount, $r.$ If you are lucky, you may find that after you set the leftmost $1$ of the $C$ scale opposite from $r$ on the $D$ scale, when you find the numbers in your list on the C scale every one of them will be opposite a number on the D scale and not shifted past the end of the D scale. Or you may not be so lucky, and you may have to move the slide to change which end of the C scale is opposite $r$ for some of the numbers.

Depending on the sequence in which you receive the numbers, and whether you can skip a number and come back to it later, this can get annoying.

You never need to have this problem with the A and B scales. You can multiply as many numbers by $r$ as you want in any sequence and never have to move the slide.

Another convenience is that after finding the square of a number from the D scale on the A scale, you can immediately use the B scale to multiply the result by a factor. You could, of course, transfer the result from the A scale to the D scale and continue there, but that's an extra step.

The fact that some manufacturers have left out the B scale goes to show that each of these uses for the B scale is merely a convenience, not a true necessity. On the other hand, don't sneer at the value of convenience.