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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.

Chifti Saidi
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  • You might be able to better explain what you are asking by referring to some documentation for the slide rule scales mentioned. See here for example. There are many arrangements of slide rules and scales. You seem to be asking why one would "need sliding A and B scales next to each other". I'm guessing that multiplication and division can be done with these, but perhaps I misunderstand the Question. – hardmath Jul 11 '18 at 14:04
  • I know how to use a slide rule, thanks, I own many of them. The question seemed clear but maybe not. I've clarified it a bit (I hope). The question is: Are there any calculations that would require (or be greatly benefited by) having A+B scales together. – Chifti Saidi Jul 14 '18 at 10:56
  • I figured out the answer! If you unblock the question I can write it below. – Chifti Saidi Jul 14 '18 at 17:19
  • @ChiftiSaidi It seems not to be on hold now. I just added an answer; yours may be better. – David K Jul 16 '18 at 04:11
  • PS: This place needs a category for this. And a "slipstick" tag. – Chifti Saidi Jul 16 '18 at 20:11
  • There seems to be a contradiction between "they always put A and B scales next to each other on a slide rule" and "I own some slide rules that don't do this." – bof Jul 17 '18 at 00:22
  • I messed up between edits. One of those statements was added later. – Chifti Saidi Jul 17 '18 at 01:44

3 Answers3

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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.

Chifti Saidi
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  • This seems a well-researched answer that makes a lot of sense. Now I will have to go through my collection to see if I have any missing A or B scales. – David K Jul 16 '18 at 16:01
  • I've got two Faber-Castells without A or B scales and a cute little Sun Hemmi #22 which only has two sticks, it has C,D where the sticks meet, A below D, CI above C. – Chifti Saidi Jul 16 '18 at 19:54
  • I just now observed that my Post Versalog has no A or B scale (though it has two copies each of C and D). It also has a K scale. – David K Jul 17 '18 at 02:32
  • I think the Versalog has R1 and R2 scales for calculating square roots. Going from D to R1 or R2 will give you the square root of D. – Chifti Saidi Jul 17 '18 at 03:14
  • I have been wondering why the A and B scales (which seem less fundamental to me) got the earlier letters of the alphabet, but this explains it. Well researched! – kqr Apr 09 '23 at 17:12
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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.

MJD
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  • Nope, it's actually the opposite way around, C+D are more accurate. A+B is a good place to teach people how to use a slide rule though - the infinite wrapping of the scales is easier to visualize there. – Chifti Saidi Jul 14 '18 at 10:53
  • Oh, whoops! I should have looked at a slide rule. I'll revise my answer. – MJD Jul 14 '18 at 14:26
  • I think you may have indirectly hit on the answer. The scales labelled A and B are the most basic scales, the C and D scales came afterwards, ie. early slide rules didn't have C+D. – Chifti Saidi Jul 14 '18 at 17:01
  • Update: I just looked for pictures of very early slide rules and can confirm that the Gunter Scale, et. al., only have what we'd call an 'A' scale today (which makes perfect sense if you only have a single scale and a pair of dividers).

    C+D were added later by Mannheim, but most people would still expect to see A+B on their slide rule for a long time after that.

    Unfortunately .. this question is "on hold" so I can't write a proper answer.

     – Chifti Saidi Jul 14 '18 at 17:17
  • @ChiftiSaidi You should still be able to edit my answer. Feel free to do that and add whatever seems useful. – MJD Jul 15 '18 at 16:38
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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.

David K
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  • I'm sure you can find a use for them but I was wondering why those scales are so universal when the calculations that could make use of them are so rare. Surely there could be something more useful there, eg. CF/DF scales (which many duplex rules have on the back) are far more useful. – Chifti Saidi Jul 16 '18 at 14:33
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    The convenience of being able to square either C or D had crossed my mind, too. – Chifti Saidi Jul 16 '18 at 14:50
  • Ironically, "all these numbers are off by a factor of $r$" is actually the most likely reason for me to pull out a slide rule nowadays, but again (ironically) I tend to use the C and D scales anyway by default unless I realize the numbers will "wrap around" too often. Of course this has little to do with what operations were most common when the slide rule's layout was decided. I think your answer is a more likely reason for the persistence of the B scale, particularly since some designers of slide rules decided it was not worth keeping. – David K Jul 16 '18 at 16:00
  • It also occurs to me that the initial motivation for the C and D scales may have been in order to take square roots rather than squares of numbers. After all, it's easy enough to square a number with the C and D scales alone (or the A and B scales alone), but hard to take a square root with only one set of scales (A and B without C or D, or vice versa). – David K Jul 16 '18 at 16:05
  • David: Looked at from another angle, A+B are the numbers and C+D are square roots of those numbers. nb. The Pickett N4 goes the other way - it uses the D scale as "square root" (=A) then has another scale split over two rows to look up the square of thast. Cube roots are done similarly with a special "cube root" scale split across three rows. – Chifti Saidi Jul 16 '18 at 19:10
  • PS: If you get a rule with CF/DF scales it's easy to avoid the scales ever wrapping around. :-) – Chifti Saidi Jul 16 '18 at 19:11