Automated enumeration of slide rule algorithms

• Idea
• Implementation
• Results
• Further improvement
• Other scales
• Hisory: previous attempts
• References

Idea

And that is just for basic ternary operation. We also have basic unary operation (vV=>R(r)), we can search with a cursor (uU->vV; |->W=T; |=>R(r)), we can do the above with slide taken out and put back reversed, finally from formulas of 3 variables r=f(u,v,w) we can get formulas of 2 variables r=f(u,v,v)=g(u,v) etc.

Implementation

Derivatives of ternary operation: r=u^av^bw^c

Reduction of ternary op to 2 variables r=u^av^b

Another way to reduce a number of variables is to substitue a constant for one of the variables. One very special constant for log scale is 1 (also known as index). To substitute one for the variable means just to drop it from the formula (1 ^{any power}=1). One thing to keep in mind when substituing 1 for the variable is that 1 on A is same as 1 on D so using A instead of D does not give new formula. There is a difference between index of a slide scale and index of stock scale, but if both scales are on stock (or slide) it does not make difference which index to use. So to avoid listing operations that look different in notation but perform same movements, this should be taken into account.

Reduction of ternary op to 1 variable r=u^a

Search: r=u^av^b

Unlike basic ternary operation, reversing the slide in search may give some additional formulas, so all this should be applied to the reversed slide for additional results.

Unary op: r=u^a

Results

usage:
 perl sr_ops3.pl [options] >output
options:
 -o{type} -- output type. t-text, h-html, l-list
 -r{rule} -- rule:
    m - mannheim,
    p - polyphase,
    o - monster,
    s - simple,
    4 - N4-ES
    t - tie-clip
 -R{rule_description} -- alternative for -r
 -n - print only 'new' monomials
 -i - print index in the end

Further improvement

Other scales

History: previous attempts

Typical slide rule manual also lists several "expressions solved with one settings of slide". For example, for K,A[B,CI,C]D layout, “The polyphase Slide Rule No. 4053” (ref) shows how to calculate the following powers: 2 3 1/2 1/3 3/2 2/3 -1 -2 -1/2 -3 -1/3 4 -4 5 -5 6 7 9 5/2 9/2 11/2 15/2 -2/3 4/3 -4/3 5/3 -5/3 7/3 8/3 -8/3 -10/3 11/3 14/3 1/6 -1/6 5/6 7/6 -7/6 11/6.

The same list (in the same order and sometimes with the same typos) can also be found in many thick books about slide rule.

For example, "A Manual of the SLIDE RULE Its History, Principle and Operation" by J. E. Thompson B.S. in E.E., A.M. (ref) on page 173 lists calculatable expressions together with their algorithms. This list, word to word, including punctuation and capitalization is identical to one found K&E manual by Bleckenridge (ref).

The question is where this list originally came from. Definitely, K&E book is not the source. Not only it is later than this Thompson's book, but it also contains some typos in the list, like item 6: x=³√a² in K&E became x=√a², or item 36: x=⁶√a⁵ became x=⁶√a. This definitely shows that W. E. Bleckenridge, A.M. just copied this table from some other source and wasn't even careful enough to do it right.

Now, I doubt that Thompson's book was the source. First of all, this table describes operation in plain english, e.g. "set 1 to a on D, over a on C read x on A", while the rest of Thompson's book incorporates "table notation" (pretty clumsy) to describe slide rule operations. Table also calls cursor "Indicator" (capitalized), while Thomson calls it "runner" in the rest of the book.

Origin must exist, but I could not find it so far. If you know who is the author of this list or where it appeared first please send me a note. I am also interested to know if there is a book about slide rules where cursor is called "Indicator" and published before 1930.

There is little doubt that this list was produced systematically using similar procedure as I have described here. Complete inpracticality of the most of the formulas (u ^14/3) speaks for itself. It was purely academic exersize on the part of unsung and yet unknown hero of automated enumeration of slide rule algorithms of the past. Unfortunately (fortunately for me) he could not (or did not wish to) afford services of a professional computer (in XIX century those used to be just guys with paper and pencil and, occasionally, slide rule) and he missed couple of gems. Now, for the first time in centuries human kind finally is capable of calculating in a single slide movement on rules with polyphase layout such powers as u^10/3 u^-5/6 u^-3/2 u^-7/3 u^-7/2 u^-9/2 u^-6 . (I might be mistaken, please correct me if you have seen mentions of these powers in some slide rule book)

Also, original author did not even mention formulas obtained by searching and searching with inverted slide as well as his list of 2 and 3 variable formulas is far more incomplete. I think he knew about them but just did not dare to undertake a complete search. Or maybe just had better things to do with his life, whatever. Nevertheless, he did good job and with the invention of electronic computers his effort may be completed.

References

1. П. М. Поливанов. Упрощённые приёмы машиностроительных рассчётов на счётной линейке. М.:Машиностроение, 1974. 48 стр. (djvu)
2. W. E. Bleckenridge. The polyphase Slide Rule No. 4053. Published 1938. 83 pages.
3. J. E. Thompson B.S. in E.E., A.M. A Manual of the SLIDE RULE Its History, Principle and Operation (second printing, NY, D. Van Nostrand Company, Inc., (C) 1930)