Calculation Before We Went Digital

We have to like [Nicola Marras]. First, he wrote a great mini-book about analog computers. Then he translated it into English. Finally, he opened with a picture of Mr. Spock using an E6-B flight slide rule. What’s not to like? We suggest you settle in when you want to read it — there are almost 60 pages of text, photos, and old ads for things like slide rules and adding machines.

There is a lot of research here. We couldn’t think of anything missed. There’s a Pascalina, Ishango’s bone, a Babylonian spreadsheet, an abacus, and even Quipu. Toward the end, he gets to nomographs, adding machines, and the early calculators.

We badly want an IBM 223-3168 — an IBM pocket calculator that uses rotary wheels to add and subtract hexadecimal. We also badly want to add a Curta to our collection. While we have many odd slide rules, we’ll confess we never saw Consul the Educated Monkey, which was a multiplying and dividing slide rule for kids. We do, however, have a Smarty Cat. We also were not familiar with the French Abaque Compteur Universelle, which was used by the French railway system.

About the only place we mildly disagreed was the statement:

[The slide rule was] always sticking out of engineers’ pocket[s].

Everyone we knew wore theirs on the belt. Engineering students looked like they were carrying a short sword all around campus. Still, maybe someone had them in their pockets.

We’ve argued that slide rules made for better engineers or, at least, helped filter out the bad ones. We have always been fascinated with this old tech, from the magic brain to the Leibniz wheel.

18 thoughts on “Calculation Before We Went Digital

  1. “Everyone we knew wore theirs on the belt. Engineering students looked like they were carrying a short sword all around campus. Still, maybe someone had them in their pockets.”

    Swiping part of was a child’s revenge on another. Try doing that with a calculator.

  2. > “It is unworthy of excellent men to lose hours like
    slaves in the labor of calculation which could be
    relegated to anyone else if machines were used” (Leibniz)

    I could agree to a certain extend but knowing a bit of “analog” (ie slide ruler) could be interesting for the mind.
    I would like to learn to use one.
    Actually more than learning how to use it, is how they were made that I would like to know. how did they ensure the accuracy of the thing?

    1. Yes, it is interesting. I was at secondary school (aka High school in US-speak) in the early to mid-1980s. By this time, slide rules had been phased out – none of the other students in my year had one; I didn’t know anyone in upper years who had one; we weren’t taught how to use them either. Almost always we were expected to use scientific calculators. However, for the sake of formality, we were taught how to use 3-figure log and trigonometry tables. (And this is relevant to you question… eventually):

      3-Figure log table were just a simple lookup, since you could cram 100 rows into 2 page and provide another 10 columns to give the full 3-digits. But in my Dad’s era they were taught 4-figure log tables, which were similar, but had an extra sets of 10 columns for the 4th digit where you just add the number for the column to get the answer. e.g.

      Log of 7589 =>Log( 10^3*7.589), so it’s 3 + Log(7.589). So you’d look up the 75th row and the 8th column to read: 0.8796 (rounding down), then you’d look up the delta column 9 which would say 5, so you’d add 5 in the last digit => 0.8801 (though 0.8802 is closer). However, if you wanted Log(7.539), you’d look up the 75th row, and 3rd column to read: 0.8768, then look up delta column 9 which (again) would say 5, so you’d add 5 to the last digit => 0.8773. The true deltas between Log(7.589)-Log(7.58) and Log(7.539)-Log(7.58) are so close you can use them for both.

      The only other complexity with Log tables was that negative logs were represented in a complement form. e.g. Log(0.7589) =. Log(10^-1* 7.589), so we would write /1.8801 instead of -0.1198 (where ‘/’ was a bar written above the number). So, that’s not how we’d do it now, but we had to, because we only had logs for the range 1.000 to 9.999. Then if we e.g. multiplied by 3.404 (whose log is 0.5320), we’d calculate: /1.8801+0.5320 = (/1+0)+(0.8801+0.5320) = (/1+0)+1.4121 = (/1+0+1)+0.4121 and /1+1 = 0, so it’s 0.4121. 10^0.4121 = 2.583, which is a good enough answer.

      This tells us how it’s possible to have an accurate enough slide rule. A slide rule as you may know is the set of original numbers 1.00 to 10.0 on a logarithmic scale, so that adding on the slide rule multiplies the original numbers. But we can treat values between the smallest graduations as being linear for the same reason that the look-up for the 4th digit on a 4-figure log table can be added to the log of any of the look-ups on the same row. So, that solves one aspect of the accuracy.

      The second thing is that early log tables were generated the slow way by multiplying by a fixed amount each time. e.g. start with 1, then multiply by 1.01 each time until you get to 10. So, then you could mark each step in 1mm intervals. It’s quite laborious, but by the time you got to 233mm, you’d get to 10.05. Then you mark the steps at readable intervals. You might think all the rounding errors would make it rubbish, but it won’t, instead it would be good enough for the reasons outlined above.

      You can also get a pdf of a slide rule Published in Scientific American, May 2006!

  3. “We badly want an IBM 223-3168 — an IBM pocket calculator that uses rotary wheels to add and subtract hexadecimal.”

    The mechanism is simple, pics are online, 3d printer is in the workshop, holidays are coming…

  4. Slide rules are rad if not only because of the excellent visual of how multiplying numbers is the same as adding logs.
    But I never bought the argument that it’s faster or better for simple multiplication though. On paper multiplication and long division is easy enough. For cube roots and stuff – sure.
    My dad had his engineering slide rule and as a pretty young kid he showed me how to use it for simple stuff. When I got to Jr High the math teacher had a gigantic demo one hanging over the blackboard and I kinda surprised her that I could use it easily. She asked me to stay after class and opened the drawer of her desk and had a big stack of slide rules in there and asked if I wanted to have one.
    I still have it, still use it occasionally for nostalgia.
    Thus the long and ongoing obsession with calculators of all types. I have a couple of circular slide rules for pilots, and an embarrassing collection of RPN early, mid and modern HP calculators.

    1. Slide rules gain ground if you practice and if the design is such that you can chain a series of operations together or read off multiple similar results for almost the same effort. It’s useful in the same situations a nomogram might be useful, if nothing else – the ones where someone needs to actively use the results but it’s helpful if the means of achieving them is both easy and visual. Otherwise, I prefer the latest (e.g. TI nspire) calculators, especially with their libraries and the ability to program your own functions when you don’t need a lot of performance. Or if you do need performance and libraries, I guess either gnu octave or mess with python for easy stuff, or just write a “real” program if you need more than that.

      For instance, if I was constantly figuring out what angle to mount a directional antenna given a certain height and distance from the tower, while I personally might program a calculator to ask for inputs and return the result, it would also be reasonable enough to get in the habit of turning the dials (my slide rule is circular) to set the height and distance, then read off angle from the back. I worked out various ways to chain things quickly or to solve particular equations while keeping a certain value to reuse. I think one thing I worked out was a very minimal number of operations for the inverse of a sum of inverses. That is the equation that tells you the value of a bunch of different things, but in my case I was thinking of parallel or series components (impedances) in a circuit. I think I may have also figured out I could do that one with only a ruler or calipers or something, though it’s not like you can’t also just perform the regular math of course.

      The industry nomograms I can think of offhand were used in telecom to get the parameters right for copper telephone and DSL circuits. I don’t remember exactly what they calculated. I can tell you that smith charts are immensely easier than mucking about with proper calculations if you want to calculate common things for wave purposes – they can be faster than a graphing calculator unless it’s been specifically programmed with those calculations in mind, which they usually aren’t by default. Any computer can find a web version though if you want it.

    2. “But I never bought the argument that it’s faster or better for simple multiplication though”

      Easy enough does not mean fast – or even easy for that matter. Consider the time line:
      – slide rule was invented in 17th century. Not really time of cheap notebooks, ball pens and common education.
      – One of the reason slide rules become so popular is that they were used by artillery. If you need to train a solider you can’t waste time (months probably – bear in mind education level and costs) to teach him write fast (or even to teach him write) and perform mathematical calculations by hand – not only this is prone to mistakes (specially under life threatening pressure of front line fire exchange) but also it would take longer than using slide rule.
      – for single calculation paper is fine. If you do a series of calculations (sometimes several times per day) you start to care if it takes you 5sec per calculation or 15sec.

  5. Thank you for the positive review of my publication, even if the English version is much smaller than the Italian original.
    This work has been used for years with the aim of communicating in schools the existence of a “pre-digital” world, a job which is unfortunately now almost impossible to carry out.
    A curiosity about slide rules and engineers: in Italy slide rules with attachable belt cases have never been sold, the standard model was too large to carry around and the engineers always had a small slide rule in their jacket pocket, it was a real badge of the category!
    Until 1975 it was forbidden to use electronic calculators at university, only slide rules were allowed. I still have my 1972 Pickett and 1974 HP 35. At the time it cost the same as a domestically produced motorcycle.
    Merry Christmas and end of year holidays, Nicola Marras

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