Homebrew Slide Rule Gets Back To Mathematical Basics

In the grand scheme of things, it really wasn’t all that long ago that a slide rule was part of an engineer’s every day equipment. Long before electronic calculators came along, a couple of sticks of wood inscribed with accurate scales was all it took to do everything from simple multiplication to logarithms and trig functions.

While finding a slide rule these days isn’t impossible, it’s still not exactly easy, and buying one off the shelf isn’t as fun or as instructive as building one yourself. [JavierL90]’s slide rule build started, ironically enough, on the computer, with a Python program designed to graphically plot the various scales needed for the fixed sections of the slide rules (the “stators”) and the moving bit (the “slide”).  His first throught was to laser-engrave the scales, but the route of printing them onto self-adhesive vinyl stock proved to be easier.

With the scale squared away, work turned to the mechanism itself. He chose walnut for the wood, aluminum for the brackets, and a 3D-printed frame holding a thin acrylic window for the sliding cursor. The woodworking is simple but well-done, as is the metalwork. We especially like the method used to create the cursor line — a simple line scored into the acrylic with a razor, which was then filled with red inks. The assembled slide rule is a thing of beauty, looking for all the world like a commercial model, especially when decked out with its custom faux leather carry case.

We have to admit that the use of a slide rule is a life skill that passed us by, but seeing this puts us in the mood for another try. We might have to start really, really simple and work up from there.

49 thoughts on “Homebrew Slide Rule Gets Back To Mathematical Basics

    1. It’s not difficult. The point is that logarithmic addition is multiplication. Log(xy) = log(x) + log(y) so when you have two slide rules with logarithmic scales, you place the zero point of the first ruler on position X on the second ruler, and then follow up the scale to position Y according to the first ruler. Likewise, division is just subtraction along the ruler.

      It’s really a simple tool that looks complicated because there’s multiple different scales on it.

      1. There is no 0 in logarithmic scales. You align “1” with the other multiplication factor. Or for division, align the numerator to the denominator and look up where the 1 on the denominator scale ends up against the nominator scale.

      2. Don’t forget that when you use one you also get a organic feel to the numbers. When you design with one it just gives you more information than you might think whereas a digital calc implies ultimate precision. Hard to explain that one but has always been one of my favorite things about using them. I still grab any I run across.

    1. That’s kind of a neat idea. You could do this easily enough, at least conceptually, but you would have to take the antilog of the position in order for it to give the right result, and you would have to do this differently for each scale that you wanted to be able to read from. Which would mean hacking or replacing the microcontroller from the digital caliper. Also, for this to be effective, you need to know the positions of TWO things: the slide and the cursor.

      This is similar to how tape counters on later VCRs could display in hours and minutes rather than arbitrary numbers. Which reminds me: back when I had my first VCR, which did NOT have this feature, and when blank tapes were kind of expensive and I wanted to put as much on each tape as I could, I reverse-engineered the counter to find the polynomial that could convert from tape counter numbers to minutes. I then made a nomogram to do the conversion, so that as long as I remembered to reset the counter at the beginning of the tape, I could always tell how much recording time was left on a tape.

    2. hmm… no… ;-) a slide rule is analog, and what will you display on the display?
      My slide rules I got from my father have at least 8 scales on it (x, x^2, 1/x, log(x), sin/cos, tan…)
      And then there is the battery, I have a caliper with digital display and the battery is always empty :-/ so I take the old one without display – and with that I can also measure length up to 0.1mm because of the nonius ;-)

      1. I don’t see how relabeling would help. The fundamental problem is that a digital caliper is a LINEAR mechanical andalog to digital converter, and with one exception, none of the scales on a slide rule are linear. (The exception, just to confuse things, is the “log” scale!)

        1. You would have to program whatever microcontroler runs the calipers to do the math for you and output the result. So basically what you would have is a digital calculator with a slider for the input rather than buttons.

          And aren’t there multiple parts that move on a sliderule? Maybe you would need multiple calipers connected together?

          If I ever learned to use a sliderule I think I would rather have the real thing because it’s one advantage over a modern calculator is it’s ability to work without electricity. But it still might be a cool project to do just for the purpose of making something that one doesn’t see every day.

  1. TL;DR: how to get quick-and-very-dirty answers without a calculator OR a slide rule, using decibels.

    The problem I always had with slide rules was that for estimating, where I only really needed one significant digit, I was more likely to get the wrong answer with a slide rule, just from messing up the decimal point, which meant that I had a result that had three digits of precision, but was off by a factor of 10!

    This is where memorizing a decibels table can be both faster and less hazardous than using logarithms OR a slide rule.

    dB linear
    0 1.0
    1 1.25
    2 1.6
    3 2.0
    4 2.5
    5 3.2
    6 4.0
    7 5.0
    8 6.4
    9 8.0
    10 10.0

    If you were a photographer in the film days, you recognize the sequence as the same sequence that was used for ASA film speeds, although you would more commonly see the sequence as 125, 160, 200, 250, 320, 400, 500, 640, 800, 1000, and so on. This is a very easy table to memorize, because once you have the first three entries, you just double each of these to get the next three, and double again for the next three.

    So to get the dB equivalent of a number, you first adjust the decimal point to get the number between 1 and 10, and the number of places you move the decimal to the left is the tens digit of the dB value. Or if the number is between 0 and 1, move it to the right, and the tens part is the negative of this. Then you just find the closest entry in the linear column, and the dB column gives you the 1s digit of the dB value.

    Of course, using dB is just a shorthand way of using logarithms, but it saves you from having to deal with a decimal point, which is very helpful if you’re doing this in your head. This is why engineers would usually rather use dB than logarithms.

    For example, to multiply 7700 by .35, first convert 7700 to dB: move the decimal point three places left, which gives us 30 dB and leaves us with 7.7. Then use the linear column to find the closest value to 7.7, which is 8, which converts to 9 dB. Add that to our 30 dB, and we have 39 dB.
    Now do the same thing with 0.35: start with the decimal point, moving it to the RIGHT one place is -10 dB, leaving us with 3.5, which is closest to 5 dB. -10 + 5 = -5 dB. NOTE: if you got -15 dB, read this again. This is the one place you can still screw this up.
    Now add the two dB values together, 39 + -5 = 34 dB.
    Now convert that back to linear numbers in the reverse way: 34 dB = 30 + 4, where 4 dB is 2.5, and 30 dB means move the decimal point three places to the right, for 2500. The actual value is 2695, but remember, we’re only looking for one significant figure, so 2500 is accurate to better than one digit.

    Division is just as quick and easy, by subtracting the dB value of the divisor instead of adding. If you can keep two 2-digit numbers in your head at the same time, and you can remember the sequence 1.0, 1.25, and 1.6, you can do all of this in your head, and never get results that are off by a factor of 10.

    As with logarithms, this is also an easy way to estimate any power, or any root of a number. With logarithms, we know that we can find the 3rd root of x by taking the antilog of ((log x) / 3). Using dB, let’s say we want the 3rd root of 385. Shifting the decimal, we get 20 dB, leaving us 3.85, which converts to 6 dB, for 26 dB. Dividing this by three gives us a little under 9 dB (because I don’t want to use a decimal point with the dBs), and converting that back to a linear number, greater than 6.4 but under 8.0. The actual calculated value is 7.27. Nowhere near as accurate as what you’d get with a log-log slide rule, but again, in the right ball park, and easier to get right on the first try.

    Not sure why I just went through all of that. Probably just to see if I could still do it.

    1. I never remebered any of the dB except 3dB = 2 and 10dB = 10. And the notion that dB+dB is multiplication, while dB-dB is division.
      Then 3dB = 2, 6dB = 3+3dB = 2*2 = 4, 9dB = 8 (or 2^3), 10dB =10, 12dB = 16 (2^4), 15dB = 32 (2^5), 18dB = 64(2^6).
      All the rest is one calculation away:
      1dB = 10dB – 9dB = 10 / 8 = 1.25
      2dB = 12dB – 10dB = 16 / 10 = 1.6
      4dB = 10dB – 6dB = 10/4 = 2.5
      5dB = 15dB – 10dB = 32/10 = 3.2
      7dB = 10dB – 3dB = 10/2 = 5
      8dB = 18dB – 10dB = 64/10 = 6.4

      But yeah, doing the calculation like you did makes, with enough excercise, estimations possible in a few seconds, which is nice when doing electronics the prototyping way, and not precise engineering.

      1. This reminds me of conversations with my mentor/Elmer K3NCO (silent key)- he repeated often: dB is ALWAYS a ratio. A “B” is just a logarithmic (base 10) ratio- the dB- a deci-Bel – 1/10th of the Bel. You could argue that anything dealing with dB is imprecise- but in the analog world, usually precise enough.

        The biggest confusion is what dB refers to, for practical purposes, you really need to include what it is relative to- which is why you might see dBV (ratio to 1V) or dBm (ratio to 1mW)- confusion between amplitude and power can get you in a world of hurt.

        Been nearly 25 years since I worked with K3NCO- an awesome man. Miss him.

        1. The admonition that dB is always a ratio is a) not true, and b) necessary because it looks like a unit of measure, but is not, and is often misused as a unit of measure. dB is a ratio when used without a unit of measure, for example a 6dB amplifier is one whose output power is 4 times its input.

          1. Nitpicking probably, but I say dB is still a ratio when used with a unit of measure. It’s just a ratio to the base unit, and the base unit obviously isn’t a ratio. For example, -3 dBuV means roughly half of a microvolt.

            The tricky part comes when you try to do math and get your dB(unit) and dB(no unit) mixed up. So in practice I do see your point—if you think of it as a ratio you’ll get the wrong answer. Examples with rough equivalent in normal math:

            -3 dBm + 6 dB = 3 dBm
            0.5 mW x 4 = 2 mW

            -3 dBm + 6 dBm = approx 6.5 dBm
            0.5 mW + 4 mW = 4.5 mW

          2. I may not have stated it as elegantly or unambiguously as I might have, but my point is that suffixing a number with “dB” doesn’t make it a ratio any more than the number itself is already a ratio. When I say a device amplifies by a factor of 2, that implies a ratio, and therefore saying that a device amplifies by 3dB, that also specifies a ratio. But if I’m just using dB as a more convenient way of specifying the logarithmic form of a number, the necessity that it be a ratio is removed. I can say that 7700 * 0.36 is a number, without it representing a ratio of two quantities, and likewise I can do the actual arithmetic for that expression using logarithms without changing that fact.

      1. And NOW you’ve got me started! The aperture scale is logarithmic, for the same reason the film speed is, but there’s a difference: the aperture is a base 2 log, while the film speed is a base 10 log. The similarity is made possible by the fact that log10(2) is .301, which means that 3dB is very close to 2 (since dB(x) is 10 * log10(x)). THe error is only about 0.3%, so for the rough calculations that dB is meant for, we can pretend that 3dB IS 2, and the error is usually smaller than your measurement accuracy anyway. The aperture scale, on the other hand, is based on the fact that the amount of light that enters a lens is proportional to the square root of the lens’s aperture, which is why we have marks for every power of the square root of 2, such as 1, 1.4, 2, 2.8, 4, 5.6, 8, … Of course you COULD use a base 2 log system to do estimations, but you would lose the feature of having the 10s digit tell you where to move the decimal point.

        This “close enough for estimation” characteristic of decibels has another interesting application: musical scales. In western music, we measure pitch in semitones, which is a logarithmic scale based on the twelfth root of 2. The frequency of each note * 1.0594 (the 12th root of 2) is the frequency of the next semitone up on the scale. This is the 12th root of 2 because it takes 12 semitones to make an octave, and an octave is exactly twice the frequency, so a semitone is 1/12 of an octave, on a logarithmic scale. The happy coincidence (like the 3dB thing) is that four semitones is a ratio of 1.26, which is close enough to a 5:4 ratio to sound “right”, and seven semitones is 1.498, which is also close enough to 3:2, and four of the other five notes of the major scale are also perfect enough that music sounds good. I’ll take the magic wherever I can get it.

  2. A nice steel ruler with a build in slide rule would be a handy tool. Rarely need a slide rule in the field but on the drafting table is where we are crunching all the less ordered systems of measures. I was just going 1:4 scale on an enclosure draft that had a lot of details to it that took up the edge of my rough draft calculating. A slide rule in a good accurate ruler would have helped tremendously more than my cell phone that kept falling all over the place.

  3. Excellent build for a first try.
    Slide rules are fascinating pieces of history and I have quite a few of them. The rule on the photo has huge gaps between the slide and the “stators”, compared to commercially made slide rules, even the cheapest ones.
    But still cudos for manufacturing one so well!

  4. A Scientific American piece about slide rules. An interesting fun read and it has a page you can print/cut to make your own. Granted it is Scientific American so it doesn’t go very deep so it is a good read for those who are kinda curious but don’t care about having 8 different scales.

  5. Most slide rules don’t have enough gauge marks. (Most have one for pi, some for e, a few for R (degrees/radian) but there should be more.) I would buy a slide rule with a well-chosen, larger than average set of gauge marks.

    Somewhere in a box in a storage unit, I have an old pamphlet explaining how to use a slide rule as a copper wire table. Gauge marks and/or scales to help with that might be useful also.

  6. Ok, so I’m old. used a slide rule all through hi-school . Had a nifty circular slide rule, made it so you didn’t have to slide the roamer back to the beginning if your answer went off the end of the rule. just had to keep track of the decimal place.
    Calculators where available , But we where not allowed to use them as they cost so much that lower income students would be at a disadvantage. One of my teachers had one, we where calculating percentages on tests, saw me using the slide rule, challenged me to a race, who could do five tests the fastest, I had all five done in the time he did one. Set the numbers (which was already done from the previous tests) move the cursor, done. By now off course I’ve forgotten the more complex function, I can still multiply and divide.

    1. Heh. I was going to college during the transition from slide rules to calculators. One day I went to a physics test and forgot my slide rule. I saw the prof had a calculator on his belt. I told him I didn’t have my slide rule, could I borrow his calculator? He smiled as he said “sure”, and handed it to me. It was an HP-35. I had never heard of RPN…suffice to say, it did’t do me any good at all.

      1. Looks like an E6-B, a flight calculator for pilots. You can’t see its most important feature, which is on the other side: a circular slide rule with an additional scale for time, so you can calculate arrival time (or run-out-of-fuel time) based on corrected airspeed, which you can also calculate. There’s also a table that takes up most of one side, that converts wind speed and direction to crosswind component. And much, much more.

  7. I always preferred the circular variety. Not as easy to hang on your belt, but much more convenient when “wrapping around the end” of a calculation. I still use a 9″ Glison Binary, with 23 scales, ©1931. The outer scale has the resolution of a 28″ slipstick!

    The Dietzgen Atlas was a unique 9″ circular with a spiral scale that could provide the resolution of a 50 foot straight slide rule, or 5-6 significant digits!

    1. When I worked in RADAR, we used an HP wavemeter to measure the frequency of our magnetrons. This was a tunable cavity coupled to a cylindrical spiral scale with a red indicator that slid in a spiral groove, for an effective scale length of about four feet.

  8. I was taking a Chemistry exam and one of my classmates bumped his slide rule which fell to the floor and broke the cursor. The professor let him make up the test later. The slide rule was an absolute must in that class.

  9. Dad taught me the slide rule because you don’t really get taught much math before 2nd grade and he was already trying to teach me electronics. I still have a couple around and they get used from time to time. USAF electronics school 1976 I came prepared with a slide rule, but QUICKLY paid an exorbitant amount for an RPN calculator. Then paid long term, over and over again, for the batteries. Still have that calculator as well.

    Thank you. Nice article. Good memories!

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