Simanaitis Says

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THERE ARE blessedly few of us these days who remember the slide rule, and fewer still who understand its workings. So why not join this hardy band and learn some arcane knowledge?

True, some students wore slide rules on their belts (not math majors, though…).

You need grapple with only two mathematical facts: Addition can be represented geometrically. And—rather more subtly—the logarithm of a product is equal to the sum of the logarithms.

I’ll expand on the first with a mathematical example. I’ll sidestep the second entirely with an example of mathematical humor.

Suppose I have one stick that’s 2 units in length and place it end to end with one that’s 3 units in length. There’s no surprise in noting that these two, end to end, equal one that’s 5 units in length. Numerically, 2 + 3 = 5. Geometrically, the same thing.

Addition can be represented geometrically.

In general, if a stick of A units in length is abutted with one of B units, the combined length is the same as a stick that’s (A + B) units long.

Okay, on to logarithms. If you’re forced to think of them at all, think of logarithms as the reverse of exponents. That is, BA = C has the logarithmic equivalent logB (C) = A. And just as the exponent rule says BA x BD = BA+D, so it is that logB (A x D) = logB (A) + logB (D).

A math joke: The Viperidae family of adders was threatened by extinction because they weren’t reproducing. A mathematician built them tables made of logs. Why? Because with log tables, adders can multiply.

And so it is with a slide rule. Its numerical scales are arranged logarithmically, the basic C and D scales reading from 1 to 10, the CI scale inverted and running from 10 to 1. There’s a sliding hairline cursor as well.

To multiple 2 x 3, say, set the cursor on the D scale’s 2. Slide the CI scale until its 3 aligns with the hairline, thus adding 2 logarithmic units on the D scale to 3 logarithmic units on the CI scale, end to end. Your answer 6 is at the end of the CI scale.

Here’s the slide rule’s calculation of 2 x 3 = 6. The other scales are for square roots, calculations using π and other applications.

Wait a second! I could’a done that in my head!

True, but what about 2.18 x 3.52? My slide rule shows it’s around 7.67. (My iPhone gets precisely 7.6736—but, no fair, we’re talking non-electronics here!)

To divide 3 by 6, align the cursor to 3 on the D scale and slide the C scale until 6 coincides with the hairline. The end of the C scale is your answer. Note, you must recognize it’s 0.5, not simply 5. That is, slide rules give numerical answers; you decide where decimal points go.

Slide rules can be circular as well. This well-used one of mine is calculating 15 ÷ 2 = 7.5.

Circular slide rules are compact. But my favorite slide rule is a cylindrical one patented by civil engineer Edwin Thacher in 1881. It’s about 20 in. overall, a rotating housing of 20 vanes containing a sliding and rotating cylinder.

This Thacher slide rule in my collection, used in a U.S. Steel plant, was made by Keuffel & Esser (“K&E,” familar to all engineers of a particular age). Oddly, it identifies both “Thacher” and “Thatcher” as inventor.

The Thacher’s vanes and enclosed cylinder contain folded logarithmic scales. Together, the result is equivalent to a slide rule that’s 30 ft. in length. Readings are to 4-place accuracy—in some ranges, 5-place.

Of course, the digital age displaced all these devices as mere curiosities. But fascinating curiosities they remain. ds

© Dennis Simanaitis,, 2012

2 comments on “WHAT’S A SLIDE RULE, GRANDPA?

  1. oldmaven
    October 29, 2012

    Been decades since Iknew how to use one, but I still salivate when I hear “log-log duplex decitrig.”

  2. Tom Tyson
    November 5, 2012

    I still have a 9″ circular slide rule in the bottom drawer of my desk just in case the batteries in both my HP-48 and my iPhone should die.

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This entry was posted on October 29, 2012 by in Sci-Tech and tagged , , , , .
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