Log-rhythmic gymnastics

I recognized the knock.  “Come on in, Jeremy, the door’s open.”

“Hi, Mr Moire.  Can you believe this weather?  Did Miss Anne like her gelato?  What’s this funny ruler thing that my Dad sent me?  He said they used it to send men to the moon.”

log rhythm

“No, yes, it’s called a slide rule, and he’s right — back in the 1960s engineers used slip-sticks like that when they couldn’t get to a four-function mechanical calculator.  Now, though, they’re about as useful as a cast-iron bath towel.  Kind of a shame, because the slide rule is based on mathematical principles that are fundamental to just about all of mathematical physics.”

“Like what?”

“The use of exponents, for one.  Add exponents to multiply, subtract to divide.  Quick — what’s 100×100×100?”

“Uhh…  Ten million?”

“Nup.  But if I recast that as 102×102×102=102+2+2?”

“106.  Oh, that’s a million.”

“See how easy?  We’ve known that kind of arithmetic since Archimedes.  The big advance was in the early 1600s when John Napier realized that the exponents didn’t have to be integers.  Take square roots, for example.  What’s the square root of 100?”

“Ten.”

“Sure — √100=√(102)=102/2=101=10.  Now write √10 with exponents.”

“Would it be 101/2?”

“Let’s see.  Do you have a spreadsheet app on that tablet you carry?”

“Sure.”

“OK, bring it up.  Poke =10^(0.5) into cell A1, and =A1^2 into A2.  What do you get?”

“Gimme a sec … the first cell says 3.162278 and the second says … exactly 10.”

“Or as exact as that software is set up for.  So what we’ve got is that 0.5 is a perfectly good power of ten, and exponent arithmetic works the same with it and all the other rational numbers that it does with integers.  Too big a leap, or are you OK with that?”

“OK, I suppose, but what does that have to do with this gadget getting people to the Moon?”

“Take a good look at at the C scale, the lowest one on the middle ruler that slides back and forth.  Are the numbers evenly spaced out?”

“No, they’re stretched out at the low end, scrunched together at the high end.”Slide rule 3“Look for 3.16 on there.  You read it like a ruler — the number before the decimal point shows as a digit, then you locate the fractional part with the high and low vertical lines.”

“Got it.  About halfway across.”

“It’s exactly on center if that’s a good slide rule.  A number’s distance along the scale should be proportional to the exponent of 10 (we call it the logarithm) that gives you that number.  The C scale’s left end is 1.0, its right end is 10.0, and 3.162 is halfway.”

“Ah, I see how it works.  Adding distances is like adding exponents.  So if I want to multiply 2 by 3 I slide the middle ruler until its 1 is against 2 on the D scale, then I look for 3 on the C scale and, yes! it’s right next to 6 on the D scale!  Oh and the A and B scales wrap twice in the same distance so they must be logarithms for squares?  Hah, there’s 10 on B right above where I found 3.16 on CK wraps three times so it must be cubes, but why did they call it K?”

“Blame the Germans, who spell ‘cube‘ with a ‘k‘.  What do you suppose CI does?”

“Hmm, it runs backwards.  Adding with CI would be like subtracting distances which would be like dividing, so … I’ll bet it’s ‘C-Inverse‘!”

“You win the mink-lined frying pan.  So you see how even a simple 5- or 6-scale device can do a lot of calculation.  The really fancy ones had as many as a dozen scales on each side, ready for doing trigonometry, compound interest, all kinds of things.  That’s the quick compute power the rocket engineers used back in the 50s.”

“Logarithms did all that, eh?”

“Yup, that and the inverse operation, exponentiation.  Of course, you don’t have to build your log and exponent system around 10.  If you’re into information theory you might use powers of 2.  If you’re doing physics or pure math you’re probably going to use a different base, Euler’s number e=2.71828.  Looks weird, but it’s really useful because calculus.”

“So logarithms do calculating.  You said something about physical principles?”

“Calculating growth, for instance…”

~~ Rich Olcott

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