Saturday, March 18, 2017

Sliding scales

The Egg bought me this slide rule, I also have the somewhat 
more sophisticated K&E slide rule that my mother-in-law, 
Gai Hamburger Donnay, bought with them money she 
earned tutoring Jackie Robinson in chemistry during her 
undergraduate days at UCLA. The log tables are from my 
father's, in his CRC Handbook of Mathematics.
I've been playing with scales off and on for the last couple of weeks. Not musical scales, or pay scales, or sliding fee schedules, but logarithmic scales on a slide rule for an essay I'm writing about computational methods.  What do we use, what did we use, and how do these options affect what we do in research and what we teach?

As part of this effort I learned to add and subtract on a slide rule.  I mentioned this to a younger colleague who seemed unimpressed, why not, weren't slide rules just the equivalent of the calculator, instead of pressing buttons, operations were done by some sort of sliding algorithm?

Ah, but the miracle1 of using logarithms to do computations was that you could multiply two numbers by adding two numbers. To multiply two numbers, say 2378 and 3467, you looked up the logarithm of each in a table — 7.774 and 8.151 respectively — added them together (15.925) and found the number corresponding to this new logarithm to arrive at the answer:  8,240,000 (to 3 significant figures, the exact answer is 8,244,526).  Put in symbolic form LOG(A x B) = LOG (A) + LOG (B).  These "logs" didn't help with addition in any way.

But you can use multiplication to add in an admittedly roundabout way.2 To add A and B:
  • Divide A by B.
  • Add 1. (Ok, yes, this is addition, but trivial to do in your head).
  • Now multiply the result by B.
  • The result is the sum of A and B.
This is supposedly handy if you are mid-calculation on a slide rule and don't want to pull a number off to paper, then return to working the rest out on the slide rule.

With a bit of practice, I'm once again getting quick with doing equilibrium problems for general chemistry, faster mid-lecture than pulling out and unlocking my phone to use the calculator on it (and my last standalone calculator bit the dust this week, after a long, long, useful life.)  Then again, when I have a quadratic to solve, these days I can just say "Hey, Siri...."

There were other methods for doing multiplication of two number by adding two numbers based on trigonometric relationships, which led me to learn the word prosthaphaeresis.

I note that one should be impressed with the tables.  It took Napier 20 years of calculations to construct those tables.

1.  And a miracle they were thought to be from the very first, John Napier's book describing his invention (published in 1614) was titled  Mirifici Logarithmorum Canonis Descriptio (A Description of the Wonderful Rule of Logarithms).
2. For the algebraically inclined,  this translates to A+B = B (A/B + 1)

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