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.
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)