I’m moving offices: just next door, but I have about 20% less shelf space so I figured it was a good opportunity to see if there were any books that I didn’t really use (answer: yes). In looking through one (Man and Number by Donald Smeltzer, which is a distracting title even though I can see that this book was published 50 years ago) I found another way to multiply! It’s not anything dramatic, but was apparently described by Nichomachus of Alexandria over 1900 years ago in his Introduction to Arithmetic. It uses the following fact:
Here’s the method: suppose first that a and b are both odd or both even. Let x be the average of the two numbers (so x is a whole number, because their sum must be even) and let y be the positive difference between x and a or b. The radius, as it were. Then:
The example given in the book is:
You still have to know your perfect squares, but if you happen to have a table of squares like the Babylonians, that’s no trouble at all. (Indeed, this is basically Formula (4) here, so I don’t know if this should count at all. But I like having an actual citation for the method.)
But what if you have an even and an odd number? Never fear, just ignore that pesky odd bit and add it on at the end. For example, if you have 24·15, you know that this is 24·14 plus an additional 24. So you find 24·14 as above, and add 24 to get 360.