I haven’t used a slide rule in ages, and even then it was as a toy. I am too young to have ever needed to use one in anger.
I’ve been trying to remember enough high school maths to reinvent the concept from first principles, without actually checking on Wikipedia to remind me how they work – just to prove to myself I still have the concepts clear enough in my mind to explain them to someone else.
My Attempt to Explain Slide Rules To Myself
Let’s make up a function f(x). Let’s make it injective, and call its inverse f-1(x), so x = f-1(f(x))
Now, let’s go into the physical world. Take two straight sticks. Mark one end of each as the origin, and at every point which is x millimetres from the origin, write down the value f(x).
Now take two arbitrary values, g1 and g2, and find where they are written on the sticks. Line up the origin of the second stick with the position of g1 on the first stick. Hold them parallel, oriented appropriately, and read off the number next to the position of g2 on the second stick.
What have you done, in mathematical terms?
Well, lining up the sticks is a rapid way to do addition. What are you adding? The distance in millimetres between the origin of the first stick and the location of g2 on the second stick is f-1(g1) + f-1(g2 ).
But, you aren’t measuring it in millimetres, you are measuring it using the first stick, so the number you read off is actually f(f-1(g1) + f-1(g1)).
Now, with a traditional slide rule, we would use the power function for f.
e.g. f(x) = 10x, f-1(x) = log10(x).
That makes a slide rule a quick way to perform the following operation:
10log (g1) + log (g2)
which is another way of saying g1 × g2.
My Sudden Realisation
Power/Log functions aren’t the only possible functions that slide rules could be used for.
If you let f(x) = sqrt, and f-1(x) = x2, then what you get is the square root of the sum of two squares. You could make a special slide rule just to work out the length of the hypotenuse of a right-angled triangle!
I have no doubt that this has been known for years, and such slide rules probably even existed, but I am happy to have discovered it myself!