# OddThinking

A blog for odd things and odd thoughts.

### Breaking the (Slide) Rules

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).

Okay, not every point, but frequently enough that you cover the important values of 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.

More excitingly, division can be performed by aligning the sticks differently, but I am skimming over that here.

#### 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!

1. The major advantage that slide rules have or had over the old desk calculators or some of our modern graphing equivalents is that they give a visceral, visible way of interacting with these (and many other) functions. You really get a sense for how f(x) grows/shrinks as x increases. Graphs have visibility going for them, but tend not to be as malleable – tweaking the equations to see how they change is harder without sliders to make the changes to the outputs more fluid.

It’s interesting that the exponential and log functions are useful for multiplication (and several other operations), but I’m not entirely sure what use the result would be if you did the same sequence of steps with squaring and square-root functions – sqrt(2x^2) isn’t all that interesting unless you’re dealing with a 45 degree right-angled triangle (or some mapping of that). Perhaps it’s just that some functions are just more versatile than others? (As an aside: would Pythagoreans think they had an exact value for sqrt(2) if they used one of these?)

Perhaps you just need to be able to combine several nth power slide rules to solve quadratic or higher power equations?

Would a slide rule work with a sine function? What about discontinuous functions like 1/x?

2. Richard,

I think there may be some confusion in your second paragraph. Such a sqrt based slide-rule (and I have now checked, and they did exist) wouldn’t (merely) calculate sqrt(2x2); it could calculate sqrt(x2 + y2) which is far more useful.

As for the other functions:

Is arcsin(sin(x) + sin(y)) a useful function to calculate? It doesn’t ring any bells for me?

1/(1/x + 1/y)) could be useful. For example, isn’t that used in electronic engineer for resistance?

The discontinuous nature isn’t an issue, per se.

There are three important issues though.

The first is that you need to be able to interpolate between points written on the stick, so if there are many discontinuities, you need space to write them all down – no Mandelbrot Set slide rules I am afraid.

The second is that that the function needs to be injective.

The third is that if the function isn’t continuous, it may be difficult to find the point corresponding to a value on the stick. Not impossible, just harder.