Napier's Bones

Let me explain:

I was playing with Apple's Make QTVR beta yesterday [12 April 1996] and suddenly had the urge to rotate a very wide image just a skosh to make its ends match exactly. (Fancy that.) But how much to rotate?

Hmmm. Looks like a job for Trig. So I asked my wife if we have a scientific calculator that works. "No," she said, "the last one broke a while ago." (So the question arises, can I even remember how to use Trig?)

There might be a billion or so scientific calculator programs out in FTPdom, but I didn't think of that. Instead I went to my workroom and started rummaging in boxes, where I found this thing.

Some of you might not know what a slide rule is. 'Way back in the dark ages, before electronic computers, calculators, or even transistors existed, folks needed some way to do math. For addition and subtraction they had mechanical adding machines, but a slide rule (a.k.a. "slipstick") was the really trick way to multiply, divide, raise powers, take roots, and do trig functions.

The essence of a slide rule is just a couple of logarithmic rulers. You might know that you can multiply or divide two numbers by adding or subtracting their logarithms. The slide rule lets you do this by adding or subtracting their lengths on two sliding logarithmic scales (labeled C and D). For trigonometry, there are extra scales (S and T) that relate angles to numbers according to the sine and tangent functions. These are the essential scales; scales for other functions are based on the same idea.

I don't know who invented the slide rule [Found out: William Oughtred, in 1622]. The mechanism, if you can call it that, might have been used for other purposes as far back as you care to imagine. A nearby book, Martin Gardner's *Logic Machines and Diagrams* says that in the 1300s a monk named Ramone Lull used something like a circular slide rule as a permutation engine to search for devine truths (sort of a secret decoder ring for the soul). As I understand it, he found permutations.

But the slide rule as we know it (or knew it) depends on logarithms, which were invented in the late 1500s by a Scottish baron named John Napier. I once heard someone call the slide rule "Napier's bones," presumably in his honor. [Readers have since informed me that Napier himself devised a set of ivory rods, the original Napier's Bones, which anticipated but were rather different from Oughtred's slide rule.]

So back to my story. I wanted to raise the right end of a 5900-pixel-wide image by 85 pixels. The angle I needed is the arctan of 8.5/(5.9X10^2), so I slid 8.5 on the C scale over to 5.9 on the D scale, then slid the hairline to the left D index. The result (on C under the hairline) is 1.44X10^-2. That's one order of magnitude low for the T scale so I looked at ST (10 times sine or tangent) for the arctan. Bang! I needed to rotate the image 0.825 degrees.

OK, so I didn't remember how to do all that immediately. But I had an answer that worked perfectly in much less time than it would have taken to go buy a new calculator... and for less money!

Slide rules like this one (more often about 12 inches long) used to be pretty handy for everyday math. Engineers who needed more precision used tables, or sometimes huge metal slide rules with very fine graduations. Nowadays, even a cheap scientific calculator is far more accurate, and I'm a big fan of having more accuracy available when I need it.

Even so, I recommend trying out a slide rule sometime if you can get your hands on one (try here or here, or try this simulation), and can find instructions for using it. When I need to understand logarithms (which, I admit, is not often), the slide rule gives me a "hands-on" grasp like nothing else can.

Also, there's nothing like working out a few practical problems on a slide rule to ease your mind about rounding. There's a natural tendency to treat a calculator's digits as if they all mattered. But if you really need more than four or five significant digits, tell your boss you need a raise.

Let's not forget slide rules entirely. They're cool, and they don't need batteries (or even solar cells).

Check out the Museum of HP Calculators and the International Slide Rule Group for lots of slide rule information and links.

Last change: Updated links, 12 February 2009

Marcus Brooks