Apollo 13 is one of my favorite movies; I’ve watched it many times. And every time I do, there’s a moment in the movie where I boggle yet again that we sent men to the moon WITH SLIDE RULES.
I know slide rules are very powerful calculation devices! But I’m used to all my complex math being done by a computer — and I don’t mean a human being, which is what the term “computer” used to refer to: a person who computes. (We also sent men into space with those, e.g. Katherine Johnson. When John Glenn was about to orbit the earth, he refused to trust the figures from the newfangled electronic computers until Johnson verified them herself.)
Over the ages, we’ve invented tons of devices for helping us perform calculations. Our hands are the most basic — but not without power; chisanbop is a method for using the hands rather like an abacus. It was invented in Korea in the 1940s, but I don’t know of any reason it couldn’t have existed sooner. You can even perform multiplication with your hands, if you know the techniques.
As for the abacus (also called a counting frame, and existing in variant forms like the Japanese soroban), it’s probably the most widespread and famous device. It’s a remarkably effective design, generally using rows of movable beads on wires; you can build it differently, but that setup allows you to operate very fast without flicking your markers off into the corners of the room. Abaci can perform not just addition and subtraction, but multiplication and division, and even complex operations like square or cubic roots. They were particularly useful before we invented positional notation, i.e. the “ones place” and “tens place” and “hundreds place” arrangement we’re all used to; you can’t really perform calculation directly with, say, Roman numerals. An abacus makes those things vastly easier.
Before we had the abacus, though, we had simpler devices. Tally sticks are more a form of numerical record than calculator, but having a record of your numbers is a key step in being able to perform operations on them. To figure out the products of things like addition and subtraction, you might use a counting board, a surface marked with lines to help you sort and track your markers: think a simpler abacus. The board could instead be a cloth, easier to fold up and transport; you can imagine the utility of that for merchants on the road.
Chinese counting rods are a particularly interesting device. I’m not well-versed in how they work, but they’re bars whose horizontal or vertical placement on a board can be used to represent particular numbers, making this a positional system. But unlike a basic counting board, these rods can represent negative numbers, because they came in two colors: red for positive, black for negative. (Apparently later versions use a go stone to stand for zero.) There were ways to transcribe the system onto paper, with a diagonal strike like our modern tally marks denoting negative numbers; looking at how different numbers were written out, it’s hard not to wonder if it influenced the design of Hangul, the Korean writing system. You could even perform a variety of operations via the methods known as rod calculus — but in the long run, that was slower and less convenient than the abacus, and so it fell out of general use.
Ropes have also been pressed into service for this job. When it comes to the Andes, we’ll probably never be fully sure how quipu worked; too few of them survive, with too few records giving us hints as to their interpretation. We do know, though, that they used knotted strings to encode numbers, with a base 10 positional system. It’s not only a numerical record, though, and color-coding also played some kind of role. (To be clear, I use the past tense because while quipu are still being made, modern iterations are different from the ancient kind. Also, yupana are in a similar boat as quipu: we’re fairly sure these geometric arrangements of boxes were used sort of like an abacus, but exactly how is a matter of debated theory.)
Meanwhile, over in medieval Europe, larger ropes with knots were also used for calculation, albeit only in a basic fashion; count five knots along the rope, then seven more, to find out what 5+7 is. On the other hand, arithmetic ropes of that kind can do something an abacus can’t, which is geometry: tie the knots at sufficiently regular intervals, and you can not only measure distances like circumference or radius, but use the rope to create triangles and circles of the necessary size. Otherwise, anybody who’s taken a geometry class will be familiar with the simple tools of the straight-edge or ruler, protractor, and compass for the construction of figures.
But calculation devices can get vastly more complex than sticks and ropes without ever needing electricity to function. The Antikythera Mechanism is a famous ancient example, though we’re still figuring out exactly how it worked. I’m particularly fond of Napier’s bones, both because they have a cool name, and because they can be super pretty! They operate on the basis of mathematical tables, which are another (much simpler) device: do a whole bunch of math by whatever means are available to you, then record the various results and just look up the answers when you need them. Napier’s bones record multiplication tables on a set of rotating rods, which you then manipulate to solve problems of multiplication or division by treating them like addition and subtraction. As with an abacus, you can also calculate square roots this way.
Get even more advanced, and you’re looking at something like Charles Babbage’s Difference Engine. This one handled polynomial functions . . . if you could build it. The engineering capabilities of 1830s England meant that producing the intricate gearing required was prohibitively expensive; Babbage himself only ever built a partial model. His Analytical Engine, which would have been even more powerful, never got past the design stages. Still, it left its mark on the world: by borrowing ideas from the jacquard looms of the time, it prefigured the punch cards of early twentieth century computing. Ada Lovelace, who worked with Babbage on the concept (and imagined many possible uses for the machine besides straightforward mathematics) is memorialized today as the world’s first computer programmer.
This isn’t the full array of techniques and devices, of course. We’ve used practically anything that came to hand to help us do math, because performing it all in your head is hard. In fiction, though, we almost never see merchants manipulating an abacus or laying out a counting cloth to calculate prices and profits, and even siege engineers seem to fling stones at cities under assault without once doing a bit of math. Even when the mathematical specifics aren’t the focus of the story, just a line of description can add verisimilitude . . . and given how much we used to protect such knowledge, the power of more sophisticated devices could be cause for outright espionage!
6 thoughts on “New Worlds: Calculation Devices”
On the TV series “Secrets of the Castle” currently streaming on Amazon Prime they deal with a 25 year acheology project to build a 13th C castle from scratch with the tools and tech of the day. In one episode our intrepid archeologist spend a week with the master masons. It all comes dowon to geometry with some very basic tools.
In another episode they get to play with a trebuchet and again use geometry to calibrate the thing.
I always like geometry better than algebra, I could see practical, everyday uses for it and not get lost in a tangle of numbers.
I always enjoy reading your articles. Very informative and enjoyable.
Never heard of Napiers Bones. Counting Bars, or the Yupana; thanks for explaining them; any books you’d suggest for further reading about them? Thank you!
My observation is that most readers run away from any mention of mathematical worldbuilding unless you’re writing very specific kinds of sci-fi. :/ I took out the mention of use of a CAS (computer algebra system – used for, hmm, certain kinds of computations/symbolic manipulations in modern mathematics, e.g. listing left cosets of a group) in Ninefox Gambit because one of the betas readers complained that relying on a computer didn’t make Cheris seem like enough of a math genius!
I saw a sliderule once. My calculus teacher brought it in so we could see one.
Slide rules taught people important skills that seem to be forgotten these days:
2. Estimating the answer.
3. Significant digits.
4. Unit conversion.
It’s not just ancient history, either. Many of those methods were in widespread use a mere half century ago.
Back in the early 70s, I was a computer programmer and consultant, and often used a slide rule to check people’s numbers – it was quicker than hand calculation but less accurate (2 digits) – more accurate slide rules were huge. Log and similar tables were OK up to about 4 digits, but slower – for a mathematician, mental calculation was often faster. As Phyllis Radford says, there are also graphical methods (again about 2 digits, but 3 on a large scale with great care) – many them are related to Euclidean methods of (say) proving Pythagoras’s theorem, and were ancient even in mediaeval times. In recent times, there was also log scale graph paper.
However, there are also a lot of mathematical tricks, some of which were used to make such operations easier for non-mathematicians such as ‘Russian multiplication’, and others to make them faster for mathematicians. Many of the former are ancient (i.e. described at least two millennia ago) and were also algorithms used on an abacus, as well as probably other mechanisms. The latter are recent (a few hundred years). Both help, depending on the user. Most of the devices used (including abacus) are actually used mainly as mneumonic aids and the actual methods used are those ancient tricks.
I have used (and occasionally taught) most of the above. But, as you say, the skills are those of my generation (I am 74) and previous ones.