Our first museum visit in Bonn was to the Arithmeum, housed in the department of mathematics at the university. It is an easy walk from the main station (through a park which has a play structure on the other side). The museum is in the atrium and stairwell of the building, a very light and open space.

I had expected a museum about mathematics in some of its higher forms, which lend themselves to artistic and other representations. And there was a bit of this in an exhibition of paintings on the walls. We noticed these but didn’t really focus on them.

The main exhibition is about arithmetic (duh, given the name), particularly the development of aids and machines to perform calculations. We ran out of steam before the development of electronic machines (the pun was not intended but is apt) but the museum does go right up to the development of computers with some interesting looking displays of the innards of some of them and explanations of how they work.

Historically, you start on the top floor and work your way down. The larger exhibition space is on the main floor. But the historical stuff is really interesting. While I have often remarked at the relatively short period of time that we have moved from computers the size of houses to computers you can fit in your pocket, I hadn’t realized that it is only about 900 years since Europeans have even had a system of written numbers that allow us to do calculations on paper.

This provides an interesting twist on the debate about whether children should be using calculators for calculations. The ability to calculate on paper is, historically, not very significant and seems to come between older forms of reckoning (using objects to count with, and more complicated arrangements of objects such as the abacus) and mechanical and electronic machines to perform calculations. In fact, it seems that the development of machines to speed up calculation is crucial to the development of higher mathematics as it enables more time and energy to be spent on the concepts.

The ability to calculate requires a system of written numbers that includes a concept of zero not only as the null set, but also as a place-holder in a system of numbers. Roman numerals do not have this. This is first develop in India, then adopted by the Arabs who bring it to Europe (through Spain) in the 12th century (I think, I may be remembering the dates badly). All that stuff about place value is actually a major advance in mathematical thought.

We also learned that the word “calculate” probably derives from the Greek word for stones (calculi – I assume calciferous stones, and the examples shown were white). These were used early on for reckoning (long before the development of a system of numbers). There were also several types of abacus, and reckoning boards. So, for a considerable time, calculation was done primarily by counting things (knots on cords, stones, beads, coins, etc).

The ability to calculate on paper, which we now consider normal, seems to quickly create a problem of speed. In order to work on more complicated mathematical concepts, complicated calculations are required. Mathematicians would rather work out the mathematics than perform calculations and a demand arises for aids and machines (the latter are distinguished by an automatic tens-carry mechanism) to make the calculations faster.

Because the early demand is from mathematicians, early developments aim to develop something that could do all 4 operations. Interestingly, as the technology developed, this became less important. When we get to the period of mass produced machines, they mostly just add (and perform multiplication as repeated addition). This is largely because the market is business people who need to do invoicing and so on. It isn’t until the development of electronic calculators that the ability to perform all four arithmetic functions returns.

Of the aids to calculation developed in the earlier period, we were particularly taken with Napier’s rods. These then form the basis of several calculating machines. Tigger even took out her notebook and made some notes. I think we might try to make some, as they are quite interesting. And I think they might be useful for those of you teaching multi-digit multiplication to your kids.

I would definitely recommend this museum. Some of the old machines (or reproductions of them) can be tried. And many have mirrors strategically placed so that you can see how they work. It would be even more interesting for those folks interested in how things work as there was a lot of detail about the kinds of gears needed to make a tens-carry mechanism work and so on. We found this interesting but only have a limited interest in such details.