The sophisticated basis of old technology

Our astronomer spent a few years in England, where he learned something of the arcane art of ringing church bells.  In the belfries of that country are wonderful examples of essentially Medieval technology that also demonstrate advanced mathematical and physical ideas.

Our astronomer writes:

It’s hard to find a device much simpler than a church bell.  A sort of rounded cone normally of bronze (which itself may be not at all simple to make) is hung, mouth down, with a pivoted clapper inside, so that when it swings back and forth it makes a sound.  The mounting may be massive, but need not be sophisticated.  The iron fittings holding it all together could even be made by the village blacksmith; I’ve seen old bells where not even threaded bolts appear, their place being taken by “feather wedges.”

And yet, this is a highly effective technology.  The sound of bells can easily carry for many miles, powered only by a pair of hands pulling on a rope.  Before gunpowder it was the only way to signal quickly to that kind of distance, and even afterward it was the most practical.  As a public address system a tower of bells could only be bettered after electric loudspeakers were invented, a much more sophisticated sort of thing.  (Even now bell-ringers are admonished not to ring them in a certain order, because that signified invasion!)

So a tower of bells is a fine example of workable primitive machinery.  As such, one would expect it to exemplify only simple things.  But an English ring of bells gives rise to some sophisticated mathematics.

A bell that swings a short distance has its own period of swing, determined only by the distance between its pivot point and its center of gravity.  This is simple harmonic motion, a subject studied by innumerable undergraduates in science and engineering.  Since bells wiegh hundreds of pounds (up to several tons), pushing them to ring more quickly or more slowly is just not an option.  In fact, to get them to ring at all one pulls on the bell rope at intervals determined by their period, building up a back-and-forth motion large enough for the clapper to fall against one side and then the other.  This is an application of resonance, exciting a system at a particular frequency to get a large response.  It’s the same thing as getting a playground swing to go high; it’s also how your radio picks out one radio station to listen to rather than another.

A set of bells ringing at their own speeds, mouth-down, produces a lot of sound, but of course there’s no structure to it.  Well, some few centuries ago the English mounted the bell-rope on a wheel, so that it would exert a pull on the bell no matter how much the bell was swinging back and forth–even if it swung all the way around.  You could then, by applying more resonant tugs, get these hundreds of pounds of bronze to balance mouth-up, an unstable equilibrium.  And as it swung through its long arc it became a physical pendulum, whose period is very sensitive to how far it’s swinging.  We’re no longer dealing with simple harmonic motion, and the mathematics of this is only given to advanced undergraduates or graduate students.

The point of adding the wheel was not, of course, to demonstrate mathematics.  Once you’re ringing to the top of the arc, or nearly so, small tugs on the rope can control when the bell rings.  This means you can sound your bells in any order you wish.  In practice, there are limits to how much you can change the period of a ten-ton bell from one clang to the next, so by convention you only change the order by one place at a time.  You don’t ring tunes on the bells.  But you do work with permutations and even group theory, branches of mathematics that also find applications in quantum mechanics.  Consider this problem: given a ring of six bells, how many different orders are there to ring them in?  And, starting in descending tones in the order 123456, how do you go through all those different orders (“ringing the changes”) by changing each bell no more than one place at a time, without repeating any?

There are several answers.  But for Medieval technology it’s a pretty sophisticated question.