The Universe of Discourse

Sat, 15 Sep 2007

The Wilkins pendulum mystery resolved
Last March, I pointed out that:

  • John Wilkins had defined a natural, decimal system of measurements,
  • that he had done this in 1668, about 110 years before the Metric System, and
  • that the basic unit of length, which he called the "standard", was almost exactly the same length as the length that was eventually adopted as the meter
("John Wilkins invents the meter", 3 March 2006.)

This article got some attention back in July, when a lot of people were Google-searching for "john wilkins metric system", because the UK Metric Association had put out a press release making the same points, this time discovered by an Australian, Pat Naughtin.

For example, the BBC Video News says:

According to Pat Naughtin, the Metric System was invented in England in 1668, one hundred and twenty years before the French adopted the system. He discovered this in an ancient and rare book...
Actually, though, he did not discover it in Wilkins' ancient and rare book. He discovered it by reading The Universe of Discourse, and then went to the ancient and rare book I cited, to confirm that it said what I had said it said. Remember, folks, you heard it here first.

Anyway, that is not what I planned to write about. In the earlier article, I discussed Wilkins' original definition of the Standard, which was based on the length of a pendulum with a period of exactly one second. Then:

Let d be the distance from the point of suspension to the center of the bob, and r be the radius of the bob, and let x be such that d/r = r/x. Then d+(0.4)x is the standard unit of measurement.
(This is my translation of Wilkins' Baroque language.)

But this was a big puzzle to me:

Huh? Why 0.4? Why does r come into it? Why not just use d? Huh?

Soon after the press release came out, I got email from a gentleman named Bill Hooper, a retired professor of physics of the University of Virginia's College at Wise, in which he explained this puzzle completely, and in some detail.

According to Professor Hooper, you cannot just use d here, because if you do, the length will depend on the size, shape, and orientation of the bob. I did not know this; I would have supposed that you can assume that the mass of the bob is concentrated at its center of mass, but apparently you cannot.

The usual Physics I calculation that derives the period of a pendulum in terms of the distance from the fulcrum to the center of the bob assumes that the bob is infinitesimal. But in real life the bob is not infinitesimal, and this makes a difference. (And Wilkins specified that one should use the most massive possible bob, for reasons that should be clear.)

No, instead you have to adjust the distance d in the formula by adding I/md, where m is the mass of the bob and I is the moment of intertia of the bob, a property which depends on the shape, size, and mass of the bob. Wilkins specified a spherical bob, so we need only calculate (or look up) the formula for the moment of inertia of a sphere. It turns out that for a solid sphere, I = 2mr2/5. That is, the distance needed is not d, but d + 2r2/5d. Or, as I put it above, d + (0.4)x, where d/r = r/x.

Well, that answers that question. My very grateful thanks to Professor Hooper for the explanantion. I think I might have figured it out myself eventually, but I am not willing to put a bound of less than two hundred years on how long it would have taken me to do so.

One lesson to learn from all this is that those early Royal Society guys were very smart, and when they say something has a mysterious (0.4)x in it, you should assume they know what they are doing. Another lesson is that mechanics was pretty well-understood by 1668.

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