## Cobbler’s wax and sirens: lessons from Kelvin’s lecturing equipment

I am never content until I have constructed a mechanical model of the subject I am studying. If I succeed in making one, I understand; otherwise I do not.

William Thomson, Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light (1902)

The Hunterian Museum at Glasgow University has an exhibition celebrating William Thomson, a.k.a. Lord Kelvin. (It lurks upstairs in the gallery and isn’t immediately obvious, but it’s well worth a visit.) An aspect that it emphasises, for obvious reasons, is Kelvin’s role as a teacher. This had two aspects, his academic lecturing and his popular lecturing: while the former seems to have received some criticism from his contemporaries, as Craik (2012) notes (and see also the concluding paragraph of Kelvin’s Mactutor entry), the evidence suggests that his popular lectures were models of exposition.

Some of the most interesting exhibits are the gizmos that Kelvin used as demonstrations in lectures — interesting not just because some, like Volta’s cannon, would never be permitted in today’s safety-conscious classrooms, but for what they say about the strengths and weaknesses of Kelvin and of the British tradition of applied mathematics to which he belonged.

The defining strength and weakness of that tradition is its insistence on the concrete and on mechanical intuition, and its relative suspicion of abstract mathematical explanation. It’s a weakness, especially for research, because it marooned the tradition from the most fertile developments in mathematical physics throughout the twentieth century; but it is a strength, especially for teaching, because it refuses to detach mathematics from other varieties of reasoning. To caricature, the characteristic cry of a mechanicist outwith Kelvin’s tradition is “yes, but what is the deeper structure of the equation?”, while the characteristic cry of a mechanicist in his tradition is “yes, but what does the equation mean?”

Here’s an example of the pedagogical strength of the approach. This faintly steampunk object has the wonderful name “Helmholtz’s Polyphonic Siren”, and essentially it’s a device for generating sounds simultaneously at two distinct but controlled frequencies. There are probably a few purposes for which this could be used, but the one closest to my heart is as an illustration of a basic trig identity:

$\cos(\omega_1t) + \cos(\omega_2t) = 2\cos\left(\dfrac{(\omega_1-\omega_2)}{2}t\right)\cos\left(\dfrac{(\omega_1+\omega_2)}{2}t\right).$

When $\omega_1$ and $\omega_2$ are rather close in value, this can be interpreted as the phenomenon of “beats”: two notes of similar pitch interfere to produce a note of intermediate pitch (the second cos term), the volume of which rises and falls on a much slower timescale (the first cos term). I recall this being demonstrated using more modern equipment in my school physics classroom, and the reinforcement of maths by physical experience made it absolutely unforgettable. One up to Kelvin, and some evidence that he was a splendid teacher.

The second picture has a more ambiguous story to tell. Kelvin seems to have been fond of cobbler’s wax — a highly viscous, non-Newtonian fluid — because its apparent physical properties vary depending on the timescale of the observation. As the caption to the exhibit points out, it responds in a solid-like, elastic manner to high-frequency forcing, while it deforms in a fluid-like manner in response to lower-frequency forcing. From Kelvin’s perspective, this was interesting because it offered a mechanical model of the properties of the luminiferous aether: it could support essentially vibrational motions (electromagnetic waves) at high frequencies, but at slower frequencies it could flow and move around bodies such as planets (and perhaps even be dragged along with them).

We all know the story of the aether, which despite brave attempts to save it eventually went the way of phlogiston and Third Lanark AC, and indeed was killed off by precisely the abstract mathematical physics of which Kelvin was always suspicious. Cobbler’s wax, though, enters another of the tales in which Kelvin backed the wrong theory. His attempt, depending how one views it, to subject geology to the established physical laws or to impose a theocratic veto on the timescales required for Darwinian evolution, is well known. A key element of the heat-transport model that he constructed was that the Earth must cool as a solid body, with no contribution from internal convection to the measured heat flux nearer the surface. Thanks to some adept positioning by Rutherford and his disciples, in the standard telling of the story the hole in Kelvin’s model is the neglect of heat generation by radioactive decay, but as England, Molnar & Richter (2007) have argued, the neglect of convection is at least as large a gap — and more surprising, because it lies well within the physics of Kelvin’s age.

An argument against a stratified Earth with a thin solid crust overlying a fluid layer was that the Earth was known to be remarkably rigid to tidal deformations; Kelvin could therefore confidently postulate that it was solid. One of Kelvin’s former subordinates, John Perry, pointed out that this was not necessarily the case: if the molten layers of the Earth had properties similar to Kelvin’s own cobbler’s wax, it could behave rigidly on the timescale of tides, yet it could accommodate slower motions by deforming as a plastic or even fluid material. It’s curious that Kelvin’s model in the Hunterian is described as a “glacier”, because these offer a classic and observable geological example of precisely this behaviour, on a much more rapid timescale than the deformation of the Earth.

Perry’s objections went largely ignored at the time, and Kelvin’s objections were dealt with by the geologists through the time-honoured method of waiting until something new came up and the anomaly went away. With the condescension of the present, it looks
like a substantial missed opportunity; had the vast age of the Earth been accepted and Perry’ s suggestions about the nature of the mantle been adopted, Wegener’s continental drift could have seemed much more plausible and the plate tectonics revolution could have occurred decades earlier. (This is indeed the moral that England et al. draw from the story.)

Of course, condescension is usually unfair to the past. A scientific debate is conducted on the terms available to it at the time, and the blind spots of science at any moment are one of the reasons it can focus on the questions that it does. It remains tempting, though, to see Kelvin’s cobbler’s wax as symbolic in a different way. His reputation by the time of the age-of-the-Earth controversy was formidable, and must have rested at least partly on his remarkable ability to communicate, and to inspire with, his mechanistic understanding of physics. Perhaps we should see the cobbler’s wax as symbolic not of a missed opportunity but of the disproportionate power that ideas can gain when they’re advocated by a supremely gifted teacher.