# “More Decimal Digits”

On occasion, somebody voices the idea that in year $N$, physicists thought they had everything basically figured out, and that all they had to do was compute more decimal digits. I won’t pretend to know whether this is actually true for any values of $N$ — when did one old man’s grumpiness become the definitive statement about a scientific age? — but it’s interesting that not every physicist with an interest in history has supported the claim.

One classic illustration of how the old guys with the beards knew their understanding of physics was incomplete involves the specific heats of gases. How much does a gas warm up when a given amount of energy is poured into it? The physics of the 1890s was unable to resolve this problem. The solution, achieved in the next century, required quantum mechanics, but the problem was far from unknown in the years before 1900. Quoting Richard Feynman’s Lectures on Physics (1964), volume 1, chapter 40, with hyperlinks added by me:

The first great paper on the dynamical theory of gases was by Maxwell in 1859. On the basis of ideas we have been discussing, he was able accurately to explain a great many known relations, such as Boyle’s law, the diffusion theory, the viscosity of gases, and things we shall talk about in the next chapter. He listed all these great successes in a final summary, and at the end he said, “Finally, by establishing a necessary relation between the motions of translation and rotation (he is talking about the 1/2 kT theorem) of all particles not spherical, we proved that a system of such particles could not possibly satisfy the known relation between the two specific heats.” He is referring to γ (which we shall see later is related to two ways of measuring specific heat), and he says we know we cannot get the right answer.

Two years later, in a lecture, he said, “I have now put before you what I consider to be the greatest difficulty yet encountered by the molecular theory.” These words represent the first discovery that the laws of classical physics were wrong. This was the first indication that there was something fundamentally impossible, because a rigorously proved theorem did not agree with experiment. About 1890, Jeans was to talk about this puzzle again [cf.]. One often hears it said that the physicists at the latter part of the nineteenth century thought they knew all the significant physical laws and that all they had to do was to calculate more decimal places. Someone may have said that once, and others copied it. But a thorough reading of the literature of the time shows they were all worrying about something.

When the history lessons handed to us are textbook cardboard, oversimplifications which leave out all the interesting parts of the story, how much insight can we honestly hope for?

UPDATE (1 December 2013): The link to the Feynman lectures now points to the online version of Volume 1’s complete text. This is a good time to point out an erratum, which I noticed when first writing this post but managed to forget to mention explicitly. If “Jeans” does refer to James Jeans (and I don’t know what other physicist it could be), then “1890” cannot be correct. Jeans was born in 1877 and did not go to Cambridge until 1896.

A better date, in the next century but still in the earliest years of the old quantum theory, would be 1904, the first publication of his book The Dynamical Theory of Gases, which has a lengthy discussion of how kinetic theory fails to account for gases’ specific heats. The nub is on p. 173: “Our theory has, then, led to a result which is in flagrant opposition to experiment.” Jeans’ own attempt at a solution was basically to deny that all the degrees of freedom were in statistical equilibrium. Additional references can be found in R. Hudson (1989), “James Jeans and radiation theory,” Studies in History and Philosophy of Science A vol. 20, pp. 57–76.

UDPATE (9 December 2013): I wrote a letter to the people in charge of revising the Feynman red books, and the incorrect year has been corrected.

UPDATE (21 March 2014): If you want a statement of this issue which really does date to “about 1890,” see Peter Guthrie Tait’s “On the Foundations of the Kinetic Theory of Gases,” Transactions of the Royal Society of Edinburgh, 14 May 1886. Tait writes of Maxwell,

He obtained, in accordance with the so-called Law of Avogadro, the result that the average energy of translation is the same per particle in each system; and he extended this in a Corollary to a mixture of any number of different systems. This proposition, if true, is of fundamental importance. It was extended by Maxwell himself to the case of rigid particles of any form, where rotations perforce come in. And it appears in such a case that the whole energy is ultimately divided equally among the various degrees of freedom. It has since been extended by Boltzmann and others to cases in which the individual particles are no longer supposed to be rigid, but are regarded as complex systems having great numbers of degrees of freedom. […] This, if accepted as true, at once raises a formidable objection to the kinetic theory. For there can be no doubt that each individual particle of a gas has a very great number of degrees of freedom besides the six which it would have if it were rigid:—the examination of its spectrum while incandescent proves this at once. But if all these degrees of freedom are to share the whole energy (on the average) equally among them, the results of theory will no longer be consistent with our experimental knowledge of the two specific heats of a gas, and the relations between them.

This passage can be found on p. 135 of the 1900 reprint collection of Tait’s papers. “On the Foundations of the Kinetic Theory of Gases” was actually a five-part series which appeared in the Transactions from 1886 to 1892, so a slightly vague dating makes a kind of sense. I wonder if Feynman had Tait’s articles in mind, and then mentally switched to Jeans while speaking.

UPDATE (18 September 2014): And here is Josiah Willard Gibbs, writing at the end of 1901:

In the present state of science, it seems hardly possible to frame a dynamic theory of molecular action which shall embrace the phenomena of thermodynamics, of radiation, and of the electrical manifestations which accompany the union of atoms. Yet any theory is obviously inadequate which does not take account of all these phenomena. Even if we confine our attention to the phenomena distinctively thermodynamic, we do not escape difficulties in as simple a matter as the number of degrees of freedom of a diatomic gas. It is well known that while theory would assign to the gas six degrees of freedom per molecule, in our experiments on specific heat we cannot account for more than five. Certainly, one is building on an insecure foundation, who rests his work on hypotheses concerning the constitution of matter.

This from the Preface to his Elementary Principles in Statistial Mechanics.