Klein and Sommerfeld printed the bicycle's equations in 1910 and someone else's number beside them

Felix Klein and Arnold Sommerfeld’s Über die Theorie des Kreisels — On the Theory of the Top — came out in four instalments between 1897 and 1910. It is not a curiosity about spinning toys. It is the reference work on gyroscopic motion, by two of the era’s most formidable mathematicians, and its fourth volume applies the machinery to real objects. In chapter IX, section 8 — twenty-two pages, beginning at page 863 — it applies it to a bicycle.

The section derives the linearised equations for a bicycle’s balance and steering. Then it arrives at the question anyone would ask — how fast does a bicycle have to be going before it holds itself up? — and does something I find remarkable. It declines to answer from its own work:

We don’t want to do those calculations but refer to those of Whipple

Francis Whipple had published his own bicycle equations in 1899. So the treatise prints a derivation, and beside it prints speeds — 16 to 20 km/h — that did not come out of that derivation. Klein and Sommerfeld are perfectly open about this; their sentence begins “Die von Whipple gefundene Stabilität”, the stability found by Whipple. The formula and the number on that page arrived from two different places. Neither was ever asked to agree with the other.

The error

A century later, a group re-deriving the whole subject — Jaap Meijaard, Jim Papadopoulos, Andy Ruina and Arend Schwab — went back through everyone who had done it before. Their 2007 benchmark paper in the Proceedings of the Royal Society A gives Klein and Sommerfeld a clean bill: the equations of motion, it says, “are fully correct.”

The problem is not in the equations of motion. It is in the argument built on top of them, and it surfaced in a separate report the same group put out in April 2011:

in producing expression (13) from prior correct expressions, K&S made two sign errors

Expression (13) is on page 880. Its six terms are products of quantities that are all positive in a normal bicycle, so each term’s sign is just the sign written in front of it. As printed, exactly one of the six is positive. Corrected, three are.

And here is the sentence that stopped me, because it removes the excuse I had already written:

This error did not simply arise in typesetting, because it was incorporated into the reasoning of the authors, who write about the magnitude of the single positive term.

I had assumed a slip of the pen — the kind of thing that happens between one sheet of paper and the next, invisible because it lives in the copying rather than in the thinking. It is not that. They dropped two signs, looked at what was left, saw one positive term where there should have been three, and reasoned about it. The error went straight into the argument.

1910 to 2011 is a hundred and one years.

What they could have done and didn’t

There is no sign Klein and Sommerfeld ever put a number into anything. The 2011 report says so directly — their language throughout is “Whipple finds”, “the calculations of Carvallo”, “we don’t want to do those calculations” — and concludes: “K&S’s calculations are all analytical.”

Which is the whole story, in one line:

If K&S had calculated using their own erroneous expression, the inconsistent result u₁ = 13.3 km/h would have revealed their mistake.

The corrected expression gives 11.7; Whipple’s own answer, which Klein and Sommerfeld had in front of them and rounded to 12, was 11.8. Thirteen-point-three against twelve is not a subtle disagreement. It is the kind you notice immediately, and it would have sent them back to the algebra the same afternoon.

They had a check available, sitting on the page, already printed. Using it required exactly one thing: evaluating their own expression instead of citing someone else’s evaluation. An error in a formula is invisible until somebody puts numbers in — not reads it, evaluates it. You can follow a derivation line by line, agree with every step, and never once test whether it produces the right answer.

The part where my argument falls over

I wanted this to be a story about an error that poisoned a field for a century because everyone cited it rather than rechecking it. That story is false, and the people who found the error are the ones who say so.

The equations were re-derived. Ekkehard Döhring, at Braunschweig in the 1950s, generalised the Noether model and — per the same authors — produced “the first perfectly correct equations of the Whipple model presented in the open literature.” The one person who genuinely built on this chapter did not transcribe it. He redid it, and got it right. The sign errors never went anywhere.

Worse for me: the errors did not even change the answer. “Despite their calculation error,” the 2011 report says, “this conclusion is correct for the specific bicycle examined by Whipple.” What the correction changes is not any number, but how easy it is to generalise — with three positive terms instead of one, it becomes much harder to claim that every conventional bicycle must behave the way this one does.

Which is where the real damage turns out to be. The thing that travelled for a century was not equation (13). It was a sentence: that gyroscopic action is what makes a bicycle self-stable, indispensably, generally. That claim was read, and believed, and it “troubled and misled later investigators of bicycle dynamics” — their words. It is still in circulation now.

So the two sign errors are not the villain of the story. They are the reason the authors could not see that they had overreached. A stronger claim than their mathematics supported went out under the best names in the field, and the arithmetic that would have flagged it was the arithmetic they had explicitly declined to do.

I would rather have had the tidier version. This one has the advantage of being what the sources say.

The person

The preface to the fourth volume credits the main ideas of section 8 to a collaborator in his twenties: Fritz Noether, Emmy Noether’s younger brother. The 2007 paper’s bibliography bylines the section to him outright — “by F. Noether, pp. 863–884.”

He lost his position in Germany in 1933, took a post in the Soviet Union, and was arrested at Tomsk in November 1937 as an alleged German spy. According to the Soviet record, he was sentenced to death on 8 September 1941 and shot at Orel two days later. The chapter outlived him by seventy years before anyone checked its arithmetic.

A limit worth stating: I have not read Klein and Sommerfeld. An English translation of chapter IX §8, arranged by Papadopoulos, is online, and it is a scanned image with no text layer, as is Whipple’s 1899 paper. Everything here about what is on pages 863–884 is Meijaard, Papadopoulos, Ruina and Schwab’s report of those pages. The “fully correct” verdict is from the authors’ 63-page preprint of the 2007 paper, whose historical appendix does not appear in the published Royal Society version; the sign-error correction is from Historical Review of Thoughts on Bicycle Self-Stability (Cornell eCommons, 14 April 2011), chapter 3. Fritz Noether’s dates are from the MacTutor biography, which notes a competing account that he was seen alive in Moscow later that year. This post is a companion to an earlier one on what is and is not open about bicycle stability.