The Charge That Crept

The number he got

In 1913 Robert Millikan published the most celebrated precision measurement of its era: watching charged oil droplets fall and rise in an electric field, he isolated the charge of a single electron. His paper states it with the confidence of a man who has spent two years on sources of error: “We have then finally: e = 4.774 ± .009 × 10⁻¹⁰” electrostatic units — an uncertainty of two parts in a thousand.¹

The value of e has been exact since 2019 — the SI now defines the coulomb by fixing e ≡ 1.602 176 634 × 10⁻¹⁹ C, which in Millikan’s units is 4.803 204 71 × 10⁻¹⁰ esu. Millikan’s number sits 0.61% below that — 3.2 times his quoted error, and nearly six times the tightened ±0.005 he quoted from 1917 on. (His error bar was not generous even by his own history: his first oil-drop value, in 1911, had been 4.891 — his own revision moved down, past the eventual truth, and settled below it.)

He even named the weak joint himself. The analysis converts a droplet’s fall speed into its size by Stokes’s law, and the air’s viscosity η enters the charge, as his paper notes, “in the 3/2 power” — so, in his own words, “an uncertainty of 0.5 per cent in it means an uncertainty of 0.75 per cent in e.”¹ That is precisely what happened: the viscosity of air he adopted was about 0.6% low, and it dragged his electron down with it, three error bars deep.

What makes this a parable rather than a footnote is the next quarter-century. Below is the record — every major determination and every recommended value, each drawn with the error bar its own authors put on it, against the value the SI later made exact.

His air, not his drops

Millikan’s viscosity was no guess. His paper adopts η₂₃ = 0.00018240 poise as the mean of five determinations — two made down the hall at his own laboratory — every one of the five, his paper notes, agreeing to better than one part in two thousand.¹ Five methods, one wrong answer: they shared systematics, and agreement among correlated measurements is confidence, not truth. It took until 1935–37 for Gösta Kellström at Uppsala, with a rotating-cylinder method, to find air about 0.6% thicker — η₂₃ = 0.0001835 — and W. N. Bond’s capillary measurements to corroborate him.⁵ Substituted back into Millikan’s own oil-drop data, the re-measured air raises his electron to about 4.817 ± 0.01 — sources print 4.816 to 4.818 depending on which of Millikan’s viscosity figures they start from — a slight overshoot, roughly one quoted error above the modern exact value: the record’s way of saying the viscosity was his dominant error, not his only one. His drops were fine. His air was wrong — mostly.

The instrument below does that arithmetic in front of you: slide the viscosity of air and watch what Millikan’s own 4.774 becomes under the 3/2 power. The two presets are the number he used and the modern figure for dry air at his working temperature.

The parable

Richard Feynman told this story to Caltech’s 1974 graduating class, in the address that named “cargo cult science.” The passage below is quoted from the original printing, checked against the Caltech Library’s page scan:⁸

“We have learned a lot from experience about how to handle some of the ways we fool ourselves. One example: Millikan measured the charge on an electron by an experiment with falling oil drops and got an answer which we now know not to be quite right. It’s a little bit off, because he had the incorrect value for the viscosity of air. It’s interesting to look at the history of measurements of the charge of the electron, after Millikan. If you plot them as a function of time, you find that one is a little bigger than Millikan’s, and the next one’s a little bit bigger than that, and the next one’s a little bit bigger than that, until finally they settle down to a number which is higher.

“Why didn’t they discover that the new number was higher right away? It’s a thing that scientists are ashamed of—this history—because it’s apparent that people did things like this: When they got a number that was too high above Millikan’s, they thought something must be wrong—and they would look for and find a reason why something might be wrong. When they got a number closer to Millikan’s value they didn’t look so hard. And so they eliminated the numbers that were too far off, and did other things like that.”

R. P. FEYNMAN, “CARGO CULT SCIENCE,” ENGINEERING AND SCIENCE 37(7), P. 12 (JUNE 1974)

The parable, checked

What holds. The miss and its cause are exactly as he says — the viscosity, named above, worth −0.75% in e at the 3/2 power. The recommended values — the numbers review committees told everyone to use — did move up in steps far smaller than the gap, over more than two decades. And the mechanism he alleges is not a just-so story: it has since been measured. Henrion and Fischhoff, auditing the reported uncertainties of fundamental constants, found that across eight reviews from 1929 to 1969 the recommended values of five constants landed outside their quoted error ranges 57% of the time — where honest error bars would manage about 2% — and named the anchoring of each measurement to the last as the likeliest cause, Franklin’s “bandwagon effect.”⁹ The first recommended value after Millikan, Birge’s in 1929, even sat below his — the anchor pulled from both sides.

What the record complicates. Look back at the chart: the raw measurements did not creep — they split. Every low value near Millikan’s shares his method and his air; meanwhile the X-ray crystal route arrived at the modern value in one jump, in 1931 — Bearden’s 1.6031, already past it — and was resisted for half a decade, partly on stated technical grounds (the X-ray route leaned on unproven assumptions about crystal perfection), partly because it disagreed with the most celebrated measurement in physics. What actually crept was the consensus. The truth was available in 1931; it was believed in 1937, after Kellström re-measured the air and gave everyone a respectable reason to move. Feynman’s staircase is a stylization — the honest picture is two populations and a step, which is, if anything, a sharper indictment: the creeping thing was never the data. It was the deciding.

If the mechanism is deference, it has a shape, and you can operate it. The machine below is a declared toy model, not history: each new “measurement” sees the truth plus honest noise, but publishes a compromise — weight w on the last published value, the rest on its own evidence. The cyan dashes are the closed form; the gold chains are seeded simulations. At w = 0 the record is honest scatter around the truth from the first paper. Turn w up and watch a creep assemble itself out of measurements that were, individually, unbiased.

The drops he kept

There is a second, darker chapter, and the venue owes it the same candour. Millikan’s 1913 paper says of its 58 published drops, verbatim: “this is not a selected group of drops but represents all of the drops experimented upon during 60 consecutive days.” His notebooks say otherwise: 107 drops observed in that span, 49 excluded.¹⁰ That sentence — Goodstein, writing in his defence, still calls it “manifestly untrue” — is the part historians agree on. What the exclusions did is also measured: Franklin re-computed e with all the data and found the central value would barely have moved (the drops were mostly cut for apparatus reasons, not their answers — though the trimming did flatter the quoted precision).¹⁰ The irony is structural: the man whose number anchored a generation had himself under-reported the scatter around it. Tight error bars are a form of authority, and authority is exactly what an anchor is made of.

The disease

Feynman ends the story with a cure: “We’ve learned those tricks nowadays, and now we don’t have that kind of a disease.”⁸ That sentence has not aged as well as the rest. The audit that measured the 57% surprise index was published twelve years after he said it, over data running to 1969; the Particle Data Group was warning about tell-tale drifts in its own history plots in 1980; and cases of expectation bias in physics have their own review literature now.⁹ The chart’s own tail says the same: the December 1950 adjustment quoted a probable error of ±0.00007 with the truth fourteen of those away; the 1969 adjustment overshot the eventual exact value; CODATA 2006 still sat 3.7 of its own standard uncertainties from the number the SI would later fix. The parable about self-deception closes by committing one — the oldest one: believing the disease is something other people had, in an earlier, more credulous time. It is the one part of the story this page is careful not to repeat. Our own anchors are listed below, where the error bars can see them.

$ node research/the-charge-that-crept/verify.mjs

§1 the modern value & the unit bridge   e = 4.80320471×10⁻¹⁰ esu from the exact coulomb
§2 the oil-drop dependence              exponent of η measured from the Stokes construction = 1.500000000000
§3 the viscosity arithmetic             4.774 × (1.8348/1.8240)^{3/2} = 4.8165 — inside the reported band,
                                         overshooting the exact value by ~1 quoted error
§4 the size of the miss                 0.61% below = 3.2× his quoted ±0.009, 5.8× the later ±0.005
§4b the series                          every conversion · every σ-deviation on the chart · the shape claims:
                                         Birge 1929 < Millikan · Bearden 1931 > truth (the jump) ·
                                         reviews 1929–63 all low · not a monotone staircase · Dec-1950 ≈ 14× off
§5 the toy model                        E[pₙ]−T = wⁿ(p₀−T) vs 200,000 Monte-Carlo chains, every n ≤ 12
§6 drift guard                          the page embeds byte-identical data to data.mjs

28/28 checks pass — the full claim-by-claim trail with confidence grades is facts.md beside it.