Washington, 1881.
In a room of the US Naval library, a man is bent over a book with a worn binding. Simon Newcomb, astronomer and mathematician, is handling for the umpteenth time the tables of logarithms that scientists of the day cannot do without for their complex calculations. Then he notices something odd: the first pages of the book are far more worn than the last ones. The tables for logarithms of numbers starting with 1 are blackened, dog-eared, almost in tatters. Those for numbers starting with 8 or 9 look almost new.
A trivial detail? Not to a mind this rigorous. Born poor in Nova Scotia, self-taught and now one of the most eminent American scientists of his time, Newcomb is in the habit of questioning anomalies. The observation leads him to a surprising conclusion: numbers starting with 1 simply seem to be more common in nature than those starting with 8 or 9.
In a short article published in the American Journal of Mathematics, he notes:
That the ten digits do not occur with the same frequency, and that the pages for numbers beginning with 1 are far more worn than the others.
Then, silence.
Newcomb’s observation is forgotten. Nobody seems interested in this curious phenomenon. After all, by any logic, digits should appear equally often, shouldn’t they? In a random world, you would expect an even spread. The 1 should come first in about 11.1% of cases, just like the 2, the 3, and so on up to the 9.
New York, 1938.
Fifty-seven years later, at the General Electric research laboratory, a physicist named Frank Benford stumbles on the same phenomenon. Like Newcomb, he uses tables of logarithms a lot and notices that they wear unevenly. But unlike his predecessor, Benford decides to dig deeper.
He embarks on a titanic, almost obsessive piece of work. He compiles 20,229 numbers from wildly different sources: river areas, city populations, physical constants, street addresses, numbers from the sports pages of newspapers. In each case, he painstakingly analyses how often each first digit appears.
The results confirm his hunch: the digit 1 comes first in about 30% of numbers, the 2 in 17.6%, and the frequencies then fall steadily down to the 9, which comes first only 4.6% of the time. A precise, mathematical, predictable distribution.
Visualising Benford’s law for the first digit
This is not a mere curiosity but a robust mathematical law. Benford formalises what will henceforth be called “Benford’s law” (much to Newcomb’s posthumous chagrin): the probability that a number drawn from a natural source starts with the digit d equals log₁₀(1 + 1/d)2.
The Zero Bullshit HQ, 2025.
Many, many years later, Benjamin Charles analyses pretty much everything, including the stats… of the Zero Bullshit and A Free Lunch (in French) newsletters, but also of his LinkedIn posts. Looking at like counts, he notices that a big chunk of them start with 1. Logical enough, since you get both the 1x and the 1xx. Same for impressions, except that there, first digits run from 1, the most frequent, down to 9, the least. Again, that can look logical, given the double presence of the 1 and how hard it gets to rack up more and more impressions.
But then he3 remembers having noticed the same thing looking at his expense claims. Thanks to Dougs, his online accountant (whom he forgot to ask to sponsor this newsletter), he exports his accounting data and finds a strange distribution. He then tests it on city populations and river lengths… Miracle.
Benford’s law applied to the populations of France’s 36,000 municipalities, from INSEE, the French national statistics office.
At the time, the findings stay confined to academic circles of mathematicians. Benford published his results in 1938 in the Proceedings of the American Philosophical Society, but his work drew only limited interest. It would take decades for its practical potential to be recognised.