This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.
A theorem can be true, useful, and widely believed for a hundred years before anyone states precisely when it actually applies. That is what happened to the central limit theorem between 1733 and 1901. Abraham de Moivre's original 1733 result covered exactly one case: the binomial distribution, the count of heads in a long run of coin flips, approximated by the normal curve as the number of flips grows. It worked, and it was useful for computing gambling odds, but it said nothing about any other kind of randomness. Pierre-Simon Laplace, working seven decades later, extended the claim far beyond coin flips, to sums of many kinds of independent random quantities, and effectively promoted it from a computational trick for one distribution into a general law about sums of randomness. But Laplace's own derivation relied on the mathematical standards of his day, which fell short of what a modern proof requires: exactly what conditions a sequence of random variables must satisfy for their sum to converge to a normal distribution was still not stated with full rigor. It took until 1901 for the Russian mathematician Aleksandr Lyapunov to supply conditions, now called the Lyapunov condition, precise enough to prove the theorem in general, opening the door to the many further variations and generalizations of the central limit theorem that followed across the twentieth century as probability theory itself became a rigorous, measure-theoretic subject. The theorem's own history is therefore a small case study in how mathematics actually develops: not usually as one person's single flash of insight, but as successive mathematicians taking a useful but locally proven claim and, often across generations, finding out how far it really extends and how firmly it can be nailed down.