The rational iteration method by Georges Lemaitre

We explain in this article how the famous Belgian astronomer and physicist Georges Lemaitre, a specialist in the theory of relativity, rediscovered, during the Second World War from Gauss' work, Aitken's (2) process. He called his discovery the rational iteration method and used it to solve an ordinary differential equation. After giving some historical elements on Aitken's (2) process and Steffensen's method, we examine in detail the genesis of the rational iteration, and then its application by Lemaitre to Picard's iterates to solve the differential equation y=dy/dx=2y2(y-x).