We explain in this article how the famous Belgian astronomer and physicist Georges Lemaître, a specialist in the theory of relativity, rediscovered, during the Second World War from Gauss’ work, Aitken’s $\Delta^2$ 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 $\Delta^2$ process and Steffensen’s method, we examine in detail the genesis of the rational iteration, and then its application by Lemaître to Picard’s iterates to solve the differential equation $y^{\prime }=dy/dx= 2y^{2}(y-x)$.