Original proofs of Stirling's series for log(n!)

Transcription into modern notations of the derivation by Stirling and De Moivre of an asymptotic series for $\log(n!)$, usually called Stirling's series. The previous discovery by Wallis of an infinite product for $\pi$, and later results on the divergence of the series are also presented. We conclude that James Stirling has priority over Abraham de Moivre for Stirling's formula and Stirling's series.