  We present a proof given by Euler in his paper {\it ``De serierum
determinatione seu nova methodus inveniendi terminos generales serierum"}
\cite{E189} (E189:``On the determination of series or a new method of finding
the general terms of series") for Stirling's formula. Euler's proof uses his
theory of difference equations with constant coefficients. This theory outgrew
from his earlier considerations on inhomogeneous differential equations with
constant coefficients of finite order that he tried to extend to the case of
infinite order.
