1. de Moivre
In 1730 Abraham de Moivre published
(Only very few copies are extant, grab one here or here.) Five years later, he included his own English translation of this Latin paper in the 1738 edition of The Doctrine of Chances, and there we can read
Sure enough, the date of 1733-12=1721 is expressly displayed in Miscellanea, where the above-mentioned problem is treated.
What de Moivre gives amounts to:
2. Stirling joins in
here) in Exemplum 2 after Propositio 28:
(logarithmo circumferentiae Circuli cujus Radius est Unitas = (to) the logarithm of the circumference of the circle whose radius is unity). De Moivre established it for himself, and delivered the finished result:
As many have remarked: this formula should be called
de Moivre-Stirling formula
3. Two-sided estimates
As with Wallis's formula, we'll give a totally elementary proof of a two-sided estimate, viz.
On the blackboard below we first deduce the estimates à la de Moivre with the constant left undetermined.
And here, at last, is Stirling's constant. We use Wallis's formula in the form given on the last line of this page.