Proof of how the normal distribution arises as the limit of the binomial — where do $\pi$ and $e$ come from?

I've always thought the emergence of the normal distribution was kind of magic, specifically that $\pi$ and $e$ both emerge in the distribution, even though they don't exist in the binomial.

I've googled heavily, but haven't found a very good guide showing how these constants arise, but would like to see it.

I'm horrible at proofs, so can someone walk me through taking binomial to the limit to arrive at the normal? Wolfram says de Moivre developed this before 1783. Also, if possible, please don't use the CLT for the proof. Perhaps I'm wrong, but my understanding is that the CLT wasn't proven until the 19th century.

• A proof (without an invocation of the CLT) is available in Feller's An Introduction to Probability Theory and Its Applications, vol. I. Apr 26, 2012 at 2:18
• $\pi$ and $e$ come from Striling's formula Apr 26, 2012 at 13:35
• My analysis at stats.stackexchange.com/a/3904/919 shows how the exponential arises naturally. DeMoivre got this far in the early 18th century. The appearance of $\pi$ is a deeper phenomenon, explained by Complex analysis. The standard elementary demonstration evaluates the Normal integral by squaring it, interpreting it as an integral over the plane, and converts to polar coordinates--and there you see $2\pi$ emerging as the circumference of the circle.
– whuber
Sep 21, 2022 at 21:16