He published a proof in the Philosophical Transactions journal in 1722. Here is an English version:
First, he introduces two infinite polynomials and their solutions, and then goes on to describe a theorem based on those, from which if you solve for $zn and z you will get de Moivre's formula. Since the proof is rather fragmented, lacks explanations and alludes to previous calculations, I have a few questions:
- What exactly do $A, B, C$, etc. (the "coefficients of the preceding terms") represent in the polynomials? Is $A = n, B = \frac{1-n^2}{2\cdot3}\cdot A, C = \frac{9-n^2}{4\cdot5}\cdot B$ etc.?
- Most importantly, how does the theorem (i.e. the two equations containing $z$) follow from the solution of the polynomials? Where do those equations come from, and why do they look as they do? And what does $z$ represent in the equation? It appears for the first time there, without explanation.
- Is $n$ assumed to be an integer, or can it be any real (or complex) number? What about $t$?