I think he was very familiar with differential equations and series expansions and the "wild west" stage of calculus in general. The frontier of cool and interesting mathematics has moved a lot in 200 years.

Also these stories are "storified" over time, the reality is always messier (same with startup founding stories etc..).

A common mistake I see in people reading mathematics (or even computer science papers) is to think the proof set out in the paper is the thought process that lead to the interesting insight. It is almost always an ex post facto formalisation.