In Euler's time they still didn't have a correct understanding of higher order differentials, so the work from this time period has genuine errors that you would need to be aware of. Might I suggest an alternative? There's a wonderful little book by Nathanial Grossman called 'The Sheer Joy of Celestial Mechanics' [1]. It assumes vector calculus of course but otherwise might be just what you're looking for. From a prepublication review:

> Don't look for axioms to memorize. Too many courses are consecrated to teaching students to play chords on a set of axioms. This book celebrates the heroic age of calculus, the time of Euler, Maclaurin, Clairault, Lagrange, and Laplace, a time before delta and epsilon. [...] mathematics was invented to do things, not just to be talked about, and today - still - its greatest triumphs are what it can do.

[1] https://www.amazon.com/Sheer-Joy-Celestial-Mechanics-dp-0817...