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.
You don't need nonstandard analysis to make differentials concrete - Cauchy showed us how to model them with ordinary variables, and this method is taught in standard undergrad textbooks like Stewart.
The issue i was bringing up is that the early use of differentials does not correspond to this. The algebraic models of differentials presented in Cauchy or Robinson's nonstandard analysis is not the same thing as the algebraic models used in Euler. The book i suggested uses differentials from beginning to end, but uses the modern, correct form that doesn't lead one to wrong answers when manipulated with standard algebraic rules. There is absolutely no reason to go back in time and inflict this confusion on yourself by intentionally unlearning the hard-won right answer.
Although I should point out that Archimedes correctly understood how to compare different orders of infinitesimals in his book 'On Spirals' (via tangents), circa 200 BCE! Wow!!!
> 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...