I've read some interesting calculus-free proofs of at least parts of the derivation of Kepler's Laws from Newton's gravitational force. One is of course Feyman's "Lost Lecture" (which was already featured here in a comprehensive post), which dervies Kepler's first law from a central inverse-square law; then there's a partial derivation using Mamikon's Theorem (MTK) of Kepler's second law from just the centrality of gravitational force.
Both of these eschew modern calculus, but they do use infinitesimals: Feynman's one uses "very small" $\Delta\theta$'s, while MTK "hides" infinitesimals within the assumption that areal velocity is "better understood" as angular momentum.
My questions are: assuming concepts such as velocity and acceleration as primitive, is there in classical mechanics a complete derivation of Kepler's Laws from Newton's inverse-square law which doesn't rely on modern calculus and infinitesimals, vanishing quantities and so on? Maybe using just indivisibles? If not - is there a way to prove that no such derivation can exist?
The proofs I mentioned above seem to hint at the possibility of the first option (MTK in particular, which basically replaces infinitesimals with indivisibles), but I'm not able to rework them in such a way that infinitesimals are totally removed (eg. to use Mamikon's Theorem to prove the relation between angular momentum and areal velocity). But maybe there is some very strong reason (ellipticity of integrals?) for which no proof based on Mamikon's Theorem or anything close to it can directly relate angular momentum and areal velocity...
I actually have no clue. Does any of you?