Florian Scheck's textbooks in the Springer Graduate Texts in Physics series could also serve as a bridge. Though very challenging for the non-mathematician I've grown quite fond of "Mechanics: From Newton's laws to Deterministic Chaos" and plan to read the other Scheck books in the series as well.
I know mathematics and I hate his series of books.
The thing that irks me most is using higher-level concepts, like the existence of a atom, to illustrate lower level concepts that led to the discovery of the atom in the first place.
