What the dear Professor forgot to mention was the kind of rigor used in proofs back then, or rather, the lack of rigor. The kind of rigor and formality used in modern proofs is relatively new - late 19th century, if I'm not mistaken.

Epsilon-Delta formalism were unknown at the time of Newton, and certainly no-one would have imagined it in the time of the Ancient Greeks, although you would think otherwise, seeing the abundant use of Greek letters in such proofs.