So, let me first describe how my doubt originated: out of curiosity I started to study Newton's Opticks, a book written more than 300 years ago. I was doing some of the experiments described on it, and had some doubts, so I asked my teacher for help, and he helped me, but also said that he thought it was good for fun but it wasn't really worth the time to study from such an old book, and that I would benefit much more from studying the more recent books.
I thought that maybe (:D) he was right (for physics), but... what about mathematics? I mean, the theorems proved more than 2000 years ago are still valid and will always be... but maybe the old way is already a closed system and doesn't serve to inspire new mathematics...
My main question: Are there examples where someone continued an old (like more than 80 years), abandoned research (or program of research) in mathematics and succeeded in extracting new, exciting, and important results (for contemporary mathematicians)? If possible, talk of examples of similar events that already have happened (and please, telling the old work that was useful!).
Could we, living in the 21st century, still be beneficed from reading the mathematical writings of the old masters (e.g. Euclid, Archimedes, Newton, Fermat, Euler, Gauss, Poincaré, and others)? -- Of course, in parallel with the contemporary literature!!!