What kind of theory of proportions (if any) before Eudoxus'/Euclides' one?

Avvertenza. Ci scusiamo per un invio precedente con data errata. La data della terza conferenza è GIOVEDI' 16 GIUGNO ( e NON 15) ========================= Warning, The present talk is the third one of the seminar. The second talk will be held on Tuesday 14 June. Till the end of the 19th century, mathematicians had to work on numbers without any good definition of them. Instead they had to rely mostly on the theory of proportions as given in book V of Euclid's Elements, a theory usually assigned to Eudoxus, a mathematician of the time of Plato. Nevertheless, well before him, as soon as mathematicians had to use irrational magnitudes, they needed some theory of proportions more general than the arithmetical one. I will firstly comment on some solutions given by modern historians of mathematics. Then I will present another one which is consistent with the knowledge of early Greek mathematics, and I will study some of its consequences, in particular for Euclid's proof of Pythagoras' theorem (proposition 47 of the first book of the Elements).