Thanks again for the interesting discussion. I am sorry that I am unable to make my point clear enough for you, but I have learned from the effort. You are of course correct about Euclid's use of ratios only in proportion, and the independent use of ratios seems to be an important development on the route to Newton's definition. However, even with Newton and beyond, the sense of proportion is always there, sometimes explicitly, as in the passage from Hamilton I previously cited.
Regards from desiccated Comer, Georgia, USA, which is being teased by the storm gods, Bob
Prof. Lueneburg wrote: > > Dear Bob and All, > > I quit the discussion, because all you are saying is to vague for me to > follow your way to the complex numbers. > > I shall be back as soon as you give a by modern standards mathematically > sound description of what your are talking about. Then we can go back to > Dedekind. --- He knew what he was talking about. > > By the way, Euclid's or rather Eudoxos' theory of ratios is a theory > on the quaternary relation of "a and b are in the same ratio as c and d". > A ratio is never defined in Euclid and, moreover, never used in the text. > (I know: V such and such. That's why I said "never used in the text".) > > All the best, Heinz Lueneburg > > >