Archimedes (in On the Sphere and Cylinder Prop. 2 ) asserts:
"Given two unequal magnitudes, it is possible to find two unequal straight lines such that the greater straight line has to the less a ratio less than the greater magnitude has to the less."
He begins his proof:
"Let AB, D be two unequal magnitudes, and let AB be the greater. "
And then says:
"By the second proposition in the first book of Euclid let BG be placed equal to D"
The proposition referred to is surely not the second in the modern numbering scheme, but perhaps the third:
"To cut off from the greater of two given unequal straight lines a straight line equal to the less."
But Euclid I.3 is about lines, not general magnitudes, and, it seems to me, is useless in this case. (But how can such a great mathematician make such an error? Perhaps the statement is an interpolation.) What is needed here is not a reference to a theorem, nor even a postulate, but to a definition of magnitude, namely, that a magnitude is such a thing that, of two unequal magnitudes of the same kind, the less can be "cut off" from the greater.
Magnitude was never defined, and one can see why - the difficulty of the thing is very great.
Another possibility is that I am really confused - not for the first time this week.
Regards from a sunny, chilly Comer, Georgia, Bob Robert Eldon Taylor philologos at mindspring dot com