What a good question, especially since I can give an answer to it!
In my book, _The Mathematics of Plato's Academy_, OUP 1987/1999, there is a Section 8.2, 'Neusis constructions in Greek geometry' (pp. 287-293/283-289} which looks at (all of?) the Greek evidence on this topic, also including the question of the 4th proportional, and comes to some strong assertions like "the scope of the Elements is not restricted to ruler-and-compass constructions", and "there are not, and there cannot be, similar [r-&-c] constructions to back up the use of the fourth proportional to two circles and a square in XII 2, and to various three-dimensional figures in XII 5, 11, 12, and 18" (on 292/287-8). If I remember correctly, I could not find anyone else who looked in detail into this question except, perhaps, Wilbur Knorr, to whom I refer.
At 7:16 pm -0500 3/1/02, Roger Cooke wrote: > I received the appended e-mail from a colleague, a first-rate mathematician > who has occasionally taught the history as well. Since I'm not an expert in > this area, I'd be grateful for the opinions of people who have studied the > issue in more detail than I have. > > Roger Cooke > > << You recall that subtle---but, as it stands, fatal---logical gap I > pointed out to you in Euclid's proof that circles are as the squares > on their diameters; namely, his assumption of the existence of a 4th > proportion, which he only proved for lengths. Am I the only person > who's noticed that? I just checked Heath, and he doesn't mention it. > I was wondering, if it *is* new, would it be worth publishing as a > note, and if so, where? Hoping for some expert advice. >>