@lavinia: Thank you. I think I understand your meaning now of solutions obtained by manipulating the coefficients, to mean that the solutions can be expressed using algebraic operations and nth roots, starting from the coefficients and rational numbers, i.e. that the equation is what we now call "solvable". My only reservation is that extraction of roots seems to me not an algebraic operation. Of course the statement that a^n = b can be regarded as an algebraic statement, but so can the statement that 3a^3 + ab +c = 0.
So I apparently misunderstood you. But I also seem to have less confidence that I understand clearly the meaning of certain familiar symbols. E.g. I myself do not feel I know precisely what number equals the square root of 2, although I also like the geometric representation as (the pair consisting of) a diagonal (and a side). But even as I say this, my skeptical self asks again, "I like visualizing those two segments, but how do I know there is a 'number' relating the side and the diagonal?" Defining the number to
be that pair, seems quite abstract and modern.
I.e. it seems to me that in Euclid, line segments (even given a unit segment) did not have "lengths" thought of as algebraically manipulable numbers. To me this would require an arithmetic of segments as Hartshorne does, but Euclid did not do. The origins of this theory is attributed by Hartshorne to Hilbert and Enriques, thousands of years after Euclid. The introduction to Hartshorne's chapter 4 discusses this interesting and provocative topic. There is of course room for differing opinions as he acknowledges.
In my notes I show that if one does even begin to do this, and defines the product of two pairs of line segments to be equal if the rectangles constructed from them are equi-decomposable, then one can skip the proportionality theory of Euclid from chapter 5, and invoke the work of earlier chapters instead. The fact that Euclid does not do this suggests again to me that he did not envision this theory of "numbers" defined by segments.
Maybe I seem to be splitting hairs, but I think it is fundamental that Euclid did not have a notion of numbers more general than rationals, and that it was the work of much later generations to construct them from his foundations. Indeed to me this is why Euclid is so important to study first, becuse it leads the way to a conception of real numbers expanded out of geometry, one that we later generations tend to take for granted, but most students today, lacking a thorough familiarity with Euclidean gometry, do not understand well.