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There is some discussion of Eudoxus' theory of ratios in the pages The first author doesn't seem to have read Book V, where the basic theory is laid out. The second doesn't seem to have read Book VI, where it is applied (and where the proof of Pythagoras' Theorem he admires is in fact to be found). To give you an idea of how subtle the theory is, the opening Theorem of Book VI is thatif you are given two rectangles with the same height, the ratio of their areas is the same as the ratio of their bases. 
If you try to prove this yourself from elementary principles, you will understand immediately what difficulty the Greeks faced, and admire them enormously for getting around it. |