Rigorous Purposes of Analysis in Greek Geometry

Analyses in Greek geometry are traditionally seen as heuristic devices. However, many occurrences of analysis in formal treatises are difficult to justify in such terms. I show that Greek analysies of geometrics can also serve formal mathematical purposes, which are arguably incomplete without which their associated syntheses are arguably incomplete. Firstly, when the solution of a problem is preceded by an analysis, the analysis latter proves rigorously that there are no other solutions to the problem than those offered in the synthesis. Secondly, whenever some construction assumption beyond ruler and compass is made, the problem is not only solvable by that assumption but is in fact equivalent to that assumption in a rigorous sense.