Abstract

In this article, we assess the heuristic power of Greek geometrical analysis by trying to reconstruct some analyses of extant propositions of which only the demonstration is found in the text. We have reconstructed the analysis of the trisection of an angle, the property of the tangent to the parabola, to the hyperbola/ellipse, and to the spiral line. In all of these cases, the results and the demonstrations can be found by the analysis alone, without arguments by analogy with other figures or kinematic considerations. We have thus concluded that analysis is a much more powerful tool than is usually assumed to be. We have also examined a part of Apollonius’ Cutting off of a ratio and confirmed that the number of possible cases to be distinguished in analysis tended to increase because the geometric objects and the magnitudes related to them were always considered in the geometric arrangement in which they appear. We suggest that the absence of an algebraic approach that would have detached the magnitudes from their geometrical context may have been one of the major obstacles to further development of Greek geometry.