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. (shrink)

Mathematical texts raise particular dilemmas for the translator. With its arm’s-length relation to verbal expression and long-standing “mathematics is written for mathematicians” ethos, mathematics lends itself awkwardly to textually centered analysis. Otherwise sound standards of historical scholarship can backfire when rigidly upheld in a mathematical context. Mathematically inclined historians have had more faith in a purported empathic sixth sense—and there is a case to be made that this is how mathematical authors have generally expected their works to be read—but it (...) is difficult to pin down exact evidentiary standards for this supposed instinct. This essay urges that both of these points of view, for all the tension between them, be kept in the historian’s toolbox. It illustrates these considerations with a case study from the Ptolemaic astronomical tradition on computing lunar model parameters from eclipse data. (shrink)