Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

126 kenneth mandersprecisely the diagram we find there would be about what was called for. Thiscould help explain why III.2 is argued by reductio rather than directly, as aBook I style fact about lines in triangles as proposed by De Morgan (HeathEuclid II 9–10). Even if part of Book III derives from an earlier work oncircles composed without the benefit of a well-developed theory of triangles,recognizing III.2 as a probing of diagram behavior for the benefit of III.13would help make its diagram and argument seem appropriate.This example should suffice to show how the notion of probing developedhere can provide an additional resource in interpretation and historical reconstructionof Euclid. As a further sample, consider I.7: triangles with the samebase, and the same sides upstanding at the same ends of the base, have thesame vertex. Imagine constructing a triangle with sides of given lengths on agiven base by circling the side lengths around the ends of the base, just as inI.1; the vertex lies where the circles intersect. But this could be subject to adiagram control challenge: maybe the circles intersect in many nearby pointsrather than in just one point. This would be similar to Protagoras’ challenge,that a circle and its tangent intersect in a segment rather than a point. I.7 couldbe read as responding to this challenge: if there were more than one triangle inthis way, we are led to a contradiction. It is especially effective in that it doesso without using any properties of the circle, which are after all in question.The greatest sensitivity of the two-circle construction is when the two circlesare nearly tangent. This occurs as the vertex angle of the triangle approaches0 or 180 ◦ . In the first situation, multiple intersections of the circles would leadto multiple triangles related as in the diagram of I.7; in the second, to multipletriangles as in Proclus’ variant diagram, in which the one triangle completelycontains the other (262–63). Taking the curvature of the circles into account,the threat of misreading whether there are one or more intersection points isgreater for small vertex angles (where the circles curve together, in the samedirection) than for large vertex angles (where the circles curve apart, in theopposite direction).While there is no direct evidence that I.7 was originally motivated by thisparticular challenge, that would explain why Euclid deals with one diagramappearance variant rather than the other added in Proclus: Euclid’s diagramdraws the consequence from the reading of the two-compass diagram thatmost compellingly competes with the standard one. By Proclus’ time, theoriginal motivation would have been lost. The statement of I.7 would then beseen as including the second variant diagram, as writers at least since Proclushave done.I.7 shows another way in which distinguishing types of probing has limitedvalue: whether a given variant diagram probes appearance control or case