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

108 kenneth mandersrecognize ‘cases’ in traditional geometry when variant geometrical practicesreconstrue traditional geometrical claims, but can proceed more uniformly withthem. The traditional geometrical text often allows more uniform formulationof claims than do diagrams; but here the scope for uniform proceedingis limited by the need for diagram-based inferences. On the other hand,Cartesian geometrical representation by equations, and others inspired by it,such as Poncelet’s, allow for such uniform treatment of results treated separatelyin traditional geometry that we regard them as analogous or as cases of ‘thesame problem’. Often, ancient texts already show, at least by their expositoryarrangement, that analogies are sensed among such distinctly treated questions;often, they do not. Notably, whereas traditional geometry handles limiting ordegenerate cases such as tangencies or coincidences separately (Heath Euclid IIp. 75), algebraic representation usually includes them in the treatment of thegeneric case; the importance of continuity in this sense was stressed by Leibnizand Poncelet.For example, when two lines through a point O meet a circle, say, one inA and B and the other in C and D, then the rectangles (products) OA byOB and OC by OD are equal (cf. Euclid III 35–37, and Heath’s summaryof related subcases and claims, (Euclid, vol. 2, pp. 71–77)). Of the variousdistinct diagrams which can arise, Euclid III.35 concerns specifically the onewhere O lies inside the circle, and III.36 thecaseinwhichO is outside butone given line is tangent to the circle. This requires separate statement as aproposition because the traditional discursive text cannot be read to includethesquareonasinglesegmentOP under the product OA by OB. Ultimately,this restriction on discursive representation seems again forced by topologicaldistinction behavior in reading diagrams, given the need for a workablecross-reference system between discursive text and diagram: OP cannot at allbe visually located in a diagram showing distinct non-tangent OA and OB,and in the opposite case one would at least have to allow double-labeling(Fig. 4.5).CAADPBBOFig. 4.5.