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

the euclidean diagram (1995) 115ADCFig. 4.9.EFBequal priority to all conditions in force from before the scope of the reductioargument, over those which arise within that scope.Again, consider Proclus’ diagram for the converse of I.15, that if, upon astraight line AB, we assume two straight lines CD and CE on opposite sides [atthesamepointC] and making the vertical angles ACD and BCE equal, theselines lie on a straight line DCE with each other (299–300). The diagram issubjected to all these straightnesses (including that of the intended conclusion)and the angle equality. For reductio, Proclus assumes DCE not straight, andenters CF at an angle to DE while stipulating in the discursive text that DCFis straight (Fig. 4.9).This may strike us as strange, for it would come to us more naturally tomake DCF straight in the diagram, enter CE off at an angle, and hash-markas equal the (visibly unequal) vertical angles ACD and BCE. But Proclusfirst gives a diagram in which the data are set out, subjected to the givenexact relations; this forces DCE to be straight (as we are out to show).Only then is the hypothesis for reductio entered, properly unpacked. Thesepriorities for subjecting the diagram to certain exact conditions rather thanothers thus become understandable if we assume that the diagram must besuitable for sequential display as the demonstration proceeds. For then wemust set out the claim first, before we announce and diagram the reductiostrategy.4.5.4 A complication in I.27: attribution blocking?According to our account so far, one is licensed to attribute angles EBG andFDC to the diagram of I.27, buttheElements refrains from doing so. Indeed,that attribution would preclude treating EFG as a triangle. That would blockthe Euclidean argument: it rests on an application to EBGDF of I.16, whichholds for triangles but fails for quadrangles (squares, for example) and beyond.For the argument to proceed unimpeded, something must be in force here,