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

114 kenneth mandersAEBCFDGFig. 4.7.subjects the diagram to a circular arc where the discursive text puts forward astraight line.Another approach would smoothly curve the segment EF so that AB andCD produced intersect at G even though the alternate angles are subject toequality and the lines are produced subject to being straight (Fig. 4.7); yetanother would have EF, EG, andFG straight but give up on equality of thealternate angles in the diagram. Proclus (313) in effect does this in a relatedapplication of I.17.Our text of Euclid does none of these. Instead, the line segments AB, CD,and EF are all subjected to straightness, and to equality of alternate angleswith EF. AnintersectionatG is indicated by inserting straight segmentsBG and DG at an angle to segments AB, CD respectively. Evidently, asa result, we would say that the diagram puts forward angles EBG andFDC. This example suggests several lessons, which appear confirmed in otherinstances.(1) It appears preferred to treat equivalently in the diagram items whichplay equivalent roles in the discursive text; even when the diagram cannot besubjected to all conditions in the discursive text. Thus, for example, both EGand FG are broken, even though the intersection could be indicated with onlyone broken and one straight (Fig. 4.8).(2) Given that a choice must be made among exact conditions, the procedurehere suggests a priority to subject the diagram to those indicated first in thediscursive text. Similarly, the diagram for I.6 is subject to all exact conditions inforce except for the one (AC = DB) arising from the hypothesis for reductio,and the subsequently drawn conclusion that the triangles DBC and ACB arecongruent. The examples do not allow us to discriminate whether the priorityarises simply from the sequence of the discursive text, or whether it accordsAEBGCFDFig. 4.8.