By Reviel Netz

The transformation of arithmetic from its historical Greek perform to its improvement within the medieval Arab-speaking global is approached by means of targeting a unmarried challenge proposed by way of Archimedes and the various recommendations provided. From a tradition of arithmetic in accordance with the localized resolution (originating within the polemical practices of early Greek science), we see a transition to a tradition of arithmetic according to the systematic strategy (grounded within the deuteronomic practices of overdue Antiquity and the center Ages). A considerably new interpretation is consequently provided of the historic trajectory of pre-modern arithmetic.

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47) for an isosceles right-angled triangle. 27, 30, 34). EO = EB through Step i. So this settles O = BE. EH = BE can be seen through Step 7. 92 (m) Let it be drawn and let it be as ϒ T; (50) therefore the point touches an ellipse given in position. ) (n) Let it be drawn and let it be as B; (58) therefore the point touches a hyperbola given in position. 99 At this point Eutocius went on to produce the synthetic solution: this was in all likelihood Eutocius’ own contribution (since, in the Arabic version, Diocles explicitly ignores the synthesis as 91 This is the Apollonian way of stating that is the parameter of the ellipse.

Hence his conic sections appear less natural, more purely quantitative, than Archimedes’ did. This may remind us of Klein’s thesis: by being dependent 38 t he pro blem solv ed by d i ocle s on a given past, mathematics becomes more second-order, more abstract. With Dionysodorus, we see no more than a suggestion of this dynamics: we shall see much more of it in the next chapters. 5 The problem solved by Diocles Following his excerpt from Dionysodorus, Eutocius went on to quote yet another solution to the problem of cutting the sphere, this time by Diocles.

11. 15. 14. 2. 14 is meant to support Step 27, not Step 28. It is probably Eutocius’ contribution and, if so, so are probably the other references to the Elements and the Conics. 34. Dionysodorus, at least as reported by Eutocius, does indeed proceed to offer this lemma. I do not reproduce it here, as it does not touch on our main theme. 62 (36) Therefore the plane produced through M, right to AB, cuts the sphere according to the given ratio; which it was required to do. To follow Dionysodorus’ line of reasoning, we should start from the problem as he had inherited it from Archimedes – that is, as stated for a particular setting arising from a sphere.