By Reviel Netz

The transformation of arithmetic from its historic Greek perform to its improvement within the medieval Arab-speaking international is approached by way of targeting a unmarried challenge proposed through Archimedes and the numerous suggestions provided. From a tradition of arithmetic in line with the localized answer (originating within the polemical practices of early Greek science), we see a transition to a tradition of arithmetic in keeping with the systematic method (grounded within the deuteronomic practices of overdue Antiquity and the center Ages). A considerably new interpretation is for that reason provided of the old trajectory of pre-modern arithmetic.

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**The Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations**

The transformation of arithmetic from its historical Greek perform to its improvement within the medieval Arab-speaking global is approached by way of concentrating on a unmarried challenge proposed by means of Archimedes and the various recommendations provided. From a tradition of arithmetic in response to the localized answer (originating within the polemical practices of early Greek science), we see a transition to a tradition of arithmetic in keeping with the systematic technique (grounded within the deuteronomic practices of overdue Antiquity and the center Ages).

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**Example text**

11 – the “symptom” of the parabola. 19 t he pro blem i n th e wo rld o f a r c hime d e s While the extension of this result to parallelepipeds has a less compelling intuitive character, we just saw Archimedes taking it for granted in some moves of his solution. If applied on his formulation of the problem, then, it could have produced the following threedimensional equality: (parallelepiped cont. by sq. X, line XZ) equals (parallelepiped cont. by sq. B , line Z ). Now we can see that the bottom side is known – both square and line are given.

M):(sq. AH). Now, as we recall, 34 the problem solved by dionysodorus arranging that two rectangles be equal is a simple task: all we need to do is to draw a hyperbola through the point H, with B, BA as asymptotes. All are given by the terms of the problem, so that we can count on being able to have the proportion (sq. AB): (sq. MB)::(sq. M):(sq. AH) when we need it. Of course, this is, once again, a “sliding” conﬁguration: we do not have a deﬁnite point M (this is what we seek to ﬁnd in this problem).

Indeed, in this case, Archimedes’ analysis itself involves special constructions, whose discovery would have been just as complicated as that of the solution to the problem itself. For one thing, already in the analysis, Archimedes constructs the parabola and the hyperbola, which are simply not given in the terms of the problem. In general, I had argued in Netz (2000) that the goal of the published versions of Greek mathematical analyses was largely expository, rather than heuristic. Analyses could serve to set out, in demonstrative form, the relations leading to the required construction.