The Transformation of Mathematics in the Early Mediterranean by Reviel Netz

By Reviel Netz

The transformation of arithmetic from its old Greek perform to its improvement within the medieval Arab-speaking global is approached by means of concentrating on a unmarried challenge proposed by way of Archimedes and the numerous ideas provided. From a convention of arithmetic according to 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 technique (grounded within the deuteronomic practices of past due Antiquity and the center Ages). A appreciably new interpretation is for that reason provided of the historic trajectory of pre-modern arithmetic.

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We shall return to the divergences between the Greek and the Arabic below. 65 For instance, if there is a sphere AB , and if it is cut by a certain plane, the circle around the diameter ,66 and if (AB being diameter, and E center) we make: as EA, ZA taken together to ZA, so HZ to ZB, and yet again, as EB, BZ taken together to ZB, so Z to ZA, it is proven: that the segment of the sphere B is equal to the cone whose base is the circle around the diameter , while its height is ZH, and that the segment A is equal to the cone whose base is the same, while its height is Z.

As we recall, in Proposition 4 of the Second Book on the Sphere and Cylinder Archimedes reduced the problem of cutting a sphere into the general problem of proportion of lines and areas. Following that, he went on as if the general problem was solved, and promised to solve it in an appendix to this Book on the Sphere and Cylinder. Whether or not such an appendix ever was attached to the main book in Antiquity, it clearly was lost from it at a very early stage, and it appears that most Hellenistic mathematicians did not have access to Archimedes’ solution.

12 to apply to the required areas). M, finally, must be tangent to the parabola at H (for the property of the parabola to apply to the required lines). In short, the constructions of the parabola and the hyperbola are intertwined. In this way, once again, a certain inherent complexity of the problem serves – without Archimedes intending so – to suggest a further, non-geometric meaning of the objects involved. The parabola and the hyperbola each arise out of a specific configurational need. In the case of the parabola, this is to simplify areas into lines.

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