A new history of greek mathematics

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Notes

1 A warning to the reader: it is obviously easy to find convincing what one has also argued oneself – cf. Jens Høyrup, ‘Hippocrates of Chios – His Elements and His Lunes: A Critique of Circular Reasoning’, AIMS Mathematics 5 (2019), 158–84.

2 At times, admittedly (and as happens to all of us), what is first announced as a free conjecture, is later referred to as a fact. Thus, on page 495: “It is said that one of the reasons attracting Descartes to Leiden was the knowledge that Golius – a Renaissance scholar of Arabic – had there a manuscript with Apollonius translated into Arabic, with more of the Conics than were known in Greek.” But on page 498 this becomes: “We are reminded […] of Descartes, relocating to Leiden so as to see the manuscript […] of Apollonius’s lost Conics.” The reader should thus remember which of Netz’s “facts” are born as hypotheses. In the present case, the reader may find an account of the “facts” that is not fully consistent with Netz’s hypothesis in Stephen Gaukroger, Descartes: An Intellectual Biography (Oxford: Clarendon Press, 1995) 187–211.

3 Even the Demotic Papyrus Cairo J.E. 89127-30, 89137-43 is drawn upon (p. 230) as evidence for what was taught to literate citizens in general. However, the papyrus in question is an obvious descendant from what had been taught to the scribes of the Pharaonic age (as we know it for instance from the Rhind Mathematical papyrus), though with certain matters shared with Mesopotamian texts prepared by the temple scribes in Seleucid times. Since the Ptolemaic rulers continued earlier administrative routines, the Demotic mathematical papyri are almost certainly linked to the education of scribes (public and temple officials), and therefore not informative about the paideia which better-class Greeks in general went through, not in Egypt and a fortiori not elsewhere.

The problem that is shown (to find the sides of a rectangle from its area and diagonal) is part of what was shared with the Mesopotamian tradition. Netz comments that now “we are back in Babylon. […] The key trick here, however, is somewhat closer to Pythagoras’s theorem than is the case in extant Babylonian mathematics.” This is a mistake. The same problem is solved in a similar way in the text Db₂-146 from early-18th-century Ešnunna, which even contains a proof building on the Pythagorean rule in partially abstract terms.

4 See Jens Høyrup, “Where and How Did Archimedes Get In? Oblique and Labyrinthine Reflections,” Interdisciplinary Science Reviews 47 (2022), 391–403. This is a response to Reviel Netz, “The Place of Archimedes in World History,” Interdisciplinary Science Reviews 47 (2022), 301–330.

5 Jacques Sesiano, (ed. and trans.), Books IV to VII of Diophantus’ Arithmetica in the Arabic Translation Attributed to Qusṭā ibn Lūqā (New York: Springer, 1982), vii. Cf. also Jan P. Hogendijk, ‘Review of Jacques Sesiano, Books IV to VII of Diophantus’ Arithmetica in the Arabic Translation Attributed to Qusṭā ibn Lūqā. New York etc.: Springer, 1982’, Historia Mathematica 12, 82–90.

6 Jens Høyrup, ‘Influences of Institutionalized Mathematics Teaching on the Development and Organization of Mathematical Thought in the Pre-Modern Period. Investigations into an Aspect of the Anthropology of Mathematics’, Materialien und Studien. Institut für Didaktik der Mathematik der Universität Bielefeld 20 (1980), 7–137, 14–29. These results were soon after badly excerpted and pirated in Johan Fauvel and Jeremy Gray (eds.), The History of Mathematics. A Reader. (London: Macmillan, 1987), 43–5.