Zeno of Sidon vindicatus: A Mereological Analysis of the Bisection of the Circle
DOI:
https://doi.org/10.12775/LLP.2022.032Keywords
Euclid’s Elements, Zeno of Sidon, mereologyAbstract
I provide a mereological analysis of Zeno of Sidon’s objection that in Euclid’s Elements we need to supplement the principle that there are no common segments of straight lines and circumferences. The objection is based on the claim that such a principle is presupposed in the proof that the diameter cuts the circle in half. Against Zeno, Posidonius attempts to prove the bisection of the circle without resorting to Zeno’s principle. I show that Posidonius’ proof is flawed as it fails to account for the case in which one of the two circumferences cut by the diameter is a proper part of the other. When such a case is considered, then either the bisection of the circle is false or it presupposes Zeno’s principle, as claimed by Zeno.
References
Acerbi, F., 2007, Euclide. Tutte le opere, Milano, Bompiani.
Angeli, A., and M. Colaizzo, 1979, “I frammenti di Zenone Sidonio”, Cronache Ercolanesi, 9: 47–119.
Apelt, O., 1891, “Die Widersacher der Mathematik im Alterum”, pages 253–271 in Beiträge zur Geschichte der Griechischen Philosophie, Leipzig, Teubner.
Cotnoir, A. and A. Varzi, 2021, Mereology, Oxford, Oxford University Press.
De Risi, V., 2014, Gerolamo Saccheri. Euclid Vindicated from Every Blemish, Basel/Boston, Birkhäuser.
De Risi, V., 2021a, “Euclid’s common notions and the theory of equivalence”, Foundations of Science, 26: 301–324.
De Risi, V., 2021b, “Gapless lines and gapless proofs: intersections and continuity in Euclid’s Elements”, Apeiron, 54: 233–259.
Heath, T., 1956, The thirteen books of the Elements, translated with introduction and commentary by Sir Thomas L. Heath, 3 volumes, 2nd edition 1926, Cambridge, Cambridge University Press (reprint: New York, Dover).
Kidd, I., 1999, Posidonius. The Translation of the Fragments, volume III, Cambridge, Cambridge University Press.
Lo Bello, A., 2003, Gerard of Cremona’s translation of the commentary of al Nayrizi on Book I of Euclid’s elements of geometry, Leiden, Brill.
Luria, S., 1933, “Die Infinitesiamallehere der antiken Atomisten”, Quellen und Studien zur Geschichte der Matematik, Astronomie, und Physik, 2: 106–185.
Maffezioli, P., 2022, “La critica di Zenone di Sidone agli Elementi di Euclide: un dibattito antico sull’unicità”, Rivista di filosofia, 113: 45–76.
Morrow, G., 1970, Proclus. A commentary on the first book of Euclid’s Elements, Princeton, Princeton University Press.
Netz, R., 2015, “Were there Epicurean mathematicians?”, Oxford Studies in Ancient Philosophy, 49: 283–319.
Simons, P., 1987, Parts. A Study in Ontology, Oxford, Clarendon Press.
Vitrac, B., 2012, “The Euclidean ideal of proof in The Elements and philological uncertainties of Heiberg’s edition of the text”, pages 69–134 in K. Chemla (ed.), The History of Mathematical Proof in Ancient Traditions, Cambridge, Cambridge University Press.
Timpanaro Cardini, M., 1970, Pseudo-Aristotele. De lineis insecabilibus, Istituto Editoriale Cisalpino, Milano-Varese.
Verde, F., 2013, Elachista. La dottrina dei minimi nell’Epicureismo, Luven, Leuven University Press.
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