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Remarks on Euclid’s Elements I,32 and the Parallel Postulate

Published online by Cambridge University Press:  01 September 2003

Ian Mueller
Ian Mueller
Affiliation:
Department of Philosophy, The University of Chicago
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Abstract

Argument

In this paper I offer some reflections on the thirty-second proposition of Book I of Euclid’s Elements, the assertion that the three interior angles of a triangle are equal to two right angles, reflections relating to the character of the theorem and the reasoning involved in it, and especially on its historical background. I reject a common view according to which there was at some time a petitio principii in the theory of parallels and argue that certain kinds of skepticism about Proclus’ historical reports are excessive. The evidence we have makes it reasonable to suppose that the so-called common notions were made explicit in the earlier fourth century BCE and the postulates, including the parallel postulate, somewhat later than that. On the other hand, it seems clear that some proof of I,32 was available by the mid-fifth century. I attempt to describe what such a proof would have been like and reflect on its significance for our understanding of early Greek geometry.

Type
Articles
Information
Science in Context , Volume 16 , Issue 3 , September 2003 , pp. 287 - 297
Copyright
2003 Cambridge University Press

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References

Aristotle.See Barnes 1984
Barnes, Jonathan,ed. 1984. The Complete Works of Aristotle: The Revised Oxford Translation. 2 vols. Princeton: Princeton University Press. (Pages cited after Bekker 1831.)
Bekker, Immanuel,ed. 1831. Aristotelis Opera. 2 vols. Berlin: Georg Reimer.
Burkert, Walter.1972. Lore and Science in Ancient Pythagoreanism. Translated by Edwin L. Minar Jr.Cambridge, Massachusetts: Harvard University Press.
Dicks, D. R.1959. “Thales.” Classical Quarterly N.S. 9:294-309.Google Scholar
Friedlein, Gottfried, ed. 1873. Procli Diadochi in Primum Euclidis Elementorum Librum Commentarii. Leipzig: B. G. Teubner.
Fritz, Kurt von.1955. “Die ARXAI in der griechischen Mathematik.” Archiv für Begriffsgeschichte 1:13-103.Google Scholar
Gillespie, Charles Coulston,ed. 1970-1990. Dictionary of Scientific Biography. 16 vols. New York: Charles Scribner’s Sons.
Heath, Thomas L.,trans. 1926. The Thirteen Books of Euclid’s Elements. vol. 1. 2nd ed. Cambridge, England: Cambridge University Press.
Heath, Thomas.1949. Mathematics in Aristotle. Oxford: Clarendon Press.
Heiberg, J. L.1904. “Mathematisches zu Aristoteles.” Abhandlungen zur Geschichte der Mathematischen Wissenschaften 18:1-49.Google Scholar
Morrow, Glenn,trans. 1992. Proclus. A Commentary on the First Book of Euclid’s Elements. Princeton: Princeton University Press. (Pages cited after Friedlein 1873.)
Mueller, Ian.1981. Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. Cambridge, Massachusetts: MIT Press.
Mueller, Ian.1991. “On the Notion of a mathematical starting point in Plato, Aristotle, and Euclid.” In Science and Philosophy in Classical Greece. Edited by Alan C. Bowen,59-97. New York and London: Garland.
Mueller, Ian.1997. “Greek arithmetic, geometry, and harmonics: Thales to Plato.” In The Routledge History of Philosophy Volume I: From the Beginning to Plato. Edited by C. C. W. Taylor,271-322. Londonand New York: Routledge.
Mugler, Charles.1958. Dictionaire historique de la terminologie géométrique des Grecs. Paris: Librairie C. Klincksieck.
Philoponus, John.See Wallies 1905.
Proclus. See Morrow 1992.
Vitrac, Bernard,trans. 1990. Euclide d’Alexandrie. Les Eléments. vol. 1. Paris: Presses Universitaires de France.
Vitrac, Bernard.1996. “Mythes (et réalités?) dans l’histoire des mathématiques grecques anciennes.” In L’Europe mathématique. Edited by Catherine Goldstein;Jeremy Gray,and Jim Ritter,31-51. Paris: Éditions de la Maison des Sciences de l’Homme.
Wallies, Maximilian,ed. 1905. Ioannis Philoponi in Aristotelis Analytica Priora Commentaria. Berlin: Georg Reimer.