Archive for History of Exact Sciences

, Volume 73, Issue 3, pp 217–242 | Cite as

There is no consequentia mirabilis in Greek mathematics

  • F. AcerbiEmail author


The paper shows that, contrary to what has been held since the sixteenth-century mathematician Christoph Clavius, there is no application of consequentia mirabilis (CM) in Greek mathematical works. This is shown by means of a detailed discussion of the logical structure of the proofs where CM is allegedly employed. The point is further enlarged to a critical assessment of the unsound methodology applied by many interpreters in seeking for specific logical rules at work in ancient mathematical texts.



  1. Acerbi, Fabio. 2003. Drowning by Multiples. Remarks on the Fifth Book of Euclid’s Elements, with Special Emphasis on Prop. 8. Archive for History of Exact Sciences 57: 175–242.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Barnes, Jonathan. 1990. Logical Form and Logical Matter. In Logica, mente e persona, ed. Antonina Alberti, 7–119. Firenze: Leo S. Olschki.Google Scholar
  3. Barnes, Jonathan. 2007. Truth, etc. Six Lectures on Ancient Logic. Oxford: Clarendon Press.Google Scholar
  4. Bellissima, Fabio, and Paolo Pagli. 1996. Consequentia mirabilis. Una regola logica tra matematica e filosofia. Firenze: Leo S. Olschki.Google Scholar
  5. Bobzien, Susanne. 1996. Stoic Syllogistic. Oxford Studies in Ancient Philosophy 14: 133–192.Google Scholar
  6. Bobzien, Susanne. 1997. The Stoics on Hypotheses and Hypothetical Arguments. Phronesis 42: 299–312.CrossRefGoogle Scholar
  7. Bobzien, Susanne. 1999. Logic. The ‘Megarics’. The Stoics. In The Cambridge History of Hellenistic Philosophy, ed. Keimpe Algra, Jonathan Barnes, Jaap Mansfeld, and Malcom Schofield, 83–157. Cambridge: Cambridge University Press.Google Scholar
  8. Busard, Hubertus Lambertus Ludovicus. 1984. The Latin Translation of the Arabic Version of Euclid’s Elements Commonly Ascribed to Gerard of Cremona. Leiden: E.J. Brill.zbMATHGoogle Scholar
  9. Castagnoli, Luca. 2000. Self-Bracketing Pyrrhonism. Oxford Studies in Ancient Philosophy 18: 263–328.Google Scholar
  10. Clavius, Christoph. 1611. Commentaria in Euclidis Elementa Geometrica. In Opera Mathematica. Tomus Primus. Moguntiae.
  11. De Young, Gregg. 1981. The Arithmetic Books of Euclid’s Elements in the Arabic Tradition: an Edition, Translation, and Commentary. PhD dissertation, Harvard University, Cambridge, MA.Google Scholar
  12. Ebrey, David. 2015. Why Are There No Conditionals in Aristotle’s Logic? Journal of the History of Philosophy 53: 185–205.CrossRefGoogle Scholar
  13. Heiberg, Johan Ludvig and Heinrich Menge. eds. 1883–1916. Euclidis opera omnia. 8 vol. Leipzig: B.G. Teubner (referred to as EOO, followed by the number of the volume).Google Scholar
  14. Łukasiewicz, Jan. 1957. Aristotle’s Syllogistic. Oxford: Clarendon Press.zbMATHGoogle Scholar
  15. Unguru, Sabetai. 1975–1976. On the Need to Rewrite the History of Greek Mathematics. Archive for History of Exact Sciences 15: 67–113.Google Scholar
  16. Vailati, Giovanni. 1904. A proposito d’un passo del Teeteto e di una dimostrazione di Euclide. Rivista di Filosofia e scienze affini 6: 5–15 (French transl. with additions: Sur une classe remarquable de raisonnements par réduction à l’absurde. Revue de Métaphysique et de Morale 12: 799–809).Google Scholar
  17. Vitrac, Bernard. 2009. Les démonstrations par l’absurde dans les Éléments d’Euclide: inventaire, formulation, usages. hal-00496748, online at Accessed 31 June 2018.

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.équipe “Monde Byzantin”, UMR 8167 Orient et MéditerranéeCNRSParisFrance

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