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The Readings of Apollonius' On the Cutting off of a Ratio

Published online by Cambridge University Press:  27 February 2012

Ioannis M. Vandoulakis
Affiliation:
University of the Aegean, Department of Cultural Technology and Communication, University Hill, 81100 Mytilene, Greece
Corresponding

Extract

During the second half of the twentieth century an attention of historians of mathematics shifted to mathematics of the Late Antiquity and its subsequent development by mathematicians of the Arabic world. Many critical editions of works of mathematicians of the Hellenistic era have made their appearance, giving rise to a new, more detailed historical picture. Among these are the critical editions of the works of Diophantus, Apollonius, Archimedes, Pappus, Diocles, and others.

Type
Essay-Review
Copyright
Copyright © Cambridge University Press 2012

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References

1 Rashed, Roshdi, L'art de l'algèbre de Diophante (in Arabic) (Cairo, 1975)Google Scholar; Diophante. Les Arithmétiques, vol. III: Book IV; vol. IV: Books V–VII (Paris, 1984)Google Scholar.

2 Roshdi Rashed (ed.), Apollonius de Perge, Coniques. Tome 1.1: Livre I. Commentaire historique et mathématique, édition et traduction du texte arabe par Roshdi Rashed, 2008. Tome 1.2: Livre I. Édition et traduction du texte grec par Micheline Decorps-Foulquier / Michel Federspiel, 2008. Tome 2.1: Livres II et III. Commentaire historique et mathématique, édition et traduction du texte arabe par Roshdi Rashed, 2010. Tome 2.2: Livre IV. Commentaire historique et mathématique, édition et traduction du texte arabe par Roshdi Rashed, 2009. Tome 2.3: Livres II-IV. Édition et traduction du texte grec par M. Decorps-Foulquier/M. Federspiel, 2010. Tome 3: Livre V. Commentaire historique et mathématique, édition et traduction du texte arabe par Roshdi Rashed, 2008. Tome 4: Livres VI et VII. Commentaire historique et mathématique, édition et traduction du texte arabe par Roshdi Rashed, 2009.

3 Al-Nadīm, , al-Fihrist, ed. Tajaddud, R. (Tehran, 1971), p. 326Google Scholar.

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7 Apollonius of Perga, On Cutting Off a Ratio. An Attempt to Recover the Original Argumentation through a Critical Translation of the Two Extant Medieval Arabic Manuscripts, transl. by Macierowski, E. M., edited by Schmidt, R. H. (Fairfield, 1987; 2nd edn 1988)Google Scholar.

8 Zeuthen, Hieronymus Georg, “Kegelsnitlaeren in Oltiden”, Kongelige Danske Videnskabernes Selskabs Skrifter, 6th ser., 3, no. 1 (1885): 1319Google Scholar, 2nd edn by O. Neugebauer (Copenhagen, 1949). German translation Fischer-Benzon, R., Die Lehre von den Kegelschnitten im Altertum (Copenhagen, 1885)Google Scholar; 2nd edn by J. E. Hofmann (Hildesheim, 1966; repr. 2010).

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13 Concerning the earlier forms of analysis in ancient geometry, there is a continuing debate as it concerns its nature and structure. See for instance, Robinson, Richard, “Analysis in Greek geometry,” Mind, 45 (1936): 464–73CrossRefGoogle Scholar; Gulley, Norman, “Greek geometrical analysis,” Phronesis, 3 (1958): 114CrossRefGoogle Scholar; Mahoney, Michael S., “Another look at Greek geometrical analysis,” Archive for History of Exact Sciences, 5 (1968–9): 319–48CrossRefGoogle Scholar; Marchi, Peggy, “The method of analysis in mathematics,” in Nickles, Thomas (ed.), Scientific Discovery, Logic, and Rationality (Dordrecht, 1980), pp. 159–72CrossRefGoogle Scholar; Knorr, Wilbur Richard, The Ancient Tradition of Geometrical Problems (Boston, 1986; repr. New York, 1993), pp. 9, 6676Google Scholar, 95, 354–60; Behboud, Ali, “Greek geometrical analysis,” Centaurus, 37 (1994): 5286CrossRefGoogle Scholar; Netz, Reviel, “Why did Greek mathematicians publish their analyses?” in Suppes, P., Moravcsik, J., and Mendell, H. (eds.), Ancient and Medieval Traditions in the Exact Sciences: Essays in Memory of Wilbur Knorr (Stanford, California, 2000), pp. 139–57Google Scholar.

14 Pappus of Alexandria, Book 7 of the Collection, ed. by Jones, Alexander, 2 vols. (New York etc., 1986), vol. 1, p. 82Google Scholar. Rashed and Bellosta use the French translation by Paul Ver Eecke (Pappus d'Alexandrie, La Collection mathématique, Œuvre traduite pour la première fois du grec en français, avec une introduction et des notes par Paul Ver Eecke, 2 vols. [Paris and Bruges, 1933; repr. Paris, 1982]).

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17 A well known reduction, which is not deductive, is Hippocrates' reduction of the problem of duplication of a cube to the problem of finding two mean proportionals between two given line segments. See also, Knorr, Wilbur Richard, The Ancient Tradition of Geometrical Problems (Boston, 1986; repr. New York, 1993), Chapter 8Google Scholar.

18 This interpretation is adopted, for instance, by Heath (Euclid, The Thirteen Books of The Elements, tr. Thomas Little Heath, 2nd edn, 3 vols. (New York, 1956; orig. publ. Cambridge, 1925, I, pp. 138–9Google Scholar). This imposes on Pappus the view that analysis and synthesis are deductively symmetrical. See also Cherniss, Harold, “Plato as mathematician,” Review of Metaphysics, 4.3 (1951): 395425Google Scholar.

19 Bellosta, “Ibrāhīm ibn Sinān: on analysis and synthesis”, pp. 221, 226 and the translated Arabic text in the Appendix.

20 Hintikka, Jaakko and Remes, Unto, The Method of Analysis: Its Geometrical Origin and its General Significance (Dordrecht, 1974)CrossRefGoogle Scholar.

21 In Euclid's Data, a geometrical object is considered as “given” in three senses: in “magnitude”, in “species”, and in “position”, but not a theorem or an axiom.

22 Marini Philosophi Commentarius in Euclidis Data, in Euclid's Data in Euclidis opera omnia, ed. H. Menge, vol. 6 (Leipzig 1896), p. 234; Taisbak, Christian Marinus (ed.), Euclid's Data: The Importance of Being Given, Acta Historica Scientiarum Naturalium Et Medicinalium 45 (Copenhagen, 2003), p. 243Google Scholar.

23 Hilbert, David, “Grundlagen der Geometrie”, in Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen, 1st edn (Leipzig, 1899), pp. 192Google Scholar. English translation: The Foundations of Geometry, 2nd edn (Chicago, 1980).

24 Gardies, Jean Louis, “Do mathematical constructions escape logic?Synthese, 134 (2003): 324CrossRefGoogle Scholar.

25 Mäenpää, Petri, “From backward reduction to configurational analysis”, in Otte, M. and Panza, M. (eds.), Analysis and Synthesis in Mathematics. History and Philosophy (Dordrecht, 1997), pp. 201–26CrossRefGoogle Scholar.

26 Per Erik Rutger Martin-Löf, Intuitionistic Type Theory (Naples, 1984).

27 Bellosta, “Ibrāhīm ibn Sinān: on analysis and synthesis”, p. 212.

28 Pappus of Alexandria, Book 7 of the Collection, ed. Jones, vol. 1, p. 67.

29 Ian Mueller, Review of: The Method of Analysis: Its Geometrical Origin and Its General Significance by Jaakko Hintikka and Unto Remes, in The Journal of Philosophy, vol. 73, no. 6 (Mar. 25, 1976): 158–62; 160–1CrossRefGoogle Scholar.

30 Rashed, Roshdi, “Analysis and synthesis according to Ibn al-Haytham”, in Gould, C.C. and Cohen, R.S. (eds.), Artifacts, Representations and Social Practice (Dordrecht, 1994), p. 122Google Scholar.

31 Rashed, Roshdi and Bellosta, Hélène, Ibrāhīm ibn Sinān. Logique et Géométrie au xe siècle (Leiden, Boston, Köln, 2000)Google Scholar.

32 Rashed, Roshdi, Les mathématiques infinitésimales du ixe au xie siècle, vol. 4: Ibn al-Haytham, Méthodes géométriques, transformations ponctuelles et philosophie des mathématiques (London, 2002).Google Scholar

33 Bellosta, “Ibrāhīm ibn Sinān: on analysis and synthesis”, p. 211.

34 See also Rashed, “Analysis and synthesis according to Ibn al-Haytham”, pp. 121–40.

35 Ibid., pp. 131–2.

Ibid.

36 Apollonius de Perge, La section des droites selon des rapports, p. 17.

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