The Transformation of Mathematics in the Early Mediterranean World

From Problems to Equations

The Transformation of Mathematics in the Early Mediterranean World

The transformation of mathematics from its ancient Greek practice to its development in the medieval Arab-speaking world is approached by focusing on a single problem proposed by Archimedes and the many solutions offered. From a practice of mathematics based on the localized solution (originating in the polemical practices of early Greek science), we see a transition to a practice of mathematics based on the systematic approach (grounded in the deuteronomic practices of Late Antiquity and the Middle Ages). A radically new interpretation is accordingly offered of the historical trajectory of pre-modern mathematics.


"For the true mathematics historian, this is a fascinating exploration, perhaps different from one's previous ideas of this time period. Highly recommended." M.D. Sanford, Felician College

"...engaging, provocative, and definitely worth reading and thinking about." MAA Reviews, Fernando Q. Gouvea

"...recommended reading--for its thought-provoking ideas and lively writing--for those with a serious interest in the mathematics of ancient Greece and medieval Islam." - Mathematical Reviews, J.L. Berggren

Arafat, W. and Winter, H. J. J. 1950. The Algebra of Umar Khayyam, Journal of the Royal Asiatic Society of Bengal, Science 16: 27–78
Benjamin, W. 1968. The Work of Art in the Age of Mechanical Reproduction, translated in Illuminations (ed. H. Arendt). New York. (Original: Das Kunstwerk im Zeitalter seiner technischen Rerodukzierbarkeit, Schriften i: 366–405, ed. T. Adorno, 1955: Frankfurt.)
Clagett, M. 1964–84. Archimedes in the Middle Ages (5 vols.). Philadelphia
Corry, L. 1996. Modern Algebra and the Rise of Mathematical Structures. Boston
Cuomo, S. 2000. Pappus of Alexandria and the Mathematics of Late Antiquity. Cambridge
Decorps-Foulquier, M. 2000. Recherches sur les Coniques d'Apollonius de Pergé et leur commentateurs grecs. Histoire de la transmission des Livres I–IV. Paris
Fowler, D. H. F. 1999. The Mathematics of Plato's Academy (2nd ed.). Oxford
Fried, M. and Unguru, S. 2001. Apollonius of Perga's Conica: Text, Context, Subtext. Boston
Gandz, S. 1936. The Sources of Al-Khowarizmi's Algebra. Osiris 1: 263–77
Goldstein, C. 1995. Un Théorème de Fermat et ses lecteurs. Saint-Denis
Heath, T. L. 1897. The Works of Archimedes. Cambridge
Heath, T. L. 1921. A History of Greek Mathematics. Oxford
Heiberg, J. L. 1885. Euclides: Opera. Vol. IV. Leipzig
Heiberg, J. L. 1891. Apollonius: Opera. Vol. I. Leipzig
Heiberg, J. L. 1893. Apollonius: Opera. Vol. II. Leipzig
Heiberg, J. L. 1898. Ptolemy: Syntaxis. Vol. I. Leipzig
Heiberg, J. L. 1910. Archimedes, Opera Omnia, Vol. I. Leipzig
Heiberg, J. L. 1913. Archimedes, Opera Omnia, Vol. II. Leipzig
Heiberg, J. L. 1915. Archimedes, Opera Omnia, Vol. III. Leipzig
Herreman, A. 2000. La Topologie et ses Signes. Paris
Hogendijk, J. P. 1989. Sharaf Al-Din Al-Tusi on the Number of Positive Roots of Cubic Equations, Historia Mathematica 16: 69–85
Hoyrup, J. 1996. ‘The Four Sides and the Area’. Oblique Light on the Prehistory of Algebra, in Vita Mathematica ed. R. Calinger, Washington DC, pp. 45–65
Hughes, B. B. 1989. Robert of Chester's Latin Translation of Al-Khwarizmi's Al-Jabr. Stuttgart
Jones, A. 1986. Pappus of Alexandria: Book 7 of the Collection. New York
Karpinski, L. C. 1915. Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi. New York
Klein, J. 1934–6/1968. Greek Mathematical Thought and the Origins of Algebra. Cambridge MA
Knorr, W. 1982. Observations on the Early History of the Conics, Centaurus 26: 1–24
Knorr, W. 1986. The Ancient Tradition of Geometric Problems. Boston
Lloyd, G. E. R. 1996. Adversaries and Authorities. Cambridge
Lorch, P. 1989. The Arabic Transmission of Archimedes' Sphere and Cylinder and Eutocius' Commentary, Zeitschrift für Geschichte der Arabische-Islamischen Wissenschaften 5: 94–114
Mansfeld, J. 1994. Prolegomena: Questions to be Settled before the Study of an Author, or a Text. Leiden
Mansfeld, J. 1998. Prolegomena Mathematica from Apollonius of Perga to the Late Neoplatonists, with an Appendix on Pappus and the History of Platonism. Leiden
Mugler, C. 1972. Archimède, Opera. Vol. iv. Paris
Musharrafa, A. and Ahmad, M. A. 1939. Al-Khwarizmi: Kitab Al-Jabr wa Al-Muqabala. Cairo
Netz, R. 1998. Deuteronomic Texts: Late Antiquity and the History of Mathematics, Revue d'Histoire des Mathématiques 4: 261–88
Netz, R. 1999. The Shaping of Deduction in Greek Mathematics: a Study in Cognitive History. Cambridge
Netz, R. 2000. Why did Greek Mathematicians Publish their Analyses? In Memorial Volume for Wilbur Knorr, eds. H. Mandel, J. Moravcsik, and P. Suppes, Stanford, pp. 139–57
Netz, R. 2001. Archimede. Encyclopedia Italiana della Storia della Scienza, Vol. I La Scienza Antica: 779–790. Rome
Netz, R. Forthcoming. The Works of Archimedes Translated into English, Volume I: The Two Books on Sphere and Cylinder. Cambridge
Netz, R., Saito, K. and N. Tchernetska. 2001. New Reading of Method Proposition 14: Preliminary Evidence from the Archimedes Palimpsest (Part 1), Sciamvs 2: 9–29
Rashed, R. 1986. Sharaf Al-Din Al-Tusi: Oeuvres Mathématiques. Algebre et Géometrie au Ⅻe siècle. Paris
Rashed, R. 2000. Les Catoptriciens Grecs: Les Mirroirs ardents. Paris
Rashed, R. and Vahabzadeh, B. 1999. Al-Khayyam Mathématicien. Paris
Rashed, R. and Vahabzadeh, B. 2000. Omar Khayyam the Mathematician (partial translation of [1999]). New York
Rome, A. 1936. Commentaires de Pappus et de Théon d'Alexandrie sur l'Almageste, Vol. II. Biblioteca Apostolica Vaticana: Vatican City
Rosen, F. 1831/1969. The Algebra of Muhammed ben Musa. New York
Saito, K. 1985. Book II of Euclid's Elements in the Light of the Theory of Conic Sections, Historia Scientiarum: 31–60
Saito, K. 1997. Index of the Propositions Used in Book vii of Pappus' Collection, The Journal of Humanities, Faculty of Letters, Chiba University 26: 155–8
Schöne, H. 1903. Hero: Opera. Vol. III. Leipzig
Schramm, M. 1965. Steps Towards the Idea of Function: a Comparison between Eastern and Western Science of the Middle Ages, History of Science 4: 70–103
Sesiano, J. 2002. Towards Omar Khayyam, Farhang 14: 193–201
Sezgin, F. 1974. Geschichte des arabischen Schriftums, Bd. 5: Mathematik Bis ca. 430 H. Leiden
Thomas, I. 1941. Greek Mathematical Works. Vol. ii. Cambridge MA
Toomer, G. J. 1976. On Burning Mirrors/Diocles. New York
Unguru, S. 1975. On the Need to Rewrite the History of Greek Mathematics, Archive for the History of Exact Sciences 15: 67–114
Unguru, S. 1979. History of Ancient Mathematics: Some Reflections on the State of the Art, Isis 555–65
Vitrac, B. 1998. Euclide/Les Éléments. Vol. iii. Paris
Woepcke, F. 1851. L'Algèbre d'Omar AlKhayyâmî. Paris
Youschkevitch, A. P. 1976. Les Mathématiques arabes. Paris
Zeuthen, H. G. 1886. Die Lehere von den Kegelschnitten im Altertum. Kopenhagen