Archive for History of Exact Sciences

, Volume 72, Issue 4, pp 353–402 | Cite as

The concept of given in Greek mathematics

  • Nathan Sidoli
Original Paper


This paper is a contribution to our understanding of the technical concept of given in Greek mathematical texts. By working through mathematical arguments by Menaechmus, Euclid, Apollonius, Heron and Ptolemy, I elucidate the meaning of given in various mathematical practices. I next show how the concept of given is related to the terms discussed by Marinus in his philosophical discussion of Euclid’s Data. I will argue that what is given does not simply exist, but can be unproblematically assumed or produced through some effective procedure. Arguments by givens are shown to be general claims about constructibility and computability. The claim that an object is given is related to our concept of an assignment—what is given is available in some uniquely determined, or determinable, way for future mathematical work.



The core ideas of this paper go back some years now to my dissertation, and I thank Alexander Jones and Jan Hogendijk for their comments on that work. I presented an overview of this argument at a conference of the SAW Project, under the direction of Karine Chemla. The discussion following this presentation helped me to clarify some of my thinking. During the time that I was a guest of the SAW Project in Paris, 2015, some of the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC Grant Agreement No. 269804. Ken Saito read an earlier draft of this paper and made a number of valuable suggestions. This paper has benefited considerably from the extensive notes made by Karine Chemla and Matthieu Husson.


Modern Scholarship

  1. Acerbi, F. 2007. Euclide, Tutte le opere. Milano: Bompiani.Google Scholar
  2. Acerbi, F. 2011a. The Language of the ‘Givens’: Its Form and Its Use as a Deductive Tool in Greek Mathematics. Archive for History of Exact Sciences 65: 119–153.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Acerbi, F. 2011b. De Polygonal Numeris, Diofanto. Pisa: Fabrio Serra.zbMATHGoogle Scholar
  4. Acerbi, F. 2012. I codici stilistici della matematica greca: dimostrazioni, procedure, algoritmi. Quaderni Urbinati Di Cultura Classica NS 101: 167–214.Google Scholar
  5. Acerbi, F., and B. Vitrac. 2014. Metrica: Héron d’Alexandrie. Pisa: Fabrizio Serra.Google Scholar
  6. Acerbi, F. 2017. The Mathematical Scholia Verera to the Almagest. SCIAMVS 18: 133–259.MathSciNetzbMATHGoogle Scholar
  7. Allard, A. 1980. Diophante d’Alexandrie, Les Arithmetiques, Histoire du text grec, édition critique, traductions et scolies. Tourpes: Chercheur du Fonds National Belge de la Recherche Scientifique.Google Scholar
  8. Artmann, B. 1999. Euclid: The Creation of Mathematics. New York: Springer.CrossRefzbMATHGoogle Scholar
  9. Berggren, J.L., and G. Van Brummelen. 2000. The Role and Development of Geometric Analysis and Synthesis in Ancient Greece and Medieval Islam. In Ancient & Medieval Traditions in the Exact Sciences: Essays in Memory of Wilbur Knorr, ed. P. Suppes, J.M. Moravcsik, and H. Mendell, 1–31. Stanford: CSLI Publications.Google Scholar
  10. Blåsjö, V. 2016. In Defence of Geometrical Algebra. Archive for History of Exact Sciences 70: 325–359.MathSciNetCrossRefGoogle Scholar
  11. Christianidis, J. 2007. The Way of Diophantus. Historia Mathematica 34: 289–305.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Christianidis, J., and J. Oaks. 2013. Practicing Algebra in Late Antiquity: The Problem-Solving of Diophantus of Alexandria. Historia Mathematica 40: 127–163.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Decorps-Foulquier, M., and M. Federspiel. 2008–2010. Apollonius de Perge, Coniques, Texte grec (et arabe) établi, traduit et commenté, tomes 1.2, 2.3. Berlin: Walter de Gruyter.Google Scholar
  14. Dijksterhuis, E.J. 1987. Archimedes. Princeton: Princeton University Press.CrossRefzbMATHGoogle Scholar
  15. Edwards, D.R. 1984. Ptolemy’s Open image in new window: An Annotated Transcription of Moebeke’s Latin Translation and of the Surviving Greek Fragments, with an English Version and Commentary. PhD Thesis, Department of Classics, Brown University.Google Scholar
  16. Federspiel, M. 2000. Notes critiques sur le livre II des Coniques d’Apollonius de Perge Seconde partie. Revue des études greques 113: 359–391.CrossRefGoogle Scholar
  17. Federspiel, M. 2008. Les problèmes des livres grecs des Coniques d’Apollonius de Perge. Les études classiques 76: 321–360.MathSciNetzbMATHGoogle Scholar
  18. Friberg, J. 2007. Amazing Traces of a Babylonian Origin in Greek Mathematics. Singapore: World Scientific.CrossRefzbMATHGoogle Scholar
  19. Fried, M., and S. Unguru. 2001. Apollonius of Perga’s Conica: Text, Context, Subtext. Leiden: Brill.CrossRefzbMATHGoogle Scholar
  20. Friedlein, G. 1873. Procli Diadochi in primum Euclidis Elementorum librum commentarii. Leipzig: Teubner.Google Scholar
  21. Fournarakis, P., and J. Christianidis. 2006. Greek Geometrical Analysis: A New Interpretation Through the ‘Givens’ Terminology. Bollettino di Storia delle Scienze Matematiche 26: 33–56.MathSciNetzbMATHGoogle Scholar
  22. Hankel, H. 1874. Zur Geschichte der Mathematik in Alterthum und Mittelalter. Leipzig: Teubner (Reprinted: Olms, Hildesheim, 1965).Google Scholar
  23. Heath, T.L. 1908. The Thirteen Books of Euclid’s Elements, vol. 3. Cambridge: Cambridge University Press (Reprinted: Dover, New York, 1956).Google Scholar
  24. Heath, T.L. 1921. A History of Greek Mathematics, vol. 2. Oxford: Oxford University Press (Reprinted: Dover, New York, 1981).Google Scholar
  25. Heiberg, J.L. 1891–1893. Apollonii Pergaei quae graece extant cum commentariis antiquis. Leipzig: Teubner.Google Scholar
  26. Heiberg, J.L. 1895. Euclidis Optica, Opticorum recensio Theonis, Catoptrica, cum scholiis antiquis. Leipzig: Teubner.zbMATHGoogle Scholar
  27. Heiberg, J.L. 1898–1903. Claudii Ptolemaei Syntaxis mathematica. Leipzig: Teubner.Google Scholar
  28. Heiberg, J.L. 1907. Claudii Ptolemaei opera astronomica minora. Leipzig: Teubner.Google Scholar
  29. Heiberg, J.L., and E.S. Stamatis. 1969–1977. Euclidis Elementa, Euclidis opera omnia, vols. 1–5. Leipzig: Teubner.Google Scholar
  30. Heiberg, J.L., and E.S. Stamatis. 1972. Archimedes opera omnia. Stuttgart: Teubner.Google Scholar
  31. Hintikka, J., and U. Remes. 1974. The Method of Analysis: Its Geometric Origin and Its General Significance. Dordrecht: Reidel.CrossRefzbMATHGoogle Scholar
  32. Hughes, B.B. 1981. Jordanus de Nemore: De numeris datis. Berkeley: University of California Press.zbMATHGoogle Scholar
  33. Jones, A. 1986. Pappus of Alexandria: Book 7 of the Collection. New York: Springer.CrossRefzbMATHGoogle Scholar
  34. Knorr, W. 1986. The Ancient Tradition of Geometric Problems. Boston: Birkhäuser (Reprinted: Dover, New York, 1993).Google Scholar
  35. Lewis, M.J.T. 2001. Surveying Instruments of Greece and Rome. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  36. Lorch, R. 2001. Thābit ibn Qurra, On the Sector Figure and Related Texts. Islamic Mathematics and Astronomy 108. Frankfurt am Main: Institut für Geschichte der arabisch-islamischen Wissenschaften (Reprinted: Dr. Erwin Rauner Verlag, Augsburg, 2008).Google Scholar
  37. Macierowski, E.M., and R.H. Schmidt. 1987. 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. Fairfield: The Golden Hind Press.Google Scholar
  38. Manitius, K. 1912. Ptolemäus, Handbuch der Astronomie. Leipzig: Teubner (Second edition revised by O. Neugebauer, 1963).Google Scholar
  39. Masià, R. 2015. On Dating Hero of Alexandria. Archive for History of Exact Sciences 69: 231–255.MathSciNetCrossRefzbMATHGoogle Scholar
  40. Mcdowell, G.L., and M.A. Sololik. 1993. The Data of Euclid. Baltimore: Union Square.Google Scholar
  41. Menge, H. 1896. Euclidis Data cum commentario Marini et scholiis antiquis, Euclidis opera omnia, 6. Leipzig: Teubner.Google Scholar
  42. Netz, R. 2000. Why Did Greek Mathematicians Publish Their Analyses? In Ancient & Medieval Traditions in the Exact Sciences: Essays in Memory of Wilbur Knorr, ed. P. Suppes, J.M. Moravcsik, and H. Mendell, 139–157. Stanford: CSLI Publications.Google Scholar
  43. Rashed, R. 1984. Diophante, Les Arithmétiques. Paris: Les Belles Lettres.Google Scholar
  44. Rashed, R. 2008–2010. Apollonius de Perge, Coniques, Texte (grec et) arabe établi, traduit et commenté, tomes 1.1, 2.1, 2.2, 3, 4. Berlin: Walter de Gruyter.Google Scholar
  45. Rashed, R. (ed.). 2009. Thābit ibn Qurra: Science and Philosophy in Ninth-Century Baghdad. Berlin: Walter de Gruyter.zbMATHGoogle Scholar
  46. Rashed, R. (ed.). 2015. Angles et grandeur, D’Euclide à Kamāl al-Dīn al-Fārisī. Berlin: Walter de Gruyter.Google Scholar
  47. Rashed, R., and H. Bellosta. 2010. Apollonius de Perge, La section des droites selon des rapports. Berlin: Walter de Gruyter.CrossRefGoogle Scholar
  48. Ritter, J. 1989. Chacun se vérité : les mathématiques en Égypte et en Mésopotamie. In Élements d’histoire des sciences, ed. M. Serres. Paris: Bordas.Google Scholar
  49. Ritter, J. 1995. Measure for Measure: Mathematics in Egypt and Mesopotamia. In A History of Scientific Thought, ed. M. Serres. Cambridge, MA: Blackwell.Google Scholar
  50. Rome A. 1931–1943. Commentaires de Pappus et de Théon d’Alexandrie sur l’Almageste, vols. I–III. Biblioteca Apostolica Vatican: Vaticana.Google Scholar
  51. Saito, K., and N. Sidoli. 2010. The Function of Diorism in Ancient Greek Analysis. Historia Mathematica 37: 579–614.MathSciNetCrossRefzbMATHGoogle Scholar
  52. Schöne, H. 1903. Herons von Alexandria, Vermessungslehre und Dioptra, Opera quae supersunt omnia, vol. 3. Leipzig: Tuebner.Google Scholar
  53. Sesiano, J. 1982. Books IV to VIII of Diophantus’ Arithmetica in the Arabic Translation Attributed to Qustā ibn Lūqā. New York: Springer.zbMATHGoogle Scholar
  54. Sidoli, N. 2004a. Ptolemy’s Mathematical Approach Applied Mathematics in the Second Century. PhD thesis, University of Toronto, Institute for the History and Philosophy of Science and Technology.Google Scholar
  55. Sidoli, N. 2004b. On the Use of the Term Diastēma in Ancient Greek Constructions. Historia Mathematica 31: 2–10.MathSciNetCrossRefzbMATHGoogle Scholar
  56. Sidoli, N. 2011. Heron of Alexandria’s Date. Centaurus 53: 55–61.MathSciNetCrossRefGoogle Scholar
  57. Sidoli, N. 2014. Mathematical Tables in Ptolemy’s Almagest. Historia Mathematica 41: 13–37.MathSciNetCrossRefzbMATHGoogle Scholar
  58. Sidoli, N. Forthcoming a. Mathematical Discourse in Philosophical Authors: Examples from Theon of Smyrna and Cleomedes on Mathematical Astronomy. In Instruments Observations Theories: Studies in the History of Early Astronomy in Honor of James Evans, ed. C. Carman, A. JonesGoogle Scholar
  59. Souffrin, P. 2000. Remarques sur la datation de la Dioptre d’Héron par l’éclipse de lune du 62. In Autour de le Dioptre d’Héron d’Alexandrie: Actes du Collogue international de Saint-Étienne, ed. G. Argoud and J.-Y. Guillaumin. Saint-Étienne: l’Université Saint-Étienne.Google Scholar
  60. Taisbak, C.M. 1991. Elements of Euclid’s Data. Apeiron 24: 135–171.MathSciNetCrossRefzbMATHGoogle Scholar
  61. Taisbak, C.M. 2003. \({\varDelta }E{\varDelta }OMENA\), Euclid’s Data: The Importance of Being Given. Copenhagen: Museum Tusculanum Press.Google Scholar
  62. Tannery, P. 1893–1895. Diophanti Alexandrini, Opera omnia cum Graecis commentariis. Stuttgart: Teubner.Google Scholar
  63. Thaer, C. 1962. Die Data von Euklid. Berlin: Springer.CrossRefGoogle Scholar
  64. Toomer, G.J. 1984. Ptolemy’s Almagest. London: Duckworth (Reprinted: Princeton University Press, Princeton, 1998).Google Scholar
  65. Toomer, G.J. 1990. Apollonius, Conics Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banū Mūsā. New York: Springer.CrossRefGoogle Scholar
  66. Toomer, G.J. 1985. Galen on the Astronomers and the Astrologers. Archive for History of Exact Sciences 32: 193–206.MathSciNetCrossRefzbMATHGoogle Scholar
  67. Tweddle, I. 2000. Simpson on Porisms: An Annotated Translation of Robert Simpson’s Posthumous Treatise on Porisms and Other Items on the Subject. London: Springer.CrossRefzbMATHGoogle Scholar
  68. Ver Eecke, P. 1959. Diophante d’Alexandrie, Les six livres Arithmétiques et Le livre des nombres polygones. Paris: Albert Blanchard.Google Scholar
  69. Vitrac, B. 1990–2001. Euclide d’Alexandrie, Les Éléments, vol. 4. Paris: Presses Universitaires de France.Google Scholar
  70. Vitrac, B. 2005. Quelques remarques sur l’usage du mouvement en géométrie dans la tradition euclidienne: de Platon et Aristote à ‘Umar Khayyam. Fahrang: Quarterly Journal of Humanities & Cultural Studies 18: 1–56.Google Scholar
  71. Zeuthen, H.G. 1885. Die Lehre von den Kegelschnitten im Altertum. A.F. Höst & Sohn, 1886 (originally published as “Kegelsnitlaeren in Oltiden,” Kongelig Danske videnskaberens Selskabs Skrifter, 6th ser., 1(3): 1–319).Google Scholar
  72. Zheng, F. 2012. Des Data d’Euclide au De numeris datis de Jordanus de Nemore : Les donneés, l’analyse et les problemes. Thèse de doctorat, Histoire des mathématiques, Université Paris 7, Denis Diderot, Laboratoire SPHERE.Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School for International Liberal StudiesWaseda UniversityTokyoJapan

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