Archive for History of Exact Sciences

, Volume 66, Issue 1, pp 1–69 | Cite as

A new analytical framework for the understanding of Diophantus’s Arithmetica I–III



This study is the foundation of a new interpretation of the introduction and the three first books of Diophantus’s Arithmetica, one that opens the way to a historically correct contextualization of the work. Its purpose, as indicated in the title, is to renew the traditional discussion on the methods of problem-solving used by Diophantus, through the detailed exposition of a new analytical framework that aims to give an account of the coherence and progressive nature of the material included in the three first books of the Arithmetica. One outcome of this new ‘toolbox’ is a new conspectus of the problems and solutions contained in the latter, which is presented in appendix. The first part of the article clarifies, as a necessary preliminary, the key notions and terminology underlying our analysis. Among these new concepts is the notion of “method of invention,” which accounts in general for any way, by which “positions” (hypostaseis) are used in the Arithmetica. The next part proposes a complete inventory of the various methods of invention found in the three first books. Finally the last part presents the above mentioned conspectus and proposes a series of preliminary conclusions that can be drawn from it.


Auxiliary Problem Lower Case Letter Previous Position Textual Unit Prospective Equality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arithm. Diophantus’s Arithmetica, quoted either in Tannery’s edition (Tannery 1893–1895) or in Allard’s 1980 edition (Allard 1980). In the second case the page number is followed by “A”, in the first case it is either followed by “T” or by nothing when the two texts are identical.Google Scholar
  2. Allard, A. 1980. Diophante d’Alexandrie. Les Arithmétiques. Histoire du texte grec, édition critique, traductions et scholies. Thèse, Université de Louvain.Google Scholar
  3. Bernard A. (2003) Sophistic Aspects of Pappus’s. Collection Archive for History of Exact Sciences 57: 93–150MathSciNetMATHCrossRefGoogle Scholar
  4. Bernard, A. 2011. Les Arithmétiques de Diophante: introduction à la lecture d’une œuvre ancrée dans différentes traditions antiques, In Circulation, Transmission, Heritage eds. E. Barbin, P. Ageron, 557–582. IREM de Basse-Normandie.Google Scholar
  5. Bernard, A., and C. Proust, eds. Scientific sources and teaching contexts throughout history: problems and perspectives. New York: Springer (forthcoming in the series Boston Studies in Philosophy of Science).Google Scholar
  6. Christianidis, J. (eds) (2004) Classics in the History of Greek Mathematics. Kluwer, DordrechtMATHGoogle Scholar
  7. Christianidis J. (2007) The way of Diophantus: some clarifications on Diophantus’s method of solution. Historia Mathematica 34: 289–305MathSciNetMATHCrossRefGoogle Scholar
  8. Diophante, 1959. Diophante d’Alexandrie: les six livres arithmétiques et le livre des nombres polygones. Œuvres traduites pour la première fois du grec en français, avec une introduction et des notes par Paul Ver Eecke. Paris: Albert Blanchard. (First edition Bruges: Desclée de Brouwer, 1926)Google Scholar
  9. Hankel, H. 1874. Beiträge zur Geschichte der Mathematik im Altertum und Mittelalter. Leipzig: B.G. Teubner. (Reprint Hildesheim: Georg Olms Verl., 1965)Google Scholar
  10. Heath, T.L. 1910/1964. Diophantus of Alexandria.A Study in the History of Greek Algebra, 2nd edition. Cambridge: Cambridge University Press. (Reprint New York: Dover, 1964; first edition, 1884)Google Scholar
  11. Klein, J. 1968/1992. Greek Mathematical Thought and the Origin of Algebra, transl. Eva Brann. Cambridge, MA: M.I.T. Press. (Reprint New York: Dover 1992)Google Scholar
  12. Knorr W.R. (1986) The Ancient Tradition of Geometric Problems. Birkhäuser, BostonGoogle Scholar
  13. Netz R. (1998) Deuteronomic texts: late antiquity and the history of mathematics. Revue d’histoire des mathématiques 4: 261–288MathSciNetMATHGoogle Scholar
  14. Oaks J.A. (2009) Polynomials and equations in Arabic algebra. Archive for History of Exact Sciences 63: 169–203MathSciNetMATHCrossRefGoogle Scholar
  15. Rashed R. (1984) Diophante, Les Arithmétiques, tome 3: Livre IV, tome 4: Livres V–VII, texte établi et traduit par R. Rashed. Les Belles Lettres, ParisGoogle Scholar
  16. Schappacher N. (1998) Wer war Diophant?. Mathematische Semesterberichte 45(2): 141–156MathSciNetMATHCrossRefGoogle Scholar
  17. Schappacher, N. 2005. Diophantus of Alexandria: a Text and its History. Enriched version of (Schappacher 1998), in English, available on line on the personal site of the author (consulted 19.4.11).Google Scholar
  18. Sesiano J. (1982) Books IV to VII of Diophantus’s Arithmetica in the Arabic translation attributed to Qustā ibn Lūqā. Springer-Verlag, New YorkGoogle Scholar
  19. Tannery, P. 1893–1895. Diophanti Alexandrini Opera Omnia cum Graeciis commentariis, edidit et latine interpretatus est P. Tannery, 2 vols. Leipzig: B.G. Teubner. (Reprint Stuttgart: B.G. Teubner 1974)Google Scholar
  20. Vitrac, B. 2008. Les formules de la “puissance” (dynamis, dynasthai) dans les mathématiques grecques et dans les dialogues de Platon. In DYNAMIS. Autour de la puissance chez Aristote, ed. M. Crubellier, A. Jaulin, D. Lefebvre, and P.-M. Morel. Louvain: Peters.Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.EHESS-Centre A. KoyréParis-Est Créteil University-IUFMCréteilFrance
  2. 2.Department of History and Philosophy of ScienceUniversity of AthensAthensGreece

Personalised recommendations