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In defence of geometrical algebra


The geometrical algebra hypothesis was once the received interpretation of Greek mathematics. In recent decades, however, it has become anathema to many. I give a critical review of all arguments against it and offer a consistent rebuttal case against the modern consensus. Consequently, I find that the geometrical algebra interpretation should be reinstated as a viable historical hypothesis.

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  1. Bos, Henk J.M. 2001. Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction. Berlin: Springer.

    Book  MATH  Google Scholar 

  2. Freudenthal, Hans. 1977. What is algebra and what has it been in history? Archive for the History of Exact Sciences 16(3): 189–200.

    MathSciNet  Article  MATH  Google Scholar 

  3. Fried, Michael N., and Sabetai Unguru. 2001. Apollonius of Perga’s conica: Text, context, subtext. Leiden: Brill.

    MATH  Google Scholar 

  4. Grattan-Guinness, Ivor. 1996. Numbers, magnitudes, ratios, and proportions in Euclid’s elements: How did he handle them? Historia Mathematica 23: 355–375.

    MathSciNet  Article  MATH  Google Scholar 

  5. Grattan-Guinness, Ivor. 2004. Decline, then recovery: An overview of activity in the history of mathematics during the twentieth century. History of Science 42(3): 279–312.

    MathSciNet  Article  Google Scholar 

  6. Heath, T.L. 1908. The thirteen books of Euclid’s elements. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  7. Høyrup, Jens. 2002. Lengths, widths, surfaces: A portrait of old Babylonian algebra and its kin. Sources and studies in the history of mathematics and physical sciences. Berlin: Springer.

  8. Mueller, Ian. 1981. Philosophy of mathematics and deductive structure in Euclid’s elements. Cambridge: MIT Press.

    MATH  Google Scholar 

  9. Netz, Reviel. 2004. The transformation of mathematics in the early Mediterranean world: From problems to equations. Cambridge Classical Studies. Cambridge: Cambridge University Press.

  10. Neugebauer, Otto. 1936. Zur geometrischen Algebra (Studien zur Geschichte der antiken Algebra III). Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, B 3: 245–259.

    MATH  Google Scholar 

  11. Rowe, David. 2012. Otto Neugebauer and Richard Courant: On exporting the Göttingen approach to the history of mathematics. Mathematical Intelligencer 34(2): 29–37.

    MathSciNet  Article  MATH  Google Scholar 

  12. Saito, Ken. 1985. Book II of Euclid’s elements in the light of the theory of conic sections. Historia Scientiarum 28: 31–60.

    MATH  Google Scholar 

  13. Saito, Ken. 1986. Compounded ratio in Euclid and Apollonius. Historia Scientiarum 31: 25–59.

    MathSciNet  MATH  Google Scholar 

  14. Sidoli, Nathan. 2013. Research on ancient greek mathematical sciences, 1998–2012. In From Alexandria, through Baghdad: Surveys and studies in the ancient Greek and medieval Islamic mathematical sciences in honor of J.L. Berggren, ed. Nathan Sidoli, and Glen Van Brummelen, 25–50. Berlin: Springer.

    Google Scholar 

  15. Szabó, Árpád. 1969. The beginnings of Greek mathematics, Synthese historical library 17, Reidel, 1978. Originally published as Anfänge griechischen Mathematik. Oldenbourg Wissenschaftsverlag.

  16. Unguru, Sabetai. 1975. On the need to rewrite the history of Greek mathematics. Archive for the History of Exact Sciences 15(1): 67–114.

    MathSciNet  Article  MATH  Google Scholar 

  17. Unguru, Sabetai. 1979. History of ancient mathematics: Some reflections of the state of the art. Isis 70(4): 555–565.

    MathSciNet  Article  MATH  Google Scholar 

  18. Unguru, Sabetai, and David E. Rowe. 1981. Does the quadratic equation have Greek roots? A study of “geometrical algebra”, “application of areas”, and related problems. Libertas Mathematica 1: 1–49.

    MathSciNet  MATH  Google Scholar 

  19. Unguru, Sabetai, and David E. Rowe. 1982. Does the quadratic equation have Greek roots? A study of “geometrical algebra”, “application of areas”, and related problems (cont.). Libertas Mathematica 2: 1–62.

    MathSciNet  MATH  Google Scholar 

  20. Van der Waerden, B.L. 1950. Science awakening, Noordhoff, Groningen, 1954. Originally published as Ontwakende wetenschap. Noordhoff, Groningen

  21. Van der Waerden, B.L. 1975. Defence of a “shocking” point of view. Archive for the History of Exact Sciences 15(3): 199–210.

    MathSciNet  Article  MATH  Google Scholar 

  22. Viète, François. 2006. The analytic art. New York: Dover Publications.

    MATH  Google Scholar 

  23. Wallis, John. 1685. A treatise of algebra, both historical and practical: shewing the original, progress, and advancement thereof, from time to time, and by what steps it hath attained to the height at which now it is, London.

  24. Weil, André. 1978. Who betrayed Euclid? Extract from a letter to the editor. Archive for History of Exact Sciences 19(2): 91–93.

    MathSciNet  Article  MATH  Google Scholar 

  25. 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

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Correspondence to Viktor Blåsjö.

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Communicated by: Len Berggren.

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Blåsjö, V. In defence of geometrical algebra. Arch. Hist. Exact Sci. 70, 325–359 (2016).

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  • Quadratic Formula
  • Geometrical Algebra
  • Quadratic Problem
  • Algebraic Thought
  • Geometric Intuition