• Davide Crippa
Part of the Frontiers in the History of Science book series (FRHIS)


In this book, I will study several attempts to prove the impossibility of solving three fundamental problems in geometry by algebraic means: the squaring of the circle, the ellipse and the hyperbola within the mathematical context of Seventeenth Century. All of these problems involve measuring areas or, in modern parlance, evaluating certain integrals. The term “quadrature” reveals the geometrical tradition in which these problems were originally conceived. Within the tradition of Greek mathematics, and in Seventeenth Century geometry as well, to find the area of a figure meant to construct, by geometrical means (the ruler and the compass, in the easiest instances, or by higher curves), a square equivalent to it: “squaring” or “quadrature” are thus just synonyms for designating this geometrical operation. Since the circle, the hyperbola and the ellipse (but not the parabola) are conic sections that possess geometrical centres, I shall refer to them as “central conic sections,” and I use the synthetic expression “quadrature of the central conic sections” for the problem of determining their areas.


  1. Archimedes. 1881. Archimedis Opera Omnia cum commentariis Eutociis. 3 vols. Ed. J.L. Heiberg. Leipzig: Teubner.Google Scholar
  2. Baron, Margaret. 1969. The Origins of infinitesimal calculus. New York: Dover.zbMATHGoogle Scholar
  3. Beeley Philip and Scriba, Christoph. 2008. Disputed Glory. John Wallis and some questions of precedence in seventeenth-century mathematics. In Kosmos und Zahl. Beiträge zur Mathematik- und Astronomiegeschichte, zur Alexander von Humboldt und Leibniz. 275–299. Ed. H. Hacht, R. Mikosch, I. Schwarz et. al, 275–299. Stuttgart: Franz Steiner Verlag.Google Scholar
  4. Bernard, Alain. 2003. Sophistic Aspects of Pappus’s Collection. Archive for History of Exact Sciences. 57: 93–150.MathSciNetCrossRefGoogle Scholar
  5. Bos, Henk. 1981. On the representation of curves in Descartes’ Géométrie. Archive for History of Exact Sciences. 24: 295–338.MathSciNetCrossRefGoogle Scholar
  6. Bos, Henk. 2001. Redefining geometrical exactness. Dordrecht Heidelberg New York London: Springer.CrossRefGoogle Scholar
  7. Clagett, Marshall. 1964. Archimedes in the Middle Ages. Volume 1. Madison: The University of Wisconsin Press.zbMATHGoogle Scholar
  8. Crippa, Davide. 2017. Review of Mathematische Schriften, Reihe 7, sechster Band, 16731676, Arithmetische Kreisquadratur, Gottfried Wilhelm Leibniz. Akademie-Verlag, Berlin (2012), Siegmund Probst and Uwe Mayer, Eds. Historia Mathematica. 44, 1, 73–76.CrossRefGoogle Scholar
  9. Cuomo, Serafina. 2000. Pappus of Alexandria and the mathematics of Late Antiquity. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  10. Descartes, René. 1897–1913. Oeuvres de Descartes. Eds. Charles Adam and Paul Tannery. 12 vols. Paris: Cerf.Google Scholar
  11. Descartes, René. 1952. The Geometry of René Descartes. Ed., transl. David E. Smith and Marcia L. Latham. La Salle: Open Court.Google Scholar
  12. Descartes, René. 1659-1661. Renati Descartes Geometria. Editio Secunda. Multis accessionibus exornata, et plus altera sua parte adaucta. Ed. Frans Van Schooten. Amsterdam: Apud Ludovicum et Danielem Elzevirios.Google Scholar
  13. Dijksterhuis, E. Jan. 1987. Archimedes, with a new bibliographic essay. ed. Wilbur R. Knorr. Princeton: Princeton University Press.Google Scholar
  14. Grootendorst, A. W., and Van Maanen, Jan. 1982. Van Heuraet’s letter (1659) on the rectification of curves. Text, translation (English, Dutch), commentary. Niew Archief voor Wiskunde. 30: 95–113.zbMATHGoogle Scholar
  15. Heath, Thomas. 1897. The Works of Archimedes. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  16. Heath, Thomas. 1921. A history of Greek mathematics. Oxford: Clarendon Press.zbMATHGoogle Scholar
  17. Hofmann, Joseph. 1942. Die Quellen der cusanischen Mathematik I: Ramon Lulls Kreisquadratur. Sitzungsberichte der Heidelberger Akademie der Wissenschaften Philosophisch-historische Klasse. 4: 1–38.zbMATHGoogle Scholar
  18. Huygens, Christiaan. 1888–1950. Oeuvres complètes publiées par la Société hollandaise des sciences, 22 vol. Ed. Bierens de Haan. The Hague: M. Nijhoff.zbMATHGoogle Scholar
  19. Israel, Giorgio. 1998. The analytic method in Descartes’ Géométrie. In Analysis and synthesis in mathematics. Eds. Michael Otte, Marco Panza, 5–30. Boston studies in the philosophy of science. Dordrecht: Kluwer.Google Scholar
  20. Jacob, Marie. 2005. La quadrature du cercle : Un problème à la mesure des Lumières. Paris: Fayard.Google Scholar
  21. Jones Arthur, Morris Sidney, Pearson Kenneth. 1991. Abstract algebra and famous impossibilities. New York: Springer.zbMATHGoogle Scholar
  22. Keyser, Paul T. 2007. Ammonius. In: The Biographical Encyclopedia of Astronomers. Eds. Hockey T. et al. New York: Springer.Google Scholar
  23. Klein, Felix. Famous problems of elementary geometry, transl. W.W. Beman and E.D. Smith. Boston, London: Ginn and Company.Google Scholar
  24. Knorr, Wilbur Richard. 1983. Construction as existence proof in Ancient Geometry. Ancient Philosophy 3: 125–148.CrossRefGoogle Scholar
  25. Knorr, Wilbur Richard. 1986. The Ancient Tradition of Geometric Problem Solving. New York, Dordrecht, London: Birkhäuser Basel.Google Scholar
  26. Lehay, Andrew. 2016. William Neile’s Contribution to Calculus. The College mathematics journal 46 (1): 42–49.MathSciNetCrossRefGoogle Scholar
  27. Leibniz, Gottfried Wilhelm. 1923-. Sämtliche Schriften und Briefe. Berlin, Göttingen: Berlin-Brandenburgische Akademie der Wissenschaften/Akademie der Wissenschaften zu Göttingen.Google Scholar
  28. Leibniz, Gottfried Wilhelm. 2011. Gottfried Wilhelm Leibniz: Die mathematischen Zeitschriftenartikel. Eds. Hans-Jürgen Hess, and Malte-Ludof Babin. Hildesheim, Zürich, New York: Georg Olms Verlag.Google Scholar
  29. Lützen, Jesper. 2009. Why was Wantzel overlooked for a century? The changing importance of an impossibility result. Historia Mathematica. 36: 374–394.MathSciNetCrossRefGoogle Scholar
  30. Lützen, Jesper. 2010. The algebra of geometric impossibilities: Descartes and Montucla on the impossibility of the duplication of the cube and the trisection of the angle. Centaurus. 52: 4–37.MathSciNetCrossRefGoogle Scholar
  31. Lützen, Jesper. 2014. 17th century arguments for the impossibility of the indefinite and the definite quadrature of the circle. Revue d’histoire des mathématiques. 20: 211–251.zbMATHGoogle Scholar
  32. Van Maanen, Jan. 1984. Hendrick van Heuraet (1634-1660?): His Life and Mathematical Work. Centaurus. 27, 3–4: 218–279.MathSciNetCrossRefGoogle Scholar
  33. Mancosu, Paolo. 1999. Philosophy of Mathematics and Mathematical Practice in XVII Century. Oxford: Oxford University Press.zbMATHGoogle Scholar
  34. Mancosu Paolo. 2007. Descartes and mathematics. In A companion to Descartes. Ed. Janet Broughton, John Carriero, 103–123. Oxford: Blackwell Publishing.Google Scholar
  35. Molland, George A. 1991. Implicit versus explicit geometrical methodologies: the case of constructions. In Mathématiques et philosophie de l’antiquité à l’âge classique: hommage à Jules Vuillemin. Ed. R. Rashed. 181–196. Paris: editions du CNRS.Google Scholar
  36. Panza, Marco. 2005. Newton et les origines de l’analyse. Paris: Blanchard.zbMATHGoogle Scholar
  37. Panza, Marco. 2011. Rethinking geometrical exactness. Historia Mathematica. 38: 42–95.MathSciNetCrossRefGoogle Scholar
  38. von Pape, Bodo. 2017. Diedrich Uhlhorn (1764-1837) und die Grössen Probleme der Antike. Mitteilungen der Hamburger Mathematischen Gesellschaft. 27: 29–59.zbMATHGoogle Scholar
  39. Pappus. 1876. Pappi Alexandrini collectionis quae supersunt, 3 vols. Ed. Friedrich Otto Hultsch. Berlin: Weidman.Google Scholar
  40. Pappus. 1986. Book 7 of the Collection, edited by A. Jones. ed., transl. Alexander Jones. Dordrecht Heidelberg New York London: Springer.Google Scholar
  41. Pappus. 2010. Pappus of Alexandria: Book 4 of the Collection. ed., transl. Heike Sefrin Weis. Dordrecht Heidelberg New York London: Springer.Google Scholar
  42. Rashed, Roshdi. 2013. Ibn al-Haytham and analytic mathematics. Trans. Daniel O’Donoghue. Ed. Geoffrey Nash. London, New York: Routledge.Google Scholar
  43. Smeur Alphons Johannes Emile Marie. 1970. On the value equivalent to π in ancient mathematical texts. A new interpretation. Archive for History of Exact Sciences. 6(4): 249–270.MathSciNetCrossRefGoogle Scholar
  44. Szabó Árpád. 1978. The beginnings of Greek mathematics. Dordrecht, Heidelberg, New York, London: Springer.Google Scholar
  45. Tropfke Johannes. 1902. Geschichte der Elementar-Mathematik in systematischer Darstellung. Zweiter Band. Leipzig: Verlag von Veit & Comp.zbMATHGoogle Scholar
  46. Yoder, Joella. 1988. Unrolling time. Christiaan Huygens and the mathematization of nature. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  47. Wallis, John. 1659. Tractatus duo, prio de cycloide et corporibus inde genitis. Posterior, epistolaris, in qua agitur de cissoide et corporibus inde genitis, Oxford: typis Academicis Lichfieldianis.Google Scholar
  48. Wallis, John. 2003. The Correspondence of John Wallis (1616-1703). Volume I. Eds. Philip Beeley, Christoph Scriba. Oxford: Oxford University Press.Google Scholar
  49. Zeuthen, H. Georg. Die geometrische Konstruktion als “Existenzbeweis” in der antiken Geometrie. Mathematische Annalen. 47: 222–228.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Davide Crippa
    • 1
  1. 1.Université Paris Diderot, SPHèreParisFrance

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