Skip to main content
SpringerLink
Log in

Poncelet’s porism: a long story of renewed discoveries, II

  • Published:
Archive for History of Exact Sciences Aims and scope Submit manuscript

Abstract

The first part of the article appeared in the previous issue of this journal. It deals with the history of the research on Poncelet’s porism, and related subjects, which were developed from the middle of the eighteenth century until the end of the nineteenth century. In this second part, we take the research developed in the twentieth century. We also offer a comparison of the main works on the subject, and we draw some conclusions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Notes

  1. See also the general introduction in the first part of the article.

  2. For a biography and the scientific production of Francesco Gerbaldi (1858–1934), see Enea (2013).

  3. This theorem, which Gerbaldi called “fundamental”, has been extended in Dragović and Radnović (2011, p. 230) to the “hyperelliptic” case, i.e. when X is of degree \(2g+2, g>1\), substantially without changing the proof. In the same book, other results of Gerbaldi on the development in continued fractions of a Halphen element are also extended to the hyperelliptic case without quoting him (pp. 231–244).

  4. We stress that this and the other definitions above are the same as in Dragović and Radnović (2011, pp. 241–243).

  5. Gerbaldi meant “intersecting in four distinct points”.

  6. From Lebesgue (1922, p. 85), we learn that this memoir was presented to the Annales de Toulouse years before its printing. In 1919 and 1920, he published two short notes, in which he stated his project, and announced a result that generalized to conics, the theorem of Chasles (recalled in section 7) on the perimeter of polygons inscribed in an ellipse and circumscribed about another one confocal to the first (Lebesgue 1919, 1920). The memoir of (1921) was included by Lebesgue in his book Les Coniques that appeared posthumously in 1942 (see Lebesgue 1942, chapter IV). Today the work of Lebesgue on the Poncelet polygons is essentially known through this book.

  7. Lebesgue was mainly referring to Cayley (1861) and also to Cayley (1853a, b).

  8. For the contribution of Gambier on Poncelet’s polygons, see the next section.

  9. This theorem asserts that if |A| is a complete linear series on a curve X, the residual divisors R of a given divisor B with respect to |A| constitute a complete linear series |R| on X, and moreover, for any divisor \(B'\), linear equivalent to B, the residual divisor of \(B'\) with resect to |A| belongs to |R|. See, for instance, (Beltrametti et al. 2009).

  10. This figure is similar to that that Lebesgue inserted at p. 62 of his memoir.

  11. Lebesgue also gave two other methods for proving this result, see pp. 71–73 of his paper.

  12. For more information on B. Gambier (1879–1954), we refer to: P. Vincensini, L’oeuvre mathématique de Bertrand Garbier (1879–1954), Ann. Fac. Sc. de Marseille, 1955; or “In memoria di Bertrand Gambier” in Boll. Un. Mat. It. 11 (1956), pp. 599–607.

  13. Here and in the following, the formulae to which we refer by (x, y) with \(x=1,\ldots ,10\) appear in the first part of the article.

  14. Theodore William Chaundy (1889–1966) was lecturer in Oxford, and he became known for the Burchnall–Chaundy theory in the field of differential operators. For more information, see the obituary published in the J. London Math. Soc. 41 (1966), pp. 755–756.

  15. John Arthur Todd (1908–1994) studied under H. F. Baker at the University of Cambridge and became lecturer in the same University. In 1948, Todd was elected a Fellow of the Royal Society of London. He is known for having introduced the so called Todd class in the theory of higher-dimensional Riemann–Roch theorem and the Coxeter–Todd lattice; see the obituary by M. Atiyah in Bull. London Math. Soc. 30 (1998).

  16. For these formulae, see for instance (Griffiths and Harris 1978b).

  17. For an extended historical account on the concept of Weierstrass point and their impact in the study of algebraic curves, see Del Centina (2008).

  18. The two authors were not aware of Halphen (1888) or Gerbaldi (1919), see Barth and Bauer (1996).

  19. Here and in the following, section or formula to which we refer by x.y, I or by x.y, II, appears respectively in the first or second part of the article.

References

  • Barth, W., and J. Michel. 1993. Modular curves and Poncelet polygons. Mathematische Annalen 295: 25–49.

    Article  MathSciNet  MATH  Google Scholar 

  • Barth, W., and Th. Bauer. 1996. Poncelet theorems. Expositiones Mathematicae 14: 125–144.

  • Barth, W., C. Peters, and A. Van de Ven. 1984. Compact complex surfaces. Berlin, Hidelberg, New York, Tokyo: Springer.

    Book  MATH  Google Scholar 

  • Beltrametti, M., E. Carletti, D. Gallarati, and Bragadin G. Monti. 2009. Lectures on curves, surfaces and projective varieties. EMS textbook. Zimmermann, Freiburg: European Mathematical Society.

    Book  Google Scholar 

  • Bos, H.J.M., C. Kers, F. Oort, and D.W. Raven. 1987. Poncelet’s closure theorem. Expositiones Mathematicae 5: 289–364.

    MathSciNet  MATH  Google Scholar 

  • Brill, A., and M. Noether. 1874. Über die algebraischen Functionen und ihre Anwendung in der Geometrie. Mathematische Annalen 7: 269–316.

    Article  MATH  Google Scholar 

  • Cayley, A. 1853a. Note on the geometrical representation of the integral \(\int \frac{dx}{\sqrt{(x+a)(x+b)(x+c)}}\). Philosophical Magazine S.4(5): 281–284; also in The Collected Mathematical Papers of Arthur Cayley, vol. 2, 1889, Cambridge University Press, pp. 53–56.

  • Cayley, A. 1853b. Note on the Porism of the in-and-circumscribed Polygon. Philosophical Magazine S.4(6): 99–103; also in The Collected Mathematical Papers of Arthur Cayley, vol. 2, 1889, Cambridge University Press, pp. 87–90.

  • Cayley, A. 1861. On the Porism of the in-and-circumscribed Polygon. Philosophical Transactions of the Royal Society of London 151: 225–239; also in The Collected Mathematical Papers of Arthur Cayley, vol. 4, 1891, Cambridge University Press, pp. 292–308.

  • Cayley, A. 1871a. On the porism of the in-and-circumscribed polygon, and the \((2,2)\)-correspondence of points on a conic. The Quarterly Journal of Mathematics 11: 83–91; also in The Collected Mathematical Papers of Arthur Cayley, vol. 8, 1895, Cambridge University Press, pp. 14–21.

  • Chaundy, T.W. 1924. Poncelet’s poristic polygons. Proceedings of the London Mathematical Society 22: 104–123.

    Article  MathSciNet  Google Scholar 

  • Chaundy, T.W. 1926. Poncelet’s poristic polygons. Proceedings of the London Mathematical Society 25: 17–44.

    Article  MathSciNet  MATH  Google Scholar 

  • Coble, A.B. 1921. Multiple binary forms with the closure property. American Journal of Mathematics 43: 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  • Coble, A.B. 1926. Double binary forms with the closure property. Transactions of the American Mathematical Society 28: 357–363.

    Article  MathSciNet  MATH  Google Scholar 

  • Del Centina, A. 2008. Weierstrass points and their impact in the study of algebraic curves: A historical account from the ”lückensatz” to the 1970s. Annali dell’Universita di Ferrara 54: 37–59.

    Article  MATH  Google Scholar 

  • Dragović, V., and M. Radnović. 2011. Poncelet Porisms and beyond. Basel: Birkäuser (Springer-Science).

    Book  MATH  Google Scholar 

  • Enea, M.R. 2013. Francesco Gerbaldi e i matematici dell’Università di Palermo, Pristem Storia 34–35, Collana a cura di A. Guerraggio, Milano, Centro Pristem Univ. Bocconi.

  • Galbura, G. 1974. Il wronskiano di un sistema di sezioni di un fibrato vettoriale di rango 1 sopra una curva algebrica ed il relativo divisore di Brill-Noether. Annali di Matematica Pura ed Applicata 98: 349–355.

    Article  MathSciNet  MATH  Google Scholar 

  • Gambier, B. 1914. Relation d’Euler entre le cercle circonscrit à un triangle et les cercles tangents aux trois côtés de ce triangle. Nouvelles Annales de Mathématiques 14: 366–368.

    Google Scholar 

  • Gambier, B. 1925. Problème de Poncelet et problème analogue. Nouvelles Annales de Mathématiques (5) 3: 256–276 (281–293).

  • Gambier, B. 1929. Polygones de Poncelet généralisés. Annales Scientifiques de l’École Normale Supérieure (3) 21: 55–71.

    Google Scholar 

  • Gerbaldi, F. 1918a. Le frazioni continue di Halphen In Scritti matematici offerti ad E. d’Ovidio, Torino: Bocca.

  • Gerbaldi, F. 1918b. Simmetria e periodicità nelle frazioni continue di Halphen. Atti della Reale Accademia delle scienze di Torino LIII: 767–784; 869–887.

  • Gerbaldi, F. 1918c. Sulle ridotte delle frazioni continue di Halphen. Rendiconti del Reale Istituto Lombardo di Scienze e Lettere LI: 523–546.

    Google Scholar 

  • Gerbaldi, F. 1919. Le frazioni continue di Halphen in relazione alle corrispondenze (2,2) involutorie e coi poligoni di Poncelet. Rendiconti del Circolo Matematico di Palermo 43: 78–104.

    Article  MATH  Google Scholar 

  • Griffiths, P.A. 1976. Variations on a theorem of Abel. Inventiones Mathematicae 35: 321–390.

    Article  MathSciNet  MATH  Google Scholar 

  • Griffiths, P.A., and J. Harris. 1977. A Poncelet theorem in space. Commentarii Mathematici Helvetici 52: 145–160.

    Article  MathSciNet  MATH  Google Scholar 

  • Griffiths, P.A., and J. Harris. 1978a. On Cayley explicit solution to Poncelet’s porism. Enseignement Mathématique 24: 31–40.

    MathSciNet  MATH  Google Scholar 

  • Griffiths, P.A., and J. Harris. 1978. Principles of algebraic geometry. New York: Wiley.

    MATH  Google Scholar 

  • Gunning, R.C. 1966. Lectures on Riemann surfaces. Princeton: Princeton University Press.

    MATH  Google Scholar 

  • Halphen, G.-H. 1886. Traité des fonctions elliptiques, premième partie. Paris: Gauthier-Villars.

    Google Scholar 

  • Halphen, G.-H. 1888. Traité des fonctions elliptiques, deuxième partie. Paris: Gauthier-Villars.

    Google Scholar 

  • Hurwitz, A. 1879. Über unendlich-vieldeutige geometrische Aufgaben insbesondere über die Schiessungsprobleme. Mathematische Annalen 15: 8–15.

    Article  MathSciNet  MATH  Google Scholar 

  • Hurwitz, A. 1893. Über algebraische Gebilde mit eindeutigen Transformationen in sich. Mathematische Annalen 41: 391–430 (323–335).

  • Jacobi, C.G.J. 1828. Ueber die Anvendung der elliptischen Transcendenten auf ein bekanntes Problem der Elementargeometrie. Journal für die reihe und Angewandte Mathematik 3: 376–389 (partly translated in French in J. de Mathématiques Pure et Appl. 10 (1845) pp. 435–444); also in Mathematische Werke, vol. 1, (1881), Berlin, Von G. Reimer, pp. 277–293.

  • Kerawala, S.M. 1947. Poncelet porism in two circles. Bulletin of the Calcutta Mathematical Society 39: 85–105.

    MathSciNet  MATH  Google Scholar 

  • Lebesgue, H. 1919. Sur le polygones de Poncelet. Comptes rendus de séances de la Société Mathématique de France 1919:99.

  • Lebesgue, H. 1920. Sur le polygones de Poncelet, Comptes rendus de séances de la Société Mathématique de France 1920:26.

  • Lebesgue, H. 1921. Exposé géometrique d’un mémoire de Cayley sur les polygones de Poncelet. Ann. Fac. Sc. Toulouse, 3e série, 13: 61–91.

  • Lebesgue, H. 1922. Notice sur les travaux scientifiques de M. Henri Lebesgue. Toulouse: E. Privat.

    Google Scholar 

  • Lebesgue, H. 1922b. Sur la théorie de residuation de Sylvester. Annales de la Faculté des Sciences de Toulouse, 3e série 14: 153–159.

  • Lebesgue, H. 1942. Les coniques. Paris: Gautier-Villars.

    MATH  Google Scholar 

  • Poncelet, J.-V. 1822. Traité sur les propriétés projectives des figures, Paris, Bachelier; 2nd edition 1865–66. Paris, Gauthier-Villars.

  • Salmon, G. 1879. A treatise on the higher plane curves, 3rd ed. Dublin: Hodges.

    Google Scholar 

  • Segre. C. 1894. Introduzione alla geometria sopra un ente algebrico semplicemente infinito. Annali di Matematica Pura ed Applicata, serie 2, XXII: 42–142.

  • Sturm, R. 1906. Die Lhere von den geometrischen Verwandtschaften, Band I. Leipzig: Teubner.

    Google Scholar 

  • Todd, J.A. 1948. Poncelet Poristic Polygons. The Mathematical Gazette 32: 274–280.

    Article  MathSciNet  MATH  Google Scholar 

  • Weyr, E. 1870. Ueber einige Sätze von Steiner und ihren Zusammenhang mit der zwei und zwiigliedrigen Verwandtschaft der Grunfgebilde ersten Grades. Journal für die Reine und Angewandte Mathematik 71: 18–28.

    Article  MathSciNet  MATH  Google Scholar 

  • White, H.S. 1916. Poncelet polygons. Science 43: 149–158.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 45, 44100, Ferrara, Italy

    Andrea Del Centina

Authors
  1. Andrea Del Centina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrea Del Centina.

Additional information

Communicated by : Jeremy Gray.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Del Centina, A. Poncelet’s porism: a long story of renewed discoveries, II. Arch. Hist. Exact Sci. 70, 123–173 (2016). https://doi.org/10.1007/s00407-015-0164-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00407-015-0164-x

Search

Navigation