Abstract
Chapter 10 of the Analyse is an exposition of the methods of Descartes and Hudde, which can be used to determine many of the same properties of curves that may be investigated using the differential calculus. L’Hôpital demonstrates how all of these methods may be easily derived and justified using Leibniz’ differential calculus. Because Leibniz’ calculus can handle transcendental curves as well as algebraic curves, and does not require removing roots in the case of algebraic curves, he concludes on the final page of the Analyse that the new calculus is vastly superior to the older methods. This chapter is an excellent way to learn about the methods of Descartes and Hudde, because l’Hôpital’s exposition of them is very clear and lucid.
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Notes
- 1.
See the discussion in Letters 25 and 26 on p. 264 and p. 269.
- 2.
Descartes used an asterisk to denote the absence of a term in a complete polynomial, see, e.g., Descartes (1954, pp. 162ff).
- 3.
For a recent exposition on the work of Hudde, see Suzuki (2005).
- 4.
In chapter 10 of L’Hôpital (1696) examples were not given numbers.
- 5.
In the calculation that follows, the term on the second row represents −atyy, a second term of the second order, even though the yy is suppressed and only the coefficient −at is written. Similar conventions are used in the remainder of this chapter.
- 6.
As in Fig. 10.6.
- 7.
As in Fig. 10.7.
- 8.
See Descartes (1954, pp. 94ff) for this construction when the point C lies on the axis.
- 9.
In L’Hôpital (1696), the French term baisant, literally “kissing,” is used for both the circle and the point. We have translated it as “osculating” because this Latin term is the standard one in English.
- 10.
Descartes (1954, pp. 100ff).
- 11.
The result known as Bézout’s Theorem implies that a circle (of degree 2) and a curve of degree n ≥ 2, generically intersect in 2n points. Although Etienne Bézout (1730–1783) lived much later, the result was widely accepted in the late seventeenth century.
References
Bernoulli, Johann, Opera Omnia, vol. 3, Bousquet, Lausanne, 1742.
Bernoulli, Johann, Der Briefwechsel von Johann I Bernoulli, vol. 1, ed. O. Spiess, Birkhäuser, Basel, 1955.
Bernoulli, Johann, Der Briefwechsel von Johann I Bernoulli, vol. 2, ed. P. Costabel, J. Peiffer, Birkhäuser, Basel, 1988.
Bernoulli, Johann, Der Briefwechsel von Johann I Bernoulli, vol. 3, ed. P. Costabel, J. Peiffer, Birkhäuser, Basel, 1992.
Bradley, Robert E., “The Curious Case of the Bird’s Beak,” International J. Math. Comp. Sci., 1 (2006), pp. 243–268.
Bossut, Charles, Histoire Générale des Mathématiques, vol. 2, 2nd ed., F. Louis, Paris, 1810.
Burton, David, The Hstory of Mathematics: An Introduction, 6th ed., Mc Graw Hill, Boston, 2007.
Cohen, I. Bernard, A Guide to Newton’s Principia, in The Principia, Newton, Isaac, University of California Press, Berkeley and Los Angeles, 1999.
Suzuki, Jeff “The Lost Calculus (1637–1670): Tangency and Optimization without Limits,” Mathematics Magazine, 78 (2005), pp. 339–353.
Descartes, René, trans. , Smith & Latham, The Geometry of René Descartes, Dover, New Yrok, 1954.
Eneström, Gustav, “Sur le part de Jean Bernoulli dans la publication de l’Analyse des infiniment petits” Bibliotecha Mathematica, 8 (1894), pp. 65–72.
Fontenelle, Bernard de, Histoire du renouvellement de l’Académie royale des sciences, Boudot, Paris, 1708.
Hahn, Alexander, “Two Historical Applications of Calculus,” The College Mathematics Journal, 29 (1998), pp. 99–103.
Hall, A. Rupert, Philosophers at War, Cambridge U. Press, Cambridge, 1980.
Huygens, Christiaan. Horologium Oscillatorium sive de motu pendulorum, F. Muguet, Paris, 1673. English translation by Richard J. Blackwell, The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks, Iowa State University Press, Ames, 1986. Page references are to the 1986 translation of Huygens’ Horologium Oscillatorium.
Katz, Victor, History of Mathematics: An Introduction, 3rd ed., Addison-Wesley, Boston, 2009.
Leibniz, Willhelm G. von, “Nova methodus pro maximis et minimis,” Acta eruditorum, 3 (1684), p. 467–473.
l’Hôpital, Guillaume F. A. de, “Méthode facile pour déterminer ler points des caustiques …,” Mémoires de mathématique et de physique, tires des registres de l’Académie Royale des Sciences, 1693, pp. 129–133.
Anonymous (Guillaume François Antoine, Marquis de l’Hôpital), Analyse des infiniment petits, Imprimerie Royale, Paris, 1696.
l’Hôpital, Guillaume F. A. de, Traité analytique des sections coniques, Boudot, Paris, 1707.
l’Hôpital, Guillaume F. A. de, Analyse des infiniment petits pour l’intellignece des lignes courbes, 2nd ed., Montalant, Paris, 1715.
l’Hôpital, Guillaume F. A. de, Analyse des infiniment petits pour l’intellignece des lignes courbes, 2nd ed. [sic], Montalant, Paris, 1716.
l’Hôpital, Guillaume F. A. de, Analyse des infiniment petits pour l’intellignece des lignes courbes, new ed. with a commentary by l’abbé Aimé-Henri Paulian, Didot le jeune, Paris, 1768.
l’Hôpital, Guillaume F. A. de, Analyse des infiniment petits pour l’intellignece des lignes courbes, new ed., revised and augmented by Arthur LeFevre, A. Jombert, Paris, 1768.
Lockwood, E. H. (1971). A Book of Curves. Cambridge, England: Cambridge University Press.
Montucla, Jean F., Histoire des Mathématiques, second ed., vol. 2, Agasse, Paris, 1799.
Schafheitlin, Paul, “Johannis (I) Bernoullii Lectiones de calculo differentialium,” Verhandlungen der Naturforschenden Gesellschaft in Basel, 34, pp. 1–32.
Stone, Edmund, An Analytick Treatise of Conick Sections, Senex et al, London, 1723.
Stone, Edmund, The Method of Fluxions, both Direct and Inverse, Innys, London, 1730.
Varignon, Pierre, Eclaircissemens sur l’analyse des infiniment petits, Rollin, Paris, 1725.
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Bradley, R.E., Petrilli, S.J., Sandifer, C.E. (2015). A New Method for Using the Differential Calculus with Geometric Curves, from Which We Deduce the Method of Messrs. Descartes and Hudde. In: L’Hôpital's Analyse des infiniments petits. Science Networks. Historical Studies, vol 50. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-17115-9_10
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