Abstract
In Chapter 1 of the Analyse, l’Hôpital gives the rules of the differential calculus. His calculus is a calculus of equations and differentials, not of functions and derivatives. Modern readers will nevertheless recognize many familiar rules, including the product rule, the quotient rule, and the power rule. L’Hôpital has no need for the chain rule, because substitutions are handled in a very natural way in this calculus. Because readers in the 1690s may not have been familiar with negative and fractional exponents, l’Hôpital also explains those ideas here.
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Notes
- 1.
In L’Hôpital (1696) the terms appliquée for ordinate and coupée for abscissa are used, literally meaning “applied” and “cut.”
- 2.
In L’Hôpital (1696), the same word différence is used for both the differential and the difference of ordinary subtraction. In this translation, it is consistently translated as “differential” when the difference is infinitely small.
- 3.
In L’Hôpital (1696), these terms are used interchangeably for the x-axis. Technically, the term diameter should only be used in the sense for a curve that is symmetric about the axis.
- 4.
Compare to Postulate 1 on p. 193.
- 5.
Compare to Postulate 2 on p. 193.
- 6.
Compare to the section “On the Addition and Subtraction of Differentials” on p. 193.
- 7.
In L’Hôpital (1696) the expression “to infinity” is frequently employed meaning roughly “of all orders.”
- 8.
Compare this with the product rule as discussed on p. 195.
- 9.
In L’Hôpital (1696), the overline is used for grouping terms. We note, however, that a negative sign does not seem to distribute through terms grouped under an overline; see §13 ff.
- 10.
Compare to the section “On the Differentials of Divided Quantities” on p. 195.
- 11.
As we will see on p. 6 (L’Hôpital 1696, p. 8), a “perfect” power is an integer and an “imperfect” power is a fraction.
- 12.
Compare to the section “On the Differentials of Surd Quantities” on p. 196.
- 13.
Compare the discussion that follows with Bernoulli’s discussion of geometric and arithmetic progressions on p. 196.
- 14.
This case is proved by Bernoulli on p. 194, without using the Product Rule.
- 15.
Bernoulli gave this example on p. 198.
- 16.
Bernoulli gave a similar example on p. 198.
- 17.
Bernoulli gave this example on p. 198.
- 18.
In L’Hôpital (1696), the + sign separating the terms of the numerator was omitted.
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). In Which We Give the Rules of This Calculus. 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_1
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