Abstract
In Chapter 2 of the Analyse, l’Hôpital describes how to use the differential calculus to find tangents to curves. In the 1690s this was not a matter of finding a slope, but rather of finding the length of a subtangent, which l’Hôpital shows to be \(\frac{y\,\mathit{dx}} {\mathit{dy}}\). He then applies this to find the tangents of a wide variety of curves, including the conic sections, the cycloid, the Spiral of Archimedes, the Quadratrix of Dinostratus, and more than a dozen other curves. This is one of the two longest chapters of the Analyse and it serves as a catalog of many of the curves that mathematicians knew about in the late 17th century.
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- 1.
- 2.
Compare this to Bernoulli’s Problem I on p. 198.
- 3.
Compare this to Bernoulli’s Problem III on p. 201.
- 4.
Compare this to the treatment of higher order parabolas beginning on p. 199.
- 5.
AMB is an ellipse with center \(\left (\frac{a} {2}, 0\right )\), horizontal axis a, and vertical axis \(\sqrt{\mathit{ab}}\). Compare this to Bernoulli’s discussion on p. 200.
- 6.
The term \(-\overline{2a\,\mathit{dx} + 2x\,\mathit{dx}} \times x^{3}\) in what follows would be written today as \((-2a\,\mathit{dx} + 2x\,\mathit{dx})x^{3}\). Thus, the negative sign before the grouping of expressions under the overline symbol is not meant to obey the distributive law. This does not affect the validity of the computed value for AT.
- 7.
In the numerator of the second line below, the expression 5x × a − x seems to mean 5x(a − x). Thus, the use of the symbol × appears to imply a grouping of expressions.
- 8.
See p. xxi for a discussion of generalized ellipses.
- 9.
In L’Hôpital (1696) the equal sign following AT was omitted.
- 10.
In L’Hôpital (1696) the last term was written as \(\frac{n} {p} \root{p}\of{}ba^{p-1}\).
- 11.
Also called the conjugate axis. See p. xxi for a discussion of generalized hyperbolas.
- 12.
This is the Folium of Descartes reflected in the x-axis.
- 13.
See p. xxvii for a discussion of this construction.
- 14.
The five terms defined here were not italicized in L’Hôpital (1696).
- 15.
See p.xxvii for a discussion of this construction of the cycloid.
- 16.
In L’Hôpital (1696) the term allongée (elongated) was used; “curtate” is the standard modern term.
- 17.
In L’Hôpital (1696) the term accourcie (shortened) was used; “prolate” is the standard modern term.
- 18.
Compare this to Bernoulli’s Problem VI on p. 204.
- 19.
See p. xxiii for a discussion of this construction of the conic sections.
- 20.
Compare this to Problem XI on p. 210.
- 21.
See p. xxix for a discussion of this construction of the Spiral of Archimedes.
- 22.
See p. xxx for a discussion of this construction of the Conchoid of Nichomedes. Compare this to Bernoulli’s Problem VII on p. 205.
- 23.
Nicomedes (ca. 280 BCE-ca. 210 BCE).
- 24.
That is, in this construction, the curve ARM moves rigidly along the axis ET as the line FPM revolves around F and that each point on CMD is the intersection of the rotated line FPM and the translated curve ARM.
- 25.
See p. xxiv for a discussion of this construction of the hyperbola.
- 26.
In L’Hôpital (1696) this is footnoted as “Géométrie, Book 3.” The “Paraboloid of Descartes” is usually called the Trident of Descartes or the Parabola of Descartes. It is a cubic curve with two branches, one of which resembles a parabola. The “companion” is the other branch.
- 27.
See p. xxxi for a discussion of this construction of the Cissoid of Diocles. Compare this to Bernoulli’s Problem VIII on p. 207.
- 28.
Diocles of Carystus (ca. 240 BCE-ca. 180 BCE).
- 29.
In L’Hôpital (1696) the numerator of the second term in the equation that follows was given as \(\mathit{yy} \times 2\overline{a - x^{2}}\).
- 30.
See p. xxxii for a discussion of this construction of the Quadratrix of Dinostratus. Compare this to Bernoulli’s Problem IX on p. 208.
- 31.
Dinostratus (ca. 390 BCE-ca. 320 BCE). Dinostratus was a Greek mathematician who discovered the quadratrix and supposedly used it to solve the problem of squaring the circle.
- 32.
Compare this to Bernoulli’s Problem X on p. 210.
- 33.
The third proportional to a and b is the value of x such that a: b:: b: x.
- 34.
See p. xxxiii for a discussion of how the Quadratrix may be used to construct π and to solve the problem of squaring the circle.
- 35.
In L’Hôpital (1696), the term foyer is used in optics to mean focus. The curves discussed in §32 are in some sense a generalization of the conic sections.
- 36.
In L’Hôpital (1696) the overline grouping of a + y was omitted here and in the next occurrence, but not in the third occurrence. They have been added for consistency.
- 37.
Medicina mentis [1687].
- 38.
Nicolas Fatio de Duillier (1664–1753).
- 39.
Journal des sçavans, March 1687.
- 40.
The quantities ax, by, and cz are products of two lines and are therefore referred to as “rectangles,” following the Euclidean tradition.
- 41.
This was first proved by Huygens and is given in Part IV, Proposition 12 of Horologium Oscillatorium sive de motu pendulorum (Huygens 1673, p. 124).
- 42.
In L’Hôpital (1696) there was a comma between a and b in the proportional relation that follows.
- 43.
See p. xxvi for a discussion of this construction of the conic sections.
- 44.
Compare this to Bernoulli’s Letter 28 p. 273.
- 45.
This follows by similar triangles that LR: OL: : VS: DV.
- 46.
In L’Hôpital (1696) the expression for RM did not have parentheses. They were added for consistency.
- 47.
In Problem V on p. 204, Bernoulli shows that the curve with constant subtangent is the curve “whose ordinates make a Geometric progression and abscissas an Arithmetic”; that is, this curve is an exponential curve. L’Hôpital uses that fact here, without proof, to identify this curve. The result had been published in Leibniz (1684). An exponential curve was called logarithmic at this time, because the logarithm of an ordinate is the corresponding abscissa. In Figure 2.25, the x-axis is vertical and the y-axis is horizontal, and curve LM has the equation \(y = y_{0}e^{-x/c}\), if we take the length of the segment AE to be y 0.
- 48.
See p. xxxiii for a discussion of the Logarithmic Spiral.
- 49.
In L’Hôpital (1696) the same word (plan) is used for both the plane that contains the two curves and the region of the plane bounded on the convex side of the curve BMC. Here and in the next sentence we translate this latter meaning by “region.”
- 50.
Compare this to Bernoulli’s Letter 28 p. 271.
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). Use of the Differential Calculus for Finding the Tangents of All Kinds of Curved Lines. 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_2
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