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.
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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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