Abstract
In Chapter 9 of the Analyse, l’Hôpital considers some miscellaneous problems based on the methods of Chapters 1–4, that do not belong to the topics covered in Chapters 5–8 This chapter begins with the celebrated theorem that now goes by the name L’Hôpital’s Rule. This rule is actually due to Bernoulli, and the version given here only covers the case in which y takes the indeterminate form \(\frac{0} {0}\) at a finite value of x. Much of the rest of the chapter is taken up with a study of the epicycloid.
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Notes
- 1.
This is the rule known as l’Hôpital ’s Rule. Compare this to Bernoulli’s Letter 28 on p. 273.
- 2.
This example is Bernoulli’s \(\frac{0} {0}\) Challenge Problem, which appears frequently in l’Hôpital ’s correspondences with Bernoulli. It first appeared in Bernoulli’s Letter 11 on p. 245.
- 3.
In his letter 28, of July 22, 1694, Bernoulli gave the similar example \(y = \frac{a\sqrt{\mathit{ax}}-\mathit{xx}} {a-\sqrt{\mathit{ax}}}\), and found that y = 3a when x = a.
- 4.
There is no mention here that y = 0 is also a root of the equation \(2\mathit{aay} -\mathit{ayy} = 0\), which is the result of substituting a for x. Bernoulli’s example in letter 28 has the spurious root \(y = -a\).
- 5.
The point E appears twice in Fig. 9.2. In the case of the point E on the right, we use AMD + ENH in the proportion. In the case of the point E on the left, we use AMD − ENH in the proportion.
- 6.
I.e., involving 3rd, 4th, etc., proportionals.
- 7.
Some seventeenth century authors used the term subtense synonymously with the term chord, e.g., Cohen (1999, p. 130).
- 8.
Pierre Varignon (1654–1722).
- 9.
In L’Hôpital (1696), the equal sign after VP(u) was omitted.
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Bradley, R.E., Petrilli, S.J., Sandifer, C.E. (2015). The Solution of Several Problems That Depend upon the Previous Methods. 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_9
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