Abstract
In Chapter 4 of the Analyse, l’Hôpital defines higher order differentials and describes how second order differentials may be used to locate inflection points and cusps on a curve. In addition to the usual rectangular coordinates, l’Hôpital also considers the case where ordinates all emanate from a single point. Although these are not the polar coordinates that came into use in later centuries, because there is no accompanying angular coordinate, they are nevertheless useful in this and subsequent chapters for describing certain curves. L’Hôpital finds the inflection points of the prolate cycloid, the Conchoid of Nicomedes and of a curve that is essentially the same as the “Witch of Agnesi.”
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Notes
- 1.
In (L’Hôpital 1696) the term grandeurs entières is used, literally “whole magnitudes,” yet even though the comparison seems to be to the use of exponents with finite quantities.
- 2.
Compare this to the definition presented in Problem XXI on p. 224.
- 3.
Following Bernoulli [letter 22], L’Hôpital used the term “point de rebroussement,” literally a “point of turning back.” The mathematical term “turning point” has a different meaning from this, so we use the term “cusp,” which is the standard English term for this type of point.
- 4.
- 5.
Compare this to the example on p. 225.
- 6.
This is essentially the curve that later became known as the “Witch of Agnesi.” This curve considered here is that curve reflected in the line \(y = \frac{a} {2}\); see p. xxxvii for more about this curve.
- 7.
Compare this to the example given on p. IV.
- 8.
If one applies the rules of the differential calculus, the value of dy would be the negative of the given value. However, the coordinates in this problem are set up so that as y increases, x decreases, so the author adjusts the sign of dy as described earlier (see §8). In any case, the value of ddy that follows has the correct sign.
- 9.
Compare this to the example given on p. 231.
- 10.
As in Example IV, in this chapter, this value of dy has its sign adjusted.
- 11.
In (L’Hôpital 1696), the denominator contained \(\overline{4\mathit{ax} - 4x^{3}}\), with a note in the Errata to replace − 4x 3 with − 4xx. In fact, that term − 4x 3 is correct, whereas for 4ax should have been replaced with 4axx.
- 12.
Compare this to the example given on p. 238.
- 13.
I.e., the product of the lengths of the arc AE and the line b.
- 14.
This curve is closely related to Spiral of Fermat. See p. xxxvii for further discussion of this curve.
- 15.
I.e., BE is the third proportional to AB and BF, or AB: BF: : BF: BE.
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Bradley, R.E., Petrilli, S.J., Sandifer, C.E. (2015). Use of the Differential Calculus for Finding Inflection Points and Cusps. 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_4
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