Abstract
This chapter contains a substantial portion of the correspondence between l’Hôpital and Bernoulli, available in English translation for the first time. Included are more than fifty pages of the correspondence, covering the period from 1692 until the publication of l’Hôpital’s Analyse. Included is the famous letter of March 17, 1694, where l’Hôpital offers Bernoulli an annual stipend of three hundred French pounds in return for his services, including the rights to publish some of Bernoulli’s discoveries. There is also the letter of July 22, 1694, where L’Hôpital’s Rule appears for the first time. There are also many details of a personal or professional nature that illuminate the complex friendship between these two men.
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Notes
- 1.
Historie des ouvrages des sçavans, June-August 1690, pp. 440–449.
- 2.
This manuscript was never published. In Bernoulli (1955, p. 158), the Editor quotes a portion of a letter written by Malebranche to a Fr. Jaquemet at about this time: “Over the holidays, I read the writings of the marquis de l’hopital which occupied two months most agreeably. I believe he will have it printed this winter, if his affairs leave him the time that is necessary for it, and you will see the finest of mathematics, explained with much clarity and few words.”
- 3.
Louis Byzance (ca. 1647–1722) and Claude Jaquement (1651–1729) were fathers in the Oratorian order and members of Malenbranche’s circle.
- 4.
This refers to Johann Bernoulli lost letter of November 1692, which was the first letter in his correspondance with L’Hôpital.
- 5.
In other words, L’Hôpital believes the curve to be transcendental and not algebraic.
- 6.
See (L’Hôpital 1696, §146–147).
- 7.
Acta Eruditorum, April 1692, p. 168.
- 8.
This is how L’Hôpital refers to Jakob Bernoulli in his correspondance with Johann Bernoulli.
- 9.
Acta Eruditorum, July 1691, p. 317.
- 10.
Historie des ouvrages des sçavans, 1690, pp. 440–449; letter from L’Hôpital to Huygens of April 1690.
- 11.
Positionum de seriebus infinitis…, Basel 1692.
- 12.
Throughout (Bernoulli 1955) this is consistently written as “actes.” We have chosen to use the proper formatting for the name of a journal.
- 13.
That letter, in response to l’Hôpital’s letter of December 8th, is lost.
- 14.
As noted in Bernoulli (1955, p. 161) the first term of what follows should be y −1 dn 2.
- 15.
See Bernoulli (1955, pp. 161–162) for a detailed explanation of this passage.
- 16.
This is a reference to the second part of Positionum de seriebus infinitis…, Basel 1692.
- 17.
That is, the Leibniz Isochrone; see Acta Eruditorum, April 1689, p. 195.
- 18.
This problem is the subject of L’Hôpital (1696, §161).
- 19.
Bernoulli’s solution, contained in the lost letter of January, 20, 1693, is the model for L’Hôpital (1696, §158).
- 20.
This is a reference to Book I, Lemma 25, Corollary 3, of the Principia.
- 21.
This is the first appearance of the problem of evaluating algebraic expressions that take the form of \(\frac{0} {0}\) for certain values of the variable by means of what is now called L’Hôpital ’s Rule. This example, which we call Bernoulli’s \(\frac{0} {0}\) Challenge, appears in L’Hôpital (1696, (see §164)). Bernoulli’s letter to Varignon is lost Bernoulli (1955, p. 173, Note 2.). Bernoulli’s solution is given in letter 28.
- 22.
Acta Eruditorum, September 1693, pp. 398–399.
- 23.
The Editor of Bernoulli (1955) believes this is the rule for integrating homogeneous differential equations.
- 24.
I.e., the Problem stated in Letter 11.
- 25.
Acta Eruditorum, May 1693, p. 234.
- 26.
This translates roughly as: “While the curve AI is, because of its degree, such that it does not have a rectification in the abstract, nevertheless the lengths of the curves themselves can be constructed and determined, as the noble Mr. Huygens has given in his own new logarithms and in what follows I will proceed in common logarithms.”
- 27.
What is meant here is the open letter of Huygens of February 1693, which prompted Bernoulli to send his second construction of the Curve of de Baune to the Acta Eruditorum.
- 28.
Presumably in the lost Letter (9, 1) in May 1693.
- 29.
This translates roughly as: “So that the placement should not give over to sinister interpretation.”
- 30.
I.e., the June issue of the Acta Eruditorum.
- 31.
Acta Eruditorum, June 1693, p. 255.
- 32.
Bernoulli had already sent this problem of the tangent to the tractrix to Varignon in May, along with his \(\frac{0} {0}\) challenge.
- 33.
L’Hôpital had asked for solutions to both of these problems in letter 15.
- 34.
In his response to letter 15, which was lost, Bernoulli presumably suggested this simplified version of his \(\frac{0} {0}\) challenge.
- 35.
Bernoulli had not written to l’Hôpital since late September 1693.
- 36.
The Editor of the Bernoulli (1955) made a correction here; 1694 was written.
- 37.
In the winter of 1693, Bernoulli became engaged to Dorothea Falkner.
- 38.
L’Hôpital had just inherited the castle St. André-de-Briord in the Department of Ain, near the Swiss border.
- 39.
This portion is illegible.
- 40.
In other words, a recurring payment.
- 41.
This currency symbol was used by L’Hôpital for pound.
- 42.
Acta Eruditorum, March 1692.
- 43.
Acta Eruditorum, September 1692.
- 44.
Bernoulli apparently made this incorrect assertion in letter (20, 1). In Letter 22, he corrects this error and gives a full account of his curve. However, this has been omitted from this volume, because of its length and lack of relevance to L’Hôpital (1696).
- 45.
This letter, in response to Letter 21, is the first surviving letter from Bernoulli to L’Hôpital. It is a copy of the original, with the note to “Mr. the Marq. De L’Hôpital” in the margin.
- 46.
This Latin quote translates roughly to: “If strength is lacking, nevertheless the desire is to be praised.”
- 47.
That is, Letter 21.
- 48.
Here Bernoulli uses the a subscript of q to represent the square of a quantity.
- 49.
See L’Hôpital (1696, §82).
- 50.
This is Bernoulli’s term, a French adjective meaning having two horns.
- 51.
This term was coined by Bernoulli and used by L’Hôpital in L’Hôpital (1696). In French rebrousser chemin means to retrace ones steps. The English word for such a point is cusp, which we will use in what follows. We will also use “bicorn” for a curve with a cusp, even though this is not standard English usage.
- 52.
The case of an extremum at a cusp was omitted from the Lectiones. L’Hôpital included it in §47 (p. 46).
- 53.
By “paraboloids,” Bernoulli means curves whose equation is x p = y q, where p and q are positive integers. So in this case, Jakob means p = 1 and q > 2.
- 54.
Dissertatio de motu musculorum, Bernoulli’s dissertation for his doctorate in Medicine, 1694.
- 55.
For example, \(y = x^{5/2}\).
- 56.
- 57.
- 58.
Memoir Mathematique, June 1693, pp. 97–101, Journal des sçavans, March 1694, p. 146. The reference in Bernoulli (1955, p. 217) was to a different article by l’Hôpital.
- 59.
Journal des sçavans, April 1694, pp. 182–183.
- 60.
L’Hôpital discovered this kind of cusp, but Bernoulli recognizes its nature and refers to it as being of the “second kind.”
- 61.
This is incorrect, as l’Hôpital will show in Letter 25.
- 62.
This translates roughly as “let the shoemaker venture no further.” These words were attributed to the painter Apelles (4th century BCE) by Pliny the Elder.
- 63.
See Bernoulli (1955, p. 224) for a discussion in modern notation.
- 64.
- 65.
See Letter 25, Footnote 64.
- 66.
Acta Eruditorum, May 1694, pp. 193–196, 196. These are translations of the articles mentioned by l’Hôpital at the end of Letter 23.
- 67.
Presumably in the lost reply to Letter 20.
- 68.
Compare to §45 on p. 41.
- 69.
Compare this to §36 on p. 35.
- 70.
I.e., That which was to be found.
- 71.
Compare this to §163 on p. 155.
- 72.
Bernoullis says les dernieres differentielles; literally “the last differentials.”
- 73.
Presumably, Bernoulli mentioned this accident in a postscript to Letter 28 that was not included in the draft. He began Letter 30 on September 9, 1694, with the sentence “It is a few days since I returned from the country; the swelling that I had in my arm from a fall is entirely cured.”
- 74.
L’Hôpital’s father.
- 75.
The French pound was divided into 20 sous or sols.
- 76.
In Letter 33, L’Hôpital had sent Bernoulli his solution to the Probloma aequilibrii, a problem posed by Sauveur, to be translated into Latin and sent to the Acts Eruditorum. In Letter 36, Bernoulli confirmed that he had done this, but also admitted to adding some remarks of his own, which substantially simplified this solution. In Letter 37, L’Hôpital sent Bernoulli a different solution asking him to sent it to the Acts Eruditorum with a request that the first piece be withdrawn. This request came too late and both pieces appeared: in February pp. 56–59 and in Supplement II, 1696 pp. 289–291.
- 77.
Nikolaus (II) Bernoulli (1695–1726).
- 78.
Letter 40.
- 79.
This may be translated roughly as: “Now concerning The Science of the Infinite, I fear that if the work lacks your contributions (as your [letter] seems to hint), then it will not come to light in good time, or else not at all.”
- 80.
The Phoenix of Mathematicians of France.
- 81.
In a lost letter, Huygens asked L’Hôpital for a recommendation regarding Bernoulli for a position at the University of Groningen.
- 82.
I.e., the Probloma aequilibrii of Sauveur.
- 83.
Craig, John. Methodus figurarum lineis rectis et curvis comprehensarum quadraturas determinandi, London, 1693. L’Hôpital spelled the name “Graiges.”
- 84.
Memoir Mathematique, August 1693, pp. 129–133.
- 85.
Letter 15.
- 86.
For a modern treatment of this argument, see Bernoulli (1955, p. 278).
- 87.
I.e., cross multiplication. This refers to a joke l’Hôpital made in Letter 32 concerning the birth of his third child.
- 88.
Charolotte Silvia (June 6, 1696-May 5, 1759).
- 89.
Letter 47.
- 90.
Letter 48.
- 91.
This was a mistake. Huygens actually died on July 8, 1695.
- 92.
Leibniz’ solution to the Drawbridge Problem appeared in Acta Eruditorum April 1695, page 184. L’Hôpital’s second solution appeared two days later in Acta Eruditorum, Supp. II, page 372–374.
- 93.
The Muses Love Alternation.
- 94.
- 95.
As noted in Bernoulli (1955, p. 295), this should say “because the curves AE and DC are given.”
- 96.
L’Hôpital combined this and the previous lemma in L’Hôpital (1696, §166). See also §168.
- 97.
In Letter 51.
- 98.
This was something that Bernoulli taught L’Hôpital in 1692 (Bernoulli 1742, pp. 447–448).
- 99.
This proposition and the full proof are given in L’Hôpital (1696, §158).
- 100.
This appeared in Acta Eruditorum, August 1695, pages 372–374.
- 101.
Notae et animadversiones tumultuariae in geometriam Cartesii. Frankfurt 1695.
- 102.
In letter 51, L’Hôpital told Bernoulli that he had learned that Huygens had died. In letter 53, he reported that this had been in error, but in fact Huygens had died on July 8th.
- 103.
Emanuel Falkner.
- 104.
Huygens died on the morning of July 8, 1695.
- 105.
This is the first mention of L’Hôpital (1696) to Bernoulli.
- 106.
Hieronymus Bernoulli (1669–1760), youngest brother of Jakob and Johann.
- 107.
That is, Jakob Bernoulli.
- 108.
The Marquis had recovered from his illness and written to Bernoulli on April 2.
- 109.
Hieronymus.
- 110.
This is the beginning of a discussion of the Problem of the Brachistochrone, or line of quickest descent. Bernoulli sent it to Varignon on May 15 and on June 11 it appeared in the Acta Eruditorum.
- 111.
This is l’Hôpital’s second mention of his book (L’Hôpital 1696).
- 112.
Hieronymus.
- 113.
In other words, the speeds of falling bodies are to one another in the same ratios as the square roots of the perpendicular distance traveled.
- 114.
Hieronymus.
- 115.
Samuel Battier (1667–1744) friend of Johann Bernoulli and student of Jakob Bernoulli.
- 116.
Niklaus (I) Bernoulli “the Younger” (1687–1769).
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.
Anonymous (Guillaume François Antoine, Marquis de l’Hôpital), Analyse des infiniment petits, Imprimerie Royale, Paris, 1696.
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Bradley, R.E., Petrilli, S.J., Sandifer, C.E. (2015). Selected Letters from the Correspondence Between the Marquis de L’Hôpital and Johann Bernoulli. 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_12
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