I recognize I owe much to Messrs. Bernoulli’s insights, above all to the young, currently a professor in Groningue. I did unceremoniously use their discoveries, as well as those of Mr. Leibniz. For this reason I consent that they claim as much credit as they please, and will content myself with what they will agree to leave me.

L’Hôpital, in the preface (page xiv) of his *Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes* (1696), the first calculus textbook, published anonymously. (A posthumous second edition, from 1716, identifies L’Hôpital as the author.)

The young Bernoulli is Johann (John I), whose jealousy is well documented. From Clifford Ambrose Truesdell, III. *The new Bernoulli edition*, Isis, **49,** (1958), 54–62. MR0090530 (19,826e):

In letters, some of which have been in print for two centuries, John Bernoulli complained to Leibniz, Varignon, and others that most of what was attributed to l’Hôpital belonged to himself. In particular, he claimed not only the solution of de Beaune’s problem and everything else of real interest in l’Hôpital’s papers, but also all but three or four pages of the

Analyse, which he said was nothing but the first part of theCourse on Differential and Integral Calculusthat he had given or dictated to l’Hôpital in Paris. Indeed, it was he who had taught the Marquis the new calculus in 1691, giving him instruction for nearly a year. In his published memoirs, Bernoulli was less positive, though his claims increased with time after the death of l’Hôpital. Since Bernoulli was far from reticent in proclaiming his own when others, even his best friends and closest relatives, were involved, the moderation and lateness of his accusations against l’Hôpital naturally caused them to be doubted. Apparently only Leibniz and some Basel friends believed Bernoulli, and in France his claims were regarded as ridiculous. In 1742 Bernoulli published Part II, the Integral Calculus, of his Course of 1691-1692, but not Part I, whose contents, it was noted, had gone into the well knownAnalyseof l’Hôpital.

The story is still more complicated. The *Analyse* has several acknowledgements to different mathematicians throughout the text for specific results, but no further mention of Bernoulli. Though Bernoulli’s accusations were discredited for centuries, in 1922, Part I of the *Course* was discovered and published, verifying that Bernoulli was indeed correct. The Bernoulli edition the paper just quoted refers to is a project to publish the complete works of the Bernoullis, including correspondence and unpublished manuscripts. Volume I,

Der Briefwechsel von Johann Bernoulli. Bd. I.Herausgegeben von der Naturforschenden Gesellschaft in Basel. Birkhäuser Verlag Basel,1955. 531 pp. (6 plates). MR0067789 (16,781h),

includes correspondence that verifies that l’Hôpital, a Marquis, paid Johannes, then twenty-four and unemployed, for private lessons that lasted for months. During this time, l’Hôpital wrote several letters and an anonymous paper, containing mathematical results that Bernoulli had shown him, without appropriate credit. Eventually, on March 17, 1694, l’Hôpital sent Bernoulli a letter with a curious proposal (the following quote is according to Truesdell’s paper):

I will be happy to give you a retainer of 300 pounds, beginning with the first of January of this year . . . I promise shortly to increase this retainer, which I know is very modest, as soon as my affairs are somewhat straightened out . . . I am not so unreasonable as to demand in return all of your time, but I will ask you to give me at intervals some hours of your time to work on what I request and also to communicate to me your discoveries, at the same time asking you not to disclose any of them to others. I ask you even not to send here to Mr. Varignon or to others any copies of the writings you have left with me; if they are published, I will not be at all pleased. Answer me regarding all this . .

Bernoulli accepted. Later correspondence has l’Hôpital stating that, while Bernoulli was to keep his results secret, he himself had no intention of publishing them, or of taking credit for himself. Bernoulli replied:

You have only to let me know your definite wishes, if I am to publish nothing more in my life, for I will follow them precisely and nothing more by me will be seen.

As Truesdell mentions, “As soon as the *Analyse* appeared, the financial arrangement lapsed” and, even while the agreement was in force, payments were not always in full, and eventually Bernoulli came to regret his decision.

It is clear now that l’Hôpital’s rule, for example, is due to Bernoulli. The proof in the *Analyse* (Section 9, article 164) comes from a letter of July 22, 1694. However ethically questionable one finds the affair, Truesdell indicates that

L’Hôpital, being a nobleman, was accustomed to pay for the services of others, and what he did would not then have been considered wrong had Bernoulli been a politician, a lawyer, perhaps even an architect.

A recent paper by Bradley,

Robert E. Bradley.

De l’Hôpital, Bernoulli, and the genesis of Analyse des infiniment petits, BSHM Bulletin: Journal of the British society for the history of mathematics,28 (1), (2013), 16-24. MR3033768,

even suggests that Bernoulli’s remark on February 26, 1707, that

To speak frankly, M. de l’Hôpital had no other part in the production of this book than to have translated into French the material that I gave him, for the most part, in Latin…

is a “strident claim,” must be “vigorously rejected,” and instead l’Hôpital must be recognized for “crafting a textbook of the first order.”

[…] 9. Sections 4.9, 4.10. L’Hôpital controversy (see also here). Homework: Section 4.9: 2, 5, 6. Section 4.10: 2, 5, 7, 14, 20. [Suggested problems from Schaum's: […]