The agreement of Bernoulli with de l'Hopital

Guillaume de l'Hopital
The section below is from the source:

C. Truesdell, The New Bernoulli Edition, Isis, Vol. 49, No. 1, 1958, pages 54-62:

It is the content of section IV "The correspondence between Bernoulli and L 'Hopital" of this article.

Johann Bernoulli
Calculus was made known to the learned world by the brilliant papers of Leibniz and the brothers Bernoulli from 1690 on. The first textbook of the new science appeared in 1696: The Analyse des infiniment petits, pour intelligence des lignes courbes, a work of some 200 pages, issued anonymously but known to be by the Marquis de l'Hopital (166I-1704). This book went through several editions and remained a standard for a century. Until the present day it has been considered as a work in large part original, and its famous rule on indeterminate forms is known as l'Hopital's. Beyond this book and a posthumous treatise on conic sections, l'Hopital's works consist in some twenty-five very short notes on special problems. He had become well known through his sixth publication, a four-page note of 1692, giving the solution of de Beaune's problem of the inverse tangent, which had been outstanding for fifty years.

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'Hopital belonged to himself. In particular, he claimed not only the solution of de Beaune's problem and everything else of real interest in l'Hopital's papers, but also all but three or four pages of the Analyse, which he said was nothing but the first part of the Course on Differential and Integral Calculus that he had given or dictated to l'Hopital in Paris. Indeed, it was he who had taught the Marquis the new calculus in 169I, 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'Hopital. 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'Hopital 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 known Analyse of l'Hopital. Indeed, in the preface there is a famous passage in which l'Hopital expresses his especial indebtedness to John Bernoulli and asserts that his book will present the discoveries of various persons without further acknowledgment. Nevertheless, in the text there are in fact many specific acknowledgments to half a dozen persons, but not one to Bernoulli.

A century ago the correspondence of l'Hopital with Leibniz and Huygens was published. Herein may be traced l'Hopital's own view, or at least the view he wished his great correspondents to entertain, of his progress in calculus and in writing his treatise. Bernoulli's name is not mentioned. After this, few if any historians allowed any credit to Bernoulli's accusations. In I922 Schafheitlin found in the Public Library of Basel and published Part I of John Bernoulli's Course. It was exactly as its author had described it, and the work of l'Hopital was at once reduced to the exposition, not the content. But the explanation is more interesting than the fact, and the explanation is to be found only in the letters of the two principals. The existence and contents of these letters has been known to a limited circle for some decades, but the general public will see them for the first time in this volume. Ten were published in a thesis by 0. J. Rebel in 1934, but these are not the most informative and in particular the amazing No. 20, from which I will quote below, is not included. A fair idea of the content of the collection has been given by Spiess in an earlier publication.2 The Course and the letters together fully substantiate John Bernoulli's claims in all but some minor matters. The entire relation between l'Hopital and John Bernoulli is traced in fascinating detail by Spiess in the special preface, pages 123-157. In this review I will give only an oversimplified summary, urging the reader to enjoy for himself Spiess's own words and the following eighty-seven letters. These spread from December 1692, a month after Bernoulli's return to Basel, to a letter from l'Hopital's widow in I707.

For Bernoulli's stay in Paris we must rely on his own autobiography, written just before his death, and on a sequence of unpublished letters detailing his recollections to Pierre de Montmort in I718. After the famous meeting of the two savants in the salon of Malebranche, when Bernoulli dramatically displayed his unpublished secret weapon, the general formula for the radius of curvature of a curve, l'Hopital immediately engaged Bernoulli to give him four lessons per week. After six months of this, the scene of instruction shifted to l'Hopital's chateau in the country for three or four more months, and then Bernoulli returned to Basel. To be brief, in the following letters we find Bernoulli giving l'Hopital full information on every current topic of research and full answers to every question. Some of these l'Hopital wrote out and sent to Huygens or Leibniz. In the case of every problem of major interest to which l'Hopital has had a claim, a lesson or letter from Bernoulli stands in the background. How did this happen? We must remember that in 1691 John Bernoulli was twenty-four, an unemployed younger son of a modest mercantile family; while a younger brother of a famous mathematician, he had himself published but one important paper. L'Hopital was a Marquis of thirty, an established savant; young enough for the ambition of learning and perhaps for learning itself, but old enough for assurance and ease in a worldly society; certain of the income of a Marquis, if somewhat improvident in the use of it. While nowadays the difference in social positions seems a trifle, in 1691 it was surely enough to impress even the ebullient self-confidence of John Bernoulli when, freed of worldly cares, he was accepted as an equal and intimate friend in the elegant establishment whose presiding deity was a charming and witty Marquise. On the other side, while l'Hopital's originality is annulled and his scientific honesty somewhat tarnished by the relation, not only was his curiosity genuine and extraordinary but also from the moment of meeting it was plain that in the face of his young friend's notorious and unconquerable tactlessness, the Marquis would have to put up with a style to which his breeding had hardly accustomed him. The precise arrangements made while Bernoulli was in France we do not know. Soon after Bernoulli returned to Basel there arose a crisis over de Beaune's problem. Bernoulli had found the solution in the course of his researches on integral calculus and had put it into his Course for l'Hopital as Lesson IX. While Bernoulli was still his guest, l'Hopital sent Bernoulli's solution to Huygens, who naturally inferred that the sender was the author, the more so since in an earlier letter l'Hopital had written that he himself had found a solution. At the same time, l'Hopital published the solution under a pseudonym. A complicated sequence of published and unpublished claims and veiled insults followed. For the plan he had in mind, l'Hopital could not afford to notice even an open affront. After some mutual explanations and a delay of more than half a year, during which Bernoulli refrained from sending l'Hopital anything of importance, l'Hopital on 17 March 1694 (letter No. 20) proposed the most extraordinary agreement in the history of science:

"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's response is lost, but the next letter from l'Hopital indicates that the acceptance was speedy. From this point on, Bernoulli was a giant enchained.

Letters 33-44 contain a scolding from l'Hopital because Bernoulli, after obediently checking, translating into Latin and transmitting to Leipzig l'Hopital's solution of a minor problem posed by Sauveur, had been unable to restrain himself from adding a note in which he generalized the problem, identified the resulting curve, and gave for the general case his own analysis consisting in one equation, replacing the 27 used by Sauveur to set the special case. L'Hopital reminded Bernoulli that he was not to publish, but to send all his works to l'Hopital, who promised to keep them secret, asserting that he had no desire to take for himself the honor of these discoveries (Letter 42). In excusing himself, Bernoulli acknowledged his faults and promised, "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." When he wrote those lines in 1695, Bernoulli was as brilliant a mathematician as any living. As soon as the Analyse appeared, the financial arrangement lapsed. We should not judge l'Hopital's procedure too harshly. While perhaps financial necessity compelled Bernoulli to accept the arrangement initially, it continued after he had settled in his professorship at Groningen in 1695. L'Hopital, 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. Certainly it was nothing for l'Hopital to be proud of. Careful examination of the letters in which l'Hopital reported his mathematical progress to Leibniz and Huygens shows that with one or two possible exceptions l'Hopital did not lie, but rather referred to Bernoulli in a condescending tone without acknowledging any debt whatever to him and in matters of provenance wrote in such a way as to suggest without actually asserting. Very soon John Bernoulli realized what he had sold away. The financial returns were ephemeral, and even for the few years the agreement was in force l'Hopital did not always pay the full sum due. (It would be unfair to suppose Bernoulli was his only disappointed creditor.) In Bernoulli's old age, he boasted of the princely sum for which l'Hopital had engaged him, magnifying both the amount and the duration.

In the development of calculus as a tool in geometry and mechanics, nearly every letter from John Bernoulli to l'Hopital is an individual achievement. What is most remarkable is the lightning speed of Bernoulli's conception. His thought and expression in French are no less masterful and far clearer and more direct than in his published works in Latin or his later letters. It would be wasteful to attempt here even to name the problems treated, since these are most easily followed by aid of an index at the end of the volume. I can find no better summary of my impression from these letters than the words l'Hopital himself wrote to Bernoulli in 1695: "I am very sure that there is scarcely a geometer in the world who can be compared to you."