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Who discovered the rule of L'Hôpital?

In the Wikipedia article, it is mentioned that it was not l'Hôpital who proved the rule but that it might have been Johann Bernoulli. I even heard something about l'Hôpital paid Bernoulli for the privilege to publish the proof instead of him. Is this true? Are there any proofs for this theory? Or is it totally nonsense?

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  • $\begingroup$ You may enjoy the brief account of the story I wrote in my blog, here. $\endgroup$
    – Andrés E. Caicedo
    Aug 7, 2016 at 19:09
  • $\begingroup$ @AndrésE.Caicedo Very interesting and detailed posting on your blog. Given the contents of L'Hopital's proposal, it is surprising from a modern perspective that Bernoulli accepted the terms. He must have been cash strapped indeed. Truesdell's description of the social context does, perhaps, provide some justification for L'Hopital's actions. $\endgroup$
    – nwr
    Aug 7, 2016 at 20:59

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According to Carl Boyer's A History of Mathematics, this is the case.

According to Boyer's account, Jean Bernoulli's father Nicolaus had intended Jean to become a merchant or physician, and Jean did in fact write his dissertation in 1690 on effervescence and fermentation...

but the following year he [Jean Bernoulli] became so deeply interested in the calculus that during 1691-1692 he composed two little textbooks on the differential and integral calculus, although neither was published until long afterward. While in Paris in 1692, he instructed a young Marquis, G.F.A. de L'Hospital (1661-1704) in the new Leibnizian discipline; and Jean Bernoulli signed a pact under which, in return for a regular salary, he agreed to send L'Hospital his discoveries in mathematics, to be used as the marquis might wish. The result was that one of Bernoulli's chief contributions, dating from 1694, has ever since been know as L'Hospital's rule of indeterminate forms. .... This well-known rule was incorporated by L'Hospital in the first textbook on differential calculus to appear in print - Analyse des infiniment petits, published in Paris in 1696. This book, the influence of which dominated most of the eighteenth century, ...

Boyer goes on to say of L'Hospital :

L'Hospital was an exceptionally effective writer, for his Traite analytique des sections coniques, published posthumously in 1707, did for analytic geometry of the eighteenth century what the Analyse did for calculus.

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    $\begingroup$ Jean and Johann Bernoulli seem to be the same person. Jean is the French variant of Johann :-) $\endgroup$
    – Chris
    Aug 5, 2016 at 8:31
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    $\begingroup$ Chris is right; and just to be clear, it's the Johann Bernoulli who lived 1667-1748. Note that there was quite some amount of Johanns in the family. $\endgroup$
    – Ben
    Aug 5, 2016 at 13:25
  • $\begingroup$ @Chris Well you learn something every day. I'll edit my answer. $\endgroup$
    – nwr
    Aug 5, 2016 at 14:27
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This is only a marginal notice but too long for a comment.

There are different opinions about the matter. It is indubitable that l'Hospital paid Johann Bernoulli for educating him in analysis. It is possible that the "rule" has been found by Johann Bernoulli. On the other hand it is fact that Johann Bernoulli was one of the first self-promoters with a fair amount of self-confidence and that he did not easily admit any negative features. (See the famous quarrel of the Bernoulli brothers about a problem posed by Jakob Bernoulli. Johann did not admit his mistake for many years and confessed it only after his brother's dead.)

To make a long story short: An author the name of whom I have forgotten wrote: The longer l'Hospital was dead, the louder sounded Johann's claim that he was the author of the rule. This aspect should not be completely overlooked.

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    $\begingroup$ You may have in mind Truesdell who wrote “[Bernoulli's] claims increased with time after the death of l'Hôpital” (p. 59 of The New Bernoulli Edition, Isis 49 (1958) 54-62, already quoted in @AndrésE.Caicedo's link). That whole long story is discussed in Bernoulli's Briefwechsel, vol. 1 (1955), Introduction, pp. 149-156. $\endgroup$
    – Francois Ziegler
    Jun 3, 2017 at 18:46
  • $\begingroup$ Thank you. It was a German text, but probably the author has copied this circumstance from Truesdell. $\endgroup$
    – Franz Kurz
    Jun 3, 2017 at 20:58
  • $\begingroup$ I remember there was a letter of Bernoulli to L'Hopital with the rule included. But I don't remember where I saw it. $\endgroup$
    – Gottfried William
    Jun 6, 2017 at 14:22
  • $\begingroup$ @Guido Jorg: That would decide the case once and for all. $\endgroup$
    – Franz Kurz
    Jun 6, 2017 at 14:30

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