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Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ (sometimes written as $y+dy=y$ as in Laugwitz 1997). I used to believe this until I looked in l'Hopital's book and did not find any such equation. What I did find is an axiom right at the beginning of the book to the effect that the $dx$ can be neglected.

What l'Hopital wrote, more precisely, was: On demande qu'on puisse prendre indifféremment l'une pour l'autre deux quantités qui ne différent entr'elles que d'une quantité infiniment petite: ou (ce qui est la même chose) qu'une quantité qui n'est augmentée ou diminuée que d'une autre quantité infiniment moindre qu'elle, puisse être considérée comme demeurant la même.

l'Hopital did not say that they are equal, but rather that "qu'on puisse prendre indifféremment l'une pour l'autre" meaning that "one can take one for the other". This viewpoint is close to the one adopted in the hyperreal formalisation of this idea in terms of the standard part function and is not known to entail contradictions like $x+dx=x$.

Does the equation perhaps appear elsewhere in the book, or is this simply an inaccuracy? A 17th century scholar I consulted with agreed that the equation is probably not in the book; it would be nice to have a reference to that effect.

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I'm not a specialist, but I'll try with a little contribute: we may presume that l'Hopital is avoiding the formula exactly because he want to avoid the inconsistency (for a mathematician, "=" means ... "equal"). Its "convoluted" axiom is reminiscent of Leibniz's principle of identity of indiscernibles : we may say that "if two quantities differ only by an infinitesimal amount", then we may "use them indifferently" ("qu'on puisse prendre indifféremment l'une pour l'autre") in the "contexts" of the calculus. –  Mauro ALLEGRANZA May 6 '14 at 13:13
    
I take it you believe the formula does not occur in the book? –  user72694 May 6 '14 at 13:19
    
I'm not certain about it; I think that it is possible that he used it in the calculations, but my "conjecture" is that he had deliberately avoided to state it among the "axioms" or "principles" of the theory... –  Mauro ALLEGRANZA May 6 '14 at 13:29

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Here is a link to the relevant page of the book. Under the heading "1. Demande ou Supposition", which one could understand to mean "Axiom 1", L'Hospital writes:

On demande qu'on puisse prendre indifféremment l'une pour l'autre deux quantités qui ne différent entr'elles que d'une quantité infiniment petite: ou (ce qui est la même chose) qu'une quantité qui n'est augmentée ou diminuée que d'une autre quantité infiniment moindre qu'elle, puisse être considérée comme demeurant la même [...].

A rough translation of this in English would be

We require that one treat identically two quantities that differ only by an infinitesimal amount; or (equivalently) that one quantity that is increased or decreased by another quantity which is infinitesimally small relative to the first be considered unchanged.

Which one could "translate" as

$$ x = x + dx $$

The left-hand side being the original quantity, and the right-hand side describing an infinitesimal change in this quantity.

EDIT This answer misses the point entirely. See the article cited in the comments below for further information.

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This is of course precisely the passage I was referring to in my question. The "translation" is precisely what is bothering me here: l'Hopital precisely did not say that they are equal, but rather that "qu'on puisse prendre indifféremment l'une pour l'autre" (which does NOT say that one "treat them identically", but only that one can take one for the other. This viewpoint is close to the one adopted in the hyperreal formalisation of this idea in terms of the standard part function and is not known to entail contradictions like $x+dx=x$ –  user72694 May 4 '14 at 14:59
    
@user72694 I am not nearly informed enough to comment on the standard part function, etc. However, I would argue that "prendre indifféremment l'une pour l'autre" is analogous to taking two distinct representatives of an equivalence class, and so one could instead say $x \equiv x + dx$. Is that more accurate? –  maxerize May 4 '14 at 15:05
    
Over the hyperreals one works with this type of equivalence relation all the time, and usually denotes it "$\approx$" with slight variations depending on the author. Of course, l'Hopital did not have the notion of an equivalence relation. At any rate, equivalence and equality are two very different things! What I have seen in the literature (including recent literature) is the claim of equality, which is an unfaithful rendering of what l'Hopital wrote in this paragraph. On the other hand, I have not read the whole book and perhaps the equality appears elsewhere in the book. –  user72694 May 4 '14 at 15:11
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Note that Laugwitz specifically excludes Leibniz from this contention. This is consistent with what we wrote in this article. I have not found equations like $y+dy=y$ in l'Hopital, though it is possible that they appear in Johann Bernoulli. I have the feeling that Laugwitz is relying on earlier authors here, contrary to his usual vigilance. –  user72694 May 4 '14 at 15:35
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Maxerize, thanks very much for helping me clarify my post. –  user72694 May 4 '14 at 16:32

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