l'Hopital's questionable premise?

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.

Solution 1:

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.