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quoted text by João Caramalho Domingues
Lazare-Nicolas-Marguerite Carnot held that differential calculus operated by means of a compensation of errors:
in the traditional process of infinitesimal calculus, we start by regarding a curve as a polygonal line; here an error is being committed; afterwords, during the calculations, the neglect of infinitesimals introduces a second error that cancels the first. (59-60)
Carnot tried to prove the compensation of errors' efficacy by means of "imperfect equations:"
The members of one of these were in fact not equal, but had the same limit, which means that they had to involve variables, or as Carnot said, “auxiliary quantities”; imperfect equations were operated upon by replacing quantities with other, infinitely close, quantities; once all the auxiliary quantities had disappeared, an exact equation would remain. (60a)
And yet, Carnot's proof did not convince other mathematicians (60b).
Domingues, João Caramalho. Lacroix and the Calculus. Basel: Birkhäuser, 2008.
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