Rolle published what we today call Rolle's theorem about 150 years before the arithmetization of the reals. Unfortunately this proof seems to have been buried in a long book [Rolle 1691] that I can't seem to find online. (Well, maybe that's fortunate because otherwise I'd have felt obligated to comb through it with my poor knowledge of French.) Rolle was an algebraist and a prominent opponent of infinitesimals. His proof of the theorem was based on something called the "method of cascades", an evolution of techniques originated by [Johannes Hudde](http://en.wikipedia.org/wiki/Johannes_Hudde). This seems to have been a method of manipulating polynomials that was equivalent to differentiating them [Itard 2008].


Today, we would consider Rolle's theorem to be a consequence of the extreme value theorem, which in turn depends on the completeness property of the reals -- stated long after Rolle was in his grave. However, there is a revisionist argument that people as early as the 17th century had quite a clear notion of what we would today call the real number system [Blaszczyk 2012].


So what did Rolle really prove when he published his proof of Rolle's theorem? Was it just a proof for polynomials? At the time, would a proof for polynomials have been considered sufficient, on the theory that any smooth function can be approximated by a polynomial?


Related: [Does Rolle's Theorem imply Dedekind completeness?](https://mathoverflow.net/questions/87848/does-rolles-theorem-imply-dedekind-completeness)


Blaszczyk, Katz, and Sherry, "Ten Misconceptions from the History of Analysis and Their Debunking", 2012, <http://arxiv.org/abs/1202.4153>


Itard, "Rolle, Michel" in Complete Dictionary of Scientific Biography, 2008, <http://www.encyclopedia.com/doc/1G2-2830903713.html>


Rolle, Démonstration d'une Méthode pour resoudre les Egalitez de tous les degrez, 1691.



What did Rolle prove when he proved Rolle's theorem?
In the lengthy review by Victor J. Katz of The Oxford Handbook of the History of Mathematics, edited by Eleanor Robson and Jacqueline Stedall, Oxford University Press, Oxford, 2009, MR2549261 (2011e:01001), it says, 


Virtually the only article in the Handbook that could be classified as "internalist'' history of a mathematical idea is June Barrow-Green's article "From cascades to calculus: Rolle's theorem'', in which she looks in detail at the theorem as stated by Michel Rolle in 1690 and then looks at how the result was treated in various textbooks up to the beginning of the twentieth century. As originally stated and proved, the theorem only concerned polynomial functions and certainly involved no calculus, especially because Rolle attacked the calculus at its beginning for its lack of rigor. 


