Rolle's "cascades" gave rise to the great theorems of analysis, which not only provide methods of proof but establish once and for all the connection between a function's behaviour and its derivative. Rolle always distrusted the "new" differential calculus, disparaged the Marquis de l'Hospital's book and entered into a dispute with Varignon on the subject. That he should owe his fame to what he fought so fiercely during his lifetime is almost his revenge on history…
The discovery of the cascade method
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The French mathematician Michel Rolle (1652–1719) made his name in 1682 by solving a problem posed by his contemporary Jacques Ozanam: "Find four numbers such that the difference between any two is a square, and the sum of any two of the first three is also a square." The author of the problem himself claimed that "the smallest of these numbers has no fewer than fifty figures [digits]", whereas Rolle found a solution using numbers of just seven digits: {2,399,057, 2,288,168, 1,873,432, 6,560,657}. His secret? Translating the problem into equations involving polynomials of degrees up to 18, then finding bounds for their roots using a method of his own invention: the celebrated méthode des cascades, which he described in his Démonstration d'une Méthode pour résoudre les Egalitez de tous les degrez (1691).
