For present day mathematicians, it may be surprising how often ancient mathematicians were right in their hunches about mathematics. Many mathematicians provided theorems that ultimately proved to be true, but at the time they gave no valid proof of how they knew the theorem to be true. For example, Fermat’s Last Theorem. The margin in which Fermat wrote was too small to contain his proof although he claims it to be remarkable. Because of this, one may wonder how these mathematicians got so lucky, so to speak. Although mathematicians are human and it can be assumed that they made mistakes, we rarely hear about one devoting a great deal of time to a theory that was false. This blog is about a man who did just that.
However, the same man is known for other contributions to mathematics, and we will address those first. Michel Rolle was born in 1652 in Ambert, France. Rolle did not receive much of a formal education. After some elementary schooling he self educated himself. Rolle worked as a transcriber for a notary and also as an assistant to several attorneys in Ambert. He then traveled to Paris in 1675 where he worked, married, and had children. Rolle did not come from wealth and he had to work very hard to support his family. For this reason, he was especially grateful to receive a monetary reward after solving a problem publicly posed by Jacquez Ozanam in 1682.
Ozanam presented the problem, “Find four numbers the difference of any two being a perfect square, in addition the sum of the first three numbers being a perfect square.” Ozanam thought the smallest of the four numbers would have at least fifty digits, but Rolle was able to find a solution of four numbers each with seven digits. Rolle gained fame for solving this problem and was rewarded a pension from Jean-Baptiste Colbert, the controller general of finance and secretary of state for the navy under King Louis XIV of France. In addition to this financial reward, François Michel le Tellier, Marquis de Louvois, the French Secretary of State for War, hired Rolle as a tutor to teach mathematics to one of his sons. He was very impressed with Rolle’s pedagogical and mathematical skills and he elected Rolle to be a member of the Academie in 1685.
In 1690 Rolle published his most famous work, Traité d'algèbre, on the theory of equations. In this work Rolle invented the famous notation still used today for the root of . In this work, Rolle also introduced his famous method of cascades. The method involves taking the derivative of a polynomial (the first cascade) and noting that between any two consecutive roots of the original polynomial exist a root of the derivative. A second derivative (the second cascade) is then taken, noting that between any two consecutive roots of the first derivative, there exists a root of the second derivative. Derivatives will be taken until a linear polynomial is reached and the cascades can be used to find approximations of the original roots.
From this method of cascades came what is now known as “Rolle’s Theorem.” The theorem was given this name by Giusto Bella in 1864. Rolle’s Theorem states: If then for some with .
In 1690 Rolle published his most famous work, Traité d'algèbre, on the theory of equations. In this work Rolle invented the famous notation still used today for the root of . In this work, Rolle also introduced his famous method of cascades. The method involves taking the derivative of a polynomial (the first cascade) and noting that between any two consecutive roots of the original polynomial exist a root of the derivative. A second derivative (the second cascade) is then taken, noting that between any two consecutive roots of the first derivative, there exists a root of the second derivative. Derivatives will be taken until a linear polynomial is reached and the cascades can be used to find approximations of the original roots.
From this method of cascades came what is now known as “Rolle’s Theorem.” The theorem was given this name by Giusto Bella in 1864. Rolle’s Theorem states: If then for some with .
Rolle was obviously a big contributor to calculus. But “he described the infinitesimal calculus as a collection of ingenious fallacies and he believed that the methods could lead to errors.” (O’Connor) Rolle was a huge critic of L’Hopital’s work. He felt there were errors in L’Hopital’s axiom, “Grant that two quantities whose difference is an infinitely small quantity may be taken (or used) indifferently for each other; or (which is the same thing) that a quantity which is increased or decreased only by an infinitesimally small quantity may be considered as remaining the same.” (O’Connor) Rolle presented a series of papers to the Academy and offered an explanation as to why the axiom was incorrect, but L’Hopital quickly noticed errors in his analysis. Eventually Rolle conceded that he was wrong.
In 1708 Rolle suffered from a stroke. Unfortunately, after this stroke his mental capacity diminished and he made no further mathematical contributions. Then in 1719 Rolle experienced another stroke, which ended up being fatal.
In 1708 Rolle suffered from a stroke. Unfortunately, after this stroke his mental capacity diminished and he made no further mathematical contributions. Then in 1719 Rolle experienced another stroke, which ended up being fatal.