# Pierre Varignon, bridge between math and physics

https://www.lemonde.fr/sciences/article/2023/02/02/pierre-varignon-passerelle-entre-maths-et-physique_6160274_1650684.html

Do you know Varignon’s theorem? The four middles of the sides of any quadrilateral always form a parallelogram. Did we really have to wait until the beginning of the 18th century to prove such an elementary result, which we sometimes find today in college mathematics textbooks? It is a simple result, and not very interesting, we must admit. It is a kind of curse: mathematicians are often credited with results that do not illustrate their work in any way. Arnold’s theorem even states that no theorem with a proper name is due to this person (and this theorem applies to himself).

Pierre Varignon was a mathematician born in 1654 and died in 1722. A colloquium, organized from January 17 to 19, on the occasion of the tercentenary of his death, allowed us to take a look at this fascinating period in the history of science. It is not about Isaac Newton (1642-1727) or Gottfried Wilhelm Leibniz (1646-1716), whose works have been extensively studied, but about a secondary character who nevertheless played an important role in French mathematics.

Varignon was both a teacher and a researcher, serving also as an intermediary between the great thinkers of his time. The polemic was raging: Newton and Leibniz both claimed the differential calculus, which they presented in very different ways. Varignon played the role of “translator” between the two variants of the same language. His posthumous work Eclaircissemens sur l’analyse des infiniment petits, published in 1725, allowed the introduction of this new differential calculus in France, at the origin of a true scientific revolution.

Premises of the vector calculus

In mechanics, we owe him clear statements on the composition of forces, only glimpsed before by Leonardo da Vinci and Galileo. His book Nouvelle Mécanique ou Statique, the draft of which was given in 1687, contains some admirable plates. We see weights suspended from cables in all sorts of configurations and the conditions of equilibrium are described. With a little imagination, one can guess the premises of the calculation of vectors, so important today, both in mathematics and in physics.

Varignon was perhaps the first professional teacher-researcher in France. He was the first professor of mathematics at the Collège Mazarin, in 1688, in the palace that would later house the Institut de France. He taught there until his death with great interest. His book Elémens de mathématique, published in 1731, contains his teaching and in particular Varignon’s theorem.

He was a true geometer, as testified by the comment of a contemporary who wrote that “he had all the difficulties in the world to say his breviary, because of the habit he had contracted of mathematical figures (…) and that it was necessary even that what he read, to be able to retain it, was susceptible of figures”. The Mazarine Library of the Institute is hosting, until April 15, a remarkable exhibition entitled “Pierre Varignon (1654-1722). Practice and transmission of mathematics at the dawn of the Enlightenment”. A visit not to be missed by lovers of old books, in an exceptional setting.

In 1694, he was elected to the Collège royal, which became the Collège de France. The new professors gave their inaugural lectures, but at the time, they were called “entrance speeches”. Here is the title of Varignon’s lecture (translated from Latin, of course): “On the mutual help that mathematics and physics provide each other: physics is uncertain without mathematics, mathematics is hardly useful without physics”. Here is a title that the writers of today’s school programs could meditate on.