In Caen he became friends with a fellow student Charles Castel, Abbé de Saint-Pierre, who arranged for Varignon to receive an income of 300 livres. The two shared lodgings and it appears that at this time Varignon continued his studies at the University of Caen. Up to this point he had taken a fairly standard route to the priesthood but his life changed course when, quite by chance, he came across Euclid's Elements and began reading the classic text. Led into mathematics by reading Euclid, he then read Descartes' Géométrie and thereafter devoted himself to the mathematical sciences. Of course as a Jesuit he belonged to an Order that valued scholarship and teaching so Varignon was able to devote the rest of his life to teaching. In 1686, together with his friend Charles Castel, Abbé de Saint-Pierre, Varignon went to Paris and immediately made contact with mathematicians and scientists there.
In 1687 Varignon published Projet d'une nouvelle méchanique which studied composition of forces using Leibniz's differential calculus in the study of mechanics. He dedicated this work to the Academy of Sciences and it was clearly highly thought of since he was elected to the Academy in the same year. Also in 1688 he became professor of mathematics at the Collège Mazarin, occupying a newly created chair, where he began to teach mathematics at the level of current research at the time. In 1704, in addition to the chair at Collège Mazarin, he became professor of mathematics at the Collège Royal. His lectures at the Collège Mazarin were published as élémens de mathématiques ... (1731) and we discuss their content at the end of this article.
Varignon's chief contributions were to graphical statics and mechanics. From the earliest of his publications such as Projet d'une nouvelle méchanique in 1687, it was clear that he understood the value of Leibniz's calculus. This was surprisingly soon after Leibniz's two articles on the new differential calculus were published in the Acta Eruditorum in October 1684 and June 1686. Although Varignon made no major mathematical contributions, he developed analytic dynamics by adapting Leibniz's calculus to the inertial mechanics of Newton's Principia being one of the first French scholars to recognise the power and importance of the calculus. While those like Huygens admired Newton as a mathematician, they did not accept a physical theory based on action at a distance. Varignon put aside these philosophical worries and began to rework large sections of the Principia into the Leibniz's approach to the differential and integral calculus.
In 1644 Torricelli published in De motu aquarum what has now became known as Torricelli's law. This states that the speed of a liquid flowing under the force of gravity out of an opening in a tank is proportional to the square root of the vertical distance between the liquid surface and the centre of the opening, and to the square root of twice the acceleration due to gravity. Experimental evidence had been found to support the law but in 1695 Varignon tried to deduce it. F Sebastiani, reviewing , writes:-
[Varignon] assumed that, during a small time interval, one can both disregard the variation of the level of the free surface and consider the motion to be essentially uniform [permanent] for the whole liquid. Varignon's proof rests on the idea that the small quantity of liquid that escapes at each moment gets all of its motion from the pressure exercised by the "weight of the columns of liquid with base equal to the opening". This extremely interesting proof nevertheless remains very uncertain in infinitesimal manipulations with respect to the use of the power and quantity of motion. For Varignon, to have derived Torricelli's law jointly from the axiom according to which "causes are always proportional to their effects", from the principles of mechanics and from the general laws of motion was to have proved it "by reason alone". He thus implicitly attributed to mechanics the same demonstrative perfection that Euclidean geometry had been thought to possess.Among his other work was a publication in 1699 on applications of the differential calculus to fluid flow and to water clocks. In 1702 he applied the calculus to clocks driven by a spring. Varignon studied the problem of the motion of projectiles in resisting media in eleven long memoirs which he presented to the Academy of Sciences between 1707 and 1711. He gave a new unified treatment of particular cases which had already been studied by Wallis, Huygens, Leibniz and Newton. Again this work is developed using Leibniz's approach to the calculus. See  for further details of his memoirs on this topic. In 1724 Varignon's Nouvelle mécanique was published which gave the best approach to geometrical statics until the work of Poinsot over 75 years later. Varignon's work was to have a major influence on Euler in his study on particle dynamics.
Varignon played a major role in defending the calculus from attacks. For example in 1700 Rolle argued against the calculus both on the grounds that it was without sound foundation and that it led to errors. Varignon argued before the Academy of Sciences that Rolle's arguments which suggested that the calculus led to errors was wrong. The two mathematicians maintained a vigorous exchange for five years until the Academy of Sciences decreed that the debate was ended.
In 1731, nine years after Varignon's death, his notes for teaching mathematics in schools was published as élémens de mathématiques ... . It :-
... is organised in two parts, the first part explaining concepts of arithmetic and elementary algebra and the larger second part covering topics in Euclidean geometry.The book contains what is today known as Varignon's parallelogram theorem: The figure formed when the mid-points of the sides of a quadrilateral are joined in order is a parallelogram. He gives a completely rigorous proof of this theorem, being the first to do so.
Varignon was elected to the French Academy of Sciences in 1688, the Berlin Academy of Science in 1713 and the Royal Society of London in 1718.
Article by: J J O'Connor and E F Robertson