- Simpson publishes Treatise on the Nature and Laws of Chance. Much of this probability treatise is based on the work of de Moivre.
- Maclaurin is awarded the Grand Prix of the Académie des Sciences for his work on gravitational theory to explain the tides.
- Maclaurin publishes Treatise on Fluxions which aims to provide a rigorous foundation for the calculus by appealing to the methods of Greek geometry. It is the first systematic exposition of Newton's methods written in reply to Berkeley's attack on the calculus for its lack of rigorous foundations.
- Goldbach conjectures, in a letter to Euler, that every even number ≥ 4 can be written as the sum of two primes. It is not yet known whether Goldbach's conjecture is true.
- D'Alembert publishes Traité de dynamique (Treatise on Dynamics). In this celebrated work he states his principle that the internal actions and reactions of a system of rigid bodies in motion are in equilibrium.
- D'Alembert publishes Traite de l'equilibre et du mouvement des fluides (Treatise on Equilibrium and on Movement of Fluids). He applies his principle to the equilibrium and motion of fluids.
- D'Alembert uses partial differential equations to study the winds in Réflexion sur la cause générale des vents (Reflection on the General Cause of Winds) which receives the prize of the Prussian Academy.
- Agnesi writes Instituzioni analitiche ad uso della gioventù italiana which is an Italian teaching text on the differential calculus. The book contains many examples which were carefully selected to illustrate the ideas. There is an investigation of a curve that becomes known as "the witch of Agnesi". (See this Famous curve.)
- Euler publishes Analysis Infinitorum (Analysis of the Infinite) which is an introduction to mathematical analysis. He defines a function and says that mathematical analysis is the study of functions. This work bases the calculus on the theory of elementary functions rather than on geometric curves, as had been done previously. The famous formula appears for the first time in this text.
- D'Alembert studies the "three-body problem" and applies calculus to celestial mechanics. Euler, Lagrange and Laplace also work on the three-body problem.
- Cramer publishes Introduction à l'analyse des lignes courbes algébraique. The work investigates curves. The third chapter looks at a classification of curves and it is in this chapter that the now famous "Cramer's rule" is given.
- Giulio Fagnano publishes much of his previous work in Produzioni matematiche. It contains remarkable properties of the lemniscate and the duplication formula for integrals. This latter result led Euler to prove the addition formula for elliptic integrals.
- Euler publishes his theory of logarithms of complex numbers.
- D'Alembert discovers the Cauchy-Riemann equations while investigating hydrodynamics.
- Euler states his theorem for polyhedra.
- Euler publishes Institutiones calculi differentialis which begins with a study of the calculus of finite differences.
- Lagrange is a founding member of a mathematical society in Italy that will eventually become the Turin Academy of Sciences.
- The appearance of "Halley's comet" on 25 December confirms Halley's predictions 15 years after his death.
- Aepinus publishes Tentamen theoriae electriciatis et magnetismi (An Attempt at a Theory of Electricity and Magnetism). It is the first work to develop a mathematical theory of electricity and magnetism.