- Monge begins the study of descriptive geometry.
- Bayes publishes An Essay Towards Solving a Problem in the Doctrine of Chances which gives Bayes theory of probability. The work contains the important "Bayes' theorem".
- Euler publishes Theory of the Motions of Rigid Bodies which lays the foundation of analytical mechanics.
- D'Alembert calls the problems to elementary geometry caused by failure to prove the parallel postulate "the scandal of elementary geometry".
- Euler publishes the first volume of his three volume work Dioptics.
- Euler makes Euler's Conjecture, namely that it is impossible to exhibit three fourth powers whose sum is a fourth power, four fifth powers whose sum is a fifth power, and similarly for higher powers.
- Lagrange proves that any integer can be written as the sum of four squares.
- Lagrange publishes Réflexions sur la résolution algébrique des équations which makes a fundamental investigation of why equations of degrees up to four can be solved by radicals. The paper is the first to consider the roots of a equation as abstract quantities rather than numbers. He studies permutations of the roots and this work leads to group theory.
- Euler publishes his textbook Algebra.
- Lagrange proves Wilson's theorem (first stated without proof by Waring) that is prime if and only if is divisible by .
- Buffon uses a mathematical and scientific approach to calculate that the age of the Earth is about 75000 years.
- Euler introduces the symbol to represent the square root of -1 in a manuscript which will not appear in print until 1794.
- Buffon carries out his probability experiment calculating π by throwing sticks over his shoulder onto a tiled floor and counting the number of times the sticks fell across the lines between the tiles.
- Bézout publishes Théorie générale des équation algébraiques on the theory of equations. The work includes a result now known as a result known as "Bézout's theorem".