Chronology

Lambert proves that π is irrational. He publishes a more general result in 1768.

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.

Lambert writes Theorie der Parallellinien which is a study of the parallel postulate. By assuming that the parallel postulate is false, he manages to deduce a large number of results about non-euclidean geometry.

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 $n$ is prime if and only if $(n - 1)! + 1$ is divisible by $n$ .

Buffon uses a mathematical and scientific approach to calculate that the age of the Earth is about 75000 years.

Euler introduces the symbol $i$ 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".