- The work unfinished by Cotes on his death is published as Harmonia mensurarum. It deals with integration of rational functions. It contains a thorough treatment of the calculus applied to logarithmic and circular functions.
- Jacapo Riccati studies the Riccati differential equation in a paper. He gives solutions for certain special cases to the equation which was first studied by Jacob Bernoulli.
- Academy of Sciences is founded in St Petersburg.
- Euler is appointed to St Petersburg. He introduces the symbol for the base of natural logarithms in a manuscript entitled Meditation upon Experiments made recently on firing of Cannon. The manuscript was not published until 1862.
- Grandi publishes Flora geometrica (Geometrical Flowers). He gives a geometrical definition of curves which resemble petals and leaves of flowers. For example the rhodonea curves are so called since they look like roses while the clelie curve is named after the Countess Clelia Borromeo to whom he dedicated his book.
- De Moivre gives further theorems concerning his trigonometric representation of complex numbers. He gives Stirling's formula.
- Clairaut publishes Recherches sur les courbes à double coubure on skew curves.
- De Moivre first describes the normal distribution curve, or law of errors, in Approximatio ad summam terminorum binomii in seriem expansi. Gauss, in 1820, also investigated the normal distribution.
- In Euclides ab Omni Naevo Vindicatus Saccheri does important early work on non-euclidean geometry, although he considers it an attempt to prove the parallel postulate of Euclid.
- Berkeley publishes The analyst: or a discourse addressed to an infidel mathematician. He argues that although the calculus led to true results its foundations were no more secure than those of religion.
- Euler introduces the notation .
- Euler solves the topographical problem known as the "Königsberg bridges problem". He proves mathematically that it is impossible to design a walk which crosses each of the seven bridges exactly once.
- Euler publishes Mechanica which is the first mechanics textbook which is based on differential equations.
- Simpson publishes his Treatise on Fluxions written as a textbook for his private students. In the book he uses infinite series to find the definite integrals of functions.
- Daniel Bernoulli publishes Hydrodynamica (Hydrodynamics). It gives for the first time the correct analysis of water flowing from a hole in a container and discusses pumps and other machines to raise water. He also gives, in Chapter 10, the basis of the kinetic theory of gases.
- D'Alembert publishes Mémoire sur le calcul intégral (Memoir on Integral Calculus).