Chronology

Gergonne publishes the first volume of his new mathematics journal Annales de mathématique pures et appliquées which became known as Annales de Gergonne.

Poisson publishes Traité de mécanique (Treatise on Mechanics). It includes Poisson's work on the applications of mathematics to topics such as electricity, magnetism and mechanics.

Laplace publishes the two volumes of Théorie Analytique des probabilités (Analytical Theory of Probabilities). The first book studies generating functions and also approximations to various expressions occurring in probability theory. The second book contains Laplace's definition of probability, Bayes's rule, and mathematical expectation.

Argand gives a beautiful proof (with some gaps) of the fundamental theorem of algebra. (See this History Topic.)
Barlow produces Barlow's Tables which give factors, squares, cubes, square roots, reciprocals and hyperbolic logs of all numbers from 1 to 10000.

Peacock, Herschel and Babbage are the leaders of the Analytical Society at Cambridge which publishes an English translation of Lacroix's textbook Traité de Calcul differéntiel et intégral.

Bessel discovers a class of integral functions, now called "Bessel functions", in his study of a problem of Kepler to determine the motion of three bodies moving under mutual gravitation.
Bolzano publishes Rein analytischer Beweis (Pure Analytical Proof) which contain an attempt to free calculus from the concept of the infinitesimal. He defines continuous functions without the use of infinitesimals. The work contains the Bolzano-Weierstrass theorem.

Inspired by the work of Laplace, Adrain publishes Investigation of the figure of the Earth and of the gravity in different latitudes.

Horner submits a paper giving "Horner's method" for solving algebraic equations to the Royal Society and was published in the same year in the Philosophical Transactions of the Royal Society.