Chronology

Babbage creates the first accurate actuarial tables for use in insurance calculations.
Poisson introduces "Poisson's ratio" in elasticity which involves stresses and strains on materials.
Peacock publishes his Treatise on Algebra which attempts to give algebra a logical treatment comparable to Euclid's Elements.

Möbius publishes Über eine besondere Art von Umkehrung der Reihen which introduces the "Möbius function" and the "Möbius inversion formula".
Cauchy gives power series expansions of analytic functions of a complex variable.

Steiner publishes Systematische Entwicklungen ... (Systematic Development of the Dependency of Geometrical Forms on One Another) which gives a treatment of projective geometry based on metric considerations.
János Bolyai's work on non-Euclidean geometry is published as an appendix to an essay by Farkas Bolyai, his father. (See this History Topic.)

Legendre points out the flaws in 12 "proofs" of the parallel postulate. (See this History Topic.)

Hamilton uses algebra in treating dynamics in On a General Method in Dynamics. This paper gives the first statement of the characteristic function applied to dynamics.

Quetelet publishes Sur l'homme et le développement de ses facultés (A treatise on Man and the Development of his Faculties). He presents his conception of the "average man" as the central value about which measurements of a human trait are grouped according to the normal curve.
Coriolis publishes Sur les équations du mouvement relatif des systèmes de corps. He introduces the "Coriolis force" and shows that the laws of motion can be used in a rotating frame of reference if an extra force called the "Coriolis acceleration" is added to the equations of motion. In the same year Coriolis publishes a work on a mathematical theory of billiards.

Ostrogradski rediscovers Green's theorem.
Liouville founds a mathematics journal Journal de Mathématiques Pures et Appliquées. This journal, sometimes known as Journal de Liouville, did much to advance mathematics in France throughout the 19th century.
Poncelet publishes Cours de mécanique appliquée aux machines (A Course in Mechanics Applied to Machines). It is the first to propose the use of mathematics in machine design.

Poisson publishes Recherches sur la probabilité des jugements (Researches on the Probabilities of Opinions). In this work he establishes the rules of probability, gives "Poisson's law of large numbers" and describes the "Poisson distribution" for a discrete random variable which is a limiting case of the binomial distribution.
The Cambridge and Dublin Mathematical Journals begins publication.
Dirichlet gives a general definition of a function.
Liouville discusses integral equations and gives the "Sturm-Liouville theory" which is used in solving such equations.
Wantzel proves that the classical problems of duplicating a cube and trisecting an angle could not be solved with ruler and compass.

Bessel measures the parallax of the star 61 Cygni, the first star for which this is calculated.
Cournot publishes Recherches sur les principes mathématiques de la théorie des richesses in which he discusses mathematical economics, in particular supply- and demand-functions.
De Morgan invents the term "mathematical induction" and makes the method precise.