Chronology

Cauchy publishes the first volume of the four volume work Exercises d'analyse et de physique mathematique.

Gauss publishes a treatise on optics in which he gives a formulae for calculating the position and size of the image formed by a lens with a given focal length.
Jacobi writes a long memoir De determinantibus functionalibus devoted to the functional determinant now called the Jacobian.
Quetelet establishes the Belgium Central Statistical Bureau.

Hesse introduces the "Hessian determinant" in a paper which investigates cubic and quadratic curves.
Stokes begins his research on fluids and publishes On the steady motion of incompressible fluids.

Hamilton discovers quaternions, which generalise complex numbers to four dimensions.
Liouville announces to the Académie des Sciences in Paris that he had found deep results in Galois's unpublished work and promises to publish Galois's papers together with his own commentary.
Kummer invents "ideal complex numbers" in his study of unique factorisation. This leads to the development of ring theory.
Cayley is the first person to investigate "geometry of $n$ dimensions" which occurs in the title of his paper of that year. He uses determinants as the major tool.

Liouville finds the first transcendental numbers - numbers that cannot be expressed as the roots of an algebraic equation with rational coefficients.
Grassmann publishes Die lineale Ausdehnundslehre, ein neuer Zweig der Mathematik in which he develops the idea of an algebra in which the symbols representing geometric entities such as points, lines and planes, are manipulated using specific rules.

Cayley publishes Theory of Linear Transformations in which he examines the composition of linear transformations.
While examining permutation groups Cauchy proves a fundamental theorem of group theory which became known as "Cauchy's theorem". (See this History Topic.)

Liouville publishes Galois' papers on the solution of algebraic equations in Liouville's Journal.
Maxwell writes his first paper at the age of 14: On the description of oval curves, and those having a plurality of foci.

Boole publishes The Mathematical Analysis of Logic, in which he shows that the rules of logic can be treated mathematically rather than metaphysically. Boole's work lays the foundation of computer logic.
De Morgan proposes two laws of set theory that are now known as "de Morgan's laws".
Von Staudt publishes Geometrie der Lage. It is the first work to completely free projective geometry from any metrical basis.

Thomson (Lord Kelvin) proposes the absolute temperature scale now named after him.