Chronology

Brianchon publishes Recherches sur la determination d'une hyperbole equilatère, au moyen de quatres conditions données which contains a statement and proof of the nine point circle theorem.

Navier gives the well known "Navier-Stokes equations" for an incompressible fluid.
Cauchy publishes Cours d'analyse (A Course in Analysis), which sets mathematical analysis on a formal footing for the first time. Designed for students at the Ecole Polytechnique it was concerned with developing the basic theorems of the calculus as rigorously as possible.

Poncelet develops the principles of projective geometry in Traité des propriétés projectives des figures (Treatise on the Projective Properties of Figures). This work contains fundamental ideas of projective geometry such as the cross-ratio, perspective, involution and the circular points at infinity.
Fourier's prize winning essay of 1811 is published as Théorie analytique de la chaleur (Analytical Theory of Heat). It makes widely available the techniques of Fourier analysis, which will have widespread applications in mathematics and throughout science.
Feuerbach publishes his discoveries on the nine point circle of a triangle.

János Bolyai completes preparation of a treatise on a complete system of non-Euclidean geometry. When Bolyai discovers that Gauss had anticipated much of his work, but not published anything, he delays publication. (See this History Topic.)
Babbage begins construction of a large "difference engine" which is able to calculate logarithms and trigonometric functions. He was using the experience gained from his small "difference engine" which he constructed between 1819 and 1822.

Sadi Carnot publishes Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (Thoughts on the Motive Power of Fire, and on Machines Suitable for Developing that Power). A book on steam engines, it will be of fundamental importance in thermodynamics. The "Carnot cycle" which forms the basis of the second law of thermodynamics also appears in the book.
Abel proves that polynomial equations of degree greater than four cannot be solved by radicals. He publishes it at his own expense as a six page pamphlet.
Bessel develops "Bessel functions" further while undertaking a study of planetary perturbations.
Steiner develops synthetic geometry. He publishes his theories on the topic in 1832.

Gompertz gives "Gompertz's Law of Mortality" which shows that the mortality rate increases in a geometric progression so when death rates are plotted on a logarithmic scale, a straight line known as the "Gompertz function" is obtained.

Ampère publishes Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience. It contains a mathematical derivation of the electrodynamic force law and describes four experiments. It lays the foundation for electromagnetic theory.
Crelle begins publication of his Journal für die reine und angewandte Mathematik which will become known as Crelle's Journal. The first volume contains several papers by Abel.
Poncelet's work on the pole and polar lines associated with conics lead him to discover the principle of duality. Gergonne, who introduced the word polar, discovers independently the principle of duality.

Jacobi writes a letter to Legendre detailing his discoveries on elliptic functions. Abel was independently working on elliptic functions at this time.
Möbius publishes Der barycentrische Calkul on analytical geometry. It becomes a classic and includes many of his results on projective and affine geometry. In it he introduces homogeneous coordinates and also discusses geometric transformations, in particular projective transformations.
Feuerbach writes a paper which, independently of Möbius, introduces homogeneous coordinates.

Gauss introduces differential geometry and publishes Disquisitiones generales circa superficies. This paper arises from his geodesic interests, but it contains such geometrical ideas as "Gaussian curvature". The paper also includes Gauss's famous theorema egregrium.
Green publishes Essay on the Application of Mathematical Analysis to the Theory of Electricity and Magnets, in which he applies mathematics to the properties of electric and magnetic fields. He introduces the term potential, develops properties of the potential function and applies them to electricity and magnetism. The formula connecting surface and volume integrals, now known as "Green's theorem", appears for the first time in the work, as does the "Green's function" which would be extensively used in the solution of partial differential equations.
Abel begins a study of doubly periodic elliptic functions.
Plücker publishes Analytisch-geometrische which develops the "Plücker abridged notation". He, independently of Möbius and Feuerbach one year earlier, discovers homogeneous coordinates.

Galois submits his first work on the algebraic solution of equations to the Académie des Sciences in Paris.
Lobachevsky develops non-euclidean geometry, in particular hyperbolic geometry, and his first account of the subject is published in the Kazan Messenger. When it was submitted for publication in the St Petersburg Academy of Sciences Ostrogradski rejects it. (See this History Topic.)