Chronology

Lacroix completes publication of his three volume textbook Traité de Calcul differéntiel et intégral.

Gauss publishes Disquisitiones Arithmeticae (Discourses on Arithmetic). It contains seven sections, the first six of which are devoted to number theory and the last to the construction of a regular 17-gon by ruler and compasses.
The minor planet Ceres is discovered but then lost. Gauss computes its orbit from the few observations that had been made leading to Ceres being rediscovered in almost exactly the position predicted by Gauss.
Gauss proves Fermat's conjecture that every number can be written as the sum of three triangular numbers.

Lazare Carnot publishes Géométrie de position in which sensed magnitudes are first used systematically in geometry.

Bessel publishes a paper on the orbit of Halley's comet using data from Harriot's observations 200 years earlier.

Argand introduces the Argand diagram as a way of representing a complex number geometrically in the plane.
Legendre develops the method of least squares to find best approximations to a set of observed data.

Fourier discovers his method of representing continuous functions by the sum of a series of trigonometric functions and uses the method in his paper On the Propagation of Heat in Solid Bodies which he submits to the Paris Academy.

Germain makes an important contribution to Fermat's last theorem. This is named "Germain's theorem" by Legendre.

Poinsot discovers two new regular polyhedra.
Gauss describes the least-squares method which he uses to find orbits of celestial bodies in Theoria motus corporum coelestium in sectionibus conicis Solem ambientium (Theory of the Movement of Heavenly Bodies).