1800

Lacroix completes publication of his three volume textbook Traité de Calcul differéntiel et intégral.
1801

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 17gon 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.
1803

Lazare Carnot publishes Géométrie de position in which sensed magnitudes are first used systematically in geometry.
1804

Bessel publishes a paper on the orbit of Halley's comet using data from Harriot's observations 200 years earlier.
1806

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.
1807

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.
1808
1809

Poinsot discovers two new regular polyhedra.

Gauss describes the leastsquares 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).