Chronology

Lagrange wins the Grand Prix of the Académie des Sciences in Paris for his work on perturbations of the orbits of comets by the planets.

Coulomb's major work on friction Théorie des machines simples wins him the Grand Prix from the Académie des Sciences.
William Herschel discovers the planet Uranus.

Legendre introduces his "Legendre polynomials" in his work Recherches sur la figure des planètes on celestial mechanics.

Condorcet publishes Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix (Essay on the Application of the Analysis to the Probability of Majority Decisions). It is a major advance in the study of probability in the social sciences.
Legendre states the law of quadratic reciprocity but his proof is incorrect.
Condorcet publishes Essay on the Application of Analysis to the Probability of Majority Decisions which is an extremely important work in the development of the theory of probability.
Lagrange begins work on elliptic functions and elliptic integrals.

Lagrange publishes Mécanique analytique (Analytical Mechanics). It summarises all the work done in the field of mechanics since the time of Newton and is notable for its use of the theory of differential equations. With this work Lagrange transforms mechanics into a branch of mathematical analysis.

De Prony begins a major task of producing the Cadastre. This consisted of logarithmic and trigonometric tables given to between 14 and 29 decimal places.

Legendre publishes Eléments de géométrie, an account of geometry which would be a leading text for 100 years. It will replace Euclid's Elements as a textbook in most of Europe and, in succeeding translations, in the United States. It becomes the prototype of later geometry texts.

Laplace presents his famous nebular hypothesis in Exposition du systeme du monde which views the solar system as originating from the contracting and cooling of a large, flattened, and slowly rotating cloud of incandescent gas.
Gauss gives the first correct proof of the law of quadratic reciprocity.

Lagrange publishes Théorie des fonctions analytiques (Theory of Analytical Functions). It is the first treatise on the theory of functions of a real variable. It uses modern notation like ${{dy}\over{dx}}$ for derivatives.
Wessel presents a paper on the vector representation of complex numbers which is published in Danish in 1799. The idea first appears in a report he wrote in 1787.
Mascheroni proves in Geometria del compasso that all Euclidean constructions can be made with compasses alone and so a ruler in not required.
Lazare Carnot publishes Réflexions sur la métaphysique du calcul infinitésimal in which he treats zero and infinity as limits. He also considers that infinitely small quantities are real objects, being representable as differences between limits.

Gauss proves the fundamental theorem of algebra and notes that earlier proofs, such as by d'Alembert in 1746, could easily be corrected. (See this History Topic.)
Laplace publishes the first volume of five-volume Traité de mécanique céleste (Celestial Mechanics). It applies calculus to study the orbits of celestial bodies and examines the stability of the Solar System.
Monge publishes Géométrie descriptive which describes orthographic projection, the graphical method used in modern mechanical drawing.
Ruffini publishes the first proof that algebraic equations of degree greater than four cannot be solved by radicals. It was largely ignored as were the further proofs he would publish in 1803, 1808 and 1813.