Translation and modern interpretation of Laplace's Théorie Analytique des Probabilités, pages 505-512, 516-520

The text of Laplace, \textit{Sur lápplication du calcul des probabilit\és \`a la philosophie naturelle,} (Th\éorie Analytique des Probabilit\és. Troisi\`eme \Édition. Premier Suppl\ément), 1820, is quoted in the context of the Gram-Schmidt algorithm. We provide an English translation of Laplace's manuscript (originally in French) and interpret the algorithms of Laplace in a contemporary context. The two algorithms given by Laplace computes the mean and the variance of two components of the solution of a linear statistical model. The first algorithm can be interpreted as {\em reverse square-root-free modified Gram-Schmidt by row} algorithm on the regression matrix. The second algorithm can be interpreted as the {\em reverse square-root-free Cholesky} algorithm.