Translation and modern interpretation of Laplace's Théorie Analytique des Probabilités, pages 505-512, 516-520
This is an incremental historical clarification for researchers interested in the origins of numerical linear algebra.
The paper translates and interprets Laplace's 1820 algorithms for computing mean and variance in linear statistical models, showing they correspond to modern reverse square-root-free modified Gram-Schmidt and Cholesky algorithms.
The text of Laplace, \textit{Sur l'application du calcul des probabilités à la philosophie naturelle,} (Théorie Analytique des Probabilités. Troisième Édition. Premier Supplément), 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.