Abstract: 

The talk is based on a letter by Gauss to his former student Gerling written in 1823. In it he introduced the Gauss-Seidel  method for solving symmetric linear systems of equations obtained from his least squares method for over-determined systems of linear equations. Further, he suggested a way of obtaining least norm solutions to under-determined linear systems.  His examples in the letter were based on geodesic survey data. The talk is partly historical and partly mathematical.