In the 1930's J. M. Thomas suggested an algorithm to split a system of algebraic equations into a finite number of systems which he called simple i.e. triangular of a rather special kind. This and the Janet algorithm for linear differential equations he applied to polynomial differential equations to obtain a triangular decomposition there as well. Up to now the algorithm has not been implemented. Recently V. Gerdt gave a more modern formulation of the algorithm. In the talk the basic ideas will be explained from a more geometric point of view and some examples will be discussed.