The above citation describes in a few words the essence of what is called today often “singularity theory”. A little bit more precisely, we can say that the subject of this relatively new area of mathematics is the study of systems of finitely many differentiable, or analytic, or algebraic, functions in the neighbourhood of a point where the Jacobian matrix of these functions is not of locally constant rank. The general notion of a “singularity” is, of course, much more comprehensive. Singularities appear in all parts of mathematics, for instance as zeroes of vector fields, or points at infinity, or points of indeterminacy of functions, but always refer to a situation which is not regular, that is, not the usual, or expected, one.