There is a `numerical' theory of Chern classes, which can be defined purely in terms of Euler characteristics. The basic observation behind this theory is that the information carried by the degrees of the Chern classes of a (possibly singular) projective variety and the Euler characteristics of general hyperplane sections of the variety are related by an involution, and hence essentially equivalent. This observation has nice consequences, including a new proof of a conjecture of Dolgachev on the degree of the gradient map of a homogeneous polynomial.