  We obtain a precise relation between the Chern-Schwartz-MacPherson class of a
subvariety of projective space and the Euler characteristics of its general
linear sections. In the case of a hypersurface, this leads to simple proofs of
formulas of Dimca-Papadima and Huh for the degrees of the polar map of a
homogeneous polynomial, extending these formula to any algebraically closed
field of characteristic~0, and proving a conjecture of Dolgachev on
'homaloidal' polynomials in the same context. We generalize these formulas to
subschemes of higher codimension in projective space.
  We also describe a simple approach to a theory of `polynomial Chern classes'
for varieties endowed with a morphism to projective space, recovering
properties analogous to the Deligne-Grothendieck axioms from basic properties
of the Euler characteristic. We prove that the polynomial Chern class defines
homomorphisms from suitable relative Grothendieck rings of varieties to Z[t].
