Orthogonal calculus is a calculus of functors, similar to Goodwillie’s calculus. The functors in question take finite dimensional real vector spaces (with an inner product) to pointed spaces. Prime example: F ( V ) = B O ( V ) F(V) = BO(V) , where O ( V ) O(V) is the orthogonal group of V V . In this example, and in general, first derivatives in the orthogonal calculus reproduce and generalize much of the theory of Stiefel-Whitney classes. Similarly, second derivatives in the orthogonal calculus reproduce and generalize much of the theory of Pontryagin classes.