the adjoint group of a simple complex lie algebra lie(g) has a unique minimal orbit which we denote by c. we describe for classical lie algebras, for any natural number k, the zariski closure of the union kc of all spaces spanned by k points on c. the image of this set in the projective space {bbb p}(lie(g)) is usually called the (k-1)-st secant variety of {bbb p}(c), and its dimension and defect are easily determined from our explicit description. we give the smallest k for which the closure of kc is equal to the lie algebra and compare these results with the upper bound on secants of general varieties given in a theorem of f. zak (eg 1993). this talk describes recent joint work with jan draisma.