Suppose that $C\subset\mathbb P^n$ is a projective curve (over an algebraically closed field of characteristic $0$) not lying in a hyperplane; denote by $C^*\subset(\mathbb P^n)^*$ the closure of the set of hyperplanes osculating to $C$ at smooth points. It is well known that $(C^*)^*=C$ and, moreover, $k$-dimensional osculating space to $C^*$ (at the generic point) coincides with the annihilator of the corresponding $n-k-1$-dimensional osculating space to $C$.

When and by whom was this first proved? Could you give a reference?

(edit: a typo corrected)

Thank you in advance,

Serge