# Osculating duality for curves (reference request)

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

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This is due to Cayley and Veronese. A reference is a book by Bertini, Geometria proiettiva degli iperspazi, available here: play.google.com/store/books/… –  Felipe Voloch Jan 5 at 17:01
@Felipe: Thanks, I will try to obtain the book. –  Serge Lvovski Jan 5 at 17:03
The reference I knew was for the 1923 edition of Bertini's book but I just noticed that the edition on google books is the 1907 edition. So it may not be there. –  Felipe Voloch Jan 5 at 17:19
Is there nothing more recent? –  Serge Lvovski Jan 5 at 17:22
I thought you wanted a historical reference. Otherwise, Piene, R., Numerical characters of a curve in projective n-space, Real and Complex Singularities, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, Thm. 5.1 –  Felipe Voloch Jan 5 at 18:54
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