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We say that a line in $\mathbb P^{n+1}_k$ is osculating to a hypersurface $Y$if they meet with contact order $n+1$. When $k=\mathbb C$, it is known thatthrough a fixed point of $Y$, there are exactly $n!$ of such lines. Under someparity condition on $n$ and $\mathrm{deg}(Y)$, we define a quadraticallyenriched count of these lines over any perfect field $k$. The count takesvalues in the Grothendieck-Witt ring of quadratic forms over $k$ and dependslinearly on $\mathrm{deg}(Y)$.

An arithmetic count of osculating lines