In Chapter $4$ of Hartshorne, the author gives some propositions about a projection $\mathbb{P}^n_k \backslash \{ O \} \to \mathbb{P}^{n-1}_k $ and secant or tangent lines. He defines the secant line determined by $P,Q$ to be the line passing through $P,Q$, and the tangent line at $P$ (to a geometrically connected smooth projective scheme $X$ in $\mathbb{P}^n_k$) to be the unique line $L$ in $\mathbb{P}^n_k$ passing through $P$ such that the tangent space of $L$ at $P$ is equal to that of $X$ at $P$, as subspaces of the tangent space of $\mathbb{P}^n_k$ at $P$.
Intuitively these definitions are clear, however I can't understand these scheme theoretically.
Please help me understand the concrete definition of secant lines and tangent lines.