Given a (smooth) projective variety $X\subset \mathbb{P}^n$, we can define a rational map $s:X\times X\rightarrow G(1,n)$ that takes a pair $(p,q)\in (X\times X)\setminus \Delta$ not on the diagonal and sends it to the line through $p$ and $q$.
We can define the secant variety $\mathscr{S}(X)$ of $X$ to be the closure in $G(1,n)$ of the image of $s$. I'm struggling with a couple of details that are supposed to be obvious in Harris's introductory book:
1) Why is it clear that every tangent line to $X$ is in $\mathscr{S}(X)$? This is Exercise 15.9, which is supposed to be "relatively straightforward." If we work over $\mathbb{C}$, I think it's intuitively clear by showing that it's in the closure in the analytic topology, but in general I don't know of a good way to show something is in the closure, without picking an arbitrarily function that vanishes on the image of $s$ and trying to show that it also vanishes on tangent lines.
2) There is a second way method mentioned in Exercise 15.11, where we take the rational map $s:X\times X\rightarrow G(1,n)$ and extend it to an honest map $Bl_\Delta(X\times X)\rightarrow G(1,n)$. Why is it clear that each fiber of $Bl_\Delta(X\times X)\rightarrow X\times X$ over a point $(p,p)$ in $\Delta$ is sent to the projective space of lines through $p$ in $T_pX$?
Googling seems to say the exceptional divisor is the projectivized tangent bundle, and, if I squint, I might be able to think that, as $q$ approaches $p$, the line through $p$ and $q$ approaches a tangent line at $p$, and $(p,q)$ approaches the point in the fiber over $(p,p)$ corresponding to that line, but I don't know how to write it out.
For part 1 if you know how to do this analytically, then maybe this helps: These varieties are constructible, the Zariski closure and the analytic closures agree. So you can just use the fact that every tangent is a limit of secants.
On the other hand, to use the vanishing of an arbitrary polynomial function on a point of the image of $s$ I think you still need to work on the image of an open set and then pass to the closure of the image of the open. Anyway, in this case one helpful trick is "polarization," which allows you to express a polynomial that vanishes on a sum as a linear combination of multilinear forms. Since it's an exercise I won't spell out all the details.