The paragraph below is taken from the book An Introduction to Special Relativity by Robert Resnick.
Let A and B be the endpoints of a rod, for example, which is at rest in the S-frame. Then the primed (S') observer, for whom the rod is moving with velocity $-\vec v$, will measure the endpoint locations as $x_B'$ and $x_A'$, whereas the unprimed observer locates them at $x_B$ and $x_A$. Using Galilean transformations, however, we find that $x'_B=x_B-vt_B$ and $x'_A=x_A-vt_A$, so that $x'_B-x'_A=x_B-x_A-v(t_B-t_A)$. Since the two endpoints, A and B, are measured at the same instant, $t_A=t_B$ and we obtain $x'_B-x'_A=x_B-x_A$.
My question is why should we compare the left-hand side to the right-hand side at $t_A=t_B$?
Since the rod is eternally at rest in the unprimed frame, the markers $x_A$ and $x_B$ will have unchanging values no matter if $x_A$ and $x_B$ measured at the same instant or at different instants in the unprimed frame. Isn't it? If so, why does it matter when $x_A$ and $x_B$ are measured in the unprimed frame (to define its rest length)?
I can define the rest length in the unprimed frame by measuring both ends at 1.00 pm noon or one end at 1.00 pm and the other end at 1.30 pm. What's wrong?
N.B. Please note that my question js about Newtonian/Galilean relativity. Not special relativity.