In the first chapter of "Rational points on Elliptic curves", Silverman, page 24 writes
The transformations we used to put the curve in normalized form do not map straight lines to straight lines. Since we defined the group law on our curve using lines connecting points, it is not at all clear that our transformation preserves the structure of the group. (That is, is our trans- transformation a homomorphism?) It is, but that is not at all obvious. The point is that our description of addition of points on the curve is not a good one, because it seems to depend on the way the curve is embedded in the plane. But in fact the addition law is an intrinsic operation which can be described on the curve and is invariant under birational transformation. This follows from basic facts about algebraic curves, but is not so easy (virtually impossible?) to prove simply by manipulating the explicit equations.
This seems really important and really surprising (at least to me)! Is this addressed later on in the book? What would be a good resource to get a at sketch of that proof, or at least to get a sense of why it works.