Here are some events which (in my mind) had to take place before humanity "knew" about elliptic curve point addition.
- Diophantus was using the tangent and secant methods. ~285 a.d.
- Bezout's Theorem was stated (nearly) by Newton and later by Bezout in 1779.
Bezout's Theorem says that two algebraic plane curves with no common component whose greatest common divisor polynomial is a constant have a number of intersection points counting multiplicity equal to the product of their degrees.
Given this, we can know that elliptic curve addition as it is defined in modern context will yield a third point. However, my question is this -- when was this realized? When was it realized that this addition operation is well defined? Who was the first to use it with claim that addition would always yield a third point? Were there any mathematicians who remarked on this as significant progress, or was it brushed off as an easy corollary to Bezout's Theorem?
I know that in order to justify the operation's status as a group operation, it took a long time to get to the Cayley–Bacharach (1886) theorem to prove the operation's associativity, but before that surely at some point between 1779 and 1886 people began to realize very concretely that addition on an elliptic curve was significant.
Could anybody link me to (english) resources which shed some light on what mathematicians knew about this in this historical interval?