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I find it amazing that the geometric construction that underlies the group law for elliptic curves gives rise to a group law.
Q: Who was the first to identify the group law for elliptic curves and, if different, who was first to prove that it is associative?
In the 17th century Bachet and Fermat gave algebraic formulas for doubling a point on a cubic, and Newton showed how to do it in terms of chords and tangents. But that is as far as geometry progressed on its own, from there the path did not go from geometry to the group law, but the other way.
In 1834 Jacobi pointed out a possible connection between cubic curves and elliptic functions, which Eisenstein proved in 1847. Clebsch made it more concrete in 1864 by suggesting parametrizations of cubic curves using elliptic functions (this where the term "elliptic curve" comes from). Weierstrass explicitly linked his addition formula for elliptic functions to the addition of points on cubic curves, and the geometry was studied by Juel in 1890s. Poincare's 1901 paper Sur les Proprietes Arithmetiques des Courbes Algebriques is considered to be the first systematization of the subject, where the chord-and-tangent addition (including the point at infinity as identity) is first shown to be a group law rigorously by modern standards, including associativity.
See Why Ellipses Are Not Elliptic Curves by Rice and Brown, Elliptic curves from Mordell to Diophantus and back by Brown, and History of elliptic curves thread on Math SE.
