# Elliptic Curve Point Addition History

Here are some events which (in my mind) had to take place before humanity "knew" about elliptic curve point addition.

1. Diophantus was using the tangent and secant methods. ~285 a.d.
2. 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?

• Bézout lived from 1730 to 1783. Where do you get 1621 from? – Henning Makholm Apr 14 '17 at 13:47
• Oops! I confused it with 1621 for Bachet's Formula. Fixed. – ctesta01 Apr 14 '17 at 13:49
• The time span from the recognition of Galois's work in group theory (when his papers were published posthumously in 1846) to the Cayley-Bacharach theorem in 1886 does not seem all that long to me. – Lee Mosher Apr 14 '17 at 14:30
• Further to @Lee's point, it was not until around 1880 that a recognizable abstract concept of group (with a group operation that does not necessarily arise from composition of functions) had been generalized out from Galois's use of permutation groups anyway. – Henning Makholm Apr 14 '17 at 18:11
• Thank you for your responses, both. I am not wondering about when the statement that the addition on elliptic curves is a group operation was formulated. I would like to know when it was known that the method of drawing the secant through P and Q to find -(P+Q) and then the vertical line through -(P+Q) would yield the fourth point P+Q. Perhaps someone first stated it as a corollary of Bezout's theorem, but I would like to know who the earliest mathematicians to know this were. – ctesta01 Apr 15 '17 at 0:50