I'm working on a long form piece on the development of elliptic curve cryptography, basically tracing the use of a point at infinity back to the discovery of the rules for linear perspective in the 1400s.
As part of my research, I'm delving into the relationship between elliptic integrals, functions and curves--both in terms of the history of these subjects as well as the mathematical connections between them.
One of the aspects I've come across involves the development of elliptic functions from the study of general addition theorems for integrals (as in Stillwell's Mathematics and its History, 12.4, 12.5 and Grattan-Guiness's Rainbow of Mathematics, 8.13).
Additionally, notes from a talk Harold M. Edwards gave at the Joint Mathematics Meetings in Baltimore in 2014 contain this statement:
Specifically, the algebraic relation among them is such that both coordinates of the point on the curve where $x = r$ can be expressed rationally in terms of the coordinates of the points where $x = p$ and $x = q$.
...
This statement extends immediately to sums of three or more integrals. For example,
$$\int_0^p \frac{dx}{y} + \int_0^q \frac{dx}{y} + \int_0^r \frac{dx}{y} = \int_0^s \frac{dx}{y}$$
where the coordinates of the point on the curve $y^2 = f(x)$* where $x > = s$ can be expressed rationally in terms of the coordinates of the points where $x = p, q,$ and $r$ (source)
*Elsewhere, Edwards states that $f(x)$ must be a polynomial of degree 4, with distinct roots and $f(0) \neq 0$
My question is whether the addition defined among, say, the group of rational points on an elliptic curve can be (is??) derived from the addition of elliptic integrals described by Euler and others.