What was the lift equation used by the Wright brothers?
The Wright brothers' equation is
$$L=kV^2AC_L.$$
where $C_L$ is the ratio of the $L/D$ of a specific wing (AoA, span, chord, camber, thickness all set and fixed) to a perpendicular board into the wind of the same $A$, $k$ is experimentally obtained, and is specific only to the medium (air or water).

In a sense, the Wright's lift equation is seeing lift from a momentum transfer perspective, i.e. the $kV^2$ term corresponds to the momentum flux of the incoming air, and the term $AC_L$ describes how much downward discharge is produced at the price of how much drag.

While the modern lift equation is 
$$L=\frac{1}{2}\rho V^2AC_L.$$
This is derived from the Kutta-Joukowski theorem:
$$L=\rho V\Gamma,$$
where $\Gamma=\int_C{\bf v}\,\mathrm{d}{\bf s}$ is a line integral of the second type around a close and simple route around the wing. $\Gamma$ is proportional to $V$. And $C_L$ is defined as
$$C_L=\frac{L}{\frac{1}{2}\rho V^2A}.$$ $C_L$ is not calculated through analytics, but derived through $A,\rho,V$ as obtained in experiments.