What was the lift equation used by the Wright brothers? NOTICE: The modern lift equation and the lift equation used by the
  Wright brothers in 1900 are slightly different. The lift coefficient
  of the modern equation is referenced to the dynamic pressure of the
  flow, while the lift coefficient of the earlier times was referenced
  to the drag of an equivalent flat plate. So the value of these two
  coefficients would be different even for the same wing and the same
  set of flow conditions.


Above is a quote from a NASA website (emphasis mine). It's new for me that the Wright brothers applied different calculations to their plane. So, what was the equation used by the two 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.