Before revision:  First , making use the Schwarz ODE, we develop an analytic method to determine a conformal mapping the unit disk onto a given circular quadrilateral. We devise a numerical method to find  the accessory parameters , solve the equation and to compute the modulus of the quadrilateral.
After revision:  First , making use the Schwarz ODE, we develop an analytic approach which reduces this problem to ODEs and devise a numeric method to find the accessory parameters , solve the equation and to compute the modulus of the quadrilateral.