Stokes conjectured in 1880 that an extreme gravity wave on water (or 'wave of greatest height') exists, has sharp crests of included angle 2 pi/3 and has a boundary that is convex between successive crests. These three conjectures have all been proved recently, but by diverse methods that are not conspicuously direct. The present paper proceeds from a first approximate solution of the extreme form of the integral equation due to Nekrasov, to a contraction mapping for a related integral equation that governs a new dependent variable in the space L (2)(0,pi). This method provides: (a) a constructive approach to an extreme wave with the sharp crests predicted by Stokes; and (b) a rather accurate second approximation. However, the method has not led (so far, at least) to the convexity.