 This was the case of like middle terms. He permitted some of the parameters \(v, v', w, w'\) to be replaced by 1, but not both \(v, v'\) and not both \(w, w'\) can be replaced by 1. One could also replace \(y\) by \(1-y\) in both equations, and independently  replace \(x\) by \(1-x\) and \(z\) by \(1-z\). The premises of many categorical syllogisms can be expressed in this form. His goal was to eliminate \(y\) and find expressions for \(x, 1-x\) and \(vx\) in terms of \(z, v, v', w, w'\). 