  Though (5)’s antecedent and consequent both state something impossible, Vaihinger contends it forms the basis of Archimedes’s proof of the area of a circle using the method of exhaustion (PAO 177). Here, then, is a case in which a proof no one would wish to call trivial allegedly proceeds from a contradictory assumption. Generalizing, Vaihinger claims, “in a hypothetical connection [sc. conditional proposition], not only real and possible but also unreal and impossible things can be introduced, because it is merely the connection between the two presuppositions and not their actual reality that is being expressed” (PAO 193–94).