The title ofthis essay is borrowed from a modern mathematical historian; its tag line is taken from an ancient philosopher. Their shared interest in questions dealing with existence has given rise to a familiar thesis about ancient geometry: that its constructions were intended to serve as proofs ofthe existence ofthe constructed figures. I propose here to examine that thesis, to argue its weakness as a historical account of ancient geon1etry, and to ofTer an alternative view ofthe role ofproblems ofconstruction: I that constructions, far from being assigned a specifically existential role, were not even the commonl) adopted format for treating of existential issues when these arose; that some central questions relating to existence were handled through postulates or tacit assumptions, rather than through explicit constructions; that, by contrast, when constructions were given, the motive lay in their intrinsic interest for the ancient geometers. On this basis I will maintain that preconceptions based on modem theories have interfered in the modem effort to interpret ancient nlathematics, thus attaching to the existential view of constructions a greater credence than the ancient evidence could justify.