ABSTRACT
Greek geometers soon noticed that ‘geometrical’ constructions were not applicable only to plane problems and that ‘constructible’ problems were not simply those that could be solved by means of straightedge and compasses. This important discovery led some mathematicians to investigate the properties of curves other than the circle, in particular conic sections. The story of these constructions has been told so many times that it need not detain us here.1 Let us merely remind ourselves that as early as the fourth century BC conics are called upon in the solution of a threedimensional problem. Menæchmus made use of a parabola and a hyperbola to solve the problem of the duplication of the cube. Was this something unusual or an established practice? Is this a procedure that lays the foundations for the theory of conics? The answers to questions like these are hidden in the mists surrounding the origins of the theory. What concerns us here is to note that from a very early date conic sections were used to construct solutions to three-dimensional problems in geometry.
