Dürer's Paradox or Why an Ellipse Is Not Egg-Shaped

Abstract

The drawing of the cone and the ellipse is taken from the 1525 work Treatise on Mensuration with the Compass and Ruler in Lines, Planes, and Whole Bodies by the German artist and mathematician Albrecht Diirer (1471-1528). Two things are likely to attract the attention of the viewer: namely, the mass of lines and arcs and-to use Diurer's own expression-the egg-curve. The purpose of this article is threefold: to describe Diurer's method, to explain why the use of the method might lead one to believe that the ellipse is egg-shaped, and to show how the analytic version of Diirer's method can be used to derive the Cartesian form of the ellipse. I have also included some historical material and suggestions for further reading in a separate section at the end of the article. The method used by Diirer is essentially equivalent to what is now called descriptive geometry. At present it is only employed to obtain graphical solutions of complicated geometrical problems such as the intersection of surfaces, and as far as I know this branch of mathematics is now taught only in engineering drawing courses. However the French geometer Gaspard Monge (1746-1818), whose Geiome'trie descriptive provided the systematic development of the subject, wrote [11, p. 1]: [One of the aims of descriptive geometry] is to give means of recognizing, based on an exact description, the forms of bodies and to deduce all the truths which are implied by their form and their respective positions. Descriptive geometry is concerned with the representation of bodies and surfaces in space by means of two-dimensional orthogonal projections. Consider for example FIGURE 1 in which a point P is shown as being a units in front of a vertical plane and b units above a horizontal plane. If we project the point P onto these planes by means of perpendicular lines then the two projections pv and pH are both determined. To avoid constant awkward repetition and notation, a point and its projections will be referred to by a single letter without super and subscripts. If the horizontal plane is now folded back about the linethe line of intersection of the two planes -until the horizontal plane is also in a vertical plane, then we will have the situation of FIGURE 2. Here the point P is represented by two two-dimensional drawings which are such that the two projections lie on a line perpendicular to the folding line. Furthermore given the two-dimensional drawings, which are usually referred to as the front and top views, then the position in space-relative to the two planes-of the point P is completely determined. The folding line is not really necessary and if we are only interested in the relative position of various points with respect to one