The question that is at the origin of this story is the following: how can we produce an image on a plane surface that can capture the depth and sense of relief that our binocular vision affords us? We need to be aware of the fact that we come to this topic today from a point of view which is the result of several centuries’education about perspective. But before the question came to be seen as concerned with geometry, artists must have come to unterstand the spatial reality of a surface in a way that presents an illusion to the eye. Transposed into geometrical terms, the question becomes: how can we represent three-dimensional space on a plane surface possessing only two dimensions? The choices that came to be made and the conventions that arose, were worked out through a long period of rationalisation of vision. The development has its origins in Ancient Greece and Rome, it became ‘liberated’during the Renaissance, it used geometrical constructions to represent three dimensional objects on the plane, and ended by being a stimulus to geometry itself, in particular in relation to what we call projective geometry. From a simple question asked about an image which appears to have depth, geometers finally drew lessons about the nature of space itself.
This article is aimed at teachers of mathematics for use as a means of introducing a historical perspective into the teaching of mathematics. It also contains exercises to be solved according to ancient and modern methods. The chapter ends with a bibliography which contains, in addition to the historical sources that have been used, a certain number of books recommended for further study of the subject. (ZDM/Mathdi)