The relation of apparent shape to apparent slant in the perception of objects.

The problem of how we perceive the shape of an object in space is complicated by the optical fact that to a particular retinal form there does not correspond a unique physical form but a whole set of possible physical forms. A given sheaf of light rays whose cross section is a particular form may be reflected from any surface whose margins coincide with the margins of the sheaf of rays; the surface does not have to be perpendicular but may be inclined to the sheaf. This set of surfaces is, mathematically, the family of perspective transformations of the cross section. A square cross section, for example, may arise from a family of variously inclined trapezoids. A trapezoidal cross section may arise not only from an inclined square but also from a different family of inclined trapezoids, of which the square is only a special case. The optical fact seems to contradict the possibility of a correspondence between variations of the retinal image (the proximal stimulus) and variations of the object (the distal 1 This paper is a revision of a thesis presented