96 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



a suggestion has been made already in this direction (Householder, 

 1940; cf. chap ix). 



Perhaps the most obvious cue for judging distance is the "ap- 

 parent size" of the object when the actual size is known or inferred 

 from previous experience. The "apparent size" is by definition the 

 solid angle subtended, but is not necessarily the size the object ap- 

 pears to have. Thus a distant man appears small, but when he is 

 fairly close the size he appears to have stays fairly constant while 

 his distance, and hence his "apparent size," varies over quite a wide 

 range. The "apparent size" is proportional to the size of the retinal 

 image and provides a distance cue. 



An object seen through a spyglass appears flattened and a pos- 

 sible explanation can be found from considerations of apparent size. 

 Thus consider a cube with one edge nearly, but not quite, sagittal in 

 direction, and suppose it is viewed through a spyglass magnifying in 

 the ratio M = 1 + u . That is, let the retinal image of the cube as 

 seen through the spyglass be M times that formed by the cube seen 

 at the same distance by the naked eye. If the actual distance of the 

 front face is d , and if the edge of the cube has length s , then the 

 back face has the actual distance d + s , but due to the magnification, 

 front and back faces appear to be only 1/M times as far. They ap- 

 pear, therefore, to have the distances d/M and (d + s)/M . But this 

 leaves for the apparent depth of the cube only the distance s/M . 



While it is well known that qualitatively the effect is present as 

 described, no quantitative data are at hand, and the theory here sug- 

 gested might fail to meet the more exacting requirements of a quan- 

 titative test. In the first place, it is assumed that, in the absence of 

 other cues, the perceived distance would be exactly 1/M times the 

 actual distance. This might not be the case. The percieved distance 

 might be, say 1/li times the true distance, where /u < M . But if so, 

 we should expect the judged depth to be 1/// times the actual depth 

 and the judged size to be M/li times the actual size. Moreover one 

 could perform an experiment in which a truncated pyramid is pre- 

 sented instead of a cube, the dimensions being such that the subject 

 would be expected to perceive it as a cube. For this, when the dis- 

 tance is judged to be 1/// times the actual distance d and the magnifi- 

 cation is M times, the retinal image is the same as would be produced 

 by a cube seen with the naked eye at a distance of dj ii . Now if a 

 cube whose edges are Ms/ t u is placed so that the nearest face is at a 

 distance of d/ n , then the visual angles subtended by an edge of the 

 front face and an edge of the back face are Ms/d- and Ms/ (d + Ms). 

 These must be the visual angles of the faces of the truncated pyra- 

 mid placed at a distance d and magnified M times. Hence, if the 



