100 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



with no parameters undetermined once the rectilinear horopter has 

 been located (Householder, 1940). 



If the geometric analysis is carried further one can see that on 

 placing" before one eye, say the left, a cylindrical size-lens, axis 90°, 

 the horopter should undergo a rotation with the result that the en- 

 tire visual field would appear to have undergone a rotation in a clock- 

 wise direction. This is indeed borne out by experiment, and the amount 

 of the rotation is, as predicted, proportional to the increment in size 

 (Ogle, 1938). 



Now so far as the horopter is concerned there is no reason to 

 suppose that any shift of the subjective space should occur when the 

 cylindrical size-lens is placed with axis 180° instead of 90°. Never- 

 theless, it turns out that there is a shift, of approximately the same 

 amount but in the opposite direction (Ogle, 1940). But none of the 

 cues so far mentioned can account for this, so another one must be 

 at work. 



The experimental procedure here is to set up before the observer 

 a plane which is free to rotate about a vertical axis and which con- 

 tains a few small circles for fusion arranged in two horizontal rows. 

 The subject is asked to adjust the plane until it is parallel to his own 

 frontal plane. Best results are obtained when the fusion-contours 

 are "restricted to relatively small areas above or below the center of 

 the plane" (Ogle, 1940). It is found that the rotation is approxi- 

 mately proportional to the magnification-increment when this is not 

 too great, but that the effect breaks down when the magnification 

 exceeds a certain critical amount. 



It is evident that the vertical disparities, rather than the hori- 

 zontal disparities, are producing the effect, and this fact, together 

 with the fact of the breakdown of the effect for higher magnifica- 

 tions, suggests that the subjects are locating the medial plane rather 

 than the frontal plane. Normally the medial plane would be the locus 

 of objects whose retinal images are equal in the vertical direction, 

 and those which produce unequal images would be on the same side 

 of the medial plane as the eye having the larger image. Normally, 

 too, the medial and the frontal planes are perpendicular. But if a 

 lens giving vertical magnification were placed before the left eye, and 

 if the magnification were not too great, the points yielding equal 

 retinal images would be to the right of the true medial plane. On the 

 basis of this cue any object would therefore appear to lie closer to 

 the left eye and farther from the right than it actually does. But 

 when the magnification is great enough, any object whatever forms 

 a larger image on the left retina, and in this extreme situation, if 

 not sooner, the localization would be accomplished by means of other 



