574 BELL SYSTEM TECHNICAL JOURNAL 



which corresponds to a difference of one limen in the shape of the figure 

 of confusion. When two images, characterized by different shapes of 

 figure of confusion, are judged by a number of observers, the proportion 

 of the observations in which one image is said to be sharper than the 

 other affords a significant measure of the evaluation of the difference 

 between them. When 25 per cent of the observations show that shape 

 A yields a sharper image than shape B, we say that shapes A and B are 

 different by one limen, and that the image A is less sharp than the 

 image B by one liminal unit. All "no choice" or "equally sharp" 

 judgments are distributed equally between the judgments for A and 

 those for B. 



In order to evaluate other than unit differences, we have assumed 

 that a normal error curve describes accurately enough the distribution 

 of sharpness differences in liminal units. Thus, image A is less sharp 

 than image B by two liminal units when it is reported to be sharper 

 than image 5 in 9 per cent of the observations. The difference is 

 three liminal units when it is reported to be sharper in 2 per cent of the 

 observations. Any difference larger than about three liminal units 

 would indicate practically complete agreement that the one image is 

 less sharp than the other, under our experimental conditions. A dis- 

 tribution of this nature was found to hold for sharpness differences 

 resulting from changes in the area of the figure of confusion, as shown 

 in Fig. 2. 



Each shape of figure of confusion was compared with each of the 

 four other selected shapes, and the sharpness differences were expressed 

 in liminal units by the procedure just discussed. A fifth difference, 

 corresponding to a null change, or a shape compared with itself, was 

 presumed to be zero. The average value of these five sharpness differ- 

 ences, averaged in liminal units, measured the relative sharpness of that 

 particular shape with respect to the average sharpness of all five shapes, 

 an unvarying reference. In Fig. 5, the sharpness scales have been 

 shifted so that zero denotes the most preferred one of the shapes, which 

 happened in each case to be the square. 



The sharpness curves are found to be slightly skewed with respect 

 to the logarithm of the width : height ratio, there being a small prefer- 

 ence for figures of confusion whose long dimension is vertical rather 

 than horizontal. This is believed to be the first evidence of an asym- 

 metric requirement on resolution. It suggests the possibility that the 

 square figure might not have been the most preferred, had we tested 

 other shapes nearer to the square than the ones we did use. With a 

 more searching experiment we might have found that the eye prefers 

 resolution in the horizontal direction to be just a little better than in 



