[Footnote Z: Since no opportunity was given us to examine Hans's eyes we do not know what their condition is in this respect. Though it would have been interesting to know, it would hardly make any difference in the views presented. If Hans should prove to be either far or near-sighted, then, if we are to make any supposition at all, it would be that the defect could not be very great, since near sightedness exceeding 2 or 3 diopters and far-sightedness exceeding one diopter is seldom found in the case of the horse. According to Mr. von Osten, Hans at one time manifested a tendency to shy easily. Be this as it may, for little could be concluded from it, since in many extremely shy horses, no kind of visual imperfection can be discovered.] But in the case of some of the horses in whom Berlin had seen the phenomenon for which we sought in vain, he himself tells us, the deflection was very slight. In that case, it would appear, no great advantage would be gained along the lines indicated. But even assuming the degree of deflection to be very great, his theory goes to pieces on the very point it was supposed to explain. A concrete example will make this clear. If Mr. von Osten, standing two feet away from the horse, raised his head 1/5 millimeter (which figure by no means represents the extreme values that were obtained), then in the horse's retinal image every point of the man's head would move through a distance of 0.0025 millimeter--assuming the horse's eye to be free from astigmatism and assuming its focal distance to be 25.5 millimeters. If, however, other conditions remaining the same, we presuppose an extreme form of astigmatism, one in which the path of the retinal image is not a straight line, but is deflected into a semicircle, then each point would pass through a distance of nearly 0.004 millimeter. If the sensitive retinal elements have a diameter of 0.002 millimeter (as Berlin, somewhat inexactly, states), then from two to four elements would be stimulated in case there were no astigmatic deflection. But in case the deflection did take place, it would not necessarily involve more elements, as can be seen by making a simple graph; indeed we can imagine cases in which the circuitous path would involve even fewer elements than the straight one. And finally, when the movement which the horse is to perceive, does not occur in a straight line but in the form of a curve, (which will generally be the rule), then the astigmatism will tend in many cases to decrease the curvature of the image's path on the retina, and sometimes even obviate it entirely. In all these cases, on Berlin's own theory, the perception of the movements would be hindered rather than aided.[AA]