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eyes look horizontally over the plane on one peg, and the others are arranged in such wise that they appear single. It is found that they must be arranged in a circle. I have tried repeatedly, but in vain, to verify this result. The difficulty is the extreme indistinctness of perception at any appreciable distance from the point of sight. But, as a general fact, the results reached by the observers thus far mentioned have been reached by the most refined mathematical calculations, based on certain premises concerning the position of corresponding points and on the laws of ocular motion. We will examine only those of Helmholtz, as being the latest and most authoritative.
Helmholtz's results are based upon the law of Listing as governing all the motions of the eye, and upon his own peculiar views concerning the relation between what he calls the apparent and the real vertical meridian of the retina. The real vertical meridian of the eye is the line traced on the retina by the image of a really vertical linear object when the median plane of the head is vertical and the eye in the primary position. The apparent vertical meridian of the eye is the line traced by the image of an apparently vertical linear object in the same position of the eye. This is also called the vertical line of demarkation, because it divides the retina into two halves which correspond each to each and point for point. Now, according to Helmholtz, the apparent vertical meridian or vertical line of demarkation does not coincide with the real vertical meridian, but makes with it in each eye an angle of 11°, and therefore with one another in the two eyes of 21°. The horizontal meridians of the eyes, both real and apparent, coincide completely. Therefore, if the two eyes were brought together in such wise that their
real vertical and horizontal meridians should coincide, their apparent horizontal meridians would also coincide; but the apparent vertical meridians would cross
each other at the central spot thus
an angle of 2°. For this reason a perfectly vertical line will appear to the right eye not vertical, but inclined to the left, and to the left eye inclined to the right. In order that a line shall appear perfectly vertical to one eye, it must incline for the right eye 11° to the right, and for the left 11° to the left. But a horizontal line appears truly horizontal. Therefore an upright rectangular cross will appear to the right eye thus
and to the left eye thus The inclination of these lines is, however, exaggerated. If, therefore, according to Helmholtz, we make a diagram of which one half is composed of black lines on white ground, and the other of white lines on black ground, like those already used, but in which, while the horizontals run straight across horizontally, the verticals on the right half are inclined 11° to the right, and on the left half the same amount to the left (Fig. 68), then, on combining these by gazing beyond the plane of the diagram (i. e., with parallel eyes), either with the naked eye or with the stereoscope, the verticals will be seen to come together parallel and unite perfectly.
Now Helmholtz's views of the form of the horopter are based wholly on this supposed relation of real and apparent vertical. Take for example his case of the eyes fixed on a distant point on the horizon. In this case, he says, “the horopter is the ground on which we stand.” This is true if the relation above mentioned is
true; for, with an interocular distance of 24 inches, two lines drawn through the optic centers, each inclined 11° with the vertical and therefore 21° with each other,
would in fact meet about 5 feet below-i. e., about the feet. If, therefore, we place two actual rods together on the ground between the feet, and the upper ends before the pupils, the eyes being parallel, it is evident that the image of the right rod on the right retina and that of the left rod on the left retina would fall exactly on Helmholtz's apparent vertical meridian, and, if Helmholtz's views be correct, on the vertical lines of demarkation and on corresponding points of the retinæ, and thus would be binocularly combined and seen as a single line lying along the ground to infinite distance. And conversely, with the eyes parallel and the lines of demarkation inclined 17° with the vertical, a rod lying on the ground to infinite distance would cast its images on these lines, and therefore be seen single throughout.
There are several curious questions which force themselves on our attention here if Helmholtz's view be true.
1. If we suppose the two eyes to be placed one on the FIG. 69.
other, so that the real vertical meridians coincide, we have already seen that Helmholtz's apparent verticals or lines of demarkation will cross each other like an X, as in Fig. 69, making with each other an angle of 21°. Now the two rods 24 inches apart at the height
of the eyes, and meeting below at the THE RETINE SUPERPOSED. feet, or the rod lying along the ground
q to infinite distance, would occupy with tion of right eye; 1l, line of demarkation of their images only the upper half of left eye.
the X. But suppose the two rods, instead of stopping opposite the eyes, to continue upward to the limits of the field of view. Obviously this upper half would cast images on the lower half of the X, and therefore would be seen single also. Where shall we
refer them? Or, to express it differently, the horopter with the eyes looking at a distant horizon, according to Helmholtz, is the ground we stand on; but this is evidently pictured on the upper halves only of the two retinæ. Where is the other half of the horopter corresponding to the lower halves of the retinæ ?
2. Again : According to Helmholtz, in looking at a distance the horopter is the ground we stand on, and he gives this as the reason why distance along the ground is more clearly perceived than in other positions.* On the contrary, it seems to me that it would have just the reverse effect. If the horopter were the ground we stand on, then relative distances on the ground could not be perceived by binocular perspective at all; for this is wholly dependent on the existence of double images, which could not occur in this case by the definition of the horopter. It would be therefore only by other forms of perspective that we could distinguish relative distance along the ground. But that we do perceive perspective of the ground binocularly-i. e., by double images—is proved by the fact that the perspective of the receding ground is very perfect in stereoscopic pictures, where the images of nearer points are necessarily double; for the camera has no such distinction between real and apparent verticality as Helmholtz attributes to the eye.
But it is useless to argue the point any further, for I am quite sure that the property which Helmholtz finds in his eye is not general, and therefore not normal. We have seen that in convergence the eyes rotate outward, so as to bring about the very condition of things temporarily which Helmholtz finds permanent in his eyes. I have therefore thought it possible, or
* Op. cit., p. 923.