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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 21°. 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 17° 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
and to the left eye thus
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. 75), 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
DIAGRAM SHOWING VERTICALS INCLINED - (Taken from Helmholtz.)
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 11° 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
other, so that the real vertical meridians coincide, we have already seen that Helmholtz's apparent verticals or lines of deniarkation will cross each other like an X, as in Fig. 76, 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 Retinæ SUPERPOSED
. feet, or the rod lying along the ground
to infinite distance, would occupy with tion of right eye; ll, line of demarkation of their images only the upper half of
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
-rr, line of demarka
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 posi
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 even probable, that the same habits in early life which, by constant adapting of the eyes to vision of near objects, finally produce myopy, may also, by constant slight rotation of the eyes outward and distortion * in convergence on near objects, finally bring about a permanent condition of slight distortion and outward rotation of 17° Helmholtz is slightly myopic.t
* Op. cit., p. 923.
However this may be, I am sure there is no such relation between real and apparent vertical meridian in my eyes as that spoken of by Helmholtz. All the experiments supposed to prove such relation fail completely with me. A vertical rectangular cross appears rectangular to either eye. The lines of Helmholtz's diagram, Fig. 75, when combined beyond the plane of the diagram, either by the naked eyes or by a stereoscope, do not come together parallel, but with a decided angle, viz., 21°. But when I turn the diagram upside down, and combine by squinting, then the vertical lines, being inclined the other way, as in my diagram, Fig. 68, combine perfectly by outward rotation of the eyes. I have constructed other diagrams with less and less inclination of the verticals, until the inclination was only 10', and still I detected the want of parallelism when combined beyond the plane of the diagram. Beyond this limit I could not detect it, but I believe only because the limit of perception was passed; for when the lines are made perfectly vertical, they come together perfectly parallel and unite absolutely. It is certain, therefore, that in my eyes the vertical line of demarkation coincides completely with the true vertical meridian.
Meissner & alone, of all writers with whom I am ac
* Simple rotation is not sufficient, because this would affect also the horizontal meridian.
+ Op. cit, p. 914. # Meissner, “Physiologie des Sehorgans”; also “Archives des Sciences,” vol. jii (1858), p. 160.