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spherical coördinates would project on a vertical wall. By calculation or by careful plotting it may be shown that at an angle of elevation or depression of 40°, and a lateral angle of the same amount, the inclination of the hyperbolic curve on the horizontals of the wall will be about 20°. Now a rectangular cross-image, if unrotated, would project as the crosses in the corners ; i. e., the vertical arm would project vertically, but the horizontal arm would be inclined 20° with the horizontal, so that the angles of the cross would be about 70° and

110°. Now rotate these crosses 15°, the Fig. 57.

right upper one to the right, the left upper one to the left, the right lower to the left, and the left lower to the right, and we have the precise phenomena represented by the diagram Fig. 55; i. e., the verticals are turned 15° right or left as the case may be, and the horizontals in

the opposite direction, but only 5o. Fig. 57 illustrates this in the case of the right-hand upper cross-image—the heavy cross representing the cross unrotated, and the lighter one the same rotated 15° to the right by extreme obliquity of the line of sight.

Therefore, the diagram which truly represents the torsion of the eye in various positions, or the torsion of the cross-image when referred to a spherical concave perpendicular to the line of sight in every position, is represented in Fig. 58. Simple inspection of this figure shows the real direction and amount of rotation both of the vertical and the horizontal image for every position of the line of sight. The crosses in the corners show that there is no distortion by projection.

We are justified therefore in formulating the laws of parallel motion of the eyes thus :

1. When the eyes move together in the primary plane to the one side or the other, or in a vertical plane up or down, there is no rotation on the optic axes, or torsion.

Fe. 58.

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DIAGRAM SHOWING THE TRUE TORSION OF TIE EYE FOB VARIOUS POSITIONS OP

THE POINT OF SIGHT.

2. When the visual plane is elevated and the eyes move to the right, they rotate to the right; when they move to the left, they rotate to the left.

3. When the visual plane is depressed, motion of the eyes to the right is accompanied with rotation to the left, and motion to the left with rotation to the right.

4. These laws may be all generalized into one, viz.: When the vertical and lateral angles have the same

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sign,

* the rotation is positive (to the right); when they have contrary signs, the rotation is negative (to the left).

The law now announced as the result of experiment is evidently identical with the law of Listing, which has been formulated by Listing himself thus :

When the line of sight passes from the primary position to any other position, the angle of torsion of the eye in its second position is the same as if the eye had come to this second position by turning about a fixed axis perpendicular both to the first and the second position of the line of sight.+

Now an axis which satisfies these conditions can be none other than an equatorial axis, or at least an axis in a plane perpendicular to the polar axis. In turning from side to side in the primary plane, it is a vertical equatorial axis. In turning up and down vertically, it is a horizontal equatorial axis. In turning obliquely, as in the experiments on torsion, it is an oblique equatorial axis. Now take a globe, and, placing the equator in a vertical plane, make a distinct vertical and horizontal mark across the pole. Then turn the globe on an oblique equatorial axis, so that the pole shall look upward and to the right. It will be seen that the polar cross is no longer vertical and horizontal, but is rotated to the right. If the globe be turned upward and to the left, the polar cross will rotate to the left; if downward and to the right, it will rotate to the left; and if to the left, it will rotate to the right. In a word, the rotation in every case is the same as given in the above laws determined by experiment.

* In reference to a vertical line, positions to the right are positive and to the left negative; in reference to a horizontal line, above is positive and below negative.

+ Helmholtz, “Optiquc Physiologique,” p. 606.

Contrary Statement by Helmholtz.—We have given these laws and their experimental proof in some detail, and have taken some pains to show that they are in complete accord with Listing's law, because Helmholtz in his great work on “Physiological Optics” has given these laws of ocular motion the very reverse of mine. I quote from the French edition of 1867, which is not only the latest but also the most authoritative edition of the work:

“When the plane of sight is directed upward, lateral displacements to the right make the eye turn to the left, and displacements to the left make it turn to the right.

“When the plane of sight is depressed, lateral displacements to the right are accompanied with torsion to the right, and vice versa.

“In other words, when the vertical and lateral angles are both of the same sign, the torsion is negative; when they are of contrary signs, the torsion is positive."' *

We have demonstrated the very reverse of every one of these propositions, and we have also shown that they are inconsistent with Listing's law as quoted by Helmholtz himself. The experiments by which Helmholtz seeks to determine the torsions of the eye are the same as those already described under experiments 1 and 2, page 165. The results which he reaches are also the same as those reached by myself, except that he makes the inclination of the vertical image on the verticals of the wall, and of the horizontal image on the horizontals of the wall, equal to each other, while I make the inclination of the verticals much greater. The diagram by which he embodies all these results is also similar to my diagram, Fig. 55, except that in his the horizontal and

*“Optique Physiologique,” p. 602.

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vertical curves are exactly similar, while in mine the curves of the verticals are much greater. He also, like myself, admits that there is a fallacy by projection. But unaccountably he imagines that the inclination of the horizontal image on the true horizontal gives true results, and the inclination of the vertical image on the true vertical deceptive results by projection; therefore he imagines the eye to turn exactly the reverse of the reality. Experiments 5 and 6, under conditions eliminating errors by projection, prove the falseness of his results. The reader who desires to follow up this subject will find it discussed in an article by the writer referred to below.*

The Rotation only Apparent.—There can be no doubt, then, that when the eye passes from its primary position to an oblique position, the vertical meridian of the retina is no longer vertical, but inclined. If we observed Fig. 59.

the iris of another person, we should see that it had turned as a wheel. In deference to the usage of other writers and to the appearance, I have spoken of this as a rotation on the optic axis, but it is so in appearance only, and not in reality; for the motion of the eye, being

always on an axis in a plane perpendicular to the polar or optic axis, can not be resolved into a rotation about that axis. A simple experiment will show the kind of rotation which takes

*“American Journal of Science and Arts,” III, vol. xx, 1880, p. 83. .

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