that the inclination or torsion of the vertical image on the true verticals, and that of the horizontal image on the true horizontals, are in opposite directions. If torsion

FIG. 55.

DIAGRAM SHOWING THE INCLINATION OF VERTICAL AND HORIZONTAL IMAGES FOR ALL POSITIONS OF THE POINT OF SIGHT.

of the images show torsion of the eye, there must be a fallacy somewhere. The one or the other must be wrong; for when one indicates torsion to the right, the other indicates torsion to the left, and vice versa. To show this contradictory testimony more clearly, and thus to prove that there is a fallacy here, we make another experiment.

Experiment 4.-Make a rectangular cross-slit in the window, gaze steadily upon it until the spectral impres

sion is made on the retina, and then cast the image on the wall. In the primary position of the eyes it is of course a perfect rectangular cross. Now turn the eyes to the extreme upper right-hand corner of the wall. The cross, by opposite rotations of the two parts, is seen distorted

manner in which the lines cross in the diagram, and we have placed crosses in the corners to indicate that fact.

Evidently the cause of the contradictory evidence of the two images is projection on a plane inclined at various angles to the line of sight. The diagram is a correct representation of the phenomena as seen projected on a vertical plane, but is not a correct representation of the torsions of the eyes. To eliminate this source of fallacy and get the true torsion of the eyes, we must project the cross-image on a plane in every case perpendicular to the line of sight.

Experiment 5.-Prepare an experimental plane a yard square, make a rectangular cross in the center, and set up a perfectly perpendicular rod at the point of crossing. Fix the plane in a position inclined 30° to 40° with the vertical, and obliquely to the right side and above, so that, when sitting before the experimental window and turning the eyes extremely upward and

to the right, the observer looks directly on the top of the rod, and this latter is projected against the plane as a round spot. We thus know that the line of sight is perpendicular to the plane. Now, after gazing at the cross-slit in the window until the spectral impression is made on the retina, without moving the head, cast the image on the center of the plane by turning the eyes obliquely upward and to the right. The rectangular cross-image rotates, both parts alike, so as to retain perfectly its rectangular symmetry, to the right, thus—

, showing unmistakably a torsion of the eyes in the same direction. If the plane be arranged similarly on the left side, the cross turns to the left, thus

If

the plane be arranged below and to the right, so that the eyes turned obliquely downward and to the right shall look perpendicularly upon it, the cross will turn

left side, the cross will turn to the right, thus—

In all cases the rectangular symmetry is perfectly preserved, a sure sign that there is no error by projection, and that they truly represent the torsion of the eyes.

Experiment 6.-In order to neglect no means of testing the truth of this conclusion, we will make one more experiment, using the sky as the plane upon which to project the image. This spatial concave is of course everywhere at right angles to the line of sight, and therefore is free from any suspicion of error from projection. Standing in the open air before a vertical

flag-staff, I gaze upon it steadily until its image is, as it were, burned into the vertical meridian of the retina. Now, without moving the head, I turn the eyes obliquely upward and to the right, and the image leans decidedly to the right; and turning to the left, the image leans to the left. In this position of the head, of course, the ground prevents us from making the same experiment with the visual plane depressed. I therefore vary the experiment slightly. Sitting directly in front of the college building, with the morning sun shining obliquely on its face, the light-colored perpendicular pilasters gleam in the sunshine, and contrast strongly with the shadows which border their northern margin. Gazing steadily at the building, I easily get a strong spectral image of the whole structure, with its vertical and its horizontal lines. Now throwing myself flat on my back, I see the image perfectly erect on the zenith. Turning the eyes upward toward the brows and to the right and left, then downward toward the feet and to the right and left, the whole image of the building rotates precisely as indicated in my previous experiments.

Evidently, then, in the diagram Fig. 55, the verticals give true results, but the horizontals deceptive results by projection. Why this is so is easily explained. Suppose an observer to stand in a room before a vertical wall; suppose him further to be surrounded by a spherical wire cage constructed of rectangular spherical coordinates, or meridians and parallels, with the eye in the center and the pole in the zenith. Evidently, the surface of this spherical concave is everywhere perpendicular to the line of sight, and therefore, like the sky, is the proper surface of projection. Evidently, also, the meridians and parallels everywhere at right angles to

each other are the true coördinates wherewith to compare the images, vertical and horizontal, in order to determine the direction and amount of their rotation. Now the simple question is, "How do these true rectangular coördinates project themselves on the wall to an eye placed in the center, or how would their shad

DIAGRAM SHOWING THE PROJECTION OF A SYSTEM OF SPHERICAL COÖRDINATES ON A VERTICAL PLANE.

ows be cast by a light in the center?" It is evident that the meridians would project as straight verticals, but the parallels not as straight lines, but as hyperbolic curves, increasing in curvature as we go upward or downward. The diagram Fig. 56 shows how the