beautiful convex ellipse, or elliptic saucer viewed from the under side. Of course these effects are exactly reversed if the combination is made beyond the object, either by the naked eyes or by means of a stereoscope.

Explanation.-It is evident that the projections, or, what amounts to the same thing, the retinal images, of the circles viewed at short distance and at high inclination will not be perfectly concentric ellipses. The spaces between the lines on the nearer or outer edges will be greater because nearer, and will gradually

DIAGRAM SHOWING CAUSE OF THE CONCAVITY. (After Stevens.)

diminish to the farther or inner edges. The inner ellipses will all be a little eccentric toward the lower or inner sides. The central dots of the innermost ellipses will be each nearer the inner edges, and therefore nearer together than the centers of the outer ellipses. Now, by reference to Figs. 52 and 56, pages 148 and 152, it will be seen that these are exactly the conditions for making a concave phantom by convergence and a convex one by use of stereoscope. In Fig. 86 (taken from Stevens) we give a simplified representation of the appearance of concentric circles viewed obliquely at short distance. If these be united by convergence, the phantom is seen to be deeply concave

far too deeply, because the eccentricity is greatly exaggerated. This is a complete explanation of the concavity from side to side.

Experiment 6.-But to complete the explanation, especially of the fore and aft curvature of the phantom saucer, one more experiment is necessary.

It will be observed in the last experiment (5) that with the cards (Fig. 85) in the normal position the phantom saucer lies level, with only its lowest point touching the line ab. If now we tip the cards one way or the other so as to make them inclined to the visual plane, but without altering their mutual relation, the phantom saucer is seen to tip in the contrary way and to a much greater degree. Thus, when the lower side of the cards is lifted up the corresponding edge of the saucer goes down, first to, and then below the line a b, more and more, until when the cards are tipped 25° to 30° the line ab pierces the center of the saucer at right angles. At the same time, the lines C, D, E, F, which in the normal position are coincident, are seen to cross one another at the center of the saucer at an angle of 25° to 30°. If the cards be tipped in the other direction-i. e., the upper end be lifted-then the lower edge of the saucer is lifted correspondingly, but in much greater degree, until it again becomes at right angles to the line a b. It is truly wonderful how sensitive the phantom is to movements of the planes. Such are the facts. Now the explanation.

Explanation. When the planes are in the normal position, the major axes of the uncombined ellipses are nearly vertical to the visual plane-they are really slightly curved (see Fig. 86)—but as soon as the planes are tipped so as to be inclined to the visual plane these major axes become inclined to the vertical lines

ab, ab in opposite directions, so that their upper ends are farther apart than their lower ends. In combining them by convergence it will require more convergence to combine the upper ends, and this end will therefore, in the resulting phantom, seem nearer than the lower end. Fig. 87 is a simplified representation of the position of the two ellipses. If these be combined by convergence it is seen at once that the resulting phantom

FIG. 87.

DIAGRAM TO SHOW THE CAUSE OF THE FORE AND AFT CURVATURE.

is strongly inclined with the upper edge nearer, and that the line ab pierces it in the center almost, if not quite, at right angles. Returning again to the phantom saucer of experiment 6, if we push the inclination of the cards still farther, the inclination of the long axes of the ellipses becomes too great to combine readily. We are plagued by a too obvious doubling of the images of the two ellipses; but the cause of the phenomenon of the tipping of the saucer-viz., the inclination of the two ellipses-becomes at once evident.

Now if we return to the fifth experiment, it becomes evident that the true cause of the fore and aft concavity

FIG. 88.

a

of the phantom saucer is to be found in this last experiment. The slightest inclination of the visual plane to the phantom plane causes a tipping in a contrary sense of the figures on the plane and to a much greater degree. Now the visual plane is at right angles only at the center of the saucer and is inclined in opposite directions above and below. Therefore the saucer is lifted both above and below, and is therefore concave fore and aft. This is implied in the figure of Prof. Stevens (Fig. 86) especially in the curved lines A CB, A' C' B', but is not explicitly stated in his paper. The two smaller circles above and below are of course inclined in opposite directions, and have been added only to make this clear. The position of all the circles in the phantom when viewed in the normal position, as in experiment 5, is shown in section in Fig. 88. The line ab touches the middle figure and pierces the other two.

b

SECTION OF BI

NOCULAR COM

BINATION OF
FIG. 85.

CHAPTER IV.

ON SOME FUNDAMENTAL PHENOMENA OF BINOCULAR VISION USUALLY OVERLOOKED, AND ON A NEW MODE OF DIAGRAMMATIC REPRESENTATION FOUNDED THEREON.*

nomena.

FIG. 89.

In all that I have said thus far, I have made use of the ordinary mode of representing binocular visual pheI have done so because I could thus make myself more easily understood. But it is evident on a little reflection that the usual diagrams do not in any case represent the real visual facts-i. e., the facts as they really seem to the binocular observer.

C

Thus, for example, if a, B, and c, Fig. 89, be three objects in the median plane, but at different distances, and the two eyes, R and L, be converged on B; as already explained, a and c will be both seen double-the former heteronymously, the latter homonymously. It will be observed that in the diagram the double images of both a and care referred to the plane of sight PP. Now every one who has ever tried the experiment knows that the double images are not thus referred in natural

*This new mode was proposed by me in 1870.-Am. Jour., vol. i, p. 33, 1871. Some years afterward I learned that something of the same kind had been previously proposed by Hering.