see the fingers as in Fig. 49, I; if we shut the right eye, we see them as in Fig. 49, II. Now these two can not be combined, because they are different. When we combine the images of the farther fingers, a and a', the b and b', will not have come together yet, and will therefore be heteronymously double, as in Fig. 50, I;

nearer,

when by greater convergence we combine the images b and b' of the nearer finger, then the images a and a' of the farther will have crossed over and become homonymously double, as in Fig. 50, II. As in previous experiments, double images are given in dotted outline because transparent, and left-eye images are marked with primed letters, and combined images with capitals.

Now, in this experiment we are distinctly conscious of a greater convergence of the optic axes necessary to combine the double images of the nearer finger, and of a less convergence to combine the double images of the farther. Thus the eyes range back and forth by greater and less convergence, combining the double images of the one and the other by transferring the point of sight from one to the other; and thus we acquire a distinct perception of distance between the two. It is literally a rapid process of triangulation, the base-line being the interocular distance.

Thus far we have explained the perception of depth of space between separate objects lying one beyond the other. We now take the case of a single object occupying considerable depth of space in the line of sight.

Experiment 2.-We take a rod about a foot long, and hold it in the median plane a little below the horizontal plane passing through the eyes, so that we can see along its upper edge, the nearer end about six or eight inches from the eyes. If now, shutting the left eye, we observe the projection of the rod against the

b

wall, it will be like this—«. -a being the nearer and the farther end. If we shut the right eye and open the left, the projection will be like this-"\

These lines are exactly like the retinal images formed by the rod in the right and left eyes respectively, except that these images are inverted. Or, to express it differently, these lines would make images on the right and left retinæ respectively exactly like those made by the rod; they are the facsimiles of the external images of the rod. If we now open both eyes and fix attention on the farther end, then the nearer end will be seen double heteronymously, and the projection will be

thus

B

If, on the contrary, we look at the

nearer end, then this of course will be single, but the farther end will now be double homonymously, and

the projection will be thus

b

If, finally, we

look at the middle point, this point will of course be scen single, but both ends double, the one homonymously, the other heteronymously, and the projection will be thus- X Or, to put it differently, the

b'

a

external images of the rod belonging to the two eyes respectively are like these lines and

a

a'

: if these two be brought together so as to unite the farther ends bb', then by greater convergence the middle points, and then by still greater convergence the nearer ends a a', the three projections above given are obtained; but it is obviously impossible to unite all parts and see single the whole rod at once. Now, if we observe attentively, we find that in looking at the rod the eyes range back and forth by greater or less convergence, uniting successively the different parts, and thus acquire a distinct perception of the difference of distance or depth of space between the nearer and the farther end.

Experiment 3.-We take next a small thin book, and hold it as before six to eight inches distant in the median plane, a little below the horizontal plane of sight, so that the back and the upper edge are visible. If we shut the left eye, we see the back, the upper edge, and the whole right side, thusThe retinal image

formed in the right eye is exactly like this figure, except that it is inverted; this figure makes exactly the same retinal image as the book does in the right eye; it is the facsimile of the external image of the book for the right eye. If we shut the right eye and open the left, we see the back, the upper edge, and the whole left side, thus- Now, if we open both eyes, we must

and do see both these images. If we look beyond the book, the two images are wholly separated, thus

If we look at the farther part, we bring these

two images together so as to unite the farther part and see it single, but the nearer part or back is double, thusIf we look at the nearer part or back,

*

then this is seen single, but the farther edge is now double, thus- But by no effort is it possible to see it single in all parts at the same time, because these dissimilar external images can not be wholly united. The eyes therefore range rapidly back and forth, successively uniting different parts by greater and less convergence, and thus acquire a distinct perception of distance between the back and front, and hence of depth of space.

The fact that two eyes are necessary for accurate estimate of distance may be illustrated by many familiar facts. (a) Using one eye only we can not dip a pen into an inkstand with the same accuracy and confidence as with both eyes open. (b) If we wish to draw the outlines of a complex object, like a chair or a table, we shut one eye, so as to destroy as much as possible the perception of depth of space and to project the object on a plane at right angles to the line of sight. (c) If two brass balls be hung by fine black threads invisible in a darkish room, one a foot or two beyond the other, the farther one a trifle larger than the nearer, and viewed nearly in a line and from such a distance that their angular diameters are equal: then, using one eye, they will seem to hang side by side, and it will be impossible to say which is the farther off; but as soon as we use both eyes the depth of space between is perceived at once.

*Of course in these figures the amount of doubling is exaggerated in order to make the principle clearer.

After these simple illustrations we are now prepared to generalize. It is evident that solid objects as seen by two eyes form different mathematical projections, and therefore form different retinal images in the two eyes, and therefore also different external images. Hence the images of the same object, whether retinal or external, formed by the two eyes, are necessarily dissimilar if the object occupies considerable depth of space. But dissimilar images can not be united wholly: for when by stronger convergence we unite the nearer parts, the farther will be double; and when by less convergence we unite the farther parts, the nearer will be double. Therefore the eyes run rapidly and unconsciously back and forth, uniting successively different parts, and thus acquire the perception of depth of space occupied by the object. But what is true of a single object is true of different objects placed one beyond the other, as the two fingers in experiment 1, page 138. We can not at the same time unite nearer and more distant objects, but the point of sight runs rapidly and unconsciously back and forth, uniting them successively, and thus we acquire a perception of depth of space lying between them. Therefore, the perception of the third dimension, viz., depth or relative distance, whether in a single object or in a scene, is the result of the successive combination of the different parts of the two dissimilar images of the object or the scene: dissimilar, because taken from different points, viz., the two eyes with the interocular distance between. This fundamental proposition will be slightly modified in our chapter on the theory of binocular perspective. In the mean time it must be clearly conceived and held fast; otherwise all that follows on stereoscopy will be unintelligible.