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is not mathematically exact for some eyes. It is probable that in some eyes the apparent vertical meridian which divides the retinæ into corresponding halves is not perfectly vertical, but slightly inclined outward at the top. This would affect all the meridians slightly; but the effect is very small, and I do not find it so in my eyes. We shall discuss this point again (page 218).

Law of Corresponding Points. After this explanation we reënunciate the law of corresponding points : Objects are seen single when their retinal images fall on corresponding points. If they do not fall on corresponding points, their external images are thrown to different places in space, and therefore are seen double.

Thus we see that the term “corresponding points" is used in two senses, which must be kept distinct in the mind of the reader. Every rod and cone in each retina has its correspondent in external space, and these exchange with each other by impression and projection. Also every rod or cone of each retina has its correspondent in a rod or cone in the other retina. Now the law of corresponding points, with which we are now dealing, states that the two external or spatial correspondents of two retinal corresponding points always coincide with each other, or the corresponding points of the two retinæ have the same spatial correspondent. In order to distinguish these two kinds of corresponding points from each other, the latter—i. e., corresponding points on the two retinæ-are often, and perhaps best, called “identical points,” because their external spatial representatives are really identical.

Thus, there is a kind of triangular correspondence between the retine and space. Every point in space has a correspondent in each retina, and the two retinal correspondents are an exact interocular distance apart; but the place of these retinal correspondents change with every movement of the eyes.

The images of spatial points, however, do not fall on their retinal correspondents except under certain conditions, viz., those which determine single vision.

Application.—We will now apply the law to the explanation of single and double vision. We have seen

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(experiment 1) that an object is seen single when looked at, but that all objects beyond or on this side the point of sight are doubled in opposite directions. Diagram Fig. 39 shows why, by the law of corresponding points, it must be so. While the two eyes, R and L, are fixed upon A, this object will be seen single, for its images, a and a', fall upon the central spots. But if, while still looking at A, we observe B and C, we shall see that both are double. The reason is, that the images of B, viz., 66', fall upon the two nasal or internal halves of the retinæ, which are non-corresponding; while the images of C, viz., c c', fall upon the two external or temporal halves of the retinæ, which are also non-corresponding. If the external double images be all referred to the plane of sight, PP (which, however, is not the fact), as is usually represented in diagrams, then the position of the double images will be correctly represented by c d', bb'. It is seen at a glance that the images c d of C are heteronymous, while the images b b' of B are homonymous. Generally, all the field of view within the lines of sight, A a, A a', belongs to the temporal halves of the retinæ, while all outside of these lines belongs to the nasal halves. Or, again, double images formed by impressions on the two nasal halves of the retinæ are homonymous, while those formed by impressions on the two temporal halves are heteronymous. Or, more generally: The central spots are, say, two and a half inches apart. All corresponding points are also two and a half inches apart. Retinal images farther apart than two and a half inches produce heteronymous external images, and therefore belong to objects nearer than the point of sight; while retinal images nearer together than two and a half inches produce homonymous external images, and belong to objects farther away than the point of sight.

Thus far the objects considered are on the median line. Next we will consider those in other positions.

Horopteric Circle of Müller.Objects at point of sight are seen single, while objects beyond or nearer than that point are seen double. But how is it with objects about the same distance as that point but not in

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R a' THE HOROPTERIC CIRCLE OF MÜLLER.-R and L, two eyes ; n n', point of crossing

of ray-lines--nodal point; A, point of sight; B, some other point in the bor. optoric circle Ann'; a a', central spots; aa', 60', retinal images of A and B.

the median line-i. e., above or below, on the right or left ? Take first the case of points lying to the right or left. In the diagram Fig. 40 the two eyes, R and L, are fixed on the object, A. This is, of course, seen single because its retinal images fall on corresponding points, viz., the central spots. Now, if a circle be drawn through the point of sight, A, and through the nodal points, n' n, of the two eyes, then by simple geometrical construction it is evident that any point, B, lying in that circle will also be seen single; for its two retinal images, 6 b', will fall on equivalent halves of the retina, and at equal distances from the central spots, a a',* and therefore are corresponding points. This is the horopteric circle of Müller. The same will not be true of points on any other line whether curved or straight. For example, an object, B', situated on a straight line tangent at the point of sight, A, will not be seen single, because its retinal images, W'1", are not on corresponding points, 1" being farther from the central spot. It will be heteronymously double. The circle of Müller is probably a true circle of single vision when the eyes are not strongly convergel.

Horopter or Surface of Single Vision. We have considered the case of objects lying right and left of the point of sight. We have yet to consider those in addition lying above or below. We have spoken of a possible horopteric circle. Is there also a horopteric surface? The surface of single vision with the point of sight fixed, or the surface passing through the point of sight all objects lying in which are seen single, is called the horopter. Whether there be such a surface at all, and if there is, what is its form, are questions upon which the acutest observers differ. Some have made it a plane, some a spherical surface. Some, by purely geometrical methods, have given it the most

* The angles A n B and A n' B are equal because they are angles at the circumference standing on the same arc A B. Their opposites, a no and a' n' b', are therefore also cqual.

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