fall on corresponding points, and are seen single; and conversely, the horopter is the surface (if it be a surface) of single vision. Is the horopter a surfact, or is it only a line ? In either case, what are its form and position ? These questions have tasked the ingenuity of physicists, mathematicians, and physiologists. If the position of corresponding points were certainly known, and if the meridians of the eye in all its motions corresponded perfectly with the spatial meridians, then the question of the horopter would be a purely mathematical one. But the position of the ocular meridians, and therefore of corresponding points, may change in ocular motions. It is evident, then, that it is only on an experimental basis that a true theory of the horopter can be constructed. And yet the experimental determination, as usually attempted, is very unsatisfactory on account of the indistinctness of perception of objects except very near the point of sight. Therefore experiments determining the laws of ocular motion, and mathematical reasoning based upon these laws, seem to be the only sure method. The most diverse views have therefore been held as to the nature and form of the horopter. Aguilonius, the inventor of the name, believed it to be a plane passing through the point of sight and perpendicular to thę median line of sight. This, as we have shown (page 117, Fig. 40), is geometrically untenable. Others have believed it to be the surface of a sphere passing through the optic centers and the point of sight; others, a torus generated by the revolution of a circle passing through the optic centers * or nodal points and the point of sight (horopteric circle of Müller), about a line joining the Optic center is here used in sense of center of the lens system, not of the ocular globe. FIG. 78. A n optic centers. The subject has been investigated with great acuteness by Prévost, Müller, Meissner, Claparède, and finally by Helmholtz. Prévost and Müller determine in it, as they think, the circumference of a circle passing through the optic centers and the point of sight (the horopteric circle), and a line passing through the point of sight and perpendicular to the plane of the circle (horopteric vertical). The horopteric circle of Müller is shown in Fig. 73, in which nn' is the line between the nodal points or points of ray-crossing, A the point of sight, and B an object to the left and situated in the circumference of the circle. Of course, the images of A fall on the central spots cc'. It is seen also that the images of B fall at bb', at equal distances from the central spots c c', one on the nasal half and one on the temporal half, and therefore on corresponding points. The horopteric vertical of Müller passes through A and perpendicular to the plane of the circle (i. e., of the diagram). Claparède * makes the horopter a surface, of such a form that it contains a straight line passing through the point of sight and perpendicular to the visual plane, and *“ Archives des Sciences,” 1858, vol. iii, p. 161. C HOROPTERIC CIRCLE OF MULLER. Fig. 74. А also such that every plane passing through the optic centers makes by intersection with this surface the circum ference of a circle. In other words, he thinks B that the horopter is a surface which contains the horopteric vertical B A B', Fig. 74, and the horopteric circle, N A N. N', and in addition is further characterized by the fact that the intersection with it of every plane passing through the optic centers N N' upward as NB N', or downward as NB' N' is also a circle. It is eviB' dent that, as these circles increase in size upward HOROPTER ACCORDING TO CLAPARÈDE. and downward, the horopter according to Claparède is a surface of singular and complex form. Helmholtz arrives at results entirely different. According to him, the horopter varies according to the position of the point of sight, and is therefore very complex. He sums up his conclusions thus : * “1. Generally the horopter is a line of double curvature produced by the intersection of two hyperboloids, which in some exceptional cases may be changed into a combination of two plane curves. “2. For example, where the point of convergence (point of sight) is situated in the median plane of the head, the horopter is composed of a straight line drawn through the point of convergence, and a conic section going through the optic centers and intersecting the straight line. * Croonian Lecture, in “Proceedings of the Royal Society,” xiii (1864), p. 197; also “Optique Physiologique,” p. 901 et seq. “3. When the point of convergence is situated in the plane which contains the primary directions of both visual lines (primary visual plane), the horopter is composed of a circle going through that point and through the optic centers (horopteric circle), and a straight line intersecting the circle. “4. When the point of convergence is situated both in the middle plane of the head and in the primary visual plane, the horopter is composed of the horopteric circle and of a straight line going through that point. “5. There is only one case in which the horopter is a plane, namely: when the point of convergence is situated in the middle plane of the head and at an infinite distance. Then the horopter is a plane parallel to the visual lines, and situated beneath them at a distance which is nearly as great as the distance of the feet of the observer from his eyes when he is standing. Therefore, when we look straight forward at a point on the horizon, the horopter is a horizontal plane going through our feet; it is the ground on which we stand. “6. When we look not at an infinite distance, but at any point on the ground on which we stand which is equally distant from the two eyes, the horopter is not a plane, but the straight line which is one of its parts coincides with the ground." Some attempts have been made to establish the existence of the horopteric circle of Müller by means of experiments. A plane is prepared and pierced with a multitude of holes into which pegs may be set. The eyes look horizontally over the plane on one peg, and the others are arranged in such wise that they appear single. It is found that they must be arranged in a circle. I have tried repeatedly, but in vain, to verify this result. The difficulty is the extreme indistinctness of perception at any appreciable distance from the point of sight to one side or the other. But, as a general fact, the results reached by the observers thus far mentioned have been reached by the most refined mathematical calculations, based on certain premises concerning the position of corresponding points and on the laws of ocular motion. We will examine only those of Helmholtz, as being the latest and most authoritative. Helmholtz’s results are based upon the law of Listing as governing all the motions of the eye, and upon his own peculiar views concerning the relation between what he calls the apparent and the real vertical meridian of the retina. According to him, the real vertical meridian of the eye is the line traced on the retina by the image of a really vertical linear object when the median plane of the head is vertical and the eye in the primary position. The apparent vertical meridian of the eye is the line traced by the image of an apparently vertical linear object in the same position of the eye. This is also called the vertical line of demarkation, because it divides the retina into two halves which correspond each to each and point for point. Now, according to Helmholtz, the apparent vertical meridian or vertical line of demarkation does not coincide with the real vertical meridian, but makes with it in each eye an angle of 11°, and therefore with one another in the two eyes of 21°. The horizontal meridians of the eyes, both real and apparent, coincide completely. Therefore, if the two eyes were brought together in such wise that their |