ing to the law of Donders; and in altering this rotation we should judge the position of external objects wrongly.

The same will take place when we change the direction of the visual line. Suppose the amplitude of such motions to be infinitely small; then we may consider this part of the field of vision, and the corresponding part of the retina on which it is projected, as plane surfaces. If during any motion of the eye the optic image is displaced so that in its new position it remains parallel to its former position on the retina, we shall have no apparent motions of the objects. When, on the contrary, the optic image of the visible objects is dislocated so that it is not parallel to its former position on the retina, we must expect to perceive an apparent rotation of the objects.

As long as the motions of the eye describe infinitely small angles, the eye can be moved in such a way that the optic image remains always parallel to its first position. For this end the eye must be turned round axes of rotation which are perpendicular to the visual line; and we see indeed that this is done, according to the law of Listing, when the eye is moving near its primary position. But it is not possible to fulfil this condition completely when the eye is moved through a wider area which comprises a larger part of the spherical field of view. For if we were to turn the eye always round an axis perpendicular to the visual line, it would come into very different positions after having been turned through different ways to the same final direction.

The fault, therefore, which we should strive to avoid in the motions of our eye, cannot be completely avoided, but it can be made as small as possible for the whole field of vision.

The problem, to find such a law for the motions of the eye that the sum of all the rotations round the visual line for all possible infinitely small motions of the eye throughout the whole field of vision becomes a minimum, is a problem to be solved by the calculus of variations. I have found that the solution for a circular field of vision, which corresponds nearly to the forms of the actual field of vision, gives indeed the law of Listing.

I conclude from these researches, that the actual mode of moving the eye is that mode by which the perception of the steadiness of the objects through the whole field of vision can be kept up the best; and I suppose, therefore, that this mode of motion is produced by experience and exercise, because it is the best suited for accurate perception of the position of external objects.

But in this mode of moving, rotations round the visual line are not completely avoided when the eye is moved in a circular direction round the primary position of the visual line; and it is easy to recognize that in such a case we are subject to optical illusions.

Turn your eyes to a horizontal line situated in the highest part of the field of vision, and let them follow this line from one end to the other.

The line will appear like a curved line, the convexity of which looks downward. When you look to its right extremity, it seems to rise from the left to the right; when you look to the left extremity of the line, the left end seems to rise. In the same way, all straight lines which go through the peripheral parts of the field of vision appear to be curved, and to change their position a little, if you look to their upper or their lower ends.

This explanation relates only to Monocular vision; we have to inquire also how it influences Binocular vision.

Each eye has its field of vision, on which the visible objects appear distributed like the objects of a picture, and the two fields with their images seem to be superimposed. Those points of both fields of view which appear to be superimposed are called corresponding (or identical) points. If we look at real objects, the accurate perception of the superimposition of two different optical images is hindered by the perception of stereoscopic form and depth; and we unite indeed, as Mr. Wheatstone has shown, two retinal images completely into the perception of one single body, without being able to perceive the duplicity of the images, even if there are very sensible differences of their form and dimensions. To avoid this, and to find those points of both fields of view which correspond with each other, it is necessary to use figures which cannot easily be united into one stereoscopic projection.

In fig. 2 you see such figures, the right of which is drawn with white lines on a black ground, the left with black lines on a white ground. The horizontal lines of both figures are parts of the same straight lines; the vertical lines are not perfectly vertical. The upper end of those of the right figure is inclined to the right, that of the left figure to the left, by about 1 degree.

Now I beg you to look alternately with the right and with the left eye at these figures. You will find that the angles of the right figure appear to the right eye equal to right angles, and those of the left figure so appear to the left eye; but the angles of the left figure appear to the right eye to deviate much from a right angle, as also do those of the right figure to the left eye.

When you draw on paper a horizontal line, and another line crossing it exactly at right angles, the right superior angle will appear to your right eye too great, to your left eye too small; the other angles show corresponding deviations. To have an apparently right angle, you must make the vertical line incline by an angle of about 14 degree for it to appear really vertical; and we must distinguish, therefore, the really vertical lines and the apparently vertical lines in our field of view.

There are several other illusions of the same kind, which I omit because they alter the images of both eyes in the same manner and have no influence upon binocular vision; for example, vertical lines appear always of greater length than horizontal lines having really the same length.

Now combine the two sides of fig. 2 into a stereoscopic combination, either by squinting, or with the help of a stereoscope, and you will see that the white lines of the one coincide exactly with the black lines of the other, as soon as the centres of both the figures coincide, although the vertical lines of the two figures are not parallel to each other.

Therefore not the really vertical meridians of both fields of view correspond, as has been supposed hitherto, but the apparently vertical meridians. On the contrary, the horizontal meridians really correspond, at least for normal eyes which are not fatigued. After having kept the eyes a long

time looking down at a near object, as in reading or writing, I found sometimes that the horizontal lines of fig. 2 crossed each other; but they became parallel again when I had looked for some time at distant objects.

In order to define the position of the corresponding points in both fields of vision, let us suppose the observer looking to the centres of the two sides of fig. 2, and uniting both pictures stereoscopically. Then planes may be laid through the horizontal and vertical lines of each picture and the centre of the corresponding eye. The planes laid through the different horizontal lines will include angles between them, which we may call angles of altitude; and we may consider as their zero the plane going through the fixed point and the horizontal meridian. The planes going through the vertical lines include other angles, which may be called angles of longitude, their zero coinciding also with the fixed point and with the apparently vertical meridian. Then the stereoscopic combination of those diagrams shows that those points correspond which have the same angles of altitude and the same angles of longitude; and we can use this result of the experiment as a definition of corresponding points.

We are accustomed to call Horopter the aggregate of all those points of the space which are projected on corresponding points of the retina. After having settled how to define the position of corresponding points, the question, what is the form and situation of the Horopter, is only a geometrical question. With reference to the results I had obtained in regard to the positions of the eye belonging to different directions of the visual lines, I have calculated the form of the Horopter, and found that generally the Horopter is a line of double curvature produced by the intersection of two hyperboloids, and that in some exceptional cases this line of double curvature can be changed into a combination of two plane curves.

That is to say, when the point of convergence is situated in the middle plane of the head, the Horopter is composed of a straight line drawn through the point of convergence, and of a conic section going through the centre of both eyes and intersecting the straight line.

When the point of convergence is situated in the plane which contains the primary directions of both the visual lines, the Horopter is a circle going through that point and through the centres of both eyes and a straight line intersecting the circle.

When the point of convergence is situated as well in the middle plane of the head as in the plane of the primary directions of the visual lines, the Horopter is composed of the circle I have just described, and a straight line going through that point.

There is only one case in which the Horopter is really a plane, as it was supposed to be in every instance by Aguilonius, the inventor of that name,— 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 certain distance which depends upon the angle between the really and apparently vertical

meridians, and 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 to a point of the horizon, the Horopter is a horizontal plane going through our feet-it is the ground upon which we are standing.

Formerly physiologists believed that the Horopter was an infinitely distant plane when we looked to an infinitely distant point. The difference of our present conclusion is consequent upon the difference between the position of the really and apparently vertical meridians, which they did not know.

When we look, not to an infinitely distant horizon, but to any point of the ground upon which we stand which is equally distant from both our eyes, the Horopter is not a plane; but the straight line which is one of its parts coincides completely with the horizontal plane upon which we are standing.

The form and situation of the Horopter is of great practical importance for the accuracy of our visual perceptions, as I have found.

Take a straight wire-a knitting-needle for instance-and bend it a little in its middle, so that its two halves form an angle of about four degrees. Hold this wire with outstretched arm in a nearly perpendicular position before you, so that both its halves are situated in the middle plane of your head, and the wire appears to both your eyes nearly as a straight line. In this position of the wire you can distinguish whether the angle of the wire is turned towards your face or away from it, by binocular vision only, as in stereoscopic diagrams; and you will find that there is one direction of the wire in which it coincides with the straight line of the Horopter, where the inflexion of the wire is more evident than in other positions. You can test if the wire really coincides with the Horopter, when you look at a point a little more or a little less distant than the wire. Then the wire appears in double images, which are parallel when it is situated in the Horopter line, and are not when the point is not so situated.

Stick three long straight pins into two little wooden boards which can slide one along the other; two pins may be fastened in one of the boards, the third pin in the second. Bring the boards into such a position that the pins are all perpendicular and parallel to each other, and situated nearly in the same plane. Hold them before your eyes and look at them, and strive to recognize if they are really in the same plane, or if their series is bent towards you or from you. You will find that you distinguish this by binocular vision with the greatest degree of certainty and accuracy (and indeed with an astonishing degree of accuracy) when the line of the three pins coincides with the direction of the circle which is a part of the Horopter.

From these observations it follows that the forms and the distances of those objects which are situated in, or very nearly in, the Horopter, are perceived with a greater degree of accuracy than the same forms and distances would be when not situated in the Horopter. If we apply this