same distance from it, as the object. When the object passes still further backwards, it again arrives at a spot at which both object and image are placed in the same plane. Still further backwards, the image remains in front of the object, moving in the same direction, although much more slowly, and finally reaches the second focal plane, when the object is at an infinite distance-i.e., when the incident rays which had been gradually becoming less convergent, have become parallel. The following summary of the properties of these points may now be intelligible: Cardinal points are formed by three pairs of imaginary points, focal, principal and knot-points. Each pair is composed of a first and second point; the first is invariably the one that has reference to the course of the ray in the first medium, the second the one that has reference to the course of the ray in the last medium. The first focal point.-Every ray passing through it before refraction is parallel with the axis after refraction. The second focal point.-Every ray that passes through it was, previously to refraction, parallel with the axis. Principal points.-Rays passing through the first principal point before refraction, will pass through the second principal point after refraction. Knot-points.-Every ray passing directly through the first knotpoint will after refraction pass through the second knot-point; the refracted will be parallel with the incident ray. Rays passing through the knot-points are called axes (Richtungs-strahlen or -linien), principal or secondary, as the case may be. Planes drawn perpendicularly to the axis through the focal, principal, and knot-points, are called respectively focal, principal, and knot-planes. The distance between the two knot-points is equal to that between the two principal points. The distance of the first principal point from the first focal point is called the first principal focal distance; the distance of the second principal point from the second focal point is the second principal focal distance. The difference of the two principal focal distances equals the distance between the corresponding principal and knot-points: the first principal focal distance bears to the second the same ratio that the index of refraction of the first medium does to that of the second. Focal planes.-Rays proceeding from a point of the first focal plane are parallel to one another after refraction-i.e., to the secondary axis. Rays parallel in the first medium have their focus where the secondary axis meets the second focal plane. Listing's ideal eye.-Examining the human eyeball from before backwards, we meet with three curved surfaces-that of the cornea, and the anterior and posterior surfaces of the lens; and four mediathe substance of the cornea, the aqueous humour, the substance of the lens, and that of the vitreous body. These curved surfaces are not exactly spherical, and their centres are not precisely on the same line, so that the principles already adduced cannot in a strict sense be applied to the eye. If, however, we take only very small portions of the surfaces, we may evidently consider them as spherical, and at the same time may regard the centres of these portions as all on the straight line drawn from the vertex of the cornea to the centre of the macula lutea (the optic or visual axis). By this means we shall be able to employ the principles already laid down, and shall obtain approximations in no way differing from those employed in other sections of physical science. Conclusions so deduced approach the true ones, just in proportion as the hypotheses on which they are founded approximate to the true facts. Such approximations furnish us with a simpler subject for consideration, in fact, with an ideal which we can afterwards compare with the real, and on which we can trace the effects of the various points previously disregarded. Indeed, by considering these (vertex) portions of the refracting surfaces of the eye as spherical segments, and their centres as placed on the visual axis, we have transformed the eye into a system of spherical refracting surfaces, of which the centres are all on the same straight line, and this may well be called an ideal or diagrammatic eye (das schematische Auge). Of course, we can apply all the principles just developed to such an ideal eye without further limitation. It must, however, be always remembered, and owing to its importance a repetition may be allowed, that all the principles, formulæ, &c., hitherto developed, are valid only for those rays that form very small angles (strictly speaking, infinitely small) with the axis, and which at the same time strike the refracting surfaces very near their vertices, so that the angles of incidence may be extremely small. An example may render this more forcible: If this page be placed eight inches from the eye, and the number of the page be the part fixed, it must not be expected that the pencils of rays proceeding from the letters immediately beneath will be refracted according to the same laws as the rays proceeding from the number fixed; the former rays would form too great angles with the axis, hence their course cannot be at all determined by the constructions previously given. If the pupil is of large size, even some of the outer rays proceeding from the point fixed will fall too obliquely to be treated by our rules. In regard to the eye, these can only include rays proceeding from points which are very near the point fixed (and which is itself in the prolonged optic axis), in comparison with the distance of the points themselves from the eye, and even such rays only in case they fall near the vertex of the cornea. We may express the latter statement in another form; the rules hitherto laid down presuppose a very small pupil. evident from the fact that the points in a plane perpendicular to the optic axis are to have their images formed in one and the same plane, and this can be true of only a very small part of the retina. more lateral images must be found by other rules; these will only apply to images formed on a piece of the retina around the extremity of the axis, and so small that it may be considered as a plane surface This is The perpendicular to the axis. Some of our laws may indeed be applicable to lateral rays; Volkmann,* for example, has concluded from his experiments, that lateral images are formed on axes which pass through the same knot-points that serve for the construction of the central images. This circumstance could not, however, have been foreseen, at all events, not from the calculations of Gauss; indeed, it is very possible that it depends on the deviation of the refracting surfaces from the spherical form. The next step in constructing such an ideal eye as may serve for a basis for further considerations in physiological optics, will be to assign definite values to the optical constants, to the radii of curvature, to the indices of refraction, &c. The values assigned should be such as really occur in normal eyes; at the same time, when combined together, they should form an optical apparatus equivalent in its action to a normal eye adapted for distant objects, or as commonly considered, at rest. Listing arranged such a system of optical constants as to form what is usually called Listing's ideal eye. He takes only four media into account air, aqueous humour, lens, and vitreous body. He thus considers the aqueous humour as reaching to the anterior surface of the cornea, a further simplification, of which the admissibility has been since confirmed by Helmholtz. A diaphragm or iris, with a central aperture or pupil, is supposed to exist in the ideal just as in the real eye. The following are the values assigned by Listing: The distance from the anterior surface of the cornea to the anterior surface of the lens = 4 mm. distance from the first to the second The thickness of the lens the distance from the refracting surface. = second to the third refracting surface = 4 mm. These values, so far as they are accessible to direct measurement on the living person, agree pretty well with those since found by Helmholtz. By means of these values the positions of the cardinal points have been calculated as follows: : The first principal point lies 2.1746 mm. behind the vertex of the cornea; the second principal point is 5-4276 mm. before the vertex of the posterior surface of the lens. Since, however, the latter is itself 8 mm. behind the vertex of the cornea, the distance between the two principal points will be 0.3978 mm. The first principal focus is 12.8326 mm. before the cornea, and the second principal focus is 14.6470 mm. behind the posterior surface of the lens. Therefore the first principal focal distance = the distance of the first principal focus from the first principal point = 12.8326 + * Volkmann in Wagner's Handwörterbuch, Band iii. Abth. 1, S. 286-9. 2-1746 15-0072 mm.; and in like manner the second principal focal distance equals 20·0746 mm. The first knot-point is situated 7-2420 mm. behind the vertex of the cornea, and 0·7580 mm. before the posterior surface of the lens : the second knot-point is 0.3602 mm. before the posterior surface of the lens, and there is therefore between the two knot-points a distance of 0.3978 mm., the same as between the two principal points. Listing considered the ideal eye to be adapted for rays proceeding from very distant objects-i.e., parallel rays have their focus on the retina; thus, the centre of the retina would coincide with the second principal focus. = Direct measurements of the antero-posterior diameter of the eye, from the vertex of the cornea to the diametrically opposite point of the outer surface of the sclerotic, have shown that its average length is about 24.25 mm. ; we have already (= second principal focal distance + distance between the two principal points + distance of the first principal point from the cornea = 20.0746 0.3978 + 2.1746 = 22-6470 distance of macula lutea from vertex of cornea) 22-6470 mm. as the distance of the second principal focus from the vertex of the cornea, and the remaining 16 mm. may be considered as the thickness of the sclerotic, &c. Fig. 1 represents a horizontal section of the right ideal eye, as seen from above, and magnified three linear dimensions. c is the centre around which the eye rotates, 12 mm. from the vertex of the cornea; F, F* are the principal foci, E, E* the principal points, D, D* the knot-points, N° the vertex of the cornea, N' that of the anterior surface, and N* that of the posterior surface of the lens. N DD Fig. 1. From Listing in Wagner, i. c., Band iv., S. 492. As already mentioned, the ideal eye of Listing is constructed on the supposition that the normal eye is accommodated for parallel rays when at rest. Should this supposition be hereafter found to be incorrect, some alterations would have to be made in the values assigned; thus, for example, Zehender* believes that the normal eye when at rest is accommodated only for a few feet (3-6 feet), or in other words, that the second principal focus is in front of the retina. If this be the case, it will be necessary to modify Listing's ideal eyet either by slightly lengthening its axis, or, what is better, by diminishing the distance between the anterior surface of the lens and the cornea (thus, from 4 mm., the value assigned by Listing, to 3.5 mm.). Helmholtz is inclined to consider, that the thickness of the lens (4 mm.) in Listing's eye would probably correspond only to that of an eye accommodated for near objects, and also that the value assigned for the distance between the lens and cornea (4 mm.) is too great. For the sake of comparison we append the values calculated by him for the optical constants and cardinal points of an ideal eye closely resembling the eyes of living persons which he examined, for two different conditions of the accommodation. The one adapted for distant objects differs from that of Listing only by the less thickness of the lens and depth of the anterior chamber. The indices of refraction are the same as those employed by Listing: 8 8 ... posterior surface of lens 10 6 5.5 Distance of the anterior surface of the lens from the anterior surface of the cornea Distance of the posterior surface of the lens from the anterior surface of the cornea Admitting that this eye, when accommodated for distant objects, would bring parallel rays to a focus on the retina, the length of its axis would be 22-231 mm. measured from the anterior surface of the cornea to the retina; and the same eye, when accommodated (as above) for near objects, would bring to a focus rays proceeding from an object which is placed 130-09 mm. before the cornea. This would correspond with the range of accommodation possessed by the normal eye. Reduced eye of Listing. In all cases where extreme accuracy is not required, we may make a further simplification in the ideal eye; for owing to the extremely small distance between the two principal * Dioptrik des Auges, i. S. 190, § 33. † Gräfe's Archiv für Ophthalm., Band. i. Abth. i. S. 133. |