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By H. KNAPP, M.D.,


In the theory of astigmatism there is in reality only one truly difficult point, namely, the demonstration of the course of the rays that are refracted by the meridians situated between the principal. As we have learned by heart, Sturm has mathematically proved that the rays refracted by asymmetrical surfaces are collected in the so-called focal interval which is bounded by two straight lines, being at right angles to each other and in the planes and foci of the principal meridians. To the student not familiar with mathematical analysis, it must be a mystery how Sturm found that out, since it seems so natural to assume that the meridians whose curvature is between that of the meridian of the strongest and that of the meridian of the weakest refractive power, have foci between the foci of these two meridians. But, gentlemen, this is utterly wrong: the intermediate meridians have no foci at all. Even if by the smallest stenopæic slit you exclude the refraction of all other meridians, the rays refracted by any intermediate, let me call them diagonal, meridians would not be collected anywhere. The reason for this proposition I can demonstrate to you in the plainest manner on this model, which, as you see, is made of beeswax, and, imitating the human eye, represents a triaxial ellipsoid. Pins are stuck in the wax along the horizontal, the vertical, and two diagonal meridians, so as to represent the normals of these meridians in a sufficient number of successive points. At once you see that the normals of the vertical and horizontal meridians only are in one plane, whereas those of the diagonal meridians are not, but curve around like a spiral, forming what is

called a “skew" surface (surface gauche in French; windschiefe Fläche in German). Now, as the incident and refracted rays must be in the same plane with the normal, it is evident that the incident and refracted rays can be in one plane only in such meridians whose normals also are in one plane. This condition, as you see, is fulfilled only by the principal meridians.

The course which the refracted rays of all meridians combined really take, i.e. Sturm's focal interval, is represented by a thread model which I constructed and described in Graefe's Archives about twenty years ago, and which has become somewhat popu. lar. I take pleasure in demonstrating it, as it shows very plainly all the intricacies of refraction by asymmetrical surfaces.

Let us now add some remarks on the practical deterinination of astigmatisin. Among the various modes of determining and correcting astigmatism, I have, for practical purposes, from the beginning preferred the use of a complete set of cylindrical glasses and the ordinary (Snellen's) test-types, as follows:

1. Determine S with and without glasses, far and near. 2. Determine ametropia with spherical glasses.

3. Put before the eye, and in ametropia before the correcting spherical glass, a concave cylivdrical glass, beginning with a's, turn it and see whether or not, how much, and in what position, S is improved. Note the direction of the axis and try stronger numbers until the greatest obtainable improvement of S is found.

4. Try convex cylindrical glasses in the same way.

5. Having determined the most suitable cylinder, put it in a frame, and for the sake of verification of the ametropia, put different spherical glasses before the cylinder, as if you had to determine a simple case of ametropia. In this way you obtain a formula of either a plano-cylindrical, or a sphero-cylindrical glass, never a bi-cylindrical, which, even for mixed astigmatism, should be avoided, as such glasses are more expensive and more difficult to grind correctly—especially in regard to centrationthan sphero-cylindrical glasses, which produce the same optical effect.

Of course, what I have just said does not exhaust the method of examination, it only lays down the principle; the prescription of glasses has to be governed by general rules, taking into account M, H, Pr, S, accommodation, the age of the patient, and the nature and duration of his work.

Only seldom, gentlemen, I use another subjective method,

though I never omit to test for astigmatism in any case of amblyopia or asthenopia by this method and the ophthalmoscope.

The DETERMINATION OF ASTIGMATISM WITH THE OPHTHALMOSCOPE is still in its infancy. The different methods that have been made known are as follows:

1. The difference of shape of the disk noticed in the erect and inverted images and at different distances, is available only in the higher degree of As. and for qualitative determinations.

2. The distortion of the retinal image: disk, blood vessels, etc., corrected by cylindrical glasses or a Stokes' lens behind the ophthalmoscope, is recommended by Schweigger, Schöler, and others. Schöler uses an ophthalmoscope with two disks above each other, the one containing spherical, the other cylindrical glasses. I think this method is very inconvenient, and have not learned anything as to its accuracy.

3. A rotable stenopæic slit, I suppose, has all the advantages and disadvantages of the stenopæic apparatus for subjective determinations. It is splendid for the description of astigmatism in the lecture-room and text-book, but tiresome and unsatisfactory in practice.

4. The determination of the refractive condition of the principal meridians with the refraction-ophthalmoscope rests on the fact that the horizontal focal line (supposing that the horizontal and vertical are the principal ineridians) is in the focal plane of the vertical meridian, and vice versa. If we look into an astigmatic eye, horizontal and vertical lines—say small blood vessels even when they are situated in the same plane, do not simultaneously appear with equal distinctness. When the ophthalmoscopist sees clearly a fine horizontal bloodvessel, he is accommodated for the vertical meridian of the eye examined, and vice versa. This fact, gentlemen, the nucleus of the method which I have cultivated for years, I beg to demonstrate on another model.

The model, roughly executed for my class of students, consists of four pieces of heavy drawing paper fastened vertically in a cigar box, over which they project about 3". The first and second are about 2" apart, the second and third about 4", the third and fourth 1". Of course, these dimensions are not essential. On the first, representing the fundus oculi, I have drawn a vertical line. Colored threads pass from three points of that

line through a circle in the second piece of paper, representing the pupil. The two threads passing from the middle point through the upper and lower ends of the vertical meridian of the pupil unite in one point in the third piece of paper, which, as is clear, represents the focal plane of the vertical meridian. After passing through, they diverge and strike the fourth paper, always remaining in the same plane, the upper ray being lowest, the lower ray highest on the fourth paper. The two rays which, emanating from the same point and passing through both ends of the horizontal meridian, pass through the third paper on both sides of the point of union of the two vertical rays, and unite in the fourth paper midway between the vertical rays. This point of union shows that the fourth paper is in the focal plane of the horizontal meridian. The rays emanating from the same point, and passing through the ends of the intermediate meridi. ans in the pupil, pass through a horizontal line of which the point of union of the vertical rays is the centre, and the horizontal rays are on its extremities; in the fourth paper they pass through a vertical line of which the point of union of the horizontal rays is the centre, and the vertical rays are the extremities.

The rays emanating from the upper point of the vertical line on the first paper formi a horizontal line on the third paper underneath that formed by the central point, and a vertical line on the fourth paper coinciding with the previous line, only not reaching quite so far as its upper end and a little farther than the lower. The rays emanating from the lower point of the vertical line in the first paper form a third horizontal line on the third paper, a little above the line formed by the first point, whereas on the fourth paper they coincide again with the same vertical line, not reaching its lower end, but a little exceeding its upper. Thus the different points of a vertical line in the fundus oculi, sending forth rays through an astigmatic refracting system (lens-cornea), form in the focal plane of the vertical meridian a number of horizontal lives, one above the other, coalescing in a broad stripe, whereas in the focal plane of the horizontal rays they form one sharp vertical line. The ophthalmoscopist, therefore, will see the vertical line in the fundus oculi of the patient clearly when the vertical dispersion line formed by the rays emerging from the patient's eye coincides with his retina, that is, when he is accommodated for the refraction of

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