Otto Eppenstein

“ Practical Elimination of Aberrations.—The existence of an optical system, which reproduces absolutely a finite plane on another with pencils of finite aperture, is doubtful; but practical systems solve this problem with an accuracy which mostly suffices for the special purpose of each species of instrument. The problem of finding a system which reproduces a given object upon a given plane with given magnification (in so far as aberrations must be taken into account) could be dealt with by means of the approximation theory; in most cases, however, the analytical difficulties are too great. ”

“ These authors proved, however, that no optical system can justify these suppositions, since they are contradictory to the fundamental laws of reflexion and refraction. Consequently the Gaussian theory only supplies a convenient method of approximating to reality; and no constructor would attempt to realize this unattainable ideal. All that at present can be attempted is, to reproduce a single plane in another plane; but even this has not been altogether satisfactorily accomplished, aberrations always occur, and it is improbable that these will ever be entirely corrected. ”

“ The Gaussian theory is only an approximation; monochromatic or spherical aberrations still occur, which will be different for different colours; and should they be compensated for one colour, the image of another colour would prove disturbing. The most important is the chromatic difference of aberration of the axis point, which is still present to disturb the image, after par-axial rays of different colours are united by an appropriate combination of glasses. ”

“ The nature of the reproduction consists in the rays proceeding from a point O being united in another point O′; in general, this will not be the case, for ξ′, η′ vary if ξ, η be constant, but x, y variable. It may be assumed that the planes I′ and II′ are drawn where the images of the planes I and II are formed by rays near the axis by the ordinary Gaussian rules; and by an extension of these rules, not, however, corresponding to reality, the Gauss image point O′0, with co-ordinates ξ′0,η′0, of the point O at some distance from the axis could be constructed. ”

“ Largest field of view; necessary corrections are—for astigmatism, curvature of field and distortion; errors of the aperture only slightly regarded; examples—photographic widest angle objectives and oculars. Between these extreme examples stands the ordinary photographic objective: the portrait objective is corrected more with regard to aperture; objectives for groups more with regard to the field of view. (3) Telescope objectives have usually not very large apertures, and small fields of view; they should, however, possess zones as small as possible, and be built in the simplest manner. ”

“ This astronomical phenomenon may be defined as an apparent motion of the heavenly bodies; the stars describing annually orbits more or less elliptical, according to the latitude of the star; consequently at any moment the star appears to be displaced from its true position. This apparent motion is due to the finite velocity of light, and the progressive motion of the observer with the earth, as it performs its yearly course about the sun. ”

“ The condition for the reproduction of a surface element in the place of a sharply reproduced point—the constant of the sine relation—must also be fulfilled with large apertures for several colours. E. Abbe succeeded in computing microscope objectives free from error of the axis point and satisfying the sine condition for several colours, which therefore, according to his definition, were “aplanatic for several colours”; such systems he termed “apochromatic”. While, however, the magnification of the individual zones is the same, it is not the same for red as for blue ”

“ In using glasses not having proportional dispersion, the deviation of a third colour can be eliminated by two lenses, if an interval be allowed between them; or by three lenses in contact, which may not all consist of the old glasses. In uniting three colours an “achromatism of a higher order” is derived; there is yet a residual “tertiary spectrum,” but it can always be neglected. ”

“ The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis (the symmetrical axis of the system) are infinitely small, i.e. with infinitesimal objects, images and lenses; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits. ”

“ If a constant of reproduction, for instance the focal length, be made equal for two colours, then it is not the same for other colours, if two different glasses are employed. For example, the condition for achromatism (4) for two thin lenses in contact is fulfilled in only one part of the spectrum, since dn2/dn1 varies within the spectrum. ”

“ This is, at the present day, the ordinary type, e.g., of telescope objective (fig. 10) ; the values of the four radii must satisfy the equations (2) and (4) . Two other conditions may also be postulated; one is always the elimination of the aberration on the axis; the second either the Herschel” or “Fraunhofer condition,” the latter being the best (vide supra, “Monochromatic Aberation”) . In practice, however, it is often more useful to avoid the second condition by making, the lenses have contact, i.e. equal radii. ”

“ Astigmatism.—A point O (fig. 6) at a finite distance from the, axis (or with an infinitely distant object, a point which subtends a finite angle at the system) is, in general, even then not sharply reproduced, if the pencil of rays issuing from it and traversing the system is made infinitely narrow by reducing the aperture stop; such a pencil consists of the rays which can pass from the object point through the now infinitely small entrance pupil. It is seen (ignoring exceptional cases) that the pencil does not meet he refracting or reflecting surface at right angles ”

“ The minor axis, on the other hand, is not constant, but, as we have already seen, depends on the latitude, being the product of the major axis into the sine of the latitude. Assured that his explanation was true, Bradley corrected his observations for aberration, but he found that there still remained a residuum which was evidently not a parallax, for it did not exhibit an annual cycle. ”

“ At the present time constructors almost always employ the inverse method: they compose a system from certain, often quite personal experiences, and test, by the trigonometrical calculation of the paths of several rays, whether the system gives the desired reproduction (examples are given in A. Gleichen, Lehrbuch der geometrischen Optik, Leipzig and Berlin, 1902) . The radii, thicknesses and distances are continually altered until the errors of the image become sufficiently small. ”

“ Imagine rain to be falling vertically, and a person carrying a thin perpendicular tube to be standing on the ground. If the bearer be stationary, rain-drops will traverse the tube without touching its sides; if, however, the person be walking, the tube must be inclined at an angle varying as his velocity in order that the rain may traverse the tube centrally. ”

“ A re-examination of his previously considered hypotheses as to the cause of these phenomena was fruitless; the true theory was ultimately discovered by a pure accident, comparable in simplicity and importance with the association of a falling apple with the discovery of the principle of universal gravitation. ”

“ When the earth is at A, in consequence of aberration, the star is displaced to a point a, its displacement sa being parallel to the earth’s motion at A; when the earth is at B, the star appears at b; and so on throughout an orbital revolution of the earth. Every star, therefore, describes an apparent orbit, which, if the line joining the sun and the star be perpendicular to the plane ABCD, will be exactly similar to that of the earth, i.e. ”

“ They determined to reinvestigate the motion of γ Draconis; the telescope, constructed by George Graham (1675–1751) , a celebrated instrument-maker, was affixed to a vertical chimney-stack, in such manner as to permit a small oscillation of the eyepiece, the amount of which, i.e. the deviation from the vertical, was regulated and measured by the introduction of a screw and a plumb-line. The instrument was set up in November 1725, and observations on γ Draconis were made on the 3rd, 5th, 11th, and 12th of December. ”

“ This aberration is quite distinct from that of the sharpness of reproduction; in unsharp, reproduction, the question of distortion arises if only parts of the object can be recognized in the figure. If, in an unsharp image, a patch of light corresponds to an object point, the “centre of gravity” of the patch may be regarded as the image point, this being the point where the plane receiving the image, e.g. ”

“ Since the maximum aperture of the pencils issuing from O is the angle u subtended by the entrance pupil at this point, the magnitude of the aberration will be determined by the position and diameter of the entrance pupil. ”