If, in the first place, monochromatic aberrations be neglected - in other words, the Gaussian theory be accepted - then every reproduction is determined by the positions of the focal planes, and the magnitude of the focal lengths, or if the focal lengths, as ordinarily happens, be equal, by three constants of reproduction.
If there are four chairs, he argued, devoted to the history of philosophy, that is to say, the minute study of all sorts of dreams and aberrations through the ages, surely there ought to be at least one to explain the formation and progress of our real knowledge?
The parallelism, which is required to avoid aberrations, otherwise introduced by the prism or grating, may often be omitted in instruments of small power.
Excessive reading of Revelation seems to have been the chief cause of the aberrations of the Minster fanatics.
The practical difficulty of constructing Gregorian telescopes of good defining quality is very considerable, because if spherical mirrors are employed their aberrations tend to increase each other, and it is extremely difficult to give a true elliptic figure to the necessarily deep concavity of the small speculum.
It is, moreover, more exactly adequate to the actual situation, for the Principe has a divine spark of patriotism yet lingering in the cinders of its frigid science, an idealistic enthusiasm surviving in its moral aberrations; whereas a great Italian critic of this decade has justly described the Ricordi as "Italian corruption codified and elevated to a rule of life."
Consequently the monochromatic class includes the aberrations at reflecting surfaces of any coloured light, and at refracting surfaces of monochromatic or light of single wave length.
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.
With a considerable aperture, the neighbouring point N will be reproduced, but attended by aberrations comparable in magnitude to ON.
This connexion is only supplied by theories which treat aberrations generally and analytically by means of indefinite series.
On account of the aberrations of all rays which pass through 0, a patch of light, depending in size on the lowest powers of E, x, y which the aberrations contain, will be formed in the plane I'.
The aberrations can also be expressed by means of the "characteristic function " of the system and its differential coefficients, instead of by the radii, &c., of the lenses; these formulae are not immediately applicable, but give, however, the relation between the number of aberrations and the order.
Sir William Rowan Hamilton (British Assoc. Report, 18 33, p. 360) thus derived the aberrations of the third order; and in later times the method was pursued by Clerk Maxwell (Proc. London Math.
1), who thus discovered the aberrations of the 5th order (of which there are nine), and possibly the shortest proof of the practical (Seidel) formulae.
Phys., 1905, 18, p. 941) founded his theory of aberrations on the differential geometry of surfaces.
The aberrations of the third order are: (1) aberration of the axis point; (2) aberration of points whose distance from the Aberra- axis is very small, less than of the third order - the tions of deviation from the sine condition and coma here fall together in one class; (3) astigmatism; (4) curvature of the field; (5) distortion.
The total aberration of two or more very thin lenses in contact, being the sum of the individual aberrations, can be zero.
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.
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.
So a black whirl and torment of rapine, violence and fraud was encircling the Western world, as a life went out which, notwithstanding some eccentricities and some aberrations, had made great tides in human destiny very luminous.
If the ordinary convex lens be employed as magnifying glass, great aberrations occur even in medium magnifications.
When the pupil regulates the aperture of the rays producing the image the aberrations of the ordinary lenses increase considerably with the magnification, or, what amounts to the same thing, with the increase in the curvature of the surfaces.
To reduce the aberrations Sir David Brewster proposed to employ in the place of glass transparent minerals of high refractive index and low dispersion.
The individual components required weaker curvatures and permitted of being more correctly manufactured, and, more particularly, the advantage of reduced aberrations was the predominant factor.
The aberrations, both spherical and chromatic, increase very rapidly with the aperture.
Owing to these aberrations, the fine structure, which in consequence of the large aperture could be resolved, could not be perceived.
In other words, a sufficiently good and distinct image as the resolving power permits cannot be arrived at, until the elimination, or a sufficient diminution, of the spherical and chromatic aberrations has been brought about.
The spherical aberrations, however, can be overcome, or at least so diminished that they are quite harmless, by forming appropriate combinations of lenses.
Thus it is possible to correct a system by combining a convex and a concave lens, if both have aberrations of the same amount but of opposite signs.
A well-corrected microscope objective with a wide aperture therefore can only represent, free from aberrations, one object-element situated on a definite spot on the axis.
By experiment Abbe proved that old, good microscope objectives, which by mere testing had become so corrected that they produced usable images, were not only free from spherical aberrations, but also fulfilled the sine-condition, and were therefore really aplanatic systems.
This refers to systems with small apertures, but still more so to systems with large ones; chromatic aberrations are exceptionally increased by large apertures.
Although such systems have been made recently for special purposes, this construction was abandoned, and a more complex one adopted, which also made the production of better objectives possible; this is the principle of the compensation of the aberrations produced in the different parts of the objective.
This method makes it specially possible to overcome the chromatic and spherical aberrations of higher orders and to fulfil – the sine-condition, and the chief merit of this improvement belongs to Amici.
In the apochromats the chromatic difference of the spherical aberrations is eliminated, for the spherical aberration is completely avoided for three colours.
The definition is better according as the chromatic and spherical aberrations are removed; there always remains in even the best constructions some slight aberration.
In consequence of these residual aberrations, every object-point is not reproduced in an ideal image-point, but as a small circle of aberration.
Once grant the above definition of disease, and even the most trivial aberrations from the normal must be regarded as diseased conditions, quite irrespective of whether, when structural, they interfere with the function of the part or not.