The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any object point unite in an image point; and therefore an object space is reproduced in an image space.

The introduction of simple auxiliary terms, due to C. F. Gauss (Dioptrische Untersuchungen, GĂ¶ttingen, 1841), named the focal lengths and focal planes, permits the determination of the image of any object for any system (see lens).

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.

The investigations of James Clerk Maxwell (Phil.Mag., 1856; Quart. Journ. Math., 1858, and Ernst Abbe) showed that the properties of these reproductions, i.e. the relative position and magnitude of the images, are not special properties of optical systems, but necessary consequences of the supposition (in Abbe) of the reproduction of all points of a space in image points (Maxwell assumes a less general hypothesis), and are independent of the manner in which the reproduction is effected.

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.

This, and related general questions, have been treated — besides the above-mentioned authors — by M. Thiesen (Berlin. Akad. Sitzber., 1890, xxxv. 799; Berlin. Phys. Ges. Verh., 1892) and H. Bruns (Leipzig. Math. Phys. Ber., 1895, xxi. 325) by means of Sir W. R. Hamilton's characteristic function (Irish Acad. Trans., Theory of Systems of Rays, 1828, et seq.). Reference may also be made to the treatise of Czapski-Eppenstein, pp. 155-161.

A review of the simplest cases of aberration will now be given.

The introduction of simple auxiliary terms, due to C. F. Gauss (Dioptrische Untersuchungen, GĂ¶ttingen, 1841), named the focal lengths and focal planes, permits the determination of the image of any object for any system (see lens).

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.

The investigations of James Clerk Maxwell (Phil.Mag., 1856; Quart. Journ. Math., 1858, and Ernst Abbe) showed that the properties of these reproductions, i.e. the relative position and magnitude of the images, are not special properties of optical systems, but necessary consequences of the supposition (in Abbe) of the reproduction of all points of a space in image points (Maxwell assumes a less general hypothesis), and are independent of the manner in which the reproduction is effected.

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.

This, and related general questions, have been treated — besides the above-mentioned authors — by M. Thiesen (Berlin. Akad. Sitzber., 1890, xxxv. 799; Berlin. Phys. Ges. Verh., 1892) and H. Bruns (Leipzig. Math. Phys. Ber., 1895, xxi. 325) by means of Sir W. R. Hamilton's characteristic function (Irish Acad. Trans., Theory of Systems of Rays, 1828, et seq.). Reference may also be made to the treatise of Czapski-Eppenstein, pp. 155-161.

A review of the simplest cases of aberration will now be given.

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