C. F. Gauss

Benjamin Apthorp Gould (September 27, 1824 – November 26, 1896) was an American astronomer. He was born in Boston, Massachusetts. Having graduated at Harvard College in 1844, he studied mathematics and astronomy under C. F. Gauss at Göttingen, Germany, during which time he published approximately 20 papers on the observation and motion of comets and asteroids. He returned to America in 1848. From 1852 to 1867 he was in charge of the longitude department of the United States Coast Survey; he developed and organized the servic ...

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In mathematics, a lemniscate is a type of curve described by a Cartesian equation of the form: (x2 + y2)2 = a2(x2 − y2) Graphing this equation produces a curve similar to . The curve has become a symbol of infinity and is widely used in math. The symbol itself is sometimes referred to as the lemniscate. Its Unicode represe ...

Including:

• Lemniscate - Other equations
• Lemniscate - Arc length and elliptic functions

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Aberration in optical systems (lenses, prisms, mirrors or series of them intended to produce a sharp image) generally leads to blurring of the image. It occurs when light from one point of an object after transmission through the system arrives in different points. Instrument-makers need to correct optical systems to compensate for aberration. The articles reflection, refraction and caustic discuss the general features of reflected and refracted rays. Aberrations fall into two classes: chromatic aberrations ...

Including:

• Aberration in optical systems - Monochromatic aberration
• Aberration in optical systems - Aberration of axial points spherical aberration in the restricted sense
• Aberration in optical systems - Aberration of elements i.e. smallest objects at right angles to the axis
• Aberration in optical systems - Aberration of lateral object points points beyond the axis with narrow pencils. Astigmatism.
• Aberration in optical systems - Aberration of lateral object points with broad pencils. Coma.
• Aberration in optical systems - Curvature of the field of the image
• Aberration in optical systems - Distortion of the image
• Aberration in optical systems - Analytic treatment of aberrations
• Aberration in optical systems - Practical elimination of aberrations
• Aberration in optical systems - Chromatic or colour aberration
• Aberration in optical systems - Authorities.

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Greek mathematics has origins that are presumed to go back to the 7th century BC, but are not easily documented. It is generally believed that it built on the computational methods of earlier Babylonian and Egyptian mathematics, and it may well have had Phoenician influences. Some of the most well-known figures in Greek mathematics are Pythagoras, a shadowy figure from the isle of Samos associated partly with number mysticism and numerology, but more commonly with his theorem, and Euclid, who is known for his Elements, a canon of geom ...

See also:

Greek mathematics, Greek mathematics - Origins, Greek mathematics - Famous Greek mathematicians

Read more here: » Greek mathematics: Encyclopedia II - Greek mathematics - Origins

A lemniscate may also be described by the polar equation r2 = a2cos2φ or the bipolar equation ...

See also:

Lemniscate, Lemniscate - Other equations, Lemniscate - Arc length and elliptic functions

Read more here: » Lemniscate: Encyclopedia II - Lemniscate - Other equations

The existence and description of a fundamental domain is in general something requiring painstaking work to establish. The diagram to the right shows part of the construction of the fundamental domain for the action of the modular group Γ on the upper half-plane H. This famous diagram appears in all classical books on elliptic modular functions. (It was probably well known to C. F. Gauss, who dealt with fundamental domains in the guise of the reduction theory of quadratic forms.) Here, each triangular region (bounded by the bl ...

See also:

Fundamental domain, Fundamental domain - Example

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In optical systems composed of lenses, the position, magnitude and errors of the image depend upon the refractive indices of the glass employed (see Lens (optics), and above, Monochromatic Aberration). Since the index of refraction varies with the colour or wavelength of the light (see dispersion), it follows that a system of lenses (uncorrected) projects images of different colours in somewhat different places and sizes and with different aberrations; i.e. there are chromatic differences of the distances of intersection, of ma ...

See also:

Aberration in optical systems, Aberration in optical systems - Monochromatic aberration, Aberration in optical systems - Aberration of axial points spherical aberration in the restricted sense, Aberration in optical systems - Aberration of elements i.e. smallest objects at right angles to the axis, Aberration in optical systems - Aberration of lateral object points points beyond the axis with narrow pencils. Astigmatism., Aberration in optical systems - Aberration of lateral object points with broad pencils. Coma., Aberration in optical systems - Curvature of the field of the image, Aberration in optical systems - Distortion of the image, Aberration in optical systems - Analytic treatment of aberrations, Aberration in optical systems - Practical elimination of aberrations, Aberration in optical systems - Chromatic or colour aberration, Aberration in optical systems - Authorities.

Read more here: » Aberration in optical systems: Encyclopedia II - Aberration in optical systems - Chromatic or colour aberration

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, Göttingen, 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 ...

See also:

Aberration in optical systems, Aberration in optical systems - Monochromatic aberration, Aberration in optical systems - Aberration of axial points spherical aberration in the restricted sense, Aberration in optical systems - Aberration of elements i.e. smallest objects at right angles to the axis, Aberration in optical systems - Aberration of lateral object points points beyond the axis with narrow pencils. Astigmatism., Aberration in optical systems - Aberration of lateral object points with broad pencils. Coma., Aberration in optical systems - Curvature of the field of the image, Aberration in optical systems - Distortion of the image, Aberration in optical systems - Analytic treatment of aberrations, Aberration in optical systems - Practical elimination of aberrations, Aberration in optical systems - Chromatic or colour aberration, Aberration in optical systems - Authorities.

Read more here: » Aberration in optical systems: Encyclopedia II - Aberration in optical systems - Monochromatic aberration

The preceding review of the several errors of reproduction belongs to the Abbe theory of aberrations, in which definite aberrations are discussed separately; it is well suited to practical needs, for in the construction of an optical instrument certain errors are sought to be eliminated, the selection of which is justified by experience. In the mathematical sense, however, this selection is arbitrary; the reproduction of a finite object with a finite aperture entails, in all probability, an infinite number of aberrations. This number ...

See also:

Aberration in optical systems, Aberration in optical systems - Monochromatic aberration, Aberration in optical systems - Aberration of axial points spherical aberration in the restricted sense, Aberration in optical systems - Aberration of elements i.e. smallest objects at right angles to the axis, Aberration in optical systems - Aberration of lateral object points points beyond the axis with narrow pencils. Astigmatism., Aberration in optical systems - Aberration of lateral object points with broad pencils. Coma., Aberration in optical systems - Curvature of the field of the image, Aberration in optical systems - Distortion of the image, Aberration in optical systems - Analytic treatment of aberrations, Aberration in optical systems - Practical elimination of aberrations, Aberration in optical systems - Chromatic or colour aberration, Aberration in optical systems - Authorities.

Read more here: » Aberration in optical systems: Encyclopedia II - Aberration in optical systems - Analytic treatment of aberrations

As Maxwell already claimed in 1858, there is no optical system, which reproduces absolutely a finite plane on another with pencils of finite aperture. A strict and general proof was given 1926 by Carathéodory. 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 ...

See also:

Aberration in optical systems, Aberration in optical systems - Monochromatic aberration, Aberration in optical systems - Aberration of axial points spherical aberration in the restricted sense, Aberration in optical systems - Aberration of elements i.e. smallest objects at right angles to the axis, Aberration in optical systems - Aberration of lateral object points points beyond the axis with narrow pencils. Astigmatism., Aberration in optical systems - Aberration of lateral object points with broad pencils. Coma., Aberration in optical systems - Curvature of the field of the image, Aberration in optical systems - Distortion of the image, Aberration in optical systems - Analytic treatment of aberrations, Aberration in optical systems - Practical elimination of aberrations, Aberration in optical systems - Chromatic or colour aberration, Aberration in optical systems - Authorities.

Read more here: » Aberration in optical systems: Encyclopedia II - Aberration in optical systems - Practical elimination of aberrations