1.
Refraction
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Refraction is the change in direction of wave propagation due to a change in its transmission medium. The phenomenon is explained by the conservation of energy and the conservation of momentum, due to the change of medium, the phase velocity of the wave is changed but its frequency remains constant. This is most commonly observed when a wave passes from one medium to another at any other than 0° from the normal. In optics, refraction is a phenomenon that occurs when waves travel from a medium with a given refractive index to a medium with another at an oblique angle. At the boundary between the media, the phase velocity is altered, usually causing a change in direction. Its wavelength increases or decreases, but its frequency remains constant, for example, a light ray will refract as it enters and leaves glass, assuming there is a change in refractive index. A ray traveling along the normal will change speed, but not direction, refraction still occurs in this case. Understanding of this led to the invention of lenses and the refracting telescope. Refraction can be seen looking into a bowl of water. Air has a index of about 1.0003. If a person looks at an object, such as a pencil or straw, which is placed at a slant, partially in the water. This is due to the bending of light rays as they move from the water to the air, once the rays reach the eye, the eye traces them back as straight lines. The lines of sight intersect at a position than where the actual rays originated. This causes the pencil to appear higher and the water to appear shallower than it really is, the depth that the water appears to be when viewed from above is known as the apparent depth. This is an important consideration for spearfishing from the surface because it will make the fish appear to be in a different place. Conversely, an object above the water has a higher apparent height when viewed from below the water, the opposite correction must be made by an archer fish. For small angles of incidence, the ratio of apparent to real depth is the ratio of the indexes of air to that of water. But, as the angle of incidence approaches 90o, the apparent depth approaches zero, albeit reflection increases, the diagram on the right shows an example of refraction in water waves
2.
Refractive index
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In optics, the refractive index or index of refraction n of a material is a dimensionless number that describes how light propagates through that medium. It is defined as n = c v, where c is the speed of light in vacuum, for example, the refractive index of water is 1.333, meaning that light travels 1.333 times faster in a vacuum than it does in water. The refractive index determines how light is bent, or refracted. The refractive indices also determine the amount of light that is reflected when reaching the interface, as well as the angle for total internal reflection. This implies that vacuum has a index of 1. The refractive index varies with the wavelength of light and this is called dispersion and causes the splitting of white light into its constituent colors in prisms and rainbows, and chromatic aberration in lenses. Light propagation in absorbing materials can be described using a refractive index. The imaginary part then handles the attenuation, while the real part accounts for refraction, the concept of refractive index is widely used within the full electromagnetic spectrum, from X-rays to radio waves. It can also be used with wave phenomena such as sound, in this case the speed of sound is used instead of that of light and a reference medium other than vacuum must be chosen. Thomas Young was presumably the person who first used, and invented, at the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers. The ratio had the disadvantage of different appearances, newton, who called it the proportion of the sines of incidence and refraction, wrote it as a ratio of two numbers, like 529 to 396. Hauksbee, who called it the ratio of refraction, wrote it as a ratio with a fixed numerator, hutton wrote it as a ratio with a fixed denominator, like 1.3358 to 1. Young did not use a symbol for the index of refraction, in the next years, others started using different symbols, n, m, and µ. For visible light most transparent media have refractive indices between 1 and 2, a few examples are given in the adjacent table. These values are measured at the yellow doublet D-line of sodium, with a wavelength of 589 nanometers, gases at atmospheric pressure have refractive indices close to 1 because of their low density. Almost all solids and liquids have refractive indices above 1.3, aerogel is a very low density solid that can be produced with refractive index in the range from 1.002 to 1.265. Moissanite lies at the end of the range with a refractive index as high as 2.65. Most plastics have refractive indices in the range from 1.3 to 1.7, for infrared light refractive indices can be considerably higher
3.
Mathematical formula
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In science, a formula is a concise way of expressing information symbolically as in a mathematical or chemical formula. The informal use of the formula in science refers to the general construct of a relationship between given quantities. The plural of formula can be spelled either as formulas or formulae, in mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language. Note that the volume V and the radius r are expressed as single instead of words or phrases. This convention, while important in a relatively simple formula, means that mathematicians can more quickly manipulate larger. Mathematical formulas are often algebraic, closed form, and/or analytical, for example, H2O is the chemical formula for water, specifying that each molecule consists of two hydrogen atoms and one oxygen atom. Similarly, O−3 denotes an ozone molecule consisting of three atoms and having a net negative charge. In a general context, formulas are applied to provide a solution for real world problems. Some may be general, F = ma, which is one expression of Newtons second law, is applicable to a range of physical situations. Other formulas may be created to solve a particular problem, for example. In all cases, however, formulas form the basis for calculations, expressions are distinct from formulas in that they cannot contain an equals sign. Whereas formulas are comparable to sentences, expressions are more like phrases, a chemical formula identifies each constituent element by its chemical symbol and indicates the proportionate number of atoms of each element. In empirical formulas, these begin with a key element and then assign numbers of atoms of the other elements in the compound. For molecular compounds, these numbers can all be expressed as whole numbers. For example, the formula of ethanol may be written C2H6O because the molecules of ethanol all contain two carbon atoms, six hydrogen atoms, and one oxygen atom. Some types of compounds, however, cannot be written with entirely whole-number empirical formulas. An example is boron carbide, whose formula of CBn is a variable non-whole number ratio with n ranging from over 4 to more than 6.5. When the chemical compound of the consists of simple molecules
4.
Angle of incidence (optics)
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In geometric optics, the angle of incidence is the angle between a ray incident on a surface and the line perpendicular to the surface at the point of incidence, called the normal. The ray can be formed by any wave, optical, acoustic, microwave, X-ray, in the figure below, the line representing a ray makes an angle θ with the normal. The angle of incidence at which light is first totally internally reflected is known as the critical angle, the angle of reflection and angle of refraction are other angles related to beams. Determining the angle of reflection with respect to a surface is trivial. The exact solution for a sphere was a problem for nearly 50 years until a closed-form result was derived by mathematicians Allen R Miller. This small angle is called an angle or grazing angle. Incidence at grazing angles is called grazing incidence, grazing incidence diffraction is used in X-ray spectroscopy and atom optics, where significant reflection can be achieved only at small values of the grazing angle. Ridged mirrors are designed for reflection of atoms coming at small grazing angle and this angle is usually measured in milliradians. In optics, there is Lloyds mirror, effect of sun angle on climate Reflection Refraction Season Total internal reflection Weisstein, Eric W. Angle of incidence. Geometry, rebound on the strip billiards Flash animation
5.
Wave
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In physics, a wave is an oscillation accompanied by a transfer of energy that travels through a medium. Frequency refers to the addition of time, wave motion transfers energy from one point to another, which displace particles of the transmission medium–that is, with little or no associated mass transport. Waves consist, instead, of oscillations or vibrations, around almost fixed locations, there are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed, restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbors, when the molecules collide, they also bounce away from each other. This keeps the molecules from continuing to travel in the direction of the wave, the second main type, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields generated by charged particles. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this varies depending on the type of wave. Further, the behavior of particles in quantum mechanics are described by waves, in addition, gravitational waves also travel through space, which are a result of a vibration or movement in gravitational fields. While mechanical waves can be transverse and longitudinal, all electromagnetic waves are transverse in free space. A single, all-encompassing definition for the wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value, however, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon as a results in a blurred line. The term wave is often understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium and it may appear that the description of waves is closely related to their physical origin for each specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a rather than an electromagnetic wave transfer caused by vibration. Concepts such as mass, momentum, inertia, or elasticity and this difference in origin introduces certain wave characteristics particular to the properties of the medium involved
6.
Isotropy
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Isotropy is uniformity in all orientations, it is derived from the Greek isos and tropos. Precise definitions depend on the subject area, exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary systematically, Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. Within mathematics, isotropy has a few different meanings, Isotropic manifolds A manifold is isotropic if the geometry on the manifold is the same regardless of direction, a manifold can be homogeneous without being isotropic, but if it is inhomogeneous, it is necessarily anisotropic. Isotropic quadratic form A quadratic form q is said to be if there is a non-zero vector v such that q =0. In complex geometry, a line through the origin in the direction of a vector is an isotropic line. Isotropic coordinates Isotropic coordinates are coordinates on a chart for Lorentzian manifolds. Isotropy group An isotropy group is the group of isomorphisms from any object to itself in a groupoid, Isotropic position A probability distribution over a vector space is in isotropic position if its covariance matrix is the identity. This follows from invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential. Kinetic theory of gases is also an example of isotropy and it is assumed that the molecules move in random directions and as a consequence, there is an equal probability of a molecule moving in any direction. Thus when there are molecules in the gas, with high probability there will be very similar numbers moving in one direction as any other hence demonstrating approximate isotropy. Fluid dynamics Fluid flow is isotropic if there is no directional preference, an example of anisotropy is in flows with a background density as gravity works in only one direction. The apparent surface separating two differing isotropic fluids would be referred to as an isotrope, thermal expansion A solid is said to be isotropic if the expansion of solid is equal in all directions when thermal energy is provided to the solid. Electromagnetics An isotropic medium is one such that the permittivity, ε, and permeability, μ, of the medium are uniform in all directions of the medium, optics Optical isotropy means having the same optical properties in all directions. The individual reflectance or transmittance of the domains is averaged if the macroscopic reflectance or transmittance is to be calculated, cosmology The Big Bang theory of the evolution of the observable universe assumes that space is isotropic. It also assumes that space is homogeneous and these two assumptions together are known as the Cosmological Principle. As of 2006, the observations suggest that, on scales much larger than galaxies, galaxy clusters are Great features. Here homogeneous means that the universe is the same everywhere and isotropic implies that there is no preferred direction, in the study of mechanical properties of materials, isotropic means having identical values of a property in all directions
7.
Ray tracing (physics)
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In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces, Ray tracing solves the problem by repeatedly advancing idealized narrow beams called rays through the medium by discrete amounts. Simple problems can be analyzed by propagating a few rays using simple mathematics, more detailed analysis can be performed by using a computer to propagate many rays. Ray theory does not describe phenomena such as interference and diffraction, the ray tracer will advance the ray over his distance, and then use a local derivative of the medium to calculate the rays new direction. From this location, a new ray is sent out and the process is repeated until a path is generated. If the simulation includes solid objects, the ray may be tested for intersection with them at each step, other properties of the ray may be altered as the simulation advances as well, such as intensity, wavelength, or polarization. The process is repeated with as many rays as are necessary to understand the behavior of the system. One particular form of ray tracing is radio signal ray tracing and this form of ray tracing involves the integration of differential equations that describe the propagation of electromagnetic waves through dispersive and anisotropic media such as the ionosphere. An example of physics-based radio signal ray tracing is shown to the right, radio communicators use ray tracing to help determine the precise behavior of radio signals as they propagate through the ionosphere. The image at the right illustrates the complexity of the situation, two sets of signals are broadcast at two different elevation angles. When the main signal penetrates into the ionosphere, the magnetic field splits the signal into two component waves which are separately ray traced through the ionosphere, the ordinary wave component follows a path completely independent of the extraordinary wave component. Sound velocity in the ocean varies with depth due to changes in density and temperature and this local minimum, called the SOFAR channel, acts as a waveguide, as sound tends to bend towards it. From this, locations of high and low signal intensity may be computed, which are useful in the fields of acoustics, underwater acoustic communication. Ray tracing may be used in the design of lenses and optical systems, such as in cameras, microscopes, telescopes, and binoculars, and its application in this field dates back to the 1900s. Geometric ray tracing is used to describe the propagation of light rays through a system or optical instrument. For the application of design, two special cases of wave interference are important to account for. In a focal point, rays from a point light source meet again, within a very small region near this point, incoming light may be approximated by plane waves which inherit their direction from the rays. The optical path length from the source is used to compute the phase
8.
Metamaterial
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A metamaterial is a material engineered to have a property that is not found in nature. They are made from assemblies of multiple elements fashioned from materials such as metals or plastics. The materials are arranged in repeating patterns, at scales that are smaller than the wavelengths of the phenomena they influence. Metamaterials derive their properties not from the properties of the base materials, appropriately designed metamaterials can affect waves of electromagnetic radiation or sound in a manner not observed in bulk materials. Those that exhibit a negative index of refraction for particular wavelengths have attracted significant research and these materials are known as negative-index metamaterials. Metamaterials offer the potential to create superlenses, such a lens could allow imaging below the diffraction limit that is the minimum resolution that can be achieved by conventional glass lenses. A form of invisibility was demonstrated using gradient-index materials, acoustic and seismic metamaterials are also research areas. Explorations of artificial materials for manipulating electromagnetic waves began at the end of the 19th century, some of the earliest structures that may be considered metamaterials were studied by Jagadish Chandra Bose, who in 1898 researched substances with chiral properties. Karl Ferdinand Lindman studied wave interaction with metallic helices as artificial chiral media in the twentieth century. Winston E. Kock developed materials that had characteristics to metamaterials in the late 1940s. In the 1950s and 1960s, artificial dielectrics were studied for lightweight microwave antennas, microwave radar absorbers were researched in the 1980s and 1990s as applications for artificial chiral media. Negative-index materials were first described theoretically by Victor Veselago in 1967 and he proved that such materials could transmit light. He showed that the phase velocity could be made anti-parallel to the direction of Poynting vector and this is contrary to wave propagation in naturally occurring materials. John Pendry was the first to identify a practical way to make a left-handed metamaterial, such a material allows an electromagnetic wave to convey energy against its phase velocity. Pendrys idea was that metallic wires aligned along the direction of a wave could provide negative permittivity, natural materials display negative permittivity, the challenge was achieving negative permeability. In 1999 Pendry demonstrated that a ring with its axis placed along the direction of wave propagation could do so. In the same paper, he showed that an array of wires. Pendry also proposed a related negative-permeability design, the Swiss roll, in 2000, Smith et al. reported the experimental demonstration of functioning electromagnetic metamaterials by horizontally stacking, periodically, split-ring resonators and thin wire structures
9.
Sine
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In mathematics, the sine is a trigonometric function of an angle. More generally, the definition of sine can be extended to any value in terms of the length of a certain line segment in a unit circle. The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy, via translation from Sanskrit to Arabic and then from Arabic to Latin. The word sine comes from a Latin mistranslation of the Arabic jiba, to define the trigonometric functions for an acute angle α, start with any right triangle that contains an angle of measure α, in the accompanying figure, angle A in triangle ABC has measure α. The three sides of the triangle are named as follows, The opposite side is the side opposite to the angle of interest, the hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle, the adjacent side is the remaining side, in this case side b. It forms a side of both the angle of interest and the right angle, once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse. As stated, the value sin appears to depend on the choice of right triangle containing an angle of measure α, however, this is not the case, all such triangles are similar, and so the ratio is the same for each of them. The trigonometric functions can be defined in terms of the rise, run, when the length of the line segment is 1, sine takes an angle and tells the rise. Sine takes an angle and tells the rise per unit length of the line segment, rise is equal to sin θ multiplied by the length of the line segment. In contrast, cosine is used for telling the run from the angle, arctan is used for telling the angle from the slope. The line segment is the equivalent of the hypotenuse in the right-triangle, in trigonometry, a unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. Let a line through the origin, making an angle of θ with the half of the x-axis. The x- and y-coordinates of this point of intersection are equal to cos θ and sin, the points distance from the origin is always 1. Unlike the definitions with the triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function. Exact identities, These apply for all values of θ. sin = cos =1 csc The reciprocal of sine is cosecant, i. e. the reciprocal of sin is csc, or cosec. Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side, the inverse function of sine is arcsine or inverse sine
10.
Pierre de Fermat
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He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermats principle for light propagation and his Fermats Last Theorem in number theory, Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a leather merchant. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth, there is little evidence concerning his school education, but it was probably at the Collège de Navarre in Montauban. He attended the University of Orléans from 1623 and received a bachelor in law in 1626. In Bordeaux he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apolloniuss De Locis Planis to one of the mathematicians there, there he became much influenced by the work of François Viète. In 1630, he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France and he held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat, fluent in six languages, Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often little or no proof of his theorems. In some of these letters to his friends he explored many of the ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession, nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus. Secrecy was common in European mathematical circles at the time and this naturally led to priority disputes with contemporaries such as Descartes and Wallis. Anders Hald writes that, The basis of Fermats mathematics was the classical Greek treatises combined with Vietas new algebraic methods, Fermats pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes famous La géométrie. This manuscript was published posthumously in 1679 in Varia opera mathematica, in these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of power functions. With his method, he was able to reduce this evaluation to the sum of geometric series, the resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pells equation, perfect numbers, amicable numbers and it was while researching perfect numbers that he discovered Fermats little theorem. Fermat developed the two-square theorem, and the polygonal number theorem, although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived
11.
Fermat's principle
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This principle is sometimes taken as the definition of a ray of light. In other words, a ray of light prefers the path such that there are paths, arbitrarily nearby on either side. Fermats principle can be used to describe the properties of light reflected off mirrors, refracted through different media. It follows mathematically from Huygens principle, Fermats text Analyse des réfractions exploits the technique of adequality to derive Snells law of refraction and the law of reflection. Fermats principle has the form as Hamiltons principle and it is the basis of Hamiltonian optics. The optical path length of a ray from a point A to a point B is defined by, S = ∫ A B n d s, the optical path length is a purely geometrical quantity since time is not considered in its calculation. An extremum in the travel time between two points A and B is equivalent to an extremum of the optical path length between those two points. The historical form proposed by Fermat is incomplete, a complete modern statement of the variational Fermat principle is that the optical length of the path followed by light between two fixed points, A and B, is an extremum. The optical length is defined as the length multiplied by the refractive index of the material. Function L is the optical Lagrangian from which the Lagrangian and Hamiltonian formulations of geometrical optics may be derived, classically, Fermats principle can be considered as a mathematical consequence of Huygens principle. Indeed, of all secondary waves the waves with the extrema paths contribute most due to constructive interference, suppose that light waves propagate from A to B by all possible routes ABj, unrestricted initially by rules of geometrical or physical optics. The various optical paths ABj will vary by amounts greatly in excess of one wavelength, waves along and close to this shortest route will thus dominate and AB0 will be the route along which the light is seen to travel. Thus, because the extremal paths cannot be canceled out. In humans, for example, Fermats principle can be demonstrated in a situation when a lifeguard has to find the fastest way to both beach and water in order to reach a drowning swimmer. In the classic mechanics of waves, Fermats principle follows from the principle of mechanics. In fact, the basic principle is in the myth of the Allegory of the Cave by Plato. Ibn al-Haytham, in his Book of Optics, expanded the principle to both reflection and refraction, and expressed an early version of the principle of least time and his experiments were based on earlier works on refraction carried out by the Greek scientist Ptolemy. The generalized principle of least time in its form was stated by Fermat in a letter dated January 1,1662
12.
Ptolemy
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Claudius Ptolemy was a Greek writer, known as a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in the city of Alexandria in the Roman province of Egypt, wrote in Koine Greek, beyond that, few reliable details of his life are known. His birthplace has been given as Ptolemais Hermiou in the Thebaid in a statement by the 14th-century astronomer Theodore Meliteniotes. This is a very late attestation, however, and there is no reason to suppose that he ever lived elsewhere than Alexandria. Ptolemy wrote several treatises, three of which were of importance to later Byzantine, Islamic and European science. The first is the astronomical treatise now known as the Almagest, although it was entitled the Mathematical Treatise. The second is the Geography, which is a discussion of the geographic knowledge of the Greco-Roman world. The third is the treatise in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day. This is sometimes known as the Apotelesmatika but more known as the Tetrabiblos from the Greek meaning Four Books or by the Latin Quadripartitum. The name Claudius is a Roman nomen, the fact that Ptolemy bore it indicates he lived under the Roman rule of Egypt with the privileges and political rights of Roman citizenship. It would have suited custom if the first of Ptolemys family to become a citizen took the nomen from a Roman called Claudius who was responsible for granting citizenship, if, as was common, this was the emperor, citizenship would have been granted between AD41 and 68. The astronomer would also have had a praenomen, which remains unknown and it occurs once in Greek mythology, and is of Homeric form. All the kings after him, until Egypt became a Roman province in 30 BC, were also Ptolemies, abu Mashar recorded a belief that a different member of this royal line composed the book on astrology and attributed it to Ptolemy. The correct answer is not known”, Ptolemy wrote in Greek and can be shown to have utilized Babylonian astronomical data. He was a Roman citizen, but most scholars conclude that Ptolemy was ethnically Greek and he was often known in later Arabic sources as the Upper Egyptian, suggesting he may have had origins in southern Egypt. Later Arabic astronomers, geographers and physicists referred to him by his name in Arabic, Ptolemys Almagest is the only surviving comprehensive ancient treatise on astronomy. Ptolemy presented his models in convenient tables, which could be used to compute the future or past position of the planets. The Almagest also contains a catalogue, which is a version of a catalogue created by Hipparchus
13.
Alexandria
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Alexandria is the second largest city and a major economic centre in Egypt, extending about 32 km along the coast of the Mediterranean Sea in the north central part of the country. Its low elevation on the Nile delta makes it vulnerable to rising sea levels. Alexandria is Egypts largest seaport, serving approximately 80% of Egypts imports and exports and it is an important industrial center because of its natural gas and oil pipelines from Suez. Alexandria is also an important tourist destination, Alexandria was founded around a small Ancient Egyptian town c.331 BC by Alexander the Great. Alexandria was the second most powerful city of the ancient world after Rome, Alexandria is believed to have been founded by Alexander the Great in April 331 BC as Ἀλεξάνδρεια. Alexanders chief architect for the project was Dinocrates, Alexandria was intended to supersede Naucratis as a Hellenistic center in Egypt, and to be the link between Greece and the rich Nile valley. The city and its museum attracted many of the greatest scholars, including Greeks, Jews, the city was later plundered and lost its significance. Just east of Alexandria, there was in ancient times marshland, as early as the 7th century BC, there existed important port cities of Canopus and Heracleion. The latter was rediscovered under water. An Egyptian city, Rhakotis, already existed on the shore also and it continued to exist as the Egyptian quarter of the city. A few months after the foundation, Alexander left Egypt and never returned to his city, after Alexanders departure, his viceroy, Cleomenes, continued the expansion. Although Cleomenes was mainly in charge of overseeing Alexandrias continuous development, the Heptastadion, inheriting the trade of ruined Tyre and becoming the center of the new commerce between Europe and the Arabian and Indian East, the city grew in less than a generation to be larger than Carthage. In a century, Alexandria had become the largest city in the world and and it became Egypts main Greek city, with Greek people from diverse backgrounds. Alexandria was not only a center of Hellenism, but was home to the largest urban Jewish community in the world. The Septuagint, a Greek version of the Tanakh, was produced there, in AD115, large parts of Alexandria were destroyed during the Kitos War, which gave Hadrian and his architect, Decriannus, an opportunity to rebuild it. On 21 July 365, Alexandria was devastated by a tsunami, the Islamic prophet, Muhammads first interaction with the people of Egypt occurred in 628, during the Expedition of Zaid ibn Haritha. He sent Hatib bin Abi Baltaeh with a letter to the king of Egypt and Alexandria called Muqawqis In the letter Muhammad said, I invite you to accept Islam, Allah the sublime, shall reward you doubly. But if you refuse to do so, you bear the burden of the transgression of all the Copts
14.
Confirmation bias
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Confirmation bias, also called confirmatory bias or myside bias, is the tendency to search for, interpret, favor, and recall information in a way that confirms ones preexisting beliefs or hypotheses. It is a type of bias and a systematic error of inductive reasoning. People display this bias when they gather or remember information selectively, the effect is stronger for emotionally charged issues and for deeply entrenched beliefs. People also tend to interpret ambiguous evidence as supporting their existing position, biased search, interpretation and memory have been invoked to explain attitude polarization, belief perseverance, the irrational primacy effect and illusory correlation. A series of experiments in the 1960s suggested that people are biased toward confirming their existing beliefs, later work re-interpreted these results as a tendency to test ideas in a one-sided way, focusing on one possibility and ignoring alternatives. In certain situations, this tendency can bias peoples conclusions, explanations for the observed biases include wishful thinking and the limited human capacity to process information. Another explanation is that people show confirmation bias because they are weighing up the costs of being wrong, rather than investigating in a neutral, confirmation biases contribute to overconfidence in personal beliefs and can maintain or strengthen beliefs in the face of contrary evidence. Poor decisions due to these biases have been found in political and organizational contexts, confirmation biases are effects in information processing. Others apply the more broadly to the tendency to preserve ones existing beliefs when searching for evidence, interpreting it. Experiments have found repeatedly that people tend to test hypotheses in a one-sided way, rather than searching through all the relevant evidence, they phrase questions to receive an affirmative answer that supports their theory. They look for the consequences that they would expect if their hypothesis were true, for example, someone using yes/no questions to find a number he or she suspects to be the number 3 might ask, Is it an odd number. People prefer this type of question, called a positive test, would yield exactly the same information. However, this does not mean that people seek tests that guarantee a positive answer, in studies where subjects could select either such pseudo-tests or genuinely diagnostic ones, they favored the genuinely diagnostic. The preference for positive tests in itself is not a bias, however, in combination with other effects, this strategy can confirm existing beliefs or assumptions, independently of whether they are true. In real-world situations, evidence is often complex and mixed, for example, various contradictory ideas about someone could each be supported by concentrating on one aspect of his or her behavior. Thus any search for evidence in favor of a hypothesis is likely to succeed, one illustration of this is the way the phrasing of a question can significantly change the answer. For example, people who are asked, Are you happy with your social life, report greater satisfaction than those asked, Are you unhappy with your social life. Even a small change in a questions wording can affect how people search through available information and this was shown using a fictional child custody case
15.
Ibn al-Haytham
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Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, also known by the Latinization Alhazen or Alhacen, was an Arab Muslim scientist, mathematician, astronomer, and philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, mathematics and he was the first to explain that vision occurs when light bounces on an object and then is directed to ones eyes. He spent most of his close to the court of the Fatimid Caliphate in Cairo and earned his living authoring various treatises. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called The Physicist and he is also sometimes called al-Baṣrī after his birthplace Basra in Iraq, or al-Miṣrī. Ibn al-Haytham was born c.965 in Basra, which was part of the Buyid emirate. Alhazen arrived in Cairo under the reign of Fatimid Caliph al-Hakim, Alhazen continued to live in Cairo, in the neighborhood of the famous University of al-Azhar, until his death in 1040. Legend has it that after deciding the scheme was impractical and fearing the caliphs anger, during this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy. Among his students were Sorkhab, a Persian from Semnan who was his student for three years, and Abu al-Wafa Mubashir ibn Fatek, an Egyptian prince who learned mathematics from Alhazen. Alhazen made significant contributions to optics, number theory, geometry, astronomy, Alhazens work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the prince of the Banu Hud dynasty of Zaragossa and author of an important mathematical text. A Latin translation of the Kitab al-Manazir was made probably in the twelfth or early thirteenth century. His research in catoptrics centred on spherical and parabolic mirrors and spherical aberration and he made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the known as Alhazens problem. Alhazen wrote as many as 200 books, although only 55 have survived, some of his treatises on optics survived only through Latin translation. During the Middle Ages his books on cosmology were translated into Latin, Hebrew, the crater Alhazen on the Moon is named in his honour, as was the asteroid 59239 Alhazen. In honour of Alhazen, the Aga Khan University named its Ophthalmology endowed chair as The Ibn-e-Haitham Associate Professor, Alhazen, by the name Ibn al-Haytham, is featured on the obverse of the Iraqi 10, 000-dinar banknote issued in 2003, and on 10-dinar notes from 1982. The 2015 International Year of Light celebrated the 1000th anniversary of the works on optics by Ibn Al-Haytham, Alhazens most famous work is his seven-volume treatise on optics Kitab al-Manazir, written from 1011 to 1021. Optics was translated into Latin by a scholar at the end of the 12th century or the beginning of the 13th century
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Book of Optics
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The Book of Optics is a seven-volume treatise on optics and other fields of study composed by the medieval Arab scholar Ibn al-Haytham, known in the West as Alhazen or Alhacen. Alhazens work extensively affected the development of optics in Europe between 1260 and 1650, before the Book of Optics was written, two theories of vision existed. The extramission or emission theory was forwarded by the mathematicians Euclid and Ptolemy, when these rays reached the object they allowed the viewer to perceive its color, shape and size. The intromission theory, held by the followers of Aristotle and Galen, argued that sight was caused by agents, al-Haytham offered many reasons against the extramission theory, pointing to the fact that eyes can be damaged by looking directly at bright lights, such as the sun. He claimed the low probability that the eye can fill the entirety of space as soon as the eyelids are opened as an observer looks up into the night sky. According to this theory, the object being viewed is considered to be a compilation of an amount of points. In the Book of Optics, al-Haytham claimed the existence of primary and secondary light, the book describes how the essential form of light comes from self-luminous bodies and that accidental light comes from objects that obtain and emit light from those self-luminous bodies. According to Ibn al-Haytham, primary light comes from self-luminous bodies, accidental light can only exist if there is a source of primary light. Both primary and secondary light travel in straight lines, transparency is a characteristic of a body that can transmit light through them, such as air and water, although no body can completely transmit light or be entirely transparent. Opaque objects are those through which light cannot pass through directly, opaque objects are struck with light and can become luminous bodies themselves which radiate secondary light. Light can be refracted by going through partially transparent objects and can also be reflected by striking smooth objects such as mirrors, al-Haytham presented many experiments in Optics that upheld his claims about light and its transmission. He also claimed that acts much like light, being a distinct quality of a form. Through experimentation he concluded that color cannot exist without air, as objects radiate light in straight lines in all directions, the eye must also be hit with this light over its outer surface. This idea presented a problem for al-Haytham and his predecessors, as if this was the case, al-Haytham solved this problem using his theory of refraction. According to al-Haytham, this causes them to be refracted and weakened and he claimed that all the rays other than the one that hits the eye perpendicularly are not involved in vision. Other parts of the eye are the aqueous humor in front of the crystalline humor and these, however, do not play as critical of a role in vision as the crystalline humor. The crystalline humor transmits the image it perceives to the brain through an optic nerve, Book I - Book I deals with al-Haythams theories on light, colors, and vision. Book II - Book II is where al-Haytham presents his theory of visual perception, Book III and Book VI - Book III and Book VI present al-Haythams ideas on the errors in visual perception with Book VI focusing on errors related to reflection
17.
Willebrord Snellius
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Willebrord Snellius was a Dutch astronomer and mathematician, known in the English-speaking world as Snell. The same law was also investigated by Ptolemy and in the Middle Ages by Witelo, the lunar crater Snellius is named after Willebrord Snellius. The Royal Netherlands Navy has named three ships after Snellius, including a currently-serving vessel. Willebrord Snellius was born in Leiden, Netherlands, in 1613 he succeeded his father, Rudolph Snel van Royen as professor of mathematics at the University of Leiden. He was helped in his measurements by two of his students, the Austrian barons Erasmus and Casparus Sterrenberg, in several cities he also received support of friends among the city leaders. In his work The terrae Ambitus vera quantitate under the authors name Snellius describes the methods he used and he came up with an estimate of 28,500 Rhineland rods - in modern units 107.37 km for one degree of latitude. 360 times 107.37 then gives a circumference of the Earth of 38,653 km, the actual circumference is 40,075 kilometers, so Snellius underestimated the circumference of the earth by 3. 5%. Snellius came to his result by calculating the distances between a number of points in the plain west and southwest of the Netherlands using triangulation. In order to carry out these measurements accurately Snellius had a large quadrant built and this quadrant can still be seen in the Museum Boerhaave in Leiden. In a network of fourteen cities a total of 53 triangulation measurements were made, in his calculations Snellius made use of a solution for what is now called the Snellius–Pothenot problem. By necessity Snellius his high points were nearly all church spires, there were hardly any other tall buildings in Snellius time in the west of the Netherlands. More or less ordered from north to south and/or in successive order of measuring Snellius used a network of fourteen points, Alkmaar. The difference in latitude between Alkmaar and Breda is 1.0436 degree, assuming Snellius corrected for this he must have calculated a distance of 107.37 *1.0436 =112.05 kilometers between the Sint-Laurenskerk in Alkmaar and the Grote Kerk in Breda. Snellius was also a mathematician, producing a new method for calculating π—the first such improvement since ancient times. He rediscovered the law of refraction in 1621, in addition to the Eratosthenes Batavus, he published Cyclometricus, de circuli dimensione, and Tiphys Batavus. He also edited Coeli et siderum in eo errantium observationes Hassiacae, a work on trigonometry authored by Snellius was published a year after his death. Snellius died at Leiden on October 1626, at the age of 46 from an illness diagnosed as colic, oConnor, John J. Robertson, Edmund F. Willebrord van Royen Snell, MacTutor History of Mathematics archive, University of St Andrews. This article incorporates text from a now in the public domain, Chisholm, Hugh, ed. Snell
18.
Baghdad
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Baghdad is the capital of the Republic of Iraq. The population of Baghdad, as of 2016, is approximately 8,765,000 making it the largest city in Iraq, the second largest city in the Arab world, and the second largest city in Western Asia. Located along the Tigris River, the city was founded in the 8th century, within a short time of its inception, Baghdad evolved into a significant cultural, commercial, and intellectual center for the Islamic world. This, in addition to housing several key institutions, garnered the city a worldwide reputation as the Centre of Learning. Throughout the High Middle Ages, Baghdad was considered to be the largest city in the world with a population of 1,200,000 people. The city was destroyed at the hands of the Mongol Empire in 1258, resulting in a decline that would linger through many centuries due to frequent plagues. With the recognition of Iraq as an independent state in 1938, in contemporary times, the city has often faced severe infrastructural damage, most recently due to the 2003 invasion of Iraq, and the subsequent Iraq War that lasted until December 2011. In recent years, the city has been subjected to insurgency attacks. As of 2012, Baghdad was listed as one of the least hospitable places in the world to live, the site where the city of Baghdad developed has been populated for millennia. By the 8th century AD, several villages had developed there, including a Persian hamlet called Baghdad, the name is of Indo-European origin and a Middle Persian compound of Bagh god and dād given by, translating to Bestowed by God or Gods gift. In Old Persian the first element can be traced to boghu and is related to Slavic bog god, a similar term in Middle Persian is the name Mithradāt, known in English by its Hellenistic form Mithridates, meaning gift of Mithra. There are a number of locations in the wider region whose names are compounds of the word bagh, including Baghlan. The name of the town Baghdati in Georgia shares the same etymological origins, when the Abbasid caliph, al-Mansur, founded a completely new city for his capital, he chose the name Madinat al-Salaam or City of Peace. This was the name on coins, weights, and other official usage. By the 11th century, Baghdad became almost the exclusive name for the world-renowned metropolis, after the fall of the Umayyads, the first Muslim dynasty, the victorious Abbasid rulers wanted their own capital whence they could rule. They chose a site north of the Sassanid capital of Ctesiphon, on 30 July 762, the caliph Al-Mansur commissioned the construction of the city, mansur believed that Baghdad was the perfect city to be the capital of the Islamic empire under the Abbasids. Mansur loved the site so much he is quoted saying, This is indeed the city that I am to found, where I am to live, and where my descendants will reign afterward. The citys growth was helped by its excellent location, based on at least two factors, it had control over strategic and trading routes along the Tigris, the abundance of water in a dry climate
19.
Thomas Harriot
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Thomas Harriot — or spelled Harriott, Hariot, or Heriot — was an English astronomer, mathematician, ethnographer, and translator. He is sometimes credited with the introduction of the potato to the British Isles, Harriot was the first person to make a drawing of the Moon through a telescope, on 26 July 1609, over four months before Galileo. After graduating from St Mary Hall, Oxford, Harriot travelled to the Americas, accompanying the 1585 expedition to Roanoke island funded by Sir Walter Raleigh and led by Sir Ralph Lane. Harriot was a member of the venture, having translated and learned the Carolina Algonquian language from two Native Americans, Wanchese and Manteo. On his return to England he worked for the 9th Earl of Northumberland, at the Earls house, he became a prolific mathematician and astronomer to whom the theory of refraction is attributed. Born in 1560 in Oxford, England, Thomas Harriot attended St Mary Hall and his name appears in the halls registry dating from 1577. Prior to his expedition with Raleigh, Harriot wrote a treatise on navigation, in addition, he made efforts to communicate with Manteo and Wanchese, two Native Americans who had been brought to England. Harriot devised an alphabet to transcribe their Carolina Algonquian language. Harriot and Manteo spent many days in one company, Harriot interrogated Manteo closely about life in the New World. In addition, he recorded the sense of awe with which the Native Americans viewed European technology, Many things they sawe with us. as mathematical instruments, as the only Englishman who had learned Algonkin prior to the voyage, Harriot was vital to the success of the expedition. His account of the voyage, named A Briefe and True Report of the New Found Land of Virginia, was published in 1588. The True Report contains an account of the Native American population encountered by the expedition, it proved very influential upon later English explorers. He wrote, Whereby it may be hoped, if means of government be used, that they may in short time be brought to civility. At the same time, his views of Native Americans industry and capacity to learn were later largely ignored in favour of the parts of the True Report about extractable minerals and resources. As a scientific adviser during the voyage, Harriot was asked by Raleigh to find the most efficient way to stack cannonballs on the deck of the ship. His ensuing theory about the close-packing of spheres shows a resemblance to atomism and modern atomic theory. His correspondence about optics with Johannes Kepler, in which he described some of his ideas, Harriott was employed for many years by Henry Percy, 9th Earl of Northumberland, with whom he resided at Syon House, which was run by Henry Percys cousin Thomas Percy. Harriot himself was interrogated and briefly imprisoned but was soon released, Walter Warner, Robert Hues, William Lower, and other scientists were present around the Earl of Northumberlands mansion as they worked for him and assisted in the teaching of the familys children
20.
Johannes Kepler
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Johannes Kepler was a German mathematician, astronomer, and astrologer. A key figure in the 17th-century scientific revolution, he is best known for his laws of motion, based on his works Astronomia nova, Harmonices Mundi. These works also provided one of the foundations for Isaac Newtons theory of universal gravitation, Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague and he was also a mathematics teacher in Linz, and an adviser to General Wallenstein. Kepler lived in an era when there was no distinction between astronomy and astrology, but there was a strong division between astronomy and physics. Kepler was born on December 27, the feast day of St John the Evangelist,1571 and his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline and his father, Heinrich Kepler, earned a precarious living as a mercenary, and he left the family when Johannes was five years old. He was believed to have died in the Eighty Years War in the Netherlands and his mother Katharina Guldenmann, an innkeepers daughter, was a healer and herbalist. Born prematurely, Johannes claimed to have weak and sickly as a child. Nevertheless, he often impressed travelers at his grandfathers inn with his phenomenal mathematical faculty and he was introduced to astronomy at an early age, and developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, in 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being called outdoors to see it and that the moon appeared quite red. However, childhood smallpox left him with vision and crippled hands. In 1589, after moving through grammar school, Latin school, there, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who also taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590. He proved himself to be a mathematician and earned a reputation as a skilful astrologer. Under the instruction of Michael Maestlin, Tübingens professor of mathematics from 1583 to 1631 and he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and he accepted the position in April 1594, at the age of 23. Keplers first major work, Mysterium Cosmographicum, was the first published defense of the Copernican system
21.
Speed of light
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The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299792458 metres per second, it is exact because the unit of length, the metre, is defined from this constant, according to special relativity, c is the maximum speed at which all matter and hence information in the universe can travel. It is the speed at which all particles and changes of the associated fields travel in vacuum. Such particles and waves travel at c regardless of the motion of the source or the reference frame of the observer. In the theory of relativity, c interrelates space and time, the speed at which light propagates through transparent materials, such as glass or air, is less than c, similarly, the speed of radio waves in wire cables is slower than c. The ratio between c and the speed v at which light travels in a material is called the index n of the material. In communicating with distant space probes, it can take minutes to hours for a message to get from Earth to the spacecraft, the light seen from stars left them many years ago, allowing the study of the history of the universe by looking at distant objects. The finite speed of light limits the theoretical maximum speed of computers. The speed of light can be used time of flight measurements to measure large distances to high precision. Ole Rømer first demonstrated in 1676 that light travels at a speed by studying the apparent motion of Jupiters moon Io. In 1865, James Clerk Maxwell proposed that light was an electromagnetic wave, in 1905, Albert Einstein postulated that the speed of light c with respect to any inertial frame is a constant and is independent of the motion of the light source. He explored the consequences of that postulate by deriving the theory of relativity and in doing so showed that the parameter c had relevance outside of the context of light and electromagnetism. After centuries of increasingly precise measurements, in 1975 the speed of light was known to be 299792458 m/s with a measurement uncertainty of 4 parts per billion. In 1983, the metre was redefined in the International System of Units as the distance travelled by light in vacuum in 1/299792458 of a second, as a result, the numerical value of c in metres per second is now fixed exactly by the definition of the metre. The speed of light in vacuum is usually denoted by a lowercase c, historically, the symbol V was used as an alternative symbol for the speed of light, introduced by James Clerk Maxwell in 1865. In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used c for a different constant later shown to equal √2 times the speed of light in vacuum, in 1894, Paul Drude redefined c with its modern meaning. Einstein used V in his original German-language papers on special relativity in 1905, but in 1907 he switched to c, sometimes c is used for the speed of waves in any material medium, and c0 for the speed of light in vacuum. This article uses c exclusively for the speed of light in vacuum, since 1983, the metre has been defined in the International System of Units as the distance light travels in vacuum in 1⁄299792458 of a second
22.
Apollonius of Perga
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Apollonius of Perga was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic and his definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. Apollonius worked on other topics, including astronomy. Most of the work has not survived except in references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, for such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Palestinian commentator, Eutocius of Ascalon, on Apollonius’ major work, Conics, states, “Apollonius, the geometrician. Came from Perga in Pamphylia in the times of Ptolemy Euergetes, the ruins of the city yet stand. It was a center of Hellenistic culture, Euergetes, “benefactor, ” identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC, times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain, the approximate times of Apollonius are thus certain, but no exact dates can be given. The figure Specific birth and death years stated by the scholars are only speculative. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt, never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. Someone designated “of Perga” might well be expected to have lived and worked there, to the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria, philip was assassinated in 336 BC. Alexander went on to fulfill his plan by conquering the vast Iranian empire, the material is located in the surviving false “Prefaces” of the books of his Conics. These are letters delivered to friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that Conics was initially requested by a house guest at Alexandria, Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” and he intended to verify and emend the books, releasing each one as it was completed
23.
Pappus of Alexandria
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Pappus of Alexandria was one of the last great Alexandrian mathematicians of Antiquity, known for his Synagoge or Collection, and for Pappuss hexagon theorem in projective geometry. Nothing is known of his life, other than, that he had a son named Hermodorus, Collection, his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a range of topics, including geometry, recreational mathematics, doubling the cube, polygons. Pappus flourished in the 4th century AD, in a period of general stagnation in mathematical studies, he stands out as a remarkable exception. In this respect the fate of Pappus strikingly resembles that of Diophantus, in his surviving writings, Pappus gives no indication of the date of the authors whose works he makes use of, or of the time at which he himself wrote. If no other information were available, all that could be known would be that he was later than Ptolemy, whom he quotes, and earlier than Proclus. The Suda states that Pappus was of the age as Theon of Alexandria. A different date is given by a note to a late 10th-century manuscript, which states, next to an entry on Emperor Diocletian. This works out as October 18,320 AD, and so Pappus must have flourished c.320 AD. The great work of Pappus, in eight books and titled Synagoge or Collection, has not survived in complete form, the first book is lost, and the rest have suffered considerably. The Suda enumerates other works of Pappus, Χωρογραφία οἰκουμενική, commentary on the 4 books of Ptolemys Almagest, Ποταμοὺς τοὺς ἐν Λιβύῃ, Pappus himself mentions another commentary of his own on the Ἀνάλημμα of Diodorus of Alexandria. Pappus also wrote commentaries on Euclids Elements, and on Ptolemys Ἁρμονικά and these discoveries form, in fact, a text upon which Pappus enlarges discursively. Heath considered the systematic introductions to the books as valuable, for they set forth clearly an outline of the contents. From these introductions one can judge of the style of Pappuss writing, heath also found his characteristic exactness made his Collection a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us. The portions of Collection which has survived can be summarized as follows and we can only conjecture that the lost Book I, like Book II, was concerned with arithmetic, Book III being clearly introduced as beginning a new subject. The whole of Book II discusses a method of multiplication from a book by Apollonius of Perga. The final propositions deal with multiplying together the values of Greek letters in two lines of poetry, producing two very large numbers approximately equal to 2*1054 and 2*1038. Book III contains geometrical problems, plane and solid, on the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure
24.
French language
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French is a Romance language of the Indo-European family. It descended from the Vulgar Latin of the Roman Empire, as did all Romance languages, French has evolved from Gallo-Romance, the spoken Latin in Gaul, and more specifically in Northern Gaul. Its closest relatives are the other langues doïl—languages historically spoken in northern France and in southern Belgium, French was also influenced by native Celtic languages of Northern Roman Gaul like Gallia Belgica and by the Frankish language of the post-Roman Frankish invaders. Today, owing to Frances past overseas expansion, there are numerous French-based creole languages, a French-speaking person or nation may be referred to as Francophone in both English and French. French is a language in 29 countries, most of which are members of la francophonie. As of 2015, 40% of the population is in Europe, 35% in sub-Saharan Africa, 15% in North Africa and the Middle East, 8% in the Americas. French is the fourth-most widely spoken mother tongue in the European Union, 1/5 of Europeans who do not have French as a mother tongue speak French as a second language. As a result of French and Belgian colonialism from the 17th and 18th century onward, French was introduced to new territories in the Americas, Africa, most second-language speakers reside in Francophone Africa, in particular Gabon, Algeria, Mauritius, Senegal and Ivory Coast. In 2015, French was estimated to have 77 to 110 million native speakers, approximately 274 million people are able to speak the language. The Organisation internationale de la Francophonie estimates 700 million by 2050, in 2011, Bloomberg Businessweek ranked French the third most useful language for business, after English and Standard Mandarin Chinese. Under the Constitution of France, French has been the language of the Republic since 1992. France mandates the use of French in official government publications, public education except in specific cases, French is one of the four official languages of Switzerland and is spoken in the western part of Switzerland called Romandie, of which Geneva is the largest city. French is the language of about 23% of the Swiss population. French is also a language of Luxembourg, Monaco, and Aosta Valley, while French dialects remain spoken by minorities on the Channel Islands. A plurality of the worlds French-speaking population lives in Africa and this number does not include the people living in non-Francophone African countries who have learned French as a foreign language. Due to the rise of French in Africa, the total French-speaking population worldwide is expected to reach 700 million people in 2050, French is the fastest growing language on the continent. French is mostly a language in Africa, but it has become a first language in some urban areas, such as the region of Abidjan, Ivory Coast and in Libreville. There is not a single African French, but multiple forms that diverged through contact with various indigenous African languages, sub-Saharan Africa is the region where the French language is most likely to expand, because of the expansion of education and rapid population growth
25.
Christiaan Huygens
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Christiaan Huygens, FRS was a prominent Dutch mathematician and scientist. He is known particularly as an astronomer, physicist, probabilist and horologist, Huygens was a leading scientist of his time. His work included early telescopic studies of the rings of Saturn and the discovery of its moon Titan and he published major studies of mechanics and optics, and pioneered work on games of chance. Christiaan Huygens was born on 14 April 1629 in The Hague, into a rich and influential Dutch family, Christiaan was named after his paternal grandfather. His mother was Suzanna van Baerle and she died in 1637, shortly after the birth of Huygens sister. The couple had five children, Constantijn, Christiaan, Lodewijk, Philips, Constantijn Huygens was a diplomat and advisor to the House of Orange, and also a poet and musician. His friends included Galileo Galilei, Marin Mersenne and René Descartes, Huygens was educated at home until turning sixteen years old. He liked to play with miniatures of mills and other machines and his father gave him a liberal education, he studied languages and music, history and geography, mathematics, logic and rhetoric, but also dancing, fencing and horse riding. In 1644 Huygens had as his mathematical tutor Jan Jansz de Jonge Stampioen, Descartes was impressed by his skills in geometry. His father sent Huygens to study law and mathematics at the University of Leiden, Frans van Schooten was an academic at Leiden from 1646, and also a private tutor to Huygens and his elder brother, replacing Stampioen on the advice of Descartes. Van Schooten brought his mathematical education up to date, in introducing him to the work of Fermat on differential geometry. Constantijn Huygens was closely involved in the new College, which lasted only to 1669, Christiaan Huygens lived at the home of the jurist Johann Henryk Dauber, and had mathematics classes with the English lecturer John Pell. He completed his studies in August 1649 and he then had a stint as a diplomat on a mission with Henry, Duke of Nassau. It took him to Bentheim, then Flensburg and he took off for Denmark, visited Copenhagen and Helsingør, and hoped to cross the Øresund to visit Descartes in Stockholm. While his father Constantijn had wished his son Christiaan to be a diplomat, in political terms, the First Stadtholderless Period that began in 1650 meant that the House of Orange was not in power, removing Constantijns influence. Further, he realised that his son had no interest in such a career, Huygens generally wrote in French or Latin. While still a student at Leiden he began a correspondence with the intelligencer Mersenne. Mersenne wrote to Constantijn on his sons talent for mathematics, the letters show the early interests of Huygens in mathematics
26.
Treatise on Light
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Treatise on Light is a 1690 book written by the Dutch polymath Christiaan Huygens on his wave theory of light. Huygens starting point was Descartes theory, as presented in the Dioptrique, Huygens theory is also seen as the historical rival of Newtons theory, which was presented in the Opticks. Huygens, Traité de la Lumière, Leiden, Pieter van der Aa,1690, Huygens, Treatise on Light, London, Macmillan,1912, archive. org/details/treatiseonlight031310mbp C. Huygens, Treatise on Light, Project Gutenberg,2005, gutenberg. org/ebooks/14725
27.
Glass
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Glass is a non-crystalline amorphous solid that is often transparent and has widespread practical, technological, and decorative usage in, for example, window panes, tableware, and optoelectronics. The most familiar, and historically the oldest, types of glass are silicate glasses based on the chemical compound silica, the primary constituent of sand. The term glass, in usage, is often used to refer only to this type of material. Many applications of silicate glasses derive from their optical transparency, giving rise to their use as window panes. Glass can be coloured by adding metallic salts, and can also be painted and printed with vitreous enamels and these qualities have led to the extensive use of glass in the manufacture of art objects and in particular, stained glass windows. Although brittle, silicate glass is extremely durable, and many examples of glass fragments exist from early glass-making cultures, because glass can be formed or moulded into any shape, it has been traditionally used for vessels, bowls, vases, bottles, jars and drinking glasses. In its most solid forms it has also used for paperweights, marbles. Some objects historically were so commonly made of glass that they are simply called by the name of the material, such as drinking glasses. Porcelains and many polymer thermoplastics familiar from everyday use are glasses and these sorts of glasses can be made of quite different kinds of materials than silica, metallic alloys, ionic melts, aqueous solutions, molecular liquids, and polymers. For many applications, like glass bottles or eyewear, polymer glasses are a lighter alternative than traditional glass, silica is a common fundamental constituent of glass. In nature, vitrification of quartz occurs when lightning strikes sand, forming hollow, fused quartz is a glass made from chemically-pure SiO2. It has excellent resistance to shock, being able to survive immersion in water while red hot. However, its high melting-temperature and viscosity make it difficult to work with, normally, other substances are added to simplify processing. One is sodium carbonate, which lowers the transition temperature. The soda makes the glass water-soluble, which is undesirable, so lime, some magnesium oxide. The resulting glass contains about 70 to 74% silica by weight and is called a soda-lime glass, soda-lime glasses account for about 90% of manufactured glass. Most common glass contains other ingredients to change its properties, lead glass or flint glass is more brilliant because the increased refractive index causes noticeably more specular reflection and increased optical dispersion. Adding barium also increases the refractive index, iron can be incorporated into glass to absorb infrared energy, for example in heat absorbing filters for movie projectors, while cerium oxide can be used for glass that absorbs UV wavelengths
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Anisotropy
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Anisotropy /ˌænaɪˈsɒtrəpi/ is the property of being directionally dependent, which implies different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, another is wood, which is easier to split along its grain than against it. In the field of graphics, an anisotropic surface changes in appearance as it rotates about its geometric normal. Anisotropic filtering is a method of enhancing the quality of textures on surfaces that are far away. Older techniques, such as bilinear and trilinear filtering, do not take account the angle a surface is viewed from. By reducing detail in one more than another, these effects can be reduced. In NMR spectroscopy, the orientation of nuclei with respect to the magnetic field determines their chemical shift. In this context, anisotropic systems refer to the distribution of molecules with abnormally high electron density. This abnormal electron density affects the magnetic field and causes the observed chemical shift to change. Anisotropy measurements reveal the average displacement of the fluorophore that occurs between absorption and subsequent emission of a photon. Physicists from University of California, Berkeley reported about their detection of the anisotropy in cosmic microwave background radiation in 1977. Their experiment demonstrated the Doppler shift caused by the movement of the earth with respect to the early Universe matter, cosmic anisotropy has also been seen in the alignment of galaxies rotation axes and polarisation angles of quasars. Physicists use the term anisotropy to describe direction-dependent properties of materials, magnetic anisotropy, for example, may occur in a plasma, so that its magnetic field is oriented in a preferred direction. Plasmas may also show filamentation that is directional, liquid crystals are examples of anisotropic liquids. Some materials conduct heat in a way that is isotropic, that is independent of spatial orientation around the heat source, heat conduction is more commonly anisotropic, which implies that detailed geometric modeling of typically diverse materials being thermally managed is required. The materials used to transfer and reject heat from the source in electronics are often anisotropic. Many crystals are anisotropic to light, and exhibit such as birefringence. Crystal optics describes light propagation in these media, an axis of anisotropy is defined as the axis along which isotropy is broken
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Crystal
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A crystal or crystalline solid is a solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, the scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification, the word crystal derives from the Ancient Greek word κρύσταλλος, meaning both ice and rock crystal, from κρύος, icy cold, frost. Examples of large crystals include snowflakes, diamonds, and table salt, most inorganic solids are not crystals but polycrystals, i. e. many microscopic crystals fused together into a single solid. Examples of polycrystals include most metals, rocks, ceramics, a third category of solids is amorphous solids, where the atoms have no periodic structure whatsoever. Examples of amorphous solids include glass, wax, and many plastics, Crystals are often used in pseudoscientific practices such as crystal therapy, and, along with gemstones, are sometimes associated with spellwork in Wiccan beliefs and related religious movements. The scientific definition of a crystal is based on the arrangement of atoms inside it. A crystal is a solid where the form a periodic arrangement. For example, when liquid water starts freezing, the change begins with small ice crystals that grow until they fuse. Most macroscopic inorganic solids are polycrystalline, including almost all metals, ceramics, ice, rocks, solids that are neither crystalline nor polycrystalline, such as glass, are called amorphous solids, also called glassy, vitreous, or noncrystalline. These have no periodic order, even microscopically, there are distinct differences between crystalline solids and amorphous solids, most notably, the process of forming a glass does not release the latent heat of fusion, but forming a crystal does. A crystal structure is characterized by its cell, a small imaginary box containing one or more atoms in a specific spatial arrangement. The unit cells are stacked in three-dimensional space to form the crystal, the symmetry of a crystal is constrained by the requirement that the unit cells stack perfectly with no gaps. There are 219 possible crystal symmetries, called space groups. These are grouped into 7 crystal systems, such as cubic crystal system or hexagonal crystal system, Crystals are commonly recognized by their shape, consisting of flat faces with sharp angles. Euhedral crystals are those with obvious, well-formed flat faces, anhedral crystals do not, usually because the crystal is one grain in a polycrystalline solid. The flat faces of a crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal. This occurs because some surface orientations are more stable than others, as a crystal grows, new atoms attach easily to the rougher and less stable parts of the surface, but less easily to the flat, stable surfaces
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Birefringence
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Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent, the birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress and this effect was first described by the Danish scientist Rasmus Bartholin in 1669, who observed it in calcite, a crystal having one of the strongest birefringences. A mathematical description of wave propagation in a birefringent medium is presented below, following is a qualitative explanation of the phenomenon. Thus rotating the material around this axis does not change its optical behavior and this special direction is known as the optic axis of the material. Light whose polarization is perpendicular to the axis is governed by a refractive index no. Light whose polarization is in the direction of the optic axis sees an optical index ne, for any ray direction there is a linear polarization direction perpendicular to the optic axis, and this is called an ordinary ray. The magnitude of the difference is quantified by the birefringence, Δ n = n e − n o, the propagation of the ordinary ray is simply described by no as if there were no birefringence involved. However the extraordinary ray, as its name suggests, propagates unlike any wave in an optical material. Its refraction at a surface can be using the effective refractive index. However it is in fact an inhomogeneous wave whose power flow is not exactly in the direction of the wave vector and this causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of calcite as photographed above. Rotating the calcite crystal will cause one of the two images, that of the ray, to rotate slightly around that of the ordinary ray. When the light propagates either along or orthogonal to the optic axis, in the first case, both polarizations see the same effective refractive index, so there is no extraordinary ray. In the second case the extraordinary ray propagates at a different phase velocity but is not an inhomogeneous wave, for instance, a quarter-wave plate is commonly used to create circular polarization from a linearly polarized source. The case of so-called biaxial crystals is substantially more complex and these are characterized by three refractive indices corresponding to three principal axes of the crystal. For most ray directions, both polarizations would be classified as extraordinary rays but with different effective refractive indices, being extraordinary waves, however, the direction of power flow is not identical to the direction of the wave vector in either case. The two refractive indices can be determined using the index ellipsoids for given directions of the polarization, note that for biaxial crystals the index ellipsoid will not be an ellipsoid of revolution but is described by three unequal principle refractive indices nα, nβ and nγ. Thus there is no axis around which a rotation leaves the optical properties invariant, for this reason, birefringent materials with three distinct refractive indices are called biaxial
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Wavefront
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In physics, a wavefront is the locus of points characterized by propagation of position of the same phase, a propagation of a line in 1D, a curve in 2D or a surface for a wave in 3D. Additionally, most optical systems and detectors are indifferent to polarization, at radio wavelengths, the polarization becomes more important, and receivers are usually phase-sensitive. Many audio detectors are also phase-sensitive, Optical systems can be described with Maxwells equations, and linear propagating waves such as sound or electron beams have similar wave equations. However, given the above simplifications, Huygens principle provides a method to predict the propagation of a wavefront through, for example. The construction is as follows, Let every point on the wavefront be considered a new point source, by calculating the total effect from every point source, the resulting field at new points can be computed. Computational algorithms are based on this approach. Specific cases for simple wavefronts can be computed directly, for example, a spherical wavefront will remain spherical as the energy of the wave is carried away equally in all directions. Such directions of flow, which are always perpendicular to the wavefront, are called rays creating multiple wavefronts. The simplest form of a wavefront is the wave, where the rays are parallel to one another. The light from this type of wave is referred to as collimated light, for many purposes, such a wavefront can be considered planar. Wavefront travel with the speed of light in all directions in an isotropic medium, methods utilizing wavefront measurements or predictions can be considered an advanced approach to lens optics, where a single focal distance may not exist due to lens thickness or imperfections. Note also that for manufacturing reasons, a lens has a spherical surface shape though, theoretically. Shortcomings such as these in an optical system cause what are called optical aberrations, the best-known aberrations include spherical aberration and coma. However there may be more complex sources of such as in a large telescope due to spatial variations in the index of refraction of the atmosphere. The deviation of a wavefront in a system from a desired perfect planar wavefront is called the wavefront aberration. Wavefront aberrations are usually described as either an image or a collection of two-dimensional polynomial terms. Minimization of these aberrations is considered desirable for applications in optical systems. A wavefront sensor is a device which measures the wavefront aberration in a coherent signal to describe the quality or lack thereof in an optical system
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Point source
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A point source is a single identifiable localised source of something. A point source has negligible extent, distinguishing it from other source geometries, sources are called point sources because in mathematical modeling, these sources can usually be approximated as a mathematical point to simplify analysis. The actual source need not be small, if its size is negligible relative to other length scales in the problem. For example, in astronomy, stars are routinely treated as point sources, in mathematics, a point source is a singularity from which flux or flow is emanating. Although singularities such as this do not exist in the universe, mathematical point sources are often used as approximations to reality in physics. Generally a source of light can be considered a point source if the resolution of the instrument is too low to resolve its apparent size. There are two types and sources of light, a point source, and an extended source. Radio emissions generated by an electrical circuit are usually polarized. Gamma ray and X-ray sources may be treated as a point source if sufficiently small, radiological contamination and nuclear sources are often point sources. This has significance in physics and radiation protection. As the pressure oscillates up and down, a point source acts in turn as a fluid point source. They are usually a capsule and are most commonly used for gamma, x-ray. In vacuum, heat escapes as radiation isotropically, if the source remains stationary in a compressible fluid such as air, flow patterns can form around the source due to convection, leading to an anisotropic pattern of heat loss. The most common form of anisotropy is the formation of a plume above the heat source. Examples, Geological hotspots on the surface of the Earth which lie at the tops of thermal plumes rising from deep inside the Earth Plumes of heat studied in thermal pollution tracking, fluid point sources are commonly used in fluid dynamics and aerodynamics. A point source of fluid is the inverse of a point sink. If the fluid is moving a plume is generated from the point source
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Index of refraction
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In optics, the refractive index or index of refraction n of a material is a dimensionless number that describes how light propagates through that medium. It is defined as n = c v, where c is the speed of light in vacuum, for example, the refractive index of water is 1.333, meaning that light travels 1.333 times faster in a vacuum than it does in water. The refractive index determines how light is bent, or refracted. The refractive indices also determine the amount of light that is reflected when reaching the interface, as well as the angle for total internal reflection. This implies that vacuum has a index of 1. The refractive index varies with the wavelength of light and this is called dispersion and causes the splitting of white light into its constituent colors in prisms and rainbows, and chromatic aberration in lenses. Light propagation in absorbing materials can be described using a refractive index. The imaginary part then handles the attenuation, while the real part accounts for refraction, the concept of refractive index is widely used within the full electromagnetic spectrum, from X-rays to radio waves. It can also be used with wave phenomena such as sound, in this case the speed of sound is used instead of that of light and a reference medium other than vacuum must be chosen. Thomas Young was presumably the person who first used, and invented, at the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers. The ratio had the disadvantage of different appearances, newton, who called it the proportion of the sines of incidence and refraction, wrote it as a ratio of two numbers, like 529 to 396. Hauksbee, who called it the ratio of refraction, wrote it as a ratio with a fixed numerator, hutton wrote it as a ratio with a fixed denominator, like 1.3358 to 1. Young did not use a symbol for the index of refraction, in the next years, others started using different symbols, n, m, and µ. For visible light most transparent media have refractive indices between 1 and 2, a few examples are given in the adjacent table. These values are measured at the yellow doublet D-line of sodium, with a wavelength of 589 nanometers, gases at atmospheric pressure have refractive indices close to 1 because of their low density. Almost all solids and liquids have refractive indices above 1.3, aerogel is a very low density solid that can be produced with refractive index in the range from 1.002 to 1.265. Moissanite lies at the end of the range with a refractive index as high as 2.65. Most plastics have refractive indices in the range from 1.3 to 1.7, for infrared light refractive indices can be considerably higher
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Derivative
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The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation, the reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration, differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small
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Stationary point
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In mathematics, particularly in calculus, a stationary point or critical point of a differentiable function of one variable is a point of the domain of the function where the derivative is zero. Informally, it is a point where the function stops increasing or decreasing, for a differentiable function of several real variables, a stationary point is an input where all its partial derivatives are zero. Stationary points are easy to visualize on the graph of a function of one variable, for a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the xy plane. The term stationary point of a function may be confused with critical point for a projection of the graph of the function. Critical point is more general, a point of a function corresponds to a critical point of its graph for the projection parallel to the x-axis. On the other hand, the points of the graph for the projection parallel to the y axis are the points where the derivative is not defined. It follows that some authors call critical point the points for any of these projections. A turning point is a point at which the derivative changes sign, a turning point may be either a relative maximum or a relative minimum. If the function is differentiable, then a point is a stationary point. If the function is differentiable, the stationary points that are not turning points are horizontal inflection points. For example, the function x ↦ x 3 has a point at x=0, which is also an inflection point. The first two options are known as local extrema. Similarly a point that is either a maximum or a global minimum is called a global extremum. The last two points that are not local extremum—are known as saddle points. By Fermats theorem, global extrema must occur on the boundary or at stationary points, determining the position and nature of stationary points aids in curve sketching of differentiable functions. Solving the equation f =0 returns the x-coordinates of all stationary points, the y-coordinates are trivially the function values at those x-coordinates. The specific nature of a point at x can in some cases be determined by examining the second derivative f, If f <0, the stationary point at x is concave down. If f >0, the point at x is concave up
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Boundary value problem
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In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them, problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems, the analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed and this means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of differential equations is devoted to proving that boundary value problems arising from scientific. Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions, boundary value problems are similar to initial value problems. Finding the temperature at all points of a bar with one end kept at absolute zero. If the problem is dependent on both space and time, one could specify the value of the problem at a point for all time or at a given time for all space. Concretely, an example of a value is the problem y ″ + y =0 to be solved for the unknown function y with the boundary conditions y =0, y =2. Without the boundary conditions, the solution to this equation is y = A sin + B cos . From the boundary condition y =0 one obtains 0 = A ⋅0 + B ⋅1 which implies that B =0, from the boundary condition y =2 one finds 2 = A ⋅1 and so A =2. One sees that imposing boundary conditions allowed one to determine a unique solution, a boundary condition which specifies the value of the function itself is a Dirichlet boundary condition, or first-type boundary condition. For example, if one end of a rod is held at absolute zero. A boundary condition which specifies the value of the derivative of the function is a Neumann boundary condition. For example, if there is a heater at one end of a rod, then energy would be added at a constant rate. If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a Cauchy boundary condition. Summary of boundary conditions for the function, y, constants c 0 and c 1 specified by the boundary conditions
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Maxwell's equations
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Maxwells equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. They underpin all electric, optical and radio such as power generation, electric motors, wireless communication, cameras, televisions. Maxwells equations describe how electric and magnetic fields are generated by charges, currents, one important consequence of the equations is the demonstration of how fluctuating electric and magnetic fields can propagate at the speed of light. This electromagnetic radiation manifests itself in ways from radio waves to light. The equations have two major variants, the microscopic Maxwell equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges, the macroscopic Maxwell equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic scale details. However, their use requires experimentally determining parameters for a description of the electromagnetic response of materials. The term Maxwells equations is used for equivalent alternative formulations. The space-time formulations, are used in high energy and gravitational physics because they make the compatibility of the equations with special and general relativity manifest. In many situations, though, deviations from Maxwells equations are immeasurably small, exceptions include nonclassical light, photon-photon scattering, quantum optics, and many other phenomena related to photons or virtual photons. In the electric and magnetic field there are four equations. The two inhomogeneous equations describe how the fields vary in space due to sources, Gausss law describes how electric fields emanate from electric charges. Gausss law for magnetism describes magnetic fields as closed field lines not due to magnetic monopoles, the two homogeneous equations describe how the fields circulate around their respective sources. A separate law of nature, the Lorentz force law, describes how the electric and magnetic field act on charged particles, a version of this law was included in the original equations by Maxwell but, by convention, is no longer. The precise formulation of Maxwells equations depends on the definition of the quantities involved. Conventions differ with the systems, because various definitions and dimensions are changed by absorbing dimensionful factors like the speed of light c. This makes constants come out differently, the vector calculus formulation below has become standard. For formulations using tensor calculus or differential forms, see alternative formulations, for relativistically invariant formulations, see relativistic formulations
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Electromagnetic radiation
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In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating through space carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, light, ultraviolet, X-, classically, electromagnetic radiation consists of electromagnetic waves, which are synchronized oscillations of electric and magnetic fields that propagate at the speed of light through a vacuum. The oscillations of the two fields are perpendicular to other and perpendicular to the direction of energy and wave propagation. The wavefront of electromagnetic waves emitted from a point source is a sphere, the position of an electromagnetic wave within the electromagnetic spectrum can be characterized by either its frequency of oscillation or its wavelength. Electromagnetic waves are produced whenever charged particles are accelerated, and these waves can interact with other charged particles. EM waves carry energy, momentum and angular momentum away from their source particle, quanta of EM waves are called photons, whose rest mass is zero, but whose energy, or equivalent total mass, is not zero so they are still affected by gravity. Thus, EMR is sometimes referred to as the far field, in this language, the near field refers to EM fields near the charges and current that directly produced them, specifically, electromagnetic induction and electrostatic induction phenomena. In the quantum theory of electromagnetism, EMR consists of photons, quantum effects provide additional sources of EMR, such as the transition of electrons to lower energy levels in an atom and black-body radiation. The energy of a photon is quantized and is greater for photons of higher frequency. This relationship is given by Plancks equation E = hν, where E is the energy per photon, ν is the frequency of the photon, a single gamma ray photon, for example, might carry ~100,000 times the energy of a single photon of visible light. The effects of EMR upon chemical compounds and biological organisms depend both upon the power and its frequency. EMR of visible or lower frequencies is called non-ionizing radiation, because its photons do not individually have enough energy to ionize atoms or molecules, the effects of these radiations on chemical systems and living tissue are caused primarily by heating effects from the combined energy transfer of many photons. In contrast, high ultraviolet, X-rays and gamma rays are called ionizing radiation since individual photons of high frequency have enough energy to ionize molecules or break chemical bonds. These radiations have the ability to cause chemical reactions and damage living cells beyond that resulting from simple heating, Maxwell derived a wave form of the electric and magnetic equations, thus uncovering the wave-like nature of electric and magnetic fields and their symmetry. Because the speed of EM waves predicted by the wave equation coincided with the speed of light. Maxwell’s equations were confirmed by Heinrich Hertz through experiments with radio waves, according to Maxwells equations, a spatially varying electric field is always associated with a magnetic field that changes over time. Likewise, a varying magnetic field is associated with specific changes over time in the electric field. In an electromagnetic wave, the changes in the field are always accompanied by a wave in the magnetic field in one direction