1.
Celestial spheres
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The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus and others. Since it was believed that the stars did not change their positions relative to one another. In modern thought, the orbits of the planets are viewed as the paths of those planets through mostly empty space, when scholars applied Ptolemys epicycles, they presumed that each planetary sphere was exactly thick enough to accommodate them. In Greek antiquity the ideas of celestial spheres and rings first appeared in the cosmology of Anaximander in the early 6th century BC, all these wheel rims had originally been formed out of an original sphere of fire wholly encompassing the Earth, which had disintegrated into many individual rings. Hence, in Anaximanderss cosmogony, in the beginning was the sphere, out of which celestial rings were formed, as viewed from the Earth, the ring of the Sun was highest, that of the Moon was lower, and the sphere of the stars was lowest. Following Anaximander, his pupil Anaximenes held that the stars, Sun, Moon, and planets are all made of fire. But whilst the stars are fastened on a crystal sphere like nails or studs, the Sun, Moon, and planets. And unlike Anaximander, he relegated the fixed stars to the region most distant from the Earth, after Anaximenes, Pythagoras, Xenophanes and Parmenides all held that the universe was spherical. And much later in the fourth century BC Platos Timaeus proposed that the body of the cosmos was made in the most perfect and uniform shape, but it posited that the planets were spherical bodies set in rotating bands or rings rather than wheel rims as in Anaximanders cosmology. Each planet is attached to the innermost of its own set of spheres. In his Metaphysics, Aristotle developed a cosmology of spheres. Aristotle considers that these spheres are made of a fifth element. Each of these concentric spheres is moved by its own god—an unchanging divine unmoved mover, by using eccentrics and epicycles, his geometrical model achieved greater mathematical detail and predictive accuracy than had been exhibited by earlier concentric spherical models of the cosmos. In Ptolemys physical model, each planet is contained in two or more spheres, but in Book 2 of his Planetary Hypotheses Ptolemy depicted thick circular slices rather than spheres as in its Book 1. The planetary spheres were arranged outwards from the spherical, stationary Earth at the centre of the universe in order, the spheres of the Moon, Mercury, Venus, Sun, Mars, Jupiter. In more detailed models the seven planetary spheres contained other secondary spheres within them, in antiquity the order of the lower planets was not universally agreed. Plato and his followers ordered them Moon, Sun, Mercury, Venus, a series of astronomers, beginning with the Muslim astronomer al-Farghānī, used the Ptolemaic model of nesting spheres to compute distances to the stars and planetary spheres. Al-Farghānīs distance to the stars was 20,110 Earth radii which, on the assumption that the radius of the Earth was 3,250 miles, came to 65,357,500 miles
2.
Earth's rotation
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Earths rotation is the rotation of the planet Earth around its own axis. The Earth rotates from the west towards east, as viewed from North Star or polestar Polaris, the Earth turns counter-clockwise. The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earths axis of rotation meets its surface and this point is distinct from the Earths North Magnetic Pole. The South Pole is the point where the Earths axis of rotation intersects its surface. The Earth rotates once in about 24 hours with respect to the sun, Earths rotation is slowing slightly with time, thus, a day was shorter in the past. This is due to the effects the Moon has on Earths rotation. Atomic clocks show that a modern-day is longer by about 1.7 milliseconds than a century ago, analysis of historical astronomical records shows a slowing trend of 2.3 milliseconds per century since the 8th century BCE. Among the ancient Greeks, several of the Pythagorean school believed in the rotation of the rather than the apparent diurnal rotation of the heavens. Perhaps the first was Philolaus, though his system was complicated, in the third century BCE, Aristarchus of Samos suggested the suns central place. However, Aristotle in the fourth century criticized the ideas of Philolaus as being based on rather than observation. He established the idea of a sphere of fixed stars that rotated about the earth and this was accepted by most of those who came after, in particular Claudius Ptolemy, who thought the earth would be devastated by gales if it rotated. In the 10th century, some Muslim astronomers accepted that the Earth rotates around its axis, treatises were written to discuss its possibility, either as refutations or expressing doubts about Ptolemys arguments against it. At the Maragha and Samarkand observatories, the Earths rotation was discussed by Tusi and Qushji, in medieval Europe, Thomas Aquinas accepted Aristotles view and so, reluctantly, did John Buridan and Nicole Oresme in the fourteenth century. Not until Nicolaus Copernicus in 1543 adopted a heliocentric world system did the earths rotation begin to be established, Copernicus pointed out that if the movement of the earth is violent, then the movement of the stars must be very much more so. He acknowledged the contribution of the Pythagoreans and pointed to examples of relative motion, for Copernicus this was the first step in establishing the simpler pattern of planets circling a central sun. Tycho Brahe, who produced accurate observations on which Kepler based his laws, in 1600, William Gilbert strongly supported the earths rotation in his treatise on the earths magnetism and thereby influenced many of his contemporaries. Those like Gilbert who did not openly support or reject the motion of the earth about the sun are often called semi-Copernicans, however, the contributions of Kepler, Galileo and Newton gathered support for the theory of the rotation of the Earth. The earths rotation implies that the bulges and the poles are flattened
3.
Geocentric model
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In astronomy, the geocentric model is a superseded description of the universe with the Earth at the center. Under the geocentric model, the Sun, Moon, stars, the geocentric model served as the predominant description of the cosmos in many ancient civilizations, such as those of Aristotle and Ptolemy. Two observations supported the idea that the Earth was the center of the Universe, first, the Sun appears to revolve around the Earth once per day. While the Moon and the planets have their own motions, they appear to revolve around the Earth about once per day. The stars appeared to be on a sphere, rotating once each day along an axis through the north and south geographic poles of the Earth. Second, the Earth does not seem to move from the perspective of an Earth-bound observer, it appears to be solid, stable, Ancient Greek, ancient Roman and medieval philosophers usually combined the geocentric model with a spherical Earth. It is not the same as the older flat Earth model implied in some mythology, the ancient Jewish Babylonian uranography pictured a flat Earth with a dome-shaped rigid canopy named firmament placed over it. The astronomical predictions of Ptolemys geocentric model were used to prepare astrological and astronomical charts for over 1500 years. The geocentric model held sway into the modern age, but from the late 16th century onward, it was gradually superseded by the Heliocentric model of Copernicus, Galileo. There was much resistance to the transition between these two theories, christian theologians were reluctant to reject a theory that agreed with Bible passages. Others felt a new, unknown theory could not subvert an accepted consensus for geocentrism, the geocentric model entered Greek astronomy and philosophy at an early point, it can be found in Pre-Socratic philosophy. In the 6th century BC, Anaximander proposed a cosmology with the Earth shaped like a section of a pillar, the Sun, Moon, and planets were holes in invisible wheels surrounding the Earth, through the holes, humans could see concealed fire. About the same time, the Pythagoreans thought that the Earth was a sphere, later these views were combined, so most educated Greeks from the 4th century BC on thought that the Earth was a sphere at the center of the universe. In the 4th century BC, two influential Greek philosophers, Plato and his student Aristotle, wrote works based on the geocentric model, according to Plato, the Earth was a sphere, stationary at the center of the universe. In his Myth of Er, a section of the Republic, Plato describes the cosmos as the Spindle of Necessity, attended by the Sirens and these spheres, known as crystalline spheres, all moved at different uniform speeds to create the revolution of bodies around the Earth. They were composed of a substance called aether. Aristotle believed that the moon was in the innermost sphere and therefore touches the realm of Earth, causing the dark spots and he further described his system by explaining the natural tendencies of the terrestrial elements, Earth, water, fire, air, as well as celestial aether. His system held that Earth was the heaviest element, with the strongest movement towards the center, the tendency of air and fire, on the other hand, was to move upwards, away from the center, with fire being lighter than air
4.
Ecliptic
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The ecliptic is the apparent path of the Sun on the celestial sphere, and is the basis for the ecliptic coordinate system. It also refers to the plane of this path, which is coplanar with the orbit of Earth around the Sun, the motions as described above are simplifications. Due to the movement of Earth around the Earth–Moon center of mass, due to further perturbations by the other planets of the Solar System, the Earth–Moon barycenter wobbles slightly around a mean position in a complex fashion. The ecliptic is actually the apparent path of the Sun throughout the course of a year, because Earth takes one year to orbit the Sun, the apparent position of the Sun also takes the same length of time to make a complete circuit of the ecliptic. With slightly more than 365 days in one year, the Sun moves a little less than 1° eastward every day, again, this is a simplification, based on a hypothetical Earth that orbits at uniform speed around the Sun. The actual speed with which Earth orbits the Sun varies slightly during the year, for example, the Sun is north of the celestial equator for about 185 days of each year, and south of it for about 180 days. The variation of orbital speed accounts for part of the equation of time, if the equator is projected outward to the celestial sphere, forming the celestial equator, it crosses the ecliptic at two points known as the equinoxes. The Sun, in its apparent motion along the ecliptic, crosses the equator at these points, one from south to north. The crossing from south to north is known as the equinox, also known as the first point of Aries. The crossing from north to south is the equinox or descending node. Likewise, the ecliptic itself is not fixed, the gravitational perturbations of the other bodies of the Solar System cause a much smaller motion of the plane of Earths orbit, and hence of the ecliptic, known as planetary precession. The combined action of two motions is called general precession, and changes the position of the equinoxes by about 50 arc seconds per year. Once again, this is a simplification, periodic motions of the Moon and apparent periodic motions of the Sun cause short-term small-amplitude periodic oscillations of Earths axis, and hence the celestial equator, known as nutation. Obliquity of the ecliptic is the used by astronomers for the inclination of Earths equator with respect to the ecliptic. It is about 23. 4° and is currently decreasing 0.013 degrees per hundred years due to planetary perturbations, the angular value of the obliquity is found by observation of the motions of Earth and other planets over many years. From 1984, the Jet Propulsion Laboratorys DE series of computer-generated ephemerides took over as the ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated, jPLs fundamental ephemerides have been continually updated. J. Laskar computed an expression to order T10 good to 0″. 04/1000 years over 10,000 years, all of these expressions are for the mean obliquity, that is, without the nutation of the equator included
5.
Right ascension
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Right ascension is the angular distance measured eastward along the celestial equator from the vernal equinox to the hour circle of the point in question. When combined with declination, these astronomical coordinates specify the direction of a point on the sphere in the equatorial coordinate system. Right ascension is the equivalent of terrestrial longitude. Both right ascension and longitude measure an angle from a direction on an equator. Right ascension is measured continuously in a circle from that equinox towards the east. Any units of measure could have been chosen for right ascension, but it is customarily measured in hours, minutes. Astronomers have chosen this unit to measure right ascension because they measure a stars location by timing its passage through the highest point in the sky as the Earth rotates. The highest point in the sky, called meridian, is the projection of a line onto the celestial sphere. A full circle, measured in units, contains 24 × 60 × 60 = 86 400s, or 24 × 60 = 1 440m. Because right ascensions are measured in hours, they can be used to time the positions of objects in the sky. For example, if a star with RA = 01h 30m 00s is on the meridian, sidereal hour angle, used in celestial navigation, is similar to right ascension, but increases westward rather than eastward. Usually measured in degrees, it is the complement of right ascension with respect to 24h and it is important not to confuse sidereal hour angle with the astronomical concept of hour angle, which measures angular distance of an object westward from the local meridian. The Earths axis rotates slowly westward about the poles of the ecliptic and this effect, known as precession, causes the coordinates of stationary celestial objects to change continuously, if rather slowly. Therefore, equatorial coordinates are inherently relative to the year of their observation, coordinates from different epochs must be mathematically rotated to match each other, or to match a standard epoch. The right ascension of Polaris is increasing quickly, the North Ecliptic Pole in Draco and the South Ecliptic Pole in Dorado are always at right ascension 18h and 6h respectively. The currently used standard epoch is J2000.0, which is January 1,2000 at 12,00 TT, the prefix J indicates that it is a Julian epoch. Prior to J2000.0, astronomers used the successive Besselian Epochs B1875.0, B1900.0, the concept of right ascension has been known at least as far back as Hipparchus who measured stars in equatorial coordinates in the 2nd century BC. But Hipparchus and his successors made their star catalogs in ecliptic coordinates, the easiest way to do that is to use an equatorial mount, which allows the telescope to be aligned with one of its two pivots parallel to the Earths axis
6.
Declination
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In astronomy, declination is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declinations angle is measured north or south of the celestial equator, the root of the word declination means a bending away or a bending down. It comes from the root as the words incline and recline. Declination in astronomy is comparable to geographic latitude, projected onto the celestial sphere, points north of the celestial equator have positive declinations, while those south have negative declinations. Any units of measure can be used for declination, but it is customarily measured in the degrees, minutes. Declinations with magnitudes greater than 90° do not occur, because the poles are the northernmost and southernmost points of the celestial sphere, the Earths axis rotates slowly westward about the poles of the ecliptic, completing one circuit in about 26,000 years. This effect, known as precession, causes the coordinates of stationary celestial objects to change continuously, therefore, equatorial coordinates are inherently relative to the year of their observation, and astronomers specify them with reference to a particular year, known as an epoch. Coordinates from different epochs must be rotated to match each other. The currently used standard epoch is J2000.0, which is January 1,2000 at 12,00 TT, the prefix J indicates that it is a Julian epoch. Prior to J2000.0, astronomers used the successive Besselian Epochs B1875.0, B1900.0, the declinations of Solar System objects change very rapidly compared to those of stars, due to orbital motion and close proximity. This similarly occurs in the Southern Hemisphere for objects with less than −90° − φ. An extreme example is the star which has a declination near to +90°. Circumpolar stars never dip below the horizon, conversely, there are other stars that never rise above the horizon, as seen from any given point on the Earths surface. Generally, if a star whose declination is δ is circumpolar for some observer, then a star whose declination is −δ never rises above the horizon, as seen by the same observer. Likewise, if a star is circumpolar for an observer at latitude φ, neglecting atmospheric refraction, declination is always 0° at east and west points of the horizon. At the north point, it is 90° − |φ|, and at the south point, from the poles, declination is uniform around the entire horizon, approximately 0°. Non-circumpolar stars are visible only during certain days or seasons of the year, the Suns declination varies with the seasons. As seen from arctic or antarctic latitudes, the Sun is circumpolar near the summer solstice, leading to the phenomenon of it being above the horizon at midnight
7.
Equatorial coordinate system
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The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. The origin at the center of the Earth means the coordinates are geocentric, a right-handed convention means that coordinates are positive toward the north and toward the east in the fundamental plane. This description of the orientation of the frame is somewhat simplified. A slow motion of Earths axis, precession, causes a slow, continuous turning of the system westward about the poles of the ecliptic. Superimposed on this is a motion of the ecliptic. In order to fix the primary direction, these motions necessitate the specification of the equinox of a particular date, known as an epoch. The three most commonly used are, Mean equinox of an epoch is a fixed standard direction. Mean equinox of date is the intersection of the ecliptic of date with the mean equator, commonly used in planetary orbit calculation. True equinox of date is the intersection of the ecliptic of date with the true equator and this is the actual intersection of the two planes at any particular moment, with all motions accounted for. A position in the coordinate system is thus typically specified true equinox and equator of date, mean equinox and equator of J2000.0. Note that there is no mean ecliptic, as the ecliptic is not subject to small periodic oscillations, a stars spherical coordinates are often expressed as a pair, right ascension and declination, without a distance coordinate. The direction of distant objects is the same for all observers. Telescopes equipped with equatorial mounts and setting circles employ the equatorial coordinate system to find objects, setting circles in conjunction with a star chart or ephemeris allow the telescope to be easily pointed at known objects on the celestial sphere. The declination symbol δ, measures the distance of an object perpendicular to the celestial equator, positive to the north. For example, the celestial pole has a declination of +90°. The origin for declination is the equator, which is the projection of the Earths equator onto the celestial sphere. Declination is analogous to terrestrial latitude, the right ascension symbol α, measures the angular distance of an object eastward along the celestial equator from the vernal equinox to the hour circle passing through the object. The vernal equinox point is one of the two where the ecliptic intersects the celestial equator, there are = 15° in one hour of right ascension, 24h of right ascension around the entire celestial equator
8.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
9.
Navigation
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Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another. The field of navigation includes four categories, land navigation, marine navigation, aeronautic navigation. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks, all navigational techniques involve locating the navigators position compared to known locations or patterns. Navigation, in a sense, can refer to any skill or study that involves the determination of position and direction. In this sense, navigation includes orienteering and pedestrian navigation, for information about different navigation strategies that people use, visit human navigation. In the European medieval period, navigation was considered part of the set of seven mechanical arts, early Pacific Polynesians used the motion of stars, weather, the position of certain wildlife species, or the size of waves to find the path from one island to another. Maritime navigation using scientific instruments such as the mariners astrolabe first occurred in the Mediterranean during the Middle Ages, the perfecting of this navigation instrument is attributed to Portuguese navigators during early Portuguese discoveries in the Age of Discovery. Open-seas navigation using the astrolabe and the compass started during the Age of Discovery in the 15th century, the Portuguese began systematically exploring the Atlantic coast of Africa from 1418, under the sponsorship of Prince Henry. In 1488 Bartolomeu Dias reached the Indian Ocean by this route, in 1492 the Spanish monarchs funded Christopher Columbuss expedition to sail west to reach the Indies by crossing the Atlantic, which resulted in the Discovery of America. In 1498, a Portuguese expedition commanded by Vasco da Gama reached India by sailing around Africa, soon, the Portuguese sailed further eastward, to the Spice Islands in 1512, landing in China one year later. The fleet of seven ships sailed from Sanlúcar de Barrameda in Southern Spain in 1519, crossed the Atlantic Ocean, some ships were lost, but the remaining fleet continued across the Pacific making a number of discoveries including Guam and the Philippines. By then, only two galleons were left from the original seven, the Victoria led by Elcano sailed across the Indian Ocean and north along the coast of Africa, to finally arrive in Spain in 1522, three years after its departure. The Trinidad sailed east from the Philippines, trying to find a path back to the Americas. He arrived in Acapulco on October 8,1565, the term stems from 1530s, from Latin navigationem, from navigatus, pp. of navigare to sail, sail over, go by sea, steer a ship, from navis ship and the root of agere to drive. Roughly, the latitude of a place on Earth is its angular distance north or south of the equator, latitude is usually expressed in degrees ranging from 0° at the Equator to 90° at the North and South poles. The height of Polaris in degrees above the horizon is the latitude of the observer, similar to latitude, the longitude of a place on Earth is the angular distance east or west of the prime meridian or Greenwich meridian. Longitude is usually expressed in degrees ranging from 0° at the Greenwich meridian to 180° east and west, sydney, for example, has a longitude of about 151° east. New York City has a longitude of 74° west, for most of history, mariners struggled to determine longitude
10.
Abstraction
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An abstraction is the product of this process — a concept that acts as a super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, in a type–token distinction, a type is more abstract than its tokens. Abstraction in its use is a material process, discussed in the themes below. Its development is likely to have been connected with the development of human language. Abstraction involves induction of ideas or the synthesis of particular facts into one theory about something. It is the opposite of specification, which is the analysis or breaking-down of an idea or abstraction into concrete facts. Thales believed that everything in the universe comes from one main substance and he deduced or specified from a general idea, everything is water, to the specific forms of water such as ice, snow, fog, and rivers. Modern scientists can use the opposite approach of abstraction, or going from particular facts collected into one general idea. This conceptual scheme emphasizes the inherent equality of both constituent and abstract data, thus avoiding problems arising from the distinction between abstract and concrete, in this sense the process of abstraction entails the identification of similarities between objects, and the process of associating these objects with an abstraction. For example, picture 1 below illustrates the concrete relationship Cat sits on Mat, for example, graph 1 below expresses the abstraction agent sits on location. This conceptual scheme entails no specific hierarchical taxonomy, only a progressive exclusion of detail, things that do not exist at any particular place and time are often considered abstract. By contrast, instances, or members, of such a thing might exist in many different places and times. Those abstract things are said to be multiply instantiated, in the sense of picture 1, picture 2. It is not sufficient, however, to abstract ideas as those that can be instantiated. Although the concepts cat and telephone are abstractions, they are not abstract in the sense of the objects in graph 1 below, perhaps confusingly, some philosophies refer to tropes as abstract particulars — e. g. the particular redness of a particular apple is an abstract particular. This is similar to qualia and sumbebekos, karl Marxs writing on the commodity abstraction recognizes a parallel process. The state as both concept and material practice exemplifies the two sides of this process of abstraction, conceptually, the current concept of the state is an abstraction from the much more concrete early-modern use as the standing or status of the prince, his visible estates. At the same time, materially, the practice of statehood is now constitutively and materially more abstract than at the time when princes ruled as the embodiment of extended power and that difference accounts for the ontological usefulness of the word abstract
11.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
12.
Concentric objects
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In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another, in the Euclidean plane, two circles that are concentric necessarily have different radii from each other. However, circles in three-dimensional space may be concentric, and have the radius as each other. For example, two different meridians of a terrestrial globe are concentric with each other and with the globe of the earth, more generally, every two great circles on a sphere are concentric with each other and with the sphere. The circumcircle and the incircle of a regular n-gon, and the regular n-gon itself, are concentric, for the circumradius-to-inradius ratio for various n, see Bicentric polygon#Regular polygons. The region of the plane between two circles is an annulus, and analogously the region of space between two concentric spheres is a spherical shell. For a given point c in the plane, the set of all circles having c as their forms a pencil of circles. Each two circles in the pencil are concentric, and have different radii, every point in the plane, except for the shared center, belongs to exactly one of the circles in the pencil. Every two disjoint circles, and every hyperbolic pencil of circles, may be transformed into a set of circles by a Möbius transformation. The ripples formed by dropping an object into still water naturally form an expanding system of concentric circles. Evenly spaced circles on the used in target archery or similar sports provide another familiar example of concentric circles. Coaxial cable is a type of cable in which the combined neutral. Johannes Keplers Mysterium Cosmographicum envisioned a system formed by concentric regular polyhedra. Concentric circles are found in diopter sights, a type of mechanic sights commonly found on target rifles. They usually feature a disk with a small-diametre hole near the shooters eye. When these sights are aligned, the point of impact will be in the middle of the front sight circle. Centered cube number Homoeoid Focaloid Circular symmetry Magic circle Geometry, Concentric circles demonstration With interactive animation
13.
Sky
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The sky is everything that lies above the surface of the Earth, including the atmosphere and outer space. In the field of astronomy, the sky is called the celestial sphere. This is viewed from Earths surface as a dome where the Sun, stars, planets. The celestial sphere is divided into regions called constellations. Usually, the sky is used informally as the point of view from the Earths surface, however. In some cases, such as in discussing the weather, the sky refers to only the lower, during daylight, the sky appears to be blue because air scatters blue sunlight more than it scatters red. At night, the sky appears to be a dark surface or region scattered with stars. During the day, the Sun can be seen in the sky obscured by clouds. In the night sky the Moon, planets and stars are visible in the sky, some of the natural phenomena seen in the sky are clouds, rainbows, and aurorae. Lightning and precipitation can also be seen in the sky during storms, birds, insects, aircraft, and kites are often considered to fly in the sky. Due to human activities, smog during the day and light pollution during the night are often seen above large cities. Except for light that directly from the sun, most of the light in the day sky is caused by scattering. The scattering due to molecule sized particles is greater in the forward and backward directions than it is in the lateral direction. Scattering is significant for light at all visible wavelengths but is stronger at the end of the visible spectrum, meaning that the scattered light is bluer than its source. The remaining sunlight, having lost some of its wavelength components. Scattering also occurs more strongly in clouds. Individual water droplets exposed to light will create a set of colored rings. If a cloud is thick enough, scattering from water droplets will wash out the set of colored rings
14.
Projective geometry
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Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry, the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry
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Projection screen
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A projection screen is an installation consisting of a surface and a support structure used for displaying a projected image for the view of an audience. Projection screens may be installed, as in a movie theater, painted on the wall. As in a room or other non-dedicated viewing space. Another popular type of portable screens are inflatable screens for movie screening. Flat or curved screens may be used depending on the used to project the image. Screens can be designed for front or back projection, the more common being front projection systems. In commercial movie theaters, the screen is a surface that may be either aluminized or a white surface with small glass beads. The screen also has hundreds of small, evenly spaced holes to allow air to and from the speakers and subwoofer, rigid wall-mounted screens maintain their geometry perfectly just like the big movie screens, which makes them suitable for applications that demand exact reproduction of image geometry. Such screens are used in home theaters, along with the pull-down screens. Pull-down screens are used in spaces where a permanently installed screen would require too much space. These commonly use painted fabric that is rolled in the case when not used. Fixed-frame screens provide the greatest level of tension on the screens surface. They are often used in theater and professional environments where the screen does not need to be recessed into the case. Electric screens can be mounted, ceiling mounted or ceiling recessed. These are often larger screens, though electric screens are available for theater use as well. Electric screens are similar to pull-down screens, but instead of the screen being pulled down manually, switchable projection screens can be switched between opaque and clear. In the opaque state, projected image on the screen can be viewed from both sides and it is very good for advertising on store windows. Mobile screens usually use either a pull-down screen on a free stand and these can be used when it is impossible or impractical to mount the screen to a wall or a ceiling
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Astronomer
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An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside of the scope of Earth. They look at stars, planets, moons, comets and galaxies, as well as other celestial objects — either in observational astronomy. Examples of topics or fields astronomers work on include, planetary science, solar astronomy, there are also related but distinct subjects like physical cosmology which studies the Universe as a whole. Astronomers usually fit into two types, Observational astronomers make direct observations of planets, stars and galaxies, and analyze the data, theoretical astronomers create and investigate models of things that cannot be observed. They use this data to create models or simulations to theorize how different celestial bodies work, there are further subcategories inside these two main branches of astronomy such as planetary astronomy, galactic astronomy or physical cosmology. Today, that distinction has disappeared and the terms astronomer. Professional astronomers are highly educated individuals who typically have a Ph. D. in physics or astronomy and are employed by research institutions or universities. They spend the majority of their time working on research, although quite often have other duties such as teaching, building instruments. The number of astronomers in the United States is actually quite small. The American Astronomical Society, which is the organization of professional astronomers in North America, has approximately 7,000 members. This number includes scientists from other such as physics, geology. The International Astronomical Union comprises almost 10,145 members from 70 different countries who are involved in research at the Ph. D. level. Before CCDs, photographic plates were a method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year, analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers time. Astronomers who serve as faculty spend much of their time teaching undergraduate and graduate classes, most universities also have outreach programs including public telescope time and sometimes planetariums as a public service to encourage interest in the field. Those who become astronomers usually have a background in maths, sciences. Taking courses that teach how to research, write and present papers are also invaluable, in college/university most astronomers get a Ph. D. in astronomy or physics. Keeping in mind how few there are it is understood that graduate schools in this field are very competitive
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Celestial coordinate system
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In astronomy, a celestial coordinate system is a system for specifying positions of celestial objects, satellites, planets, stars, galaxies, and so on. Coordinate systems can specify a position in 3-dimensional space, or merely the direction of the object on the celestial sphere, the coordinate systems are implemented in either spherical coordinates or rectangular coordinates. Spherical coordinates, projected on the sphere, are analogous to the geographic coordinate system used on the surface of the Earth. These differ in their choice of fundamental plane, which divides the sphere into two equal hemispheres along a great circle. Rectangular coordinates, in units, are simply the cartesian equivalent of the spherical coordinates, with the same fundamental plane. Each coordinate system is named after its choice of fundamental plane, the following table lists the common coordinate systems in use by the astronomical community. The fundamental plane divides the sphere into two equal hemispheres and defines the baseline for the latitudinal coordinates, similar to the equator in the geographic coordinate system. The poles are located at ±90° from the fundamental plane, the primary direction is the starting point of the longitudinal coordinates. The origin is the distance point, the center of the celestial sphere. The horizontal, or altitude-azimuth, system is based on the position of the observer on Earth, the positioning of a celestial object by the horizontal system varies with time, but is a useful coordinate system for locating and tracking objects for observers on Earth. It is based on the position of relative to an observers ideal horizon. The equatorial coordinate system is centered at Earths center, but fixed relative to the celestial poles, the coordinates are based on the location of stars relative to Earths equator if it were projected out to an infinite distance. The equatorial describes the sky as seen from the solar system, the equatorial system is the normal coordinate system for most professional and many amateur astronomers having an equatorial mount that follows the movement of the sky during the night. Celestial objects are found by adjusting the telescopes or other instruments scales so that they match the equatorial coordinates of the object to observe. There are also subdivisions into mean of date coordinates, which average out or ignore nutation, and true of date, the fundamental plane is the plane of the Earths orbit, called the ecliptic plane. The geocentric ecliptic system was the coordinate system for ancient astronomy and is still useful for computing the apparent motions of the Sun, Moon. The heliocentric ecliptic system describes the orbital movement around the Sun. The system is used for computing the positions of planets and other solar system bodies
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Astronomical object
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An astronomical object or celestial object is a naturally occurring physical entity, association, or structure that current astronomy has demonstrated to exist in the observable universe. In astronomy, the object and body are often used interchangeably. Examples for astronomical objects include planetary systems, star clusters, nebulae and galaxies, while asteroids, moons, planets, and stars are astronomical bodies. A comet may be identified as both body and object, It is a body when referring to the nucleus of ice and dust. The universe can be viewed as having a hierarchical structure, at the largest scales, the fundamental component of assembly is the galaxy. Galaxies are organized groups and clusters, often within larger superclusters. Disc galaxies encompass lenticular and spiral galaxies with features, such as spiral arms, at the core, most galaxies have a supermassive black hole, which may result in an active galactic nucleus. Galaxies can also have satellites in the form of dwarf galaxies, the constituents of a galaxy are formed out of gaseous matter that assembles through gravitational self-attraction in a hierarchical manner. At this level, the fundamental components are the stars. The great variety of forms are determined almost entirely by the mass, composition. Stars may be found in systems that orbit about each other in a hierarchical organization. A planetary system and various objects such as asteroids, comets and debris. The various distinctive types of stars are shown by the Hertzsprung–Russell diagram —a plot of stellar luminosity versus surface temperature. Each star follows a track across this diagram. If this track takes the star through a region containing a variable type. An example of this is the instability strip, a region of the H-R diagram that includes Delta Scuti, RR Lyrae, the table below lists the general categories of bodies and objects by their location or structure. International Astronomical Naming Commission List of light sources List of Solar System objects Lists of astronomical objects SkyChart, Sky & Telescope Monthly skymaps for every location on Earth
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Brooklyn Museum
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The Brooklyn Museum is an art museum located in the New York City borough of Brooklyn. At 560,000 square feet, the museum is New York Citys third largest in physical size, the museum initially struggled to maintain its building and collection, only to be revitalized in the late 20th century, thanks to major renovations. Significant areas of the collection include antiquities, specifically their collection of Egyptian antiquities spanning over 3,000 years, African, Oceanic, and Japanese art make for notable antiquities collections as well. American art is represented, starting at the Colonial period. Artists represented in the collection include Mark Rothko, Edward Hopper, Norman Rockwell, Winslow Homer, Edgar Degas, Georgia OKeeffe, the museum also has a Memorial Sculpture Garden which features salvaged architectural elements from throughout New York City. The roots of the Brooklyn Museum extend back to the 1823 founding by Augustus Graham of the Brooklyn Apprentices’ Library in Brooklyn Heights, in 1890, under its director Franklin Hooper, Institute leaders reorganized as the Brooklyn Institute of Arts and Sciences and began planning the Brooklyn Museum. The initial design for the Brooklyn Museum was four times as large as the actualized version, Daniel Chester French, the noted sculptor of the Lincoln Memorial, was the principal designer of the pediment sculptures and the monolithic 12. 5-foot figures along the cornice. The figures were created by 11 sculptors and carved by the Piccirilli Brothers, by 1920, the New York City Subway reached the museum with a subway station, this greatly improved access to the once-isolated museum from Manhattan and other outer boroughs. The Brooklyn Institutes director Franklin Hooper was the museums first director and he was followed by Philip Newell Youtz, Laurance Page Roberts, Isabel Spaulding Roberts, Charles Nagel, Jr. and Edgar Craig Schenck. Thomas S. Buechner became the director in 1960, making him one of the youngest directors in the country. Buechner oversaw a major transformation in the way the museum displayed art and brought some one thousand works that had languished in the museums archives and put them on display. Buechner played a role in rescuing the Daniel Chester French sculptures from destruction due to an expansion project at the Manhattan Bridge in the 1960s. The Brooklyn Museum changed its name to Brooklyn Museum of Art in 1997, on March 12,2004, the museum announced that it would revert to its previous name. In April 2004, the museum opened the James Polshek-designed entrance pavilion on the Eastern Parkway façade, in September 2014, Lehman announced that he was planning to retire around June 2015. In May 2015, Creative Time president and artistic director Anne Pasternak was named the Museums next director, member institutions occupy land or buildings owned by the City of New York and derive part of their yearly funding from the City. The Brooklyn Museum also supplements its earned income with funding from Federal and State governments, as well as donations by individuals. Major benefactors include Frank Lusk Babbott, the museum is the site of the annual Brooklyn Artists Ball which has included celebrity hosts such as Sarah Jessica Parker and Liv Tyler. The Brooklyn Museum exhibits collections that seek to embody the rich heritage of world cultures
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Equidistant
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A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal. In two-dimensional Euclidean geometry the locus of points equidistant from two points is their perpendicular bisector. For a triangle the circumcentre is a point equidistant from each of the three vertices, every non-degenerate triangle has such a point. This result can be generalised to cyclic polygons, the circumcentre is equidistant from each of the vertices, likewise, the incentre of a triangle or any other tangential polygon is equidistant from the points of tangency of the polygons sides with the circle. Every point on a perpendicular bisector of the side of a triangle or other polygon is equidistant from the two vertices at the ends of that side, every point on the bisector of an angle of any polygon is equidistant from the two sides that emanate from that angle. The center of a rectangle is equidistant from all four vertices, a point on the axis of symmetry of a kite is equidistant between two sides. The center of a circle is equidistant from every point on the circle, likewise the center of a sphere is equidistant from every point on the sphere. A parabola is the set of points in a plane equidistant from a point and a fixed line. In shape analysis, the skeleton or medial axis of a shape is a thin version of that shape that is equidistant from its boundaries. In Euclidean geometry, parallel lines are equidistant in the sense that the distance of any point on one line from the nearest point on the line is the same for all points. In hyperbolic geometry the set of points that are equidistant from and on one side of a given line form an hypercycle
