1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Plane curve
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In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves, and algebraic plane curves, a smooth plane curve is a curve in a real Euclidean plane R2 and is a one-dimensional smooth manifold. This means that a plane curve is a plane curve which locally looks like a line, in the sense that near every point. An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation f =0 Algebraic curves were studied extensively since the 18th century, for example, the circle given by the equation x2 + y2 =1 has degree 2. The non-singular plane algebraic curves of degree 2 are called conic sections, the plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree four are called plane curves. Algebraic curve Differential geometry Algebraic geometry Plane curve fitting Projective varieties Two-dimensional graph Coolidge, a Treatise on Algebraic Plane Curves, Dover Publications, ISBN 0-486-49576-0. A handbook on curves and their properties, J. W. Edwards, ASIN B0007EKXV0
3.
Reflection symmetry
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Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry, in 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its image is called mirror symmetric. The set of operations that preserve a property of the object form a group. Two objects are symmetric to each other with respect to a group of operations if one is obtained from the other by some of the operations. Another way to think about the function is that if the shape were to be folded in half over the axis. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match, a circle has infinitely many axes of symmetry. Triangles with reflection symmetry are isosceles, quadrilaterals with reflection symmetry are kites, deltoids, rhombuses, and isosceles trapezoids. All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, and one through edges, for an arbitrary shape, the axiality of the shape measures how close it is to being bilaterally symmetric. It equals 1 for shapes with reflection symmetry, and between 2/3 and 1 for any convex shape, for each line or plane of reflection, the symmetry group is isomorphic with Cs, one of the three types of order two, hence algebraically C2. The fundamental domain is a half-plane or half-space, in certain contexts there is rotational as well as reflection symmetry. Then mirror-image symmetry is equivalent to inversion symmetry, in contexts in modern physics the term parity or P-symmetry is used for both. For more general types of reflection there are more general types of reflection symmetry. For example, with respect to a non-isometric affine involution with respect to circle inversion, most animals are bilaterally symmetric, likely because this supports forward movement and streamlining. Mirror symmetry is used in architecture, as in the facade of Santa Maria Novella. It is also found in the design of ancient structures such as Stonehenge, Symmetry was a core element in some styles of architecture, such as Palladianism. Patterns in nature Point reflection symmetry Stewart, Ian, weidenfeld & Nicolson. is potty Weyl, Hermann. Mapping with symmetry - source in Delphi Reflection Symmetry Examples from Math Is Fun
4.
Shape
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A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material composition. Psychologists have theorized that humans mentally break down images into simple geometric shapes called geons, examples of geons include cones and spheres. Some simple shapes can be put into broad categories, for instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into categories, triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can be rectangles, rhombi, trapezoids, squares. Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, among the most common 3-dimensional shapes are polyhedra, which are shapes with flat faces, ellipsoids, which are egg-shaped or sphere-shaped objects, cylinders, and cones. If an object falls into one of these categories exactly or even approximately, thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk. Similarity, Two objects are similar if one can be transformed into the other by a scaling, together with a sequence of rotations, translations. Isotopy, Two objects are isotopic if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it. Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the b and d are a reflection of each other, and hence they are congruent and similar. Sometimes, only the outline or external boundary of the object is considered to determine its shape, for instance, an hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to determine whether or not two objects have the shape, or to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same. Simple shapes can often be classified into basic objects such as a point, a line, a curve, a plane. However, most shapes occurring in the world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals. In geometry, two subsets of a Euclidean space have the shape if one can be transformed to the other by a combination of translations, rotations. In other words, the shape of a set of points is all the information that is invariant to translations, rotations
5.
Point (geometry)
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In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, in particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects, Euclid originally defined the point as that which has no part. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by a triplet with the additional third number representing depth. Further generalizations are represented by an ordered tuplet of n terms, many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points, As an example, a line is a set of points of the form L =. Similar constructions exist that define the plane, line segment and other related concepts, a line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, in spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics, in all of the common definitions, a point is 0-dimensional. The dimension of a space is the maximum size of a linearly independent subset. In a vector space consisting of a point, there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero,1 ⋅0 =0, if no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a open set. The Hausdorff dimension of X is defined by dim H , = inf, a point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although the notion of a point is considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e. g. noncommutative geometry. More precisely, such structures generalize well-known spaces of functions in a way that the operation take a value at this point may not be defined
6.
Focus (geometry)
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In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse. An ellipse can be defined as the locus of points for each of which the sum of the distances to two given foci is a constant, a circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, a parabola is a limiting case of an ellipse in which one of the foci is a point at infinity. A hyperbola can be defined as the locus of points for each of which the value of the difference between the distances to two given foci is a constant. It is also possible to describe all conic sections in terms of a focus and a single directrix. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a positive constant. If e is zero and one the conic is an ellipse, if e=1 the conic is a parabola. If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero and it is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. The ellipse thus generated has its focus at the center of the directrix circle. For the parabola, the center of the moves to the point at infinity. The directrix circle becomes a curve with zero curvature, indistinguishable from a straight line. To generate a hyperbola, the radius of the circle is chosen to be less than the distance between the center of this circle and the focus, thus, the focus is outside the directrix circle. The two branches of a hyperbola are thus the two halves of a curve closed over infinity, in projective geometry, all conics are equivalent in the sense that every theorem that can be stated for one can be stated for the others. Plutos ellipse is entirely inside Charons ellipse, as shown in animation of the system. The barycenter is about three-quarters of the distance from Earths center to its surface, moreover, the Pluto-Charon system moves in an ellipse around its barycenter with the Sun, as does the Earth-Moon system
7.
Line (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
8.
Directrix (conic section)
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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
9.
Locus (mathematics)
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In geometry, a locus is a set of points, whose location satisfies or is determined by one or more specified conditions. Until the beginning of 20th century, a shape was not considered as an infinite set of points, rather. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a distance of a fixed point. In contrast to the view, the old formulation avoids considering infinite collections. Once set theory became the universal basis over which the mathematics is built. Examples from plane geometry include, The set of points equidistant from two points is a perpendicular bisector to the segment connecting the two points. The set of points equidistant from two lines cross is the angle bisector. All conic sections are loci, Parabola, the set of points equidistant from a single point, Circle, the set of points for which the distance from a single point is constant. The set of points for each of which the ratio of the distances to two given foci is a constant is referred to as a Circle of Apollonius. Hyperbola, the set of points for each of which the value of the difference between the distances to two given foci is a constant. Ellipse, the set of points for each of which the sum of the distances to two given foci is a constant, the circle is the special case in which the two foci coincide with each other. Other examples of loci appear in areas of mathematics. For example, in dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps. Proof that all the points on the given shape satisfy the conditions and we find the locus of the points P that have a given ratio of distances k = d1/d2 to two given points. In this example we choose k=3, A and B as the fixed points and it is the circle of Apollonius defined by these values of k, A, and B. A triangle ABC has a side with length c. We determine the locus of the third vertex C such that the medians from A and C are orthogonal and we choose an orthonormal coordinate system such that A, B. C is the third vertex
10.
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
11.
Conic section
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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
12.
Conical surface
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Each of those lines is called a generatrix of the surface. Every conic surface is ruled and developable, in general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex, sometimes the term conical surface is used to mean just one nappe. If the directrix is a circle C, and the apex is located on the circles axis and this special case is often called a cone, because it is one of the two distinct surfaces that bound the geometric solid of that name. The aperture of the cone is the angle 2 θ, a cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry a cylindrical surface is just a case of a conical surface. A conical surface S can be described parametrically as S = v + u q, in implicit form, the same surface is described by S =0 where S =2 − z 22. Conic section Developable surface Quadric Ruled surface
13.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
14.
Parallel (geometry)
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In geometry, parallel lines are lines in a plane which do not meet, that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same space that never meet. Parallel lines are the subject of Euclids parallel postulate, parallelism is primarily a property of affine geometries and Euclidean space is a special instance of this type of geometry. Some other spaces, such as space, have analogous properties that are sometimes referred to as parallelism. For example, A B ∥ C D indicates that line AB is parallel to line CD, in the Unicode character set, the parallel and not parallel signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation equal and parallel to, given parallel straight lines l and m in Euclidean space, the following properties are equivalent, Every point on line m is located at exactly the same distance from line l. Line m is in the plane as line l but does not intersect l. When lines m and l are both intersected by a straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Thus, the property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are consequences of Euclids Parallel Postulate. Another property that also involves measurement is that parallel to each other have the same gradient. The definition of parallel lines as a pair of lines in a plane which do not meet appears as Definition 23 in Book I of Euclids Elements. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate, proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius definition as well as its modification by the philosopher Aganis, at the end of the nineteenth century, in England, Euclids Elements was still the standard textbook in secondary schools. A major difference between these texts, both between themselves and between them and Euclid, is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson, wrote a play, Euclid and His Modern Rivals, one of the early reform textbooks was James Maurice Wilsons Elementary Geometry of 1868
15.
Tangent
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In geometry, the tangent line to a plane curve at a given point is the straight line that just touches the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve, a similar definition applies to space curves and curves in n-dimensional Euclidean space. Similarly, the tangent plane to a surface at a point is the plane that just touches the surface at that point. The concept of a tangent is one of the most fundamental notions in geometry and has been extensively generalized. The word tangent comes from the Latin tangere, to touch, euclid makes several references to the tangent to a circle in book III of the Elements. In Apollonius work Conics he defines a tangent as being a line such that no other straight line could fall between it and the curve, archimedes found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself and these methods led to the development of differential calculus in the 17th century. Many people contributed, Roberval discovered a method of drawing tangents. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents, further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was a line which touches a curve. This old definition prevents inflection points from having any tangent and it has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve. The tangent at A is the limit when point B approximates or tends to A, the existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as differentiability. At most points, the tangent touches the curve without crossing it, a point where the tangent crosses the curve is called an inflection point. Circles, parabolas, hyperbolas and ellipses do not have any point, but more complicated curves do have, like the graph of a cubic function. Conversely, it may happen that the curve lies entirely on one side of a line passing through a point on it. This is the case, for example, for a passing through the vertex of a triangle. In convex geometry, such lines are called supporting lines, the geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century, suppose that a curve is given as the graph of a function, y = f
16.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
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Graph of a function
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In mathematics, the graph of a function f is the collection of all ordered pairs. If the function x is a scalar, the graph is a two-dimensional graph. If the function x is an ordered pair of real numbers, the graph is the collection of all ordered triples. Graphing on a Cartesian plane is referred to as curve sketching. The graph of a function on real numbers may be mapped directly to the representation of the function. The concept of the graph of a function is generalized to the graph of a relation, note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real numbers, to test whether a graph of a curve is a function of x, one uses the vertical line test. To test whether a graph of a curve is a function of y, if the function has an inverse, the graph of the inverse can be found by reflecting the graph of the original function over the line y = x. In science, engineering, technology, finance, and other areas, in the simplest case one variable is plotted as a function of another, typically using rectangular axes, see Plot for details. In the modern foundation of mathematics known as set theory, a function, F = { a, if x =1, d, if x =2, c, if x =3, is. The graph of the polynomial on the real line f = x 3 −9 x is. If this set is plotted on a Cartesian plane, the result is a curve, the graph of the trigonometric function f = sin cos is. If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface, oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function, f = −2 Given a function f of n variables, x 1, …, x n, the normal to the graph is. This is seen by considering the graph as a set of the function g = f − z. The graph of a function is contained in a Cartesian product of sets, fibre bundles arent cartesian products, but appear to be up close. There is a notion of a graph on a fibre bundle called a section
18.
Quadratic function
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A univariate quadratic function has the form f = a x 2 + b x + c, a ≠0 in the single variable x. The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis. If the quadratic function is set equal to zero, then the result is a quadratic equation, the solutions to the univariate equation are called the roots of the univariate function. In general there can be a large number of variables, in which case the resulting surface is called a quadric. The adjective quadratic comes from the Latin word quadrātum, a term like x2 is called a square in algebra because it is the area of a square with side x. In general, a prefix indicates the number 4. Quadratum is the Latin word for square, the coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring. When using the quadratic polynomial, authors sometimes mean having degree exactly 2. If the degree is less than 2, this may be called a degenerate case, usually the context will establish which of the two is meant. Sometimes the word order is used with the meaning of degree, a quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z. Any single-variable quadratic polynomial may be written as a x 2 + b x + c, where x is the variable, and a, b, and c represent the coefficients. In elementary algebra, such polynomials often arise in the form of a quadratic equation a x 2 + b x + c =0, each quadratic polynomial has an associated quadratic function, whose graph is a parabola. Such polynomials are fundamental to the study of sections, which are characterized by equating the expression for f to zero. Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurfaces, in linear algebra, quadratic polynomials can be generalized to the notion of a quadratic form on a vector space. F = a 2 + k is called the vertex form, the coefficient a is the same value in all three forms. To convert the standard form to factored form, one only the quadratic formula to determine the two roots r1 and r2. To convert the standard form to form, one needs a process called completing the square. To convert the factored form to form, one needs to multiply
19.
Axis of symmetry
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Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
20.
Vertex (curve)
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In the geometry of planar curves, a vertex is a point of where the first derivative of curvature is zero. This is typically a maximum or minimum of curvature. However, other cases may occur, for instance when the second derivative is also zero. For space curves, on the hand, a vertex is a point where the torsion vanishes. A hyperbola has two vertices, one on each branch, they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis, on a parabola, the sole vertex lies on the axis of symmetry. On an ellipse, two of the four vertices lie on the axis and two lie on the minor axis. For a circle, which has constant curvature, every point is a vertex, vertices are points where the curve has 4-point contact with the osculating circle at that point. In contrast, generic points on a curve typically only have 3-point contact with their osculating circle. The symmetry set of a curve has endpoints at the corresponding to the vertices, and the medial axis. According to the classical four-vertex theorem, every simple closed planar smooth curve must have at least four vertices. A more general fact is that every simple closed curve which lies on the boundary of a convex body, or even bounds a locally convex disk. If a planar curve is bilaterally symmetric, it will have a vertex at the point or points where the axis of symmetry crosses the curve. Thus, the notion of a vertex for a curve is related to that of an optical vertex. Computer Graphics and Geometric Modelling, Mathematics, Springer, ISBN9781852338176, fuks, D. B.07626 Gibson, C. G. Elementary Geometry of Differentiable Curves, An Undergraduate Introduction, Cambridge University Press, ISBN9780521011075. Four vertices of a space curve, Bull
21.
Similarity (geometry)
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Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling and this means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other zoomed in or out at some level. For example, all circles are similar to other, all squares are similar to each other. On the other hand, ellipses are not all similar to other, rectangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure and it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem, due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. There are several statements each of which is necessary and sufficient for two triangles to be similar,1, the triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is, If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′, all the corresponding sides have lengths in the same ratio, AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle is an enlargement of the other, two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance, AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′ and this is known as the SAS Similarity Criterion. When two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′, there are several elementary results concerning similar triangles in Euclidean geometry, Any two equilateral triangles are similar. Two triangles, both similar to a triangle, are similar to each other. Corresponding altitudes of similar triangles have the ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, the statement that the point F satisfying this condition exists is Walliss Postulate and is logically equivalent to Euclids Parallel Postulate
22.
Reflection (physics)
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Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves, the law of reflection says that for specular reflection the angle at which the wave is incident on the surface equals the angle at which it is reflected. In acoustics, reflection causes echoes and is used in sonar, in geology, it is important in the study of seismic waves. Reflection is observed with surface waves in bodies of water, Reflection is observed with many types of electromagnetic wave, besides visible light. Reflection of VHF and higher frequencies is important for radio transmission, even hard X-rays and gamma rays can be reflected at shallow angles with special grazing mirrors. Reflection of light is either specular or diffuse depending on the nature of the interface, a mirror provides the most common model for specular light reflection, and typically consists of a glass sheet with a metallic coating where the reflection actually occurs. Reflection is enhanced in metals by suppression of wave propagation beyond their skin depths, Reflection also occurs at the surface of transparent media, such as water or glass. In the diagram, a light ray PO strikes a vertical mirror at point O, by projecting an imaginary line through point O perpendicular to the mirror, known as the normal, we can measure the angle of incidence, θi and the angle of reflection, θr. The law of reflection states that θi = θr, or in other words, in fact, reflection of light may occur whenever light travels from a medium of a given refractive index into a medium with a different refractive index. In the most general case, a fraction of the light is reflected from the interface. This is analogous to the way impedance mismatch in a circuit causes reflection of signals. Total internal reflection of light from a denser medium occurs if the angle of incidence is above the critical angle, total internal reflection is used as a means of focusing waves that cannot effectively be reflected by common means. X-ray telescopes are constructed by creating a tunnel for the waves. As the waves interact at low angle with the surface of this tunnel they are reflected toward the focus point, a conventional reflector would be useless as the X-rays would simply pass through the intended reflector. When light reflects off a material denser than the external medium, in contrast, a less dense, lower refractive index material will reflect light in phase. This is an important principle in the field of thin-film optics, specular reflection at a curved surface forms an image which may be magnified or demagnified, curved mirrors have optical power. Such mirrors may have surfaces that are spherical or parabolic, if the reflecting surface is very smooth, the reflection of light that occurs is called specular or regular reflection. The laws of reflection are as follows, The incident ray, the reflected ray, the angle which the incident ray makes with the normal is equal to the angle which the reflected ray makes to the same normal
23.
Light
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Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to light, which is visible to the human eye and is responsible for the sense of sight. Visible light is defined as having wavelengths in the range of 400–700 nanometres, or 4.00 × 10−7 to 7.00 × 10−7 m. This wavelength means a range of roughly 430–750 terahertz. The main source of light on Earth is the Sun, sunlight provides the energy that green plants use to create sugars mostly in the form of starches, which release energy into the living things that digest them. This process of photosynthesis provides virtually all the used by living things. Historically, another important source of light for humans has been fire, with the development of electric lights and power systems, electric lighting has effectively replaced firelight. Some species of animals generate their own light, a process called bioluminescence, for example, fireflies use light to locate mates, and vampire squids use it to hide themselves from prey. Visible light, as all types of electromagnetic radiation, is experimentally found to always move at this speed in a vacuum. In physics, the term sometimes refers to electromagnetic radiation of any wavelength. In this sense, gamma rays, X-rays, microwaves and radio waves are also light, like all types of light, visible light is emitted and absorbed in tiny packets called photons and exhibits properties of both waves and particles. This property is referred to as the wave–particle duality, the study of light, known as optics, is an important research area in modern physics. Generally, EM radiation, or EMR, is classified by wavelength into radio, microwave, infrared, the behavior of EMR depends on its wavelength. Higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths, when EMR interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries. There exist animals that are sensitive to various types of infrared, infrared sensing in snakes depends on a kind of natural thermal imaging, in which tiny packets of cellular water are raised in temperature by the infrared radiation. EMR in this range causes molecular vibration and heating effects, which is how these animals detect it, above the range of visible light, ultraviolet light becomes invisible to humans, mostly because it is absorbed by the cornea below 360 nanometers and the internal lens below 400. Furthermore, the rods and cones located in the retina of the eye cannot detect the very short ultraviolet wavelengths and are in fact damaged by ultraviolet. Many animals with eyes that do not require lenses are able to detect ultraviolet, by quantum photon-absorption mechanisms, various sources define visible light as narrowly as 420 to 680 to as broadly as 380 to 800 nm
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Collimated
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Collimated light is light whose rays are parallel, and therefore will spread minimally as it propagates. A perfectly collimated beam, with no divergence, would not disperse with distance, such a beam cannot be created, due to diffraction. Light can be collimated by a number of processes, for instance by means of a collimator. Perfectly collimated light is said to be focused at infinity. Thus, as the distance from a point source increases, the spherical wavefronts become flatter and closer to plane waves, the word collimate comes from the Latin verb collimare, which originated in a misreading of collineare, to direct in a straight line. In practice, gas lasers can use concave mirrors, flat mirrors, the divergence of high-quality laser beams is commonly less than 1 milliradian, and can be much less for large-diameter beams. Laser diodes emit less-collimated light due to their short cavity, synchrotron light is very well collimated. It is produced by bending relativistic electrons around a circular track, when the electrons are at relativistic speeds, the resulting radiation is highly collimated, a result which does not occur at slower speeds. The light from stars can be considered collimated for almost any purpose, because they are so far away they have almost no angular size, light from the Sun is nearly collimated by the time it reaches Earth because of its distance from the Earth. A perfect parabolic mirror will bring parallel rays to a focus at a single point, conversely, a point source at the focus of a parabolic mirror will produce a beam of collimated light creating a Collimator. Since the source needs to be small, such a system cannot produce much optical power. Spherical mirrors are easier to make than parabolic mirrors and they are used to produce approximately collimated light. Many types of lenses can also produce collimated light from point-like sources and this principle is used in full flight simulators, that have specially designed systems for displaying imagery of the Out-The-Window scene to the pilots in the replica aircraft cabin. Collimation refers to all the elements in an instrument being on their designed optical axis. It also refers to the process of adjusting an optical instrument so that all its elements are on that designed axis. With regards to a telescope, the term refers to the fact that the axis of each optical component should be centered and parallel. Most amateur reflector telescopes need to be re-collimated every few years to maintain optimum performance, collimation can also be tested using a shearing interferometer, which is often used to test laser collimation. Decollimation is any mechanism or process which causes a beam with the minimum possible ray divergence to diverge or converge from parallelism, decollimation must be accounted for to fully treat many systems such as radio, radar, sonar, and optical communications
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Sound
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In physics, sound is a vibration that propagates as a typically audible mechanical wave of pressure and displacement, through a transmission medium such as air or water. In physiology and psychology, sound is the reception of such waves, humans can hear sound waves with frequencies between about 20 Hz and 20 kHz. Sound above 20 kHz is ultrasound and below 20 Hz is infrasound, other animals have different hearing ranges. Acoustics is the science that deals with the study of mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound. A scientist who works in the field of acoustics is an acoustician, an audio engineer, on the other hand, is concerned with the recording, manipulation, mixing, and reproduction of sound. Auditory sensation evoked by the oscillation described in, sound can propagate through a medium such as air, water and solids as longitudinal waves and also as a transverse wave in solids. The sound waves are generated by a source, such as the vibrating diaphragm of a stereo speaker. The sound source creates vibrations in the surrounding medium, as the source continues to vibrate the medium, the vibrations propagate away from the source at the speed of sound, thus forming the sound wave. At a fixed distance from the source, the pressure, velocity, at an instant in time, the pressure, velocity, and displacement vary in space. Note that the particles of the medium do not travel with the sound wave and this is intuitively obvious for a solid, and the same is true for liquids and gases. During propagation, waves can be reflected, refracted, or attenuated by the medium, the behavior of sound propagation is generally affected by three things, A complex relationship between the density and pressure of the medium. This relationship, affected by temperature, determines the speed of sound within the medium, if the medium is moving, this movement may increase or decrease the absolute speed of the sound wave depending on the direction of the movement. For example, sound moving through wind will have its speed of propagation increased by the speed of the if the sound and wind are moving in the same direction. If the sound and wind are moving in opposite directions, the speed of the wave will be decreased by the speed of the wind. Medium viscosity determines the rate at which sound is attenuated, for many media, such as air or water, attenuation due to viscosity is negligible. When sound is moving through a medium that does not have constant physical properties, the mechanical vibrations that can be interpreted as sound can travel through all forms of matter, gases, liquids, solids, and plasmas. The matter that supports the sound is called the medium, sound cannot travel through a vacuum. Sound is transmitted through gases, plasma, and liquids as longitudinal waves and it requires a medium to propagate
26.
Parabolic antenna
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A parabolic antenna is an antenna that uses a parabolic reflector, a curved surface with the cross-sectional shape of a parabola, to direct the radio waves. The most common form is shaped like a dish and is called a dish antenna or parabolic dish. The main advantage of an antenna is that it has high directivity. It functions similarly to a searchlight or flashlight reflector to direct the waves in a narrow beam. Parabolic antennas have some of the highest gains, that is, they can produce the narrowest beamwidths and they are also used in radio telescopes. The other large use of antennas is for radar antennas, in which there is a need to transmit a narrow beam of radio waves to locate objects like ships, airplanes. With the advent of satellite television receivers, parabolic antennas have become a common feature of the landscapes of modern countries. The parabolic antenna was invented by German physicist Heinrich Hertz during his discovery of radio waves in 1887 and he used cylindrical parabolic reflectors with spark-excited dipole antennas at their focus for both transmitting and receiving during his historic experiments. Conversely, a plane wave parallel to the axis will be focused to a point at the focal point. A typical parabolic antenna consists of a parabolic reflector with a small feed antenna suspended in front of the reflector at its focus. The reflector is a surface formed into a paraboloid of revolution. In a transmitting antenna, radio frequency current from a transmitter is supplied through a transmission line cable to the feed antenna, the radio waves are emitted back toward the dish by the feed antenna and reflect off the dish into a parallel beam. The reflector can be of metal, metal screen, or wire grill construction. Large dishes often require a supporting structure behind them to provide the required stiffness. A reflector made of a grill of parallel wires or bars oriented in one direction acts as a polarizing filter as well as a reflector and it only reflects linearly polarized radio waves, with the electric field parallel to the grill elements. This type is used in radar antennas. Combined with a linearly polarized feed horn, it helps filter out noise in the receiver, the feed antenna at the reflectors focus is typically a low-gain type such as a half-wave dipole or more often a small horn antenna called a feed horn. The feed antenna is connected to the associated radio-frequency transmitting or receiving equipment by means of a cable transmission line or waveguide
27.
Parabolic microphone
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A parabolic microphone is a microphone that uses a parabolic reflector to collect and focus sound waves onto a receiver, in much the same way that a parabolic antenna does with radio waves. Parabolic microphones have great sensitivity to sounds in one direction, along the axis of the dish, parabolic microphones were used in many parts of the world as early as World War II, especially by the Japanese. Parabolic microphones are not used for high fidelity recording applications because dishes small enough to be portable have poor low frequency response. This is because, from the Rayleigh criterion, parabolic dishes can only focus waves with a much smaller than the diameter of their aperture. The wavelength of sound waves at the low end of human hearing is about 17 metres, a typical parabolic microphone dish with a diameter of one metre would have little directivity for sound waves longer than 30 cm, corresponding to frequencies below 1 kHz. A gain of about 15 dB can be expected, however, a shotgun microphone may be used as an alternative for applications requiring high fidelity
28.
Ballistic missiles
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A ballistic missile is a missile that follows a ballistic trajectory with the objective of delivering one or more warheads to a predetermined target. A ballistic missile is only guided during relatively brief periods of flight and this contrasts to a cruise missile, which is aerodynamically guided in powered flight. Long range intercontinental ballistic missiles are launched on a flight trajectory. Shorter range ballistic missiles stay within the Earths atmosphere, the earliest use of rockets as a weapon date to the 13th Century. A pioneer ballistic missile was the A-4, commonly known as the V-2 rocket developed by Nazi Germany in the 1930s and 1940s under the direction of Wernher von Braun. The first successful launch of a V-2 was on October 3,1942, by the end of World War II in May 1945, over 3,000 V-2s had been launched. The R-7 Semyorka was the first intercontinental ballistic missile, a total of 30 nations have deployed operational ballistic missiles. Development continues with around 100 ballistic missile tests in 2007, mostly by China, Iran. In 2010, the U. S. and Russian governments signed a treaty to reduce their inventory of intercontinental ballistic missiles over a period to 1550 units each. Ballistic missiles can be launched from fixed sites or mobile launchers, including vehicles, aircraft, ships, the powered flight portion can last from a few tenths of seconds to several minutes and can consist of multiple rocket stages. When in space and no more thrust is provided, the missile enters free-flight, the re-entry stage begins at an altitude where atmospheric drag plays a significant part in missile trajectory, and lasts until missile impact. The course taken by ballistic missiles has two significant desirable properties, first, ballistic missiles that fly above the atmosphere have a much longer range than would be possible for cruise missiles of the same size. Powered rocket flight through thousands of kilometers of air would require greater amounts of fuel, making the launch vehicles larger and easier to detect. Powered missiles that can cover similar ranges, such as missiles, do not use rocket motors for the majority of their flight. Despite this, cruise missiles have not made ballistic missiles obsolete, due to the major advantage. An ICBM can strike a target within a 10,000 km range in about 30 to 35 minutes, with terminal speeds of over 5,000 m/s, ballistic missiles are much harder to intercept than cruise missiles, due to the much shorter time available to intercept them. This is why ballistic missiles are some of the most feared weapons available, despite the fact that cruise missiles are cheaper, more mobile, Ballistic missiles can vary widely in range and use, and are often divided into categories based on range. A comparable missile would be the decommissioned Chinas JL-1 SLBM with a range of less than 2, tactical, short- and medium-range missiles are often collectively referred to as tactical and theatre ballistic missiles, respectively
29.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
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Engineering
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The term Engineering is derived from the Latin ingenium, meaning cleverness and ingeniare, meaning to contrive, devise. Engineering has existed since ancient times as humans devised fundamental inventions such as the wedge, lever, wheel, each of these inventions is essentially consistent with the modern definition of engineering. The term engineering is derived from the engineer, which itself dates back to 1390 when an engineer originally referred to a constructor of military engines. In this context, now obsolete, a referred to a military machine. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, the word engine itself is of even older origin, ultimately deriving from the Latin ingenium, meaning innate quality, especially mental power, hence a clever invention. The earliest civil engineer known by name is Imhotep, as one of the officials of the Pharaoh, Djosèr, he probably designed and supervised the construction of the Pyramid of Djoser at Saqqara in Egypt around 2630–2611 BC. Ancient Greece developed machines in both civilian and military domains, the Antikythera mechanism, the first known mechanical computer, and the mechanical inventions of Archimedes are examples of early mechanical engineering. In the Middle Ages, the trebuchet was developed, the first steam engine was built in 1698 by Thomas Savery. The development of this gave rise to the Industrial Revolution in the coming decades. With the rise of engineering as a profession in the 18th century, similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. The inventions of Thomas Newcomen and the Scottish engineer James Watt gave rise to mechanical engineering. The development of specialized machines and machine tools during the revolution led to the rapid growth of mechanical engineering both in its birthplace Britain and abroad. John Smeaton was the first self-proclaimed civil engineer and is regarded as the father of civil engineering. He was an English civil engineer responsible for the design of bridges, canals, harbours and he was also a capable mechanical engineer and an eminent physicist. Smeaton designed the third Eddystone Lighthouse where he pioneered the use of hydraulic lime and his lighthouse remained in use until 1877 and was dismantled and partially rebuilt at Plymouth Hoe where it is known as Smeatons Tower. The United States census of 1850 listed the occupation of engineer for the first time with a count of 2,000, there were fewer than 50 engineering graduates in the U. S. before 1865. In 1870 there were a dozen U. S. mechanical engineering graduates, in 1890 there were 6,000 engineers in civil, mining, mechanical and electrical. There was no chair of applied mechanism and applied mechanics established at Cambridge until 1875, the theoretical work of James Maxwell and Heinrich Hertz in the late 19th century gave rise to the field of electronics
31.
Leonardo da Vinci
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He has been variously called the father of palaeontology, ichnology, and architecture, and is widely considered one of the greatest painters of all time. Sometimes credited with the inventions of the parachute, helicopter and tank, many historians and scholars regard Leonardo as the prime exemplar of the Universal Genius or Renaissance Man, an individual of unquenchable curiosity and feverishly inventive imagination. Much of his working life was spent in the service of Ludovico il Moro in Milan. He later worked in Rome, Bologna and Venice, and he spent his last years in France at the home awarded to him by Francis I of France, Leonardo was, and is, renowned primarily as a painter. Among his works, the Mona Lisa is the most famous and most parodied portrait, Leonardos drawing of the Vitruvian Man is also regarded as a cultural icon, being reproduced on items as varied as the euro coin, textbooks, and T-shirts. Perhaps fifteen of his paintings have survived, Leonardo is revered for his technological ingenuity. He conceptualised flying machines, a type of armoured fighting vehicle, concentrated power, an adding machine. Some of his inventions, however, such as an automated bobbin winder. A number of Leonardos most practical inventions are nowadays displayed as working models at the Museum of Vinci. He made substantial discoveries in anatomy, civil engineering, geology, optics, and hydrodynamics, today, Leonardo is widely considered one of the most diversely talented individuals ever to have lived. Leonardo was born on 15 April 1452 at the hour of the night in the Tuscan hill town of Vinci. He was the son of the wealthy Messer Piero Fruosino di Antonio da Vinci, a Florentine legal notary, and Caterina. Leonardo had no surname in the modern sense – da Vinci simply meaning of Vinci, his birth name was Lionardo di ser Piero da Vinci, meaning Leonardo. The inclusion of the title ser indicated that Leonardos father was a gentleman, little is known about Leonardos early life. He spent his first five years in the hamlet of Anchiano in the home of his mother and his father had married a sixteen-year-old girl named Albiera Amadori, who loved Leonardo but died young in 1465 without children. When Leonardo was sixteen, his father married again to twenty-year-old Francesca Lanfredini, pieros legitimate heirs were born from his third wife Margherita di Guglielmo and his fourth and final wife, Lucrezia Cortigiani. Leonardo received an education in Latin, geometry and mathematics. In later life, Leonardo recorded only two childhood incidents, one, which he regarded as an omen, was when a kite dropped from the sky and hovered over his cradle, its tail feathers brushing his face
32.
Doubling the cube
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Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a cube whose volume is double that of the first, using only the tools of a compass. As with the problems of squaring the circle and trisecting the angle. The Egyptians, Indians, and particularly the Greeks were aware of the problem and made futile attempts at solving what they saw as an obstinate. However, the nonexistence of a solution was finally proven by Pierre Wantzel in 1837, in algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x3 =2, in other words, x = 3√2. This is because a cube of side length 1 has a volume of 13 =1, the impossibility of doubling the cube is therefore equivalent to the statement that 3√2 is not a constructible number. This implies that the degree of the extension generated by a constructible point must be a power of 2. The field extension generated by 3√2, however, is of degree 3 and we begin with the unit line segment defined by points and in the plane. We are required to construct a line segment defined by two separated by a distance of 3√2. Any newly defined point either arises as the result of the intersection of two circles, as the intersection of a circle and a line, or as the intersection of two lines. Restated in more abstract terminology, the new x- and y-coordinates have minimal polynomials of degree at most 2 over the subfield of ℝ generated by the previous coordinates, therefore, the degree of the field extension corresponding to each new coordinate is 2 or 1. By Gausss Lemma, p is irreducible over ℚ, and is thus a minimal polynomial over ℚ for 3√2. The field extension ℚ, ℚ is therefore of degree 3. But this is not a power of 2, so by the above, 3√2 is not the coordinate of a point, and thus a line segment of 3√2 cannot be constructed. The problem owes its name to a story concerning the citizens of Delos, the oracle responded that they must double the size of the altar to Apollo, which was a regular cube. This may be why the problem is referred to in the 350s BC by the author of the pseudo-Platonic Sisyphus as still unsolved, however another version of the story says that all three found solutions but they were too abstract to be of practical value. In modern notation, this means that given segments of lengths a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so that a r = r s = s 2 a. In turn, this means that r = a ⋅23 But Pierre Wantzel proved in 1837 that the root of 2 is not constructible
33.
Compass-and-straightedge construction
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, more formally, the only permissible constructions are those granted by Euclids first three postulates. It turns out to be the case that every point constructible using straightedge, the ancient Greek mathematicians first conceived compass-and-straightedge constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in cases were unable to do so. Gauss showed that some polygons are constructible but that most are not, some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems, in terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be using the four basic arithmetic operations. Circles can only be starting from two given points, the centre and a point on the circle. The compass may or may not collapse when its not drawing a circle, the straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature and it would appear that the modern compass is a more powerful instrument than the ancient collapsing compass. However, by Proposition 2 of Book 1 of Euclids Elements, although the proposition is correct, its proofs have a long and checkered history. Eyeballing it and getting close does not count as a solution and that is, it must have a finite number of steps, and not be the limit of ever closer approximations. One of the purposes of Greek mathematics was to find exact constructions for various lengths, for example. The Greeks could not find constructions for these three problems, among others, Squaring the circle, Drawing a square the same area as a given circle, doubling the cube, Drawing a cube with twice the volume of a given cube. Trisecting the angle, Dividing a given angle into three smaller angles all of the same size, for 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be generally impossible, the ancient Greek mathematicians first attempted compass-and-straightedge constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of an angle, a square whose area is twice that of another square, a square having the same area as a given polygon
34.
Archimedes
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Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε
35.
Method of exhaustion
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The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes small, the possible values for the area of the shape are systematically exhausted by the lower bound areas successively established by the sequence members. The method of exhaustion typically required a form of proof by contradiction and this amounts to finding an area of a region by first comparing it to the area of a second region. The idea originated in the late 5th century BC with Antiphon, the theory was made rigorous a few decades later by Eudoxus of Cnidus, who used it to calculate areas and volumes. It was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle, the first use of the term was in 1647 by Grégoire de Saint-Vincent in Opus geometricum quadraturae circuli et sectionum. The method of exhaustion is seen as a precursor to the methods of calculus, the development of analytical geometry and rigorous integral calculus in the 17th-19th centuries subsumed the method of exhaustion so that it is no longer explicitly used to solve problems. Euclid used the method of exhaustion to prove the following six propositions in the book 12 of his Elements, proposition 2 The area of a circle is proportional to the square of its radius. Proposition 5 The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases, proposition 10 The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. Proposition 11 The volume of a cone of the height is proportional to the area of the base. Proposition 12 The volume of a cone that is the similar to another is proportional to the cube of the ratio of the diameters of the bases, proposition 18 The volume of a sphere is proportional to the cube of its diameter. Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater area and greater number of sides. He also provided the bounds 3 + 10/71 < π <3 + 10/70, the Method of Mechanical Theorems The Quadrature of the Parabola Trapezoidal rule
36.
The Quadrature of the Parabola
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The Quadrature of the Parabola is a treatise on geometry, written by Archimedes in the 3rd century BC. The statement of the problem used the method of exhaustion, Archimedes may have dissected the area into infinitely many triangles whose areas form a geometric progression. He computes the sum of the geometric series, and proves that this is the area of the parabolic segment. A parabolic segment is the bounded by a parabola and line. To find the area of a segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the chord of the parabola. By Proposition 1, a line from the third vertex drawn parallel to the axis divides the chord into equal segments, the main theorem claims that the area of the parabolic segment is 4/3 that of the inscribed triangle. Archimedes gives two proofs of the main theorem, the first uses abstract mechanics, with Archimedes arguing that the weight of the segment will balance the weight of the triangle when placed on an appropriate lever. The second, more famous proof uses pure geometry, specifically the method of exhaustion, of the twenty-four propositions, the first three are quoted without proof from Euclids Elements of Conics. The main idea of the proof is the dissection of the segment into infinitely many triangles. Each of these triangles is inscribed in its own segment in the same way that the blue triangle is inscribed in the large segment. In propositions eighteen through twenty-one, Archimedes proves that the area of each triangle is one eighth of the area of the blue triangle. Using the method of exhaustion, it follows that the area of the parabolic segment is given by Area = T +2 +4 +8 + ⋯. This simplifies to give Area = T, to complete the proof, Archimedes shows that 1 +14 +116 +164 + ⋯ =43. The formula above is a geometric series—each successive term is one fourth of the previous term, in modern mathematics, that formula is a special case of the sum formula for a geometric series. Archimedes evaluates the sum using a geometric method, illustrated in the adjacent picture. This picture shows a square which has been dissected into an infinity of smaller squares. Each successive purple square has one fourth the area of the previous square, however, the purple squares are congruent to either set of yellow squares, and so cover 1/3 of the area of the unit square
37.
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
38.
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
39.
Galileo Galilei
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Galileo Galilei was an Italian polymath, astronomer, physicist, engineer, philosopher, and mathematician. He played a role in the scientific revolution of the seventeenth century. Galileo also worked in applied science and technology, inventing an improved military compass, Galileos championing of heliocentrism and Copernicanism was controversial during his lifetime, when most subscribed to either geocentrism or the Tychonic system. He met with opposition from astronomers, who doubted heliocentrism because of the absence of a stellar parallax. He was tried by the Inquisition, found vehemently suspect of heresy and he spent the rest of his life under house arrest. He has been called the father of observational astronomy, the father of modern physics, the father of scientific method, and the father of science. Galileo was born in Pisa, Italy, on 15 February 1564, the first of six children of Vincenzo Galilei, a famous lutenist, composer, and music theorist, and Giulia, three of Galileos five siblings survived infancy. The youngest, Michelangelo, also became a noted lutenist and composer although he contributed to financial burdens during Galileos young adulthood, Michelangelo was unable to contribute his fair share of their fathers promised dowries to their brothers-in-law, who would later attempt to seek legal remedies for payments due. Michelangelo would also occasionally have to borrow funds from Galileo to support his musical endeavours and these financial burdens may have contributed to Galileos early fire to develop inventions that would bring him additional income. When Galileo Galilei was eight, his family moved to Florence and he then was educated in the Vallombrosa Abbey, about 30 km southeast of Florence. Galileo Bonaiuti was buried in the church, the Basilica of Santa Croce in Florence. It was common for mid-sixteenth century Tuscan families to name the eldest son after the parents surname, hence, Galileo Galilei was not necessarily named after his ancestor Galileo Bonaiuti. The Italian male given name Galileo derives from the Latin Galilaeus, meaning of Galilee, the biblical roots of Galileos name and surname were to become the subject of a famous pun. In 1614, during the Galileo affair, one of Galileos opponents, in it he made a point of quoting Acts 1,11, Ye men of Galilee, why stand ye gazing up into heaven. Despite being a genuinely pious Roman Catholic, Galileo fathered three children out of wedlock with Marina Gamba and they had two daughters, Virginia and Livia, and a son, Vincenzo. Their only worthy alternative was the religious life, both girls were accepted by the convent of San Matteo in Arcetri and remained there for the rest of their lives. Virginia took the name Maria Celeste upon entering the convent and she died on 2 April 1634, and is buried with Galileo at the Basilica of Santa Croce, Florence. Livia took the name Sister Arcangela and was ill for most of her life, Vincenzo was later legitimised as the legal heir of Galileo and married Sestilia Bocchineri
40.
Parabolic reflector
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A parabolic reflector is a reflective surface used to collect or project energy such as light, sound, or radio waves. Its shape is part of a paraboloid, that is. The parabolic reflector transforms an incoming plane wave traveling along the axis into a spherical wave converging toward the focus, conversely, a spherical wave generated by a point source placed in the focus is reflected into a plane wave propagating as a collimated beam along the axis. Parabolic reflectors are used to collect energy from a distant source, since the principles of reflection are reversible, parabolic reflectors can also be used to focus radiation from an isotropic source into a narrow beam. In optics, parabolic mirrors are used to light in reflecting telescopes and solar furnaces, and project a beam of light in flashlights, searchlights, stage spotlights. In acoustics, parabolic microphones are used to record faraway sounds such as calls, in sports reporting. Strictly, the shape of the reflector is called a paraboloid. A parabola is the two-dimensional figure, however, in informal language, the word parabola and its associated adjective parabolic are often used in place of paraboloid and paraboloidal. Correspondingly, the dimensions of a symmetrical paraboloidal dish are related by the equation,4 F D = R2, where F is the length, D is the depth of the dish. All units must be the same, if two of these three quantities are known, this equation can be used to calculate the third. A more complex calculation is needed to find the diameter of the dish measured along its surface. This is sometimes called the diameter, and equals the diameter of a flat, circular sheet of material, usually metal. Two intermediate results are useful in the calculation, P =2 F and Q = P2 + R2, where F, D, and R are defined as above. The diameter of the dish, measured along the surface, is given by, R Q P + P ln . The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom, is given by 12 π R2 D, where the symbols are defined as above. This can be compared with the formulae for the volumes of a cylinder, a hemisphere (23 π R2 D, {\displaystyle \scriptstyle, and a cone. π R2 is the area of the dish, the area enclosed by the rim. The area of the surface of the dish can be found using the area formula for a surface of revolution which gives A = π R6 D2
41.
Reflecting telescope
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A reflecting telescope is an optical telescope which uses a single or combination of curved mirrors that reflect light and form an image. The reflecting telescope was invented in the 17th century as an alternative to the telescope which. Although reflecting telescopes produce other types of aberrations, it is a design that allows for very large diameter objectives. Almost all of the telescopes used in astronomy research are reflectors. Reflecting telescopes come in many variations and may employ extra optical elements to improve image quality or place the image in a mechanically advantageous position. Since reflecting telescopes use mirrors, the design is referred to as a catoptric telescope. The idea that curved mirrors behave like lenses dates back at least to Alhazens 11th century treatise on optics, the potential advantages of using parabolic mirrors, primarily reduction of spherical aberration with no chromatic aberration, led to many proposed designs for reflecting telescopes. The most notable being James Gregory, who published a design for a ‘reflecting’ telescope in 1663. It would be ten years, before the experimental scientist Robert Hooke was able to build this type of telescope, Isaac Newton has been generally credited with building the first reflecting telescope in 1668. It used a spherically ground metal primary mirror and a diagonal mirror in an optical configuration that has come to be known as the Newtonian telescope. A curved primary mirror is the reflector telescopes basic optical element that creates an image at the focal plane, the distance from the mirror to the focal plane is called the focal length. The primary mirror in most modern telescopes is composed of a glass cylinder whose front surface has been ground to a spherical or parabolic shape. A thin layer of aluminum is deposited onto the mirror. Some telescopes use primary mirrors which are made differently, molten glass is rotated to make its surface paraboloidal, and is kept rotating while it cools and solidifies. The resulting mirror shape approximates a desired paraboloid shape that requires grinding and polishing to reach the exact figure needed. Reflecting telescopes, just like any other system, do not produce perfect images. The use of mirrors avoids chromatic aberration but they produce other types of aberrations, to avoid this problem most reflecting telescopes use parabolic shaped mirrors, a shape that can focus all the light to a common focus. Field curvature - The best image plane is in general curved and it is sometimes corrected by a field flattening lens
42.
Mathematician
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A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and with whom the study of mathematics for its own sake begins, the first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked. Science and mathematics in the Islamic world during the Middle Ages followed various models and it was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences, an example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham, the Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking. ”Thus, seminars, overall, science became the focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems
43.
Marin Mersenne
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Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the father of acoustics. Mersenne, an ordained priest, had contacts in the scientific world and has been called the center of the world of science. Marin Mersenne was born of peasant parents near Oizé, Maine and he was educated at Le Mans and at the Jesuit College of La Flèche. On 17 July 1611, he joined the Minim Friars, and, after studying theology, between 1614 and 1618, he taught theology and philosophy at Nevers, but he returned to Paris and settled at the convent of LAnnonciade in 1620. There he studied mathematics and music and met with other kindred spirits such as René Descartes, Étienne Pascal, Pierre Petit, Gilles de Roberval and he corresponded with Giovanni Doni, Constantijn Huygens, Galileo Galilei, and other scholars in Italy, England and the Dutch Republic. He was a defender of Galileo, assisting him in translations of some of his mechanical works. For four years, Mersenne devoted himself entirely to philosophic and theological writing and it is sometimes incorrectly stated that he was a Jesuit. He was educated by Jesuits, but he never joined the Society of Jesus and he taught theology and philosophy at Nevers and Paris. He was not afraid to cause disputes among his friends in order to compare their views. In 1635 Mersenne met with Tommaso Campanella, but concluded that he could teach nothing in the sciences but still he has a good memory, Mersenne asked if René Descartes wanted Campanella to come to Holland to meet him, but Descartes declined. He visited Italy fifteen times, in 1640,1641 and 1645, in 1643–1644 Mersenne also corresponded with the German Socinian Marcin Ruar concerning the Copernican ideas of Pierre Gassendi, finding Ruar already a supporter of Gassendis position. Among his correspondents were Descartes, Galilei, Roberval, Pascal, Beeckman and he died September 1 through complications arising from a lung abscess. Some history scientists suggest he died for having drunk a huge quantity of water, along with Descartes. It was written as a commentary on the Book of Genesis, at first sight the book appears to be a collection of treatises on various miscellaneous topics. However Robert Lenoble has shown that the principle of unity in the work is a polemic against magical and divinatory arts, cabalism and he mentions Martin Del Rios Investigations into Magic and criticises Marsilio Ficino for claiming power for images and characters. He condemns astral magic and astrology and the anima mundi, a popular amongst Renaissance neo-platonists. Whilst allowing for an interpretation of the Cabala, he wholeheartedly condemned its magical application—particularly to angelology. He also criticises Pico della Mirandola, Cornelius Agrippa and Francesco Giorgio with Robert Fludd as his main target, Fludd responded with Sophia cum moria certamen, wherein Fludd admits his involvement with the Rosicrucians
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James Gregory (mathematician)
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James Gregory FRS was a Scottish mathematician and astronomer. His surname is spelt as Gregorie, the original Scottish spelling. In his book Geometriae Pars Universalis Gregory gave both the first published statement and proof of the theorem of the calculus, for which he was acknowledged by Isaac Barrow. It was his mother who endowed Gregory with his appetite for geometry, her uncle – Alexander Anderson – having been a pupil, after his fathers death in 1651 his elder brother David took over responsibility for his education. He attended Aberdeen Grammar School, and then Marischal College from 1653–1657, in 1663 he went to London, meeting John Collins and fellow Scot Robert Moray, one of the founders of the Royal Society. In 1664 he departed for the University of Padua, in the Venetian Republic, passing through Flanders, Paris, at Padua he lived in the house of his countryman James Caddenhead, the professor of philosophy, and he was taught by Stefano Angeli. He was successively professor at the University of St Andrews and the University of Edinburgh and he had married Mary, daughter of George Jameson, painter, and widow of John Burnet of Elrick, Aberdeen, their son James was Professor of Physics at Kings College, Aberdeen. He was the grandfather of John Gregory, uncle of David Gregorie and brother of David Gregory, about a year after assuming the Chair of Mathematics at Edinburgh, James Gregory suffered a stroke while viewing the moons of Jupiter with his students. He died a few days later at the age of 36, in the Optica Promota, published in 1663, Gregory described his design for a reflecting telescope, the Gregorian telescope. In 1667, Gregory issued his Vera Circuli et Hyperbolae Quadratura, in which he showed how the areas of the circle, nevertheless Gregory was effectively among the first to speculate about the existence of what are now termed transcendental numbers. In addition the first proof of the theorem of calculus. The book also contains series expansions of sin, cos, arcsin, Gregory was probably unaware that the earliest enunciations of these expansions were made by Madhava in India in the 14th century. The book was reprinted in 1668 with an appendix, Geometriae Pars, in his 1663 Optica Promota, James Gregory described his reflecting telescope which has come to be known by his name, the Gregorian telescope. Gregory pointed out that a telescope with a parabolic mirror would correct spherical aberration as well as the chromatic aberration seen in refracting telescopes. According to his own confession, Gregory had no practical skill, the Gregorian telescope design is rarely used today, as other types of reflecting telescopes are known to be more efficient for standard applications. Gregorian optics are used in radio telescopes such as Arecibo. The following excerpt is from the Pantologia, in 1671, or perhaps earlier, he established the theorem that θ = tan θ − tan 3 θ + tan 5 θ − …, the result being true only if θ lies between −π and π. This formula was used to calculate digits of π, although more efficient formulas were later discovered
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Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. Newtons Principia formulated the laws of motion and universal gravitation that dominated scientists view of the universe for the next three centuries. Newtons work on light was collected in his influential book Opticks. He also formulated a law of cooling, made the first theoretical calculation of the speed of sound. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He was knighted by Queen Anne in 1705 and he spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint and his father, also named Isaac Newton, had died three months before. Born prematurely, he was a child, his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Newtons mother had three children from her second marriage. From the age of twelve until he was seventeen, Newton was educated at The Kings School, Grantham which taught Latin and Greek. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, Henry Stokes, master at the Kings School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a bully, he became the top-ranked student. In June 1661, he was admitted to Trinity College, Cambridge and he started as a subsizar—paying his way by performing valets duties—until he was awarded a scholarship in 1664, which guaranteed him four more years until he would get his M. A. He set down in his notebook a series of Quaestiones about mechanical philosophy as he found it, in 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his B. A. degree in August 1665, in April 1667, he returned to Cambridge and in October was elected as a fellow of Trinity. Fellows were required to become ordained priests, although this was not enforced in the restoration years, however, by 1675 the issue could not be avoided and by then his unconventional views stood in the way. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II. A and he was elected a Fellow of the Royal Society in 1672. Newtons work has been said to distinctly advance every branch of mathematics then studied and his work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers
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Newton's reflector
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The first reflecting telescope built by Sir Isaac Newton in 1668 is a landmark in the history of telescopes, being the first known successful reflecting telescope. It was the prototype for a design that came to be called a newtonian telescope. Isaac Newton built his reflecting telescope as a proof for his theory that light is composed of a spectrum of colours. He had concluded that the lens of any refracting telescope would suffer from the dispersion of light into colours, the telescope he constructed used mirrors as the objective which bypass that problem. To create the primary mirror Newton used a composition of metal consisting of six parts copper to two parts tin, an early composition of speculum metal. He devised means for shaping and grinding the mirror and may have been the first to use a lap to polish the optical surface. This unique addition allowed the image to be viewed with minimal obstruction of the objective mirror and he also made all the tube, mount, and fittings. The Eye-glass was Plano-convex, and the diameter of the Sphere to which the side was ground was about 1/5 of an Inch, or a little less. By another way of measuring I found it magnified 35 times, for this Circle being placed here, stoppd much of the erroneous Light, which other wise would have disturbed the Vision. By comparing it with a pretty good Perspective of four Feet in length, made with a concave Eye-glass, had it magnified but 30 or 25 times, it would have made the Object appear more brisk and pleasant. The object-metal was two inches broad, and about one-third part of a thick, to keep it from bending. I had two of these metals, and when I had polished them both I tried which was best, and ground the other again, to see if I could make it better, than that which I kept. The mirror was aperture reduced to an aperture of 1.3 inches by placing a disk with a hole in it between the observers eye and the eyepiece. The telescope had a diagonal secondary mirror bouncing the light at a 90° angle to a Plano-convex eyepiece with a probable focal length of 4. 5mm yielding his observed 35 times magnification. Newton said the telescope was 6.25 inches long, this matches the length of the instrument pictured in his monograph Opticks, Newton completed his first reflecting telescope in late 1668 and first wrote about it in a February 23,1669 letter to Henry Oldenburg. Newton found that he could see the four Galilean moons of Jupiter, Newtons friend Isaac Barrow showed the telescope to small group from the Royal Society of London at the end of 1671. They were so impressed with it they demonstrated it for Charles II in January 1672 and this telescope remained in the repository of the Royal Society until it disintegrated and then disappeared from their records. The last reference to it was in 1731 saying that only two mirrors remained of it, there has been much confusion and dispute concerning the telescopes that Newton built, but it is now clear that his first telescope was a prototype that he constructed in 1668
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Spherical mirror
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A curved mirror is a mirror with a curved reflecting surface. The surface may be convex or concave. Most curved mirrors have surfaces that are shaped like part of a sphere, one advantage that mirror optics have over lens optics is that mirrors do not introduce chromatic aberration. A convex mirror, fish eye mirror or diverging mirror, is a mirror in which the reflective surface bulges toward the light source. Convex mirrors reflect light outwards, therefore they are not used to focus light, such mirrors always form a virtual image, since the focal point and the centre of curvature are both imaginary points inside the mirror, that cannot be reached. As a result, images formed by these mirrors cannot be projected on a screen, the image is smaller than the object, but gets larger as the object approaches the mirror. A collimated beam of light diverges after reflection from a convex mirror, the passenger-side mirror on a car is typically a convex mirror. In some countries, these are labeled with the safety warning Objects in mirror are closer than they appear, convex mirrors are preferred in vehicles because they give an upright, though diminished, image. Also they provide a field of view as they are curved outwards. These mirrors are often found in the hallways of buildings, including hospitals, hotels, schools, stores. They are usually mounted on a wall or ceiling where hallways intersect each other and they are useful for people accessing the hallways, especially at locations having blind spots or where visibility may be limited. They are also used on roads, driveways, and alleys to provide safety for motorists where there is a lack of visibility, especially at curves and turns. Convex mirrors are used in automated teller machines as a simple and handy security feature. Similar devices are sold to be attached to ordinary computer monitors, convex mirrors make everything seem smaller but cover a larger area of surveillance. Round convex mirrors called Oeil de Sorcière were a luxury item from the 15th century onwards. With 15th century technology, it was easier to make a curved mirror than a perfectly flat one. They were also known as bankers eyes due to the fact that their field of vision was useful for security. Famous examples in art include the Arnolfini Portrait by Jan van Eyck, the image on a convex mirror is always virtual, diminished, and upright
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Locus of points
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In geometry, a locus is a set of points, whose location satisfies or is determined by one or more specified conditions. Until the beginning of 20th century, a shape was not considered as an infinite set of points, rather. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a distance of a fixed point. In contrast to the view, the old formulation avoids considering infinite collections. Once set theory became the universal basis over which the mathematics is built. Examples from plane geometry include, The set of points equidistant from two points is a perpendicular bisector to the segment connecting the two points. The set of points equidistant from two lines cross is the angle bisector. All conic sections are loci, Parabola, the set of points equidistant from a single point, Circle, the set of points for which the distance from a single point is constant. The set of points for each of which the ratio of the distances to two given foci is a constant is referred to as a Circle of Apollonius. Hyperbola, the set of points for each of which the value of the difference between the distances to two given foci is a constant. Ellipse, the set of points for each of which the sum of the distances to two given foci is a constant, the circle is the special case in which the two foci coincide with each other. Other examples of loci appear in areas of mathematics. For example, in dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps. Proof that all the points on the given shape satisfy the conditions and we find the locus of the points P that have a given ratio of distances k = d1/d2 to two given points. In this example we choose k=3, A and B as the fixed points and it is the circle of Apollonius defined by these values of k, A, and B. A triangle ABC has a side with length c. We determine the locus of the third vertex C such that the medians from A and C are orthogonal and we choose an orthonormal coordinate system such that A, B. C is the third vertex