1.
Reuleaux triangle
–
A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. It is a curve of constant width, the simplest and best known such curve other than the circle itself, Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same, the Reuleaux triangle is one answer to the question Other than a circle, Reuleaux triangles have also been called spherical triangles, but that term more properly refers to triangles on the curved surface of a sphere. Among constant-width shapes with a width, the Reuleaux triangle has the minimum area. By several numerical measures it is the farthest from being centrally symmetric and it provides the largest constant-width shape avoiding the points of an integer lattice, and is closely related to the shape of the quadrilateral maximizing the ratio of perimeter to diameter. It can perform a rotation within a square while at all times touching all four sides of the square. However, although it covers most of the square in this process, it fails to cover a small fraction of the squares area. Because of this property of rotating within a square, the Reuleaux triangle is sometimes known as the Reuleaux rotor. The Reuleaux triangle is the first of a sequence of Reuleaux polygons, some of these curves have been used as the shapes of coins. Alternatively, the surface of revolution of the Reuleaux triangle also has constant width, the Reuleaux triangle may be constructed either directly from three circles, or by rounding the sides of an equilateral triangle. The three-circle construction may be performed with a compass alone, not even needing a straightedge, by the Mohr–Mascheroni theorem the same is true more generally of any compass-and-straightedge construction, but the construction for the Reuleaux triangle is particularly simple. The first step is to mark two points of the plane, and use the compass to draw a circle centered at one of the marked points. Next, one draws a circle, of the same radius, centered at the other marked point. Finally, one draws a circle, again of the same radius. The central region in the arrangement of three circles will be a Reuleaux triangle. Alternatively, a Reuleaux triangle may be constructed from an equilateral triangle T by drawing three arcs of circles, each centered at one vertex of T and connecting the two vertices. Or, equivalently, it may be constructed as the intersection of three disks centered at the vertices of T, with equal to the side length of T. In any pair of parallel supporting lines, one of the two lines will necessarily touch the triangle at one of its vertices, the other supporting line may touch the triangle at any point on the opposite arc, and their distance equals the radius of this arc
2.
Spherical geometry
–
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean, two practical applications of the principles of spherical geometry are navigation and astronomy. In plane geometry, the concepts are points and lines. On a sphere, points are defined in the usual sense, the equivalents of lines are not defined in the usual sense of straight line in Euclidean geometry, but in the sense of the shortest paths between points, which are called geodesics. On a sphere, the geodesics are the circles, other geometric concepts are defined as in plane geometry. Spherical geometry is not elliptic geometry, but is rather a subset of elliptic geometry, for example, it shares with that geometry the property that a line has no parallels through a given point. An important geometry related to that of the sphere is that of the projective plane. Locally, the plane has all the properties of spherical geometry. In particular, it is non-orientable, or one-sided, Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Higher-dimensional spherical geometries exist, see elliptic geometry, the earliest mathematical work of antiquity to come down to our time is On the rotating sphere by Autolycus of Pitane, who lived at the end of the fourth century BC. The book of unknown arcs of a written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the law of sines. The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe, however, Gerolamo Cardano noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah. L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15,1771, pp. 195–216, Opera Omnia, Series 1, Volume 28, pp. 142–160. L. Euler, De mensura angulorum solidorum, Acta academiae scientarum imperialis Petropolitinae 2,1781, p. 31–54, Opera Omnia, Series 1, vol. L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientarum imperialis Petropolitinae 4,1783, p. 91–96, Opera Omnia, Series 1, vol. L. Euler, Geometrica et sphaerica quaedam, Mémoires de lAcademie des Sciences de Saint-Petersbourg 5,1815, p. 96–114, Opera Omnia, Series 1, vol. L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientarum imperialis Petropolitinae 3,1782, p. 72–86, Opera Omnia, Series 1, vol
3.
Trigonometric functions
–
In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
4.
Edge (geometry)
–
For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007
5.
Angle
–
In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
6.
Great circle
–
A great circle, also known as an orthodrome or Riemannian circle, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. This partial case of a circle of a sphere is opposed to a circle, the intersection of the sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, a great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean 3-space is a circle of exactly one sphere. For most pairs of points on the surface of a sphere, there is a great circle through the two points. The exception is a pair of points, for which there are infinitely many great circles. The minor arc of a circle between two points is the shortest surface-path between them. In this sense, the arc is analogous to “straight lines” in Euclidean geometry. The length of the arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry. The great circles are the geodesics of the sphere, in higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn+1. To prove that the arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider the class of all paths from a point p to another point q. Introduce spherical coordinates so that p coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by θ = θ, ϕ = ϕ, a ≤ t ≤ b provided we allow φ to take on arbitrary real values. The infinitesimal arc length in these coordinates is d s = r θ ′2 + ϕ ′2 sin 2 θ d t. So the length of a curve γ from p to q is a functional of the curve given by S = r ∫ a b θ ′2 + ϕ ′2 sin 2 θ d t. Note that S is at least the length of the meridian from p to q, S ≥ r ∫ a b | θ ′ | d t ≥ r | θ − θ |. Since the starting point and ending point are fixed, S is minimized if and only if φ =0, so the curve must lie on a meridian of the sphere φ = φ0 = constant
7.
Sphere
–
A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
8.
Astronomy
–
Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
9.
Geodesy
–
Geodesists also study geodynamical phenomena such as crustal motion, tides, and polar motion. For this they design global and national networks, using space and terrestrial techniques while relying on datums. Geodesy — from the Ancient Greek word γεωδαισία geodaisia — is primarily concerned with positioning within the temporally varying gravity field, such geodetic operations are also applied to other astronomical bodies in the solar system. It is also the science of measuring and understanding the earths geometric shape, orientation in space and this applies to the solid surface, the liquid surface and the Earths atmosphere. For this reason, the study of the Earths gravity field is called physical geodesy by some, the geoid is essentially the figure of the Earth abstracted from its topographical features. It is an idealized surface of sea water, the mean sea level surface in the absence of currents, air pressure variations etc. The geoid, unlike the ellipsoid, is irregular and too complicated to serve as the computational surface on which to solve geometrical problems like point positioning. The geometrical separation between the geoid and the ellipsoid is called the geoidal undulation. It varies globally between ±110 m, when referred to the GRS80 ellipsoid, a reference ellipsoid, customarily chosen to be the same size as the geoid, is described by its semi-major axis a and flattening f. The quantity f = a − b/a, where b is the axis, is a purely geometrical one. The mechanical ellipticity of the Earth can be determined to high precision by observation of satellite orbit perturbations and its relationship with the geometrical flattening is indirect. The relationship depends on the density distribution, or, in simplest terms. The 1980 Geodetic Reference System posited a 6,378,137 m semi-major axis and this system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics. It is essentially the basis for geodetic positioning by the Global Positioning System and is also in widespread use outside the geodetic community. The locations of points in space are most conveniently described by three cartesian or rectangular coordinates, X, Y and Z. Since the advent of satellite positioning, such systems are typically geocentric. The X-axis lies within the Greenwich observatorys meridian plane, the coordinate transformation between these two systems is described to good approximation by sidereal time, which takes into account variations in the Earths axial rotation. A more accurate description also takes polar motion into account, a closely monitored by geodesists
10.
Navigation
–
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another. The field of navigation includes four categories, land navigation, marine navigation, aeronautic navigation. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks, all navigational techniques involve locating the navigators position compared to known locations or patterns. Navigation, in a sense, can refer to any skill or study that involves the determination of position and direction. In this sense, navigation includes orienteering and pedestrian navigation, for information about different navigation strategies that people use, visit human navigation. In the European medieval period, navigation was considered part of the set of seven mechanical arts, early Pacific Polynesians used the motion of stars, weather, the position of certain wildlife species, or the size of waves to find the path from one island to another. Maritime navigation using scientific instruments such as the mariners astrolabe first occurred in the Mediterranean during the Middle Ages, the perfecting of this navigation instrument is attributed to Portuguese navigators during early Portuguese discoveries in the Age of Discovery. Open-seas navigation using the astrolabe and the compass started during the Age of Discovery in the 15th century, the Portuguese began systematically exploring the Atlantic coast of Africa from 1418, under the sponsorship of Prince Henry. In 1488 Bartolomeu Dias reached the Indian Ocean by this route, in 1492 the Spanish monarchs funded Christopher Columbuss expedition to sail west to reach the Indies by crossing the Atlantic, which resulted in the Discovery of America. In 1498, a Portuguese expedition commanded by Vasco da Gama reached India by sailing around Africa, soon, the Portuguese sailed further eastward, to the Spice Islands in 1512, landing in China one year later. The fleet of seven ships sailed from Sanlúcar de Barrameda in Southern Spain in 1519, crossed the Atlantic Ocean, some ships were lost, but the remaining fleet continued across the Pacific making a number of discoveries including Guam and the Philippines. By then, only two galleons were left from the original seven, the Victoria led by Elcano sailed across the Indian Ocean and north along the coast of Africa, to finally arrive in Spain in 1522, three years after its departure. The Trinidad sailed east from the Philippines, trying to find a path back to the Americas. He arrived in Acapulco on October 8,1565, the term stems from 1530s, from Latin navigationem, from navigatus, pp. of navigare to sail, sail over, go by sea, steer a ship, from navis ship and the root of agere to drive. Roughly, the latitude of a place on Earth is its angular distance north or south of the equator, latitude is usually expressed in degrees ranging from 0° at the Equator to 90° at the North and South poles. The height of Polaris in degrees above the horizon is the latitude of the observer, similar to latitude, the longitude of a place on Earth is the angular distance east or west of the prime meridian or Greenwich meridian. Longitude is usually expressed in degrees ranging from 0° at the Greenwich meridian to 180° east and west, sydney, for example, has a longitude of about 151° east. New York City has a longitude of 74° west, for most of history, mariners struggled to determine longitude
11.
History of trigonometry
–
Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy, in Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata. During the Middle Ages, the study of continued in Islamic mathematics. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics, the term trigonometry was derived from Greek τρίγωνον trigōnon, triangle and μέτρον metron, measure. Our modern word sine is derived from the Latin word sinus, the Arabic term is in origin a corruption of Sanskrit jīvā, or chord. Sanskrit jīvā in learned usage was a synonym of jyā chord, Sanskrit jīvā was loaned into Arabic as jiba. Particularly Fibonaccis sinus rectus arcus proved influential in establishing the term sinus, the words minute and second are derived from the Latin phrases partes minutae primae and partes minutae secundae. These roughly translate to first small parts and second small parts, the ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, based on one interpretation of the Plimpton 322 cuneiform tablet, some have even asserted that the ancient Babylonians had a table of secants. There is, however, much debate as to whether it is a table of Pythagorean triples, the Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC. Ahmes solution to the problem is the ratio of half the side of the base of the pyramid to its height, in other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face. Ancient Greek and Hellenistic mathematicians made use of the chord, given a circle and an arc on the circle, the chord is the line that subtends the arc. A chords perpendicular bisector passes through the center of the circle and bisects the angle. One half of the chord is the sine of one half the bisected angle, that is, c h o r d θ =2 sin θ2. Due to this relationship, a number of identities and theorems that are known today were also known to Hellenistic mathematicians. For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosines for obtuse and acute angles, respectively, theorems on the lengths of chords are applications of the law of sines. And Archimedes theorem on broken chords is equivalent to formulas for sines of sums, the first trigonometric table was apparently compiled by Hipparchus of Nicaea, who is now consequently known as the father of trigonometry. Hipparchus was the first to tabulate the corresponding values of arc and it seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords
12.
Mathematics in medieval Islam
–
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries, the study of algebra, the name of which is derived from the Arabic word meaning completion or reunion of broken parts, flourished during the Islamic golden age. Muhammad ibn Musa al-Khwarizmi, a scholar in the House of Wisdom in Baghdad, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi deals with ways to solve for the roots of first. He also introduces the method of reduction, and unlike Diophantus, Al-Khwarizmis algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the work of Diophantus, which was syncopated. The transition to symbolic algebra, where symbols are used, can be seen in the work of Ibn al-Banna al-Marrakushi. It is important to understand just how significant this new idea was and it was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a theory which allowed rational numbers, irrational numbers, geometrical magnitudes. It gave mathematics a whole new development path so much broader in concept to that which had existed before, another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. Several other mathematicians during this time expanded on the algebra of Al-Khwarizmi. Omar Khayyam, along with Sharaf al-Dīn al-Tūsī, found several solutions of the cubic equation, omar Khayyam found the general geometric solution of a cubic equation. Omar Khayyám wrote the Treatise on Demonstration of Problems of Algebra containing the solution of cubic or third-order equations. Khayyám obtained the solutions of equations by finding the intersection points of two conic sections. This method had used by the Greeks, but they did not generalize the method to cover all equations with positive roots. Sharaf al-Dīn al-Ṭūsī developed an approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. His surviving works give no indication of how he discovered his formulae for the maxima of these curves, various conjectures have been proposed to account for his discovery of them. The earliest implicit traces of mathematical induction can be found in Euclids proof that the number of primes is infinite, the first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique
13.
John Napier
–
John Napier of Merchiston, also signed as Neper, Nepair, nicknamed Marvellous Merchiston) was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston and his Latinized name was Ioannes Neper. John Napier is best known as the discoverer of logarithms and he also invented the so-called Napiers bones and made common the use of the decimal point in arithmetic and mathematics. Napiers birthplace, Merchiston Tower in Edinburgh, is now part of the facilities of Edinburgh Napier University, Napier died from the effects of gout at home at Merchiston Castle and his remains were buried in the kirkyard of St Giles. Following the loss of the kirkyard there to build Parliament House, archibald Napier was 16 years old when John Napier was born. As was the practice for members of the nobility at that time, he was privately tutored and did not have formal education until he was 13. He did not stay in very long. It is believed that he dropped out of school in Scotland, in 1571, Napier, aged 21, returned to Scotland, and bought a castle at Gartness in 1574. On the death of his father in 1608, Napier and his family moved into Merchiston Castle in Edinburgh and he died at the age of 67. In such conditions, it is surprising that many mathematicians were acutely aware of the issues of computation and were dedicated to relieving practitioners of the calculation burden. In particular, the Scottish mathematician John Napier was famous for his devices to assist with computation and he invented a well-known mathematical artifact, the ingenious numbering rods more quaintly known as “Napiers bones, ” that offered mechanical means for facilitating computation. He appreciated that, for the most part, practitioners who had laborious computations generally did them in the context of trigonometry, therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant. His work, Mirifici Logarithmorum Canonis Descriptio contained fifty-seven pages of explanatory matter, the book also has an excellent discussion of theorems in spherical trigonometry, usually known as Napiers Rules of Circular Parts. Modern English translations of both Napiers books on logarithms, and their description can be found on the web, as well as a discussion of Napiers Bones and his invention of logarithms was quickly taken up at Gresham College, and prominent English mathematician Henry Briggs visited Napier in 1615. Among the matters discussed were a re-scaling of Napiers logarithms. Napier delegated to Briggs the computation of a revised table, the computational advance available via logarithms, the converse of powered numbers or exponential notation, was such that it made calculations by hand much quicker. The way was opened to later scientific advances, in astronomy, dynamics and he improved Simon Stevins decimal notation. Lattice multiplication, used by Fibonacci, was more convenient by his introduction of Napiers bones
14.
Jean Baptiste Joseph Delambre
–
Jean Baptiste Joseph, chevalier Delambre was a French mathematician and astronomer. He was also director of the Paris Observatory, and author of books on the history of astronomy from ancient times to the 18th century. After a childhood fever, he suffered from very sensitive eyes, for fear of losing his ability to read, he devoured any book available and trained his memory. Delambres quickly achieved success in his career in astronomy, such that in 1788 and this portion of the meridian, which also passes through Paris, was to serve as the basis for the length of the quarter meridian, connecting the North Pole with the Equator. In April 1791, the academys Metric Commission confided this mission to Jean-Dominique de Cassini, Cassini was chosen to head the northern expedition but, as a royalist, he refused to serve under the revolutionary government after the arrest of King Louis XVI on his Flight to Varennes. Pierre Méchain headed the expedition, measuring from Barcelona to Rodez. The measurements were finished in 1798, the gathered data were presented to an international conference of savants in Paris the following year. After Méchains death in 1804, he was appointed director of the Paris Observatory and he was also professor of Astronomy at the Collège de France. The same year he married Elisabeth-Aglaée Leblanc de Pommard, a widow with whom he had lived already for a long time and he was a knight of the Order of Saint Michael and of the Légion dhonneur. His name is one of the 72 names inscribed on the Eiffel tower. He was elected a Foreign Honorary Member of the American Academy of Arts, Delambre died in 1822 and was interred in Père Lachaise Cemetery in Paris. The crater Delambre on the Moon is named after him,1, lxxii,556 pp.1 folded plate, vol. 2, viii,639 pp.16 folded plates, Reprinted by New York and London, Johnson Reprint Corporation,1965, with a new preface by Otto Neugebauer. Histoire de lastronomie du moyen age, Paris, Mme Ve Courcier,1819, lxxxiv,640 pp.17 folded plates. Reprinted by New York and London, Johnson Reprint Corporation,1965 OCLC647834, also reprinted by Paris, J. Gabay,2006. Histoire de lastronomie moderne, Paris, Mme Ve Courcier,1821,1, lxxxii,715 pp.9 folded plates, vol. 2,804 pp.8 folded plates, Reprinted by New York and London, Johnson Reprint Corporation,1969, with a new introduction and tables of contents by I. Also reprinted by Paris, Editions Jacques Gabay,2006 and this takes the history to the 17th century
15.
Polygon
–
In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
16.
Arc (geometry)
–
In Euclidean geometry, an arc is a closed segment of a differentiable curve. A common example in the plane, is a segment of a circle called a circular arc, in space, if the arc is part of a great circle, it is called a great arc. Every pair of points on a circle determines two arcs. The length, L, of an arc of a circle with radius r and this is because L c i r c u m f e r e n c e = θ2 π. Substituting in the circumference L2 π r = θ2 π, and, with α being the angle measured in degrees, since θ = α/180π. For example, if the measure of the angle is 60 degrees and this is so because the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportional. The area of the sector formed by an arc and the center of a circle is A =12 r 2 θ. The area A has the proportion to the circle area as the angle θ to a full circle. We can cancel π on both sides, A r 2 = θ2, by multiplying both sides by r2, we get the final result, A =12 r 2 θ. Using the conversion described above, we find that the area of the sector for an angle measured in degrees is A = α360 π r 2. The area of the bounded by the arc and the straight line between its two end points is 12 r 2. To get the area of the arc segment, we need to subtract the area of the triangle, determined by the circles center and the two end points of the arc, from the area A. Using the intersecting chords theorem it is possible to calculate the radius r of a circle given the height H and its perpendicular bisector is another chord, which is a diameter of the circle. The length of the first chord is W, and it is divided by the bisector into two halves, each with length W/2. The total length of the diameter is 2r, and it is divided into two parts by the first chord, the length of one part is the sagitta of the arc, H, and the other part is the remainder of the diameter, with length 2r − H. Applying the intersecting chords theorem to these two chords produces H =2, whence 2 r − H = W24 H, so r = W28 H + H2
17.
Lune (geometry)
–
In plane geometry, a lune is the concave-convex area bounded by two circular arcs, while a convex-convex area is termed a lens. The word lune derives from luna, the Latin word for Moon, formally, a lune is the relative complement of one disk in another. Alternatively, if A and B are disks, then L = A − A ∩ B is a lune, in the 5th century BC, Hippocrates of Chios showed that certain lunes could be exactly squared by straightedge and compass. Arbelos Crescent Gauss–Bonnet theorem Weisstein, Eric W. Lune, the Five Squarable Lunes at MathPages
18.
Digon
–
In geometry, a digon is a polygon with two sides and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved, a regular digon has both angles equal and both sides equal and is represented by Schläfli symbol. It may be constructed on a sphere as a pair of 180 degree arcs connecting antipodal points, the digon is the simplest abstract polytope of rank 2. A truncated digon, t is a square, an alternated digon, h is a monogon. A straight-sided digon is regular even though it is degenerate, because its two edges are the length and its two angles are equal. As such, the regular digon is a constructible polygon, some definitions of a polygon do not consider the digon to be a proper polygon because of its degeneracy in the Euclidean case. A digon as a face of a polyhedron is degenerate because it is a degenerate polygon, but sometimes it can have a useful topological existence in transforming polyhedra. A spherical lune is a digon whose two vertices are antipodal points on the sphere, a spherical polyhedron constructed from such digons is called a hosohedron. The digon is an important construct in the theory of networks such as graphs. Topological equivalences may be established using a process of reduction to a set of polygons. The digon represents a stage in the simplification where it can be removed and substituted by a line segment. The cyclic groups may be obtained as rotation symmetries of polygons, monogon Demihypercube Herbert Busemann, The geometry of geodesics. New York, Academic Press,1955 Coxeter, Regular Polytopes, Dover Publications Inc,1973 ISBN 0-486-61480-8 Weisstein, a. B. Ivanov, Digon, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Media related to Digons at Wikimedia Commons
19.
Spherical law of cosines
–
In spherical trigonometry, the law of cosines is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a triangle on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere. Since this is a sphere, the lengths a, b. As a special case, for C = π/2, then cos C =0, if the law of cosines is used to solve for c, the necessity of inverting the cosine magnifies rounding errors when c is small. In this case, the formulation of the law of haversines is preferable. It can be obtained from consideration of a spherical triangle dual to the given one, a proof of the law of cosines can be constructed as follows. Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle. Then, the angle C is given by, cos C = t a ⋅ t b = cos c − cos a cos b sin a sin b from which the law of cosines immediately follows. To the diagram above, add a plane tangent to the sphere at u and we then have two plane triangles with a side in common, the triangle containing u, y and z and the one containing O, y and z. Sides of the first triangle are tan a and tan b, so − tan a tan b cos C =1 − sec a sec b cos c Multiply both sides by cos a cos b and rearrange. The angles and distances do not change if the sphere is rotated, so we can rotate the sphere so that u is at the north pole, with this rotation, the spherical coordinates for v are = and the spherical coordinates for w are =. The Cartesian coordinates for v are = and the Cartesian coordinates for w are =
20.
Trigonometry
–
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies, Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles, thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees, the ancient Nubians used a similar method. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, in the 2nd century AD, the Greco-Egyptian astronomer Ptolemy printed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta and these Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, at about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond, Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for maps of large geographic areas. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595, gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry, the works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general Taylor series, if one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees, they are complementary angles, the shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, if the length of one of the sides is known, the other two are determined. Sin A = opposite hypotenuse = a c, Cosine function, defined as the ratio of the adjacent leg to the hypotenuse
21.
Law of sines
–
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle to the sines of its angles. When the last of these equations is not used, the law is sometimes stated using the reciprocals, the law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. Numerical calculation using this technique may result in an error if an angle is close to 90 degrees. It can also be used when two sides and one of the angles are known. In some such cases, the triangle is not uniquely determined by this data, the law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines. The law of sines can be generalized to higher dimensions on surfaces with constant curvature, the area T of any triangle can be written as one half of its base times its height. Thus, depending on the selection of the base the area of the triangle can be written as any of, multiplying these by 2/abc gives 2 T a b c = sin A a = sin B b = sin C c. When using the law of sines to find a side of a triangle, in the case shown below they are triangles ABC and AB′C′. Given a general triangle the following conditions would need to be fulfilled for the case to be ambiguous, The only information known about the triangle is the angle A, the side a is shorter than the side c. The side a is longer than the altitude h from angle B, without further information it is impossible to decide which is the triangle being asked for. The following are examples of how to solve a problem using the law of sines, given, side a =20, side c =24, and angle C = 40°. Using the law of sines, we conclude that sin A20 = sin 40 ∘24, note that the potential solution A =147. 61° is excluded because that would necessarily give A + B + C > 180°. The second equality above readily simplifies to Herons formula for the area, the law of sines takes on a similar form in the presence of curvature. In the spherical case, the formula is, sin A sin α = sin B sin β = sin C sin γ. Here, α, β, and γ are the angles at the center of the sphere subtended by the three arcs of the spherical surface triangle a, b, and c, respectively, a, B, and C are the surface angles opposite their respective arcs. See also Spherical law of cosines and Half-side formula, in hyperbolic geometry when the curvature is −1, the law of sines becomes sin A sinh a = sin B sinh b = sin C sinh c. Define a generalized function, depending also on a real parameter K. The law of sines in constant curvature K reads as sin A sin K a = sin B sin K b = sin C sin K c
22.
Triple product
–
In vector algebra, a branch of mathematics, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name triple product is used for two different products, the scalar triple product and, less often, the vector-valued vector triple product. The scalar triple product is defined as the dot product of one of the vectors with the product of the other two. Geometrically, the triple product a ⋅ is the volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first, if it were, it would leave the cross product of a scalar and a vector, which is not defined. This follows from the property and the commutative property of the dot product. A ⋅ = ⋅ c Swapping any two of the three operands negates the triple product and this follows from the circular-shift property and the anticommutativity of the cross product. If the scalar product is equal to zero, then the three vectors a, b, and c are coplanar, since the parallelepiped defined by them would be flat and have no volume. Although the scalar product gives the volume of the parallelepiped, it is the signed volume. This means the product is negated if the orientation is reversed, for example by a parity transformation and this also relates to the handedness of the cross product, the cross product transforms as a pseudovector under parity transformations and so is properly described as a pseudovector. The dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar, so the scalar triple product must be pseudoscalar-valued. If T is an operator, then T a ⋅ = a ⋅. In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the product of three vectors is a trivector. A bivector is a plane element and a trivector is an oriented volume element. Given vectors a, b and c, the product a ∧ b ∧ c is a trivector with magnitude equal to the triple product. As the exterior product is associative brackets are not needed as it does not matter which of a ∧ b or b ∧ c is calculated first, though the order of the vectors in the product does matter. Geometrically the trivector a ∧ b ∧ c corresponds to the parallelepiped spanned by a, b, and c, with bivectors a ∧ b, b ∧ c, the triple product is identical to the volume form of the Euclidean 3-space applied to the vectors via interior product. It also can be expressed as a contraction of vectors with a rank-3 tensor equivalent to the form, the vector triple product is defined as the cross product of one vector with the cross product of the other two
23.
Mnemonic
–
A mnemonic device, or memory device is any learning technique that aids information retention in the human memory. Mnemonics make use of encoding, retrieval cues, and imagery as specific tools to encode any given information in a way that allows for efficient storage. Mnemonics aid original information in becoming associated with something more meaningful—which, in turn, the word mnemonic is derived from the Ancient Greek word μνημονικός, meaning of memory, or relating to memory and is related to Mnemosyne, the name of the goddess of memory in Greek mythology. Both of these words are derived from μνήμη, remembrance, memory, mnemonics in antiquity were most often considered in the context of what is today known as the art of memory. Ancient Greeks and Romans distinguished between two types of memory, the memory and the artificial memory. The former is inborn, and is the one that everyone uses instinctively, the latter in contrast has to be trained and developed through the learning and practice of a variety of mnemonic techniques. Mnemonic systems are techniques or strategies consciously used to improve memory and they help use information already stored in long-term memory to make memorisation an easier task. Mnemonic devices were much cultivated by Greek sophists and philosophers and are referred to by Plato. In later times the poet Simonides was credited for development of these techniques, the Romans valued such helps in order to support facility in public speaking. The Greek and the Roman system of mnemonics was founded on the use of mental places and signs or pictures, to recall these, an individual had only to search over the apartments of the house until discovering the places where images had been placed by the imagination. Except that the rules of mnemonics are referred to by Martianus Capella, among the voluminous writings of Roger Bacon is a tractate De arte memorativa. Ramon Llull devoted special attention to mnemonics in connection with his ars generalis, about the end of the 15th century, Petrus de Ravenna provoked such astonishment in Italy by his mnemonic feats that he was believed by many to be a necromancer. His Phoenix artis memoriae went through as many as nine editions, about the end of the 16th century, Lambert Schenkel, who taught mnemonics in France, Italy and Germany, similarly surprised people with his memory. He was denounced as a sorcerer by the University of Louvain, the most complete account of his system is given in two works by his pupil Martin Sommer, published in Venice in 1619. In 1618 John Willis published Mnemonica, sive ars reminiscendi, containing a statement of the principles of topical or local mnemonics. Giordano Bruno included a memoria technica in his treatise De umbris idearum, other writers of this period are the Florentine Publicius, Johannes Romberch, Hieronimo Morafiot, Ars memoriae, and B. The philosopher Gottfried Wilhelm Leibniz adopted a very similar to that of Wennsshein for his scheme of a form of writing common to all languages. Wennssheins method was adopted with slight changes afterward by the majority of subsequent original systems and it was modified and supplemented by Richard Grey, a priest who published a Memoria technica in 1730
24.
Solution of triangles
–
Solution of triangles is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere, applications requiring triangle solutions include geodesy, astronomy, construction, and navigation. A general form triangle has six characteristics, three linear and three angular. The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three, a side and the two angles adjacent to it A side, the angle opposite to it and an angle adjacent to it. For all cases in the plane, at least one of the side lengths must be specified, If only the angles are given, the side lengths cannot be determined, because any similar triangle is a solution. The standard method of solving the problem is to use fundamental relations, there are other universal relations, the law of cotangents and Mollweides formula. To find an angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle, for example, if sin β =0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem, within the interval from 0° to 180° the cosine value unambiguously determines its angle. On the other hand, if the angle is small, then it is more robust numerically to determine it from its sine than its cosine because the function has a divergent derivative at 1. We assume that the position of specified characteristics is known. If not, the reflection of the triangle will also be a solution. For example, three side lengths uniquely define either a triangle or its reflection, let three side lengths a, b, c be specified. Then angle γ = 180° − α − β, some sources recommend to find angle β from the law of sines but there is a risk of confusing an acute angle value with an obtuse one. Another method of calculating the angles from known sides is to apply the law of cotangents, here the lengths of sides a, b and the angle γ between these sides are known. The third side can be determined from the law of cosines, now we use law of cosines to find the second angle, α = arccos b 2 + c 2 − a 22 b c. Finally, β = 180° − α − γ and this case is not solvable in all cases, a solution is guaranteed to be unique only if the side length adjacent to the angle is shorter than the other side length. Assume that two sides b, c and the angle β are known, the equation for the angle γ can be implied from the law of sines, sin γ = c b sin β
25.
Solid angle
–
In geometry, a solid angle is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large the object appears to an observer looking from that point, in the International System of Units, a solid angle is expressed in a dimensionless unit called a steradian. A small object nearby may subtend the same angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, indeed, as viewed from any point on Earth, both objects have approximately the same solid angle as well as apparent size. This is evident during a solar eclipse, an objects solid angle in steradians is equal to the area of the segment of a unit sphere, centered at the angles vertex, that the object covers. A solid angle in steradians equals the area of a segment of a sphere in the same way a planar angle in radians equals the length of an arc of a unit circle. Solid angles are used in physics, in particular astrophysics. The solid angle of an object that is far away is roughly proportional to the ratio of area to squared distance. Here area means the area of the object when projected along the viewing direction. The solid angle of a sphere measured from any point in its interior is 4π sr, Solid angles can also be measured in square degrees, in square minutes and square seconds, or in fractions of the sphere, also known as spat. In spherical coordinates there is a formula for the differential, d Ω = sin θ d θ d φ where θ is the colatitude, at the equator you see all of the celestial sphere, at either pole only one half. Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where a →, b →, c → are the positions of the vertices A, B and C. Define the vertex angle θa to be the angle BOC and define θb, let φab be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define φac, φbc correspondingly. When implementing the above equation care must be taken with the function to avoid negative or incorrect solid angles. One source of errors is that the scalar triple product can be negative if a, b, c have the wrong winding. Computing abs is a sufficient solution since no other portion of the equation depends on the winding, the other pitfall arises when the scalar triple product is positive but the divisor is negative. Indices are cycled, s0 = sn and s1 = sn +1, the solid angle of a latitude-longitude rectangle on a globe is s r, where φN and φS are north and south lines of latitude, and θE and θW are east and west lines of longitude. Mathematically, this represents an arc of angle φN − φS swept around a sphere by θE − θW radians, when longitude spans 2π radians and latitude spans π radians, the solid angle is that of a sphere
26.
Thomas Harriot
–
Thomas Harriot — or spelled Harriott, Hariot, or Heriot — was an English astronomer, mathematician, ethnographer, and translator. He is sometimes credited with the introduction of the potato to the British Isles, Harriot was the first person to make a drawing of the Moon through a telescope, on 26 July 1609, over four months before Galileo. After graduating from St Mary Hall, Oxford, Harriot travelled to the Americas, accompanying the 1585 expedition to Roanoke island funded by Sir Walter Raleigh and led by Sir Ralph Lane. Harriot was a member of the venture, having translated and learned the Carolina Algonquian language from two Native Americans, Wanchese and Manteo. On his return to England he worked for the 9th Earl of Northumberland, at the Earls house, he became a prolific mathematician and astronomer to whom the theory of refraction is attributed. Born in 1560 in Oxford, England, Thomas Harriot attended St Mary Hall and his name appears in the halls registry dating from 1577. Prior to his expedition with Raleigh, Harriot wrote a treatise on navigation, in addition, he made efforts to communicate with Manteo and Wanchese, two Native Americans who had been brought to England. Harriot devised an alphabet to transcribe their Carolina Algonquian language. Harriot and Manteo spent many days in one company, Harriot interrogated Manteo closely about life in the New World. In addition, he recorded the sense of awe with which the Native Americans viewed European technology, Many things they sawe with us. as mathematical instruments, as the only Englishman who had learned Algonkin prior to the voyage, Harriot was vital to the success of the expedition. His account of the voyage, named A Briefe and True Report of the New Found Land of Virginia, was published in 1588. The True Report contains an account of the Native American population encountered by the expedition, it proved very influential upon later English explorers. He wrote, Whereby it may be hoped, if means of government be used, that they may in short time be brought to civility. At the same time, his views of Native Americans industry and capacity to learn were later largely ignored in favour of the parts of the True Report about extractable minerals and resources. As a scientific adviser during the voyage, Harriot was asked by Raleigh to find the most efficient way to stack cannonballs on the deck of the ship. His ensuing theory about the close-packing of spheres shows a resemblance to atomism and modern atomic theory. His correspondence about optics with Johannes Kepler, in which he described some of his ideas, Harriott was employed for many years by Henry Percy, 9th Earl of Northumberland, with whom he resided at Syon House, which was run by Henry Percys cousin Thomas Percy. Harriot himself was interrogated and briefly imprisoned but was soon released, Walter Warner, Robert Hues, William Lower, and other scientists were present around the Earl of Northumberlands mansion as they worked for him and assisted in the teaching of the familys children
27.
Legendre's theorem on spherical triangles
–
In geometry, Legendres theorem on spherical triangles, named after Adrien-Marie Legendre, is stated as follows, Let ABC be a spherical triangle on the unit sphere with small sides a, b, c. Let ABC be the triangle with the same sides. Then the angles of the spherical triangle exceed the corresponding angles of the triangle by approximately one third of the spherical excess. The theorem was very important in simplifying the heavy work in calculating the results of traditional geodetic surveys from about 1800 until the middle of the twentieth century. The theorem was stated by Legendre who provided a proof in a supplement to the report of the measurement of the French meridional arc used in the definition of the metre, Legendre does not claim that he was the originator of the theorem despite the attribution to him. Tropfke maintains that the method was in use by surveyors at the time. The excess, or area, of small triangles is very small and this result was proved by Buzengeiger —an extended proof may be found in Osborne. Other results are surveyed by Nádeník, general Investigations of Curved Surfaces of 1827 and 1825. English translation of Disquisitiones generales circa superficies curvas
28.
Angular defect
–
In geometry, the defect means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the excess, classically the defect arises in two ways, the defect of a vertex of a polyhedron, the defect of a hyperbolic triangle, and the excess also arises in two ways, the excess of a toroidal polyhedron. The excess of a triangle, In the plane, angles about a point add up to 360°. In modern terms, the defect at a vertex or over a triangle is precisely the curvature at that point or the total over the triangle, for a polyhedron, the defect at a vertex equals 2π minus the sum of all the angles at the vertex. If a polyhedron is convex, then the defect of each vertex is always positive, if the sum of the angles exceeds a full turn, as occurs in some vertices of many non-convex polyhedra, then the defect is negative. The concept of defect extends to higher dimensions as the amount by which the sum of the angles of the cells at a peak falls short of a full circle. The defect of any of the vertices of a dodecahedron is 36°, or π/5 radians. Each of the angles measures 108°, three of these meet at each vertex, so the defect is 360° − = 36°, the polyhedron need not be convex. A generalization says the number of circles in the total defect equals the Euler characteristic of the polyhedron and this is a special case of the Gauss–Bonnet theorem which relates the integral of the Gaussian curvature to the Euler characteristic. Here the Gaussian curvature is concentrated at the vertices, on the faces and edges the Gaussian curvature is zero and this can be used to calculate the number V of vertices of a polyhedron by totaling the angles of all the faces, and adding the total defect. This total will have one circle for every vertex in the polyhedron. Care has to be taken to use the correct Euler characteristic for the polyhedron and it is tempting to think that every non-convex polyhedron must have some vertices whose defect is negative, but this need not be the case. Two counterexamples to this are the small stellated dodecahedron and the great stellated dodecahedron, now consider the same cube where the square pyramid goes into the cube, this is concave, but the defects remain the same and so are all positive. Negative defect indicates that the vertex resembles a saddle point, whereas positive defect indicates that the vertex resembles a local maximum or minimum, richeson, D. Eulers Gem, The Polyhedron Formula and the Birth of Topology, Princeton, Pages 220-225
29.
Hyperbolic geometry
–
In mathematics, hyperbolic geometry is a non-Euclidean geometry. Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature, a modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space. In Russia it is commonly called Lobachevskian geometry, named one of its discoverers. This page is mainly about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry, Hyperbolic geometry can be extended to three and more dimensions, see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is closely related to Euclidean geometry than it seems. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry, there are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of geometry, including the first 28 propositions of book one of Euclids Elements, are valid in Euclidean. Propositions 27 and 28 of Book One of Euclids Elements prove the existence of parallel/non-intersecting lines and this difference also has many consequences, concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry, new concepts need to be introduced. Further, because of the angle of parallelism hyperbolic geometry has an absolute scale, single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points define a line, and lines can be infinitely extended. Two intersecting lines have the properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, when we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are many lines that do not intersect either of the given lines. While in some models lines look different they do have these properties, non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry, For any line R and any point P which does not lie on R. In the plane containing line R and point P there are at least two lines through P that do not intersect R. This implies that there are through P an infinite number of lines that do not intersect R. All other non-intersecting lines have a point of distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting. Some geometers simply use parallel lines instead of limiting parallel lines and these limiting parallels make an angle θ with PB, this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism
30.
Air navigation
–
The basic principles of air navigation are identical to general navigation, which includes the process of planning, recording, and controlling the movement of a craft from one place to another. Successful air navigation involves piloting an aircraft from place to place without getting lost, breaking the laws applying to aircraft, air navigation differs from the navigation of surface craft in several ways, Aircraft travel at relatively high speeds, leaving less time to calculate their position on route. Aircraft normally cannot stop in mid-air to ascertain their position at leisure, Aircraft are safety-limited by the amount of fuel they can carry, a surface vehicle can usually get lost, run out of fuel, then simply await rescue. There is no rescue for most aircraft. Additionally, collisions with obstructions are usually fatal, therefore, constant awareness of position is critical for aircraft pilots. The techniques used for navigation in the air will depend on whether the aircraft is flying under visual flight rules or instrument flight rules. In the latter case, the pilot will navigate exclusively using instruments and radio navigation such as beacons. In the VFR case, a pilot will largely navigate using dead reckoning combined with visual observations and this may be supplemented using radio navigation aids. The first step in navigation is deciding where one wishes to go, a private pilot planning a flight under VFR will usually use an aeronautical chart of the area which is published specifically for the use of pilots. This map will depict controlled airspace, radio aids and airfields prominently, as well as hazards to flying such as mountains, tall radio masts. It also includes sufficient ground detail - towns, roads, wooded areas - to aid visual navigation, in the UK, the CAA publishes a series of maps covering the whole of the UK at various scales, updated annually. The information is updated in the notices to airmen, or NOTAMs. The pilot will choose a route, taking care to avoid controlled airspace that is not permitted for the flight, restricted areas, danger areas, the chosen route is plotted on the map, and the lines drawn are called the track. The aim of all subsequent navigation is to follow the track as accurately as possible. Occasionally, the pilot may elect on one leg to follow a clearly visible feature on the such as a railway track, river, highway. The pilot must adjust heading to compensate for the wind, in order to follow the ground track and these figures are generally accurate and updated several times per day, but the unpredictable nature of the weather means that the pilot must be prepared to make further adjustments in flight. A general aviation pilot will make use of either the E6B flight computer – a type of slide rule – or a purpose-designed electronic navigational computer to calculate initial headings. The primary instrument of navigation is the magnetic compass, the needle or card aligns itself to magnetic north, which does not coincide with true north, so the pilot must also allow for this, called the magnetic variation
31.
Schwarz triangle
–
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. These can be defined generally as tessellations of the sphere. Each Schwarz triangle on a sphere defines a group, while on the Euclidean or hyperbolic plane they define an infinite group. A Schwarz triangle is represented by three rational numbers each representing the angle at a vertex, the value n/d means the vertex angle is d/n of the half-circle. When these are numbers, the triangle is called a Möbius triangle, and corresponds to a non-overlapping tiling. A Schwarz triangle is represented graphically by a triangular graph, each node represents an edge of the Schwarz triangle. Each edge is labeled by a value corresponding to the reflection order. Order-2 edges represent perpendicular mirrors that can be ignored in this diagram, the Coxeter-Dynkin diagram represents this triangular graph with order-2 edges hidden. A Coxeter group can be used for a simpler notation, as for graphs, and = for. Density 10, The Schwarz triangle is the smallest hyperbolic Schwarz triangle and its triangle group is the triangle group, which is the universal group for all Hurwitz groups – maximal groups of isometries of Riemann surfaces. All Hurwitz groups are quotients of the group, and all Hurwitz surfaces are tiled by the Schwarz triangle. The smallest Hurwitz group is the group of order 168, the second smallest non-abelian simple group, which is isomorphic to PSL. The triangle tiles the Bolza surface, a highly symmetric surface of genus 2, the triangles with one noninteger angle, listed above, were first classified by Anthony W. Knapp in. A list of triangles with multiple noninteger angles is given in, 3D The general Schwarz triangle and the generalized incidence matrices of the corresponding polyhedra
32.
Spherical polyhedron
–
In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the ball, thought of as a hosohedron. Some improper polyhedra, such as the hosohedra and their duals the dihedra, in the examples below, is a hosohedron and is the dual dihedron. The first known man-made polyhedra are spherical polyhedra carved in stone, many have been found in Scotland, and appear to date from the neolithic period. During the European Dark Age, the Islamic scholar Abū al-Wafā Būzjānī wrote the first serious study of spherical polyhedra, two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra. In the middle of the 20th Century, Coxeter used them to all but one of the uniform polyhedra. All the regular, semiregular polyhedra and their duals can be projected onto the sphere as tilings, given by their Schläfli symbol or vertex figure a. b. c. Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as, and dihedra, regular figures as
33.
Isaac Todhunter
–
Isaac Todhunter FRS, was an English mathematician who is best known today for the books he wrote on mathematics and its history. The son of George Todhunter, a Nonconformist minister, and Mary née Hume, he was born at Rye and he was educated at Hastings, where his mother had opened a school after the death of his father in 1826. He became an assistant master at a school at Peckham, attending at the time evening classes at the University College. In 1842 he obtained a scholarship and graduated as B. A. at London University. About this time he became master at a school at Wimbledon. In 1862 he was made a fellow of the Royal Society, in 1871 he gained the Adams Prize and was elected to the council of the Royal Society. He was elected fellow of St Johns in 1874, having resigned his fellowship on his marriage in 1864. In 1880 his eyesight began to fail, and shortly afterwards he was attacked with paralysis and he is buried in the Mill Road cemetery, Cambridge. Todhunter married 13 August 1864 to Louisa Anna Maria, eldest daughter of Captain George Davies and he died on 1 March 1884, at his residence,6 Brookside, Cambridge. A mural tablet and medallion portrait were placed in the ante-chapel of his college by his widow and he was a sound Latin and Greek scholar, familiar with French, German, Spanish, Italian, and also Russian, Hebrew, and Sanskrit. He was well versed in the history of philosophy, and on three occasions acted as examiner for the moral sciences tripos, an unfinished work, The History of the Theory of Elasticity, was edited and published posthumously in 1886 by Karl Pearson. A biographical work on William Whewell was published in 1876, in addition to original papers in scientific journals. Todhunters major historical works include a history of the Probability theory from the time of Blaise Pascal to that of Pierre-Simon Laplace first published in 1865, some of these are available at Isaac Todhunters publications at Google Books. Attribution This article incorporates text from a now in the public domain, Lee, Sidney, ed. Todhunter. 37, p. xxvvii A digital version of the obituary is at the Gallica site. Obituary, The Eagle, A Magazine Support by Members of St. Johns College, oConnor, John J. Robertson, Edmund F. Isaac Todhunter, MacTutor History of Mathematics archive, University of St Andrews, the conflict of studies, and other essays on subjects connected with education. Master of Trinity College, Cambridge, An Account of His Writings with Selections from His Literary, Works by Isaac Todhunter at Project Gutenberg Works by or about Isaac Todhunter at Internet Archive
34.
OCLC
–
The Online Computer Library Center is a US-based nonprofit cooperative organization dedicated to the public purposes of furthering access to the worlds information and reducing information costs. It was founded in 1967 as the Ohio College Library Center, OCLC and its member libraries cooperatively produce and maintain WorldCat, the largest online public access catalog in the world. OCLC is funded mainly by the fees that libraries have to pay for its services, the group first met on July 5,1967 on the campus of the Ohio State University to sign the articles of incorporation for the nonprofit organization. The group hired Frederick G. Kilgour, a former Yale University medical school librarian, Kilgour wished to merge the latest information storage and retrieval system of the time, the computer, with the oldest, the library. The goal of network and database was to bring libraries together to cooperatively keep track of the worlds information in order to best serve researchers and scholars. The first library to do online cataloging through OCLC was the Alden Library at Ohio University on August 26,1971 and this was the first occurrence of online cataloging by any library worldwide. Membership in OCLC is based on use of services and contribution of data, between 1967 and 1977, OCLC membership was limited to institutions in Ohio, but in 1978, a new governance structure was established that allowed institutions from other states to join. In 2002, the structure was again modified to accommodate participation from outside the United States. As OCLC expanded services in the United States outside of Ohio, it relied on establishing strategic partnerships with networks, organizations that provided training, support, by 2008, there were 15 independent United States regional service providers. OCLC networks played a key role in OCLC governance, with networks electing delegates to serve on OCLC Members Council, in early 2009, OCLC negotiated new contracts with the former networks and opened a centralized support center. OCLC provides bibliographic, abstract and full-text information to anyone, OCLC and its member libraries cooperatively produce and maintain WorldCat—the OCLC Online Union Catalog, the largest online public access catalog in the world. WorldCat has holding records from public and private libraries worldwide. org, in October 2005, the OCLC technical staff began a wiki project, WikiD, allowing readers to add commentary and structured-field information associated with any WorldCat record. The Online Computer Library Center acquired the trademark and copyrights associated with the Dewey Decimal Classification System when it bought Forest Press in 1988, a browser for books with their Dewey Decimal Classifications was available until July 2013, it was replaced by the Classify Service. S. The reference management service QuestionPoint provides libraries with tools to communicate with users and this around-the-clock reference service is provided by a cooperative of participating global libraries. OCLC has produced cards for members since 1971 with its shared online catalog. OCLC commercially sells software, e. g. CONTENTdm for managing digital collections, OCLC has been conducting research for the library community for more than 30 years. In accordance with its mission, OCLC makes its research outcomes known through various publications and these publications, including journal articles, reports, newsletters, and presentations, are available through the organizations website. The most recent publications are displayed first, and all archived resources, membership Reports – A number of significant reports on topics ranging from virtual reference in libraries to perceptions about library funding
35.
Wolfram Demonstrations Project
–
It is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. At its launch, it contained 1300 demonstrations but has grown to over 10,000, the site won a Parents Choice Award in 2008. Each Demonstration also has a description of the concept. Demonstrations are now easily embeddable into any website or blog, each Demonstration page includes a snippet of JavaScript code in the Share section of the sidebar. The website is organized by topic, for example, science, mathematics, computer science, art, biology and they cover a variety of levels, from elementary school mathematics to much more advanced topics such as quantum mechanics and models of biological organisms. The site is aimed at educators and students, as well as researchers who wish to present their ideas to the broadest possible audience. Wolfram Researchs staff organizes and edits the Demonstrations, which may be created by any user of Mathematica, then freely published, the Demonstrations are open-source, which means that they not only demonstrate the concept itself but also show how to implement it. The use of the web to transmit small interactive programs is reminiscent of Suns Java applets, Adobes Flash, however, those creating Demonstrations have access to the algorithmic and visualization capabilities of Mathematica making it more suitable for technical demonstrations. The Demonstrations Project also has similarities to user-generated content websites like Wikipedia and its business model is similar to Adobes Acrobat and Flash strategy of charging for development tools but providing a free reader
36.
Integrated Authority File
–
The Integrated Authority File or GND is an international authority file for the organisation of personal names, subject headings and corporate bodies from catalogues. It is used mainly for documentation in libraries and increasingly also by archives, the GND is managed by the German National Library in cooperation with various regional library networks in German-speaking Europe and other partners. The GND falls under the Creative Commons Zero license, the GND specification provides a hierarchy of high-level entities and sub-classes, useful in library classification, and an approach to unambiguous identification of single elements. It also comprises an ontology intended for knowledge representation in the semantic web, available in the RDF format
37.
National Diet Library
–
The National Diet Library is the only national library in Japan. It was established in 1948 for the purpose of assisting members of the National Diet of Japan in researching matters of public policy, the library is similar in purpose and scope to the United States Library of Congress. The National Diet Library consists of two facilities in Tokyo and Kyoto, and several other branch libraries throughout Japan. The Diets power in prewar Japan was limited, and its need for information was correspondingly small, the original Diet libraries never developed either the collections or the services which might have made them vital adjuncts of genuinely responsible legislative activity. Until Japans defeat, moreover, the executive had controlled all political documents, depriving the people and the Diet of access to vital information. The U. S. occupation forces under General Douglas MacArthur deemed reform of the Diet library system to be an important part of the democratization of Japan after its defeat in World War II. In 1946, each house of the Diet formed its own National Diet Library Standing Committee, hani Gorō, a Marxist historian who had been imprisoned during the war for thought crimes and had been elected to the House of Councillors after the war, spearheaded the reform efforts. Hani envisioned the new body as both a citadel of popular sovereignty, and the means of realizing a peaceful revolution, the National Diet Library opened in June 1948 in the present-day State Guest-House with an initial collection of 100,000 volumes. The first Librarian of the Diet Library was the politician Tokujirō Kanamori, the philosopher Masakazu Nakai served as the first Vice Librarian. In 1949, the NDL merged with the National Library and became the national library in Japan. At this time the collection gained a million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, in 1986, the NDLs Annex was completed to accommodate a combined total of 12 million books and periodicals. The Kansai-kan, which opened in October 2002 in the Kansai Science City, has a collection of 6 million items, in May 2002, the NDL opened a new branch, the International Library of Childrens Literature, in the former building of the Imperial Library in Ueno. This branch contains some 400,000 items of literature from around the world. Though the NDLs original mandate was to be a library for the National Diet. In the fiscal year ending March 2004, for example, the library reported more than 250,000 reference inquiries, in contrast, as Japans national library, the NDL collects copies of all publications published in Japan. The NDL has an extensive collection of some 30 million pages of documents relating to the Occupation of Japan after World War II. This collection include the documents prepared by General Headquarters and the Supreme Commander of the Allied Powers, the Far Eastern Commission, the NDL maintains a collection of some 530,000 books and booklets and 2 million microform titles relating to the sciences