Sorø Academy is a boarding school and gymnasium located in the small town of Sorø, Denmark. It traces its history back to the 12th century when Bishop Absalon founded a monastery at the site, confiscated by the Crown after the Reformation, since, on and off, it has served as an educational institution, in a variety of forms, including as a knight academy founded by Christian IV and a venue for higher learning during the Danish Golden Age. Danish writer and academian Ludvig Holberg bequested most of his fortune to re-establishing the academy in 1750 after a devastating fire. Sorø Academy traces its history back to 1140 when Archbishop Absalon founded the Cistercian Sorø Abbey in a remote woodlands setting on the shores of Lake Sorø on the island of Zealand, it developed into the most wealthy monastery in Denmark. After the Reformation in 1536, the Crown confiscated the Catholic Church's properties and the former abbey served first as an educational institution for Protestant priests before Frederick II turned it into a boarding school for an equal number of noble and commoner boys.
Sorø Academy was founded in 1623 when Christian IV turned the boarding school into a Equestrian Academy. Attempts were made to transform it into a university proper but it only existed as such for about 20 years before closing in 1665. After the closure the premises continued as a school until 1737. Efforts were made to reestablish the academy and around 1740, under the reign of Christian VI, the old buildings were rebuilt by Lauritz de Thurah, yet the plans did not materialize until Ludvig Holberg, who had no heirs, was persuaded to bequest his considerable fortune to the institution; the agreement, settled upon exempted Holberg from paying taxes from the proceeds of his lands and to reach this end he was ennobled with title of Baron. Holberg was consulted on the organization of the academy and the appointment of professors. Jens Schielderup Sneedorff was appointed professor in political sciences on his recommendation in 1751; the main wing burnt down in a fire in 1813 but was rebuilt from 1822 to 1827 to the design of Peder Malling.
In 1825, before the rebuilding had been completed, the Sorø Academy reopened once again. Over the next decades it became a central venue of the Danish Golden Age with Bernhard Severin Ingemann as a central figure. Both N. F. S. Grundtvig, Hans Christian Andersen and Bertel Thorvaldsen visited the Academy during this period; the current main wing is designed by Peder Malling in a Neoclassical style which relies more on Greek than Roman architecture for its inspiration. It interior has decorative works by Georg Hilker; the Academy is surrounded by an English-style park known as the Academy Garden. Located in the park is the Vænget building which contains Adam Wilhelm Hauch's Physical Cabinet, one of the largest collections of scientific instruments in Europe; the conventual church is an example of Cistercian craftsmanship. It is the third longest church in Denmark, is one of the first Danish churches built of brick; the Reformation whitewashed the traditional decorations of the church. Holberg is buried in the church, as are King Valdemar Atterdag and his father King Christopher II.
The gatehouse is the oldest inhabited building in Denmark today. It is where Saxo Grammaticus wrote the famous chronicles'Gesta Danorum', a medieval historical work recounting the early Christian history of Scandinavia. Two former professor's residences, today known as Molbech's House and Ingemann's House, survived the fire in 1813 and date from Lauritz de Thurah's rebuilding of the Academy in 1740; the old well, stemming from the original abbey, was in 1915 topped by a well house designed by Martin Nyrop, one of the schools former students. Other buildings are the Alumnatet and the Library Building; the current school has 630 students, of which 140 are boarders and the rest day students from Sorø, Ringsted and the surrounding countryside. The library has a large collection of rare books. Wilhelm Hauch's physical Physical Cabinet, one of the largest collections of scientific instruments in Europe; the Sorø Academy Foundation owns 6000 hectares of land covered by forest. The foundation owns a number of properties in the town of Sorø.
Reinhold Timm, painter Abraham Wuchters, painter Johann Elias Schlegel, political sciences, trade sciences Jens Schielderup Sneedorff, political sciences Johann Bernhard Basedow, moral philosophy Ove Høegh-Guldberg, historian, de facto prime minister Johan Theodor Holmskjold and natural history Bernhard Severin Ingemann, Danish literature Frederik Johnstrup, natural science Christen Dalsgaard, painter Ulrik of Denmark, administrator of the Prince-Bishopric of Schwerin, military Esaias Fleischer, printmaker Hinrich Johannes Rink, geologist Frederik Vermehren, painter Carl Steen Andersen Bille, journalist and civil servant Fredrik Bajer H. R. Hiort-Lorenzen and writer Christian Henrik Arendrup, governor of the Danish West Indies Martin Nyrop, architect Kristian Zahrtmann, painter Hans Egede Budtz, actor Herman Bang, writer Poul Rasmusen, politician Sigurd Langberg, actor Ebbe Hamerik, composer Hans Kirk, writer Jørgen-Frantz Jacobsen, writer Aage Kann Rasmussen, engineer Ove Arup, structural engineer Erik Seidenfaden, journalist Gunnar Seidenfaden and botanist, Mogens Boisen and translator Dan Fink, businessman Villum Kann Rasmussen, engineer Ha
A planimeter known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape. There are several kinds of planimeters; the precise way in which they are constructed varies, with the main types of mechanical planimeter being polar and Prytz or "hatchet" planimeters. The Swiss mathematician Jakob Amsler-Laffon built the first modern planimeter in 1854, the concept having been pioneered by Johann Martin Hermann in 1814. Many developments followed Amsler's famous planimeter, including electronic versions; the Amsler type consists of a two-bar linkage. At the end of one link is a pointer, used to trace around the boundary of the shape to be measured; the other end of the linkage pivots on a weight that keeps it from moving. Near the junction of the two links is a measuring wheel of calibrated diameter, with a scale to show fine rotation, worm gearing for an auxiliary turns counter scale; as the area outline is traced, this wheel rolls on the surface of the drawing.
The operator sets the wheel, turns the counter to zero, traces the pointer around the perimeter of the shape. When the tracing is complete, the scales at the measuring wheel show the shape's area; when the planimeter's measuring wheel moves perpendicular to its axis, it rolls, this movement is recorded. When the measuring wheel moves parallel to its axis, the wheel skids without rolling, so this movement is ignored; that means the planimeter measures the distance that its measuring wheel travels, projected perpendicularly to the measuring wheel's axis of rotation. The area of the shape is proportional to the number of turns through which the measuring wheel rotates; the polar planimeter is restricted by design to measuring areas within limits determined by its size and geometry. However, the linear type has no restriction in one dimension, its wheels must not slip. Developments of the planimeter can establish the position of the first moment of area, the second moment of area; the images show the principles of a polar planimeter.
The pointer M at one end of the planimeter follows the contour C of the surface S to be measured. For the linear planimeter the movement of the "elbow" E is restricted to the y-axis. For the polar planimeter the "elbow" is connected to an arm with its other endpoint O at a fixed position. Connected to the arm ME is the measuring wheel with its axis of rotation parallel to ME. A movement of the arm ME can be decomposed into a movement perpendicular to ME, causing the wheel to rotate, a movement parallel to ME, causing the wheel to skid, with no contribution to its reading; the working of the linear planimeter may be explained by measuring the area of a rectangle ABCD. Moving with the pointer from A to B the arm EM moves through the yellow parallelogram, with area equal to PQ×EM; this area is equal to the area of the parallelogram A"ABB". The measuring wheel measures the distance PQ. Moving from C to D the arm EM moves through the green parallelogram, with area equal to the area of the rectangle D"DCC".
The measuring wheel now moves in the opposite direction. The movements along BC and DA are the same but opposite, so they cancel each other with no net effect on the reading of the wheel; the net result is the measuring of the difference of the yellow and green areas, the area of ABCD. The operation of a linear planimeter can be justified by applying Green's theorem onto the components of the vector field N, given by: N =, where b is the y-coordinate of the elbow E; this vector field is perpendicular to the measuring arm EM: E M → ⋅ N = x N x + N y = 0 and has a constant size, equal to the length m of the measuring arm: ‖ N ‖ = 2 + x 2 = m Then: ∮ C = ∬ S d x d y = ∬ S d x d
Magnus Gustaf Mittag-Leffler was a Swedish mathematician. His mathematical contributions are connected chiefly with the theory of functions, which today is called complex analysis. Mittag-Leffler was born in Stockholm, son of the school principal John Olof Leffler and Gustava Wilhelmina Mittag, his sister was the writer Anne Charlotte Leffler. He matriculated at Uppsala University in 1865, completed his Ph. D. in 1872 and became docent at the university the same year. He was curator of the Stockholms nation, he next traveled to Göttingen and Berlin, studying under Weierstrass in the latter place. He took up a position as professor of mathematics at the University of Helsinki from 1877 to 1881 and as the first professor of mathematics at the University College of Stockholm. Mittag-Leffler went into business and became a successful businessman in his own right, but an economic collapse in Europe wiped out his fortune in 1922, he was a member of the Royal Swedish Academy of Sciences, the Finnish Society of Sciences and Letters, the Royal Swedish Society of Sciences in Uppsala, the Royal Physiographic Society in Lund and about 30 foreign learned societies, including the Royal Society of London and Académie des sciences in Paris.
He held honorary doctorates from the University of several other universities. Mittag-Leffler was a convinced advocate of women's rights and was instrumental in making Sofia Kovalevskaya a full professor of mathematics in Stockholm, as the first woman anywhere in the world to hold that position; as a member of the Nobel Prize Committee in 1903, he was responsible for inducing the committee to relent and award the prize for Physics to Marie Curie as well as her husband Pierre. Mittag-Leffler founded the mathematical journal Acta Mathematica, with the help of King Oscar's sponsorship, paid for with the fortune of his wife Signe Lindfors, who came from a wealthy Finnish family, he collected a large mathematical library in his villa in the Stockholm suburb of Djursholm. The house and its contents was donated to the Academy of Sciences as the Mittag-Leffler Institute. Mittag-Leffler function Mittag-Leffler star Mittag-Leffler summation Mittag-Leffler theorem Mittag-Leffler Institute Mittag-Lefflerbreen Works by Gösta Mittag-Leffler at Project Gutenberg Works by or about Gösta Mittag-Leffler at Internet Archive O'Connor, John J..
Gösta Mittag-Leffler at the Mathematics Genealogy Project
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics. By extension, use of complex analysis has applications in engineering fields such as nuclear, aerospace and electrical engineering; as a differentiable function of a complex variable is equal to the sum of its Taylor series, complex analysis is concerned with analytic functions of a complex variable. Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler, Riemann, Cauchy and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is used throughout analytic number theory. In modern times, it has become popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions.
Another important application of complex analysis is in string theory which studies conformal invariants in quantum field theory. A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are supposed to have a domain that contains a nonempty open subset of the complex plane. For any complex function, the values z from the domain and their images f in the range may be separated into real and imaginary parts: z = x + i y and f = f = u + i v, where x, y, u, v are all real-valued. In other words, a complex function f: C → C may be decomposed into u: R 2 → R and v: R 2 → R, i.e. into two real-valued functions of two real variables. Any complex-valued function f on an arbitrary set X can be considered as an ordered pair of two real-valued functions: or, alternatively, as a vector-valued function from X into R 2; some properties of complex-valued functions are nothing more than the corresponding properties of vector valued functions of two real variables.
Other concepts of complex analysis, such as differentiability are direct generalizations of the similar concepts for real functions, but may have different properties. In particular, every differentiable complex function is analytic, two differentiable functions that are equal in a neighborhood of a point are equal on the intersection of their domain; the latter property is the basis of the principle of analytic continuation which allows extending every real analytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions are defined in this way, including exponential functions, logarithmic functions, trigonometric functions. Complex functions that are differentiable at every point of an open subset Ω of the complex plane are said to be holomorphic on Ω. In the context of complex analysis, the derivative of f at z 0 is defined to be f ′ = lim z → z 0 f − f z − z 0, z ∈ C.
Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we
Straightedge and compass construction
Straightedge and compass construction known as ruler-and-compass construction or classical construction, is the construction of lengths and other geometric figures using only an idealized ruler and compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, no markings on it; the compass is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. More formally, the only permissible constructions are those granted by Euclid's first three postulates, it turns out to be the case that every point constructible using straightedge and compass may be constructed using compass alone. The ancient Greek mathematicians first conceived straightedge and compass constructions, a number of ancient problems in plane geometry impose this restriction; the ancient Greeks developed many constructions. Gauss showed 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. Many of these problems are solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone. In terms of algebra, a length is constructible if and only if it represents a constructible number, 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 written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots; the "straightedge" and "compass" of straightedge and compass constructions are idealizations of rulers and compasses in the real world: 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 extend an existing segment; the compass can be opened arbitrarily wide.
Circles can only be drawn starting from two given points: a point on the circle. The compass may not collapse when it is not drawing a circle. Actual compasses do not collapse and modern geometric constructions use this feature. A'collapsing compass' would appear to be a less powerful instrument. However, by the compass equivalence theorem in Proposition 2 of Book 1 of Euclid's Elements, no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a checkered history; each construction must be exact. "Eyeballing" it and getting close does not count as a solution. Each construction must terminate; that is, it must have a finite number of steps, not be the limit of closer approximations. Stated this way and compass constructions appear to be a parlour game, rather than a serious practical problem; the ancient Greek mathematicians first attempted straightedge and compass constructions, they discovered how to construct sums, products and square roots of given lengths.
They could construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, a regular polygon with 3, 4, or 5 sides. But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or a regular polygon with other numbers of sides. Nor could they construct the side of a cube whose volume would be twice the volume of a cube with a given side. Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas, but these cannot be constructed by straightedge and compass. In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square the circle, Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle. No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed.
In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths. He showed that Gauss's sufficient constructibility condition for regular polygons is necessary. In 1882 Lindemann showed that π is a transcendental number, thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle. All straightedge and compass constructions consist of repeated application of five basic constructions using the points and circles that have been constructed; these are: Creating the line through two existing points Creating the circle through one point with centre another point Creating the point, th
Royal Danish Academy of Sciences and Letters
The Royal Danish Academy of Sciences and Letters is a Danish non-governmental science Academy, founded in 1742 for the advancement of science in Denmark. It is based in the Carlsberg Foundation's building at the corner of H. C. Andersens Boulevard and Dantes Plads in central Copenhagen; the Society was founded on 13 November 1742 by permission of King Christian VI, as a historical Collegium Antiquitatum. It was founded by secretary of state, Count Johan Ludvig Holstein and the history professor Hans Gram; the building at 35 H. C. Andersens Boulevard was designed by Vilhelm Petersen in a Neoclassical style; the Carlsberg Foundation is based in the ground floor while the Royal Academy has the three upper floors. First floor contains the Old Meeting Hall, it is decorated with a large oil painting by Peder Severin Krøyer depicting A meeting in the Royal Scientific Society. There is a library and rooms for researchers. Second floor contain the Academy's secretariate and archives as the President's, General Secretary's and Editor's offices.
Third floor now contains New Meeting Room. It was created in the former attic and book storage in connection with an adaption of the building in 1976. Since 2009, it has access to a roof terrace. Members of the Academy are researching and publishing in nearly all fields of science; the Academy has 250 national and 260 foreign members. In 2011 the division The Young Academy was added, which counts 34 young researchers as members. Olaf Pedersen, Lovers of Learning - A History of the Royal Danish Academy of Sciences and Letters 1742-1992, Munksgaard, 1992. ISBN 87-7304-236-6. Royal Danish Academy of Sciences and Letters Digital archive of Matematisk-fysiske meddelelser from the University of Southern Denmark Library
Peter Tait (physicist)
Peter Guthrie Tait FRSE was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook Treatise on Natural Philosophy, which he co-wrote with Kelvin, his early investigations into knot theory, His work on knot theory contributed to the eventual formation of topology as a mathematical discipline, his name is known in graph theory for Tait's conjecture. Tait was born in Dalkeith on 28 April 1831 the only son of Mary Ronaldson and John Tait, secretary to the 5th Duke of Buccleuch, he was educated at Dalkeith Grammar School Edinburgh Academy. He studied Maths and Physics at the University of Edinburgh, went to Peterhouse, graduating as senior wrangler and first Smith's prizeman in 1852; as a fellow and lecturer of his college he remained at the University for a further two years, before leaving to take up the professorship of mathematics at Queen's College, Belfast. There he made the acquaintance of Thomas Andrews, whom he joined in researches on the density of ozone and the action of the electric discharge on oxygen and other gases, by whom he was introduced to Sir William Rowan Hamilton and quaternions.
In 1860, Tait succeeded his old master, James D. Forbes, as professor of natural philosophy at the University of Edinburgh, occupied the Chair until shortly before his death; the first scientific paper under Tait's name only was published in 1860. His earliest work dealt with mathematical subjects, with quaternions, of which he was the leading exponent after their originator, William Rowan Hamilton, he was the author of two text-books on them—one an Elementary Treatise on Quaternions, written with the advice of Hamilton, though not published till after his death, the other an Introduction to Quaternions, in which he was aided by Philip Kelland, one of his teachers at the University of Edinburgh. Quaternions was one of the themes of his address as president of the mathematical section of the British Association for the Advancement of Science in 1871, he produced original work in mathematical and experimental physics. In 1864, he published a short paper on thermodynamics, from that time his contributions to that and kindred departments of science became frequent and important.
In 1871, he emphasised the significance and future importance of the principle of the dissipation of energy. In 1873 he took thermoelectricity for the subject of his discourse as Rede lecturer at Cambridge, in the same year he presented the first sketch of his well-known thermoelectric diagram before the Royal Society of Edinburgh. Two years researches on "Charcoal Vacua" with James Dewar led him to see the true dynamical explanation of the Crookes radiometer in the large mean free path of the molecule of the rarefied air. From 1879 to 1888, he engaged in difficult experimental investigations; these began with an inquiry into what corrections were required for thermometers operating at great pressure. This was for the benefit of thermometers employed by the Challenger expedition for observing deep-sea temperatures, were extended to include the compressibility of water and mercury; this work led to the first formulation of the Tait equation, used to fit liquid density to pressure. Between 1886 and 1892 he published a series of papers on the foundations of the kinetic theory of gases, the fourth of which contained what was, according to Lord Kelvin, the first proof given of the Waterston-Maxwell theorem of the average equal partition of energy in a mixture of two gases.
About the same time he carried out investigations into its duration. Many other inquiries conducted by him might be mentioned, some idea may be gained of his scientific activity from the fact that a selection only from his papers, published by the Cambridge University Press, fills three large volumes; this mass of work was done in the time he could spare from his professorial teaching in the university. For example, in 1880 he worked on the Four color theorem and proved that it was true if and only if no snarks were planar. In addition, he was the author of a number of articles. Of the former, the first, published in 1856, was on the dynamics of a particle. With Lord Kelvin, he collaborated in writing the well-known Treatise on Natural Philosophy. "Thomson and Tait," as it is familiarly called, was planned soon after Lord Kelvin became acquainted with Tait, on the latter's appointment to his professorship in Edinburgh, it was intended to be an all-comprehensive treatise on physical science, the foundations being laid in kinematics and dynamics, the structure completed with the properties of matter, light and magnetism.
But the literary partnership ceased in about eighteen years, when only the first portion of the plan had been completed, because each of the members felt he could work to better advantage separately than jointly. The friendship, endured for the remaining twenty-three years of Tait's life. Tait collaborated with Balfour Stewart in the Unseen Universe, followed by Paradoxical Philosophy, it was in his 1875 review of The Unseen Universe, that William James first put forth his Will to Believe Doctrine. Tait's articles include those he wrote for the ninth edition of the Encyclopædia Britannica on light, quaternions and thermodynamics, the biographical notices of Hamilton an