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
In algebra, a cubic function is a function of the form f = a x 3 + b x 2 + c x + d in which a is nonzero. Setting f = 0 produces a cubic equation of the form a x 3 + b x 2 + c x + d = 0; the solutions of this equation are called roots of the polynomial f. If all of the coefficients a, b, c, d of the cubic equation are real numbers it has at least one real root. All of the roots of the cubic equation can be found algebraically; the roots can be found trigonometrically. Alternatively, numerical approximations of the roots can be found using root-finding algorithms such as Newton's method; the coefficients do not need to be complex numbers. Much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3; the solutions of the cubic equation do not belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are non-rational complex numbers. Cubic equations were known to the ancient Babylonians, Chinese and Egyptians.
Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, but no evidence exists to confirm that they did; the problem of doubling the cube involves the simplest and oldest studied cubic equation, one for which the ancient Egyptians did not believe a solution existed. In the 5th century BC, Hippocrates reduced this problem to that of finding two mean proportionals between one line and another of twice its length, but could not solve this with a compass and straightedge construction, a task, now known to be impossible. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BC and commented on by Liu Hui in the 3rd century. In the 3rd century AD, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations. Hippocrates and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections, though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations.
Some others like T. L. Heath, who translated all Archimedes' works, putting forward evidence that Archimedes solved cubic equations using intersections of two conics, but discussed the conditions where the roots are 0, 1 or 2. In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved numerically 25 cubic equations of the form x3 + px2 + qx = N, 23 of them with p, q ≠ 0, two of them with q = 0. In the 11th century, the Persian poet-mathematician, Omar Khayyam, made significant progress in the theory of cubic equations. In an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions, he found a geometric solution. In his work, the Treatise on Demonstration of Problems of Algebra, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.
In the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of a cubic equation: x3 + 12x = 6x2 + 35. In the 12th century, another Persian mathematician, Sharaf al-Dīn al-Tūsī, wrote the Al-Muʿādalāt, which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions, he used what would be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He used the concepts of maxima and minima of curves in order to solve cubic equations which may not have positive solutions, he understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations. In his book Flos, Leonardo de Pisa known as Fibonacci, was able to approximate the positive solution to the cubic equation x3 + 2x2 + 10x = 20. Writing in Babylonian numerals he gave the result as 1,22,7,42,33,4,40, which has a relative error of about 10−9.
In the early 16th century, the Italian mathematician Scipione del Ferro found a method for solving a class of cubic equations, namely those of the form x3 + mx = n. In fact, all cubic equations can be reduced to this form if we allow m and n to be negative, but negative numbers were not known to him at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fiore about it. In 1530, Niccolò Tartaglia received two problems in cubic equations from Zuanne da Coi and announced that he could solve them, he was soon challenged by Fiore. Each contestant had to put up a certain amoun
George Alfred Leon Sarton, was a Belgian-born American chemist and historian. He is considered the founder of the discipline of the history of science, he has a significant importance in the history of science and his most influential work was the Introduction to the History of Science, which consists of three volumes and 4,296 pages. Sarton aimed to achieve an integrated philosophy of science that provided a connection between the sciences and the humanities, which he referred to as "the new humanism", he gives his name to the George Sarton Medal. George Alfred Leon Sarton was born in Ghent, Belgium on August 31, 1884, his parents were Alfred Sarton and Léonie Van Halmé. His mother died, he graduated from the University of Ghent in 1906 and two years won a gold medal for one of his papers on chemistry. He received his PhD in mathematics at the University of Ghent in 1911, he emigrated to the United States from Belgium due to the First World War, worked there the rest of his life and writing about the history of science.
In 1911, he married an English artist. Their daughter Eleanore Marie was born the following year in 1912. Although he and his family emigrated to England after World War I broke out, they immigrated to the United States in 1915, where they would live for the rest of their lives, he worked for the Carnegie Foundation for International Peace and lectured at Harvard University, 1916–18. At Harvard, he became a lecturer in 1920, a professor of the history of science from 1940 until his retirement in 1951, he was a research associate of the Carnegie Institution of Washington from 1919 until 1948. Sarton intended to complete an exhaustive nine-volume history of science. By the time of his death, he had completed only the first three volumes: I. From Homer to Omar Khayyam. From Rabbi Ben Ezra to Roger Bacon, pt. 1–2. Science and learning in the fourteenth-century, pt. 1–2. Sarton had been inspired for his project by his study of Leonardo da Vinci, but he had not reached this period in history before dying.
After his death, a representative selection of his papers was edited by Dorothy Stimson. It was published by Harvard University Press in 1962. In honor of Sarton's achievements, the History of Science Society created the award known as the George Sarton Medal, it is the most prestigious award of the History of Science Society. It has been awarded annually since 1955 to an outstanding historian of science selected from the international scholarly community; the medal honors a scholar for lifetime scholarly achievement. Sarton was the founder of this society and of its journals: Isis and Osiris, which publish articles on science and culture. Sarton, George. "The New Humanism". Isis. 6: 9–42. Doi:10.1086/358203. JSTOR 223969. Sarton, Introduction to the History of Science, Carnegie Institution of Washington Publication no. 376. Baltimore: Williams and Wilkins, Co. George Sarton, "The Incubation of Western Culture in the Middle East: a George C. Keiser Foundation Lecture", March 29, 1950, Washington, D. C.
Introduction to the History of Science. Baltimore: Williams & Wilkins. A History of Science. Ancient science through the Golden Age of Greece, Massachusetts: Harvard University Press, 1952. A History of Science. Hellenistic science and culture in the last three centuries B. C. Cambridge, Massachusetts: Harvard University Press, 1959; the Study of the History of Science. Quotations related to George Sarton at Wikiquote Full-text works of George Sarton on Internet Archive Biography of George Sarton from BookRags
Sławnikowice, Lower Silesian Voivodeship
Sławnikowice is a village in the administrative district of Gmina Zgorzelec, within Zgorzelec County, Lower Silesian Voivodeship, in south-western Poland, close to the German border. The Waldhufendorf is located in the Polish part of the Upper Lusatia historical region 12 kilometres east of Zgorzelec and 131 km west of the regional capital Wrocław; the A4 autostrada runs about 4 km north of the village. Keselingswalde was first mentioned in a 1301 deed; the manor was held by the noble Tschirnhaus dynasty from 1483 onwards, residing here until 1714. Afterwards, the estates changed hands several times, they were acquired by the Witzleben dynasty in 1862. Following World War II east of the Oder-Neisse line fell to the Republic of Poland according to the Potsdam Agreement; the native German population was replaced by Poles. Ehrenfried Walther von Tschirnhaus, physicist and philosopher Georg Mohr, died in Kieslingswalde
The Netherlands is a country located in Northwestern Europe. The European portion of the Netherlands consists of twelve separate provinces that border Germany to the east, Belgium to the south, the North Sea to the northwest, with maritime borders in the North Sea with Belgium and the United Kingdom. Together with three island territories in the Caribbean Sea—Bonaire, Sint Eustatius and Saba— it forms a constituent country of the Kingdom of the Netherlands; the official language is Dutch, but a secondary official language in the province of Friesland is West Frisian. The six largest cities in the Netherlands are Amsterdam, The Hague, Utrecht and Tilburg. Amsterdam is the country's capital, while The Hague holds the seat of the States General and Supreme Court; the Port of Rotterdam is the largest port in Europe, the largest in any country outside Asia. The country is a founding member of the EU, Eurozone, G10, NATO, OECD and WTO, as well as a part of the Schengen Area and the trilateral Benelux Union.
It hosts several intergovernmental organisations and international courts, many of which are centered in The Hague, dubbed'the world's legal capital'. Netherlands means'lower countries' in reference to its low elevation and flat topography, with only about 50% of its land exceeding 1 metre above sea level, nearly 17% falling below sea level. Most of the areas below sea level, known as polders, are the result of land reclamation that began in the 16th century. With a population of 17.30 million people, all living within a total area of 41,500 square kilometres —of which the land area is 33,700 square kilometres —the Netherlands is one of the most densely populated countries in the world. It is the world's second-largest exporter of food and agricultural products, owing to its fertile soil, mild climate, intensive agriculture; the Netherlands was the third country in the world to have representative government, it has been a parliamentary constitutional monarchy with a unitary structure since 1848.
The country has a tradition of pillarisation and a long record of social tolerance, having legalised abortion and human euthanasia, along with maintaining a progressive drug policy. The Netherlands abolished the death penalty in 1870, allowed women's suffrage in 1917, became the world's first country to legalise same-sex marriage in 2001, its mixed-market advanced economy had the thirteenth-highest per capita income globally. The Netherlands ranks among the highest in international indexes of press freedom, economic freedom, human development, quality of life, as well as happiness; the Netherlands' turbulent history and shifts of power resulted in exceptionally many and varying names in different languages. There is diversity within languages; this holds for English, where Dutch is the adjective form and the misnomer Holland a synonym for the country "Netherlands". Dutch comes from Theodiscus and in the past centuries, the hub of Dutch culture is found in its most populous region, home to the capital city of Amsterdam.
Referring to the Netherlands as Holland in the English language is similar to calling the United Kingdom "Britain" by people outside the UK. The term is so pervasive among potential investors and tourists, that the Dutch government's international websites for tourism and trade are "holland.com" and "hollandtradeandinvest.com". The region of Holland consists of North and South Holland, two of the nation's twelve provinces a single province, earlier still, the County of Holland, a remnant of the dissolved Frisian Kingdom. Following the decline of the Duchy of Brabant and the County of Flanders, Holland became the most economically and politically important county in the Low Countries region; the emphasis on Holland during the formation of the Dutch Republic, the Eighty Years' War and the Anglo-Dutch Wars in the 16th, 17th and 18th century, made Holland serve as a pars pro toto for the entire country, now considered either incorrect, informal, or, depending on context, opprobrious. Nonetheless, Holland is used in reference to the Netherlands national football team.
The region called the Low Countries and the Country of the Netherlands. Place names with Neder, Nieder and Nedre and Bas or Inferior are in use in places all over Europe, they are sometimes used in a deictic relation to a higher ground that consecutively is indicated as Upper, Oben, Superior or Haut. In the case of the Low Countries / Netherlands the geographical location of the lower region has been more or less downstream and near the sea; the geographical location of the upper region, changed tremendously over time, depending on the location of the economic and military power governing the Low Countries area. The Romans made a distinction between the Roman provinces of downstream Germania Inferior and upstream Germania Superior; the designation'Low' to refer to the region returns again in the 10th century Duchy of Lower Lorraine, that covered much of the Low Countries. But this time the corresponding Upper region is Upper Lorraine, in nowadays Northern France; the Dukes of Burgundy, who ruled the Low Countries in the 15th century, used the term les pays de par deçà for the Low Countries as opposed to les pays de par delà for their original