Euclidean space

Euclidean space is the fundamental space of classical geometry. It was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane, it was introduced by the Ancient Greek mathematician Euclid of Alexandria, the qualifier Euclidean is used to distinguish it from other spaces that were discovered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical universe, their great innovation was to prove all properties of the space as theorems by starting from a few fundamental properties, called postulates, which either were considered as evident, or seemed impossible to prove. After the introduction at the end of 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition.

It is this definition, more used in modern mathematics, detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space. There is only one Euclidean space of each dimension. Therefore, in many cases, it is possible to work with a specific Euclidean space, the real n-space R n, equipped with the dot product. An isomorphism from a Euclidean space to R n associates with each point an n-tuple of real numbers which locate that point in the Euclidean space and are called the Cartesian coordinates of that point. Euclidean space was introduced by ancient Greeks as an abstraction of our physical space, their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few basic properties, which are abstracted from the physical world, cannot be mathematically proved because of the lack of more basic tools. These properties are called axioms in modern language.

This way of defining Euclidean space is still in use under the name of synthetic geometry. In 1637, René Descartes introduced Cartesian coordinates and showed that this allows reducing geometric problems to algebraic computations with numbers; this reduction of geometry to algebra was a major change of point of view, as, until the real numbers—that is, rational numbers and non-rational numbers together–were defined in terms of geometry, as lengths and distance. Euclidean geometry was not applied in spaces of more than three dimensions until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of n dimensions using both synthetic and algebraic methods, discovered all of the regular polytopes that exist in Euclidean spaces of any number of dimensions. Despite the wide use of Descartes' approach, called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century; the introduction of abstract vector spaces allowed their use in defining Euclidean spaces with a purely algebraic definition.

This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition, now most used for introducing Euclidean spaces. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance; the other is rotation around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures of the plane should be considered equivalent if one can be transformed into the other by some sequence of translations and reflections. In order to make all of this mathematically precise, the theory must define what is a Euclidean space, the related notions of distance, angle and rotation.

When used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, so on. A purely mathematical definition of Euclidean space ignores questions of units of length and other physical dimensions: the distance in a "mathematical" space is a number, not something expressed in inches or metres; the standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is to define a Euclidean space as a set of points on which acts a real vector space, the space of translations, equipped with an inner product. The action of translations makes the space an affine space, this allows defining lines, subspaces and parallelism; the inner product allows defining distance and angles. The set R n of n-tuples of real numbers equipped with the dot product is a Euclidean space of dimension n. Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism b

William McFarland

William McFarland was an American politician who served in the United States House of Representatives from 1875 to 1877, representing the 1st congressional district of Tennessee. He is one of only two Democrats to have won this district's seat since the Civil War. McFarland served as a state court judge in 1869, as mayor of Morristown, from 1882 to 1885. A Southern Unionist, he was a member of the Jefferson County delegation at the pro-Union East Tennessee Convention in 1861. McFarland was born in Jefferson County, the son of Robert and Mary Ann McFarland, his grandfather named Robert McFarland, was a Revolutionary War veteran, his father was a War of 1812 recruiting officer, militia colonel, justice of the peace. While still a child, William moved with his family to Springvale Farm near Morristown in what was northern Jefferson County, but is now part of Hamblen County, he was educated in the common schools, attended Tusculum College near Greeneville. McFarland worked as a salesman for a Tazewell businessman, but returned home to manage his family's affairs following his father's death in 1844.

He operated a mercantile tannery throughout the 1850s. During this period, he began to take an interest in railroad construction, helping to raise funds for the East Tennessee and Virginia Railroad. McFarland remained loyal to the Union during the secession crisis of 1860–1861, he attended the Knoxville session of the East Tennessee Convention in May 1861, represented Jefferson County on the Convention's powerful business committee. During the war, he studied law under Judge Robert M. Barton, was admitted to the bar in 1863. In 1866, he moved to Morristown to practice law. During the years following the Civil War, McFarland supported President Andrew Johnson and the conservative faction of the state government, he was a delegate to the pro-Johnson National Union Convention in August 1866. In April 1869, Governor Dewitt Clinton Senter appointed McFarland judge of the state's second judicial circuit to finish out the term of James P. Swann, who had resigned; that same year, he ran for state attorney general, but was defeated by the incumbent, Thomas M. Coldwell.

In subsequent years, he was active at the municipal level in Hamblen County, created in 1870, included his Springvale Farm and Morristown. In 1874, McFarland ran on the Democratic ticket against four-term Republican incumbent Roderick R. Butler for the 1st district congressional seat. While Republicans controlled the district, Butler was caught up in a scandal over the selling of cadetships at U. S. military academies that year among Republican voters. On election day, McFarland defeated Butler, 8,783 votes to 6,995. During his lone term in Congress, McFarland sought to limit federal prosecutions for illegal whiskey distilling in East Tennessee, which many of his constituents felt had gotten out of control, he introduced legislation that would allow farmers to sell the first $100 of their annual tobacco crop tax-free, sought appropriations for improvements to the Tennessee River and its tributaries. He favored the use of silver as legal tender. Running for reelection in 1876, McFarland was ruthlessly assailed for his affiliation with the Democratic Party, with his opponents suggesting he was conspiring with former Confederates and Southern Democrats to destroy the federal government and reopen the slave trade.

He defended himself by pointing out he had remained loyal to the Union throughout the war, had supported every measure regarding pensions and other aid for former Union soldiers and their families. With Republicans no longer boycotting the vote, he stood little chance of winning a second term. On election day, he was defeated by the Republican candidate, James H. Randolph, 12,349 votes to 11,215. McFarland was again considered for the Democratic nomination for the 1st district seat in 1878, but was outpolled at the district convention that year by rising politician Robert Love Taylor. While McFarland campaigned for presidential candidates Winfield Scott Hancock in 1880 and Grover Cleveland in 1884, he turned his attention to local politics, he was elected to Morristown's Board of Aldermen in 1880, was reelected in 1881, winning more votes than any of the other fourteen candidates in this second election. In 1882, McFarland was elected mayor, defeating Major W. D. Gammon for the office after the incumbent, A.

H. Gregg, declined to seek reelection, he served as mayor until 1885. McFarland died in Morristown on April 27, 1900, was interred at City Cemetery, he was reinterred at Emma Jarnagin Cemetery in Morristown. United States Congress. "William McFarland". Biographical Directory of the United States Congress

Irish Commemorative Stone

The Irish Commemorative Stone is a monument in Pointe-Saint-Charles, island of Montreal, Quebec commemorating the deaths from "ship fever" of 6,000 Irish immigrants to Canada during the immigration following the Great Irish Famine in 1847-48. It is a 10-foot high boulder. Named the Irish Commemorative Stone, it is more known as the Black Rock and has been referred to as the Ship Fever Monument or the Boulder Stone. During the mid-19th century, workers constructing the Victoria Bridge across the St. Lawrence River discovered a mass grave in Windmill Point where victims of the typhus epidemic of 1847 had been quarantined in fever sheds; the workers, many of whom were of Irish descent, were unsettled by the discovery and wanted to create a memorial to ensure the grave, which held the coffins of 6,000 Irish immigrants, would not be forgotten. Erected on December 1, 1859, the stone was the first Canadian monument to represent the famine; the inscription on the stone reads: "To Preserve from Desecration the Remains of 6000 Immigrants Who died of Ship Fever A.

D. 1847-48 This Stone is erected by the Workmen of Messrs. Peto and Betts Employed in the Construction of the Victoria Bridge A. D. 1859" Located in the median of Bridge St. at 45°29'12.3"N 73°32'46.6"W. Google maps have blurred out the inscription at a close distance but it still can be seen at a distance. On ACME Mapper the location is written N 45.48683 W 73.54638. About 75,000 Irish people are believed to have emigrated to Canada during the famine; the official figures, gave the figures of 5,293 deaths at sea, "Dr. Douglas, the medical Superintendent of Grosse Isle, estimated that 8,000 died at sea in 1847." However, the Montreal Gazette reported in 1934 that 18,000 Irish men and children died on the trip to Canada. The Black Rock continues to be a significant icon within the Montreal Irish community led by the Ancient Order of Hibernians Canada; each year at the end of May, the Canadian Irish community hosts a walk from St. Gabriel's church in Pointe St. Charles to the stone to commemorate those lives that were lost.

Ireland Park Gallagher, The Reverend John A. "The Irish Emigration of 1847 and Its Canadian Consequences" CCHA Report, University of Manitoba Web site. Retrieved February 07, 2011. Montreal Irish walk in remembrance of coffin ship victims Irish Central, July 23, 2009