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. Its boundary 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, what shape can a manhole cover be made so that it cannot fall down through the hole?"Reuleaux triangles have been called spherical triangles, but that term more properly refers to triangles on the curved surface of a sphere. They are named after Franz Reuleaux, a 19th-century German engineer who pioneered the study of machines for translating one type of motion into another, who used Reuleaux triangles in his designs. However, these shapes were known before his time, for instance by the designers of Gothic church windows, by Leonardo da Vinci, who used it for a map projection, by Leonhard Euler in his study of constant-width shapes.
Other applications of the Reuleaux triangle include giving the shape to guitar picks and drill bits for drilling square holes, as well as in graphic design in the shapes of some signs and corporate logos. Among constant-width shapes with a given width, the Reuleaux triangle has the minimum area and the sharpest possible angle at its corners. By several numerical measures it is the farthest from being centrally symmetric, it provides the largest constant-width shape avoiding the points of an integer lattice, is related to the shape of the quadrilateral maximizing the ratio of perimeter to diameter. It can perform a complete rotation within a square while at all times touching all four sides of the square, has the smallest possible area of shapes with this property. However, although it covers most of the square in this rotation process, it fails to cover a small fraction of the square's area, near its corners; 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, whose boundaries are curves of constant width formed from regular polygons with an odd number of sides. Some of these curves have been used as the shapes of coins; the Reuleaux triangle can be generalized into three dimensions in multiple ways: the Reuleaux tetrahedron does not have constant width, but can be modified by rounding its edges to form the Meissner tetrahedron, which does. Alternatively, the surface of revolution of the Reuleaux triangle 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 needing a straightedge. By the Mohr–Mascheroni theorem the same is true more of any compass-and-straightedge construction, but the construction for the Reuleaux triangle is simple; the first step is to mark two arbitrary points of the plane, use the compass to draw a circle centered at one of the marked points, through the other marked point.
Next, one draws a second circle, of the same radius, centered at the other marked point and passing through the first marked point. One draws a third circle, again of the same radius, with its center at one of the two crossing points of the two previous circles, passing through both marked points; the central region in the resulting 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 other two vertices. Or, equivalently, it may be constructed as the intersection of three disks centered at the vertices of T, with radius equal to the side length of T; the most basic property of the Reuleaux triangle is that it has constant width, meaning that for every pair of parallel supporting lines the two lines have the same Euclidean distance from each other, regardless of the orientation of these lines. In any pair of parallel supporting lines, one of the two lines will touch the triangle at one of its vertices.
The other supporting line may touch the triangle at any point on the opposite arc, their distance equals the radius of this arc. The first mathematician to discover the existence of curves of constant width, to observe that the Reuleaux triangle has constant width, may have been Leonhard Euler. In a paper that he presented in 1771 and published in 1781 entitled De curvis triangularibus, Euler studied curvilinear triangles as well as the curves of constant width, which he called orbiforms. By many different measures, the Reuleaux triangle is one of the most extreme curves of constant width. By the Blaschke–Lebesgue theorem, the Reuleaux triangle has the smallest possible area of any curve of given constant width; this area is 1 2. One method for deriving this area formula is to partition the Reuleaux triangle into an inner equilateral triangle and three curvilinear regions between this inner triangle and the arcs forming the Reuleaux triangle, add the
West Indies Federation
The West Indies Federation known as the West Indies, the Federation of the West Indies or the West Indian Federation, was a short-lived political union that existed from 3 January 1958 to 31 May 1962. Various islands in the Caribbean that were colonies of the United Kingdom, including Trinidad and Tobago, Barbados and those on the Leeward and Windward Islands, came together to form the Federation, with its capital in Port of Spain and Tobago; the expressed intention of the Federation was to create a political unit that would become independent from Britain as a single state—possibly similar to the Canadian Confederation, Australian Commonwealth, or Central African Federation. The territories that would have become part of the Federation became the nine contemporary sovereign states of Antigua and Barbuda, Dominica, Jamaica, Saint Kitts and Nevis, Saint Lucia, Saint Vincent and the Grenadines, Trinidad and Tobago. British Guiana and British Honduras held observer status within the West Indies Federation.
The total population of the West Indies Federation was between 3 and 4 million people, with the majority being of black West African descent. Minorities included Indians from the subcontinent, Europeans and Caribs. There was a large population of mixed descent. In terms of religion, most of the population was Protestant, with significant numbers of Catholics and some Hindus and Muslims; the West Indies Federation consisted of around 24 main inhabited islands and 220–230 minor offshore islands and cays. The largest island was Jamaica, located in the far northwest of the Federation. To the southeast lay the second largest island, followed by Barbados, located at the eastern extremity of the Federation; the Federation spanned all the island groupings in the Caribbean: The Greater Antilles: Jamaica and the Cayman Islands and the Turks and Caicos Islands The Lesser Antilles: Barbados, east of the Windward Islands Leeward Islands: Antigua and Barbuda, Saint Christopher-Nevis-Anguilla, Montserrat Windward Islands: Dominica, Saint Lucia, Saint Vincent and the Grenadines, Grenada Trinidad and TobagoAt its widest, from the Cayman Islands to Barbados it spanned some 2,425 kilometres and from the Turks and Caicos Islands in the north, to the Icacos Point, Trinidad in the south it extended 1,700 kilometres.
However, most of the area along either of these distances was taken up by open water. By comparison Great Britain stretches across nearly 10 degrees of latitude and Spain extends across 20 degrees of longitude. Though the West Indies was spread across such a vast area, most of its provinces were contiguous and clustered close together in the Eastern Caribbean, with the obvious exceptions of Jamaica, the Cayman Islands, Turks and Caicos Islands. Most of the islands have mountainous interiors surrounded by narrow coastal plains; the exceptions were Anguilla, Barbuda, the Cayman Islands, the Turks and Caicos Islands, Trinidad. The narrow coastal plains as well as historical trade is the main reason why all of the major settlements of the Federation were located on the coast. Chief towns included Kingston, Port of Spain, Spanish Town, Montego Bay, Castries, Roseau, St. George's, Kingstown, St. John's, Basseterre; the climate in all the islands is tropical, with hot and humid weather, although inland regions in the larger islands have more temperate climates.
Regions falling within the rain shadows are drier. There are two seasons annually: the dry season for the first six months of the year, the rainy season in the second half of the year. Many of the islands fall within the traditional hurricane belt, with the exception of Trinidad and thus are at risk from potential wind and flood damage. Britain classified the Federation as being part of its "Caribbean and North Atlantic Territories" region, shared alongside other possessions such as Bermuda; the Federation today is geographically considered to be part of the North American continent as all of its islands are in and around the Caribbean though Trinidad is located just offshore from South America and lies on the same continental shelf. See Bicontinental countries. "West Indian" nations The Bahamas, Belize, the British Virgin Islands, Guyana opted not to join because they believed that their future lay with association with North America, Central America, the United States Virgin Islands. Guyana opted not to join at that time due to its ongoing political and internal struggles for independence from the UK, started in the 1950s.
At issue were the newly formed political party with socialist leanings
Shilling
The shilling is a unit of currency used in Austria, the United Kingdom, New Zealand, United States and other British Commonwealth countries. The shilling is used as a currency in four east African countries: Kenya, Tanzania and Somalia, it is the proposed currency that the east African community plans to introduce. The word shilling comes from old English "Scilling", a monetary term meaning twentieth of a pound, from the Proto-Germanic root skiljaną meaning'to separate, divide.' The word "Scilling" is mentioned in the earliest recorded Germanic law codes, those of Æthelberht of Kent. Slang terms for the old shilling coins include "bob" and "hog". While the derivation of "bob" is uncertain, John Camden Hotten in his 1864 Slang Dictionary says the original version was "bobstick" and speculates that it may be connected with Sir Robert Walpole. One abbreviation for shilling is s, it was represented by a solidus symbol, which may have stood for a long s or ſ, thus 1/9 would be one shilling and ninepence.
A price with no pence was sometimes written with a solidus and a dash: 11/–. The solidus symbol is still used for the Kenyan shilling, rather than sh. During the Great Recoinage of 1816, the mint was instructed to coin one troy pound of standard silver into 66 shillings, or its equivalent in other denominations; this set the weight of the shilling, its subsequent decimal replacement 5 new pence coin, at 87.2727 grains or 5.655 grams from 1816 until 1990, when a new smaller 5p coin was introduced. In the past, the English world has had various myths about the shilling. One myth was that it was deemed to be the value of a cow in a sheep elsewhere. A shilling was a coin used in England from the reign of Henry VII; the shilling continued in use after the Acts of Union of 1707 created a new United Kingdom from the Kingdoms of England and Scotland, under Article 16 of the Articles of Union, a common currency for the new United Kingdom was created. The term shilling was in use in Scotland from early medieval times.
The common currency created in 1707 by Article 16 of the Articles of Union continued in use until decimalisation in 1971. In the traditional pounds and pence system, there were 20 shillings per pound and 12 pence per shilling, thus there were 240 pence in a pound. Three coins denominated in multiple shillings were in circulation at this time, they were: two shillings, which adopted the value of 10 new pence at decimalisation. At decimalisation in 1971, the shilling coin was superseded by the new five-pence piece, of identical size and weight and had the same value, inherited the shilling's slang name of a bob. Shillings remained in circulation until the five pence coin was reduced in size in 1991. Between 1701 and the unification of the currencies in 1825, the Irish shilling was valued at 13 pence and known as the "black hog", as opposed to the 12-pence English shillings which were known as "white hogs". In the Irish Free State and Republic of Ireland the shilling coin was issued as scilling in Irish.
It was worth 1/20th of an Irish pound, was interchangeable at the same value to the British coin, which continued to be used in Northern Ireland. The coin featured a bull on the reverse side; the first minting, from 1928 until 1941, contained 75% silver, more than the equivalent British coin. The original Irish shilling coin ) was withdrawn from circulation on 1 January 1993, when a smaller five pence coin was introduced. Australian shillings, twenty of which made up one Australian pound, were first issued in 1910, with the Australian coat of arms on the reverse and King Edward VII on the face; the coat of arms design was retained through the reign of King George V until a new ram's head design was introduced for the coins of King George VI. This design continued until the last year of issue in 1963. In 1966, Australia's currency was decimalised and the shilling was replaced by a ten cent coin, where 10 shillings made up one Australian dollar; the slang term for a shilling coin in Australia was "deener".
The slang term for a shilling as currency unit was "bob", the same as in the United Kingdom. After 1966, shillings continued to circulate, as they were replaced by 10-cent coins of the same size and weight. New Zealand shillings, twenty of which made up one New Zealand pound, were first issued in 1933 and featured the image of a Maori warrior carrying a taiaha "in a warlike attitude" on the reverse. In 1967, New Zealand's currency was decimalised and the shilling was replaced by a ten cent coin of the same size and weight. Ten cent coins minted through the remainder of the 1960s included the legend "ONE SHILLING" on the reverse. Smaller 10-cent coins were introduced in 2006. Shillings were used in Malta, prior to decimalisation in 1972, had a face value of five Maltese cents. In British Ceylon, an shilling was equivalent to eight fanams. With the replacement of the rixdollar by the rupee in 1852, a shilling was deemed to be equivalent to half a rupee. On the decimalisation of the currency
Brass
Brass is an alloy of copper and zinc, in proportions which can be varied to achieve varying mechanical and electrical properties. It is a substitutional alloy: atoms of the two constituents may replace each other within the same crystal structure. Bronze is an alloy containing copper, but instead of zinc it has tin. Both bronze and brass may include small proportions of a range of other elements including arsenic, phosphorus, aluminium and silicon; the distinction is historical. Modern practice in museums and archaeology avoids both terms for historical objects in favour of the all-embracing "copper alloy". Brass is used for decoration for its bright gold-like appearance, it is used in zippers. Brass is used in situations in which it is important that sparks not be struck, such as in fittings and tools used near flammable or explosive materials. Brass has higher malleability than zinc; the low melting point of brass and its flow characteristics make it a easy material to cast. By varying the proportions of copper and zinc, the properties of the brass can be changed, allowing hard and soft brasses.
The density of brass is 8.4 to 8.73 grams per cubic centimetre. Today 90% of all brass alloys are recycled; because brass is not ferromagnetic, it can be separated from ferrous scrap by passing the scrap near a powerful magnet. Brass scrap is transported to the foundry where it is melted and recast into billets. Billets are extruded into the desired form and size; the general softness of brass means that it can be machined without the use of cutting fluid, though there are exceptions to this. Aluminium makes brass more corrosion-resistant. Aluminium causes a beneficial hard layer of aluminium oxide to be formed on the surface, thin and self-healing. Tin has a similar effect and finds its use in seawater applications. Combinations of iron, aluminium and manganese make brass wear and tear resistant. To enhance the machinability of brass, lead is added in concentrations of around 2%. Since lead has a lower melting point than the other constituents of the brass, it tends to migrate towards the grain boundaries in the form of globules as it cools from casting.
The pattern the globules form on the surface of the brass increases the available lead surface area which in turn affects the degree of leaching. In addition, cutting operations can smear the lead globules over the surface; these effects can lead to significant lead leaching from brasses of comparatively low lead content. In October 1999 the California State Attorney General sued 13 key manufacturers and distributors over lead content. In laboratory tests, state researchers found the average brass key, new or old, exceeded the California Proposition 65 limits by an average factor of 19, assuming handling twice a day. In April 2001 manufacturers agreed to reduce lead content to 1.5%, or face a requirement to warn consumers about lead content. Keys plated with other metals are not affected by the settlement, may continue to use brass alloys with higher percentage of lead content. In California, lead-free materials must be used for "each component that comes into contact with the wetted surface of pipes and pipe fittings, plumbing fittings and fixtures."
On January 1, 2010, the maximum amount of lead in "lead-free brass" in California was reduced from 4% to 0.25% lead. The so-called dezincification resistant brasses, sometimes referred to as CR brasses, are used where there is a large corrosion risk and where normal brasses do not meet the standards. Applications with high water temperatures, chlorides present, or deviating water qualities play a role. DZR-brass is excellent in water boiler systems; this brass alloy must be produced with great care, with special attention placed on a balanced composition and proper production temperatures and parameters to avoid long-term failures. The high malleability and workability good resistance to corrosion, traditionally attributed acoustic properties of brass, have made it the usual metal of choice for construction of musical instruments whose acoustic resonators consist of long narrow tubing folded or coiled for compactness. Collectively known as brass instruments, these include the trombone, trumpet, baritone horn, tenor horn, French horn, many other "horns", many in variously-sized families, such as the saxhorns.
Other wind instruments may be constructed of brass or other metals, indeed most modern student-model flutes and piccolos are made of some variety of brass a cupronickel alloy similar to nickel silver/German silver. Clarinets low clarinets such as the contrabass and subcontrabass, are sometimes made of metal because of limited supplies of the dense, fine-grained tropical hardwoods traditionally preferred for smaller woodwinds. For the same reason, some low clarinets and contrabassoons feature a hybrid construction, with long, straight sections of wood, curved joints, and/or bell of metal; the use of metal avoids the risks of exposing wooden instruments to changes in temperature or humid
George VI
George VI was King of the United Kingdom and the Dominions of the British Commonwealth from 11 December 1936 until his death on 6 February 1952. He was the first Head of the Commonwealth. Known publicly as Albert until his accession, "Bertie" among his family and close friends, George VI was born in the reign of his great-grandmother Queen Victoria, was named after his great-grandfather Albert, Prince Consort; as the second son of King George V, he was not expected to inherit the throne and spent his early life in the shadow of his elder brother, Edward. He attended naval college as a teenager, served in the Royal Navy and Royal Air Force during the First World War. In 1920, he was made Duke of York, he married Lady Elizabeth Bowes-Lyon in 1923 and they had two daughters and Margaret. In the mid-1920s, he had speech therapy for a stammer, which he never overcame. George's elder brother ascended the throne as Edward VIII upon the death of their father in 1936; however that year Edward revealed his desire to marry divorced American socialite Wallis Simpson.
British prime minister Stanley Baldwin advised Edward that for political and religious reasons he could not marry a divorced woman and remain king. Edward abdicated to marry Simpson, George ascended the throne as the third monarch of the House of Windsor. During George's reign, the break-up of the British Empire and its transition into the Commonwealth of Nations accelerated; the parliament of the Irish Free State removed direct mention of the monarch from the country's constitution on the day of his accession. The following year, a new Irish constitution changed the name of the state to Ireland and established the office of President. From 1939, the Empire and Commonwealth – except Ireland – was at war with Nazi Germany. War with Italy and Japan followed in 1941, respectively. Though Britain and its allies were victorious in 1945, the United States and the Soviet Union rose as pre-eminent world powers and the British Empire declined. After the independence of India and Pakistan in 1947, George remained king of both countries, but relinquished the title of Emperor of India in June 1948.
Ireland formally declared itself a republic and left the Commonwealth in 1949, India became a republic within the Commonwealth the following year. George adopted the new title of Head of the Commonwealth, he was beset by smoking-related health problems in the years of his reign. He was succeeded by his elder daughter, Elizabeth II. George was born at York Cottage, on the Sandringham Estate in Norfolk, during the reign of his great-grandmother Queen Victoria, his father was Prince George, Duke of York, the second and eldest-surviving son of the Prince and Princess of Wales. His mother was the Duchess of York, the eldest child and only daughter of the Duke and Duchess of Teck, his birthday, 14 December 1895, was the 34th anniversary of the death of his great-grandfather, Prince Consort. Uncertain of how the Prince Consort's widow, Queen Victoria, would take the news of the birth, the Prince of Wales wrote to the Duke of York that the Queen had been "rather distressed". Two days he wrote again: "I think it would gratify her if you yourself proposed the name Albert to her".
Queen Victoria was mollified by the proposal to name the new baby Albert, wrote to the Duchess of York: "I am all impatience to see the new one, born on such a sad day but rather more dear to me as he will be called by that dear name, a byword for all, great and good". He was baptised "Albert Frederick Arthur George" at St. Mary Magdalene's Church near Sandringham three months later. Within the family, he was known informally as "Bertie", his maternal grandmother, the Duchess of Teck, did not like the first name the baby had been given, she wrote prophetically that she hoped the last name "may supplant the less favoured one". Albert was fourth in line to the throne at birth, after his grandfather and elder brother, Edward, he suffered from ill health and was described as "easily frightened and somewhat prone to tears". His parents were removed from their children's day-to-day upbringing, as was the norm in aristocratic families of that era, he had a stammer. Although left-handed, he was forced to write with his right hand, as was common practice at the time.
He suffered from chronic stomach problems as well as knock knees, for which he was forced to wear painful corrective splints. Queen Victoria died on 22 January 1901, the Prince of Wales succeeded her as King Edward VII. Prince Albert moved up to third in line after his father and elder brother. From 1909, Albert attended Osborne, as a naval cadet. In 1911 he came bottom of the class in the final examination, but despite this he progressed to the Royal Naval College, Dartmouth; when his grandfather, Edward VII, died in 1910, Albert's father became King George V. Edward became Prince of Wales, with Albert second in line to the throne. Albert spent the first six months of 1913 on the training ship HMS Cumberland in the West Indies and on the east coast of Canada, he was rated as a midshipman aboard HMS Collingwood on 15 September 1913, spent three months in the Mediterranean. His fellow officers gave him the nickname "Mr. Johnson"; the First World War broke out a year after his commission. Three weeks after the outbreak of war he was medically evacuated from the ship to Aberdeen where his appendix was removed by Sir John Marnoch.
He was mentioned in despatches for his action as a turret officer aboard Collingwood i
Canadian Bank of Commerce
The Canadian Bank of Commerce was a Canadian bank, founded in 1867, had hundreds of branches throughout Canada. It merged in 1961 with the Imperial Bank of Canada to form the Canadian Imperial Bank of Commerce. In 1866 a group of businessmen, including William McMaster, purchased a charter from the defunct Bank of Canada, which had folded in 1858; the Canadian Bank of Commerce was founded the following year issued stock, opened its headquarters in Toronto, Ontario. The bank soon opened branches in St. Catharines and Barrie. During the following years, the bank opened more branches in Ontario, took over the business of the local Gore Bank, before expanding across Canada through the acquisition of the Bank of British Columbia in 1901 and the Halifax Banking Company in 1903. By 1907 the Canadian Bank of Commerce had 172 branches. By the beginning of World War II, this had expanded to 379 branches, including a large building at Darling and Pearson, Manitoba, built in 1910 in beaux-arts classic style.
During World War I, 1,701 staff from the Canadian Bank of Commerce enlisted in the war effort. A memorial on the East and West Memorial Buildings in Ottawa, Ontario is dedicated to the memory of 1701 Men of the Canadian Bank of Commerce who served in the First World War A War Memorial at Commerce Court in Toronto, Ontario commemorates their service. In 1931, the Toronto headquarters of the bank, designed by architects John Pearson and Frank Darling, was completed. At 34 stories, for many years it was the tallest building in the British Empire. Once again, during World War II, 2,300 staff members enlisted in the armed forces; the Canadian Bank of Commerce merged with the Imperial Bank of Canada in 1961 to form the Canadian Imperial Bank of Commerce, now one of the Big Five Canadian banks. The following are on the Registry of Historical Places of Canada; the Bank of Commerce in Nanaimo, British Columbia, built in 1914. The Canadian Bank of Commerce in New Westminster, British Columbia built in 1910 to 1911.
The Bank of Commerce in Vancouver, British Columbia, built in 1914 to 1915 the Canadian Bank of Commerce in Watson, Saskatchewan built in 1906 to 1907. The Bank of Commerce in Nokomis, built in 1910; the Bank of Commerce in Winnipeg, completed in 1912. The Bank of Commerce in Kelsey, built in The Pas in 1912; the Canadian Bank of Commerce in Innisfree, built in 1905. The Canadian Bank of Commerce in Dawson, built in 1901; the Canadian Bank of Commerce grew through acquisitions of other banks in Canada: Halifax Banking Company Established in 1825 and merged with the Commerce in 1903. Gore Bank Formed in 1836 and merged with the Commerce in 1870. Eastern Townships Bank Formed in 1859 and merged with the Commerce in 1912. Bank of British Columbia Established with a Royal Charter in 1862 and merged with the Commerce in 1901. Merchants Bank of Prince Edward Island Formed Oct 6, 1871 and merged with the Commerce in 1906. Bank of Hamilton Bank of Hamilton merged with the Commerce in 1924; the Standard Bank of Canada Formed in 1876 and merged with the Commerce in 1928.
List of Canadian banks Charles Peers Davidson `A Compilation Of The Statutes Passed Since Confederation Relating To Banks And Banking, Government And Other Savings Banks, Promissory Notes And Bills` BiblioLife | January 10, 2010
Barbados
