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Trigonometry

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies; the Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, celestial mechanics, navigation. Trigonometry is known for its many identities, which are equations used for rewriting trigonometrical expressions to solve equations, to find a more useful expression, or to discover new relationships. Sumerian astronomers studied angle measure. They, the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles; the ancient Nubians used a similar method.

In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, used them to solve problems in trigonometry and spherical trigonometry. In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy constructed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. Centuries passed before more detailed tables were produced, Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine and Western European worlds; the modern sine convention is first attested in the Surya Siddhanta, its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata.

These Greek and Indian works were expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, were applying them to problems in spherical geometry; the Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right. Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, he developed spherical trigonometry into its present form, he listed the six distinct cases of a right-angled triangle in spherical trigonometry, in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, provided proofs for both these laws. Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi.

One of the earliest works on trigonometry by a northern European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus, encouraged to write, provided with a copy of the Almagest, by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years. 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 accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics. 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 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. In the 18th century, Brook Taylor defined the general Taylor series. Trigonometric ratios are the ratios between edges of a right triangle; these ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure: Sine function, defined as the ratio of the side opposite the angle to the hypotenuse. Sin ⁡ A = opposite hypotenuse = a c. Cosine function, defined as the ratio of the adjacent leg to the hypotenuse. Cos ⁡ A = adjacent hypotenuse = b c. Tangent function, defined as the ratio of the opposite leg to

Meggitt

Meggitt PLC is a UK based international company specialising in components and sub-systems for the aerospace and selected energy markets. It is a FTSE 100 company, listed on the London Stock Exchange; the company's history spans back to multiple preceding businesses that were founded in both the nineteenth and twentieth centuries. Meggitt's own official history claims that the company's roots can be traced through to 1850 via the scientific instrumentation business Negretti & Zambra, which had, amongst other innovations, invented the world’s first altimeter for the hot air balloon. During 1947, a new business was founded under the trading name Willson Lathes; that same year, Willson Lathes became a quoted public company. During 1964, Meggitt, a Dorset based light engineering business, was wholly acquired by Willson Lathes. During 1983, Nigel McCorkell and Ken Coates, together with 3i Group, took control of Meggitt via a management buy-in; the new management team soon embarked on a series of acquisitions, aimed at increasing the business' geographical diversity to become an international engineering company.

During 1985, Meggitt Holdings acquired Zambra. In 1986, the company bought the engineering interest Bestobell, which had focused on aviation air ducting and sealing solutions. Following the appointment of Michael Stacey as the CEO of Meggitt Holdings in 1990, the company was reorganised to focus its efforts around three core markets: aerospace, defence systems and electronics. Further acquisitions were conducted during the 1990s. In 1992, Meggitt acquired sensor specialist firm Endevco, specialists in sensors for test and measurement applications. During 1998, engine diagnostics specialist Vibro-Meter was acquired to improve the company's portfolio of condition monitoring capabilities. In the following year, Californian aviation aftermarket support firm Whittaker Corporation was acquired by Meggitt for $380m. During July 1997, Meggitt received their first contract from American commercial airline manufacturer Boeing to provide solid-state clocks for the Boeing 737; that same year Spanish aviation company CASA appointed it to supply the air ducting system for its new C-295 utility transport aircraft.

In 1998, Raytheon Aircraft awarded the company a contract to supply solid state altimeters and secondary flight display systems for numerous business aircraft, including the Beech King Air, Beech 1900D, Hawker 800XP and Hawker Horizon, in its lineup. That year, Boeing announced that Meggitt would be its sole supplier of solid-state electronic standby instrumentation for all of its airlines. During 2001, a new CEO, Terry Twigger, took over at Meggitt; the company continued to expand through numerous acquisitions throughout the 2000s. In 2002, it arranged to acquire Lodge, a British manufacturer of speed and temperature sensors for aero engines, from Smiths Aerospace. During the following year, Meggitt bought Western Design, which manufactured automated ammunition-handling apparatus and environmental control systems. In 2003, it acquired a provider of high-tech live fire training systems. During 2004, the company, in cooperation with The Carlyle Group, bought the Dunlop Standard Aerospace Group's design and manufacturing divisions.

This rapid acquisition rate was attributed with Meggitt's high increases in annual revenue around this period. During 2005, Meggitt acquired electronics specialist Sensorex. In 2006, the company purchased both simulation provider Firearms Training Systems and compressor producer Airdynamics. By this point, the North American market comprised around 50 per cent of the firm's revenue stream. A year Meggitt acquired K&F Industries, the parent company to the Aircraft Braking Systems Corporation. During 2008, it bought Ferroperm Piezoceramics A/S, which manufactured high quality piezoceramic materials for sensors; the company launched several products throughout the decade, such as the Meggitt Avionics new Generation Integrated Cockpit for business aircraft, bleed air leak detection system, its Electro-Thermal-based Ice Protection system choosing to promote its latest entries at the annual Farnborough Air Show. Meggitt has been a long time supplier of Brazillian aircraft manufacturer Embraer. Meggitt's vibration monitoring system was integrated into multiple platforms, including the CFM International CFM56, General Electric GEnx, Rolls-Royce Trent and PowerJet SaM146 turbofan engines, amongst others.

In 2010, the company restructured itself into five new divisions. In 2011, Meggitt acquired the Pacific Scientfic Aerospace Group, a component supplier of both civilian and military aerospace sectors.

Stoke St Michael

Stoke St Michael is a village and civil parish on the Mendip Hills 4 miles north east of Shepton Mallet, 8 miles west of Frome, in the Mendip district of Somerset, England. Since the 14th century the village has been known as Stoke Lane, although the origin of the alternative name is unclear, but may be connected to John de Lison who gave lands in the village to Glastonbury Abbey in 1253; the parish of Stoke Lane was part of the Whitstone Hundred. The village became a centre for cloth manufacture with fulling mills being established on the River Frome to the north of the village. Henry Fussell established paper mills in 1803, his family, who came from the village, including James Fussell established their iron works and edge-tool business in Mells; the Knatchbull Arms was built in the late 17th century, is named after the Knatchbulls of Babington who held the manor in the late 18th century. The manor house on Tower Hill, known as the old vicarage, was built around 1700; the parish council has responsibility for local issues, including setting an annual precept to cover the council’s operating costs and producing annual accounts for public scrutiny.

The parish council evaluates local planning applications and works with the local police, district council officers, neighbourhood watch groups on matters of crime and traffic. The parish council's role includes initiating projects for the maintenance and repair of parish facilities, as well as consulting with the district council on the maintenance and improvement of highways, footpaths, public transport, street cleaning. Conservation matters and environmental issues are the responsibility of the council; the village falls within the Non-metropolitan district of Mendip, formed on 1 April 1974 under the Local Government Act 1972, having been part of Shepton Mallet Rural District, responsible for local planning and building control, local roads, council housing, environmental health and fairs, refuse collection and recycling and crematoria, leisure services and tourism. Somerset County Council is responsible for running the largest and most expensive local services such as education, social services, main roads, public transport and fire services, trading standards, waste disposal and strategic planning.

It is part of the Somerton and Frome county constituency represented in the House of Commons of the Parliament of the United Kingdom. It elects one Member of Parliament by the first past the post system of election. Several significant caves of the Mendip Hills are close to the village including Stoke Lane Slocker, many falling within the St. Dunstan's Well Catchment and those at the disused Fairy Cave Quarry. Moon's Hill Quarry is a basalt quarry. Cook's Wood Quarry is a geological Site of Special Scientific Interest and Geological Conservation Review Site; the main exposures are cut in steeply-dipping Carboniferous Limestone. This was the original locality for the type section of the proposed ‘Cookswoodian Stage’. 9 species of Bat, Dormice and 4 species of Newts including the rare Great Crested Newt reside in Cooks Wood Quarry. Edford Woods and Meadows is a biological Site of Special Scientific Interest, important for the occurrence of a wide range of types of semi-natural ancient woodland and for unimproved meadows and pastures of a type, now uncommon in Britain.

The Church of St Michael has a western tower of c. 1400, the remainder being built in 1838 by Jesse Gane. It is a Grade II* listed building; the church was a chapelry of Doulting. The former non-conformist chapel on Stoke Hill is now a private residence. Stoke St Michael village website

John Duncan Lowe

John Duncan Lowe CB was a Scottish lawyer, Crown Agent for Scotland and Sheriff of Glasgow and Strathkelvin. Lowe was born at Alloa, Scotland, in 1948 and received his education at Hamilton Academy and the University of Glasgow, graduating MA, LLB. Following an apprenticeship with a firm of solicitors and a short period in local government, in 1974 Lowe joined the Procurator Fiscal Service and worked in Procurator Fiscal offices at Kilmarnock, Glasgow and at Edinburgh. Lowe was subsequently appointed Deputy Crown Agent for Scotland and in 1988 took up the post of Regional Procurator Fiscal at Edinburgh, a position he held till 1991. At the early age of 42, in 1991 Lowe was appointed Crown Agent for Scotland and in 1995 was invested CB, In 1997 Lowe was appointed Sheriff of Glasgow and Strathkelvin, a position he held until his death at Edinburgh the following year

Zunich–Kaye syndrome

Zunich–Kaye syndrome known as Zunich neuroectodermal syndrome, is a rare congenital ichthyosis first described in 1983. It is referred to as CHIME syndrome, after its main symptoms, it is a congenital syndrome with only a few cases published. Associated symptoms range from things such as colobomas of the eyes, heart defects, ichthyosiform dermatosis, intellectual disability, ear abnormalities. Further symptoms that may be suggested include characteristic facies, hearing loss, cleft palate. Zunich–Kay syndrome is considered to have an autosomal recessive inheritance pattern; this means the defective gene is located on an autosome, two copies of the gene, one from each parent, are required to inherit the disorder. The parents of an individual with autosomal recessive disorder both carry one copy of the defective gene, but do not have the disorder. Treatment with isotretinoin may induce substantial resolution of skin lesions, but the risk of secondary infection remains. List of cutaneous conditions Schnur RE, Greenbaum BH, Heymann WR, Christensen K, Buck AS, Reid CS.

"Acute lymphoblastic leukemia in a child with the CHIME neuroectodermal dysplasia syndrome". Am. J. Med. Genet. 72: 24–9. Doi:10.1002/1096-862872:1<24::AID-AJMG5>3.0. CO. PMID 9295069. Shashi V, Zunich J, Kelly TE, Fryburg JS. "Neuroectodermal syndrome: an additional case with long term follow up of all reported cases". J. Med. Genet. 32: 465–9. Doi:10.1136/jmg.32.6.465. PMC 1050487. PMID 7666399. Zunich J, Esterly NB, Holbrook KA, Kaye CI. "Congenital migratory ichthyosiform dermatosis with neurologic and ophthalmologic abnormalities". Arch Dermatol. 121: 1149–56. Doi:10.1001/archderm.121.9.1149. PMID 4037840. Zunich J, Esterly NB, Kaye CI. "Autosomal recessive transmission of neuroectodermal syndrome". Arch Dermatol. 124: 1188–9. Doi:10.1001/archderm.124.8.1188. PMID 3041916. Zunich J, Kaye CI. "Additional case report of new neuroectodermal syndrome". Am. J. Med. Genet. 17: 707–10. Doi:10.1002/ajmg.1320170324. PMID 6711621

Armscye

In sewing, the armscye is the armhole, the fabric edge to which the sleeve is sewn. The length of the armscye is the total length of this edge. Multiple theories for the etymology of "armscye" have been proposed; the scholarly etymology has the origin as "arm" + "scye." The first documented use of "scye" in print is by Jamieson Suppl.: "sey," a Scots and Ulster dialect word meaning ‘the opening of a gown, etc. into which the sleeve is inserted. A more fanciful folk etymology is; because the expression "arm's eye" was used in some older sewing texts it is conjectured that in poor prints the apostrophe and the crossbar of the lower case "e" were indistinct, the neologism "armscye" was created by readers who concatenated the orphaned fragments "arm" and "s" with the corrupt "cye". According to this undocumented theory, until the beginning of the 20th century writers favoured the original term or at least a more logical variation, but as self-proclaimed experts copied each other, the term "armscye" became enough used by home sewers to gain general acceptance.

The latter theory contradicts evidence that the term "scye" was in use at least as early as 1825. Therefore, the erroneous folk analysis was not in the direction from "arm's eye" to "armcye", but rather from the original "armcye" to "arm's eye". Armscye & Armhole measurement in sewing Dolman Jenna Wilson. "Ravellings on the knitted sleeve: Part II - Creating a set-in sleeve for a sleeveless body". Knitty. — Wilson discusses the sizes and shapes of armscyes