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British brass band

In Britain, a brass band (known regionally as a silver band or colliery band) is a musical ensemble comprising a standardized range of brass and percussion instruments. The modern form of the brass band in the United Kingdom dates back to the 19th century, with a vibrant tradition of competition based around communities and local industry, with colliery bands being particularly notable. The Stalybridge Old Band (still in existence) was formed in 1809 and was perhaps the first civilian brass band in the world.

Bands using the British instrumentation are the most common form of brass band in the United Kingdom, Australia and New Zealand, and are also widespread in continental Europe, Japan and North America. The tradition for brass bands in the UK is continuing, and local communities and schools have brass bands. British band contests are highly competitive, with bands organized into five sections much like a football league. Competitions are held throughout the year at local, regional, and national levels, and at the end of each year there are promotions and relegations. The 2019 holder of the National Brass Band Championships of Great Britain was the Cory Band from South Wales.

A selection of brass bands can be experienced at the annual Durham Miners' Gala. There are also hotly-contested annual events held on Whit Friday in the Saddleworth area of Greater Manchester in which hundreds of bands compete.

Etymology
The terms brass band and silver band are synonymous in that the vast majority of bands termed either brass or silver incorporate musicians playing a mix of lacquered brass and silver-plated instruments. However, in the days when plain or lacquered brass instruments were not as costly as silver-plated ones, the term silver band implied an ensemble that could afford the latter and were thus perceived to be more successful. With the cost of each type now being similar, the distinction between brass and silver bands is generally not made. There are, however, some brass bands which carried the name silver prize band as a result of their successes in contests and competitions. As time went on, some kept the name silver but not necessarily the instruments. Nowadays, the only real difference is that silver bands are located mainly in the south of England and brass bands in the north, although some silver bands do still exist in the North and many brass bands exist in the South.

Instrumentation
British brass bands are limited to specific instruments, excluding, for instance, trumpets or French horns, which are found in orchestras and concert bands.

The standard instrumentation is as follows:
 * 1 soprano cornet (E♭)
 * 9 cornets (B♭) –
 * Front row: one principal cornet, three solo cornets
 * Back row: one repiano cornet, two 2nd cornets, two 3rd cornets
 * 1 flugelhorn (B♭)
 * 3 tenor horns (E♭) called alto horn in many countries – solo, 1st, 2nd
 * 2 baritone horns (B♭) – 1st, 2nd
 * 2 tenor trombones (B♭) – 1st, 2nd
 * 1 bass trombone (B♭)
 * 2 euphoniums (B♭)
 * 2 E♭ basses, also known as E♭ tubas
 * 2 B♭ basses, also known as B♭ tubas
 * 2 to 4 percussion

The above totals 27–29 players, although in practice a band often has fewer than this. Spare seats may be filled for concerts and contests by players brought in from other bands, commonly known as deputising players or deps.

Notation
With the exception of percussion, bass trombone and some older tenor trombone music, all parts are transposing and written in the treble clef with the instrument's lowest open note (B♭ or E♭) notated as middle C. This means that for every valved instrument, from the basses to the soprano cornet, a given note on the stave corresponds to the same valve fingering, enabling players to move more easily between parts. This system is unique to UK-style brass bands, though historically the North American drum and bugle corps activity followed the brass band convention of all-treble-clef writing.

Tenor trombone music is usually in treble clef like the other instruments in the band, though older scores or marches sometimes use tenor clef.

Bass trombone music is written at concert pitch in bass clef. This was historically due to this part being taken by a G bass trombone, rather than the modern B♭ bass trombone. As instrument technology modernized, the need for a bass trombone in G diminished, with the introduction of the B♭ bass trombone with F rotary valve. The larger bore and open wrap of the F valve gave the B♭ bass trombone a lower available playing register than the straight G bass trombone.

Tuned percussion is written in concert pitch (with the appropriate octave transpositions) with the appropriate clef for the instrument – e.g. bass clef for timpani, treble clef for glockenspiel (sounding two octaves higher). Unpitched percussion parts (including drum kit) are written in neutral clef on a 1- to 5-line stave using standard percussion notation, though some older scores may use bass and/or treble clefs.

History
Most of the instruments used in British brass bands had been in use for some time in village, church and military bands, and in the 1840s and 1850s the brass band emerged from these as a popular pastime. Brass bands were a response to industrialisation, which produced a large working class population, technological advances in instrument design, and the mass production to manufacture and distribute the instruments. A major improvement to the old designs was the development of efficient piston valves, which were easier to play and produced a more accurate, consistent sound. Arguably brass bands were an expression of the local solidarity and aspirations of newly formed or rapidly growing communities. This was seen, for instance, in the creation of brass band competitions by the late 1850s. In 1853 John Jennison, owner of Belle Vue Zoological Gardens in Manchester, agreed to stage the first British open brass band championships. The event was attended by a crowd of over 16,000 and continued annually until 1981. Brass bands reached their peak in the 1890s, when there were around 5,000 brass bands in the country. Today, there are 1,200 brass bands in Britain with around 30,000 players.

Many UK bands were originally either works bands or sponsored by various industrial concerns. This was particularly evident in coal mining areas, for example, bands such as the Grimethorpe Colliery Band and Carlton Main Frickley Colliery Band in Yorkshire. Bands sponsored by factories include The Black Dyke Mills Band, Yorkshire Imperial Band (originally the Yorkshire Copperworks Band), Foden's sponsored by the truck manufacturer, Fairey sponsored by the aircraft manufacturer, and Leyland Band sponsored by the vehicle manufacturer. One of the reasons for this was to keep the workers from organizing in radical groups. With the decline of these industries several bands have dissolved and others now draw their membership from other industries and other parts of the community. The Brighouse and Rastrick Band, is unique in having operated continually at the highest level without sponsorship, drawing its income from regular concerts, public donations and sales of recordings and merchandise.

The leading bands in Wales include the Cory Band, Tredegar Town Band and Tongwynlais Temperance Band The leading bands in Scotland are The Cooperation Band (formally Co-operative Funeralcare), Whitburn Band and the Kirkintilloch Band. All these bands compete at the highest level in the banding movement. In Northern Ireland the Brass Band League currently has 26 members.

There are also many non-contesting or 'community' brass bands in the UK providing entertainment for audiences and enjoyment for musicians of all ages.

Contesting in the United Kingdom
British banding is highly competitive, with bands organized into five sections much like a football league – Championship section, 1st section, 2nd section, 3rd section and 4th section. Competitions are held throughout the year at local, regional, and national levels, and at the end of each year there are promotions and relegations.

At a national level the main contest is the National Brass Band Championship, and this determines a band's section. For this, the UK is split into 8 regions: London and Southern Counties, Midlands, North of England, North West, Scotland, Wales, West of England, and Yorkshire. Each year in Spring the bands compete in a regional contest for their section, and the top two or three in each section go on to the "National Finals" in Autumn. Recently (2011), the finals for Sections 1 to 4 have been held in Cheltenham, and the finals for the Championship Section at the Royal Albert Hall in London (as featured in the film Brassed Off).

The bands are awarded points for their result (1st gets 1 point, 9th gets 9 points), and this is added to the previous two years to give a three-year total. Two or three bands with the best total are promoted, and two or three bands are relegated.

There are also separate contests for university brass bands, including the annual UniBrass contest, founded at Lancaster University in 2010 and hosted in 2023 by the University of Warwick. The Cambridge and Oxford University brass bands also compete against each other in annual 'varsity' contests. The National Youth Band Brass Championships held annually has a Youth Champs' non-competitive section offering the opportunity for young people new to brass to take part and contribute a performance without the pressure of competition. Bands of all standards play in the Debut Section and join in the fun.

The Salvation Army
In the UK, the brass bands of The Salvation Army have run parallel to the main brass band movement since the 1870s. Bands of The Salvation Army range from small church bands to staff bands composed of the best bandsmen of The Salvation Army in the area. Their instrumentation is almost identical except for a minor difference in the cornet section whereby the repiano is dropped and the remainder of the row is made up of parts designated 1st and 2nd (two players each) rather than 2nd and 3rd; and that some major pieces have a split first trombone part, the lower part usually cued elsewhere in the band. Bands of The Salvation Army can be found in most countries around the world in which it operates.

Australia
The Australian derivation of a brass band is the same as the UK brass band (i.e. standard instrumentation with no woodwind). Contesting bands in Australia are graded from A Grade to D Grade. National Contests are held each year at Easter, with the location moving from state to state, and often including bands from New Zealand or other locations around the Asia Pacific region.

Each state also conducts their own championships. National and state contests are generally of the same format: a set test piece for each grade, a hymn, an own choice, a stage march (march or concert march performed on the stage without marching) and a light entertainment concert program. Smaller regional contests often replace the major works with an own choice concert program.

Among the country's most famous brass band identities have been conductor and composer Percy Code; and David King and Frank Wright who made their mark in Britain. Early Australian brass composers include Thomas Bulch, Alex Lithgow and Charles Trussell. In later years, leading composers have included Brenton Broadstock ("Winds of Change", "Rutherford Variations", "Valiant Take All My Sins Away" and many more), Barrie Gott ("Mumbo Jumbo", "Gospel Train", "Glasshouse Sketches") and Joe Cook ("Taskforce", "Keighley Moor").

Belgium
In Belgium, there are about 30 brass bands. The best known is Brass Band Willebroek, who were crowned European Brass Band Champions in 1993, 2006 and 2007. The average banding level is still rising, as the brass movement was only introduced in Belgium some 30 years ago.

The national brass band Championships are held annually at CC Zwanenberg in Heist-Op-Den-Berg. The winner represents Belgium at the European Championships.

Participants are divided into four categories going from Championship section to third section. Currently there are seven bands competing in Championship section. These are: Brass Band Willebroek Brass Band Buizingen, Metropole Brass Band, Festival Brass Band, Kortrijk Brass Band, Brass Band Heist, and De Noord-Limburgse Brassband,

Other Belgian bands include: Mercator Brassband, Brassband Kempenzonen Tielen, Brass Band Leieland, Brass Band Zele, Brass Band De Kempengalm, Brass Band de Thudinie and many more.

Some of the brass bands in Belgium are primarily based around recreational music, do not conform to British brass band instrumentation, and therefore do not compete.

Germany
Brass bands in Germany are not as widespread as in other European countries like Switzerland or the Netherlands. There are just a few bands (about 40) playing in authentic British instrumentation. This is primarily due to the popularity of wind bands, particularly in Southern Germany, and brass groups (composed of soprano to bass instruments) known as "Posaunenchöre" (trombone choirs) related to churches.

National Brass Band "competitions" took place as part of the German Festival of Wind Music in Würzburg 2007 and 2008, in Duisburg 2010, in Coesfeld 2012 and in Bad Kissingen 2014.

Ireland
Brass bands in Ireland can primarily trace their origins to the past influence of British Army bands during the period between 1801 and 1922. As well as military bands however, community led groups were also in existence around this time. While concert bands are more prevalent today, possibly due to the association of brass bands with British rule, brass bands are still popular, especially on the east coast. Dungarvan, Arklow and Drogheda are all examples of towns with brass bands of note. Dublin has a number of Brass Bands. The oldest is the Blanchardstown Brass Band (Blanchardstown) formed in 1826, St.Georges Brass Band Formed 1936 (in Dublin City Centre) and the Stedfast Brass Band (Blackrock) Brass competitions take place annually, both regionally and nationally.

Netherlands
In the years 1870–1900 the Salvation Army spread out their missionary activities in the Netherlands, at first the main port city Amsterdam, and then to the then still important northern seaport of Harlingen, Friesland around the 1900.

Brass Band contesting began in Friesland after World War II. The Frisian conductor Sierd de Boer made a great effort to develop the British Brass Band standardisation, and the Frisians nowadays continue to set the standard of brass bands in the Netherlands. Brass Bands in Friesland include "De Wâldsang" Buitenpost, "Pro Rege" Heerenveen, "De Bazuin" Oenkerk and "De Spijkerpakkenband" Opsterland. Other well known brass bands across the country are Brassband Rijnmond (Rotterdam) and Brass Band Schoonhoven.

Norway
The Norwegian Band Federation is the largest voluntary music organisation in Norway. Through their membership, the 1712 member bands with their 70,318 players (official figures as of 9/1/04) are offered good conditions for their hobby. A network rich in tradition also gives them a strong and influential position in the cultural life of Norway. Many primary schools have their own bands.

Crown Prince Haakon is the patron of the Norwegian Band Federation. The Federation is granted an annual audience with the Crown Prince in order to report to him and receive advice for the future. The King has awarded a royal trophy to the championships for adult bands.

Canada
Brass bands in the British tradition, sometimes sponsored by employers, existed in Canada in the late 19th and early 20th centuries. The geography of Canada (e.g. large distances between communities, making regular contests and migration of players difficult) was a key factor among many challenges that led to the demise of most such bands.

Today, excepting the Salvation Army bands, there are few British-style brass bands in Canada. Most operate as recreational, amateur, "community" bands such as the Oshawa Civic Band, King Street Brass from Kitchener, and the Whitby Brass Band, who celebrated their 160th anniversary in 2023. There are some semi-professional groups, like the Intrada Brass of Oakville, Ontario and the Weston Silver Band of Toronto. One of the most successful brass bands in Canada is the Hannaford Street Silver Band.

There are hopes for growth in the education field of brass bands in Canada, and in particular southern Ontario. In 1999, the professional Hannaford Street Silver Band launched the Hannaford Street Youth Band, which is internationally recognized for its success. In 2005, another youth band was created for beginning brass players known as the Hannaford Junior Band. Beginning in September 2006, the Hannaford organization plans to launch a third, intermediate, band known as the Hannaford Community Youth Band. These three ensembles target a wide range of youth from ages 11 to people well into their twenties.

Latin America
Brass bands long enjoyed popularity in many parts of Latin America as well.

Mexican banda music has similar origins to the British brass band tradition and was influenced by it in its early days. In 19th century Mexico very large bands were formed, such as that of composer Juventino Rosas and in parts of Mexico brass band concerts remain a popular entertainment.

United States
Brass bands in the British tradition are becoming more popular through the efforts of the North American Brass Band Association (NABBA), which organizes an annual brass band convention and set piece style contest. The US Open Brass Band Championships are held in early November and serves as the countries leading entertainment competition. Brass Band Festivals such as The Great American Brass Festival, Ohio Brass Arts Festival, Gettysburg Brass Band Festival, Dublin Festival of Brass and the Mid-Atlantic Brass Festival occur throughout the year.

Brass bands were very popular throughout the United States in the late 19th century and the early decades of the 20th century. Well known bands of virtuoso musicians toured widely, and most towns had their own bands that put on weekend music concerts. One notable example of this was The Ohio State University Marching Band which formed in 1879 and still performs with only brass and percussion instruments. Other groups, ranging from benevolent societies to large factories, would often have a band. The brass band movement has undergone a resurgence in the late twentieth century, led by the North American Brass Band Association. The United States boasts a number of professional brass bands, including the Brass Band of Battle Creek, the River City Brass Band, and the Orchid City Brass Band (West Palm Beach, FL); community brass bands such as the Central Ohio Brass Band, Dublin Silver Band, Motor City Brass Band, Atlantic Brass Band, Nebraska Brass Band, Imperial Brass, Brass of the Potomac, Chesapeake Silver Cornet Brass Band (Delaware), Five Lakes Silver Band, Allegheny Brass Band, The Brass Band of Central Florida, the Sheldon Theatre Brass Band (Red Wing, MN), the Lake Wobegon Brass Band (Anoka, MN), the Twin Cities Brass Band (Bloomington, MN), the Fountain City Brass Band, the Triangle Brass Band, the Smoky Mountain Brass Band the River Brass, and the Mission Peak Brass Band (Fremont, CA), and the Rockville Brass Band (the oldest competing American band in Maryland, founded in 1976); several collegiate brass bands, including the James Madison University Brass Band, the Mansfield University Brass Band, the Slippery Rock University Brass Ensemble, the Pennwest-Clarion University Brass Ensemble, and the University of Georgia British Brass Band; and various youth brass bands, including the Triangle Youth Brass Band and the Fountain City Youth Brass Bands also exist.

Repertoire
There are several notable composers in the brass band world. The current favourite is possibly Philip Sparke, who has written many pieces, including Music of the Spheres. A noted Welsh Composer was T. J. (Tom) Powell, born in Tredegar in 1897, also known as "The Welsh Sousa". Powell composed over 500 pieces for brass bands, including marches (such as "The Castle Marches"), tone poems and suites.

Other notable composers include:

Several classical composers have written music specifically for brass band. These include:
 * Malcolm Arnold: Little Suite for Brass
 * Granville Bantock: Prometheus Unbound
 * Judith Bingham: Prague
 * Harrison Birtwistle: Grimethorpe Aria
 * Arthur Bliss: Kenilworth
 * Derek Bourgeois: Blitz, Concertos Nos. 1 & 2 for Brass Band, Concerto Grosso & Apocalypse
 * Arthur Butterworth: Three Impressions for Brass, Odin
 * Simon Dobson: The Drop, Lydian Pictures & Penlee
 * Edward Elgar: The Severn Suite
 * Edward Gregson: Of Men and Mountains, Connotations & Variations on Laudate Dominum
 * Hans Werner Henze: Ragtimes and Habaneras
 * Joseph Holbrooke: Clive of India
 * Gustav Holst: A Moorside Suite
 * Joseph Horovitz: Ballet for Band & Sinfonietta
 * Herbert Howells: Three Figures Suite
 * John Ireland: A Comedy Overture & A Downland Suite
 * George Lloyd: Diversions on a Bass Theme & English Heritage
 * Frederik Magle: The Hope for brass band, choir, organ and percussion
 * William Mathias: Vivat Regina
 * John McCabe: Cloudcatcher Fells, Salamander & The Maunsell Forts
 * John Pickard: Gaia Symphony & Eden
 * Edmund Rubbra: Variations on 'The Shining River'
 * Robert Simpson: Energy & Volcano
 * Phyllis Tate: Illustrations
 * Bramwell Tovey: Coventry Variations, Pictures in the Smoke & The Night To Sing
 * Ralph Vaughan Williams: Overture – Henry the Fifth, Prelude on Three Welsh Hymn Tunes & Variations for Brass Band
 * Philip Wilby: Paganini Variations, A Breathless Alleluia, ...Dove Descending..., Vienna Nights, Revelation, Masquerade, Northern Lights, New Jerusalem & Red Priest
 * Thomas Wilson: Processional, Sinfonietta, Refrains and Cadenzas, Cartoon for cornet and brass band, Cartoon for solo cornet and brass band, Cartoon for brass ensemble and optional percussion, Cartoon for cornet and piano

Outdoor banding

An advantage of the brass band is that it is mobile and capable of playing at any time and anywhere, even on the march. Marching and outdoor functions have been the preserve of the brass band for well over a century. Many UK brass bands are closely woven into the local community, performing outdoor civic duties throughout the year.

The bandstand became popular in the Victorian era, typically associated with the British brass band or military band. It is a simple construction which not only creates an ornamental focal point, but also serves acoustic requirements whilst providing shelter from the changeable British weather.

Band associations

 * Brass Bands England
 * Swiss Brass Band Association
 * Brass Bands' Association of New Zealand
 * European Brass Band Association
 * National Association of Brass Band Conductors
 * National Band Council of Australia
 * North American Brass Band Association
 * Scottish Brass Band Association
 * Southern Counties Amateur Bands Association

Europe

 * Brass Band Willebroek, brass band from Belgium.
 * Brass Band Buizingen, brass band from Belgium.
 * Eikanger-Bjørsvik Musikklag, a Norwegian brass band.
 * Manger Musikklag, a Norwegian brass band (European champion in 2011).

Scotland

 * The Cooperation Band (formerly Co-operative Funeralcare), 34 times Scottish champions and twice British champions

Northern England

 * Reg Vardy Band, 33 times North of England champions and 2011 Grand Shield champions
 * Fairey Band, brass bands and the current 2011 English Champions
 * Black Dyke Band, Reigning National Champions of Great Britain, 2017 British Open Champions and English National Champions in 2014; the most successful contesting band in the world
 * Brighouse and Rastrick Brass Band
 * Carlton Main Frickley Colliery Band, current Yorkshire Area Champions, Based in South Elmsall West Yorkshire
 * Rothwell Temperance Band, 2009 Yorkshire region champions based in Rothwell, West Yorkshire
 * Foden's Band, brass band based in Cheshire
 * Rode Hall Silver Band, third section band based in Cheshire
 * Grimethorpe Colliery Band, one of the most successful and successful brass bands in England, and one of the few to be widely known outside brass banding circles, due to their role in the film Brassed Off.
 * The Harrogate Band, brass band based in North Yorkshire
 * Hammonds Saltaire Band formerly also known as YBS Band, Yorkshire Building Society Band, or Hammonds Sauce Works Band. A brass band based in Huddersfield, West Yorkshire
 * Yorkshire Imperial Band, National Winners 1978, British Open Winners on three occasions

The Midlands

 * The GUS Band, championship section band from Kettering.

Southern England

 * Callender's Cableworks Band (active 1898–1961), prolific BBC Radio broadcasters in 1920s and 1930s
 * City of Oxford Silver Band founded in 1887
 * Friary Brass Band, based in Guildford, London & Southern Counties champions in five consecutive years (2015–2019)
 * Helston Town Band, a renowned Cornish band which plays the famous Floral Dance
 * Lewes, Glynde & Beddingham Brass Band, brass band originally from the village of Glynde in East Sussex, now based in the neighbouring town of Lewes

Wales

 * Tredegar Town Band, a Welsh brass band
 * Cory Band, formed in the Rhondda Valley in 1884, they have been the number one ranked brass band in the world for over 10 years. In 2016, they won all four major contests in an unprecedented "Grand Slam", an accolade they achieved again in 2019.

Ohio

 * Dublin Silver Band

Midwest

 * Eastern Iowa Brass Band, founded in 1985, a traditional, British-styled brass band.
 * Bend in the River Brass Band, a British style brass band in Evansville, Indiana.

Oceania

 * Wellington Brass Band, current national champions of New Zealand and Australia
 * Dalewool Auckland Brass, New Zealand Band of the Year 2011.

title: British brass band


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Trigonometric substitution

In mathematics, a trigonometric substitution replaces a trigonometric function for another expression. In calculus, trigonometric substitutions are a technique for evaluating integrals. In this case, an expression involving a radical function is replaced with a trigonometric one. Trigonometric identities may help simplify the answer. Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.

Case I: Integrands containing a2 − x2
Let $$x = a \sin \theta,$$ and use the identity $$1-\sin^2 \theta = \cos^2 \theta.$$

Example 1
In the integral

$$\int\frac{dx}{\sqrt{a^2-x^2}},$$

we may use

$$x=a\sin \theta,\quad dx=a\cos\theta\, d\theta, \quad \theta=\arcsin\frac{x}{a}.$$

Then, $$\begin{align} \int\frac{dx}{\sqrt{a^2-x^2}} &= \int\frac{a\cos\theta \,d\theta}{\sqrt{a^2-a^2\sin^2 \theta}} \\[6pt] &= \int\frac{a\cos\theta\,d\theta}{\sqrt{a^2(1 - \sin^2 \theta )}} \\[6pt] &= \int\frac{a\cos\theta\,d\theta}{\sqrt{a^2\cos^2\theta}} \\[6pt] &= \int d\theta \\[6pt] &= \theta + C \\[6pt] &= \arcsin\frac{x}{a}+C. \end{align}$$

The above step requires that $$a > 0$$ and $$\cos \theta > 0.$$ We can choose $$a$$ to be the principal root of $$a^2,$$ and impose the restriction $$-\pi /2 < \theta < \pi /2$$ by using the inverse sine function.

For a definite integral, one must figure out how the bounds of integration change. For example, as $$x$$ goes from $$0$$ to $$a/2,$$ then $$\sin \theta$$ goes from $$0$$ to $$1/2,$$ so $$\theta$$ goes from $$0$$ to $$\pi / 6.$$ Then,

$$\int_0^{a/2}\frac{dx}{\sqrt{a^2-x^2}}=\int_0^{\pi/6} d\theta = \frac{\pi}{6}.$$

Some care is needed when picking the bounds. Because integration above requires that $$-\pi /2 < \theta < \pi /2$$, $$\theta$$ can only go from $$0$$ to $$\pi / 6.$$ Neglecting this restriction, one might have picked $$\theta$$ to go from $$\pi$$ to $$5\pi /6,$$ which would have resulted in the negative of the actual value.

Alternatively, fully evaluate the indefinite integrals before applying the boundary conditions. In that case, the antiderivative gives

$$\int_{0}^{a/2} \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin \left( \frac{x}{a} \right) \Biggl|_{0}^{a/2} = \arcsin \left ( \frac{1}{2}\right) - \arcsin (0) = \frac{\pi}{6}$$ as before.

Example 2
The integral

$$\int\sqrt{a^2-x^2}\,dx,$$

may be evaluated by letting $x=a\sin \theta,\, dx=a\cos\theta\, d\theta,\, \theta=\arcsin\dfrac{x}{a},$ where $$a > 0$$ so that $\sqrt{a^2}=a,$  and $-\pi/2 \le \theta \le \pi/2$  by the range of arcsine, so that $$\cos \theta \ge 0$$ and $\sqrt{\cos^2 \theta} = \cos \theta.$

Then, $$\begin{align} \int\sqrt{a^2-x^2}\,dx &= \int\sqrt{a^2-a^2\sin^2\theta}\,(a\cos\theta) \,d\theta \\[6pt] &= \int\sqrt{a^2(1-\sin^2\theta)}\,(a\cos\theta) \,d\theta \\[6pt] &= \int\sqrt{a^2(\cos^2\theta)}\,(a\cos\theta) \,d\theta \\[6pt] &= \int(a\cos\theta)(a\cos\theta) \,d\theta \\[6pt] &= a^2\int\cos^2\theta\,d\theta \\[6pt] &= a^2\int\left(\frac{1+\cos 2\theta}{2}\right)\,d\theta \\[6pt] &= \frac{a^2}{2} \left(\theta+\frac{1}{2}\sin 2\theta \right) + C \\[6pt] &= \frac{a^2}{2}(\theta+\sin\theta\cos\theta) + C \\[6pt] &= \frac{a^2}{2}\left(\arcsin\frac{x}{a}+\frac{x}{a}\sqrt{1-\frac{x^2}{a^2}}\right) + C \\[6pt] &= \frac{a^2}{2}\arcsin\frac{x}{a}+\frac{x}{2}\sqrt{a^2-x^2}+C. \end{align}$$

For a definite integral, the bounds change once the substitution is performed and are determined using the equation $\theta = \arcsin\dfrac{x}{a},$ with values in the range $-\pi/2 \le \theta \le \pi/2.$  Alternatively, apply the boundary terms directly to the formula for the antiderivative.

For example, the definite integral

$$\int_{-1}^1\sqrt{4-x^2}\,dx,$$

may be evaluated by substituting $$x = 2\sin\theta, \,dx = 2\cos\theta\,d\theta,$$ with the bounds determined using $\theta = \arcsin\dfrac{x}{2}.$

Because $$\arcsin(1/{2}) = \pi/6$$ and $$\arcsin(-1/2) = -\pi/6,$$ $$\begin{align} \int_{-1}^1\sqrt{4-x^2}\,dx &= \int_{-\pi/6}^{\pi/6}\sqrt{4-4\sin^2\theta}\,(2\cos\theta) \,d\theta \\[6pt] &= \int_{-\pi/6}^{\pi/6}\sqrt{4(1-\sin^2\theta)}\,(2\cos\theta) \,d\theta \\[6pt] &= \int_{-\pi/6}^{\pi/6}\sqrt{4(\cos^2\theta)}\,(2\cos\theta) \,d\theta \\[6pt] &= \int_{-\pi/6}^{\pi/6}(2\cos\theta)(2\cos\theta) \,d\theta \\[6pt] &= 4\int_{-\pi/6}^{\pi/6}\cos^2\theta\,d\theta \\[6pt] &= 4\int_{-\pi/6}^{\pi/6}\left(\frac{1+\cos 2\theta}{2}\right)\,d\theta \\[6pt] &= 2 \left[\theta+\frac{1}{2} \sin 2\theta \right]^{\pi/6}_{-\pi/6} = [2\theta+\sin 2\theta] \Biggl |^{\pi/6}_{-\pi/6} \\[6pt] &= \left(\frac{\pi}{3}+\sin\frac{\pi}{3}\right)-\left(-\frac{\pi}{3}+\sin\left(-\frac{\pi}{3}\right)\right) = \frac{2\pi}{3}+\sqrt{3}. \end{align}$$

On the other hand, direct application of the boundary terms to the previously obtained formula for the antiderivative yields $$\begin{align} \int_{-1}^1\sqrt{4-x^2}\,dx &= \left[ \frac{2^2}{2}\arcsin\frac{x}{2}+\frac{x}{2}\sqrt{2^2-x^2} \right]_{-1}^{1}\\[6pt] &= \left( 2 \arcsin \frac{1}{2} + \frac{1}{2}\sqrt{4-1}\right) - \left( 2 \arcsin \left(-\frac{1}{2}\right) + \frac{-1}{2}\sqrt{4-1}\right)\\[6pt] &= \left( 2 \cdot \frac{\pi}{6} + \frac{\sqrt{3}}{2}\right) - \left( 2\cdot \left(-\frac{\pi}{6}\right) - \frac{\sqrt 3}{2}\right)\\[6pt] &= \frac{2\pi}{3} + \sqrt{3} \end{align} $$ as before.

Case II: Integrands containing a2 + x2
Let $$x = a \tan \theta,$$ and use the identity $$1+\tan^2 \theta = \sec^2 \theta.$$

Example 1
In the integral

$$\int\frac{dx}{a^2+x^2}$$

we may write

$$x=a\tan\theta,\quad dx=a\sec^2\theta\, d\theta, \quad \theta=\arctan\frac{x}{a},$$

so that the integral becomes

$$\begin{align} \int\frac{dx}{a^2+x^2} &= \int\frac{a\sec^2\theta\, d\theta}{a^2 + a^2\tan^2\theta} \\[6pt] &= \int\frac{a\sec^2\theta\, d\theta}{a^2(1+\tan^2\theta)} \\[6pt] &= \int\frac{a\sec^2\theta\, d\theta}{a^2\sec^2\theta} \\[6pt] &= \int\frac{d\theta}{a} \\[6pt] &= \frac{\theta}{a}+C \\[6pt] &= \frac{1}{a} \arctan \frac{x}{a} + C, \end{align}$$

provided $$a \neq 0.$$

For a definite integral, the bounds change once the substitution is performed and are determined using the equation $$\theta = \arctan\frac{x}{a},$$ with values in the range $$-\frac{\pi}{2} < \theta < \frac{\pi}{2}.$$ Alternatively, apply the boundary terms directly to the formula for the antiderivative.

For example, the definite integral

$$\int_0^1\frac{4\, dx}{1+x^2}\,$$

may be evaluated by substituting $$x = \tan\theta, \,dx = \sec^2\theta\,d\theta,$$ with the bounds determined using $$\theta = \arctan x.$$

Since $$\arctan 0 = 0$$ and $$\arctan 1 = \pi/4,$$ $$\begin{align} \int_0^1\frac{4\,dx}{1+x^2} &= 4\int_0^1\frac{dx}{1 + x^2} \\[6pt] &= 4\int_0^{\pi/4}\frac{\sec^2\theta\, d\theta}{1+\tan^2\theta} \\[6pt] &= 4\int_0^{\pi/4}\frac{\sec^2\theta\, d\theta}{\sec^2\theta} \\[6pt] &= 4\int_0^{\pi/4}d\theta \\[6pt] &= (4\theta)\Bigg|^{\pi/4}_0 = 4 \left (\frac{\pi}{4} - 0 \right) = \pi. \end{align}$$

Meanwhile, direct application of the boundary terms to the formula for the antiderivative yields $$\begin{align} \int_0^1\frac{4\,dx}{1+x^2}\, &= 4\int_0^1\frac{dx}{1+x^2} \\[6pt] &= 4\left[\frac{1}{1} \arctan \frac{x}{1} \right]^1_0 \\[6pt] &= 4(\arctan x)\Bigg|^1_0 \\[6pt] &= 4(\arctan 1 - \arctan 0) \\[6pt] &= 4 \left (\frac{\pi}{4} - 0 \right) = \pi, \end{align}$$ same as before.

Example 2
The integral

$$\int\sqrt{a^2+x^2}\,{dx}$$

may be evaluated by letting $$x=a\tan\theta,\, dx=a\sec^2\theta\, d\theta, \, \theta=\arctan\frac{x}{a},$$

where $$a > 0$$ so that $$\sqrt{a^2}=a,$$ and $$-\frac{\pi}{2}<\theta<\frac{\pi}{2}$$ by the range of arctangent, so that $$\sec \theta > 0$$ and $$\sqrt{\sec^2 \theta} = \sec \theta.$$

Then, $$\begin{align} \int\sqrt{a^2+x^2}\,dx &= \int\sqrt{a^2 + a^2\tan^2\theta}\,(a \sec^2\theta)\, d\theta \\[6pt] &= \int\sqrt{a^2 (1+\tan^2\theta)}\,(a \sec^2\theta)\, d\theta \\[6pt] &= \int\sqrt{a^2 \sec^2\theta}\,(a \sec^2\theta)\, d\theta \\[6pt] &= \int(a \sec\theta)(a \sec^2\theta)\, d\theta \\[6pt] &= a^2\int \sec^3\theta\, d\theta. \\[6pt] \end{align}$$ The integral of secant cubed may be evaluated using integration by parts. As a result, $$\begin{align} \int\sqrt{a^2+x^2}\,dx &= \frac{a^2}{2}(\sec\theta \tan\theta + \ln|\sec\theta+\tan\theta|)+C \\[6pt] &= \frac{a^2}{2}\left(\sqrt{1+\frac{x^2}{a^2}}\cdot\frac{x}{a} + \ln\left|\sqrt{1+\frac{x^2}{a^2}}+\frac{x}{a}\right|\right)+C \\[6pt] &= \frac{1}{2}\left(x\sqrt{a^2+x^2} + a^2\ln\left|\frac{x+\sqrt{a^2+x^2}}{a}\right|\right)+C. \end{align}$$

Case III: Integrands containing x2 − a2
Let $$x = a \sec \theta,$$ and use the identity $$\sec^2 \theta -1 = \tan^2 \theta.$$

Examples of Case III

Integrals such as

$$\int\frac{dx}{x^2 - a^2}$$

can also be evaluated by partial fractions rather than trigonometric substitutions. However, the integral

$$\int\sqrt{x^2 - a^2}\, dx$$

cannot. In this case, an appropriate substitution is: $$x = a \sec\theta,\, dx = a \sec\theta\tan\theta\, d\theta, \, \theta = \arcsec\frac{x}{a},$$

where $$a > 0$$ so that $$\sqrt{a^2}=a,$$ and $$0 \le \theta < \frac{\pi}{2}$$ by assuming $$x > 0,$$ so that $$\tan \theta \ge 0$$ and $$\sqrt{\tan^2 \theta} = \tan \theta.$$

Then, $$\begin{align} \int\sqrt{x^2 - a^2}\, dx &= \int\sqrt{a^2 \sec^2\theta - a^2} \cdot a \sec\theta\tan\theta\, d\theta \\ &= \int\sqrt{a^2 (\sec^2\theta - 1)} \cdot a \sec\theta\tan\theta\, d\theta \\ &= \int\sqrt{a^2 \tan^2\theta} \cdot a \sec\theta\tan\theta\, d\theta \\ &= \int a^2 \sec\theta\tan^2\theta\, d\theta \\ &= a^2 \int (\sec\theta)(\sec^2\theta - 1)\, d\theta \\ &= a^2 \int (\sec^3\theta - \sec\theta)\, d\theta. \end{align}$$

One may evaluate the integral of the secant function by multiplying the numerator and denominator by $$( \sec \theta + \tan \theta)$$ and the integral of secant cubed by parts. As a result, $$\begin{align} \int\sqrt{x^2-a^2}\,dx &= \frac{a^2}{2}(\sec\theta \tan\theta + \ln|\sec\theta+\tan\theta|)-a^2\ln|\sec\theta+\tan\theta|+C \\[6pt] &= \frac{a^2}{2}(\sec\theta \tan\theta - \ln|\sec\theta+\tan\theta|)+C \\[6pt] &= \frac{a^2}{2}\left(\frac{x}{a}\cdot\sqrt{\frac{x^2}{a^2}-1} - \ln\left|\frac{x}{a}+\sqrt{\frac{x^2}{a^2}-1}\right|\right)+C \\[6pt] &= \frac{1}{2}\left(x\sqrt{x^2-a^2} - a^2\ln\left|\frac{x+\sqrt{x^2-a^2}}{a}\right|\right)+C. \end{align}$$

When $$\frac{\pi}{2} < \theta \le \pi,$$ which happens when $$x < 0$$ given the range of arcsecant, $$\tan \theta \le 0,$$ meaning $$\sqrt{\tan^2 \theta} = -\tan \theta$$ instead in that case.

Substitutions that eliminate trigonometric functions
Substitution can be used to remove trigonometric functions.

For instance,

$$\begin{align} \int f(\sin(x), \cos(x))\, dx &=\int\frac1{\pm\sqrt{1-u^2}} f\left(u,\pm\sqrt{1-u^2}\right)\, du && u=\sin (x) \\[6pt] \int f(\sin(x), \cos(x))\, dx &=\int\frac{1}{\mp\sqrt{1-u^2}} f\left(\pm\sqrt{1-u^2},u\right)\, du && u=\cos (x) \\[6pt] \int f(\sin(x), \cos(x))\, dx &=\int\frac2{1+u^2} f \left(\frac{2u}{1+u^2},\frac{1-u^2}{1+u^2}\right)\, du && u=\tan\left (\frac{x}{2} \right ) \\[6pt] \end{align}$$

The last substitution is known as the Weierstrass substitution, which makes use of tangent half-angle formulas.

For example,

$$\begin{align} \int\frac{4 \cos x}{(1+\cos x)^3}\, dx &= \int\frac2{1+u^2}\frac{4\left(\frac{1-u^2}{1+u^2}\right)}{\left(1+\frac{1-u^2}{1+u^2}\right)^3}\, du = \int (1-u^2)(1+u^2)\, du \\&= \int (1-u^4)\,du = u - \frac{u^5}{5} + C = \tan \frac{x}{2} - \frac{1}{5} \tan^5 \frac{x}{2} + C. \end{align}$$

Hyperbolic substitution
Substitutions of hyperbolic functions can also be used to simplify integrals.

For example, to integrate $$1/\sqrt{a^2+x^2}$$, introduce the substitution $$x=a\sinh{u}$$ (and hence $$dx=a\cosh u \,du$$), then use the identity $\cosh^2 (x) - \sinh^2 (x) = 1$ to find:

$$\begin{align} \int \frac{dx}{\sqrt{a^2+x^2}} &= \int \frac{a\cosh u \,du}{\sqrt{a^2+a^2\sinh^2 u}} \\[6pt] &=\int \frac{\cosh{u} \,du}{\sqrt{1+\sinh^2{u}}} \\[6pt] &=\int \frac{\cosh{u}}{\cosh u} \,du \\[6pt] &=u+C \\[6pt] &=\sinh^{-1}{\frac{x}{a}} + C. \end{align}$$

If desired, this result may be further transformed using other identities, such as using the relation $\sinh^{-1}{z} = \operatorname{arsinh}{z} = \ln(z + \sqrt{z^2 + 1})$: $$\begin{align} \sinh^{-1}{\frac{x}{a}} + C &=\ln\left(\frac{x}{a} + \sqrt{\frac{x^2}{a^2} + 1}\,\right) + C \\[6pt] &=\ln\left(\frac{x + \sqrt{x^2+a^2}}{a}\,\right) + C. \end{align}$$

title: Trigonometric substitution


text:
Worshipful Company of Framework Knitters

The Worshipful Company of Framework Knitters is one of the livery companies of the City of London. It was incorporated by letters patent issued by Oliver Cromwell in 1657, with a Royal Charter from Charles II in 1663. It was granted livery status in 1713. For a period it had its own hall in Red Cross Street, however for various reasons it was sold.

The Framework Knitters' Company ranks sixty-fourth in the order of precedence for livery companies. Its motto is Speed, Strength and Truth United.

It maintains almshouses known as the Cottage Homes in Oadby, Leicestershire - and has an active bursary awards scheme for up-and-coming students in fashion and textiles.

Quarterly dinners are held, normally at Mansion House or livery halls within the square mile of the City of London. These are well attended with guests including the Lord Mayor of London and the sheriffs.


text:
Chhatrapati Shivaji Maharaj Vastu Sangrahalaya

Chhatrapati Shivaji Maharaj Vastu Sangrahalaya, (CSMVS) formerly named the Prince of Wales Museum of Western India, is a museum in Mumbai (Bombay) which documents the history of India from prehistoric to modern times.

It was founded during British rule of India in the early years of the 20th century by prominent citizens of the city then called Bombay, with the help of the government, to commemorate the visit of the Prince of Wales (later George V, king of the United Kingdom and emperor of India). It is located in the heart of South Mumbai near the Gateway of India. The museum was renamed in 1998 after Shivaji, the founder of the Maratha Kingdom.

The building is built in the Indo-Saracenic style of architecture, incorporating elements of other styles of architecture like the Mughal, Maratha and Jain. The museum building is surrounded by a garden of palm trees and formal flower beds.

The museum houses approximately 50,000 exhibits of ancient Indian history as well as objects from foreign lands, categorised primarily into three sections: Art, Archaeology and Natural History. The museum houses Indus Valley civilization artefacts, and other relics from ancient India from the time of the Guptas, Mauryas, Chalukyas and Rashtrakuta.

History
In 1904, some leading citizens of Bombay decided to provide a museum to commemorate the visit of the Prince of Wales, the future King George V. On 14 August 1905, the committee passed a resolution saying: "The museum building embodies the pomp and height at which the British raj was moving ahead with their ambitious plans, in building the great metropolis Bombay". "In keeping pace with the best style of local architecture, many buildings were built, among which, Bombay High Court building, and later, Gateway of India buildings were the most notable ones".

The foundation stone was laid by the Prince of Wales on 11 November 1905 and the museum was formally named "Prince of Wales Museum of Western India". On 1 March 1907, the government of the Bombay Presidency granted the museum committee a piece of land called the "Crescent Site", where the museum now stands. Following an open design competition, in 1909 the architect George Wittet was commissioned to design the Museum building. Wittet had already worked on the design of the General Post Office and in 1911 would design one of Mumbai's most famous landmarks, the Gateway of India.

The museum was funded by the Royal Visit (1905) Memorial Funds. Additionally, the Government and the Municipality granted Rs. 300,000 and Rs. 250,000 respectively. Sir Currimbhoy Ibrahim (first Baronet) donated another Rs. 300,000 and Sir Cowasji Jehangir gave Rs. 50,000. The Museum was established under Bombay Act No. III of 1909. The museum is now maintained by annual grants from the Government and the Bombay Municipal Corporation. The latter pays for these grants from the interest accruing on the funds at the disposal of the Trust of the Museum.

The museum building was completed in 1915, but was used as a Children's Welfare Centre and a Military Hospital during the First World War, before being handed over to the committee in 1920. The Prince of Wales Museum was inaugurated on 10 January 1922, by Lady Lloyd, the wife of George Lloyd, Governor of Bombay.

The museum building is a Grade I Heritage Building of the city and was awarded first prize (Urban Heritage Award) by the Bombay Chapter of the Indian Heritage Society for heritage building maintenance in 1990. In 1998 the Museum was renamed Chhatrapati Shivaji Maharaj Vastu Sangrahalaya after the warrior king and founder of the Maratha Kingdom, Shivaji. The museum was renamed after the renaming of the city in 1995, when the colonial name "Bombay" was replaced by the native "Mumbai".

Architecture
The museum building is situated in 3 acre area, having a built up area of 12,142.23m sq. It is surrounded by a garden of palm trees and formal flower beds.

The museum building, built of locally quarried grey Kurla basalt and buff coloured trachyte Malad stone. It is a three-storied rectangular structure, capped by a dome set upon a base, which adds an additional storey in the centre of the building. Built in the Western Indian and Indo-Saracenic style of architecture, the building accommodates a central entrance porch, above which rises a dome, tilled and modified well "tiled in white and blue flecks, supported on a lotus - petal base". A cluster of pinnacles, topped with miniature domes surround the central dome. The building incorporates features like Islamic dome with a finial along with protruding balconies and inlaid floors, inspired by Mughal palace architecture. The architect, George Wittet, modelled the dome on that of Golconda Fort and the inner vaulting arches on those at the Gol Gumbaz in Bijapur. The interior of the museum combines the columns, railings and balcony of an 18th-century Wada (a Maratha mansion) with Jain style interior columns, which form the main body of the central pavilion below the Maratha balcony. In its recent modernisation programme (2008), the museum created 30000 sqft space for installation of five new galleries, a conservation studio, a visiting exhibition gallery and a seminar room, in the East Wing of the Museum. The museum also houses a library.

Collections

The museum collection comprises approximately 50,000 artefacts. The collection of the museum is categorised primarily into three sections: art, archaeology and natural history. The museum also houses a forestry section, which has specimens of timbers grown in the Bombay Presidency (British India), and one exhibiting a small local geological collection of rocks, minerals and fossils. The Maritime Heritage Gallery, which displays objects relating to navigation, is the "first of its kind in India". In 2008, the museum installed two new galleries, displaying the "Karl and Meherbai Khandalavala collection" and "the Coins of India".

Art section
The art section displays the collections of Sir Purushottam Mavji, acquired in 1915, and the art collections of Sir Ratan Tata and Sir Dorab Tata, donated in 1921 and 1933 respectively.

The museum's miniature collection encompasses representations of the main schools of Indian painting namely, Mughal, Rajasthani, Pahari and Deccani. It features palm leaf manuscripts dating to the 11th-12th centuries to the early 19th century pahari paintings, as well as paintings from the Sultanate period. Notable manuscripts housed in the museum include the Anwar-Suhayli painted in Mughal emperor Akbar’s studio and a 17th-century manuscript of the Hindu epic Ramayana from Mewar.

The ivory section has artefacts dating as early as the Gupta era. The museum also has decorative artefacts such as textiles, ivories, Mughal jades, silver, gold and artistic metal ware. It also has a collection of European paintings, Chinese and Japanese porcelain, ivory and jade artefacts. The museum also has sections dedicated to arms and armour and another to Nepali and Tibetan art. The arms and armour section contains a finely decorated armour of Akbar dating to 1581 CE, consisting of a steel breastplate and a shield, the former inscribed with religious verses.

Archaeological section
Sculptures and coins transferred from the Poona Museum in Pune and the collections of the Bombay branch of the Royal Asiatic Society resulted in the development of an archaeological section, with precious sculptures and epigrams. The Indus Valley Culture Gallery houses fishing hooks, weapons, ornaments and weights and measures from the Indus Valley civilization (2600–1900 BCE). Artefacts from the excavation of the Buddhist stupa of Mirpurkhas, were housed in the Museum in 1919. The sculpture collection holds Gupta (280 to 550 CE) terracotta figures from Mirpurkhas in Sind of the early 5th century, artefacts dating to the Chalukyan era (6th-12th century, Badami Chalukyas and Western Chalukyas), and sculptures of the Rashtrakuta period (753 – 982 CE) from Elephanta, near Mumbai.

Natural history section
The Bombay Natural History Society aided the Museum Trust in creating the natural history section. The museum's natural history section makes use of habitat group cases and dioramas, along with diagrams and charts, to illustrate Indian wildlife, including flamingoes, great hornbills, Indian bison, and tigers.

New galleries
An exhibition entitled Pravaha highlighting early phase of Sir J. J. School of Art and the Progressive Art Movement was launched on 24 July 2017. The exhibition covered a range of paintings from the 1880s to 1950s through works of Pestonji Bomanji, Rustom Siodia, Sawlaram Haldankar, António Xavier Trindade, S. N. Gorakshakar, Govind Mahadev Solegaonkar, G. H. Nagarkar, J. M. Ahivasi, Raghunath Dhondopant Dhopeshwarkar, Raghuveer Govind Chimulkar, Rasiklal Parikh and Y. K. Shukla, Abalal Rahiman, Keshav Bhavanrao Chudekar, Lakshman Narayan Taskar, Syed Haider Raza, and K. H. Ara.

A prints gallery was launched with an exhibition entitled Bombay to Mumbai - Door of the East with its face to the West on 29 January 2015. The gallery was inaugurated by Neil MacGregor, Director of the British Museum in London, who also gave an illustrated lecture on 'World Cultures' at the central foyer of the museum.

As part of the renovation project initiated in October 2008, the Krishna Gallery holding artworks related to the Hindu god Krishna, a Hindu deity of the preserver-god Vishnu, was opened in March 2009.

A textile gallery, the first gallery in the city, was opened in April 2010. It illustrates "various techniques of textile manufacturing, regional collections and traditional Indian costumes".

Matrika Design Collaborative is currently designing the museum's Indian miniature painting gallery. The content developed for the gallery will be converted into Braille text and tactile labels for the blind with help from designers, fabricators and consultants from the Helen Keller Institute.

A new gallery on traditional Indian jewellery will be opening in 2020. At the gallery, there will be an exhibit on Golconda diamonds - replicas of which have been presented to the museum.

Galleries
CSMVS Museum has various galleries. They are related to Art, History, Natural History and Indian Culture. They are as follows:
 * Sculpture gallery
 * Pre and Proto History gallery
 * Natural History Section
 * Indian Miniature Painting gallery
 * Krishna gallery
 * Himalayan Art gallery
 * Decorative Metalware gallery
 * House of Laxmi- Coin gallery
 * Karl and Meherbai Khandalavala gallery
 * Chinese and Japanese Art gallery
 * Sir Ratan Tata and Sir Dorab Tata gallery of European Paintings
 * Arms and Armour gallery
 * Jehangir Nicholson gallery
 * Premchand Roychand gallery
 * Key gallery
 * First Floor Circle gallery
 * Second Floor Circle gallery
 * European Decorative Art gallery
 * Bombay School gallery
 * Jahangir Sabawala gallery
 * Textile Gallery
 * Prints gallery
 * Curators gallery and Conservation Centre

Sculpture Gallery
The archaeological collections were originally started by pioneering archaeologists Sir Henry Cousens and Sir John Marshall. Amongst the important sculptures are the Gupta period terracottas and bricks from Mirpurkhas excavated by Cousens, a large number of Buddhist images from Gandhara and ceiling panels from a dilapidated temple at Aihole. The early examples are from Pauni and Pitalkhora. Mumbai itself has a rich tradition exemplified by the colossal Parel Relief of Shiva (represented here by a cast) and a Matrika from Baijanath Temple at Sewri near Parel belonging to the same phase as Elephanta. Other noteworthy images from Maharashtra are a Vishnu and a Ganesha of the eleventh century CE. Some well known sculptures are:
 * Brahma, from Elephanta Caves
 * Mahishasuramardini, from Elephanta
 * Parel Relief of Shiva (plaster cast) from Parel
 * Sculptures from Aihole and Pattadakal
 * Dvarapala, from Shamalji, Gujarat
 * Garuda, from the Konark Sun Temple
 * Yaksha, from Pitalkhora
 * Buddha and Devotee from Mirpur Khas
 * Ashthamahesha Replica bust


text:
Talk:Metallica (album)

To-do list
I'd love to see this article become GA, but I don't know what still needs improved. A lot could've been fixed since the nearly 4-year-old GA review, but there's no list as to how its progress is coming. What needs done? DannyMusicEditor (talk) 23:36, 11 June 2015 (UTC)

Actually, I looked this over, and the article looks like it's in really nice shape. I made a few obvious tweaks, and I've received recommendations to nominate it. It's been recently copyedited, so I trust that did some really well-done work. I'm going to take a chance and see if this qualifies for GA. Here we go! DannyMusicEditor (talk) 18:13, 23 June 2015 (UTC)

There is some conflicting info on which singles where released from the album. In the infobox on the right there are five singles listed. In the bodytext under "Promotion", six singles are listed. The one that's missing in the infobox is "Don't Tread On Me". Which list is the right one? — Preceding unsigned comment added by 81.225.103.245 (talk) 10:58, 15 January 2017 (UTC)

This article is atrocious. Citing an online petition that not even 2,000 fans signed to hate on Bob Rock because they were no longer making pretentious prog and nine minute songs? Very poorly written, it could use a rewrite by someone who has a broader vocabulary and grasp on writing for an audience. Due to the “controversies between bob and the band” made me cringe, controversial is typically used in a sociological or media sense not personal, bette words are antagonisms, tension, and difficulties. Shhsbavavaa (talk) 00:14, 3 November 2022 (UTC)

"Sad but True" audio sample
I don't think a 2-minute sample is minimal use of the work as described in non-free use policy. If the song is 5:24 (324 secs), and this is 2:00 (120 secs), that's about 37% of the whole song, compared to the 5% (30/329) from Enter Sandman and 7% from The Unforgiven (30/386). Either we need a smaller audio sample, or we should remove it altogether and put the caption's information in. And do we really need this many audio samples anyway? While they're formatted correctly, I'm not sure they're all completely necessary. In fact, I've removed the audio sample of Sad but True for now because I'm afraid it may not sit well with the GA nomination. DannyMusicEditor (talk) 20:48, 23 June 2015 (UTC), updated 21:08
 * DannyMusicEditor, the length of File:Metallica - Sad But True.ogg is 30 seconds, not 2 minutes. Some IP vandal changed the duration two years ago.--Retrohead (talk) 20:41, 24 June 2015 (UTC)

Nice catch. But I'm still wondering if we need this many samples... DannyMusicEditor (talk) 20:55, 24 June 2015 (UTC)

Things I (the nominator) have found so far
I notice the accessdates are improper in the certification section. (#98-116, at the time this was posted) I will fix these during the review. So far this is the only obvious thing I see missing, so I won't sweat it at the moment. DannyMusicEditor (talk) 00:11, 27 June 2015 (UTC)

Less harsh?
I would not say that this album is any less harsh than their previous albums. It is slower, and heavier, like groove metal. It also still contains many thrash metal elements. — Preceding unsigned comment added by Iron Wizard13 (talk • contribs) 05:58, 9 July 2015 (UTC)

I think this counts as groove metal.
Many of the songs on here sound like groove metal, mainly Sad But True, but others as well. Personally groove metal should be added to the genres. TheEarthboundFan2001 (talk) 23:55, 16 May 2016 (UTC)

 * Do you have reliable sources to support your opinion ? Please see previous discussions on this topic  Mlpearc Phone  ( open channel ) 00:06, 17 May 2016 (UTC)
 * Is this a good source: https://metalstuff.net/2016/12/14/groove-metal/. Go to the Metallica paragraph with the Sad But True sample. Yungstirjoey666 (talk) 17:34, 27 January 2023 (UTC)

genre changes
genre changes (diff) Consensus is needed for this change. Mlpearc Phone ( open channel ) 18:18, 22 May 2016 (UTC)

...And Justice for All
I disagree with this inclusion and rational and am looking for consensus. If it's the "groups background" then it needs to be at Metallica or ...And Justice for All I've started the discussion for you   Mlpearc  ( open channel ) 19:15, 30 September 2016 (UTC)
 * If you read the GA review, the reviewer suggests adding detail about this album's predecessor and its accompanying tour (which is what I was doing). Furthermore, explaining the album's background is vital for the reader to understand why Metallica opted to write simpler songs, and how that affected the album's sales. I also meant to write that the album was released in a six week span with albums by Nirvana, Red Hot Chili Peppers, Pearl Jam, and Guns N' Roses, but that would be sufficient because we would have been writing about rock's background in the early 90s, right?--Retrohead (talk) 19:22, 30 September 2016 (UTC)
 * I agree the background is important but it should be addressed here not on a article about a completely different album.  Mlpearc  ( open channel ) 19:28, 30 September 2016 (UTC)
 * It is already addressed there. Also, why would FAs such as Love It to Death be talking about "the group's first two albums, Pretties for You (1969) and Easy Action (1970)" in the background when it can be included in Alice Cooper?--Retrohead (talk) 19:31, 30 September 2016 (UTC)
 * Let's talk about this article and not what's going on somewhere else. Obviously we need others to chime in, I wait for further input from others.  Mlpearc  ( open channel ) 19:38, 30 September 2016 (UTC)
 * Fine by me. I'm just saying it's a common practice for albums to feature information about the artist's background. The reader won't bother going around Metallica's discography to figure out how this album is related to its predecessors and followups.--Retrohead (talk) 19:42, 30 September 2016 (UTC)
 * Who mentioned Metallica's discography ? I suggested here and you said it is covered there, or here in the article about the information you want to post.  Mlpearc  ( open channel ) 20:03, 30 September 2016 (UTC)
 * Another thing, how the timing when the album was recorded and mixed, and its commercial performance is not related to this article? Why did you deleted that?--Retrohead (talk) 19:25, 30 September 2016 (UTC)
 * A bit late, but seeing as I was the nominator (the noob I was), I agree with Retrohead. Things covered in a band's bio can be on their albums too.  danny music editor  Speak up! 18:15, 25 November 2016 (UTC)

So you're rejecting a Rolling Stone article?
I've seen it all now. I'm with the warrior on this one. Whether it's a "Did you know?" article where the band doesn't explicitly say that this is the case is irrelevant. Perhaps it shouldn't be exactly how it was requested to be, but it should be reported that Kirk didn't play according to Roling Stone, because they're always reliable. Would highly suggest taking the protection down a notch too.  danny music editor  Speak up! 19:35, 21 January 2018 (UTC)

 * , yes, I will reject a Rolling Stone article if (as is completely relevant) it makes an assumption simply based on an interview where a band member doesn't explicitly say what the magazine says it does, and doesn't say or source anything else – I will happily continue to argue this, as nothing else has been presented and, as far as I'm concerned, it's not reliable in this instance. The words "It was reported" do not belong anywhere in the article (or any article, for that matter, as it violates WP:RUMOUR), regardless of how reliable the magazine might be. I've linked to a discussion that the editor began with me at my talk page. To the editor him/herself,, you have been asked more than once to discuss here, but you have instead continued to revert, resulting in this overzealous (level of) protection; I know it takes two to tango, but I'm also not the only editor to have asked you.  4TheWynne (talk) (contribs) 00:29, 22 January 2018 (UTC)

 * This is what is being used to source the edit in question, as it appears in the Rolling Stone article:

 * '...[James] Hetfield then recorded the song's acoustic intro and bluesy guitar solo himself, making it one of the few Metallica tracks that Kirk Hammett doesn't play on. "I had to relearn that whole intro part to play by myself onstage," Hammett recalled in 2012, "which was a little bit intimidating for me at that point, [because] we never had a song that started that way."'

 * If the magazine knows something that the rest of us don't – which we clearly don't, as nothing else has been sourced to this point and we can't just make assumptions or go by what we may or may not already know – then it should at least say where it's getting its information from, because if the magazine is saying what it's saying only going by the interview alone, then that's not enough. I don't believe that this is the case, but the magazine doesn't say where else it's getting its information from, either (specifically, that Hammett didn't record anything for the song, if this is true), and therefore it can't be considered reliable just because of how respected it might be or how reliable it might have been in the past. Hammett barely says anything more on the subject during the full interview being sourced to go into any further depth, so it's not like the magazine left anything out.  4TheWynne (talk) (contribs) 00:58, 22 January 2018 (UTC)

 * I have no problem accepting the article. RS is not an academic journal citing all its information all the time; that's just not how they work. That we don't (yet?) have another source saying this is beside the point: RS is an RS. Drmies (talk) 03:33, 26 January 2018 (UTC)

Track listing
Why is the track listing in vinyl format and not CD? I'm just curious. CelestialWeevil (talk) 00:24, 18 March 2018 (UTC)

 * I honestly don't know – it's only a recent change. I've gone and changed it back.  4TheWynne (talk) (contribs) 00:37, 18 March 2018 (UTC)

 * Nice. It looks better this way. CelestialWeevil (talk) 00:40, 18 March 2018 (UTC)

Source for some personnel?
On the album's liner notes (at least on the 1991, Elektra Records, which is the cited one) and on Metallica's official webpage, George Marino, Peter Mensch, Don Brautigam, Ross Halfin, Rick Likong and Rob Ellis aren't mentioned. If they are included because of some third-party sources, shouldn't these be cited, as per WP:PERSONNEL? Jocafus (talk) 02:07, 25 August 2021 (UTC)

Since nobody responded, I'm going to delete the unsourced material from the personnel section. Jocafus (talk) 00:23, 26 September 2021 (UTC)

Old name
Wasn’t the album originally called “the black album”? Why isn't that here? (unless I'm wrong.) 2600:1002:B009:651E:F517:70AB:B888:560E (talk) 14:34, 1 December 2022 (UTC)

“Slower, heavier, and more refined” is fanboying. We need more objectivity and perception not fawning....
A better sentence would be something to the effect of wanting to change direction from the progressive influenced thrash of justice to shorter and simpler songs. I don’t think they got “heavier” or more “refined”, those are subjective valuations, what is a fact is they were shorter, more traditional pop structure, and simpler, as well as more melodic singing and a change of lyric subject matter to be more personalized instead of the proselytizing of “master of puppets” and “justice for all” blah blah blah. 2600:1012:B127:6F06:F522:92CA:41D4:E43B (talk) 11:58, 10 June 2023 (UTC)

Not Heavy Metal
Metallica's black album was the first in there 90s transition to hard Rock. The whole album is far too clean in both instrumentals and vocals to be considered heavy metal in any kind. a key difference the genres is how distorted the guitars and vocals get. Metallica black does show a couple hints of nearly reaching heavy metal such as the god that failed but as a whole the album is most certainly not Heavy Metal in any way and much more closely fits in with the Hard Rock genre with James Hetfield's singing being far too melodic to ever be considered anything heavier.(talk) 21:13, 9 January 2024 (UTC)

title: Talk:Metallica (album)