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Episcopus vagans

In Christianity, an episcopus vagans (plural episcopi vagantes; Latin for 'wandering bishops' or 'stray bishops') is a person consecrated, in a "clandestine or irregular way", as a bishop outside the structures and canon law of the established churches; a person regularly consecrated but later excommunicated, and not in communion with any generally recognized diocese; or a person who has in communion with them small groups that appear to exist solely for the bishop's sake.

David V. Barrett, in the Encyclopedia of New Religious Movements, specifies that now episcopi vagantes are "those independent bishops who collect several different lines of transmission of apostolic succession, and who will happily (and sometimes for a fee) consecrate anyone who requests it". Those described as wandering bishops often see the term as pejorative. The general term for "wandering" clerics, as were common in the Middle Ages, is clerici vagantes; the general term for those recognising no leader is acephali.

The Oxford Dictionary of the Christian Church mentions as the main lines of succession deriving from episcopi vagantes in the 20th century those founded by Arnold Mathew, Joseph René Vilatte and Leon Chechemian.

History
According to Buchanan, "the real rise of the problem" happened in the 19th century, in the "wake of the Anglo-Catholic movement", "through mischievous activities of a tiny number of independently acting bishops". They exist worldwide, he writes, "mostly without congregations", and "many in different stages of delusion and fantasy, not least in the Episcopal titles they confer on themselves"; "the distinguishing mark" to "specifically identif[y]" an episcopus vagans is "the lack of a true see or the lack of a real church life to oversee".

Paul Halsall, on the Internet History Sourcebooks Project, did not list a single church edifice of independent bishops, in a 1996–1998 New York City building architecture survey of religious communities, which maintain bishops claiming apostolic succession and claim cathedral status but noted there "are now literally hundreds of these 'episcopi vagantes', of lesser or greater spiritual probity. They seem to have a tendency to call living room sanctuaries 'cathedrals';" those buildings were not perceived as cultural symbols and did not meet the survey criteria. David V. Barrett wrote, in A Brief Guide to Secret Religions, that "one hallmark of such bishops is that they often collect as many lineages as they can to strengthen their Episcopal legitimacy—at least in their own eyes", and adds that their groups have more clergy than members.

Barrett wrote that leaders "of some esoteric movements, are also priests or bishops in small non-mainstream Christian Churches"; he explains, this type of "independent or autocephalous" group has "little in common with the Church it developed from, the Old Catholic Church, and even less in common with the Roman Catholic Church" but still claims its authority from apostolic succession.

Buchanan writes that based the criteria of having "a true see" or having "a real church life to oversee", the bishops of most forms of the Continuing Anglican movement are not necessarily classified as vagantes, but "are always in danger of becoming such".

Catholic
A Roman or Eastern Catholic ordained to the episcopacy without a mandate from the Pope is automatically excommunicated and is thereby forbidden to celebrate the sacraments, according to Catholic canon law. Through the concept of "valid but illicit" ordinations however, and the dogma of sacramental character, though excommunicated and forbidden to celebrate sacraments within any church in communion with the Holy See, the person still holds a valid episcopacy though unrecognized at large.

According to a theological view affirmed, for instance, by the International Bishops' Conference of the Old Catholic Church with regard to ordinations by Arnold Mathew, an episcopal ordination is for service within a specific Christian church, and an ordination ceremony that concerns only the individual himself does not make him truly a bishop. The Holy See has not commented on the validity of this theory, but has declared with regard to ordinations of this kind carried out, for example, by Emmanuel Milingo toward Peter Paul Brennan and others, that the Roman Church "does not recognize and does not intend to recognize in the future those ordinations or any of the ordinations derived from them and therefore the canonical state of the alleged bishops remains that in which they were before the ordination conferred by Mr Milingo".

Eastern Orthodox
Vlassios Pheidas, on an official Church of Greece site, uses the canonical language of the Eastern Orthodox tradition, to describe the conditions in ecclesial praxis when sacraments, including Holy Orders, are real, valid, and efficacious. He notes language is itself part of the ecclesiological problem. "If […] divine grace is granted to all, […] then it stands to reason that it is bestowed also in those believers outside the [Eastern] Orthodox Church, even if such persons belong to a heresy of schism. Thus, the sacraments performed outside the Church are not only real (υποστατά), but also valid (έγκυρα), because they only lack the efficacy (ενέργεια) of the bestowed divine grace, which is operative through the Holy Spirit only within the [Eastern] Orthodox Church.

[...]

Through such a teaching […] one finds himself face to face with the […] principle of 'extra Ecclesia nulla salus', which strictly determines the canonical limits of the Church. Thus, the [Eastern] Orthodox Church, while accepting the canonical possibility of recognising the existence (υποστατόν) of sacraments performed outside herself, it questions their validity (έγκυρον) and certainly rejects their efficacy (ενεργόν). It is already well-known that in the ecclesial praxis, the Orthodox Church moves, according to the specific circumstances, between canonical 'acribeia' and ecclesial economy, recognising by economy the validity (κύρος) of the sacraments of those ecclesiastical bodies. Yet, such a practice of economy does not overthrow the canonical 'acribeia', which also remains in force and expresses the exclusive character of orthodox ecclesiology.

This observation is really important, because it reveals that the canonical recognition (αναγνώρισις) of the validity of sacraments performed outside the [Eastern] Orthodox Church: (a) is done by economy, (b) covers only specific cases in certain given instances, and (c) refers to the validity of the sacraments only of those who join the  [Eastern] Orthodox Church, and not of the ecclesiastical bodies to which belong those who join the  [Eastern] Orthodox Church. There is, […] a variety of opinions or reservations concerning this question. No one, […] could propose or support the view that the mutual recognition of the validity of sacraments among the Churches is an ecclesiastical act consistent with orthodox ecclesiology, or an act which is not rejected by the orthodox canonical tradition. […]

[…]

[…] the mutual recognition of the validity of certain sacraments, […] is for an [Eastern] Orthodox an act of inconsistency, when it is assessed with orthodox ecclesiological principles. These ecclesiological principles manifest in a strict fashion the organic unity of the orthodox ecclesial body and differentiate those who do not belong to its body as either schismatics or heretics.

The relation of schismatics or heretics to the body of the [Eastern] Orthodox Church is strictly defined by the canonical tradition. However, orthodox canonical tradition and praxis appraises and classifies these ecclesiastical bodies into various categories, […] in which some form of ecclesiality is recognised. This type of ecclesiality is not easily determined, because the orthodox tradition […] does not recognise the efficacy of the divine grace outside the canonical boundaries of the [Eastern] Orthodox Church."

This applies to the validity and efficacy of the ordination of bishops and the other sacraments, not only of the Independent Catholic churches, but also of all other Christian churches, including the Roman Catholic Church, Oriental Orthodoxy and the Assyrian Church of the East. However, although strict adherence to law supersedes economy by their Cyprian understanding, some mainstream Eastern Orthodox bodies recognize Roman Catholic orders and don't conditionally ordain clergy as each autocephalous church determines the validity of another's ordination. The Ecumenical Patriarchate of Constantinople likewise teaches that through "extreme oikonomia [economy]", those who are baptized in the Oriental Orthodox, Roman Catholic, Lutheran, Old Catholic, Moravian, Anglican, Methodist, Reformed, Presbyterian, Church of the Brethren, Assemblies of God, or Baptist traditions can be received into the Eastern Orthodox Church through the sacrament of Chrismation and not through re-baptism.

Anglican
Anglican bishop Colin Buchanan, in the Historical Dictionary of Anglicanism, says that the Anglican Communion has held an Augustinian view of orders, by which "the validity of Episcopal ordinations (to whichever order) is based solely upon the historic succession in which the ordaining bishop stands, irrespective of their contemporary ecclesial context".

He describes the circumstances of Archbishop Matthew Parker's consecration as one of the reasons why this theory is "generally held". Parker was chosen by Queen Elizabeth I of England to be the first Church of England Archbishop of Canterbury after the death of the previous office holder, Cardinal Reginald Pole, the last Roman Catholic Archbishop of Canterbury. Buchanan notes the Roman Catholic Church also focuses on issues of intention and not just breaks in historical succession. He does not explain whether intention has an ecclesiological role, for Anglicans, in conferring or receiving sacraments.

Particular consecrations
Arnold Mathew, according to Buchanan, "lapsed into the vagaries of an episcopus vagans". Stephen Edmonds, in the Oxford Dictionary of National Biography, wrote that in 1910 Mathew's wife separated from him; that same year, he declared himself and his church seceded from the Union of Utrecht. Within a few months, on 2 November 1911, he was excommunicated by the Roman Catholic Church. He later sued The Times for libel based on the words "pseudo-bishop" used to describe him in the newspaper's translation from the Latin text "pseudo-episcopus", and, lost his case in 1913.

Henry R.T. Brandreth wrote, in Episcopi Vagantes and the Anglican Church, "[o]ne of the most regrettable features of Mathew's episcopate was the founding of the Order of Corporate Reunion (OCR) in 1908. This claimed to be a revival of Frederick George Lee's movement, but was in fact unconnected with it". Brandreth thought it "seems still to exist in a shadowy underground way" in 1947, but disconnected. Colin Holden, in Ritualist on a Tricycle, places Mathew and his OCR into perspective, he wrote Mathew was an episcopus vagans, lived in a cottage provided for him, and performed his conditional OCR acts, sometimes called according to Holden "bedroom ordinations", in his cottage. Mathew questioned the validity of Anglican ordinations and became involved with the OCR, in 1911 according to Edmonds, and he openly advertised his offer to reordain Anglican clergy who requested it. This angered the Church of England.

In 1912, D. J. Scannell O'Neill wrote in The Fortnightly Review that London "seems to have more than her due share of bishops" and enumerates what he refers to as "these hireling shepherds". He also announces that one of them, Mathew, revived the OCR and published The Torch, a monthly review, advocating the reconstruction of Western Christianity and reunion with Eastern Christianity. The Torch stated "that the ordinations of the Church of England are not recognized by any church claiming to be Catholic" so the promoters involved Mathew to conditionally ordain group members who are "clergy of the Established Church" and "sign a profession of the Catholic Faith". It stipulated Mathew's services were not a system of simony and given without simoniac expectations. The group sought to enroll "earnest-minded Catholics who sincerely desire to help forward the work of [c]orporate [r]eunion with the Holy See". Nigel Yates, in Anglican Ritualism in Victorian Britain, 1830-1910, described it as "an even more bizarre scheme to promote a Catholic Uniate Church in Britain" than Lee and Ambrose Lisle March Phillipps de Lisle's Association for the Promotion of the Unity of Christendom. It was editorialized by O'Neill that the "most charitable construction to be placed on this latest move of Mathew is that he is not mentally sound. Being an Irishman, it is strange that he has not sufficient humor to see the absurdity of falling away from the Catholic Church in order to assist others to unite with the Holy See". Edmonds reports that "anything between 4 and 265 was suggested" as to how many took up his offer of reordination.

title: Episcopus vagans


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Elizabeth Garrett Anderson

Elizabeth Garrett Anderson (9 June 1836 – 17 December 1917) was an English physician and suffragist. She is known for being the first woman to qualify in Britain as a physician and surgeon and as a co-founder and dean of the London School of Medicine for Women, which was the first medical school in Britain to train women as doctors. She was the first female dean of a British medical school, the first woman in Britain to be elected to a school board and, as mayor of Aldeburgh, the first female mayor in Britain.

Early life

Elizabeth was born in Whitechapel, London, and was the second of eleven children of Newson Garrett (1812–1893), from Leiston, Suffolk, and his wife, Louisa (born Dunnell; c. 1813–1903), from London.

Her paternal ancestors had been ironworkers in East Suffolk since the early seventeenth century. Newson was the youngest of three sons and not academically inclined, although he possessed the family's entrepreneurial spirit. When he finished school, Newson found few opportunities in Leiston, so he moved to London to make his fortune. There, he fell in love with his brother's sister-in-law, Louisa Dunnell, the daughter of an innkeeper of Suffolk origin. After their wedding, the couple went to live in a pawnbroker's shop at 1 Commercial Road, Whitechapel. The Garretts had their first three children in quick succession: Louie, Elizabeth, and a son, Dunnell, who died at the age of six months. When Garrett was three years old, the family moved to 142 Long Acre, where they lived for two years, while one more child was born and her father advanced in his career, becoming not only the manager of a larger pawnbroker's shop, but also a silversmith. Garrett's grandfather, owner of the family engineering works, Richard Garrett & Sons, had died in 1837, leaving the business to his eldest son, Garrett's uncle. Despite his lack of capital, Newson was determined to be successful and in 1841, at the age of 29, he moved his family to Suffolk, where he bought a barley and coal merchants business in Snape, constructing Snape Maltings from 1846.

The Garretts lived in a square Georgian house opposite the church in Aldeburgh until 1852. Newson's malting business expanded and more children were born, Edmund (1840), Alice (1842), Agnes (1845), Millicent (1847), who was to become a leader in the constitutional campaign for women's suffrage, Sam (1850), Josephine (1853) and George (1854). By 1850, Newson was a prosperous businessman and was able to build Alde House, a mansion on a hill behind Aldeburgh. A "by-product of the industrial revolution", Garrett grew up in an atmosphere of "triumphant economic pioneering" and the Garrett children were to grow up to become achievers in the professional classes of late-Victorian England. Elizabeth was encouraged to take an interest in local politics and, contrary to practices at the time, was allowed the freedom to explore the town with its nearby salt-marshes, beach and the small port of Slaughden with its boatbuilders' yards and sailmakers' lofts.

Early education
There was no school in Aldeburgh, so Garrett learned reading, writing, and arithmetic from her mother. When she was 10 years old, a governess, Miss Edgeworth, a poor gentlewoman, was employed to educate Garrett and her sister. Mornings were spent in the schoolroom; there were regimented afternoon walks; educating the young ladies continued at mealtimes when Edgeworth ate with the family; at night, the governess slept in a curtained off area in the girls' bedroom. Garrett reportedly despised her governess and sought to outwit the teacher in the classroom. When Garrett was 13 and her sister 15, they were sent to a private school, the Boarding School for Ladies in Blackheath, London, which was run by the step aunts of the poet Robert Browning. There, English literature, French, Italian and German as well as deportment, were taught.

Later in life, Garrett recalled the stupidity of her teachers there, though her schooling there did help establish a love of reading. Her main complaint about the school was the lack of science and mathematics instruction. Her reading there included works of Tennyson, Wordsworth, Milton, Coleridge, Trollope, Thackeray and George Eliot. Elizabeth and Louie were known as "the bathing Garretts", as their father had insisted they be allowed a hot bath once a week. However, they made what were to be lifelong friends there. When they finished in 1851, they were sent on a short tour abroad, ending with a memorable visit to the Great Exhibition in Hyde Park, London.

After this formal education, Garrett spent the next nine years tending to domestic duties, but she continued to study Latin and arithmetic in the mornings and also read widely. Her sister Millicent recalled Garrett's weekly lectures, "Talks on Things in General", when her younger siblings would gather while she discussed politics and current affairs from Garibaldi to Macaulay's History of England. In 1854, when she was eighteen, Garrett and her sister went on a long visit to their school friends, Jane and Anne Crow, in Gateshead where she met Emily Davies, the early feminist and future co-founder of Girton College, Cambridge. Davies was to be a lifelong friend and confidante, always ready to give sound advice during the important decisions of Garrett's career. It may have been in the English Woman's Journal, first issued in 1858, that Garrett first read of Elizabeth Blackwell, who had become the first female doctor in the United States in 1849. When Blackwell visited London in 1859, Garrett travelled to the capital. By then, her sister Louie was married and living in London. Garrett joined the Society for Promoting the Employment of Women, which organised Blackwell's lectures on "Medicine as a Profession for Ladies" and set up a private meeting between Garrett and the doctor. It is said that during a visit to Alde House around 1860, one evening while sitting by the fireside, Garrett and Davies selected careers for advancing the frontiers of women's rights; Garrett was to open the medical profession to women, Davies the doors to a university education for women, while 13-year-old Millicent was allocated politics and votes for women. At first Newson was opposed to the radical idea of his daughter becoming a physician but came round and agreed to do all in his power, both financially and otherwise, to support Garrett.

Medical education
After an initial unsuccessful visit to leading doctors in Harley Street, Garrett decided to first spend six months as a surgery nurse at Middlesex Hospital, London in August 1860. On proving to be a good nurse, she was allowed to attend an outpatients' clinic, then her first operation. She unsuccessfully attempted to enroll in the hospital's Medical School but was allowed to attend private tuition in Latin, Greek and pharmacology with the hospital's apothecary, while continuing her work as a nurse. She also employed a tutor to study anatomy and physiology three evenings a week. Eventually she was allowed into the dissecting room and the chemistry lectures. Gradually, Garrett became an unwelcome presence among the male students, who in 1861 presented a memorial to the school against her admittance as a fellow student, despite the support she enjoyed from the administration. She was obliged to leave the Middlesex Hospital but she did so with an honours certificate in chemistry and materia medica. Garrett then applied to several medical schools, including Oxford, Cambridge, Glasgow, Edinburgh, St Andrews and the Royal College of Surgeons, all of which refused her admittance.

A companion to Garrett in this effort was the lesser known Sophia Jex-Blake. While both are considered "outstanding" medical figures of the late 19th century, Garrett was able to obtain her credentials by way of a "side door" through a loophole in admissions at the Worshipful Society of Apothecaries. Having privately obtained a certificate in anatomy and physiology, she was admitted in 1862 by the Society of Apothecaries who, as a condition of their charter, could not legally exclude her on account of her sex. She was the only woman in the Apothecaries Hall who sat the exam that year. Among the 51 male candidates was William Heath Strange, who went on to found the Hampstead General Hospital, which was on the site now occupied by the Royal Free Hospital. She continued her battle to qualify by studying privately with various professors, including some at the University of St Andrews, the Edinburgh Royal Maternity and the London Hospital Medical School.

In 1865, Garrett finally took her exam and obtained a licence (LSA) from the Society of Apothecaries to practise medicine, the first woman qualified in Britain to do so openly (previously there was Dr James Barry who was born and raised female but presented as male from the age of 20). On the day, three out of seven candidates passed the exam, Garrett with the highest marks. The Society of Apothecaries immediately amended its regulations to prevent other women obtaining a licence meaning that Jex-Blake could not follow this same path; the new rule disallowed privately educated women to be eligible for examination. It was not until 1876 that the new Medical Act (39 and 40 Vict, Ch. 41) passed, which allowed British medical authorities to license all qualified applicants whatever their gender.

Career
Though she was now a licentiate of the Society of Apothecaries, as a woman, Garrett could not hold a medical post in any hospital. So in late 1865, Garrett opened her own practice at 20 Upper Berkeley Street, London. At first patients were scarce, but the practice gradually grew. After six months in practice, she wished to open an outpatients dispensary, to enable poor women to obtain medical help from a qualified practitioner of their own gender. In 1865, there was an outbreak of cholera in Britain, affecting both rich and poor, and in their panic, some people forgot any prejudices they had in relation to a female physician. The first death due to cholera occurred in 1866, but by then Garrett had already opened St Mary's Dispensary for Women and Children, at 69 Seymour Place. In the first year, she tended to 3,000 new patients, who made 9,300 outpatient visits to the dispensary. On hearing that the Dean of the faculty of medicine at the University of Sorbonne, Paris was in favour of admitting women as medical students, Garrett studied French so that she could apply for a medical degree, which she obtained in 1870 after some difficulty.

The same year she was elected to the first London School Board, an office newly opened to women; Garrett's was the highest vote among all the candidates. Also in that year, she was made a visiting physician of the East London Hospital for Children (later the Queen Elizabeth Hospital for Children), becoming the first woman in Britain to be appointed to a medical post, but she found the duties of these two positions to be incompatible with her principal work in her private practice and the dispensary, as well as her role as a new mother, so she resigned from these posts by 1873. In 1872, the dispensary became the New Hospital for Women and Children, treating women from all over London for gynaecological conditions; the hospital moved to new premises in Marylebone Street in 1874. Around this time, Garrett also entered into discussion with male medical views regarding women. In 1874, Henry Maudsley's article on Sex and Mind in Education appeared, which argued that education for women caused over-exertion and thus reduced their reproductive capacity, sometimes causing "nervous and even mental disorders". Garrett's counter-argument was that the real danger for women was not education but boredom and that fresh air and exercise were preferable to sitting by the fire with a novel. In the same year, she co-founded the London School of Medicine for Women with Sophia Jex-Blake and became a lecturer in what was then the only teaching hospital in Britain to offer courses for women. She continued to work there for the rest of her career and was dean of the school from 1883 to 1902. This school was later called the Royal Free Hospital School of Medicine, and became part of the medical school of University College London.

BMA membership
In 1873, Garrett gained membership of the British Medical Association (BMA). In 1878, a motion was proposed to exclude women following the election of Garrett Anderson and Frances Hoggan. The motion was opposed by Dr Norman Kerr who maintained the equal rights of members. This was "one of several instances where Garrett, uniquely, was able to enter a hitherto all male medical institution which subsequently moved formally to exclude any women who might seek to follow her." In 1892, women were again admitted to the British Medical Association. In 1897, Garrett Anderson was elected president of the East Anglian branch of the BMA.

Garrett Anderson worked steadily at the development of the New Hospital for Women and Children and in 1874 co-founded and served as dean of the London School of Medicine for Women (LSMW). Both institutions were handsomely and suitably housed and equipped. The New Hospital for Women commissioned a building in the Euston Road; the architect was J. M. Brydon, who took into his employment Anderson's sister Agnes Garrett and her cousin Rhoda Garrett, who contributed to its design. For many years, the hospital was staffed entirely by medical women. The schools (in Hunter Street, WC1) had over 200 students, most of them preparing for the medical degree of London University, which was opened to women in 1877.

Women’s suffrage movement

Garrett Anderson was also active in the women's suffrage movement. In 1866, Garrett Anderson and Davies presented petitions with more than 1,500 signatures asking that female heads of household be given the right to vote. That year, Garrett Anderson joined the first British Women's Suffrage Committee. She was not as active as her sister, Millicent Garrett Fawcett, though Garrett Anderson became a member of the Central Committee of the National Society for Women's Suffrage in 1889. After her husband's death in 1907, she became more active. As mayor of Aldeburgh, she gave speeches for suffrage, before the increasing militant activity in the movement led to her withdrawal in 1911. Her daughter Louisa, also a physician, was more active and more militant, spending time in prison in 1912 for her suffrage activities.

Personal life
Elizabeth Garrett Anderson once remarked that "a doctor leads two lives, the professional and the private, and the boundaries between the two are never traversed". In 1871, she married James George Skelton Anderson (died 1907) of the Orient Steamship Company, but she did not give up her medical practice. She had three children, Louisa (1873–1943), Margaret (1874–1875), who died of meningitis, and Alan (1877–1952). Louisa also became a pioneering doctor of medicine and feminist activist.

They retired to Aldeburgh in 1902, moving to Alde House in 1903, after the death of Elizabeth's mother. Skelton died of a stroke in 1907. She enjoyed a happy marriage and in later life, devoted time to Alde House, gardening, and travelling with younger members of the extended family.

On 9 November 1908, she was elected mayor of Aldeburgh, the first female mayor in England. Her father had been mayor in 1889.

She died in 1917 and is buried in the churchyard of St Peter and St Paul's Church, Aldeburgh.

Legacy
The New Hospital for Women was renamed the Elizabeth Garrett Anderson Hospital in 1918 and amalgamated with the Obstetric Hospital in 2001 to form the Elizabeth Garrett Anderson and Obstetric Hospital before relocating to become the University College Hospital Elizabeth Garrett Anderson Wing at UCH.

The former Elizabeth Garrett Anderson Hospital buildings are incorporated into the new National Headquarters for the public service trade union UNISON. The Elizabeth Garrett Anderson Gallery, a permanent installation set within the restored hospital building, uses a variety of media to set the story of Garrett Anderson, her hospital, and women's struggle to achieve equality in the field of medicine within the wider framework of 19th and 20th century social history.

The critical care centre at Ipswich Hospital was named the Garrett Anderson Centre in her honour and in recognition of her connection to the county of Suffolk.

The new medical school at the University of Worcester, due to accept its first students in 2023, is to be called the Elizabeth Garrett Anderson Building.

Elizabeth Garrett Anderson School, a secondary school for girls in Islington, London, is named after her.

The archives of Elizabeth Garrett Anderson are held at the Women's Library at the London School of Economics. The archives of the Elizabeth Garrett Anderson Hospital (formerly the New Hospital for Women) are held at the London Metropolitan Archives.

On 9 June 2016, Google Doodle commemorated her 180th birthday.

The Elizabeth Garrett Anderson Programme of the NHS Leadership Academy is a master's degree in leadership and management.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Elizabeth Garrett Anderson


text:
Erosion

Erosion is the action of surface processes (such as water flow or wind) that removes soil, rock, or dissolved material from one location on the Earth's crust and then transports it to another location where it is deposited. Erosion is distinct from weathering which involves no movement. Removal of rock or soil as clastic sediment is referred to as physical or mechanical erosion; this contrasts with chemical erosion, where soil or rock material is removed from an area by dissolution. Eroded sediment or solutes may be transported just a few millimetres, or for thousands of kilometres.

Agents of erosion include rainfall; bedrock wear in rivers; coastal erosion by the sea and waves; glacial plucking, abrasion, and scour; areal flooding; wind abrasion; groundwater processes; and mass movement processes in steep landscapes like landslides and debris flows. The rates at which such processes act control how fast a surface is eroded. Typically, physical erosion proceeds the fastest on steeply sloping surfaces, and rates may also be sensitive to some climatically controlled properties including amounts of water supplied (e.g., by rain), storminess, wind speed, wave fetch, or atmospheric temperature (especially for some ice-related processes). Feedbacks are also possible between rates of erosion and the amount of eroded material that is already carried by, for example, a river or glacier. The transport of eroded materials from their original location is followed by deposition, which is arrival and emplacement of material at a new location.

While erosion is a natural process, human activities have increased by 10–40 times the rate at which soil erosion is occurring globally. At agriculture sites in the Appalachian Mountains, intensive farming practices have caused erosion at up to 100 times the natural rate of erosion in the region. Excessive (or accelerated) erosion causes both "on-site" and "off-site" problems. On-site impacts include decreases in agricultural productivity and (on natural landscapes) ecological collapse, both because of loss of the nutrient-rich upper soil layers. In some cases, this leads to desertification. Off-site effects include sedimentation of waterways and eutrophication of water bodies, as well as sediment-related damage to roads and houses. Water and wind erosion are the two primary causes of land degradation; combined, they are responsible for about 84% of the global extent of degraded land, making excessive erosion one of the most significant environmental problems worldwide.

Intensive agriculture, deforestation, roads, anthropogenic climate change and urban sprawl are amongst the most significant human activities in regard to their effect on stimulating erosion. However, there are many prevention and remediation practices that can curtail or limit erosion of vulnerable soils.

Rainfall and surface runoff
Rainfall, and the surface runoff which may result from rainfall, produces four main types of soil erosion: splash erosion, sheet erosion, rill erosion, and gully erosion. Splash erosion is generally seen as the first and least severe stage in the soil erosion process, which is followed by sheet erosion, then rill erosion and finally gully erosion (the most severe of the four).

In splash erosion, the impact of a falling raindrop creates a small crater in the soil, ejecting soil particles. The distance these soil particles travel can be as much as 0.6 m vertically and 1.5 m horizontally on level ground.

If the soil is saturated, or if the rainfall rate is greater than the rate at which water can infiltrate into the soil, surface runoff occurs. If the runoff has sufficient flow energy, it will transport loosened soil particles (sediment) down the slope. Sheet erosion is the transport of loosened soil particles by overland flow. Rill erosion refers to the development of small, ephemeral concentrated flow paths which function as both sediment source and sediment delivery systems for erosion on hillslopes. Generally, where water erosion rates on disturbed upland areas are greatest, rills are active. Flow depths in rills are typically of the order of a few centimetres (about an inch) or less and along-channel slopes may be quite steep. This means that rills exhibit hydraulic physics very different from water flowing through the deeper, wider channels of streams and rivers.

Gully erosion occurs when runoff water accumulates and rapidly flows in narrow channels during or immediately after heavy rains or melting snow, removing soil to a considerable depth. A gully is distinguished from a rill based on a critical cross-sectional area of at least one square foot, i.e. the size of a channel that can no longer be erased via normal tillage operations.

Extreme gully erosion can progress to formation of badlands. These form under conditions of high relief on easily eroded bedrock in climates favorable to erosion. Conditions or disturbances that limit the growth of protective vegetation (rhexistasy) are a key element of badland formation.

Rivers and streams
Valley or stream erosion occurs with continued water flow along a linear feature. The erosion is both downward, deepening the valley, and headward, extending the valley into the hillside, creating head cuts and steep banks. In the earliest stage of stream erosion, the erosive activity is dominantly vertical, the valleys have a typical V-shaped cross-section and the stream gradient is relatively steep. When some base level is reached, the erosive activity switches to lateral erosion, which widens the valley floor and creates a narrow floodplain. The stream gradient becomes nearly flat, and lateral deposition of sediments becomes important as the stream meanders across the valley floor. In all stages of stream erosion, by far the most erosion occurs during times of flood when more and faster-moving water is available to carry a larger sediment load. In such processes, it is not the water alone that erodes: suspended abrasive particles, pebbles, and boulders can also act erosively as they traverse a surface, in a process known as traction.

Bank erosion is the wearing away of the banks of a stream or river. This is distinguished from changes on the bed of the watercourse, which is referred to as scour. Erosion and changes in the form of river banks may be measured by inserting metal rods into the bank and marking the position of the bank surface along the rods at different times.

Thermal erosion is the result of melting and weakening permafrost due to moving water. It can occur both along rivers and at the coast. Rapid river channel migration observed in the Lena River of Siberia is due to thermal erosion, as these portions of the banks are composed of permafrost-cemented non-cohesive materials. Much of this erosion occurs as the weakened banks fail in large slumps. Thermal erosion also affects the Arctic coast, where wave action and near-shore temperatures combine to undercut permafrost bluffs along the shoreline and cause them to fail. Annual erosion rates along a 100 km segment of the Beaufort Sea shoreline averaged 5.6 m per year from 1955 to 2002.

Most river erosion happens nearer to the mouth of a river. On a river bend, the longest least sharp side has slower moving water. Here deposits build up. On the narrowest sharpest side of the bend, there is faster moving water so this side tends to erode away mostly.

Rapid erosion by a large river can remove enough sediments to produce a river anticline, as isostatic rebound raises rock beds unburdened by erosion of overlying beds.

Coastal erosion
Shoreline erosion, which occurs on both exposed and sheltered coasts, primarily occurs through the action of currents and waves but sea level (tidal) change can also play a role.

Hydraulic action takes place when the air in a joint is suddenly compressed by a wave closing the entrance of the joint. This then cracks it. Wave pounding is when the sheer energy of the wave hitting the cliff or rock breaks pieces off. Abrasion or corrasion is caused by waves launching sea load at the cliff. It is the most effective and rapid form of shoreline erosion (not to be confused with corrosion). Corrosion is the dissolving of rock by carbonic acid in sea water. Limestone cliffs are particularly vulnerable to this kind of erosion. Attrition is where particles/sea load carried by the waves are worn down as they hit each other and the cliffs. This then makes the material easier to wash away. The material ends up as shingle and sand. Another significant source of erosion, particularly on carbonate coastlines, is boring, scraping and grinding of organisms, a process termed bioerosion.

Sediment is transported along the coast in the direction of the prevailing current (longshore drift). When the upcurrent supply of sediment is less than the amount being carried away, erosion occurs. When the upcurrent amount of sediment is greater, sand or gravel banks will tend to form as a result of deposition. These banks may slowly migrate along the coast in the direction of the longshore drift, alternately protecting and exposing parts of the coastline. Where there is a bend in the coastline, quite often a buildup of eroded material occurs forming a long narrow bank (a spit). Armoured beaches and submerged offshore sandbanks may also protect parts of a coastline from erosion. Over the years, as the shoals gradually shift, the erosion may be redirected to attack different parts of the shore.

Erosion of a coastal surface, followed by a fall in sea level, can produce a distinctive landform called a raised beach.

Chemical erosion
Chemical erosion is the loss of matter in a landscape in the form of solutes. Chemical erosion is usually calculated from the solutes found in streams. Anders Rapp pioneered the study of chemical erosion in his work about Kärkevagge published in 1960.

Formation of sinkholes and other features of karst topography is an example of extreme chemical erosion.

Glaciers
Glaciers erode predominantly by three different processes: abrasion/scouring, plucking, and ice thrusting. In an abrasion process, debris in the basal ice scrapes along the bed, polishing and gouging the underlying rocks, similar to sandpaper on wood. Scientists have shown that, in addition to the role of temperature played in valley-deepening, other glaciological processes, such as erosion also control cross-valley variations. In a homogeneous bedrock erosion pattern, curved channel cross-section beneath the ice is created. Though the glacier continues to incise vertically, the shape of the channel beneath the ice eventually remain constant, reaching a U-shaped parabolic steady-state shape as we now see in glaciated valleys. Scientists also provide a numerical estimate of the time required for the ultimate formation of a steady-shaped U-shaped valley—approximately 100,000 years. In a weak bedrock (containing material more erodible than the surrounding rocks) erosion pattern, on the contrary, the amount of over deepening is limited because ice velocities and erosion rates are reduced.

Glaciers can also cause pieces of bedrock to crack off in the process of plucking. In ice thrusting, the glacier freezes to its bed, then as it surges forward, it moves large sheets of frozen sediment at the base along with the glacier. This method produced some of the many thousands of lake basins that dot the edge of the Canadian Shield. Differences in the height of mountain ranges are not only being the result tectonic forces, such as rock uplift, but also local climate variations. Scientists use global analysis of topography to show that glacial erosion controls the maximum height of mountains, as the relief between mountain peaks and the snow line are generally confined to altitudes less than 1500 m. The erosion caused by glaciers worldwide erodes mountains so effectively that the term glacial buzzsaw has become widely used, which describes the limiting effect of glaciers on the height of mountain ranges. As mountains grow higher, they generally allow for more glacial activity (especially in the accumulation zone above the glacial equilibrium line altitude), which causes increased rates of erosion of the mountain, decreasing mass faster than isostatic rebound can add to the mountain. This provides a good example of a negative feedback loop. Ongoing research is showing that while glaciers tend to decrease mountain size, in some areas, glaciers can actually reduce the rate of erosion, acting as a glacial armor. Ice can not only erode mountains but also protect them from erosion. Depending on glacier regime, even steep alpine lands can be preserved through time with the help of ice. Scientists have proved this theory by sampling eight summits of northwestern Svalbard using Be10 and Al26, showing that northwestern Svalbard transformed from a glacier-erosion state under relatively mild glacial maxima temperature, to a glacier-armor state occupied by cold-based, protective ice during much colder glacial maxima temperatures as the Quaternary ice age progressed.

These processes, combined with erosion and transport by the water network beneath the glacier, leave behind glacial landforms such as moraines, drumlins, ground moraine (till), glaciokarst, kames, kame deltas, moulins, and glacial erratics in their wake, typically at the terminus or during glacier retreat.

The best-developed glacial valley morphology appears to be restricted to landscapes with low rock uplift rates (less than or equal to 2mm per year) and high relief, leading to long-turnover times. Where rock uplift rates exceed 2mm per year, glacial valley morphology has generally been significantly modified in postglacial time. Interplay of glacial erosion and tectonic forcing governs the morphologic impact of glaciations on active orogens, by both influencing their height, and by altering the patterns of erosion during subsequent glacial periods via a link between rock uplift and valley cross-sectional shape.

Floods
At extremely high flows, kolks, or vortices are formed by large volumes of rapidly rushing water. Kolks cause extreme local erosion, plucking bedrock and creating pothole-type geographical features called rock-cut basins. Examples can be seen in the flood regions result from glacial Lake Missoula, which created the channeled scablands in the Columbia Basin region of eastern Washington.

Wind erosion

Wind erosion is a major geomorphological force, especially in arid and semi-arid regions. It is also a major source of land degradation, evaporation, desertification, harmful airborne dust, and crop damage—especially after being increased far above natural rates by human activities such as deforestation, urbanization, and agriculture.

Wind erosion is of two primary varieties: deflation, where the wind picks up and carries away loose particles; and abrasion, where surfaces are worn down as they are struck by airborne particles carried by wind. Deflation is divided into three categories: (1) surface creep, where larger, heavier particles slide or roll along the ground; (2) saltation, where particles are lifted a short height into the air, and bounce and saltate across the surface of the soil; and (3) suspension, where very small and light particles are lifted into the air by the wind, and are often carried for long distances. Saltation is responsible for the majority (50–70%) of wind erosion, followed by suspension (30–40%), and then surface creep (5–25%).

Wind erosion is much more severe in arid areas and during times of drought. For example, in the Great Plains, it is estimated that soil loss due to wind erosion can be as much as 6100 times greater in drought years than in wet years.

Mass wasting

Mass wasting or mass movement is the downward and outward movement of rock and sediments on a sloped surface, mainly due to the force of gravity.

Mass wasting is an important part of the erosional process and is often the first stage in the breakdown and transport of weathered materials in mountainous areas. It moves material from higher elevations to lower elevations where other eroding agents such as streams and glaciers can then pick up the material and move it to even lower elevations. Mass-wasting processes are always occurring continuously on all slopes; some mass-wasting processes act very slowly; others occur very suddenly, often with disastrous results. Any perceptible down-slope movement of rock or sediment is often referred to in general terms as a landslide. However, landslides can be classified in a much more detailed way that reflects the mechanisms responsible for the movement and the velocity at which the movement occurs. One of the visible topographical manifestations of a very slow form of such activity is a scree slope.

Slumping happens on steep hillsides, occurring along distinct fracture zones, often within materials like clay that, once released, may move quite rapidly downhill. They will often show a spoon-shaped isostatic depression, in which the material has begun to slide downhill. In some cases, the slump is caused by water beneath the slope weakening it. In many cases it is simply the result of poor engineering along highways where it is a regular occurrence.

Surface creep is the slow movement of soil and rock debris by gravity which is usually not perceptible except through extended observation. However, the term can also describe the rolling of dislodged soil particles 0.5 to 1.0 mm in diameter by wind along the soil surface.

Submarine sediment gravity flows
On the continental slope, erosion of the ocean floor to create channels and submarine canyons can result from the rapid downslope flow of sediment gravity flows, bodies of sediment-laden water that move rapidly downslope as turbidity currents. Where erosion by turbidity currents creates oversteepened slopes it can also trigger underwater landslides and debris flows. Turbidity currents can erode channels and canyons into substrates ranging from recently deposited unconsolidated sediments to hard crystalline bedrock. Almost all continental slopes and deep ocean basins display such channels and canyons resulting from sediment gravity flows and submarine canyons act as conduits for the transfer of sediment from the continents and shallow marine environments to the deep sea. Turbidites, which are the sedimentary deposits resulting from turbidity currents, comprise some of the thickest and largest sedimentary sequences on Earth, indicating that the associated erosional processes must also have played a prominent role in Earth's history.

Climate
The amount and intensity of precipitation is the main climatic factor governing soil erosion by water. The relationship is particularly strong if heavy rainfall occurs at times when, or in locations where, the soil's surface is not well protected by vegetation. This might be during periods when agricultural activities leave the soil bare, or in semi-arid regions where vegetation is naturally sparse. Wind erosion requires strong winds, particularly during times of drought when vegetation is sparse and soil is dry (and so is more erodible). Other climatic factors such as average temperature and temperature range may also affect erosion, via their effects on vegetation and soil properties. In general, given similar vegetation and ecosystems, areas with more precipitation (especially high-intensity rainfall), more wind, or more storms are expected to have more erosion.

In some areas of the world (e.g. the mid-western US), rainfall intensity is the primary determinant of erosivity (for a definition of erosivity check, ) with higher intensity rainfall generally resulting in more soil erosion by water. The size and velocity of rain drops is also an important factor. Larger and higher-velocity rain drops have greater kinetic energy, and thus their impact will displace soil particles by larger distances than smaller, slower-moving rain drops.

In other regions of the world (e.g. western Europe), runoff and erosion result from relatively low intensities of stratiform rainfall falling onto the previously saturated soil. In such situations, rainfall amount rather than intensity is the main factor determining the severity of soil erosion by water. According to the climate change projections, erosivity will increase significantly in Europe and soil erosion may increase by 13–22.5% by 2050

In Taiwan, where typhoon frequency increased significantly in the 21st century, a strong link has been drawn between the increase in storm frequency with an increase in sediment load in rivers and reservoirs, highlighting the impacts climate change can have on erosion.

Vegetative cover
Vegetation acts as an interface between the atmosphere and the soil. It increases the permeability of the soil to rainwater, thus decreasing runoff. It shelters the soil from winds, which results in decreased wind erosion, as well as advantageous changes in microclimate. The roots of the plants bind the soil together, and interweave with other roots, forming a more solid mass that is less susceptible to both water and wind erosion. The removal of vegetation increases the rate of surface erosion.

Topography
The topography of the land determines the velocity at which surface runoff will flow, which in turn determines the erosivity of the runoff. Longer, steeper slopes (especially those without adequate vegetative cover) are more susceptible to very high rates of erosion during heavy rains than shorter, less steep slopes. Steeper terrain is also more prone to mudslides, landslides, and other forms of gravitational erosion processes.

Tectonics
Tectonic processes control rates and distributions of erosion at the Earth's surface. If the tectonic action causes part of the Earth's surface (e.g., a mountain range) to be raised or lowered relative to surrounding areas, this must necessarily change the gradient of the land surface. Because erosion rates are almost always sensitive to the local slope (see above), this will change the rates of erosion in the uplifted area. Active tectonics also brings fresh, unweathered rock towards the surface, where it is exposed to the action of erosion.

However, erosion can also affect tectonic processes. The removal by erosion of large amounts of rock from a particular region, and its deposition elsewhere, can result in a lightening of the load on the lower crust and mantle. Because tectonic processes are driven by gradients in the stress field developed in the crust, this unloading can in turn cause tectonic or isostatic uplift in the region. In some cases, it has been hypothesised that these twin feedbacks can act to localize and enhance zones of very rapid exhumation of deep crustal rocks beneath places on the Earth's surface with extremely high erosion rates, for example, beneath the extremely steep terrain of Nanga Parbat in the western Himalayas. Such a place has been called a "tectonic aneurysm".

Development
Human land development, in forms including agricultural and urban development, is considered a significant factor in erosion and sediment transport, which aggravate food insecurity. In Taiwan, increases in sediment load in the northern, central, and southern regions of the island can be tracked with the timeline of development for each region throughout the 20th century. The intentional removal of soil and rock by humans is a form of erosion that has been named lisasion.

Mountain ranges
Mountain ranges are known to take many millions of years to erode to the degree they effectively cease to exist. Scholars Pitman and Golovchenko estimate that it takes probably more than 450 million years to erode a mountain mass similar to the Himalaya into an almost-flat peneplain if there are no major sea-level changes. Erosion of mountains massifs can create a pattern of equally high summits called summit accordance. It has been argued that extension during post-orogenic collapse is a more effective mechanism of lowering the height of orogenic mountains than erosion.

Examples of heavily eroded mountain ranges include the Timanides of Northern Russia. Erosion of this orogen has produced sediments that are now found in the East European Platform, including the Cambrian Sablya Formation near Lake Ladoga. Studies of these sediments indicate that it is likely that the erosion of the orogen began in the Cambrian and then intensified in the Ordovician.

Soils
If the rate of erosion is higher than the rate of soil formation the soils are being destroyed by erosion. Where soil is not destroyed by erosion, erosion can in some cases prevent the formation of soil features that form slowly. Inceptisols are common soils that form in areas of fast erosion.

While erosion of soils is a natural process, human activities have increased by 10-40 times the rate at which erosion is occurring globally. Excessive (or accelerated) erosion causes both "on-site" and "off-site" problems. On-site impacts include decreases in agricultural productivity and (on natural landscapes) ecological collapse, both because of loss of the nutrient-rich upper soil layers. In some cases, the eventual result is desertification. Off-site effects include sedimentation of waterways and eutrophication of water bodies, as well as sediment-related damage to roads and houses. Water and wind erosion are the two primary causes of land degradation; combined, they are responsible for about 84% of the global extent of degraded land, making excessive erosion one of the most significant environmental problems.

In the United States, farmers cultivating highly erodible land must comply with a conservation plan to be eligible for certain forms of agricultural assistance.

title: Erosion


text:
Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his Elements, with the great innovation of proving all properties of the space as theorems, by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate).

After the introduction at the end of the 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space.

There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore it is usually possible to work with a specific Euclidean space, denoted $$\mathbf{E}^n$$ or $$\mathbb{E}^n$$, which can be represented using Cartesian coordinates as the real $n$-space $$\R^n$$ equipped with the standard dot product.

History of the definition
Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called postulates, or axioms in modern language. This way of defining Euclidean space is still in use under the name of synthetic geometry.

In 1637, René Descartes introduced Cartesian coordinates, and showed that these allow reducing geometric problems to algebraic computations with numbers. This reduction of geometry to algebra was a major change in point of view, as, until then, the real numbers were defined in terms of lengths and distances.

Euclidean geometry was not applied in spaces of dimension more than three until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of dimension $n$, using both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any dimension.

Despite the wide use of Descartes' approach, which was called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract vector spaces allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces.

Motivation of the modern definition
One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as motions) on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually considered as subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections (see below).

In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions: the distance in a "mathematical" space is a number, not something expressed in inches or metres.

The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a real vector space acts — the space of translations which is equipped with an inner product. The action of translations makes the space an affine space, and this allows defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles.

The set $$\R^n$$ of $n$-tuples of real numbers equipped with the dot product is a Euclidean space of dimension $n$. Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism between a Euclidean space of dimension $n$ and $$\R^n$$ viewed as a Euclidean space.

It follows that everything that can be said about a Euclidean space can also be said about $$\R^n.$$ Therefore, many authors, especially at elementary level, call $$\R^n$$ the standard Euclidean space of dimension $n$, or simply the Euclidean space of dimension $n$.

A reason for introducing such an abstract definition of Euclidean spaces, and for working with it instead of $$\R^n$$ is that it is often preferable to work in a coordinate-free and origin-free manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no origin nor any basis in the physical world.

Technical definition
A Euclidean vector space is a finite-dimensional inner product space over the real numbers.

A Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called Euclidean affine spaces to distinguish them from Euclidean vector spaces.

If $E$ is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted $$\overrightarrow E.$$ The dimension of a Euclidean space is the dimension of its associated vector space.

The elements of $E$ are called points, and are commonly denoted by capital letters. The elements of $$\overrightarrow E$$ are called Euclidean vectors or free vectors. They are also called translations, although, properly speaking, a translation is the geometric transformation resulting from the action of a Euclidean vector on the Euclidean space.

The action of a translation $v$ on a point $P$ provides a point that is denoted $P + v$. This action satisfies

$$P+(v+w)= (P+v)+w.$$

Note: The second $+$ in the left-hand side is a vector addition; each other $+$ denotes an action of a vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of $+$, it suffices to look at the nature of its left argument.

The fact that the action is free and transitive means that, for every pair of points $(P, Q)$, there is exactly one displacement vector $v$ such that $P + v = Q$. This vector $v$ is denoted $Q − P$ or $$\overrightarrow {PQ}\vphantom{\frac){}}.$$

As previously explained, some of the basic properties of Euclidean spaces result from the structure of affine space. They are described in and its subsections. The properties resulting from the inner product are explained in and its subsections.

Prototypical examples
For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a Euclidean space that has itself as the associated vector space.

A typical case of Euclidean vector space is $$\R^n$$ viewed as a vector space equipped with the dot product as an inner product. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is isomorphic to it. More precisely, given a Euclidean space $E$ of dimension $n$, the choice of a point, called an origin and an orthonormal basis of $$\overrightarrow E$$ defines an isomorphism of Euclidean spaces from $E$ to $$\R^n.$$

As every Euclidean space of dimension $n$ is isomorphic to it, the Euclidean space $$\R^n$$ is sometimes called the standard Euclidean space of dimension $n$.

Affine structure
Some basic properties of Euclidean spaces depend only on the fact that a Euclidean space is an affine space. They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections.

Subspaces
Let $E$ be a Euclidean space and $$\overrightarrow E$$ its associated vector space.

A flat, Euclidean subspace or affine subspace of $E$ is a subset $F$ of $E$ such that

$$\overrightarrow F = \Bigl\{\overrightarrow {PQ} \mathrel{\Big|} P\in F, Q\in F \Bigr\}\vphantom{\frac({}}$$

as the associated vector space of $F$ is a linear subspace (vector subspace) of $$\overrightarrow E.$$ A Euclidean subspace $F$ is a Euclidean space with $$\overrightarrow F$$ as the associated vector space. This linear subspace $$\overrightarrow F$$ is also called the direction of $F$.

If $P$ is a point of $F$ then

$$F = \Bigl\{P+v \mathrel{\Big|} v\in \overrightarrow F \Bigr\}.$$

Conversely, if $P$ is a point of $E$ and $$\overrightarrow V$$ is a linear subspace of $$\overrightarrow E,$$ then

$$P + \overrightarrow V = \Bigl\{P + v \mathrel{\Big|} v\in \overrightarrow V \Bigr\}$$

is a Euclidean subspace of direction $$\overrightarrow V$$. (The associated vector space of this subspace is $$\overrightarrow V$$.)

A Euclidean vector space $$\overrightarrow E$$ (that is, a Euclidean space that is equal to $$\overrightarrow E$$) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.

Lines and segments
In a Euclidean space, a line is a Euclidean subspace of dimension one. Since a vector space of dimension one is spanned by any nonzero vector, a line is a set of the form

$$\Bigl\{P + \lambda \overrightarrow{PQ} \mathrel{\Big|} \lambda \in \R \Bigr\},\vphantom{\frac({}}$$

where $P$ and $Q$ are two distinct points of the Euclidean space as a part of the line.

It follows that there is exactly one line that passes through (contains) two distinct points. This implies that two distinct lines intersect in at most one point.

A more symmetric representation of the line passing through $P$ and $Q$ is

$$\Bigl\{O + (1-\lambda)\overrightarrow{OP} + \lambda \overrightarrow{OQ} \mathrel{\Big|} \lambda \in \R \Bigr\},\vphantom{\frac({}}$$

where $O$ is an arbitrary point (not necessary on the line).

In a Euclidean vector space, the zero vector is usually chosen for $O$; this allows simplifying the preceding formula into

$$\bigl\{(1-\lambda) P + \lambda Q \mathrel{\big|} \lambda \in \R\bigr\}.$$

A standard convention allows using this formula in every Euclidean space, see.

The line segment, or simply segment, joining the points $P$ and $Q$ is the subset of points such that $0 ≤ 𝜆 ≤ 1$ in the preceding formulas. It is denoted $PQ$ or $QP$; that is

$$PQ = QP = \Bigl\{P+\lambda \overrightarrow{PQ} \mathrel{\Big|} 0 \le \lambda \le 1\Bigr\}.\vphantom{\frac({}}$$

Parallelism
Two subspaces $S$ and $T$ of the same dimension in a Euclidean space are parallel if they have the same direction (i.e., the same associated vector space). Equivalently, they are parallel, if there is a translation vector $v$ that maps one to the other:

$$T= S+v.$$

Given a point $P$ and a subspace $S$, there exists exactly one subspace that contains $P$ and is parallel to $S$, which is $$P + \overrightarrow S.$$ In the case where $S$ is a line (subspace of dimension one), this property is Playfair's axiom.

It follows that in a Euclidean plane, two lines either meet in one point or are parallel.

The concept of parallel subspaces has been extended to subspaces of different dimensions: two subspaces are parallel if the direction of one of them is contained in the direction to the other.

Metric structure
The vector space $$\overrightarrow E$$ associated to a Euclidean space $E$ is an inner product space. This implies a symmetric bilinear form

$$\begin{align} \overrightarrow E \times \overrightarrow E &\to \R\\ (x,y)&\mapsto \langle x,y \rangle \end{align}$$

that is positive definite (that is $$\langle x,x \rangle$$ is always positive for $x ≠ 0$).

The inner product of a Euclidean space is often called dot product and denoted $x ⋅ y$. This is specially the case when a Cartesian coordinate system has been chosen, as, in this case, the inner product of two vectors is the dot product of their coordinate vectors. For this reason, and for historical reasons, the dot notation is more commonly used than the bracket notation for the inner product of Euclidean spaces. This article will follow this usage; that is $$\langle x,y \rangle$$ will be denoted $x ⋅ y$ in the remainder of this article.

The Euclidean norm of a vector $x$ is

$$\|x\| = \sqrt {x \cdot x}.$$

The inner product and the norm allows expressing and proving metric and topological properties of Euclidean geometry. The next subsection describe the most fundamental ones. In these subsections, $E$ denotes an arbitrary Euclidean space, and $$\overrightarrow E$$ denotes its vector space of translations.

Distance and length
The distance (more precisely the Euclidean distance) between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is

$$d(P,Q) = \Bigl\|\overrightarrow {PQ}\Bigr\|.\vphantom{\frac({}}$$

The length of a segment $PQ$ is the distance $d(P, Q)$ between its endpoints P and Q. It is often denoted $$|PQ|$$.

The distance is a metric, as it is positive definite, symmetric, and satisfies the triangle inequality

$$d(P,Q)\le d(P,R) + d(R, Q).$$

Moreover, the equality is true if and only if a point $R$ belongs to the segment $PQ$. This inequality means that the length of any edge of a triangle is smaller than the sum of the lengths of the other edges. This is the origin of the term triangle inequality.

With the Euclidean distance, every Euclidean space is a complete metric space.

Orthogonality
Two nonzero vectors $u$ and $v$ of $$\overrightarrow E$$ (the associated vector space of a Euclidean space $E$) are perpendicular or orthogonal if their inner product is zero:

$$ u \cdot v =0$$

Two linear subspaces of $$\overrightarrow E$$ are orthogonal if every nonzero vector of the first one is perpendicular to every nonzero vector of the second one. This implies that the intersection of the linear subspaces is reduced to the zero vector.

Two lines, and more generally two Euclidean subspaces (A line can be considered as one Euclidean subspace.) are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect are said perpendicular.

Two segments $AB$ and $AC$ that share a common endpoint $A$ are perpendicular or form a right angle if the vectors $$\overrightarrow {AB}\vphantom{\frac){}}$$ and $$\overrightarrow {AC}\vphantom{\frac){}}$$ are orthogonal.

If $AB$ and $AC$ form a right angle, one has

$$|BC|^2 = |AB|^2 + |AC|^2.$$

This is the Pythagorean theorem. Its proof is easy in this context, as, expressing this in terms of the inner product, one has, using bilinearity and symmetry of the inner product:

$$\begin{align} &=\Bigl(\overrightarrow {BA}+\overrightarrow {AC}\Bigr) \cdot \Bigl(\overrightarrow {BA}+\overrightarrow {AC}\Bigr)\\[4mu] &=\overrightarrow {BA}\cdot \overrightarrow {BA}+ \overrightarrow {AC}\cdot \overrightarrow {AC} -2 \overrightarrow {AB}\cdot \overrightarrow {AC}\\[6mu] &=\overrightarrow {AB}\cdot \overrightarrow {AB} + \overrightarrow {AC}\cdot\overrightarrow {AC}\\[6mu] &=|AB|^2 + |AC|^2. \end{align}$$
 * BC|^2 &= \overrightarrow {BC}\cdot \overrightarrow {BC} \vphantom{\frac({}}\\[2mu]

Here, $$\overrightarrow {AB}\cdot \overrightarrow {AC} = 0 \vphantom{\frac({}}$$ is used since these two vectors are orthogonal.

Angle
The (non-oriented) angle $θ$ between two nonzero vectors $x$ and $y$ in $$\overrightarrow E$$ is

$$\theta = \arccos\left(\frac{x\cdot y}{|x|\,|y|}\right)$$

where $arccos$ is the principal value of the arccosine function. By Cauchy–Schwarz inequality, the argument of the arccosine is in the interval $[−1, 1]$. Therefore $θ$ is real, and $0 ≤ θ ≤ π$ (or $0 ≤ θ ≤ 180$ if angles are measured in degrees).

Angles are not useful in a Euclidean line, as they can be only 0 or $\pi$.

In an oriented Euclidean plane, one can define the oriented angle of two vectors. The oriented angle of two vectors $x$ and $y$ is then the opposite of the oriented angle of $y$ and $x$. In this case, the angle of two vectors can have any value modulo an integer multiple of $2π$. In particular, a reflex angle $π < θ < 2π$ equals the negative angle $−π < θ − 2π < 0$.

The angle of two vectors does not change if they are multiplied by positive numbers. More precisely, if $x$ and $y$ are two vectors, and $λ$ and $μ$ are real numbers, then

$$\operatorname{angle}(\lambda x, \mu y)= \begin{cases} \operatorname{angle}(x, y) \qquad\qquad \text{if } \lambda \text{ and } \mu \text{ have the same sign}\\ \pi - \operatorname{angle}(x, y)\qquad \text{otherwise}. \end{cases}$$

If $A$, $B$, and $C$ are three points in a Euclidean space, the angle of the segments $AB$ and $AC$ is the angle of the vectors $$\overrightarrow {AB}\vphantom{\frac({}}$$ and $$\overrightarrow {AC}.\vphantom{\frac({}}$$ As the multiplication of vectors by positive numbers do not change the angle, the angle of two half-lines with initial point $A$ can be defined: it is the angle of the segments $AB$ and $AC$, where $B$ and $C$ are arbitrary points, one on each half-line. Although this is less used, one can define similarly the angle of segments or half-lines that do not share an initial point.

The angle of two lines is defined as follows. If $θ$ is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is either $θ$ or $π − θ$. One of these angles is in the interval $[0, π/2]$, and the other being in $[π/2, π]$. The non-oriented angle of the two lines is the one in the interval $[0, π/2]$. In an oriented Euclidean plane, the oriented angle of two lines belongs to the interval $[−π/2, π/2]$.

Cartesian coordinates
Every Euclidean vector space has an orthonormal basis (in fact, infinitely many in dimension higher than one, and two in dimension one), that is a basis $$(e_1, \dots, e_n) $$ of unit vectors ($$\|e_i\| = 1$$) that are pairwise orthogonal ($$e_i\cdot e_j = 0$$ for $i ≠ j$). More precisely, given any basis $$(b_1, \dots, b_n),$$ the Gram–Schmidt process computes an orthonormal basis such that, for every $i$, the linear spans of $$(e_1, \dots, e_i)$$ and $$(b_1, \dots, b_i)$$ are equal.

Given a Euclidean space $E$, a Cartesian frame is a set of data consisting of an orthonormal basis of $$\overrightarrow E,$$ and a point of $E$, called the origin and often denoted $O$. A Cartesian frame $$(O, e_1, \dots, e_n)$$ allows defining Cartesian coordinates for both $E$ and $$\overrightarrow E$$ in the following way.

The Cartesian coordinates of a vector $v$ of $$\overrightarrow E$$ are the coefficients of $v$ on the orthonormal basis $$e_1, \dots, e_n.$$ For example, the Cartesian coordinates of a vector $$v$$ on an orthonormal basis $$(e_1,e_2,e_3)$$ (that may be named as $$(x,y,z)$$ as a convention) in a 3-dimensional Euclidean space is $$(\alpha_1,\alpha_2,\alpha_3)$$ if $$v = \alpha_1 e_1 + \alpha_2 e_2 + \alpha_3 e_3$$. As the basis is orthonormal, the $i$-th coefficient $$\alpha_i$$ is equal to the dot product $$v\cdot e_i.$$

The Cartesian coordinates of a point $P$ of $E$ are the Cartesian coordinates of the vector $$\overrightarrow {OP}.\vphantom{\frac({}}$$

Other coordinates

As a Euclidean space is an affine space, one can consider an affine frame on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define affine coordinates, sometimes called skew coordinates for emphasizing that the basis vectors are not pairwise orthogonal.

An affine basis of a Euclidean space of dimension $n$ is a set of $n + 1$ points that are not contained in a hyperplane. An affine basis define barycentric coordinates for every point.

Many other coordinates systems can be defined on a Euclidean space $E$ of dimension $n$, in the following way. Let $f$ be a homeomorphism (or, more often, a diffeomorphism) from a dense open subset of $E$ to an open subset of $$\R^n.$$ The coordinates of a point $x$ of $E$ are the components of $f(x)$. The polar coordinate system (dimension 2) and the spherical and cylindrical coordinate systems (dimension 3) are defined this way.

For points that are outside the domain of $f$, coordinates may sometimes be defined as the limit of coordinates of neighbour points, but these coordinates may be not uniquely defined, and may be not continuous in the neighborhood of the point. For example, for the spherical coordinate system, the longitude is not defined at the pole, and on the antimeridian, the longitude passes discontinuously from –180° to +180°.

This way of defining coordinates extends easily to other mathematical structures, and in particular to manifolds.

Isometries
An isometry between two metric spaces is a bijection preserving the distance, that is

$$d(f(x), f(y))= d(x,y).$$

In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm

$$\|f(x)\| = \|x\|,$$

since the norm of a vector is its distance from the zero vector. It preserves also the inner product

$$f(x)\cdot f(y)=x\cdot y,$$

since

$$x \cdot y=\tfrac 1 2 \left(\|x+y\|^2 - \|x\|^2 - \|y\|^2\right).$$

An isometry of Euclidean vector spaces is a linear isomorphism.

An isometry $$f\colon E\to F$$ of Euclidean spaces defines an isometry $$\overrightarrow f \colon \overrightarrow E \to \overrightarrow F$$ of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if $f$ and $E$ are Euclidean spaces, $O ∈ E$, $O ∈ F$, and $$\overrightarrow f\colon \overrightarrow E\to \overrightarrow F$$ is an isometry, then the map $$f\colon E\to F$$ defined by

$$f(P)=O' + \overrightarrow f\Bigl(\overrightarrow{OP}\Bigr)\vphantom{\frac({}}$$

is an isometry of Euclidean spaces.

It follows from the preceding results that an isometry of Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces.

Isometry with prototypical examples
If $F$ is a Euclidean space, its associated vector space $$\overrightarrow E$$ can be considered as a Euclidean space. Every point $O ∈ E$ defines an isometry of Euclidean spaces

$$P\mapsto \overrightarrow {OP},\vphantom{\frac({}}$$

which maps $E$ to the zero vector and has the identity as associated linear map. The inverse isometry is the map

$$v\mapsto O+v.$$

A Euclidean frame $O$ allows defining the map

$$\begin{align} E&\to \R^n\\ P&\mapsto \Bigl(e_1\cdot \overrightarrow {OP}, \dots, e_n\cdot\overrightarrow {OP}\Bigr),\vphantom{\frac({}} \end{align}$$

which is an isometry of Euclidean spaces. The inverse isometry is

$$\begin{align} \R^n&\to E \\ (x_1\dots, x_n)&\mapsto \left(O+x_1e_1+ \dots + x_ne_n\right). \end{align}$$

This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension.

This justifies that many authors talk of $$\R^n$$ as the Euclidean space of dimension $(O, e_1, \dots, e_n)$.

Euclidean group
An isometry from a Euclidean space onto itself is called Euclidean isometry, Euclidean transformation or rigid transformation. The rigid transformations of a Euclidean space form a group (under composition), called the Euclidean group and often denoted $E(n)$ of $ISO(n)$.

The simplest Euclidean transformations are translations

$$P \to P+v.$$

They are in bijective correspondence with vectors. This is a reason for calling space of translations the vector space associated to a Euclidean space. The translations form a normal subgroup of the Euclidean group.

A Euclidean isometry $n$ of a Euclidean space $f$ defines a linear isometry $$\overrightarrow f$$ of the associated vector space (by linear isometry, it is meant an isometry that is also a linear map) in the following way: denoting by $Q – P$ the vector $$\overrightarrow {PQ},\vphantom{\frac({}}$$ if $E$ is an arbitrary point of $O$, one has

$$\overrightarrow f\Bigl(\overrightarrow {OP}\Bigr)= f(P)-f(O).\vphantom{\frac({}}$$

It is straightforward to prove that this is a linear map that does not depend from the choice of $E$

The map $$f \to \overrightarrow f$$ is a group homomorphism from the Euclidean group onto the group of linear isometries, called the orthogonal group. The kernel of this homomorphism is the translation group, showing that it is a normal subgroup of the Euclidean group.

The isometries that fix a given point $O.$ form the stabilizer subgroup of the Euclidean group with respect to $P$. The restriction to this stabilizer of above group homomorphism is an isomorphism. So the isometries that fix a given point form a group isomorphic to the orthogonal group.

Let $P$ be a point, $P$ an isometry, and $f$ the translation that maps $t$ to $f(P)$. The isometry $$g=t^{-1}\circ f$$ fixes $P$. So $$f= t\circ g,$$ and the Euclidean group is the semidirect product of the translation group and the orthogonal group.

The special orthogonal group is the normal subgroup of the orthogonal group that preserves handedness. It is a subgroup of index two of the orthogonal group. Its inverse image by the group homomorphism $$f \to \overrightarrow f$$ is a normal subgroup of index two of the Euclidean group, which is called the special Euclidean group or the displacement group. Its elements are called rigid motions or displacements.

Rigid motions include the identity, translations, rotations (the rigid motions that fix at least a point), and also screw motions.

Typical examples of rigid transformations that are not rigid motions are reflections, which are rigid transformations that fix a hyperplane and are not the identity. They are also the transformations consisting in changing the sign of one coordinate over some Euclidean frame.

As the special Euclidean group is a subgroup of index two of the Euclidean group, given a reflection $P$, every rigid transformation that is not a rigid motion is the product of $r$ and a rigid motion. A glide reflection is an example of a rigid transformation that is not a rigid motion or a reflection.

All groups that have been considered in this section are Lie groups and algebraic groups.

Topology
The Euclidean distance makes a Euclidean space a metric space, and thus a topological space. This topology is called the Euclidean topology. In the case of $$\mathbb R^n,$$ this topology is also the product topology.

The open sets are the subsets that contains an open ball around each of their points. In other words, open balls form a base of the topology.

The topological dimension of a Euclidean space equals its dimension. This implies that Euclidean spaces of different dimensions are not homeomorphic. Moreover, the theorem of invariance of domain asserts that a subset of a Euclidean space is open (for the subspace topology) if and only if it is homeomorphic to an open subset of a Euclidean space of the same dimension.

Euclidean spaces are complete and locally compact. That is, a closed subset of a Euclidean space is compact if it is bounded (that is, contained in a ball). In particular, closed balls are compact.

Axiomatic definitions
The definition of Euclidean spaces that has been described in this article differs fundamentally of Euclid's one. In reality, Euclid did not define formally the space, because it was thought as a description of the physical world that exists independently of human mind. The need of a formal definition appeared only at the end of 19th century, with the introduction of non-Euclidean geometries.

Two different approaches have been used. Felix Klein suggested to define geometries through their symmetries. The presentation of Euclidean spaces given in this article, is essentially issued from his Erlangen program, with the emphasis given on the groups of translations and isometries.

On the other hand, David Hilbert proposed a set of axioms, inspired by Euclid's postulates. They belong to synthetic geometry, as they do not involve any definition of real numbers. Later G. D. Birkhoff and Alfred Tarski proposed simpler sets of axioms, which use real numbers (see Birkhoff's axioms and Tarski's axioms).

In Geometric Algebra, Emil Artin has proved that all these definitions of a Euclidean space are equivalent. It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Artin's proof is the following. In Hilbert's axioms, congruence is an equivalence relation on segments. One can thus define the length of a segment as its equivalence class. One must thus prove that this length satisfies properties that characterize nonnegative real numbers. Artin proved this with axioms equivalent to those of Hilbert.

Usage
Since the ancient Greeks, Euclidean space has been used for modeling shapes in the physical world. It is thus used in many sciences, such as physics, mechanics, and astronomy. It is also widely used in all technical areas that are concerned with shapes, figure, location and position, such as architecture, geodesy, topography, navigation, industrial design, or technical drawing.

Space of dimensions higher than three occurs in several modern theories of physics; see Higher dimension. They occur also in configuration spaces of physical systems.

Beside Euclidean geometry, Euclidean spaces are also widely used in other areas of mathematics. Tangent spaces of differentiable manifolds are Euclidean vector spaces. More generally, a manifold is a space that is locally approximated by Euclidean spaces. Most non-Euclidean geometries can be modeled by a manifold, and embedded in a Euclidean space of higher dimension. For example, an elliptic space can be modeled by an ellipsoid. It is common to represent in a Euclidean space mathematical objects that are a priori not of a geometrical nature. An example among many is the usual representation of graphs.

Other geometric spaces
Since the introduction, at the end of 19th century, of non-Euclidean geometries, many sorts of spaces have been considered, about which one can do geometric reasoning in the same way as with Euclidean spaces. In general, they share some properties with Euclidean spaces, but may also have properties that could appear as rather strange. Some of these spaces use Euclidean geometry for their definition, or can be modeled as subspaces of a Euclidean space of higher dimension. When such a space is defined by geometrical axioms, embedding the space in a Euclidean space is a standard way for proving consistency of its definition, or, more precisely for proving that its theory is consistent, if Euclidean geometry is consistent (which cannot be proved).

Affine space
A Euclidean space is an affine space equipped with a metric. Affine spaces have many other uses in mathematics. In particular, as they are defined over any field, they allow doing geometry in other contexts.

As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the complex numbers as an extension of Euclidean spaces. For example, a circle and a line have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically closed fields. The shapes that are studied in algebraic geometry in these affine spaces are therefore called affine algebraic varieties.

Affine spaces over the rational numbers and more generally over algebraic number fields provide a link between (algebraic) geometry and number theory. For example, the Fermat's Last Theorem can be stated "a Fermat curve of degree higher than two has no point in the affine plane over the rationals."

Geometry in affine spaces over a finite fields has also been widely studied. For example, elliptic curves over finite fields are widely used in cryptography.

Projective space
Originally, projective spaces have been introduced by adding "points at infinity" to Euclidean spaces, and, more generally to affine spaces, in order to make true the assertion "two coplanar lines meet in exactly one point". Projective space share with Euclidean and affine spaces the property of being isotropic, that is, there is no property of the space that allows distinguishing between two points or two lines. Therefore, a more isotropic definition is commonly used, which consists as defining a projective space as the set of the vector lines in a vector space of dimension one more.

As for affine spaces, projective spaces are defined over any field, and are fundamental spaces of algebraic geometry.

Non-Euclidean geometries
Non-Euclidean geometry refers usually to geometrical spaces where the parallel postulate is false. They include elliptic geometry, where the sum of the angles of a triangle is more than 180°, and hyperbolic geometry, where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is consistent (if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the foundational crisis in mathematics of the beginning of 20th century, and motivated the systematization of axiomatic theories in mathematics.

Curved spaces
A manifold is a space that in the neighborhood of each point resembles a Euclidean space. In technical terms, a manifold is a topological space, such that each point has a neighborhood that is homeomorphic to an open subset of a Euclidean space. Manifolds can be classified by increasing degree of this "resemblance" into topological manifolds, differentiable manifolds, smooth manifolds, and analytic manifolds. However, none of these types of "resemblance" respect distances and angles, even approximately.

Distances and angles can be defined on a smooth manifold by providing a smoothly varying Euclidean metric on the tangent spaces at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a Riemannian manifold. Generally, straight lines do not exist in a Riemannian manifold, but their role is played by geodesics, which are the "shortest paths" between two points. This allows defining distances, which are measured along geodesics, and angles between geodesics, which are the angle of their tangents in the tangent space at their intersection. So, Riemannian manifolds behave locally like a Euclidean space that has been bent.

Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a sphere. In this case, geodesics are arcs of great circle, which are called orthodromes in the context of navigation. More generally, the spaces of non-Euclidean geometries can be realized as Riemannian manifolds.

Pseudo-Euclidean space
An inner product of a real vector space is a positive definite bilinear form, and so characterized by a positive definite quadratic form. A pseudo-Euclidean space is an affine space with an associated real vector space equipped with a non-degenerate quadratic form (that may be indefinite).

A fundamental example of such a space is the Minkowski space, which is the space-time of Einstein's special relativity. It is a four-dimensional space, where the metric is defined by the quadratic form

$$x^2+y^2+z^2-t^2,$$

where the last coordinate (t) is temporal, and the other three (x, y, z) are spatial.

To take gravity into account, general relativity uses a pseudo-Riemannian manifold that has Minkowski spaces as tangent spaces. The curvature of this manifold at a point is a function of the value of the gravitational field at this point.

title: Euclidean space