text:
Definable real number

Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, $$\sqrt{2}$$, can be defined as the unique positive solution to the equation $$x^2 = 2$$, and it can be constructed with a compass and straightedge.

Different choices of a formal language or its interpretation give rise to different notions of definability. Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers. Because formal languages can have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument, there are uncountably many real numbers, so almost every real number is undefinable.

Constructible numbers
One way of specifying a real number uses geometric techniques. A real number $$r$$ is a constructible number if there is a method to construct a line segment of length $$r$$ using a compass and straightedge, beginning with a fixed line segment of length 1.

Each positive integer, and each positive rational number, is constructible. The positive square root of 2 is constructible. However, the cube root of 2 is not constructible; this is related to the impossibility of doubling the cube.

Real algebraic numbers

A real number $$r$$ is called a real algebraic number if there is a polynomial $$p(x)$$, with only integer coefficients, so that $$r$$ is a root of $$p$$, that is, $$p(r)=0$$. Each real algebraic number can be defined individually using the order relation on the reals. For example, if a polynomial $$q(x)$$ has 5 real roots, the third one can be defined as the unique $$r$$ such that $$q(r)=0$$ and such that there are two distinct numbers less than $$r$$ at which $$q$$ is zero.

All rational numbers are constructible, and all constructible numbers are algebraic. There are numbers such as the cube root of 2 which are algebraic but not constructible.

The real algebraic numbers form a subfield of the real numbers. This means that 0 and 1 are algebraic numbers and, moreover, if $$a$$ and $$b$$ are algebraic numbers, then so are $$a+b$$, $$a-b$$, $$ab$$ and, if $$b$$ is nonzero, $$a/b$$.

The real algebraic numbers also have the property, which goes beyond being a subfield of the reals, that for each positive integer $$n$$ and each real algebraic number $$a$$, all of the $$n$$th roots of $$a$$ that are real numbers are also algebraic.

There are only countably many algebraic numbers, but there are uncountably many real numbers, so in the sense of cardinality most real numbers are not algebraic. This nonconstructive proof that not all real numbers are algebraic was first published by Georg Cantor in his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers".

Non-algebraic numbers are called transcendental numbers. The best known transcendental numbers are $\pi$ and $e$.

Computable real numbers
A real number is a computable number if there is an algorithm that, given a natural number $$n$$, produces a decimal expansion for the number accurate to $$n$$ decimal places. This notion was introduced by Alan Turing in 1936.

The computable numbers include the algebraic numbers along with many transcendental numbers including $$\pi$$ and $e$. Like the algebraic numbers, the computable numbers also form a subfield of the real numbers, and the positive computable numbers are closed under taking $$n$$th roots for each positive $n$.

Not all real numbers are computable. Specific examples of noncomputable real numbers include the limits of Specker sequences, and algorithmically random real numbers such as Chaitin's Ω numbers.

Definability in arithmetic
Another notion of definability comes from the formal theories of arithmetic, such as Peano arithmetic. The language of arithmetic has symbols for 0, 1, the successor operation, addition, and multiplication, intended to be interpreted in the usual way over the natural numbers. Because no variables of this language range over the real numbers, a different sort of definability is needed to refer to real numbers. A real number $$a$$ is definable in the language of arithmetic (or arithmetical) if its Dedekind cut can be defined as a predicate in that language; that is, if there is a first-order formula $$\varphi$$ in the language of arithmetic, with three free variables, such that $$\forall m \, \forall n \, \forall p \left (\varphi(n,m,p)\iff\frac{(-1)^p\cdot n}{m+1}<a \right ).$$ Here m, n, and p range over nonnegative integers.

The second-order language of arithmetic is the same as the first-order language, except that variables and quantifiers are allowed to range over sets of naturals. A real that is second-order definable in the language of arithmetic is called analytical.

Every computable real number is arithmetical, and the arithmetical numbers form a subfield of the reals, as do the analytical numbers. Every arithmetical number is analytical, but not every analytical number is arithmetical. Because there are only countably many analytical numbers, most real numbers are not analytical, and thus also not arithmetical.

Every computable number is arithmetical, but not every arithmetical number is computable. For example, the limit of a Specker sequence is an arithmetical number that is not computable.

The definitions of arithmetical and analytical reals can be stratified into the arithmetical hierarchy and analytical hierarchy. In general, a real is computable if and only if its Dedekind cut is at level $$\Delta^0_1$$ of the arithmetical hierarchy, one of the lowest levels. Similarly, the reals with arithmetical Dedekind cuts form the lowest level of the analytical hierarchy.

Definability in models of ZFC
A real number $$a$$ is first-order definable in the language of set theory, without parameters, if there is a formula $$\varphi$$ in the language of set theory, with one free variable, such that $$a$$ is the unique real number such that $$\varphi(a)$$ holds. This notion cannot be expressed as a formula in the language of set theory.

All analytical numbers, and in particular all computable numbers, are definable in the language of set theory. Thus the real numbers definable in the language of set theory include all familiar real numbers such as 0, 1, $$\pi$$, $$e$$, et cetera, along with all algebraic numbers. Assuming that they form a set in the model, the real numbers definable in the language of set theory over a particular model of ZFC form a field. Each set model $$M$$ of ZFC set theory that contains uncountably many real numbers must contain real numbers that are not definable within $$M$$ (without parameters). This follows from the fact that there are only countably many formulas, and so only countably many elements of $$M$$ can be definable over $$M$$. Thus, if $$M$$ has uncountably many real numbers, one can prove from "outside" $$M$$ that not every real number of $$M$$ is definable over $$M$$. This argument becomes more problematic if it is applied to class models of ZFC, such as the von Neumann universe. The assertion "the real number $$x$$ is definable over the class model $$N$$" cannot be expressed as a formula of ZFC. Similarly, the question of whether the von Neumann universe contains real numbers that it cannot define cannot be expressed as a sentence in the language of ZFC. Moreover, there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable.

title: Definable real number


text:
Diego de Almagro

Diego de Almagro (c. 1475 – July 8, 1538), also known as El Adelantado and El Viejo, was a Spanish conquistador known for his exploits in western South America. He participated with Francisco Pizarro in the Spanish conquest of Peru. While subduing the Inca Empire he laid the foundation for Quito and Trujillo as Spanish cities in present-day Ecuador and Peru respectively. From Peru, Almagro led the first Spanish military expedition to central Chile. Back in Peru, a longstanding conflict with Pizarro over the control of the former Inca capital of Cuzco erupted into a civil war between the two bands of conquistadores. In the battle of Las Salinas in 1538, Almagro was defeated by the Pizarro brothers and months later he was executed.

Early years
The origins of Diego de Almagro were humble. He was born in 1475 in the village of Almagro or in Malagón, in Ciudad Real, where he was given the name of the village for his surname as he was the illegitimate son of Juan de Montenegro and Elvira Gutiérrez. In order to preserve the honor of his mother, her relatives took the infant Diego and took him to the nearby town of Bolaños de Calatrava, where he was raised by Sancha López del Peral, later moving to Aldea del Rey.

At the age of four he returned to Almagro, and was placed under the tutelage of an uncle named Hernán Gutiérrez. At age fifteen he ran away from home because of his uncle's harshness. He went to the home of his mother, who was now living with her new husband, to tell her what had happened and that he was going to travel the world, and asked for some bread. His mother, anguished, gave him a piece of bread and some coins and said: "Take, son, and do not give me more trouble, and go, and God help you in your adventure."

He went to Seville and after probably stealing to survive, Almagro became a criado or servant of Don Luis Gonzalez de Polanco, one of the four Alcaldes de la Casa y Corte de Su Majestad and later a Counselor of the Catholic Monarchs. While living in Seville, Almagro stabbed another servant in an argument, inflicting serious enough injuries that he was to be tried in court.

Don Luis, using his influence, prevailed upon Don Pedro Arias Dávila to allow Almagro to embark in one of the ships going to the New World from the port of Sanlucar de Barrameda. The Casa de Contratacion (royal agency for the Spanish Empire) required that the men who crossed the Atlantic provide their own weapons, clothes, and farming tools, which Don Polanco provided to his servant.

Arrival in America
Diego de Almagro, now in his late thirties, arrived in the New World on June 30, 1514, with the expedition that Ferdinand II of Aragon had sent under the leadership of Dávila. The expedition arrived at the city of Santa María la Antigua del Darién, Panama, where many other future conquistadors were already assembled, among them Francisco Pizarro.

There are not many details of Almagro's activities during this period, but it is known that he accompanied various sailors who departed from Darien between 1514 and 1515. He eventually returned and settled in Darien, where he was granted an encomienda, building a house and making a living from agriculture.

Almagro undertook his first independent conquest on November 1515, commanding 260 men as he founded Villa del Acla, named after the Indian place. Due to illness he had to hand over command to Gaspar de Espinosa.

Espinosa decided to undertake a new expedition, which departed in December 1515 with 200 men, including Almagro and Francisco Pizarro, who for the first time was designated as a captain. During this expedition, which lasted 14 months, Almagro, Pizarro and Hernando de Luque became close friends.

Also during this time Almagro established a friendship with Vasco Núñez de Balboa, who was in charge of Acla. Almagro wanted to have a ship built with the remaining materials of the Espinosa expedition, to be finished on the coast of the "Great South Sea", as the Pacific Ocean was first called by the Spanish. Current historians do not believe that Almagro was expected to participate in Balboa's expedition and probably returned to Darien.

Almagro took part in the various expeditions that took place in the Gulf of Panama, including those of Espinosa, which were supported by Balboa's ships. Almagro was recorded as a witness on the lists of natives whom Espinosa ordered to be carried. He remained as an early settler in the newly founded city of Panama, staying there for four years, managing his properties and those of Pizarro. He took Ana Martínez, an indigenous woman, as a common-law wife. In this period, his first son, El Mozo, was born to them.

Conquest of Peru
By 1524 an association of conquest regarding South America was formalized among De Almagro, Pizarro and Luque. By the beginning of August 1524, they had received the requisite permission to discover and conquer lands further south. In the first expedition, De Almagro lost his eye to an arrow shot at the Battle of Punta Quemada. He subsequently remained in Panama to recruit men and gather supplies for the expeditions led by Pizarro.

After several expeditions to South America, Pizarro secured his stay in Peru with the Capitulation on 6 July 1529. During Pizarro's continued exploration of Incan territory, he and his men succeeded in defeating the Inca army under Emperor Atahualpa during the Battle of Cajamarca in 1532. De Almagro joined Pizarro soon afterward, bringing more men and arms.

After Peru fell to the Spanish, both Pizarro and De Almagro initially worked together in the founding of new cities to consolidate their dominions. As such, Pizarro dispatched De Almagro to pursue Quizquiz, fleeing to the Inca Empire's northern city of Quito. Their fellow conquistador Sebastián de Belalcázar, who had gone forth without Pizarro's approval, had already reached Quito and witnessed the destruction of the city by Inca general Rumiñawi. The Inca warrior had ordered the city to be burned and its gold to be buried at an undisclosed location where the Spanish could never find it. The arrival of Pedro de Alvarado from Guatemala, in search of Inca gold further complicated the situation for Almagro and Belalcázar. Alvarado's presence, however, did not last long as he left South America in exchange for monetary compensation from Pizarro.

In an attempt to claim Quito ahead of Belalcázar, in August 1534 De Almagro founded a city on the shores of Laguna de Colta (Colta Lake) in the foothills of Chimborazo, some 200 km south of present-day Quito, and named it "Santiago de Quito." Four months later would come the foundation of the Peruvian city of Trujillo, which Almagro named as "Villa Trujillo de Nueva Castilla" (the Village of Trujillo in New Castille) in honor of Francisco Pizarro's birthplace, Trujillo in Extremadura, Spain. These events were the height of the Pizarro-Almagro friendship, which historians describe as one of the last events in which their friendship soon faded and entered a period of turmoil for the control of the Incan capital of Cuzco.

Conflict with Pizarro
After splitting the treasure of Inca emperor Atahualpa, both Pizarro and Almagro left towards Cuzco and took the city in 1533. However, De Almagro's friendship with Pizarro showed signs of deterioration in 1526 when Pizarro, in the name of the rest of the conquistadors, called forth the "Capitulacion de Toledo" law in which King Charles I of Spain had laid out his authorization for the conquest of Peru and the awards every conquistador would receive from it. Long before, however, each conquistador had promised to equally split the benefits. Pizarro managed to have a larger stake and awards for himself. Despite this, De Almagro still obtained an important fortune for his services, and the King awarded him in November 1532 the noble title of "Don" and he was assigned a personal coat of arms.

Although by this time Diego de Almagro had already acquired sufficient wealth in the conquest of Peru and was living a luxurious life in Cuzco, the prospect of conquering the lands further south was very attractive to him. Given that the dispute with Pizarro over Cuzco had kept intensifying, Almagro spent a great deal of time and money equipping a company of 500 men for a new exploration south of Peru.

By 1534 the Spanish crown had determined to split the region in two parallel lines, forming the governorship of "Nueva Castilla" (from the 1° to the 14° latitude, close to Pisco), and that of "Nueva Toledo" (from the 14° to the 25° latitude, in Taltal, Chile), assigning the first to Francisco Pizarro and the second to Diego de Almagro. The crown had previously assigned Almagro the governorship of Cuzco, and as such De Almagro was heading there when Charles V divided the territory between Nueva Castilla and Nuevo Toledo. This might have been the reason why Almagro did not immediately confront Pizarro for Cuzco, and promptly decided to embark on his new quest for the discovery of the riches of Chile.

The preparations
Charles V had given Diego a grant extending two hundred leagues south of Francisco Pizarro's. Francisco and Diego concluded a new contract on 12 June 1535, in which they agreed to share future discoveries equally. Diego raised an expedition for Chile, expecting it "would lead to even greater riches than they had found in Peru." Almagro prepared the way by sending ahead three of his Spanish soldiers, the religious chief of the Inca empire, Willaq Umu, and Paullo Topa, brother of Manco Inca Yupanqui. De Almagro sent Juan de Saavedra forward with one hundred and fifty men, and soon followed them with additional forces. Saavedra established on January 23, 1535, the first Spanish settlement in Bolivia near the Inca regional capital of Paria.

Following the Inca Trail and crossing the Andes
Almagro left Cuzco on July 3, 1535, with his supporters and stopped at Moina until the 20th of that month. Meanwhile, Francisco Pizarro's brother, Juan Pizarro, had arrested Inca Manco Inca Yupanqui, further complicating De Almagro's plans as it heavily increased the dissatisfaction of the Indians submitted to Spanish rule. Not having formally been appointed governor of any territories in the Capitulation of Toledo in 1528, however, forcing him to declare himself adelantado (governor) of Nueva Toledo, or southern Peru and present-day Chile. Some sources suggest Almagro received such a requirement in 1534 by the Spanish king and was officially declared governor of New Toledo.

Once he left Moina, De Almagro followed the Inca trail followed by 750 Spaniards deciding to join him in quest for the gold lost in the ransom of Atahualpa, which had mainly benefited the Pizarro brothers and their supporters. After crossing the Bolivian mountain range and traveling past Lake Titicaca, Almagro arrived on the shores of the Desaguadero River and finally set up camp in Tupiza. From there, the expedition stopped at Chicoana and then turned to the southeast to cross the Andes mountains.

The expedition turned out to be a difficult and exhausting endeavor. The hardest phase was the crossing of the Andean cordilleras: the cold, hunger and tiredness meant the death of various Spanish and natives, but mainly slaves who were not accustomed to such rigorous climate.

Upon this point, De Almagro determined everything was a failure. He ordered a small group under Rodrigo Orgóñez on a reconnaissance of the country to the south.

By luck, these men found the Valley of Copiapó, where Gonzalo Calvo Barrientos, a Spanish soldier whom Pizarro had expelled from Peru for stealing objects the Inca had offered for his ransom, had already established a friendship with the local natives. There, in the valley of the river Copiapó, Almagro took official possession of Chile and claimed it in the name of King Charles V.

Dismayed in Chile
De Almagro promptly initiated the exploration of the new territory, starting up the valley the Aconcagua River, where he was well received by the natives. However, the intrigues of his interpreter, Felipillo, who had previously helped Pizarro in dealing with Atahualpa, almost thwarted De Almagro's efforts. Felipillo had secretly urged the local natives to attack the Spanish, but they desisted, not understanding the dangers that they posed. De Almagro directed Gómez de Alvarado along with 100 horsemen and 100 foot to continue the exploration, which ended in the confluence of the Ñuble and Itata rivers. The Battle of Reinohuelén between the Spanish and Mapuche indigenous peoples forced the explorers to return to the north.

De Almagro's own reconnaissance of the land and the bad news of Gómez de Alvarado's encounter with the fierce Mapuche, along with the bitter cold winter that settled ferociously upon them, only served to confirm that everything had failed. He never found gold or the cities which Incan scouts had told him lay ahead, only communities of the indigenous population who lived from subsistence agriculture. Local tribes put up fierce resistance to the Spanish forces. The exploration of the territories of Nueva Toledo, which lasted 2 years, was marked by a complete failure for De Almagro. Despite this, at first he thought staying and founding a city would serve well for his honor. The initial optimism that led Almagro to bring his son he had with the indigenous Panamanian Ana Martínez to Chile had faded.

Some historians have suggested that, but for the urging of his senior explorers, De Almagro would probably have stayed permanently in Chile. He was urged to return to Peru and this time take definitive possession of Cuzco, so as to consolidate an inheritance for his son. Dismayed with his experience in the south, Almagro made plans of return to Peru. He never officially founded a city in the territory of what is now Chile.

The withdrawal of the Spanish from valleys of Chile was violent: Almagro authorized his soldiers to ransack the natives' properties, leaving their soil desolate. In addition, the Spanish soldiers took natives captive to serve as slaves. The locals were captured, tied together, and forced to carry the heavy loads belonging to the conquistadors.

Return to Peru
After the exhausting crossing of the Atacama Desert, mainly due to the harsh weather conditions, Almagro finally reached Cuzco, Peru, in 1537. According to some authors, it was during this time that the Spanish term "roto" (torn), used by Peruvians to refer to Chileans, was first coined. De Almagro's disappointed troops returned to Cuzco with their "torn clothes" due to the extensive and laborious passage on foot by the Atacama Desert.

After his return, De Almagro was surprised to learn of the Inca Manco's rebellion. Diego de Almagro sent an embassy to the Inca, but they mistrusted all of the Spaniards by this time. Hernando Pizarro's men formed an uneasy truce with De Almagro's men, surveying to determine the boundaries of their leaders' royal grants. They needed to determine in which portion the city of Cuzco was located. However, De Almagro's troops quickly took the city and imprisoned the Pizarro brothers, Hernando and Gonzalo, on the night of 8 April 1537.

After occupying Cuzco, De Almagro confronted an army sent by Francisco Pizarro to liberate his brothers. Alonso de Alvarado commanded it and was defeated during the Battle of Abancay on July 12, 1537. He and some of his men were imprisoned. Later, Gonzalo Pizarro and De Alvarado escaped prison. Subsequent negotiations between Francisco Pizarro and De Almagro concluded with the liberation of Hernando, the third Pizarro brother, in return for conceding control and administration of Cuzco to De Almagro. Pizarro never intended to give up the city permanently, but was buying time to organize an army strong enough to defeat Almagro's troops.

During this time De Almagro fell ill, and Pizarro and his brothers grabbed the opportunity to defeat him and his followers. The Almagristas were defeated at Las Salinas in April 1538, with Orgóñez being killed on the field of battle. De Almagro fled to Cuzco, still in the hands of his loyal supporters, but found only temporary refuge; the forces of the Pizarro brothers entered the city without resistance. Once captured, Almagro was humiliated by Hernando Pizarro and his requests to appeal to the King were ignored.

When Diego de Almagro begged for his life, Hernando responded:

"'-he was surprised to see Almagro demean himself in a manner so unbecoming a brave cavalier, that his fate was no worse than had befallen many a soldier before him; and that, since God had given him the grace to be a Christian, he should employ his remaining moments in making up his account with Heaven!'"

Almagro was condemned to death and executed by garrote in his dungeon, and then decapitated, on July 8, 1538. His corpse was taken to the public Plaza Mayor of Cuzco, where a herald proclaimed his crimes. Hernán Ponce de León took his body and buried him in the church of Our Lady of Mercy in Cuzco.

El Mozo
Diego de Almagro II (1520–1542), known as El Mozo (The Lad), son of Diego de Almagro I, whose mother was an Indian girl of Panama, became the foil of the conspirators who had put Pizarro to the sword. Pizarro was murdered on June 26, 1541; the conspirators promptly proclaimed the lad De Almagro Governor of Peru. From various causes, all of the conspirators either died or were killed except for one, who was executed after the lad Almagro gave an order. The lad De Almagro fought the desperate battle of Chupas on September 16, 1542, escaped to Cuzco, but was arrested, immediately condemned to death, and executed in the great square of the city.

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

title: Diego de Almagro


text:
Divinity

Divinity or the divine are things that are either related to, devoted to, or proceeding from a deity. What is or is not divine may be loosely defined, as it is used by different belief systems. Under monotheism and polytheism this is clearly delineated. However, in pantheism and animism this becomes synonymous with concepts of sacredness and transcendence.

Etymology
The root of the word divinity is the Latin divus meaning of or belonging to a God (deus). The word entered English from Medieval Latin in the 14th century.

Usages
Divinity as a quality has two distinct usages:
 * Divine force or power – Powers or forces that are universal, or transcend human capacities
 * Divinity applied to mortals – Qualities of individuals who are considered to have some special access or relationship to the divine.

Overlap occurs between these usages because deities or godly entities are often identical with or identified by the powers and forces that are credited to them — in many cases, a deity is merely a power or force personified — and these powers and forces may then be extended or granted to mortal individuals. For instance, Jehovah is closely associated with storms and thunder throughout much of the Old Testament. He is said to speak in thunder, and thunder is seen as a token of his anger. This power was then extended to prophets like Moses and Samuel, who caused thunderous storms to rain down on their enemies. Divinity always carries connotations of goodness, beauty, beneficence, justice, and other positive, pro-social attributes. In monotheistic faiths there is an equivalent cohort of malefic supernatural beings and powers, such as demons, devils, afreet, etc., which are not conventionally referred to as divine; demonic is often used instead. Polytheistic and animistic systems of belief make no such distinction; gods and other beings of transcendent power often have complex, ignoble, or even incomprehensible motivations for their acts. Note that while the terms demon and demonic are used in monotheistic faiths as antonyms to divine, they are in fact derived from the Greek word daimón (δαίμων), which itself translates as divinity.

Uses in religious discourse
There are three distinct usages of divinity and divine in religious discourse:

Entity
In monotheistic faiths, the word divinity is often used to refer to the singular God central to that faith. Often the word takes the definite article and is capitalized — "the Divinity" — as though it were a proper name or definitive honorific. Divine — capitalized — may be used as an adjective to refer to the manifestations of such a Divinity or its powers: e.g. "basking in the Divine presence..."

The terms divinity and divine — uncapitalized, and lacking the definite article — are sometimes used to denote 'god(s) or certain other beings and entities which fall short of absolute Godhood but lie outside the human realm.

Divine force or power
As previously noted, divinities are closely related to the transcendent force(s) or power(s) credited to them, so much so that in some cases the powers or forces may themselves be invoked independently. This leads to the second usage of the word divine (and less common usage of divinity): to refer to the operation of transcendent power in the world.

In its most direct form, the operation of transcendent power implies some form of divine intervention. For monotheistic and polytheistic faiths this usually implies the direct action of one god or another on the course of human events. In Greek legend, for instance, it was Poseidon (god of the sea) who raised the storms that blew Odysseus's craft off course on his return journey, and Japanese tradition holds that a god-sent wind saved them from Mongol invasion. Prayers or propitiations are often offered to specific gods to garner favorable interventions in particular enterprises: e.g. safe journeys, success in war, or a season of bountiful crops. Many faiths around the world — from Japanese Shinto and Chinese traditional religion, to certain African practices and the faiths derived from those in the Caribbean, to Native American beliefs — hold that ancestral or household deities offer daily protection and blessings. In monotheistic religions, divine intervention may take very direct forms: miracles, visions, or intercessions by blessed figures.

Transcendent force or power may also operate through more subtle and indirect paths. Monotheistic faiths generally support some version of divine providence, which acknowledges that the divinity of the faith has a profound but unknowable plan always unfolding in the world. Unforeseeable, overwhelming, or seemingly unjust events are often thrown on 'the will of the Divine', in deferences like the Muslim inshallah ('as God wills it') and Christian 'God works in mysterious ways'. Often such faiths hold out the possibility of divine retribution as well, where the divinity will unexpectedly bring evil-doers to justice through the conventional workings of the world; from the subtle redressing of minor personal wrongs to such large-scale havoc as the destruction of Sodom and Gomorrah or the biblical Great Flood. Other faiths are even more subtle: the doctrine of karma shared by Buddhism and Hinduism is a divine law similar to divine retribution but without the connotation of punishment: our acts, good or bad, intentional or unintentional, reflect back on us as part of the natural working of the universe. Philosophical Taoism also proposes a transcendent operant principle — transliterated in English as tao or dao, meaning 'the way' — which is neither an entity nor a being per se, but reflects the natural ongoing process of the world. Modern western mysticism and new age philosophy often use the term 'the Divine' as a noun in this latter sense: a non-specific principle or being that gives rise to the world, and acts as the source or wellspring of life. In these latter cases, the faiths do not promote deference, as happens in monotheisms; rather each suggests a path of action that will bring the practitioner into conformance with the divine law: ahimsa — 'no harm' — for Buddhist and Hindu faiths; de or te — 'virtuous action' — in Taoism; and any of numerous practices of peace and love in new age thinking.

Mortal
In the third usage, extensions of divinity and divine power are credited to living, mortal individuals. Political leaders are known to have claimed actual divinity in certain early societies — the ancient Egyptian Pharaohs being the premier case — taking a role as objects of worship and being credited with superhuman status and powers. More commonly, and more pertinent to recent history, leaders merely claim some form of divine mandate, suggesting that their rule is in accordance with the will of God. The doctrine of the divine right of kings was introduced as late as the 17th century, proposing that kings rule by divine decree; Japanese Emperors ruled by divine mandate until the inception of the Japanese constitution after World War II.

Less politically, most faiths have any number of people that are believed to have been touched by divine forces: saints, prophets, heroes, oracles, martyrs, and enlightened beings, among others. Saint Francis of Assisi, in Catholicism, is said to have received instruction directly from God and it is believed that he grants plenary indulgence to all who confess their sins and visit his chapel on the appropriate day. In Greek mythology, Achilles' mother bathed him in the river Styx to give him immortality, and Hercules — as the son of Zeus — inherited near-godly powers. In religious Taoism, Laozi is venerated as a saint with his own powers. Various individuals in the Buddhist faith, beginning with Siddhartha, are considered to be enlightened, and in religious forms of Buddhism they are credited with divine powers. Christ in the Bible is said to be God's Son and is said to have performed divine miracles.

In general, mortals with divine qualities are carefully distinguished from the deity or deities in their religion's main pantheon. Even the Christian faith, which generally holds Christ to be identical to God, distinguishes between God the Father and Christ the begotten Son. There are, however, certain esoteric and mystical schools of thought, present in many faiths — Sufis in Islam, Gnostics in Christianity, Advaitan Hindus, Zen Buddhists, as well as several non-specific perspectives developed in new age philosophy — which hold that all humans are in essence divine, or unified with the Divine in a non-trivial way. Such divinity, in these faiths, would express itself naturally if it were not obscured by the social and physical worlds we live in; it needs to be brought to the fore through appropriate spiritual practices.

Christianity
In the New Testament the Greek word θεῖον (theion) in the Douay Version, is translated as "divinity". Examples are below:
 * Acts 17:29
 * "Being therefore the offspring of God, we must not suppose the divinity to be like unto gold, or silver, or stone, the graving of art, and device of man."

 * Romans 1:20
 * "For the invisible things of him, from the creation of the world, are clearly seen, being understood by the things that are made; his eternal power also, and divinity: so that they are inexcusable."

 * Revelation 5:12
 * "Saying with a loud voice: The Lamb that was slain is worthy to receive power, and divinity, and wisdom, and strength, and honour, and glory, and benediction."

The word translated as either "deity", "Godhead", or "divinity" in the Greek New Testament is also the Greek word θεότητος (theotētos), and the one verse that contains it is this: Colossians 2:9
 * "Quia in ipso inhabitat omnis plenitudo divinitatis [divinity] corporaliter." (Vulgate)
 * "For in him dwelleth all the fulness of the Godhead bodily." (KJV)
 * "Because it is in him that all the fullness of the divine quality dwells bodily." (NWT)
 * "For in him all the fullness of deity lives in bodily form." (NET)
 * "For the full content of divine nature lives in Christ." (TEV)

The word "divine" in the New Testament is the Greek word θείας (theias), and is the adjective form of "divinity". Biblical examples from the King James Bible are below:
 * 2 Peter 1:3
 * "According as his divine power hath given unto us all things that pertain unto life and godliness, through the knowledge of him that hath called us to glory and virtue."

 * 2 Peter 1:4
 * "Whereby are given unto us exceeding great and precious promises: that by these ye might be partakers of the divine nature, having escaped the corruption that is in the world through lust."

Latter-day Saints
The most prominent conception of divine entities in the Church of Jesus Christ of Latter-day Saints (LDS Church) is the Godhead, a divine council of three distinct beings: Elohim (the Father), Jehovah (the Son, or Jesus), and the Holy Spirit. Joseph Smith described a nontrinitarian Godhead, with God the Father and Jesus Christ each having individual physical bodies, and the Holy Spirit as a distinct personage with a spirit body. Smith also introduced the existence of a Heavenly Mother in the King Follett Discourse, but very little is acknowledged or known beyond her existence.

Mormons hold a belief in the divine potential of humanity; Smith taught a form of divinization where mortal men and women can become like god through salvation and exaltation. Lorenzo Snow succinctly summarized this using a couplet, which is often repeated within the LDS Church: "As man now is, God once was: As God now is, man may be."

Wicca
Wiccan views of divinity are generally theistic, and revolve around a Goddess and a Horned God, thereby being generally dualistic. In traditional Wicca, as expressed in the writings of Gerald Gardner and Doreen Valiente, the emphasis is on the theme of divine gender polarity, and the God and Goddess are regarded as equal and opposite divine cosmic forces. In some newer forms of Wicca, such as feminist or Dianic Wicca, the Goddess is given primacy or even exclusivity. In some forms of traditional witchcraft that share a similar duotheistic theology, the Horned God is given precedence over the Goddess.

title: Divinity


text:
Talk:Dynamical systems and chaos theory

I always felt that "dynamical" was a retarded word. "al" and "ic" mean the same thing! The word literally breaks down to: Pertaining to pertaining to... Ok, I don't know how the dynam bit breaks down. If you look up "dynamical" in mirriam webster online, it takes you to the entry for "dynamic" and starts the first definition with "Also dynamical." It's not proper English! I say we change the name of the article to "Dynamic Systems and Chaos."

It would be grand actually to have an article. :-) --LMS

Concerning "dynamical" vs "dynamic" I think "dynamical" is OK in this context; see:

http://www.xrefer.com/entry.jsp?xrefid=592879&secid

Yes, "Dynamical systems" is the name of the field. See for instance the Mathematical Atlas.

title: Talk:Dynamical systems and chaos theory


text:
Depth of field

The depth of field (DOF) is the distance between the nearest and the furthest objects that are in acceptably sharp focus in an image captured with a camera. See also the closely related depth of focus.

Factors affecting depth of field

For cameras that can only focus on one object distance at a time, depth of field is the distance between the nearest and the farthest objects that are in acceptably sharp focus in the image. "Acceptably sharp focus" is defined using a property called the "circle of confusion".

The depth of field can be determined by focal length, distance to subject (object to be imaged), the acceptable circle of confusion size, and aperture. Limitations of depth of field can sometimes be overcome with various techniques and equipment. The approximate depth of field can be given by:

$$ \text{DOF} \approx \frac{2u^2Nc}{f^2} $$

for a given maximum acceptable circle of confusion $c$, focal length $f$, f-number $N$, and distance to subject $u$.

As distance or the size of the acceptable circle of confusion increases, the depth of field increases; however, increasing the size of the aperture (i.e., reducing f-number) or increasing the focal length reduces the depth of field. Depth of field changes linearly with f-number and circle of confusion, but changes in proportion to the square of the distance to the subject and inversely in proportion to the square of the focal length. As a result, photos taken at extremely close range (i.e., so small $u$) have a proportionally much smaller depth of field.

Rearranging the DOF equation shows that it is the ratio between distance and focal length that affects DOF;

$$ \text{DOF} \approx 2Nc\left(\frac{u}{f}\right)^2=2Nc\left(1 - \frac{1}{M_T}\right)^2 $$

Note that $M_T = -\frac{f}{u - f}$ is the transverse magnification which is the ratio of the lateral image size to the lateral subject size.

Image sensor size affects DOF in counterintuitive ways. Because the circle of confusion is directly tied to the sensor size, decreasing the size of the sensor while holding focal length and aperture constant will the depth of field (by the crop factor). The resulting image however will have a different field of view. If the focal length is altered to maintain the field of view, the change in focal length will counter the decrease of DOF from the smaller sensor and the depth of field (also by the crop factor).

Effect of lens aperture
For a given subject framing and camera position, the DOF is controlled by the lens aperture diameter, which is usually specified as the f-number (the ratio of lens focal length to aperture diameter). Reducing the aperture diameter (increasing the f-number) increases the DOF because only the light travelling at shallower angles passes through the aperture so only cones of rays with shallower angles reach the image plane. In other words, the circles of confusion are reduced or increasing the DOF.

For a given size of the subject's image in the focal plane, the same f-number on any focal length lens will give the same depth of field. This is evident from the above DOF equation by noting that the ratio $u/f$ is constant for constant image size. For example, if the focal length is doubled, the subject distance is also doubled to keep the subject image size the same. This observation contrasts with the common notion that "focal length is twice as important to defocus as f/stop", which applies to a constant subject distance, as opposed to constant image size.

Motion pictures make limited use of aperture control; to produce a consistent image quality from shot to shot, cinematographers usually choose a single aperture setting for interiors (e.g., scenes inside a building) and another for exteriors (e.g., scenes in an area outside a building), and adjust exposure through the use of camera filters or light levels. Aperture settings are adjusted more frequently in still photography, where variations in depth of field are used to produce a variety of special effects.

Effect of circle of confusion
Precise focus is only possible at an exact distance from a lens; at that distance, a point object will produce a small spot image. Otherwise, a point object will produce a larger or blur spot image that is typically and approximately a circle. When this circular spot is sufficiently small, it is visually indistinguishable from a point, and appears to be in focus. The diameter of the largest circle that is indistinguishable from a point is known as the acceptable circle of confusion, or informally, simply as the circle of confusion.

The acceptable circle of confusion depends on how the final image will be used. The circle of confusion as 0.25 mm for an image viewed from 25 cm away is generally accepted.

For 35mm motion pictures, the image area on the film is roughly 22 mm by 16 mm. The limit of tolerable error was traditionally set at 0.05 mm diameter, while for 16 mm film, where the size is about half as large, the tolerance is stricter, 0.025 mm. More modern practice for 35 mm productions set the circle of confusion limit at 0.025 mm.

Camera movements
The term "camera movements" refers to swivel (swing and tilt, in modern terminology) and shift adjustments of the lens holder and the film holder. These features have been in use since the 1800s and are still in use today on view cameras, technical cameras, cameras with tilt/shift or perspective control lenses, etc. Swiveling the lens or sensor causes the plane of focus (POF) to swivel, and also causes the field of acceptable focus to swivel with the POF; and depending on the DOF criteria, to also change the shape of the field of acceptable focus. While calculations for DOF of cameras with swivel set to zero have been discussed, formulated, and documented since before the 1940s, documenting calculations for cameras with non-zero swivel seem to have begun in 1990.

More so than in the case of the zero swivel camera, there are various methods to form criteria and set up calculations for DOF when swivel is non-zero. There is a gradual reduction of clarity in objects as they move away from the POF, and at some virtual flat or curved surface the reduced clarity becomes unacceptable. Some photographers do calculations or use tables, some use markings on their equipment, some judge by previewing the image.

When the POF is rotated, the near and far limits of DOF may be thought of as wedge-shaped, with the apex of the wedge nearest the camera; or they may be thought of as parallel to the POF.

Object-field calculation methods
Traditional depth-of-field formulas can be hard to use in practice. As an alternative, the same effective calculation can be done without regard to the focal length and f-number. Moritz von Rohr and later Merklinger observe that the effective absolute aperture diameter can be used for similar formula in certain circumstances.

Moreover, traditional depth-of-field formulas assume equal acceptable circles of confusion for near and far objects. Merklinger suggested that distant objects often need to be much sharper to be clearly recognizable, whereas closer objects, being larger on the film, do not need to be so sharp. The loss of detail in distant objects may be particularly noticeable with extreme enlargements. Achieving this additional sharpness in distant objects usually requires focusing beyond the hyperfocal distance, sometimes almost at infinity. For example, if photographing a cityscape with a traffic bollard in the foreground, this approach, termed the object field method by Merklinger, would recommend focusing very close to infinity, and stopping down to make the bollard sharp enough. With this approach, foreground objects cannot always be made perfectly sharp, but the loss of sharpness in near objects may be acceptable if recognizability of distant objects is paramount.

Other authors such as Ansel Adams have taken the opposite position, maintaining that slight unsharpness in foreground objects is usually more disturbing than slight unsharpness in distant parts of a scene.

Overcoming DOF limitations
Some methods and equipment allow altering the apparent DOF, and some even allow the DOF to be determined after the image is made. These are based or supported by computational imaging processes. For example, focus stacking combines multiple images focused on different planes, resulting in an image with a greater (or less, if so desired) apparent depth of field than any of the individual source images. Similarly, in order to reconstruct the 3-dimensional shape of an object, a depth map can be generated from multiple photographs with different depths of field. Xiong and Shafer concluded, in part, "...the improvements on precisions of focus ranging and defocus ranging can lead to efficient shape recovery methods."

Another approach is focus sweep. The focal plane is swept across the entire relevant range during a single exposure. This creates a blurred image, but with a convolution kernel that is nearly independent of object depth, so that the blur is almost entirely removed after computational deconvolution. This has the added benefit of dramatically reducing motion blur.

Light Scanning Photomacrography (LSP) is another technique used to overcome depth of field limitations in macro and micro photography. This method allows for high-magnification imaging with exceptional depth of field. LSP involves scanning a thin light plane across the subject that is mounted on a moving stage perpendicular to the light plane. This ensures the entire subject remains in sharp focus from the nearest to the farthest details, providing comprehensive depth of field in a single image. Initially developed in the 1960s and further refined in the 1980s and 1990s, LSP was particularly valuable in scientific and biomedical photography before digital focus stacking became prevalent.

Other technologies use a combination of lens design and post-processing: Wavefront coding is a method by which controlled aberrations are added to the optical system so that the focus and depth of field can be improved later in the process.

The lens design can be changed even more: in colour apodization the lens is modified such that each colour channel has a different lens aperture. For example, the red channel may be, green may be , whilst the blue channel may be. Therefore, the blue channel will have a greater depth of field than the other colours. The image processing identifies blurred regions in the red and green channels and in these regions copies the sharper edge data from the blue channel. The result is an image that combines the best features from the different f-numbers.

At the extreme, a plenoptic camera captures 4D light field information about a scene, so the focus and depth of field can be altered after the photo is taken.

Diffraction and DOF
Diffraction causes images to lose sharpness at high f-numbers (i.e., narrow aperture stop opening sizes), and hence limits the potential depth of field. (This effect is not considered in the above formula giving approximate DOF values.) In general photography this is rarely an issue; because large f-numbers typically require long exposure times to acquire acceptable image brightness, motion blur may cause greater loss of sharpness than the loss from diffraction. However, diffraction is a greater issue in close-up photography, and the overall image sharpness can be degraded as photographers are trying to maximize depth of field with very small apertures.

Hansma and Peterson have discussed determining the combined effects of defocus and diffraction using a root-square combination of the individual blur spots. Hansma's approach determines the f-number that will give the maximum possible sharpness; Peterson's approach determines the minimum f-number that will give the desired sharpness in the final image and yields a maximum depth of field for which the desired sharpness can be achieved. In combination, the two methods can be regarded as giving a maximum and minimum f-number for a given situation, with the photographer free to choose any value within the range, as conditions (e.g., potential motion blur) permit. Gibson gives a similar discussion, additionally considering blurring effects of camera lens aberrations, enlarging lens diffraction and aberrations, the negative emulsion, and the printing paper. Couzin gave a formula essentially the same as Hansma's for optimal f-number, but did not discuss its derivation.

Hopkins, Stokseth, and Williams and Becklund have discussed the combined effects using the modulation transfer function.

DOF scales
Many lenses include scales that indicate the DOF for a given focus distance and f-number; the 35 mm lens in the image is typical. That lens includes distance scales in feet and meters; when a marked distance is set opposite the large white index mark, the focus is set to that distance. The DOF scale below the distance scales includes markings on either side of the index that correspond to f-numbers. When the lens is set to a given f-number, the DOF extends between the distances that align with the f-number markings.

Photographers can use the lens scales to work backwards from the desired depth of field to find the necessary focus distance and aperture. For the 35 mm lens shown, if it were desired for the DOF to extend from 1 m to 2 m, focus would be set so that index mark was centered between the marks for those distances, and the aperture would be set to.

On a view camera, the focus and f-number can be obtained by measuring the depth of field and performing simple calculations. Some view cameras include DOF calculators that indicate focus and f-number without the need for any calculations by the photographer.

Near:far distribution
The DOF beyond the subject is always greater than the DOF in front of the subject. When the subject is at the hyperfocal distance or beyond, the far DOF is infinite, so the ratio is 1:∞; as the subject distance decreases, near:far DOF ratio increases, approaching unity at high magnification. For large apertures at typical portrait distances, the ratio is still close to 1:1.

DOF formulae
This section covers some additional formula for evaluating depth of field; however they are all subject to significant simplifying assumptions: for example, they assume the paraxial approximation of Gaussian optics. They are suitable for practical photography, lens designers would use significantly more complex ones.

Focus and f-number from DOF limits
For given near and far DOF limits $DN$ and $DF$, the required f-number is smallest when focus is set to

$$s = \frac{2 D_{\mathrm N} D_{\mathrm F}}{D_{\mathrm N} + D_{\mathrm F}},$$

the harmonic mean of the near and far distances. In practice, this is equivalent to the arithmetic mean for shallow depths of field. Sometimes, view camera users refer to the difference $vN − vF$ as the focus spread.

Foreground and background blur
If a subject is at distance $s$ and the foreground or background is at distance $D$, let the distance between the subject and the foreground or background be indicated by

$$x_{\mathrm d} = |D - s|.$$

The blur disk diameter $b$ of a detail at distance $xd$ from the subject can be expressed as a function of the subject magnification $ms$, focal length $f$, f-number $N$, or alternatively the aperture $d$, according to

$$b = \frac{fm_\mathrm s}{N} \frac{x_\mathrm{d}}{s \pm x_\mathrm{d}} = dm_\mathrm{s} \frac{x_\mathrm{d}}{D}.$$

The minus sign applies to a foreground object, and the plus sign applies to a background object.

The blur increases with the distance from the subject; when $b$ is less than the circle of confusion, the detail is within the depth of field.

General and cited references

 * Available in PDF online.
 * "Available as GIF images on the Large Format page".
 * Version 1.03e available in PDF at http://www.trenholm.org/hmmerk/.
 * Version 1.6.1 available in PDF at http://www.trenholm.org/hmmerk/.
 * Available as GIF images on the Large Format page.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Version 1.03e available in PDF at http://www.trenholm.org/hmmerk/.
 * Version 1.6.1 available in PDF at http://www.trenholm.org/hmmerk/.
 * Available as GIF images on the Large Format page.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.
 * Reprinted 2002, Bellingham, WA: SPIE Press, ISBN 0-8194-4336-0.

title: Depth of field


text:
Dumnonii

The Dumnonii or Dumnones were a British tribe who inhabited Dumnonia, the area now known as Cornwall and Devon (and some areas of present-day Dorset and Somerset) in the further parts of the South West peninsula of Britain, from at least the Iron Age up to the early Saxon period. They were bordered to the east by the Durotriges tribe.

Etymology
William Camden, in his 1607 edition of Britannia, describes Cornwall and Devon as being two parts of the same 'country' which:"was in ancient time inhabited by those Britains whom Solinus called Dunmonii, Ptolomee Damnonii, or (as we find in some other copies) more truly Danmonii. ... . But... the Country of this nation is at this day divided into two parts, known by later names of Cornwall and Denshire [Devonshire] ... The near or hithermore region of the Danmonians that I spake of is now commonly called Denshire, [or] by the Cornish-Britains 'Dewnan', and by the Welsh Britains 'Duffneint' [sic], that is, 'low valleys', for that the people dwell for the most part beneath in Vales; by the English Saxons [it is known as] 'Deven-schire', whereof grew the Latin name 'Devonia', and by that contraction which the vulgar people useth, 'Denshire'."

Camden had learnt some Welsh during the course of his studies and it would appear that he is the origin of the interpretation of Dumnonii as "deep valley dwellers" from his understanding of the Welsh of his time. The modern Welsh term is Dyfnaint. John Rhŷs later theorized that the tribal name was derived from the name of a goddess, Domnu, probably meaning "the goddess of the deep". The proto-Celtic root *dubno- or *dumno- meaning "the deep" or "the earth" (or alternatively meaning "dark" or "gloomy" ) appears in personal names such as Dumnorix and Dubnovellaunus. Another group with a similar name but with no known links were the Fir Domnann of Connacht.

The Roman name of the town of Exeter, Isca Dumnoniorum ("Isca of the Dumnonii"), contains the root *iska- "water" for "Water of the Dumnonii". The Latin name suggests that the city was already an oppidum, or walled town, on the banks on the River Exe before the foundation of the Roman city, in about AD 50. The Dumnonii gave their name to the English county of Devon, and their name is represented in Britain's two extant Brythonic languages as Dewnans in Cornish and Dyfnaint in Welsh. Amédée Thierry (Histoire des Gaulois, 1828), one of the inventors of the "historic race" of Gauls, could confidently equate them with the Cornish ("les Cornouailles").

Victorian historians often referred to the tribe as the Damnonii, which is also the name of another people from lowland Scotland, although there are no known links between the two populations.

Language
The people of Dumnonia spoke a Southwestern Brythonic dialect of Celtic similar to the forerunner of more recent Cornish and Breton. Irish immigrants, the Déisi, are evidenced by the Ogham-inscribed stones they have left behind, confirmed and supplemented by toponymical studies. The stones are sometimes inscribed in Latin, sometimes in both scripts. Tristram Risdon suggested the continuance of a Brythonic dialect in the South Hams, Devon, as late as the 14th century, in addition to its use in Cornwall.

Territory
Ptolemy's 2nd century Geography places the Dumnonii to the west of the Durotriges. The name purocoronavium that appears in the Ravenna Cosmography implies the existence of a sub-tribe called the Cornavii or Cornovii, perhaps the ancestors of the Cornish people.

Gaius Iulius Solinus, probably in the 3rd century, remarks: "This turbid strait also divides the island Silura from the shore which is held by the Dumnonii, a British tribe. The men of this island even now preserve an old custom: they do not use coins. They give and accept, obtaining the necessities of life by exchange rather than by money. They reverence gods, and the men and women equally declare knowledge of the future."

In the sub-Roman period a Brythonic kingdom called Dumnonia emerged, covering the entire peninsula, although it is believed by some to have effectively been a collection of sub-kingdoms.

A kingdom of Domnonée (and of Cornouaille alongside) was established in the province of Armorica directly across the English Channel, and has apparent links with the British population, suggesting an ancient connection of peoples along the western Atlantic seaboard which is also borne out by the modern genetics of Devonian and Cornish populations.

Isca Dumnoniorum
The Latin name for Exeter is Isca Dumnoniorum ("Water of the Dumnonii"). This oppidum (a Latin term meaning an important town) on the banks of River Exe certainly existed prior to the foundation of the Roman city in about AD 50. Isca is derived from the Brythonic word for flowing water, which was given to the River Exe. The Gaelic term for water is uisce/uisge. This is reflected in the Welsh name for Exeter: Caerwysg meaning "fortified settlement on the river Uisc".

Isca Dumnoniorum originated with a settlement that developed around the Roman fortress of the Legio II Augusta and is one of the four poleis (cities) attributed to the tribe by Ptolemy. It is also listed in two routes of the late 2nd century Antonine Itinerary.

A legionary bath-house was built inside the fortress sometime between 55 and 60 and underwent renovation shortly afterwards (c. 60-65) but by c. 68 (perhaps even 66) the legion had transferred to a newer fortress at Gloucester. This saw the dismantling of the Isca fortress, and the site was then abandoned. Around AD 75, work on the civitas forum and basilica had commenced on the site of the former principia and by the late 2nd century the civitas walls had been completed. They were 3 metres thick and 6 metres high and enclosed exactly the same area as the earlier fortress. However, by the late 4th century the civitas was in decline.

Other settlements
As well as Isca Dumnoniorum, Ptolemy's 2nd century Geography names three other towns:
 * Voliba, which remains unidentified,
 * Uxella, possibly on the River Axe, or at Launceston, and
 * Tamara, generally considered to be somewhere on the River Tamar.

The Ravenna Cosmography includes the last two names (in slightly different forms, as "Tamaris" and "Uxelis"), and adds several more names which may be settlements in the territory. These include:
 * Nemetostatio, a name relating to nemeton, signifying "sanctuary' or "sacred grove". Probably to be identified with North Tawton in Devon where there is a Roman earthwork that may be military, or possibly a tax collection station.
 * Purocoronavis, which may refer to an important native hill fort, such as Carn Brea or Tintagel. The name has led to speculation about the Cornish Cornovii.

Other Romano-British sites in Dumnonia include:
 * Topsham, Devon - a settlement and harbour that served Isca Dumnoniorum to which it was connected by road and river.
 * Nanstallon (Cornwall) - a square military enclosure, seemingly associated with tin workings at nearby Boscarne.
 * Mount Batten (Devon) - an Iron Age tin port that continued into Roman times.
 * Plymouth (Devon) - evidence of a Roman settlement has been found on the north side of the harbour.
 * Ictis - an ancient port trading in tin.

New settlements continued to be built throughout the Roman period, including sites at Chysauster and Trevelgue Head. The style is native in form with no Romanised features. Near Padstow, a site of some importance that was inhabited from the late Bronze/early Iron Age to the mid 6th century now lies buried under the sands on the opposite side of the Camel estuary near St. Enodoc's Church, and may have been a western coastal equivalent of a Saxon Shore Fort. Byzantine and African pottery has been discovered at the site. At Magor Farm in Illogan, near Camborne, an archaeological site has been identified as being a villa.

Archaeology
The Dumnonii are thought to have occupied relatively isolated territory in Cornwall, Devon, Somerset and possibly part of Dorset. Their cultural connections, as expressed in their ceramics, were with the peninsula of Armorica across the Channel, rather than with the southeast of Britain. They do not seem to have been politically centralised: coins are relatively rare, none of them locally minted, and the structure, distribution and construction of Bronze Age and Iron Age hill forts, Cornish rounds, and defensible farmsteads in the south west point to a number of smaller tribal groups living alongside each other.

Dumnonia is noteworthy for its many settlements that have survived from the Romano-British period, but also for its lack of a villa system. Local archaeology has revealed instead the isolated enclosed farmsteads known locally as rounds. These seem to have survived the Roman abandonment of Britain, but were subsequently replaced, in the 6th and 7th centuries, by the unenclosed farms taking the Brythonic toponymic tre-.

As in most other Brythonic areas, Iron Age hill forts, such as Hembury Castle, were refortified for the use of chieftains or kings. Other high-status settlements such as Tintagel seem to have been reconstructed during this period. Post-Roman imported pottery has been excavated from many sites across the region, and the apparent surge in late 5th century Mediterranean and/or Byzantine imports is yet to be explained satisfactorily.

Industries
Apart from fishing and agriculture, the main economic resource of the Dumnonii was tin mining. The area of Dumnonia had been mined since ancient times, and the tin was exported from the ancient trading port of Ictis (St Michael's Mount). Tin extraction (mainly by streaming) had existed here from the early Bronze Age around the 22nd century BC. West Cornwall, around Mount's Bay, was traditionally thought to have been visited by metal traders from the eastern Mediterranean

During the first millennium BC trade became more organised, first with the Phoenicians, who settled Gades (Cadiz) around 1100 BC, and later with the Greeks, who had settled Massilia (Marseilles) and Narbo (Narbonne) around 600 BC. Smelted Cornish tin was collected at Ictis whence it was conveyed across the Bay of Biscay to the mouth of the Loire and then to Gades via the Loire and Rhone valleys. It went then through the Mediterranean Sea in ships to Gades.

During the period c. 500-450 BC, the tin deposits seem to have become more important, and fortified settlements appear such as at Chun Castle and Kenidjack Castle, to protect both the tin smelters and mines.

The earliest account of Cornish tin mining was written by Pytheas of Massilia late in the 4th century BC after his circumnavigation of the British Isles. Underground mining was described in this account, although it cannot be determined when it had started. Pytheas's account was noted later by other writers including Pliny the Elder and Diodorus Siculus.

It is likely that tin trade with the Mediterranean was later on under the control of the Veneti. Britain was one of the places proposed for the Cassiterides, that is Tin Islands. Tin working continued throughout Roman occupation although it appears that output declined because of new supplies brought in from the deposits discovered in Iberia (Spain and Portugal). However, when these supplies diminished, production in Dumnonia increased and appears to have reached a peak during the 3rd century AD.

Sub-Roman and post-Roman Dumnonia
The Sub-Roman or Post-Roman history of Dumnonia comes from a variety of sources and is considered exceedingly difficult to interpret given that historical fact, legend and confused pseudo-history are compounded by a variety of sources in Middle Welsh and Latin. The main sources available for discussion of this period include Gildas's De Excidio Britanniae and Nennius's Historia Brittonum, the Annales Cambriae, Anglo-Saxon Chronicle, William of Malmesbury's Gesta Regum Anglorum and De Antiquitate Glastoniensis Ecclesiae, along with texts from the Black Book of Carmarthen and the Red Book of Hergest, and Bede's Historia ecclesiastica gentis Anglorum as well as "The Descent of the Men of the North" (Bonedd Gwŷr y Gogledd, in Peniarth MS 45 and elsewhere) and the Book of Baglan.

Annales Cambriae

 * Phillimore, Egerton, ed., "The Annales Cambriae and Old Welsh Genealogies from Harleian MS. 3859", Y Cymmrodor; 9 (1888) pp. 141–183.
 * Remfry, P. M., Annales Cambriae: a Translation of Harleian 3859; PRO E.164/1; Cottonian Domitian, A 1; Exeter Cathedral Library MS. 3514 and MS Exchequer DB Neath, PRO E (ISBN 1-899376-81-X)
 * Williams (ab Ithel), John, ed. (1860), Annales Cambriae (4441288), London: Longman, Green, Longman, and Roberts.

title: Dumnonii