For the video game, go to Divinity: Original Sin In religion, divinity or Godhead is the state of things that are believed to come from a supernatural power or deity, such as God, the supreme being, creator deity, or spirits, are therefore regarded as sacred and holy. Such things are regarded as divine due to their transcendental origins or because their attributes or qualities are superior or supreme relative to things of the Earth. Divine things are regarded as eternal and based in truth, while material things are regarded as ephemeral and based in illusion; such things that may qualify as divine are apparitions, prophecies, in some views the soul, or more general things like resurrection, immortality and salvation. Otherwise what is or is not divine may be loosely defined, as it is used by different belief systems; the root of the word "divine" is "godly", but the use varies depending on which deity is being discussed. This article outlines the major distinctions in the conventional use of the terms.
For specific related academic terms, see Divinity, or Divine. 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. DV Divinity always carries connotations of goodness, beneficence and other positive, pro-social attributes. In monotheistic faiths there is an equivalent cohort of malefic supernatural beings and powers, such as demons, afreet, etc. which are not conventionally referred to as divine. Pantheistic and polytheistic faiths make no such distinction. 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, which itself translates as divinity. There are three distinct usages of divinity and divine in religious discourse: In monotheistic faiths, the word divinity is used to refer to the singular God central to that faith.
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, lacking the definite article — are sometimes used as to denote'god or certain other beings and entities which fall short of absolute Godhood but lie outside the human realm. These include: As noted, divinities are related to the transcendent force or power 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: 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 pan- and polytheistic faiths this implies the direct action of one god or another on the course of human events.
In Greek legend, for instance, it was Poseidon who raised the storms that blew Odysseus's craft off course on his return journey, Japanese tradition holds that a god-sent wind saved them from Mongol invasion. Prayers or propitiations are offered to specific gods of pantheisms 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 direct forms: miracles, visions, or intercessions by blessed figures. Transcendent force or power may operate through more subtle and indirect paths. Monotheistic faiths 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 unjust events are thrown on'the will of the Divine', in deferences like the Muslim inshallah and Christian'God works in mysterious ways'. 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. Other faiths are 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 proposes a transcendent operant principle — transliterated in English as tao or dao, meaning'the way' —, neither an entity or a being per se, but reflects the natural ongoing process of the world. Modern western mysticism and new age philosophy use the term'the Divine' as a noun in this latter sense: a non-specific principle or being that gives rise to the world, acts as the source o
Dagon (short story)
"Dagon" is a short story by American author H. P. Lovecraft, it is one of the first stories that Lovecraft wrote as an adult. It was first published in the November 1919 edition of The Vagrant. Dagon was published in Weird Tales, it is considered by many to be one of Lovecraft's most forward-looking stories. The story is the testament of a tortured, morphine-addicted man who relates an incident that occurred during his service as an officer during World War I. In the unnamed narrator's account, his cargo ship is captured by an Imperial German sea-raider in "one of the most open and least frequented parts of the broad Pacific", he escapes on a lifeboat and drifts aimlessly, south of the equator, until he finds himself stranded on "a slimy expanse of hellish black mire which extended about in monotonous undulations as far as could see.... The region was putrid with the carcasses of decaying fish and less describable things which saw protruding from the nasty mud of the unending plain." He theorizes that this area was a portion of the ocean floor thrown to the surface by volcanic activity, "exposing regions which for innumerable millions of years had lain hidden under unfathomable watery depths."After waiting three days for the seafloor to dry out sufficiently to walk on, he ventures out on foot to find the sea and possible rescue.
After two days of walking, he reaches his goal, a hill which turns out to be a mound on the edge of an "immeasurable pit or canyon". Descending the slope, he sees a gigantic white stone object that he soon perceives to be a "well-shaped monolith whose massive bulk had known the workmanship and the worship of living and thinking creatures." The monolith, situated next to a channel of water in the bottom of the chasm, is covered in unfamiliar hieroglyphs "consisting for the most part of conventionalized aquatic symbols such as fishes, octopuses, mollusks and the like." There are "crude sculptures" depicting: men—at least, a certain sort of men. Curiously enough, they seemed to have been chiseled badly out of proportion with their scenic background; as the narrator looks at the monolith, a creature emerges from the water: With only a slight churning to mark its rise to the surface, the thing slid into view above the dark waters. Vast, Polyphemus-like, loathsome, it darted like a stupendous monster of nightmares to the monolith, about which it flung its gigantic scaly arms, the while it bowed its hideous head and gave vent to certain measured sounds.
Horrified, the mariner flees back to his stranded boat and vaguely recalls a "great storm". His next memory is of a San Francisco hospital, where he was taken after being rescued in mid-ocean by a U. S. ship. There are no reports of any Pacific upheavals, he does not expect anyone to believe his incredible story, he mentions one abortive attempt to gain understanding of his experience: Once I sought out a celebrated ethnologist, amused him with peculiar questions regarding the ancient Philistine legend of Dagon, the Fish-God. Haunted by visions of the creature, "especially when the moon is gibbous and waning", he describes his fears for the future of humanity: I cannot think of the deep sea without shuddering at the nameless things that may at this moment be crawling and floundering on its slimy bed, worshipping their ancient stone idols and carving their own detestable likenesses on submarine obelisks of water-soaked granite. I dream of a day when they may rise above the billows to drag down in their reeking talons the remnants of puny, war-exhausted mankind --of a day when the land shall sink, the dark ocean floor shall ascend amidst universal pandemonium.
With the drug that has given him "transient surcease" running out, he declares himself ready to do himself in. The story ends with the narrator rushing to the window as he hears "a noise at the door, as of some immense slippery body lumbering against it." After reading Lovecraft's juvenilia in 1917, W. Paul Cook, editor of the amateur press journal The Vagrant, encouraged him to resume writing fiction; that summer, Lovecraft wrote two stories: "The Tomb" and "Dagon". The story was inspired in part by a dream. "I dreamed that whole hideous crawl, can yet feel the ooze sucking me down!" he wrote. Critic William Fulwiler indicates that Lovecraft may have been influenced by Irvin S. Cobb's "Fishhead", a story about a strange fish-like human. Fulwiler has suggested that Lovecraft took the story's theme of "an ancient prehuman race that will someday rise to conquer humanity" from Edgar Rice Burroughs' At the Earth's Core; the story mentions the Piltdown Man, which had not been exposed by the scientific community as a fraud and hoax at the time of writing.
As to the name of the story, Lovecraft seems to be referring to the ancient Sumerian god named Dagon, the fertility god of grains and fish, because in the story, the main character makes inquiries "....regarding the ancient Philistine legend of Dagon, the Fish-God." The Sumerian deity is sometimes depicted as being part fish, or wearing a fish. Since Lovecraft was fond of references to actual archaeological discoveries in his writings from time
Optimism is a mental attitude reflecting a belief or hope that the outcome of some specific endeavor, or outcomes in general, will be positive and desirable. A common idiom used to illustrate optimism versus pessimism is a glass filled with water to the halfway point: an optimist is said to see the glass as half full, while a pessimist sees the glass as half empty; the term derives from the Latin optimum, meaning "best". Being optimistic, in the typical sense of the word, is defined as expecting the best possible outcome from any given situation; this is referred to in psychology as dispositional optimism. It thus reflects a belief. For this reason, it is seen as a trait. Theories of optimism include dispositional models, models of explanatory style. Methods to measure optimism have been developed within both theoretical systems, such as various forms of the Life Orientation Test, for the original definition of optimism, or the Attributional Style Questionnaire designed to test optimism in terms of explanatory style.
Variation in optimism and pessimism is somewhat heritable and reflects biological trait systems to some degree. It is influenced by environmental factors, including family environment, with some suggesting it can be learned. Optimism may be linked to health. Researchers operationalize the term differently depending on their research; as with any trait characteristic, there are several ways to evaluate optimism, such as the Life Orientation Test. Dispositional optimism and pessimism are assessed by asking people whether they expect future outcomes to be beneficial or negative; the LOT returns separate pessimism scores for each individual. Behaviourally, these two scores correlate around r = 0.5. Optimistic scores on this scale predict better outcomes in relationships, higher social status, reduced loss of well-being following adversity. Health preserving behaviors are associated with optimism while health-damaging behaviors are associated with pessimism; some have argued that pessimism and optimism are ends of a single dimension, with any distinction between them reflecting factors such as social desirability.
Confirmatory modelling, supports a two-dimensional model and the two dimensions predict different outcomes. Genetic modelling confirms this independence, showing that pessimism and optimism are inherited as independent traits, with the typical correlation between them emerging as a result of a general well-being factor and family environment influences. Explanatory style is distinct from dispositional theories of optimism. While related to life-orientation measures of optimism, attributional style theory suggests that dispositional optimism and pessimism are reflections of the ways people explain events, i.e. that attributions cause these dispositions. Measures of attributional style distinguish three dimensions among explanations for events: Whether these explanations draw on internal versus external causes. In addition, the measures distinguish attributions for negative events. An optimistic person attributes internal and global explanations to good things. Pessimistic explanations attribute these traits of stability and internality to negative events, such as difficulty in relationships.
Models of Optimistic and Pessimistic attributions show that attributions themselves are a cognitive style – individuals who tend to focus on the global explanations do so for all types of events, the styles correlate among each other. In addition to this, individuals vary in how optimistic their attributions are for good events, on how pessimistic their attributions are for bad events, but these two traits of optimism and pessimism are un-correlated. There is much debate about the relationship between explanatory optimism; some researchers argue that optimism is the lay-term for what researchers know as explanatory style. More it is found that explanatory style is quite distinct from dispositional optimism, the two should not be used interchangeably as they are marginally correlated at best. More research is required to "bridge" or further differentiate these concepts; as with all psychological traits, differences in both dispositional optimism and pessimism and in attributional style are heritable.
Both optimism and pessimism are influenced by environmental factors, including family environment. It has been suggested that optimism may be indirectly inherited as a reflection of underlying heritable traits such as intelligence and alcoholism. There is evidence from twin studies that show, for instance, that the inherited component of the dispositional optimism is about 25 percent, making this trait a stable personality dimension and a predictor of life outcomes, its genetic origin, which interacts with environmental influences and other risks determines the vulnerability to depression across the lifespan. Many theories assume optimism can be learned, research supports a modest role of family-environment acting to raise optimism and lower neuroticism and pessimism. Work utilising brain imaging and biochemistry suggests that at a biological trait level and pessimism reflect brain systems specialised for the tasks of processing and incorporating beliefs regarding good and bad information respectively.
The Life Orientation Test was designed by Scheier and Carver to assess dispositional optimism – expecting positive or negative outcomes, is one of the more popular tests of optimism and pessimism. This was often use
The Dunwich Horror
"The Dunwich Horror" is a horror short story by American writer H. P. Lovecraft. Written in 1928, it was first published in the April 1929 issue of Weird Tales, it takes place in a fictional town in Massachusetts. It is considered one of the core stories of the Cthulhu Mythos. In the isolated, decrepit village of Dunwich, Wilbur Whateley is the hideous son of Lavinia Whateley, a deformed and unstable albino mother, an unknown father. Strange events surround his birth and precocious development. Wilbur matures at an abnormal rate. Locals shun him and his family, animals fear and despise him due to his odor. All the while, his sorcerer grandfather indoctrinates him into certain dark rituals and the study of witchcraft. Various locals grow suspicious after Old Whateley buys more and more cattle, yet the number of his herd never increases, the cattle in his field become mysteriously afflicted with severe open wounds. Wilbur and his grandfather have sequestered an unseen presence at their farmhouse. Year by year, this unseen entity grows to monstrous proportions, requiring the two men to make frequent modifications to their residence.
People begin to notice a trend of cattle mysteriously disappearing. Wilbur's grandfather dies, his mother disappears soon afterwards; the colossal entity occupies the whole interior of the farmhouse. Wilbur ventures to Miskatonic University in Arkham to procure their copy of the Necronomicon – Miskatonic's library is one of only a handful in the world to stock an original; the Necronomicon has spells that Wilbur can use to summon the Old Ones, but his family's copy is damaged and lacks the page he needs to open the "door." When the librarian, Dr. Henry Armitage, refuses to release the university's copy to him, Wilbur breaks into the library at night to steal it. A guard dog, maddened by Wilbur's alien body odor and kills him with unusual ferocity; when Dr. Armitage and two other professors, Warren Rice and Francis Morgan, arrive on the scene, they see Wilbur's semi-human corpse before it melts leaving no evidence. With Wilbur dead, no one attends to the mysterious presence growing in the Whateley farmhouse.
Early one morning, the farmhouse explodes and the thing, an invisible monster, rampages across Dunwich, cutting a path through fields and ravines, leaving huge "prints" the size of tree trunks. The monster makes forays into inhabited areas; the invisible creature terrorizes Dunwich for several days, killing two families and several policemen, until Armitage and Morgan arrive with the knowledge and weapons needed to kill it. The use of a magic powder renders it visible; the barn-sized monster screams for help – in English – just before the spell destroys it, leaving a huge burned area. In the end, its nature is revealed: it is Wilbur's twin brother, though it "looked more like the father than Wilbur did." Old Whateley Lavinia Whateley's "aged and half-insane father, about whom the most frightful tales of magic had been whispered in his youth". He has a large collection of "rotting ancient books and parts of books" which he uses to "instructs and catechise" his grandson Wilbur, he dies of natural causes on August 2, 1924.
Whateley was given no certain first name by Lovecraft, although Fungi from Yuggoth mentions a John Whateley. According to S. T. Joshi, "It is not certain where Lovecraft got the name Whateley," though there is a small town called Whately in northwestern Massachusetts near the Mohawk Trail, which Lovecraft hiked several times, including in the summer of 1928. Robert M. Price's short story "Wilbur Whateley Waiting" emphasizes the obvious pun in the name. Lavinia Whateley One of Lovecraft's few female characters. Born circa 1878, Lavinia Whateley is the spinster daughter of Old Whateley whose mother met an "unexplained death by violence" when Lavinia was 12, she is described as a somewhat deformed, unattractive albino woman...a lone creature given to wandering amidst thunderstorms in the hills and trying to read the great odorous books which her father had inherited through two centuries of Whateleys.... She had never been to school, but was filled with disjointed scraps of ancient lore that Old Whateley had taught her....
Isolated among strange influences, Lavinia was fond of wild and grandiose day-dreams and singular occupations. Elsewhere, she is called "slatternly crinkly-haired". In 1913, she gave birth to Wilbur Whately by an unknown father revealed to be Yog-Sothoth. On Halloween night in 1926, she disappeared under mysterious circumstances killed or sacrificed by her son. Wilbur Whateley Born February 1913 at 5 a.m. to Lavinia Whateley and Yog-Sothoth. Described as a "dark, goatish-looking infant"—neighbors refer to him as "Lavinny's black brat"—he shows extreme precocity: "Within three months of his birth, he had attained a size and muscular power not found in infants under a full year of age.... At seven months, he began to walk unassisted," and he "commenced to talk...at the age of only eleven months." At three years of age, "he looked like a boy of ten," while at four and a half, he "looked like a lad of fifteen. His lips and cheeks were fuzzy with a coarse dark down, his voice had begun to break." "Though he shared his mother's and grandfather's chinlessness, his firm and precociously shaped nose united with the expression of his large, dark Latin eyes to give him an air of..well-nigh pretern
Miscegenation is the mixing of different racial groups through marriage, sexual relations, or procreation mixing, perceived to negatively impact the purity of a particular race or culture. Anti-miscegenation is a prominent theme of white supremacy. Though the notion that racial mixing is undesirable has arisen at different points in history, it gained particular prominence in Europe during the era of colonialism; the term miscegenation entered the English language in the 19th century as racial segregation began to become more formalized in the United States. It was used to refer to interracial marriage and interracial sexual relations; the term came to be associated with laws banning interracial marriage and sex, known as anti-miscegenation laws. The term miscegenation is always used to refer to racist ideologies; when speaking about mixed-race relationships in a more neutral context, terms such as interracial, interethnic, or cross-cultural are more common in contemporary usage. In the present day, the word miscegenation is avoided by many scholars, because the term suggests that race is a concrete biological phenomenon, rather than a categorization imposed on certain relationships.
The term's historical use in contexts that implied disapproval is a reason why more unambiguously neutral terms such as interracial, interethnic or cross-cultural are more common in contemporary usage. The term remains in use among scholars when referring to past practices concerning multiraciality, such as anti-miscegenation laws that banned interracial marriages. In Spanish and French, the words used to describe the mixing of races are mestizaje, mestiçagem and métissage; these words, much older than the term miscegenation, are derived from the Late Latin mixticius for "mixed", the root of the Spanish word mestizo. These non-English terms for "race-mixing" are not considered as offensive as "miscegenation", although they have been tied to the caste system, established during the colonial era in Spanish-speaking Latin America. Today, the mixes among races and ethnicities are diverse, so it is considered preferable to use the term "mixed-race" or "mixed". In Portuguese-speaking Latin America, a milder form of caste system existed, although it provided for legal and social discrimination among individuals belonging to different races, since slavery for blacks existed until the late 19th century.
Intermarriage occurred from the first settlements, with their descendants achieving high rank in government and society. To this day, there are controversies if Brazilian class system would be drawn around socio-economic lines, not racial ones. Conversely, people classified in censuses as black, brown or indigenous have disadvantaged social indicators in comparison to the white population; the concept of miscegenation is tied to concepts of racial difference. As the different connotations and etymologies of miscegenation and mestizaje suggest, definitions of race, "race mixing" and multiraciality have diverged globally as well as depending on changing social circumstances and cultural perceptions. Mestizo are people of mixed white and indigenous Amerindian ancestry, who do not self-identify as indigenous peoples or Native Americans. In Canada, the Métis, who have Amerindian and white French-Canadian, have identified as an ethnic group and are a constitutionally recognized aboriginal people; the differences between related terms and words which encompass aspects of racial admixture show the impact of different historical and cultural factors leading to changing social interpretations of race and ethnicity.
Thus the Comte de Montlosier, in exile during the French Revolution, equated class difference in 18th-century France with racial difference. Borrowing Boulainvilliers' discourse on the "Nordic race" as being the French aristocracy that invaded the plebeian "Gauls", he showed his contempt for the lowest social class, the Third Estate, calling it "this new people born of slaves... mixture of all races and of all times". Miscegenation comes from the Latin miscere, "to mix" and genus, "kind"; the word was coined in the U. S. in 1863, the etymology of the word is tied up with political conflicts during the American Civil War over the abolition of slavery and over the racial segregation of African-Americans. The reference to genus was made to emphasize the distinct biological differences between whites and non-whites, though all humans belong to the same genus and the same species, Homo sapiens; the word was coined in an anonymous propaganda pamphlet published in New York City in December 1863, during the American Civil War.
The pamphlet was entitled Miscegenation: The Theory of the Blending of the Races, Applied to the American White Man and Negro. It purported to advocate the intermarriage of whites and blacks until they were indistinguishably mixed, as a desirable goal, further asserted that this was the goal of the Republican Party; the pamphlet was a hoax, concocted by Democrats, to discredit the Republicans by imputing to them what were radical views that offended against the attitudes of the vast majority of whites, including those who opposed slavery. There was much opposition to the war effort; the pamphlet and variations on it were reprinted in both the north and south by Democrats and Confederates. Only in November 1864 was the pamphlet exposed as a hoax; the hoax pamphlet was written by David Goodman Croly, managing editor of the New York World, a De
Michel Houellebecq is a French author and poet. His first book was a biographical essay on the horror writer H. P. Lovecraft. Houellebecq published his first novel, Whatever, in 1994, his next novel, published in 1998, brought him international fame as well as controversy. Platform followed in 2001, he published several books of poems, including The Art of Struggle in 1996. After a publicity tour for Platform led to his being taken to court for inciting racial hatred, he moved to Ireland for several years, he resides in France, where he has been described as "France’s biggest literary export and, some say, greatest living writer." In 2010 he published La Carte et le Territoire. Houellebecq was born in 1956 on the French island of Réunion, the son of Lucie Ceccaldi, a French doctor born in Algeria of Corsican descent and René Thomas, a ski instructor and mountain guide, he lived in Algeria from the age of five months with his maternal grandmother. His website states that his parents "lost interest in his existence pretty quickly" and at the age of six, he was sent to France to live with his paternal grandmother, a communist, while his mother left to live a hippie lifestyle in Brazil with her recent boyfriend.
His grandmother's maiden name was Houellebecq. He went to Lycée Henri Moissan, a high school at Meaux in the north-east of Paris, as a boarder, he went to Lycée Chaptal in Paris to follow preparation courses in order to qualify for grandes écoles. He began attending the Institut National Agronomique Paris-Grignon in 1975, he started a literary review wrote poetry. Houellebecq graduated from the Institut national agronomique Paris-Grignon in 1980, married and had a son, his first poems appeared in 1985 in the magazine La Nouvelle Revue. Six years in 1991, he published a biographical essay on the horror writer H. P. Lovecraft, a teenage passion, with the prophetic subtitle Against the World, Against Life. A short poetical essay named Rester vivant: méthode appeared the same year, dealing with the art of writing as a way of life. Meanwhile, he worked as a computer administrator in Paris, including at the French National Assembly, before he became the so-called "pop star of the single generation", starting to gain fame with his debut novel Extension du domaine de la lutte in 1994, published by Maurice Nadeau.
Throughout the 1990s Houellebecq published several books of poetry and articles in magazines such as L'Infini edited by Philippe Sollers. He lived at this time at the same address as writer Marc-Edouard Nabe, at 103, rue de la Convention in Paris. Nabe wrote about this proximity in Le Vingt-Septieme Livre, comparing both neighbours' careers and the way their writings were met by critics and audiences, his second novel, Les Particules Élémentaires proved to be his breakthrough, bringing him national, soon international fame and controversy for its intricate mix of brutally honest social commentary and pornographic depictions. It won the 1998 Prix Novembre, missing the more prestigious Prix Goncourt for which it was the favorite; the novel became an instant "nihilistic classic", was praised for the boldness of its ideas and thought-provoking qualities, although it was heavily criticized for its relentless bleakness as well as its vivid depictions of racism, paedophilia and for being an apology of eugenics.
The novel won Houellebecq the International Dublin Literary Award in 2002. In 2000, Houellebecq published the short fiction Lanzarote, in which he develops a number of the themes he would explore in novels, including sex tourism, fringe religions and cult leaders, his subsequent novel, confirmed him as a prominent writer. The book is a romance told in the first-person by a 40-year-old male arts administrator named Michel, who shares many real life characteristics with the author, including his apathy and low self-esteem; the novel's depiction of life and its explicit criticism of Islam, together with an interview its author gave to the magazine Lire, led to accusations against Houellebecq by several organisations, including France's Human Rights League, the Mecca-based World Islamic League as well as the mosques of Paris and Lyon. Charges were brought to trial, but a panel of three judges, delivering their verdict to a packed Paris courtroom, acquitted the author of having provoked'racial' hatred, ascribing Houellebecq's opinions to the legitimate right of criticizing r
In mathematics, non-Euclidean geometry consists of two geometries based on axioms related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries; when the metric requirement is relaxed there are affine planes associated with the planar algebras which give rise to kinematic geometries that have been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line ℓ and a point A, not on ℓ, there is one line through A that does not intersect ℓ. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ.
Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: In Euclidean geometry, the lines remain at a constant distance from each other and are known as parallels. In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular. In elliptic geometry, the lines intersect. Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, geometries that deviated from this were not accepted as legitimate until the 19th century; the debate that led to the discovery of the non-Euclidean geometries began as soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions and sought to prove all the other results in the work; the most notorious of the postulates is referred to as "Euclid's Fifth Postulate," or the "parallel postulate", which in Euclid's original formulation is: If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it appears to be more complicated than Euclid's other postulates: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, believed it could be proved as a theorem from the other four. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī, Giovanni Girolamo Saccheri; the theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were "the first few theorems of the hyperbolic and the elliptic geometries." These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the development of non-Euclidean geometry.
These early attempts at challenging the fifth postulate had a considerable influence on its development among European geometers, including Witelo, Levi ben Gerson, John Wallis and Saccheri. All of these early attempts made at trying to formulate non-Euclidean geometry, provided flawed proofs of the parallel postulate, containing assumptions that were equivalent to the parallel postulate; these early attempts did, provide some early properties of the hyperbolic and elliptic geometries. Khayyam, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher": "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge." Khayyam considered the three cases right and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid which he didn't realize was equivalent to his own postulate.
Another example is al-Tusi's son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusi's thoughts, which presented another hypothesis equivalent to the parallel postulate. "He revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements." His work was published in Rome in 1594 and was studied by European geometers, including Saccheri who criticised this work as well as that of Wallis. Giordano Vitale, in his book Euclide restituo, used the Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summ