:: wikimiki.org ::

Trivium

Trivium

:This article is about the University syllabus. :
- For the metalcore band, see Trivium (band). :
- For the stream cipher, see Trivium (cipher). In medieval universities, the trivium comprised the three subjects taught first, before the quadrivium. The word is Latin, meaning "the three ways" or "the three roads", the beginning of the liberal arts. It also serves as a root for the concept of triviality. At many medieval universities, such as Oxford, this would have been the principal undergraduate course. In medieval educational theory, the trivium consisted of grammar, rhetoric, and logic (or dialectic - logic and dialectic were synonymous at the time). (As Latin was both a second language and the international language of scholarship and thought, it had to be learned intentionally and thoroughly.) Grammar is the mechanics of a language; logic is the "mechanics" of thought and analysis; rhetoric is the use of language to instruct and persuade. These were considered preparatory fields for the quadrivium, which was made up of arithmetic, geometry, music, and astronomy. In turn, the quadrivium was considered preparatory work for the serious study of philosophy and theology. This schema is sometimes referred to as classical education, but it is more accurately a development of the 12th and 13th centuries rather than a direct descendant of the educational systems of antiquity.

External links

- [http://www.classical-homeschooling.org/trivium.html Classical Christian Homeschooling - Trivium] Category:History of education

Metalcore

Metalcore is a musical genre consisting of a mix between heavy metal and hardcore. Although the genre has risen in popularity since the turn of the millennium, it is not a recent genre, as many would believe, since bands such as Integrity have been around since the late 1980s. Defining the metalcore sound is not an easy task as bands have often fused hardcore-influenced sound and attitude with almost any imaginable type of metal. In fact, the earliest signs of this genre before a name could be put to it was called 'crossover'. The band that declared the crossover was Dirty Rotten Imbeciles on their late 1980's album of the same name. Other bands like Nuclear Assault are the bleeding edge pioneers themselves.

The early scene

Nuclear Assault was amongst the first to call their music a Heavy Metal-Hardcore Punk Hybrid. Although not thought of as a metalcore band today, Judge was arguably one of the earliest bands to start fusing heavy metal-influenced riffing with more traditional hardcore sound without being a thrash metal band. This idea obviously spread outwards, and although their first real release ("Those Who Fear Tomorrow") wasn't until 1991, "Integrity" was formed in 1989. Most songwriting by metalcore bands at this time was similar to New York hardcore bands, but differed in their harder sound thanks to use of double bass drums, harder distortion and louder, more gruff vocal shouts. This basic sound of metalcore has received the epithets - which can be used both with and without derision - "tough guy hardcore" due to the lyrical focus, which is often similar to older hardcore in that they call for moral and mental strength and integrity, but may also have a slight focus on violence, or "moshcore" due to the often breakdown-centric, mosh-friendly songwriting that some bands use. During the middle of the 1990s, bands started expanding the metalcore sound, prime examples being All Out War who used straightforward thrash riffing, as well as bands such as Rorschach, Starkweather, Orange County's Adamantium, and Deadguy, who experimented with looser, often discordant songwriting as well as more untraditional rhythm. Converge, although starting out as self-confessed "hardcore kids with leftover Slayer riffs", have since bloomed into a hybrid of hardcore, metal and progressive instrumental and electronic experimentation. Zao is another band that left a mark upon the genre with their Carcass-like vocals and varied songwriting, particularly the Christian bands of the genre.

The later scene

From the late 1990s and particularly after the turn of the millennium, metalcore has grown immensely, to the point where major record companies are taking interest in the genre. Recent (2005) releases, such as As I Lay Dying's "Shadows Are Security" and Norma Jean's "O' God, the Aftermath" have managed to sell well enough to make it onto Billboard charts. One sound that has become immensely popular is to mix Gothenburg melodic death metal, popularized by more traditional metal bands such as At the Gates and In Flames and newer bands such as Children of Bodom, together with a more traditional metalcore sound and occasionally slow, melodic breakdowns. Today, many famous metalcore bands play this style, such as Killswitch Engage, The Agony Scene, As I Lay Dying and Unearth. Poison the Well are given some credit for making this sound popular, even though they do not strictly fit into that sound. Some bands, such as Botch, expanded on the blueprints of Rorschach and Deadguy, bringing forth intense mathematic influenced guitar riffs, as well as songs. This style is popularized by few recent bands. The sounds of metalcore today are so varied, that one can almost use a mix-and-match approach of aural particulars and still arrive at a particular band's sound. For example, on later releases, Shai Hulud were able to mix extremely varied, melodic, near-mathematical songwriting with the more traditional hardcore sound. In recent years, Christian lyrics, or at least Christian band members have grown in numbers. It has known to be a "trend" and an easier way to get the band signed to a record label, although most people call tell which bands are real and which arent. Wheather real or not, many of the known Christian bands include As I Lay Dying, Haste the Day, MyChildren MyBride, Norma Jean, Zao, The Chariot, Underoath, and many others.

The breakdown

Central to many bands of the genre, quite a few of which eschew traditional verse-chorus-verse songwriting, is the breakdown. Stereotypically, a breakdown consists of slowing a song down, giving the guitars room to play a set of rhythmically oriented riffs, usually on open strings so as to achieve the lowest sound for which the guitars are tuned. These riffs are often accented by the drummer through double bass drums. Breakdowns are usually responded to by an audience by hardcore dancing. Vocalists also tend to throw in a single, repeated statement throughout the breakdown, giving those who are not dancing an opportunity to sing along. Many metalcore bands rely on having memorable breakdowns rather than memorable choruses. Songs with breakdowns have become more common, and some bands have used them far more often than was previously the norm, with some songs even resembling one elongated breakdown.

Bands

Trivium (band)

Trivium is a metalcore band with melodic death metal and thrash metal influences band from Altamonte Springs, Florida. Bands that have inspired their sound include Unearth and Atreyu and to some extent Avenged Sevenfold, Pantera, and Metallica. Recently, they won the Kerrang award for best International Newcomer. Trivium's music includes the traditional "riffing" and screaming of metalcore, but also has impressive instrumental breaks including solos and guitar work more attributed to melodic metal. Also found in their music are slow, epic choruses, with clean singing, a la Avenged Sevenfold (circa 2002-2003). The band formed in 2000 when Corey Beaulieu, Brent Young and Travis Smith witnessed Matt Heafy perform at a high school talent show, singing The Offspring's Self Esteem. He was immediately asked to join the group. Trivium's debut with Roadrunner Records, Ascendancy, scanned a 7,054 on the Neilsen SoundScan charts. The album also debuted at #4 on the US HeatSeeker Charts and #151 on the US Billboard Top 200 Charts, which landed Trivium the title of 4th highest new band debut in Roadrunner's history.

Band Members

- Matt Heafy: Guitar, Lead Vocals
- Corey Beaulieu: Guitar, Vocals
- Paolo Gregoletto: Bass, Vocals
- Travis Smith: Drums

Previous Members

- Brent Young: Bass

Discography

- EP (2003)
- Ember to Inferno (2003)
- Ascendancy (2005)
- The Crusade (2006) The demo version of song "Like Light To The Flies" is included in the MTV2 Headbangers Ball Volume 2 compilation. Pull Harder On The Strings of Your Martyr was on the movie The Cave's soundtrack

Video Game Appearances

- Rainbow Six: Lockdown
- The Sims 2 (Non-PC Versions)
- Infected

Videos

- Like Light To The Flies
- Pull Harder On The Strings Of Your Martyr
- A Gunshot To The Head Of Trepidation
- Dying in Your Arms - Shooting in December 2005

External links

- [http://www.trivium.org Trivium.org Official Trivium Website]
- [http://www.myspace.com/trivium Official Trivium Myspace Page]
- [http://www.livejournal.com/community/triviumfans/ Livejournal Trivium Community Blog]
- [http://www.triviumworld.com Trivium world.com Trivium's #1 Fansite]
- [http://www.gotrivium.com goTRIVIUM.com Open-Public US Promotions] Category:Heavy metal musical groups Category:American musical groups

Stream cipher

In cryptography, a stream cipher is a symmetric cipher in which the plaintext digits are encrypted one at a time, and in which the transformation of successive digits varies during the encryption. An alternative name is a state cipher, as the encryption of each digit is dependent on the current state. In practice, the digits are typically single bits or bytes. Stream ciphers represent a different approach to symmetric encryption from block ciphers. Block ciphers operate on large blocks of digits with a fixed, unvarying transformation. This distinction is not always clear-cut: some modes of operation use a block cipher primitive in such a way that it then acts effectively as a stream cipher. Stream ciphers typically execute at a higher speed than block ciphers and have lower hardware complexity. However, stream ciphers can be susceptible to serious security problems if used incorrectly: see stream cipher attacks — in particular, the same starting state must never be used twice. NSA documents sometimes use the term combiner-type algorithms, referring to algorithms that use some function to combine a pseudorandom number generator (PRNG) with a plaintext stream.

Loose inspiration from the one-time pad

Stream ciphers can be viewed as approximating the action of a theoretically unbreakable cipher, the one-time pad (OTP), sometimes known as the Vernam cipher. A one-time pad uses a key stream of completely random digits. The key stream is combined with the plaintext digits one at a time to form the ciphertext. This system was proved to be theoretically secure by Shannon in 1949. However, the key stream must be (at least) the same length as the plaintext, and generated completely at random. This makes the system very cumbersome to implement in practice, and as a result the one-time pad has not been widely used, except for the most critical applications. A stream cipher makes use of a much smaller and convenient key — 128 bits, for example. Based on this key, it generates a pseudorandom key stream which can be combined with the plaintext digits in a similar fashion to the one-time pad. However, this comes at a cost: because the keystream is now pseudorandom, and not truly random, the proof of security associated with the one-time pad no longer holds: it is quite possible for a stream cipher to be completely insecure.

Types of stream cipher

A stream cipher generates successive elements of the keystream based on an internal state. This state is updated in essentially two ways: if the state changes independently of the plaintext or ciphertext messages, the cipher is classified as a synchronous stream cipher. By contrast, self-synchronising stream ciphers update their state based on previous ciphertext digits.

Synchronous stream ciphers

In a synchronous stream cipher a stream of pseudo-random digits is generated independently of the plaintext and ciphertext messages, and then combined with the plaintext (to encrypt) or the ciphertext (to decrypt). In the most common form, binary digits are used (bits), and the keystream is combined with the plaintext using the exclusive or operation (XOR). This is termed a binary additive stream cipher. In a synchronous stream cipher, the sender and receiver must be exactly in step for decryption to be successful. If digits are added or removed from the message during transmission, synchronisation is lost. To restore synchronisation, various offsets can be tried systematically to obtain the correct decryption. Another approach is to tag the ciphertext with markers at regular points in the output. If, however, a digit is corrupted in transmission, rather than added or lost, only a single digit in the plaintext is affected and the error does not propagate to other parts of the message. This property is useful when the transmission error rate is high; however, it makes it less likely the error would be detected without further mechanisms. Moreover, because of this property synchronous stream ciphers are very susceptible to active attacks — if an attacker can change a digit in the ciphertext, he might be able to make predictable changes to the corresponding plaintext bit; for example, flipping a bit in the ciphertext causes the same bit to be flipped in the plaintext.

Self-synchronising stream ciphers

Another approach uses several of the previous N ciphertext digits to compute the keystream. Such schemes are known as self-synchronizing stream ciphers, asynchronous stream ciphers or ciphertext autokey (CTAK). The idea of self-synchronization was patented in 1946 (), and has the advantage that the receiver will automatically synchronise with the keystream generator after receiving N ciphertext digits, making it easier to recover if digits are dropped or added to the message stream. Single-digit errors are limited in their effect, affecting only up to N plaintext digits. It is somewhat more difficult to perform active attacks on self-synchronising stream ciphers by comparison with their synchronous counterparts. An example of a self-synchronising stream cipher is a block cipher in cipher-feedback mode (CFB).

LFSR-based stream ciphers

block cipher in cipher-feedback mode Binary stream ciphers are often constructed using linear feedback shift registers (LFSR)s because they can be easily implemented in hardware and can be readily analysed mathematically. The use of LFSRs on their own, however, is insufficient to provide good security. Various schemes have been proposed to increase the security of LFSRs.

Non-linear combining functions

hardware Because LFSRs are inherently linear, one technique for removing the linearity is to feed the outputs of several parallel LFSRs into a non-linear Boolean function to form a combination generator. Various properties of such a combining function are critical for ensuring the security of the resultant scheme, for example, in order to avoid correlation attacks.

Clock-controlled generators

Normally LFSRs are stepped regularly. One approach to introducting non-linearity is to have the LFSR clocked irregularly, controlled by the output of a second LFSR. Such generators include the stop-and-go generator, the alternating step generator and the shrinking generator. The stop-and-go generator (Beth and Piper, 1984) consists of two LFSRs. One LFSR is clocked if the output of a second is a "1", otherwise it repeats its previous output. This output is then (in some versions) combined with the output of a third LFSR clocked at a regular rate. The shrinking generator takes a different approach. Two LFSRs are used, both clocked regularly. If the output of the first LFSR is "1", the output of the second LFSR becomes the output of the generator. If the first LFSR outputs "0", however, the output of the second is discarded, and no bit is output by the generator. This mechanism suffers from timing attacks on the second generator, since the speed of the output is variable in a manner that depends on the second generator's state. This can be alleviated by buffering the output.

Filter generator

Another approach to improving the security of an LFSR is to pass the entire state of a single LFSR into a non-linear filtering function.

Other designs

shrinking generator

Security

:Main article: Stream cipher attack To be secure, the period of the keystream, that is, the number of digits output before the stream repeats itself, needs to be sufficiently large. If the sequence repeats, then the overlapping ciphertexts can be aligned against each other "in depth", and there are techniques which could allow the plaintext to be extracted. This can be a practical concern: for example, the DES block cipher was initially allowed to be used in a certain mode (OFB) with a varying parameter. However, for most choices of this parameter, the resulting stream had a period of only 2³² — for many applications, this period is far too low. For example, if encryption is being performed at a rate of 1 megabyte per second, a stream of period 2³² will repeat after around 8.5 minutes.

Usage

Stream ciphers are often used in applications where plaintext comes in quantities of unknowable length—for example, a secure wireless connection. If a block cipher were to be used in this type of application, the designer would need to choose either transmission efficiency or implementation complexity, since block ciphers cannot directly work on blocks shorter than their block size. For example, if a 128-bit block cipher received separate 32-bit bursts of plaintext, three quarters of the data transmitted would be padding. Block ciphers must be used in ciphertext stealing or residual block termination mode to avoid padding, while stream ciphers eliminate this issue by naturally operating on the smallest unit that can be transmitted (usually bytes). Another advantage of stream ciphers in military cryptography is that the cipher stream can be generated in a separate box that is subject to strict security measures and fed to other devices, e.g. a radio set, which will perform the xor operation as part of their function. The latter device can then be designed and used in less stringent environments. RC4 is the most widely-used stream cipher in software; others include: A5/1, A5/2, Chameleon, FISH, Helix, ISAAC, MUGI, Panama, Pike, QUISCI, SEAL, SOBER, SOBER-128 and WAKE.

References

- Matt J. B. Robshaw, Stream Ciphers Technical Report TR-701, version 2.0, RSA Laboratories, 1995 [http://security.ece.orst.edu/koc/ece575/rsalabs/tr-701.pdf (PDF)].
- Thomas Beth and Fred Piper, The Stop-and-Go Generator. EUROCRYPT 1984, pp88-92.

Medieval

The Middle Ages formed the middle period in a traditional schematic division of European history into three "ages": the classical civilization of Antiquity, the Middle Ages, and modern times. The Middle Ages of Western Europe are commonly dated from the end of the Western Roman Empire (5th century) until the rise of national monarchies, the start of European overseas exploration, the humanist revival, and the Protestant Reformation starting in 1517. These various changes all mark the beginning of the Early Modern period that preceded the Industrial Revolution. The Middle Ages are commonly referred to as the medieval period or medieval times or simply medieval.

The Early Middle Ages

medieval flourished in the early Middle Ages: Hildesheim.]] As the authority of the Roman Empire dwindled in Western Europe, its territories were entered and settled by succeeding waves of "barbarian" tribal confederations, some of whom distrusted and rejected the classical culture of Rome, while others, like the Goths admired it and considered themselves the legatees and heirs of Rome. Prominent among these peoples in the movement were the Huns and Avars and Magyars with the large number of Germanic and later Slavic peoples. The era of the migrations is referred to as the Migration Period. It has historically been termed the "Dark Ages" by Western European historians, and as Völkerwanderung ("wandering of the peoples") by German historians. The term "Dark Ages" has now fallen from favour, partly to avoid the entrenched stereotypes associated with the phrase, but also partly because more recent research into the period has in fact revealed its surprising artistic sophistication, though its political and social senses were unevolved and its technologies undeveloped, compared to the preceding culture. Although the settled population of the Roman period were not everywhere decimated, the new peoples greatly altered established society, and with it, law, culture and religion, and patterns of property ownership. The Pax Romana, with its accompanying benefits of safe conditions for trade and manufacture, and a unified cultural and educational milieu of far-ranging connections, had already been in decline for some time as the 5th century drew to a close. Now it was largely lost, to be replaced by the rule of local potentates, and the gradual break-down of economic and social linkages and infrastructure. This break-down was often fast and dramatic as it became unsafe to travel or carry goods over any distance and there was a consequent collapse in trade and manufacture for export. Major industries that depended on trade, such as large-scale pottery manufacture, vanished almost overnight in places like Britain. The Islamic invasions of the 7th and 8th centuries, which conquered the Levant, North Africa, Spain, Portugal and some of the Mediterranean islands (including Sicily), increased localization by halting much of what remained of seaborne commerce. So where sites like Tintagel in Cornwall had managed to obtain supplies of Mediterranean luxury goods well into the 6th century, this connection too was lost. Administrative, educational and military infrastructure quickly vanished, leading to the rise of illiteracy among leadership.

A new order

Until recently it has been common to speak of "barbarian invasions" sweeping in from beyond Imperial borders and bringing about the end of the Roman Empire. Modern historians now acknowledge that this presents an incomplete portrait of a complex time of migration. In some important cases, such as that of the Franks entering Gaul, settlement of the newcomers took place over many decades, as groups seeking new economic opportunities crossed into Roman territory, retaining their own tribal leadership, and acculturating to or displacing the Gallo-Roman society, often without widespread violence. Other outsiders, like Theodoric of the Ostrogoths, were civilized, though illiterate patrons, who saw themselves successors to the Roman tradition, employing cultured Roman ministers, like Cassiodorus. Like the Goths, many of the outsiders were foederati, military allies of the Empire, who had earned rights of settlement, including among others the Franks and the Burgundians. Between the 5th and 8th centuries a completely new political and social infrastructure developed across the lands of the former empire, based upon powerful regional noble families, and the newly established kingdoms of the Ostrogoths in Italy, Visigoths in Spain and Portugal, Franks and Burgundians in Gaul and western Germany, and Saxons in England. These lands remained Christian, and their Arian conquerors were soon converted, following the example of the pagan Frank Clovis I. The interaction between the culture of the newcomers, the remnants of classical culture, and Christian influences, produced a new model for society. The centralized administrative systems of the Romans did not withstand the changes, and the institutional support for large scale chattel slavery largely disappeared. However beyond these areas of Europe were many people with little or no contact with Christianity or with classic Roman culture. Warrior people such as the Avars and the Vikings were still capable of causing major disruption to the newly emerging societies of Western Europe. The Christian Church, the only centralized institution to survive the fall of the western Roman Empire intact, was the sole unifying cultural influence, preserving its selection from Latin learning, maintaining the art of writing, and a centralized administration through its network of bishops. The Early Middle Ages are characterized by the urban control of bishops and the territorial control exercised by dukes and counts. The rise of urban communes marked the beginning of the High Middle Ages. bishop Outside the de-urbanized remains of cities, the power of central government was greatly reduced. Consequently government authority, and responsibility for military organization, taxation and law and order, was delegated to provincial and local lords, who supported themselves directly from the proceeds of the territories over which they held military, political and judicial power. In this lay the beginnings of the feudal system. The High Middle Ages would see the regrowth of centralized power, and the growth of new "national" identities, as strong rulers sought to eliminate competition (and potential threat to their rule) from powerful feudal nobles. Well known examples of such consolidation include the Albigensian Crusade and the Wars of the Roses. This hierarchy of reciprocal obligations, known as feudalism or the feudal system, binding each man to serve his superior in return for the latter's protection, made for a confusion of territorial sovereignty (since allegiances were subject to change over time, and were sometimes mutually contradictory). The benefit of feudalism however, was its resiliency, and the ability of local arrangements to provide stable government in the absence of a strong royal power in a political order distinguished by its lack of uniformity. Territoriality was reduced to a network of personal allegiances. In the east, the Eastern Roman Empire (called by historians the "Byzantine Empire"), maintained a form of Christianised Roman rule in the lands of Asia Minor, Greece and the Slavic territories bordering Greece, and in Sicily and southern Italy. The eastern emperors had maintained a nominal claim to rule over the west, reconquered by Belisarius, but this was a political fiction under Lombard rule and became strongly disputed from 800, with the creation of the so-called Holy Roman Empire, under Charlemagne, briefly uniting much of modern day France, western Germany and northern Italy. From now on, Europe was to be bi-polar, with east and west competing for power and influence in the largely un-christianized expanses of northern Europe. The spread of Christianity in the Migrations Period, both from the Mediterranean area and from Ireland, occasioned a pre-eminent cultural and ideological role for its abbots, and the collapse of a res publica meant that the bishops became identified with the remains of urban government. Christianity provided the basis for a first European "identity," Christendom, unified until the separation of Orthodox Churches from the Catholic Church in the Great Schism of 1054, one of the dates that marks the onset of the High Middle Ages.

A Carolingian renaissance

See also the careers of Charlemagne and Otto I, Holy Roman Emperor.
The High Middle Ages
:Main article: High Middle Ages From beginnings roughly about the year 1000, greater stability came to the lands of western Europe. With the brief exception of the Mongol incursions, major barbarian invasions had ceased. The advance of Christian kingdoms and military orders into previously pagan regions in the Baltic and Finnic northeast brought the forced assimilation of numerous native peoples to the European entity. The "High Middle Ages" describes the expansionist culture and intellectual revival from the late 11th century to the beginning of the 14th. In central and northern Italy and in Flanders the rise of towns that were self-governing to some degree within their territories marked a beginning for re-urbanization in Western Europe. In Spain and Portugal, a slow reconquest of the urban and literate Muslim-ruled territories began. One consequence of this was that the Latin-literate world gained access to libraries that included classical literature and philosophy. Through translations these libraries gave rise to a vogue for the philosophy of Aristotle. Meanwhile, trade grew throughout Europe as the dangers of travel were reduced, and steady economic growth resumed. This period saw the formation of the Hanseatic league and other trading and banking institutions that operated across western Europe. The first universities were established in major European cities from 1080 onwards, bringing in a new interest and inquisitiveness about the world. Literacy began to grow, and there were major advances in art, sculpture, music and architecture. Large cathedrals were built across Europe, first in the romanesque, and later in the more decorative gothic style.
The Crusades
:Main article: Crusade Following the Great Schism, prime examples of the force of the divided cultural identities of Christendom can be found in the unfolding developments of the Crusades, during which Popes, kings, and emperors drew on the concept of Christian unity to inspire the population of Western Europe to unite and defend Christendom from the aggression of Islam, often at the expense of the Byzantine Empire. From the 7th century onward, Islam had been gaining ground along Europe's southern and eastern borders. Muslim armies conquered Egypt, the rest of North Africa, Jerusalem, Spain, Sicily, and most of Anatolia (in modern Turkey), although they were finally turned back in western Europe by Christian armies at the Battle of Tours in southern France. Political unanimity in Europe was less secure than it appeared, however, and the military support for most crusades was drawn from limited regions of Europe. Substantial areas of northern Europe also remained outside Christendom until the twelfth century or later; these areas also became crusading venues during the expansionist High Middle Ages.
Technology
:Main article: Medieval technology During the 12th and 13th century in Europe there was a radical change in the rate of new inventions, innovations in the ways of managing traditional means of production, and economic growth. The period saw major technological advances, including the invention of cannons, spectacles and artesian wells; and the cross-cultural introduction of gunpowder, silk, compass and astrolabe from the east. There was also great improvements with ships and upon the clock. The latter advances made possible the dawn of the Age of Exploration.
The Late Middle Ages (circa 1300-1500)
:Main article: Late Middle Ages The 14th century witnessed a decline that began with the first economic retrenchment after the long, gently inflationary rise of a unified economy that had been under way since the 11th century. The European climate itself was worsening, after the long Medieval Warm Period, leading to the onset of the Little Ice Age. In the Black Death, large areas of Western Europe lost up to a third of their population, especially in the crowded conditions of the towns, where the heart of innovations lay. The Black Death sealed a sudden end to the previous period of massive change, which resumed centuries later in the Early Modern Period. Politically, the later Middle Ages were typified by the decline of feudal power replaced by the development of strong, royalty-based nation-states. Wars between kingdoms, such as the Hundred Years' War between England and France, weakened the Christian nations in their confrontations with Islam. Religously Christendom was increasingly divided during the Western Schism, which resulted in greater loyalty to national churches, though lay piety rarely wavered. The Great Famine of 1315-1317, the Black Death of 1348, popular uprisings all produced stresses while encouraging creative social, economic, and technological responses that signalled the end of the old medieval order and laying the groundwork for further great changes in the Early Modern Period. In the east, the Byzantine Empire followed a separate destiny, with its strongest period coinciding with the Western collapse during the Early Medieval period. After the Battle of Manzikert (1071), the former empire was reduced to a shell; it survived until 1453, but in a diminished and weakened form.
Historiography

Middle Ages in history
:Main article: Middle Ages in history After the Middle Ages ended subsequent generations imagined, portrayed and interpreted the Middle Ages in different ways. Every century has created its own vision of the Middle Ages, the 18th century view of the Middle Ages was entirely different from the 19th century which was different from the 16th century view. The reality of these images remains with us today in the form of film, architecture, literature, art and popular conception.
Medieval and Middle Ages

"Middle Age"
The term "Middle Age" ("medium ævum") was first coined by Flavio Biondo, an Italian humanist, in the early 15th Century. Until the Renaissance (and some time after) the standard scheme of history was to divide history into six ages, inspired by the biblical six days of creation, or four monarchies based on Daniel 2:40. The early Renaissance historians instead talked about two periods in history, that of Ancient times and that of the period referred to as the "Dark Age". In the early 15th Century it was believed history had evolved from the Dark Age to a Modern period and scholars began to write about a middle period between the Ancient and Modern, which became known as the Middle Age. This is known as the three period view of history. The plural form of the term, Middle "Ages", is used in English, Dutch, Russian and Icelandic while all other European languages uses the singular form. This difference originates in different Neo-Latin terms used for the Middle Ages before media aetas became the standard term. Some were singular (media aetas, media antiquitas, medium saeculum and media tempestas), others plural (media saecula and media tempora). There seem to be no simple reason why a particular language ended up with the singular or the plural form. Further information can be found in Fred C. Robinson: "Medieval, the Middle Ages" in Speculum, Vol. 59:4 (Oct. 1984), p. 745-56. The common subdivision Early, High and Late Middle Ages came into use after World War I. It was caused by the works of Henri Pirenne (in particular the article "Les periodes de l'historie du capitalism" in Academie Royale de Belgique. Bulletin de la Classe des Lettres, 1914) and Johan Huizinga (The Autumn of the Middle Ages, 1919). A medieval era can also be applied to other parts of the world that historians have seen as embodying the same feudal characteristics as Europe in this period. The pre-westernization period in the history of Japan is sometimes referred to as medieval. The pre-colonial period in the developed parts of sub-Saharan Africa is also sometimes termed medieval. Today historians are far more reluctant to try to fit the history of other regions to the European model and these terms are less often used.
"Medieval"
The term "medieval" was first contracted from the Latin medium ævum, or more precisely "middle epoch", by Enlightenment thinkers as a pejorative descriptor of the Middle Ages. The spelling of "medieval" may depend on context. Medieval is the modern English spelling, used in normal discourse in England and elsewhere. Mediaeval is a legacy of the Latin spelling Mediæval, which uses the diphthong ae rendered as a ligature; it is an antiquated spelling found in older works, or those that emphesis the words Latin origins. Medieval was originally a pejorative description, and as such it has taken on broader meanings that usually impart some kind of value judgement, such as things that are old, "byzantine", "gothic", crude, heavy, harsh, or dark in nature.
Periodization issues
:See also: Periodization It is extremely difficult to decide when the Middle Ages ended, and in fact scholars assign different dates in different parts of Europe. Most scholars who work in 15th century Italian history, for instance, consider themselves Renaissance or Early Modern historians, while anyone working on England in the early 15th century is considered a medievalist. Others choose specific events, such as the Turkish capture of Constantinople or the end of the Anglo-French Hundred Years' War (both 1453), the invention of printing by Johann Gutenberg (around 1455) or the fall of Muslim Spain or Columbus's voyage to America (both 1492), or the Protestant Reformation starting 1517 to mark the period's end. In England the change of monarchs which occurred on 22 August 1485 at the Battle of Bosworth is often considered to mark the end of the period, Richard III representing the old medieval world and the Tudors, a new royal house and a new historical period. Similar differences are now emerging in connection with the start of the period. Traditionally, the Middle Ages is said to begin when the West Roman Empire formally ceased to exist in 476. However, that date is not important in itself, since the West Roman Empire had been very weak for some time, while Roman culture was to survive at least in Italy for yet a few decades or more. Today, some date the beginning of the Middle Ages to the division and Christianization of the Roman Empire (4th century) while others, like Henri Pirenne see the period to the rise of Islam (7th century) as "late Classical". The Middle Ages are often subdivided into an early period (sometimes called the "Dark Ages", at least from the fifth to eighth centuries) of shifting polities, a relatively low level of economic activity and successful incursions by non-Christian peoples (Slavs, Arabs, Scandinavians, Magyars); a middle period (the High Middle Ages) of developed institutions of lordship and vassalage, castle-building and mounted warfare, and reviving urban and commercial life; and a later period of growing royal power, the rise of commercial interests and weakening customary ties of dependence, especially after the 14th-century plague.
Religion in the Middle Ages

- Holy Roman Empire
- The Crusades
- Pilgrimage
- Papacy
- Medieval Inquisition
- Heresy (for example, Arian; Cathar; John Wyclif)
- Alchemy
- Monastic orders
- Benedictines
- Carthusians
- Cistercians
- Mendicant friars
- Franciscans
- Dominicans
- Carmelites
- Augustinians
- Judaism
- Islam (Western Europe): Moors
- Islam (Eastern Europe): Sultanate of Rum & Ottoman Empire
See also

- Medieval art
- Medieval architecture
- Medieval climate optimum
- Medieval communes
- Medieval Chronological Timeline
- Medieval demography
- Middle Ages in film
- Medieval guilds
- Medieval hunting
- Medieval medicine
- Medieval music
- Medieval tournament
- Slave trade in the Middle Ages
- History of the Jews in the Middle Ages
Selected bibliography

- Monumenta Germaniae Historica
- Migne's Patrologiae
- Liber Pontificalis
- C. Warren Hollister and Judith M. Bennett, Medieval Europe, A Short History. McGraw-Hill, 2002.
- David Abulafia et al., The New Cambridge Medieval History. Cambridge, 1995.
External links

- [http://www.fordham.edu/halsall/ Internet Medieval Sourcebook Project] Primary source archive of the Middle Ages. See also Internet Medieval Sourcebook.
- [http://www.the-orb.net/ The Online Reference Book of Medieval Studies] Academic peer reviewed articles. See also Online Reference Book of Medieval Studies.
- [http://the-orb.net/ The Labyrinth] Resources for Medieval Studies.
- [http://www.netserf.org/ NetSERF] The Internet Connection for Medieval Resources.
- [http://www.medievalmap.net Interactive Medieval Map] (Flash Plug-in Required.)
- [http://www.sca.org.au/cunnan/ Cunnan: A Wiki collecting information for re-enactors of the Middle Ages and Renaissance] with a heavy slant towards members of the SCA
- [http://www.shadowedrealm.com/ Shadowed Realm - Medieval Content and Discussion] Contains hundreds of glossary terms, a timeline, quotations, quizzes, a wiki, forums, and more.
- [http://www.medieval-castles.org Contains Medieval Castles and their history.] ja:中世 simple:Middle Ages

Quadrivium

The quadrivium was comprised of the four subjects taught in medieval universities after the trivium. The word is Latin, meaning "the four ways" or "the four roads": the completion of the liberal arts. At many medieval universities, such as Oxford, this would have been the principal postgraduate course leading to the degree of Master of Arts. After the MA the student could enter for Bachelor's degrees of the higher faculties, such as Music. To this day some of the postgraduate degree courses lead to the degree of Bachelor (the B.Phil and B.Litt degrees are examples in the field of philosophy). In medieval educational theory, the quadrivium consisted of arithmetic, geometry, music, and astronomy. These followed the preparatory work of the trivium, made up of grammar, logic (or dialectic) and rhetoric. In turn, the quadrivium was considered preparatory work for the serious study of philosophy and theology. The quadrivium could be considered as the study of number: arithmetic was pure number, geometry was number in space, music number in time, and astronomy number in space and time. This schema is sometimes referred to as classical education, but it is more accurately a development of the 12th and 13th centuries rather than an organic growth from the educational systems of antiquity.

Trivial

The term trivia is widely used to refer to tidbits of unimportant (or trivial) information, but it can also refer to basic or elementary knowledge. Generally, the latter definition prevails when people "play trivia". The term comes from medieval universities, where trivia were the things learned in the trivium, comprising the three subjects that were studied first: grammar, logic, and rhetoric. Before the trivia subculture became widespread, via radio and TV quiz shows and books, the term commonly referred to bits of information to which most adults in the culture had at one time been exposed, via standard education or via popular culture. In time the term came also to comprise more obscure and arcane bits of knowledge. In 1974, a former Sacramento air traffic controller named Fred L. Worth published The Trivia Encyclopedia, which he followed in 1977 with The Complete Unabridged Super Trivia Encyclopedia, and in 1981 with Super Trivia, vol. II. The popularity of these books (one appears as a prop in movie Almost Famous) laid the groundwork for the first edition of Trivial Pursuit in the early 1980s. The enormous success of this game led, in the United States, to the re-launch of Jeopardy!, reviving a quiz show genre that had been dormant since the scandal of the 1950s. In the early 1990s, a Holiday Inn DJ named Curtis Earth began a trivia show night, mixing trivia and light comedy, in Lancaster, Pennsylvania. He would go on to host similar pub quiz nights all over the state and eventually Florida too. The Disney Cruise Line now features "Curtis Earth" trivia questions for their interactive family quizzes. ABC had a surprise hit with Who Wants to be a Millionaire, an import of a successful British quiz format which launched another wave of interest in trivia. In the UK, the quiz format has enjoyed continuous success since the 1950s, untouched by the scandals that dogged the American format. In addition to the mass media trivia, there have also been two entrenched trivia subcultures. One is the pub quiz phenomenon, which is especially prevalent in Great Britain and in select US cities, particularly in pubs that serve a large Irish-American community. (The US pub quiz scene is crimped by the popularity of NTN, a satellite-based game.) The other subculture is the quiz bowl format found in high schools and universities in the US; the Canadian equivalent is competition geared toward Reach for the Top, among high schools, whereas Canadian universities are beginning to participate in US quiz bowl leagues.

Etymology

A number of theories have been put forward as to the etymology of the word "trivia". One variation dates to early Latin, from the prefix tri-, "three", and via, "road". Trivium thus meant "the meeting place of three roads, especially as a place of public resort." In the Roman empire, a trivium would often have a tavern (Latin: taverna). In Roman times, such a place was viewed as common and vulgar, in the sense that we express in the phrase the gutter, as in "His manners were formed in the gutter." The Latin adjective triviālis, derived from trivium, thus meant "appropriate to the street corner, commonplace, vulgar." The first known usage of the word "trivial" in Modern English is from 1589; it was used with a sense identical to that of triviālis. Shortly after that trivial is recorded in the sense most familiar to us: "of little importance or significance." Gradually, the word trivia came to be applied for any information that is of fleeting importance and of general interest. Another, slightly different use of trivium may be more directly related to the modern meaning of the word, the earliest known use of which in English is in a work from 1432-1450. This work mentions the "arte trivialle", a reference to the three liberal arts that made up the first three subjects taught in medieval universities, namely grammar, rhetoric, and logic. The remaining four liberal arts of the quadrivium, namely arithmetic, geometry, music, and astronomy, were more challenging. Hence, trivial in this sense would have been "of interest only to an undergraduate". The first known usage of the word trivial in Modern English is from 1589; it was used with a sense identical to that of triviālis. Shortly after that trivial is recorded in the sense most familiar to us: "of little importance or significance."

External links

- [http://www.curtisearthtrivia.com/ Welcome to the world of the american quiz master, Curtis Earth.
- [http://www.absurdtrivia.com/ Absurd Trivia] Thousands of user submitted online trivia quiz games
- [http://www.allabouttrivia.com/ All About Trivia](Thousands of questions in flash animated games. Get bonus points for answering quickly)
- [http://www.the-trivia.com/ENGsite/ the-trivia.com] (Online trivia quiz games and tournaments)
- [http://www.interactrivia.com/ InteracTrivia.com] (Real-Time interactive multiplayer trivia game)
- [http://www.quicktrivia.com/ QuickTrivia.com] (Example of a large-scale trivia game)
- [http://www.triviahalloffame.com/essays.htm/ Trivia on Trivia] (Essays on the world of trivia and interviews with trivia-world celebrities)
- [http://www.triviatribute.com/ TV and Movie Trivia Tribute] (Scored quizzes related to TV shows and movies)
- [http://www.completetrivia.com.au/ Complete Interactive Trivia] (Trivia for the Trivia Buff including photo and sound clues. More than just multiple choice. Win money and prizes! Something for everyone.)
- [http://www.quizquizquiz.biz Trivia Quiz Nights] (Trivia Quiz Nights in the UK)
- [http://www.triviabox.com/Affiliate.aspx?cs=1363 Triviabox.com] (Online trivia games where you can win real cash prizes)

Resources

- American Heritage Dictionaries (2000). The American Heritage® Dictionary of the English Language (4th ed.). Houghton Mifflin Company. ISBN 0395825172. Category:English phrases ja:トリヴィア

University of Oxford

The University of Oxford, located in the city of Oxford, England, is the oldest university in the English-speaking world. The university traces its roots back to at least the end of the 11th century, although the exact date of foundation remains unclear. According to legend, after riots between scholars and townsfolk broke out in 1209, some of the academics at Oxford fled north-east to the town of Cambridge, where the University of Cambridge was founded. The two universities have since had a long history of competition with each other, and are widely seen as the most prestigious universities in the United Kingdom (see Oxbridge rivalry). Oxford has recently topped two university-ranking league tables produced by British newspapers: it came first according to The Guardian and, for the fourth consecutive year, in The Times table. Although widely contested (as with most league tables) on the basis of their ranking criteria, recent international tables produced by Shanghai Jiao Tong University rated Oxford tenth[http://ed.sjtu.edu.cn/rank/2005/ARWU2005_Top100.htm] in the world. Oxford is a member of the Russell Group of research-led British universities, the Coimbra Group (a network of leading European universities), the LERU (League of European Research Universities), and is also a core member of the Europaeum.

History

Europaeum The date of the University's foundation is unknown, and indeed it may not have been a single event, but there is evidence of teaching there as early as 1096. When Henry II of England forbade English students to study at the University of Paris in 1167, Oxford began to grow very quickly. The foundation of the first halls of residence, which later became colleges, dates from this period. Rioting in 1209 led many scholars to leave Oxford for other parts of the country, leading to the establishment of a university in Cambridge. On June 20 1214, a charter of liberties was granted to the University by Nicholas de Romanis, the papal legate, which authorised the appointment of a chancellor of the University. Riots between townsmen and scholars ("town and gown") were common until the St Scholastica Day riot in 1355 led to the king confirming the supremacy of the University over the town. In 1555 - 6 the Protestant Oxford Martyrs, Latimer, Ridley and Cranmer were burned at Oxford. The University's status was formally confirmed by an Act for the Incorporation of Both Universities in 1571, in which the University's formal title is given as The Chancellor, Masters and Scholars of the University of Oxford. In 1603 the University granted the right to appoint two Members of Parliament, a right which lasted until the abolition of university constituencies in 1949. The comprehensive set of statutes, known as the Laudian Code, was drawn up by Archbishop William Laud in 1636 and ratified by Charles I. The University supported the king during the English Civil War, and was the site of his court and parliament, but clashed with his grandson, the Roman Catholic James II, who was later overthrown in the Glorious Revolution. In the 1830s the University was the site of the Oxford Movement in the Church of England. A Royal Commission to reform the University was appointed in 1850 and its proposals, accepted by Parliament, revolutionised the medieval workings of the University, until then still governed by the code of 1636. Later royal commissions were appointed in 1872 and 1919. In 1871 the Universities Tests Act opened the University to Dissenters and Roman Catholics. The first women's halls were established in 1878, and women were admitted to degrees in 1920.

Organisation

Oxford is a collegiate university, consisting of the University's central facilities, such as departments and faculties, libraries and science facilities, and 39 colleges and 7 Permanent Private Halls (PPHs). All teaching staff and degree students must belong to one of the colleges (or PPHs). These colleges are not only houses of residence, but have substantial responsibility for the teaching of undergraduates and postgraduates. Some colleges only accept postgraduate students. Only one of the colleges, St Hilda's, remains single-sex, accepting only women (though several of the religious PPHs are male-only). Oxford's collegiate system springs from the fact that the University came into existence through the gradual agglomeration of independent institutions in the city of Oxford. : See also: Colleges of Oxford University, and a list of Cambridge sister colleges. Image:brasnose.JPG
Brasenose College in the 1670s As well as the collegiate level of organisation, the University is subdivided into departments on a subject basis, much like most other universities. Departments take a major role in graduate education and an increasing role in undergraduate education, providing lectures and classes and organising examinations. Departments are also a centre of research, funded by outside bodies including major research councils; while colleges have an interest in research, few are subject-specialized in organisa

Grammar

:This article is about grammar from a linguistic perspective. For English grammar rules, see English grammar or Disputed English grammar Grammar is the study of rules governing the use of language. The set of rules governing a particular language is also called the grammar of the language; thus, each language can be said to have its own distinct grammar. Grammar is part of the general study of language called linguistics. The subfields of modern grammar are phonetics, phonology, morphology, syntax, and semantics. Traditional grammars include only morphology and syntax.

Types of grammar

- A prescriptive grammar presents authoritative norms for a particular language, and tends to deprecate non-standard constructions. Traditional grammars are typically prescriptive. Prescriptive grammars are usually based on the prestige dialects of a speech community, and often specifically condemn certain constructions which are common only among lower socioeconomic groups, such as the use of "ain't" and double negatives in English. Though prescriptive grammars remain common in pedagogy and foreign language teaching, they have fallen out of favor in modern academic linguistics, as they describe only a subset of actual language usage.
- A descriptive grammar attempts to describe actual usage, avoiding prescriptive judgements. Descriptive grammars are bound to a particular speech community, and attempt to provide rules for any utterance considered grammatically correct within that community. For example, in many dialects of English, the use of double negatives is very common, though ungrammatical from the point of view of a prescriptive English grammar. A descriptive grammar of a speech community where "I didn't do nothing" is acceptable will treat that sentence as grammatical, and provide rules that account for it. A descriptive grammar of formal English would rather provide rules for "I didn't do anything."
- Traditional grammar is the collection of ideas about grammar that Western societies have received from Greek and Roman sources. Prescriptive grammar is always formulated in terms of the descriptive concepts inherited from traditional grammar. Modern descriptive grammar aims to correct the errors of traditional grammar, and generalize them, so as to avoid shoehorning all languages to the model of Latin. Nearly all materials used in teaching language, however, are still based on traditional grammar.
- A formal grammar is a precisely defined grammar, typically used for computer programming languages.
- A generative grammar is a formal grammar that can in some sense "generate" the well-formed expressions of a natural language. An entire branch of linguistic theory is based on generative grammars. Generative grammars were popularized by Noam Chomsky.

Development of grammars

Grammars evolve through usage and human population separations. With the advent of written representations, formal rules about language usage tend to appear also. Formal grammars are codifications of usage that are developed by observation. As the rules become established and developed, the prescriptive concept of grammatical correctness can arise. This often creates a gulf between contemporary usage and that which is accepted as correct. Linguists normally consider that prescriptive grammars do not have any justification beyond their authors' aesthetic tastes. However, prescriptions are considered in sociolinguistics as part of the explanation for why some people say "I didn't do nothing", some say "I didn't do anything", and some say one or the other depending on social context. The formal study of grammar is an important part of education from a young age through advanced learning, though the rules taught in schools are not a "grammar" in the sense most linguists use the term, as they are often prescriptive rather than descriptive. Planned languages are more common in the modern day. Many have been designed to aid human communication (such as Esperanto or the intercultural, highly logic-compatible artificial language Lojban) or created as part of a work of fiction (such as the Klingon language and Elvish languages). Each of these artificial languages has its own grammar. It is a myth that analytic languages have simpler grammar than synthetic languages. Analytic languages use syntax to convey information that is encoded via inflection in synthetic languages. In other words, word order is not significant and morphology is highly significant in a purely synthetic language, whereas morphology is not significant and syntax is highly significant in an analytic language. Chinese and Afrikaans, for example, are highly analytic and meaning is therefore very context dependent. (Both do have some inflections, and had more in the past; thus, they are becoming even less synthetic and more "purely" analytic over time.) Latin, which is highly synthetic, uses affixes and inflections to convey the same information that Chinese does with syntax. Because Latin words are quite (though not completely) self-contained, an intelligible Latin sentence can be made from elements placed in largely arbitrary order. Latin has a complex affixation and a simple syntax, while Chinese has the opposite. ----- In computer science, the syntax of each programming language is defined by a formal grammar. In theoretical computer science and mathematics, formal grammars define formal languages. The Chomsky hierarchy defines several important classes of formal grammars.

External links

- [http://www.krysstal.com/grammar.html Grammar Terms]
- [http://www.gramster.com/ English Grammar Software]
- [http://www.figarospeech.com/ It Figures-Figures of Speech]
- als:Grammatik ja:文法 simple:Grammar th:ไวยากรณ์

Quadrivium

Geometry

Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematics and algebraic geometry.)

The earliest geometry

The earliest recorded beginnings of geometry may be traced to Ancient Egypt (see geometry in Egypt) and Ancient Babylon (see Babylonian mathematics) around 3000 B.C. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras; the Egyptians had a correct formula for the volume of a frustum of a square pyramid; the Babylonians had a trigonometry table. Chinese culture at this same time period was equally advanced, so it is likely that they had an equally advanced mathematics, but no artifacts have survived from which we could learn about it. This may be partly due to their early use of paper, rather than clay tablets or stone, to record their achievements.

The Greek period (c. 600 B.C. – 600 A.D.)

The Greek Period must be considered in detail, since geometry, for most of its history, was what the Greeks made it. For the Ancient Greeks, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies “eternal forms”, or abstractions, of which physical objects are only approximations; and they developed the idea of an “axiomatic theory”, which, for more than 2000 years, was regarded to be the ideal paradigm for all scientific theories.

Thales and Pythagoras

Thales (635-543 B.C.) of Ionia (now southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 B.C.) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and probably traveled to Babylon and Egypt. The theorem that bears his name was not his discovery, but he was the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers.

Plato

Plato (427-347 B.C.), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, “Let none enter here who are ignorant of geometry.” Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but a compass and straight edge – never measuring instruments such as a marked ruler or a protractor, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of the possible ruler and compass constructions, and three classic ruler-and-compass problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 B.C.), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century.

Euclid

Euclid (365?-275? B.C.), probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in the ideal axiomatic form. The treatise is not a compendium of all that the Greeks knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt. The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read. # Any two points can be joined by a straight line. # Any finite straight line can be extended in a straight line. # A circle can be drawn with any center and any radius. # All right angles are equal to each other. # If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate). It was soon observed, and no doubt Euclid himself knew, that his fifth axiom could be replaced by the shorter statement “Given a line and a point not on the line, there is only one line through the given point and in the same plane with the given line that does not intersect the given line.” This is called Playfair’s Axiom, after the British teacher who proposed to make the replacement in all the school textbooks. The axioms, according to Plato, should be simple and self-evident principles, so clearly true that they need no proof. Euclid’s first four axioms meet this criterion, but the fifth, even if replaced by Playfair’s Axiom, is not simple, and most would say not self-evident like the first four. The fifth resembled more the theorems that Euclid proved from the axioms. Furthermore, Euclid developed a substantial part of his theory of triangles without using the Fifth Axiom. The speculation arose, probably during Euclid’s lifetime, that the Fifth Axiom can and should be proved as a theorem from the first four, and thus is unnecessary as an axiom. Thus began many centuries of attempts to prove the Fifth Axiom, and the question was not settled until the 19th century.

Archimedes

Archimedes (287-212 B.C.), of Syracuse, Sicily, when it was a Greek city-state, was the greatest of the Greek mathematicians, and often named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.

After Archimedes

After Archimedes, Greek mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of Commentary on the First Book of Euclid, was one of the last important players in Greek geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

The Middle Ages, Renaissance, and Reformation

The great library of Alexandria was burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later. Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign. The Islamic ascendency in the Middle East, north Africa, and Spain began about 640 A.D. Original Arab mathematics during this period was primarily algebraic rather than geometric, though there were important commentaries on geometry. Omar Khayyám, for example, was a geometer as well as a poet. Scholarship in Europe declined until even the great works of antiquity were lost to them, and survived only in the Islamic centers of learning. When Europe started to emerge from the intellectual darkness of the Middle Ages, the writers of Ancient Greece and Rome were rediscovered in Islamic libraries and translated from Arabic into Latin. Euclid’s Elements of Geometry was recovered, and the rigorous deductive methods of geometry were relearned. Development of geometry in the style of Euclid resumed, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.

The 17th and early 18th centuries

In the early 17th century, there were two important developments in geometry. The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by Rene Descartes (1596-1650) and Pierre de Fermat (1601-1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591-1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by Greek geometers, notably Pappus (c. 340). The greatest flowering of the field occurred with Jean-Victor Poncelet (1788-1867). In the late 17th century, calculus was developed independently and almost simultaneously by Isaac Newton (1642-1727) and Gottfried Wilhelm von Leibniz (1646-1716). This was the beginning of a new field of mathematics now called analysis. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.

The late 18th and 19th centuries

Non-Euclidean geometry

The old problem of proving Euclid’s Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. Saccheri, Lambert, and Legendre each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, Gauss, Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for Einstein's theory of relativity. It remained to prove mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry. While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earth-bound) distances. With the development of relativity theory in physics, this question became vastly more complicated.

Introduction of mathematical rigor

All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms were given by David Hilbert in 1894 in his dissertation Grundlagen der Geometrie (Foundations of Geometry). Some other complete sets of axioms had been given a few years earlier, but did not match Hilbert's in economy, elegance, and similarity to Euclid's axioms.

Analysis situs, or topology

In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as topology. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Euclidean geometry. Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.

The 20th century

External links

- [http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml What Is Geometry?] at cut-the-knot
- [http://www.elvenkids.com/tools/geometria/Geometria.php Geometria] An online tool to compute lines, surfaces and volumes of the main plane and solid figures, through direct and indirect formulas.
- [http://www.geogebra.at/ Geogebra] A free dynamic geometry tool, useful for exploring geometry.
- [http://agutie.homestead.com Geometry Step by Step from the Land of the Incas] by Antonio Gutierrez.
- [http://www.cut-the-knot.org/geometry.shtml Geometry] at cut-the-knot
- [http://www.islamicarchitecture.org/art/islamic-geometry-and-floral-patterns.html Islamic Geometry]
- Stanford Encyclopedia of Philosophy:
- [http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry]
- [http://plato.stanford.edu/entries/geometry-19th/ Geometry in the 19th Century]
- [http://www.egwald.com/geometry/index.php Online Interactive Geometric Objects] by Elmer G. Wiens Category:Geometry ko:기하학 ja:幾何学 simple:Geometry zh-min-nan:Kí-hô-ha̍k

Astronomy

:This article is about the science branch. For information about the magazine, see Astronomy (magazine). Astronomy (magazine) as they circled the Moon in 1969. Located near the center of the far side of Earth's Moon, its diameter is about 58 miles (93 km).]] Astronomy (Greek: αστρονομία = άστρον + νόμος, astronomia = astron + nomos, literally, "law of the stars") is the science of celestial objects and phenomena that originate outside the Earth's atmosphere, such as stars, planets, comets, galaxies, and the cosmic background radiation. It is concerned with the formation and development of the universe, the evolution and physical and chemical properties of celestial objects and the calculation of their motions. Astronomical observations are not only relevant for astronomy as such, but provide essential information for the verification of fundamental theories in physics, such as general relativity theory. Complementary to observational astronomy, theoretical astrophysics seeks to explain astronomical phenomena. Astronomy is one of the oldest sciences, with a scientific methodology existing at the time of Ancient Greece and advanced observation techniques possibly much earlier (see archaeoastronomy). Historically, amateurs have contributed to many important astronomical discoveries, and astronomy is one of the few sciences where amateurs can still play an active role, especially in the discovery and observation of transient phenomena. Astronomy is not to be confused with astrology, which assumes that people's destiny and human affairs in general are correlated to the apparent positions of astronomical objects in the sky -- although the two fields share a common origin, they are quite different; astronomers embrace the scientific method, while astrologers do not. In other words, there is no proof that the stars predict our future, but there is proof that our planet is 93 million miles from the sun.

Divisions

In ancient Greece

Trivium

See also

External links

Metalcore

The early scene

The later scene

The breakdown

Bands

See also

Trivium (band)

Band Members

Previous Members

Discography

Video Game Appearances

Videos

External links

Stream cipher

Loose inspiration from the one-time pad

Types of stream cipher

Synchronous stream ciphers

Self-synchronising stream ciphers

LFSR-based stream ciphers

Non-linear combining functions

Clock-controlled generators

Filter generator

Other designs

Security

Usage

References

See also

Medieval

The Early Middle Ages

A new order

A Carolingian renaissance

The High Middle Ages

The Crusades

Technology

The Late Middle Ages (circa 1300-1500)

Historiography

Middle Ages in history

Medieval and Middle Ages

"Middle Age"

"Medieval"

Periodization issues

Religion in the Middle Ages

See also

Selected bibliography

External links

Quadrivium

See also

Trivial

Etymology

See also

External links

Resources

University of Oxford

History

Organisation

Grammar

Types of grammar

Development of grammars

See also

Grammatical devices

Grammatical terms

Related topics

References

External links

Quadrivium

See also

Geometry

The earliest geometry

The Greek period (c. 600 B.C. – 600 A.D.)

Thales and Pythagoras

Plato

Euclid

Archimedes

After Archimedes

The Middle Ages, Renaissance, and Reformation

The 17th and early 18th centuries

The late 18th and 19th centuries