The Ottoman Empire known in Western Europe as the Turkish Empire or Turkey, was a state that controlled much of Southeast Europe, Western Asia and North Africa between the 14th and early 20th centuries. It was founded at the end of the 13th century in northwestern Anatolia in the town of Söğüt by the Oghuz Turkish tribal leader Osman I. After 1354, the Ottomans crossed into Europe, with the conquest of the Balkans, the Ottoman beylik was transformed into a transcontinental empire; the Ottomans ended the Byzantine Empire with the 1453 conquest of Constantinople by Mehmed the Conqueror. During the 16th and 17th centuries, at the height of its power under the reign of Suleiman the Magnificent, the Ottoman Empire was a multinational, multilingual empire controlling most of Southeast Europe, parts of Central Europe, Western Asia, parts of Eastern Europe and the Caucasus, North Africa and the Horn of Africa. At the beginning of the 17th century, the empire contained numerous vassal states; some of these were absorbed into the Ottoman Empire, while others were granted various types of autonomy during the course of centuries.
With Constantinople as its capital and control of lands around the Mediterranean basin, the Ottoman Empire was at the centre of interactions between the Eastern and Western worlds for six centuries. While the empire was once thought to have entered a period of decline following the death of Suleiman the Magnificent, this view is no longer supported by the majority of academic historians; the empire continued to maintain a flexible and strong economy and military throughout the 17th and much of the 18th century. However, during a long period of peace from 1740 to 1768, the Ottoman military system fell behind that of their European rivals, the Habsburg and Russian empires; the Ottomans suffered severe military defeats in the late 18th and early 19th centuries, which prompted them to initiate a comprehensive process of reform and modernisation known as the Tanzimat. Thus, over the course of the 19th century, the Ottoman state became vastly more powerful and organised, despite suffering further territorial losses in the Balkans, where a number of new states emerged.
The empire allied with Germany in the early 20th century, hoping to escape from the diplomatic isolation which had contributed to its recent territorial losses, thus joined World War I on the side of the Central Powers. While the Empire was able to hold its own during the conflict, it was struggling with internal dissent with the Arab Revolt in its Arabian holdings. During this time, atrocities were committed by the Young Turk government against the Armenians and Pontic Greeks; the Empire's defeat and the occupation of part of its territory by the Allied Powers in the aftermath of World War I resulted in its partitioning and the loss of its Middle Eastern territories, which were divided between the United Kingdom and France. The successful Turkish War of Independence against the occupying Allies led to the emergence of the Republic of Turkey in the Anatolian heartland and the abolition of the Ottoman monarchy; the word Ottoman is a historical anglicisation of the name of Osman I, the founder of the Empire and of the ruling House of Osman.
Osman's name in turn was the Turkish form of the Arabic name ʿUthmān. In Ottoman Turkish, the empire was referred to as Devlet-i ʿAlīye-yi ʿOsmānīye, or alternatively ʿOsmānlı Devleti. In Modern Turkish, it is known as Osmanlı Devleti; the Turkish word for "Ottoman" referred to the tribal followers of Osman in the fourteenth century, subsequently came to be used to refer to the empire's military-administrative elite. In contrast, the term "Turk" was used to refer to the Anatolian peasant and tribal population, was seen as a disparaging term when applied to urban, educated individuals. In the early modern period, an educated urban-dwelling Turkish-speaker, not a member of the military-administrative class would refer to himself neither as an Osmanlı nor as a Türk, but rather as a Rūmī, or "Roman", meaning an inhabitant of the territory of the former Byzantine Empire in the Balkans and Anatolia; the term Rūmī was used to refer to Turkish-speakers by the other Muslim peoples of the empire and beyond.
In Western Europe, the two names "Ottoman Empire" and "Turkey" were used interchangeably, with "Turkey" being favoured both in formal and informal situations. This dichotomy was ended in 1920–23, when the newly established Ankara-based Turkish government chose Turkey as the sole official name. Most scholarly historians avoid the terms "Turkey", "Turks", "Turkish" when referring to the Ottomans, due to the empire's multinational character; as the Seljuk Sultanate of Rum declined in the 13th century, Anatolia was divided into a patchwork of independent Turkish principalities known as the Anatolian Beyliks. One of these beyliks, in the region of Bithynia on the frontier of the Byzantine Empire, was led by the Turkish tribal leader Osman I, a figure of obscure origins from whom the name Ottoman is derived. Osman's early followers consisted both of Turkish tribal groups and Byzantine renegades, many but not all converts to Islam. Osman extended the control of his principality by conquering Byzantine towns along the Sakarya River.
It is not well understood how the early Ottomans came to dominate their
North Macedonia the Republic of North Macedonia, is a country in the Balkan Peninsula in Southeast Europe. It is one of the successor states of the former Yugoslavia, from which it declared independence in September 1991 under the name Republic of Macedonia; the country became a member of the United Nations in April 1993, but as a result of a dispute with Greece over the name, it was admitted under the provisional description the former Yugoslav Republic of Macedonia, a term, used by some other international organisations. In June 2018, Macedonia and Greece resolved the conflict with an agreement that the country should rename itself Republic of North Macedonia; this renaming came into effect in February 2019, with a several-months-long transition for passports, licence plates, customs, border signs, government websites, among other things. A landlocked country, North Macedonia has borders with Kosovo to the northwest, Serbia to the northeast, Bulgaria to the east, Greece to the south, Albania to the west.
It constitutes the northern third of the larger geographical region of Macedonia, which comprises the neighbouring parts of northern Greece and southwestern Bulgaria. The country's geography is defined by mountains and rivers; the capital and largest city, Skopje, is home to a quarter of the nation's 2.06 million inhabitants. The majority of the residents are a South Slavic people. Albanians form a significant minority at around 25%, followed by Turks, Serbs, Bosniaks and Bulgarians; the history of the region dates back to antiquity, beginning with the kingdom of Paeonia a mixed Thraco-Illyrian polity. In the late sixth century BC, the area was incorporated into the Persian Achaemenid Empire annexed by the kingdom of Macedonia in the fourth century BC; the Romans conquered the region in the second century BC and made it part of the much larger province of Macedonia. Τhe area remained part of the Byzantine Empire, but was raided and settled by Slavic tribes beginning in the sixth century of the Christian era.
Following centuries of contention between the Bulgarian and Serbian Empire, it was part of the Ottoman dominion from the mid-14th until the early 20th century, when following the Balkan Wars of 1912 and 1913, the modern territory of North Macedonia came under Serbian rule. During the First World War it was ruled by Bulgaria, but after the end of the war, it returned under Serbian rule as part of the newly formed Kingdom of Serbs and Slovenes. During the Second World War, it was ruled by Bulgaria again, in 1945 it was established as a constituent communist republic into the Federal People's Republic of Yugoslavia, which it remained until its peaceful secession in 1991. North Macedonia is of the Council of Europe. Since 2005, it has been a candidate for joining the European Union and has applied for NATO membership. One of the poorest countries in Europe, North Macedonia has made significant progress in developing an open, market-based economy; the state's name derives from a kingdom named after the ancient Macedonians.
Their name, Μακεδόνες, derives from the ancient Greek adjective μακεδνός, meaning tall or taper, which shares the same root as the adjective μακρός, meaning long, tall, or high, in ancient Greek. The name is believed to have meant either highlanders or the tall ones descriptive of the people. According to linguist Robert S. P. Beekes, both terms are of Pre-Greek substrate origin and cannot be explained in terms of Indo-European morphology. Prior to June 2018, the use of the name Macedonia was disputed between Greece and the then-Republic of Macedonia; the Prespa agreement, signed by Macedonia and Greece on 17 June, saw the country change its name to the Republic of North Macedonia eight months later. A non-binding national referendum on the matter passed with 90% approval but did not reach the required 50% turnout due to a boycott, leaving the final decision with parliament to ratify the result. Parliament approved of the name change on 19 October, reaching the required two-thirds majority needed to enact constitutional changes.
The vote to amend the constitution and change the name of the country passed on 11 January 2019 in favour of the amendment. The amendment entered into force on 12 February, following the ratification of the Prespa agreement and the Protocol on the Accession of North Macedonia to NATO by the Greek Parliament. On 25 January, the Greek parliament had narrowly voted to back the agreement, with 153 approving and 146 against. Prior to February 2019, in Macedonian the country name was Македонија Република Македонија. North Macedonia geographically corresponds to the ancient kingdom of Paeonia, located north of the ancient kingdom of Macedonia. Paeonia was inhabited by the Paeonians, a Thracian people, whilst the northwest was inhabited by the Dardani and the southwest by tribes known as the Enchelae and Lyncestae. In the late 6th century BC, the Achaemenid Persians under Darius the Great conquered the Paeonians, incorporating w
Albania the Republic of Albania, is a country in Southeast Europe on the Adriatic and Ionian Sea within the Mediterranean Sea. It shares land borders with Montenegro to the northwest, Kosovo to the northeast, North Macedonia to the east, Greece to the south and a maritime border with Italy to the west. Geographically, the country displays varied climatic, geological and morphological conditions, defined in an area of 28,748 km2, it possesses remarkable diversity with the landscape ranging from the snow-capped mountains in the Albanian Alps as well as the Korab, Skanderbeg and Ceraunian Mountains to the hot and sunny coasts of the Albanian Adriatic and Ionian Sea along the Mediterranean Sea. The area of Albania was populated by various Illyrian and Ancient Greek tribes as well as several Greek colonies established in the Illyrian coast; the area was annexed in the 3rd century by Romans and became an integral part of the Roman provinces of Dalmatia and Illyricum. The autonomous Principality of Arbër emerged in 1190, established by archon Progon in the Krujë, within the Byzantine Empire.
In the late thirteenth century, Charles of Anjou conquered Albanian territories from the Byzantines and established the medieval Kingdom of Albania, which at its maximal extension was extending from Durrës along the coast to Butrint in the south. In the mid-fifteenth century, it was conquered by the Ottomans; the modern nation state of Albania emerged in 1912 following the defeat of the Ottomans in the Balkan Wars. The modern Kingdom of Albania was invaded by Italy in 1939, which formed Greater Albania, before becoming a Nazi German protectorate in 1943. After the defeat of Nazi Germany, a Communist state titled the People's Socialist Republic of Albania was founded under the leadership of Enver Hoxha and the Party of Labour; the country experienced widespread social and political transformations in the communist era, as well as isolation from much of the international community. In the aftermath of the Revolutions of 1991, the Socialist Republic was dissolved and the fourth Republic of Albania was established.
Politically, the country is a unitary parliamentary constitutional republic and developing country with an upper-middle income economy dominated by the tertiary sector followed by the secondary and primary sector. It went through a process of transition, following the end of communism in 1990, from a centralized to a market-based economy, it provides universal health care and free primary and secondary education to its citizens. The country is a member of the United Nations, World Bank, UNESCO, NATO, WTO, COE, OSCE and OIC, it is an official candidate for membership in the European Union. In addition it is one of the founding members of the Energy Community, including the Organization of the Black Sea Economic Cooperation and Union for the Mediterranean; the term Albania is the medieval Latin name of the country. It may be derived from the Illyrian tribe of Albani recorded by Ptolemy, the geographer and astronomer from Alexandria, who drafted a map in 150 AD, which shows the city of Albanopolis located northeast of the city of Durrës.
The term may have a continuation in the name of a medieval settlement called Albanon or Arbanon, although it is not certain that this was the same place. In his history written in the 10th century, the Byzantine historian Michael Attaliates was the first to refer to Albanoi as having taken part in a revolt against Constantinople in 1043 and to the Arbanitai as subjects of the Duke of Dyrrachium. During the Middle Ages, the Albanians called their country Arbëri or Arbëni and referred to themselves as Arbëreshë or Arbëneshë. Nowadays, Albanians call their country Shqipëria; as early as the 17th century the placename Shqipëria and the ethnic demonym Shqiptarë replaced Arbëria and Arbëresh. The two terms are popularly interpreted as "Land of the Eagles" and "Children of the Eagles"; the first traces of human presence in Albania, dating to the Middle Paleolithic and Upper Paleolithic eras, were found in the village of Xarrë close to Sarandë and Dajti near Tiranë. The objects found in a cave near Xarrë include flint and jasper objects and fossilized animal bones, while those found at Mount Dajt comprise bone and stone tools similar to those of the Aurignacian culture.
The Paleolithic finds of Albania show great similarities with objects of the same era found at Crvena Stijena in Montenegro and north-western Greece. Several Bronze Age artefacts from tumulus burials have been unearthed in central and southern Albania that show close connection with sites in south-western Macedonia and Lefkada, Greece. Archaeologists have come to the conclusion that these regions were inhabited from the middle of the third millennium BC by Indo-European people who spoke a Proto-Greek language. A part of this population moved to Mycenae around 1600 BC and founded the Mycenaean civilisation there. In ancient times, the territory of modern Albania was inhabited by a number of Illyrian tribes; the Illyrian tribes never collectively regarded themselves as'Illyrians', it is unlikely that they used any collective nomenclature for themselves. The name Illyrians seems to be the name applied to a specific Illyrian tribe, the first to come in contact with the ancient Greeks during the Bronze Age, causing the name Illyrians to be applied pars pro toto to all people of similar language and customs.
The territory known as Illyria corresponded to the area east of the Adriatic sea, extending in the south to the mouth of the Vjosë river. The first accou
The Roman Empire was the post-Roman Republic period of the ancient Roman civilization. Ruled by emperors, it had large territorial holdings around the Mediterranean Sea in Europe, North Africa, the Middle East, the Caucasus. From the constitutional reforms of Augustus to the military anarchy of the third century, the Empire was a principate ruled from the city of Rome; the Roman Empire was ruled by multiple emperors and divided in a Western Roman Empire, based in Milan and Ravenna, an Eastern Roman Empire, based in Nicomedia and Constantinople. Rome remained the nominal capital of both parts until 476 AD, when Odoacer deposed Romulus Augustus after capturing Ravenna and the Roman Senate sent the imperial regalia to Constantinople; the fall of the Western Roman Empire to barbarian kings, along with the hellenization of the Eastern Roman Empire into the Byzantine Empire, is conventionally used to mark the end of Ancient Rome and the beginning of the Middle Ages. The previous Republic, which had replaced Rome's monarchy in the 6th century BC, became destabilized in a series of civil wars and political conflict.
In the mid-1st century BC Julius Caesar was appointed as perpetual dictator and assassinated in 44 BC. Civil wars and proscriptions continued, culminating in the victory of Octavian, Caesar's adopted son, over Mark Antony and Cleopatra at the Battle of Actium in 31 BC; the following year Octavian conquered Ptolemaic Egypt, ending the Hellenistic period that had begun with the conquests of Alexander the Great of Macedon in the 4th century BC. Octavian's power was unassailable and in 27 BC the Roman Senate formally granted him overarching power and the new title Augustus making him the first emperor; the first two centuries of the Empire were a period of unprecedented stability and prosperity known as the Pax Romana. It reached its greatest territorial expanse during the reign of Trajan. A period of increasing trouble and decline began with the reign of Commodus. In the 3rd century, the Empire underwent a crisis that threatened its existence, but was reunified under Aurelian. In an effort to stabilize the Empire, Diocletian set up two different imperial courts in the Greek East and Latin West.
Christians rose to power in the 4th century following the Edict of Milan in 313 and the Edict of Thessalonica in 380. Shortly after, the Migration Period involving large invasions by Germanic peoples and the Huns of Attila led to the decline of the Western Roman Empire. With the fall of Ravenna to the Germanic Herulians and the deposition of Romulus Augustulus in 476 AD by Odoacer, the Western Roman Empire collapsed and it was formally abolished by emperor Zeno in 480 AD; the Eastern Roman Empire, known in the post-Roman West as the Byzantine Empire, collapsed when Constantinople fell to the Ottoman Turks of Mehmed II in 1453. Due to the Roman Empire's vast extent and long endurance, the institutions and culture of Rome had a profound and lasting influence on the development of language, architecture, philosophy and forms of government in the territory it governed Europe; the Latin language of the Romans evolved into the Romance languages of the medieval and modern world, while Medieval Greek became the language of the Eastern Roman Empire.
Its adoption of Christianity led to the formation of Christendom during the Middle Ages. Greek and Roman art had a profound impact on the late medieval Italian Renaissance, while Rome's republican institutions influenced the political development of republics such as the United States and France; the corpus of Roman law has its descendants in many legal systems of the world today, such as the Napoleonic Code. Rome's architectural tradition served as the basis for Neoclassical architecture. Rome had begun expanding shortly after the founding of the republic in the 6th century BC, though it did not expand outside the Italian peninsula until the 3rd century BC, it was an "empire" long before it had an emperor. The Roman Republic was not a nation-state in the modern sense, but a network of towns left to rule themselves and provinces administered by military commanders, it was ruled, not by annually elected magistrates in conjunction with the senate. For various reasons, the 1st century BC was a time of political and military upheaval, which led to rule by emperors.
The consuls' military power rested in the Roman legal concept of imperium, which means "command". Successful consuls were given the honorary title imperator, this is the origin of the word emperor since this title was always bestowed to the early emperors upon their accession. Rome suffered a long series of internal conflicts and civil wars from the late second century BC onward, while extending its power beyond Italy; this was the period of the Crisis of the Roman Republic. Towards the end of this era, in 44 BC, Julius Caesar was perpetual dictator before being assassinated; the faction of his assassins was driven from Rome and defeated at the Battle of Philippi in 42 BC by an army led by Mark Antony and Caesar's adopted son Octavian. Antony and Octavian's division of the Roman world between themselves did not last and Octavian's forces defeated those of Mark Antony and Cleopatra at the Battle of Actium in 31 BC, ending the Final War of the Roman Republic. In 27 BC the Senate and People of Rome made Octavian princeps ("first citi
Athens is the capital and largest city of Greece. Athens dominates the Attica region and is one of the world's oldest cities, with its recorded history spanning over 3,400 years and its earliest human presence starting somewhere between the 11th and 7th millennium BC. Classical Athens was a powerful city-state that emerged in conjunction with the seagoing development of the port of Piraeus, a distinct city prior to its 5th century BC incorporation with Athens. A center for the arts and philosophy, home of Plato's Academy and Aristotle's Lyceum, it is referred to as the cradle of Western civilization and the birthplace of democracy because of its cultural and political impact on the European continent, in particular the Romans. In modern times, Athens is a large cosmopolitan metropolis and central to economic, industrial, maritime and cultural life in Greece. In 2012, Athens was ranked the world's 39th richest city by purchasing power and the 67th most expensive in a UBS study. Athens is a global one of the biggest economic centres in southeastern Europe.
It has a large financial sector, its port Piraeus is both the largest passenger port in Europe, the second largest in the world. While at the same time being the sixth busiest passenger port in Europe; the Municipality of Athens had a population of 664,046 within its administrative limits, a land area of 38.96 km2. The urban area of Athens extends beyond its administrative municipal city limits, with a population of 3,090,508 over an area of 412 km2. According to Eurostat in 2011, the functional urban area of Athens was the 9th most populous FUA in the European Union, with a population of 3.8 million people. Athens is the southernmost capital on the European mainland; the heritage of the classical era is still evident in the city, represented by ancient monuments and works of art, the most famous of all being the Parthenon, considered a key landmark of early Western civilization. The city retains Roman and Byzantine monuments, as well as a smaller number of Ottoman monuments. Athens is home to two UNESCO World Heritage Sites, the Acropolis of Athens and the medieval Daphni Monastery.
Landmarks of the modern era, dating back to the establishment of Athens as the capital of the independent Greek state in 1834, include the Hellenic Parliament and the so-called "architectural trilogy of Athens", consisting of the National Library of Greece, the National and Kapodistrian University of Athens and the Academy of Athens. Athens is home to several museums and cultural institutions, such as the National Archeological Museum, featuring the world's largest collection of ancient Greek antiquities, the Acropolis Museum, the Museum of Cycladic Art, the Benaki Museum and the Byzantine and Christian Museum. Athens was the host city of the first modern-day Olympic Games in 1896, 108 years it welcomed home the 2004 Summer Olympics, making it one of only a handful of cities to have hosted the Olympics more than once. In Ancient Greek, the name of the city was Ἀθῆναι a plural. In earlier Greek, such as Homeric Greek, the name had been current in the singular form though, as Ἀθήνη, it was rendered in the plural on, like those of Θῆβαι and Μυκῆναι.
The root of the word is not of Greek or Indo-European origin, is a remnant of the Pre-Greek substrate of Attica. In antiquity, it was debated whether Athens took its name from its patron goddess Athena or Athena took her name from the city. Modern scholars now agree that the goddess takes her name from the city, because the ending -ene is common in names of locations, but rare for personal names. During the medieval period, the name of the city was rendered once again in the singular as Ἀθήνα. However, after the establishment of the modern Greek state, due to the conservatism of the written language, Ἀθῆναι became again the official name of the city and remained so until the abandonment of Katharevousa in the 1970s, when Ἀθήνα, Athína, became the official name. According to the ancient Athenian founding myth, the goddess of wisdom, competed against Poseidon, the god of the seas, for patronage of the yet-unnamed city. According to the account given by Pseudo-Apollodorus, Poseidon struck the ground with his trident and a salt water spring welled up.
In an alternative version of the myth from Vergil's Georgics, Poseidon instead gave the Athenians the first horse. In both versions, Athena offered the Athenians the first domesticated olive tree. Cecrops declared Athena the patron goddess of Athens. Different etymologies, now rejected, were proposed during the 19th century. Christian Lobeck proposed as the root of the name the word ἄθος or ἄνθος meaning "flower", to denote Athens as the "flowering city". Ludwig von Döderlein proposed the stem of the verb θάω, stem θη- to denote Athens as having fertile soil. In classical literature, the city was sometimes referred to as the City of the Violet Crown, first documented in Pindar's ἰοστέφανοι Ἀθᾶναι, or as τὸ κλεινὸν ἄστυ. In medieval texts, variant names include Setines and Astines, all derivations involving false splitting of p
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Olympic Games ceremony
Olympic Games ceremonies of the Ancient Olympic Games were an integral part of these Games. Some of the elements of the modern ceremonies harken back to the Ancient Games from which the Modern Olympics draw their ancestry. An example of this is the prominence of Greece in closing ceremonies. During the 2004 Games, the medal winners received a crown of olive branches, a direct reference to the Ancient Games, in which the victor's prize was an olive wreath; the various elements of the ceremonies are mandated by the Olympic Charter, cannot be changed by the host nation. This requirement of seeking the approval of the International Olympic Committee includes the artistic portion of opening and closing ceremonies; the ceremonies have evolved over the centuries. Ancient Games incorporated ceremonies to mark the ending of each successive game. There are both similarities and differences between the ancient Olympic ceremonies and their modern counterparts. While the presentation of the Games has evolved with improvements in technology and the desire of the host nations to showcase their own artistic expression, the basic events of each ceremony have remained unchanged.
The presentation of the Opening and Closing Ceremonies continue to increase in scope and expense with each successive celebration of the Games, but they are still steeped in tradition. The 2028 Los Angeles Olympics organisers propose that the opening and closing ceremonies will each, for the first time, be staged across two different stadiums; the Ancient Games, held in Greece from ca. 776 BC to ca. 393 AD, provide the first examples of Olympic ceremonies. The victory celebration, elements of which are in evidence in the modern-day medal and closing ceremonies involved elaborate feasts, drinking and the recitation of poetry; the wealthier the victor the more extravagant the celebration. The victors were presented with an olive wreath or crown harvested from a special tree in Olympia by a boy, specially selected for this purpose, using a golden sickle; the festival would conclude with the victors making solemn vows and performing ritual sacrifices to the various gods to which they were beholden.
There is evidence of dramatic changes in the format of the Ancient Games over the nearly 12 centuries that they were celebrated. By the 77th Olympiad, a standard 18-event program was established. In order to open a Games in ancient Greece the organizers would hold an Inauguration Festival; this was followed by a ceremony. The first competition, an artistic competition of trumpeters and heralds, concluded the opening festivities; the Olympic opening ceremonies represent the official commencement of an Olympic Games. In recent Olympics, athletic competition began prior to the opening ceremonies. For example, the football competitions for both men and women at the 2008 Summer Olympics began two days prior to the opening ceremonies; the 2014 Winter Olympics became the first Winter Games to hold competitions before the opening ceremony. As mandated by the Olympic Charter, various elements frame the Opening Ceremonies of a celebration of the Olympic Games. Most of these rituals were canonized at the 1920 Summer Olympics in Belgium.
The artistic program is. Coubertin's initial vision of the Modern Olympics featured both athletic competitions and artistic achievements; as the modern Olympics have evolved into a celebration of sport, it is in the opening ceremonies that one can see the most of Coubertin's ideal. The opening ceremonies are an important ritual of the Olympic games that represent a wide variety of features such as similar qualities and messages that link together local and global issues, as well as cultural similarities at the same scopes; the artistic program of the ceremonies allows the host country to showcase its past and future in a comprehensive way. The ceremonies start with the raising of the host country's flag and a performance of its national anthem; the host nation presents artistic displays of music, singing and theater representative of its culture and the current Olympic game motto. Since the 1980 Summer Olympics in Moscow, the artistic presentations have continued to grow in scale and complexity.
The opening ceremony at the Beijing Games, for example cost US$100 million, with much of the cost incurring in the artistic portion of the ceremony. The traditional part of the ceremonies starts with a "Parade of Nations", during which most participating athletes march into the stadium, country by country, it is not compulsory for athletes to participate in the opening ceremonies. Because some of the first events of the Games may start on the day before, on the day, or the day after the ceremonies, athletes competing in these early events may elect not to participate. For every Opening Ceremony, each host country has a theme. During the "Parade of Nations", the host country’s goal is to represent their cultural identity and to show the world their place in society. For example, in the 2008 Beijing Olympics the theme was “unity”. On May 12, 2008, a devastating earthquake erupted in Sichuan; as the host country, China wanted to remember this tragic event by having Yao Ming, a Chinese basketball legend, walk hand-in-hand with Lin Hao, a nine-year-old boy who saved some of his classmates during the earthquake.
Each country's delegation is led by a sign with the name of their country and by their nation's flag. Traditionally, Greece always enters first and leads the parade due to the historical status as the progenitor of the Olympics, the host nation enters last. All other particip