1.
Republic of Florence
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The Republic of Florence, also known as the Florentine Republic, was a medieval and early modern state that was centered on the Italian city of Florence in Tuscany. The republic originated in 1115, when the Florentine people rebelled against the Margraviate of Tuscany upon the death of Matilda, the Florentines formed a commune in her successors place. The republic was ruled by a council, known as the signoria, the signoria was chosen by the gonfaloniere, who was elected every two months by Florentine guild members. The republic had a history of coups and counter-coups against various factions. The Medici faction gained governance of the city in 1434, upon Cosimo de Medicis counter-coup against the faction that had sent him into exile the previous year, the Medici kept control of Florence until 1494. Giovanni de Medici re-conquered the republic in 1512, Florence repudiated Medici authority for a second time in 1527, during the War of the League of Cognac. The Medici re-assumed their rule in 1531, after an 11-month siege of the city, the republican government was disestablished in 1532, when Pope Clement VII appointed Alessandro de Medici Duke of the Florentine Republic, making the republic a hereditary monarchy. The city of Florence was established in 59 B. C. by Julius Caesar, the city had been part of the Marquisate of Tuscany before the death of Margravine Matilda in 1115. The city did not submit readily to her successor, Rabodo, the first official mention of the republic was in 1138 when several cities around Tuscany formed a league against Henry X of Bavaria. The country was part of the Holy Roman Empire. Florence prospered in the 12th century, trading extensively with foreign countries and this, in turn, provided a platform for demographic growth of the city. The growth of Florences population mirrored the rate of construction, many churches and this prosperity was shattered when Emperor Frederick I Barbarossa invaded the Italian peninsula in 1185. The Margraves of Tuscany re-acquired Florence and its townlands, the Florentines re-asserted their independence when Holy Roman Emperor Henry VI died in 1197. Florences population continued to grow into the 13th century, reaching 30,000 inhabitants, as has been said, the extra inhabitants supported the citys trade and vice versa. Several new bridges and churches were built, most prominently the cathedral of Santa Maria del Fiore, the buildings from the era serve as Florences best example of Gothic Architecture. Politically, Florence was barely able to maintain peace between factions, the precarious peace that existed at the beginning of the century was destroyed in 1216 when two factions known as the Guelphs and the Ghibellines began to war. The Ghibellines were the rulers of Florence. The Ghibellines, who under Frederick of Antioch had ruled the city since 1244, were deposed in 1250 by the Guelphs, the Guelphs led Florence to prosper further
2.
Italy
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Italy, officially the Italian Republic, is a unitary parliamentary republic in Europe. Located in the heart of the Mediterranean Sea, Italy shares open land borders with France, Switzerland, Austria, Slovenia, San Marino, Italy covers an area of 301,338 km2 and has a largely temperate seasonal climate and Mediterranean climate. Due to its shape, it is referred to in Italy as lo Stivale. With 61 million inhabitants, it is the fourth most populous EU member state, the Italic tribe known as the Latins formed the Roman Kingdom, which eventually became a republic that conquered and assimilated other nearby civilisations. The legacy of the Roman Empire is widespread and can be observed in the distribution of civilian law, republican governments, Christianity. The Renaissance began in Italy and spread to the rest of Europe, bringing a renewed interest in humanism, science, exploration, Italian culture flourished at this time, producing famous scholars, artists and polymaths such as Leonardo da Vinci, Galileo, Michelangelo and Machiavelli. The weakened sovereigns soon fell victim to conquest by European powers such as France, Spain and Austria. Despite being one of the victors in World War I, Italy entered a period of economic crisis and social turmoil. The subsequent participation in World War II on the Axis side ended in defeat, economic destruction. Today, Italy has the third largest economy in the Eurozone and it has a very high level of human development and is ranked sixth in the world for life expectancy. The country plays a prominent role in regional and global economic, military, cultural and diplomatic affairs, as a reflection of its cultural wealth, Italy is home to 51 World Heritage Sites, the most in the world, and is the fifth most visited country. The assumptions on the etymology of the name Italia are very numerous, according to one of the more common explanations, the term Italia, from Latin, Italia, was borrowed through Greek from the Oscan Víteliú, meaning land of young cattle. The bull was a symbol of the southern Italic tribes and was often depicted goring the Roman wolf as a defiant symbol of free Italy during the Social War. Greek historian Dionysius of Halicarnassus states this account together with the legend that Italy was named after Italus, mentioned also by Aristotle and Thucydides. The name Italia originally applied only to a part of what is now Southern Italy – according to Antiochus of Syracuse, but by his time Oenotria and Italy had become synonymous, and the name also applied to most of Lucania as well. The Greeks gradually came to apply the name Italia to a larger region, excavations throughout Italy revealed a Neanderthal presence dating back to the Palaeolithic period, some 200,000 years ago, modern Humans arrived about 40,000 years ago. Other ancient Italian peoples of undetermined language families but of possible origins include the Rhaetian people and Cammuni. Also the Phoenicians established colonies on the coasts of Sardinia and Sicily, the Roman legacy has deeply influenced the Western civilisation, shaping most of the modern world
3.
Franciscans
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The Franciscans are a group of related mendicant religious orders within the Catholic Church, founded in 1209 by Francis of Assisi. These orders include the Order of Friars Minor, the Order of Saint Clare, Francis began preaching around 1207 and traveled to Rome to seek approval from the Pope in 1209. The original Rule of Saint Francis approved by the Pope disallowed ownership of property, the austerity was meant to emulate the life and ministry of Jesus Christ. Franciscans traveled and preached in the streets, while boarding in church properties, Saint Clare, under Franciss guidance, founded the Poor Clares in 1212, which remains a Second Order of the Franciscans. The extreme poverty required of members was relaxed in final revision of the Rule in 1223, the degree of observance required of members remained a major source of conflict within the order, resulting in numerous secessions. The Order of Friars Minor, previously known as the Observant branch, is one of the three Franciscan First Orders within the Catholic Church, the others being the Capuchins and Conventuals. The Order of Friars Minor, in its current form, is the result of an amalgamation of smaller orders completed in 1897 by Pope Leo XIII. The latter two, the Capuchin and Conventual, remain distinct religious institutes within the Catholic Church, observing the Rule of Saint Francis with different emphases, Franciscans are sometimes referred to as minorites or greyfriars because of their habit. In Poland and Lithuania they are known as Bernardines, after Bernardino of Siena, the name of original order, Friars Minor, means lesser brothers, and stems from Francis of Assisis rejection of extravagance. Francis was the son of a cloth merchant, but gave up his wealth to pursue his faith more fully. Francis adopted of the tunic worn by peasants as the religious habit for his order. Those who joined him became the original Order of Friars Minor and they all live according to a body of regulations known as the Rule of St Francis. First Order The First Order or the Order of Friars Minor are commonly called simply the Franciscans and this Order is a mendicant religious order of men, some of whom trace their origin to Francis of Assisi. Their official Latin name is the Ordo Fratrum Minorum, St. Francis thus referred to his followers as Fraticelli, meaning Little Brothers. Franciscan brothers are informally called friars or the Minorites and they all live according to a body of regulations known as the Rule of St Francis. These are The Order of Friars Minor, known as the Observants, most commonly simply called Franciscan friars, official name, the Order of Friars Minor Capuchin or simply Capuchins, official name, Friars Minor Capuchin. The Conventual Franciscans or Minorites, official name, Friars Minor Conventual, Second Order The Second Order, most commonly called Poor Clares in English-speaking countries, consists of religious sisters. The order is called the Order of St. Clare, but in the century, prior to 1263, this order was referred to as The Poor Ladies, The Poor Enclosed Nuns
4.
Leonardo da Vinci
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He has been variously called the father of palaeontology, ichnology, and architecture, and is widely considered one of the greatest painters of all time. Sometimes credited with the inventions of the parachute, helicopter and tank, many historians and scholars regard Leonardo as the prime exemplar of the Universal Genius or Renaissance Man, an individual of unquenchable curiosity and feverishly inventive imagination. Much of his working life was spent in the service of Ludovico il Moro in Milan. He later worked in Rome, Bologna and Venice, and he spent his last years in France at the home awarded to him by Francis I of France, Leonardo was, and is, renowned primarily as a painter. Among his works, the Mona Lisa is the most famous and most parodied portrait, Leonardos drawing of the Vitruvian Man is also regarded as a cultural icon, being reproduced on items as varied as the euro coin, textbooks, and T-shirts. Perhaps fifteen of his paintings have survived, Leonardo is revered for his technological ingenuity. He conceptualised flying machines, a type of armoured fighting vehicle, concentrated power, an adding machine. Some of his inventions, however, such as an automated bobbin winder. A number of Leonardos most practical inventions are nowadays displayed as working models at the Museum of Vinci. He made substantial discoveries in anatomy, civil engineering, geology, optics, and hydrodynamics, today, Leonardo is widely considered one of the most diversely talented individuals ever to have lived. Leonardo was born on 15 April 1452 at the hour of the night in the Tuscan hill town of Vinci. He was the son of the wealthy Messer Piero Fruosino di Antonio da Vinci, a Florentine legal notary, and Caterina. Leonardo had no surname in the modern sense – da Vinci simply meaning of Vinci, his birth name was Lionardo di ser Piero da Vinci, meaning Leonardo. The inclusion of the title ser indicated that Leonardos father was a gentleman, little is known about Leonardos early life. He spent his first five years in the hamlet of Anchiano in the home of his mother and his father had married a sixteen-year-old girl named Albiera Amadori, who loved Leonardo but died young in 1465 without children. When Leonardo was sixteen, his father married again to twenty-year-old Francesca Lanfredini, pieros legitimate heirs were born from his third wife Margherita di Guglielmo and his fourth and final wife, Lucrezia Cortigiani. Leonardo received an education in Latin, geometry and mathematics. In later life, Leonardo recorded only two childhood incidents, one, which he regarded as an omen, was when a kite dropped from the sky and hovered over his cradle, its tail feathers brushing his face
5.
Tuscany
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Tuscany is a region in central Italy with an area of about 23,000 square kilometres and a population of about 3.8 million inhabitants. Tuscany is known for its landscapes, traditions, history, artistic legacy, Tuscany produces wines, including Chianti, Vino Nobile di Montepulciano, Morellino di Scansano and Brunello di Montalcino. Having a strong linguistic and cultural identity, it is considered a nation within a nation. Tuscany is traditionally a popular destination in Italy, and the main tourist destinations by number of tourist arrivals are Florence, Pisa, Montecatini Terme, Castiglione della Pescaia and Grosseto. The village of Castiglione della Pescaia is also the most visited destination in the region. Additionally, Siena, Lucca, the Chianti region, Versilia and Val dOrcia are also internationally renowned, Tuscany has over 120 protected nature reserves, making Tuscany and its capital Florence popular tourist destinations that attract millions of tourists every year. In 2012, the city of Florence was the worlds 89th most visited city, roughly triangular in shape, Tuscany borders the regions of Liguria to the northwest, Emilia-Romagna to the north and east, Umbria to the east and Lazio to the southeast. The comune of Badia Tedalda, in the Tuscan Province of Arezzo, has an exclave named Ca Raffaello within Emilia-Romagna, Tuscany has a western coastline on the Tyrrhenian Sea, containing the Tuscan Archipelago, of which the largest island is Elba. Tuscany has an area of approximately 22,993 square kilometres, surrounded and crossed by major mountain chains, and with few plains, the region has a relief that is dominated by hilly country used for agriculture. Hills make up nearly two-thirds of the total area, covering 15,292 square kilometres, and mountains. Plains occupy 8. 4% of the total area—1,930 square kilometres —mostly around the valley of the River Arno, many of Tuscanys largest cities lie on the banks of the Arno, including the capital Florence, Empoli and Pisa. The pre-Etruscan history of the area in the late Bronze and Iron Ages parallels that of the early Greeks, following this, the Villanovan culture saw Tuscany, and the rest of Etruria, taken over by chiefdoms. City-states developed in the late Villanovan before Orientalization occurred and the Etruscan civilization rose, the Etruscans created the first major civilization in this region, large enough to establish a transport infrastructure, to implement agriculture and mining and to produce vibrant art. The Etruscans lived in Etruria well into prehistory, throughout their existence, they lost territory to Magna Graecia, Carthage and Celts. Despite being seen as distinct in its manners and customs by contemporary Greeks, the cultures of Greece, one reason for its eventual demise was this increasing absorption by surrounding cultures, including the adoption of the Etruscan upper class by the Romans. Soon after absorbing Etruria, Rome established the cities of Lucca, Pisa, Siena, and Florence, endowed the area with new technologies and development, and ensured peace. These developments included extensions of existing roads, introduction of aqueducts and sewers, however, many of these structures have been destroyed by erosion due to weather. The Roman civilization in the West collapsed in the 5th century AD, in the years following 572, the Longobards arrived and designated Lucca the capital of their Duchy of Tuscia
6.
Venice
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Venice is a city in northeastern Italy and the capital of the Veneto region. It is situated across a group of 118 small islands that are separated by canals and these are located in the shallow Venetian Lagoon, an enclosed bay that lies between the mouths of the Po and the Piave Rivers. Parts of Venice are renowned for the beauty of their settings, their architecture, the lagoon and a part of the city are listed as a World Heritage Site. In 2014,264,579 people resided in Comune di Venezia, together with Padua and Treviso, the city is included in the Padua-Treviso-Venice Metropolitan Area, with a total population of 2.6 million. PATREVE is a metropolitan area without any degree of autonomy. The name is derived from the ancient Veneti people who inhabited the region by the 10th century BC, the city was historically the capital of the Republic of Venice. Venice has been known as the La Dominante, Serenissima, Queen of the Adriatic, City of Water, City of Masks, City of Bridges, The Floating City, and City of Canals. The City State of Venice is considered to have been the first real international financial center which gradually emerged from the 9th century to its peak in the 14th century and this made Venice a wealthy city throughout most of its history. It is also known for its several important artistic movements, especially the Renaissance period, Venice has played an important role in the history of symphonic and operatic music, and it is the birthplace of Antonio Vivaldi. Venice has been ranked the most beautiful city in the world as of 2016, the name Venetia, however, derives from the Roman name for the people known as the Veneti, and called by the Greeks Eneti. The meaning of the word is uncertain, although there are other Indo-European tribes with similar-sounding names, such as the Celtic Veneti, Baltic Veneti, and the Slavic Wends. Linguists suggest that the name is based on an Indo-European root *wen, so that *wenetoi would mean beloved, lovable, a connection with the Latin word venetus, meaning the color sea-blue, is also possible. The alternative obsolete form is Vinegia, some late Roman sources reveal the existence of fishermen on the islands in the original marshy lagoons. They were referred to as incolae lacunae, the traditional founding is identified with the dedication of the first church, that of San Giacomo on the islet of Rialto — said to have taken place at the stroke of noon on 25 March 421. Beginning as early as AD166 to 168, the Quadi and Marcomanni destroyed the center in the area. The Roman defences were again overthrown in the early 5th century by the Visigoths and, some 50 years later, New ports were built, including those at Malamocco and Torcello in the Venetian lagoon. The tribuni maiores, the earliest central standing governing committee of the islands in the Lagoon, the traditional first doge of Venice, Paolo Lucio Anafesto, was actually Exarch Paul, and his successor, Marcello Tegalliano, was Pauls magister militum. In 726 the soldiers and citizens of the Exarchate rose in a rebellion over the controversy at the urging of Pope Gregory II
7.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
8.
Louis XII of France
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Louis XII was a monarch of the House of Valois who ruled as King of France from 1498 to 1515 and King of Naples from 1501 to 1504. The son of Charles, Duke of Orléans, and Maria of Cleves, he succeeded his cousin Charles VIII, who died without a closer heir in 1498. Before his accession to the throne of France, he was known as Louis of Orléans and was compelled to be married to his disabled and supposedly sterile cousin Joan by his second cousin, king Louis XI. By doing so, Louis XI hoped to extinguish the Orléans cadet branch of the House of Valois, Louis of Orléans was one of the great feudal lords who opposed the French monarchy in the conflict known as the Mad War. At the royal victory in the Battle of Saint-Aubin-du-Cormier in 1488, Louis was captured and he subsequently took part in the Italian War of 1494–1498 as one of the French commanders. When Louis XII became king in 1498, he had his marriage with Joan annulled by Pope Alexander VI and instead married Anne of Brittany and this marriage allowed Louis to reinforce the personal Union of Brittany and France. Louis persevered in the Italian Wars, initiating a second Italian campaign for the control of the Kingdom of Naples, Louis conquered the Duchy of Milan in 1500 and pushed forward to the Kingdom of Naples, which fell to him in 1501. Proclaimed King of Naples, Louis faced a new coalition gathered by Ferdinand II of Aragon and was forced to cede Naples to Spain in 1504. A popular king, Louis was proclaimed Father of the People in 1506 by the Estates-General of Tours for his reduction of the tax known as taille, legal reforms, Louis XII died in 1515 without a male heir. He was succeeded by his cousin Francis from the Angoulême cadet branch of the House of Valois, Louis was born on 27 June 1462 in the Château de Blois, Touraine. The son of Charles, Duke of Orléans, and Marie of Cleves, however, Louis XI may have been more influenced in this opinion by his opposition to the entire Orleanist faction of the royal family than by the actual facts of this paternity case. Despite any alleged doubts that King Louis XI may have had, King Louis XI died on 30 August 1483. He was succeeded to the throne of France by his thirteen year-old son, nobody knew the direction which the new king would take in leading the kingdom. Accordingly, on 24 October 1483, a call went out for a convocation of the Estates General of the French kingdom, in January 1484, deputies of the Estates General began to arrive in Tours, France. The deputies represented three different estates in society, the First Estate was the Church, in France this meant the Roman Catholic Church. The Second Estate was composed of the nobility and the royalty of France, the Third Estate was generally composed of commoners and the class of traders and merchants in France. Louis, the current Duke of Orleans and future Louis XII, each estate brought their chief complaints to the Estates General in hopes to have some impact on the policies that the new King would pursue. The First Estate wanted a return to the Pragmatic Sanction, the Pragmatic Sanction had been first instituted by King Charles VII, the current King Charles VIIIs grandfather
9.
Mathematics and art
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Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty, Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts, Mathematics and art have a long historical relationship. Artists have used mathematics since the 5th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1, persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione, illustrated with woodcuts by Leonardo da Vinci, another Italian painter, Piero della Francesca, developed Euclids ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I, in modern times, the graphic artist M. C. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jaali pierced stone screens, and widespread muqarnas vaulting. Mathematics has directly influenced art with conceptual tools such as perspective, the analysis of symmetry, and mathematical objects such as polyhedra. Magnus Wenninger creates colourful stellated polyhedra, originally as models for teaching, mathematical concepts such as recursion and logical paradox can be seen in paintings by Rene Magritte and in engravings by M. C. Computer art often makes use of including the Mandelbrot set. Polykleitos the elder was a Greek sculptor from the school of Argos, and his works and statues consisted mainly of bronze and were of athletes. While his sculptures may not be as famous as those of Phidias, Polykleitos uses the distal phalanx of the little finger as the basic module for determining the proportions of the human body. Next, he takes the length and multiplies that by √2 to get the length of the palm from the base of the finger to the ulna. This geometric series of measurements progresses until Polykleitos has formed the arm, chest, body, the influence of the Canon of Polykleitos is immense in Classical Greek, Roman, and Renaissance sculpture, many sculptors following Polykleitoss prescription. While none of Polykleitoss original works survive, Roman copies demonstrate his ideal of physical perfection, some scholars argue that Pythagorean thought influenced the Canon of Polykleitos. In classical times, rather than making distant figures smaller with linear perspective, painters sized objects, in the Middle Ages, some artists used reverse perspective for special emphasis. The Muslim mathematician Alhazen described a theory of optics in his Book of Optics in 1021, the Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand nature and the arts
10.
John Napier
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John Napier of Merchiston, also signed as Neper, Nepair, nicknamed Marvellous Merchiston) was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston and his Latinized name was Ioannes Neper. John Napier is best known as the discoverer of logarithms and he also invented the so-called Napiers bones and made common the use of the decimal point in arithmetic and mathematics. Napiers birthplace, Merchiston Tower in Edinburgh, is now part of the facilities of Edinburgh Napier University, Napier died from the effects of gout at home at Merchiston Castle and his remains were buried in the kirkyard of St Giles. Following the loss of the kirkyard there to build Parliament House, archibald Napier was 16 years old when John Napier was born. As was the practice for members of the nobility at that time, he was privately tutored and did not have formal education until he was 13. He did not stay in very long. It is believed that he dropped out of school in Scotland, in 1571, Napier, aged 21, returned to Scotland, and bought a castle at Gartness in 1574. On the death of his father in 1608, Napier and his family moved into Merchiston Castle in Edinburgh and he died at the age of 67. In such conditions, it is surprising that many mathematicians were acutely aware of the issues of computation and were dedicated to relieving practitioners of the calculation burden. In particular, the Scottish mathematician John Napier was famous for his devices to assist with computation and he invented a well-known mathematical artifact, the ingenious numbering rods more quaintly known as “Napiers bones, ” that offered mechanical means for facilitating computation. He appreciated that, for the most part, practitioners who had laborious computations generally did them in the context of trigonometry, therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant. His work, Mirifici Logarithmorum Canonis Descriptio contained fifty-seven pages of explanatory matter, the book also has an excellent discussion of theorems in spherical trigonometry, usually known as Napiers Rules of Circular Parts. Modern English translations of both Napiers books on logarithms, and their description can be found on the web, as well as a discussion of Napiers Bones and his invention of logarithms was quickly taken up at Gresham College, and prominent English mathematician Henry Briggs visited Napier in 1615. Among the matters discussed were a re-scaling of Napiers logarithms. Napier delegated to Briggs the computation of a revised table, the computational advance available via logarithms, the converse of powered numbers or exponential notation, was such that it made calculations by hand much quicker. The way was opened to later scientific advances, in astronomy, dynamics and he improved Simon Stevins decimal notation. Lattice multiplication, used by Fibonacci, was more convenient by his introduction of Napiers bones
11.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
12.
Golden ratio
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio
