1.
Fibonacci number
–
The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently
Fibonacci number
–
A page of
Fibonacci 's
Liber Abaci from the
Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Latin and Roman numerals and the value in Hindu-Arabic numerals.
Fibonacci number
–
A tiling with squares whose side lengths are successive Fibonacci numbers
2.
Otto Fibonacci
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This article contains a list of minor characters in the American television series Prison Break. The listed characters are those who are played by guest stars, the characters are listed alphabetically by their last name or by the name which appears in the episode credits. Jonathan Krantz, or the Pad Man to fans, is the main antagonist in the series and he is the leader of The Company and holds the rank of General. He appears only fleetingly in his first two seasons, as William Kims shadowy superior and architect of Michael Scofields incarceration at Sona. The character plays a large and important role in season 4, in earlier seasons, he is identified by his tendency to issue orders on notepads, to avoid being recorded, a tendency which he fully abandons by the fourth season. In his appearance in the season 4 premiere, he discovers that James Whistler, after Whistlers betrayal, he orders drastic measures to be taken against The Companys enemies, dispatching the ruthless assassin Wyatt to kill anyone and everyone necessary to achieve those ends. When it is reported that Agent Don Self did an identity search on the General, he seems to be very shocked, in Safe and Sound, it is revealed to the viewers that the Generals name is Jonathan Krantz, as seen from the images of him. Learning that the General has one of the Scylla Cards, Michaels team plan an attack against his limo in The Price, due to the betrayal of team member Roland Glenn, the General is alerted and the attack is not successful. Now knowing that Scylla is in danger, the General orders it to be moved and he is tricked in the next episode into thinking that Wyatt has killed the brothers, but decides to go ahead with moving Scylla anyway. He plays a role in Selfless, where he comes face to face with the Scylla team for the first time. When he goes to confront Michael in the bunker, he is forced to hand over his card by Michaels team at gunpoint. Thinking it is useless without the other cards, hes shocked when Michael unlocks Scylla with copies of the five cards. Under duress and the threat of his daughter Lisas life, he allows the Scylla team to escape, in the following episodes, the General is put under immense pressure as he must direct efforts to retrieve Scylla while dealing with dissent amongst his own ranks. After learning in Just Business that Gretchen Morgan and Self has Scylla, he sends his men to retrieve it and his men fail, but capture Michael instead. Discovering that he is in a condition, he is taken back to Company HQ. The General orders the doctors to look after Michael and keep him stable, the General has a somewhat reduced role in these final episodes, often having only a couple of scenes per episode. This is in due to the introduction of a new prominent antagonist. A former employee of The Company, Christina turns rogue and acquires Scylla to set her own plan in motion
Otto Fibonacci
–
Patricia Wettig, the actress who played Caroline Reynolds.
Otto Fibonacci
–
John Billingsley played Terrence Steadman in the first season.
Otto Fibonacci
–
Haywire in Season 1.
Otto Fibonacci
–
Callie Thorne played Pamela Mahone.
3.
Pisa
–
Pisa is a city in Tuscany, Central Italy, straddling the Arno just before it empties into the Tyrrhenian Sea. It is the city of the Province of Pisa. Although Pisa is known worldwide for its tower, the city of over 90,834 residents contains more than 20 other historic churches, several medieval palaces. Much of the architecture was financed from its history as one of the Italian maritime republics. The origin of the name, Pisa, is a mystery, while the origin of the city had remained unknown for centuries, the Pelasgi, the Greeks, the Etruscans, and the Ligurians had variously been proposed as founders of the city. Archaeological remains from the 5th century BC confirmed the existence of a city at the sea, trading with Greeks, the presence of an Etruscan necropolis, discovered during excavations in the Arena Garibaldi in 1991, confirmed its Etruscan origins. Ancient Roman authors referred to Pisa as an old city, strabo referred Pisas origins to the mythical Nestor, king of Pylos, after the fall of Troy. Virgil, in his Aeneid, states that Pisa was already a center by the times described. The Virgilian commentator Servius wrote that the Teuti, or Pelops, the maritime role of Pisa should have been already prominent if the ancient authorities ascribed to it the invention of the naval ram. Pisa took advantage of being the port along the western coast from Genoa to Ostia. Pisa served as a base for Roman naval expeditions against Ligurians, Gauls, in 180 BC, it became a Roman colony under Roman law, as Portus Pisanus. In 89 BC, Portus Pisanus became a municipium, Emperor Augustus fortified the colony into an important port and changed the name in Colonia Iulia obsequens. It is supposed that Pisa was founded on the shore, however, due to the alluvial sediments from the Arno and the Serchio, whose mouth lies about 11 kilometres north of the Arnos, the shore moved west. Strabo states that the city was 4.0 kilometres away from the coast, currently, it is located 9.7 kilometres from the coast. However it was a city, with ships sailing up the Arno. In the 90s AD, a complex was built in the city. During the later years of the Roman Empire, Pisa did not decline as much as the cities of Italy, probably thanks to the complexity of its river system. After Charlemagne had defeated the Lombards under the command of Desiderius in 774, Pisa went through a crisis, politically it became part of the duchy of Lucca
Pisa
–
Pisa
Pisa
–
Coat of arms
Pisa
–
Hypothetical map of Pisa in the 5th century AD
Pisa
–
Hypothetical map of Pisa in the 11th century AD
4.
Mathematician
–
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and with whom the study of mathematics for its own sake begins, the first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked. Science and mathematics in the Islamic world during the Middle Ages followed various models and it was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences, an example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham, the Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking. ”Thus, seminars, overall, science became the focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems
Mathematician
–
Euclid (holding
calipers), Greek mathematician, known as the "Father of Geometry"
Mathematician
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In 1938 in the United States, mathematicians were desired as teachers, calculating machine operators, mechanical engineers, accounting auditor bookkeepers, and actuary statisticians
Mathematician
–
Archimedes, c. 287 – 212 BC
Mathematician
–
Brahmagupta, c. 598 - 670
5.
Liber Abaci
–
Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. Liber Abaci was among the first Western books to describe Hindu–Arabic numbers traditionally described as Arabic Numerals, by addressing the applications of both commercial tradesmen and mathematicians, it contributed to convincing the public of the superiority of the Hindu–Arabic numeral system. The title of Liber Abaci means The Book of Calculation, the second version of Liber Abaci was dedicated to Michael Scot in 1227 CE. No versions of the original 1202 CE book have been found, the first section introduces the Hindu–Arabic numeral system, including methods for converting between different representation systems. The second section presents examples from commerce, such as conversions of currency and measurements, another example in this chapter, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots, the book also includes proofs in Euclidean geometry. Fibonaccis method of solving algebraic equations shows the influence of the early 10th-century Egyptian mathematician Abū Kāmil Shujāʿ ibn Aslam, there are three key differences between Fibonaccis notation and modern fraction notation. We generally write a fraction to the right of the number to which it is added. Fibonacci instead would write the same fraction to the left, i. e.132. That is, b a d c = a c + b c d, the notation was read from right to left. For example, 29/30 could be written as 124235 and this can be viewed as a form of mixed radix notation, and was very convenient for dealing with traditional systems of weights, measures, and currency. Sigler also points out an instance where Fibonacci uses composite fractions in which all denominators are 10, Fibonacci sometimes wrote several fractions next to each other, representing a sum of the given fractions. For instance, 1/3+1/4 = 7/12, so a notation like 14132 would represent the number that would now more commonly be written as the mixed number 2712, or simply the improper fraction 3112. Notation of this form can be distinguished from sequences of numerators and denominators sharing a fraction bar by the break in the bar. If all numerators are 1 in a written in this form, and all denominators are different from each other. This notation was also combined with the composite fraction notation. The complexity of this notation allows numbers to be written in different ways. In the Liber Abaci, Fibonacci says the following introducing the Modus Indorum or the method of the Indians, today known as Hindu–Arabic numerals or traditionally, just Arabic numerals
Liber Abaci
–
A page of the Liber Abaci from the
Biblioteca Nazionale di Firenze showing (on right) the numbers of the
Fibonacci sequence.
6.
Italians
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Italians are a nation and ethnic group native to Italy who share a common culture, ancestry and speak the Italian language as a native tongue. The majority of Italian nationals are speakers of Standard Italian. Italians have greatly influenced and contributed to the arts and music, science, technology, cuisine, sports, fashion, jurisprudence, banking, Italian people are generally known for their localism and their attention to clothing and family values. The term Italian is at least 3,000 years old and has a history that goes back to pre-Roman Italy. According to one of the common explanations, the term Italia, from Latin, Italia, was borrowed through Greek from the Oscan Víteliú. 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 Etruscan civilization reached its peak about the 7th century BC, but by 509 BC, when the Romans overthrew their Etruscan monarchs, its control in Italy was on the wane. By 350 BC, after a series of wars between Greeks and Etruscans, the Latins, with Rome as their capital, gained the ascendancy by 272 BC, and they managed to unite the entire Italian peninsula. This period of unification was followed by one of conquest in the Mediterranean, in the course of the century-long struggle against Carthage, the Romans conquered Sicily, Sardinia and Corsica. Finally, in 146 BC, at the conclusion of the Third Punic War, with Carthage completely destroyed and its inhabitants enslaved, octavian, the final victor, was accorded the title of Augustus by the Senate and thereby became the first Roman emperor. After two centuries of rule, in the 3rd century AD, Rome was threatened by internal discord and menaced by Germanic and Asian invaders. Emperor Diocletians administrative division of the empire into two parts in 285 provided only temporary relief, it became permanent in 395, in 313, Emperor Constantine accepted Christianity, and churches thereafter rose throughout the empire. However, he moved his capital from Rome to Constantinople. The last Western emperor, Romulus Augustulus, was deposed in 476 by a Germanic foederati general in Italy and his defeat marked the end of the western part of the Roman Empire. During most of the period from the fall of Rome until the Kingdom of Italy was established in 1861, Odoacer ruled well for 13 years after gaining control of Italy in 476. Then he was attacked and defeated by Theodoric, the king of another Germanic tribe, Theodoric and Odoacer ruled jointly until 493, when Theodoric murdered Odoacer. Theodoric continued to rule Italy with an army of Ostrogoths and a government that was mostly Italian, after the death of Theodoric in 526, the kingdom began to grow weak
Italians
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Amerigo Vespucci, the notable geographer and traveller from whose name the word
America is derived.
Italians
Italians
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Christopher Columbus, the discoverer of the
New World.
Italians
–
Laura Bassi, the first chairwoman of a university in a scientific field of studies.
7.
Republic of Pisa
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The Republic of Pisa was a de facto independent state centered on the Tuscan city of Pisa during the late 10th and 11th centuries. It rose to become a powerhouse, a commercial center whose merchants dominated Mediterranean and Italian trade for a century before being surpassed and superseded by the Republic of Genoa. The power of Pisa as a mighty maritime nation began to grow, during the High Middle Ages the city grew into a very important commercial and naval center and controlled a significant Mediterranean merchant fleet and navy. It expanded its influence through the sack of Reggio di Calabria in the south of Italy in 1005, Pisa was in continuous conflict with the Saracens, whose bases were in the Italian astersa, for control of the Mediterranean. In alliance with Genoa, Sardinia was captured in 1016 with the defeat of the Saracen leader Mujāhid al-‘Āmirī and this victory gave Pisa supremacy in the Tyrrhenian Sea. When the Pisans subsequently ousted the Genoese from Sardinia, a new conflict, between 1030 and 1035 Pisa went on to successfully defeat several rival towns in the Emirate of Sicily and conquer Carthage in North Africa. In 1051-1052 Admiral Jacopo Ciurini conquered Corsica, provoking more resentment from the Genoese. In 1063, the Pisans approached the Norman Roger I of Sicily, Roger declined due to other commitments. With no land support, the Pisan attack against Palermo failed, in 1060 Pisa engaged in its first battle against Genoa and the Pisan victory helped to consolidate its position in the Mediterranean. This was simply a confirmation of the present situation, because at the time the marquis of Tuscany had already excluded from power. Pisa sacked the Zirid city of Mahdia in 1088, four years later, Pisan and Genoese ships helped Alfonso VI of Castile force El Cid out of Valencia. In 1092 Pope Urban II awarded Pisa supremacy over Corsica and Sardinia, a Pisan fleet of 120 ships participated in the First Crusade and the Pisans were instrumental in the siege of Jerusalem in 1099. On their way to the Holy Land the Pisan ships did not miss the opportunity to sack several Byzantine islands, the Pisan crusaders were led by their archbishop, Dagobert, the future Latin Patriarch of Jerusalem. Pisa and the maritime republics took advantage of the crusade to establish trading posts and colonies in the eastern coastal regions of Syria, Lebanon. In particular the Pisans founded colonies in Antioch, Acre, Jaffa, Tripoli, Tyre, in all these cities the Pisans were granted privileges and immunity from taxation, but had to contribute to their defence in case of attack. In the 12th century the Pisan quarter in the part of Constantinople had grown to 1,000 people. For some years of that century Pisa was the most prominent merchant and military ally of the Byzantine Empire, Pisa, as an international power, was destroyed forever by the crushing defeat of its navy in the Battle of Meloria against Genoa in 1284. In this battle, most of the Pisan galleys were destroyed, in 1290, an assault by Genoese ships against the Porto Pisano sealed the fate of the independent Pisan state
Republic of Pisa
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Map of Pisa in the 11th century
Republic of Pisa
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Flag
Republic of Pisa
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The
Cathedral of Pisa was built during the Republic's heydays (11th and 12th century) and financed by the spoils and loot from the
Sack of Mahdia (1087)
Republic of Pisa
–
Relief of the
seaport of Pisa on the
Tower of Pisa
8.
Middle Ages
–
In the history of Europe, the Middle Ages or Medieval Period lasted from the 5th to the 15th century. It began with the fall of the Western Roman Empire and merged into the Renaissance, the Middle Ages is the middle period of the three traditional divisions of Western history, classical antiquity, the medieval period, and the modern period. The medieval period is subdivided into the Early, High. Population decline, counterurbanisation, invasion, and movement of peoples, the large-scale movements of the Migration Period, including various Germanic peoples, formed new kingdoms in what remained of the Western Roman Empire. In the seventh century, North Africa and the Middle East—once part of the Byzantine Empire—came under the rule of the Umayyad Caliphate, although there were substantial changes in society and political structures, the break with classical antiquity was not complete. The still-sizeable Byzantine Empire survived in the east and remained a major power, the empires law code, the Corpus Juris Civilis or Code of Justinian, was rediscovered in Northern Italy in 1070 and became widely admired later in the Middle Ages. In the West, most kingdoms incorporated the few extant Roman institutions, monasteries were founded as campaigns to Christianise pagan Europe continued. The Franks, under the Carolingian dynasty, briefly established the Carolingian Empire during the later 8th, the Crusades, first preached in 1095, were military attempts by Western European Christians to regain control of the Holy Land from Muslims. Kings became the heads of centralised nation states, reducing crime and violence, intellectual life was marked by scholasticism, a philosophy that emphasised joining faith to reason, and by the founding of universities. Controversy, heresy, and the Western Schism within the Catholic Church paralleled the conflict, civil strife. Cultural and technological developments transformed European society, concluding the Late Middle Ages, the Middle Ages is one of the three major periods in the most enduring scheme for analysing European history, classical civilisation, or Antiquity, the Middle Ages, and the Modern Period. Medieval writers divided history into periods such as the Six Ages or the Four Empires, when referring to their own times, they spoke of them as being modern. In the 1330s, the humanist and poet Petrarch referred to pre-Christian times as antiqua, leonardo Bruni was the first historian to use tripartite periodisation in his History of the Florentine People. Bruni and later argued that Italy had recovered since Petrarchs time. The Middle Ages first appears in Latin in 1469 as media tempestas or middle season, in early usage, there were many variants, including medium aevum, or middle age, first recorded in 1604, and media saecula, or middle ages, first recorded in 1625. The alternative term medieval derives from medium aevum, tripartite periodisation became standard after the German 17th-century historian Christoph Cellarius divided history into three periods, Ancient, Medieval, and Modern. The most commonly given starting point for the Middle Ages is 476, for Europe as a whole,1500 is often considered to be the end of the Middle Ages, but there is no universally agreed upon end date. English historians often use the Battle of Bosworth Field in 1485 to mark the end of the period
Middle Ages
–
The
Cross of Mathilde, a
crux gemmata made for
Mathilde, Abbess of Essen (973–1011), who is shown kneeling before the Virgin and Child in the
enamel plaque. The body of Christ is slightly later. Probably made in
Cologne or
Essen, the cross demonstrates several medieval techniques:
cast figurative sculpture,
filigree, enamelling, gem polishing and setting, and the reuse of Classical
cameos and
engraved gems.
Middle Ages
–
A late Roman statue
depicting the four Tetrarchs, now in
Venice
Middle Ages
–
Coin of
Theodoric
Middle Ages
–
Mosaic showing
Justinian with
the bishop of
Ravenna, bodyguards, and courtiers
9.
Consul
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Consul was the title of one of the chief magistrates of the Roman Republic, and subsequently a somewhat significant title under the Roman Empire. The title was used in other city states and also revived in modern states. The relating adjective is consular, from the consularis, in modern terminology, a consul is a type of diplomat. The American Heritage Dictionary defines consul as an appointed by a government to reside in a foreign country. Throughout most of southern France, a consul was an equivalent to the échevins of the north. The most prominent were those of Bordeaux and Toulouse, which came to be known as jurats and capitouls, the capitouls of Toulouse were granted transmittable nobility. In many other towns the first consul, was the equivalent of a mayor today, assisted by a variable number of secondary consuls. His main task was to levy and collect tax, the Dukes of Gaeta often used also the title of consul in its Greek form Hypatos. The city-state of Genoa, unlike ancient Rome, bestowed the title of consul on various state officials, among these were Genoese officials stationed in various Mediterranean ports, whose role included helping Genoese merchants and sailors in difficulties with the local authorities. This institution, with its name, was emulated by other powers and is reflected in the modern usage of the word. In reality, the first consul, Bonaparte, dominated his two colleagues and held power, soon making himself consul for life and eventually, in 1804. Chief magistrate, an office held for four months by one of the consuls. As noted above, Bologna already had consuls at some parts of its Medieval history, while many cities had a double-headed chief magistracy, often another title was used, such as Duumvir or native styles such as Meddix, but consul was used in some. It was not uncommon for an organization under Roman private law to copy the terminology of state, the founding statute, or contract, of such an organisation was called lex, law. The people elected each year were patricians, members of the upper class. org, see each present country
Consul
–
A portrait of the three Consuls,
Jean Jacques Régis de Cambacérès,
Napoleon Bonaparte and
Charles-François Lebrun (left to right)
10.
Bejaia
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Béjaïa, formerly Bougie and Bugia, is a Mediterranean port city on the Gulf of Béjaïa in Algeria, it is the capital of Béjaïa Province, Kabylia. Béjaïa is the largest principally Kabyle-speaking city in the Kabylie region of Algeria, the history of Béjaïa explains the diversity of the local population. All three of these features are contained in the Gouraya National Park. The Soummam river runs past the town, under French rule, it was formerly known under various European names, such as Budschaja in German, Bugia in Italian, and Bougie. It was an important town and a bishopric in the province of Mauretania Caesariensis, after the 7th-century Muslim conquest, it was refounded as Béjaïa, the Hammadid dynasty made it their capital, and it became an important port and centre of culture. The son of a Pisan merchant, posthumously known as Fibonacci, there learned under the Almohad dynasty about Muslim mathematics and he introduced these and modern mathematics into medieval Europe. In 1315, Raymond Lully died as a result of being stoned at Béjaïa, where, after a Spanish occupation, the city was taken by the Ottoman Turks in the Capture of Bougie in 1555. For nearly three centuries, Béjaïa was a stronghold of the Barbary pirates, city landmarks include a 16th-century mosque and a casbah built by the Spanish in 1545. A picture of the Orientalist painter Maurice Boitel, who painted in the city for a while and it was captured by the French in 1833 and became a part of colonial Algeria. Most of the time it was the seat of an arrondissement in the Département of Constantine, during World War II, Operation Torch landed forces in North Africa, including a battalion of the British Royal West Kent Regiment at Béjaïa on November 11,1942. That same day, at 4,40 PM, a German Luftwaffe air raid struck Béjaïa with thirty Ju-88 bombers, the transports Awatea and Cathay were sunk and the monitor HMS Roberts was damaged. The following day, the anti-aircraft ship SS Tynwald was torpedoed and sank, after Algerian independence, it became the eponymous capital of Béjaïa Province, covering part of the eastern Berber region Kabylia. With the spread of Christianity, Saldae became a bishopric and its bishop Paschasius was one of the Catholic bishops whom the Arian Vandal king Huneric summoned to Carthage in 484 and then exiled. Christianity survived the Arab conquest, the disappearance of the old city of Saldae, a letter of Pope Gregory VII exists, addressed to clero et populo Buzee, in which he writes of the consecration of a bishop named Servandus for Christian North Africa. No longer a residential bishopric, Saldae is today listed by the Catholic Church as a titular see. and this titular see was for a long time, alternatively and concurrently with the citys authentic Roman Latin name Saldae, called Bugia, the Italian language form of Béjaïa. The population of the city in 2008 in the latest census was 177,988, the northern terminus of the Hassi Messaoud oil pipeline from the Sahara, Béjaïa is the principal oil port of the Western Mediterranean. Exports, aside from crude petroleum, include iron, phosphates, wines, dried figs, the city also has textile and cork industries. Cevital has its office in the city. Net History of Béjaïa GigaCatholic
Bejaia
–
Béjaïa ⴱⴳⴰⵢⴻⵜ
Bejaia
–
Monkey Peak (
Pic des singes).
Bejaia
–
The Western Roman empire, in the second century AD, during the reign of Hadrian. Saldae can be seen on the south coast of the Mediterranean
Bejaia
–
Coin of the
Hafsids, with ornamental
Kufic script, from Béjaïa, 1249-1276.
11.
Almohad dynasty
–
The Almohad Caliphate was a Moroccan Berber Muslim movement founded in the 12th century. The Almohad movement was founded by Ibn Tumart among the Berber Masmuda tribes of southern Morocco, around 1120, the Almohads first established a Berber state in Tinmel in the Atlas Mountains. They succeeded in overthrowing the ruling Almoravid dynasty governing Morocco by 1147 and they then extended their power over all of the Maghreb by 1159. Al-Andalus followed the fate of North Africa and all Islamic Iberia was under Almohad rule by 1172, nearly all of the Moorish dominions in Iberia were lost soon after, with the great Moorish cities of Cordova and Seville falling to the Christians in 1236 and 1248 respectively. The Almohads continued to rule in Africa until the loss of territory through the revolt of tribes and districts enabled the rise of their most effective enemies. The Almohad movement originated with Ibn Tumart, a member of the Masmuda, at the time, Morocco, and much of the rest of North Africa and Spain, was under the rule of the Almoravids, a Sanhaja Berber dynasty. Early in his life, Ibn Tumart went to Spain to pursue his studies, in Baghdad, Ibn Tumart attached himself to the theological school of al-Ashari, and came under the influence of the teacher al-Ghazali. He soon developed his own system, combining the doctrines of various masters, Ibn Tumarts main principle was a strict unitarianism, which denied the independent existence of the attributes of God as being incompatible with His unity, and therefore a polytheistic idea. Ibn Tumart represented a revolt against what he perceived as anthropomorphism in Muslim orthodoxy and his followers would become known as the al-Muwahhidun, meaning those who affirm the unity of God. After his return to the Maghreb c.1117, Ibn Tumart spent some time in various Ifriqiyan cities, preaching and agitating, heading riotous attacks on wine-shops and on other manifestations of laxity. He laid the blame for the latitude on the dynasty of the Almoravids. His antics and fiery preaching led fed-up authorities to him along from town to town. After being expelled from Bejaia, Ibn Tumart set up camp in Mellala, in the outskirts of the city, where he received his first disciples - notably, al-Bashir and Abd al-Mumin. In 1120, Ibn Tumart and his band of followers proceeded to Morocco, stopping first in Fez. He even went so far as to assault the sister of the Almoravid emir Ali ibn Yusuf, in the streets of Fez, because she was going about unveiled, after the manner of Berber women. After the debate, the scholars concluded that Ibn Tumarts views were blasphemous and the man dangerous, but the emir decided merely to expel him from the city. Ibn Tumart took refuge among his own people, the Hargha, in his village of Igiliz. He retreated to a cave, and lived out an ascetic lifestyle, coming out only to preach his program of puritan reform
Almohad dynasty
–
The Almohads transferred the capital of Al-Andalus to
Seville.
Almohad dynasty
–
Flag
Almohad dynasty
–
Almohads after 1212
Almohad dynasty
–
La
Giralda.
12.
North Africa
–
North Africa or Northern Africa is the northernmost region of Africa. The United Nationss definition of Northern Africa is, Algeria, Egypt, Libya, Morocco, Sudan, Tunisia, the countries of Algeria, Morocco, Tunisia, and Libya are often collectively referred to as the Maghreb, which is the Arabic word for sunset. Egypt lies to the northeast and encompasses part of West Asia, while Sudan is situated on the edge of the Sahel, Egypt is a transcontinental country because of the Sinai Peninsula, which geographically lies in Western Asia. North Africa also includes a number of Spanish possessions, the Canary Islands and Madeira in the North Atlantic Ocean northwest of the African mainland are included in considerations of the region. From 3500 BC, following the abrupt desertification of the Sahara due to changes in the Earths orbit. The Islamic influence in the area is significant, and North Africa is a major part of the Muslim world. Some researchers have postulated that North Africa rather than East Africa served as the point for the modern humans who first trekked out of the continent in the Out of Africa migration. The Atlas Mountains extend across much of Morocco, northern Algeria and Tunisia, are part of the mountain system that also runs through much of Southern Europe. They recede to the south and east, becoming a steppe landscape before meeting the Sahara desert, the sediments of the Sahara overlie an ancient plateau of crystalline rock, some of which is more than four billion years old. Sheltered valleys in the Atlas Mountains, the Nile Valley and Delta, a wide variety of valuable crops including cereals, rice and cotton, and woods such as cedar and cork, are grown. Typical Mediterranean crops, such as olives, figs, dates and citrus fruits, the Nile Valley is particularly fertile, and most of the population in Egypt and Sudan live close to the river. Elsewhere, irrigation is essential to improve yields on the desert margins. The inhabitants of Saharan Africa are generally divided in a manner corresponding to the principal geographic regions of North Africa, the Maghreb, the Nile valley. The edge of the Sahel, to the south of Egypt has mainly been inhabited by Nubians, Ancient Egyptians record extensive contact in their Western desert with people that appear to have been Berber or proto-Berber, as well as Nubians from the south. They have contributed to the Arabized Berber populations, the official language or one of the official languages in all of the countries in North Africa is Arabic. The people of the Maghreb and the Sahara regions speak Berber languages and several varieties of Arabic, the Arabic and Berber languages are distantly related, both being members of the Afroasiatic language family. The Tuareg Berber languages are more conservative than those of the coastal cities. Over the years, Berbers have been influenced by contact with cultures, Greeks, Phoenicians, Egyptians, Romans, Vandals, Arabs, Europeans
North Africa
–
Market of
Biskra in Algeria, 1899
North Africa
–
Northern Africa (UN subregion)
North Africa
–
The
kasbah of
Aït Benhaddou in Morocco
North Africa
–
The first
Roman emperor native to North Africa was
Septimius Severus, born in Leptis Magna in present-day Libya.
13.
Algeria
–
Algeria, officially the Peoples Democratic Republic of Algeria, is a sovereign state in North Africa on the Mediterranean coast. Its capital and most populous city is Algiers, located in the far north. With an area of 2,381,741 square kilometres, Algeria is the tenth-largest country in the world, the country is a semi-presidential republic consisting of 48 provinces and 1,541 communes. Abdelaziz Bouteflika has been President since 1999, Berbers are the indigenous inhabitants of Algeria. Algeria is a regional and middle power, the North African country supplies large amounts of natural gas to Europe, and energy exports are the backbone of the economy. According to OPEC Algeria has the 16th largest oil reserves in the world, Sonatrach, the national oil company, is the largest company in Africa. Algeria has one of the largest militaries in Africa and the largest defence budget on the continent, most of Algerias weapons are imported from Russia, with whom they are a close ally. Algeria is a member of the African Union, the Arab League, OPEC, the countrys name derives from the city of Algiers. The citys name in turn derives from the Arabic al-Jazāir, a form of the older Jazāir Banī Mazghanna. In the region of Ain Hanech, early remnants of hominid occupation in North Africa were found, neanderthal tool makers produced hand axes in the Levalloisian and Mousterian styles similar to those in the Levant. Algeria was the site of the highest state of development of Middle Paleolithic Flake tool techniques, tools of this era, starting about 30,000 BC, are called Aterian. The earliest blade industries in North Africa are called Iberomaurusian and this industry appears to have spread throughout the coastal regions of the Maghreb between 15,000 and 10,000 BC. Neolithic civilization developed in the Saharan and Mediterranean Maghreb perhaps as early as 11,000 BC or as late as between 6000 and 2000 BC and this life, richly depicted in the Tassili nAjjer paintings, predominated in Algeria until the classical period. The amalgam of peoples of North Africa coalesced eventually into a native population that came to be called Berbers. These settlements served as market towns as well as anchorages, as Carthaginian power grew, its impact on the indigenous population increased dramatically. Berber civilization was already at a stage in which agriculture, manufacturing, trade, by the early 4th century BC, Berbers formed the single largest element of the Carthaginian army. In the Revolt of the Mercenaries, Berber soldiers rebelled from 241 to 238 BC after being unpaid following the defeat of Carthage in the First Punic War. They succeeded in obtaining control of much of Carthages North African territory, the Carthaginian state declined because of successive defeats by the Romans in the Punic Wars
Algeria
–
Ancient
Roman Empire ruins of
Timgad. Street leading to the Arch of Trajan.
Algeria
–
Flag
Algeria
–
Ancient Roman theatre in Djémila
Algeria
–
Numidia along with Egypt, Rome, and Carthage 200 BCE
14.
Mediterranean
–
The sea is sometimes considered a part of the Atlantic Ocean, although it is usually identified as a separate body of water. The name Mediterranean is derived from the Latin mediterraneus, meaning inland or in the middle of land and it covers an approximate area of 2.5 million km2, but its connection to the Atlantic is only 14 km wide. The Strait of Gibraltar is a strait that connects the Atlantic Ocean to the Mediterranean Sea and separates Gibraltar. In oceanography, it is called the Eurafrican Mediterranean Sea or the European Mediterranean Sea to distinguish it from mediterranean seas elsewhere. The Mediterranean Sea has a depth of 1,500 m. The sea is bordered on the north by Europe, the east by Asia and it is located between latitudes 30° and 46° N and longitudes 6° W and 36° E. Its west-east length, from the Strait of Gibraltar to the Gulf of Iskenderun, the seas average north-south length, from Croatia’s southern shore to Libya, is approximately 800 km. The Mediterranean Sea, including the Sea of Marmara, has an area of approximately 2,510,000 square km. The sea was an important route for merchants and travelers of ancient times that allowed for trade, the history of the Mediterranean region is crucial to understanding the origins and development of many modern societies. In addition, the Gaza Strip and the British Overseas Territories of Gibraltar and Akrotiri, the term Mediterranean derives from the Latin word mediterraneus, meaning amid the earth or between land, as it is between the continents of Africa, Asia and Europe. The Ancient Greek name Mesogeios, is similarly from μέσο, between + γη, land, earth) and it can be compared with the Ancient Greek name Mesopotamia, meaning between rivers. The Mediterranean Sea has historically had several names, for example, the Carthaginians called it the Syrian Sea and latter Romans commonly called it Mare Nostrum, and occasionally Mare Internum. Another name was the Sea of the Philistines, from the people inhabiting a large portion of its shores near the Israelites, the sea is also called the Great Sea in the General Prologue by Geoffrey Chaucer. In Ottoman Turkish, it has also been called Bahr-i Sefid, in Modern Hebrew, it has been called HaYam HaTikhon, the Middle Sea, reflecting the Seas name in ancient Greek, Latin, and modern languages in both Europe and the Middle East. Similarly, in Modern Arabic, it is known as al-Baḥr al-Mutawassiṭ, in Turkish, it is known as Akdeniz, the White Sea since among Turks the white colour represents the west. Several ancient civilisations were located around the Mediterranean shores, and were influenced by their proximity to the sea. It provided routes for trade, colonisation, and war, as well as food for numerous communities throughout the ages, due to the shared climate, geology, and access to the sea, cultures centered on the Mediterranean tended to have some extent of intertwined culture and history. Two of the most notable Mediterranean civilisations in classical antiquity were the Greek city states, later, when Augustus founded the Roman Empire, the Romans referred to the Mediterranean as Mare Nostrum
Mediterranean
–
Circa the 6th century BCE: In ancient times the Mediterranean provided sources of food and local commerce and direct routes for trade and communications, colonisation, and war. Numerous cities and colonies were situated at its shores or within the basin: Greek (red) and Phoenician (yellow)
colonies in antiquity; and other cities (grey), including the provincial
"Rom".
Mediterranean
–
Map of the Mediterranean Sea
Mediterranean
–
With its highly indented coastline and large number of islands, Greece has the longest Mediterranean coastline.
Mediterranean
–
The
Battle of Lepanto, 1571, ended in victory for the European
Holy League against the Ottoman Turks.
15.
Frederick II, Holy Roman Emperor
–
Frederick II was a Holy Roman Emperor and King of Sicily in the Middle Ages, a member of the House of Hohenstaufen. His political and cultural ambitions, based in Sicily and stretching through Italy to Germany, however, his enemies, especially the popes, prevailed, and his dynasty collapsed soon after his death. As such, he was King of Germany, of Italy, at the age of three, he was crowned King of Sicily as a co-ruler with his mother, Constance of Hauteville, the daughter of Roger II of Sicily. His other royal title was King of Jerusalem by virtue of marriage, Pope Gregory IX went so far as to call him an Antichrist. Speaking six languages, Frederick was a patron of science. He played a role in promoting literature through the Sicilian School of poetry. His Sicilian royal court in Palermo, from around 1220 to his death, saw the first use of a form of an Italo-Romance language. The poetry that emanated from the school had a significant influence on literature and he was also the first king who explicitly outlawed trials by ordeal as they were considered irrational. After his death, his line died out and the House of Hohenstaufen came to an end. Born in Iesi, near Ancona, Italy, Frederick was the son of the emperor Henry VI and he was known as the puer Apuliae. Some chronicles say that his mother, the forty-year-old Constance, gave birth to him in a square in order to forestall any doubt about his origin. In 1196 at Frankfurt am Main the infant Frederick was elected King of the Germans and his rights in Germany were disputed by Henrys brother Philip of Swabia and Otto of Brunswick. At the death of his father in 1197, Frederick was in Italy travelling towards Germany when the bad news reached his guardian, Conrad of Spoleto. Frederick was hastily brought back to his mother Constance in Palermo, Sicily, Constance of Sicily was in her own right queen of Sicily, and she established herself as regent. Upon Constances death in 1198, Pope Innocent III succeeded as Fredericks guardian, Fredericks tutor during this period was Cencio, who would become Pope Honorius III. However, Markward of Annweiler, with the support of Henrys brother, Philip of Swabia, reclaimed the regency for himself, in 1200, with the help of Genoese ships, he landed in Sicily and one year later seized the young Frederick. He thus ruled Sicily until 1202, when he was succeeded by another German captain, William of Capparone, Frederick was subsequently under tutor Walter of Palearia, until, in 1208, he was declared of age. His first task was to reassert his power over Sicily and southern Italy, Otto of Brunswick had been crowned Holy Roman Emperor by Pope Innocent III in 1209
Frederick II, Holy Roman Emperor
–
Frederick II
Frederick II, Holy Roman Emperor
–
The birth of Frederick II
Frederick II, Holy Roman Emperor
–
Gold
augustale of Emperor Frederik II, as King of Sicily 1198–1250.
Frederick II, Holy Roman Emperor
–
Frederick II (left) meets
Al-Kamil (right).
16.
National Central Library (Florence)
–
The library was founded in 1714 when scholar Antonio Magliabechi bequeathed his entire collection of books, encompassing approximately 30,000 volumes, to the city of Florence. By 1743, it was required that a copy of work published in Tuscany be submitted to the library. Originally known as the Magliabechiana, the library was opened to the public in 1747 and its holdings were combined with those of the Biblioteca Palatina in 1861, and by 1885, the library had been renamed as the National Central Library of Florence, or the BNCF. Since 1870, the library has collected copies of all Italian publications, since 1935, the collections have been housed in a building designed by Cesare Bazzani and V. Mazzei, located along the Arno River in the quarter of Santa Croce. Before this, they were found in various rooms belonging to the Uffizi Gallery, the National Library System, located in the BNCF, is responsible for the automation of library services and the indexing of national holdings. Unfortunately, a flood of the Arno River in 1966 damaged nearly one-third of the librarys holdings, most notably its periodicals and Palatine. The Restoration Center was subsequently established and may be credited with saving many of these priceless artifacts, however, much work remains to be done and some items are forever lost
National Central Library (Florence)
–
The Library from Piazza dei Cavalleggeri
National Central Library (Florence)
–
Arno river facade
National Central Library (Florence)
–
The library beside Santa Croce
National Central Library (Florence)
–
The distribution hall
17.
Place value
–
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100
Place value
–
Numeral systems
18.
Numeral system
–
A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
Numeral system
–
Numeral systems
19.
Bookkeeping
–
Bookkeeping is the recording of financial transactions, and is part of the process of accounting in business. Transactions include purchases, sales, receipts, and payments by a person or an organization/corporation. Bookkeeping is usually performed by a bookkeeper, a bookkeeper is a person who records the day-to-day financial transactions of a business. He or she is responsible for writing the daybooks, which contain records of purchases, sales, receipts. The bookkeeper brings the books to the trial stage, an accountant may prepare the income statement and balance sheet using the trial balance. The origin of book-keeping is lost in obscurity, but recent researches would appear to show that some method of keeping accounts has existed from the remotest times. Babylonian records have been found dating back as far as 2600 B. C. written with a stylus on small slabs of clay, the term waste book was used in colonial America referring to bookkeeping. The purpose was to document daily transactions including receipts and expenditures and this was recorded in chronological order, and the purpose was for temporary use only. The daily transactions would then be recorded in a daybook or account ledger in order to balance the accounts, the name waste book comes from the fact that once the waste books data were transferred to the actual journal, the waste book could be discarded. The bookkeeping process primarily records the effects of transactions. The difference between a manual and any electronic accounting system results from the formers latency between the recording of a transaction and its posting in the relevant account. In the normal course of business, a document is produced each time a transaction occurs, Sales and purchases usually have invoices or receipts. Deposit slips are produced when lodgements are made to a bank account, checks are written to pay money out of the account. Bookkeeping first involves recording the details of all of these documents into multi-column journals. For example, all sales are recorded in the sales journal. Each column in a journal normally corresponds to an account, in the single entry system, each transaction is recorded only once. Most individuals who balance their check-book each month are using such a system, after a certain period, typically a month, each column in each journal is totalled to give a summary for that period. Using the rules of double-entry, these summaries are then transferred to their respective accounts in the ledger
Bookkeeping
–
Portrait of the Italian
Luca Pacioli, painted by
Jacopo de' Barbari, 1495, (
Museo di Capodimonte). Pacioli is regarded as the Father of Accounting and Bookkeeping.
20.
Abacus
–
The exact origin of the abacus is still unknown. Today, abaci are often constructed as a frame with beads sliding on wires. The use of the word abacus dates before 1387 AD, when a Middle English work borrowed the word from Latin to describe a sandboard abacus, the Latin word came from Greek ἄβαξ abax which means something without base, and improperly, any piece of rectangular board or plank. Alternatively, without reference to ancient texts on etymology, it has suggested that it means a square tablet strewn with dust. Whereas the table strewn with dust definition is popular, there are those that do not place credence in this at all, Greek ἄβαξ itself is probably a borrowing of a Northwest Semitic, perhaps Phoenician, word akin to Hebrew ʾābāq, dust. The preferred plural of abacus is a subject of disagreement, with both abacuses and abaci in use, the user of an abacus is called an abacist. The period 2700–2300 BC saw the first appearance of the Sumerian abacus, some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus. Archaeologists have found ancient disks of various sizes that are thought to have used as counters. However, wall depictions of this instrument have not been discovered, during the Achaemenid Empire, around 600 BC the Persians first began to use the abacus. The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC, also Demosthenes talked of the need to use pebbles for calculations too difficult for your head. The Greek abacus was a table of wood or marble, pre-set with small counters in wood or metal for mathematical calculations and this Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, Ancient Rome and, until the French Revolution, the Western Christian world. A tablet found on the Greek island Salamis in 1846 AD, dates back to 300 BC and it is a slab of white marble 149 cm long,75 cm wide, and 4.5 cm thick, on which are 5 groups of markings. Below these lines is a space with a horizontal crack dividing it. Also from this frame the Darius Vase was unearthed in 1851. It was covered with pictures including a holding a wax tablet in one hand while manipulating counters on a table with the other. The earliest known documentation of the Chinese abacus dates to the 2nd century BC. The Chinese abacus, known as the suanpan, is typically 20 cm tall and it usually has more than seven rods. There are two beads on each rod in the deck and five beads each in the bottom for both decimal and hexadecimal computation
Abacus
–
A Chinese abacus
Abacus
–
Calculating-Table by
Gregor Reisch: Margarita Philosophica, 1503. The woodcut shows Arithmetica instructing an
algorist and an abacist (inaccurately represented as
Boethius and
Pythagoras). There was keen competition between the two from the introduction of the
Algebra into Europe in the 12th century until its triumph in the 16th.
Abacus
–
Copy of a
Roman abacus
Abacus
–
Japanese
soroban
21.
Banking
–
A bank is a financial institution that accepts deposits from the public and creates credit. Lending activities can be performed either directly or indirectly through capital markets, due to their importance in the financial stability of a country, banks are highly regulated in most countries. Most nations have institutionalized a system known as fractional reserve banking under which banks hold liquid assets equal to only a portion of their current liabilities. In the history of banking, a number of banking dynasties — notably, the Medicis, the Fuggers, the Welsers, the Berenbergs, the oldest existing retail bank is Banca Monte dei Paschi di Siena, while the oldest existing merchant bank is Berenberg Bank. Banking began with the first prototype banks of merchants of the ancient world and this began around 2000 BC in Assyria and Babylonia. Later, in ancient Greece and during the Roman Empire, lenders based in temples made loans, archaeology from this period in ancient China and India also shows evidence of money lending activity. The origins of modern banking can be traced to medieval and early Renaissance Italy, to the cities in the centre and north like Florence, Lucca, Siena, Venice. The Bardi and Peruzzi families dominated banking in 14th-century Florence, establishing branches in other parts of Europe. One of the most famous Italian banks was the Medici Bank, the earliest known state deposit bank, Banco di San Giorgio, was founded in 1407 at Genoa, Italy. Modern banking practices, including fractional reserve banking and the issue of banknotes, merchants started to store their gold with the goldsmiths of London, who possessed private vaults, and charged a fee for that service. The goldsmith paid interest on these deposits, since the promissory notes were payable on demand, and the advances to the goldsmiths customers were repayable over a longer time period, this was an early form of fractional reserve banking. Thus, the goldsmiths of London became the forerunners of banking by creating new money based on credit, the Bank of England was the first to begin the permanent issue of banknotes, in 1695. The Royal Bank of Scotland established the first overdraft facility in 1728, by the beginning of the 19th century a bankers clearing house was established in London to allow multiple banks to clear transactions. The Rothschilds pioneered international finance on a scale, financing the purchase of the Suez canal for the British government. The word bank was borrowed in Middle English from Middle French banque, from Old Italian banca, meaning table, from Old High German banc, bank bench, counter. Benches were used as makeshift desks or exchange counters during the Renaissance by Jewish Florentine bankers, the definition of a bank varies from country to country. See the relevant country pages under for more information, the business of banking is in many English common law countries not defined by statute but by common law, the definition above. In other English common law there are statutory definitions of the business of banking or banking business
Banking
–
The
Bank of England, established in 1694.
Banking
–
Personal finance
Banking
–
Among many other things, the
Code of Hammurabi from 1754 BC recorded interest-bearing loans.
Banking
–
The sealing of the
Bank of England Charter (1694).
22.
Accounting
–
Accounting or accountancy is the measurement, processing and communication of financial information about economic entities such as businesses and corporations. The modern field was established by the Italian mathematician Luca Pacioli in 1494, practitioners of accounting are known as accountants. The terms accounting and financial reporting are often used as synonyms, Accounting can be divided into several fields including financial accounting, management accounting, external auditing, and tax accounting. Accounting information systems are designed to support accounting functions and related activities, Accounting is facilitated by accounting organizations such as standard-setters, accounting firms and professional bodies. Financial statements are audited by accounting firms, and are prepared in accordance with generally accepted accounting principles. GAAP is set by various standard-setting organizations such as the Financial Accounting Standards Board in the United States, as of 2012, all major economies have plans to converge towards or adopt the International Financial Reporting Standards. The history of accounting is thousands of old and can be traced to ancient civilizations. By the time of the Emperor Augustus, the Roman government had access to detailed financial information, double-entry bookkeeping developed in medieval Europe, and accounting split into financial accounting and management accounting with the development of joint-stock companies. The first work on a double-entry bookkeeping system was published in Italy, both the words accounting and accountancy were in use in Great Britain by the mid-1800s, and are derived from the words accompting and accountantship used in the 18th century. In Middle English the verb to account had the form accounten, which was derived from the Old French word aconter, which is in turn related to the Vulgar Latin word computare, meaning to reckon. The base of computare is putare, which meant to prune, to purify, to correct an account, hence, to count or calculate. The word accountant is derived from the French word compter, which is derived from the Italian. Accountancy refers to the occupation or profession of an accountant, particularly in British English, Accounting has several subfields or subject areas, including financial accounting, management accounting, auditing, taxation and accounting information systems. Financial accounting focuses on the reporting of a financial information to external users of the information, such as investors. It calculates and records business transactions and prepares financial statements for the users in accordance with generally accepted accounting principles. GAAP, in turn, arises from the agreement between accounting theory and practice, and change over time to meet the needs of decision-makers. This branch of accounting is also studied as part of the exams for qualifying as an actuary. It is interesting to note that two professionals, accountants and actuaries, have created a culture of being archrivals
Accounting
–
Accounting
Accounting
–
Portrait of Luca Pacioli, painted by
Jacopo de' Barbari, 1495, (
Museo di Capodimonte).
23.
Irrational numbers
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In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and c are in the smallest possible terms. By the Pythagorean theorem, c2 = a2+b2 = b2+b2 = 2b2, since c2 = 2b2, c2 is divisible by 2, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on
Irrational numbers
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The mathematical constant
pi (π) is an irrational number that is much represented in popular culture.
24.
Prime numbers
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
Prime numbers
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The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.
25.
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
Golden ratio
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Michael Maestlin, first to publish a decimal approximation of the golden ratio, in 1597
Golden ratio
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Line segments in the golden ratio
Golden ratio
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Many of the proportions of the
Parthenon are alleged to exhibit the golden ratio.
Golden ratio
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The drawing of a man's body in a
pentagram suggests relationships to the golden ratio.
26.
Camposanto di Pisa
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The Campo Santo, also known as Camposanto Monumentale or Camposanto Vecchio, is a historical edifice at the northern edge of the Cathedral Square in Pisa, Italy. A legend claims that bodies buried in that ground will rot in just 24 hours, the burial ground lies over the ruins of the old baptistery of the church of Santa Reparata, the church that once stood where the cathedral now stands. The term monumental serves to differentiate it from the urban cemetery in Pisa. The building was the fourth and last one to be raised in the Cathedral Square and it dates from a century after the bringing of the soil from Golgotha, and was erected over the earlier burial ground. The construction of huge, oblong Gothic cloister was begun in 1278 by the architect Giovanni di Simone. He died in 1284 when Pisa suffered a defeat in the battle of Meloria against the Genoans. The cemetery was completed in 1464. It seems that the building was not meant to be a cemetery, but a church called Santissima Trinità. However we know that the part was the western one. The outer wall is composed of 43 blind arches, the one on the right is crowned by a gracious Gothic tabernacle. It contains the Virgin Mary with Child, surrounded by four saints and it is the work from the second half of the 14th century by a follower of Giovanni Pisano. This was the entrance door. Most of the tombs are under the arcades, although a few are on the central lawn, the inner court is surrounded by elaborate round arches with slender mullions and plurilobed tracery. In the Aulla chapel we can see also the original incense lamp that Galileo Galilei used for calculation of pendular movements and this lamp is the one Galileo saw inside the cathedral, now replaced by a larger more elaborate one. The last chapel was Dal Pozzo, commissioned by archbishop of Pisa Carlo Antonio Dal Pozzo in 1594, it has a dedicated to St. Jerome. Also in the Dal Pozzo chapel sometimes a Mass is celebrated, the sarcophagi were initially all around the cathedral, often attached to the building itself. That until the cemetery was built, then they were collected in the all over the meadow. Carlo Lasinio, in the years he was the curator of the Campo Santo, nowadays the sarcophagi are inside the galleries, near the walls
Camposanto di Pisa
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The Campo Santo
Camposanto di Pisa
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Interior courtyard
Camposanto di Pisa
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Interior hallway, pictured in 2012
Camposanto di Pisa
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The Triumph of Death
27.
Camposanto Monumentale
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The Campo Santo, also known as Camposanto Monumentale or Camposanto Vecchio, is a historical edifice at the northern edge of the Cathedral Square in Pisa, Italy. A legend claims that bodies buried in that ground will rot in just 24 hours, the burial ground lies over the ruins of the old baptistery of the church of Santa Reparata, the church that once stood where the cathedral now stands. The term monumental serves to differentiate it from the urban cemetery in Pisa. The building was the fourth and last one to be raised in the Cathedral Square and it dates from a century after the bringing of the soil from Golgotha, and was erected over the earlier burial ground. The construction of huge, oblong Gothic cloister was begun in 1278 by the architect Giovanni di Simone. He died in 1284 when Pisa suffered a defeat in the battle of Meloria against the Genoans. The cemetery was completed in 1464. It seems that the building was not meant to be a cemetery, but a church called Santissima Trinità. However we know that the part was the western one. The outer wall is composed of 43 blind arches, the one on the right is crowned by a gracious Gothic tabernacle. It contains the Virgin Mary with Child, surrounded by four saints and it is the work from the second half of the 14th century by a follower of Giovanni Pisano. This was the entrance door. Most of the tombs are under the arcades, although a few are on the central lawn, the inner court is surrounded by elaborate round arches with slender mullions and plurilobed tracery. In the Aulla chapel we can see also the original incense lamp that Galileo Galilei used for calculation of pendular movements and this lamp is the one Galileo saw inside the cathedral, now replaced by a larger more elaborate one. The last chapel was Dal Pozzo, commissioned by archbishop of Pisa Carlo Antonio Dal Pozzo in 1594, it has a dedicated to St. Jerome. Also in the Dal Pozzo chapel sometimes a Mass is celebrated, the sarcophagi were initially all around the cathedral, often attached to the building itself. That until the cemetery was built, then they were collected in the all over the meadow. Carlo Lasinio, in the years he was the curator of the Campo Santo, nowadays the sarcophagi are inside the galleries, near the walls
Camposanto Monumentale
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The Campo Santo
Camposanto Monumentale
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Interior courtyard
Camposanto Monumentale
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Interior hallway, pictured in 2012
Camposanto Monumentale
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The Triumph of Death
28.
Piazza dei Miracoli
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Considered sacred by the Catholic Church, its owner, the square is dominated by four great religious edifices, the Pisa Cathedral, the Pisa Baptistry, the Campanile, and the Camposanto Monumentale. Partly paved and partly grassed, the Piazza dei Miracoli is also the site of the Ospedale Nuovo di Santo Spirito, which houses the Sinopias Museum, the square is sometimes called the Campo dei Miracoli. In 1987, the square was declared a UNESCO World Heritage Site. The heart of the Piazza del Duomo is the Duomo, the cathedral of the Archdiocese of Pisa. The cathedral has two aisles on either side of the nave, the transept consists of three aisles. The church is also as the Primatial, the archbishop of Pisa being a Primate since 1092. Its construction began in 1064 by the architect Buscheto and it set the model for the distinctive Pisan Romanesque style of architecture. The mosaics of the interior, as well as the pointed arches, the façade, of grey marble and white stone set with discs of coloured marble, was built by a master named Rainaldo, as indicated by an inscription above the middle door, Rainaldus prudens operator. The massive bronze doors were made in the workshops of Giambologna. The original central door was of bronze, made around 1180 by Bonanno Pisano, however, worshippers have never used the façade doors to enter, instead entering by way of the Porta di San Ranieri, in front of the Leaning Tower, built around 1180 by Bonanno Pisano. Above the doors are four rows of galleries with, on top, statues of Madonna with Child and, on the corners. Also in the façade is found the tomb of Buscheto and an inscription about the foundation of the Cathedral, the interior is faced with black and white marble and has a gilded ceiling and a frescoed dome. It was largely redecorated after a fire in 1595, which destroyed most of the Renaissance art works, fortunately, the impressive mosaic of Christ in Majesty, in the apse, flanked by the Blessed Virgin and St. John the Evangelist, survived the fire. It evokes the mosaics in the church of Monreale, Sicily, although it is said that the mosaic was done by Cimabue, only the head of St. John was done by the artist in 1302, his last work, since he died in Pisa the same year. The cupola, at the intersection of the nave and transept, was decorated by Riminaldi showing the assumption of the Blessed Virgin. Galileo is believed to have formulated his theory about the movement of a pendulum by watching the swinging of the lamp hanging from the ceiling of the nave. That lamp, smaller and simpler than the present one, is now kept in the Camposanto, the granite Corinthian columns between the nave and the aisle came originally from the mosque of Palermo, captured by the Pisans in 1063. The coffer ceiling of the nave was replaced after the fire of 1595, the present gold-decorated ceiling carries the coat of arms of the Medici
Piazza dei Miracoli
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UNESCO World Heritage Site
Piazza dei Miracoli
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Pisa Cathedral façade
Piazza dei Miracoli
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Pisa Cathedral with the Leaning Tower of Pisa
Piazza dei Miracoli
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Pisa Cathedral interior and Galileo's Lamp
29.
The Fibonaccis
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The Fibonaccis were an American art rock band formed in 1981 in Los Angeles. The band consisted of songwriters John Dentino and Ron Stringer, Magie Song, Joe Berardi, the Fibonaccis were formed out of the Los Angeles art punk scene which included bands such as Wall of Voodoo, Oingo Boingo and Sparks. Lyrically, the band regularly explored dark and esoteric subject matter ranging from serial killers to UFOs, presented in a satirical, the Fibonaccis released their debut EP in 1982, following up with a 12 single/EP, Tumor/Psycho/Slow Beautiful Sex, the next year. In 1984, the group filmed a music video for an unreleased cover of Jimi Hendrixs Purple Haze. Throughout their career, The Fibonaccis regularly contributed their music to independent film soundtracks, in 1986, the band collaborated with composer Richard Band on the score for the horror-comedy TerrorVision, recording five tracks including the movies theme song. Their song Sergio Leone was used for the credits of 1982s Android. The Fibonaccis appeared onscreen as the band Sexy Holiday in the 1987 comedy Valet Girls, lipsynching to Slow Beautiful Sex, in 1987, the band released their sole studio LP, Civilization and Its Discotheques on the Blue Yonder Sounds label. In explaining the reason for the LPs delay, the said that various hassles and difficulties with record companies had plagued a more timely release. Their frustration over the recording, added with a lack of media recognition. In 1992, Restless Records released a 26-track retrospective of the work called Repressed - The Best of the Fibonaccis. To celebrate the release of the album, The Fibonaccis performed a reunion show in Los Angeles on November 19,1992. Following The Fibonaccis disbandment, John Dentino continued to compose music independently and has recently been working on independent documentary films. Joe Berardi went on to perform and tour with Wall of Voodoos Stan Ridgway, Magie Song acted in a number of independent films in the early 1990s, including Gregg Arakis The Living End and Stephen Sayadians Dr. Caligari, and currently works as an acupuncturist in Los Angeles. Ron Stringer most recently served as editor and critic for the LA Weekly. Tom Corey died from an aneurysm in late 2001. Currently, the entire Fibonaccis discography is out of print, with Repressed being their only released on compact disc. In 2006, John Dentino created a website for the band. The Ordinary Women -2,234, rice Song -2,41 Tumor 1
The Fibonaccis
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The Fibonaccis, circa 1982. Left to right: Berardi, Dentino, Corey and Song.
30.
Surveying
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Surveying or land surveying is the technique, profession, and science of determining the terrestrial or three-dimensional position of points and the distances and angles between them. A land surveying professional is called a land surveyor, Surveyors work with elements of geometry, trigonometry, regression analysis, physics, engineering, metrology, programming languages and the law. Surveying has been an element in the development of the environment since the beginning of recorded history. The planning and execution of most forms of construction require it and it is also used in transport, communications, mapping, and the definition of legal boundaries for land ownership. It is an important tool for research in other scientific disciplines. Basic surveyance has occurred since humans built the first large structures, the prehistoric monument at Stonehenge was set out by prehistoric surveyors using peg and rope geometry. In ancient Egypt, a rope stretcher would use simple geometry to re-establish boundaries after the floods of the Nile River. The almost perfect squareness and north-south orientation of the Great Pyramid of Giza, built c.2700 BC, the Groma instrument originated in Mesopotamia. The mathematician Liu Hui described ways of measuring distant objects in his work Haidao Suanjing or The Sea Island Mathematical Manual, the Romans recognized land surveyors as a profession. They established the basic measurements under which the Roman Empire was divided, Roman surveyors were known as Gromatici. In medieval Europe, beating the bounds maintained the boundaries of a village or parish and this was the practice of gathering a group of residents and walking around the parish or village to establish a communal memory of the boundaries. Young boys were included to ensure the memory lasted as long as possible, in England, William the Conqueror commissioned the Domesday Book in 1086. It recorded the names of all the owners, the area of land they owned, the quality of the land. It did not include maps showing exact locations, abel Foullon described a plane table in 1551, but it is thought that the instrument was in use earlier as his description is of a developed instrument. Gunters chain was introduced in 1620 by English mathematician Edmund Gunter and it enabled plots of land to be accurately surveyed and plotted for legal and commercial purposes. Leonard Digges described a Theodolite that measured horizontal angles in his book A geometric practice named Pantometria, joshua Habermel created a theodolite with a compass and tripod in 1576. Johnathon Sission was the first to incorporate a telescope on a theodolite in 1725, in the 18th century, modern techniques and instruments for surveying began to be used. Jesse Ramsden introduced the first precision theodolite in 1787 and it was an instrument for measuring angles in the horizontal and vertical planes
Surveying
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A surveyor at work with an infrared reflector used for distance measurement.
Surveying
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Table of Surveying, 1728
Cyclopaedia
Surveying
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A map of India showing the Great Trigonometrical Survey, produced in 1870
Surveying
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A German engineer surveying during the
First World War, 1918
