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120-746: In ancient Greek geometry , the Ostomachion , also known as loculus Archimedius (from Latin  'Archimedes' box') or syntomachion , is a mathematical treatise attributed to Archimedes . This work has survived fragmentarily in an Arabic version and a copy, the Archimedes Palimpsest , of the original ancient Greek text made in Byzantine times. The word Ostomachion ( Ὀστομάχιον ) comes from Greek ὀστέον (osteon)  'bone' and μάχη (mache)  'fight, battle, combat'. The manuscripts refer to

240-465: A , b , c {\displaystyle a,b,c} and rational area is of the form: for some rational numbers u , v , {\displaystyle u,v,} and w {\displaystyle w} . Parameshvara Nambudiri was the first mathematician to give a formula for the radius of the circle circumscribing a cyclic quadrilateral. The expression is sometimes attributed to Lhuilier [1782], 350 years later. With

360-469: A frustum of a square pyramid. The ancient Egyptians knew that they could approximate the area of a circle as follows: Problem 50 of the Ahmes papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter. This assumes that π is 4×(8/9) (or 3.160493...), with an error of slightly over 0.63 percent. This value

480-683: A Chalcedonian, as Patriarch. Cyrus was determined to convert the Egyptian Miaphysites by any means. He expelled Coptic monks and bishops from their monasteries and sees. Many died in the chaos, and the resentment of the Egyptians against their Byzantine conquerors reached a peak. Meanwhile, the new religion of Islam was making headway in Arabia , culminating in the Muslim conquests that took place following Muhammad 's death. In AD 639,

600-635: A city which had by then become a center of culture and learning in the entire Mediterranean oikoumene . St. Mark is said to have founded the Holy Apostolic See of Alexandria and to have become its first Patriarch . Within 50 years of St. Mark's arrival in Alexandria, a fragment of New Testament writings appeared in Oxyrhynchus (Bahnasa), which suggests that Christianity already began to spread south of Alexandria at an early date. By

720-410: A computer search by William H. Cutler. However, this count has been disputed because surviving images of the puzzle show it in a rectangle, not a square, and rotations or reflections of pieces may not have been allowed. Ancient Greek geometry Classic geometry was focused in compass and straightedge constructions . Geometry was revolutionized by Euclid , who introduced mathematical rigor and

840-570: A cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). Brahmagupta's formula: The area, A , of a cyclic quadrilateral with sides of lengths a , b , c , d , respectively, is given by where s , the semiperimeter , given by: s = a + b + c + d 2 . {\displaystyle s={\frac {a+b+c+d}{2}}.} Brahmagupta's Theorem on rational triangles: A triangle with rational sides

960-504: A cylinder. Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for π of 3.111... The two problems together indicate a range of values for π between 3.11 and 3.16. Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding

1080-493: A dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhma Sphuṭa Siddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In

1200-594: A few minor stars yet to come, but the golden age of geometry was over. Proclus (410–485), author of Commentary on the First Book of Euclid , was one of the last important players in Hellenistic geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed

1320-460: A marked ruler or a protractor , because these were a workman's tools, not worthy of a scholar. This dictum led to a deep study of possible compass and straightedge constructions, and three classic construction problems: how to use these tools to trisect an angle , to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in

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1440-478: A minority. The Mamluks were mainly ethnic Circassians and Turks who had been captured as slaves then recruited into the army fighting on behalf of the Islamic empire. Native Egyptians were not allowed to serve in the army until the reign of Mohamed Ali . Historian James Jankwoski writes: Ultimately, Mamluk rule rested on force. The chronicles of the period are replete with examples of Mamluk violence against

1560-646: A modern mathematician might be hard put to derive some of them without the use of calculus and algebra. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras around 800 BC contained the first statements of the theorem; the Egyptians had a correct formula for the volume of

1680-575: A new flourishing of mathematics. One of the oldest Chinese mathematical texts to present geometric progressions was the Suàn shù shū of 186 BC, during the Western Han era. The mathematician, inventor, and astronomer Zhang Heng (78–139 AD) used geometrical formulas to solve mathematical problems. Although rough estimates for pi ( π ) were given in the Zhou Li (compiled in the 2nd century BC), it

1800-663: A prudish "Saudization" of their culture due to Saudi Arabian petrodollar-flush investment in the Egyptian entertainment industry . Twice Libya was on the brink of war with Egypt due to mistreatment of Egyptian workers and after the signing of the peace treaty with Israel. When the Gulf War ended, Egyptian workers in Iraq were subjected to harsh measures and expulsion by the Iraqi government and to violent attacks by Iraqis returning from

1920-590: A reduction in the body of material available in the Library, especially in the 4th century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign. In the Bakhshali manuscript , there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with

2040-730: A separate entity...No longer even spiritually linked with Constantinople, theologians began to write more in Coptic and less in Greek. Coptic art developed its own national character, and the Copts stood united against the imperial power." Before the Muslim conquest of Egypt , the Byzantine Emperor Heraclius was able to reclaim the country after a brief Persian invasion in AD 616, and subsequently appointed Cyrus of Alexandria ,

2160-541: A series of foreign occupations, identified the story of the mother-goddess Isis protecting her child Horus with that of the Virgin Mary and her son Jesus escaping the emperor Herod . Consequently, many sites believed to have been the resting places of the holy family during their sojourn in Egypt became sacred to the Egyptians. The visit of the holy family later circulated among Egyptian Christians as fulfillment of

2280-485: A small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point. Much like Euclid 's first and third definitions and Plato 's 'beginning of a line', the Mo Jing stated that "a point may stand at

2400-543: A society and a state. Regular inundations were a cause for celebration; low waters often meant famine and starvation. The ancient Egyptians personified the river flood as the god Hapy and dedicated a Hymn to the Nile to celebrate it. km.t , the Black Land, was as Herodotus observed, "the gift of the river." When Alexander died, a story began to circulate that Nectanebo II was Alexander's father. This made Alexander in

2520-661: A spiritual void as Egyptians became increasingly disillusioned and weary due to successive waves of foreign invasions. When the Romans annexed Egypt in 30 BC, the social structure created by the Greeks was largely retained, though the power of the Egyptian priesthood diminished. The Roman emperors lived abroad and did not perform the ceremonial functions of Egyptian kingship as the Ptolemies had. The art of mummy portraiture flourished, but Egypt became further stratified with Romans at

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2640-725: A tradition of geometry, mostly expressed in the construction of elaborate altars. Early Indian texts (1st millennium BC) on this topic include the Satapatha Brahmana and the Śulba Sūtras . The Śulba Sūtras has been described as "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians." They make use of Pythagorean triples , which are particular cases of Diophantine equations . According to mathematician S. G. Dani,

2760-462: A treatise in 13 books (chapters), titled The Elements of Geometry , in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry . The treatise is not a compendium of all that the Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid's was not

2880-463: A wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge. He also figured out that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. Furthermore, Liu Hui described Cavalieri's principle on volume, as well as Gaussian elimination . From the Nine Chapters , it listed

3000-498: Is credited with having introduced the concept of Gnosis into Islam, as well as of being able to decipher a number of hieroglyphic characters due to his knowledge of Coptic . He was keenly interested in ancient Egyptian sciences, and claimed to have received his knowledge of alchemy from Egyptian sources. In the years to follow the Arab occupation of Egypt, a social hierarchy was created whereby Egyptians who converted to Islam acquired

3120-475: Is estimated as 3. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3 and 1/8. The Babylonians are also known for

3240-575: Is sometimes outlandish, but the attachment flows through all Egyptians, as the Nile through Egypt. A sizable Egyptian diaspora did not begin to form until well into the 1980s, when political and economic conditions began driving Egyptians out of the country in significant numbers. Today, the diaspora numbers nearly 4 million (2006 est). Generally, those who emigrate to the United States and western European countries tend to do so permanently, with 93% and 55.5% of Egyptians (respectively) settling in

3360-502: Is taken to end with his death in 323 BC. The Ptolemaic dynasty ruled Egypt from 305 BC to 30 BC and introduced Hellenic culture to Egyptians. 4,000 Celtic mercenaries under Ptolemy II had even attempted an ambitious but doomed coup d'état around the year 270 BC. Throughout the Pharaonic epoch (viz., from 2920 BC to 525 BC in conventional Egyptian chronology ), divine kingship was the glue which held Egyptian society together. It

3480-675: Is the Basilica di San Lorenzo in Florence by Filippo Brunelleschi (1377–1446). In c. 1413 Filippo Brunelleschi demonstrated the geometrical method of perspective, used today by artists, by painting the outlines of various Florentine buildings onto a mirror. Soon after, nearly every artist in Florence and in Italy used geometrical perspective in their paintings, notably Masolino da Panicale and Donatello . Melozzo da Forlì first used

3600-530: Is the Egyptian attitude toward foreigners—those they considered not fortunate enough to be part of the community of rmṯ or "the people" (i.e., Egyptians.) This attitude was facilitated by the Egyptians' more frequent contact with other peoples during the New Kingdom when Egypt expanded to an empire that also encompassed Nubia through Jebel Barkal and parts of the Levant . The Egyptian sense of superiority

3720-532: Is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000–70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later. Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to

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3840-710: Is the possibility that even older written material existed. However, due to the infamous Burning of the Books in a political maneuver by the Qin dynasty ruler Qin Shihuang (r. 221–210 BC), multitudes of written literature created before his time were purged. In addition, the Mo Jing presents geometrical concepts in mathematics that are perhaps too advanced not to have had a previous geometrical base or mathematic background to work upon. The Mo Jing described various aspects of many fields associated with physical science, and provided

3960-653: The Almagest as well as several works by Euclid from Greek to Latin. Although the Sicilians generally translated directly from the Greek, when Greek texts were not available, they would translate from Arabic. Eugenius of Palermo (d. 1202) translated Ptolemy's Optics into Latin, drawing on his knowledge of all three languages in the task. The rigorous deductive methods of geometry found in Euclid's Elements of Geometry were relearned, and further development of geometry in

4080-672: The Arab states of the Persian Gulf , Iraq and Libya have been reported by the Egyptian Human Rights Organization and different media outlets. Arab nationals have in the past expressed fear over an "'Egyptianization' of the local dialects and culture that were believed to have resulted from the predominance of Egyptians in the field of education" (see also Egyptian Arabic – Geographics ). A Newsweek article in 2008 featured Egyptian citizens objecting to

4200-689: The Coptic Orthodox Church , whose liturgical language , Coptic , is the most recent stage of the ancient Egyptian language and is still used in prayers along with Egyptian Arabic. Egyptians have received several names: There are an estimated 105.3 million Egyptians. Most are native to Egypt, where Egyptians constitute around 99.6% of the population. Approximately 84–90% of the population of Egypt are Muslim adherents and 10–15% are Christian adherents (10–15% Coptic Christian , 1% other Christian Sects (mainly Greek Orthodox )) according to estimates. Most of Egypt's people live along

4320-746: The New Kingdom . During this period, the Egyptian civilization rose to the status of an empire under Pharaoh Thutmose III of the 18th Dynasty . It remained a super-regional power throughout the Amarna Period as well as during the 19th and 20th dynasties (the Ramesside Period), lasting into the Early Iron Age . The Bronze Age collapse that had afflicted the Mesopotamian empires reached Egypt with some delay, and it

4440-648: The Nile Valley in Egypt . Egyptian identity is closely tied to geography . The population is concentrated in the Nile Valley , a small strip of cultivable land stretching from the First Cataract to the Mediterranean and enclosed by desert both to the east and to the west . This unique geography has been the basis of the development of Egyptian society since antiquity . The daily language of

4560-761: The Penrose tilings . The transmission of the Greek Classics to medieval Europe via the Arabic literature of the 9th to 10th century " Islamic Golden Age " began in the 10th century and culminated in the Latin translations of the 12th century . A copy of Ptolemy 's Almagest was brought back to Sicily by Henry Aristippus (d. 1162), as a gift from the Emperor to King William I (r. 1154–1166). An anonymous student at Salerno travelled to Sicily and translated

4680-538: The Sulvasutras . The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily. Thales (635–543 BC) of Miletus (now in southwestern Turkey),

4800-484: The axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of

4920-404: The comparison of lengths and for parallels , along with principles of space and bounded space. It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. The book provided definitions for circumference, diameter, and radius, along with the definition of volume. The Han dynasty (202 BC – 220 AD) period of China witnessed

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5040-400: The 1470s. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective. Della Francesca fleshed it out, explicitly covering solids in any area of the picture plane. Della Francesca also started the now common practice of using illustrated figures to explain the mathematical concepts, making his treatise easier to understand than Alberti's. Della Francesca

5160-434: The 1973 boom in oil prices and again in 1979, but it was only in the second half of the 1980s that Egyptian migration became prominent. Egyptian emigration today is motivated by even higher rates of unemployment, population growth and increasing prices. Political repression and human rights violations by Egypt's ruling régime are other contributing factors (see Egypt § Human rights ). Egyptians have also been impacted by

5280-455: The 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384–322 BC), Plato's greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic ) which was not substantially improved upon until the 19th century. Euclid (c. 325–265 BC), of Alexandria , probably a student at the Academy founded by Plato, wrote

5400-698: The Arab general 'Amr ibn al-'As marched into Egypt, facing off with the Byzantines in the Battle of Heliopolis that ended with the Byzantines' defeat. The relationship between the Greek Melkites and the Egyptian Copts had grown so bitter that most Egyptians did not put up heavy resistance against the Arabs. The new Muslim rulers moved the capital to Fustat and, through the 7th century, retained

5520-557: The Ayyubids, the country for the most part continued to prosper. The Mamluks of Egypt (AD 1258–1517) as a whole were, some of the most enlightened rulers of Egypt, not only in the arts and in providing for the welfare of their subjects, but also in many other ways, such as efficient organisation of law and order and postal services, and the building of canals, roads, bridges and aqueducts. Though turbulent, often treacherous and brutal in their feuds, and politically and economically inept,

5640-582: The Babylonian cuneiform tablet Plimpton 322 written c. 1850 BC "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple, indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BC. "Since these tablets predate the Sulbasutras period by several centuries, taking into account

5760-462: The Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a time-mile used for measuring the travel of the Sun, therefore, representing time. There have been recent discoveries showing that ancient Babylonians may have discovered astronomical geometry nearly 1400 years before Europeans did. The Indian Vedic period had

5880-729: The Biblical prophecy "When Israel was a child, then I loved him, and called my son out of Egypt" (Hosea 11:1). The feast of the coming of the Lord of Egypt on June 1 became an important part of Christian Egyptian tradition. According to tradition, Christianity was brought to Egypt by Saint Mark the Evangelist in the early 40s of the 1st century, under the reign of the Roman emperor Nero . The earliest converts were Jews residing in Alexandria ,

6000-511: The Circle and On Conoids and Spheroids . His work On Floating Bodies is the first known work on hydrostatics, of which Archimedes is recognized as the founder. Renaissance translations of his works, including the ancient commentaries, were enormously influential in the work of some of the best mathematicians of the 17th century, notably René Descartes and Pierre de Fermat . After Archimedes, Hellenistic mathematics began to decline. There were

6120-412: The Egyptian reaction was swift, rejecting the decrees of the council as incompatible with the Miaphysite doctrine of Coptic Orthodoxy. The Copts' upholding of the Miaphysite doctrine against the pro-Chalcedonian Greek Melkites had both theological and national implications. As Coptologist Jill Kamil notes, the position taken by the Egyptians "paved [the way] for the Coptic church to establish itself as

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6240-458: The Egyptian side, literary works from both the Mamluk and Ottoman eras indicate that literate Egyptians had not totally submerged their identity within Islam, but retained an awareness of Egypt's distinctiveness as a uniquely fertile region of the Muslim world, as a land of great historical antiquity and splendor... At least for some Egyptians, 'the land of Egypt' ( al-diyar al-misriyya ) was an identifiable and emotionally meaningful entity within

6360-446: The Egyptians is a continuum of the local varieties of Arabic ; the most famous dialect is known as Egyptian Arabic or Masri . Additionally, a sizable minority of Egyptians living in Upper Egypt speak Sa'idi Arabic . Egyptians are predominantly adherents of Sunni Islam with a small Shia minority and a significant proportion who follow native Sufi orders . A considerable percentage of Egyptians are Coptic Christians who belong to

6480-478: The English easier to read. Concepts, that are now understood as algebra , were expressed geometrically by Euclid, a method referred to as Greek geometric algebra . Archimedes (287–212 BC), of Syracuse , Sicily , when it was a Greek city-state , was one of the most famous mathematicians of the Hellenistic period . He is known for his formulation of a hydrostatic principle (known as Archimedes' principle ) and for his works on geometry, including Measurement of

6600-419: The Greek city-states produced excellent engineers, but no mathematicians of note. The great Library of Alexandria was later burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction

6720-413: The Greeks. There is a story that he had inscribed above the entrance to his famous school, "Let none ignorant of geometry enter here." However, the story is considered to be untrue. Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as

6840-403: The Hyksos, Assyrians , Libyans , Persians and Greeks—their rulers assumed the role of the Egyptian Pharaoh and were often depicted praying to Egyptian gods. The ancient Egyptians used a solar calendar that divided the year into 12 months of 30 days each, with five extra days added. The calendar revolved around the annual Nile Inundation ( akh.t ), the first of three seasons into which the year

6960-425: The Mamluks was not a very secure place to live. Egyptians under the Ottoman Turks from the 16th to the 18th centuries lived within a social hierarchy similar to that of the Mamluks, Arabs, Romans, Greeks and Persians before them. Native Egyptians applied the term atrak (Turks) indiscriminately to the Ottomans and Mamluks, who were at the top of the social pyramid, while Egyptians, most of whom were farmers, were at

7080-421: The Mediterranean, the Egyptians are a hinterland people with little appetite for travel, even inside their own country. They glorify their national dishes, including the variety of concoctions surrounding the simple bean . Most of all, they have a sense of all-encompassing familiarity at home and a sense of alienation when abroad ... There is something particularly excruciating about Egyptian nostalgia for Egypt: it

7200-416: The Muslim Arabs took place in the 8th and 9th centuries under the reign of the Umayyads and Abbasids . The greatest was one in which disaffected Muslim Egyptians joined their Christian compatriots around AD 830 in an unsuccessful attempt to repel the Arabs. The Egyptian Muslim historian Ibn Abd al-Hakam spoke harshly of the Abbasids—a reaction that according to Egyptologist Okasha El-Daly can be seen "within

7320-400: The addition of a number of demotic characters. This stage of the Egyptian language would later come to be known as Coptic along with its alphabet . The third theologian to head the Catachetical School was a native Egyptian by the name of Origen . Origen was an outstanding theologian and one of the most influential Church Fathers . He traveled extensively to lecture in various churches around

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7440-414: The apex of the social pyramid, Greeks and Jews occupied the middle stratum, while Egyptians, who constituted the vast majority, were at the bottom. Egyptians paid a poll tax at full rate, Greeks paid at half-rate and Roman citizens were exempt. The Roman emperor Caracalla advocated the expulsion of all ethnic Egyptians from the city of Alexandria, saying "genuine Egyptians can easily be recognized among

7560-567: The astronomical gnomon , the circle and square, as well as measurements of heights and distances. The editor Liu Hui listed pi as 3.141014 by using a 192 sided polygon , and then calculated pi as 3.14159 using a 3072 sided polygon. This was more accurate than Liu Hui's contemporary Wang Fan , a mathematician and astronomer from Eastern Wu , would render pi as 3.1555 by using ⁄ 45 . Liu Hui also wrote of mathematical surveying to calculate distance measurements of depth, height, width, and surface area. In terms of solid geometry, he figured out that

7680-414: The banks of the Nile River , and more than two-fifths of the population lives in urban areas. Along the Nile, the population density is one of the highest in the world, in excess of 5,000 persons per square mile (1,900 persons/km ) in a number of riverine governorates. The rapidly growing population is young, with roughly one-third of the total under age 15 and about three-fifths under 30. In response to

7800-482: The beginning of a distinct Egyptian or Coptic Church . It became known as the 'Era of the Martyrs' and is commemorated in the Coptic calendar in which dating of the years began with the start of Diocletian's reign. When Egyptians were persecuted by Diocletian, many retreated to the desert to seek relief. The practice precipitated the rise of monasticism , for which the Egyptians, namely St. Antony , St. Bakhum , St. Shenouda and St. Amun , are credited as pioneers. By

7920-425: The bottom. Frequent revolts by the Egyptian peasantry against the Ottoman-Mamluk Beys took place throughout the 18th century, particularly in Upper Egypt where the peasants at one point wrested control of the region and declared a separatist government. The only segment of Egyptian society which appears to have retained a degree of power during this period were the Muslim ' ulama or religious scholars, who directed

8040-405: The context of the struggle between proud native Egyptians and the central Abbasid caliphate in Iraq." The form of Islam that eventually took hold in Egypt was Sunni , though very early in this period Egyptians began to blend their new faith with indigenous beliefs and practices that had survived through Coptic Christianity. Just as Egyptians had been pioneers in early monasticism so they were in

8160-399: The contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India." Dani goes on to say: As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in

8280-467: The country's agricultural economy by ensuring regular Nile floods . Ascendancy to the Egyptian throne reflected the myth of Horus who assumed kingship after he buried his murdered father Osiris . The king of Egypt, as a living personification of Horus, could claim the throne after burying his predecessor, who was typically his father. When the role of the king waned, the country became more susceptible to foreign influence and invasion. The attention paid to

8400-819: The dead, and the veneration with which they were held, were one of the hallmarks of ancient Egyptian society . Egyptians built tombs for their dead that were meant to last for eternity. This was most prominently expressed by the Great Pyramids . The ancient Egyptian word for tomb pr nḥḥ means ' House of Eternity '. The Egyptians also celebrated life, as is shown by tomb reliefs and inscriptions, papyri and other sources depicting Egyptians farming, conducting trade expeditions, hunting, holding festivals, attending parties and receptions with their pet dogs, cats and monkeys, dancing and singing, enjoying food and drink, and playing games. The ancient Egyptians were also known for their engaging sense of humor, much like their modern descendants. Another important continuity during this period

8520-423: The defining moments in the history of the Church in Egypt is a controversy that ensued over the nature of Jesus, which culminated in the final split of the Coptic Church from both the Byzantine and Roman Catholic Churches. The Council of Chalcedon convened in AD 451, signaling the Byzantine Empire's determination to assert its hegemony over Egypt. When it declared that Jesus was of two natures embodied in his person,

8640-518: The desert. The cult of Isis , like those of Osiris and Serapis , had been popular in Egypt and throughout the Roman Empire at the coming of Christianity, and continued to be the main competitor with Christianity in its early years. The main temple of Isis remained a major center of worship in Egypt until the reign of the Byzantine emperor Justinian I in the 6th century, when it was finally closed down. Egyptians, disaffected and weary after

8760-639: The development of the mystical form of Islam, Sufism . Various Sufi orders were founded in the 8th century and flourished until the present day. One of the earliest Egyptian Sufis was Dhul-Nun al-Misri (i.e., Dhul-Nun the Egyptian). He was born in Akhmim in AD 796 and achieved political and social leadership over the Egyptian people. Dhul-Nun was regarded as the Patron Saint of the Physicians and

8880-411: The end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it." Similar to the atomists of Democritus , the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved. It stated that two lines of equal length will always finish at the same place, while providing definitions for

9000-544: The end of the 4th century, it is estimated that the mass of the Egyptians had either embraced Christianity or were nominally Christian. The Catachetical School of Alexandria was founded in the 3rd century by Pantaenus , becoming a major school of Christian learning as well as science, mathematics and the humanities. The Psalms and part of the New Testament were translated at the school from Greek to Egyptian, which had already begun to be written in Greek letters with

9120-594: The existing Byzantine administrative structure with Greek as its language. Native Egyptians filled administrative ranks and continued to worship freely so long as they paid the jizya poll tax, in addition to a land tax that all Egyptians irrespective of religion also had to pay. The authority of the Miaphysite doctrine of the Coptic Church was for the first time nationally recognized. According to al-Ya'qubi , repeated revolts by Egyptian Christians against

9240-545: The eyes of the Egyptians a legitimate heir to the native pharaohs. The new Ptolemaic rulers, however, exploited Egypt for their own benefit and a great social divide was created between Egyptians and Greeks. The local priesthood continued to wield power as they had during the Dynastic age. Egyptians continued to practice their religion undisturbed and largely maintained their own separate communities from their foreign conquerors. The language of administration became Greek , but

9360-646: The field are barely recognizable as the descendants of early geometry. (See Areas of mathematics and Algebraic geometry .) The earliest recorded beginnings of geometry can be traced to early peoples, such as the ancient Indus Valley (see Harappan mathematics ) and ancient Babylonia (see Babylonian mathematics ) from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. Among these were some surprisingly sophisticated principles, and

9480-439: The figure, or the game. Victorinus , Bassus Ennodius and Lucretius have also discussed the game. The game is a 14-piece dissection puzzle forming a square. One form of play to which classical texts attest is the creation of different objects, animals, plants etc. by rearranging the pieces: an elephant, a tree, a barking dog, a ship, a sword, a tower etc. Another suggestion is that it exercised and developed memory skills in

9600-497: The first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I , King of Egypt. The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates ), and general quantitative principles (called common notions ) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make

9720-546: The following geometrical formulas that were known by the time of the Former Han dynasty (202 BCE – 9 CE). Areas for the Volumes for the Continuing the geometrical legacy of ancient China, there were many later figures to come, including the famed astronomer and mathematician Shen Kuo (1031–1095 CE), Yang Hui (1238–1298) who discovered Pascal's Triangle , Xu Guangqi (1562–1633), and many others. Thābit ibn Qurra , using what he called

9840-469: The indigenous population of Egypt...From horseback, they simply terrorized those lesser breeds who crossed their paths. The sudden and arbitrary use of force by the government and its dominant military elite; frequent resort to cruelty to make a point; ingenious methods of torture employed both for exemplary purpose and to extract wealth from others: all these measures were routine in the Mamluk era. Egypt under

9960-468: The intangible pull of Egypt is the ever present Nile, which is more than a constant backdrop. Its varying colors and changing water levels signal the coming and going of the Nile flood that sets the rhythm of farming in a rainless country and holds the attention of all Egyptians. No Egyptian is ever far from his river and, except for the Alexandrines whose personality is split by looking outward toward

10080-419: The later Mameluks maintained the splendour and artistic traditions of their predecessors. The reign of Kait Bey (1468–1496) was one of high achievement in architecture, showing great refinement of taste in the building of elegant tombs, mosques and palaces. It was a period in which learning flourished. By the 15th century most Egyptians had already been converted to Islam, while Coptic Christians were reduced to

10200-404: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral : Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side. Chapter 12 also included a formula for the area of

10320-435: The linen-weavers by their speech." This attitude lasted until AD 212 when Roman citizenship was finally granted to all the inhabitants of Egypt, though ethnic divisions remained largely entrenched. The Romans, like the Ptolemies, treated Egypt like their own private property, a land exploited for the benefit of a small foreign elite. The Egyptian peasants, pressed for maximum production to meet Roman quotas, suffered and fled to

10440-422: The mass of the Egyptian population was Egyptian -speaking and concentrated in the countryside, while most Greeks lived in Alexandria and only few had any knowledge of Egyptian. The Ptolemaic rulers all retained their Greek names and titles, but projected a public image of being Egyptian pharaohs. Much of this period's vernacular literature was composed in the demotic phase and script of the Egyptian language. It

10560-520: The mathematician Toscanelli ), but did not publish, the mathematics behind perspective. Decades later, his friend Leon Battista Alberti wrote De pictura (1435/1436), a treatise on proper methods of showing distance in painting based on Euclidean geometry. Alberti was also trained in the science of optics through the school of Padua and under the influence of Biagio Pelacani da Parma who studied Alhazen's Optics . Piero della Francesca elaborated on Della Pittura in his De Prospectiva Pingendi in

10680-399: The method of reduction and composition, provided two different general proofs of the Pythagorean theorem for all triangles , before which proofs only existed for the theorem for the special cases of a special right triangle . A 2007 paper in the journal Science suggested that girih tiles possessed properties consistent with self-similar fractal quasicrystalline tilings such as

10800-649: The mid-third century, a sizable number of Egyptians were persecuted by the Romans on account of having adopted the new Christian faith, beginning with the Edict of Decius . Christianity was tolerated in the Roman Empire until AD 284, when the Emperor Diocletian persecuted and put to death a great number of Christian Egyptians. This event became a watershed in the history of Egyptian Christianity, marking

10920-450: The new country. On the other hand, Egyptians migrating to Arab countries almost always only go there with the intention of returning to Egypt; virtually none settle in the new country on a permanent basis. Prior to 1974, only few Egyptian professionals had left the country in search for employment. Political, demographic and economic pressures led to the first wave of emigration after 1952. Later more Egyptians left their homeland first after

11040-637: The other notable approximation. In comparison to later works, the formula for pi given by the French mathematician Franciscus Vieta (1540–1603) fell halfway between Zu's approximations. The Nine Chapters on the Mathematical Art , the title of which first appeared by 179 AD on a bronze inscription, was edited and commented on by the 3rd century mathematician Liu Hui from the Kingdom of Cao Wei . This book included many problems where geometry

11160-549: The religious and social affairs of the native Egyptian population and interceded on their behalf when dealing with the Turko-Circassian elite. It is also believed that during the late periods of the Ottoman era of Egypt, native Egyptians were allowed and required to join the army for the first time since the Roman period of Egypt, including Coptic Christians who were civil servants at the time of Mohammed Ali Pasha. From

11280-693: The rest elsewhere in the region) and the remaining 30% are living mostly in Europe and North America (318,000 in the United States , 110,000 in Canada and 90,000 in Italy ). Their characteristic rootedness as Egyptians, commonly explained as the result of centuries as a farming people clinging to the banks of the Nile , is reflected in sights, sounds and atmosphere that are meaningful to all Egyptians. Dominating

11400-536: The rest of the empire. Egyptians continued to live socially and politically separate from their foreign conquerors, but their rulers like the Ptolemies before them were able to stabilize the country and bring renewed economic prosperity. It was under the Shiite Fatimids from the 10th to the 12th centuries that Muslim Egyptian institutions began to take form along with the Egyptian dialect of Arabic, which

11520-593: The sides of the cyclic quadrilateral being a, b, c, and d , the radius R of the circumscribed circle is: The first definitive work (or at least oldest existent) on geometry in China was the Mo Jing , the Mohist canon of the early philosopher Mozi (470–390 BC). It was compiled years after his death by his followers around the year 330 BC. Although the Mo Jing is the oldest existent book on geometry in China, there

11640-902: The specific provinces , towns and villages from which they hail. Therefore, return migrants, such as temporary workers abroad, come back to their region of origin in Egypt. According to the International Organization for Migration , an estimated 2.7 million Egyptians live abroad and contribute actively to the development of their country through remittances (US$ 7.8 billion in 2009), circulation of human and social capital, as well as investment. Approximately 70% of Egyptian migrants live in Arab countries (923,600 in Saudi Arabia , 332,600 in Libya , 226,850 in Jordan , 190,550 in Kuwait with

11760-549: The status of mawali or "clients" to the ruling Arab elite, while those who remained Christian, the Copts, became dhimmis , but the Egyptians who converted to Islam were also called Copts until the Mamluk period. In time the power of the Arabs waned throughout the Islamic Empire so that in the 10th century, the Turkish Ikhshids were able to take control of Egypt and made it an independent political unit from

11880-595: The strain put on Egypt's economy by the country's burgeoning population, a national family planning program was initiated in 1964, and by the 1990s it had succeeded in lowering the birth rate. Improvements in health care also brought the infant mortality rate well below the world average by the turn of the 21st century. Life expectancy averages about 72 years for men and 74 years for women. Egyptians also form smaller minorities in neighboring countries, North America, Europe and Australia. Egyptians also tend to be provincial, meaning their attachment extends not only to Egypt but to

12000-615: The styles of both Euclid ( Euclidean geometry ) and Khayyam ( algebraic geometry ) continued, resulting in an abundance of new theorems and concepts, many of them very profound and elegant. Advances in the treatment of perspective were made in Renaissance art of the 14th to 15th century which went beyond what had been achieved in antiquity. In Renaissance architecture of the Quattrocento , concepts of architectural order were explored and rules were formulated. A prime example of

12120-490: The technique of upward foreshortening (in Rome, Loreto , Forlì and others), and was celebrated for that. Not only was perspective a way of showing depth, it was also a new method of composing a painting. Paintings began to show a single, unified scene, rather than a combination of several. As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend

12240-420: The volume of a frustum of a pyramid, describing the correct formula: where a and b are the base and top side lengths of the truncated pyramid and h is the height. The Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π

12360-532: The war to fill the workforce. Ancient Egypt saw a succession of thirty dynasties spanning three millennia. During this period, Egyptian culture underwent significant development in terms of religion , arts , language , and customs. Egypt fell under Hyksos rule in the Middle Bronze Age . The native nobility managed to expel the conquerors by the Late Bronze Age , thereby initiating

12480-547: The wars between Egypt and Israel , particularly after the Six-Day War in 1967, when migration rates began to rise. In August 2006, Egyptians made headlines when 11 students from Mansoura University failed to show up at their American host institutions for a cultural exchange program in the hope of finding employment. Egyptians in neighboring countries face additional challenges. Over the years, abuse, exploitation and/or ill-treatment of Egyptian workers and professionals in

12600-419: The word as " Stomachion ", an apparent corruption of the original Greek. Ausonius gives us the correct name "Ostomachion" ( quod Graeci ostomachion vocavere , "which the Greeks called ostomachion"). The Ostomachion which he describes was a puzzle similar to tangrams and was played perhaps by several persons with pieces made of bone. It is not known which is older, Archimedes' geometrical investigation of

12720-522: The world and has many important texts to his credit including the Hexapla , an exegesis of various translations of the Hebrew Bible . At the threshold of the Byzantine period, the New Testament had been entirely translated into Coptic. But while Christianity continued to thrive in Egypt, the old pagan beliefs which had survived the test of time were facing mounting pressure. The Byzantine period

12840-561: The young. James Gow, in his Short History of Greek Mathematics (1884), footnotes that the purpose was to put the pieces back in their box, and this was also a view expressed by W. W. Rouse Ball in some intermediate editions of Mathematical Essays and Recreations , but edited out from 1939. The number of different ways to arrange the parts of the Stomachions within a square were determined to be 17,152 by Fan Chung , Persi Diaconis , Susan P. Holmes , and Ronald Graham , and confirmed by

12960-503: Was Zhang Heng who was the first to make a concerted effort at creating a more accurate formula for pi. Zhang Heng approximated pi as 730/232 (or approx 3.1466), although he used another formula of pi in finding a spherical volume, using the square root of 10 (or approx 3.162) instead. Zu Chongzhi (429–500 AD) improved the accuracy of the approximation of pi to between 3.1415926 and 3.1415927, with ⁄ 113 (密率, Milü, detailed approximation) and ⁄ 7 (约率, Yuelü, rough approximation) being

13080-525: Was also the first to accurately draw the Platonic solids as they would appear in perspective. Egyptians This is an accepted version of this page Egyptians ( Arabic : مِصرِيُّون , romanized :  Miṣriyyūn , IPA: [mɪsˤrɪjˈjuːn] ; Egyptian Arabic : مَصرِيِّين , romanized:  Maṣriyyīn , IPA: [mɑsˤɾɪjˈjiːn] ; Coptic : ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ , romanized:  remenkhēmi ) are an ethnic group native to

13200-483: Was applied, such as finding surface areas for squares and circles, the volumes of solids in various three-dimensional shapes, and included the use of the Pythagorean theorem . The book provided illustrated proof for the Pythagorean theorem, contained a written dialogue between of the earlier Duke of Zhou and Shang Gao on the properties of the right angle triangle and the Pythagorean theorem, while also referring to

13320-405: Was divided. The other two were Winter and Summer, each lasting for four months. The modern Egyptian fellahin calculate the agricultural seasons, with the months still bearing their ancient names, in much the same manner. The importance of the Nile in Egyptian life, ancient and modern, cannot be overemphasized. The rich alluvium carried by the Nile inundation was the basis of Egypt's formation as

13440-526: Was especially pronounced in the Old Kingdom and Middle Kingdom and continued until the Roman conquest. The societal structure created by this system of government remained virtually unchanged up to modern times. The role of the king was considerably weakened after the 20th Dynasty . The king in his role as Son of Ra was entrusted to maintain Ma'at , the principle of truth, justice, and order, and to enhance

13560-542: Was focused on earlier stages of Egyptian history when Egyptians were independent and ruled by great native pharaohs such as Ramesses II . Prophetic writings circulated among Egyptians promising expulsion of the Greeks, and frequent revolts by the Egyptians took place throughout the Ptolemaic period. A revival in animal cults, the hallmark of the Predynastic and Early Dynastic periods, is said to have come about to fill

13680-545: Was given religious validation, as foreigners in the land of Ta-Meri (Egypt) were anathema to the maintenance of Maat—a view most clearly expressed by the admonitions of Ipuwer in reaction to the chaotic events of the Second Intermediate Period . Foreigners in Egyptian texts were described in derogatory terms, e.g., 'wretched Asiatics' (Semites), 'vile Kushites' (Nubians), and 'Ionian dogs' (Greeks). Egyptian beliefs remained unchallenged when Egypt fell to

13800-596: Was only in the 11th century BC that the Empire declined, falling into the comparative obscurity of the Third Intermediate Period of Egypt . The 25th Dynasty of Nubian rulers was again briefly replaced by native nobility in the 7th century BC, and in 525 BC, Egypt fell under Persian rule . Egypt fell under Greek control after Alexander the Great 's conquest in 332 BC. The Late Period of ancient Egypt

13920-408: Was particularly brutal in its zeal to erase any traces of ancient Egyptian religion. Under emperor Theodosius I , Christianity had already been proclaimed the religion of the Empire and all pagan cults were forbidden. When Egypt fell under the jurisdiction of Constantinople after the split of the Roman Empire, many ancient Egyptian temples were either destroyed or converted into monasteries. One of

14040-414: Was probably one of the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers . Plato (427–347 BC) was a philosopher, highly esteemed by

14160-474: Was slightly less accurate than the calculations of the Babylonians (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed until Archimedes ' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000. Ahmes knew of the modern 22/7 as an approximation for π , and used it to split a hekat, hekat x 22/x x 7/22 = hekat; however, Ahmes continued to use the traditional 256/81 value for π for computing his hekat volume found in

14280-399: Was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582–496 BC) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and traveled to Babylon and Egypt . The theorem that bears his name may not have been his discovery, but he

14400-607: Was to eventually slowly supplant native Egyptian or Coptic as the spoken language. Al-Azhar was founded in AD 970 in the new capital Cairo , not very far from its ancient predecessor in Memphis. It became the preeminent Muslim center of learning in Egypt and by the Ayyubid period it had acquired a Sunni orientation. The Fatimids with some exceptions were known for their religious tolerance and their observance of local Muslim, Coptic and indigenous Egyptian festivals and customs. Under

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