1.
Ancient Egypt
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Ancient Egypt was a civilization of ancient Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. It is one of six civilizations to arise independently, Egyptian civilization followed prehistoric Egypt and coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh Narmer. In the aftermath of Alexander the Greats death, one of his generals, Ptolemy Soter and this Greek Ptolemaic Kingdom ruled Egypt until 30 BC, when, under Cleopatra, it fell to the Roman Empire and became a Roman province. The success of ancient Egyptian civilization came partly from its ability to adapt to the conditions of the Nile River valley for agriculture, the predictable flooding and controlled irrigation of the fertile valley produced surplus crops, which supported a more dense population, and social development and culture. Its art and architecture were widely copied, and its antiquities carried off to far corners of the world and its monumental ruins have inspired the imaginations of travelers and writers for centuries. The Nile has been the lifeline of its region for much of human history, nomadic modern human hunter-gatherers began living in the Nile valley through the end of the Middle Pleistocene some 120,000 years ago. By the late Paleolithic period, the climate of Northern Africa became increasingly hot and dry. In Predynastic and Early Dynastic times, the Egyptian climate was less arid than it is today. Large regions of Egypt were covered in treed savanna and traversed by herds of grazing ungulates, foliage and fauna were far more prolific in all environs and the Nile region supported large populations of waterfowl. Hunting would have been common for Egyptians, and this is also the period when many animals were first domesticated. The largest of these cultures in upper Egypt was the Badari, which probably originated in the Western Desert, it was known for its high quality ceramics, stone tools. The Badari was followed by the Amratian and Gerzeh cultures, which brought a number of technological improvements, as early as the Naqada I Period, predynastic Egyptians imported obsidian from Ethiopia, used to shape blades and other objects from flakes. In Naqada II times, early evidence exists of contact with the Near East, particularly Canaan, establishing a power center at Hierakonpolis, and later at Abydos, Naqada III leaders expanded their control of Egypt northwards along the Nile. They also traded with Nubia to the south, the oases of the desert to the west. Royal Nubian burials at Qustul produced artifacts bearing the oldest-known examples of Egyptian dynastic symbols, such as the crown of Egypt. They also developed a ceramic glaze known as faience, which was used well into the Roman Period to decorate cups, amulets, and figurines. During the last predynastic phase, the Naqada culture began using written symbols that eventually were developed into a system of hieroglyphs for writing the ancient Egyptian language. The Early Dynastic Period was approximately contemporary to the early Sumerian-Akkadian civilisation of Mesopotamia, the third-century BC Egyptian priest Manetho grouped the long line of pharaohs from Menes to his own time into 30 dynasties, a system still used today
Ancient Egypt
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The Great Sphinx and the pyramids of Giza are among the most recognizable symbols of the civilization of ancient Egypt.
Ancient Egypt
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A typical Naqada II jar decorated with gazelles. (Predynastic Period)
Ancient Egypt
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The Narmer Palette depicts the unification of the Two Lands.
2.
Abydos, Egypt
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Abydos /əˈbaɪdɒs/ is one of the oldest cities of ancient Egypt, and also of the eighth nome in Upper Egypt, of which it was the capital city. It is located about 11 kilometres west of the Nile at latitude 26°10 N, in the ancient Egyptian language, the city was called Abdju. The English name Abydos comes from the Greek Ἄβυδος, a name borrowed by Greek geographers from the city of Abydos on the Hellespont. These tombs began to be seen as extremely significant burials and in times it became desirable to be buried in the area. Today, Abydos is notable for the temple of Seti I. It is a chronological list showing cartouches of most dynastic pharaohs of Egypt from Menes until Seti Is father, the Great Temple and most of the ancient town are buried under the modern buildings to the north of the Seti temple. Many of the structures and the artifacts within them are considered irretrievable and lost. Abydos was occupied by the rulers of the Predynastic period, whose town, temple, the temple and town continued to be rebuilt at intervals down to the times of the thirtieth dynasty, and the cemetery was used continuously. The pharaohs of the first dynasty were buried in Abydos, including Narmer, who is regarded as founder of the first dynasty and it was in this time period that the Abydos boats were constructed. Some pharaohs of the dynasty were also buried in Abydos. The temple was renewed and enlarged by these pharaohs as well, funerary enclosures, misinterpreted in modern times as great forts, were built on the desert behind the town by three kings of the second dynasty, the most complete is that of Khasekhemwy. From the fifth dynasty, the deity Khentiamentiu, foremost of the Westerners, Pepi I constructed a funerary chapel which evolved over the years into the Great Temple of Osiris, the ruins of which still exist within the town enclosure. Abydos became the centre of the worship of the Isis and Osiris cult, during the First Intermediate Period, the principal deity of the area, Khentiamentiu, began to be seen as an aspect of Osiris, and the deities gradually merged and came to be regarded as one. Khentiamentius name became an epithet of Osiris, King Mentuhotep II was the first one building a royal chapel. In the twelfth dynasty a gigantic tomb was cut into the rock by Senusret III, associated with this tomb was a cenotaph, a cult temple and a small town known as Wah-Sut, that was used by the workers for these structures. Next to that cenotaph were buried kings of the Thirteenth Dynasty, the building during the eighteenth dynasty began with a large chapel of Ahmose I. The Pyramid of Ahmose I was also constructed at Abydos—the only pyramid in the area, thutmose III built a far larger temple, about 130 ft ×200 ft. He also made a way leading past the side of the temple to the cemetery beyond
Abydos, Egypt
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Façade of the Temple of Seti I in Abydos
Abydos, Egypt
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Part of the Abydos King List
Abydos, Egypt
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Tomb relief depicting the vizier Nespeqashuty and his wife, KetjKetj, making the journey of the dead to the holy city of Abydos – from Deir el-Bahri, Late Period, twenty-sixth dynasty of Egypt, reign of Psammetichus I
Abydos, Egypt
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Panel from the Osiris temple: Horus presents royal regalia to a worshipping pharaoh.
3.
Narmer Macehead
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The Narmer macehead is an ancient Egyptian decorative stone mace head. It was found during a dig at Kom al Akhmar, the site of Hierakonpolis and it is dated to the Early Dynastic Period reign of king Narmer whose serekh is engraved on it. The macehead is now kept at the Ashmolean Museum, Oxford, the Narmer macehead is better preserved than the Scorpion Macehead and has had various interpretations. On the left side of this macehead we see a king wearing the Red Crown sitting under a canopy on a dais, covered in a long cloth or cloak. He is holding the flail and above the canopy a vulture hovers with spread wings, possibly Nekhbet, the goddess of Nekhen. Directly in front of him is another dais or possibly litter on which sits facing him a cloaked figure and this figure has been interpreted as a princess being presented to the king for marriage, kings child or a deity. The dais is covered by a structure and behind it are three registers. In the center register attendants are walking or running behind the dais, behind the enclosure four standard-bearers approach the throne. In the bottom register, in front of the fan-bearers, are seen what looks like a collection of offerings and he is followed by a man carrying a long pole. Above him three men are walking, two of them likewise carrying long poles, the serekh displaying the signs for Narmer can be seen above these. The top field to the right of the field shows a building, perhaps a shrine. Below this, an enclosure shows three animals, probably antelopes and this has been suggested as signifying the ancient town of Buto, the place where the events described on the macehead might have taken place
Narmer Macehead
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Narmer Macehead Centre left: Pharaoh Narmer seated in a naos
4.
Old Kingdom
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The term itself was coined by eighteenth-century historians and the distinction between the Old Kingdom and the Early Dynastic Period is not one which would have been recognized by Ancient Egyptians. The Old Kingdom is most commonly regarded as the period from the Third Dynasty through to the Sixth Dynasty, many Egyptologists also include the Memphite Seventh and Eighth Dynasties in the Old Kingdom as a continuation of the administration centralized at Memphis. During the Old Kingdom, the king of Egypt became a god who ruled absolutely and could demand the services. Under King Djoser, the first king of the Third Dynasty of the Old Kingdom, the capital of Egypt was moved to Memphis. A new era of building was initiated at Saqqara under his reign, King Djosers architect, Imhotep is credited with the development of building with stone and with the conception of the new architectural form—the Step Pyramid. Indeed, the Old Kingdom is perhaps best known for the number of pyramids constructed at this time as burial places for Egypts kings. For this reason, the Old Kingdom is frequently referred to as the Age of the Pyramids, the first king of the Old Kingdom was Djoser of the third dynasty, who ordered the construction of a pyramid in Memphis necropolis, Saqqara. An important person during the reign of Djoser was his vizier and it was in this era that formerly independent ancient Egyptian states became known as nomes, under the rule of the king. The former rulers were forced to assume the role of governors or otherwise work in tax collection, Egyptians in this era worshipped their king as a god, believing that he ensured the annual flooding of the Nile that was necessary for their crops. Egyptian views on the nature of time during this period held that the worked in cycles. They also perceived themselves as a specially selected people, the Old Kingdom and its royal power reached a zenith under the Fourth Dynasty, which began with Sneferu. Using more stones than any king, he built three pyramids, a now collapsed pyramid in Meidum, the Bent Pyramid at Dahshur. However, the development of the pyramid style of building was reached not at Saqqara. Sneferu was succeeded by his son, Khufu who built the Great Pyramid of Giza, after Khufus death, his sons Djedefra and Khafra may have quarrelled. The latter built the pyramid and the Sphinx in Giza. Recent reexamination of evidence has led Egyptologist Vassil Dobrev to propose that the Sphinx had been built by Djedefra as a monument to his father Khufu, alternatively, the Sphinx has been proposed to be the work of Khafra and Khufu himself. There were military expeditions into Canaan and Nubia, with Egyptian influence reaching up the Nile into what is today the Sudan, the later kings of the Fourth Dynasty were king Menkaure, who built the smallest pyramid in Giza, Shepseskaf and, perhaps, Djedefptah. The Fifth Dynasty began with Userkaf and was marked by the importance of the cult of sun god Ra
Old Kingdom
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During the Old Kingdom of Egypt (circa 2686 B.C.E. — circa 2181 B.C.E.), Egypt consisted of the Nile River region and the area along the river south to Elephantine.
Old Kingdom
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The Pyramid of Djoser at Saqqara.
Old Kingdom
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Temple of Djoser at Saqqara
5.
Mastaba
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A mastaba or pr-djt is a type of ancient Egyptian tomb in the form of a flat-roofed, rectangular structure with inward sloping sides, constructed out of mud-bricks or stone. These edifices marked the sites of many eminent Egyptians during Egypts Early Dynastic Period. In the Old Kingdom epoch, local kings began to be buried in pyramids instead of in mastabas, egyptologists call these tombs mastaba, which is the Arabic word for stone bench. The afterlife was a focus of Egyptian civilization and ruled every aspect of the society. This is reflected in their architecture and most prominently by the amounts of time, money. Ancient Egyptians believed the soul could live only if the body was preserved from corruption and depredation as well as fed, starting from the Predynastic era and into the later dynasties, the ancient Egyptians developed increasingly complex and effective methods for preserving and protecting the bodies of the dead. The Ancient Egyptians initially began by burying their dead in pit graves dug out from the sand, the body of the deceased was buried inside the pit on a mat, usually along with some items believed to help them in the afterlife. The first tomb structure that the Egyptians built was the mastaba, mastabas provided better protection from scavenging animals and grave robbers. However, the remains were not in contact with the dry desert sand. Use of the more secure mastabas required Ancient Egyptians to devise a system of artificial mummification, until at least the Old Period or First Intermediate Period, only high officials and royalty would be buried in these mastabas. The word mastaba comes from the Arabic word for a bench of mud, historians speculate that the Egyptians may have borrowed architectural ideas from Mesopotamia since at the time they were both building similar structures. The above-ground structure of a mastaba is rectangular in shape with inward-sloping sides, the exterior building materials were initially bricks made of sun dried mud, which was readily available from the Nile River. Even after more durable materials like stone came into use, all, mastabas were often about four times as long as they were wide, and many rose to at least 30 feet in height. The mastaba was built with an orientation, which the Ancient Egyptians believed was essential for access to the afterlife. This above-ground structure had space for an offering chapel equipped with a false door. Inside the mastaba, a chamber was dug into the ground and lined with stone. The burial chambers were cut deep, until they passed the bedrock, the mastaba housed a statue of the deceased that was hidden within the masonry for its protection. High up the walls of the serdab were small openings that would allow the ba to leave and return to the body, Ancient Egyptians believed the ba had to return to its body or it would die
Mastaba
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Example of a mastaba
6.
Meidum
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Meidum, Maydum or Maidum is an archaeological site in Lower Egypt. It contains a pyramid and several mud-brick mastabas. The area is located around 62 miles south of modern Cairo, the architect was a successor to the famous Imhotep, the inventor of the stone built pyramid. The collapse of the pyramid is likely due to the made to Imhoteps pyramid design as well as the decisions taken twice during construction to extend the pyramid. Because of its appearance, the pyramid is called el-heram el-kaddaab — in Egyptian Arabic. The second extension turned the original step pyramid design into a pyramid by filling in the steps with limestone encasing. While this approach is consistent with the design of the other true pyramids, firstly, the outer layer was founded on sand and not on rock, like the inner layers. Secondly, the step pyramids had been designed as the final stage. Thus the outer surface was polished and the platforms of the steps were not horizontal and this severely compromised the stability and is likely to have caused the collapse of the Meidum Pyramid in a downpour while the building was still under construction. Some believe the pyramid not to have collapsed until the New Kingdom, the Meidum Pyramid seems never to have been completed. Beginning with Sneferu and to the 12th dynasty all pyramids had a valley temple, the mortuary temple, which was found under the rubble at the base of the pyramid, apparently never was finished. Two Steles inside, usually bearing the names of the pharaoh, are missing inscriptions, the burial chamber inside the pyramid itself is uncompleted, with raw walls and wooden supports still in place which are usually removed after construction. Affiliated mastabas were never used or completed and none of the burials have been found. Finally, the first examinations of the Meidum Pyramid found everything below the surface of the rubble mound fully intact, stones from the outer cover were stolen only after they were exposed by the excavations. This makes a catastrophic collapse more probable than a gradual one, the collapse of this pyramid during the reign of Sneferu is the likely reason for the change from the usual 52 to 43 degrees of his second pyramid at Dahshur, the Bent Pyramid. By the time it was investigated by Napoleons Expedition in 1799 the Meidum Pyramid had its present three steps. The Meidum Pyramid was excavated by John Shae Perring in 1837, Lepsius in 1843 and then by Flinders Petrie later in the nineteenth century, in 1920 Ludwig Borchardt studied the area further, followed by Alan Rowe in 1928 and then Ali el-Kholi in the 1970s. In its ruined state, the structure is 213 feet high, the chamber is unlikely to have been used for any burial
Meidum
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View of the Pyramid
Meidum
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Lantern Slide Collection: Views, Objects: Egypt. Meidum. Old Kingdom. Step Pyramid of Meidum, 4th Dyn., n.d. Brooklyn Museum Archives
Meidum
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Passageway in the Meidum Pyramid
Meidum
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Another view of Meidum Pyramid
7.
Cubit
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The cubit is an ancient unit based on the forearm length from the middle finger tip to the elbow bottom. Cubits of various lengths were employed in many parts of the world in antiquity, during the Middle Ages, the term is still used in hedge laying, the length of the forearm being frequently used to determine the interval between stakes placed within the hedge. The English word cubit comes from the Latin noun cubitus elbow, from the verb cubo, cubare, cubui, cubitum to lie down, the ancient Egyptian royal cubit is the earliest attested standard measure. Cubit rods were used for the measurement of length, a number of these rods have survived, two are known from the tomb of Maya, the treasurer of the 18th dynasty pharaoh Tutankhamun, in Saqqara, another was found in the tomb of Kha in Thebes. Fourteen such rods, including one double cubit rod, were described and compared by Lepsius in 1865. These cubit rods range from 523.5 to 529.2 mm in length, and are divided into seven palms, each palm is divided into four fingers and the fingers are further subdivided. Use of the royal cubit is also known from Old Kingdom architecture, in 1916, during the last years of the Ottoman Empire and in the middle of World War I, the German assyriologist Eckhard Unger found a copper-alloy bar while excavating at Nippur. The bar dates from c.2650 BC and Unger claimed it was used as a measurement standard and this irregularly formed and irregularly marked graduated rule supposedly defined the Sumerian cubit as about 518.6 mm. The Near Eastern or Biblical cubit is usually estimated as approximately 457.2 mm, in ancient Greek units of measurement, the standard forearm cubit measured approximately 0.46 m. The short forearm cubit, from the wrist to the elbow, in ancient Rome, according to Vitruvius, a cubit was equal to 1 1⁄2 Roman feet or 6 palm widths. Other measurements based on the length of the forearm include some lengths of ell, the Chinese chi, the Japanese shaku, the Indian hasta, the Thai sok, the Tamil, the Telugu, a cubit arm in heraldry may be dexter or sinister. It may be vested and may be shown in positions, most commonly erect. It is most often used erect as a crest, for example by the families of Poyntz of Iron Acton, Rolle of Stevenstone, the Encyclopaedia of Ancient Egyptian Architecture. The Cubit, A History and Measurement Commentary, Journal of Anthropology doi,10. 1155/2014/489757,2014 Media related to Cubit arms at Wikimedia Commons The dictionary definition of cubit at Wiktionary
Cubit
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Egyptian cubit rod in the Liverpool World Museum
Cubit
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Cubit rod of Maya, 1336-1327 BC (Eighteenth Dynasty)
Cubit
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Cubit rod from the Turin Museum.
Cubit
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The Nippur cubit-rod in the Archeological Museum of Istanbul, Turkey
8.
Ancient Egyptian units of measurement
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Egyptian Circle Egyptian units of length are attested from the Early Dynastic Period, when the Palermo stone recorded the level of the Nile River. During the reign of Pharaoh Djer, the height of the Nile was recorded as 6 cubits and 1 palm, a 3rd-dynasty diagram shows how to construct an elliptical vault using simple measures along an arc. The ostracon depicting this diagram was found near the Step Pyramid of Saqqara, a curve is divided into five sections and the height of the curve is given in cubits, palms, and digits in each of the sections. At some point, lengths were standardized by cubit rods, examples have been found in the tombs of officials, noting lengths up to remen. Royal cubits were used for land measures such as roads and fields, fourteen rods, including one double-cubit rod, were described and compared by Lepsius. Two examples are known from the Saqqara tomb of Maya, the treasurer of Tutankhamun, another was found in the tomb of Kha in Thebes. These cubits are about 52.5 cm long and are divided into palms and hands, each palm is divided into four fingers from left to right and the fingers are further subdivided into ro from right to left. The rules are divided into hands so that for example one foot is given as three hands and fifteen fingers and also as four palms and sixteen fingers. Surveying and itinerant measurement were undertaken using rods, poles, a scene in the tomb of Menna in Thebes shows surveyors measuring a plot of land using rope with knots tied at regular intervals. Similar scenes can be found in the tombs of Amenhotep-Sesi, Khaemhat, the balls of rope are also shown in New Kingdom statues of officials such as Senenmut, Amenemhet-Surer, and Penanhor. The digit was also subdivided into smaller fractions of ½, ⅓, ¼, minor units include the Middle Kingdom reed of 2 royal cubits, the Ptolemaic xylon of three royal cubits, the Ptolemaic fathom of four lesser cubits, and the kalamos of six royal cubits. Records of land area also date to the Early Dynastic Period, the Palermo stone records grants of land expressed in terms of kha and setat. Mathematical papyri also include units of area in their problems. The setat was the unit of land measure and may originally have varied in size across Egypts nomes. Later, it was equal to one square khet, where a khet measured 100 cubits, the setat could be divided into strips one khet long and ten cubit wide.25 m². A36 sq. cubit area was known as a kalamos, the uncommon bikos may have been 1½ hammata or another name for the cubit strip. The Coptic shipa was a unit of uncertain value, possibly derived from Nubia. Units of volume appear in the mathematical papyri, for example, computing the volume of a circular granary in RMP42 involves cubic cubits, khar, heqats, and quadruple heqats
Ancient Egyptian units of measurement
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Cubit rod from the Turin Museum.
Ancient Egyptian units of measurement
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Problem 80 on the Rhind Mathematical Papyrus
9.
Moscow Mathematical Papyrus
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Golenishchev bought the papyrus in 1892 or 1893 in Thebes. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, approximately 5½ m long and varying between 3.8 and 7.6 cm wide, its format was divided into 25 problems with solutions by the Soviet Orientalist Vasily Vasilievich Struve in 1930. It is a well-known mathematical papyrus along with the Rhind Mathematical Papyrus, the Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two. The problems in the Moscow Papyrus follow no particular order, the papyrus is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a frustum respectively, the remaining problems are more common in nature. Problems 2 and 3 are ships part problems, one of the problems calculates the length of a ships rudder and the other computes the length of a ships mast given that it is 1/3 + 1/5 of the length of a cedar log originally 30 cubits long. Aha problems involve finding unknown quantities if the sum of the quantity, the Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1,19, and 25 of the Moscow Papyrus are Aha problems, for instance problem 19 asks one to calculate a quantity taken 1 and ½ times and added to 4 to make 10. In other words, in mathematical notation one is asked to solve 3 /2 × x +4 =10 Most of the problems are pefsu problems,10 of the 25 problems. A pefsu measures the strength of the beer made from a heqat of grain pefsu = number loaves of bread number of heqats of grain A higher pefsu number means weaker bread or beer, the pefsu number is mentioned in many offering lists. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain. Calculate 1/2 of 5 heqat, the result will be 2 1/2 Take this 2 1/2 four times The result is 10, then you say to him, Behold. The beer quantity is found to be correct, problems 11 and 23 are Baku problems. These calculate the output of workers, problem 11 asks if someone brings in 100 logs measuring 5 by 5, then how many logs measuring 4 by 4 does this correspond to. Problem 23 finds the output of a given that he has to cut. Seven of the problems are geometry problems and range from computing areas of triangles, to finding the surface area of a hemisphere. The 10th problem of the Moscow Mathematical Papyrus asks for a calculation of the area of a hemisphere or possibly the area of a semi-cylinder. Below we assume that the problem refers to the area of a hemisphere, the text of problem 10 runs like this, Example of calculating a basket
Moscow Mathematical Papyrus
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14th problem of the Moscow Mathematical Papyrus (V. Struve, 1930)
Moscow Mathematical Papyrus
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The neutrality of this article is disputed. Relevant discussion may be found on the talk page. Please do not remove this message until the dispute is resolved. (July 2015)
10.
Kahun Papyri
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The Kahun Papyri are a collection of ancient Egyptian texts discussing administrative, mathematical and medical topics. Its many fragments were discovered by Flinders Petrie in 1889 and are kept at the University College London and this collection of papyri is one of the largest ever found. Most of the texts are dated to ca.1825 BC, in general the collection spans the Middle Kingdom of Egypt. The texts span a variety of topics, Business papers of the cult of Senusret II Hymns to king Senusret III, the Kahun Gynaecological Papyrus, which deals with gynaecological illnesses and conditions. The Lahun Mathematical Papyri are a collection of mathematical texts A veterinarian papyrus A late Middle Kingdom account, listing festivals A Kahun Mathematical Fragment, legon PlanetMath, Kahun Papyrus and Arithmetic Progressions
Kahun Papyri
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Fragments of the Kahun Papyrus on veterinary medicine
11.
Berlin Papyrus 6619
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The Berlin Papyrus 6619, simply called the Berlin Papyrus when the context makes it clear, is an ancient Egyptian papyrus document from the Middle Kingdom, second half of the 12th or 13th dynasty. The two readable fragments were published by Hans Schack-Schackenburg in 1900 and 1902, the papyrus is one of the primary sources of ancient Egyptian mathematics. The Berlin Papyrus contains two problems, the first stated as the area of a square of 100 is equal to that of two smaller squares, the side of one is ½ + ¼ the side of the other. The interest in the question may suggest some knowledge of the Pythagorean theorem, though the papyrus only shows a straightforward solution to a single second degree equation in one unknown. In modern terms, the simultaneous equations x2 + y2 =100 and x = y reduce to the equation in y,2 + y2 =100. Papyrology Timeline of mathematics Egyptian fraction Simultaneous equation examples from the Berlin papyrus Two algebra problems compared to RMP algebra Two suggested solutions
Berlin Papyrus 6619
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Berlin Papyrus 6619, as reproduced in 1900 by Schack-Schackenburg
12.
Rhind Mathematical Papyrus
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The Rhind Mathematical Papyrus is one of the best known examples of Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian and it dates to around 1550 BC. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus, the Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former. The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt and it was copied by the scribe Ahmes, from a now-lost text from the reign of king Amenemhat III. Written in the script, this Egyptian manuscript is 33 cm tall. The papyrus began to be transliterated and mathematically translated in the late 19th century, the mathematical translation aspect remains incomplete in several respects. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later historical note on its verso likely dating from the period of his successor, Khamudi. In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving Accurate reckoning for inquiring into things, the scribe Ahmose writes this copy. Several books and articles about the Rhind Mathematical Papyrus have been published, a more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute. The first part of the Rhind papyrus consists of reference tables, the problems start out with simple fractional expressions, followed by completion problems and more involved linear equations. The first part of the papyrus is taken up by the 2/n table, the fractions 2/n for odd n ranging from 3 to 101 are expressed as sums of unit fractions. For example,2 /15 =1 /10 +1 /30. The decomposition of 2/n into unit fractions is never more than 4 terms long as in for example 2 /101 =1 /101 +1 /202 +1 /303 +1 /606. This table is followed by a smaller, tiny table of fractional expressions for the numbers 1 through 9 divided by 10. Problems 1-7, 7B and 8-40 are concerned with arithmetic and elementary algebra, problems 1–6 compute divisions of a certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 = 7/4 and 1 + 2/3 + 1/3 =2 by different fractions, problems 21–23 are problems in completion, which in modern notation are simply subtraction problems. Problems 24–34 are ‘’aha’’ problems, these are linear equations, problem 32 for instance corresponds to solving x + 1/3 x + 1/4 x =2 for x. Problems 35–38 involve divisions of the heqat, which is an ancient Egyptian unit of volume, problems 39 and 40 compute the division of loaves and use arithmetic progressions
Rhind Mathematical Papyrus
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A portion of the Rhind Papyrus
Rhind Mathematical Papyrus
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Building
13.
Second Intermediate Period
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The Second Intermediate Period marks a period when Ancient Egypt fell into disarray for a second time, between the end of the Middle Kingdom and the start of the New Kingdom. It is best known as the period when the Hyksos made their appearance in Egypt, the Twelfth Dynasty of Egypt came to an end at the end of the 19th century BC with the death of Queen Sobekneferu. Apparently she had no heirs, causing the twelfth dynasty to come to an end, and, with it. Retaining the seat of the dynasty, the thirteenth dynasty ruled from Itjtawy near Memphis and Lisht. The Thirteenth Dynasty is notable for the accession of the first formally recognised Semitic-speaking king, the Fifteenth Dynasty dates approximately from 1650 to 1550 BC. Known rulers of the Fifteenth Dynasty are as follows, Salitis Sakir-Har Khyan Apophis, 1550–1540 BC The Fifteenth Dynasty of Egypt was the first Hyksos dynasty, ruled from Avaris, without control of the entire land. The Hyksos preferred to stay in northern Egypt since they infiltrated from the north-east, the names and order of kings is uncertain. The Turin King list indicates that there were six Hyksos kings, the surviving traces on the X figure appears to give the figure 8 which suggests that the summation should be read as 6 kings ruling 108 years. Some scholars argue there were two Apophis kings named Apepi I and Apepi II, but this is due to the fact there are two known prenomens for this king, Awoserre and Aqenenre. However, the Danish Egyptologist Kim Ryholt maintains in his study of the Second Intermediate Period that these prenomens all refer to one man, Apepi and this is also supported by the fact that this king employed a third prenomen during his reign, Nebkhepeshre. Apepi likely employed several different prenomens throughout various periods of his reign and this scenario is not unprecedented, as later kings, including the famous Ramesses II and Seti II, are known to have used two different prenomens in their own reigns. The Sixteenth Dynasty ruled the Theban region in Upper Egypt for 70 years, of the two chief versions of Manethos Aegyptiaca, Dynasty XVI is described by the more reliable Africanus as shepherd kings, but by Eusebius as Theban. For this reason other scholars do not follow Ryholt and see only insufficient evidence for the interpretation of the Sixteenth Dynasty as Theban, the continuing war against Dynasty XV dominated the short-lived 16th dynasty. The armies of the 15th dynasty, winning town after town from their enemies, continually encroached on the 16th dynasty territory, eventually threatening. Famine, which had plagued Upper Egypt during the late 13th dynasty, from Ryholts reconstruction of the Turin canon,15 kings of the dynasty can now be named, five of whom appear in contemporary sources. While most likely based in Thebes itself, some may have been local rulers from other important Upper Egyptian towns, including Abydos, El Kab. By the reign of Nebiriau I, the controlled by the 16th dynasty extended at least as far north as Hu. Not listed in the Turin canon is Wepwawetemsaf, who left a stele at Abydos and was likely a local kinglet of the Abydos Dynasty, Ryholt gives the list of kings of the 16th dynasty as shown in the table below
Second Intermediate Period
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The political situation in the Second Intermediate Period of Egypt (circa 1650 B.C.E. — circa 1550 B.C.E.) Thebes was briefly conquered by the Hyksos circa 1580 B.C.E.
Second Intermediate Period
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Thebes (Luxor Temple pictured) was the capital of many of the Dynasty XVI pharaohs.
14.
Ancient Egyptian
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Ancient Egypt was a civilization of ancient Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. It is one of six civilizations to arise independently, Egyptian civilization followed prehistoric Egypt and coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh Narmer. In the aftermath of Alexander the Greats death, one of his generals, Ptolemy Soter and this Greek Ptolemaic Kingdom ruled Egypt until 30 BC, when, under Cleopatra, it fell to the Roman Empire and became a Roman province. The success of ancient Egyptian civilization came partly from its ability to adapt to the conditions of the Nile River valley for agriculture, the predictable flooding and controlled irrigation of the fertile valley produced surplus crops, which supported a more dense population, and social development and culture. Its art and architecture were widely copied, and its antiquities carried off to far corners of the world and its monumental ruins have inspired the imaginations of travelers and writers for centuries. The Nile has been the lifeline of its region for much of human history, nomadic modern human hunter-gatherers began living in the Nile valley through the end of the Middle Pleistocene some 120,000 years ago. By the late Paleolithic period, the climate of Northern Africa became increasingly hot and dry. In Predynastic and Early Dynastic times, the Egyptian climate was less arid than it is today. Large regions of Egypt were covered in treed savanna and traversed by herds of grazing ungulates, foliage and fauna were far more prolific in all environs and the Nile region supported large populations of waterfowl. Hunting would have been common for Egyptians, and this is also the period when many animals were first domesticated. The largest of these cultures in upper Egypt was the Badari, which probably originated in the Western Desert, it was known for its high quality ceramics, stone tools. The Badari was followed by the Amratian and Gerzeh cultures, which brought a number of technological improvements, as early as the Naqada I Period, predynastic Egyptians imported obsidian from Ethiopia, used to shape blades and other objects from flakes. In Naqada II times, early evidence exists of contact with the Near East, particularly Canaan, establishing a power center at Hierakonpolis, and later at Abydos, Naqada III leaders expanded their control of Egypt northwards along the Nile. They also traded with Nubia to the south, the oases of the desert to the west. Royal Nubian burials at Qustul produced artifacts bearing the oldest-known examples of Egyptian dynastic symbols, such as the crown of Egypt. They also developed a ceramic glaze known as faience, which was used well into the Roman Period to decorate cups, amulets, and figurines. During the last predynastic phase, the Naqada culture began using written symbols that eventually were developed into a system of hieroglyphs for writing the ancient Egyptian language. The Early Dynastic Period was approximately contemporary to the early Sumerian-Akkadian civilisation of Mesopotamia, the third-century BC Egyptian priest Manetho grouped the long line of pharaohs from Menes to his own time into 30 dynasties, a system still used today
Ancient Egyptian
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The Great Sphinx and the pyramids of Giza are among the most recognizable symbols of the civilization of ancient Egypt.
Ancient Egyptian
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A typical Naqada II jar decorated with gazelles. (Predynastic Period)
Ancient Egyptian
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The Narmer Palette depicts the unification of the Two Lands.
15.
Egyptian fraction
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An Egyptian fraction is a finite sum of distinct unit fractions, such as 12 +13 +116. That is, each fraction in the expression has an equal to 1 and a denominator that is a positive integer. The value of an expression of type is a positive rational number a/b. Every positive rational number can be represented by an Egyptian fraction, in modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern theory and recreational mathematics. Beyond their historical use, Egyptian fractions have some advantages over other representations of fractional numbers. For instance, Egyptian fractions can help in dividing a number of objects into equal shares, for more information on this subject, see Egyptian numerals, Eye of Horus, and Egyptian mathematics. Egyptian fraction notation was developed in the Middle Kingdom of Egypt, five early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner Papyrus, the Kahun Papyrus and the Akhmim Wooden Tablet. A later text, the Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions, the Rhind papyrus was written by Ahmes and dates from the Second Intermediate Period, it includes a table of Egyptian fraction expansions for rational numbers 2/n, as well as 84 word problems. Solutions to each problem were written out in scribal shorthand, with the answers of all 84 problems being expressed in Egyptian fraction notation. 2/n tables similar to the one on the Rhind papyrus also appear on some of the other texts, however, as the Kahun Papyrus shows, vulgar fractions were also used by scribes within their calculations. To write the unit used in their Egyptian fraction notation, in hieroglyph script. Similarly in hieratic script they drew a line over the letter representing the number. For example, The Egyptians had special symbols for 1/2, 2/3, the remaining number after subtracting one of these special fractions was written using as a sum of distinct unit fractions according to the usual Egyptian fraction notation. These have been called Horus-Eye fractions after a theory that they were based on the parts of the Eye of Horus symbol, the unit fraction 1/n is expressed as n, and the fraction 2/n is expressed as n, and the plus sign “＋” is omitted. For example, 2/3 = 1/2 + 1/6 is expressed as 3 =26, modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form 2/n in the Rhind papyrus, although these expansions can generally be described as algebraic identities, the methods used by the Egyptians may not correspond directly to these identities. This method is available for not only odd prime denominators but also all odd denominators, for larger prime denominators, an expansion of the form 2/p = 1/A + 2A − p/Ap was used, where A is a number with many divisors between p/2 and p
Egyptian fraction
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Eye of Horus
16.
New Kingdom
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Radiocarbon dating places the exact beginning of the New Kingdom between 1570–1544 BC. The New Kingdom followed the Second Intermediate Period and was succeeded by the Third Intermediate Period and it was Egypt’s most prosperous time and marked the peak of its power. The later part of period, under the Nineteenth and Twentieth Dynasties is also known as the Ramesside period. It is named after the pharaohs that took the name of Ramesses I. Egyptian armies fought Hittite armies for control of modern-day Syria, the Eighteenth Dynasty contained some of Egypts most famous Pharaohs, including Ahmose I, Hatshepsut, Thutmose III, Amenhotep III, Akhenaten and Tutankhamun. Queen Hatshepsut concentrated on expanding Egypts external trade by sending an expedition to the land of Punt. Thutmose III expanded Egypts army and wielded it with success to consolidate the empire created by his predecessors. This resulted in a peak in Egypts power and wealth during the reign of Amenhotep III, during the reign of Thutmose III, Pharaoh, originally referring to the kings palace, became a form of address for the person who was king. Akhenatens religious fervor is cited as the reason why he was written out of Egyptian history. Under his reign, in the 14th century BC, Egyptian art flourished and attained a level of realism. Towards the end of the 18th Dynasty, the situation had changed radically, Ramesses II sought to recover territories in the Levant that had been held by the 18th Dynasty. His campaigns of reconquest culminated in the Battle of Kadesh, where he led Egyptian armies against those of the Hittite king Muwatalli II. Ramesses was caught in historys first recorded military ambush, although he was able to rally his troops, the outcome of the battle was undecided with both sides claiming victory at their home front, ultimately resulting in a peace treaty between the two nations. The last great pharaoh from the New Kingdom is widely considered to be Ramesses III, in the eighth year of his reign the Sea Peoples invaded Egypt by land and sea. Ramesses III defeated them in two great land and sea battles and he incorporated them as subject peoples and settled them in Southern Canaan although there is evidence that they forced their way into Canaan. Their presence in Canaan may have contributed to the formation of new states, such as Philistia and he was also compelled to fight invading Libyan tribesmen in two major campaigns in Egypts Western Delta in his sixth year and eleventh year respectively. The heavy cost of this warfare slowly drained Egypts treasury and contributed to the decline of the Egyptian Empire in Asia. Something in the air prevented much sunlight from reaching the ground, one proposed cause is the Hekla 3 eruption of the Hekla volcano in Iceland but the dating of this remains disputed
New Kingdom
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New Kingdom at its maximum territorial extent in the 15th century BC.
New Kingdom
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Relief of a Nobleman, ca. 1295-1070 B.C.E. Brooklyn Museum
New Kingdom
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Queen Ahmose-Nefertari
17.
Papyrus Anastasi I
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Papyrus Anastasi I is an ancient Egyptian papyrus containing a satirical text used for the training of scribes during the Ramesside Period. One scribe, a scribe, Hori, writes to his fellow scribe, Amenemope, in such a way as to ridicule the irresponsible. The papyrus was purchased from Giovanni Anastasi in 1839. In a long section Hori discusses the geography of the Mediterranean coast as far north as the Lebanon and this papyrus is important to historians and Bible scholars above all for the information it supplies about towns in Syria and Canaan during the New Kingdom. The border lands of Egypts province of Caanan with Kadesh are defined in the Gardiner translation p.19, Hori goes on to show that Amenemope is not skilled in the role of a maher. The word maher is discussed in Gardiners Egyptian Grammar under Messenger, Hori then relates what appears to be an actual anecdote for which Amenemope is apparently infamous. It contains a lot of detail reflecting discreditably on his name and comparing him to Qedjerdi and this touches on the concept of gossip amongst the scribes for which the idiom is Much in the mouths of. Amenemope gets ambushed in a pass, possibly at a battle in the campaigns against Kadesh which go on throughout the 18th and 19th dynasties. Hori makes clear that these routes that should be well known to the scribes operating as mahers or messengers. Illustrations from the battle of Kadesh provide an excellent background for Horis tale showing the form of the chariots, Amenemopes lack of experience causes him not to be apprehensive when he should be and then panicking when he should remain calm. Amenmopes chariot is on a mountain pass above a ravine in which some four or five cubit tall Shashu are lurking. The road is rough and tangled vegetation and the Shashu look dangerous. Amenmope wrecks his rig and has to cut it loose with a knife from some trees it is tangled up in and he cuts himself trying to get the traces free of the branches. His self abuse is much in the mouths of his followers, the scribe Hori says
Papyrus Anastasi I
18.
Ramesses III
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Usimare Ramesses III was the second Pharaoh of the Twentieth Dynasty and is considered to be the last New Kingdom king to wield any substantial authority over Egypt. His long reign saw the decline of Egyptian political and economic power, linked to a series of invasions, Ramesses III was the son of Setnakhte and Queen Tiy-Merenese. He was probably murdered by an assassin in a conspiracy led by one of his wives, Tiye. Ramesses two main names transliterate as wsr-mꜢʿt-rʿ–mry-ỉmn rʿ-ms-s–ḥḳꜢ-ỉwnw and they are normally realised as Usermaatre-Meryamun Rameses-Heqaiunu, meaning The Maat of Ra is strong, Beloved of Amun, Born of Ra, Ruler of Heliopolis. Ramesses III is believed to have reigned from March 1186 to April 1155 BC and this is based on his known accession date of I Shemu day 26 and his death on Year 32 III Shemu day 15, for a reign of 31 years,1 month and 19 days. Alternate dates for his reign are 1187 to 1156 BC, in Year 8 of his reign, the Sea Peoples, including Peleset, Denyen, Shardana, Meshwesh of the sea, and Tjekker, invaded Egypt by land and sea. Ramesses III defeated them in two great land and sea battles, although the Egyptians had a reputation as poor seamen, they fought tenaciously. Rameses lined the shores with ranks of archers who kept up a continuous volley of arrows into the ships when they attempted to land on the banks of the Nile. Then, the Egyptian navy attacked using grappling hooks to haul in the enemy ships, in the brutal hand-to-hand fighting which ensued, the Sea People were utterly defeated. The Harris Papyrus states, As for those who reached my frontier, their seed is not, their heart and their presence in Canaan may have contributed to the formation of new states in this region such as Philistia after the collapse of the Egyptian Empire in Asia. Ramesses III was also compelled to fight invading Libyan tribesmen in two campaigns in Egypts Western Delta in his Year 5 and Year 11 respectively. The heavy cost of these battles slowly exhausted Egypts treasury and contributed to the decline of the Egyptian Empire in Asia. Something in the air prevented much sunlight from reaching the ground, the result in Egypt was a substantial increase in grain prices under the later reigns of Ramesses VI–VII, whereas the prices for fowl and slaves remained constant. Thus the cooldown affected Ramesses IIIs final years and impaired his ability to provide a constant supply of rations to the workmen of the Deir el-Medina community. No temple in the heart of Egypt prior to Ramesses reign had ever needed to be protected in such a manner. Thanks to the discovery of papyrus trial transcripts, it is now known there was a plot against his life as a result of a royal harem conspiracy during a celebration at Medinet Habu. The conspiracy was instigated by Tiye, one of his three wives, over whose son would inherit the throne. Tytis son, Ramesses Amonhirkhopshef, was the eldest and the chosen by Ramesses III in preference to Tiyes son Pentaweret
Ramesses III
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Relief from the sanctuary of the Temple of Khonsu at Karnak depicting Ramesses III
Ramesses III
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Ramses III offering incense, wall painting in KV11.
Ramesses III
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Osirid statues of Ramses III at his temple in Karnak (in the first courtyard of the Great Temple of Amun).
Ramesses III
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Red granite sarcophagus of Ramesses III (Louvre)
19.
Deir el-Medina
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During the Christian era, the temple of Hathor was converted into a church from which the Arabic name Deir el-Medina is derived. At the time when the press was concentrating on Howard Carters discovery of the Tomb of Tutankhamun in 1922. This work has resulted in one of the most thoroughly documented accounts of community life in the ancient world that spans almost four hundred years. There is no site in which the organisation, social interactions. The site is located on the west bank of the Nile, the village may have been built apart from the wider population in order to preserve secrecy in view of sensitive nature of the work carried out in the tombs. A significant find of papyri was made in the 1840s in the vicinity of the village, the archaeological site was first seriously excavated by Ernesto Schiaparelli between 1905–1909 which uncovered large amounts of ostraca. A French team directed by Bernard Bruyère excavated the site, including village, dump and cemetery. Unfortunately through lack of control it is now thought that half of the papyri recovered was removed without the knowledge or authorization of the team director. Around five thousand ostraca of assorted works of commerce and literature were found in a close to the village. Jaroslav Černý, who was part of Bruyères team, went on to study the village for almost fifty years until his death in 1970 and was able to name, the peak overlooking the village was renamed Mont Cernabru in recognition of Černý and Bruyères work on the village. The main road through the village may have been covered to shelter the villagers from the intense glare, the size of the habitations varied, with an average floor space of 70 m2, but the same construction methods were used throughout the village. Walls were made of mudbrick, built on top of stone foundations, mud was applied to the walls which were then painted white on the external surfaces with some of the inner surfaces whitewashed up to a height of around one metre. A wooden front door might have carried the occupants name, houses consisted of four to five rooms comprising an entrance, main room, two smaller rooms, kitchen with cellar and staircase leading to the roof. The full glare of the sun was avoided by situating the windows high up on the walls, the main room contained a mudbrick platform with steps which may have been used as a shrine or a birthing bed. Nearly all houses contained niches for statues and small altars, the tombs built by the community for their own use include small rock-cut chapels and substructures adorned with small pyramids. 1110–1080 BCE during the reign of Ramesses XI due to increasing threats of Libyan raids, the Ptolemies later built a temple to Hathor on the site of an ancient shrine dedicated to her. The surviving texts record the events of life rather than major historical incidents. Personal letters reveal much about the relations and family life of the villagers
Deir el-Medina
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Ruins of Deir el-Medina. A UNESCO World Heritage Site
Deir el-Medina
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Statue from the intact Tomb of Kha and Merit (Turin Museum)
Deir el-Medina
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Ra slays Apep (tomb scene in Deir el-Medina)
Deir el-Medina
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A worker's tomb
20.
Egyptian numerals
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The system of ancient Egyptian numerals was used in Ancient Egypt around 3000 BC until the early first millennium AD. It was a system of numeration based on the scale of ten, often rounded off to the power, written in hieroglyphs. The hieratic form of numerals stressed an exact finite series notation, the Ancient Egyptian system used bases of ten. The following hieroglyphics were used to denote powers of ten, Multiples of these values were expressed by repeating the symbol as many times as needed, for instance, a stone carving from Karnak shows the number 4622 as Egyptian hieroglyphs could be written in both directions. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. Rational numbers could also be expressed, but only as sums of fractions, i. e. sums of reciprocals of positive integers, except for 2⁄3. The hieroglyph indicating a fraction looked like a mouth, which meant part, Fractions were written with this fractional solidus, i. e. the numerator 1, and the positive denominator below. As with most modern day languages, the ancient Egyptian language could also write out numerals as words phonetically, just like one can write thirty instead of 30 in English. The word, for instance, was written as while the numeral was This was, however, uncommon for most numbers other than one, instances of numerals written in hieratic can be found as far back as the Early Dynastic Period. The Old Kingdom Abusir Papyri are an important corpus of texts that utilize hieratic numerals. A large number like 9999 could thus be written only four signs—combining the signs for 9000,900,90. Boyer saw the new hieratic numerals as ciphered, mapping one number onto one Egyptian letter for the first time in human history, greeks adopted the new system, mapping their counting numbers onto two of their alphabets, the Doric and Ionian. In the oldest hieratic texts the individual numerals were written in a ciphered relationship to the Egyptian alphabet. But during the Old Kingdom a series of standardized writings had developed for sign-groups containing more than one numeral, however, repetition of the same numeral for each place-value was not allowed in the hieratic script. As the hieratic writing system developed over time, these sign-groups were further simplified for quick writing, two famous mathematical papyri using hieratic script are the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus. The majuscule letter A in some reconstructed forms means that the quality of that remains uncertain, Ancient Egypt Egyptian language Egyptian mathematics Allen. Middle Egyptian, An Introduction to the Language and Culture of Hieroglyphs, Egyptian Grammar, Being an Introduction to the Study of Hieroglyphs. Hieratische Paläographie, Die aegyptische Buchschrift in ihrer Entwicklung von der Fünften Dynastie bis zur römischen Kaiserzeit, Introduction Egyptian numerals Numbers and dates http, //egyptianmath. blogspot. com
Egyptian numerals
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Numeral systems
21.
Egyptian hieroglyphs
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Egyptian hieroglyphs were the formal writing system used in Ancient Egypt. It combined logographic, syllabic and alphabetic elements, with a total of some 1,000 distinct characters, cursive hieroglyphs were used for religious literature on papyrus and wood. The later hieratic and demotic Egyptian scripts are derived from hieroglyphic writing, the writing system continued to be used throughout the Late Period, as well as the Persian and Ptolemaic periods. Late survivals of hieroglyphic use are found well into the Roman period, with the closing of pagan temples in the 5th century, knowledge of hieroglyphic writing was lost, and the script remained undeciphered throughout the medieval and early modern period. The decipherment of hieroglyphs would only be solved in the 1820s by Jean-François Champollion, the word hieroglyph comes from the Greek adjective ἱερογλυφικός, a compound of ἱερός and γλύφω, supposedly a calque of an Egyptian phrase mdw·w-nṯr gods words. The glyphs themselves were called τὰ ἱερογλυφικὰ γράμματα the sacred engraved letters, the word hieroglyph has become a noun in English, standing for an individual hieroglyphic character. As used in the sentence, the word hieroglyphic is an adjective. Hieroglyphs emerged from the artistic traditions of Egypt. For example, symbols on Gerzean pottery from c.4000 BC have been argued to resemble hieroglyphic writing, proto-hieroglyphic symbol systems develop in the second half of the 4th millennium BC, such as the clay labels of a Predynastic ruler called Scorpion I recovered at Abydos in 1998. The first full sentence written in hieroglyphs so far discovered was found on a seal found in the tomb of Seth-Peribsen at Umm el-Qaab. There are around 800 hieroglyphs dating back to the Old Kingdom, Middle Kingdom, by the Greco-Roman period, there are more than 5,000. However, given the lack of evidence, no definitive determination has been made as to the origin of hieroglyphics in ancient Egypt. Since the 1990s, and discoveries such as the Abydos glyphs, as writing developed and became more widespread among the Egyptian people, simplified glyph forms developed, resulting in the hieratic and demotic scripts. These variants were more suited than hieroglyphs for use on papyrus. Hieroglyphic writing was not, however, eclipsed, but existed alongside the other forms, especially in monumental, the Rosetta Stone contains three parallel scripts – hieroglyphic, demotic, and Greek. Hieroglyphs continued to be used under Persian rule, and after Alexander the Greats conquest of Egypt, during the ensuing Ptolemaic and Roman periods. It appears that the quality of comments from Greek and Roman writers about hieroglyphs came about, at least in part. Some believed that hieroglyphs may have functioned as a way to distinguish true Egyptians from some of the foreign conquerors, another reason may be the refusal to tackle a foreign culture on its own terms, which characterized Greco-Roman approaches to Egyptian culture generally
Egyptian hieroglyphs
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A section of the Papyrus of Ani showing cursive hieroglyphs.
Egyptian hieroglyphs
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Hieroglyphs on a funerary stela in Manchester Museum
Egyptian hieroglyphs
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The Rosetta Stone in the British Museum
Egyptian hieroglyphs
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Hieroglyphs typical of the Graeco-Roman period
22.
Hobble (device)
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A hobble or spancel is a device that prevents or limits the locomotion of a human or an animal, by tethering one or more legs. Although hobbles are most commonly used on horses, they are used also on other animals. On dogs, they are used especially during force-fetch training to limit the movement of a dogs front paws when training it to stay still and they are made from leather, rope, or synthetic materials such as nylon or Neoprene. There are various designs for breeding, casting, and mounting horses, western-style horse hobbles are tied around the pasterns or cannon bones of the horses front legs. They comprise three basic types, The vaquero or braided hobble, which is often of a quite fancy plaiting and lighter than other varieties, and is therefore only suitable for short term use. The figure eight hobble or Queensland Utility Strap, a style of hobble that stockmen wear as a belt and can be used neck strap, lunch-time hobble. This hobble is made with three pieces of leather and two rings, plus a buckle fastening, the twist hobble, made of soft leather or rope, with a twist between the horses legs. The above patterns are unsuitable for training as they can tighten around a leg, hobbles also allow a horse to graze and move short and slow distances, yet prevent the horse from running off too far. This is handy at night if the rider has to get some sleep, hobble training a horse is a form of sacking out and desensitizing a horse to accept restraints on its legs. This helps a horse accept pressure on its legs in case it becomes entangled in barbed wire or fencing. A hobble trained horse is likely to pull, struggle. Breeding or service hobbles usually fasten around a mares hocks, pass between her front legs to a neck strap and they are used to protect a stallion from kicks. Casting hobbles are the same as the above, but with another rope or strap attached to the hind foot. When these straps or ropes are pulled up together, the horse will fall, cattle hobbles are a strong strap with a metal keeper in the middle and a buckle at the end. They are used on the legs for a short period when capturing feral cattle. Drovers’ or grazing hobbles have a buckle on a wide double redhide or chrome leather strap and they are placed around the pasterns. Hind leg pull up strap passes from a strap and around a hind pastern to draw up a hind foot for shoeing or treatment. Hopples are a piece of equipment used by Standardbred pacers to help the horse maintain its pacing gait, mounting hobbles are knee hobbles that are made with a quick release, on a lead that passes to the rider
Hobble (device)
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A hobbled donkey in Sardinia
Hobble (device)
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Hind leg pull up strap
Hobble (device)
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Drovers' and cattle hobbles
Hobble (device)
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Pacing hopples
23.
Louvre
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The Louvre or the Louvre Museum is the worlds largest museum and a historic monument in Paris, France. A central landmark of the city, it is located on the Right Bank of the Seine in the citys 1st arrondissement, approximately 38,000 objects from prehistory to the 21st century are exhibited over an area of 72,735 square metres. The Louvre is the second most visited museum after the Palace Museum in China. The museum is housed in the Louvre Palace, originally built as a fortress in the late 12th century under Philip II, remnants of the fortress are visible in the basement of the museum. Due to the expansion of the city, the fortress eventually lost its defensive function and. The building was extended many times to form the present Louvre Palace, in 1692, the building was occupied by the Académie des Inscriptions et Belles Lettres and the Académie Royale de Peinture et de Sculpture, which in 1699 held the first of a series of salons. The Académie remained at the Louvre for 100 years, during the French Revolution, the National Assembly decreed that the Louvre should be used as a museum to display the nations masterpieces. The museum opened on 10 August 1793 with an exhibition of 537 paintings, because of structural problems with the building, the museum was closed in 1796 until 1801. The collection was increased under Napoleon and the museum renamed Musée Napoléon, the collection was further increased during the reigns of Louis XVIII and Charles X, and during the Second French Empire the museum gained 20,000 pieces. Holdings have grown steadily through donations and bequests since the Third Republic, whether this was the first building on that spot is not known, it is possible that Philip modified an existing tower. According to the authoritative Grand Larousse encyclopédique, the name derives from an association with wolf hunting den, in the 7th century, St. Fare, an abbess in Meaux, left part of her Villa called Luvra situated in the region of Paris to a monastery. This territory probably did not correspond exactly to the modern site, the Louvre Palace was altered frequently throughout the Middle Ages. In the 14th century, Charles V converted the building into a residence and in 1546, Francis acquired what would become the nucleus of the Louvres holdings, his acquisitions including Leonardo da Vincis Mona Lisa. After Louis XIV chose Versailles as his residence in 1682, constructions slowed, however, on 14 October 1750, Louis XV agreed and sanctioned a display of 96 pieces from the royal collection, mounted in the Galerie royale de peinture of the Luxembourg Palace. Under Louis XVI, the museum idea became policy. The comte dAngiviller broadened the collection and in 1776 proposed conversion of the Grande Galerie of the Louvre – which contained maps – into the French Museum, many proposals were offered for the Louvres renovation into a museum, however, none was agreed on. Hence the museum remained incomplete until the French Revolution, during the French Revolution the Louvre was transformed into a public museum. In May 1791, the Assembly declared that the Louvre would be a place for bringing together monuments of all the sciences, on 10 August 1792, Louis XVI was imprisoned and the royal collection in the Louvre became national property
Louvre
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the Richelieu wing (2005)
Louvre
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The only portion of the medieval Louvre still visible
Louvre
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Antonio Canova 's Psyche Revived by Cupid's Kiss was commissioned in 1787, donated in 1824.
Louvre
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The Venus de Milo was added to the Louvre's collection during the reign of Louis XVIII.
24.
Predynastic Egypt
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This Predynastic era is traditionally equivalent to the final part of the Neolithic period beginning c.6000 BC and corresponds to the Naqada III period. The Predynastic period is divided into cultural periods, each named after the place where a certain type of Egyptian settlement was first discovered. The Late Paleolithic in Egypt started around 30,000 BC, the Nazlet Khater skeleton was found in 1980 and dated in 1982 from nine samples ranging between 35,100 and 30,360 years. This specimen is the only complete human skeleton from the earliest Late Stone Age in Africa. Excavation of the Nile has exposed early stone tools, the earliest of these lithic industries were located within the 100-foot terrace, and were Chellean, primitive Acheulean and an Egyptian form of the Clactonian. Within the 50-foot terrace was developed Acheulean, originally reported as Early Mousterian but since changed to Levalloisean, other implements were located in the 30-foot terrace. The 15- and 10-foot terraces saw a more developed version of the Levalloisean, finally, tools of the Egyptian Sebilian technology and an Egyptian version of the Aterian technology were also located. Some of the oldest known buildings were discovered in Egypt by archaeologist Waldemar Chmielewski along the border near Wadi Halfa. They were mobile structures—easily disassembled, moved, and reassembled—providing hunter-gatherers with semi-permanent habitation, Aterian tool-making reached Egypt c.40,000 BC. The Khormusan industry in Egypt began between 40,000 and 30,000 BC, khormusans developed advanced tools not only from stone but also from animal bones and hematite. They also developed small arrow heads resembling those of Native Americans, the end of the Khormusan industry came around 16,000 B. C. with the appearance of other cultures in the region, including the Gemaian. The Halfan culture flourished along the Nile Valley of Egypt and Nubia between 18,000 and 15,000 BC, though one Halfan site dates to before 24,000 BC, people survived on a diet of large herd animals and the Khormusan tradition of fishing. Greater concentrations of artifacts indicate that they were not bound to seasonal wandering and they are viewed as the parent culture of the Ibero-Maurusian industry, which spread across the Sahara and into Spain. The Halfan culture was derived in turn from the Khormusan, which depended on specialized hunting, fishing, the primary material remains of this culture are stone tools, flakes, and a multitude of rock paintings. Qadan peoples developed sickles and grinding stones to aid in the collecting and processing of plant foods prior to consumption. However, there are no indications of the use of these tools after around 10,000 BC, in Egypt, analyses of pollen found at archaeological sites indicate that the Sebilian culture were gathering wheat and barley. It has been hypothesized that the sedentary lifestyle used by farmers led to increased warfare, continued expansion of the desert forced the early ancestors of the Egyptians to settle around the Nile more permanently and adopt a more sedentary lifestyle. The period from 9000 to 6000 BC has left little in the way of archaeological evidence
Predynastic Egypt
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Predynastic artifacts: clockwise from top left: a Bat figurine, a Naqada jar, an ivory figurine, cosmetic palette, a flint knife, and a diorite vase.
Predynastic Egypt
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Female Figure, ca. 3500-3400 B.C.E. Terracotta, painted, 11 1/2 x 5 1/2 x 2 1/4 in. (29.2 x 14 x 5.7 cm). Brooklyn Museum
Predynastic Egypt
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A typical Naqada II pot with ship theme
25.
Cattle count
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In Ancient Egypt, the cattle count was one of the two main means of evaluating the amount of taxes to be levied, the other one being the height of the annual inundation. A very important economic event, the count was controlled by high officials. In addition it served as a means of dating other events, the frequency of cattle counts varied through the history of Ancient Egypt, in the Old Kingdom it was most likely biennial, i. e. occurring every two years, and became more frequent subsequently. To perform the count, all cattles were rounded up. Following the count, the percentage of cattles to be taxed by the state would be calculated, the cattle count was performed in every nomes of Egypt. From the 2nd dynasty onwards, the count was connected with the Following of Horus which occurred every two years. The Shemsu Hor consisted of a journey by the king and his court throughout Egypt which facilitated the assessment and levying of taxes by the central administration. The cattle count is of importance to Egyptologists and historians. Thus these inscriptions are used to assess the minimum duration of the reign of the pharaoh and this last point being of paramount importance for correct datation of reign lengths, it is highly disputed up to this day. After this period, however, it was performed frequently and finally yearly. The first pharaoh during whose reign yearly cattles counts are known to have taken place with certainty is king Pepy I of the 6th dynasty and this does not exclude that the cattle count necessarily took place every second year before Pepi I. An example of conflicting evaluations for a reign duration via cattle count is the case of king Khufu, the highest known numbers of cattle counts under Khufu are found in workmens graffiti inside the relieving chambers of the Khufu pyramid. The ink inscription reports the 17th occasion of the cattle count, since the Palermo stone inscriptions hold that the cattle count was performed every second year during the 4th dynasty, it would prove that Khufu ruled at least 34 years. This calculation is rejected by several Egyptologists, because another ancient Egyptian source, at the opposite, the ancient Greek historian Herodotus claims that Khufu ruled for 50 years, which is now seen as an exaggeration
Cattle count
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Cattle count after a relief in Mastaba tomb G75 at Giza.
26.
Carl Richard Lepsius
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Karl or Carl Richard Lepsius was a pioneering Prussian Egyptologist and linguist and pioneer of modern archaeology. He was born in Naumburg an der Saale, Saxony, the son of Friedericke Glaser and Peter Carl Lepsius. Karl Richards grandfather was Johann August Lepsius, the mayor of Naumburg upon Saale and he studied Greek and Roman archaeology at the University of Leipzig, the George Augustus University of Göttingen, and the Frederick William University of Berlin. After the death of Champollion, Lepsius made a study of the French scholars Grammaire égyptienne. In that year, Lepsius travelled to Tuscany to meet with Ippolito Rosellini, in a series of letters to Rosellini, Lepsius expanded on Champollions explanation of the use of alphabetic signs in hieroglyphic writing, emphasizing that vowels were not written. In 1842, Lepsius was commissioned by King Frederich Wilhelm IV of Prussia to lead an expedition to Egypt, the Prussian expedition was modelled after the earlier Napoleonic mission, with surveyors, draftsmen, and other specialists. The mission reached Giza in November 1842 and spent six months making some of the first scientific studies of the pyramids of Giza, Abusir, Saqqara and they discovered 67 pyramids recorded in the pioneering Lepsius list of pyramids and more than 130 tombs of noblemen in the area. While at the Great Pyramid of Giza, Lepsius inscribed a graffito written in Egyptian hieroglyphs that honours Friedrich Wilhelm IV above the original entrance. In 1843 he visited Naqa and copied some of the inscriptions and representations of the temple standing there, afterwards they stopped at Coptos, the Sinai, and sites in the Egyptian Delta, such as Tanis, before returning to Europe in 1846. In 1866 Lepsius returned to Egypt, where he discovered the Decree of Canopus at Tanis, a closely related to the Rosetta Stone. Lepsius was president of the German Archaeological Institute in Rome from 1867–1880, and from 1873 until his death in 1884, the head of the Royal Library at Berlin. He was the editor of the Zeitschrift für ägyptische Sprache und Altertumskunde, a scientific journal for the new field of Egyptology. While at the helm, Lepsius commissioned typographer Ferdinand Theinhardt to cut the first hieroglyphic typeface, the so-called Theinhardt font. Much of his work is fundamental to the field, indeed, Lepsius even coined the phrase Totenbuch. He was also a leader in the field of African linguistics, on 5 July 1846, he married Elisabeth Klein, daughter of the composer Bernhard Klein and great-granddaughter of Friedrich Nicolai. They had six children, including the geologist and Rector of the Darmstadt University of Technology G.1842, das Todtenbuch der Ägypter nach dem hieroglyphischen Papyrus in Turin mit einem Vorworte zum ersten Male Herausgegeben. Translated into English 1853 Discoveries in Egypt, Ethiopia and the Peninsular of Sinai, das allgemeine linguistische Alphabet, Grundsätze der Übertragung fremder Schriftsysteme und bisher noch ungeschriebener Sprachen in europäische Buchstaben. Berlin, Verlag von Wilhelm Hertz 1856, ägyptische königsdynastie nebst einigen bemerkungen zu der XXVI
Carl Richard Lepsius
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Karl Richard Lepsius
Carl Richard Lepsius
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Plates of El-Lahun and Tura from Denkmäler aus Aegypten und Aethiopien.
27.
Rhind mathematical papyrus
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The Rhind Mathematical Papyrus is one of the best known examples of Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian and it dates to around 1550 BC. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus, the Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former. The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt and it was copied by the scribe Ahmes, from a now-lost text from the reign of king Amenemhat III. Written in the script, this Egyptian manuscript is 33 cm tall. The papyrus began to be transliterated and mathematically translated in the late 19th century, the mathematical translation aspect remains incomplete in several respects. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later historical note on its verso likely dating from the period of his successor, Khamudi. In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving Accurate reckoning for inquiring into things, the scribe Ahmose writes this copy. Several books and articles about the Rhind Mathematical Papyrus have been published, a more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute. The first part of the Rhind papyrus consists of reference tables, the problems start out with simple fractional expressions, followed by completion problems and more involved linear equations. The first part of the papyrus is taken up by the 2/n table, the fractions 2/n for odd n ranging from 3 to 101 are expressed as sums of unit fractions. For example,2 /15 =1 /10 +1 /30. The decomposition of 2/n into unit fractions is never more than 4 terms long as in for example 2 /101 =1 /101 +1 /202 +1 /303 +1 /606. This table is followed by a smaller, tiny table of fractional expressions for the numbers 1 through 9 divided by 10. Problems 1-7, 7B and 8-40 are concerned with arithmetic and elementary algebra, problems 1–6 compute divisions of a certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 = 7/4 and 1 + 2/3 + 1/3 =2 by different fractions, problems 21–23 are problems in completion, which in modern notation are simply subtraction problems. Problems 24–34 are ‘’aha’’ problems, these are linear equations, problem 32 for instance corresponds to solving x + 1/3 x + 1/4 x =2 for x. Problems 35–38 involve divisions of the heqat, which is an ancient Egyptian unit of volume, problems 39 and 40 compute the division of loaves and use arithmetic progressions
Rhind mathematical papyrus
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A portion of the Rhind Papyrus
Rhind mathematical papyrus
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Building
28.
Moscow mathematical papyrus
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Golenishchev bought the papyrus in 1892 or 1893 in Thebes. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, approximately 5½ m long and varying between 3.8 and 7.6 cm wide, its format was divided into 25 problems with solutions by the Soviet Orientalist Vasily Vasilievich Struve in 1930. It is a well-known mathematical papyrus along with the Rhind Mathematical Papyrus, the Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two. The problems in the Moscow Papyrus follow no particular order, the papyrus is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a frustum respectively, the remaining problems are more common in nature. Problems 2 and 3 are ships part problems, one of the problems calculates the length of a ships rudder and the other computes the length of a ships mast given that it is 1/3 + 1/5 of the length of a cedar log originally 30 cubits long. Aha problems involve finding unknown quantities if the sum of the quantity, the Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1,19, and 25 of the Moscow Papyrus are Aha problems, for instance problem 19 asks one to calculate a quantity taken 1 and ½ times and added to 4 to make 10. In other words, in mathematical notation one is asked to solve 3 /2 × x +4 =10 Most of the problems are pefsu problems,10 of the 25 problems. A pefsu measures the strength of the beer made from a heqat of grain pefsu = number loaves of bread number of heqats of grain A higher pefsu number means weaker bread or beer, the pefsu number is mentioned in many offering lists. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain. Calculate 1/2 of 5 heqat, the result will be 2 1/2 Take this 2 1/2 four times The result is 10, then you say to him, Behold. The beer quantity is found to be correct, problems 11 and 23 are Baku problems. These calculate the output of workers, problem 11 asks if someone brings in 100 logs measuring 5 by 5, then how many logs measuring 4 by 4 does this correspond to. Problem 23 finds the output of a given that he has to cut. Seven of the problems are geometry problems and range from computing areas of triangles, to finding the surface area of a hemisphere. The 10th problem of the Moscow Mathematical Papyrus asks for a calculation of the area of a hemisphere or possibly the area of a semi-cylinder. Below we assume that the problem refers to the area of a hemisphere, the text of problem 10 runs like this, Example of calculating a basket
Moscow mathematical papyrus
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14th problem of the Moscow Mathematical Papyrus (V. Struve, 1930)
Moscow mathematical papyrus
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The neutrality of this article is disputed. Relevant discussion may be found on the talk page. Please do not remove this message until the dispute is resolved. (July 2015)
29.
Linear equation
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A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. A simple example of an equation with only one variable, x, may be written in the form, ax + b =0, where a and b are constants. The constants may be numbers, parameters, or even functions of parameters. Linear equations can have one or more variables. An example of an equation with three variables, x, y, and z, is given by, ax + by + cz + d =0, where a, b, c, and d are constants and a, b. Linear equations occur frequently in most subareas of mathematics and especially in applied mathematics, an equation is linear if the sum of the exponents of the variables of each term is one. Equations with exponents greater than one are non-linear, an example of a non-linear equation of two variables is axy + b =0, where a and b are constants and a ≠0. It has two variables, x and y, and is non-linear because the sum of the exponents of the variables in the first term and this article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients, a linear equation in one unknown x may always be rewritten a x = b. If a ≠0, there is a solution x = b a. The origin of the name comes from the fact that the set of solutions of such an equation forms a straight line in the plane. Linear equations can be using the laws of elementary algebra into several different forms. These equations are referred to as the equations of the straight line. In what follows, x, y, t, and θ are variables, in the general form the linear equation is written as, A x + B y = C, where A and B are not both equal to zero. The equation is written so that A ≥0, by convention. The graph of the equation is a line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, if B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis, is C/B, and the slope of the line is −A/B. The general form is written as, a x + b y + c =0
Linear equation
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Graph sample of linear equations.
30.
Method of false position
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False position method and regula falsi method are two early, and still current, names for a very old method for solving an equation in one unknown. To solve an equation means to write, or determine the value of. Many equations, including most of the more complicated ones, can be solved only by numerical approximation. That consists of trial and error, in various values of the unknown quantity. That trial-and-error may be informed by an estimate for the solution. Basic procedure Terminology for this section By moving all of an equation’s terms to one side, we can get an equation that says, f =0 and that transforms the problem into one of finding the x-value at which f =0. That x-value is the equation’s solution, in this section, the symbol “y” will be used interchangeably with f when that improves brevity, clarity, and reduces clutter. Here, “y” means “y” means “f”, the expressions “y” and “f” will both be used here, and they mean the same thing. The symbol “y” is familiar, as the name for the vertical co-ordinate on a graph, often a function of “x”. Example Lets solve the equation x + x/4 =15 by false position and we get 4 + 4/4 =5, note 4 is not the solution. Lets now multiply with 3 on both sides to get 12 + 12/4 =15, obtaining the solution x =12, the example is problem 26 on the Rhind papyrus. A History of Mathematics, 3rd edition, by Victor J. Katz categorizes the problem as false position, many methods for the calculated-estimate are used. The oldest and simplest class of methods, and the class that contains the most reliable method, are the two-point bracketing methods. Those methods start with two x-values, initially found by trial-and-error, at which f has opposite signs, in other words, Two x-values such that, for one of them, f is positive, and for the other, f is negative. In that way, those two f values can be said to “bracket” zero, because they’re on opposite sides of zero, …a guarantee not available with such other methods as Newton’s method or the Secant method. When f is evaluated at a certain x-value, call it x1, resulting in f, that combination of x and y values is called a “data-point”, the data point. The two-point bracketing methods use, for each step, two such data points, from which to get a calculated estimate for the solution. F is then evaluated for that estimated x, to get a new point, from which to calculate a new
Method of false position
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The first two iterations of the false position method. The red curve shows the function f and the blue lines are the secants.
31.
Quadratic equation
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If a =0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the coefficient, the linear coefficient. Because the quadratic equation involves only one unknown, it is called univariate, solutions to problems equivalent to the quadratic equation were known as early as 2000 BC. A quadratic equation with real or complex coefficients has two solutions, called roots and these two solutions may or may not be distinct, and they may or may not be real. It may be possible to express a quadratic equation ax2 + bx + c =0 as a product =0. In some cases, it is possible, by inspection, to determine values of p, q, r. If the quadratic equation is written in the form, then the Zero Factor Property states that the quadratic equation is satisfied if px + q =0 or rx + s =0. Solving these two linear equations provides the roots of the quadratic, for most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. As an example, x2 + 5x +6 factors as, the more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where b =0 or c =0 and this means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection. The process of completing the square makes use of the identity x 2 +2 h x + h 2 =2. Starting with an equation in standard form, ax2 + bx + c =0 Divide each side by a. Subtract the constant term c/a from both sides, add the square of one-half of b/a, the coefficient of x, to both sides. This completes the square, converting the left side into a perfect square, write the left side as a square and simplify the right side if necessary. Produce two linear equations by equating the square root of the side with the positive and negative square roots of the right side. Completing the square can be used to derive a formula for solving quadratic equations. The mathematical proof will now be briefly summarized and it can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation,2 = b 2 −4 a c 4 a 2. Taking the square root of both sides, and isolating x, gives, x = − b ± b 2 −4 a c 2 a and these result in slightly different forms for the solution, but are otherwise equivalent
Quadratic equation
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The trajectory of the cliff jumper is parabolic because horizontal displacement is a linear function of time, while vertical displacement is a quadratic function of time. As a result the path follows quadratic equation, where and are horizontal and vertical components of the original velocity, a is gravity and h is original height.
Quadratic equation
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Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0)
32.
Babylonian mathematics
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Babylonian mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited, in respect of time they fall in two distinct groups, one from the Old Babylonian period, the other mainly Seleucid from the last three or four centuries BC. In respect of content there is any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for two millennia. In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, the majority of recovered clay tablets date from 1800 to 1600 BCE, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC7289 gives an approximation to 2 accurate to three significant sexagesimal digits, Babylonian mathematics is a range of numeric and more advanced mathematical practices in the ancient Near East, written in cuneiform script. Study has historically focused on the Old Babylonian period in the second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages. Babylonian mathematics is perhaps an unhelpful term since the earliest suggested origins date to the use of accounting devices, such as bullae and tokens, the Babylonian system of mathematics was sexagesimal numeral system. From this we derive the modern day usage of 60 seconds in a minute,60 minutes in an hour, the Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a highly composite number, having factors of 1,2,3,4,5,6,10,12,15,20,30,60. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, the ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period. Most clay tablets that describe Babylonian mathematics belong to the Old Babylonian, some clay tablets contain mathematical lists and tables, others contain problems and worked solutions. The Babylonians used pre-calculated tables to assist with arithmetic, for example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the squares of numbers up to 59 and the cubes of numbers up to 32. The Babylonians used the lists of squares together with the formulae a b =2 − a 2 − b 22 a b =2 −24 to simplify multiplication, the Babylonians did not have an algorithm for long division. Instead they based their method on the fact that a b = a ×1 b together with a table of reciprocals
Babylonian mathematics
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Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, 1 24 51 10, which is good to about six decimal digits. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296... The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888...
33.
Hellenistic mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
Hellenistic mathematics
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Statue of Euclid in the Oxford University Museum of Natural History
Hellenistic mathematics
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An illustration of Euclid 's proof of the Pythagorean Theorem
Hellenistic mathematics
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The Antikythera mechanism, an ancient mechanical calculator.
34.
Roman math
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The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and it greatly reduced the time needed to perform the basic operations of arithmetic using Roman numerals. But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more, above all, it has preserved the fact of the unattached counters so faithfully that we can discern this more clearly than if we possessed an actual counting board. What the Greeks called psephoi, the Romans called calculi, the Latin word calx means pebble or gravel stone, calculi are thus little stones. Both the Roman abacus and the Chinese suanpan have been used since ancient times, the rightmost two grooves were for fractional counting. The abacus was made of a metal plate where the ran in slots. The size was such that it could fit in a shirt pocket. The beads in the shorter grooves denote fives—five units, five tens. Computations are made by means of beads which would probably have been slid up and these latter two slots are for mixed-base math, a development unique to the Roman hand abacus described in following sections. The longer slot with five beads below the Ө position allowed for the counting of 1/12 of a unit called an uncia, making the abacus useful for Roman measures. The first column was either a single slot with 4 beads or 3 slots with one, one, in either case, three symbols were included beside the single slot version or one symbol per slot for the three slot version. Many measures were aggregated by twelfths, thus the Roman pound, consisted of 12 ounces. A measure of volume, congius, consisted of 12 heminae, the Roman foot, was 12 inches. The actus, the furrow length when plowing, was 120 pedes. There were however other measures in common use - for example the sextarius was two heminae, the as, the principal copper coin in Roman currency, was also divided into 12 unciae. Again, the abacus was ideally suited for counting currency, the first column was arranged either as a single slot with three different symbols or as three separate slots with one, one and two beads or counters respectively and a distinct symbol for each slot. It is most likely that the rightmost slot or slots were used to enumerate fractions of an uncia, the upper character in this slot is the character most closely resembling that used to denote a semuncia or 1/24. The name semuncia denotes 1/2 of an uncia or 1/24 of the base unit, likewise, the next character is that used to indicate a sicilicus or 1/48 of an As, which is 1/4 of an uncia
Roman math
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A reconstruction of a Roman hand abacus, made by the RGZ Museum in Mainz, 1977. The original is bronze and is held by the Bibliothèque nationale de France, in Paris.
35.
Islamic mathematics
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Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries, the study of algebra, the name of which is derived from the Arabic word meaning completion or reunion of broken parts, flourished during the Islamic golden age. Muhammad ibn Musa al-Khwarizmi, a scholar in the House of Wisdom in Baghdad, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi deals with ways to solve for the roots of first. He also introduces the method of reduction, and unlike Diophantus, Al-Khwarizmis algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the work of Diophantus, which was syncopated. The transition to symbolic algebra, where symbols are used, can be seen in the work of Ibn al-Banna al-Marrakushi. It is important to understand just how significant this new idea was and it was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a theory which allowed rational numbers, irrational numbers, geometrical magnitudes. It gave mathematics a whole new development path so much broader in concept to that which had existed before, another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. Several other mathematicians during this time expanded on the algebra of Al-Khwarizmi. Omar Khayyam, along with Sharaf al-Dīn al-Tūsī, found several solutions of the cubic equation, omar Khayyam found the general geometric solution of a cubic equation. Omar Khayyám wrote the Treatise on Demonstration of Problems of Algebra containing the solution of cubic or third-order equations. Khayyám obtained the solutions of equations by finding the intersection points of two conic sections. This method had used by the Greeks, but they did not generalize the method to cover all equations with positive roots. Sharaf al-Dīn al-Ṭūsī developed an approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. His surviving works give no indication of how he discovered his formulae for the maxima of these curves, various conjectures have been proposed to account for his discovery of them. The earliest implicit traces of mathematical induction can be found in Euclids proof that the number of primes is infinite, the first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique
Islamic mathematics
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A page from the The Compendious Book on Calculation by Completion and Balancing by Al-Khwarizmi.
Islamic mathematics
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Engraving of Abū Sahl al-Qūhī 's perfect compass to draw conic sections.
Islamic mathematics
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The theorem of Ibn Haytham.
36.
Egyptian hieroglyphics
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Egyptian hieroglyphs were the formal writing system used in Ancient Egypt. It combined logographic, syllabic and alphabetic elements, with a total of some 1,000 distinct characters, cursive hieroglyphs were used for religious literature on papyrus and wood. The later hieratic and demotic Egyptian scripts are derived from hieroglyphic writing, the writing system continued to be used throughout the Late Period, as well as the Persian and Ptolemaic periods. Late survivals of hieroglyphic use are found well into the Roman period, with the closing of pagan temples in the 5th century, knowledge of hieroglyphic writing was lost, and the script remained undeciphered throughout the medieval and early modern period. The decipherment of hieroglyphs would only be solved in the 1820s by Jean-François Champollion, the word hieroglyph comes from the Greek adjective ἱερογλυφικός, a compound of ἱερός and γλύφω, supposedly a calque of an Egyptian phrase mdw·w-nṯr gods words. The glyphs themselves were called τὰ ἱερογλυφικὰ γράμματα the sacred engraved letters, the word hieroglyph has become a noun in English, standing for an individual hieroglyphic character. As used in the sentence, the word hieroglyphic is an adjective. Hieroglyphs emerged from the artistic traditions of Egypt. For example, symbols on Gerzean pottery from c.4000 BC have been argued to resemble hieroglyphic writing, proto-hieroglyphic symbol systems develop in the second half of the 4th millennium BC, such as the clay labels of a Predynastic ruler called Scorpion I recovered at Abydos in 1998. The first full sentence written in hieroglyphs so far discovered was found on a seal found in the tomb of Seth-Peribsen at Umm el-Qaab. There are around 800 hieroglyphs dating back to the Old Kingdom, Middle Kingdom, by the Greco-Roman period, there are more than 5,000. However, given the lack of evidence, no definitive determination has been made as to the origin of hieroglyphics in ancient Egypt. Since the 1990s, and discoveries such as the Abydos glyphs, as writing developed and became more widespread among the Egyptian people, simplified glyph forms developed, resulting in the hieratic and demotic scripts. These variants were more suited than hieroglyphs for use on papyrus. Hieroglyphic writing was not, however, eclipsed, but existed alongside the other forms, especially in monumental, the Rosetta Stone contains three parallel scripts – hieroglyphic, demotic, and Greek. Hieroglyphs continued to be used under Persian rule, and after Alexander the Greats conquest of Egypt, during the ensuing Ptolemaic and Roman periods. It appears that the quality of comments from Greek and Roman writers about hieroglyphs came about, at least in part. Some believed that hieroglyphs may have functioned as a way to distinguish true Egyptians from some of the foreign conquerors, another reason may be the refusal to tackle a foreign culture on its own terms, which characterized Greco-Roman approaches to Egyptian culture generally
Egyptian hieroglyphics
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A section of the Papyrus of Ani showing cursive hieroglyphs.
Egyptian hieroglyphics
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Hieroglyphs on a funerary stela in Manchester Museum
Egyptian hieroglyphics
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The Rosetta Stone in the British Museum
Egyptian hieroglyphics
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Hieroglyphs typical of the Graeco-Roman period
37.
Ancient Egyptian technology
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Ancient Egyptian technology describes devices and technologies invented or used in Ancient Egypt. The Egyptians invented and used many simple machines, such as the ramp and they used rope trusses to stiffen the beam of ships. Egyptian paper, made from papyrus, and pottery were mass-produced and exported throughout the Mediterranean basin, the wheel was used for a number of purposes, but chariots only came into use after the Second Intermediate period. The Egyptians also played an important role in developing Mediterranean maritime technology including ships, significant advances in ancient Egypt during the dynastic period include astronomy, mathematics, and medicine. Their geometry was a outgrowth of surveying to preserve the layout and ownership of farmland. The 3,4,5 right triangle and other rules of thumb served to represent rectilinear structures, Egypt also was a center of alchemy research for much of the western world. The word paper comes from the Greek term for the ancient Egyptian writing material called papyrus, Papyrus was produced as early as 3000 BC in Egypt, and sold to ancient Greece and Rome. The establishment of the Library of Alexandria limited the supply of papyrus for others, as a result, according to the Roman historian Pliny, parchment was invented under the patronage of Eumenes II of Pergamon to build his rival library at Pergamon. This however is a myth, parchment had been in use in Anatolia, Egyptian hieroglyphs, a phonetic writing system, served as the basis for the Phoenician alphabet from which later alphabets were derived. With this ability, writing and record keeping, the Egyptians developed one of the —if not the— first decimal system, the city of Alexandria retained preeminence for its records and scrolls with its library. That ancient library was damaged by fire when it fell under Roman rule, with it, a huge amount of antique literature, history, and knowledge was lost. Some of the tools used in the construction of Egyptian housing included reeds. According to Lucas and Harris, “reeds were plastered with clay in order to out of heat. Other tools that were used were limestone, chiseled stones, wooden mallets, with these tools, ancient Egyptians were able to create more than just housing, but also sculptures of their gods, goddesses, pyramids, etc. Many temples from Ancient Egypt are not standing today, some are in ruin from wear and tear, while others have been lost entirely. The Egyptian structures are among the largest constructions ever conceived and built by humans and they constitute one of the most potent and enduring symbols of Ancient Egyptian civilization. Temples and tombs built by a famous for her projects, Hatshepsut, were massive. Pharaoh Tutankamuns rock-cut tomb in the Valley of the Kings was full of jewellery, in some late myths, Ptah was identified as the primordial mound and had called creation into being, he was considered the deity of craftsmen, and in particular, of stone-based crafts
Ancient Egyptian technology
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Ancient Egyptian depiction of women engaged in mechanical rope making, the first graphic evidence of the craft, shown in the two lower rows of the illustration
Ancient Egyptian technology
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A section of the Egyptian Book of the Dead, which is written and drawn on papyrus
Ancient Egyptian technology
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The Lighthouse of Alexandria on the island of Pharos.
Ancient Egyptian technology
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Giza Plateau, Cairo. Khafre's pyramid in the background
38.
Mathematics and architecture
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Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. In Ancient Egypt, Ancient Greece, India, and the Islamic world, buildings including pyramids, temples, mosques, palaces, in Islamic architecture, geometric shapes and geometric tiling patterns are used to decorate buildings, both inside and outside. Some Hindu temples have a structure where parts resemble the whole. In Chinese architecture, the tulou of Fujian province are circular, in the twenty-first century, mathematical ornamentation is again being used to cover public buildings. In the twentieth century, styles such as architecture and Deconstructivism explored different geometries to achieve desired effects. But, they argue, the two are connected, and have been since antiquity. A master builder at the top of his profession was given the title of architect or engineer, in the Renaissance, the quadrivium of arithmetic, geometry, music and astronomy became an extra syllabus expected of the Renaissance man such as Leon Battista Alberti. Similarly in England, Sir Christopher Wren, known today as an architect, was firstly a noted astronomer and they argue that architects have avoided looking to mathematics for inspiration in revivalist times. This would explain why in revivalist periods, such as the Gothic Revival in 19th century England, equally, they note that in reactionary times such as the Italian Mannerism of about 1520 to 1580, or the 17th century Baroque and Palladian movements, mathematics was barely consulted. In contrast, the revolutionary early 20th century movements such as Futurism and Constructivism actively rejected old ideas, embracing mathematics and leading to Modernist architecture. Towards the end of the 20th century, too, fractal geometry was quickly seized upon by architects, as was aperiodic tiling, to provide interesting, Architects use mathematics for several reasons, leaving aside the necessary use of mathematics in the engineering of buildings. Firstly, they use geometry because it defines the form of a building. Secondly, they use mathematics to design forms that are considered beautiful or harmonious, thirdly, they may use mathematical objects such as tessellations to decorate buildings. Fourthly, they may use mathematics in the form of computer modelling to meet environmental goals, the influential Ancient Roman architect Vitruvius argued that the design of a building such as a temple depends on two qualities, proportion and symmetria. Proportion ensures that part of a building relates harmoniously to every other part. Symmetria in Vitruviuss usage means something closer to the English term modularity than mirror symmetry, in his Basilica at Fano, he uses ratios of small integers, especially the triangular numbers to proportion the structure into modules. Thus the Basilicas width to length is 1,2, the aisle around it is as high as it is wide,1,1, the columns are five feet thick and fifty feet high,1,10. Vitruvius named three qualities required of architecture in his De architectura, c.15 B. C. firmness, usefulness and these can be used as categories for classifying the ways in which mathematics is used in architecture
Mathematics and architecture
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"The Gherkin", 30 St Mary Axe, London, completed 2003, is a parametrically designed solid of revolution.
Mathematics and architecture
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Kandariya Mahadeva Temple, Khajuraho, India, is an example of religious architecture with a fractal -like structure which has many parts that resemble the whole. c. 1030
Mathematics and architecture
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In the Renaissance, an architect like Leon Battista Alberti was expected to be knowledgeable in many disciplines, including arithmetic and geometry.
Mathematics and architecture
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Plan of a Greek house by Vitruvius
39.
Raymond Clare Archibald
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Raymond Clare Archibald was a prominent Canadian-American mathematician. He is known for his work as an historian of mathematics, his editorships of mathematical journals, Raymond Clare Archibald was born in South Branch, Stewiacke, Nova Scotia on 7 October 1875. He was the son of Abram Newcomb Archibald and Mary Mellish Archibald and he was the fourth cousin twice removed of the famous Canadian-American astronomer and mathematician Simon Newcomb. Archibald graduated in 1894 from Mount Allison College with B. A. degree in mathematics and teachers certificate in violin. After teaching mathematics and violin for a year at the Mount Allison Ladies’ College he went to Harvard where he received a B. A.1896 and a M. A. in 1897. He then traveled to Europe where he attended the University of Berlin during 1898 and his advisor was Karl Theodor Reye and title of his dissertation was The Cardioide and Some of its Related Curves. He returned to Canada in 1900 and taught mathematics and violin at the Mount Allison Ladies’ College until 1907, after a one-year appointment at Acadia University he accepted an invitation of join the mathematics department at Brown University. He stayed at Brown for the rest of his becoming a Professor Emeritus in 1943. While at Brown he created one of the finest mathematical libraries in the western hemisphere, Archibald returned to Mount Allison in 1954 to curate the Mary Mellish Archibald Memorial Library, the library he had founded in 1905 to honor his mother. At his death the library contained 23,000 volumes,2,700 records, Raymond Clare Archibald was a world-renowned historian of mathematics with a lifelong concern for the teaching of mathematics in secondary schools. This knowledge and an untiring energy he dedicated to the upbuilding of the library at Brown University. From modest beginnings he has developed this essential equipment of the investigator to a point where it has no superior, in completeness. Archibald received honorary degrees from the University of Padua, Mount Allison University and he contributed to over 20 different journals, mathematical, scientific, educational and literary. S. Government Printing Office,1917 Benjamin Peirce, 1809—1880, 1951—1960,1964 Who Was Who in America. Fiftieth Anniversary Report,1946 Jim Tattersall and Shawnee McMurran, Raymond Clare Archibald, A Euterpean Historian of Mathematics, New England Math J. v. ~36, Works by Raymond Clare Archibald at Project Gutenberg Works by or about Raymond Clare Archibald at Internet Archive
Raymond Clare Archibald
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Raymond Clare Archibald
40.
Bharath Sriraman
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Bharath Sriraman is a professor of mathematics at the University of Montana – Missoula with an adjunct appointment in the department of Central and Southwest-Asian Studies. Sriraman is the founder and editor-in-chief of The Mathematics Enthusiast, an independent open access journal hosted by University of Montana and he previously received the School Science and Mathematics Association Early Scholar Award in 2007. In 2016 he was the recipient of the University of Montana Distinguished Scholar Award Sriraman, the influence of Platonism on mathematics research and theological beliefs. 2, no.1, pp. 131–147 Sriraman. B, & Benesch, W. Consciousness and Science, theology and Science, vol.3, no.1, pp. 39–54. Theories of Mathematics Education, Seeking New Frontiers, berlin, New York, Springer Science+Business Media. Crossroads in the History of Mathematics and Mathematics Education, the interfaces of innovation in mathematics and the arts. ”International Handbook on Innovation Education Routledge, Taylor & Francis Chernoff, E. Sriraman, B. Springer Science+Business Media, Dodrecht, Netherlands, ISBN 978-94-007-7154-3 Ambrose, D, Sriraman, Sense Publishers, Rotterdam, Netherlands, ISBN 978-9-46209771-1 Huaman-Sumida, L. & Sriraman, B. Proceedings of The Second International Symposium of Mathematics and its Connections to the Arts and Sciences, Odense, Denmark 29-31, May 2007, University of Southern Denmark Press Sriraman, B. & Freiman, V. Interdisciplinarity for the 21st Century, Proceedings of Third International Symposium on Mathematics and its Connections to the Arts, english, L. Bartolini-Bussi, M. Lesh, R. Jones, G. Sriraman, B. The Handbook of International Research in Mathematics Education, Routledge, Taylor & Francis Sriraman, B. Goodchild, S. Palsdottir, G. et al, the First Sourcebook on Nordic Research in Mathematics Education. Cai, J. Lee, K. et al, The First Sourcebook on Asian Research in Mathematics Education, China, Korea, Japan, Singapore, information Age Publishing, Charlotte, NC.1782 pages Sriraman, B. The Elements of Creativity and Giftedness in Mathematics, Sense Publishers, The Netherlands Leikin, R. & Sriraman, B. Creativity and Giftedness, Interdisciplinary perspectives from mathematics and beyond. Springer International, Switzerland, ISBN 978-3-319-38838-0 Beghetto, R. &Sriraman, creative Contradictions in Education, Cross disciplinary paradoxes and perspectives, Springer International, Switzerland, ISBN 978-3-319-21924-0 Sriraman, B. International Perspectives on Social Justice in Mathematics Education, Monograph in The Montana Mathematics Monograph Series in Mathematics Education. Information Age Publishing, Charlotte, NC Ambrose, D. Sternberg, R. Sriraman, Taylor and Francis London, New York Ambrose, D. Sriraman, B. The Roeper School, A Model for Holistic Development of High Ability, Sense Publishers, Rotterdam, Netherlands, mistele, J. Sriraman, B, Mathematics Teacher Education in the Public Interest. Ernest, N. Critical Mathematics Education, Theory, Praxis, official website Bharath Sriraman at the Mathematics Genealogy Project
Bharath Sriraman
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Sriraman, Universidad Antonio Narino, 2015
41.
Ancient Egyptian architecture
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The core of the pyramids consisted of locally quarried stone, mudbricks, sand or gravel. For the casing stones were used that had to be transported from farther away, predominantly white limestone from Tura, Ancient Egyptian houses were made out of mud collected from the Nile river. It was placed in molds and left to dry in the hot sun to harden for use in construction, others are inaccessible, new buildings having been erected on ancient ones. Fortunately, the dry, hot climate of Egypt preserved some mud brick structures, examples include the village Deir al-Madinah, the Middle Kingdom town at Kahun, and the fortresses at Buhen and Mirgissa. Also, many temples and tombs have survived because they were built on high ground unaffected by the Nile flood and were constructed of stone, in a similar manner, the incised and flatly modeled surface adornment of the stone buildings may have derived from mud wall ornamentation. Exterior and interior walls, as well as the columns and piers, were covered with hieroglyphic and pictorial frescoes, many motifs of Egyptian ornamentation are symbolic, such as the scarab, or sacred beetle, the solar disk, and the vulture. Other common motifs include leaves, the papyrus plant. Hieroglyphs were inscribed for decorative purposes as well as to record historic events or spells, in addition, these pictorial frescoes and carvings allow us to understand how the Ancient Egyptians lived, statuses, wars that were fought and their beliefs. This was especially true when exploring the tombs of Ancient Egyptian officials in recent years, Ancient Egyptian temples were aligned with astronomically significant events, such as solstices and equinoxes, requiring precise measurements at the moment of the particular event. Measurements at the most significant temples may have been undertaken by the Pharaoh himself. The Giza Necropolis stands on the Giza Plateau, on the outskirts of Cairo and this complex of ancient monuments is located some 8 kilometers inland into the desert from the old town of Giza on the Nile, some 20 kilometers southwest of Cairo city center. The pyramids, which were built in the Fourth Dynasty, testify to the power of the pharaonic religion and they were built to serve both as grave sites and also as a way to make their names last forever. The size and simple design show the skill level of Egyptian design. The pyramid of Khafre is believed to have been completed around 2532 BC, Khafre ambitiously placed his pyramid next to his fathers. It is not as tall as his fathers pyramid but he was able to give it the impression of appearing taller by building it on a site with a foundation 33 feet higher than his fathers. Along with building his pyramid, Chefren commissioned the building of the giant Sphinx as guardian over his tomb, the face of a human, possibly a depiction of the pharaoh, on a lions body was seen as a symbol of divinity among the Greeks fifteen hundred years later. The Great Sphinx is carved out the bedrock and stands about 65 feet tall. Menkaures pyramid dates to circa 2490 BC and stands 213 feet high making it the smallest of the Great Pyramids, popular culture leads people to believe that Pyramids are highly confusing, with many tunnels within the pyramid to create confusion for grave robbers
Ancient Egyptian architecture
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The well preserved Temple of Horus at Edfu is an example of Egyptian architecture and architectural sculpture.
Ancient Egyptian architecture
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Drawings of the types of the architectural capitals specific for the Ancient Egyptian civilization.
Ancient Egyptian architecture
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The Pyramids of Giza
Ancient Egyptian architecture
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The hypostyle hall of Karnak Temple
42.
Egyptian Revival architecture
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Egyptian revival is an architectural style that uses the motifs and imagery of ancient Egypt. It is attributed generally to the awareness of ancient Egyptian monuments generated by Napoleons conquest of Egypt. Napoleon took an expedition with him to Egypt. Publication of the work, the Description de lÉgypte, began in 1809 and was published as a series through 1826. However, works of art and architecture in the Egyptian style had made or built occasionally on the European continent. The most important example is probably Gian Lorenzo Berninis obelisk in the Piazza Navona in Rome, others may be found at Belan, County Kildare, and Dangan, County Meath. The Casteltown Folly in County Kildare is probably the best known, Egyptian buildings had also been built as garden follies. The most elaborate was probably the one built by Frederick I and it included an Egyptian bridge across which guests walked to reach an island with an elaborate Egyptian-influenced bath house. Designed by the court architect, Jean Baptiste Kleber, the building had a billiards room. New after the Napoleonic invasion was a increase of the number of works of art. In France and Britain this was at least partially inspired by successful war campaigns undertaken by each country while in Egypt, according to David Brownlee, the 1798 Karlsruhe Synagogue, an early building by the influential Friedrich Weinbrenner was the first large Egyptian building to be erected since antiquity. According to Diana Muir Appelbaum, it was the first public building in the Egyptian revival style, a street or passage named the Place du Caire or Foire du Caire was built in Paris in 1798 on the former site of the convent of the Filles de la Charité. In 1828 a building with Egyptianizing detail including large Hathor heads, among the earliest monuments of the Egyptian revival in Paris is the Fontaine du Fellah in Paris, built in 1806. It was designed by François-Jean Bralle, a well-documented example, destroyed after Napoleon was deposed, was the monument to General Louis Desaix in the Place des Victoires was built in 1810. It featured a statue of the general and an obelisk. Another example of a still standing site of Egyptian Revival is the Egyptian Gate of Tsarskoe Selo, one of the first British buildings to show an Egyptian revival interior was the newspaper office of the Courier on the Strand in London. It was built in 1804 and featured a cavetto cornice and Egyptian-influenced columns with palmiform capitals, Egyptian revival architecture enjoyed considerable popularity in other countries as well. The most notable Egyptian structure in the United States was the Washington Monument, begun in 1848, the South African College in the then-British Cape Colony features an Egyptian building constructed in 1841, the Egyptian revival building of the Cape Town Hebrew Congregation is also still standing
Egyptian Revival architecture
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Egyptian Hall (1812) in London
Egyptian Revival architecture
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Foire du Caire building (1828) in Paris
Egyptian Revival architecture
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The Egyptian Gates (1827–30) in Tsarskoe Selo, St. Petersburg
Egyptian Revival architecture
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4th Precinct Police Station (1836) in New Orleans
43.
Art of ancient Egypt
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Ancient Egyptian art is the painting, sculpture, architecture and other arts produced by the civilization of ancient Egypt in the lower Nile Valley from about 3000 BC to 30 AD. Ancient Egyptian art reached a level in painting and sculpture. It was famously conservative, and Egyptian styles changed remarkably little over more than three thousand years, much of the surviving art comes from tombs and monuments and thus there is an emphasis on life after death and the preservation of knowledge of the past. Ancient Egyptian art included paintings, sculpture in wood, stone and ceramics, drawings on papyrus, faience, jewelry, ivories and it displays an extraordinarily vivid representation of the ancient Egyptians socioeconomic status and belief systems. This appears as early as the Narmer Palette from Dynasty I, other conventions make statues of males darker than females ones. Egyptian art uses hierarchical proportion, where the size of figures indicates their relative importance, symbolism can be observed throughout Egyptian art and played an important role in establishing a sense of order. The pharaohs regalia, for example, represented his power to maintain order, animals were also highly symbolic figures in Egyptian art. Not all Egyptian reliefs were painted, and less prestigious works in tombs, Stone surfaces were prepared by whitewash, or if rough, a layer of coarse mud plaster, with a smoother gesso layer above, some finer limestones could take paint directly. Pigments were mostly mineral, chosen to withstand sunlight without fading. The binding medium used in painting remains unclear, egg tempera and various gums and it is clear that true fresco, painted into a thin layer of wet plaster, was not used. Instead the paint was applied to dried plaster, in what is called fresco a secco in Italian, small objects including wooden statuettes were often painted using similar techniques. Many ancient Egyptian paintings have survived in tombs, and sometimes temples, the paintings were often made with the intent of making a pleasant afterlife for the deceased. The themes included journey through the afterworld or protective deities introducing the deceased to the gods of the underworld, some tomb paintings show activities that the deceased were involved in when they were alive and wished to carry on doing for eternity. In the New Kingdom and later, the Book of the Dead was buried with the entombed person and it was considered important for an introduction to the afterlife. Egyptian paintings are painted in such a way to show a profile view, for example, the painting to the right shows the head from a profile view and the body from a frontal view. Their main colors were red, blue, green, gold, black, the monumental sculpture of ancient Egypts temples and tombs is world-famous, but refined and delicate small works exist in much greater numbers. The Egyptians used the technique of sunk relief, which is well suited to very bright sunlight. The distinctive pose of standing statues facing forward with one foot in front of the other was helpful for the balance and it was adopted very early and remained unchanged until the arrival of the Greeks
Art of ancient Egypt
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Thutmose, Bust of Nefertiti, 1345 BC, Egyptian Museum of Berlin
Art of ancient Egypt
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Sunk relief of the crocodile god Sobek
Art of ancient Egypt
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Tomb of Sarenput II.
Art of ancient Egypt
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Wall painting of Nefertari
44.
Egyptian astronomy
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Egyptian astronomy begins in prehistoric times, in the Predynastic Period. In the 5th millennium BCE, the circles at Nabta Playa may have made use of astronomical alignments. The Egyptian pyramids were aligned towards the pole star. Roman Egypt produced the greatest astronomer of the era, Ptolemy and his works on astronomy, including the Almagest, became the most influential books in the history of Western astronomy. Following the Muslim conquest of Egypt, the region came to be dominated by Arabic culture, the astronomer Ibn Yunus observed the suns position for many years using a large astrolabe, and his observations on eclipses were still used centuries later. In 1006, Ali ibn Ridwan observed the SN1006, a regarded as the brightest stellar event in recorded history. In the 14th century, Najm al-Din al-Misri wrote a treatise describing over 100 different types of scientific and astronomical instruments, Egyptian astronomy begins in prehistoric times. The presence of stone circles at Nabta Playa in Upper Egypt dating from the 5th millennium BCE show the importance of astronomy to the life of ancient Egypt even in the prehistoric period. The constellation system used among the Egyptians also appears to have been essentially of native origin, the precise orientation of the Egyptian pyramids serves as a lasting demonstration of the high degree of technical skill in watching the heavens attained in the 3rd millennium BCE. It has been shown the pyramids were aligned towards the star, which, because of the precession of the equinoxes, was at that time Thuban. The length of the corridor down which sunlight would travel would have limited illumination at other times of the year, Astronomy played a considerable part in religious matters for fixing the dates of festivals and determining the hours of the night. The titles of several books are preserved recording the movements and phases of the sun, moon. The rising of Sirius at the beginning of the inundation was an important point to fix in the yearly calendar. One of the most important Egyptian astronomical texts was the Book of Nut, beginning with the 9th Dynasty, ancient Egyptians produced Diagonal star tables, which were usually painted on the inside surface of wooden coffin lids. This practice continued until the 12th dynasty and these Diagonal star tables or star charts are also known as diagonal star clocks, in the past they have also been known as star calendars, or decanal clocks. These star charts featuring the paintings of Egyptian deities, decans, constellations, according to the texts, in founding or rebuilding temples the north axis was determined by the same apparatus, and we may conclude that it was the usual one for astronomical observations. In careful hands, it might give results of a degree of accuracy. He named it the Egyptian System, and stated that it did not escape the skill of the Egyptians and he must know by heart the Hermetic astrological books, which are four in number
Egyptian astronomy
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Chart from Senemut's tomb, 18th dynasty
Egyptian astronomy
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Nut, Egyptian goddess of the sky, with the star chart in the tomb of Ramses VI
Egyptian astronomy
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' Star clock ' method from the tomb of Rameses VI
Egyptian astronomy
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An Egyptian 30th-dynasty (Ptolemaic) terracotta astrological disc at the Los Angeles County Museum of Art.
45.
Ancient Egyptian burial customs
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The ancient Egyptians had an elaborate set of funerary practices that they believed were necessary to ensure their immortality after death. These rituals and protocols included mummifying the body, casting of magic spells, the burial process used by the ancient Egyptians evolved throughout time as old customs were discarded and new ones adopted, but several important elements of the process persisted. Although specific details changed over time, the preparation of the body, the rituals involved. Though no writing survives from Predynastic Egypt, scholars believe the importance of the physical body and this would explain why people of that time did not follow the common practice of cremation, but rather buried the dead. Some also believe they may have feared the bodies would rise again if mistreated after death, early bodies were buried in simple, shallow oval pits, with a few burial goods. Sometimes multiple people and animals were placed in the same grave, over time, graves became more complex, with the body placed in a wicker basket, then later in wooden or terracotta coffins. The latest tombs Egyptians made were sarcophaguses and these graves contained burial goods like jewelry, food, games and sharpened splint. This demonstrates that this ancient period had a sense of the afterlife and this may be because admission required that the deceased must be able to serve a purpose there. The pharaoh was allowed in because of his role in life, human sacrifices found in early royal tombs reinforce this view. These people were meant to serve the pharaoh during his eternal life. Eventually, figurines and wall paintings begin to replace human victims, some of these figurines may have been created to resemble certain people, so they could follow the pharaoh after their lives ended. Note that not only the classes had to rely on the pharaoh’s favor. They believed that when he died, the became a type of god. This belief existed from the period through the Old Kingdom. In the First Intermediate Period, however, the importance of the pharaoh declined, funerary texts, previously restricted to royal use, became more widely available. The first farmers in Egypt are known from the villages of Omari, the people of these villages buried their dead in a simple, round graves with one pot. The body was neither treated nor arranged in a way as would be the case later in the historical period. Without any written evidence, there is little to provide information about contemporary beliefs concerning the afterlife except for the inclusion of a single pot in the grave
Ancient Egyptian burial customs
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Professional mourners in an eloquent gesture of mourning.
Ancient Egyptian burial customs
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Female Figure, ca. 3500-3400 B.C.E., Brooklyn Museum
Ancient Egyptian burial customs
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Mask from a Coffin. Cartonnage, 37.1387E, Brooklyn Museum
46.
Egyptian chronology
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The majority of Egyptologists agree on the outline and many details of the chronology of Ancient Egypt. Scholarly consensus on the outline of the conventional chronology current in Egyptology has not fluctuated much over the last 100 years. For the Old Kingdom, consensus fluctuates by as much as a few centuries and this is illustrated by comparing the chronology as given by two Egyptologists, the first writing in 1906, the second in 2000. The disparities between the two sets of result from additional discoveries and refined understanding of the still very incomplete source evidence. For example, Breasted adds a ruler in the Twentieth dynasty that further research showed did not exist, following Manetho, Breasted also believed all the dynasties were sequential, whereas it is now known that several existed at the same time. These revisions have resulted in a lowering of the chronology by up to 400 years at the beginning of Dynasty I. The backbone of Egyptian chronology are the years as recorded in Ancient Egyptian king lists. In addition, some Egyptian dynasties may have overlapped, with different pharaohs ruling in different regions at the same time, not knowing whether monarchies were simultaneous or sequential results in widely differing chronological interpretations. However, further research has shown that these censuses were taken in consecutive years. The sed festival was celebrated on the thirtieth anniversary of the Pharaohs ascension. However, once again, this may not be the practice in all cases. In the early days of Egyptology, the compilation of regnal periods may also have been hampered due to bias on the part of the Egyptologists. This was most pervasive before the mid 19th century, when Manethos figures were recognized as conflicting with biblical chronology based on Old Testament references to Egypt, in the 20th century, such biblical bias has mostly been confined to alternative chronologies outside of scholarly mainstream. A useful way to work around these gaps in knowledge is to find chronological synchronisms, over the past decades, a number of these have been found, although they are of varying degrees of usefulness and reliability. While this does not fix a person or event to a specific year, another example are blocks from the Old Kingdom bearing the names of several kings, which were reused in the construction of Middle Kingdom pyramid-temples at Lisht in the structures of Amenemhat I. The poor documentation of these finds in the Serapeum also compounds the difficulties in using these records. The best known of these is the Sothic cycle, and careful study of this led Richard A. Parker to argue that the dates of the Twelfth dynasty could be fixed with absolute precision. More recent research has eroded this confidence, questioning many of the assumptions used with the Sothic Cycle and this is useful especially for the Early Dynastic period, where Egyptological consensus has only been possible within a range of about three or four centuries
Egyptian chronology
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Astronomical ceiling from the tomb of Seti I showing stars and constellations used in calendar calculations
Egyptian chronology
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'Diagonal star table' from the 11th Dynasty coffin lid; found at Asyut, Egypt. Roemer- und Pelizaeus-Museum Hildesheim
47.
Clothing in ancient Egypt
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Ancient Egyptian clothes refers to clothing worn in ancient Egypt from the end of the Neolithic period to the collapse of the Ptolemaic dynasty with the death of Cleopatra VII in 30 BC. Egyptian clothing was filled with a variety of colors, adorned with precious gems and jewels, the fashions of the Ancient Egyptians were made for not only beauty but also comfort. Egyptian fashion was created to keep cool while in the hot desert, in ancient Egypt, linen was by far the most common textile. It helped people to be comfortable in the subtropical heat, linen is made from the flax plant by spinning the fibers from the stem of the plant. Spinning, weaving and sewing were very important techniques for all Egyptian societies, plant dyes could be applied to clothing but the clothing was usually left in its natural color. Wool was known, but considered impure, only the wealthy wore animal fibers that were the object of taboos. They were used on occasion for overcoats, but were forbidden in temples and sanctuaries, peasants, workers and other people of modest condition often wore nothing, but the shenti was worn by all people. The most common headdress was the khat or nemes, a cloth worn by men. From about 2130 BC during the Old Kingdom, garments were simple, the men wore wrap around skirts known as the shendyt, which were belted at the waist, sometimes pleated or gathered in the front. During this time, mens skirts were short, as the Middle Kingdom of Egypt,1600 B. C. came, the skirt was worn longer. Then, around 1420 BC, there was a tunic or blouse with sleeves. During the Old, Middle and New Kingdom, Ancient Egyptian women often wore simple sheath dresses called kalasiris, womens clothing in ancient Egypt was more conservative than mens clothing. The dresses were held up by one or two straps and were worn down to the ankle, while the edge could be worn above or below the breasts. The length of the dress denoted the social class of the wearer, beading or feathers were also used as an embellishment on the dress. Over the dress, women had a choice of wearing shawls, capes, the shawl was a piece of cloth around 4 feet wide by 13 or 14 feet long. This was mostly worn pleated as well, female clothes only changed slightly through the millennia. Draped clothing sometimes gave the impression of different clothing. It was made of haïk, a fine muslin
Clothing in ancient Egypt
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The clothing of men and women of several social levels of ancient Egypt are depicted in this tomb mural from the fifteenth century BC.
48.
Ancient Egyptian cuisine
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The cuisine of ancient Egypt covers a span of over three thousand years, but still retained many consistent traits until well into Greco-Roman times. The staples of both poor and wealthy Egyptians were bread and beer, often accompanied by green-shooted onions, other vegetables, depictions of banquets can be found in paintings from both the Old Kingdom and New Kingdom. They usually started sometime in the afternoon, men and women were separated unless they were married. Seating varied according to status, with those of the highest status sitting on chairs, those slightly lower sat on stools. Before the food was served, basins were provided along with perfumes and cones of scented fat were lit to spread pleasant smells or to repel insects, depending on the type. Lily flowers and flower collars were handed out and professional dancers entertained, accompanied by musicians playing harps, lutes, drums, tambourines, and clappers. There were usually considerable amounts of alcohol and abundant quantities of foods, there were whole roast oxen, ducks, geese, pigeons, the dishes frequently consisted of stews served with great amounts of bread, fresh vegetables and fruit. For sweets there were cakes baked with dates and sweetened with honey, the goddess Hathor was often invoked during feasts. Food could be prepared by stewing, baking, boiling, grilling, frying, spices and herbs were added for flavor, though the former were expensive imports and therefore confined to the tables of the wealthy. Food such as meats was mostly preserved by salting, and dates, the staples bread and beer were usually prepared in the same locations, as the yeast used for bread was also used for brewing. The two were prepared either in special bakeries or, more often, at home, and any surplus would be sold, Egyptian bread was made almost exclusively from emmer wheat, which was more difficult to turn into flour than most other varieties of wheat. The chaff does not come off through threshing, but comes in spikelets that needed to be removed by moistening and pounding with a pestle to avoid crushing the grains inside. It was then dried in the sun, winnowed and sieved and finally milled on a saddle quern, the baking techniques varied over time. In the Old Kingdom, heavy pottery molds were filled with dough, during the Middle Kingdom tall cones were used on square hearths. In the New Kingdom a new type of a large open-topped clay oven, cylindrical in shape, was used, dough was then slapped on the heated inner wall and peeled off when done, similar to how a tandoor oven is used for flatbreads. Tombs from the New Kingdom show images of bread in many different shapes and sizes, loaves shaped like human figures, fish, various animals and fans, all of varying dough texture. Flavorings used for bread included coriander seeds and dates, but it is not known if this was used by the poor. Other than emmer, barley was grown to make bread and also used for making beer, and so were lily seeds and roots, and tiger nut
Ancient Egyptian cuisine
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An early Ramesside Period mural painting from Deir el-Medina tomb depicts an Egyptian couple harvesting crops
Ancient Egyptian cuisine
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A depiction of the royal bakery from an engraving in the tomb of Ramesses III in the Valley of the Kings. There are many types of loaves, including ones that are shaped like animals. 20th dynasty.
Ancient Egyptian cuisine
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Hunting game birds and plowing a field. Depiction on a burial chamber from c. 2700 BC. Tomb of Nefermaat I and his wife Itet.
49.
List of ancient Egyptian dynasties
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In Ancient Egyptian history, dynasties are series of rulers sharing a common origin who are usually of the same family. Ancient Egypts historical period is divided into thirty-one pharaonic dynasties. The thirty-first dynastys name is not due to Manetho and is a later coining, while widely used and useful, the system does have its shortcomings. Some dynasties only ruled part of Egypt and existed concurrently with other dynasties based in other cities. The Seventh might not have existed at all, the Tenth seems to be a continuation of the Ninth and this page lists articles on dynasties of Ancient Egypt. The cities in power was held during these dynasties follow their names
List of ancient Egyptian dynasties
50.
Great Royal Wife
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Great Royal Wife, or alternatively Chief Kings Wife, is the term that was used to refer to the principal wife of the pharaoh of Ancient Egypt. The former is also, in the form of the simplification Great Wife, while most Ancient Egyptians were monogamous, the pharaoh would have had other, lesser wives and concubines in addition to the Great Royal Wife. This arrangement would allow the pharaoh to enter into marriages with the daughters of allies. In the past the order of succession in Ancient Egypt was thought to pass through the royal women and this theory, referred to as the Heiress Theory, has been rejected ever since the 1980s and is now not accepted by Egyptologists. The throne likely just passed to the eldest living son of the pharaoh, examples include Iset, the mother of Thutmose III, Tiaa, the mother of Thutmose IV and Mutemwia, the mother of Amenhotep III. Meretseger, the wife of Senusret III, is the earliest queen whose name appears with this title. However, she is attested in the New Kingdom so the title might be an anachronism. Perhaps the first holder of its title was Nubkhaes of the Second Intermediate Period, a special place in the history of great royal wives was taken by Hatshepsut. She was Great Royal Wife to her half-brother Thutmose II, during this time Hatshepsut also became a Gods Wife of Amun. After the death of her husband, she became regent because of the minority of her stepson, the male heir. While he was very young, however, Hatshepsut was crowned as pharaoh. When she became pharaoh, she designated her daughter, Neferure and her daughter may have been the great royal wife of Thutmose III but there is no clear evidence for this proposed marriage. Elsewhere, in Kush and other states of ancient Africa. The practice of creating great wives has continued to the present day, in addition to the said queen of the Zulus, contemporary holders of the title include the numerous bearers of the Iyaafin chieftaincy of Yorubaland and the future Ndlovukati of Swaziland. List of consorts of the Muhammad Ali Dynasty, for the modern queens and sultanas of Egypt Gods Wife of Amun Divine Adoratrice of Amun Interregnum queen
Great Royal Wife
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Nefertari, the Great Royal Wife of Ramasses II, from the temple he built to her at Abu Simbel, she holds a sistrum and a sacred lotus
Great Royal Wife
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Hatshepsut was Great Royal Wife to Thutmose II, then regent for her stepson Thutmose III (Museum of Fine Arts, Boston)
Great Royal Wife
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Royal titulary
51.
History of ancient Egypt
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The history of ancient Egypt spans the period from the early prehistoric settlements of the northern Nile valley to the Roman conquest, in 30 BC. The Pharaonic Period is dated from the 32nd century BC, when Upper and Lower Egypt were unified, until the country fell under Macedonian rule, note For alternative revisions to the chronology of Egypt, see Egyptian chronology. Egypts history is split into different periods according to the ruling dynasty of each pharaoh. The dating of events is still a subject of research, the conservative dates are not supported by any reliable absolute date for a span of about three millennia. The following is the list according to conventional Egyptian chronology, traces of these early people appear in the form of artifacts and rock carvings along the terraces of the Nile and in the oases. To the Egyptians the Nile meant life and the desert meant death, evidence also indicates human habitation and cattle herding in the southwestern corner of Egypt near the Sudan border before the 8th millennium BC. Despite this, the idea of an independent bovine domestication event in Africa must be abandoned because subsequent evidence gathered over a period of thirty years has failed to corroborate this, the oldest-known domesticated cattle remains in Africa are from the Faiyum c.4400 BC. Continued desiccation forced the early ancestors of the Egyptians to settle around the Nile more permanently, however, the period from 9th to the 6th millennium BC has left very little in the way of archaeological evidence. The Nile valley of Egypt was basically uninhabitable until the work of clearing and irrigating the land along the banks was started, however it appears that this clearance and irrigation was largely under way by the 6th millennium. By that time, Nile society was already engaged in organized agriculture, at this time, Egyptians in the southwestern corner of Egypt were herding cattle and also constructing large buildings. Mortar was in use by the 4th millennium, the people of the valley and the Nile Delta were self-sufficient and were raising barley and emmer, an early variety of wheat, and stored it in pits lined with reed mats. They raised cattle, goats and pigs and they wove linen, prehistory continues through this time, variously held to begin with the Amratian culture. Between 5500 BC and the 31st century BC, small settlements flourished along the Nile, the Tasian culture was the next to appear, it existed in Upper Egypt starting about 4500 BC. This group is named for the burials found at Deir Tasa, the Tasian culture is notable for producing the earliest blacktop-ware, a type of red and brown pottery painted black on its top and interior. The Badari culture, named for the Badari site near Deir Tasa, followed the Tasian, however, the Badari culture continued to produce the kind of pottery called blacktop-ware, and was assigned the sequence dating numbers between 21 and 29. The Amratian culture is named after the site of el-Amreh, about 120 kilometres south of Badari, el-Amreh was the first site where this culture was found unmingled with the later Gerzeh culture. However, this period is attested at Nagada, and so is also referred to as the Naqada I culture. The Amratian period falls between S. D.30 and 39, newly excavated objects indicate that trade between Upper and Lower Egypt existed at this time
History of ancient Egypt
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A Naqada II vase decorated with gazelles, on display at the Louvre.
History of ancient Egypt
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An Osiride statue of Mentuhotep II, the founder of the Middle Kingdom
History of ancient Egypt
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Statuette of Merankhre Mentuhotep VI, a minor king of the 16th Dynasty, reigning over the Theban region c. 1585 BC.
52.
Ancient Egyptian literature
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Ancient Egyptian literature was written in the Egyptian language from ancient Egypts pharaonic period until the end of Roman domination. It represents the oldest corpus of Egyptian literature, along with Sumerian literature, it is considered the worlds earliest literature. Writing in ancient Egypt—both hieroglyphic and hieratic—first appeared in the late 4th millennium BC during the phase of predynastic Egypt. It was not until the early Middle Kingdom that a narrative Egyptian literature was created and this was a media revolution which, according to Richard B. However, it is possible that the literacy rate was less than one percent of the entire population. The creation of literature was thus an elite exercise, monopolized by a scribal class attached to government offices, However, there is no full consensus among modern scholars concerning the dependence of ancient Egyptian literature on the sociopolitical order of the royal courts. Middle Egyptian, the language of the Middle Kingdom, became a classical language during the New Kingdom. Scribes of the New Kingdom canonized and copied many literary texts written in Middle Egyptian, some genres of Middle Kingdom literature, such as teachings and fictional tales, remained popular in the New Kingdom, although the genre of prophetic texts was not revived until the Ptolemaic period. Popular tales included the Story of Sinuhe and The Eloquent Peasant, while important teaching texts include the Instructions of Amenemhat and The Loyalist Teaching. By the New Kingdom period, the writing of graffiti on sacred temple and tomb walls flourished as a unique genre of literature. The acknowledgment of rightful authorship remained important only in a few genres, while texts of the genre were pseudonymous. Ancient Egyptian literature has been preserved on a variety of media. This includes papyrus scrolls and packets, limestone or ceramic ostraca, wooden writing boards, monumental stone edifices, Texts preserved and unearthed by modern archaeologists represent a small fraction of ancient Egyptian literary material. The area of the floodplain of the Nile is under-represented because the moist environment is unsuitable for the preservation of papyri, on the other hand, hidden caches of literature, buried for thousands of years, have been discovered in settlements on the dry desert margins of Egyptian civilization. By the Early Dynastic Period in the late 4th millennium BC, Egyptian hieroglyphs, Egyptian hieroglyphs are small artistic pictures of natural objects. The Narmer Palette, dated c.3100 BC during the last phase of Predynastic Egypt, combines the hieroglyphs for catfish and chisel to produce the name of King Narmer. The Egyptians called their hieroglyphs words of god and reserved their use for exalted purposes, such as communicating with divinities, each hieroglyphic word represented both, a specific object and embodied the essence of that object, recognizing it as divinely made and belonging within the greater cosmos. Through acts of priestly ritual, like burning incense, the priest allowed spirits, mutilating the hieroglyph of a venomous snake, or other dangerous animal, removed a potential threat
Ancient Egyptian literature
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Egyptian hieroglyphs with cartouches for the name " Ramesses II ", from the Luxor Temple, New Kingdom
Ancient Egyptian literature
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The slab stela of the Old Kingdom Egyptian princess Neferetiabet (dated c. 2590–2565 BC), from her tomb at Giza, with hieroglyphs carved and painted on limestone
Ancient Egyptian literature
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Abbott Papyrus, a record written in hieratic script; it describes an inspection of royal tombs in the Theban Necropolis and is dated to the 16th regnal year of Ramesses IX, ca. 1110 BCE.
Ancient Egyptian literature
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An ostracon with hieratic script mentioning officials involved in the inspection and clearing of tombs during the Twenty-first dynasty of Egypt, c. 1070–945 BC
53.
Ancient Egyptian medicine
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The medicine of the ancient Egyptians is some of the oldest documented. Egyptian medical thought influenced later traditions, including the Greeks, until the 19th century, the main sources of information about ancient Egyptian medicine were writings from later in antiquity. The Greek historian Herodotus visited Egypt around 440 BC and wrote extensively of his observations of their medicinal practice, pliny the Elder also wrote favourably of them in historical review. Hippocrates, Herophilos, Erasistratus and later Galen studied at the temple of Amenhotep, in 1822, the translation of the Rosetta stone finally allowed the translation of ancient Egyptian hieroglyphic inscriptions and papyri, including many related to medical matters. The Edwin Smith Papyrus is a textbook on surgery and details anatomical observations and the examination, diagnosis, treatment and it was probably written around 1600 BC, but is regarded as a copy of several earlier texts. Medical information in it dates from as early as 3000 BC and it is thus viewed as a learning manual. Treatments consisted of ointments made from animal, vegetable or fruit substances or minerals, the earliest known surgery to be performed in Egypt occurred around 2750 BC. The Ebers papyrus c.1550 BC is full of incantations and foul applications meant to turn away disease-causing demons and it may also contain the earliest documented awareness of tumors, if the poorly understood ancient medical terminology has been correctly interpreted. The Kahun Gynaecological Papyrus treats womens complaints, including problems with conception, thirty four cases detailing diagnosis and treatment survive, some of them fragmentarily. Dating to 1800 BC, it is the oldest surviving text of any kind. Other documents such as the Hearst papyrus, and Berlin Papyrus also provide insight into ancient Egyptian medicine. Other information comes from the images that often adorn the walls of Egyptian tombs, advances in modern medical technology also contributed to the understanding of ancient Egyptian medicine. Paleopathologists were able to use X-Rays and later CAT Scans to view the bones, electron microscopes, mass spectrometry and various forensic techniques allowed scientists unique glimpses of the state of health in Egypt 4000 years ago. The ancient Egyptians were at least partially aware of the importance of diet, owing to Egypts great endowment of fertile land, food production was never a major issue although of course no matter how bountiful the land, paupers and starvation still exist. The main crops for most of ancient Egyptian history were emmer wheat, barley was also used in beer. Vegetables and fruits of many types were widely grown, oil was produced from the linseed plant and there was a limited selection of spices and herbs. Offerings to King Unas were recorded as and it is clear that the Egyptian diet was not lacking for the upper classes and that even the lower classes may have had some selection. Like many civilizations in the past the ancient Egyptians amply discovered the properties of the plant life around them
Ancient Egyptian medicine
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The Edwin Smith Papyrus documents ancient Egyptian medicine, including the diagnosis and treatment of injuries.
Ancient Egyptian medicine
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Ebers Papyrus treatment for cancer: recounting a " tumor against the god Xenus", it recommends "do thou nothing there against"
Ancient Egyptian medicine
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Ancient Egyptian medical instruments depicted in a Ptolemaic period inscription on the Temple of Kom Ombo.
Ancient Egyptian medicine
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This wood and leather prosthetic toe was used by an amputee to facilitate walking
54.
Military of ancient Egypt
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Ancient Egypt was an ancient civilization of eastern North Africa, concentrated along the Northern reaches of the Nile River in Egypt. The civilization coalesced around 3150 BC with the unification of Upper and Lower Egypt under the first pharaoh. Its history occurred in a series of stable Kingdoms, separated by periods of relative instability known as Intermediate Periods, ancient Egypt reached its pinnacle during the New Kingdom, after which it entered a period of slow decline. Egypt was conquered by a succession of foreign powers in late period. Although the Egyptian military forces in the Old and Middle kingdoms were well maintained, for most parts of its long history, ancient Egypt was unified under one government. The main military concern for the nation was to keep enemies out, the arid plains they wanted to get rid of and deserts surrounding Egypt were inhabited by nomadic tribes who occasionally tried to raid or settle in the fertile Nile river valley. Nevertheless, the expanses of the desert formed a barrier that protected the river valley and was almost impossible for massive armies to cross. The Egyptians built fortresses and outposts along the borders east and west of the Nile Delta, in the Eastern Desert, small garrisons could prevent minor incursions, but if a large force was detected a message was sent for the main army corps. Most Egyptian cities lacked city walls and other defenses, the history of ancient Egypt is divided into three kingdoms and two intermediate periods. During the three Kingdoms Egypt was unified under one government, during the Intermediate periods government control was in the hands of the various nomes and various foreigners. The geography of Egypt served to isolate the country and allowed it to thrive and this circumstance set the stage for many of Egypts military conquests. They enfeebled their enemies by using small projectile weapons, like bows and arrows and they also had chariots which they used to charge at the enemy. The Old Kingdom was one of the most prosperous times in Egypts history, because of this affluence, it allowed the government to stabilize and in turn organize a functioning military. Before Egypts New Kingdom, there were four major causes for military conflict, the Old Kingdoms military was most marked by their construction of forts along the Nile River. At this time, the conflict was with Nubia and Egypt felt the urge to defend their borders by building forts deep into this country. These forts were never attacked, but they acted as a deterring factor towards potential invaders. Many are currently underwater in Lake Nasser, but while they were visible they were a testament to the affluence. During the Old Kingdom there was no army in Egypt
Military of ancient Egypt
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The Hyksos of Ancient Egypt drove chariots
Military of ancient Egypt
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A New Kingdom khopesh
Military of ancient Egypt
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Wooden figures found in the tomb of Mesehti: Egyptian army of the 11th Dynasty
Military of ancient Egypt
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Ancient Egyptian chariot
55.
Music of Egypt
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Music has been an integral part of Egyptian culture since antiquity. The Bible documents the instruments played by the ancient Hebrews, all of which are correlated in Egyptian archaeology, Egyptian music probably had a significant impact on the development of ancient Greek music, and via the Greeks was important to early European music well into the Middle Ages. The modern music of Egypt is considered Arabic music as it has been a source for or influence on regional styles. The ancient Egyptians credited the goddess Bat with the invention of music, the cult of Bat was eventually syncretised into that of Hathor because both were depicted as cows. Hathors music was believed to have used by Osiris as part of his effort to civilize the world. The lion-goddess Bastet was also considered a goddess of music, in prehistoric Egypt, music and chanting were commonly used in magic and rituals. Rhythms during this time were ovular and music served to create rhythm, small shells were used as whistles. During the predynastic period of Egyptian history, funerary chants continued to play an important role in Egyptian religion and were accompanied by clappers or a flute, the evidence is for instruments played more securely attested in the Old Kingdom when harps, flutes and double clarinets were played. Percussion instruments and lutes were added to orchestras by the Middle Kingdom, cymbals frequently accompanied music and dance, much as they still do in Egypt today. Typically ancient Egyptian music was composed from the dominant scale. The phrygian dominant scale may often feature a note or two in parts to create tension. For instance the music could typically be in the key of E phrygian dominant using the notes E, F, G sharp, A, B, C, D and then have an A sharp, B, A sharp, G natural and E to create tension. Arabic music is said to have begun in the 7th century in Syria during the Umayyad dynasty. Early Arabic music was influenced by Byzantine, Indian and Persian forms, which were heavily influenced by earlier Greek, Semitic. Egyptians in Medieval Cairo believed that music exercised too powerful an effect upon the passions, however, Egyptians generally were very fond of music. Though, according to E. W. Lane, no man of sense would ever become a musician, tradesmen of every occupation used music during work and schools taught the Quran by chanting. The music of Medieval Egypt was derived from Greek, Persian and Indian traditions, the songs of this period were similar in sound and simple, within a small range of tones. Egyptian song, though simple in form, is embellished by the singer, distinct enunciation and a quavering voice are also characteristics of Egyptian singing
Music of Egypt
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Musicians of Amun, Tomb of Nakht, 18th Dynasty, Western Thebes.
Music of Egypt
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Lute and double pipe players from a painting found in the Theban tomb of Nebamun, a nobleman of the 18th Dynasty of the New Kingdom, c. 1350 BC
Music of Egypt
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Broken Egyptian Sistrum
Music of Egypt
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Collection of sistrums at the Louvre
56.
Egyptian mythology
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Egyptian mythology is the collection of myths from ancient Egypt, which describe the actions of the Egyptian gods as a means of understanding the world. The beliefs that these myths express are an important part of ancient Egyptian religion, Myths appear frequently in Egyptian writings and art, particularly in short stories and in religious material such as hymns, ritual texts, funerary texts, and temple decoration. These sources rarely contain an account of a myth and often describe only brief fragments. Inspired by the cycles of nature, the Egyptians saw time in the present as a series of recurring patterns, Myths are set in these earliest times, and myth sets the pattern for the cycles of the present. Present events repeat the events of myth, and in doing so renew maat, events from the present that might be regarded as myths include Ras daily journey through the world and its otherworldly counterpart, the Duat. The details of these sacred events differ greatly from one text to another, Egyptian myths are primarily metaphorical, translating the essence and behavior of deities into terms that humans can understand. Each variant of a myth represents a different symbolic perspective, enriching the Egyptians understanding of the gods and it inspired or influenced many religious rituals and provided the ideological basis for kingship. Scenes and symbols from myth appeared in art in tombs, temples, in literature, myths or elements of them were used in stories that range from humor to allegory, demonstrating that the Egyptians adapted mythology to serve a wide variety of purposes. The development of Egyptian myth is difficult to trace, Egyptologists must make educated guesses about its earliest phases, based on written sources that appeared much later. One obvious influence on myth is the Egyptians natural surroundings, thus the Egyptians saw water and the sun as symbols of life and thought of time as a series of natural cycles. This orderly pattern was at constant risk of disruption, unusually low floods resulted in famine, the hospitable Nile valley was surrounded by harsh desert, populated by peoples the Egyptians regarded as uncivilized enemies of order. For these reasons, the Egyptians saw their land as an place of stability, or maat. These themes—order, chaos, and renewal—appear repeatedly in Egyptian religious thought, another possible source for mythology is ritual. Many rituals make reference to myths and are based directly on them. But it is difficult to determine whether a cultures myths developed before rituals or vice versa, questions about this relationship between myth and ritual have spawned much discussion among Egyptologists and scholars of comparative religion in general. In ancient Egypt, the earliest evidence of religious practices predates written myths, rituals early in Egyptian history included only a few motifs from myth. For these reasons, some scholars have argued that, in Egypt, but because the early evidence is so sparse, the question may never be resolved for certain. In private rituals, which are often called magical, the myth, many of the myth-like stories that appear in the rituals texts are not found in other sources
Egyptian mythology
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Nun, the embodiment of the primordial waters, lifts the barque of the sun god Ra into the sky at the moment of creation.
Egyptian mythology
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The sky depicted as a cow goddess supported by other deities. This image combines several coexisting visions of the sky: as a roof, as the surface of a sea, as a cow, and as a goddess in human form.
Egyptian mythology
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Temple decoration at Dendera, depicting the goddesses Isis and Nephthys watching over the corpse of their brother Osiris
Egyptian mythology
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The air god Shu, assisted by other gods, holds up Nut, the sky, as Geb, the earth, lies beneath.
57.
Pharaoh
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The word pharaoh ultimately derive from the Egyptian compound pr-ˤ3 great house, written with the two biliteral hieroglyphs pr house and ˤ3 column, here meaning great or high. It was used only in larger phrases such as smr pr-ˤ3 Courtier of the High House, with specific reference to the buildings of the court or palace. From the twelfth dynasty onward, the word appears in a wish formula Great House, may it live, prosper, and be in health, but again only with reference to the royal palace and not the person. During the reign of Thutmose III in the New Kingdom, after the rule of the Hyksos during the Second Intermediate Period. During the eighteenth dynasty the title pharaoh was employed as a designation of the ruler. From the nineteenth dynasty onward pr-ˤ3 on its own was used as regularly as hm. f, the term, therefore, evolved from a word specifically referring to a building to a respectful designation for the ruler, particularly by the twenty-second dynasty and twenty-third dynasty. For instance, the first dated appearance of the pharaoh being attached to a rulers name occurs in Year 17 of Siamun on a fragment from the Karnak Priestly Annals. Here, an induction of an individual to the Amun priesthood is dated specifically to the reign of Pharaoh Siamun and this new practice was continued under his successor Psusennes II and the twenty-second dynasty kings. Shoshenq I was the successor of Siamun. Meanwhile, the old custom of referring to the sovereign simply as pr-ˤ3 continued in traditional Egyptian narratives, by this time, the Late Egyptian word is reconstructed to have been pronounced *par-ʕoʔ whence Herodotus derived the name of one of the Egyptian kings, Φερων. In the Bible, the title also occurs as פרעה, from that, Septuagint φαραώ pharaō and then Late Latin pharaō, both -n stem nouns. The Quran likewise spells it فرعون firawn with n, interestingly, the Arabic combines the original pharyngeal ayin sound from Egyptian, along with the -n ending from Greek. English at first spelt it Pharao, but the King James Bible revived Pharaoh with h from the Hebrew, meanwhile in Egypt itself, *par-ʕoʔ evolved into Sahidic Coptic ⲡⲣ̅ⲣⲟ prro and then rro. Scepters and staves were a sign of authority in ancient Egypt. One of the earliest royal scepters was discovered in the tomb of Khasekhemwy in Abydos, kings were also known to carry a staff, and Pharaoh Anedjib is shown on stone vessels carrying a so-called mks-staff. The scepter with the longest history seems to be the heqa-scepter, the earliest examples of this piece of regalia dates to pre-dynastic times. A scepter was found in a tomb at Abydos that dates to the late Naqada period, another scepter associated with the king is the was-scepter. This is a long staff mounted with an animal head, the earliest known depictions of the was-scepter date to the first dynasty
Pharaoh
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Den
Pharaoh
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Narmer Palette
Pharaoh
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Nomen and prenomen of Ramesses III
Pharaoh
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Royal titulary
58.
List of pharaohs
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Note that the dates given are approximate. Modern lists of pharaohs are based on records, Ancient Egyptian king lists and later histories, such as Manethos Aegyptiaca. An additional problem is that ancient king lists are often damaged, the following ancient king lists are known, Den seal impressions, found on a cylinder seal in Dens tomb. It lists all 1st dynasty kings from Narmer to Den by their Horus names, palermo stone, carved on an olivin-basalt slab. Broken into pieces and thus today incomplete, giza King List, painted with red, green and black ink on gypsum and cedar wood. South Saqqara Stone, carved on a black basalt slab, karnak King List, carved on limestone. Abydos King List of Seti I, carved on limestone, very detailed, but omitting the First Intermediate Period. Abydos King List of Ramses II, carved on limestone, Saqqara King List, carved on limestone. Very detailed, but omitting most kings of the 1st dynasty for unknown reasons, Turin King List, written with red and black ink on papyrus. Most possibly the most complete king list in history, today damaged, Manethos Aegyptiaca, possibly written on papyrus. The original writings are lost today and many anecdotes assigned to certain kings seem fictitious, Lower Egypt geographically consisted of the northern Nile and the Nile delta. The following list may not be complete, Regrouped here are predynastic rulers of Upper Egypt belonging to the late Naqada III period, the following list of predynastic rulers may be incomplete. Since these kings precede the First Dynasty, they have been grouped as Dynasty 0. The Fifth Dynasty ruled from 2498 to 2345 BC, the Sixth Dynasty ruled from 2345 to 2181 BC. The First Intermediate Period is a period of disarray and chaos between the end of the Old Kingdom and the advent of the Middle Kingdom, the Old Kingdom rapidly collapsed after the death of Pepi II. He had reigned for more than 64 and likely up to 94 years, the latter years of his reign were marked by inefficiency because of his advanced age. The union of the Two Kingdoms fell apart and regional leaders had to cope with the resulting famine. The kings of the 7th and 8th Dynasties, who represented the successors of the 6th Dynasty, tried to hold some power in Memphis
List of pharaohs
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Iry-Hor
List of pharaohs
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The Pschent combined the Red Crown of Lower Egypt and the White Crown of Upper Egypt.
List of pharaohs
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Ka
List of pharaohs
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Scorpion II
59.
Ancient Egyptian religion
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Ancient Egyptian religion was a complex system of polytheistic beliefs and rituals which were an integral part of ancient Egyptian society. It centered on the Egyptians interaction with many deities who were believed to be present in, and in control of, rituals such as prayers and offerings were efforts to provide for the gods and gain their favor. Formal religious practice centered on the pharaoh, the king of Egypt and he acted as the intermediary between his people and the gods and was obligated to sustain the gods through rituals and offerings so that they could maintain order in the universe. The state dedicated enormous resources to Egyptian rituals and to the construction of the temples, individuals could interact with the gods for their own purposes, appealing for their help through prayer or compelling them to act through magic. These practices were distinct from, but closely linked with, the formal rituals, the popular religious tradition grew more prominent in the course of Egyptian history as the status of the Pharaoh declined. Another important aspect was the belief in the afterlife and funerary practices, the Egyptians made great efforts to ensure the survival of their souls after death, providing tombs, grave goods, and offerings to preserve the bodies and spirits of the deceased. The religion had its roots in Egypts prehistory and lasted for more than 3,000 years, the details of religious belief changed over time as the importance of particular gods rose and declined, and their intricate relationships shifted. At various times, certain gods became preeminent over the others, including the sun god Ra, the creator god Amun, for a brief period, in the theology promulgated by the Pharaoh Akhenaten, a single god, the Aten, replaced the traditional pantheon. Ancient Egyptian religion and mythology left behind many writings and monuments, along with significant influences on ancient, the beliefs and rituals now referred to as ancient Egyptian religion were integral within every aspect of Egyptian culture. Their language possessed no single term corresponding to the modern European concept of religion, the characteristics of the gods who populated the divine realm were inextricably linked to the Egyptians understanding of the properties of the world in which they lived. The Egyptians believed that the phenomena of nature were divine forces in and these deified forces included the elements, animal characteristics, or abstract forces. The Egyptians believed in a pantheon of gods, which were involved in all aspects of nature and their religious practices were efforts to sustain and placate these phenomena and turn them to human advantage. This polytheistic system was complex, as some deities were believed to exist in many different manifestations. Conversely, many forces, such as the sun, were associated with multiple deities. The diverse pantheon ranged from gods with vital roles in the universe to minor deities or demons with very limited or localized functions. It could include gods adopted from foreign cultures, and sometimes humans, deceased Pharaohs were believed to be divine, and occasionally, distinguished commoners such as Imhotep also became deified. The depictions of the gods in art were not meant as representations of how the gods might appear if they were visible. Instead, these depictions gave recognizable forms to the deities by using symbolic imagery to indicate each gods role in nature
Ancient Egyptian religion
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The gods Osiris, Anubis, and Horus, in order from left to right
Ancient Egyptian religion
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Amun-Ra kamutef, wearing the plumed headdress of Amun and the sun disk representing Ra
Ancient Egyptian religion
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The air god Shu, assisted by other gods, holds up Nut, the sky, as Geb, the earth, lies beneath.
Ancient Egyptian religion
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Colossal statue of the Pharaoh Ramesses II
60.
Giza Necropolis
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The Giza pyramid complex is an archaeological site on the Giza Plateau, on the outskirts of Cairo, Egypt. It is located in the Libyan Desert, approximately 9 km west of the Nile river at the old town of Giza and it is by far the oldest of the ancient Wonders and the only one still in existence. The Great Sphinx lies on the east side of the complex, current consensus among Egyptologists is that the head of the Great Sphinx is that of Khafre. Along with these monuments are a number of smaller satellite edifices, known as queens pyramids, causeways. The valley temple was connected to a causeway which was destroyed when the village was constructed. The causeway led to the Mortuary Temple of Khufu, from this temple the basalt pavement is the only thing that remains. The mortuary temple was connected to the king’s pyramid, the king’s pyramid has three smaller queen’s pyramids associated with it and five boat pits. The boat pits contained a ship, and the 2 pits on the side of the pyramid still contained intact ships. One of these ships has been restored and is on display, khufus pyramid still has a limited collection of casing stones at its base. These casing stones were made of white limestone quarried from the nearby range. Khafre’s pyramid complex consists of a temple, the Sphinx temple, a causeway, a mortuary temple. The valley temple yielded several statues of Khafre, several were found in a well in the floor of the temple by Mariette in 1860. Others were found during excavations by Sieglin, Junker, Reisner. Khafre’s complex contained five boat-pits and a pyramid with a serdab. Khafres pyramid retains a prominent display of casing stones at its apex, menkaure’s pyramid complex consists of a valley temple, a causeway, a mortuary temple, and the king’s pyramid. The valley temple once contained several statues of Menkaure, during the 5th dynasty, a smaller ante-temple was added on to the valley temple. The mortuary temple also yielded several statues of Menkaure, the king’s pyramid has three subsidiary or queen’s pyramids. Of the four monuments, only Menkaures pyramid is seen today without any of its original polished limestone casing
Giza Necropolis
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All of the six pyramids of the Giza pyramid complex
Giza Necropolis
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Views, Objects: Egypt. Gizeh [selected images]. View 05: Sphinx and Pyramids., n.d., New York. Brooklyn Museum Archives
Giza Necropolis
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Pyramids of Ghizeh. 1893. Egypt; heliogravures after original views. Wilbour Library of Egyptology. Brooklyn Museum
Giza Necropolis
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The Great Sphinx partially excavated, photo taken between 1867 and 1899
61.
Ancient Egyptian mathematics
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Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt c.3000 to c.300 BC. Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found in Tomb U-j at Abydos and these labels appear to have been used as tags for grave goods and some are inscribed with numbers. Further evidence of the use of the base 10 number system can be found on the Narmer Macehead which depicts offerings of 400,000 oxen,1,422,000 goats and 120,000 prisoners. The evidence of the use of mathematics in the Old Kingdom is scarce, the lines in the diagram are spaced at a distance of one cubit and show the use of that unit of measurement. The earliest true mathematical documents date to the 12th dynasty, the Rhind Mathematical Papyrus which dates to the Second Intermediate Period is said to be based on an older mathematical text from the 12th dynasty. The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so-called mathematical problem texts and they consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems, an interesting feature of Ancient Egyptian mathematics is the use of unit fractions. Scribes used tables to help work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain 2 n tables and these tables allowed the scribes to rewrite any fraction of the form 1 n as a sum of unit fractions. During the New Kingdom mathematical problems are mentioned in the literary Papyrus Anastasi I, in the workers village of Deir el-Medina several ostraca have been found that record volumes of dirt removed while quarrying the tombs. Our understanding of ancient Egyptian mathematics is impeded by the paucity of available sources. The Reisner Papyrus dates to the early Twelfth dynasty of Egypt and was found in Nag el-Deir, the Rhind Mathematical Papyrus dates from the Second Intermediate Period, but its author, Ahmes, identifies it as a copy of a now lost Middle Kingdom papyrus. The RMP is the largest mathematical text, from the New Kingdom we have a handful of mathematical texts and inscription related to computations, The Papyrus Anastasi I is a literary text from the New Kingdom. It is written as a written by a scribe named Hori. A segment of the letter describes several mathematical problems, ostracon Senmut 153 is a text written in hieratic. Ostracon Turin 57170 is a written in hieratic. Ostraca from Deir el-Medina contain computations, ostracon IFAO1206 for instance shows the calculations of volumes, presumably related to the quarrying of a tomb
Ancient Egyptian mathematics
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Slab stela of Old Kingdom princess Neferetiabet (dated 2590–2565 BC) from her tomb at Giza, painting on limestone, now in the Louvre.
Ancient Egyptian mathematics
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Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.