1.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
–
Euclid (holding
calipers), Greek mathematician, 3rd century BC, as imagined by
Raphael in this detail from
The School of Athens.
Mathematics
–
Greek mathematician
Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the
Pythagorean theorem
Mathematics
–
Leonardo Fibonacci, the
Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians
2.
Ancient Egypt
–
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
–
The
Great Sphinx and the
pyramids of Giza are among the most recognizable symbols of the civilization of ancient Egypt.
Ancient Egypt
–
A typical Naqada II jar decorated with gazelles. (Predynastic Period)
Ancient Egypt
–
The
Narmer Palette depicts the unification of the Two Lands.
3.
Abydos, Egypt
–
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
–
Façade of the
Temple of Seti I in Abydos
Abydos, Egypt
–
Part of the Abydos King List
Abydos, Egypt
–
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
–
Panel from the
Osiris temple:
Horus presents royal regalia to a worshipping pharaoh.
4.
Narmer Macehead
–
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
–
Narmer Macehead Centre left: Pharaoh Narmer seated in a
naos
5.
Old Kingdom
–
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
–
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
–
The
Pyramid of Djoser at
Saqqara.
Old Kingdom
–
Temple of Djoser at
Saqqara
6.
Mastaba
–
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
–
Example of a mastaba
7.
Meidum
–
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
–
View of the Pyramid
Meidum
–
Lantern Slide Collection: Views, Objects: Egypt. Meidum. Old Kingdom. Step Pyramid of Meidum, 4th Dyn., n.d. Brooklyn Museum Archives
Meidum
–
Passageway in the Meidum Pyramid
Meidum
–
Another view of Meidum Pyramid
8.
Cubit
–
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
–
Egyptian cubit rod in the
Liverpool World Museum
Cubit
–
Cubit rod of
Maya, 1336-1327 BC (
Eighteenth Dynasty)
Cubit
–
Cubit rod from the Turin Museum.
Cubit
–
The Nippur cubit-rod in the
Archeological Museum of
Istanbul, Turkey
9.
Ancient Egyptian units of measurement
–
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
–
Cubit rod from the Turin Museum.
Ancient Egyptian units of measurement
–
Problem 80 on the Rhind Mathematical Papyrus
10.
Moscow Mathematical Papyrus
–
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
–
14th problem of the Moscow Mathematical Papyrus (V. Struve, 1930)
Moscow Mathematical Papyrus
–
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)
11.
Kahun Papyri
–
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
–
Fragments of the Kahun Papyrus on
veterinary medicine
12.
Berlin Papyrus 6619
–
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
–
Berlin Papyrus 6619, as reproduced in 1900 by Schack-Schackenburg
13.
Rhind Mathematical Papyrus
–
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
14.
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.
15.
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.
16.
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
17.
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
18.
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
19.
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)
20.
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
21.
Ostracon
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An ostracon is a piece of pottery, usually broken off from a vase or other earthenware vessel. In an archaeological or epigraphical context, ostraca refer to sherds or even pieces of stone that have writing scratched into them. Anything with a surface could be used as a writing surface. But limestone sherds, being flaky and of a color, were most common. The importance of ostraca for Egyptology is immense, the combination of their physical nature and the Egyptian climate have preserved texts, from the medical to the mundane, which in other cultures were lost. These can often serve as witnesses of everyday life than literary treatises preserved in libraries. The many ostraca found at Deir el-Medina provide a compelling view into the medical workings of the New Kingdom. These ostraca have shown that, like other Egyptian communities, the workmen and inhabitants of Deir el-Medina received care through a combination of treatment, prayer. The ostraca from Deir el-Medina also differed in their circulation, magical spells and remedies were widely distributed among the workmen, there are even several cases of spells being sent from one worker to another, with no “trained” intermediary. There are also documents that show the writer sending for medical ingredients. From 1964–1971, Bryan Emery excavated at Saqqara in search of Imhoteps tomb, instead, apparently it was a pilgrim site, with as many as 1½ million ibis birds interred. This 2nd-century BC site contained extensive pottery debris from the offerings of the pilgrims. Emerys excavations uncovered the Dream Ostraca, created by a scribe named Hor of Sebennytos, a devotee of the god Thoth, he lived adjacent to Thoths sanctuary at the entrance to the North Catacomb and worked as a proto-therapist, advising and comforting clients. He transferred his divinely-inspired dreams onto ostraca, the Dream Ostraca are 65 Demotic texts written on pottery and limestone. In October 2008, Israeli archaeologist, Yosef Garfinkel of the Hebrew University of Jerusalem, has discovered what he says to be the earliest known Hebrew text. This text was written on an Ostracon shard, Garfinkel believes this shard dates to the time of King David from the Old Testament, about 3,000 years ago. Carbon dating of the Ostracon and analysis of the pottery have dated the inscription to be about 1,000 years older than the Dead Sea Scrolls, the inscription has yet to be deciphered, however, some words, such as king, slave and judge have been translated. The shard was found about 20 miles southwest of Jerusalem at the Elah Fortress in Khirbet Qeiyafa, some Christian texts are preserved upon ostraca
Ostracon
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Ostrakon of
Cimon, an
Athenian statesman, showing his name (as "Kimon [son] of Miltiades")
Ostracon
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Ostrakon of
Megacles, son of Hippocrates (inscription: ΜΕΓΑΚΛΕΣ ΗΙΠΠΟΚΡΑΤΟΣ), 487 BC. On display in the Ancient Agora Museum in Athens, housed in the
Stoa of Attalus
Ostracon
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Voting ostraca (for
ostracism, Ancient Greece)
Ostracon
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One of four official letters to
vizier Khay copied onto a
limestone ostracon, in Egyptian
Hieratic
22.
Middle Kingdom of Egypt
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Some scholars also include the Thirteenth Dynasty of Egypt wholly into this period as well, in which case the Middle Kingdom would finish c. 1650, while others only include it until Merneferre Ay c.1700 BC, during the Middle Kingdom period, Osiris became the most important deity in popular religion. The period comprises two phases, the 11th Dynasty, which ruled from Thebes and the 12th Dynasty onwards which was centered on el-Lisht, after the collapse of the Old Kingdom, Egypt entered a period of weak Pharaonic power and decentralization called the First Intermediate Period. Towards the end of period, two rival dynasties, known in Egyptology as the Tenth and Eleventh, fought for power over the entire country. The Theban 11th Dynasty only ruled southern Egypt from the first cataract to the Tenth Nome of Upper Egypt, to the north, Lower Egypt was ruled by the rival 10th Dynasty from Herakleopolis. The struggle was to be concluded by Mentuhotep II, who ascended the Theban throne in 2055 B. C, during Mentuhotep IIs fourteenth regnal year, he took advantage of a revolt in the Thinite Nome to launch an attack on Herakleopolis, which met little resistance. After toppling the last rulers of the 10th Dynasty, Mentuhotep began consolidating his power over all Egypt, for this reason, Mentuhotep II is regarded as the founder of the Middle Kingdom. Mentuhotep II commanded military campaigns south as far as the Second Cataract in Nubia and he also restored Egyptian hegemony over the Sinai region, which had been lost to Egypt since the end of the Old Kingdom. He also sent the first expedition to Punt during the Middle Kingdom, by means of ships constructed at the end of Wadi Hammamat, Mentuhotep III was succeeded by Mentuhotep IV, whose name significantly is omitted from all ancient Egyptian king lists. The Turin Papyrus claims that after Mentuhotep III came seven kingless years, despite this absence, his reign is attested from a few inscriptions in Wadi Hammamat that record expeditions to the Red Sea coast and to quarry stone for the royal monuments. The leader of expedition was his vizier Amenemhat, who is widely assumed to be the future pharaoh Amenemhet I. Mentuhotep IVs absence from the king lists has prompted the theory that Amenemhet I usurped his throne, while there are no contemporary accounts of this struggle, certain circumstantial evidence may point to the existence of a civil war at the end of the 11th dynasty. Inscriptions left by one Nehry, the Haty-a of Hermopolis, suggest that he was attacked at a place called Shedyet-sha by the forces of the reigning king, but his forces prevailed. Khnumhotep I, an official under Amenemhet I, claims to have participated in a flotilla of 20 ships to pacify Upper Egypt, donald Redford has suggested these events should be interpreted as evidence of open war between two dynastic claimants. What is certain is that, however he came to power, from the 12th dynasty onwards, pharaohs often kept well-trained standing armies, which included Nubian contingents. These formed the basis of larger forces which were raised for defence against invasion, however, the Middle Kingdom was basically defensive in its military strategy, with fortifications built at the First Cataract of the Nile, in the Delta and across the Sinai Isthmus. Early in his reign, Amenemhet I was compelled to campaign in the Delta region, in addition, he strengthened defenses between Egypt and Asia, building the Walls of the Ruler in the East Delta region. Perhaps in response to this perpetual unrest, Amenemhat I built a new capital for Egypt in the north, known as Amenemhet Itj Tawy, or Amenemhet, the location of this capital is unknown, but is presumably near the citys necropolis, the present-day el-Lisht
Middle Kingdom of Egypt
–
An
Osiride statue of the first pharaoh of the Middle Kingdom, Mentuhotep II
Middle Kingdom of Egypt
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The head of a statue of Senusret I.
Middle Kingdom of Egypt
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Statue head of Senusret III
23.
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
24.
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
25.
Hieratic
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Hieratic is a cursive writing system used in the provenance of the pharaohs in Egypt and Nubia. It developed alongside cursive hieroglyphs, from which it is separate yet intimately related and it was primarily written in ink with a reed brush on papyrus, allowing scribes to write quickly without resorting to the time-consuming hieroglyphs. In the 2nd century AD, the term hieratic was first used by Saint Clement of Alexandria. It derives from the Greek phrase γράμματα ἱερατικά, as at time, hieratic was used only for religious texts, as had been the case for the previous eight. Hieratic can also be an adjective meaning f or associated with sacred persons or offices, in the Proto-Dynastic Period of Egypt, hieratic first appeared and developed alongside the more formal hieroglyphic script. It is an error to view hieratic as a derivative of hieroglyphic writing, indeed, the earliest texts from Egypt are produced with ink and brush, with no indication their signs are descendants of hieroglyphs. True monumental hieroglyphs carved in stone did not appear until the 1st Dynasty, the two writing systems, therefore, are related, parallel developments, rather than a single linear one. Hieratic was used throughout the period and into the Graeco-Roman Period. Around 660 BC, the Demotic script replaced hieratic in most secular writing, through most of its long history, hieratic was used for writing administrative documents, accounts, legal texts, and letters, as well as mathematical, medical, literary, and religious texts. During the Græco-Roman period, when Demotic had become the chief administrative script, in general, hieratic was much more important than hieroglyphs throughout Egypts history, being the script used in daily life. It was also the system first taught to students, knowledge of hieroglyphs being limited to a small minority who were given additional training. In fact, it is possible to detect errors in hieroglyphic texts that came about due to a misunderstanding of an original hieratic text. Most often, hieratic script was written in ink with a brush on papyrus, wood. Thousands of limestone ostraca have been found at the site of Deir al-Madinah, besides papyrus, stone, ceramic shards, and wood, there are hieratic texts on leather rolls, though few have survived. There are also hieratic texts written on cloth, especially on linen used in mummification, there are some hieratic texts inscribed on stone, a variety known as lapidary hieratic, these are particularly common on stelae from the 22nd Dynasty. During the late 6th Dynasty, hieratic was sometimes incised into mud tablets with a stylus, similar to cuneiform. About five hundred of these tablets have been discovered in the palace at Ayn Asil. At the time the tablets were made, Dakhla was located far from centers of papyrus production and these tablets record inventories, name lists, accounts, and approximately fifty letters
Hieratic
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One of four official letters to
vizier Khay copied onto fragments of limestone (an
ostracon).
Hieratic
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Hieratic
Hieratic
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Exercise tablet with hieratic excerpt from
The Instructions of Amenemhat.
Dynasty XVIII, reign of
Amenhotep I, c. 1514–1493 BC. Text reads: "Be on your guard against all who are subordinate to you... Trust no brother, know no friend, make no intimates."
26.
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
27.
Neferetiabet
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Nefertiabet was an ancient Egyptian princess of the 4th dynasty. She was possibly a daughter of Pharaoh Khufu and her tomb at Giza is known. The mastaba is about 24.25 x 11.05 m. in size, a statue of her, now in Munich, probably originates from her tomb. She is best known from her beautiful slab stela, now in the Louvre, Nefertiabet is shown seated facing to right. She is depicted with a wig and a panther skin garment. Her right hand is extended to table, a table in front of her is piled with bread. Under the table offerings are depicted including linen and ointment on the left, and on the offerings of bread, beer, oryx. On the right of the slab a linen list is depicted, the tomb originally contained one shaft which contained the burial of Nefertiabet. The shaft contains a passage and a chamber, fragments of a white limestone coffin with a flat lid were found. A canopic pit had been dug in one of the corners of the chamber, the chamber contained some bowls and jars. An annex with one additional burial shaft was added later, but was completely plundered
Neferetiabet
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Nefertiabet, stela
Neferetiabet
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Nefertiabet's stela from her tomb in Giza.
28.
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.
29.
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
