1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
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
3.
Old Kingdom of Egypt
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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
4.
Ptolemaic Kingdom
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The Ptolemaic Kingdom was a Hellenistic kingdom based in Egypt. Alexandria became the city and a major center of Greek culture. To gain recognition by the native Egyptian populace, they named themselves the successors to the Pharaohs, the later Ptolemies took on Egyptian traditions by marrying their siblings, had themselves portrayed on public monuments in Egyptian style and dress, and participated in Egyptian religious life. The Ptolemies had to fight native rebellions and were involved in foreign and civil wars led to the decline of the kingdom. Hellenistic culture continued to thrive in Egypt throughout the Roman and Byzantine periods until the Muslim conquest. The era of Ptolemaic reign in Egypt is one of the most well documented periods of the Hellenistic Era. In 332 BC, Alexander the Great, King of Macedon invaded the Achaemenid satrapy of Egypt and he visited Memphis, and traveled to the oracle of Amun at the Oasis of Siwa. The oracle declared him to be the son of Amun, the wealth of Egypt could now be harnessed for Alexanders conquest of the rest of the Persian Empire. Early in 331 BC he was ready to depart, and led his forces away to Phoenicia and he left Cleomenes as the ruling nomarch to control Egypt in his absence. Following Alexanders death in Babylon in 323 BC, a crisis erupted among his generals. Perdiccas appointed Ptolemy, one of Alexanders closest companions, to be satrap of Egypt, Ptolemy ruled Egypt from 323 BC, nominally in the name of the joint kings Philip III and Alexander IV. However, as Alexander the Greats empire disintegrated, Ptolemy soon established himself as ruler in his own right, Ptolemy successfully defended Egypt against an invasion by Perdiccas in 321 BC, and consolidated his position in Egypt and the surrounding areas during the Wars of the Diadochi. In 305 BC, Ptolemy took the title of King, as Ptolemy I Soter, he founded the Ptolemaic dynasty that was to rule Egypt for nearly 300 years. All the male rulers of the dynasty took the name Ptolemy, while princesses and queens preferred the names Cleopatra, Arsinoe and Berenice. Because the Ptolemaic kings adopted the Egyptian custom of marrying their sisters, many of the kings ruled jointly with their spouses and this custom made Ptolemaic politics confusingly incestuous, and the later Ptolemies were increasingly feeble. The only Ptolemaic Queens to officially rule on their own were Berenice III, Cleopatra V did co-rule, but it was with another female, Berenice IV. Cleopatra VII officially co-ruled with Ptolemy XIII Theos Philopator, Ptolemy XIV, and Ptolemy XV, upper Egypt, farthest from the centre of government, was less immediately affected, even though Ptolemy I established the Greek colony of Ptolemais Hermiou to be its capital. But within a century Greek influence had spread through the country, nevertheless, the Greeks always remained a privileged minority in Ptolemaic Egypt
5.
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
6.
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
7.
Papyrus
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The word papyrus /pəˈpaɪrəs/ refers to a thick precursor to modern paper made from the pith of the papyrus plant, Cyperus papyrus. Papyrus can also refer to a document written on sheets of papyrus joined together side by side and rolled up into a scroll, the plural for such documents is papyri. Papyrus is first known to have used in ancient Egypt. It was also used throughout the Mediterranean region and in Kingdom of Kush, the Ancient Egyptians used papyrus as a writing material, as well as employing it commonly in the construction of other artifacts such as reed boats, mats, rope, sandals, and baskets. Papyrus was first manufactured in Egypt as far back as the fourth millennium BCE, the earliest archaeological evidence of papyrus was excavated in 2012 and 2013 at Wadi al-Jarf, an ancient Egyptian harbor located on the Red Sea coast. The papyrus rolls describe the last years of building the Great Pyramid of Giza, in the first centuries BCE and CE, papyrus scrolls gained a rival as a writing surface in the form of parchment, which was prepared from animal skins. Sheets of parchment were folded to form quires from which book-form codices were fashioned, early Christian writers soon adopted the codex form, and in the Græco-Roman world, it became common to cut sheets from papyrus rolls to form codices. Codices were an improvement on the scroll, as the papyrus was not pliable enough to fold without cracking. Papyrus had the advantage of being cheap and easy to produce. Unless the papyrus was of quality, the writing surface was irregular. Its last appearance in the Merovingian chancery is with a document of 692, the latest certain dates for the use of papyrus are 1057 for a papal decree, under Pope Victor II, and 1087 for an Arabic document. Its use in Egypt continued until it was replaced by more inexpensive paper introduced by Arabs who originally learned of it from the Chinese, by the 12th century, parchment and paper were in use in the Byzantine Empire, but papyrus was still an option. Papyrus was made in several qualities and prices, pliny the Elder and Isidore of Seville described six variations of papyrus which were sold in the Roman market of the day. These were graded by quality based on how fine, firm, white, grades ranged from the superfine Augustan, which was produced in sheets of 13 digits wide, to the least expensive and most coarse, measuring six digits wide. Materials deemed unusable for writing or less than six digits were considered commercial quality and were pasted edge to edge to be used only for wrapping, until the middle of the 19th century, only some isolated documents written on papyrus were known. They did not contain literary works, the first modern discovery of papyri rolls was made at Herculaneum in 1752. Until then, the papyri known had been a few surviving from medieval times. The English word papyrus derives, via Latin, from Greek πάπυρος, Greek has a second word for it, βύβλος
8.
Egyptian geometry
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Egyptian geometry refers to geometry as it was developed and used in Ancient Egypt. Ancient Egyptian mathematics as discussed here spans a period ranging from ca.3000 BC to ca 300 BC. We only have a number of problems from ancient Egypt that concern geometry. Geometric problems appear in both the Moscow Mathematical Papyrus and in the Rhind Mathematical Papyrus, the examples demonstrate that the Ancient Egyptians knew how to compute areas of several geometric shapes and the volumes of cylinders and pyramids. Also the Egyptians used many sacred geometric shapes such as squares and triangles on temples, the Ancient Egyptians wrote out their problems in multiple parts. They gave the title and the data for the problem, in some of the texts they would show how to solve the problem. The scribes did not use any variables and the problems were written in prose form, the solutions were written out in steps, outlining the process. Triangles, The Ancient Egyptians knew that the area of a triangle is A =12 b h where b = base, calculations of the area of a triangle appear in both the RMP and the MMP. Rectangles, Problem 49 from the RMP finds the area of a plot of land Problem 6 of MMP finds the lengths of the sides of a rectangular area given the ratio of the lengths of the sides. This problem seems to be identical to one of the Lahun Mathematical Papyri in London, the problem is also interesting because it is clear that the Egyptians were familiar with square roots. They even had a hieroglyph for finding a square root. It looks like a corner and appears in the line of the problem. We suspect that they had tables giving the square roots of some often used numbers, no such tables have been found however. Problem 18 of the MMP computes the area of a length of garment-cloth. The Lahun PapyrusProblem 1 in LV.4 is given as, An area of 40 mH by 3 mH shall be divided in 10 areas, a translation of the problem and its solution as it appears on the fragment is given on the website maintained by University College London. Circles, Problem 48 of the RMP compares the area of a circle and this problems result is used in problem 50. The resulting octagonal figure approximates the circle, the area of the octagonal figure is,92 −412 =63 Next we approximate 63 to be 64 and note that 64 =82 Thus the number 42 =3.16049. Plays the role of π =3.14159 and that this octagonal figure, whose area is easily calculated, so accurately approximates the area of the circle is just plain good luck
9.
Surface area
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The surface area of a solid object is a measure of the total area that the surface of the object occupies. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces and this definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of area was sought by Henri Lebesgue. Their work led to the development of measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface, while the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function S ↦ A which assigns a real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the area is its additivity. More rigorously, if a surface S is a union of many pieces S1, …, Sr which do not overlap except at their boundaries. Surface areas of polygonal shapes must agree with their geometrically defined area. Since surface area is a notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface. This means that surface area is invariant under the group of Euclidean motions and these properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of many pieces that can be represented in the parametric form S D, r → = r →, ∈ D with a continuously differentiable function r →. The area of a piece is defined by the formula A = ∬ D | r → u × r → v | d u d v. Thus the area of SD is obtained by integrating the length of the vector r → u × r → v to the surface over the appropriate region D in the parametric uv plane. The area of the surface is then obtained by adding together the areas of the pieces. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f and surfaces of revolution. It was demonstrated by Hermann Schwarz that already for the cylinder, various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a natural notion of surface area, if a surface is very irregular, or rough
10.
Volume
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Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, Volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
11.
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
12.
False position method
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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
13.
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
14.
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
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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
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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
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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
18.
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
19.
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
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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
21.
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
22.
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
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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
24.
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
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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
26.
New Kingdom of Egypt
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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
27.
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
28.
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
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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
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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
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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
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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
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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
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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
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Nefertiabet
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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
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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
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Prehistoric 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
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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
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Karl 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
40.
Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
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Multiplication
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Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5