1.
British Museum
–
The British Museum is dedicated to human history, art and culture, and is located in the Bloomsbury area of London. The British Museum was established in 1753, largely based on the collections of the physician, the museum first opened to the public on 15 January 1759, in Montagu House, on the site of the current building. Although today principally a museum of art objects and antiquities. Its foundations lie in the will of the Irish-born British physician, on 7 June 1753, King George II gave his formal assent to the Act of Parliament which established the British Museum. They were joined in 1757 by the Old Royal Library, now the Royal manuscripts, together these four foundation collections included many of the most treasured books now in the British Library including the Lindisfarne Gospels and the sole surviving copy of Beowulf. The British Museum was the first of a new kind of museum – national, belonging to neither church nor king, freely open to the public, sloanes collection, while including a vast miscellany of objects, tended to reflect his scientific interests. The addition of the Cotton and Harley manuscripts introduced a literary, the body of trustees decided on a converted 17th-century mansion, Montagu House, as a location for the museum, which it bought from the Montagu family for £20,000. The Trustees rejected Buckingham House, on the now occupied by Buckingham Palace, on the grounds of cost. With the acquisition of Montagu House the first exhibition galleries and reading room for scholars opened on 15 January 1759. During the few years after its foundation the British Museum received several gifts, including the Thomason Collection of Civil War Tracts. A list of donations to the Museum, dated 31 January 1784, in the early 19th century the foundations for the extensive collection of sculpture began to be laid and Greek, Roman and Egyptian artefacts dominated the antiquities displays. Gifts and purchases from Henry Salt, British consul general in Egypt, beginning with the Colossal bust of Ramesses II in 1818, many Greek sculptures followed, notably the first purpose-built exhibition space, the Charles Towneley collection, much of it Roman Sculpture, in 1805. In 1816 these masterpieces of art, were acquired by The British Museum by Act of Parliament. The collections were supplemented by the Bassae frieze from Phigaleia, Greece in 1815, the Ancient Near Eastern collection also had its beginnings in 1825 with the purchase of Assyrian and Babylonian antiquities from the widow of Claudius James Rich. The neoclassical architect, Sir Robert Smirke, was asked to draw up plans for an extension to the Museum. For the reception of the Royal Library, and a Picture Gallery over it, and put forward plans for todays quadrangular building, much of which can be seen today. The dilapidated Old Montagu House was demolished and work on the Kings Library Gallery began in 1823, the extension, the East Wing, was completed by 1831. The Museum became a site as Sir Robert Smirkes grand neo-classical building gradually arose
British Museum
–
British Museum
British Museum
–
The centre of the museum was redeveloped in 2001 to become the
Great Court, surrounding the original
Reading Room.
British Museum
–
Sir
Hans Sloane
British Museum
–
Montagu House, c. 1715
2.
Second Intermediate Period of Egypt
–
The Second Intermediate Period marks a period when Ancient Egypt fell into disarray for a second time, between the end of the Middle Kingdom and the start of the New Kingdom. It is best known as the period when the Hyksos made their appearance in Egypt, the Twelfth Dynasty of Egypt came to an end at the end of the 19th century BC with the death of Queen Sobekneferu. Apparently she had no heirs, causing the twelfth dynasty to come to an end, and, with it. Retaining the seat of the dynasty, the thirteenth dynasty ruled from Itjtawy near Memphis and Lisht. The Thirteenth Dynasty is notable for the accession of the first formally recognised Semitic-speaking king, the Fifteenth Dynasty dates approximately from 1650 to 1550 BC. Known rulers of the Fifteenth Dynasty are as follows, Salitis Sakir-Har Khyan Apophis, 1550–1540 BC The Fifteenth Dynasty of Egypt was the first Hyksos dynasty, ruled from Avaris, without control of the entire land. The Hyksos preferred to stay in northern Egypt since they infiltrated from the north-east, the names and order of kings is uncertain. The Turin King list indicates that there were six Hyksos kings, the surviving traces on the X figure appears to give the figure 8 which suggests that the summation should be read as 6 kings ruling 108 years. Some scholars argue there were two Apophis kings named Apepi I and Apepi II, but this is due to the fact there are two known prenomens for this king, Awoserre and Aqenenre. However, the Danish Egyptologist Kim Ryholt maintains in his study of the Second Intermediate Period that these prenomens all refer to one man, Apepi and this is also supported by the fact that this king employed a third prenomen during his reign, Nebkhepeshre. Apepi likely employed several different prenomens throughout various periods of his reign and this scenario is not unprecedented, as later kings, including the famous Ramesses II and Seti II, are known to have used two different prenomens in their own reigns. The Sixteenth Dynasty ruled the Theban region in Upper Egypt for 70 years, of the two chief versions of Manethos Aegyptiaca, Dynasty XVI is described by the more reliable Africanus as shepherd kings, but by Eusebius as Theban. For this reason other scholars do not follow Ryholt and see only insufficient evidence for the interpretation of the Sixteenth Dynasty as Theban, the continuing war against Dynasty XV dominated the short-lived 16th dynasty. The armies of the 15th dynasty, winning town after town from their enemies, continually encroached on the 16th dynasty territory, eventually threatening. Famine, which had plagued Upper Egypt during the late 13th dynasty, from Ryholts reconstruction of the Turin canon,15 kings of the dynasty can now be named, five of whom appear in contemporary sources. While most likely based in Thebes itself, some may have been local rulers from other important Upper Egyptian towns, including Abydos, El Kab. By the reign of Nebiriau I, the controlled by the 16th dynasty extended at least as far north as Hu. Not listed in the Turin canon is Wepwawetemsaf, who left a stele at Abydos and was likely a local kinglet of the Abydos Dynasty, Ryholt gives the list of kings of the 16th dynasty as shown in the table below
Second Intermediate Period of Egypt
–
Thebes (
Luxor Temple pictured) was the capital of many of the Dynasty XVI pharaohs.
3.
Thebes, Egypt
–
Thebes, known to the ancient Egyptians as Waset, was an ancient Egyptian city located east of the Nile about 800 kilometers south of the Mediterranean. Its ruins lie within the modern Egyptian city of Luxor, Thebes was the main city of the fourth Upper Egyptian nome. It was close to Nubia and the desert, with their valuable mineral resources. It was a center and the wealthiest city of ancient Egypt at its heyday. The Ancient Egyptians originally knew Thebes as Wose or Wase A was was the scepter of the pharaohs, a staff with an animals head. Thebes is the Latinized form of the Greek Thebai, the form of the Demotic Egyptian Ta-pe. This was the name not for the city itself but for the Karnak temple complex on the northern east bank of the city. As early as Homers Iliad, the Greeks distinguished the Egyptian Thebes as Thebes of the Hundred Gates, as opposed to the Thebes of the Seven Gates in Boeotia, from the end of the New Kingdom, Thebes was known in Egyptian as Niwt-Imn, the City of Amun. Amun was the chief of the Theban Triad of gods whose other members were Mut and this name appears in the Bible as the Nōʼ ʼĀmôn of the Book of Nahum and probably also as the No mentioned in Ezekiel and Jeremiah. In the interpretatio graeca, Amun was seen as a form of Zeus, the name was therefore translated into Greek as Diospolis, the City of Zeus. To distinguish it from the other cities by this name. The Greek names came into use after the conquest of Egypt by Alexander the Great. Thebes was located along the banks of the Nile River in the part of Upper Egypt about 800 km from the Delta. It was built largely on the plains of the Nile Valley which follows a great bend of the Nile. As a natural consequence, the city was laid in a northeast-southwest axis parallel to the river channel. Thebes had an area of 93 km2 which included parts of the Theban Hills in the west that culminates at the sacred 420-meter al-Qurn, in the east lies the mountainous Eastern Desert with its wadis draining into the valley. Significant of these wadis is Wadi Hammamat near Thebes and it was used as an overland trade route going to the Red Sea coast. In the fourth Upper Egyptian nome, Thebes was found to have neighboring towns such as Per-Hathor, Madu, Djerty, Iuny, Sumenu, according to George Modelski, Thebes had about 40,000 inhabitants in 2000 BC
Thebes, Egypt
–
Egypt - Temple of Seti, east entrance, Thebes. Brooklyn Museum Archives, Goodyear Archival Collection
Thebes, Egypt
–
Luxor Temple
Thebes, Egypt
–
The Theban Necropolis
4.
Egyptian language
–
The language spoken in ancient Egypt was a branch of the Afroasiatic language family. The earliest known complete sentence in the Egyptian language has been dated to about 2690 BCE, making it one of the oldest recorded languages known. Egyptian was spoken until the seventeenth century in the form of Coptic. The national language of modern Egypt is Egyptian Arabic, which gradually replaced Coptic as the language of life in the centuries after the Muslim conquest of Egypt. Coptic is still used as the language of the Coptic Orthodox Church of Alexandria. It has several hundred fluent speakers today, the Egyptian language belongs to the Afroasiatic language family. Of the other Afroasiatic branches, Egyptian shows its greatest affinities with Semitic, in Egyptian, the Proto-Afroasiatic voiced consonants */d z ð/ developed into pharyngeal ⟨ꜥ⟩ /ʕ/, e. g. Eg. Afroasiatic */l/ merged with Egyptian ⟨n⟩, ⟨r⟩, ⟨ꜣ⟩, and ⟨j⟩ in the dialect on which the language was based. Original */k g ḳ/ palatalize to ⟨ṯ j ḏ⟩ in some environments and are preserved as ⟨k g q⟩ in others, Egyptian has many biradical and perhaps monoradical roots, in contrast to the Semitic preference for triradical roots. Egyptian probably is more archaic in this regard, whereas Semitic likely underwent later regularizations converting roots into the triradical pattern, scholars group the Egyptian language into six major chronological divisions, Archaic Egyptian language Old Egyptian language Middle Egyptian language, characterizing Middle Kingdom. Demotic Coptic The earliest Egyptian glyphs date back to around 3300 BC and these early texts are generally lumped together under the general term Archaic Egyptian. They record names, titles and labels, but a few of them show morphological and syntactic features familiar from later, more complete, Old Egyptian is dated from the oldest known complete sentence, found in the tomb of Seth-Peribsen and dated to around 2690 BCE. It reads, dmḏ. n. f t3wj n z3. f nswt-bjt pr-jb. snj He has united the Two Lands for his son, extensive texts appear from about 2600 BCE. Demotic first appears about 650 BCE and survived as a written language until the fifth century CE and it probably survived in the Egyptian countryside as a spoken language for several centuries after that. Bohairic Coptic is still used by the Coptic Churches, Old, Middle, and Late Egyptian were all written using hieroglyphs and hieratic. Demotic was written using a script derived from hieratic, its appearance is similar to modern Arabic script and is also written from right to left. Coptic is written using the Coptic alphabet, a form of the Greek alphabet with a number of symbols borrowed from Demotic for sounds that did not occur in ancient Greek. Arabic became the language of Egypts political administration soon after the early Muslim conquests in the seventh century, today, Coptic survives as the sacred language of the Coptic Orthodox Church of Alexandria and the Coptic Catholic Church
Egyptian language
–
Seal impression from the tomb of
Seth-Peribsen, containing the oldest known complete sentence in Egyptian
Egyptian language
–
Ebers Papyrus detailing treatment of
asthma
Egyptian language
–
3rd-century Coptic inscription
5.
Hieratic
–
Hieratic is a cursive writing system used in the provenance of the pharaohs in Egypt and Nubia. It developed alongside cursive hieroglyphs, from which it is separate yet intimately related and it was primarily written in ink with a reed brush on papyrus, allowing scribes to write quickly without resorting to the time-consuming hieroglyphs. In the 2nd century AD, the term hieratic was first used by Saint Clement of Alexandria. It derives from the Greek phrase γράμματα ἱερατικά, as at time, hieratic was used only for religious texts, as had been the case for the previous eight. Hieratic can also be an adjective meaning f or associated with sacred persons or offices, in the Proto-Dynastic Period of Egypt, hieratic first appeared and developed alongside the more formal hieroglyphic script. It is an error to view hieratic as a derivative of hieroglyphic writing, indeed, the earliest texts from Egypt are produced with ink and brush, with no indication their signs are descendants of hieroglyphs. True monumental hieroglyphs carved in stone did not appear until the 1st Dynasty, the two writing systems, therefore, are related, parallel developments, rather than a single linear one. Hieratic was used throughout the period and into the Graeco-Roman Period. Around 660 BC, the Demotic script replaced hieratic in most secular writing, through most of its long history, hieratic was used for writing administrative documents, accounts, legal texts, and letters, as well as mathematical, medical, literary, and religious texts. During the Græco-Roman period, when Demotic had become the chief administrative script, in general, hieratic was much more important than hieroglyphs throughout Egypts history, being the script used in daily life. It was also the system first taught to students, knowledge of hieroglyphs being limited to a small minority who were given additional training. In fact, it is possible to detect errors in hieroglyphic texts that came about due to a misunderstanding of an original hieratic text. Most often, hieratic script was written in ink with a brush on papyrus, wood. Thousands of limestone ostraca have been found at the site of Deir al-Madinah, besides papyrus, stone, ceramic shards, and wood, there are hieratic texts on leather rolls, though few have survived. There are also hieratic texts written on cloth, especially on linen used in mummification, there are some hieratic texts inscribed on stone, a variety known as lapidary hieratic, these are particularly common on stelae from the 22nd Dynasty. During the late 6th Dynasty, hieratic was sometimes incised into mud tablets with a stylus, similar to cuneiform. About five hundred of these tablets have been discovered in the palace at Ayn Asil. At the time the tablets were made, Dakhla was located far from centers of papyrus production and these tablets record inventories, name lists, accounts, and approximately fifty letters
Hieratic
–
One of four official letters to
vizier Khay copied onto fragments of limestone (an
ostracon).
Hieratic
–
Hieratic
Hieratic
–
Exercise tablet with hieratic excerpt from
The Instructions of Amenemhat.
Dynasty XVIII, reign of
Amenhotep I, c. 1514–1493 BC. Text reads: "Be on your guard against all who are subordinate to you... Trust no brother, know no friend, make no intimates."
6.
Egyptian mathematics
–
Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt c.3000 to c.300 BC. Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found in Tomb U-j at Abydos and these labels appear to have been used as tags for grave goods and some are inscribed with numbers. Further evidence of the use of the base 10 number system can be found on the Narmer Macehead which depicts offerings of 400,000 oxen,1,422,000 goats and 120,000 prisoners. The evidence of the use of mathematics in the Old Kingdom is scarce, the lines in the diagram are spaced at a distance of one cubit and show the use of that unit of measurement. The earliest true mathematical documents date to the 12th dynasty, the Rhind Mathematical Papyrus which dates to the Second Intermediate Period is said to be based on an older mathematical text from the 12th dynasty. The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so-called mathematical problem texts and they consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems, an interesting feature of Ancient Egyptian mathematics is the use of unit fractions. Scribes used tables to help work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain 2 n tables and these tables allowed the scribes to rewrite any fraction of the form 1 n as a sum of unit fractions. During the New Kingdom mathematical problems are mentioned in the literary Papyrus Anastasi I, in the workers village of Deir el-Medina several ostraca have been found that record volumes of dirt removed while quarrying the tombs. Our understanding of ancient Egyptian mathematics is impeded by the paucity of available sources. The Reisner Papyrus dates to the early Twelfth dynasty of Egypt and was found in Nag el-Deir, the Rhind Mathematical Papyrus dates from the Second Intermediate Period, but its author, Ahmes, identifies it as a copy of a now lost Middle Kingdom papyrus. The RMP is the largest mathematical text, from the New Kingdom we have a handful of mathematical texts and inscription related to computations, The Papyrus Anastasi I is a literary text from the New Kingdom. It is written as a written by a scribe named Hori. A segment of the letter describes several mathematical problems, ostracon Senmut 153 is a text written in hieratic. Ostracon Turin 57170 is a written in hieratic. Ostraca from Deir el-Medina contain computations, ostracon IFAO1206 for instance shows the calculations of volumes, presumably related to the quarrying of a tomb
Egyptian mathematics
–
Slab stela of
Old Kingdom princess
Neferetiabet (dated 2590–2565 BC) from her tomb at Giza, painting on limestone, now in the
Louvre.
Egyptian mathematics
–
Image of Problem 14 from the
Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
7.
Alexander Henry Rhind
–
Alexander Henry Rhind was a Scottish antiquarian and archaeologist. Born in Wick on 26 July 1833 in the Highlands, Rhind studied at the University of Edinburgh and he has often been erroneously referred to as a lawyer, but he never actually studied law. Rhind excavated and published a number of sites in northern Scotland in the early 1850s. Suffering from pulmonary disease, he travelled to Egypt in the winters of 1855-1857 with the intention of excavating and collecting for the newly formed National Museum of Antiquities of Scotland. He collected material for his book entitled Thebes, its Tombs and their Tenants and he was a prolific writer with a methodical research style, despite continuing to battle ill health. Among the items that he collected was the Rhind Papyrus, also known as the Ahmes Papyrus after its Egyptian scribe, Rhind acquired it in 1863, and following his death shortly afterwards, it was sold to the British Museum, along with the similar Egyptian Mathematical Leather Roll. Both are mathematical treatises and both were purchased in the Luxor market, and may have previously been stolen from the Ramesseum. When chemically softened and decoded years afterward, they show the Egyptians had computed the value of π as 3.1605, Rhind died in his sleep on 3 July 1863 in Cadenabbia at the age of 30. Rhind directed that a sum from his estate at Sibster, Caithness, be used for purpose, once the interests of living parties was extinguished. Thebes, its tombs and their tenants, ancient and present Memoir of the late Alexander Henry Rhind, of Sibster by John Stuart
Alexander Henry Rhind
–
Engraving from photo in
Stuart 's Memoir by Robert C. Bell
8.
Scotland
–
Scotland is a country that is part of the United Kingdom and covers the northern third of the island of Great Britain. It shares a border with England to the south, and is surrounded by the Atlantic Ocean, with the North Sea to the east. In addition to the mainland, the country is made up of more than 790 islands, including the Northern Isles, the Kingdom of Scotland emerged as an independent sovereign state in the Early Middle Ages and continued to exist until 1707. By inheritance in 1603, James VI, King of Scots, became King of England and King of Ireland, Scotland subsequently entered into a political union with the Kingdom of England on 1 May 1707 to create the new Kingdom of Great Britain. The union also created a new Parliament of Great Britain, which succeeded both the Parliament of Scotland and the Parliament of England. Within Scotland, the monarchy of the United Kingdom has continued to use a variety of styles, titles, the legal system within Scotland has also remained separate from those of England and Wales and Northern Ireland, Scotland constitutes a distinct jurisdiction in both public and private law. Glasgow, Scotlands largest city, was one of the worlds leading industrial cities. Other major urban areas are Aberdeen and Dundee, Scottish waters consist of a large sector of the North Atlantic and the North Sea, containing the largest oil reserves in the European Union. This has given Aberdeen, the third-largest city in Scotland, the title of Europes oil capital, following a referendum in 1997, a Scottish Parliament was re-established, in the form of a devolved unicameral legislature comprising 129 members, having authority over many areas of domestic policy. Scotland is represented in the UK Parliament by 59 MPs and in the European Parliament by 6 MEPs, Scotland is also a member nation of the British–Irish Council, and the British–Irish Parliamentary Assembly. Scotland comes from Scoti, the Latin name for the Gaels, the Late Latin word Scotia was initially used to refer to Ireland. By the 11th century at the latest, Scotia was being used to refer to Scotland north of the River Forth, alongside Albania or Albany, the use of the words Scots and Scotland to encompass all of what is now Scotland became common in the Late Middle Ages. Repeated glaciations, which covered the land mass of modern Scotland. It is believed the first post-glacial groups of hunter-gatherers arrived in Scotland around 12,800 years ago, the groups of settlers began building the first known permanent houses on Scottish soil around 9,500 years ago, and the first villages around 6,000 years ago. The well-preserved village of Skara Brae on the mainland of Orkney dates from this period and it contains the remains of an early Bronze Age ruler laid out on white quartz pebbles and birch bark. It was also discovered for the first time that early Bronze Age people placed flowers in their graves, in the winter of 1850, a severe storm hit Scotland, causing widespread damage and over 200 deaths. In the Bay of Skaill, the storm stripped the earth from a large irregular knoll, when the storm cleared, local villagers found the outline of a village, consisting of a number of small houses without roofs. William Watt of Skaill, the laird, began an amateur excavation of the site, but after uncovering four houses
Scotland
–
Edinburgh Castle. Human habitation of the site is dated back as far as the 9th century BC, although the nature of this early settlement is unclear.
Scotland
–
Flag
Scotland
–
The class I
Pictish stone at
Aberlemno known as Aberlemno 1 or the Serpent Stone.
Scotland
–
The
Wallace Monument commemorates
William Wallace, the 13th-century Scottish hero.
9.
Papyrus
–
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
Papyrus
–
Papyrus (P. BM EA 10591 recto column IX, beginning of lines 13-17)
Papyrus
–
An official letter on a papyrus of the
third century B.C.
Papyrus
–
A section of the Egyptian
Book of the Dead written on papyrus
Papyrus
–
Bill of sale for a donkey, papyrus; 19.3 by 7.2 cm, MS Gr SM2223, Houghton Library, Harvard University
10.
Luxor, Egypt
–
Luxor is a city in Upper Egypt and the capital of Luxor Governorate. The population numbers 487,896, with an area of approximately 416 square kilometres, immediately opposite, across the River Nile, lie the monuments, temples and tombs of the West Bank Necropolis, which includes the Valley of the Kings and Valley of the Queens. Thousands of tourists from all around the world arrive annually to visit these monuments, the name Luxor comes from the Arabic al-ʾuqṣur, lit. the palaces, from the collective pl. of qaṣr, which may be a loanword from the Latin castrum fortified camp. Luxor was the ancient city of Thebes, the capital of Egypt during the New Kingdom. Montuhotep II who united Egypt after the troubles of the first intermediate period brought stability to the lands as the city grew in stature. The city attracted peoples such as the Babylonians, the Mitanni, the Hittites of Anatolia, the Canaanites of Ugarit, the Phoenicians of Byblos and Tyre, a Hittite prince from Anatolia even came to marry with the widow of Tutankhamun, Ankhesenamun. However, as the city of the god Amon-Ra, Thebes remained the capital of Egypt until the Greek period. The main god of the city was Amon, who was worshipped together with his wife, the Goddess Mut, and their son Khonsu, the God of the moon. With the rise of Thebes as the foremost city of Egypt and his great temple, at Karnak just north of Thebes, was the most important temple of Egypt right until the end of antiquity. Later, the city was attacked by Assyrian emperor Assurbanipal who installed the Libyan prince on the throne, the city of Thebes was in ruins and fell in significance. However, Alexander the Great did arrive at the temple of Amun, where the statue of the god was transferred from Karnak during the Opet Festival, Aswan and Luxor have the hottest summer days of any other city in Egypt. Aswan and Luxor have nearly the same climate, Luxor is one of the hottest, sunniest and driest cities in the world. Average high temperatures are above 40 °C during summer while average low temperatures remain above 22 °C, during the coldest month of the year, average high temperatures remain above 22.0 °C while average low temperatures remain above 5 °C. The climate of Luxor has precipitation levels lower than even most other places in the Sahara, the desert city is one of the driest ones in the world, and rainfall does not occur every year. The air is dry in Luxor but much more humid than in Aswan. There is a relative humidity of 39. 9%, with a maximum mean of 57% during winter. In addition, Luxor, Minya, Sohag, Qena and Asyut have the widest difference of temperatures between days and nights of any city in Egypt, with almost 16 °C difference. The hottest temperature recorded was on May 15,1991 which was 50 °C, the Coptic Catholic minority established on November 26,1895 an Eparchy of Luqsor alias Thebes, on territory split off from the Apostolic Vicariate of Egypt
Luxor, Egypt
–
Top: First pylon in Precinct of Amun-Re, 2nd left: Night view in Luxor Temple, 2nd right: Colossi of Memnon Statue, Middle left: Pillars of Great Hypostyle Hall ancient site, Middle right: Hatshepsut Temple in Deir el-Bchari, 4th left: Statue of Ramses Ⅱ in Karnak Temple, 4th right: Needle Monument in Karnak Temple, Bottom: View of Pillars of Great Hypostyle Hall ancient site
Luxor, Egypt
–
Luxor Temple
Luxor, Egypt
–
Streets of Luxor in 2004
Luxor, Egypt
–
Luxor
souq
11.
Ramesseum
–
The Ramesseum is the memorial temple of Pharaoh Ramesses II. It is located in the Theban necropolis in Upper Egypt, across the River Nile from the city of Luxor. It was originally called the House of millions of years of Usermaatra-setepenra that unites with Thebes-the-city in the domain of Amon, Usermaatra-setepenra was the prenomen of Ramesses II. Surviving records indicate that work on the project shortly after the start of his reign. The design of Ramessess mortuary temple adheres to the canons of New Kingdom temple architecture. Oriented northwest and southeast, the temple itself comprised two stone pylons, one after the other, each leading into a courtyard, beyond the second courtyard, at the centre of the complex, was a covered 48-column hypostyle hall, surrounding the inner sanctuary. An enormous pylon stood before the first court, with the palace at the left. As was customary, the pylons and outer walls were decorated with scenes commemorating pharaohs military victories and leaving due record of his dedication to, and kinship with, the gods. The scenes of the pharaoh and his army triumphing over the Hittite forces fleeing before Kadesh, as portrayed in the canons of the epic poem of Pentaur. Only fragments of the base and torso remain of the statue of the enthroned pharaoh,62 feet high. This was alleged to have been transported 170 miles over land and this is the largest remaining colossal statue in the world. However fragments of 4 granite Colossi of Ramses were found in Tanis, estimated height is 69 to 92 feet. Like four of the six colossi of Amenhotep III there are no longer complete remains so it is based partly on unconfirmed estimates, remains of the second court include part of the internal façade of the pylon and a portion of the Osiride portico on the right. Scenes of war and the rout of the Hittites at Kadesh are repeated on the walls, in the upper registers, feast and honour of the phallic god Min, god of fertility. On the opposite side of the court the few Osiride pillars, scattered remains of the two statues of the seated king can also be seen, one in pink granite and the other in black granite, which once flanked the entrance to the temple. The head of one of these has been removed to the British Museum, thirty-nine out of the forty-eight columns in the great hypostyle hall still stand in the central rows. They are decorated with the scenes of the king before various gods. Part of the ceiling decorated with stars on a blue ground has also been preserved
Ramesseum
–
Aerial view of Thebes' Ramesseum, showing pylons and secondary buildings
Ramesseum
–
Osirid statues
Ramesseum
–
The "other" granite head
Ramesseum
–
Laser scanned point cloud image of a headless Osiris pillar, second court, from a
CyArk /
Supreme Council of Antiquities research partnership
12.
Brooklyn Museum
–
The Brooklyn Museum is an art museum located in the New York City borough of Brooklyn. At 560,000 square feet, the museum is New York Citys third largest in physical size, the museum initially struggled to maintain its building and collection, only to be revitalized in the late 20th century, thanks to major renovations. Significant areas of the collection include antiquities, specifically their collection of Egyptian antiquities spanning over 3,000 years, African, Oceanic, and Japanese art make for notable antiquities collections as well. American art is represented, starting at the Colonial period. Artists represented in the collection include Mark Rothko, Edward Hopper, Norman Rockwell, Winslow Homer, Edgar Degas, Georgia OKeeffe, the museum also has a Memorial Sculpture Garden which features salvaged architectural elements from throughout New York City. The roots of the Brooklyn Museum extend back to the 1823 founding by Augustus Graham of the Brooklyn Apprentices’ Library in Brooklyn Heights, in 1890, under its director Franklin Hooper, Institute leaders reorganized as the Brooklyn Institute of Arts and Sciences and began planning the Brooklyn Museum. The initial design for the Brooklyn Museum was four times as large as the actualized version, Daniel Chester French, the noted sculptor of the Lincoln Memorial, was the principal designer of the pediment sculptures and the monolithic 12. 5-foot figures along the cornice. The figures were created by 11 sculptors and carved by the Piccirilli Brothers, by 1920, the New York City Subway reached the museum with a subway station, this greatly improved access to the once-isolated museum from Manhattan and other outer boroughs. The Brooklyn Institutes director Franklin Hooper was the museums first director and he was followed by Philip Newell Youtz, Laurance Page Roberts, Isabel Spaulding Roberts, Charles Nagel, Jr. and Edgar Craig Schenck. Thomas S. Buechner became the director in 1960, making him one of the youngest directors in the country. Buechner oversaw a major transformation in the way the museum displayed art and brought some one thousand works that had languished in the museums archives and put them on display. Buechner played a role in rescuing the Daniel Chester French sculptures from destruction due to an expansion project at the Manhattan Bridge in the 1960s. The Brooklyn Museum changed its name to Brooklyn Museum of Art in 1997, on March 12,2004, the museum announced that it would revert to its previous name. In April 2004, the museum opened the James Polshek-designed entrance pavilion on the Eastern Parkway façade, in September 2014, Lehman announced that he was planning to retire around June 2015. In May 2015, Creative Time president and artistic director Anne Pasternak was named the Museums next director, member institutions occupy land or buildings owned by the City of New York and derive part of their yearly funding from the City. The Brooklyn Museum also supplements its earned income with funding from Federal and State governments, as well as donations by individuals. Major benefactors include Frank Lusk Babbott, the museum is the site of the annual Brooklyn Artists Ball which has included celebrity hosts such as Sarah Jessica Parker and Liv Tyler. The Brooklyn Museum exhibits collections that seek to embody the rich heritage of world cultures
Brooklyn Museum
–
Brooklyn Museum
Brooklyn Museum
–
Replica of the
Statue of Liberty in back lot.
Brooklyn Museum
–
Book of the Dead of the Goldworker of Amun, Sobekmose, 31.1777e
Brooklyn Museum
–
Brooklyn Papyrus 664-332 BCE
13.
New York City
–
The City of New York, often called New York City or simply New York, is the most populous city in the United States. With an estimated 2015 population of 8,550,405 distributed over an area of about 302.6 square miles. Located at the tip of the state of New York. Home to the headquarters of the United Nations, New York is an important center for international diplomacy and has described as the cultural and financial capital of the world. Situated on one of the worlds largest natural harbors, New York City consists of five boroughs, the five boroughs – Brooklyn, Queens, Manhattan, The Bronx, and Staten Island – were consolidated into a single city in 1898. In 2013, the MSA produced a gross metropolitan product of nearly US$1.39 trillion, in 2012, the CSA generated a GMP of over US$1.55 trillion. NYCs MSA and CSA GDP are higher than all but 11 and 12 countries, New York City traces its origin to its 1624 founding in Lower Manhattan as a trading post by colonists of the Dutch Republic and was named New Amsterdam in 1626. The city and its surroundings came under English control in 1664 and were renamed New York after King Charles II of England granted the lands to his brother, New York served as the capital of the United States from 1785 until 1790. It has been the countrys largest city since 1790, the Statue of Liberty greeted millions of immigrants as they came to the Americas by ship in the late 19th and early 20th centuries and is a symbol of the United States and its democracy. In the 21st century, New York has emerged as a node of creativity and entrepreneurship, social tolerance. Several sources have ranked New York the most photographed city in the world, the names of many of the citys bridges, tapered skyscrapers, and parks are known around the world. Manhattans real estate market is among the most expensive in the world, Manhattans Chinatown incorporates the highest concentration of Chinese people in the Western Hemisphere, with multiple signature Chinatowns developing across the city. Providing continuous 24/7 service, the New York City Subway is one of the most extensive metro systems worldwide, with 472 stations in operation. Over 120 colleges and universities are located in New York City, including Columbia University, New York University, and Rockefeller University, during the Wisconsinan glaciation, the New York City region was situated at the edge of a large ice sheet over 1,000 feet in depth. The ice sheet scraped away large amounts of soil, leaving the bedrock that serves as the foundation for much of New York City today. Later on, movement of the ice sheet would contribute to the separation of what are now Long Island and Staten Island. The first documented visit by a European was in 1524 by Giovanni da Verrazzano, a Florentine explorer in the service of the French crown and he claimed the area for France and named it Nouvelle Angoulême. Heavy ice kept him from further exploration, and he returned to Spain in August and he proceeded to sail up what the Dutch would name the North River, named first by Hudson as the Mauritius after Maurice, Prince of Orange
New York City
–
Clockwise, from top:
Midtown Manhattan,
Times Square, the
Unisphere in
Queens, the
Brooklyn Bridge,
Lower Manhattan with
One World Trade Center,
Central Park, the
headquarters of the United Nations, and the
Statue of Liberty
New York City
–
New Amsterdam, centered in the eventual
Lower Manhattan, in 1664, the year
England took control and renamed it "New York".
New York City
–
The
Battle of Long Island, the largest battle of the
American Revolution, took place in
Brooklyn in 1776.
New York City
–
Broadway follows the Native American Wickquasgeck Trail through Manhattan.
14.
Moscow Mathematical Papyrus
–
Golenishchev bought the papyrus in 1892 or 1893 in Thebes. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, approximately 5½ m long and varying between 3.8 and 7.6 cm wide, its format was divided into 25 problems with solutions by the Soviet Orientalist Vasily Vasilievich Struve in 1930. It is a well-known mathematical papyrus along with the Rhind Mathematical Papyrus, the Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two. The problems in the Moscow Papyrus follow no particular order, the papyrus is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a frustum respectively, the remaining problems are more common in nature. Problems 2 and 3 are ships part problems, one of the problems calculates the length of a ships rudder and the other computes the length of a ships mast given that it is 1/3 + 1/5 of the length of a cedar log originally 30 cubits long. Aha problems involve finding unknown quantities if the sum of the quantity, the Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1,19, and 25 of the Moscow Papyrus are Aha problems, for instance problem 19 asks one to calculate a quantity taken 1 and ½ times and added to 4 to make 10. In other words, in mathematical notation one is asked to solve 3 /2 × x +4 =10 Most of the problems are pefsu problems,10 of the 25 problems. A pefsu measures the strength of the beer made from a heqat of grain pefsu = number loaves of bread number of heqats of grain A higher pefsu number means weaker bread or beer, the pefsu number is mentioned in many offering lists. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain. Calculate 1/2 of 5 heqat, the result will be 2 1/2 Take this 2 1/2 four times The result is 10, then you say to him, Behold. The beer quantity is found to be correct, problems 11 and 23 are Baku problems. These calculate the output of workers, problem 11 asks if someone brings in 100 logs measuring 5 by 5, then how many logs measuring 4 by 4 does this correspond to. Problem 23 finds the output of a given that he has to cut. Seven of the problems are geometry problems and range from computing areas of triangles, to finding the surface area of a hemisphere. The 10th problem of the Moscow Mathematical Papyrus asks for a calculation of the area of a hemisphere or possibly the area of a semi-cylinder. Below we assume that the problem refers to the area of a hemisphere, the text of problem 10 runs like this, Example of calculating a basket
Moscow Mathematical Papyrus
–
14th problem of the Moscow Mathematical Papyrus (V. Struve, 1930)
Moscow Mathematical Papyrus
–
The
neutrality of this article is
disputed. Relevant discussion may be found on the
talk page. Please do not remove this message until the
dispute is resolved. (July 2015)
15.
Second Intermediate Period
–
The Second Intermediate Period marks a period when Ancient Egypt fell into disarray for a second time, between the end of the Middle Kingdom and the start of the New Kingdom. It is best known as the period when the Hyksos made their appearance in Egypt, the Twelfth Dynasty of Egypt came to an end at the end of the 19th century BC with the death of Queen Sobekneferu. Apparently she had no heirs, causing the twelfth dynasty to come to an end, and, with it. Retaining the seat of the dynasty, the thirteenth dynasty ruled from Itjtawy near Memphis and Lisht. The Thirteenth Dynasty is notable for the accession of the first formally recognised Semitic-speaking king, the Fifteenth Dynasty dates approximately from 1650 to 1550 BC. Known rulers of the Fifteenth Dynasty are as follows, Salitis Sakir-Har Khyan Apophis, 1550–1540 BC The Fifteenth Dynasty of Egypt was the first Hyksos dynasty, ruled from Avaris, without control of the entire land. The Hyksos preferred to stay in northern Egypt since they infiltrated from the north-east, the names and order of kings is uncertain. The Turin King list indicates that there were six Hyksos kings, the surviving traces on the X figure appears to give the figure 8 which suggests that the summation should be read as 6 kings ruling 108 years. Some scholars argue there were two Apophis kings named Apepi I and Apepi II, but this is due to the fact there are two known prenomens for this king, Awoserre and Aqenenre. However, the Danish Egyptologist Kim Ryholt maintains in his study of the Second Intermediate Period that these prenomens all refer to one man, Apepi and this is also supported by the fact that this king employed a third prenomen during his reign, Nebkhepeshre. Apepi likely employed several different prenomens throughout various periods of his reign and this scenario is not unprecedented, as later kings, including the famous Ramesses II and Seti II, are known to have used two different prenomens in their own reigns. The Sixteenth Dynasty ruled the Theban region in Upper Egypt for 70 years, of the two chief versions of Manethos Aegyptiaca, Dynasty XVI is described by the more reliable Africanus as shepherd kings, but by Eusebius as Theban. For this reason other scholars do not follow Ryholt and see only insufficient evidence for the interpretation of the Sixteenth Dynasty as Theban, the continuing war against Dynasty XV dominated the short-lived 16th dynasty. The armies of the 15th dynasty, winning town after town from their enemies, continually encroached on the 16th dynasty territory, eventually threatening. Famine, which had plagued Upper Egypt during the late 13th dynasty, from Ryholts reconstruction of the Turin canon,15 kings of the dynasty can now be named, five of whom appear in contemporary sources. While most likely based in Thebes itself, some may have been local rulers from other important Upper Egyptian towns, including Abydos, El Kab. By the reign of Nebiriau I, the controlled by the 16th dynasty extended at least as far north as Hu. Not listed in the Turin canon is Wepwawetemsaf, who left a stele at Abydos and was likely a local kinglet of the Abydos Dynasty, Ryholt gives the list of kings of the 16th dynasty as shown in the table below
Second Intermediate Period
–
The political situation in the Second Intermediate Period of Egypt (circa 1650 B.C.E. — circa 1550 B.C.E.) Thebes was briefly conquered by the
Hyksos circa 1580 B.C.E.
Second Intermediate Period
–
Thebes (
Luxor Temple pictured) was the capital of many of the Dynasty XVI pharaohs.
16.
History of ancient Egypt
–
The history of ancient Egypt spans the period from the early prehistoric settlements of the northern Nile valley to the Roman conquest, in 30 BC. The Pharaonic Period is dated from the 32nd century BC, when Upper and Lower Egypt were unified, until the country fell under Macedonian rule, note For alternative revisions to the chronology of Egypt, see Egyptian chronology. Egypts history is split into different periods according to the ruling dynasty of each pharaoh. The dating of events is still a subject of research, the conservative dates are not supported by any reliable absolute date for a span of about three millennia. The following is the list according to conventional Egyptian chronology, traces of these early people appear in the form of artifacts and rock carvings along the terraces of the Nile and in the oases. To the Egyptians the Nile meant life and the desert meant death, evidence also indicates human habitation and cattle herding in the southwestern corner of Egypt near the Sudan border before the 8th millennium BC. Despite this, the idea of an independent bovine domestication event in Africa must be abandoned because subsequent evidence gathered over a period of thirty years has failed to corroborate this, the oldest-known domesticated cattle remains in Africa are from the Faiyum c.4400 BC. Continued desiccation forced the early ancestors of the Egyptians to settle around the Nile more permanently, however, the period from 9th to the 6th millennium BC has left very little in the way of archaeological evidence. The Nile valley of Egypt was basically uninhabitable until the work of clearing and irrigating the land along the banks was started, however it appears that this clearance and irrigation was largely under way by the 6th millennium. By that time, Nile society was already engaged in organized agriculture, at this time, Egyptians in the southwestern corner of Egypt were herding cattle and also constructing large buildings. Mortar was in use by the 4th millennium, the people of the valley and the Nile Delta were self-sufficient and were raising barley and emmer, an early variety of wheat, and stored it in pits lined with reed mats. They raised cattle, goats and pigs and they wove linen, prehistory continues through this time, variously held to begin with the Amratian culture. Between 5500 BC and the 31st century BC, small settlements flourished along the Nile, the Tasian culture was the next to appear, it existed in Upper Egypt starting about 4500 BC. This group is named for the burials found at Deir Tasa, the Tasian culture is notable for producing the earliest blacktop-ware, a type of red and brown pottery painted black on its top and interior. The Badari culture, named for the Badari site near Deir Tasa, followed the Tasian, however, the Badari culture continued to produce the kind of pottery called blacktop-ware, and was assigned the sequence dating numbers between 21 and 29. The Amratian culture is named after the site of el-Amreh, about 120 kilometres south of Badari, el-Amreh was the first site where this culture was found unmingled with the later Gerzeh culture. However, this period is attested at Nagada, and so is also referred to as the Naqada I culture. The Amratian period falls between S. D.30 and 39, newly excavated objects indicate that trade between Upper and Lower Egypt existed at this time
History of ancient Egypt
–
A Naqada II vase decorated with gazelles, on display at the
Louvre.
History of ancient Egypt
–
An
Osiride statue of Mentuhotep II, the founder of the Middle Kingdom
History of ancient Egypt
–
Statuette of
Merankhre Mentuhotep VI, a minor king of the
16th Dynasty, reigning over the Theban region c. 1585 BC.
17.
Pharaoh
–
The word pharaoh ultimately derive from the Egyptian compound pr-ˤ3 great house, written with the two biliteral hieroglyphs pr house and ˤ3 column, here meaning great or high. It was used only in larger phrases such as smr pr-ˤ3 Courtier of the High House, with specific reference to the buildings of the court or palace. From the twelfth dynasty onward, the word appears in a wish formula Great House, may it live, prosper, and be in health, but again only with reference to the royal palace and not the person. During the reign of Thutmose III in the New Kingdom, after the rule of the Hyksos during the Second Intermediate Period. During the eighteenth dynasty the title pharaoh was employed as a designation of the ruler. From the nineteenth dynasty onward pr-ˤ3 on its own was used as regularly as hm. f, the term, therefore, evolved from a word specifically referring to a building to a respectful designation for the ruler, particularly by the twenty-second dynasty and twenty-third dynasty. For instance, the first dated appearance of the pharaoh being attached to a rulers name occurs in Year 17 of Siamun on a fragment from the Karnak Priestly Annals. Here, an induction of an individual to the Amun priesthood is dated specifically to the reign of Pharaoh Siamun and this new practice was continued under his successor Psusennes II and the twenty-second dynasty kings. Shoshenq I was the successor of Siamun. Meanwhile, the old custom of referring to the sovereign simply as pr-ˤ3 continued in traditional Egyptian narratives, by this time, the Late Egyptian word is reconstructed to have been pronounced *par-ʕoʔ whence Herodotus derived the name of one of the Egyptian kings, Φερων. In the Bible, the title also occurs as פרעה, from that, Septuagint φαραώ pharaō and then Late Latin pharaō, both -n stem nouns. The Quran likewise spells it فرعون firawn with n, interestingly, the Arabic combines the original pharyngeal ayin sound from Egyptian, along with the -n ending from Greek. English at first spelt it Pharao, but the King James Bible revived Pharaoh with h from the Hebrew, meanwhile in Egypt itself, *par-ʕoʔ evolved into Sahidic Coptic ⲡⲣ̅ⲣⲟ prro and then rro. Scepters and staves were a sign of authority in ancient Egypt. One of the earliest royal scepters was discovered in the tomb of Khasekhemwy in Abydos, kings were also known to carry a staff, and Pharaoh Anedjib is shown on stone vessels carrying a so-called mks-staff. The scepter with the longest history seems to be the heqa-scepter, the earliest examples of this piece of regalia dates to pre-dynastic times. A scepter was found in a tomb at Abydos that dates to the late Naqada period, another scepter associated with the king is the was-scepter. This is a long staff mounted with an animal head, the earliest known depictions of the was-scepter date to the first dynasty
Pharaoh
–
Den
Pharaoh
–
Narmer Palette
Pharaoh
–
Nomen and prenomen of
Ramesses III
Pharaoh
–
Royal titulary
18.
Amenemhat III
–
See Amenemhat, for other individuals with this name. Amenemhat III, also spelled Amenemhet III, was a pharaoh of the Twelfth Dynasty of Egypt. He ruled from c.1860 BC to c.1814 BC and his reign is regarded as the golden age of the Middle Kingdom. He may have had a coregency with his father, Senusret III. His daughter, Sobekneferu, later succeeded Amenemhat IV, as the last ruler of the twelfth dynasty, Amenemhat IIIs throne name, Nimaatre, means Belonging to the Justice of Re. He built his first pyramid at Dahshur, but there were construction problems, around Year 15 of his reign the king decided to build a new pyramid at Hawara, near the Faiyum. The pyramid at Dahshur was used as ground for several royal women. The mortuary temple attached to the Hawara pyramid may have known to Herodotus and Diodorus Siculus as the Labyrinth. Strabo praised it as a wonder of the world, nevertheless, the kings burial was robbed in antiquity. His daughter or sister, Neferuptah, was buried in a separate pyramid 2 km southwest of the kings, the pyramidion of Amenemhet IIIs pyramid tomb was found toppled from the peak of its structure and preserved relatively intact, it is today located in the Cairo Egyptian Museum. There is very little evidence for military expeditions in the reign of the king, there is only one record for a small mission in year nine of the king. The evidence for that was found in an inscription in Nubia. The short text reports that a mission was guided by the mouth of Nekhen Zamonth who reports that he went north with a small troop. Many expeditions to mining areas are recorded under the king, there are two expeditions known to the Wadi el-Hudi at the southern border of Egypt, where Amethyst was collected. One of the dates to year 11, of the king. Two further to year 20 and to year 28, there were further mining expeditions to the Wadi Hammamat. These are dated to year 2,3,19,20 and 33 of the kings reign, the inscriptions of year 19 and 20 might be related to the building start of the pyramid complex at Hawara. They report the breaking of stone for statues, at the Red Sea coast, at Mersa was discovered a stela mentioning an expedition to Punt under Amenemhat III
Amenemhat III
–
Statuette head of Amenemhat III, now in the
Louvre
Amenemhat III
–
Statue from the
Egyptian Collection of the Hermitage Museum
Amenemhat III
–
Pectoral of Amenemhat III, tomb of Mereret.
Amenemhat III
–
Pyramidion or Capstone of Amenemhat III's pyramid.
19.
Manuscript
–
A manuscript is any document written by hand or typewritten, as opposed to being mechanically printed or reproduced in some automated way. More recently, it is understood to be a written, typed, or word-processed copy of a work. Before the arrival of printing, all documents and books were manuscripts, manuscripts are not defined by their contents, which may combine writing with mathematical calculations, maps, explanatory figures or illustrations. Manuscripts may be in form, scrolls or in codex format. Illuminated manuscripts are enriched with pictures, border decorations, elaborately embossed initial letters or full-page illustrations. The traditional abbreviations are MS for manuscript and MSS for manuscripts, while the forms MS. ms or ms. for singular, and MSS. mss or mss. for plural are also accepted. The second s is not simply the plural, by an old convention, it doubles the last letter of the abbreviation to express the plural, just as pp. means pages. Before the invention of printing in China or by moveable type in a printing press in Europe. Historically, manuscripts were produced in form of scrolls or books, manuscripts were produced on vellum and other parchment, on papyrus, and on paper. In Russia birch bark documents as old as from the 11th century have survived, in India, the palm leaf manuscript, with a distinctive long rectangular shape, was used from ancient times until the 19th century. Paper spread from China via the Islamic world to Europe by the 14th century, when Greek or Latin works were published, numerous professional copies were made simultaneously by scribes in a scriptorium, each making a single copy from an original that was declaimed aloud. Manuscripts in Tocharian languages, written on leaves, survived in desert burials in the Tarim Basin of Central Asia. Volcanic ash preserved some of the Roman library of the Villa of the Papyri in Herculaneum, ironically, the manuscripts that were being most carefully preserved in the libraries of antiquity are virtually all lost. Originally, all books were in manuscript form, in China, and later other parts of East Asia, woodblock printing was used for books from about the 7th century. The earliest dated example is the Diamond Sutra of 868, in the Islamic world and the West, all books were in manuscript until the introduction of movable type printing in about 1450. Manuscript copying of books continued for a least a century, as printing remained expensive, private or government documents remained hand-written until the invention of the typewriter in the late 19th century. In the Philippines, for example, as early as 900AD, specimen documents were not inscribed by stylus and this type of document was rare compared to the usual leaves and bamboo staves that were inscribed. However, neither the leaves nor paper were as durable as the document in the hot
Manuscript
–
Christ Pantocrator seated in a capital "U" in an
illuminated manuscript from the Badische Landesbibliothek, Germany.
Manuscript
–
10th-century minuscule manuscript of
Thucydides 's
History of the Peloponnesian War
Manuscript
–
Armenian Manuscript
Manuscript
20.
Hyksos
–
The Hyksos were a people of mixed origins from Western Asia, who settled in the eastern Nile Delta, some time before 1650 BC. The arrival of the Hyksos led to the end of the Thirteenth Dynasty of Egypt, in the context of Ancient Egypt, the term Asiatic – which is often used of the Hyksos – may refer to any people native to areas east of Egypt. Immigration by Canaanite populations preceded the Hyksos, canaanites first appeared in Egypt towards the end of the 12th Dynasty c.1800 BC, and either around that time or c.1720 BC, established an independent realm in the eastern Nile Delta. The Canaanite rulers of the Delta, regrouped in the Fourteenth Dynasty, coexisted with the Egyptian Thirteenth Dynasty, the power of the 13th and 14th Dynasties progressively waned, perhaps due to famine and plague. In about 1650 BC, both dynasties were invaded by the Hyksos, who formed the Fifteenth Dynasty. The collapse of the Thirteenth Dynasty created a vacuum in the south, which may have led to the rise of the Sixteenth Dynasty, based in Thebes. The Hyksos eventually conquered both, albeit for only a time in the case of Thebes. From then on, the 17th Dynasty took control of Thebes and reigned for some time in peaceful coexistence with the Hyksos kings, eventually, Seqenenre Tao, Kamose and Ahmose waged war against the Hyksos and expelled Khamudi, their last king, from Egypt c.1550 BC. The Hyksos practiced horse burials, and their deity, their native storm god, Baal, became associated with the Egyptian storm and desert god. The Hyksos were a people of mixed Asiatic origin with mainly Semitic-speaking components, although some scholars have suggested that the Hyksos contained a Hurrian component, most other scholars have dismissed this possibility. The Hyksos brought several innovations to Egypt, as well as cultural infusions such as new musical instruments. The changes introduced include new techniques of working and pottery, new breeds of animals. In warfare, they introduced the horse and chariot, the bow, improved battle axes. Because of these advances, Hyksos rule became decisive for Egypt’s later empire in the Middle East. There are various hypotheses as to the Hyksos ethnic identity, most archaeologists describe the Hyksos as multi-ethnic, to include all of the peoples who occupied the Nile Delta. The origin of the term Hyksos derives from the Egyptian expression hekau khaswet, the German Egyptologist Wolfgang Helck once argued that the Hyksos were part of massive and widespread Hurrian and Indo-Aryan migrations into the Near East. According to Helck, the Hyksos were Hurrians and part of a Hurrian empire that, most scholars have rejected this theory, and Helck himself abandoned this hypothesis in a 1993 article. The Hyksos were likely Semites who came from the Eastern Mediterranean, khyans name has generally been interpreted as Amorite Hayanu which the Egyptian form represents perfectly, and this is in all likelihood the correct interpretation
Hyksos
–
A group of Asiatic peoples (perhaps the future Hyksos) depicted entering Egypt c.1900 BC from the tomb of a 12th Dynasty official
Khnumhotep II under pharaoh
Senusret II at
Beni Hasan. The glyphs above are above the head of the first animal
Hyksos
–
Mummified head of
Seqenenre Tao, bearing axe-blade wounds. The common theory is that he died in a battle against the Hyksos.
21.
Apepi I
–
Apepi or Apophis was a ruler of Lower Egypt during the fifteenth dynasty and the end of the Second Intermediate Period that was dominated by this foreign dynasty of rulers called the Hyksos. Neb-khepesh-Re, A-qenen-Re and A-user-Re are three praenomina or throne names used by this same ruler during various parts of his reign, in the final decade or so of his reign, Apepi chose Auserre as his last prenomen. While the prenomen was altered, there is no difference in the translation of both Aqenenre and Auserre and his Horus name Shetep-tawy is attested only twice. It appears on a table and on blocks found at Bubastis. Rather than building his own monuments, Apepi generally usurped the monuments of previous pharaohs by inscribing his own name over two sphinxes of Amenemhat II and two statues of Imyremeshaw and he was succeeded by Khamudi, the last Hyksos ruler. Ahmose I, who drove out the Hyksos kings from Egypt, in the Ramesside era, he is recorded as worshiping Seth in a monolatric way, chose for his Lord the god Seth. He didnt worship any other deity in the land except Seth. There is some discussion in Egyptology concerning whether Apepi also ruled Upper Egypt, there are indeed several objects with the kings name most likely coming from Thebes and Upper Egypt. These include a dagger with the name of the bought on the art market in Luxor. There is an axe of unknown provenance where the king is called beloved of Sobek, Sumenu is nowadays identified with Mahamid Qibli, about 24 kilometers south of Thebes and there is a fragment of a stone vessel found in a Theban tomb. For all these objects it is arguable that they were traded to Upper Egypt, more problematic is a block with the kings name found at Gebelein. The block had been taken as evidence for building activity of the king in Upper Egypt and, hence, seen as proof that the Hyksos also ruled in Upper Egypt. However, the block is not very big and many scholars argue today, two sisters are known, Tani and Ziwat. Tani is mentioned on a door of a shrine in Avaris and she was the sister of the king. Ziwat is mentioned on a found in Spain. A Prince Apepi, named on a seal is likely to have been his son, the vase, however, could have been an item which was looted from Avaris after the eventual victory over the Hyksos by Ahmose I
Apepi I
–
Dagger with the names Neb-Khepesh-Re Apepi.
Apepi I
–
Scarab bearing the final prenomen of the Hyksos pharaoh Apepi
22.
Verso
–
The terms recto and verso refer to the text written on the front and back sides of a leaf of paper in a bound item such as a codex, book, broadsheet, or pamphlet. The terms are shortened from Latin rectō foliō and versō foliō, translating to on the side of the page and on the turned side of the page. The page faces themselves are called folium rectum and folium versum in Latin, in codicology, each physical sheet of a manuscript is numbered and the sides are referred to as rectum and folium versum, abbreviated as r and v respectively. Editions of manuscripts will thus mark the position of text in the manuscript in the form fol. 1r, sometimes with the r and v in superscript, as in 1r, or with a superscript o indicating the ablative recto, verso and this terminology has been standard since the beginnings of modern codicology in the 17th century. The use of the recto and verso are also used in the codicology of manuscripts written in right-to-left scripts, like Syriac, Arabic. However, as these scripts are written in the direction to the scripts witnessed in European codices. The reading order of each folio remains first recto, then verso regardless of writing direction, the distinction between recto and verso can be convenient in the annotation of scholarly books, particularly in bilingual edition translations. The recto and verso terms can also be employed for the front and back of a one-sheet artwork, a recto-verso drawing is a sheet with drawings on both sides, for example in a sketchbook—although usually in these cases there is no obvious primary side. Some works are planned to exploit being on two sides of the piece of paper, but usually the works are not intended to be considered together. Paper was relatively expensive in the past, indeed good drawing paper still is more expensive than normal paper. In some early printed books, it is the rather than the pages. Thus each folium carries a number on its recto side. This was also common in e. g. internal company reports in the 20th century. Book design Obverse and reverse in coins Page spread
Verso
–
Left-to-right language books (such as English)
23.
Khamudi
–
Khamudi was the last Hyksos ruler of the Fifteenth Dynasty of Egypt. Khamudi came to power in 1534 BC or 1541 BC, ruling the northern portion of Egypt from his capital Avaris and his ultimate defeat at the hands of Ahmose I, after a short reign, marks the end of the Second Intermediate Period. Khamudi is listed on the Turin canon, column 10, line 28 as the last Hyksos king, beyond this, only two scarab seals are firmly attributed to him, both from Jericho. Additionally, a seal of unknown provenance but possibly from Byblos is inscribed with a cartouche which may read Khamudi. This reading is contested by the egyptologist Kim Ryholt who proposed that the cartouche reads Kandy instead and refers to an hitherto unknown king. In any case, even if the cartouche bears Khamudis name, the seal is currently housed in the Petrie Museum, catalog number UC11616. Based on the scarcity of material dating to Khamudis reign, Ryholt has proposed that his reign must have been short, amounting to no more than a year. In this situation, Khamudi would have inherited more than the Hyksos throne, being possibly already besieged in Sharuhen. This is contested by scholars, such as Manfred Bietak. Bietak and many believe that this year 11 belongs to Khamudi since the text of the papyrus refers to Ahmose I. Since he of the South must denote the Theban ruler Ahmose, the Hyksos capital Avaris will have fallen to Ahmose not much later. Another date on the papyrus is dated to Year 33 of Khamudis predecessor Apepi. It is generally believed that Ahmose I defeated the Hyksos king by his 18th or 19th regnal year and this is suggested by a graffito in the quarry at Tura whereby oxen from Canaan were used at the opening of the quarry in Ahmoses regnal year 22
Khamudi
–
Cylinder seal with a cartouche possibly reading "Khamudi".
24.
Egyptian fraction
–
An Egyptian fraction is a finite sum of distinct unit fractions, such as 12 +13 +116. That is, each fraction in the expression has an equal to 1 and a denominator that is a positive integer. The value of an expression of type is a positive rational number a/b. Every positive rational number can be represented by an Egyptian fraction, in modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern theory and recreational mathematics. Beyond their historical use, Egyptian fractions have some advantages over other representations of fractional numbers. For instance, Egyptian fractions can help in dividing a number of objects into equal shares, for more information on this subject, see Egyptian numerals, Eye of Horus, and Egyptian mathematics. Egyptian fraction notation was developed in the Middle Kingdom of Egypt, five early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner Papyrus, the Kahun Papyrus and the Akhmim Wooden Tablet. A later text, the Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions, the Rhind papyrus was written by Ahmes and dates from the Second Intermediate Period, it includes a table of Egyptian fraction expansions for rational numbers 2/n, as well as 84 word problems. Solutions to each problem were written out in scribal shorthand, with the answers of all 84 problems being expressed in Egyptian fraction notation. 2/n tables similar to the one on the Rhind papyrus also appear on some of the other texts, however, as the Kahun Papyrus shows, vulgar fractions were also used by scribes within their calculations. To write the unit used in their Egyptian fraction notation, in hieroglyph script. Similarly in hieratic script they drew a line over the letter representing the number. For example, The Egyptians had special symbols for 1/2, 2/3, the remaining number after subtracting one of these special fractions was written using as a sum of distinct unit fractions according to the usual Egyptian fraction notation. These have been called Horus-Eye fractions after a theory that they were based on the parts of the Eye of Horus symbol, the unit fraction 1/n is expressed as n, and the fraction 2/n is expressed as n, and the plus sign “＋” is omitted. For example, 2/3 = 1/2 + 1/6 is expressed as 3 =26, modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form 2/n in the Rhind papyrus, although these expansions can generally be described as algebraic identities, the methods used by the Egyptians may not correspond directly to these identities. This method is available for not only odd prime denominators but also all odd denominators, for larger prime denominators, an expansion of the form 2/p = 1/A + 2A − p/Ap was used, where A is a number with many divisors between p/2 and p
Egyptian fraction
–
Eye of Horus
25.
Linear equations
–
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. A simple example of an equation with only one variable, x, may be written in the form, ax + b =0, where a and b are constants. The constants may be numbers, parameters, or even functions of parameters. Linear equations can have one or more variables. An example of an equation with three variables, x, y, and z, is given by, ax + by + cz + d =0, where a, b, c, and d are constants and a, b. Linear equations occur frequently in most subareas of mathematics and especially in applied mathematics, an equation is linear if the sum of the exponents of the variables of each term is one. Equations with exponents greater than one are non-linear, an example of a non-linear equation of two variables is axy + b =0, where a and b are constants and a ≠0. It has two variables, x and y, and is non-linear because the sum of the exponents of the variables in the first term and this article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients, a linear equation in one unknown x may always be rewritten a x = b. If a ≠0, there is a solution x = b a. The origin of the name comes from the fact that the set of solutions of such an equation forms a straight line in the plane. Linear equations can be using the laws of elementary algebra into several different forms. These equations are referred to as the equations of the straight line. In what follows, x, y, t, and θ are variables, in the general form the linear equation is written as, A x + B y = C, where A and B are not both equal to zero. The equation is written so that A ≥0, by convention. The graph of the equation is a line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, if B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis, is C/B, and the slope of the line is −A/B. The general form is written as, a x + b y + c =0
Linear equations
–
Graph sample of linear equations.
26.
Ancient Egyptian units of measurement
–
Egyptian Circle Egyptian units of length are attested from the Early Dynastic Period, when the Palermo stone recorded the level of the Nile River. During the reign of Pharaoh Djer, the height of the Nile was recorded as 6 cubits and 1 palm, a 3rd-dynasty diagram shows how to construct an elliptical vault using simple measures along an arc. The ostracon depicting this diagram was found near the Step Pyramid of Saqqara, a curve is divided into five sections and the height of the curve is given in cubits, palms, and digits in each of the sections. At some point, lengths were standardized by cubit rods, examples have been found in the tombs of officials, noting lengths up to remen. Royal cubits were used for land measures such as roads and fields, fourteen rods, including one double-cubit rod, were described and compared by Lepsius. Two examples are known from the Saqqara tomb of Maya, the treasurer of Tutankhamun, another was found in the tomb of Kha in Thebes. These cubits are about 52.5 cm long and are divided into palms and hands, each palm is divided into four fingers from left to right and the fingers are further subdivided into ro from right to left. The rules are divided into hands so that for example one foot is given as three hands and fifteen fingers and also as four palms and sixteen fingers. Surveying and itinerant measurement were undertaken using rods, poles, a scene in the tomb of Menna in Thebes shows surveyors measuring a plot of land using rope with knots tied at regular intervals. Similar scenes can be found in the tombs of Amenhotep-Sesi, Khaemhat, the balls of rope are also shown in New Kingdom statues of officials such as Senenmut, Amenemhet-Surer, and Penanhor. The digit was also subdivided into smaller fractions of ½, ⅓, ¼, minor units include the Middle Kingdom reed of 2 royal cubits, the Ptolemaic xylon of three royal cubits, the Ptolemaic fathom of four lesser cubits, and the kalamos of six royal cubits. Records of land area also date to the Early Dynastic Period, the Palermo stone records grants of land expressed in terms of kha and setat. Mathematical papyri also include units of area in their problems. The setat was the unit of land measure and may originally have varied in size across Egypts nomes. Later, it was equal to one square khet, where a khet measured 100 cubits, the setat could be divided into strips one khet long and ten cubit wide.25 m². A36 sq. cubit area was known as a kalamos, the uncommon bikos may have been 1½ hammata or another name for the cubit strip. The Coptic shipa was a unit of uncertain value, possibly derived from Nubia. Units of volume appear in the mathematical papyri, for example, computing the volume of a circular granary in RMP42 involves cubic cubits, khar, heqats, and quadruple heqats
Ancient Egyptian units of measurement
–
Cubit rod from the Turin Museum.
Ancient Egyptian units of measurement
–
Problem 80 on the Rhind Mathematical Papyrus
27.
Dimensional analysis
–
Converting from one dimensional unit to another is often somewhat complex. Dimensional analysis, or more specifically the method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra. The concept of physical dimension was introduced by Joseph Fourier in 1822, Physical quantities that are measurable have the same dimension and can be directly compared to each other, even if they are originally expressed in differing units of measure. If physical quantities have different dimensions, they cannot be compared by similar units, hence, it is meaningless to ask whether a kilogram is greater than, equal to, or less than an hour. Any physically meaningful equation will have the dimensions on their left and right sides. Checking for dimensional homogeneity is an application of dimensional analysis. Dimensional analysis is routinely used as a check of the plausibility of derived equations and computations. It is generally used to categorize types of quantities and units based on their relationship to or dependence on other units. Many parameters and measurements in the sciences and engineering are expressed as a concrete number – a numerical quantity. Often a quantity is expressed in terms of other quantities, for example, speed is a combination of length and time. Compound relations with per are expressed with division, e. g.60 mi/1 h, other relations can involve multiplication, powers, or combinations thereof. A base unit is a unit that cannot be expressed as a combination of other units, for example, units for length and time are normally chosen as base units. Units for volume, however, can be factored into the units of length. Sometimes the names of units obscure that they are derived units, for example, an ampere is a unit of electric current, which is equivalent to electric charge per unit time and is measured in coulombs per second, so 1 A =1 C/s. Similarly, one newton is 1 kg⋅m/s2, percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as 1/100, derivatives with respect to a quantity add the dimensions of the variable one is differentiating with respect to on the denominator. Thus, position has the dimension L, derivative of position with respect to time has dimension LT−1 – length from position, time from the derivative, the second derivative has dimension LT−2. In economics, one distinguishes between stocks and flows, a stock has units of units, while a flow is a derivative of a stock, in some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions
Dimensional analysis
–
Base quantity
28.
Geometry
–
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
Geometry
–
Visual checking of the
Pythagorean theorem for the (3, 4, 5)
triangle as in the
Chou Pei Suan Ching 500–200 BC.
Geometry
–
An illustration of
Desargues' theorem, an important result in
Euclidean and
projective geometry
Geometry
–
Geometry lessons in the 20th century
Geometry
–
A
European and an
Arab practicing geometry in the 15th century.
29.
Eye of Horus
–
The Eye of Horus is an ancient Egyptian symbol of protection, royal power and good health. The eye is personified in the goddess Wadjet, the Eye of Horus is similar to the Eye of Ra, which belongs to a different god, Ra, but represents many of the same concepts. Wadjet was one of the earliest of Egyptian deities who later associated with other goddesses such as Bast, Sekhmet, Mut. She was the deity of Lower Egypt and the major Delta shrine the per-nu was under her protection. Hathor is also depicted with this eye, funerary amulets were often made in the shape of the Eye of Horus. The Wadjet or Eye of Horus is the element of seven gold, faience, carnelian. The Wedjat was intended to protect the pharaoh in the afterlife, Ancient Egyptian and Middle-Eastern sailors would frequently paint the symbol on the bow of their vessel to ensure safe sea travel. Horus was the ancient Egyptian sky god who was depicted as a falcon. His right eye was associated with the sun god, Ra, the eye symbol represents the marking around the eye of the falcon, including the teardrop marking sometimes found below the eye. The mirror image, or left eye, sometimes represented the moon, in one myth, when Set and Horus were fighting for the throne after Osiriss death, Set gouged out Horuss left eye. The majority of the eye was restored by either Hathor or Thoth, when Horuss eye was recovered, he offered it to his father, Osiris, in hopes of restoring his life. Hence, the eye of Horus was often used to sacrifice, healing, restoration. There are seven different hieroglyphs used to represent the eye, most commonly ir. t in Egyptian, in Egyptian myth the eye was not the passive organ of sight but more an agent of action, protection or wrath. The Eye of Horus was represented as a hieroglyph, designated D10 in Gardiners sign list and it is represented in the Unicode character block for Egyptian hieroglyphs as U+13080. In Ancient Egyptian most fractions were written as the sum of two or more unit fractions, with scribes possessing tables of answers, thus instead of 3⁄4, one would write 1⁄2 + 1⁄4. Studies from the 1970s to this day in Egyptian mathematics have clearly shown this theory was fallacious, the evolution of the symbols used in mathematics, although similar to the different parts of the Eye of Horus, is now known to be distinct. Wadjet eye tatoos associated with Hathor depicted on 3, 000-year-old mummy
Eye of Horus
–
An Eye of Horus or Wedjat
pendant
Eye of Horus
–
The
Wedjat, later called The Eye of Horus
Eye of Horus
–
The crown of a
Nubian king
Eye of Horus
–
Wooden case decorated with bronze, silver, ivory and gold
30.
Area
–
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
Area
–
A square metre
quadrat made of PVC pipe.
Area
–
The combined area of these three
shapes is
approximately 15.57
squares.
31.
Area of a disk
–
In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents a constant, approximately equal to 3.14159, one method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. The area of a polygon is half its perimeter multiplied by the distance from its center to its sides. Therefore, the area of a disk is the precise phrase for the area enclosed by a circle. Modern mathematics can obtain the area using the methods of calculus or its more sophisticated offspring. However the area of a disk was studied by the Ancient Greeks, eudoxus of Cnidus in the fifth century B. C. had found that the area of a disk is proportional to its radius squared. The circumference is 2πr, and the area of a triangle is half the times the height. A variety of arguments have been advanced historically to establish the equation A = π r 2 of varying degrees of mathematical rigor, the area of a regular polygon is half its perimeter times the apothem. As the number of sides of the regular polygon increases, the polygon tends to a circle, and this suggests that the area of a disk is half the circumference of its bounding circle times the radius. Following Archimedes, compare the area enclosed by a circle to a triangle whose base has the length of the circles circumference. If the area of the circle is not equal to that of the triangle and we eliminate each of these by contradiction, leaving equality as the only possibility. We use regular polygons in the same way, suppose that the area C enclosed by the circle is greater than the area T = 1⁄2cr of the triangle. Let E denote the excess amount, inscribe a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments, if the total area of those gaps, G4, is greater than E, split each arc in half. This makes the square into an inscribed octagon, and produces eight segments with a smaller total gap. Continue splitting until the total gap area, Gn, is less than E, now the area of the inscribed polygon, Pn = C − Gn, must be greater than that of the triangle. E = C − T > G n P n = C − G n > C − E P n > T But this forces a contradiction, as follows. Draw a perpendicular from the center to the midpoint of a side of the polygon, its length, also, let each side of the polygon have length s, then the sum of the sides, ns, is less than the circle circumference
Area of a disk
–
Circle with square and octagon inscribed, showing area gap
32.
Pi
–
The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
Pi
–
The constant π is represented in this
mosaic outside the Mathematics Building at the
Technical University of Berlin.
Pi
–
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.
Pi
–
Archimedes developed the polygonal approach to approximating π.
Pi
–
Isaac Newton used
infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".
33.
Pyramid
–
A pyramid is a structure whose outer surfaces are triangular and converge to a single point at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, as such, a pyramid has at least three outer triangular surfaces. The square pyramid, with base and four triangular outer surfaces, is a common version. A pyramids design, with the majority of the closer to the ground. This distribution of weight allowed early civilizations to create stable monumental structures and it has been demonstrated that the common shape of the pyramids of antiquity, from Egypt to Central America, represents the dry-stone construction that requires minimum human work. Pyramids have been built by civilizations in many parts of the world, the largest pyramid by volume is the Great Pyramid of Cholula, in the Mexican state of Puebla. Khufus Pyramid is built mainly of limestone, and is considered an architectural masterpiece. It contains over 2,000,000 blocks ranging in weight from 2.5 tonnes to 15 tonnes and is built on a base with sides measuring about 230 m. Its four sides face the four cardinal points precisely and it has an angle of 52 degrees and it is still the tallest pyramid. The Mesopotamians built the earliest pyramidal structures, called ziggurats, in ancient times, these were brightly painted in gold/bronze. Since they were constructed of sun-dried mud-brick, little remains of them, ziggurats were built by the Sumerians, Babylonians, Elamites, Akkadians, and Assyrians for local religions. Each ziggurat was part of a complex which included other buildings. The precursors of the ziggurat were raised platforms that date from the Ubaid period during the fourth millennium BC, the earliest ziggurats began near the end of the Early Dynastic Period. The latest Mesopotamian ziggurats date from the 6th century BC, built in receding tiers upon a rectangular, oval, or square platform, the ziggurat was a pyramidal structure with a flat top. Sun-baked bricks made up the core of the ziggurat with facings of fired bricks on the outside, the facings were often glazed in different colors and may have had astrological significance. Kings sometimes had their names engraved on these glazed bricks, the number of tiers ranged from two to seven. It is assumed that they had shrines at the top, but there is no evidence for this. Access to the shrine would have been by a series of ramps on one side of the ziggurat or by a ramp from base to summit
Pyramid
–
The
Egyptian pyramids of the
Giza Necropolis, as seen from the air
Pyramid
–
Pyramid of the Moon,
Teotihuacan
Pyramid
–
Prasat Thom temple at
Koh Ker
Pyramid
–
Pyramids of Güímar,
Tenerife (
Spain)
34.
Seked
–
Seked is an ancient Egyptian unit of measure for the inclination of the triangular faces of a right pyramid. The system was based on the Egyptians length measure known as the royal cubit and it was subdivided into seven palms, each of which was sub-divided into four digits. The inclination of measured slopes was therefore expressed as the number of horizontal palms, a form of our modern measure of gradient, is thus a measure equivalent to our modern cotangent of the angle of slope. The most famous example of a slope is of the Great Pyramid of Giza in Egypt built around 2,550 B. C. The most widely quoted example is perhaps problem 56 from the Rhind Mathematical Papyrus, the most famous of all the pyramids of Egypt is the Great Pyramid of Giza built around 2,550 B. C. 84° from the horizontal, using the modern 360 degree system. Furthermore, according to Petries survey data in The Pyramids and Temples of Gizeh the mean slope of the Great Pyramids entrance passage is 26°3123 ±5. This is less than 1/20th of one degree in deviation from a slope of 1 in 2. This equates to a seked of 14, and is considered to have been the intentional designed slope applied by the Old Kingdom builders for internal passages. It is thus an equivalent to our modern cotangent of the angle of slope. In general, the seked of a pyramid is a kind of fraction, given as so many palms horizontally for each cubit of vertically, the Egyptian word seked is thus related to our modern word gradient. The Great Pyramid scholar Professor I. E. S Edwards considered this to have been the normal or most typical choice for pyramids. Petrie wrote. these relations of areas and of circular ratio are so systematic that we should grant that they were in the builders design, cubit Great Pyramid of Giza Rhind papyrus Edwards, I. E. S. Mathematics in the Time of the Pharaohs, Egyptian Tomb Architecture, The Archaeological Facts of Pharaonic Circular Symbolism. British Archaeological Reports International Series S1852, the Pyramids and Temples of Gizeh. British School of Archaeology in Egypt and B, verner, Miroslav, The Pyramids – Their Archaeology and History, Atlantic Books,2001, ISBN 1-84354-171-8 Arnold, Dieter. Building In Egypt, Pharaonic Stone Masory,1991, Oxford, Oxford University Press Jackson, K & J. Stamp
Seked
–
Seked slope of Great Pyramid
Seked
–
casing stone
35.
Geometric progression
–
For example, the sequence 2,6,18,54. is a geometric progression with common ratio 3. Similarly 10,5,2.5,1.25. is a sequence with common ratio 1/2. Examples of a sequence are powers rk of a fixed number r, such as 2k. The general form of a sequence is a, a r, a r 2, a r 3, a r 4, … where r ≠0 is the common ratio. The n-th term of a sequence with initial value a. Such a geometric sequence also follows the relation a n = r a n −1 for every integer n ≥1. Generally, to whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. The common ratio of a sequence may be negative, resulting in an alternating sequence, with numbers switching from positive to negative. For instance 1, −3,9, −27,81, the behaviour of a geometric sequence depends on the value of the common ratio. If the common ratio is, Positive, the terms will all be the sign as the initial term. Negative, the terms will alternate between positive and negative, greater than 1, there will be exponential growth towards positive or negative infinity. 1, the progression is a constant sequence, between −1 and 1 but not zero, there will be exponential decay towards zero. −1, the progression is an alternating sequence Less than −1, for the absolute values there is exponential growth towards infinity, due to the alternating sign. Geometric sequences show exponential growth or exponential decay, as opposed to the growth of an arithmetic progression such as 4,15,26,37,48. This result was taken by T. R. Malthus as the foundation of his Principle of Population. A geometric series is the sum of the numbers in a geometric progression, for example,2 +10 +50 +250 =2 +2 ×5 +2 ×52 +2 ×53. The formula works for any real numbers a and r. For example, −2 π +4 π2 −8 π3 = −2 π +2 +3 = −2 π1 − = −2 π1 +2 π ≈ −214.855
Geometric progression
–
Diagram illustrating three basic geometric sequences of the pattern 1(r n −1) up to 6 iterations deep. The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.
36.
Horus eye
–
The Eye of Horus is an ancient Egyptian symbol of protection, royal power and good health. The eye is personified in the goddess Wadjet, the Eye of Horus is similar to the Eye of Ra, which belongs to a different god, Ra, but represents many of the same concepts. Wadjet was one of the earliest of Egyptian deities who later associated with other goddesses such as Bast, Sekhmet, Mut. She was the deity of Lower Egypt and the major Delta shrine the per-nu was under her protection. Hathor is also depicted with this eye, funerary amulets were often made in the shape of the Eye of Horus. The Wadjet or Eye of Horus is the element of seven gold, faience, carnelian. The Wedjat was intended to protect the pharaoh in the afterlife, Ancient Egyptian and Middle-Eastern sailors would frequently paint the symbol on the bow of their vessel to ensure safe sea travel. Horus was the ancient Egyptian sky god who was depicted as a falcon. His right eye was associated with the sun god, Ra, the eye symbol represents the marking around the eye of the falcon, including the teardrop marking sometimes found below the eye. The mirror image, or left eye, sometimes represented the moon, in one myth, when Set and Horus were fighting for the throne after Osiriss death, Set gouged out Horuss left eye. The majority of the eye was restored by either Hathor or Thoth, when Horuss eye was recovered, he offered it to his father, Osiris, in hopes of restoring his life. Hence, the eye of Horus was often used to sacrifice, healing, restoration. There are seven different hieroglyphs used to represent the eye, most commonly ir. t in Egyptian, in Egyptian myth the eye was not the passive organ of sight but more an agent of action, protection or wrath. The Eye of Horus was represented as a hieroglyph, designated D10 in Gardiners sign list and it is represented in the Unicode character block for Egyptian hieroglyphs as U+13080. In Ancient Egyptian most fractions were written as the sum of two or more unit fractions, with scribes possessing tables of answers, thus instead of 3⁄4, one would write 1⁄2 + 1⁄4. Studies from the 1970s to this day in Egyptian mathematics have clearly shown this theory was fallacious, the evolution of the symbols used in mathematics, although similar to the different parts of the Eye of Horus, is now known to be distinct. Wadjet eye tatoos associated with Hathor depicted on 3, 000-year-old mummy
Horus eye
–
An Eye of Horus or Wedjat
pendant
Horus eye
–
The
Wedjat, later called The Eye of Horus
Horus eye
–
The crown of a
Nubian king
Horus eye
–
Wooden case decorated with bronze, silver, ivory and gold
37.
Arnold Buffum Chace
–
Arnold Buffum Chace was a textile businessman, mathematics scholar, and eleventh chancellor of Brown University in Providence, Rhode Island. Arnold was born November 10,1845 in Cumberland, Rhode Island and his grandfather Oliver Chace was founder of the Valley Falls textile company, which later became Berkshire Hathaway. His parents Samuel Buffington Chace and Elizabeth Buffum Chace were Quakers, arnold married Eliza Chace Greene, daughter of Christopher A. and Sarah A. Greene on October 24,1871. Their three children were, Malcolm Greene Chace, Edward Gould Chace, cotton manufacturer, and Margaret Chace, wife of Russell S. Rowland, M. D. of Detroit, MI. Arnold Buffum Chace received his bachelors degree from Brown University in 1866 and he also studied for one year at the École de Médecine in Paris. Chace taught physics and mathematics for one term at Brown, before having to interrupt his career to handle the family textile business and he remained involved in leadership at Brown for most of his life. In 1876 he was elected trustee, in 1882 he became treasurer and he wrote many articles on mathematical subjects, including one called A Certain Class of Cubic Surfaces Treated by Quaternions in the Journal of Mathematics. He attended the International Mathematical Congress at Cambridge, England in 1912, Chace published his work on the Egyptian Rhind Papyrus in 1927 and 1929, at age 87. His academic career was interrupted in 1869, when he became responsible for his familys cotton mill on the death of a family member, in 1871 he became a director of Westminster Bank, and in 1894 he became its president. He was also a director of the National Bank of North America, during this time he managed to attend mathematics classes at Harvard once a week. Chace died in Providence, Rhode Island, on February 28,1932 and is buried at Swan Point Cemetery, arnold Buffum Chace at Find a Grave
Arnold Buffum Chace
–
Arnold Buffum Chace
38.
Recto and verso
–
The terms recto and verso refer to the text written on the front and back sides of a leaf of paper in a bound item such as a codex, book, broadsheet, or pamphlet. The terms are shortened from Latin rectō foliō and versō foliō, translating to on the side of the page and on the turned side of the page. The page faces themselves are called folium rectum and folium versum in Latin, in codicology, each physical sheet of a manuscript is numbered and the sides are referred to as rectum and folium versum, abbreviated as r and v respectively. Editions of manuscripts will thus mark the position of text in the manuscript in the form fol. 1r, sometimes with the r and v in superscript, as in 1r, or with a superscript o indicating the ablative recto, verso and this terminology has been standard since the beginnings of modern codicology in the 17th century. The use of the recto and verso are also used in the codicology of manuscripts written in right-to-left scripts, like Syriac, Arabic. However, as these scripts are written in the direction to the scripts witnessed in European codices. The reading order of each folio remains first recto, then verso regardless of writing direction, the distinction between recto and verso can be convenient in the annotation of scholarly books, particularly in bilingual edition translations. The recto and verso terms can also be employed for the front and back of a one-sheet artwork, a recto-verso drawing is a sheet with drawings on both sides, for example in a sketchbook—although usually in these cases there is no obvious primary side. Some works are planned to exploit being on two sides of the piece of paper, but usually the works are not intended to be considered together. Paper was relatively expensive in the past, indeed good drawing paper still is more expensive than normal paper. In some early printed books, it is the rather than the pages. Thus each folium carries a number on its recto side. This was also common in e. g. internal company reports in the 20th century. Book design Obverse and reverse in coins Page spread
Recto and verso
–
Left-to-right language books (such as English)
39.
Number theory
–
Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
Number theory
–
A
Lehmer sieve, which is a primitive
digital computer once used for finding
primes and solving simple
Diophantine equations.
Number theory
–
The Plimpton 322 tablet
Number theory
–
Title page of the 1621 edition of Diophantus' Arithmetica, translated into
Latin by
Claude Gaspard Bachet de Méziriac.
Number theory
–
Leonhard Euler
40.
Arithmetic
–
Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are still used to refer to a wider part of number theory. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC and these artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, in both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the system for Babylonian numerals. Because of this concept, the ability to reuse the same digits for different values contributed to simpler. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols, one set for the units place, one for the tens place, and one for the hundreds. Then for the place they would reuse the symbols for the units place. Their addition algorithm was identical to ours, and their multiplication algorithm was very slightly different. Their long division algorithm was the same, and the square root algorithm that was taught in school was known to Archimedes. He preferred it to Heros method of successive approximation because, once computed, a digit doesnt change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a part, such as 546.934. The ancient Chinese used a positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the place
Arithmetic
–
Arithmetic tables for children, Lausanne, 1835
Arithmetic
–
A scale calibrated in imperial units with an associated cost display.
41.
Linear Equation
–
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. A simple example of an equation with only one variable, x, may be written in the form, ax + b =0, where a and b are constants. The constants may be numbers, parameters, or even functions of parameters. Linear equations can have one or more variables. An example of an equation with three variables, x, y, and z, is given by, ax + by + cz + d =0, where a, b, c, and d are constants and a, b. Linear equations occur frequently in most subareas of mathematics and especially in applied mathematics, an equation is linear if the sum of the exponents of the variables of each term is one. Equations with exponents greater than one are non-linear, an example of a non-linear equation of two variables is axy + b =0, where a and b are constants and a ≠0. It has two variables, x and y, and is non-linear because the sum of the exponents of the variables in the first term and this article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients, a linear equation in one unknown x may always be rewritten a x = b. If a ≠0, there is a solution x = b a. The origin of the name comes from the fact that the set of solutions of such an equation forms a straight line in the plane. Linear equations can be using the laws of elementary algebra into several different forms. These equations are referred to as the equations of the straight line. In what follows, x, y, t, and θ are variables, in the general form the linear equation is written as, A x + B y = C, where A and B are not both equal to zero. The equation is written so that A ≥0, by convention. The graph of the equation is a line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, if B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis, is C/B, and the slope of the line is −A/B. The general form is written as, a x + b y + c =0
Linear Equation
–
Graph sample of linear equations.
42.
Linear algebra
–
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, the set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns, such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics, for instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces, combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Because linear algebra is such a theory, nonlinear mathematical models are sometimes approximated by linear models. The study of linear algebra first emerged from the study of determinants, determinants were used by Leibniz in 1693, and subsequently, Gabriel Cramer devised Cramers Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, the study of matrix algebra first emerged in England in the mid-1800s. In 1844 Hermann Grassmann published his Theory of Extension which included foundational new topics of what is called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for womb, while studying compositions of linear transformations, Arthur Cayley was led to define matrix multiplication and inverses. Crucially, Cayley used a letter to denote a matrix. In 1882, Hüseyin Tevfik Pasha wrote the book titled Linear Algebra, the first modern and more precise definition of a vector space was introduced by Peano in 1888, by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its form in the first half of the twentieth century. The use of matrices in quantum mechanics, special relativity, the origin of many of these ideas is discussed in the articles on determinants and Gaussian elimination. Linear algebra first appeared in American graduate textbooks in the 1940s, following work by the School Mathematics Study Group, U. S. high schools asked 12th grade students to do matrix algebra, formerly reserved for college in the 1960s. In France during the 1960s, educators attempted to teach linear algebra through finite-dimensional vector spaces in the first year of secondary school and this was met with a backlash in the 1980s that removed linear algebra from the curriculum. To better suit 21st century applications, such as mining and uncertainty analysis
Linear algebra
–
The three-dimensional
Euclidean space R 3 is a vector space, and lines and planes passing through the
origin are vector subspaces in R 3.
43.
Cubit
–
The cubit is an ancient unit based on the forearm length from the middle finger tip to the elbow bottom. Cubits of various lengths were employed in many parts of the world in antiquity, during the Middle Ages, the term is still used in hedge laying, the length of the forearm being frequently used to determine the interval between stakes placed within the hedge. The English word cubit comes from the Latin noun cubitus elbow, from the verb cubo, cubare, cubui, cubitum to lie down, the ancient Egyptian royal cubit is the earliest attested standard measure. Cubit rods were used for the measurement of length, a number of these rods have survived, two are known from the tomb of Maya, the treasurer of the 18th dynasty pharaoh Tutankhamun, in Saqqara, another was found in the tomb of Kha in Thebes. Fourteen such rods, including one double cubit rod, were described and compared by Lepsius in 1865. These cubit rods range from 523.5 to 529.2 mm in length, and are divided into seven palms, each palm is divided into four fingers and the fingers are further subdivided. Use of the royal cubit is also known from Old Kingdom architecture, in 1916, during the last years of the Ottoman Empire and in the middle of World War I, the German assyriologist Eckhard Unger found a copper-alloy bar while excavating at Nippur. The bar dates from c.2650 BC and Unger claimed it was used as a measurement standard and this irregularly formed and irregularly marked graduated rule supposedly defined the Sumerian cubit as about 518.6 mm. The Near Eastern or Biblical cubit is usually estimated as approximately 457.2 mm, in ancient Greek units of measurement, the standard forearm cubit measured approximately 0.46 m. The short forearm cubit, from the wrist to the elbow, in ancient Rome, according to Vitruvius, a cubit was equal to 1 1⁄2 Roman feet or 6 palm widths. Other measurements based on the length of the forearm include some lengths of ell, the Chinese chi, the Japanese shaku, the Indian hasta, the Thai sok, the Tamil, the Telugu, a cubit arm in heraldry may be dexter or sinister. It may be vested and may be shown in positions, most commonly erect. It is most often used erect as a crest, for example by the families of Poyntz of Iron Acton, Rolle of Stevenstone, the Encyclopaedia of Ancient Egyptian Architecture. The Cubit, A History and Measurement Commentary, Journal of Anthropology doi,10. 1155/2014/489757,2014 Media related to Cubit arms at Wikimedia Commons The dictionary definition of cubit at Wiktionary
Cubit
–
Egyptian cubit rod in the
Liverpool World Museum
Cubit
–
Cubit rod of
Maya, 1336-1327 BC (
Eighteenth Dynasty)
Cubit
–
Cubit rod from the Turin Museum.
Cubit
–
The Nippur cubit-rod in the
Archeological Museum of
Istanbul, Turkey
44.
Pyramid (geometry)
–
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face and it is a conic solid with polygonal base. A pyramid with a base has n +1 vertices, n +1 faces. A right pyramid has its apex directly above the centroid of its base, nonright pyramids are called oblique pyramids. A regular pyramid has a polygon base and is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid. A triangle-based pyramid is often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base, in a tetrahedron these qualifiers change based on which face is considered the base. Pyramids are a subclass of the prismatoids, pyramids can be doubled into bipyramids by adding a second offset point on the other side of the base plane. A right pyramid with a base has isosceles triangle sides, with symmetry is Cnv or. It can be given an extended Schläfli symbol ∨, representing a point, a join operation creates a new edge between all pairs of vertices of the two joined figures. The trigonal or triangular pyramid with all equilateral triangles faces becomes the regular tetrahedron, a lower symmetry case of the triangular pyramid is C3v, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of convex polygons. Right pyramids with regular star polygon bases are called star pyramids, for example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ∨P, where is the point, ∨ is a join operator. It has C1v symmetry from two different base-apex orientations, and C2v in its full symmetry, a rectangular right pyramid, written as ∨, and a rhombic pyramid, as ∨, both have symmetry C2v. The volume of a pyramid is V =13 b h and this works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base
Pyramid (geometry)
–
Regular-based right pyramids
45.
Dihedral angle
–
A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common, in solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimension, a dihedral angle represents the angle between two hyperplanes, a dihedral angle is an angle between two intersecting planes on a third plane perpendicular to the line of intersection. A torsion angle is an example of a dihedral angle. In stereochemistry every set of three atoms of a molecule defines a plane, when two such planes intersect, the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation, stereochemical arrangements corresponding to angles between 0° and ±90° are called syn, those corresponding to angles between ±90° and 180° anti. Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called clinal and those between 0° and ±30° or ±150° and 180° are called periplanar. The synperiplanar conformation is also known as the syn- or cis-conformation, antiperiplanar as anti or trans, for example, with n-butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The syn-conformation shown above, with an angle of 60° is less stable than the anti-configuration with a dihedral angle of 180°. For macromolecular usage the symbols T, C, G+, G−, A+, a Ramachandran plot, originally developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan, is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein structure, the figure at right illustrates the definition of the φ and ψ backbone dihedral angles. In a protein chain three dihedral angles are defined as φ, ψ and ω, as shown in the diagram, the planarity of the peptide bond usually restricts ω to be 180° or 0°. The distance between the Cα atoms in the trans and cis isomers is approximately 3.8 and 2.9 Å, the cis isomer is mainly observed in Xaa–Pro peptide bonds. The sidechain dihedral angles tend to cluster near 180°, 60°, and −60°, which are called the trans, gauche+, the stability of certain sidechain dihedral angles is affected by the values φ and ψ. For instance, there are steric interactions between the Cγ of the side chain in the gauche+ rotamer and the backbone nitrogen of the next residue when ψ is near -60°. An alternative method is to calculate the angle between the vectors, nA and nB, which are normal to the planes. Cos φ = − n A ⋅ n B | n A | | n B | where nA · nB is the dot product of the vectors and |nA| |nB| is the product of their lengths. Any plane can also be described by two non-collinear vectors lying in that plane, taking their cross product yields a vector to the plane
Dihedral angle
–
Free energy diagram of butane as a function of dihedral angle.
Dihedral angle
46.
Cotangent
–
In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
Cotangent
–
Trigonometric functions in the complex plane
Cotangent
–
Trigonometry
Cotangent
Cotangent
47.
Trigonometry
–
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies, Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles, thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees, the ancient Nubians used a similar method. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, in the 2nd century AD, the Greco-Egyptian astronomer Ptolemy printed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta and these Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, at about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond, Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for maps of large geographic areas. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595, gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry, the works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general Taylor series, if one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees, they are complementary angles, the shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, if the length of one of the sides is known, the other two are determined. Sin A = opposite hypotenuse = a c, Cosine function, defined as the ratio of the adjacent leg to the hypotenuse
Trigonometry
–
Hipparchus, credited with compiling the first
trigonometric table, is known as "the father of trigonometry".
Trigonometry
–
All of the
trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
Trigonometry
–
Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a
marine chronometer, the position of the ship can be determined from such measurements.
48.
Commutative property
–
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says 3 +4 =4 +3 or 2 ×5 =5 ×2, the property can also be used in more advanced settings. The name is needed there are operations, such as division and subtraction. The commutative property is a property associated with binary operations and functions. If the commutative property holds for a pair of elements under a binary operation then the two elements are said to commute under that operation. The term commutative is used in several related senses, putting on socks resembles a commutative operation since which sock is put on first is unimportant. Either way, the result, is the same, in contrast, putting on underwear and trousers is not commutative. The commutativity of addition is observed when paying for an item with cash, regardless of the order the bills are handed over in, they always give the same total. The multiplication of numbers is commutative, since y z = z y for all y, z ∈ R For example,3 ×5 =5 ×3. Some binary truth functions are also commutative, since the tables for the functions are the same when one changes the order of the operands. For example, the logical biconditional function p ↔ q is equivalent to q ↔ p and this function is also written as p IFF q, or as p ≡ q, or as Epq. Further examples of binary operations include addition and multiplication of complex numbers, addition and scalar multiplication of vectors. Concatenation, the act of joining character strings together, is a noncommutative operation, rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order. The twists of the Rubiks Cube are noncommutative and this can be studied using group theory. Some non-commutative binary operations, Records of the use of the commutative property go back to ancient times. The Egyptians used the property of multiplication to simplify computing products. Euclid is known to have assumed the property of multiplication in his book Elements
Commutative property
–
This image illustrates that addition is commutative.
49.
Algebra
–
Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
Algebra
–
A page from
Al-Khwārizmī 's
al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Algebra
–
Italian mathematician
Girolamo Cardano published the solutions to the
cubic and
quartic equations in his 1545 book
Ars magna.