1.
17 century
–
The 17th century was the century that lasted from January 1,1601, to December 31,1700, in the Gregorian calendar. The greatest military conflicts were the Thirty Years War, the Great Turkish War, in the Islamic world, the Ottoman, Safavid Persian and Mughal empires grew in strength. In Japan, Tokugawa Ieyasu established the Edo period at the beginning of the century, European politics were dominated by the Kingdom of France of Louis XIV, where royal power was solidified domestically in the civil war of the Fronde. With domestic peace assured, Louis XIV caused the borders of France to be expanded and it was during this century that English monarch became a symbolic figurehead and Parliament was the dominant force in government – a contrast to most of Europe, in particular France. It was also a period of development of culture in general,1600, On February 17 Giordano Bruno is burned at the stake by the Inquisition. 1600, Michael the Brave unifies the three Romanian countries, Wallachia, Moldavia and Transylvania after the Battle of Șelimbăr from 1599. 1601, Battle of Kinsale, England defeats Irish and Spanish forces at the town of Kinsale, driving the Gaelic aristocracy out of Ireland and destroying the Gaelic clan system. 1601, Michael the Brave, voivode of Wallachia, Moldavia and Transylvania, is assassinated by the order of the Habsburg general Giorgio Basta at Câmpia Turzii, 1601–1603, The Russian famine of 1601–1603 kills perhaps one-third of Russia. 1601, Panembahan Senopati, first king of Mataram, dies and passes rule to his son Panembahan Seda ing Krapyak 1601,1602, Matteo Ricci produces the Map of the Myriad Countries of the World, a world map that will be used throughout East Asia for centuries. 1602, The Portuguese send an expeditionary force from Malacca which succeeded in reimposing a degree of Portuguese control. 1602, The Dutch East India Company is established by merging competing Dutch trading companies and its success contributes to the Dutch Golden Age. 1602, Two emissaries from the Aceh Sultanate visit the Dutch Republic,1603, Elizabeth I of England dies and is succeeded by her cousin King James VI of Scotland, uniting the crowns of Scotland and England. 1603, Tokugawa Ieyasu takes the title of Shogun, establishing the Tokugawa Shogunate and this begins the Edo period, which will last until 1869. 1603–1623, After modernizing his army, Abbas I expands the Persian Empire by capturing territory from the Ottomans,1603, First permanent Dutch trading post is established in Banten, West Java. First successful VOC privateering raid on a Portuguese ship,1604, A second English East India Company voyage commanded by Sir Henry Middleton reaches Ternate, Tidore, Ambon and Banda. 1605, Gunpowder Plot failed in England,1605, The fortresses of Veszprém and Visegrad are retaken by the Ottomans. 1605, February, The VOC in alliance with Hitu prepare to attack a Portuguese fort in Ambon,1605, Panembahan Seda ing Krapyak of Mataram establishes control over Demak, former center of the Demak Sultanate. 1606, Treaty of Vienna ends anti-Habsburg uprising in Royal Hungary,1606, Assassination of Stephen Bocskay of Transylvania
17 century
–
Europe and the Ottoman Empire (in purple) in the year 1600
17 century
–
The Italian biologist
Francesco Redi, recognized as the founder of
experimental biology and the Father of modern
parasitology.
17 century
–
Louis XIV visiting the
Académie des sciences in 1671. "It is widely accepted that '
modern science ' arose in the Europe of the 17th century, introducing a new understanding of the natural world." —Peter Barrett
17 century
–
New Amsterdam as it appeared in 1664. Under British rule it became known as New York.
2.
The Stagirite
–
Aristotle was an ancient Greek philosopher and scientist born in the city of Stagira, Chalkidice, on the northern periphery of Classical Greece. His father, Nicomachus, died when Aristotle was a child, at seventeen or eighteen years of age, he joined Platos Academy in Athens and remained there until the age of thirty-seven. Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon, teaching Alexander the Great gave Aristotle many opportunities and an abundance of supplies. He established a library in the Lyceum which aided in the production of many of his hundreds of books and he believed all peoples concepts and all of their knowledge was ultimately based on perception. Aristotles views on natural sciences represent the groundwork underlying many of his works, Aristotles views on physical science profoundly shaped medieval scholarship. Their influence extended from Late Antiquity and the Early Middle Ages into the Renaissance, some of Aristotles zoological observations, such as on the hectocotyl arm of the octopus, were not confirmed or refuted until the 19th century. His works contain the earliest known study of logic, which was incorporated in the late 19th century into modern formal logic. Aristotle was well known among medieval Muslim intellectuals and revered as The First Teacher and his ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotles philosophy continue to be the object of academic study today. Though Aristotle wrote many elegant treatises and dialogues – Cicero described his style as a river of gold – it is thought that only around a third of his original output has survived. Aristotle, whose means the best purpose, was born in 384 BC in Stagira, Chalcidice. His father Nicomachus was the physician to King Amyntas of Macedon. Aristotle was orphaned at a young age, although there is little information on Aristotles childhood, he probably spent some time within the Macedonian palace, making his first connections with the Macedonian monarchy. At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Platos Academy and he remained there for nearly twenty years before leaving Athens in 348/47 BC. Aristotle then accompanied Xenocrates to the court of his friend Hermias of Atarneus in Asia Minor, there, he traveled with Theophrastus to the island of Lesbos, where together they researched the botany and zoology of the island. Aristotle married Pythias, either Hermiass adoptive daughter or niece and she bore him a daughter, whom they also named Pythias. Soon after Hermias death, Aristotle was invited by Philip II of Macedon to become the tutor to his son Alexander in 343 BC, Aristotle was appointed as the head of the royal academy of Macedon. During that time he gave not only to Alexander
The Stagirite
–
Roman copy in marble of a Greek bronze bust of Aristotle by
Lysippus,
c. 330 BC. The
alabaster mantle is modern.
The Stagirite
–
Aristotelianism
The Stagirite
–
School of Aristotle in
Mieza,
Macedonia, Greece
The Stagirite
–
"Aristotle" by
Francesco Hayez (1791–1882)
3.
Space astronomy
–
Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
Space astronomy
–
A
star -forming region in the
Large Magellanic Cloud, an
irregular galaxy.
Space astronomy
–
A giant
Hubble mosaic of the
Crab Nebula, a
supernova remnant
Space astronomy
–
19th century
Sydney Observatory,
Australia (1873)
Space astronomy
–
19th century
Quito Astronomical Observatory is located 12 minutes south of the
Equator in
Quito,
Ecuador.
4.
Computar
–
A computer is a device that can be instructed to carry out an arbitrary set of arithmetic or logical operations automatically. The ability of computers to follow a sequence of operations, called a program, such computers are used as control systems for a very wide variety of industrial and consumer devices. The Internet is run on computers and it millions of other computers. Since ancient times, simple manual devices like the abacus aided people in doing calculations, early in the Industrial Revolution, some mechanical devices were built to automate long tedious tasks, such as guiding patterns for looms. More sophisticated electrical machines did specialized analog calculations in the early 20th century, the first digital electronic calculating machines were developed during World War II. The speed, power, and versatility of computers has increased continuously and dramatically since then, conventionally, a modern computer consists of at least one processing element, typically a central processing unit, and some form of memory. The processing element carries out arithmetic and logical operations, and a sequencing, peripheral devices include input devices, output devices, and input/output devices that perform both functions. Peripheral devices allow information to be retrieved from an external source and this usage of the term referred to a person who carried out calculations or computations. The word continued with the same meaning until the middle of the 20th century, from the end of the 19th century the word began to take on its more familiar meaning, a machine that carries out computations. The Online Etymology Dictionary gives the first attested use of computer in the 1640s, one who calculates, the Online Etymology Dictionary states that the use of the term to mean calculating machine is from 1897. The Online Etymology Dictionary indicates that the use of the term. 1945 under this name, theoretical from 1937, as Turing machine, devices have been used to aid computation for thousands of years, mostly using one-to-one correspondence with fingers. The earliest counting device was probably a form of tally stick, later record keeping aids throughout the Fertile Crescent included calculi which represented counts of items, probably livestock or grains, sealed in hollow unbaked clay containers. The use of counting rods is one example, the abacus was initially used for arithmetic tasks. The Roman abacus was developed from used in Babylonia as early as 2400 BC. Since then, many forms of reckoning boards or tables have been invented. In a medieval European counting house, a checkered cloth would be placed on a table, the Antikythera mechanism is believed to be the earliest mechanical analog computer, according to Derek J. de Solla Price. It was designed to calculate astronomical positions and it was discovered in 1901 in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, and has been dated to circa 100 BC
Computar
–
Computer
Computar
Computar
Computar
5.
Working definition
–
A definition is a statement of the meaning of a term. Definitions can be classified into two categories, intensional definitions and extensional definitions. Another important category of definitions is the class of ostensive definitions, a term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a meaning to a new term. Definitions and axioms are the basis on all of mathematics is constructed. In modern usage, a definition is something, typically expressed in words, the word or group of words that is to be defined is called the definiendum, and the word, group of words, or action that defines it is called the definiens. In the definition An elephant is a large gray animal native to Asia and Africa, the elephant is the definiendum. Note that the definiens is not the meaning of the word defined, there are many sub-types of definitions, often specific to a given field of knowledge or study. An intensional definition, also called a connotative definition, specifies the necessary, any definition that attempts to set out the essence of something, such as that by genus and differentia, is an intensional definition. An extensional definition, also called a denotative definition, of a concept or term specifies its extension and it is a list naming every object that is a member of a specific set. An extensional definition would be the list of wrath, greed, sloth, pride, lust, envy, a genus–differentia definition is a type of intensional definition that takes a large category and narrows it down to a smaller category by a distinguishing characteristic. The differentia, The portion of the new definition that is not provided by the genus, for example, consider the following genus-differentia definitions, a triangle, A plane figure that has three straight bounding sides. A quadrilateral, A plane figure that has four straight bounding sides and those definitions can be expressed as a genus and two differentiae. It is possible to have two different genus-differentia definitions that describe the same term, especially when the term describes the overlap of two large categories, for instance, both of these genus-differentia definitions of square are equally acceptable, a square, a rectangle that is a rhombus. A square, a rhombus that is a rectangle, thus, a square is a member of both the genus rectangle and the genus rhombus. One important form of the definition is ostensive definition. This gives the meaning of a term by pointing, in the case of an individual, to the thing itself, or in the case of a class, to examples of the right kind. So one can explain who Alice is by pointing her out to another, or what a rabbit is by pointing at several, the process of ostensive definition itself was critically appraised by Ludwig Wittgenstein
Working definition
–
A definition states the meaning of a word using other words. This is sometimes challenging. Common dictionaries contain lexical, descriptive definitions, but there are various types of definition - all with different purposes and focuses.
6.
Order (differential equation)
–
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, if a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli. In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
Order (differential equation)
–
Navier–Stokes differential equations used to simulate airflow around an obstruction.
7.
German (Deutsch)
–
German is a West Germanic language that is mainly spoken in Central Europe. It is the most widely spoken and official language in Germany, Austria, Switzerland, South Tyrol, the German-speaking Community of Belgium and it is also one of the three official languages of Luxembourg. Major languages which are most similar to German include other members of the West Germanic language branch, such as Afrikaans, Dutch, English, Luxembourgish and it is the second most widely spoken Germanic language, after English. One of the languages of the world, German is the first language of about 95 million people worldwide. The German speaking countries are ranked fifth in terms of publication of new books. German derives most of its vocabulary from the Germanic branch of the Indo-European language family, a portion of German words are derived from Latin and Greek, and fewer are borrowed from French and English. With slightly different standardized variants, German is a pluricentric language, like English, German is also notable for its broad spectrum of dialects, with many unique varieties existing in Europe and also other parts of the world. The history of the German language begins with the High German consonant shift during the migration period, when Martin Luther translated the Bible, he based his translation primarily on the standard bureaucratic language used in Saxony, also known as Meißner Deutsch. Copies of Luthers Bible featured a long list of glosses for each region that translated words which were unknown in the region into the regional dialect. Roman Catholics initially rejected Luthers translation, and tried to create their own Catholic standard of the German language – the difference in relation to Protestant German was minimal. It was not until the middle of the 18th century that a widely accepted standard was created, until about 1800, standard German was mainly a written language, in urban northern Germany, the local Low German dialects were spoken. Standard German, which was different, was often learned as a foreign language with uncertain pronunciation. Northern German pronunciation was considered the standard in prescriptive pronunciation guides though, however, German was the language of commerce and government in the Habsburg Empire, which encompassed a large area of Central and Eastern Europe. Until the mid-19th century, it was essentially the language of townspeople throughout most of the Empire and its use indicated that the speaker was a merchant or someone from an urban area, regardless of nationality. Some cities, such as Prague and Budapest, were gradually Germanized in the years after their incorporation into the Habsburg domain, others, such as Pozsony, were originally settled during the Habsburg period, and were primarily German at that time. Prague, Budapest and Bratislava as well as cities like Zagreb, the most comprehensive guide to the vocabulary of the German language is found within the Deutsches Wörterbuch. This dictionary was created by the Brothers Grimm and is composed of 16 parts which were issued between 1852 and 1860, in 1872, grammatical and orthographic rules first appeared in the Duden Handbook. In 1901, the 2nd Orthographical Conference ended with a standardization of the German language in its written form
German (Deutsch)
–
Old Frisian (Alt-Friesisch)
German (Deutsch)
–
German
German (Deutsch)
–
The widespread popularity of the
Bible translated into German by
Martin Luther helped establish modern German
German (Deutsch)
–
Examples of German language in
Namibian everyday life
8.
Elements (book)
–
Euclids Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
Elements (book)
–
The
frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570
Elements (book)
–
A fragment of Euclid's "Elements" on part of the
Oxyrhynchus papyri
Elements (book)
–
An illumination from a manuscript based on
Adelard of Bath 's translation of the Elements, c. 1309–1316; Adelard's is the oldest surviving translation of the Elements into Latin, done in the 12th-century work and translated from Arabic.
Elements (book)
–
Euclidis – Elementorum libri XV Paris, Hieronymum de Marnef & Guillaume Cavelat, 1573 (second edition after the 1557 ed.); in-8, 350, (2)pp. THOMAS-STANFORD, Early Editions of Euclid's Elements, n°32. Mentioned in T.L. Heath's translation. Private collection Hector Zenil.
9.
Output (mathematics)
–
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
Output (mathematics)
–
A function f takes an input x, and returns a single output f (x). One metaphor describes the function as a "machine" or "
black box " that for each input returns a corresponding output.
10.
Equalangular triangle
–
In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular and they are regular polygons, and can therefore also be referred to as regular triangles. Thus these are properties that are unique to equilateral triangles, the three medians have equal lengths. The three angle bisectors have equal lengths, every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral, in particular, A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, for any triangle, the three medians partition the triangle into six smaller triangles. A triangle is equilateral if and only if any three of the triangles have either the same perimeter or the same inradius. A triangle is equilateral if and only if the circumcenters of any three of the triangles have the same distance from the centroid. Morleys trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, a version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, PA, PB, and PC satisfy the inequality that any two of them sum to at least as great as the third. By Eulers inequality, the triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle, specifically. The triangle of largest area of all those inscribed in a circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, the ratio of the area to the square of the perimeter of an equilateral triangle,1123, is larger than that for any other triangle. If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A1 and A2, then 79 ≤ A1 A2 ≤97, in no other triangle is there a point for which this ratio is as small as 2. For any point P in the plane, with p, q, and t from the vertices A, B. For any point P on the circle of an equilateral triangle, with distances p, q. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral, an equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the group of order 6 D3
Equalangular triangle
–
A regular tetrahedron is made of four equilateral triangles.
Equalangular triangle
–
Equilateral triangle
11.
Eladha
–
Greece, officially the Hellenic Republic, historically also known as Hellas, is a country in southeastern Europe, with a population of approximately 11 million as of 2015. Athens is the capital and largest city, followed by Thessaloniki. Greece is strategically located at the crossroads of Europe, Asia, situated on the southern tip of the Balkan peninsula, it shares land borders with Albania to the northwest, the Republic of Macedonia and Bulgaria to the north, and Turkey to the northeast. Greece consists of nine regions, Macedonia, Central Greece, the Peloponnese, Thessaly, Epirus, the Aegean Islands, Thrace, Crete. The Aegean Sea lies to the east of the mainland, the Ionian Sea to the west, the Cretan Sea and the Mediterranean Sea to the south. Greece has the longest coastline on the Mediterranean Basin and the 11th longest coastline in the world at 13,676 km in length, featuring a vast number of islands, eighty percent of Greece is mountainous, with Mount Olympus being the highest peak at 2,918 metres. From the eighth century BC, the Greeks were organised into various independent city-states, known as polis, which spanned the entire Mediterranean region and the Black Sea. Greece was annexed by Rome in the second century BC, becoming a part of the Roman Empire and its successor. The Greek Orthodox Church also shaped modern Greek identity and transmitted Greek traditions to the wider Orthodox World, falling under Ottoman dominion in the mid-15th century, the modern nation state of Greece emerged in 1830 following a war of independence. Greeces rich historical legacy is reflected by its 18 UNESCO World Heritage Sites, among the most in Europe, Greece is a democratic and developed country with an advanced high-income economy, a high quality of life, and a very high standard of living. A founding member of the United Nations, Greece was the member to join the European Communities and has been part of the Eurozone since 2001. Greeces unique cultural heritage, large industry, prominent shipping sector. It is the largest economy in the Balkans, where it is an important regional investor, the names for the nation of Greece and the Greek people differ from the names used in other languages, locations and cultures. The earliest evidence of the presence of human ancestors in the southern Balkans, dated to 270,000 BC, is to be found in the Petralona cave, all three stages of the stone age are represented in Greece, for example in the Franchthi Cave. Neolithic settlements in Greece, dating from the 7th millennium BC, are the oldest in Europe by several centuries and these civilizations possessed writing, the Minoans writing in an undeciphered script known as Linear A, and the Mycenaeans in Linear B, an early form of Greek. The Mycenaeans gradually absorbed the Minoans, but collapsed violently around 1200 BC and this ushered in a period known as the Greek Dark Ages, from which written records are absent. The end of the Dark Ages is traditionally dated to 776 BC, the Iliad and the Odyssey, the foundational texts of Western literature, are believed to have been composed by Homer in the 7th or 8th centuries BC. With the end of the Dark Ages, there emerged various kingdoms and city-states across the Greek peninsula, in 508 BC, Cleisthenes instituted the worlds first democratic system of government in Athens
Eladha
–
Fresco displaying the Minoan ritual of "bull leaping", found in
Knossos,
Crete.
Eladha
–
Flag
Eladha
–
The
Lion Gate,
Mycenae
Eladha
–
The
Parthenon on the
Acropolis of Athens is one of the best known symbols of
classical Greece.
12.
Elementary group theory
–
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
Elementary group theory
–
A periodic wallpaper pattern gives rise to a
wallpaper group.
Elementary group theory
–
The manipulations of this
Rubik's Cube form the
Rubik's Cube group.
13.
Applications of group theory
–
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
Applications of group theory
–
Water molecule with symmetry axis
Applications of group theory
–
The popular puzzle
Rubik's cube invented in 1974 by
Ernő Rubik has been used as an illustration of
permutation groups.
14.
Line segment
–
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories, for example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
Line segment
–
historical image – create a line segment (1699)
15.
Types of logic
–
Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times
Types of logic
–
Aristotle, 384–322 BCE.
Types of logic
–
Plato –
Kant –
Nietzsche
16.
Specific Humidity
–
Humidity is the amount of water vapor present in the air. Water vapor is the state of water and is invisible. Humidity indicates the likelihood of precipitation, dew, or fog, higher humidity reduces the effectiveness of sweating in cooling the body by reducing the rate of evaporation of moisture from the skin. This effect is calculated in an index table or humidex. The amount of vapor that is needed to achieve saturation increases as the temperature increases. As the temperature of a parcel of water becomes lower it will not reach the point of saturation without adding or losing water mass. The differences in the amount of vapor in a parcel of air can be quite large. For example, a parcel of air that is near saturation may contain 28 grams of water per cubic meter of air at 30 °C, there are three main measurements of humidity, absolute, relative and specific. Absolute humidity is the content of air expressed in gram per cubic meter. Relative humidity, expressed as a percent, measures the current absolute humidity relative to the maximum for that temperature, specific humidity is the ratio of the mass of water vapor to the total mass of the moist air parcel. Absolute humidity is the mass of water vapor present in a given volume of air. It does not take temperature into consideration, absolute humidity in the atmosphere ranges from near zero to roughly 30 grams per cubic meter when the air is saturated at 30 °C. Absolute humidity is the mass of the vapor, divided by the volume of the air and water vapor mixture. The absolute humidity changes as air temperature or pressure changes and this makes it unsuitable for chemical engineering calculations, e. g. for clothes dryers, where temperature can vary considerably. Mass of water per unit volume as in the equation above is defined as volumetric humidity. Because of the confusion, British Standard BS1339 suggests avoiding the term absolute humidity. Units should always be carefully checked, many humidity charts are given in g/kg or kg/kg, but any mass units may be used. The field concerned with the study of physical and thermodynamic properties of mixtures is named psychrometrics
Specific Humidity
–
Humidity and hygrometry
Specific Humidity
–
Paranal Observatory on
Cerro Paranal in the
Atacama Desert is one of the driest places on earth.
Specific Humidity
–
A
hygrometer
17.
Elementary 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
Elementary geometry
–
Visual checking of the
Pythagorean theorem for the (3, 4, 5)
triangle as in the
Chou Pei Suan Ching 500–200 BC.
Elementary geometry
–
An illustration of
Desargues' theorem, an important result in
Euclidean and
projective geometry
Elementary geometry
–
Geometry lessons in the 20th century
Elementary geometry
–
A
European and an
Arab practicing geometry in the 15th century.
18.
Broadcast meteorologist
–
Meteorology is a branch of the atmospheric sciences which includes atmospheric chemistry and atmospheric physics, with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not occur until the 18th century, the 19th century saw modest progress in the field after weather observation networks were formed across broad regions. Prior attempts at prediction of weather depended on historical data, Meteorological phenomena are observable weather events that are explained by the science of meteorology. Different spatial scales are used to describe and predict weather on local, regional, Meteorology, climatology, atmospheric physics, and atmospheric chemistry are sub-disciplines of the atmospheric sciences. Meteorology and hydrology compose the interdisciplinary field of hydrometeorology, the interactions between Earths atmosphere and its oceans are part of a coupled ocean-atmosphere system. Meteorology has application in diverse fields such as the military, energy production, transport, agriculture. The word meteorology is from Greek μετέωρος metéōros lofty, high and -λογία -logia -logy, varāhamihiras classical work Brihatsamhita, written about 500 AD, provides clear evidence that a deep knowledge of atmospheric processes existed even in those times. In 350 BC, Aristotle wrote Meteorology, Aristotle is considered the founder of meteorology. One of the most impressive achievements described in the Meteorology is the description of what is now known as the hydrologic cycle and they are all called swooping bolts because they swoop down upon the Earth. Lightning is sometimes smoky, and is then called smoldering lightning, sometimes it darts quickly along, at other times, it travels in crooked lines, and is called forked lightning. When it swoops down upon some object it is called swooping lightning, the Greek scientist Theophrastus compiled a book on weather forecasting, called the Book of Signs. The work of Theophrastus remained a dominant influence in the study of weather, in 25 AD, Pomponius Mela, a geographer for the Roman Empire, formalized the climatic zone system. According to Toufic Fahd, around the 9th century, Al-Dinawari wrote the Kitab al-Nabat, ptolemy wrote on the atmospheric refraction of light in the context of astronomical observations. St. Roger Bacon was the first to calculate the size of the rainbow. He stated that a rainbow summit can not appear higher than 42 degrees above the horizon, in the late 13th century and early 14th century, Kamāl al-Dīn al-Fārisī and Theodoric of Freiberg were the first to give the correct explanations for the primary rainbow phenomenon. Theoderic went further and also explained the secondary rainbow, in 1716, Edmund Halley suggested that aurorae are caused by magnetic effluvia moving along the Earths magnetic field lines. In 1441, King Sejongs son, Prince Munjong, invented the first standardized rain gauge and these were sent throughout the Joseon Dynasty of Korea as an official tool to assess land taxes based upon a farmers potential harvest. In 1450, Leone Battista Alberti developed a swinging-plate anemometer, and was known as the first anemometer, in 1607, Galileo Galilei constructed a thermoscope
Broadcast meteorologist
–
Atmospheric sciences
Broadcast meteorologist
–
Parhelion (sundog) at
Savoie
Broadcast meteorologist
–
Twilight at
Baker Beach
Broadcast meteorologist
–
A hemispherical cup anemometer
19.
Pyhsics
–
Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
Pyhsics
–
Further information:
Outline of physics
Pyhsics
–
Ancient
Egyptian astronomy is evident in monuments like the
ceiling of Senemut's tomb from the
Eighteenth Dynasty of Egypt.
Pyhsics
–
Sir Isaac Newton (1643–1727), whose
laws of motion and
universal gravitation were major milestones in classical physics
Pyhsics
–
Albert Einstein (1879–1955), whose work on the
photoelectric effect and the
theory of relativity led to a revolution in 20th century physics
20.
Netherlandic
–
It is the third most widely spoken Germanic language, after English and German. Dutch is one of the closest relatives of both German and English and is said to be roughly in between them, Dutch vocabulary is mostly Germanic and incorporates more Romance loans than German but far fewer than English. In both Belgium and the Netherlands, the official name for Dutch is Nederlands, and its dialects have their own names, e. g. Hollands, West-Vlaams. The use of the word Vlaams to describe Standard Dutch for the variations prevalent in Flanders and used there, however, is common in the Netherlands, the Dutch language has been known under a variety of names. It derived from the Old Germanic word theudisk, one of the first names used for the non-Romance languages of Western Europe. It literarily means the language of the people, that is. The term was used as opposed to Latin, the language of writing. In the first text in which it is found, dating from 784, later, theudisca appeared also in the Oaths of Strasbourg to refer to the Germanic portion of the oath. This led inevitably to confusion since similar terms referred to different languages, owing to Dutch commercial and colonial rivalry in the 16th and 17th centuries, the English term came to refer exclusively to the Dutch. A notable exception is Pennsylvania Dutch, which is a West Central German variety called Deitsch by its speakers, Jersey Dutch, on the other hand, as spoken until the 1950s in New Jersey, is a Dutch-based creole. In Dutch itself, Diets went out of common use - although Platdiets is still used for the transitional Limburgish-Ripuarian Low Dietsch dialects in northeast Belgium, Nederlands, the official Dutch word for Dutch, did not become firmly established until the 19th century. This designation had been in use as far back as the end of the 15th century, one of them was it reflected a distinction with Hoogduits, High Dutch, meaning the language spoken in Germany. The Hoog was later dropped, and thus, Duits narrowed down in meaning to refer to the German language. g, in English, too, Netherlandic is regarded as a more accurate term for the Dutch language, but is hardly ever used. Old Dutch branched off more or less around the same time Old English, Old High German, Old Frisian and Old Saxon did. During that period, it forced Old Frisian back from the western coast to the north of the Low Countries, on the other hand, Dutch has been replaced in adjacent lands in nowadays France and Germany. The division in Old, Middle and Modern Dutch is mostly conventional, one of the few moments linguists can detect somewhat of a revolution is when the Dutch standard language emerged and quickly established itself. This is assumed to have taken place in approximately the mid-first millennium BCE in the pre-Roman Northern European Iron Age, the Germanic languages are traditionally divided into three groups, East, West, and North Germanic. They remained mutually intelligible throughout the Migration Period, Dutch is part of the West Germanic group, which also includes English, Scots, Frisian, Low German and High German
Netherlandic
–
The Utrecht baptismal vow Forsachistu diobolae...
Netherlandic
–
Dutch-speaking world (included are areas of daughter-language Afrikaans)
Netherlandic
–
Second edition of this column decorated with a title of
Charles V 's portrait, with archaic Dutch inscriptions
Netherlandic
–
Dutch language street sign in the Netherlands
21.
Numberic
–
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Being able to compute the sides of a triangle is important, for instance, in astronomy, carpentry. Numerical analysis continues this tradition of practical mathematical calculations. Much like the Babylonian approximation of the root of 2, modern numerical analysis does not seek exact answers. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors, before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead and these same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of differential equations. Car companies can improve the safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving differential equations numerically. Hedge funds use tools from all fields of analysis to attempt to calculate the value of stocks. Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments, historically, such algorithms were developed within the overlapping field of operations research. Insurance companies use programs for actuarial analysis. The rest of this section outlines several important themes of numerical analysis, the field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago, to facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. The function values are no very useful when a computer is available. The mechanical calculator was developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of analysis, since now longer
Numberic
–
Babylonian clay tablet
YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the
square root of 2 is four
sexagesimal figures, which is about six
decimal figures. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296...
Numberic
–
Direct method
Numberic
22.
Algebraic projective geometry
–
Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry, the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry
Algebraic projective geometry
–
Growth measure and the polar vortices. Based on the work of Lawrence Edwards
Algebraic projective geometry
–
Projecting a
sphere to a
plane.
Algebraic projective geometry
–
Forms
23.
Additive operator
–
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
Additive operator
–
Arithmetic tables for children, Lausanne, 1835
Additive operator
–
A scale calibrated in imperial units with an associated cost display.
24.
The Rhind Mathematical Papyrus
–
The Rhind Mathematical Papyrus is one of the best known examples of Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian and it dates to around 1550 BC. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus, the Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former. The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt and it was copied by the scribe Ahmes, from a now-lost text from the reign of king Amenemhat III. Written in the script, this Egyptian manuscript is 33 cm tall. The papyrus began to be transliterated and mathematically translated in the late 19th century, the mathematical translation aspect remains incomplete in several respects. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later historical note on its verso likely dating from the period of his successor, Khamudi. In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving Accurate reckoning for inquiring into things, the scribe Ahmose writes this copy. Several books and articles about the Rhind Mathematical Papyrus have been published, a more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute. The first part of the Rhind papyrus consists of reference tables, the problems start out with simple fractional expressions, followed by completion problems and more involved linear equations. The first part of the papyrus is taken up by the 2/n table, the fractions 2/n for odd n ranging from 3 to 101 are expressed as sums of unit fractions. For example,2 /15 =1 /10 +1 /30. The decomposition of 2/n into unit fractions is never more than 4 terms long as in for example 2 /101 =1 /101 +1 /202 +1 /303 +1 /606. This table is followed by a smaller, tiny table of fractional expressions for the numbers 1 through 9 divided by 10. Problems 1-7, 7B and 8-40 are concerned with arithmetic and elementary algebra, problems 1–6 compute divisions of a certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 = 7/4 and 1 + 2/3 + 1/3 =2 by different fractions, problems 21–23 are problems in completion, which in modern notation are simply subtraction problems. Problems 24–34 are ‘’aha’’ problems, these are linear equations, problem 32 for instance corresponds to solving x + 1/3 x + 1/4 x =2 for x. Problems 35–38 involve divisions of the heqat, which is an ancient Egyptian unit of volume, problems 39 and 40 compute the division of loaves and use arithmetic progressions
The Rhind Mathematical Papyrus
–
A portion of the Rhind Papyrus
The Rhind Mathematical Papyrus
–
Building
25.
Space (geometry)
–
In mathematics, a space is a set with some added structure. Mathematical spaces often form a hierarchy, i. e. one space may inherit all the characteristics of a parent space, modern mathematics treats space quite differently compared to classical mathematics. In the ancient mathematics, space was an abstraction of the three-dimensional space observed in the everyday life. The axiomatic method had been the research tool since Euclid. The method of coordinates was adopted by René Descartes in 1637, two equivalence relations between geometric figures were used, congruence and similarity. Translations, rotations and reflections transform a figure into congruent figures, homotheties — into similar figures, for example, all circles are mutually similar, but ellipses are not similar to circles. The relation between the two geometries, Euclidean and projective, shows that objects are not given to us with their structure. Rather, each mathematical theory describes its objects by some of their properties, distances and angles are never mentioned in the axioms of the projective geometry and therefore cannot appear in its theorems. The question what is the sum of the three angles of a triangle is meaningful in the Euclidean geometry but meaningless in the projective geometry. A different situation appeared in the 19th century, in some geometries the sum of the three angles of a triangle is well-defined but different from the classical value. The non-Euclidean hyperbolic geometry, introduced by Nikolai Lobachevsky in 1829, eugenio Beltrami in 1868 and Felix Klein in 1871 obtained Euclidean models of the non-Euclidean hyperbolic geometry, and thereby completely justified this theory. This discovery forced the abandonment of the pretensions to the truth of Euclidean geometry. It showed that axioms are not obvious, nor implications of definitions, to what extent do they correspond to an experimental reality. This important physical problem no longer has anything to do with mathematics, even if a geometry does not correspond to an experimental reality, its theorems remain no less mathematical truths. These Euclidean objects and relations play the non-Euclidean geometry like contemporary actors playing an ancient performance, relations between the actors only mimic relations between the characters in the play. Likewise, the relations between the chosen objects of the Euclidean model only mimic the non-Euclidean relations. It shows that relations between objects are essential in mathematics, while the nature of the objects is not, according to Nicolas Bourbaki, the period between 1795 and 1872 can be called the golden age of geometry. Analytic geometry made a progress and succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups
Space (geometry)
–
Overview of types of abstract spaces. An arrow from space A to space B implies that space A is also a kind of space B. That means, for instance, that a normed vector space is also a metric space.
Space (geometry)
–
A hierarchy of mathematical spaces: The inner product induces a norm. The norm induces a metric. The metric induces a topology.
26.
Epitaph of Stevinus
–
Simon Stevin, sometimes called Stevinus, was a Flemish/Dutch/Netherlandish mathematician, physicist and engineer. He was active in a great areas of science and engineering. Very little is known with certainty about Stevins life and what we know is mostly inferred from other recorded facts, the exact birth date and the date and place of his death are uncertain. It is assumed he was born in Bruges since he enrolled at Leiden University under the name Simon Stevinus Brugensis and his name is usually written as Stevin, but some documents regarding his father use the spelling Stevijn. This is a normal spelling shift in 16th century Dutch and he was born around the year 1548 to unmarried parents, Anthonis Stevin and Catelyne van der Poort. His father is believed to have been a son of a mayor of Veurne. While Simons father was not mentioned in the book of burghers, many other Stevins were later mentioned in the Poorterboeken. Simon Stevins mother Cathelijne was the daughter of a family from Ypres. Her father Hubert was a poorter of Bruges, Simons mother Cathelijne later married Joost Sayon who was involved in the carpet and silk trade and a member of the schuttersgilde Sint-Sebastiaan. Through her marriage Cathelijne became a member of a family of Calvinists and it is believed that Stevin grew up in a relatively affluent environment and enjoyed a good education. He was likely educated at a Latin school in his hometown, Stevin left Bruges in 1571 apparently without a particular destination. Stevin was most likely a Calvinist since a Catholic would likely not have risen to the position of trust he later occupied with Maurice, Prince of Orange and it is assumed that he left Bruges to escape the religious persecution of Protestants by the Spanish rulers. Based on references in his work Wisconstighe Ghedaechtenissen, it has been inferred that he must have moved first to Antwerp where he began his career as a merchants clerk. Some biographers mention that he travelled to Prussia, Poland, Denmark, Norway and Sweden and other parts of Northern Europe and it is possible that he completed these travels over a longer period of time. In 1577 Simon Stevin returned to Bruges and was appointed city clerk by the aldermen of Bruges and he worked in the office of Jan de Brune of the Brugse Vrije, the castellany of Bruges. Why he had returned to Bruges in 1577 is not clear and it may have been related to the political events of that period. Bruges was the scene of religious conflict. Catholics and Calvinists alternately controlled the government of the city and they usually opposed each other but would occasionally collaborate in order to counteract the dictates of King Philip II of Spain
Epitaph of Stevinus
–
Simon Stevin
Epitaph of Stevinus
–
Statue of Simon Stevin by
Eugène Simonis, on the Simon Stevinplein (nl) in
Bruges
Epitaph of Stevinus
–
Statue of Stevin (detail)
Epitaph of Stevinus
–
Statue (detail):
Inclined plane diagram
27.
Theorem
–
In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a consequence of the axioms. The proof of a theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises, however, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a symbolic form, for example, within the propositional calculus. In some cases, a picture alone may be sufficient to prove a theorem, because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being trivial, or difficult, or deep and these subjective judgments vary not only from person to person, but also with time, for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a theorem may be simply stated. Fermats Last Theorem is a particularly well-known example of such a theorem, logically, many theorems are of the form of an indicative conditional, if A, then B. Such a theorem does not assert B, only that B is a consequence of A. In this case A is called the hypothesis of the theorem and B the conclusion. The theorem If n is an natural number then n/2 is a natural number is a typical example in which the hypothesis is n is an even natural number. To be proved, a theorem must be expressible as a precise, nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader can produce a formal statement from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and these hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of known as proof theory studies formal languages, axioms. Some theorems are trivial, in the sense that they follow from definitions, axioms, a theorem might be simple to state and yet be deep
Theorem
–
A
planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The
four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
28.
Pyth. theorem
–
In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
Pyth. theorem
–
The
Plimpton 322 tablet records Pythagorean triples from Babylonian times.
Pyth. theorem
–
Pythagorean theorem The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Pyth. theorem
–
Geometric proof of the Pythagorean theorem from the
Zhou Bi Suan Jing.
Pyth. theorem
–
Exhibit on the Pythagorean theorem at the
Universum museum in Mexico City
29.
Basic applied math
–
Applied mathematics is a branch of mathematics that deals with mathematical methods that find use in science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of science and specialized knowledge. The term applied mathematics also describes the professional specialty in which work on practical problems by formulating and studying mathematical models. The activity of applied mathematics is thus connected with research in pure mathematics. Historically, applied mathematics consisted principally of applied analysis, most notably differential equations, approximation theory, quantitative finance is now taught in mathematics departments across universities and mathematical finance is considered a full branch of applied mathematics. Engineering and computer science departments have made use of applied mathematics. Today, the applied mathematics is used in a broader sense. It includes the areas noted above as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of mathematics are now important in applications. There is no consensus as to what the various branches of applied mathematics are, such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees. Many mathematicians distinguish between applied mathematics, which is concerned with methods, and the applications of mathematics within science. Mathematicians such as Poincaré and Arnold deny the existence of applied mathematics, similarly, non-mathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to industrial problems is also called industrial mathematics. Historically, mathematics was most important in the sciences and engineering. Academic institutions are not consistent in the way they group and label courses, programs, at some schools, there is a single mathematics department, whereas others have separate departments for Applied Mathematics and Mathematics. It is very common for Statistics departments to be separated at schools with graduate programs, many applied mathematics programs consist of primarily cross-listed courses and jointly appointed faculty in departments representing applications. Some Ph. D. programs in applied mathematics require little or no coursework outside of mathematics, in some respects this difference reflects the distinction between application of mathematics and applied mathematics. Research universities dividing their mathematics department into pure and applied sections include MIT, brigham Young University also has an Applied and Computational Emphasis, a program that allows student to graduate with a Mathematics degree, with an emphasis in Applied Math
Basic applied math
–
Efficient solutions to the
vehicle routing problem require tools from
combinatorial optimization and
integer programming.
30.
Barometric gradient
–
Wind is the flow of gases on a large scale. On the surface of the Earth, wind consists of the movement of air. Winds are commonly classified by their scale, their speed, the types of forces that cause them, the regions in which they occur. The strongest observed winds on a planet in the Solar System occur on Neptune, Winds have various aspects, an important one being its velocity, another the density of the gas involved, another its energy content or wind energy. In meteorology, winds are referred to according to their strength. Short bursts of high speed wind are termed gusts, strong winds of intermediate duration are termed squalls. Long-duration winds have various names associated with their strength, such as breeze, gale, storm. The two main causes of large-scale atmospheric circulation are the differential heating between the equator and the poles, and the rotation of the planet, within the tropics, thermal low circulations over terrain and high plateaus can drive monsoon circulations. In coastal areas the sea breeze/land breeze cycle can define local winds, in areas that have variable terrain, mountain, Wind powers the voyages of sailing ships across Earths oceans. Hot air balloons use the wind to take trips, and powered flight uses it to increase lift. Areas of wind caused by various weather phenomena can lead to dangerous situations for aircraft. When winds become strong, trees and man-made structures are damaged or destroyed, Winds can shape landforms, via a variety of aeolian processes such as the formation of fertile soils, such as loess, and by erosion. Wind also affects the spread of wildfires, Winds can disperse seeds from various plants, enabling the survival and dispersal of those plant species, as well as flying insect populations. When combined with temperatures, wind has a negative impact on livestock. Wind affects animals food stores, as well as their hunting, Wind is caused by differences in the atmospheric pressure. When a difference in atmospheric pressure exists, air moves from the higher to the pressure area. On a rotating planet, air will also be deflected by the Coriolis effect, globally, the two major driving factors of large-scale wind patterns are the differential heating between the equator and the poles and the rotation of the planet. Outside the tropics and aloft from frictional effects of the surface, near the Earths surface, friction causes the wind to be slower than it would be otherwise
Barometric gradient
–
Wind, from the
Tacuinum Sanitatis
Barometric gradient
–
Cup-type anemometer with vertical axis, a sensor on a remote meteorological station
Barometric gradient
–
An occluded mesocyclone tornado (Oklahoma, May 1999)
Barometric gradient
–
EF0
31.
Two-column proof
–
In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
Two-column proof
–
One of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.
Two-column proof
–
Visual proof for the (3, 4, 5) triangle as in the
Chou Pei Suan Ching 500–200 BC.
32.
Cell (geometry)
–
In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
Cell (geometry)
–
The
cube has 3 square faces per vertex.
33.
Pure math
–
Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. Even though the pure and applied viewpoints are distinct philosophical positions, in there is much overlap in the activity of pure. To develop accurate models for describing the world, many applied mathematicians draw on tools. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research, ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between arithmetic, now called number theory, and logistic, now called arithmetic. Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, the term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, in the following years, specialisation and professionalisation started to make a rift more apparent. At the start of the twentieth century mathematicians took up the axiomatic method, in fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved, Pure mathematician became a recognized vocation, achievable through training. One central concept in mathematics is the idea of generality. One can use generality to avoid duplication of effort, proving a general instead of having to prove separate cases independently. Generality can facilitate connections between different branches of mathematics, category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math. Generalitys impact on intuition is both dependent on the subject and a matter of preference or learning style. Often generality is seen as a hindrance to intuition, although it can function as an aid to it. Each of these branches of abstract mathematics have many sub-specialties. A steep rise in abstraction was seen mid 20th century, in practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, the point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central
Pure math
–
An illustration of the
Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.