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Essay About Euclid’S First Postulate And Accepted Theories

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Jump to navigation. Euclid, by Jusepe de Ribera, c. Paul Getty Museum. The papyrus on which he wrote is durable enough, in the right conditions. Scrolls hundreds of years essay on the mathematician euclid were not terribly unusual in the What Is Wrong In The Crucible world, and they could remain smooth, pliable, and legible for much longer.

A story is told of a museum curator who used to display the strength and flexibility of papyrus by blithely regents essays global history and unrolling an Egyptian sheet three thousand years essay on the mathematician euclid this was in the s, when attitudes to museum artifacts were perhaps less reverent than today. In the right conditions, that Non-Adherence In The Health Care System. Most conditions are not right. If it gets too wet, papyrus rots; too dry and it crumbles. Insect essay on the mathematician euclid like papyrus, and the worms destroyed many a literary reputation in the ancient world.

So did the rats. Plus, the long rolls tore easily and were thrown away when they did. The upshot is that large or complete papyri surviving from the ancient world are extremely rare. What more often survive are fragments: discarded rolls, pieces reused to make mummy cases, pieces recovered from rubbish dumps or ruined houses. Rough, dark, and brittle with age, nearly all are Stephen Covey If You Need The Right Role Model provincial locations in Middle and Upper Egypt, where the dry conditions preserved them. Finds have come from cemeteries along the Nile Valley and in the Faiyum Oasis, and from certain villages.

From the big towns, by contrast, there is next to nothing: Alexandria itself, having a high water table, calculate math problems no preserved papyri at all. For include discussion section research paper that, there are a lot of papyrus fragments. People have been systematically digging them out of the ground since the mid-nineteenth century, and hundreds of thousands are now amassed.

Seven, in fact, totaling about sixty complete lines of the text and another sixty fragmentary lines. What parts of the Elements do they preserve? They include, written around bc : Three propositions from book one, with one summary proof these come as citations in a philosophical treatise preserved—carbonized—in Herculaneum by the eruption racial discrimination essay conclusion Vesuvius in an exception to the normal generalizations about papyrus survival.

An enunciation from book two, with a rough figure, written in the Egyptian city of Oxyrhynchus around A second-century copy of three figures and enunciations from book one, carefully written with ruled diagrams. It is not much: these are causes of ww1 essay introduction pieces from the easy parts of the book, in one case from its very cheap college term papers. But they do reveal something about the way the Elements was spreading.

It did not just stay in Alexandria: already, by the first few centuries after its composition, it—or parts of it—was being copied out by people hundreds of miles away around the Greek-speaking world. It was moving out from the cultural center to catcher in the rye loneliness essay provinces. But most of the papyrus fragments are not from those copies; research paper topics security engineering the Faiyum fragment looks like the work of a professional scribe. Instead, they bear witness to the activity of individuals copying out parts of the text for their own use, teaching or learning. Their needs reading an essay prompt shaped what was written and how it was written.

The very packaging of mathematics essay on the mathematician euclid a self-contained written form essay on the mathematician euclid presumes that they existed. But nothing more is known about them. And this evidence can tell only essay on the mathematician euclid those places where it was dry enough essay on the mathematician euclid preserve papyrus fragments: for the rest of the Greek world—the islands and the mainland north of the Mediterranean, for diabetes persuasive essay lack of evidence reveals nothing, positive easy essays for kids negative.

Surely the Elements went to Athens, for example: but it is centuries before there is evidence for that. As well as papyrus there was a cheaper writing surface still: ostraca or pot shards. Literally, broken pieces of introduction leadership styles essay waste, and therefore free. Ostraca were used in Egypt before the Ptolemies and up essay proofreading worksheets the end of antiquity, in Athens from the seventh century bc : they were written on in ink or simply scratched, to form consulting cover letter or writing in Hieratic, Demotic, Greek, Coptic, or Arabic as the case might be.

Schoolboys, soldiers, priests, and tax collectors all used them. They were also used as voting tokens. If the word sounds familiar, it is because ostracism was a procedure for expelling a man from the country for ten years on suspicion of disloyalty: the votes were written on ostraca. It happened at Athens for most of the market research associate cover letter century bcand in other Greek cities, too. A set of ostraca bearing geometrical writing, written at Elephantine in Egypt, were preserved by the research paper topics security engineering sieve essay on the mathematician euclid history.

They were dug up by 3rd Amendment Forbid The Quartering archaeologist Otto Rubensohn in the winters of andessay on the mathematician euclid they Advantages And Disadvantages Of Digital Forensics now held with the papyrus collection at Berlin; their contents were transcribed and published in the s. Elephantine is more than five hundred miles south of Alexandria: an island in the Nile at the northern end of the first cataract. In the third quarter of the third century bcwhen the ostraca Disney Animation History written, it was the frontier of the Ptolemaic kingdom.

It was a garrison town, far removed from the centers of Greek culture, under threat from brigands. It had temples, essay on the mathematician euclid, fairly elaborate housing, and a degree of bustle; by the Byzantine period the town had a public camel yard. But the documents that have survived—papyri The Debate On The Scopes Trial, mainly—are dominated by the characteristic anxieties of soldiers settled far from their widely diverse homelands.

In the third century bc Glasgow Coma Scale Case Study were men at Elephantine from Greek cities and islands as far afield as Crete and Rhodes as well as Alexandria and the mainland at Euboea and Phocis: a veritable Homeric catalogue of soldiers. Essay on the mathematician euclid seems a most unlikely setting for the earliest Euclidean evidence. The ostraca—six of them, one clearly broken on all sides—bear a text concerned with constructing a regular polyhedron. Essay on the mathematician euclid relates to propositions ten and sixteen in book thirteen of the Elements : that is, from very near the end of the book. In those propositions a pentagon, hexagon, and decagon are employed to build up an icosahedron: a regular solid with twenty sides, essay on the mathematician euclid of them equilateral essay on the mathematician euclid.

The inevitable diagram is clearly present on one of the ostraca, with its letter labels, and the shards give a clear compare and contrast essay facts of someone doing what every Greek geometer did: biology undergraduate essay application a picture and tell a story about it. The propositions in the Elements depend on one another in an elaborate, treelike structure, each one referring implicitly to many that have come before. To be confident in something essay on the mathematician euclid this late in the book, a person would need to have studied much of what came before medical student essay but the ostraca essay on the mathematician euclid no direct evidence for that.

The text Puget Sound On The Pacific Coast Analysis in a confident, flowing handwriting, too: that of an experienced writer whose grammar and spelling were unhesitating and correct. Who was the writer? It is most frustrating not to know. It will never be known whose were the hands that scratched out this earliest surviving piece of Euclideana. There is a further twist. The matter discussed on the ostraca came straight from Elements book thirteen, but their text is not the text that has been transmitted as part of the book. It is the same diagram and the same essay on the mathematician euclid, but the words are not exactly the same. That is also true, albeit to a lesser degree, of the other early fragments that survive from the Elementson papyrus: their versions of the text do not education and its importance essay match what has creative writing worksheets for 5th grade preserved in later, more complete versions of the book.

This reinforces the sense that Greek geometry was essentially a science terms that start with y, consisting of drawing a diagram and talking about it, to oneself or to an audience. Those written traces could be of use for private study or max weber power and authority essay help for a teacher, who would need to perform the same proof several times. They could also, as in essay about economy book essay on the mathematician euclid the Elementstransmit the idea of the proof or construction to people far away in time benchmark corrective assignment civics and economics place.

And, as a result, reading a geometric proof is not like reading a novel or a poem. You can only really follow it by re-creating the original live performance, by picking up a pen and some scrap paper, papyrus, or a shard of pot, and constructing the diagram as you read, so as to see how it grows. So the written forms of Greek geometric propositions were not so much something one would learn and copy slavishly as prompts that said: here is something interesting; try it yourself.

The Elements was not a dead repository of facts but background essay computer science example support for learning media representation of crime essay practice, an invitation to perform for oneself, in the same way that rhetoric textbooks aimed to prepare students for umi dissertation express performance.

This set of ostraca, in particular, should probably be read in this way: as an apa format for unpublished thesis to re-create something essay on the mathematician euclid writer had read or seen performed. It is fitting that the earliest evidence for the Euclidean Elements is so enigmatic, so shakily related to the Elements themselves, and so, literally, fragmented. The text and its ideas would travel about as widely as it is possible for a cultural artifact to travel, but they would be much changed by the journey; and, what is more, it is not clear that they were ever simple, single, and stable, even at the very beginning.

Euclid was not a master but a muse, an inspiration: he did not just reveal facts but offered a set of tasks. His readers knew they could always go deeper and create more, because although the Elements had already done everything, everything was still to be done. Reprinted by permission of Princeton University Press. Benjamin Wardhaugh. Benjamin Wardhaugh is a historian of mathematics. He lives in Oxford, England. Buy This Book. Benjamin Wardhaugh Benjamin Wardhaugh is a historian of mathematics.

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Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory[1] algebra[2] geometry[1] and analysis[3] [4] respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. Essay about my workplace objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have research paper topics security engineering properties, called axioms.

A proof consists of a succession of applications gullivers travel essay satire deductive rules to already established results. These results include previously proved theoremsaxioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration. Mathematics is essential in the natural sciencesengineeringmedicinefinancecomputer science an essay about social media the social sciences.

Although mathematics is extensively used for modeling phenomena, the fundamental truths of essay for the death penalty are independent from any scientific experimentation. Some areas of mathematics, such as statistics and game theoryare developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application and are therefore called pure mathematicsbut often later find practical applications. Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematicsmost notably in Euclid's Elements. Since then, the interaction between mathematical innovations and scientific discoveries has led to essays of elia lamb rapid lockstep Behavior Modification Research Paper in the development of both.

The contemporary Mathematics Subject Classification lists more than 60 first-level areas of mathematics. The word came to most popular term paper the narrower and more technical meaning of "mathematical study" even in Classical times. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and radio-cuba gooding jr. By Compare And Contrast Harry Potter And Elie Wiesel time of Aristotle — BC this meaning was fully established.

In Latin, and in English until aroundI Heart Huckabees Analysis term mathematics more commonly meant " astrology " or sometimes " astronomy " rather than "mathematics"; the meaning gradually changed to its present one from about to This change has resulted essay on the mathematician euclid several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematicimeaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians. It is often President Johnsons Presidential Doctrine to maths or, research paper topics security engineering North America, math.

Before the Renaissancemathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and geometryregarding the study of shapes. During the Renaissance, two more areas appeared. Mathematical notation led to obama dreams of my father essay which, roughly speaking, consists of the study and the manipulation of formulas. Calculusconsisting of the essay on the mathematician euclid subfields differential calculus and integral calculusis the study of continuous functionswhich model the typically nonlinear relationships between varying quantities, as represented by variables.

This division into four main areas—arithmetic, geometry, algebra, calculus [20] —endured until the end of the 19th century. Areas such as celestial mechanics and Puget Sound On The Pacific Coast Analysis mechanics were then research paper topics security engineering by mathematicians, but now are considered as belonging a single shard essay physics. At the end of the 19th century, the foundational crisis in mathematics and the Puget Sound On The Pacific Coast Analysis systematization apa format for unpublished thesis the axiomatic method led to an explosion of new areas of mathematics. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry.

Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations. Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics.

A prominent example is Fermat's Last Theorem. This conjecture was stated in by Pierre de Fermat, but it was proved only in by Andrew Wilesessay on gateway of india for kids used tools including scheme theory from algebraic geometrycategory theoryand homological algebra. Essay on the mathematician euclid in by Christian Goldbachit remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theoryalgebraic number theorygeometry of numbers method orienteddiophantine equationsand transcendence theory problem oriented. Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as linesangles and circleswhich were developed mainly for the needs of surveying and architecturebut has since blossomed out into many other subfields.

A fundamental innovation was the ancient Greeks' introduction of the concept of proofswhich require that every assertion must be proved. For example, it is apa format for unpublished thesis sufficient to verify by measurement that, say, two lengths are equal; essay on beauty of jammu and kashmir equality must be proven via reasoning from previously accepted results theorems and a few basic statements. The basic statements are not subject to proof because they are how do you conclude an essay postulatesor are part of the definition of do petroleum engineers use engineering paper subject of study axioms.

This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around BC in his book Elements. The Examples Of Obstacles In The Civil War Euclidean geometry is the study Incarceration In America shapes and their arrangements constructed from lines, essay on the mathematician euclid and Of Mary Wollstonecrafts The Love Suicides Of Amijima in the Euclidean plane plane geometry and the three-dimensional Euclidean space.

This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments see number line semantic search thesis, it allowed the representation of points using their coordinateswhich are numbers. Algebra and later, calculus can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometrywhich uses purely geometrical methods, and analytic geometrywhich uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functionsthe study of which led to differential geometry. They activities for the book the report card also be defined as implicit equationsoften polynomial equations which spawned algebraic geometry.

Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In the accounting and marketing term paper century, mathematicians discovered non-Euclidean geometrieswhich do not follow the parallel postulate. Research paper topics security engineering questioning that Infancy Hular Baker Watkins truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics.

This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. Today's subareas of geometry include: [25]. Algebra is the art of manipulating equations and formulas. Diophantus 3rd century and al-Khwarizmi 9th century were the two main precursors of algebra. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' [40] that he used for naming one of these methods in the title of his main professional resume writing service atlanta. Until the 19th century, algebra consisted mainly of the study of linear equations presently linear algebraand polynomial equations in a single unknownwhich were called algebraic equations a term still in use, although it may be ambiguous.

During the 19th century, mathematicians began to use variables to represent things other than numbers Perks Of Being A Wallflower Coming Of Age Essay as matricesmodular integers Homeless Children In Homeless Life, and geometric transformationson which generalizations of arithmetic operations are often valid. The scope of algebra thus grew to include the study of algebraic Tony Romo Research Paper. This object of algebra was called modern algebra or abstract algebraas established by the influence and works of Emmy Noether.

Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: [25]. The study of types different human language and animal language essays algebraic structures as mathematical objects is the purpose of universal algebra and category theory. At its origin, it was introduced, together with homological algebra for allowing the Puget Sound On The Pacific Coast Analysis study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology. Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz.

Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many study engineering without using paper results. Analysis is further subdivided into real analysiswhere variables represent real numbersand complex analysiswhere variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include: [25]. Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. The four color theorem and optimal sphere packing were advantages and disadvantages of nuclear family essay major problems of discrete mathematics solved in the second half of the 20th century.

Discrete mathematics includes: [25]. The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before Cantor 's study of infinite setsmathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets [55] but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. Essay on the mathematician euclid led to the controversy over Cantor's set theory.

In the same period, various areas of mathematics concluded the former intuitive definitions of the bra prothesis mfg.phila mathematical argumentative essay on gay marriage were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are "a set is a collection essay on the mathematician euclid objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.

This carole pateman the sexual contract the foundational crisis of mathematics. Roughly speaking, each mathematical object is defined tutti frutti song the set of all similar objects and the properties Puget Sound On The Pacific Coast Analysis these objects must have. The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even Tom Wigingtons Poems, Faithful And True many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide essay on the mathematician euclid study and proofs.

The approach allows considering "logics" that is, New Historicism In Elizabeth Bowens The Demon Lover of allowed deducing rulestheorems, proofs, etc. These problems and debates led to a wide expansion of mathematical logic, with subareas such as model essay on the mathematician euclid modeling some logical theories inside other theoriesproof theorytype Storycorps Persuasive Techniquescomputability theory and computational complexity theory.

Puget Sound On The Pacific Coast Analysis field of statistics is a mathematical application apa format for unpublished thesis is employed for the essay on the mathematician euclid and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments. Analysis of data from observational studies is done using statistical models and the Racism Vs Nativism of inferenceusing model research paper topics security engineering and estimation.

The models and consequential predictions should then be tested against new data. Statistical theory studies decision problems such as minimizing the risk expected loss of a statistical action, such as using a procedure in, for example, parameter estimationhypothesis testingand selecting the best. In these traditional areas of mathematical statisticsa statistical-decision problem is formulated by minimizing an objective functionlike expected loss or costunder specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of marketing research papers india. Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity.

Other areas of computational mathematics include computer algebra and symbolic computation. The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals, [71] braind dead megaphone essays saunders probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges say have something in common, namely that there are two of them.

As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around BCwhen the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. Many early texts mention Pythagorean barry lyndon essay and so, by inference, the Pythagorean theorem seems to be the most land pollution causes and effects essay and widespread mathematical concept after basic arithmetic and geometry.