21. Greek Mathematics And Its Modern Heirs Greek Mathematics and its Modern Heirs Classical Roots of the Scientific Revolution For over a thousand yearsfrom the fifth century B.C. to the fifth century A.D.Greek http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Greek_math.htm

Greek Mathematics and its Modern Heirs Classical Roots of the Scientific Revolution Euclid, Elements In Greek, Ninth century Euclid's "Elements," written about 300 B.C., a comprehensive treatise on geometry, proportions, and the theory of numbers, is the most long-lived of all mathematical works. This manuscript preserves an early version of the text. Shown here is Book I Proposition 47, the Pythagorean Theorem: the square on the hypotenuse of a right triangle is equal to the sum of the squares on the sides. This is a famous and important theorem that receives many notes in the manuscript. Vat. gr. 190, vol. 1 fols. 38 verso - 39 recto math01 NS.01 Archimedes, Works In Latin, Translated by Jacobus Cremonensis, ca. 1458 In the early 1450's, Pope Nicholas V commissioned Jacobus de Sancto Cassiano Cremonensis to make a new translation of Archimedes with the commentaries of Eutocius. This became the standard version and was finally printed in 1544. This early and very elegant manuscript may have been in the possession of Piero della Francesca before coming to the library of the Duke of Urbino. The pages displayed here show the beginning of Archimedes' "On Conoids and Spheroids" with highly ornate, and rather curious, illumination. Urb. lat. 261 fol. 44 verso - 45 recto math02 NS.17

22. Ancient Greek Mathematics - Crystalinks Greek mathematics, as that term is used in this article, is the mathematics developed from the 6th century BC to the 5th century AD around the shores of the http://www.crystalinks.com/greekmath.html

Greek mathematics, as that term is used in this article, is the mathematics developed from the 6th century BC to the 5th century AD around the shores of the Mediterranean. It constitutes a major period of the history of mathematics, fundamental in respect of geometry and the idea of formal proof. Greek mathematics also contributed importantly to ideas on number theory, mathematical analysis, applied mathematics, and, at times, approached close to integral calculus. Mathematical developments took place in Greek-speaking centers as far apart as Sicily and Egypt, and with a high estimation of the intellectual and cultural status of mathematics (for example in the school of Plato). Greek mathematics has origins that are presumed to go back to the 7th century BC, but are not easily documented. It is generally believed that it built on the computational methods of earlier Babylonian and Egyptian mathematics, and it may well have had Phoenician influences. Some of the most well-known figures in Greek mathematics are Pythagoras, a shadowy figure from the isle of Samos associated partly with number mysticism and numerology, but more commonly with his theorem, and Euclid, who is known for his Elements, a canon of geometry for centuries. The Sand Reckoner by Archimedes bespeaks a man who made major discoveries, and whose originality and accomplishments are commonly reckoned to be on par with those of Isaac Newton and C. F. Gauss.

23. Greece - Greek Math Resources on ancient Greek mathematics, calculations, geometry, and on Zeno, Archimedes, and Roman numerals. http://ancienthistory.about.com/od/greekmath/Greece_Greek_Math.htm

zWASL=1 zGL='0';zGR='ca-about-radlink'; zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') zDO=0 Home Education Ancient / Classical History Ancient / Classical History Search Ancient History Wars Filed In: Philosophy / Science Mathematics Resources on ancient Greek mathematics, calculations, geometry, and on Zeno, Archimedes, and Roman numerals. Hippocrates of Chios Hippocrates of Chios, who lived about the same time as the medical Hippocrates, wrote the first known work on geometry. Archimedes Archimedes probably studied mathematics in Alexandria with the successors of Euclid. The name Archimedes is connected to a pumping device now known as a Archimedes Screw, which he may have seen in operation in Egypt. zSB(3,3) Abacus - Base 10 and Greek Counting If the ancient Greeks used letters for their numbers, were they able to and did they use a counting system like an abacus that relies on a base like the decimal system? Finger Counting Counting on one's fingers seems a natural way to compute numbers, but the Greeks and Romans didn't just count "on" their fingers. They counted with their fingers, and not to be quick and accurate with the finger symbols could be embarrassing.

24. Mathematics :: Greek Mathematics -- Britannica Online Encyclopedia mathematics, Greek mathematics, Britannica Online Encyclopedia, The Greeks divided the field of mathematics into arithmetic (the study of “multitude,” or discrete quantity) and http://www.britannica.com/EBchecked/topic/369194/mathematics/65977/Greek-mathema

document.write(''); Search Site: Advanced Search With all of these words With the exact phrase With any of these words Without these words Home SHOP BROWSE BLOG ... HELP Username: Password: Remember me Forgot your password? Help Email " is the e-mail address you used when you registered. Password " is case sensitive. If you need additional assistance, please contact customer support Enter the e-mail address you used when enrolling for Britannica Premium Service and we will e-mail your password to you. mathematics My Britannica CREATE MY mathematics NEW ARTICLE ... SAVE mathematics Table of Contents: mathematics Article Article Ancient mathematical sources Ancient mathematical sources Mathematics in ancient Mesopotamia Mathematics in ancient Mesopotamia - The numeral system and arithmetic ope... The numeral system and arithmetic operations - Geometric and algebraic problems Geometric and algebraic problems - Mathematical astronomy Mathematical astronomy Mathematics in ancient Egypt Mathematics in ancient Egypt - The numeral system and arithmetic ope... The numeral system and arithmetic operations - Geometry Geometry - Assessment of Egyptian mathematics Assessment of Egyptian mathematics Greek mathematics Greek mathematics - The development of pure mathematics The development of pure mathematics - - The pre-Euclidean period The pre-Euclidean period - - The Elements The Elements - - The three classical problems The three classical problems - Geometry in the 3rd century BC Geometry in the 3rd century BC - - Archimedes

25. A History Of Greek Mathematics, Vol. 2 Volume 2 of an authoritative twovolume set that covers the essentials of mathematics and features every landmark innovation and every important figure, including Euclid http://store.doverpublications.com/0486240746.html

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26. Classical Greek Mathematics During the period from about 600 B.C. to 300 B.C. , known as the classical period of Greek mathematics, mathematics was transformed from an ecclectic http://www.rbjones.com/rbjpub/maths/math005.htm

Classical Greek Mathematics During the period from about 600 B.C. to 300 B.C. , known as the classical period of Greek mathematics, mathematics was transformed from an ecclectic collection of practical techniques into a coherent structure of deductive knowledge. For many mathematicians, the discipline we call mathematics was founded in this period. Here we briefly survey the achievements from a logical point of view From Procedural to Declarative Knowledge The change of focus from practical problem solving methods to knowledge of general mathematical truths and the development of a body of theory transforms mathematics into a scientific discipline. Abstraction Pythagorean abstraction and Plato's "ideals" make the subject matter of mathematics out of this world Logic The cannons of deductive reasoning are systematised by Aristotle in his syllogistic logic Foundations The Greeks showed concern for the logical structure of mathematics. The Pythagorean's sought to found all of mathematics on number but were confounded by the discovery of incommensurable ratios in geometry. This prevented them from giving an account of geometric magnitudes in terms of their numbers (what we now call the natural numbers or positive integers). By the end of the Pythagorean period geometry has come to be regarded as fundamental. The problem of incommensurable ratios will remain unresolved for more than two millenia. Deduction From very early in the classical period deduction is perceived as the primary method of arriving at mathematical truths. This contrasts with (but does not entirely displace) non-deductive generalisation from particulars.

27. ANCIENT GREEK MATHEMATICS If you are interested in learning more about Greek mathematics, you can browse this website or chat (below) with others about the influence of Greek http://www.angelfire.com/me/Huffamoose/

ANCIENT GREEK MATHEMATICS Space provided by Angelfire Communications. Best viewed at 640 X 480 (256 colors). If you are interested in learning more about Greek mathematics, you can browse this website or chat (below) with others about the influence of Greek mathematics on western civilization. For those who have little knowledge about this subject, ask yourself: "How will I ever use these postulates and theorems, and who has come up with all of these interesting ideas? Where did all of this come from?".... If you guessed Ancient Greece, you're right! While Alexander the Great was out on his conquests, mathematicians like Euclid and Aristotle were coming up with new ideas which would benefit western civilization 2300 years later. If you look at the Parthenon (bottom of page) from a distance away, it looks perfectly rectangular. But in fact, the floor, walls, and columns are actually bowed. The ancient Greek architects who calculated the measurements that would be used in the construction of the Parthenon, used mathematics to come up with their designs. Buildings, like the Parthenon, have influenced many modern buildings like the Philadelphia Art Museum. Greek mathematics has made an astonishing impact on our world. By scrolling down this page, you will be able to see the influence that Ancient Greek mathematics has had on Western Civilization.... Your journey begins here.

28. Greek Sources I There are two separate articles in this archive How do we know about Greek mathematics? and How do we know about Greek mathematicians? http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Greek_sources_1.html

How do we know about Greek mathematics? Ancient Greek index History Topics Index Version for printing There are two separate articles in this archive: "How do we know about Greek mathematics?" and How do we know about Greek mathematicians? . There is a common belief that the question posed in this article, about Greek mathematics rather than Greek mathematicians, is easy to answer. Perhaps all we need to do to answer it is to read the mathematical treatises which the Greek mathematicians wrote. We might think, very naively, that although some of the origainal texts have been lost there should be plenty left for us to be able to gain an excellent picture of Greek mathematics. The truth, however, is not nearly so simple and we will illustrate the way that Greek mathematical texts have come down to us by looking first at perhaps the most famous example, namely Euclid 's Elements . When we read Heath's The Thirteen Books of Euclid's Elements are we reading an English translation of the words which Euclid wrote in 300 BC? In order to answer this question we need to examine the way the Elements has reached us, and, more generally, how the writings of the ancient Greek mathematicians have been preserved.

29. The Origins Of Greek Mathematics Jan 31, 1997 In actual fact, our direct knowledge of Greek mathematics is less reliable than that of the older Egyptian and Babylonian mathematics, http://www.math.tamu.edu/~dallen/history/greekorg/greekorg.html

Next: About this document The Origins of Greek Mathematics Though the Greeks certainly borrowed from other civilizations, they built a culture and civilization on their own which is The most impressive of all civilizations, The most influential in Western culture, The most decisive in founding mathematics as we know it. Basic facts about the origin of Greek civilization and its mathematics. The best estimate is that the Greek civilization dates back to 2800 B.C. just about the time of the construction of the great pyramids in Egypt. The Greeks settled in Asia Minor, possibly their original home, in the area of modern Greece, and in southern Italy, Sicily, Crete, Rhodes, Delos, and North Africa. About 775 B.C. they changed from a hieroglyphic writing to the Phoenician alphabet. This allowed them to become more literate, or at least more facile in their ability to express conceptual thought. The ancient Greek civilization lasted until about 600 B.C. The Egyptian and Babylonian influence was greatest in Miletus, a city of Ionia in Asia Minor and the birthplace of Greek philosophy, mathematics and science. From the viewpoint of its mathematics, it is best to distinguish between the two periods: the

30. Mathematics Greek Mathematics. Ptolemy's Geography. Greek Astronomy. Also, someone left a note on the wall. When you have seen everything, walk back to the Main Hall. http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Mathematics.ht

Mathematics Ancient Science and Its Modern Fates Until recently, historians of the Scientific Revolution of the 16th and 17th centuries treated it as a kind of rebellion against the authority of ancient books and humanist scholarship. In fact, however, it began with the revival of several tremendously important and formidably difficult works of Greek science. Scholarship supported science in this world where faith and science were not yet seen as two, irreconcilable cultures. The three ancient doors to the next rooms all have signs written on them in Greek and Latin. Luckily for you we created modern metal plates with the translations, next to the doors. So you can pick any of: Greek Mathematics Ptolemy's Geography Greek Astronomy Also, someone left a note on the wall. When you have seen everything, walk back to the Main Hall

31. Greek Mathematics Greek mathematicsPeriodization Classical Greek mathematics refers to the mathematics studied before the Hellenistic period, when Greek mathematics was limited to Greece. http://www.docstoc.com/docs/15285800/Greek-mathematics

32. 2. Greek Mathematics And Pythagoras. Next 3. Geometric algebra. Up BEING AND SIGN Previous 1. Mesopotamian and . 2. Greek mathematics and Pythagoras. 'Mathema' means ''study subject'' and in the PythagoreanPlatonic http://www.dm.uniba.it/~psiche/bas2/node3.html

Next: 3. Geometric algebra. Up: BEING AND SIGN Previous: 1. Mesopotamian and 2. Greek mathematics and Pythagoras. 'Mathema' means ''study subject'' and in the Pythagorean-Platonic tradition the term ''mathematics'' comprehended geometry, arithmetic, music and astronomy, the so called ''quadrivium'', that in the Platonic Academia was not an introductory discipline, but a sort of 'final training' for the aristocratic Guardians described in the Platonic Respublica. They appear organised as two pairs: geometry/astronomy and arithmetic/music, each pair including both the abstract and the concrete aspects, and describing respectively the being and signs worlds. In the Greek mathematics it is noteworthy the distinction between arithmetic and logistics. The former is a ''number theory'' and concerns with the numbers 'in themselves', by distinguishing even and odd, linear, plane and solid numbers, with a likely evident genetic reference to their ancient representation as pebbles on a surface, according to the ancient Pythagorean tradition. Following definitions are those of prime, square, cubic, oblong numbers. Proofs are geometric too, representing the numbers as figures, squares, rectangles, triangles, gnomons, and so on, according to the so called ''dot-algebra'' ((KNORR 1975)), and are the bases of that ''geometric algebra'' which we are going to analyse in the following sections. The latter copes with the practical aspects in measurement and trade, and deals with ''specific numbers of perceived and counted things'', using also special number-names depending on the numbered things: as, for example, 'phialites' ''number of bowls'', from 'phiale', ''bowl''. From this point of view, it is worthwhile to remind the adjectival nature of the Greek numbers (in Greek 1, 2, 3, 4, hundreds, thousands, ten thousands are inflected for genders and cases). Logistics comprehends elementary arithmetic operations and simple equations solution too.

33. Basic Ideas In Greek Mathematics This is again reminiscent of the Greek strategy in approximating the square root of 2. The result of all his efforts was the inequality 3 10/71 pi 3 http://galileoandeinstein.physics.virginia.edu/lectures/greek_math.htm

previous index next PDF Basic Ideas in Greek Mathematics Michael Fowler UVa Physics Department Closing in on the Square Root of 2 m n . Specifically, 3 + 1. These results were also noted by the Greeks, and set down in tabular form as follows: After staring at this pattern of numbers for a while, the pattern emerges: 3 + 2 = 5 and 7 + 5 = 12, so the number in the right-hand column, after the first row, is the sum of the two numbers in the row above. Furthermore, 2 + 5 = 7 and 5 + 12 = 17, so the number in the left-hand column is the sum of the number to its right and the number immediately above that one. The question is: does this pattern continue? To find out, we use it to find the next pair. The right hand number should be 17 + 12 = 29, the left-hand 29 + 12 = 41. Now 41 = 1681, and 29 = 841, so 41 - 1. Repeating the process gives 41 + 29 = 70 and 70 + 29 = 99. It is easy to check that 99 + 1. So 99 . In other words, the difference between the square root of 2 and the rational number 99/70 is approximately of the magnitude 1/70 . (You can check this with your calculator). Zeno of Elea (495-435 BC) is said to have been a self-taught country boy. He was a friend of a well-known philosopher, Parmenides, and visited

34. Aristotle And Mathematics > Aristotle And Greek Mathematics (Stanford Encycloped by H Mendell 2004 - Cited by 6 - Related articles http://plato.stanford.edu/entries/aristotle-mathematics/supplement4.html

Cite this entry Search the SEP Advanced Search Tools ... Stanford University Supplement to Aristotle and Mathematics Aristotle and Greek Mathematics This supplement provides some general indications of Aristotle's awareness and participation in mathematical activities of his time. Greek mathematics in Aristotle's Works Here are twenty-five of his favorite propositions (the list is not exhaustive). Where a proposition occurs in Euclid's Elements , the number is given, * indicates that we can reconstruct from what Aristotle says a proof different from that found in Euclid). Where the attribution is in doubt, I cite the scholar who endorses it. In many cases, the theorem is inferred from the context. In a given circle equal chords form equal angles with the circumference of the circle ( Prior Analytics i.24; not at all Euclidean in conception) The angles at the base of an isosceles triangle are equal ( Prior Analytics i.24; Eucl. i.5*). The angles about a point are two right angles ( Metaphysics ix 9; Eucl. follows from i def. 10). If two straight-lines are parallel and a straight-line intersects them, the interior angle is equal to the exterior angle (

35. Greek Mathematics Index How do we know about Greek mathematics? How do we know about Greek mathematicians? Greek number systems The teaching of mathematics in Ancient Greece http://www-history.mcs.st-and.ac.uk/Indexes/Greeks.html

History Topics: Index of Ancient Greek mathematics Articles about Greek mathematics. Squaring the circle Doubling the cube Trisecting an angle Greek Astronomy ... Geography Various lists of Greek mathematicians. Full list Mathematicans/Philosophers Mathematicians/Astronomers Mathematicians/Astronomers/Philosophers ... Later circle squarers Click on a name below to go to that biography. Full List of Greek Mathematicians in our archive Anaxagoras Anthemius Antiphon Apollonius ... Zenodorus Greek Mathematicans/Philosophers Anaxagoras Antiphon Archytas Aristotle ... Zeno of Elea Greek Mathematicians/Astronomers Apollonius Archimedes Aristarchus Aristotle ... Theon of Smyrna Greek Mathematicians/Astronomers/Philosophers Aristotle Cleomedes Democritus Eudoxus ... Thales Greek Circle squarers Anaxagoras Antiphon Apollonius Archimedes ... Bryson Carpus Dinostratus Hippias Hippocrates Nicomedes ... Sporus Later Circle squarers Al-Haytham Johann Bernoulli Cusa James Gregory ... Search Form JOC/EFR May 2010 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/Indexes/Greeks.html

36. Perseus Digital Library Welcome to Perseus 4.0, also known as the Perseus Hopper. Read more on the Perseus version history. New to Perseus? Click here for a short tutorial. http://www.perseus.tufts.edu/hopper?redirect=true

37. Greek Mathematics File Format PDF/Adobe Acrobat Quick View http://faculty.unlv.edu/bellomo/Math714/Notes/10_Greek.pdf

38. Explore: Greece - Mathematics The two pioneers of Greek mathematics, Thales and Pythagoras, therefore have no surviving texts to their credit. Much of what we know from them is http://library.thinkquest.org/C0122667/greece/maths.html

Explore Gallery Interactive Forum ... Warfare A n c i e n t F a c t s Greece Mathematics Introduction Early Greek Mathematics (650-400 BCE) Thales Of Miletus (ca. 624-548 BCE) Thales was, by legend, a clever man, who was said to have learned much from the Egyptians and Babylonians. He is reputed to have demonstrated that the angle inscribed in a semicircle is a right angle (Theorem of Thales) and put down a series of rules regarding the angles of triangles. He was reported to have measured the height of the Pyramids by comparing the length of their shadows to that of a vertical stick. At the moment the length of the stick�s shadow was equal to its height, the length of the Pyramids� shadow would indicate their height. Though much of the knowledge attributed to Thales originated from Egypt and Babylon, he is credited with organising them in a rational manner. Thales also broke away from the rigidity of using geometry solely for measurement and tried to apply it in practical methods. This logical structure he provided to geometry set forth a great idea followed by later mathematicians such as Pythagoras and Plato. Pythagoras of Samos (ca. 580-500 BCE)

39. Perseus Digital Library www.perseus.tufts.edu/GreekScience/Students/Chris/GreekMath.html SimilarHistory of Mathematics Greeceby D Laertius - Related articles http://www.perseus.tufts.edu/GreekScience/Students/Chris/GreekMath.html

40. The Origins Of Greek Mathematics About this document . The Origins of Greek Mathematics. Though the Greeks certainly borrowed from other civilizations, they built a culture and civilization on their own which is http://www.math.tamu.edu/~don.allen/history/greekorg/greekorg.html

Next: About this document The Origins of Greek Mathematics Though the Greeks certainly borrowed from other civilizations, they built a culture and civilization on their own which is The most impressive of all civilizations, The most influential in Western culture, The most decisive in founding mathematics as we know it. Basic facts about the origin of Greek civilization and its mathematics. The best estimate is that the Greek civilization dates back to 2800 B.C. just about the time of the construction of the great pyramids in Egypt. The Greeks settled in Asia Minor, possibly their original home, in the area of modern Greece, and in southern Italy, Sicily, Crete, Rhodes, Delos, and North Africa. About 775 B.C. they changed from a hieroglyphic writing to the Phoenician alphabet. This allowed them to become more literate, or at least more facile in their ability to express conceptual thought. The ancient Greek civilization lasted until about 600 B.C. The Egyptian and Babylonian influence was greatest in Miletus, a city of Ionia in Asia Minor and the birthplace of Greek philosophy, mathematics and science. From the viewpoint of its mathematics, it is best to distinguish between the two periods: the

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