21. Read This: The Shaping Of Deduction In Greek Mathematics There are differences, though greek mathematicians were not illiterateoral performers. RN gives a competent analysis of the characteristic http://www.maa.org/reviews/netz.html

Read This! The MAA Online book review column The Shaping of Deduction in Greek Mathematics A study in cognitive history by Reviel Netz Reviewed by Christian Marinus Taisbak Reviel Netz has written an stimulating book about diagrams and mathematics, telling us facts that we all know, but hardly ever thought of. Thus he sets himself in the best of company, for isn't that what Euclid did from the very first proposition in the Elements? "The diagram is the metonym of mathematics" is RN's main claim. To understand what he means by that, think of two typical situations in the circus of conferences: if a philosopher or historian gives a talk, he will read aloud for half an hour, facing his audience without moving from his chair. If a mathematician gives a talk, he will dance around the platform talking to the blackboard while writing figures and letters on it, most of the time ignoring his audience and concentrating on his written deductions as they emerge out of sheer necessity. Years ago David Fowler (of Plato's Academy ) coined a motto: "Greek mathematics is to draw a figure and tell a story about it." RN has widened and deepened this into "Deductive mathematics grew out of the Greeks drawing lettered diagrams and telling stories by means of them, not only about them." The diagram and the argument live in such a close symbiosis that one cannot be understood without the other. The diagram is the metonym of mathematics.

22. MAA Tour Of Greece The greek mathematicians that we met were very welcoming and kind. They all seemeddelighted to host a group of visiting mathematicians from the United States. http://www.maa.org/features/greecetrip03.html

Search MAA Online MAA Home MAA Tour of Greece Victor Katz and Lisa Kolbe, tour organizers, at the Acropolis. Thirty members (and guests) of the MAA traveled to Greece in late May to participate in the first MAA Study Tour abroad. There was a wide variety of participants, from high school teachers to retired professors, including two former presidents of the MAA as well as the current and former executive directors. All agreed that the trip met, and even exceeded, their expectations. Five of them agreed to write brief comments for FOCUS about their experiences. From Joel and Linda Haack University of Northern Iowa, Cedar Falls, IA: There were quite a few reasons that we were eager to participate on the MAA Study Tour of Greece. First, of course, as I have been teaching the history of mathematics for parts of the past 15 years, I was very eager to see Athens, Delphi, Miletus, and Samos. Second, there are a few places in the world that my wife Linda and I have always wanted to see. Greece has been one of those, ever since learning about it in our ancient history class way back in ninth grade. Finally, this trip provided us the perfect occasion to celebrate our thirtieth wedding anniversary. Temple at Delphi.

23. ThinkQuest : Library : Go Forth & Multiply: A Mathematics Adventure or postulates, assumptions accepted without justification. Otherfamous greek mathematicians include Archimedes and Apollonius. http://library.thinkquest.org/C0110248/geometry/history2.htm

Index Education Want to learn more about the world of mathematics? Then go forth, and enter the wildest math adventure you've ever been! Learn new math concepts and refresh your knowledge for those you've already known. Understand how the formulae you use were derived from. Or, you can take a step back into the past and read about how mathematics and its concepts originated. Go forth and multiply! Visit Site 2001 ThinkQuest Internet Challenge Awards Achievement Award Students Teow Lim Raffles Junior College, Singapore, Singapore Vee San Raffles Girls' School (Secondary), Singapore, Singapore Coaches Poh Kheng Pioneer Junior College, Singapore, Singapore Jee Wah Raffles Girls' School (Sec), Hougang, Singapore Want to build a ThinkQuest site? The ThinkQuest site above is one of thousands of educational web sites built by students from around the world. Click here to learn how you can build a ThinkQuest site. Privacy Policy

24. Interesting Facts - Ancient Greek Science And Philosophy When greek mathematicians first proved the the square root of two isan irrational number, they celebrated by sacrificing 100 oxen. http://www.sentex.net/~ajy/facts/greeksci.html

Ancient Greek Science and Philosophy "No fact is so simple that it is not harder to believe than to doubt at the first presentation." - Lucretius The Greek philosopher Thales (624-546 B.C.) is considered to be the first philosopher, as he was the first man in history to ask questions such as "Of what is the Universe made?", and to answer without introducing gods and demons. In later centuries, when the Greeks made up lists of the "seven wise men", Thales invariably was placed first. [ Philosophy Ancient Greek Science and Philosophy The Lydians (allies of the Greek Spartans) and the Medes (dominated by Cyrus the Persian) had been locked in a five-year war in Asia Minor in 585 B.C. On May 28 th , the two armies were preparing for a crucial daytime battle when a solar eclipse occurred, one that is believed to have been predicted by Thales, a Greek mathematician. When the Medes and Lydians observed the eclipse, they ceased fighting and signed a peace treaty. Incidentally, this is the earliest event in human history that we are able to assign an exact date to, due to the eclipse. [ Calendars Ancient Greek Science and Philosophy The first person we know who realized the Earth couldn't be flat was the Greek philosopher Anaximander. Around 560 B.C., he suggested that the Earth had a cylindrical shape. By 350 B.C., the concept of a spherical Earth was so satisfying and free of paradox that it was generally accepted by scholars even in the absence of direct proof. Eighteen more centuries were to pass before that direct proof occurred - the lone surviving ship in an expedition once commanded by Ferdinand Magellan sailed completely around the globe.

25. Euclid --  Encyclopædia Britannica Life. Of Euclid s life nothing is known except what the Greek philosopher Proclus(c. AD 410–485) reports in his “summary” of famous greek mathematicians. http://www.britannica.com/eb/article?eu=33758&tocid=2173&query=euclid

26. Euclid --  Encyclopædia Britannica Of Euclid s life nothing is known except what the Greek philosopher Proclus (c.AD 410–485) reports in his “summary” of famous greek mathematicians. http://www.britannica.com/eb/article?eu=33761&tocid=0&query=euclid

27. The Origins Of Greek Mathematics The School of Eudoxus founded by Eudoxus (c. 408 BC), the most famous ofall the classical greek mathematicians and second only to Archimedes. 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

28. Harmony Harmony and Dissonance. Many of the greek mathematicians also usedHarmony and Dissonance in their studies of mathematics. One Greek http://www.springfield.k12.il.us/schools/southeast/pprojects/harmony.html

Harmony and Dissonance Many of the Greek mathematicians also used Harmony and Dissonance in their studies of mathematics. One Greek mathematician, Pythagoras, noticed that vibrating strings produced harmonious tones when the ratios of the lengths of the strings were whole numbers. Pythagoras aslo noticed that these ratios could be extended to other instruments, which allowed him to make remarkable contributions to the mathematical elements of music. Euclid, a Greek mathematician who specialized in Geometry, also conmtributed to the properties of harmony and dissonance. He found that if you take two strings in the same degree of tension, and then divide one of them exactly in half, when they are plucked, the pitch of the shorter string is exactly one octave higher than the longer. He also discovered that if the length of the two strings are in relation to each other 2:3, the difference in pitch is called a fifth. Also if the length of the strings are in relation to each other 3:4, then the difference is called a fourth. Thus the musical notation of the Greeks, which we have inherited can be expressed mathematically as 1:2:3:4 Musical harmonies are numerical ratios. A string or flute shortened to half of its original length produces a tone which is one octave higher. Ratios of 3 : 2 give a fifth and 4 : 3 give a fourth. The ratio of 3 : 4 : 5 gives the sides of a right-angled triangle, which established a connection of numbers to angles. Mathematicians classified numbers into categories of odd, even, prime, composite, perfect and amicable numbers. They used stones or pebbles in groups to form different patterns, which they classified as figurate, triangular, or square numbers.

29. Calculus.html It is the most elegant of all Greek mathematical manuscripts. Both of thesegreek mathematicians contributed to today s study of calculus. 3. http://www.springfield.k12.il.us/schools/southeast/pprojects/calculus.html

Calculus C alculus n a method of computation or calculation in a special notation (as of logic or symbolic logic.) the mathematical methods comprising differential and integral calculus. T he main theorem of calculus is used to unite the differential and integral calculus into one system that can be used to solve a broad range of problems. A nother Greek mathematicin was a man named Apollonius's. Apollonius wrote "Conics," about 200 B.C., about conic sections, the ellipse, parabola, and hyperbola. "Conics" is the most complex and difficult single work of all Greek mathematics and was unknown until the fifteenth century. It is the most elegant of all Greek mathematical manuscripts. This illustrates the book "Conics" that Apollonius wrote about calculus. T he Greek mathematician Archimedes also contributed to the findings of calculus. Archimedes used the methods for calculus that are the same methods that we use today. Both of these Greek mathematicians contributed to today's study of calculus. Home Mythology and Creation Math Daily Life ... Families of Mythology

30. Aristotle And Greek Mathematics: A Supplement To Aristotle And Mathematics use of angles formed by straight lines and circles, neither of which is admittedin Euclid s Elements, is important evidence that greek mathematicians were not http://plato.stanford.edu/entries/aristotle-mathematics/supplement4.html

Stanford Encyclopedia of Philosophy Supplement to Aristotle and Mathematics Citation Information 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 (

31. Gnist.no: Fagbokhandelen PÃ¥ Internett 4. Formulae; 5. The shaping of necessity; 6. The shaping of generality; 7. The historicalsetting; Appendix the main greek mathematicians cited in the book. http://www.gnist.no/kategori.php?varenummer=1550404&side=PBX

32. Theuth-hilaire greek mathematicians a Group Picture. Matapli. forthcoming. Why did greek mathematiciansPublish their Analyses? Memorial Volume for Wilbur Knorr, eds. http://name.math.univ-rennes1.fr/theuth/biblios/theuth-netz.html

Reviel Netz netz@leland.stanford.edu (janvier 2001) Academic Books: The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History . Cambridge University Press 1999. Archimedes: Translation and Commentary, with a Critical Edition of the Diagrams and a Translation of Eutocius' commentaries . Cambridge University Press [forthcoming]. Barbed Wire . Picador [forthcoming]. Articles: How does a Geometrical Problem become a Cubic Equation? Farhang. Archimedes and Mar Saba: a Preliminary Notice. The Sabaite Heritage: The Sabaite Factor in the Orthodox Church: Monastic life, Liturgy, Theology, Literature, Art and Archaeology , ed. J. Patrich. The Limits of Text in Greek Mathematics. In History of Science, History of Text , ed. K. Chemla, Dordrecht: Reidel [forthcoming]. Greek Mathematicians: a Group Picture. In Science in the Ancient World , ed. C. Tuplin, Oxford University Press [forthcoming]. Eudemus of Rhodes, Hippocrates of Chios and the Earliest Form of a Greek Mathematical Text. In Eudemus of Rhodes , eds. W. W. Fortenbaugh and I. Bodnar, Rutgers University studies in classical humanities [forthcoming].

33. Greek Authors A Mathematician (b. c287 BC) Eureka ? You would have thought that there wouldbe plenty of material on this, the most famous, of greek mathematicians. http://www.hellenicbookservice.com/classics/greek_authors_a.htm

Achilles Tatius-Archimedes Aeschylus Loeb editions Oxford Classical Texts Green and Yellows Various Notes: (York, Cliff, Max etc.) Penguin translations Clarendon Texts Oxford World Classics Aris and Phillips Chicago Translations Bristol Classical Press The Icons against the books refer to their edition and in most cases the language in which they are written. Click on the images above for an explanation as what to expect from these particular editions. I have tried to scan images of other books, but this is a very slow process. Other Authors Compiled by Andrew Stoddart Achilles Tatius Novelist. (b.2nd century AD?) Achilles Tatius - Leucippe and Clitophon Achilles Tatius Loeb Leucippe and Cleitophon in parallel Text Collected Greek Novels Achilles Tatius wrote 'Leucippe and Cleitophon' which appears in Reardon's superb 'Collected Ancient Greek Novel' £24.00 Aeschines Orator (b. c397BC) Aeschines and Athenian Politics Aeschines - Against Timarchos Ed by Fisher Ed by Fisher £19.99 X Aeschines Loeb Aeschylus Aeschylus (Ted Hughes, trans.)

34. Term Papers (model), Term Papers (model) And More Term Papers (model) Mathematic The Importance of Mathematics in Early Greek Culture A 12 page comprehensivestudy of early greek mathematicians and their cultural significance. http://www.termpapers-on-file.com/mathmatics.htm

MATHEMATICIANS Mathematics Papers - MORE! Back to Main Categories Back to Main ... Categories NOW! ALL PAPERS ON FILE ARE ONLY $9.95/PAGE!!! MORE EXAMPLE TERM PAPERS ON MATHEMATICS A 15 page paper that provides an overview of the history and development of the abacus. The report essentially compares the Chinese, Roman, Greek, Russian and Indian counting methods utilizing similar instruments. Bibliography lists 6 sources. Abacus.doc Benefits Of Computer-Taught Math Over Standard Textbook Practices A 10 page study that provides support for the hypothesis that computer taught math provides significant beneficial outcomes for learners in terms of test scores. Bibliography lists 10 sources. Mtcomp.wps Differential Equations An 18 page research paper on every available aspect of differential equations including Laplace Transforms and much more. A number of graphical illustrations are provided and the bibliography lists more than 8 sources. Diffequa.wps Linear Algebra A 15 page research paper on various concepts in linear algebra. The writer details multivariables, vectors, determinants, gaussian elimination, and other elements of linear algebra. Bibliography lists 6 sources. Linalgeb.wps

35. Mathematics (Rome Reborn: The Vatican Library & Renaissance Culture) Nicholas V supported translations of the greatest of greek mathematicians,Archimedes, and the greatest of Greek astronomers, Ptolemy. http://www.loc.gov/exhibits/vatican/math.html

The Library of Congress Exhibitions HOME Exhibition Sections: Introduction The Vatican Library Archaeology Humanism ... Credits MATHEMATICS Greek Mathematics and its Modern Heirs Ptolemy's Geography Greek Astronomy Greek Mathematics and its Modern Heirs Euclid, Elements In Greek Parchment 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. Archimedes, Works In Latin Translated by Jacobus Cremonensis ca. 1458 In the early 1450s, 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.

36. Fermat S Last Theorem - The Birth Of The Problem Many books of greek mathematicians suffered a similar fate, and someof them survived only through their translation to Arabic. http://fermat.workjoke.com/flt1.htm

chapter: Proofs for special cases Table of contents The birth of the problem "I have discovered a marvelous proof to this theorem, that this margin is too narrow to contain", so had scribbled Pierre de Fermat, about 350 years ago, in the margin of a mathematics book he read. That theorem came to be known as Fermat's Last Theorem (which will be abbreviated hence as FLT), and the attempt to prove it had baffled many mathematicians, both professionals and amateurs. The equation x +y =z has many solutions where x, y, and z are integers, i.e. ,3 , or 5 . Such a solution is called a Pythagorean triplet, because according to Pythagoras' Theorem such a triplet represents the sides of a right-angled triangle. On a Babylonian clay tablet dated about 3,500 years ago (about 1,000 years preceding Pythagoras) there was found a list of fifteen Pythagorean triplets. It is reasonable to assume, according to the size of the numbers on the list, that its creator had a systematic way of finding Pythagorean triplets, but we do not know his method. A technique to create the infinite list of all Pythagorean triplets appears in Euclid's famous book, the Elements , which was written in the fourth century BC The problem of finding Pythagorean triplets is an example to a type of problems that occupied Diophantus, whom after are named "Diophantine equations". In these equations, the number of unknowns is greater than the number of equations, and the required solution must be in integers only. Diophantus was a Greek mathematician of the third century, who lived in Alexandria (Egypt). Only half of the thirteen volumes of his book

37. Historia Matematica Mailing List Archive: [HM] Even The Greatest Greek Mathemati HM Even the greatest greek mathematicians couldn t do this! Why couldn t eventhe greatest ancient greek mathematicians factor the Trojan monarch? http://sunsite.utk.edu/math_archives/.http/hypermail/historia/dec98/0011.html

[HM] Even the greatest Greek mathematicians couldn't do this! Samuel S. Kutler s-kutler@sjca.edu Tue, 1 Dec 1998 19:55:08 -0500 (EST) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: John F Harper: "Re: [HM] Even the greatest Greek mathematicians couldn't do this!" Previous message: Martin Davis: "Re: [HM] Goethe's Remark" Friends: Why couldn't even the greatest ancient Greek mathematicians factor the Trojan monarch? Best wishes, Sam Kutler Next message: John F Harper: "Re: [HM] Even the greatest Greek mathematicians couldn't do this!" Previous message: Martin Davis: "Re: [HM] Goethe's Remark"

38. Historia Matematica Mailing List Archive: Re: [HM] Even The Greatest Greek Mathe Re HM Even the greatest greek mathematicians couldn t do this! Why couldn teven the greatest ancient greek mathematicians factor the Trojan monarch? http://sunsite.utk.edu/math_archives/.http/hypermail/historia/dec98/0012.html

Re: [HM] Even the greatest Greek mathematicians couldn't do this! John F Harper John.Harper@vuw.ac.nz Wed, 2 Dec 1998 14:45:35 +1300 (NZD) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: Walter Felscher: "[HM] translations German/English" Previous message: Samuel S. Kutler: "[HM] Even the greatest Greek mathematicians couldn't do this!" In reply to: John F Harper: "Re: [HM] Even the greatest Greek mathematicians couldn't do this!" Next in thread: David Wilkins: "Re: [HM] translations German/English" On Tue, 1 Dec 1998, Samuel S. Kutler wrote: Because he wasn't for sale. (One meaning of the verb "factor" is of course to deal with goods, money etc as a factor, who in this context is a mercantile agent or commission merchant) John Harper, School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand e-mail john.harper@vuw.ac.nz phone (+64)(4)471 5341 fax (+64)(4)495 5045 Next message: Walter Felscher: "[HM] translations German/English" Previous message: Samuel S. Kutler: "[HM] Even the greatest Greek mathematicians couldn't do this!"

39. Real Numbers them. It wasn t until about 200 BC., however, that greek mathematiciansmade the jump from finite numbers to infinite numbers. This http://www.personal.psu.edu/faculty/j/x/jxt18/Math4_WEB/Math4 Math Assignments/R

40. Assign115/#5B/98 Archimedes overestimate of p, p » 3 1/7 = 22/7, was used for hundreds of yearsafterwards by greek mathematicians and by mathematicians of the next great http://newton.uor.edu/facultyfolder/beery/math115/m115_activ_est_pi.htm

Archimedes' Estimate of Activity The formula On the Measurement of the Circle, Proposition 3. The ratio of the circumference of any circle to its diameter is less than 3 1/7 but greater than 3 10/71 (Dunham, 97; Katz, 109). p p p was the first in history that was correct to two places after the decimal point! p C is the circumference of the circle, r is its radius, and P insc and P circ are the perimeters of the inscribed and circumscribed polygons, respectively, then P insc C P circ , or P insc p r P circ , so that P insc r p P circ If we take the radius of the circle to be 1 ( r = 1), then P insc p P circ Archimedes started with inscribed and circumscribed regular hexagons. Since each of the six sides of a regular hexagon inscribed in a circle of radius 1 has length 1, then P insc = 6 in this case (see Problem 1a). Likewise, since each side of a regular hexagon circumscribed about a circle of radius 1 has length , then P circ (see Problem 1b). Hence, P insc P circ /2 yields , or Archimedes then doubled the number of sides of each polygon to 12, obtaining an inscribed regular dodecagon of perimeter P insc (see Problem 1c), and a circumscribed regular dodecagon of perimeter

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