21. Ancient Greece Mathematics Timeline About 430440 BC hippocrates of chios squared the lune, a major step toward squaringthe circle, probably using the theorem that circles are to one another as http://www.mlahanas.de/Greeks/TLMathematics.htm

Timeline Ancient Greece Mathematics Around 600 BC the Cretan poet Epimenides is attributed to have invented the linguistic paradox with his phrase "Cretans are ever liars" - the Liar's Paradox. 2500 years later, the mathematician Kurt Gödel invents an adaptation of the Liar's Paradox that reveals serious axiomatic problems at the heart of modern mathematics. About 600 BC Thales of Miletus , He brings Babylonian mathematical knowledge to Greece and uses geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. About 530 BC Pythagoras no common rational measure is discoverable About 480 BC Parmenides of Elea founded the Eleatic School where he taught that 'all is one,' not an aggregation of units as Pythagoras had said, and that to arrive at a true statement, logical argument is necessary. Truth "is identical with the thought that recognizes it" (Lloyd 1963:327). Change or movement and non-being, he held, are impossibilities since everything is 'full' and 'nothing' is a contradiction which, as such, cannot exist. "Parmenides is said to have been the first to assert that the Earth is spherical in shape...; there was, however, an alternative tradition stating that it was Pythagoras" (Heath 1913:64). Corollary to Parmenides' rejection of the existence of 'nothing' is the Greek number system which, like the later Roman system, refused to use the Babylonian positional number system with its marker for 'nothing.' Making no clear distinction between nature and geometry, "mathematics, instead of being a science of possible relations, was to [the Greeks] the study of situations thought to subsist in nature" (Boyer 1949:25). Moreover, "almost everything in [Greek] philosophy became subordinated to the problem of change.... All temporal changes observed by the senses were mere permutations and combinations of 'eternal principles,' [and] the historical sequence of events (which formed part of the 'flux') lost all fundamental significance" (Toulmin and Goodfield 1965:40).

22. §Æªi§J©Ô©³¡]Hippocrates Of Chios, ¤½¤¸«e5¥@¬ö¤U¥b¸­¡^ The summary for this Chinese (Traditional) page contains characters that cannot be correctly displayed in this language/character set. http://www.edp.ust.hk/math/history/3/3_79.htm

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23. Lecture 3 Hippocrates Quadrature Of Lunes hippocrates of chios, c. 410 ( Plato -400, Euclid -360, Archimedes -250)was a failed merchant of Athens who took to mathematics as consolation. http://www.maths.uwa.edu.au/~schultz/3M3/L3Hippocrates.html

Lecture 3 Hippocrates Quadrature of Lunes Greek mathematics The distinguishing feature of Greek mathematics is that it is concerned with logical development, not problem solving. We use the term Greek Mathematics to denote mathematics written in the Greek language between about -600 (Thales) and about 250 (Diophantos). The mathematicians were not necessarily ethnically Greek nor living in the region we now call Greece. In fact the major developments occurred in the Greek colonies now known as Turkey, Egypt and Italy. The Greeks did not have a sophisticated number system. The integers were expressed by concatenating the letters a-k for 19, and l-u for1090 etc. Special letters were invented for larger numbers. Later, Archimedes in the "Sand Reckoner", (in which he calculated the number of grains of sand needed to fill the Universe) developed an exponential system for arbitrarily large numbers. The Greeks used a decimal system for common purposes and a sexigesimal system for scientific purposes, for example astronomy. Concatenations of unit fractions were used for rationals, although later Diophantos developed special symbols for rationals. In Greek mathematics the numbers were 2,3,4,.. The unity 1 was not a number, but the unit in which the numbers were measured. There were no negative numbers or zero. Geometrical quantities such as line segments, angles, areas and volumes were called

24. Apronyms: HIPPOCRATES OF CHIOS A Possibly Ribald Offering, Now Your Mates Smirk. APRONYM. 95%, HIPPOCRATESOF CHIOS, His InterPersonal Proficiency s Often Considered Rusty http://acronyms.co.nz/gonym.php?ap=HIPPOCRATES OF CHIOS

25. Hippocrats' Quadrature Of The Lune Indroduction. hippocrates of chios taught in Athens and worked on theclassical problems of squaring the circle and duplicating the cube. http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Lunefolder/Lune.h

Hippocrates' Quadrature of the lune Indroduction Hippocrates of Chios taught in Athens and worked on the classical problems of squaring the circle and duplicating Iamblichus [4] writes:- One of the Pythagorean [Hippocrates] lost his property, and when this misfortune befell him he was allowed to make money by teaching geometry. Heath [6] recounts two versions of this story:- One version of the story is that [Hippocrates] was a merchant, but lost all his property through being captured by a pirate vessel. He then came to Athens to persecute the offenders and, during a long stay, attended lectures, finally attaining such proficiency in geometry that he tried to square the circle. Heath also recounts a different version of the story as told by Aristotle:- ... he allowed himself to be defrauded of a large sum by custom-house officers at Byzantium, thereby proving, in Aristotle's opinion, that, though a good geometer, he was stupid and incompetent in the business of ordinary life. The suggestion is that this 'long stay' in Athens was between about 450 BC and 430 BC. In his attempts to square the circle, Hippocrates was able to find the areas of lunes, certain crescent-shaped figures, using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii. We describe this

26. Department Of Mathematics Think lunar like the phases of the moon. There are five squarable lunes,three of which hippocrates of chios found around the year 440 BC. http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/White/JanEssay1/JanEssay1.html

Department of Mathematics J. Wilson, EMAT 6690 Five Quadratures of Lunes By Jan White First we need to know what quadrature and lune actually means so here are the definitions. Quadrature: We mean that we can construct a square of some plane figure having the same area as the plane figure with only a compass and straightedge. Lune: A lune is the area between two curves. Think lunar like the phases of the moon. There are five squarable lunes, three of which Hippocrates of Chios found around the year 440 BC. The other two quadratures of lunes are attributed to Leonhard Euler (1707-1783) around 1771, but all five may have been given in a dissertation in 1766 by Martin Johan Wallenius. Before discussing the three quadratures of lunes found by Hippocrates let's first look at how a quadrature of a rectangle and triangle can be shown. Let ABCD be a rectangle. First we extend AB and equal distance to BD by using a compass and marking this segment BF. Next we find the midpoint of AF and mark off a semicircle as shown. We then construct a line perpendicular to AF and mark the intersection with the semicircle L. From this we construct the square BLMK. See figure 1. Does the area of ABCD = BLMK? Proof: By the Pythagorean theorem, GL^2=BL^2 +GB^2 or GL^2-GB^2=BL^2

27. Antiphon.html Simplicius identifies the squaring through segments with the construction oflunules by hippocrates of chios, as suggested by Aristotle, Sophistical http://www.calstatela.edu/faculty/hmendel/Ancient Mathematics/Philosophical Text

Antiphon's Squaring of the Circle by Henry Mendell (Cal. State U., L.A.) Return to Vignettes of Ancient Mathematics In the text in Aristotle discussed by Simplicius, Aristotle claims that he does not have to refute Parmenides' view that what is is just one and unchangeable. In a book on nature, he does not have to concern himself with hypotheses which reject nature altogether. He then draws a contrast between two attempts to square the circle, one through segments, and one by Antiphon. The mathematician needs to concern himself with a refutation of squaring by segments, but does not need to be concerned with refuting Antiphon's, which rejects mathematical principles. Elsewhere, in Met . K 1 (assuming Aristotle to be the author), he appears to hold that such a refutation belongs to first philosophy. Simplicius identifies the squaring through segments with the construction of lunules by Hippocrates of Chios, as suggested by Aristotle, Sophistical Refutations b Characters of the Drama Antiphon lived in the late 5th century B.C.E. He may or not be the Athenian orator of that name. Fragments of his work on nature as well as another comment in Aristotle's Physics a Simplicius wrote his commentary on the Physics sometime around the 540's.

28. Hippocrates hippocrates of chios. Born about 470 BC in Chios (now Khios),Greece Died about 410 BC. Show birthplace location. http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/Hppcrts.htm

Hippocrates of Chios Born: about 470 BC in Chios (now Khios), Greece Died: about 410 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Welcome page Hippocrates taught in Athens and worked on squaring the circle and duplicating the cube. In his attempts to square the circle, Hippocrates was able to find the areas of certain crescent-shaped figures using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii. Hippocrates also showed that a cube can be doubled if two mean proportionals can be determined between a number and its double. He was the first to write an Elements of Geometry and although his work is now lost it must have contained much of what Euclid later included in Books 1 and 2 of the Elements . Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration. Little is known of his life but he is reported to be an excellent geometer who, in other respects, was stupid and lacking in sense. He was defrauded of a large sum of money because of his naivety. References (5 books/articles) Other Web sites: Simon Fraser, Canada

29. References For Hippocrates References for hippocrates of chios. Biography in Dictionary of ScientificBiography (New York 19701990). Biography in Encyclopaedia Britannica. http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/~DZCFC8.htm

References for Hippocrates of Chios Biography in Dictionary of Scientific Biography (New York 1970-1990). Biography in Encyclopaedia Britannica. Articles: J. Recreational Math. T L Heath, A History of Greek Mathematics I (Oxford, 1921), 182-202. B B Hughes, Hippocrates and Archytas double the cube : a heuristic interpretation, College Math. J. Close this window or click this link to go back to Hippocrates Welcome page Biographies Index History Topics Index Famous curves index ... Search Suggestions JOC/EFR December 1996 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Hippocrates.html

30. Absolut Et Besøg Værd - Samos - Et Paradis På Jorden Mathematicians/Hippocrates.html hippocrates of chios taught in Athens and workedon the classical problems of squaring the circle and duplicating the cube. http://www.visit-samos.dk/chios.htm

http://www.chios.com/ Welcome to chios.com! "Chios.com" is not enterprise, a tourist office or an internet service provider. It came as a labor of love, and it was made to serve as a communication bridge for Chians all over the world. At the same time, it is a complete and nice informative guide for those who are interested in our island or wish to visit it. A side from the historical, geographical and folkloric elements, will find also registrations, selected with base the high quality of offered services. Through "Chios.com" someone can attempt a visit to Chios island of Greece and travel around the picturesque villages and the beautiful, sun -drenched coastlines... http://www.chiosonline.gr/ http://www.chios.gr/end.swf http://www.chiosnet.gr/home.asp http://www.chiosnet.gr/tourism/ Planning to visit a Greek island on the beautiful and sunny Aegean Sea? Seeking for the original flavour of Greek tradition and local culture?Perhaps the Island of Chios is just the right place you've been looking for. A place where the tourist feels more like a guest and has the opportunity to discover many values and traditions the mass tourism has driven away from other places.The rich history, the various monuments and the crystal-clear beaches are just few of the things that make Chios unique among other islands.We invite you to visit Chios through these pages, and discover yourself the experiences Chios can offer to its visitors... http://www.hcaa-eleng.gr/chios.htm

31. Grecia Heroica study. Anaxagoras of Clazomenae (Athens) hippocrates of chios (Athens).squaring the success. 3. hippocrates of chios (430 BC). He spent http://descartes.cnice.mecd.es/ingles/maths_workshop/A_history_of_Mathematics/Gr

THE GREEK HEROIC AGE History THE HEROIC AGE (Vth century B.C.) One of the most important personalities of this century is Pericles Athens attracted intellectuals from all parts of the Greek world wanting to satisfy their thirst for knowledge. Rather than coming up with necessary solutions to practical problems at that time, the scholars were more interested in developing their own personal intellect. This desire for wisdom lead them to focus their learning on theoretical issues. During this period the three famous (or classical) problems were dealt with and two methods of reasoning were put into use The table below lists the mathematicians who lived during this period and the problems that formed the focus of their study. Anaxagoras of Clazomenae (Athens) Hippocrates of Chios (Athens) squaring the circle or how to draw a square whose area is the same as that of a circle using a ruler and compass. Hippias de Elis (Attic peninsular) the trisection of the angle or how to construct an angle equal to a third of another given angle Philolaus of Tarentum (Southern Italy) Archytas of Tarentum the duplication of the cube or how to construct another cube whose volume is double that of the given cube Hippasus of Metapontum (Southern Italy) Incommensurability or line segments which are not in rational proportion to one another (THE GOLDEN SECTION)

32. Malaspina.com - Hippocrates (ca. 460-377 BC) HTML, Internet Classics Archive; On Ulcers HTML, Internet ClassicsArchive. MacTutor Entry on hippocrates of chios. Top of Page. http://www.mala.bc.ca/~mcneil/hippo1.htm

Hippocrates (ca. 460-377 B.C.) [Biography, SFU] Etexts by this Author [Athena] Great Books Biography [Malaspina] Amazon Search Form] Library of Canada Online Citations [NLC] Library of Congress Online Citations [LC] Library of Congress Offline Citations [MGB] COPAC UK Online Citations [COPAC] Free Online Practice Exams [Grad Links] Canadian Book Orders! Chapters-Indigo Save on Textbooks! [Study Abroad] Used Books Search Form Alibris Dummies Books Amazon Books from Amazon Amazon EBay! Ebay Books from Amazon UK Amazon UK Books from Chapters Canada Chapters Amazon's 100 Hot Books Amazon Hippocrates Amazon Greek Medicine Amazon Hippocrates in a World of Pagans and Christians Amazon Works by Hippocrates [HTML, Internet Classics Archive] Oath and Law of Hippocrates [Text, Wiretap] On Airs, Waters, and Places [HTML, Internet Classics Archive] On Ancient Medicine [HTML, Internet Classics Archive] Aphorisms [HTML, Internet Classics Archive] On the Articulations [HTML, Internet Classics Archive] The Book of Prognostics [HTML, Internet Classics Archive] On Fistulae [HTML, Internet Classics Archive]

33. History Of Geometry hippocrates of chios (470410 BC) wrote the first Elements of Geometry which Euclid may have used as a model for Books I and II. http://geometryalgorithms.com/history.htm

History Home Overview [History] Algorithms Books Gifts Web Sites A Short History of Geometry Ancient This is a short outline of geometry's history, exemplified by major geometers responsible for it's evolution. Click on a person's picture or name for an expanded biography at the excellent: History of Mathematics Archive (Univ of St Andrews, Scotland) Also, Click these links for recommended: Greek Medieval Modern History Books ... History Web Sites Ancient Geometry (2000 BC - 500 BC) Babylon Egypt The geometry of Babylon (in Mesopotamia) and Egypt was mostly experimentally derived rules used by the engineers of those civilizations. They knew how to compute areas, and even knew the "Pythagorian Theorem" 1000 years before the Greeks (see: Pythagoras's theorem in Babylonian mathematics ). But there is no evidence that they logically deduced geometric facts from basic principles. Nevertheless, they established the framework that inspired Greek geometry. A detailed analysis of Egyptian mathematics is given in the book: Mathematics in the Time of the Pharaohs India (1500 BC - 200 BC) The Sulbasutras Baudhayana (800-740 BC) Apastamba (600-540 BC) Greek Geometry (600 BC - 400 AD) Time Line of Greek Mathematicians Major Greek Geometers (listed cronologically) [click on a name or picture for an expanded biography].

34. Title hippocrates of chios Ca. 470 BCE to 410 BCE A Pythagorean, Hippocrates ofChios spent many years in Athens studying and teaching mathematics. http://www.math.uvic.ca/courses/math415/Math415Web/greece/gmen/hippotext.html

HIPPOCRATES OF CHIOS Ca. 470 BCE to 410 BCE A Pythagorean , Hippocrates of Chios spent many years in Athens studying and teaching mathematics. Hippocrates is said to have been a brilliant mathematician, but naïve and absent-minded when dealing with day to day affairs. Hippocrates' major contribution to mathematics was his book, entitled Elements . Undoubtedly one of the influences for Euclid 's Elements , Hippocrates was the first person to compile a comprehensive book on elementary demonstrative geometry. In doing so, he helped to popularize the ideas of an axiomatic treatment of geometry, which would last until the present day. Hippocrates was also interested in squaring the circle, and in duplicating the cube. He discovered that lunes, or crescent shapes could be squared, and is suspected to be partly responsible for Eudoxus ' method of exhaustion, although this can not be proved due to the fact that Hippocrates' work has not survived.

35. JMM HM DICIONÁRIO Translate this page Hípias de Eleia Hipócrates de Quios (c. 425?) Hipsicles Hórus Ísis, HerodotusHipparchus, Hero Herodotos Hypatia Hipparchos hippocrates of chios, hekat Heron http://phoenix.sce.fct.unl.pt/jmmatos/HISTMAT/HMHTM/HMDIC.HTM

Bibliografia Recursos na rede bem vindos em latim Anaximandro (-611-545) Antifonte Aristarco de Samos (-310-230?) Aristeo (c. -330) Arquimedes de Siracusa (-287?-212) Arquitas de Tarento (c. -375) Apollonius Archimedes Boetius Apollonios of Perga Aristarchos Aristaeus Aristotle Archimedes of Syracuse Archytas Apollonius of Perga Aristarchus Aristaeus Aristotle Archimedes of Syracuse Archytas Boethius Apollonios Diofanto de Alexandria (c. 250) Diophantus Democritos Dinostratos Diophantos Diocles Democritos Dinostratus Diophantus Diocles Diogenes Laertius Euclides de Alexandria (c. -300) Filolaos Endemus Eudoxus Philolaus Eratosthenes Euclid of Alexandria Endemos Eudoxos of Cnidos Eratosthenes Euclid of Alexandria Endemus Eudoxus of Cnidos Philolaus Euclide Hiparco de Alexandria (-190-120) Hipasos Hipsicles Herodotus Hipparchus Hero Herodotos Hypatia Hipparchos Hippocrates of Chios hekat Heron Herodotus Hypatia Hipparchus Hippocrates of Chios Iamblichus Iamblichos Iamblichus Menecmo (c. -350)

36. ThinkQuest : Library : Elysium Chios is the main city of Lesbos. Chios was the birthplace of the great mathematician,hippocrates of chios (not the same as the physician Hippocrates of Cos). http://library.thinkquest.org/26264/tools/map/site003.htm

Index Greek Elysium "Elysium" takes as its starting point Greek Mythology and Antiquity. The site is divided into three sections in which Elysium is illustrated like a cartoon (original drawings especially made for this site). Since the section is interactive, you play an active part in what happens. The Newcomer Section introduces you to the mythical world of Ancient Greece, The Student Section contains all the myths in detail, and the Teacher Section features guidelines for classroom teaching . Visit Site 1999 ThinkQuest Internet Challenge Languages English Students Sandra Aarhus Katedralskole, Aarhus, Denmark Sxren Aarhus Katedralskole, Risskov, Denmark Jesper Aarhus Katedralskole, Solbjerg, Denmark Coaches Thomas Maarslet, Denmark Frantz Lars Scheibel, Odder, Denmark 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

37. ThinkQuest : Library : A Taste Of Mathematic Leucippus (c. 450); hippocrates of chios (c. 450); Meton (c. 430) *SB;Hippias of Elis (c. 425); Theodorus of Cyrene (c. 425); Socrates (469 http://library.thinkquest.org/C006364/ENGLISH/history/historygreece.htm

Index Math A Taste of Mathematic Welcome to A Taste of Mathematics.You will find the taste of mathematics here.The history of Mathematics,famous mathematicians,cxciting knowledge,the world difficult problems and also mathematics in our life... Browsing,thinking,enjoying,and have a good time here! Visit Site 2000 ThinkQuest Internet Challenge Languages English Chinese Students fangfei Beijing No.4 High School, Beijing, China ziyan Beijing No.4 High School, Beijing, China Coaches Tife Zesps3 Szks3 Ogslnokszta3c9cych Numer 1, Beijing, China xueshun Beijing No.4 High School, Beijing, China 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

38. The Beginnings Of Early Greek Sciene c = (p 2 + q 2 )/2 = (3 2 + 1 2 )/2 = 5. EARLY GREEK GEOMETRY. The quadratureof the lune was accomplished by hippocrates of chios (c. 440 BC). http://departments.weber.edu/physics/carroll/Greeks/Greeks.htm

THE BEGINNINGS OF SCIENCE IN EARLY GREECE (c. 600 - 400 BC) "Atlas Brings Heracles the Apples of the Hesperides in the Presence of Athena," from the Temple of Zeus at Olympia, c. 460 BC. During this time, when gods embodied nature and interfered in human lives, the journey from mythos to logos began. WHAT DO WE KNOW OF THE PRESOCRATIC PHILOSOPHERS? Not much ... not a single original work has survived intact for us to study! The only sources are 1. Fragments - a few quotations from Presocratic works that have survived in works written later. 2. Testimonia - comments in the writings of Plato and Aristotle on Presocratic ideas. 3. Doxography - summaries and (summaries) of Presocratic works. Milesians Pythagoreans Eleatics Independent Atomists Physiologists Thales of Miletus Pythagoras of Samos Parmenides of Elea Heraclitus of Ephesus Democritus 624 - 546 BC 570 - 500 BC 540 - 480 BC c.500 BC c.460 - 370 BC Water Number Eon (Being) Pyr and Logos (Fire and Rule) Atom Anaximander of Miletus Philolaus Zeno of Elea Empedocles Leucippus 610 - 540 BC c.470 - 390 BC

39. The Five Squarable Lunes hippocrates of chios was the first to demonstrate such quadratures (around 440BC) for lunes. It turns out that only five particular lunes can be squared . http://www.mathpages.com/home/kmath171.htm

The Five Squarable Lunes Return to MathPages Main Menu

40. Mathematics: Development Of Mathematics BC), hippocrates of chios made the beginnings of an axiomatic approach to geometryand Zeno of Elea proposed his famous paradoxes concerning the infinite and http://www.infoplease.com/cgi-bin/id/A0859534.html

HOTWORDS CITE PRINT EMAIL in All Infoplease Almanacs Biographies Dictionary Encyclopedia Home Almanacs Atlas Dictionary ... D-Day Infoplease Tools Periodic Table Conversion Tool Perpetual Calendar Year by Year ... Site Map Also from Infoplease Search Infoplease Info search tips Search Biographies Bio search tips Encyclopedia mathematics Development of Mathematics The earliest records of mathematics show it arising in response to practical needs in agriculture, business, and industry. In Egypt and Mesopotamia, where evidence dates from the 2d and 3d millennia B.C. , it was used for surveying and mensuration; estimates of the value of pi ) are found in both locations. There is some evidence of similar developments in India and China during this same period, but few records have survived. This early mathematics is generally empirical, arrived at by trial and error as the best available means for obtaining results, with no proofs given. However, it is now known that the Babylonians were aware of the necessity of proofs prior to the Greeks, who had been presumed the originators of this important step. Greek Contributions A profound change occurred in the nature and approach to mathematics with the contributions of the Greeks. The earlier (Hellenic) period is represented by

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