The Origins Of Greek Mathematics hippocrates of chios, though probably a Pythagorean computed the quadrature ofcertain lunes. (This is the first correct proof of the area of a curvilinear http://www.math.tamu.edu/~don.allen/history/greekorg/greekorg.html
Extractions: The Origins of Greek Mathematics Though the Greeks certainly borrowed from other civilizations, they built a culture and civilization on their own which is 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
The Beginnings Of Early Greek Sciene The quadrature of the lune was accomplished by hippocrates of chios (c. 440 BC).Using only a compass (divider) and straightedge, Hippocrates determined http://departments.weber.edu/physics/carroll/Greeks/Greeks.htm
Extractions: 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
Dupcubfin.html In addition to Eudoxus solution, hippocrates of chios also developed a workingsolution of the Delian problem. Hippocrates claimed that this problem was http://www.ms.uky.edu/~carl/ma330/projects/dupcubfin1.html
Extractions: Duplication of the Cube : Darrell Mattingly, Cateryn Kiernan The ancient Greeks originated numerous mathematical questions, most of which they learned to solve using simple mathematical tools, such as the straight edge and the collapsable compass. Three of these problems persist today, challenging students in contemporary classrooms. This triology of problems, the trisection of a given angle, the squaring of a circle, and the duplication of the cube, have since been proved impossible using exclusively the straight edge and the compass. In the quest to solve these problems using those specific tools, however, mathematicians developed numerous alternate solutions using other mathematical tools. The last problem of the trilogy is the focus of this discussion, and it challenged mathematicians for centuries, due to the restriction of using only the aforementioned tools. Origin of the Problem Proof that NO Platoic Solution Exists for the "Delian" Problem After centuries of mathematicians had worked on this problem, a proof developed that it could not be done using exclusively the straight edge and compass. This proof is based on theorems about the powers of degrees of subfields generated by the x and y coordinates of the side of the cube to be duplicated. Although the desired point can be approximated, it cannot in fact be found based on these theorems.
Euclid's Elements In Greek but were the work of earlier Greek mathematicians such as Pythagoras (and hisschool), hippocrates of chios, Theaetetus, and Eudoxus of Cnidos. http://farside.ph.utexas.edu/euclid.html
Extractions: Euclid's Elements in Greek Euclid's Elements is by far the most famous mathematical work of classical antiquity, and also has the distinction of being the world's oldest continuously used mathematical textbook. Little is known about the author, beyond the fact that he lived in Alexandria around 300 BCE. The subject matter of this work is Geometry, which was something of an obsession for the Ancient Greeks. Most of the theorems appearing in Euclid's Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from five simple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of previously discovered theorems: e.g., Theorem 48 in Book 1.
History Of Geometry hippocrates of chios (470410 BC) wrote the first Elements of Geometry whichEuclid may have used as a model for his own Books I and II more than a http://geometryalgorithms.com/history.htm
Extractions: Home Overview [History] Algorithms Books Web Sites Gift Shop 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 the following links for recommended: Books about Geometry History Web Sites about Geometry History Greek Medieval ... 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 . One of the few surviving documents was written by: Ahmes 1680-1620 BC)
Axiom hippocrates of chios (the doubling the cube guy, 470400 BCE) (according toProclus) was the first to compose Elements Euclid (according to Proclus) writing http://darkwing.uoregon.edu/~mwilson/axiom.html
ITS Advisory the mathematician hippocrates of chios (about 70). Socrates (70). About this ageone might feel confident, though Diogenes Laertius 2.44 also informs us http://www.uq.edu.au/~uqtparki/oldancients.html
Extractions: Information Technology Services Advisory You have requested the document http://www.uq.edu.au/~uqtparki/oldancients.html . This is a staff web area hosted on a University of Queensland web server. Please be advised that the web pages within this area are NOT officially endorsed by The University of Queensland. The University accepts no responsibility or liability for the contents of this area. This message has been displayed in accordance with the University's Internet Code of Practice , which forms a part of the Please note that you will need to enable cookies in your browser in order to proceed.
MSN Encarta - Mathematics Among great geometers of the 5th century bc were the philosopher Democritus andthe mathematician hippocrates of chios. Democritus, who is better known for http://encarta.msn.com/encyclopedia_761578291_6/Mathematics.html
Extractions: Search for books and more related to Mathematics Encarta Search Search Encarta about Mathematics Editors' Picks Great books about your topic, Mathematics ... Click here Advertisement document.write(' Page 6 of 12 Encyclopedia Article Multimedia 45 items Article Outline Introduction Mathematics: The Language of Science Branches of Mathematics History of Mathematics a Number System The Greek number system was based on the alphabet. The Attic system, in use from 600 bc to 200 bc , used a stroke for 1 and the initial letters of the words for 5, 10, 100, 1,000, and 10,000ânamely, the initials of pente deka hekaton khilioi , and myrioi âto represent the respective numbers. A later system assigned number values to the 24 letters of the Greek alphabet and to 3 other letters that were no longer used. The letters could be combined to form numbers through 999. For higher numbers, a stroke preceding the initial letter (1 through 9) indicated a multiple of 1,000 (1,000 through 9,000). For 10,000 and above, the symbol M indicated that the numeral below should be multiplied by 10,000. See also Numerals b Pythagoras and the Pythagoreans Pythagoras taught the importance of studying numbers in order to understand the world. We know of his achievements only from his disciples, the Pythagoreans, who made important discoveries about number theory and geometry. The Pythagoreans represented whole numbers by using arrangements of dots or pebbles, and classified these numbers according to the shapes produced. (The English word
Conic Sections In Ancient Greece A breakthrough of a kind occurred when hippocrates of chios reduced the problemto the equivalent problem of two mean proportionals , http://www.math.rutgers.edu/~cherlin/History/Papers1999/schmarge.html
Extractions: The knowledge of conic sections can be traced back to Ancient Greece. Menaechmus is credited with the discovery of conic sections around the years 360-350 B.C.; it is reported that he used them in his two solutions to the problem of "doubling the cube". Following the work of Menaechmus, these curves were investigated by Aristaeus and of Euclid. The next major contribution to the growth of conic section theory was made by the great Archimedes. Though he obtained many theorems concerning the conics, it does not appear that he published any work devoted solely to them. Apollonius, on the other hand, is known as the "Great Geometer" on the basis of his text Conic Sections , an eight-"book" (or in modern terms, "chapter") series on the subject. The first four books have come down to us in the original Ancient Greek, but books V-VII are known only from an Arabic translation, while the eighth book has been lost entirely. In the years following Apollonius the Greek geometric tradition started to decline, though there were developments in astronomy, trigonometry, and algebra (Eves, 1990, p. 182). Pappus, who lived about 300 A.D., furthered the study of conic sections somewhat in minor ways. After Pappus, however, conic sections were nearly forgotten for 12 centuries. It was not until the sixteenth century, in part as a consequence of the invention of printing and the resulting dissemination of Apollonius' work, that any significant progress in the theory or applications of conic sections occurred; but when it did occur, in the work of Kepler, it was as part of one of the major advances in the history of science.
Assignment 4 hippocrates of chios (ca. 460 380 BC) , who worked on the problems of squaringthe circle and duplicating a cube was able to obtain some interesting http://mtl.math.uiuc.edu/modules/module13/Unit 1.4/assgn-4.html
Extractions: Assignment 1.4 Assignment Completion and Submission Directions s: Prepare a single Geometer's Sketchpad document (or a MSWord document with GSP figures inserted) that presents the problems and their solutions in Assignment 1.4 in a manner that is Your grade on this assignment will be based on the extent to which the GSP file you submit meets both of these criteria. If you are using Version 4.0 or later, use a separate page for each problem. When you complete Assignment 1.4, submit your file through the Module Working Environment (also referred to as ClassComm). Select Module 13 and enter your log-in and password. Then follow the directions there for submitting your assignment. Problem 1: Express cos(3 m) as a combinantion of positive integer powers of cos(m). Use this expression to obtain a cubic polynomial p(x) with rational number coefficients such that x = cos(m) is a root. Graph the resulting polynomial for m = 20 degrees on a window that displays all three of the roots. Problem 2: Trisecting Angles with a Marked Ruler.
Introduction To The Works Of Euclid It is known that hippocrates of chios (fl. c. 440 BC) and others had composedbooks of elements before him. 39 However, Euclid s treatise was quickly http://www.obkb.com/dcljr/euclid_orig.html
Extractions: This is a paper I wrote as an undergrad for a History of Science course. Although it's not publishable or anything, it's one of my favorite papers because it was so difficult to do. In fact, the whole History of Science course was quite an experience. Footnotes (actually, endnotes) appear in square-brackets, like this: . After following the link to the footnote, a similar link brings you back to where you started. Try it with the footnote above. Okay, here's an outline of the paper. You may go directly to a section by choosing it in the list below. Note: You can also see my High school Euclid paper , which was more or less the original version of this paper. The name of Euclid is often considered synonymous with geometry. His
PHIL 320 FORMAL LOGIC 2 Sep 9, 11 Building up to Euclid hippocrates of chios and the quadrature ofthe lune. Euclids first proofs. Gaps in the rigor of Euclids arguments. http://vms.cc.wmich.edu/~mcgrew/flog.htm
Untitled Document those of Pythagoras (c.560480B-C), hippocrates of chios (fl.440B. Hippocrates like pythagoreans believed in the concept of a single comet. http://www.vigyanprasar.com/dream/mar2001/comets.htm
Extractions: Development of Cometary Thought PART - I Subodh Mahanti Lucius Annaeus Seneca (4B.C.-A.D.65) in Natural Questions ... In thick smoke of human sins, rising every day, every hour, every moment full of stench and horror, before the face of God and becoming gradually so thick as to form a comet, with curled and plaited tresses, which at last is kindled by hot and fiery anger of the supreme Heavenly Judge. Andreas Celichius in The Theologial Reminder of the New Comet (1578) Donald K.Yeomans in Comets : A Chronological History of observations, Science, Myth and Folklore (1991). The development of the scientific understanding about comets has a long and intriguing history. For centuries people (common people and scientists alike) have pondered the appearance of these mysterious apparitions. People's fascination for them, as seneca pointed out, was because they were unusual strange phenomena. They appear rarely. Before the seventeenth century comets were not considered as celestial bodies but as signals at a sinful Earth from God. celichius as quoted above was no doubt expressing the majority view of the comet prevalent in the 16th century. of course, there were opponents, though their number were few. for example Andreas Dudith (1533-89), the Hungarian scholar, countered celichius views by stating that if comets were caused by the sins of the mortals then they would never be absent from the sky.
Pedro Pablo Fuentes González authors as diverse as the Stoics Cornutus and Epictetus, the polygraph andscholar Eratosthenes of Cyrene or the mathematician hippocrates of chios. http://www.ugr.es/~odiseo/Fuentes.html
Family Reunion I presented my family members with a proof of doubling the square byhippocrates of chios (400 BC), which provoked Is that important? http://www.bayarea.net/~kins/Personal-FAMILY_REUNION.html
Extractions: Kins trying to understand why people are led to the idea of God. From: Kins Collins (04/29/97) To: Robin Smith, John Dunne Subject: Kins back from trip Here's some unexpected mail for you, Robin (and John you too)! I thought both of you might enjoy this personal note written for my colleagues here at Apple. Kins - To all As you may know, I was away for a few days to attend a family reunion at my mother's place in La Jolla. My mother is 98 yrs. old and doing fine. I showed her how to use the Mac (PowerBook 2300c), and together we wrote a letter, balanced her checkbook, drew some KidPix pictures, explored the Internet, etc. Then she played a solitaire card game on it by herself. On the other hand, my composer brother idea of proof. From: John Dunne (04/30/97) To: Kins Collins Subject: Kins back from trip Kins, Ok, its my turn on the soapbox. I have to respond. You are welcome to toss it out as religious nonsense, but please read it. [... text temporarily omitted ...] John From: Robin Smith (05/02/97) To: Kins Collins On 30 Apr 97 at 14:36, Kins Collins wrote: > I thought you might enjoy this personal note written for my colleagues > here at Apple. > Kins KinsI'm glad to see you are still at Apple, in these parlous times. I assume the mention of your brother as being back in the Middle Ages relates not to his music but rather to his attitude toward technology. ========================== /Robin Smith/ Smith@robin.cat.com ==========================
Extractions: 2 Knoche, Untersuchungen ber die neuaufgefundenen Schol-ien des Prokius Diadochus su Euclids Elementen, pp. 20 and 23 (Herford, 1865). PYTHAGORAS (OF RHEGIUM)PYTHEAS (OF MARSEILLES) 703 combination of arithmetic with geometry. The notions of an equation and a proportionwhich are common to both, and contain the first germ of algebrawere introduced among the Greeks by Thales. These notions, especially the latter, were elaborated by Pythagoras and his school, so that they reached the rank of a true scientific method in their theory of proportion. To Pythagoras, then, is due the honor of having supplied a method which is common to all branches of mathematics, and in this respect he is fully comparable to Descartes, to whom we owe the decisive combination of algebra with geometry. See G. J. Allman, Greek Geometry from Thales to Euclid (Cambridge, 1889); M. Cantor, Vorlesungen uber Geschichte der Mathematik (Leipzig, 1894); James Gow, Short History of Greek Mathematics (Cambridge, 1884). (G. J. A.) PYTHAGORAS, of Rhegium, a noted Greek sculptor of the 5th century B.C., a contemporary of Myron and Polyclitus, and their rival in making statues of athletes. He was born at Samos and migrated in his youth to Rhegium in Italy. He made a statue of Philoctetes notable for the physical expression of pain, an Apollo shooting the Python at Delphi, and a man singing to the lyre. He is said to have introduced improvements in the rendering of muscles, veins and hair.
Ancient Coins Are What I Collect Athens was a mecca for hippocrates of chios and Hippocrates of Cos, Anaximander,Aristotle, et. al, and home to Socrates, Meton, et al. http://www.limunltd.com/numismatica/articles/ancients-what-i-collect.html
Extractions: by Michael E. Marotta , 4 Jun 1994 Like most libertarians, I have always held on to some silver and gold in preference to other forms of saving. After a while, one Kennedy half looks pretty much like the next. Just two years ago, my daughter worked as a page at a state coin show. Dropping her off and picking her up, I walked around the bourse room. It was all very nice and all, with American 19th Century Liberties being far lovelier than most others . . . until I sat down to a tray of ancients. Today, I have a Whitman for Mercuries that lacks only the 1916-D to be complete. Many of the entries have been upgraded to Fine and above. I have some Hard Times Tokens, 19th century world bronzes featuring Liberty, political silver bars, phone cards, Barber Dimes, and a lot more of this and that. However, my formal answer to what I collect is: Ancients. Greeks. Archaic to Hellenistic, from 650 to 38 BC: From the rise of Croesus to the fall of Cleopatra. Here is what I have and why: Miletus; 1/12 stater; 6thC; SGCV 3532(var); SNG vonA 2080
Extractions: Er wurde um 470 v.Chr. auf Chios geboren und ist etwa 410 v.Chr. in Athen gestorben. Hippokrates war angefänglich Kaufmann, aber Aristoteles soll gesagt haben, dass er kein sehr raffinierter Kaufmann war. Er wurde von den Steuerbeamten in Byzanz betrogen und andere Quellen behaupten das er durch Piraten beraubt wurde. Hippocrates war ein brillanter Mathematiker und Astronom. Einer der grossen Geometriker des Altertums. Nach seinem finanziellen Fiasco ging er etwa 430 v.Chr. nach Athen. Dort hat er Vorträge in seiner Freizeit besucht und ist schließlich ein Lehrer der Geometrie geworden. Eines von Hippokrates Fachgebieten war die Quadratur des Kreises, eines der drei grossen mathematischen Fragen der Antike. In seinen Wegen zur Lösung entdeckte und beschrieb er die nach ihm genannten Möndchen über den Katheten eines rechtwinkligen Dreiecks. Für das Delische Problem, die Verdoppelung des Würfels, reduzierte Hippokrates das Problem auf das einfachere, zwei mittlere Proportionalen zu bestimmen. Mehr über Hippocrates auf die seite von der university of St Andrews: Hippocrates von Chios Oenopides Theopompus Chios home The Fragrant Island Hippocrates Hippocrates In einer schönen Umgebung mit einer atemberaubenden Aussicht.
UNSW Handbook Course - History Of Mathematics - GENS2005 The Greek miracle round earth, logic, Pythagoras All is number ; Hippocratesof chios on areas of lunes proof; Euclid on axioms, http://www.handbook.unsw.edu.au/undergraduate/courses/2005/GENS2005.html
Extractions: History of Mathematics - GENS2005 PRINT THIS PAGE Faculty: Faculty of Science School: School of Mathematics Campus: Kensington Campus Career: Undergraduate Units of Credit: Contact Hours per Week: Enrolment Requirements: Prerequisite: Any Level 1 Mathematics course or ECON1202 or ECON1203; Excluded: MATH3560. Offered: Semester 2 2005 Fee Band: Description Classifications of mathematics, ancient and modern; Counting, navigation and measurement in pre-literate societies; Babylonian mathematics on calculating lengths and areas; The 'Greek miracle': round earth, logic, Pythagoras' 'All is number'; Hippocrates of Chios on areas of lunes: proof; Euclid on axioms, on idea of deductive structure; Ptolemy's geocentric astronomy; Ancient Chinese simultaneous linear equations; 16th C solution of cubic equations; Copernicus' heliocentric astronomy; 17th C mathematical laws: Galileo, Kepler, Snell, Hooke, Boyle; development of calculus: Topology: Euler on the bridges of Konigsberg; Statistical inference, 'average man', Galton and correlation; Abstract set theory; Formal (symbolic) logic in 19-20th C, and its role in computing software; Operations research, e.g., stock-cutting and hunting submarines; Chaos, fractals and self-organisation; Social context of mathematics.
Hippias Concise scholarly biography of this discoverer of the quadratrix, from the MacTutorHistory of Mathematics. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hippias.html
Extractions: Version for printing Hippias of Elis was a statesman and philosopher who travelled from place to place taking money for his services. He lectured on poetry, grammar, history, politics, archaeology, mathematics and astronomy. Plato describes him as a vain man being both arrogant and boastful, having a wide but superficial knowledge. Heath tells us something of this character when he writes in [3]:- He claimed ... to have gone once to the Olympian festival with everything that he wore made by himself, ring and sandal engraved , oil-bottle, scraper, shoes, clothes, and a Persian girdle of expensive type; he also took poems, epics, tragedies, dithyrambs, and all sorts of prose works. As to Hippias's academic achievements, Heath writes:- He was a master of the science of calculation, geometry, astronomy, 'rhythms and harmonies and correct writing'. He also had a wonderful system of mnemonics enabling him, if he once heard a string of fifty names to remember them all. A rather nice story, which says more of the Spartans than it does of Hippias, is that it was reported that he received no payment for the lectures he gave in Sparta since [3]:-
