21. Eudoxus Of Cnidus eudoxus of cnidus. Eudoxus was born in 400 BC in Cnidos. He studied mathematics withArchytus in Tarentum. Later he studies medicine with Philistium on Sicily. http://www.cas.muohio.edu/~devriepl/phy211/greeks/eudoxus.htm

Eudoxus of Cnidus All information taken from: http://www.math.tamu.edu/~don.allen/history/eudoxus/eudoxus.html

22. Eudoxus Eudoxus. We're here with the ghost of the famous mathematician, eudoxus of cnidus. Allen,Don. “eudoxus of cnidus.” February 1997. 20 February 2002. http://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/eudoxus.html

Eudoxus We're here with the ghost of the famous mathematician, Eudoxus of Cnidus. Responsible for many mathematical feats, he was also a famous astronomer and legislator. Q. When were you born? A. About 400 BC. My father was Aischimes. Q. Where were you born? A. Cnidus, on the Black Sea. It is located in Asia Minor, or current day Turkey. Q. Did you receive any education? A. First I studied under Archytas, a follower of Pythagoras, who influenced my math career. I later studied medicine with Philistium on Sicily. Then when I was twenty-three years old, I went to Plato's academy in Athens where I studied philosophy and rhetoric. I also studied astronomy in Egypt at Helopolis. Q. After receiving your education, what did you do? A. I established a school at Cyzicus, located in northwestern Asia Minor, on the shore of the Marmora Sea. I returned with my pupils to Athens in 365 BC. There, I became a colleague of Plato and a respected legislator. Q. Were you close friends with Plato? A. Actually, he was jealous of how popular my school was. So, as you can imagine, we weren't exactly great friends. Q. When did you die?

23. Eudoxus Systems - Biography Of Eudoxus eudoxus of cnidus. eudoxus of cnidus (c.408 c.355 BC) was one of the greatestGreek mathematicians. He was also an astronomer, philosopher and legislator. http://www.eudoxus.com/eudoxus.html

Home Search About us Eudoxus Tools We Use What is Optimization? MP in Action Lecture Notes ... Site Map Eudoxus of Cnidus Eudoxus of Cnidus (c.408 - c.355 BC) was one of the greatest Greek mathematicians. He was also an astronomer, philosopher and legislator. His main contributions to mathematics were: the theory of proportion, which resolved the crisis in Greek mathematics caused by the discovery of irrational numbers; the method of exhaustion, which was a precursor (by 2000 years) of the integral calculus. He may also have been responsible for the development of the axiomatic method, the foundation of modern mathematics. His work in astronomy has stood the test of time less well. He developed a model of the universe which sought to explain the motions of the sun, the moon and the planets by fixing them to a system of 27 (or according to some authorities, 55) concentric spheres. These rotated on assorted axes at various speeds with the earth at the centre. Even with all this ingenuity he was unable to explain the motions of Venus and Mars nor the variation in brightness of the moon. His scheme was a magnificent attempt to explain observed phenomena, but wrong. More long lasting in its influence was 'the sphere of Eudoxus'. This was an engraved celestial globe which showed the constellations together with their names. Eudoxus did not invent these, but carried them over from an earlier civilisation, most probably the Babylonians of c. 2500 BC. These names have remained in use to this day and are also the names we use as the signs of the Zodiac.

24. Eudoxus Systems - Site Map Site map. Contact, Site Map. Home. Search. About us. eudoxus of cnidus. Recruitment.Services. Consultancy and Modelling. Support and Maintenance. Seminars and Training. http://www.eudoxus.com/map.html

Home Search About us Tools We Use ... Lecture Notes Site Map Site Map Home Search About us Eudoxus of Cnidus ... 4. Sensitivity Analysis 4.1 The Oil Blending Model 4.2 Drawing Inferences from the Solution 4.3 Two-variable Problem 4.4 Reduced Cost 4.5 Alternative Optimal Solutions 4.6 Shadow Prices ... 5. Ranging Information 5.1 Introduction 5.2 Objective Ranging 5.3 Right Hand Side Ranging 5.4 Ranging the Simple Oil Blending Model Practical Integer Programming 1. Introduction ... Privacy © Eudoxus Systems Ltd 1995 - 2003

25. Eudoxus- Germantown Academy Mathematical Biographies eudoxus of cnidus Mitch Please we all don't speakancient Greek here. Eudo Its eudoxus of cnidus, and you are very welcome. http://www.ga.k12.pa.us/academics/US/Math/Millar/Eudoxus/Beer.htm

"Edoxus of Cnidus " by Mitchell Beer '00 Mitch of Springfield : Pleasure to speak with you today Eudoxus. Eudoxus of Cnidus Mitch: Please we all don't speak ancient Greek here. Eudoxus: Sorry about that, it is a pleasure to have returned from the grave to speak with you today Mitch. Mitch: Firstly before we start Eudoxus, can I call you Eudo? Eudo : No. Mitch: Well then Eudo, how about we start with you and Cnidus. Eudo: I said you couldn't call me Eudo, but Cnidus is a quaint little town on the Resadiye peninsula in Asia Minor. I was born in 408, not a particularly interesting year. Actually the only reason it is interesting is because I was born then. Died in the same place too in 355. Mitch: How about your daddy. He too had a strange name. Eudo: What do you expect; we were ancient Greeks. He was Aischines. Mitch: A pleasant name as well. Let's now move to your travels and studies throughout your life. Were did you go? What and whom did you know? Eudo: I have been to a plethora of places throughout my philosophical career, most notable being Tarentum and Athens and Sicily, as well as Heliopolis in Egypt. I started with Tarentum, where I studied the mathematics of geometry under the beneficent Archytas, who studied under the great Pythagoras. Archytas's interest in the Duplication of the Cube led me to be interested in it as well. Mitch: Fascinating.

26. Eudoxus Of Cnidus - Acapedia - Free Knowledge, For All Friends of Acapedia eudoxus of cnidus. eudoxus of cnidus (ca.408 BC ca.347 BC)was a Greek astronomer, mathematician, physician, scholar and friend of Plato. http://acapedia.org/aca/Eudoxus_of_Cnidus

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27. St. Mark School - Venice Tuesday, July 24, 2001. eudoxus of cnidus Mathemitician Jamie DittmarJamie Dittmar eudoxus of cnidus by Jamie Dittmar. The mathematician http://www.venicebeach.com/sms/news/jamiedittmar.shtml

Saint Mark Elementary School Venice, California HOME SIGN GUESTBOOK VIEW GUESTBOOK FORUM Tuesday, July 24, 2001 Eudoxus of Cnidus: Mathemitician Jamie Dittmar Jamie Dittmar Eudoxus of Cnidus by Jamie Dittmar The mathematician I am researching is Euxodus of Cnidus. Eudoxus made important contributions to the theory of proportion, where he made a definition allowing possible irrational lengths to be compared in a similar way to the method of cross multiplying used today. During this time, there was a mathematical problem that needed to be resolved. The way to measure lengths that were not comparable had not been discovered. The existing Pythagorean method for comparing two lengths failed to work for certain lines. Eudoxus’ theory is called the Axiom of Eudoxus and states: “Magnitudes are said to have a ratio to one another which is capable, when of multiplied exceeding one another.” Eudoxus meant that an area and length do not have a capable ratio, therefore both rational or irrational lengths can be solved using his theory. I hope to learn more about this in my future math classes. Monday, July 16, 2001

28. Encyclopædia Britannica eudoxus of cnidus Greek mathematician and astronomer who substantially advanced proportiontheory, contributed to the identification of constellations and thus http://search.britannica.com/search?query=Eudoxus

29. The Homocentric Spheres Of Eudoxus eudoxus of cnidus was the first individual to successfully ascribe a geometricalmodel to the heavens, using a complex system of rotating spheres to describe http://www.cco.caltech.edu/~deborahe/core1.htm

A Revolution of Thought: The Homocentric Spheres of Eudoxus To appreciate the significance of their achievements, we must first throw ourselves back 2500 years and look at the sky through ancient eyes, setting aside our current view of the universe. We must disregard our heliocentric model and throw away the outer three planets of the solar system, for they cannot be seen with the naked eye and optical aids like telescopes do not exist. Forget basic Newtonian mechanics, ignore the current understanding of gravity, and toss out the knowledge that the planets are all made of the same matter as the earth. We must temporarily erase from our mind the pictures of Earth taken from space and remember that no one in ancient Greece could see anything more than what can be observed from the ground with the naked eye. From this perspective, the night sky looks like a very different place. The Sun is now a very unique object, not just one star among billions. The Moon is the only object that shows any surface features, the other planets being points of light far more similar to stars than anything else in the heavens. From our vantage point on Earth, the heavens appear to revolve around us as the stars rotate east to west with the sun lagging just slightly behind (Fig. 1).

30. History Of Ancient World Mathematics Page eudoxus of cnidus (408 355 BC). Eudoxus was born in Cnidus where hespent his youth in poverty like many of his fellow mathematicians. http://www.roma.unisa.edu.au/07305/ancmm.htm

Ancient World Mathematics Written by Paul Dickson (University of South Australia, 1996) Thales of Miletus (640 - 546 BC) Thales was born in Miletus in 640 BC and became a merchant as soon as his skills allowed, actual history concerning Thales is scarce but some stories about him have filtered down through the ages, whether they are true or not..... no one really knows. Thales' major mathematical contribution is believed to be the theory of a triangle inscribed within a semi-circle being right angled at the corner touching the arc if one side is the diameter of the circle. Figure 1: A Triangle inscribed in a Semi-circle makes a right angle. Thales and the Salt Caravan It is believed that while transporting salt which was loaded on mules, one of the animals slipped in a stream. The mule's load of salt was slightly dissolved by the water and it's load became lightened. This mule being smart at ways to get out of work rolled over at the next ford it came to and found it's load lighter again. Whether these mules were Thales or not is unclear bu the was consulted and came up with a plan to break the mule of this bad habit. The mule was loaded with sponges and rags, which when the mule rolled over, absorbed the water and made the load heavier. This eventually cured the mule of it's troublesome habit. Thales and the Olive Oil Empire In the ancient world of the mediterranean Olive Oil was an important commodity, as important as wheat or sugar is in todays. The Olive crop was a bumper havest one year and fearing that supply would outgrow demand for the coming Olive Oil production Thales quietly bought all the Olive presses he could afford to (no small task considering he was a very wealthy merchant by this time). Thus Thales controlled most of the Olive Oil production and 'cornered the market' of Olive Oil, a man much before his time Thales therefore became the first recorded man at about 600 BC to create a monopoly.

31. Title eudoxus of cnidus Ca. 410 BCE to 355 BCE Eudoxus was one of the most famousstudents of Archytas, and also studied under Plato in Athens. http://www.math.uvic.ca/courses/math415/Math415Web/greece/gmen/eudoxtext.html

EUDOXUS OF CNIDUS Ca. 410 BCE to 355 BCE Eudoxus was one of the most famous students of Archytas , and also studied under Plato in Athens. As such, he was almost certainly influenced by Pythagorean ideas. Eudoxus was an excellent mathematician and astronomer. Unfortunately, none of Eudoxus' works have survived, and our information about his work comes to us indirectly through other sources. Eudoxus made two major contributions to mathematics. The first was his theory of proportions. After Theaetetus discovered irrational numbers, a crisis arose in the mathematical community because many of the Pythagorean proofs did not account for the existence of irrationals, an assumption that rendered these proofs invalid. Eudoxus' theory of proportions, well documented in Euclid 's Elements , solved this problem, thus ending the crisis of irrationals. It is not entirely clear to what extent Eudoxus got his theory of proportions from Theaetetus' work, but history has generally credited it to Eudoxus. Eudoxus' second major contribution to mathematics was his method of exhaustion. It was well known by Eudoxus' time that the circumference of a circle can be approximated by inscribing a polygon in the circle and measuring the perimeter of that polygon. Eudoxus took this idea one step further and invented the method of exhaustion. He reasoned that by adding more sides to the inscribed polygon, a better approximation of the circumference, and hence p, is possible. The method of exhaustion involves inscribing polygons with successively more sides into a circle, thereby exhausting the small area between the circle and polygon. Although there is some speculation that

32. Title 430 to 350 BCE. Plato, Ca. 427 to 347 BCE. Theaetetus of Athens, Ca. 415 to 369BCE. eudoxus of cnidus, Ca. 410 to 355 BCE. Menaechmus, Ca. 380 to 320 BCE. Euclid,Ca. http://www.math.uvic.ca/courses/math415/Math415Web/greece/gmen.html

Important Greek Mathematicians The following some of the most influencial mathematicians of Ancient Greek times. Thales of Miletus Ca. 625 to 550 BCE Pythagoras of Samos Ca. 572 to 495 BCE Zeno of Elea Ca. 490 to 430 BCE Hippocrates of Chios Ca. 470 to 410 BCE Archytas of Tarentum Ca. 430 to 350 BCE Plato Ca. 427 to 347 BCE Theaetetus of Athens Ca. 415 to 369 BCE Eudoxus of Cnidus Ca. 410 to 355 BCE Menaechmus Ca. 380 to 320 BCE Euclid Ca. 325 to 265 BCE Archimedes of Syracuse Ca. 287 to 212 BCE Eratosthenes Ca. 275 to 200 BCE Apollonius of Perga Ca. 260 to 190 BCE Hipparchus of Rhodes Ca. 190 to 120 BCE Claudius Ptolemy Ca. 86 to 165 AD Diophantus of Alexandria Ca. 200 to 285 AD Hypatia of Alexandria Ca. 370 to 415 AD

33. I Need Help Writing A Thesis Statemant On Eudoxus Of Cnidus I need help writing a thesis statemant on eudoxus of cnidus. From HELPDate 23 Feb 2003 Time 125750 Remote Name 152.163.189.234. Comments. http://www.powa.org/stafpost/_disc4/0000016c.htm

Conversation Board Contents Search Post Reply ... Previous Up I need help writing a thesis statemant on Eudoxus of Cnidus From: HELP Date: 23 Feb 2003 Time: Remote Name: Comments hurry my paper is due tuesday Last changed: March 28, 2003

34. Online Book 1990). This page is part of Eric Weisstein's World of Scientific Biography.eudoxus of cnidus eudoxus of cnidus (ca. 400ca. 347 http://www.physics.sfasu.edu/astro/astronomylinks/all1.html

Ancient Astronomy A Brief Introduction to Archaeoastronomy The study of the astronomical practices, celestial lore, mythologies, religions and world-views of all ancient cultures we call archaeoastronomy. America's Stonehenge Along with it's astronomical alignments, America's Stonehenge continues to be one of the most fascinating archaeological discoveries of the century. Ancient Astronomy Links Some selected links to pages with information on Ancient Astronomy Calendars and Ancient Astronomy in Egypt, India, Maya, and Mesopotamia. Ancient Astronomy Timeline A timeline of ancient astronomy from 4000 B.C. to around 140 A.D. Ancient Mesopotamia: Astronomy and Calendry Part of a history of astronomy course with links to photographs and explanations. Babylonian Planetary Theory and the Heliocentric Concept Gathered ideas regarding the view of the cosmos by ancient Babylonians as well as new interpretations are on display here. With data on periods and planetary motions. Chinese Astronomy Astronomy truly is an ancient science in China. In fact, mankind's first record of an eclipse of the Sun was made in China in 2136 BC. Egyptian Estimates of the Size and Shape of the Earth How did ancient Egyptians measure the size of the Earth?

35. Eudoxus Of Cnidus (ca. 400-ca. 347 BC) -- From Eric Weisstein's World Of Scienti Similar pages Universal Library writings of, for example, Herodotus, Plato (esp. Phaedo and Timaeus),Theophrastus (Athen.2. 42b) and eudoxus of cnidus (DL 8. 8690). http://www.astro.virginia.edu/~eww6n/bios/Eudoxus.html

Branch of Science Astronomers Branch of Science Mathematicians ... Greek Eudoxus of Cnidus (ca. 400-ca. 347 BC) Greek philosopher, astronomer, and mathematician who accepted Plato's notion of the rotation of the planets around the Earth on crystalline spheres, but noticed discrepancies with observations. He tried to adjust Plato's model by postulating that each crystalline sphere had its poles set to the next sphere. His model contained no mechanical explanation; it was simply a mathematical description. There were problems, however, with his model. First of all, each "hippopede" produced by the superposition of the motions of two spheres produced the same curve, yet the retrogressions of planets were observed to exhibit differing shapes. Secondly, although his models predicted tolerable retrogressions for Jupiter and Saturn and not for Mars or Venus Thirdly, his model in no way accounted for the observed differences in the lengths of the seasons Finally, the model failed to account for variations in the observed diameter of the Moon or changes in the brightness of planets, which were correctly interpreted to indicate that their distances were changing. Eudoxus was the first Greek to make a map of the stars.

36. My Favoriate Mathematicians: Eudoxus Eudoxus. 408 BC355 BC eudoxus of cnidus solved the foundational crisis arisingfrom the existence of irrational numbers (perhaps uncovered by Hippasus). http://homepages.feis.herts.ac.uk/~nehaniv/eudoxus.html

Eudoxus 408 BC-355 BC Eudoxus of Cnidus solved the foundational crisis arising from the existence of irrational numbers (perhaps uncovered by Hippasus ). His solution, constructing real numbers as limits of sequences of ratios of commensurables, was given a "uniqueness" part by Dedekind. Eudoxus' treatment of irrational numbers comprises Book X of Euclid, first chairman of the Mathematics Dept. at the University of Alexandria. More Links: Short Biography of Eudoxus Document by C. Nehaniv, February 2, 1996 e-mail: nehaniv@u-aizu.ac.jp

37. Links SQL: Physical Science/Astronomy/1. Fundamentals/(b) Greek Astronomy http//scienceworld.wolfram.com/biography/Eratosthenes.html (AddedSat Oct 26 2002). eudoxus of cnidus eudoxus of cnidus (ca. 400ca. http://www.mhhe.com/links/1258/1226/1388/1453/

HOME SEARCH Looking for something in particular? the entire directory only this category More search options Home Physical Science Astronomy ... 1. Fundamentals : (b) Greek Astronomy LINKS: Anaximander of Miletus Anaximander of Miletus (610-ca. 546 BC) conceived the idea that the stars were fixed on a crystalline sphere rotating around the Earth. Anaximander thought the Earth to be cylindrical with a diameter three times its height, and the center of the universe. http://scienceworld.wolfram.com/biography/Anaximander.html (Added: Sun Oct 27 2002) Anaximenes of Miletus Anaximenes was the first Greek to distinguish clearly between planets and stars. He believed the primary substance of the universe to be air, which could form the other elements of water, Earth, and fire by rarefaction and condensation. This page is part of Eric Weisstein's World of Scientific Biography. http://scienceworld.wolfram.com/biography/Anaximenes.html (Added: Sun Oct 27 2002) Aristarchus and the Size of the Moon The jump Aristarchus made from terrestrial measurements of scale to the celestial is truly remarkable. Without any measures of the sizes of or distances to any celestial objects, he was able to measure both for the Moon. http://www.hastings.edu/Courses/physics/sivron/astronomy/specials/aristarchus.html

38. Hipparchus On A Poem eudoxus of cnidus (c. 390c. 340 BC) produced a work known as the Phenomena, inwhich he described a calendar with references to the risings and settings of http://www.hps.cam.ac.uk/starry/hipppoem.html

Links Hipparchus Tour (Next) Previous Hipparchus Pages Hipparchus Astrology Calendars and Weather Prediction Mathematical Techniques ... Index Hipparchus on a Poem Title page of Aratus and Eudoxus Image by kind permission of the Master and Fellows of Trinity College Cambridge. Large image (78K). Very large image (4.2M). The sole surviving work of Hipparchus (who flourished during the second half of second century BC) is known as the Commentary on the Phenomena of Aratus and Eudoxus. Eudoxus of Cnidus (c. 390-c. 340 BC) produced a work known as the Phenomena, in which he described a calendar with references to the risings and settings of constellations. Aratus (c. 315 - before 240 BC) produced an enormously popular poem, also known as the Phenomena , which utilised Eudoxus' work. Although he wrote many other poems, Aratus' Phenomena is his only extant work. The Phenomena quickly became one of the most widely read poems in the ancient world, after the Homeric poems, the Iliad and the Odyssey.

39. History Of Geometry eudoxus of cnidus (408355 BC) foreshadowed algebra by developing a theory of proportionwhich is presented in Book V of Euclid's Elements in which Definitions http://geometryalgorithms.com/history.htm

A Short History of Geometry Ancient This page gives 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 our 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].

40. 600 Years From Thales To Cleopatra, Greek And Egyptian Intertwined Archytas of Tarentum (c. 428350 BC) (a close friend by the way ofPlato who taught at Athens) taught eudoxus of cnidus. (Cnidus http://hometown.aol.com/befree2byourself/myhomepage/collection.html

htmlAdWH('7002816', '120', '30'); htmlAdWH('7002528', '234', '60'); Main Create Edit Help 600 Years From Thales To Cleopatra, Greek And Egyptian Intertwined 600 YEARS FROM THALES TO CLEOPATRA GREEK AND EGYPTIAN INTERTWINED by Mark Edward Westerfield The purpose of this letter is to trace the continuous line(s) teacher to student who became teacher to student... from Thales to Alexander the Great and thereby to the Ptolemy Dynasty and to Queen Cleopatra in an unbroken line of appreciation for ancient mysteries and learning passed down from generation to generation for 600 years among these Greek world figures and to see their connections to the more ancient Egypt. The ancient Greeks settled the western coast of what is now Turkey about 1000 BC where trade flourished with success connecting many cultures. It is here that the earliest great minds of mathematics and astronomy developed in the Greek world. Among these was Thales of Miletus. So many various estimated dates exist for Thales that I have no idea which to put down here. So I will give a few samples: 624-546 BC, 620-555 BC, 636-546 BC.

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