61. Plato And Platonism eudoxus of cnidusauthor of the doctrine of proportion expounded in Euclid's Elements,inventor of the method of finding the areas and volumes of curvilinear http://www.msu.org/ethics/content_ethics/texts/plato/plato_eb.htm

Plato and Platonism "Plato and Platonism: Bibliography" Britannica Online. http://www.eb.com:180/cgi-bin/g?DocF=macro/5005/10/28.html Plato and his thought LIFE Plato was born, the son of Ariston and Perictione, in Athens, or perhaps in Aegina, in about 428 BC, the year after the death of the great statesman Pericles. His family, on both sides, was among the most distinguished in Athens. Ariston is said to have claimed descent from the god Poseidon through Codrus, the last king of Athens; on the mother's side, the family was related to the early Greek lawmaker Solon. Nothing is known about Plato 's father's death. It is assumed that he died when Plato was a boy. Perictione apparently married as her second husband her uncle Pyrilampes, a prominent supporter of Pericles; and Plato was probably brought up chiefly in his house. Critias and Charmides, leaders among the extremists of the oligarchic terror of 404, were, respectively, cousin and brother of Perictione; both were friends of Socrates, and through them Plato must have known the philosopher from boyhood.

62. History Of Mathematics Although eudoxus of cnidus later solved the dilemma by working out a theory ofproportion, after Pythagoras's time Greek mathematics became essentially http://pratt.edu/~arch543p/help/history_of_mathematics.html

Note: the following has been abstracted from the Grolier Encyclopedia. History of Mathematics Mathematics is as old as civilization itself. By the Neolithic Period, as life became settled and villages began to appear, writing and counting became increasingly useful, if not necessary. With counting, the history of mathematics began. To count the passage of time, to weave intricate patterns in baskets or fabrics, and to apportion goods, crops, and livestock required a basic sense of arithmetic. Similarly, even in the most rudimentary cultures, the ability to decorate pottery with intricate designs, to distinguish constellations among the stars, or to arrange stones, obelisks, and tombs in ritualistic formations indicates a sense of space and geometry. Egyptian, Babylonian, and Greek Mathematics Arithmetic and Algebra Geometry and Trigonometry Islamic and Medieval Mathematics ... Philosophy of Mathematics Egyptian, Babylonian, and Greek Mathematics The earliest knowledge of mathematics is preserved in Egyptian papyruses, Babylonian cuneiform tablets, and Greek manuscripts. They indicate that the first mathematical concerns involved ARITHMEtic, Algebra, Geometry, and Trigonometry. Arithmetic and Algebra Among the earliest surviving mathematical texts are the famous Rhind papyrus (c.1750 BC) and the Golonishev papyrus. They reveal that the Egyptians used a decimal system; the unit was represented by a single line, and tens, hundreds, and thousands by hieroglyphic symbols. Arithmetic for the Egyptians was essentially additive; repeated doubling was used for multiplication. Except for the fraction 2/3, for which there was a special hieroglyph, all fractions were expressed as unit fractions of the form 1/n; a relatively simple fraction like 2/59 was always handled in the more complex though equivalent form 1/36 + 1/236 + 1/531 = 2/59.

63. 370 BCE eudoxus of cnidus devises calendars using zodiac with 12 equal zodiacsigns. Invents geometrical theory of proportion. He had built http://www.astarotology.com/timeline/timeline/370_BCEx.html

Eudoxus of Cnidus devises calendars using zodiac with 12 equal zodiac signs. Invents geometrical theory of proportion. He had built an observatory on Cnidus and we know that from there he observed the star Canopus. He spent over a year in Egypt where he studied astronomy with the priests at Heliopolis. At this time Eudoxus made astronomical observations from an observatory which was situated between Heliopolis and Cercesura. From Egypt Eudoxus travelled to Cyzicus in northwestern Asia Minor on the south shore of the sea of Marmara. There he established a School which proved very popular and he had many followers.

64. Sample Template eudoxus of cnidus eudoxus of cnidus was the son of Aischines. Hewas born on 408 BC in Cnidus, Asia Minor (now Turkey.) He died http://www.fallcreek.k12.wi.us/fcmiddle/departments/Math/Mlsna/7/Eudoxus4.htm

65. Winter Constellations: Asterisms And Constellations The first complete description of the constellations was by eudoxus of cnidus in366 BC Most people think Eudoxus' writings are the main source of the legends http://members.aol.com/ckckside/reports/constellation/astrob.htm

ASTERISMS AND CONSTELLATIONS What is the difference between asterisms and constellations? Many people get them confused with each other. They are similar, but they are also different. Asterisms are groups of stars that make a shape or form of something. The Big Dipper is an example of an asterism. It is made of seven stars, four make up the bowl and three are the handle. Constellations are a group of stars that often include asterisms. They are usually outlined by some imaginary line for them to have a shape. The Big Dipper is an asterism in the constellation Ursa Major. Often it is easier to find the asterism in the constellation than it is to find the constellation itself. For instance, I can find the Big Dipper very easily. Once I find the Big Dipper, then I can see Ursa Major. But if I tried to find Ursa Major without seeing the Big Dipper, I might not ever find Ursa Major. Since constellations include other asterisms, star clusters, nebulae and meteorites, their outline is not as clear as the asterisms which have fewer objects. A nuclear reaction something like the hydrogen bomb causes stars to shine. Hydrogen is transformed into helium and about one percent of its mass is transferred into heat energy. This energy keeps the temperature at the center of the star at millions of degrees.

66. LUNATIC REORGANIZES HEAVENS Some lunatic named eudoxus of cnidus, (there have been whisperings from the oracleat Delphi that Cnidus will one day be called Turkey), thinks that the http://www.rainbowkids.de/unterhaltung/Geschichten/AWinghaven/lunatic.htm

LUNATIC REORGANIZES HEAVENS! (This article is written as though it were being published in the days of Eudoxus.) Some lunatic named Eudoxus of Cnidus, (there have been whisperings from the oracle at Delphi that Cnidus will one day be called Turkey), thinks that the planets are carried on spheres. He claims that these spheres are nested around the earth in mountings like compass gimbals. "Rotations on these explain observed motions of stars," says Eudoxus. He also says that the solar year is six hours longer than three hundred and sixty five days. He has been a pupil of the Greek philosopher Archytas and also has studied under Plato. Eudoxus is a Greek geometer and astronomer; Now he has founded a school at Cyzicus. The priests of Helios are outraged. They say that we should stick to the proper ways of reckoning the movements of the heavens, which is that Helios, the sun, mounts his glowing chariot after Eos, the dawn goddess, opens the gates of morning, and rides across the sky to light the day. Then at night Selene, or Diana, the moon goddess comes out to bring the moon to

67. Astronomical Games: June 2001 explicit attempt at answering Plato's question in the affirmative was made by theGreek philosopher, astronomer, and mathematician eudoxus of cnidus (c. 400 http://astro.isi.edu/games/kepler.html

Astronomical Games: June 2001 Music of the Ellipses Our understanding of the solar system took some unplanned detours Mankind is not a circle with a single center but an ellipse with two focal points of which facts are one and ideas are the other. Victor Hugo, A SEMI-RECENT survey [ ] showed that about a quarter of American adults believe that the Sun goes around the Earth. You can imagine the uproar that rose up in educational institutions around the country. (Actually, it was pretty subdued, and if you were of a cynical bent, you could draw some pretty depressing conclusions about what higher education thinks of the American mandatory educational system. But let's not get into that.) How is it possible that so many Americans could believe such a thing? Well, they believe it for the same reason that the ancient Greeks and everyone else up to about the 16th century believed it. All you have to do is look up, and if you have the common sense God granted the garden snail, it is plain to see that the Sun goes around the Earth. After all, astronomers claim the Moon goes around the Earth, and no one laughs at them for that Granted, appearances were not all that mattered to the Greeks. They had their theory, too. Aristotelian physics held that the Earth was all that was base and ignoble, and it therefore sank to the very center of the universe. The celestial objects, howevereverything up in the skywere good and noble, and therefore light and airy, and they all travelled in great circular arcs around the lowly center, maintaining a cordial distance at all times.

68. Mercator's World Online eudoxus of cnidus, a contemporary of Plato’s, elaborated on the classical notionthat the sun, moon, planets, and stars occupied a series of concentric http://www.mercatormag.com/article.php3?i=38

69. Chapter 15, Golden Mean 8 Perhaps the most gifted geometer to study there was eudoxus of cnidus, who finallybroke the deadlock of the irrationals, and freed geometry for the great http://www.anselm.edu/homepage/dbanach/pyth4.htm

Selections from Julia E. Diggins, String, Straightedge, and Shadow Viking Press, New York , 1965. (Illustrations by Corydon Bell) 15. THE GOLDEN AGE AND THE GOLDEN MEAN The second half of the 5th century B.C. was the Golden Age of Greece. This was the period of her most beautiful art and architecture, and some of her wisest thinkers besides. Both owed much to the popular new study of geometry. By the start of the next century, geometry itself was entering its own classic age with a series of great developments, including the Golden Mean. The times were glorious in many ways. The Persian invaders had been driven out of Hellas forever, and Pericles was rebuilding Athens into the most beautiful city in the world. At his invitation, Greek mathematicians from elsewhere flocked into the new capital. From Ionia came Anaxa- goras, nicknamed "the mind." From southern Italy and Sicily came learned Pythagoreans and the noted Zeno of Elea. And their influence was felt over all Athens. High on the hill of the Acropolis rose new marble temples and bronze and painted statues. Crowds thronged the vast new open-air theater nearby, to hear immortal tragedies and comedies by the greatest Greek playwrights. These splendid public works were completed under the direction of the sculptor Phidias and several architects, all of whom knew and used the principles of geometry and optics. "Success in art," they insisted, "is achieved by meticulous accuracy in a multitude of mathematical proportions." And their buildings had a dazzling perfection never seen before-the beauty of calculated geometric harmony.

70. Chapter 16: Archimedes many of the irrationals. In Plato's own time, the two greatest wereTheaetetus of Athens and eudoxus of cnidus. And at the Lyceum http://www.anselm.edu/homepage/dbanach/arch.htm

Selections from Julia E. Diggins, String, Straightedge, and Shadow Viking Press, New York , 1965. (Illustrations by Corydon Bell) 16. A ROYAL ROAD, AFTER ALL During the 4th century B.C., Greek geometry burst its bonds and went on to the tremendous discoveries of the "age of giants." And Greek culture, too, burst from the mainland of Hellas and spread to most of the eastern Mediterranean. Both developments were connected with the romantic figure of Alexander the Great. After Plato's time, teachers and alumni from the Academy had gone on to found schools of their own. In particular, Plato's most famous associate, the great philosopher Aristotle, had set up the Lyceum in Athens, and started the systematic classification of human knowledge. And Aristotle's most renowned pupil was the warrior king Alexander of Macedon, who tried to conquer the world. In thirteen years, Alexander extended his rule over Greece proper, and Ionia, Phoenicia, Egypt, and the vast Persian domains as far as India. Then he died, and his empire broke up. But throughout those far-flung lands, he had founded Greek cities and planted the seeds of Greek civilization-the Greek language, Greek art, and, of course, Greek mathematics. Mathematicians traveled with his armies. And there is even a

71. History Of Astronomy Thales. · Plato. · Aristarchus. · eudoxus of cnidus. · Aristotle. · Eratosthenesof Alexandria. · Hipparchus of Nicaea. · Ptolemy of Alexandria. Medieval. http://www.cerritos.edu/ladkins/a106/history_list_4.htm

History of Astronomy Course Notes and overviews from other Institutions Galileo and Einstein Nick Strobel's Textbook History of Science History of Cosmology (UCR) ... Archaeoastronomy Links Specific Ancient Cultures Mayan Astronomy Aztec Calendar Stonehenge Another Stonehenge ... How the Shaman Stole the Moon C lassical Mediterranean Antiquity Greek Astronomy (Overview) Early Greek Astronomy Thales Plato ... Ptolemy of Alexandria Medieval Omar Kahyyam Omar Kahyyam's The Rubaiyat William of Ockham ... " Renaissance - 18th Century Nicolas Copernicus Johannes Kepler Tycho Brahe Galileo Galilei ... Antique Celestial Atlases 19th Century Annie Jump Cannon Henrietta Leavitt Lord Rosse Josef Fraunhofer ... Henry Draper 20th Century Mount Wilson Observatory Shapley's Milky Way The “Great Debate” Cecilia Payne ... Edwin Hubble (photo) Edwin Hubble (bio) Scale of the Universe 1996 Einstein Other Links of Interest Timeline Women in Astronomy BACK TO HOME PAGE

72. Models Of Planetary Motion -- McConnell The Eudoxan Solution eudoxus of cnidus (c. 390 c. 337 BC) envisioned a system ofspheres whose combined uniform motion would resemble a hippopede, a figure http://faculty.fullerton.edu/cmcconnell/Planets.html

Models of Planetary Motion from Antiquity to the Renaissance Craig Sean McConnell Assistant Professor of Liberal Studies California State University, Fullerton Contents Introduction Retrograde Motion of the Planets The Eudoxan Solution From Hippopede to Retrograde Loops ... For Further Study Introduction Since antiquity, astronomers have attempted to explain the motions they observed in the heavens with geometrical models. This web site has been designed to help students in history of astronomy courses who are encountering these models for the first time. Often, students struggle to visualize how the static drawings in their textbook relate to the complex motions of the planets. By animating these images, I hope students will be able to more completely "see" how combinations of circles and spheres produced the distinctive retrograde motions exhibited by the planets. These images are not drawn to scale; they are meant only to serve as an aid to understanding how these models account for the motions in the heavens. Though this site includes a narrative description of the elements of these astronomical models, it is not intended to serve as a complete introduction to the history of ancient astronomy. For such an introduction, please consult one of the texts recommended

73. Plato eudoxus of cnidus author of the doctrine of proportion expounded in Euclid's Elements,inventor of the method of finding the areas and volumes of curvilinear http://www.kat.gr/kat/history/Greek/Ph/Plato.htm

Plato b. 428/427 BC, Athens, or Aegina, Greece d. 348/347, Athens Ancient Greek philosopher, the second of the great trio of ancient Greeks Socrates , Plato, and Aristotle who between them laid the philosophical foundations of Western culture. Building on the life and thought of Socrates, Plato developed a profound and wide-ranging system of philosophy. His thought has logical, epistemological, and metaphysical aspects; but its underlying motivation is ethical. It sometimes relies upon conjectures and myth, and it is occasionally mystical in tone; but fundamentally Plato is a rationalist, devoted to the proposition that reason must be followed wherever it leads. Thus the core of Plato's philosophy, resting upon a foundation of eternal Ideas, or Forms, is a rationalistic ethics. Life Plato was born, the son of Ariston and Perictione, in about 428 BC, the year after the death of the great statesman Pericles . His family, on both sides, was among the most distinguished in Athens. Ariston is said to have claimed descent from the god Poseidon through Codrus, the last king of Athens; on the mother's side, the family was related to the early Greek lawmaker Solon . Nothing is known about Plato's father's death. It is assumed that he died when Plato was a boy. Perictione apparently married as her second husband her uncle Pyrilampes, a prominent supporter of Pericles; and Plato was probably brought up chiefly in his house. Critias and Charmides, leaders among the extremists of the oligarchic terror of 404, were, respectively, cousin and brother of Perictione; both were friends of Socrates, and through them Plato must have known the philosopher from boyhood.

74. Henry Mendell 343. Article on eudoxus of cnidus, Encyclopaedia Britannica online, (2002) http//www.britannica.com/eb/article?eu=33776 . http://www.ceu.hu/sun/sun 2003 modmod/CV/henry_mendell_2003.htm

Central European University A Program for University Teachers, Researchers and Professionals in the Social Sciences and Humanities Summer University you are visitor no. Henry Mendell Philosophy Department, California State University, Los Angeles5151 State U. Dr. Education 1977-85: Stanford University (Ph.D. Jan., 1986) 1972-74: St. John's College, Cambridge, England (B.A. 1974, M.A. 1980in Philosophy) 1968-72: Cornell University (A.B. 1971 in Classics (Magna cum laude) and Philosophy) Dissertation Topic: Aristotle and the Mathematicians: Some Cross-Currents in the Fourth Century Principal Thesis Adviser: Julius Moravcsik AOS: Ancient Philosophy, Early Greek Mathematics and Astronomy AOC: Philosophy of Science, Metaphysics Publications Book with Pat Suppesand Julius Moravcsik (eds.). Ancient and Medieval Traditions in the Exact Sciences: Essays in Memory of Wilbur Knorr. Stanford: CSLI (distr. University of Chicago Press), 2001. Articles "The Trouble withEudoxus". In Pat Suppes, Julius Moravcsik, and Henry Mendell (eds.), Ancient and Medieval Traditions in the Exact Sciences: Essays in Memory of Wilbur Knorr (Stanford: CSLI (distr. University of Chicago Press), 2001), 59-138 "Making Sense of Aristotelian Demonstration". Oxford Studies in Ancient Philosophy, 16 (1998), 160-225.

75. COSMOLOGY DAY. BOTH PLATO AND AFTERWARDS ARISTOTLE WERE FAMILIAR WITH AND DEVELOPEDTHE WORK OF eudoxus of cnidus (APPROX. 390337 BC). EUDOXUS http://mac01.eps.pitt.edu/courses/GEO0870-CGS/COSMOLOGY.htm

COSMOLOGY: THE BRANCH OF PHILOSOPHY, OFTEN CONSIDERED A BRANCH OF METAPHYSICS THAT DEALS WITH THE UNIVERSE AS A TOTALITY OF PHENOMENA, ATTEMPTING TO COMBINE METAPHYSICAL SPECULATION AND SCIENTIFIC EVIDENCE WITHIN A COHERENT FRAMEWORK. AND THE MODERN SCIENTIFIC STUDY OF THE ORIGIN AND STRUCTURE OF THE UNIVERSE BASED ON SUCH THINGS AS THE SPECTRAL INVESTIGATION OF THE DISTRIBUTION OF ELEMENTS AND STRUCTURES THROUGHOUT THE UNIVERSE. MANY ANCIENT CIVILIZATIONS DEVELOPED QUITE SOPHISTICATED OBSERVATIONAL CAPACITIES FOR RECORDING THE PROPERTIES OF THE HEAVENS. THESE INCLUDED SUCH CIVILIZATIONS AS THE SUMERIANS, BABYLONIANS, EGYPTIANS, CHINESE, AND THE VARIOUS EARLY INDIAN CIVILIZATIONS IN THE OLD WORLD IN THE NEW WORLD, INCLUDED THE MAYA, INCA, AND AZTEC. THE ANCIENT CIVILIZATIONS DEVELOPED RECORDS THAT OFTEN WERE AS PRECISE AS THE TECHNOLOGIES OPEN TO THEM COULD MAKE THEM, AND WERE RELATIVELY COMPLETE FOR HUNDREDS OF YEARS. THIS PRECISION WAS NOT GREAT COMPARED TO OURS. THE ANCIENT GREEK ASTRONOMERS WERE INTERESTED PRIMARILY IN A RELATIVELY QUALITATIVE VISUAL MATCH AT FIRST, BUT SUMERIAN AND BABYLONIAN ASTRONOMY WAS MORE PRECISE AND PREDICTIVE. THE BABYLONIANS TACKLED PLANETARY MOTIONS AND THE ECLIPSES OF VARIOUS BODIES SUCH AS THE MOON.

76. Planet Mars Chronology 4th century BC eudoxus of cnidus (c. 400350 BC), a Greek astronomerand mathematician, developed a system of 27 concentric spheres. http://humbabe.arc.nasa.gov/mgcm/fun/ancient_mars.html

Planet Mars and the Ancients Compiled by Paul Karol and David Catling This page is part of the Planet Mars Chronology from ancient to present day. You may also be interested in Planet Mars in Popular Culture A ncient Civilizations c. 3000 BC Egyptians recognize the apparent retrograde motion of Mars calling it Sekded-ef em khetkhet, one "who travels backward". c. 23 century BC A series of tablets are written in the time of King Sargon of Akkad (died c. 2279 BC). The text relates to many astrological descriptions of the heavens, including the planets. Sargon was a ruler of Mesopotamia (modern Iraq), who is reported to have built the capital city of Agade, which unfortunately has never been located or excavated probably because it was destroyed. Modern astronomy derives from the work of these early Mesopotamian astronomers, such as Sargon's daughter, En Hedu'anna . However, in early times, astronomy (science) was always subsumed by astrology (superstition). c. 1000 BC Ancient Chaldeans are reported to call Mars Nergal (or Nirgal). Chaldea was a land in southern Babylonia frequently mentioned in the Old Testament and identified with modern Iraq. Strictly speaking, Chaldea is the land bordering the head of the Persian Gulf between the Arabian desert and the Euphrates delta. Nergal (Mars) was the great hero, the king of conflicts, the master of battles, and the champion of gods. On the banks of the Euphrates the planet Mars was also known under the names of Allamou and Almou. c. 1200-300 BC

77. Ulearn Today - Magazine Book V shifts from plane geometry to expand on a general theory of ratiosand proportions, generally attributed to eudoxus of cnidus. http://www.ulearntoday.com/magazine/physics_article1.jsp?FILE=euclid

78. GET YOUR DOMAIN .com .net .org .biz .info eudoxus of cnidus. Born 408 BC in Cnidus (on Resadiye BC in Cnidus, AsiaMinor (now Turkey). eudoxus of cnidus was the son of Aischines. http://math.5u.com/Eudoxus.htm

Cheap Web Site Hosting Eudoxus of Cnidus Born: 408 BC in Cnidus (on Resadiye peninsula), Asia Minor (now Turkey) Died: 355 BC in Cnidus, Asia Minor (now Turkey) Eudoxus of Cnidus was the son of Aischines. As to his teachers, we know that he travelled to Tarentum, now in Italy, where he studied with Archytas who was a follower of Pythagoras . The problem of duplicating the cube was one which interested Archytas and it would be reasonable to suppose that Eudoxus's interest in that problem was stimulated by his teacher. Other topics that it is probable that he learnt about from Archytas include number theory and the theory of music. Eudoxus also visited Sicily, where he studied medicine with Philiston, before making his first visit to Athens in the company of the physician Theomedon. Eudoxus spent two months in Athens on this visit and he certainly attended lectures on philosophy by Plato and other philosophers at the Academy which had only been established a short time before. Heath [3] writes of Eudoxus as a student in Athens:- ... so poor was he that he took up his abode at the Piraeus and trudged to Athens and back on foot each day. After leaving Athens, he spent over a year in Egypt where he studied astronomy with the priests at Heliopolis. At this time Eudoxus made astronomical observations from an observatory which was situated between Heliopolis and Cercesura. From Egypt Eudoxus travelled to Cyzicus in northwestern Asia Minor on the south shore of the sea of Marmara. There he established a School which proved very popular and he had many followers.

79. EUDOXIAN MAGNITUDES In modern parlance these magnitudes are called Archimedean, but Archimedes(c.298 BC212 BC) got it from eudoxus of cnidus (c.408 BC-c.355 BC). http://www.ida.liu.se/~rosgr/eudoxianmag.html

EUDOXIAN MAGNITUDES by Ross Lee Graham This concept is fundamental to quantitative reference-frames. In Physics, it is fundamental to the concept of invariant magnitudes. In modern parlance these magnitudes are called Archimedean, but Archimedes (c.298 BC-212 BC) got it from Eudoxus of Cnidus (c.408 BC-c.355 BC). Euclid (c.300 BC) systemized this concept in its most general form in Book V in his Elements . Euclid used a synthetic mode of exposition. The following outline is predominantly analytic. For a characteristic magnitude to be Eudoxian it must be possible to add any instance of itself to itself enough times such that it will exceed any given greater instance of the same characteristic. Thus: If a b, where multiplying by n represents repeated addition. From this we see that for a characteristic magnitude to be Eudoxian the magnitude must be additive. Any instances of the characteristic magnitude can be added together to produce another instance of the same characteristic magnitude. This is known as closure for addition of a characteristic magnitude. For an algorithmic convenience, addition is usually defined as dyadic. (There is no necessary reason for addition to be dyadic.) For addition to be Eudoxian it must obey the following arithmetic rules: 1. Rule of ISOMETRY: If equals be added to equals the results are equal.

80. Escaut To Euphuism. Alphabetic Index To Entries. The Columbia Encyclopedia, Sixt Euclid of Megara. Eudes. Eudocia. Eudocia Macrembolitissa. Eudoxia. eudoxusof cnidus. eudoxus of Cyzicus. Eugene III. Eugene IV. Eugene. Eugene of Savoy. http://www.bartleby.com/65/index80.html

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