41. Eudoxus eudoxus of cnidus. Born 408 BC in Cnidus (on Resadiye peninsula), AsiaMinor (now Turkey) Died 355 BC in Cnidus, Asia Minor (now Turkey). http://sfabel.tripod.com/mathematik/database/Eudoxus.html

Eudoxus of Cnidus Born: 408 BC in Cnidus (on Resadiye peninsula), Asia Minor (now Turkey) Died: 355 BC in Cnidus, Asia Minor (now Turkey) Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Welcome page Eudoxus studied at Plato 's Academy and was a student of Archytas of Tarentum. He spent over a year in Egypt and then, on his return, established a school that competed with Plato . There is ample evidence to suggest that Eudoxus had little respect for Plato 's analytic ability. Eudoxus proposed a heliocentric system for the solar system; a very important contribution. Other important contributions were to the theory of proportion, where he made a definition of equal ratios similar to cross multiplying, and early work on integration with the theory of exhaustion. The kampyle curve was studied by Eudoxus also in relation to the classical problem of duplication of the cube. Eudoxus found formulas for measuring pyramids cones and cylinders. Books V and XII of Euclid 's Elements are attributed to Eudoxus by some experts.

42. Lecture 6 Eudoxus' Proof That The Ratio Of Circles Is The Ratio Of Diameter Squa The most important was eudoxus of cnidus (408 - -355), who inventedthe Method of Exhaustion , the Greek version of limits. At http://www.maths.uwa.edu.au/~schultz/3M3/L6Euclid.html

Lecture 6: Eudoxus' proof that the area of a circle is a constant times diameter squared. Classical Greek mathematics The century after Plato -320 - -220 was the classical age of Greek mathematics. There were 20 or 30 mathematicians whose names have been recorded, although their writings don't all survive. The most important was Eudoxus of Cnidus (-408 - -355), who invented the "Method of Exhaustion", the Greek version of limits. At the end of this period, the major achievments were put in textbook form by the great teacher Euclid of Alexandria, whose dates are uncertain probably around -300, (Alexander -356 - -323). His "Elements" of 13 books was in use for 2000 years, and is the second most translated book. Books 16: plane geometry Books 79: number theory Books 10 : magnitudes Books 1113: solid geometry. For a more detailed index to Euclid, click here. These books contain definitions, postulates, theorems etc, much like today's mathematical texts, which are all descended from them. Euclid did not record the names of the discoverers, but his commentators did. The text Euclid Book XII, Proposition 2.

43. Sample Chapter For Dolling, L.M., Statile, G.N., Gianelli, A.F.,: The Tests Of T One of Plato's pupils, eudoxus of cnidus (409 BC356 BC), was the firstastronomer to follow Plato's recommendation. Blending careful http://pup.princeton.edu/chapters/i7432.html

PRINCETON University Press SEARCH: Keywords Author Title More Options Power Search Search Hints E-MAIL NOTICES NEW IN PRINT E-BOOKS ... HOME PAGE The Tests of Time: Readings in the Development of Physical Theory Edited by Lisa M. Dolling, Arthur F. Gianelli, and Glenn N. Statile Book Description Endorsements Class Use and other Permissions . For more information, send e-mail to permissions@pupress.princeton.edu This file is also available in Adobe Acrobat PDF format INTRODUCTION Although Heliocentric Theory is well known, describing it without the use of unwarranted or unjustified assumptions is not easy. Simply put, the theory suggests that the earth has two motions, a rotation on an axis and an orbital motion about the sun. Further, it maintains that the sun is central to, although not exactly in the center of, the orbits of all those heavenly bodies known as the planets, of which the earth is one. The physical reference frame used to determine the motions of this "solar" system is the frame of the fixed stars, bodies that do not appear to change their positions relative to one another. In this theory the dual motions attributed to the earth are considered to be in some sense real. The Geocentric View of Eudoxus Although not really interested in astronomy, the philosopher Plato had a great influence on the course of its early history. Because he perceived the heavens to be more perfect than the earth, Plato urged astronomers to describe celestial motions in terms of the most perfect of geometrical shapes, the circle. In fact, for Plato, the most perfect motion would be uniform circular motion, motion in a circle at a constant rate of speed.

44. 46 Lessons In Early Geometry, Part 4/10 This would be done for them by eudoxus of cnidus ; they were not, however tosuppose that it was this the god desired, but rather that he was ordering the http://www.seshat.ch/home/geom04.htm

46 Lessons in Early Geometry, part 4/10 / provisional version in my own freestyle English / a corrected version will follow in March, April or May (hopefully) / Franz Gnaedinger / February 2003 / www.seshat.ch early geometry 1 early geometry 2 early geometry 3 / early geometry 4 / early geometry 5 early geometry 6 early geometry 7 early geometry 8 ... early geometry 10 Lesson 15 The Babylonians used an excellent value for the square root of 2, found on the clay tablet YBC 7289 from around 1650 or 1700 BC: YBC 7289 exact value How did the Babylonians possibly find this value? Divide 1393 by 985 and you obtain 1;24,51,10,3,2... Leave out the small numbers ...3,2... and keep the value 1;24,51,10. Archimedes used the ratios 265/153 and 1351/780 as boundaries for the square root of 3. Lesson 16 Victor J. Katz, A HISTORY OF MATHEMATICS, Addison-Wesley 1998, page 46, The Beginnings of Mathematics in Greece 'A report from a visit to Egypt with Plato by Simmias of Thebes in 379 B.C.E. (from a dramatization by Plutarch of Chaeronea (first/second century C.E.)): "On our return from Egypt a party of Delians met us ... and requested Plato, as a geometer, to solve a problem set them by the god in a strange oracle. The oracle was to this effect. The present troubles of the Delians and the rest of the Greeks would be at an end when they had doubled the altar at Delos How did Eudoxus solve the problem? He is known for a ladder of numbers that allows to double the square:

45. EUCLID eudoxus of cnidus, mathematician and astronomer, who studied at Plato'sacademy, and lived for some times in Egypt. He advanced http://www.hyperhistory.com/online_n2/people_n2/persons2_n2/eudoxus.html

EUDOXUS c.408 - c.355 BC Greek Mathematician Eudoxus of Cnidus, mathematician and astronomer, who studied at Plato's academy, and lived for some times in Egypt. He advanced subsantially number theory, and constructed a model of 27 spheres to explain the motions of the Moon, stars, and planets. www link : From the University of St. Andrews, Scotland School of Mathematics Biography

46. Mathematicians Archytas, Greek, 428350 BC, Ar, As, G, M. eudoxus of cnidus, Greek,408-335 BC, Eudoxan Theory of Proportion, Method of Exhaustion, As. http://members.fortunecity.com/kokhuitan/mathematicians.html

Great Mathematicians and Their Achievements Mathematics exist before 1900 BC, in great civilizations everywhere, including China, India, Babylon etc. However, the first record of Mathematical manuscripts is found in Egypt, namely, the Moscow Papyrus and the Rhind Papyrus. In the 'Achievement' column below, the notations are as follows: AG = Analytic Geometry Al = Algebra Ar = Arithmetic As = Astronomy C = Calculus DE = Differential Equation FM = Foundation of Mathematics G = Geometry GT = Group Theory L = Logic M = Mechanics N = Number Theory P = Probability RM = Recreational Mathematics S = Statistic ST = Set Theory T = Topology The list here is not exhaustive. The mathematicians listed here are either pioneers in various fields of Mathematics, or those who have contributed to almost all fields, or those who have settled unsolved problems. For a more complete list of mathematicians, click on index of mathematicians Name Nationality Year Achievements Egyptian 1900 BC Moscow Papyrus (25 problems on G Ahmes Egyptian 1700 BC Rhind Papyrus (84 problems on Ar, Al, G

47. The Ten Greatest Mathematicians Conversely Euclid's work has great historical significance, but much of it wasprobably borrowed from the more brilliant eudoxus of cnidus (408355 BC). http://freepages.computers.rootsweb.com/~jamesdow/mathmen.htm

The Ten Greatest Mathematicians of All Time ranked in approximate order of ``greatness.'' To qualify, the mathematician's work must have breadth depth , and historical importance. Carl F. Gauss Archimedes of Syracuse Leonhard Euler Sir Isaac Newton Augustin Cauchy Euclid of Alexandria Pierre de Fermat Gottfried Wilhelm Leibniz Bernhard Riemann These are my personal rankings, but most of the selections should have wide acceptance among mathematical historians. Most of the selections were easy, but choosing Bernhard Riemann and Euclid to round out the list was difficult. Riemann was undoubtedly an outstanding genius, but several great mathematicians might be better choices based on their work's historical importance, for example Joseph-Louis Lagrange Georg Cantor (1845-1918) or (1906-1978). Conversely Euclid's work has great historical significance, but much of it was probably borrowed from the more brilliant Eudoxus of Cnidus (408-355 BC). Other notable ancient Greeks include Apollonius of Perga (262?-190? BC) (the ``Great Geometer'') and, for his historical importance, Pythagoras of Samos (570?-500? BC).

48. Early Observations Of Mars eudoxus of cnidus. eudoxus of cnidus was a brilliant mathematicianthat lived during the fourth century BC Eudoxus, who was a pupil http://hyperion.advanced.org/19455/early_observations.htm

Please obtain a browser with java support to view this interactive menu Early Observations of Mars Much of what we know today about Mars would have never been possible if not for the work of early astronomers from the ancient civilizations of Babylonia, Egypt, and Greece, to the veritable contributions of Copernicus, Brahe, Kepler, and Lowell. Babylonians The Babylonians made careful astronomical observations and developed a sophisticated system of arithmetical computations for predicting astronomical phenomena such as eclipses. Their purposes were strictly calendrical and religious; however, they never attempted to explain the reasons for any of the movements they observed. Superstition was widespread, and many astronomical events were regarded as ominous-not just eclipses but even the risings of Venus were viewed as omens. Eudoxus of Cnidus Eudoxus of Cnidus was a brilliant mathematician that lived during the fourth century B.C. Eudoxus, who was a pupil at Plato's Academy, sought to reconcile Greek misconceptions concering planetary movement. The Ancient Greeks took for granted that the Earth was the center of the universe. They also assumed that the planets moved in a uniform circular motion. However, uniform motion around simple circles did not account for the complicated movements the Greeks actually observed; thus, they encountered the dilemma of showing how the observed movements could be reconciled with their principle of uniform circular motions. Eudoxus attacked this problem by developing an ingenious system of homocentric spheres, according to which the observed motion of a planet was produced by the independent motions of several internested spheres, each centered on the Earth. The significance of this system was that it was able to account for the

49. Biography-center - Letter E project/eudoxus/eudoxus.htm; eudoxus of cnidus, wwwhistory.mcs.st-and.ac.uk/~history/Mathematicians/Eudoxus.html;Euler, Leonhard www http://www.biography-center.com/e.html

Visit a random biography ! Any language Arabic Bulgarian Catalan Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Norwegian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Turkish E 272 biographies Eadred, www.britannia.com/history/monarchs/mon10.html Eadwig, www.britannia.com/history/monarchs/mon11.html Eagan, Edward www.olympic.org/uk/athletes/heroes/bio_uk.asp?PAR_I_ID=52407 Eagles, www.hotelcal.com Eakins, Thomas www.ibiblio.org/wm/paint/auth/eakins/ Eakins, Thomas www.getty.edu/art/collections/bio/a1729-1.html Eakins, Thomas www.pbs.org/wnet/americanmasters/database/eakins_t.html Eales, Henry www.whonamedit.com/doctor.cfm/1429.html Earl of Aberdeen, www.spartacus.schoolnet.co.uk/PRaberdeen.htm Earl of Derby, www.spartacus.schoolnet.co.uk/PRderby1.htm Early, Jubal Anderson civilwarhome.com/earlybio.htm Earnhardt (Ralph), Dale www.hickoksports.com/biograph/earnhardtd.shtml Easterly, Thomas Martin

50. Eudoxus Biographical Information. eudoxus of cnidus Biography of his life and accomplishments.eudoxus of cnidus Biography of his life and accomplishments. http://www.csce.uark.edu/~crane/people/eudoxus.htm

Eudoxus Born: 408 BC in Cnidus (on Resadiye peninsula), Asia Minor (now Turkey) Died: 355 BC in Cnidus, Asia Minor (now Turkey) Demonstrations Eudoxus' Planetary Model Contains discussion and a diagram Biographical Information Eudoxus of Cnidus Biography of his life and accomplishments. Eudoxus of Cnidus Biography of his life and accomplishments. From Plato's Time and Dialogues. Eudoxus of Cnidus Biography of his life. Eudoxus Biography of his life from Ball's . A Short Account of the History of Mathematics Problems Eudoxus Sequences Gives the first 15 values of the sequence Kampyle of Eudoxus Contains a picture of the curve that Eudoxus studied

51. Untitled However, the original mathematical model of the cosmos was developed in thefourthcentury by the geometer eudoxus of cnidus (ca. 400 -347 BC). http://www.ouc.bc.ca/phys/dkay/eudoxus.htm

The Homocentric Spheres of Eudoxus, Callippus, and Aristotle (Note: This site is under construction last updated on 30 July 99) Ever since the publication of the Principia by Isaac Newton in 1687 one of the principal goals of physics has been the construction of mathematical models to describe how nature works. However, this idea did not originate with Newton. The most famous earlier examples are the laws of planetary motion of Johannes Kepler (1571 - 1630) and the planetary systems of Claudius Ptolemy (2nd century A.D.) and Nicolas Copernicus (1473 - 1543). As Copernicus drew on the methods of Ptolemy, so too did Ptolemy draw on earlier work, particularly that of Apollonius of Perga and Hipparchus of the third and second centuries B.C., respectively. However, the original mathematical model of the cosmos was developed in the fourth-century by the geometer Eudoxus of Cnidus (ca. 400 -347 B.C.). His geometrical model for describing the motions of the planets has come to be known as the Homocentric Spheres. By the beginning of the fourth century B.C. the most widely held view of the universe was that it consisted of a spherical earth at rest in the center of a rotating "celestial sphere" carrying the fixed stars. The planets moved in the region between these two spheres. Outside the sphere of stars was nothing, not even space. The complete celestial catalog was short. There were the fixed stars, and seven "wanderers", the planets Sun, Moon, Mercury (Hermes), Venus (Aphrodite), Mars (Ares), Jupiter (Zeus), and Saturn (Kronos).

52. Scientists: Physical Sciences Sir Frank Watson; Eddington, Sir Arthur Stanley; Encke, Johann Franz;eudoxus of cnidus; Galileo; Galle, Johann Gottfried; Hale, George http://www.factmonster.com/spot/scibio6.html

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53. Eudoxus Of Cyzicus He undertook a second voyage down the west coast of Africa and was neverheard from again. Previous eudoxus of cnidus, Next Eugene III. http://www.factmonster.com/ce6/people/A0817825.html

Encyclopedia Eudoxus of Cyzicus [y OO u u s] Pronunciation Key Eudoxus of Cyzicus , fl. 130 B.C. Eudoxus of Cnidus Eugene III AD AD AD AD AD Print this page Cite this page Awards and Press Link to Fact Monster Add Fact Monster search ... Privacy

54. The Geocentric Hierarchy The geocentric school of astronomy began with eudoxus of cnidus (409356 BC), aneminent resident at the academy of Plato (427-347 BC), several decades after http://www.neo-tech.com/zero/part3.html

NT Home Page Neo-Tech Related Items Neo-Tech: The Philosophical Zero Search WWW Search neo-tech.com Neo-Tech Home Page The Ultimate Leverage for Riches Next Page Contents ... Previous Page 3. THE GEOCENTRIC HIERARCHY History reveals that around 200 B.C, less than five decades after Aristarchus' exquisite formulation of the celestial system, the geocentric concept of the universe, despite its inherent theoretical difficulty, became more and more adopted by the power structure of the Western world by the master neocheaters operating through their governments. The geocentric concept achieved prominence over the heliocentric system not because it was superior theoretically but because it was more expedient politically. It was not a scientific decision but a political strategy that made the geocentric system the "official" picture of the universe. The geocentric school of astronomy began with Eudoxus of Cnidus (409-356 B.C.), an eminent resident at the academy of Plato (427-347 B.C.), several decades after Philolaus had postulated his distinctively non-geocentric theory. Eudoxus' theory was further developed by Callipus (c. 325 B.C.), Aristotle (384-322 B.C.), Apollonius (c. 220 B.C.), Hipparchus (190-120 B.C.), and finally Claudius Ptolemy of Alexandria (A.D. 85-165).

55. Index Of Ancient Greek Mathematicians And Astronomers He is said to have invented a water clock with an alarm. eudoxus of cnidus (408360BC). Greek astronomer and mathematician. Belonged to the school of Plato. http://www.ics.forth.gr/~vsiris/ancient_greeks/classical_period.html

Classical period (5th - 4th century B.C.) Within this period Athens flourishes under Pericles, the Parthenon is built on the Acropolis, the tragedies of Sophocles, Aeschylus and Euripides are created, the phisolophical schools of Socrates and Plato (known as Academy) are established, and the Lyceum of Athens is founded by Aristotle. In science, the importance of the experimental method is accepted. Philosophers-Scientists Socrates (Athens, 470-399 B.C.). Died from poison after the state found him guilty for corrupting the youth. Theodorus of Cyrene (4th century B.C.). . Pythagorean. Plato's teacher in mathematics. Shows that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, and 17 are irrational. Archytas of Tarentum (420-350 B.C.). Greek mathematician, astronomer and engineer. Last of the Pythagorians. Plato and Eudoxus was his pupils. Built a series of toys, among them a mechanical pigeon propelled by a steam jet. Developed the theory for the pulley. Plato (Athens, 430-350 B.C.) . Greek philosopher. He was the founder of the Academy (named from the hero Academos owner of the grove where the Academy was built). Believed that mathematics played an important role in education. Disregarded practicality, a belief he passed to his students such as Eucledes. He started a three part trilogy :

56. Index Of Ancient Greek Scientists Wrote the History of Mathematics, which is lost however. eudoxus of cnidus (408360BC). Greek astronomer and mathematician. Belonged to the school of Plato. http://www.ics.forth.gr/~vsiris/ancient_greeks/whole_list.html

not complete Agatharchos. Greek mathematician. Discovered the laws of perspectives. Anaxagoras of Clazomenae (480-430 B.C.). Greek philosopher. Believed that a large number of seeds make up the properties of materials, that heavenly bodies are made up of the same materials as Earth and that the sun is a large, hot, glowing rock. Discovered that the moon reflected light and formulated the correct theory for the eclipses. Erroneously believed that the Earth was flat. Links: Anaxagoras of Clazomenae, MIT Anaximander (610-545 B.C.). Greek astronomer and philosopher, pupil of Thales. Introduced the apeiron (infinity). Formulated a theory of origin and evolution of life, according to which life originated in the sea from the moist element which evaporated from the sun ( On Nature ). Was the first to model the Earth according to scientific principles. According to him, the Earth was a cylinder with a north-south curvature, suspended freely in space, and the stars where attached to a sphere that rotated around Earth. Links: Anaximander, Internet Encyclopedia of Philosophy

57. 1Up Info > Eudoxus Of Cnidus (Astronomy, Biographies) - Encyclopedia You are here 1Up Info Encyclopedia Astronomy, Biographies Eudoxusof Cnidus, 1Up Info A Portal with a Difference. eudoxus of cnidus. http://www.1upinfo.com/encyclopedia/E/EudoxusCn.html

You are here 1Up Info Encyclopedia Astronomy, Biographies Eudoxus of Cnidus ... News Search 1Up Info ENCYCLOPEDIA Astronomy, Biographies Eudoxus of Cnidus Related Category: Astronomy, Biographies Eudoxus of Cnidus [y d s, n s] Pronunciation Key B.C. , Greek astronomer, mathematician, and physician. From the accounts of various ancient writers, he appears to have studied with Plato in Athens, spent some time in Heliopolis, Egypt, founded a school in Cyzicus, and spent his later years in Cnidus, where he had an observatory. It is claimed that he calculated the length of the solar year, indicating a calendar reform like that made later by Julius Caesar, and that he was the discoverer of some parts of geometry included in the work of Euclid. He was the first Greek astronomer to explain the movements of the planets in a scientific manner. His system involved a number of concentric spheres supporting the planets in their paths. Some scientists still held this belief at the time of Copernicus. Related Resources and Utilities AMAZON All Products Books Magazines Popular Music Classical Music Video DVD Electronics Software Outdoor Living Wireless Phones Computers Outlet Read articles on eLibrary: Encyclopedia Topics History People Places Content on this web site is provided for informational purposes only. We accept no responsibility for any loss, injury or inconvenience sustained by any person resulting from information published on this site. We encourage you to verify any critical information with the relevant authorities.

58. 1Up Info > Astronomy, Biographies - Encyclopedia Dreyer, Johan Ludwig Emil • Dyson, Sir Frank Watson • Eddington, Sir ArthurStanley • Encke, Johann Franz • eudoxus of cnidus • Flammarion, Camille http://www.1upinfo.com/encyclopedia/categories/astrobio.html

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59. Astronomers-Zoom Astronomy Glossary eudoxus of cnidus eudoxus of cnidus (408355 BC) was a Greek scholar (perhapsa student of Plato) who theorized that the Earth was at the center of the http://www.enchantedlearning.com/subjects/astronomy/glossary/Astronomers.shtml

EnchantedLearning.com is a user-supported site. Click here to learn more. Zoom Astronomy Astronomy Dictionary A B C D ... Z Click on an underlined word for more information on that subject. If the astronomy term you are looking for is not in the dictionary, please e-mail us and we'll add it. Zoom Astronomy HALL OF ASTRONOMERS An astronomer is someone who studies astronomy. The following night-owls are important astronomers, astrophysicists, mathematicians, and other scientists who have contributed greatly to our knowledge of the universe. ADAMS, JOHN C. John Couch Adams (1819-1892) was an English astronomer and mathematician who, at 24 years old, predicted the existence of the planet Neptune (Le Verrier also predicted its existence, independently). AIRY, GEORGE Sir George Bidell Airy (1801-1892) was the director of Greenwich Observatory/Astronomer Royal of England from 1835 to 1881. Airy installed a transit (a precise surveying device) at Greenwich, England, which was used to define the zero degree meridian of the Earth (zero-degrees longitude). A crater on Mars about 5 degrees south of the equator and on what is defined as Mars' prime meridian (zero-degrees longitude) is call Airy. A small crater within this crater (which is called Airy-0) is where the meridian line (zero-degrees longitude) crosses. A crater on the moon is also named for him (latitude 18.1 degrees, longitude 354.3 degrees, diameter 36 km). Airy is supposed to have stated incorrectly that Charles Babbage's new "analytical engine" (the predecessor of the computer) was "worthless," effectively ending Babbage's government funding.

60. The Planet Mars: A History Of Observation And Discovery. Chapter 1: Motions Of M One ingenious scheme was introduced by eudoxus of cnidus, a mathematician and contemporaryof Plato (indeed, he stayed two months in Athens as a pupil at the http://www.uapress.arizona.edu/online.bks/mars/chap01.htm

The Planet Mars: A History of Observation and Discovery William Sheehan Chapter 1 Motions of Mars For thousands of years it was only a blood-red dot among the starry host-a nameless denizen of the trackless night. Sometimes, when it veered closer to the Earth and shone like a burning coal in the darkness, it must have roused terror among primitive sky watchers, only to fade away into relative obscurity and be forgotten once more. By the time the Egyptians settled their civilization along the banks of the Nile, it had become familiar enough to receive a name-Har décher, the Red One. The Babylonians referred to it as Nergal, the Star of Death, and the Greeks too associated it with warfare and bloodshed-it was the Fiery One, or the war god, Ares-one and the same with the Roman god Mars. The Babylonians made careful astronomical observations and developed a sophisticated system of arithmetical computations for predicting astronomical phenomena such as eclipses. Their purposes were strictly calendrical and religious, however, and they never attempted to explain the reasons for any of the movements they observed. Superstition was widespread, and many astronomical events were regarded as ominous-not just eclipses but even the risings of Venus were viewed as omens. It is among the early Greeks that we must look for the first stirrings of a more rational perspective. They identified Mars as one of the five "wandering" stars, or planets, which move relative to the "fixed" stars. Two of the planets-Mercury and Venus-always remain close to the Sun in the sky; their distances never exceed 28° and 47°, respectively, and they may pass between the Earth and the Sun (inferior conjunction) or behind the Sun (superior conjunction). This behavior, as we now know, is due to the fact that their orbits lie inside that of the Earth.

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