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1784:, a Greek astronomer of the 4th century, added seven spheres to Eudoxus's original 27 (in addition to the planetary spheres, Eudoxus included a sphere for the fixed stars). Aristotle described both systems, but insisted on adding "unrolling" spheres between each set of spheres to cancel the motions of the outer set. Aristotle was concerned about the physical nature of the system; without unrollers, the outer motions would be transferred to the inner planets.
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1683:, another work by Aristotle. According to a story reported by Simplicius, Plato posed a question for Greek astronomers: "By the assumption of what uniform and orderly motions can the apparent motions of the planets be accounted for?" Plato proposed that the seemingly chaotic wandering motions of the planets could be explained by combinations of uniform circular motions centered on a spherical Earth, apparently a novel idea in the 4th century BC.
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1600:, astronomy was a branch of mathematics; astronomers sought to create geometrical models that could imitate the appearances of celestial motions. Identifying the astronomical work of Eudoxus as a separate category is therefore a modern convenience. Some of Eudoxus's astronomical texts whose names have survived include:
1768:, when a planet appears to slow down, then briefly reverse its motion through the zodiac. By inclining the axes of the two spheres with respect to each other, and rotating them in opposite directions but with equal periods, Eudoxus could make a point on the inner sphere trace out a figure-eight shape, or
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Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short
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Eudoxus's model of planetary motion. Each of his homocentric spheres is represented here as a ring which rotates on the axis shown. The outermost (yellow) sphere rotates once per day; the second (blue) describes the planet's motion through the zodiac; the third (green) and fourth (red) together move
528:
which was also used in a masterly way by
Archimedes in the following century. In applying the method, Eudoxus proved such mathematical statements as: areas of circles are to one another as the squares of their radii, volumes of spheres are to one another as the cubes of their radii, the volume of a
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Ancient Greek mathematicians calculated not with quantities and equations as we do today; instead, a proportionality expressed a relationship between geometric magnitudes. The ratio of two magnitudes was not a numerical value, as we think of it today; the ratio of two magnitudes was a primitive
567:
cannot be expressed as the ratio of two integers. This discovery had heralded the existence of incommensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole. For example,
560:. The change in focus by Eudoxus stimulated a divide in mathematics which lasted two thousand years. In combination with a Greek intellectual attitude unconcerned with practical problems, there followed a significant retreat from the development of techniques in arithmetic and algebra.
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A major flaw in the
Eudoxian system is its inability to explain changes in the brightness of planets as seen from Earth. Because the spheres are concentric, planets will always remain at the same distance from Earth. This problem was pointed out in Antiquity by
1720:
Animation depicting
Eudoxus's model of retrograde planetary motion. The two innermost homocentric spheres of his model are represented as rings here, each turning with the same period but in opposite directions, moving the planet along a figure-eight curve, or
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The complexity of the definition reflects the deep conceptual and methodological innovation involved. The
Eudoxian definition of proportionality uses the quantifier, "for every ..." to harness the infinite and the infinitesimal, similar to the modern
1712:
The Sun is also assigned three spheres. The second completes its motion in a year instead of a month. The inclusion of a third sphere implies that
Eudoxus mistakenly believed that the Sun had motion in latitude.
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All things, rational and irrational, aim at pleasure; things aim at what they believe to be good; a good indication of what the chief good is would be the thing that most things aim at.
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1989:
Nikolić, Milenko (1974). "The
Relation between Eudoxus' Theory of Proportions and Dedekind's Theory of Cuts". In Cohen, Robert S.; Stachel, John J.; Wartofsky, Marx W. (eds.).
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The
Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with the sides of the square; this is the famous discovery that the
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413:. He eventually returned to his native Cnidus, where he served in the city assembly. While in Cnidus, he built an observatory and continued writing and lecturing on
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emphasis on number and arithmetic, focusing instead on geometrical concepts as the basis of rigorous mathematics. Some
Pythagoreans, such as Eudoxus's teacher
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The third also completes its revolution in a month, but its axis is tilted at a slightly different angle, explaining motion in latitude (deviation from the
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Similarly, pleasure's opposite—pain—is universally avoided, which provides additional support for the idea that pleasure is universally considered good.
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Any other good that you can think of would be better if pleasure were added to it, and it is only by good that good can be increased.
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to describe and work with continuous geometrical entities such as lines, angles, areas and volumes, thereby avoiding the use of
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Of all of the things that are good, happiness is peculiar for not being praised, which may show that it is the crowning good.
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that define a real number by the set of rational numbers that are larger, equal or smaller than the number to be defined.
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556:, Eudoxus established what may have been the first deductive organization of mathematics on the basis of explicit
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with the same base and altitude, and the volume of a cone is one-third that of the corresponding cylinder.
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for 16 months, to pursue his study of astronomy and mathematics. From Egypt, he then traveled north to
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The second rotates eastward once in a month, explaining the monthly motion of the Moon through the
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and others in the 16th century, it became the basis for quantitative work in science, and inspired
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Eudoxus is credited with defining equality between two ratios, the subject of Book V of the
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In most modern reconstructions of the
Eudoxan model, the Moon is assigned three spheres:
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295:, but 20th century scholars found their choices contradictory and prefer a birth year of
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People don't seek pleasure as a means to something else, but as an end in its own right.
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VI.31) a simpler proof from similar triangles, which relies on ratios of line segments.
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Around 368 BC, Eudoxus returned to Athens with his students. According to some sources,
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2016:
1991:
For Dirk Struik: Scientific, Historical and
Political Essays in Honor of Dirk J. Struik
1962:
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Knorr, Wilbur Richard (1978). "Archimedes and the Pre-Euclidean Proportion Theory".
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1850:(a fairly recently discovered construction of the real numbers, named in his honor)
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the planet along a figure-eight curve (or hippopede) to explain retrograde motion.
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421:. He had one son, Aristagoras, and three daughters, Actis, Philtis, and Delphis.
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The outermost rotates westward once in 24 hours, explaining rising and setting.
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375:' lectures—then returned home to Cnidus. His friends then paid to send him to
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233:. All of his original works are lost, though some fragments are preserved in
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2100:(English 2nd ed.). London and Glasgow: Blackie & Son, Ltd. p.
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2406:, a documentary on Eudoxus, including a description of his planetary model
1945:
De Santillana, George (1940). "Eudoxus and Plato. A Study in Chronology".
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Eudoxos of Knidos (Eudoxus of Cnidus): astronomy and homocentric spheres
1641:, for Eudoxus's prose text was the basis for a poem of the same name by
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1646:
1568: in this section. Unsourced material may be challenged and removed.
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of, the latter equimultiples respectively taken in corresponding order.
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234:
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2454:, Craig McConnell, Ph.D., Cal State, Fullerton (archived 19 July 2011)
1966:
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In mathematical astronomy, his fame is due to the introduction of the
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447:: it allows rigorous treatment of continuous quantities and not just
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1627:, probably based on observations made by Eudoxus in Egypt and Cnidus
1613:(Ὀκταετηρίς), on an eight-year lunisolar-Venus cycle of the calendar
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1384:, and likewise for "equal" and "smaller". This can be compared with
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428:, and his early contributions to understanding the movement of the
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Hipparchi in Arati et Eudoxi Phaenomena Commentariorum Libri Tres
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and walking 7 miles (11 km) each way every day to attend the
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2040:. Oak Park, Illinois: Moore Publishing Company, Inc. p. 75.
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118:
2410:
Dennis Duke, "Statistical dating of the Phaenomena of Eudoxus",
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Eudoxus' definition depends on comparing the similar quantities
271:. The years of Eudoxus' birth and death are not fully known but
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Working model and complete explanation of the Eudoxus's Spheres
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409:) of the Academy during Plato's period in Syracuse, and taught
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267:(also transliterated Knidos), a city on the southwest coast of
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quoted from the text of Eudoxus in his commentary on Aratus.
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The second explains the planet's motion through the zodiac.
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536:
Eudoxus introduced the idea of non-quantified mathematical
395:. During his travels he gathered many students of his own.
180:
171:
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2232:(in German). Vol. 6.1. pp. 930–950 – via
509:. Eudoxus was probably the source for most of book V of
924:
of the first and third; likewise form the equimultiples
607:, this can be made more explicit. Given four quantities
572:
provides an elaborate proof of the Pythagorean theorem (
387:, located on the south shore of the Sea of Marmara, the
302:. His name Eudoxus means "honored" or "of good repute" (
576:
I.47), by using addition of areas and only much later (
2447:
Herodotus Project: Extensive B+W photo essay of Cnidus
2229:
Realencyclopädie der classischen Altertumswissenschaft
5242:
2443:
Henry Mendell, Cal State U, LA (archived 16 May 2011)
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Eudoxus is considered by some to be the greatest of
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This particular argument is referenced in Book One.
355:. At the age of 23, he traveled with the physician
162:
16:
Greek astronomer and mathematician (c.390–c.340 BC)
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1926:
1637:We are fairly well informed about the contents of
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275:gave several biographical details, mentioned that
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2437:Donald Allen, Professor, Texas A&M University
2077:Mathematical Thought from Ancient to Modern Times
2054:
1904:
5362:
2269:Archives Internationales d'Histoire des Sciences
2247:the Dictionary of Scientific Biography, volume 4
1817:, attributes to Eudoxus an argument in favor of
316:"opinion, belief, fame", analogous to the Latin
263:Eudoxus, son of Aeschines, was born and died in
2370:
1776:
1530:Book V, was credited to Eudoxus by Archimedes.
2340:(Two volume ed.). Loeb Classical Library.
1352:are the same as the ones that are larger than
359:—who was his patron and possibly his lover—to
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3978:
3470:
2483:
2192:The History and Practice of Ancient Astronomy
2172:A Short Account of the History of Mathematics
1944:
984:of the second and fourth. If it happens that
813:, can be defined by the following condition:
703:, take the ratio of the first to the second,
735:, and the ratio of the third to the fourth,
594:In Definition 5 of Euclid's Book V we read:
225:, doctor, and lawmaker. He was a student of
3484:
2354:A history of ancient mathematical astronomy
2286:The Ancient tradition of geometric problems
2226:. In Pauly, August; Wissowa, Georg (eds.).
1792:. Astronomers responded by introducing the
1652:
3985:
3971:
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325:According to Diogenes Laërtius, crediting
2254:Huxley, G. L. (1963). "Eudoxian Topics".
2096:Theory and Application of Infinite Series
2082:
1993:. Dordrecht: Springer. pp. 225–243.
1584:Learn how and when to remove this message
2512:
2320:
2119:Early Greek Science: Thales to Aristotle
2071:
2069:
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1758:The outermost explains the daily motion.
1724:
1715:
816:For any two arbitrary positive integers
767:. That the two ratios are proportional,
2216:
2079:Oxford University Press, 1972 pp. 48–50
2029:
1988:
1932:
1915:MacTutor History of Mathematics Archive
1898:
5363:
5014:Latin translations of the 12th century
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2240:
1764:The third and fourth together explain
367:. He spent two months there—living in
4744:Straightedge and compass construction
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529:pyramid is one-third the volume of a
4709:Incircle and excircles of a triangle
2309:Die Fragmente des Eudoxos von Knidos
2175:(4th ed.). Dover Publications.
2165:
2060:
1893:Die Fragmente des Eudoxos von Knidos
1566:adding citations to reliable sources
1537:
1227:{\displaystyle m\cdot c<n\cdot d}
1185:{\displaystyle m\cdot a<n\cdot b}
1059:{\displaystyle m\cdot c>n\cdot d}
1017:{\displaystyle m\cdot a>n\cdot b}
391:. He traveled south to the court of
2464:applet) (archived 21 November 2007)
2256:Greek, Roman, and Byzantine Studies
335:, Eudoxus studied mathematics with
255:may be based on a work by Eudoxus.
13:
2452:Models of Planetary Motion—Eudoxus
2375:(5th ed.). Leiden: Noordhoff.
1754:) are assigned four spheres each:
14:
5432:
5386:4th-century BC Greek philosophers
2458:The Universe According to Eudoxus
2380:
2333:Lives of the Eminent Philosophers
1663:A general idea of the content of
1143:{\displaystyle m\cdot c=n\cdot d}
1101:{\displaystyle m\cdot a=n\cdot b}
239:Commentaries on the Phenomena of
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5381:4th-century BC Greek physicians
3099:Pseudo-Dionysius the Areopagite
2371:Van der Waerden, B. L. (1988).
2336:. Vol. 2:8. Translated by
2159:
2146:
2137:
2108:
1553:needs additional citations for
363:to study with the followers of
5046:A History of Greek Mathematics
4559:The Quadrature of the Parabola
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1982:
1973:
1938:
1885:
1876:
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1:
5416:4th-century BC mathematicians
3875:Inferior and superior planets
3232:Giovanni Pico della Mirandola
2327:"Pythagoreans: Eudoxus"
2021:: CS1 maint: date and year (
1869:
1451:, and the similar quantities
544:. In doing so, he reversed a
399:
296:
204:
72:
39:
5396:Ancient Greek mathematicians
4827:Intersecting secants theorem
1999:10.1007/978-94-010-2115-9_19
1777:Importance of Eudoxan system
1675:XII, 8, and a commentary by
1533:
584:relationship between them.
347:) and studied medicine with
7:
4822:Intersecting chords theorem
4689:Doctrine of proportionality
2356:. Berlin: Springer-Verlag.
2194:. Oxford University Press.
2123:. W.W. Norton. p. 84.
1841:
516:. He rigorously developed
501:mathematicians, and in all
489:) is also named after him.
481:are named in his honor. An
10:
5437:
5421:People from Muğla Province
4518:On the Sphere and Cylinder
4471:On the Sizes and Distances
3952:Medieval Islamic astronomy
3749:On the Sizes and Distances
2223:"Eudoxos Von Knidos"
2167:Ball, Walter William Rouse
1979:Diogenes Laertius; VIII.87
1891:Lasserre, François (1966)
1882:Diogenes Laertius; VIII.86
1859:Incommensurable magnitudes
1734:The five visible planets (
1656:
554:incommensurable quantities
304:
196:
18:
5391:Ancient Greek astronomers
5220:Ancient Greece portal
5209:
5159:
5037:
5024:Philosophy of mathematics
4994:
4987:
4961:
4939:Ptolemy's table of chords
4883:
4865:
4764:
4757:
4613:
4575:
4392:
4000:
3994:Ancient Greek mathematics
3942:Medieval European science
3934:
3913:
3822:
3771:
3710:
3672:Sosigenes the Peripatetic
3492:
3415:
3329:
3322:
3258:
3198:
3191:
3115:
2861:
2753:
2707:Metrodorus of Stratonicea
2689:
2649:
2640:
2521:
2505:
2036:Calinger, Ronald (1982).
1803:
1704:), and the motion of the
1605:Disappearances of the Sun
1519:of limit and continuity.
1517:epsilon-delta definitions
864:, form the equimultiples
455:. When it was revived by
147:
106:
99:
85:
68:
35:
28:
4891:Aristarchus's inequality
4464:On Conoids and Spheroids
2903:Eustathius of Cappadocia
1920:University of St Andrews
1653:Eudoxan planetary models
1502:{\displaystyle n\cdot d}
1472:{\displaystyle m\cdot c}
1442:{\displaystyle n\cdot b}
1412:{\displaystyle m\cdot a}
975:{\displaystyle n\cdot d}
945:{\displaystyle n\cdot b}
915:{\displaystyle m\cdot c}
885:{\displaystyle m\cdot a}
19:Not to be confused with
5401:Ancient Greek ethicists
4999:Ancient Greek astronomy
4812:Inscribed angle theorem
4802:Greek geometric algebra
4457:Measurement of a Circle
3667:Sosigenes of Alexandria
3486:Ancient Greek astronomy
2968:Gaius Marius Victorinus
2563:Demetrius of Amphipolis
2352:Neugebauer, O. (1975).
2038:Classics of Mathematics
1276:{\displaystyle a/b=c/d}
804:{\displaystyle a/b=c/d}
258:
5233:Mathematics portal
5019:Non-Euclidean geometry
4974:Mouseion of Alexandria
4847:Tangent-secant theorem
4797:Geometric mean theorem
4782:Exterior angle theorem
4777:Angle bisector theorem
4481:On Sizes and Distances
3739:On Sizes and Distances
2587:Lastheneia of Mantinea
2582:Hestiaeus of Perinthus
2289:. Boston: Birkhäuser.
1731:
1722:
1633:, on planetary motions
1607:, possibly on eclipses
1503:
1473:
1443:
1413:
1376:
1344:
1312:
1277:
1228:
1186:
1144:
1102:
1060:
1018:
976:
946:
916:
886:
856:
832:
805:
759:
727:
695:
671:
647:
623:
601:
524:, a precursor to the
253:Theodosius of Bithynia
5376:4th-century BC deaths
5371:4th-century BC births
4921:Pappus's area theorem
4857:Theorem of the gnomon
4734:Quadratrix of Hippias
4657:Circles of Apollonius
4605:Problem of Apollonius
4583:Constructible numbers
4407:Archimedes Palimpsest
3850:Deferent and epicycle
3779:Antikythera mechanism
3398:Nicholas Wolterstorff
3168:Theodoric of Freiberg
2838:Clement of Alexandria
2768:Eudorus of Alexandria
2722:Aeschines of Neapolis
2190:Evans, James (1998).
1794:deferent and epicycle
1728:
1719:
1677:Simplicius of Cilicia
1504:
1474:
1444:
1414:
1377:
1345:
1320:that are larger than
1313:
1278:
1229:
1187:
1145:
1103:
1061:
1019:
977:
947:
917:
887:
857:
833:
806:
760:
728:
696:
672:
648:
624:
596:
405:he assumed headship (
5137:prehistoric counting
4934:Ptolemy's inequality
4875:Apollonius's theorem
4714:Method of exhaustion
4684:Diophantine equation
4674:Circumscribed circle
4491:On the Moving Sphere
3921:Babylonian astronomy
3612:Hippocrates of Chios
3178:Berthold of Moosburg
3123:John Scotus Eriugena
3094:David the Invincible
2803:Alexander Peloplaton
2344:Manitius, C. (1894)
2311:(de Gruyter: Berlin)
2143:Largely in Book Ten.
1907:Robertson, Edmund F.
1895:(de Gruyter: Berlin)
1679:(6th century AD) on
1667:can be gleaned from
1562:improve this article
1524:Archimedean property
1487:
1457:
1427:
1397:
1358:
1326:
1294:
1245:
1200:
1158:
1116:
1074:
1032:
990:
960:
930:
900:
870:
846:
822:
773:
741:
709:
685:
661:
637:
613:
522:method of exhaustion
279:said he reached his
5223: •
5029:Neusis construction
4949:Spiral of Theodorus
4842:Pythagorean theorem
4787:Euclidean algorithm
4729:Lune of Hippocrates
4598:Squaring the circle
4354:Theon of Alexandria
4029:Aristaeus the Elder
3692:Theon of Alexandria
3299:Ralph Waldo Emerson
2773:Philo of Alexandria
2612:Menedemus of Pyrrha
2607:Heraclides Ponticus
2592:Timolaus of Cyzicus
2568:Euaeon of Lampsacus
2397:Eudoxus (and Plato)
2241:Huxley, GL (1980).
2115:Lloyd, GER (1970).
1910:"Eudoxus of Cnidus"
1905:O'Connor, John J.;
1790:Autolycus of Pitane
1625:spherical astronomy
1375:{\displaystyle c/d}
1343:{\displaystyle a/b}
1311:{\displaystyle n/m}
758:{\displaystyle c/d}
726:{\displaystyle a/b}
439:shows insight into
4916:Menelaus's theorem
4906:Irrational numbers
4719:Parallel postulate
4694:Euclidean geometry
4662:Apollonian circles
4204:Isidore of Miletus
3926:Egyptian astronomy
3840:Circle of latitude
3294:Emanuel Swedenborg
3227:Cristoforo Landino
3209:Florentine Academy
2999:Plutarch of Athens
2948:Eusebius of Myndus
2943:Maximus of Ephesus
2928:Theodorus of Asine
2823:Numenius of Apamea
2602:Axiothea of Phlius
2558:Erastus of Scepsis
2433:1997-07-23 at the
2402:2018-08-16 at the
2338:Hicks, Robert Drew
2322:Laërtius, Diogenes
2303:Lasserre, François
2243:Eudoxus of Cnidus
2218:Hultsch, Friedrich
1814:Nicomachean Ethics
1732:
1723:
1659:Concentric spheres
1526:, definition 4 of
1499:
1469:
1439:
1409:
1372:
1340:
1308:
1273:
1224:
1182:
1140:
1098:
1056:
1014:
972:
942:
912:
882:
852:
828:
801:
755:
723:
691:
667:
643:
619:
542:irrational numbers
487:Kampyle of Eudoxus
441:irrational numbers
426:concentric spheres
201:Eúdoxos ho Knídios
94:Concentric spheres
90:Kampyle of Eudoxus
21:Eudoxus of Cyzicus
5411:Students of Plato
5240:
5239:
5205:
5204:
4957:
4956:
4944:Ptolemy's theorem
4817:Intercept theorem
4667:Apollonian gasket
4593:Doubling the cube
4566:The Sand Reckoner
3960:
3959:
3835:Celestial spheres
3452:
3451:
3448:
3447:
3393:Peter van Inwagen
3358:Roderick Chisholm
3284:
3283:
3240:
3239:
3111:
3110:
3107:
3106:
2953:Priscus of Epirus
2755:Middle Platonists
2749:
2748:
2745:
2744:
2737:Dio of Alexandria
2662:Diocles of Cnidus
2428:Eudoxus of Cnidus
2422:Eudoxus of Cnidus
2417:see pages 7 to 23
2373:Science Awakening
2008:978-90-277-0379-8
1800:is considerable.
1594:
1593:
1586:
855:{\displaystyle n}
831:{\displaystyle m}
694:{\displaystyle d}
670:{\displaystyle c}
646:{\displaystyle b}
622:{\displaystyle a}
526:integral calculus
417:, astronomy, and
273:Diogenes Laërtius
197:Εὔδοξος ὁ Κνίδιος
154:Eudoxus of Cnidus
151:
150:
101:Scientific career
30:Eudoxus of Cnidus
5428:
5406:Ancient Cnidians
5353:
5352:
5341:
5340:
5339:
5329:
5328:
5327:
5317:
5316:
5315:
5305:
5304:
5293:
5292:
5291:
5281:
5280:
5269:
5268:
5267:
5257:
5256:
5255:
5248:
5231:
5230:
5218:
5217:
5216:
4992:
4991:
4979:Platonic Academy
4926:Problem II.8 of
4896:Crossbar theorem
4852:Thales's theorem
4792:Euclid's theorem
4762:
4761:
4679:Commensurability
4640:Axiomatic system
4588:Angle trisection
4553:
4543:
4505:
4495:
4485:
4475:
4451:
4441:
4424:
3987:
3980:
3973:
3964:
3963:
3947:Indian astronomy
3900:Sublunary sphere
3870:Hipparchic cycle
3809:Mural instrument
3784:Armillary sphere
3763:
3753:
3743:
3733:
3723:
3479:
3472:
3465:
3456:
3455:
3327:
3326:
3205:
3204:
3196:
3195:
3044:Ammonius Hermiae
2989:
2988:
2848:Origen the Pagan
2727:Philo of Larissa
2702:Hagnon of Tarsus
2647:
2646:
2632:Crates of Athens
2519:
2518:
2510:
2509:
2492:
2485:
2478:
2469:
2468:
2376:
2367:
2341:
2329:
2318:
2317:
2300:
2281:Knorr, Wilbur R.
2276:
2263:
2250:
2237:
2225:
2213:
2186:
2153:
2150:
2144:
2141:
2135:
2134:
2122:
2112:
2106:
2105:
2099:
2086:
2080:
2073:
2064:
2058:
2052:
2051:
2033:
2027:
2026:
2020:
2012:
1986:
1980:
1977:
1971:
1970:
1942:
1936:
1930:
1924:
1923:
1902:
1896:
1889:
1883:
1880:
1619:(Φαινόμενα) and
1589:
1582:
1578:
1575:
1569:
1546:
1538:
1510:
1508:
1506:
1505:
1500:
1480:
1478:
1476:
1475:
1470:
1450:
1448:
1446:
1445:
1440:
1420:
1418:
1416:
1415:
1410:
1383:
1381:
1379:
1378:
1373:
1368:
1351:
1349:
1347:
1346:
1341:
1336:
1319:
1317:
1315:
1314:
1309:
1304:
1284:
1282:
1280:
1279:
1274:
1269:
1255:
1239:This means that
1235:
1233:
1231:
1230:
1225:
1193:
1191:
1189:
1188:
1183:
1151:
1149:
1147:
1146:
1141:
1109:
1107:
1105:
1104:
1099:
1067:
1065:
1063:
1062:
1057:
1025:
1023:
1021:
1020:
1015:
983:
981:
979:
978:
973:
953:
951:
949:
948:
943:
923:
921:
919:
918:
913:
893:
891:
889:
888:
883:
863:
861:
859:
858:
853:
839:
837:
835:
834:
829:
812:
810:
808:
807:
802:
797:
783:
766:
764:
762:
761:
756:
751:
734:
732:
730:
729:
724:
719:
702:
700:
698:
697:
692:
678:
676:
674:
673:
668:
654:
652:
650:
649:
644:
630:
628:
626:
625:
620:
565:square root of 2
461:Richard Dedekind
453:rational numbers
445:linear continuum
404:
401:
307:
306:
301:
298:
294:
290:
213:
209:
206:
198:
190:
189:
186:
185:
182:
179:
176:
173:
170:
167:
164:
80:Cnidus, Anatolia
77:
74:
44:
41:
26:
25:
5436:
5435:
5431:
5430:
5429:
5427:
5426:
5425:
5361:
5360:
5359:
5347:
5337:
5335:
5325:
5323:
5313:
5311:
5299:
5289:
5287:
5275:
5265:
5263:
5253:
5251:
5243:
5241:
5236:
5225:
5214:
5212:
5201:
5167:Arabian/Islamic
5155:
5144:numeral systems
5033:
4983:
4953:
4901:Heron's formula
4879:
4861:
4753:
4749:Triangle center
4739:Regular polygon
4616:and definitions
4615:
4609:
4571:
4551:
4541:
4503:
4493:
4483:
4473:
4449:
4439:
4422:
4388:
4359:Theon of Smyrna
4004:
3996:
3991:
3961:
3956:
3930:
3909:
3895:Spherical Earth
3830:Callippic cycle
3818:
3799:Equatorial ring
3767:
3761:
3751:
3741:
3731:
3721:
3706:
3697:Theon of Smyrna
3488:
3483:
3453:
3444:
3411:
3408:Edward N. Zalta
3388:Alvin Plantinga
3383:Jan Łukasiewicz
3363:Michael Dummett
3318:
3309:Bernard Bolzano
3280:
3254:
3236:
3222:Marsilio Ficino
3187:
3173:Meister Eckhart
3103:
3084:John Philoponus
2987:
2857:
2828:Ammonius Saccas
2818:Maximus of Tyre
2741:
2685:
2636:
2577:Python of Aenus
2501:
2496:
2435:Wayback Machine
2404:Wayback Machine
2383:
2364:
2315:
2297:
2202:
2183:
2162:
2157:
2156:
2151:
2147:
2142:
2138:
2131:
2113:
2109:
2087:
2083:
2074:
2067:
2059:
2055:
2048:
2034:
2030:
2014:
2013:
2009:
1987:
1983:
1978:
1974:
1943:
1939:
1931:
1927:
1903:
1899:
1890:
1886:
1881:
1877:
1872:
1844:
1806:
1798:Greek astronomy
1779:
1661:
1655:
1623:(Ἔνοπτρον), on
1590:
1579:
1573:
1570:
1559:
1547:
1536:
1488:
1485:
1484:
1482:
1458:
1455:
1454:
1452:
1428:
1425:
1424:
1422:
1398:
1395:
1394:
1392:
1364:
1359:
1356:
1355:
1353:
1332:
1327:
1324:
1323:
1321:
1300:
1295:
1292:
1291:
1289:
1265:
1251:
1246:
1243:
1242:
1240:
1201:
1198:
1197:
1195:
1159:
1156:
1155:
1153:
1117:
1114:
1113:
1111:
1075:
1072:
1071:
1069:
1033:
1030:
1029:
1027:
991:
988:
987:
985:
961:
958:
957:
955:
931:
928:
927:
925:
901:
898:
897:
895:
871:
868:
867:
865:
847:
844:
843:
841:
823:
820:
819:
817:
793:
779:
774:
771:
770:
768:
747:
742:
739:
738:
736:
715:
710:
707:
706:
704:
686:
683:
682:
680:
662:
659:
658:
656:
638:
635:
634:
632:
614:
611:
610:
608:
605:modern notation
505:second only to
499:classical Greek
495:
483:algebraic curve
463:'s work on the
402:
299:
292:
288:
261:
211:
207:
161:
157:
143:
92:
81:
78:
75:
64:
54:
45:
42:
31:
24:
17:
12:
11:
5:
5434:
5424:
5423:
5418:
5413:
5408:
5403:
5398:
5393:
5388:
5383:
5378:
5373:
5358:
5357:
5345:
5333:
5321:
5309:
5297:
5285:
5273:
5261:
5238:
5237:
5210:
5207:
5206:
5203:
5202:
5200:
5199:
5194:
5189:
5184:
5179:
5174:
5169:
5163:
5161:
5160:Other cultures
5157:
5156:
5154:
5153:
5152:
5151:
5141:
5140:
5139:
5129:
5128:
5127:
5117:
5116:
5115:
5105:
5104:
5103:
5093:
5092:
5091:
5081:
5080:
5079:
5069:
5068:
5067:
5057:
5056:
5055:
5041:
5039:
5035:
5034:
5032:
5031:
5026:
5021:
5016:
5011:
5009:Greek numerals
5006:
5004:Attic numerals
5001:
4995:
4989:
4985:
4984:
4982:
4981:
4976:
4971:
4965:
4963:
4959:
4958:
4955:
4954:
4952:
4951:
4946:
4941:
4936:
4931:
4923:
4918:
4913:
4908:
4903:
4898:
4893:
4887:
4885:
4881:
4880:
4878:
4877:
4871:
4869:
4863:
4862:
4860:
4859:
4854:
4849:
4844:
4839:
4834:
4832:Law of cosines
4829:
4824:
4819:
4814:
4809:
4804:
4799:
4794:
4789:
4784:
4779:
4773:
4771:
4759:
4755:
4754:
4752:
4751:
4746:
4741:
4736:
4731:
4726:
4724:Platonic solid
4721:
4716:
4711:
4706:
4704:Greek numerals
4701:
4696:
4691:
4686:
4681:
4676:
4671:
4670:
4669:
4664:
4654:
4649:
4648:
4647:
4637:
4636:
4635:
4630:
4619:
4617:
4611:
4610:
4608:
4607:
4602:
4601:
4600:
4595:
4590:
4579:
4577:
4573:
4572:
4570:
4569:
4562:
4555:
4545:
4535:
4532:Planisphaerium
4528:
4521:
4514:
4507:
4497:
4487:
4477:
4467:
4460:
4453:
4443:
4433:
4426:
4416:
4409:
4404:
4396:
4394:
4390:
4389:
4387:
4386:
4381:
4376:
4371:
4366:
4361:
4356:
4351:
4346:
4341:
4336:
4331:
4326:
4321:
4316:
4311:
4306:
4301:
4296:
4291:
4286:
4281:
4276:
4271:
4266:
4261:
4256:
4251:
4246:
4241:
4236:
4231:
4226:
4221:
4216:
4211:
4206:
4201:
4196:
4191:
4186:
4181:
4176:
4171:
4166:
4161:
4156:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4111:
4106:
4101:
4096:
4091:
4086:
4081:
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4056:
4051:
4046:
4041:
4036:
4031:
4026:
4021:
4016:
4010:
4008:
4002:Mathematicians
3998:
3997:
3990:
3989:
3982:
3975:
3967:
3958:
3957:
3955:
3954:
3949:
3944:
3938:
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3923:
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3801:
3796:
3791:
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3781:
3775:
3773:
3769:
3768:
3766:
3765:
3759:On the Heavens
3755:
3745:
3735:
3732:(Eratosthenes)
3725:
3714:
3712:
3708:
3707:
3705:
3704:
3699:
3694:
3689:
3684:
3679:
3674:
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3654:
3649:
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3639:
3637:Philip of Opus
3634:
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3459:
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3449:
3446:
3445:
3443:
3442:
3437:
3435:Roman Ingarden
3432:
3430:Edmund Husserl
3427:
3421:
3419:
3413:
3412:
3410:
3405:
3403:Crispin Wright
3400:
3395:
3390:
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3375:
3370:
3368:W. V. O. Quine
3365:
3360:
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3333:
3324:
3320:
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3311:
3306:
3301:
3296:
3291:
3285:
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3281:
3279:
3274:
3269:
3267:Ralph Cudworth
3264:
3262:
3256:
3255:
3253:
3252:
3247:
3245:Giordano Bruno
3241:
3238:
3237:
3235:
3234:
3229:
3224:
3219:
3213:
3211:
3202:
3193:
3189:
3188:
3186:
3185:
3183:Paul of Venice
3180:
3175:
3170:
3165:
3160:
3158:Henry of Ghent
3155:
3150:
3145:
3140:
3135:
3130:
3125:
3119:
3117:
3113:
3112:
3109:
3108:
3105:
3104:
3102:
3101:
3096:
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3076:
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2609:
2604:
2599:
2594:
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2584:
2579:
2570:
2565:
2560:
2551:
2546:
2544:Philip of Opus
2541:
2536:
2531:
2525:
2523:
2516:
2507:
2503:
2502:
2495:
2494:
2487:
2480:
2472:
2466:
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2455:
2449:
2444:
2438:
2425:
2424:Britannica.com
2419:
2407:
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2382:
2381:External links
2379:
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2107:
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2075:Morris Kline,
2065:
2053:
2046:
2028:
2007:
1981:
1972:
1959:10.1086/347693
1953:(2): 248–262.
1937:
1925:
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1854:Delian problem
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1778:
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1773:
1766:retrogradation
1762:
1759:
1710:
1709:
1698:
1691:
1657:Main article:
1654:
1651:
1635:
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1628:
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1598:ancient Greece
1592:
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1574:September 2022
1550:
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1371:
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1307:
1303:
1299:
1286:if and only if
1272:
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1254:
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1214:
1211:
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1166:
1163:
1152:. Finally, if
1139:
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260:
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136:
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97:
96:
87:
86:Known for
83:
82:
79:
70:
66:
65:
55:(now Yazıköy,
46:
37:
33:
32:
29:
15:
9:
6:
4:
3:
2:
5433:
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5142:
5138:
5135:
5134:
5133:
5130:
5126:
5123:
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5121:
5118:
5114:
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5110:
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5102:
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5017:
5015:
5012:
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5007:
5005:
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4986:
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4864:
4858:
4855:
4853:
4850:
4848:
4845:
4843:
4840:
4838:
4837:Pons asinorum
4835:
4833:
4830:
4828:
4825:
4823:
4820:
4818:
4815:
4813:
4810:
4808:
4807:Hinge theorem
4805:
4803:
4800:
4798:
4795:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4775:
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4710:
4707:
4705:
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4697:
4695:
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4680:
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4502:
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4476:
4474:(Aristarchus)
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4459:
4458:
4454:
4452:
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4408:
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4379:Zeno of Sidon
4377:
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4370:
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4362:
4360:
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3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3880:Metonic cycle
3878:
3876:
3873:
3871:
3868:
3866:
3865:Heliocentrism
3863:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3845:Counter-Earth
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3827:
3825:
3821:
3815:
3812:
3810:
3807:
3805:
3802:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3782:
3780:
3777:
3776:
3774:
3770:
3764:
3760:
3756:
3754:
3752:(Aristarchus)
3750:
3746:
3744:
3740:
3736:
3734:
3730:
3726:
3724:
3720:
3716:
3715:
3713:
3709:
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3700:
3698:
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3648:
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3640:
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3618:
3615:
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3600:
3598:
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3560:
3558:
3555:
3553:
3550:
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3535:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3505:
3503:
3500:
3499:
3497:
3495:
3491:
3487:
3480:
3475:
3473:
3468:
3466:
3461:
3460:
3457:
3441:
3438:
3436:
3433:
3431:
3428:
3426:
3425:Henri Bergson
3423:
3422:
3420:
3418:
3414:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3353:Alonzo Church
3351:
3349:
3346:
3344:
3341:
3339:
3338:Gottlob Frege
3336:
3334:
3332:
3328:
3325:
3321:
3315:
3314:Aleksei Losev
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3295:
3292:
3290:
3289:Thomas Taylor
3287:
3286:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3261:
3257:
3251:
3250:Blaise Pascal
3248:
3246:
3243:
3242:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3214:
3212:
3210:
3206:
3203:
3201:
3197:
3194:
3190:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3138:Peter Abelard
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3120:
3118:
3114:
3100:
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3049:Asclepiodotus
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2996:
2994:
2990:
2984:
2981:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2877:
2874:
2873:
2872:
2869:
2868:
2866:
2864:
2863:Neoplatonists
2860:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2783:Justin Martyr
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2760:
2758:
2756:
2752:
2738:
2735:
2733:
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2713:
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2374:
2369:
2365:
2363:0-387-06995-X
2359:
2355:
2350:
2347:
2343:
2339:
2335:
2334:
2328:
2323:
2313:
2310:
2306:
2302:
2298:
2296:0-8176-3148-8
2292:
2288:
2287:
2282:
2278:
2274:
2270:
2265:
2261:
2257:
2252:
2248:
2244:
2239:
2235:
2231:
2230:
2224:
2219:
2215:
2211:
2207:
2203:
2201:0-19-509539-1
2197:
2193:
2188:
2184:
2182:9780486206301
2178:
2174:
2173:
2168:
2164:
2163:
2149:
2140:
2132:
2130:9780393005837
2126:
2121:
2120:
2111:
2103:
2098:
2097:
2091:
2090:Knopp, Konrad
2085:
2078:
2072:
2070:
2063:, p. 54.
2062:
2057:
2049:
2047:0-935610-13-8
2043:
2039:
2032:
2024:
2018:
2010:
2004:
2000:
1996:
1992:
1985:
1976:
1968:
1964:
1960:
1956:
1952:
1948:
1941:
1934:
1929:
1921:
1917:
1916:
1911:
1908:
1901:
1894:
1888:
1879:
1875:
1865:
1862:
1860:
1857:
1855:
1852:
1849:
1848:Eudoxus reals
1846:
1845:
1836:
1833:
1830:
1827:
1824:
1823:
1822:
1820:
1816:
1815:
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1622:
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1612:
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1606:
1603:
1602:
1601:
1599:
1588:
1585:
1577:
1567:
1563:
1557:
1556:
1551:This section
1549:
1545:
1540:
1539:
1531:
1529:
1525:
1520:
1518:
1512:
1496:
1493:
1490:
1466:
1463:
1460:
1436:
1433:
1430:
1406:
1403:
1400:
1389:
1387:
1386:Dedekind cuts
1369:
1365:
1361:
1337:
1333:
1329:
1305:
1301:
1297:
1287:
1270:
1266:
1262:
1259:
1256:
1252:
1248:
1237:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1068:. If instead
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1011:
1008:
1005:
1002:
999:
996:
993:
969:
966:
963:
939:
936:
933:
909:
906:
903:
879:
876:
873:
849:
825:
814:
798:
794:
790:
787:
784:
780:
776:
752:
748:
744:
720:
716:
712:
688:
664:
640:
616:
606:
600:
595:
592:
590:
585:
581:
579:
575:
571:
566:
561:
559:
555:
551:
547:
543:
539:
534:
532:
527:
523:
519:
515:
512:
508:
504:
500:
490:
488:
484:
480:
476:
472:
468:
466:
462:
458:
454:
450:
449:whole numbers
446:
442:
438:
433:
431:
427:
422:
420:
416:
412:
408:
396:
394:
390:
386:
382:
378:
374:
370:
366:
362:
358:
354:
350:
346:
345:Magna Graecia
342:
338:
334:
333:
328:
323:
321:
320:
315:
311:
300: 390 BC
286:
283:in the 103rd
282:
278:
274:
270:
266:
256:
254:
250:
249:
244:
242:
236:
232:
228:
224:
223:mathematician
220:
217:
216:ancient Greek
212: 340 BC
202:
194:
193:Ancient Greek
188:
155:
146:
140:
137:
135:
132:
130:
127:
125:
122:
120:
117:
115:
112:
111:
109:
105:
102:
98:
95:
91:
88:
84:
76: 340 BC
71:
67:
62:
58:
53:
49:
43: 390 BC
38:
34:
27:
22:
5343:Solar System
5224:
5211:
5053:Thomas Heath
5044:
4927:
4911:Law of sines
4767:
4699:Golden ratio
4564:
4557:
4548:
4542:(Theodosius)
4538:
4530:
4523:
4516:
4509:
4500:
4490:
4484:(Hipparchus)
4480:
4470:
4462:
4455:
4446:
4436:
4428:
4423:(Apollonius)
4419:
4411:
4399:
4374:Zeno of Elea
4148:
4134:Eratosthenes
4124:Dionysodorus
3758:
3748:
3742:(Hipparchus)
3738:
3729:Catasterismi
3728:
3718:
3586:
3577:Eratosthenes
3373:David Kaplan
3323:Contemporary
3304:Josiah Royce
3089:Olympiodorus
3004:Asclepigenia
2923:Chrysanthius
2538:
2416:
2411:
2372:
2353:
2345:
2331:
2308:
2304:
2285:
2272:
2268:
2259:
2255:
2246:
2245:p. 465-7 in:
2242:
2227:
2191:
2171:
2160:Bibliography
2148:
2139:
2118:
2110:
2095:
2084:
2076:
2056:
2037:
2031:
1990:
1984:
1975:
1950:
1946:
1940:
1933:Hultsch 1907
1928:
1913:
1900:
1892:
1887:
1878:
1812:
1807:
1786:
1780:
1733:
1711:
1685:
1680:
1672:
1664:
1662:
1638:
1636:
1630:
1620:
1616:
1610:
1604:
1595:
1580:
1571:
1560:Please help
1555:verification
1552:
1527:
1521:
1513:
1390:
1238:
1194:, then also
1110:, then also
1026:, then also
815:
602:
597:
593:
588:
586:
582:
577:
573:
562:
535:
496:
469:
465:real numbers
435:His work on
434:
423:
397:
330:
324:
317:
313:
309:
262:
246:
238:
200:
153:
152:
100:
5331:Outer space
5319:Spaceflight
5283:Mathematics
5120:mathematics
4928:Arithmetica
4525:Ostomachion
4494:(Autolycus)
4413:Arithmetica
4189:Hippocrates
4119:Dinostratus
4104:Dicaearchus
4034:Aristarchus
3860:Geocentrism
3772:Instruments
3762:(Aristotle)
3567:Cleostratus
3532:Aristarchus
3512:Anaximander
3494:Astronomers
3440:Leo Strauss
3417:Continental
3378:Saul Kripke
3343:G. E. Moore
3277:Anne Conway
3200:Renaissance
3163:Bonaventure
2712:Clitomachus
2549:Aristonymus
2414:, volume 15
1706:lunar nodes
1673:Metaphysics
1288:the ratios
546:Pythagorean
493:Mathematics
437:proportions
419:meteorology
327:Callimachus
312:"good" and
277:Apollodorus
243:and Eudoxus
114:Mathematics
5365:Categories
5172:Babylonian
5072:arithmetic
5038:History of
4867:Apollonius
4552:(Menelaus)
4511:On Spirals
4430:Catoptrics
4369:Xenocrates
4364:Thymaridas
4349:Theodosius
4334:Theaetetus
4314:Simplicius
4304:Pythagoras
4289:Posidonius
4274:Philonides
4234:Nicomachus
4229:Metrodorus
4219:Menaechmus
4174:Hipparchus
4164:Heliodorus
4114:Diophantus
4099:Democritus
4079:Chrysippus
4049:Archimedes
4044:Apollonius
4014:Anaxagoras
4006:(timeline)
3935:Influenced
3914:Influences
3885:Octaeteris
3814:Triquetrum
3702:Timocharis
3687:Theodosius
3647:Posidonius
3607:Hipparchus
3597:Heraclides
3537:Aristyllus
3522:Apollonius
3517:Andronicus
3348:Kurt Gödel
3272:Henry More
3074:Simplicius
2893:Iamblichus
2657:Arcesilaus
2617:Xenocrates
2597:Speusippus
2573:Heraclides
2499:Platonists
2389:(video on
2275:: 183–244.
2234:Wikisource
1870:References
1864:Speusippus
1721:hippopede.
1647:Hipparchus
1639:Phaenomena
1617:Phaenomena
1611:Oktaeteris
507:Archimedes
403: 367
377:Heliopolis
319:Benedictus
235:Hipparchus
219:astronomer
210: – c.
208: 390
139:Philosophy
5295:Astronomy
5259:Biography
4633:Inscribed
4393:Treatises
4384:Zenodorus
4344:Theodorus
4319:Sosigenes
4264:Philolaus
4249:Oenopides
4244:Nicoteles
4239:Nicomedes
4199:Hypsicles
4094:Ctesibius
4084:Cleomedes
4069:Callippus
4054:Autolycus
4039:Aristotle
4019:Anthemius
3789:Astrolabe
3722:(Ptolemy)
3642:Philolaus
3632:Oenopides
3617:Hypsicles
3562:Cleomedes
3557:Callippus
3547:Autolycus
3502:Aglaonice
3260:Cambridge
3128:Al-Farabi
3069:Damascius
3059:Zenodotus
3009:Hierocles
2978:Macrobius
2973:Augustine
2958:Antoninus
2908:Sosipatra
2853:Calcidius
2763:Antiochus
2717:Charmadas
2697:Carneades
2681:Hegesinus
2534:Aristotle
2514:Academics
2348:(Teubner)
2210:185509676
2061:Ball 1908
2017:cite book
1811:, in the
1809:Aristotle
1782:Callippus
1770:hippopede
1669:Aristotle
1665:On Speeds
1631:On Speeds
1534:Astronomy
1494:⋅
1464:⋅
1434:⋅
1404:⋅
1219:⋅
1207:⋅
1177:⋅
1165:⋅
1135:⋅
1123:⋅
1093:⋅
1081:⋅
1051:⋅
1039:⋅
1009:⋅
997:⋅
967:⋅
937:⋅
907:⋅
877:⋅
538:magnitude
511:Euclid's
503:Antiquity
457:Tartaglia
411:Aristotle
407:scholarch
389:Propontis
357:Theomedon
349:Philiston
214:) was an
129:Astronomy
124:Geography
5197:Japanese
5182:Egyptian
5125:timeline
5113:timeline
5101:timeline
5096:geometry
5089:timeline
5084:calculus
5077:timeline
5065:timeline
4768:Elements
4614:Concepts
4576:Problems
4549:Spherics
4539:Spherics
4504:(Euclid)
4450:(Euclid)
4447:Elements
4440:(Euclid)
4401:Almagest
4309:Serenus
4284:Porphyry
4224:Menelaus
4179:Hippasus
4154:Eutocius
4129:Domninus
4024:Archytas
3890:Solstice
3823:Concepts
3719:Almagest
3662:Seleucus
3622:Menelaus
3582:Euctemon
3331:Analytic
3116:Medieval
3079:Priscian
3014:Syrianus
2983:Boethius
2938:Salutius
2918:Dexippus
2913:Aedesius
2888:Porphyry
2876:Students
2871:Plotinus
2833:Longinus
2808:Apuleius
2798:Alcinous
2778:Plutarch
2672:Telecles
2642:Skeptics
2554:Coriscus
2431:Archived
2400:Archived
2324:(1925).
2283:(1986).
2262:: 83–96.
2220:(1907).
2169:(1908).
2092:(1951).
1842:See also
1819:hedonism
1702:ecliptic
1681:De caelo
1621:Enoptron
1528:Elements
589:Elements
578:Elements
574:Elements
550:Archytas
518:Antiphon
513:Elements
477:and the
451:or even
443:and the
415:theology
393:Mausolus
373:Sophists
365:Socrates
353:Sicilian
341:Tarentum
337:Archytas
285:Olympiad
269:Anatolia
248:Spherics
227:Archytas
134:Medicine
52:Anatolia
5355:Science
5245:Portals
5177:Chinese
5132:numbers
5060:algebra
4988:Related
4962:Centers
4758:Results
4628:Central
4299:Ptolemy
4294:Proclus
4259:Perseus
4214:Marinus
4194:Hypatia
4184:Hippias
4159:Geminus
4149:Eudoxus
4139:Eudemus
4109:Diocles
3794:Dioptra
3657:Pytheas
3652:Ptolemy
3602:Hicetas
3592:Geminus
3587:Eudoxus
3542:Attalus
3507:Agrippa
3217:Plethon
3153:Thierry
3148:Gilbert
3143:Bernard
3064:Agapius
3039:Isidore
3034:Marinus
3029:Proclus
3024:Aedesia
3019:Hermias
2992:Academy
2963:Hypatia
2898:Sopater
2883:Amelius
2813:Atticus
2793:Albinus
2676:Evander
2667:Lacydes
2627:Polemon
2622:Crantor
2539:Eudoxus
2506:Ancient
2391:YouTube
1748:Jupiter
1736:Mercury
1509:
1483:
1479:
1453:
1449:
1423:
1419:
1393:
1382:
1354:
1350:
1322:
1318:
1290:
1283:
1241:
1234:
1196:
1192:
1154:
1150:
1112:
1108:
1070:
1066:
1028:
1024:
986:
982:
956:
952:
926:
922:
896:
892:
866:
862:
842:
838:
818:
811:
769:
765:
737:
733:
705:
701:
681:
677:
657:
653:
633:
629:
609:
471:Craters
430:planets
385:Cyzicus
369:Piraeus
332:Pinakes
308:, from
305:εὔδοξος
119:Physics
5271:Greece
5192:Indian
4969:Cyrene
4501:Optics
4420:Conics
4339:Theano
4329:Thales
4324:Sporus
4269:Philon
4254:Pappus
4144:Euclid
4074:Carpus
4064:Bryson
3905:Zodiac
3855:Equant
3804:Gnomon
3682:Thales
3677:Strabo
3527:Aratus
3192:Modern
3133:Anselm
3054:Hegias
2933:Julian
2843:Origen
2732:Cicero
2650:Middle
2360:
2319:
2293:
2208:
2198:
2179:
2127:
2044:
2005:
1967:226242
1965:
1804:Ethics
1752:Saturn
1750:, and
1695:zodiac
1643:Aratus
679:, and
603:Using
570:Euclid
558:axioms
361:Athens
293:355 BC
289:365 BC
265:Cnidus
241:Aratus
107:Fields
61:Turkey
48:Cnidus
5307:Stars
5187:Incan
5108:logic
4884:Other
4652:Chord
4645:Axiom
4623:Angle
4279:Plato
4169:Heron
4089:Conon
3711:Works
3627:Meton
3572:Conon
2788:Gaius
2529:Plato
2307:1966)
1963:JSTOR
1740:Venus
531:prism
485:(the
381:Egypt
287:(368–
231:Plato
57:Muğla
5149:list
4437:Data
4209:Leon
4059:Bion
3552:Bion
2674:and
2575:and
2556:and
2462:Java
2358:ISBN
2291:ISBN
2206:OCLC
2196:ISBN
2177:ISBN
2125:ISBN
2042:ISBN
2023:link
2003:ISBN
1947:Isis
1744:Mars
1522:The
1481:and
1421:and
1213:<
1171:<
1045:>
1003:>
954:and
894:and
840:and
479:Moon
475:Mars
351:the
339:(of
314:doxa
281:acme
259:Life
229:and
69:Died
36:Born
5051:by
4765:In
2690:New
2522:Old
2412:DIO
1995:doi
1955:doi
1671:'s
1596:In
1564:by
520:'s
473:on
322:).
251:by
166:juː
5367::
2330:.
2273:28
2271:.
2258:.
2204:.
2068:^
2019:}}
2015:{{
2001:.
1961:.
1951:32
1949:.
1918:.
1912:.
1746:,
1742:,
1738:,
1645:.
1236:.
655:,
631:,
591:.
467:.
432:.
400:c.
379:,
343:,
329:'
310:eu
297:c.
245:.
237:'
221:,
205:c.
203:;
199:,
195::
191:;
73:c.
59:,
50:,
40:c.
5247::
3986:e
3979:t
3972:v
3478:e
3471:t
3464:v
2491:e
2484:t
2477:v
2460:(
2393:)
2366:.
2305:(
2299:.
2260:4
2249:.
2236:.
2212:.
2185:.
2133:.
2104:.
2102:7
2050:.
2025:)
2011:.
1997::
1969:.
1957::
1935:.
1922:.
1772:.
1708:.
1697:.
1587:)
1581:(
1576:)
1572:(
1558:.
1497:d
1491:n
1467:c
1461:m
1437:b
1431:n
1407:a
1401:m
1370:d
1366:/
1362:c
1338:b
1334:/
1330:a
1306:m
1302:/
1298:n
1271:d
1267:/
1263:c
1260:=
1257:b
1253:/
1249:a
1222:d
1216:n
1210:c
1204:m
1180:b
1174:n
1168:a
1162:m
1138:d
1132:n
1129:=
1126:c
1120:m
1096:b
1090:n
1087:=
1084:a
1078:m
1054:d
1048:n
1042:c
1036:m
1012:b
1006:n
1000:a
994:m
970:d
964:n
940:b
934:n
910:c
904:m
880:a
874:m
850:n
826:m
799:d
795:/
791:c
788:=
785:b
781:/
777:a
753:d
749:/
745:c
721:b
717:/
713:a
689:d
665:c
641:b
617:a
187:/
184:s
181:ə
178:s
175:k
172:ə
169:d
163:ˈ
160:/
156:(
63:)
23:.
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