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f from similar triangles which relies on ratios of line segments 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 relationship between them eudoxus is credited with defining equality between two ratios the subject of book v of the elements in definition 5 of euclid s book v we read 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 of the latter equimultiples respectively taken in corresponding order using modern notation this can be made more explicit given four quantities a displaystyle a b displaystyle b c displaystyle c and d displaystyle d take the ratio of the first to the second a b displaystyle a b and the ratio of the third to the fourth c d displaystyle c d that the two ratios are proportional a b c d displaystyle a b c d can be defined by the following condition for any two arbitrary positive integers m displaystyle m and n displaystyle n form the equimultiples m a displaystyle m cdot a and m c displaystyle m cdot c of the first and third likewise form the equimultiples n b displaystyle n cdot b and n d displaystyle n cdot d of the second and fourth if it happens that m a n b displaystyle m cdot a n cdot b then also m c n d displaystyle m cdot c n cdot d if instead m a n b displaystyle m cdot a n cdot b then also m c n d displaystyle m cdot c n cdot d finally if m a n b displaystyle m cdot a n cdot b then also m c n d displaystyle m cdot c n cdot d this means that a b c d displaystyle a b c d if and only if the ratios n m displaystyle n m that are larger than a b displaystyle a b are the same as the ones that are larger than c d displaystyle c d and likewise for equal and smaller this can be compared with dedekind cuts that define a real number by the set of rational numbers that are larger equal or smaller than the number to be defined eudoxus s definition depends on comparing the similar quantities m a displaystyle m cdot a and n b displaystyle n cdot b and the similar quantities m c displaystyle m cdot c and n d displaystyle n cdot d and does not depend on the existence of a common unit for measuring these quantities 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 epsilon delta definitions of limit and continuity the archimedean property definition 4 of elements book v was credited to eudoxus by archimedes 12 astronomy edit this section needs more citations please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed september 2022 learn how and when to remove this message in ancient greece 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 disappearances of the sun possibly on eclipses oktaeteris ὀκταετηρίς on an eight year lunisolar venus cycle of the calendar phaenomena φαινόμενα and enoptron ἔνοπτρον on spherical astronomy probably based on observations made by eudoxus in egypt and cnidus on speeds on planetary motions we are fairly well informed about the contents of phaenomena for eudoxus s prose text was the basis for a poem of the same name by aratus hipparchus quoted from the text of eudoxus in his commentary on aratus eudoxan planetary models edit main article concentric spheres a general idea of the content of on speeds can be gleaned from aristotle s metaphysics xii 8 and a commentary by simplicius of cilicia 6th century ad on de caelo 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 13 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 in most modern reconstructions of the eudoxan model the moon is assigned three spheres the outermost rotates westward once in 24 hours explaining rising and setting the second rotates eastward once in a month explaining the monthly motion of the moon through the zodiac 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 ecliptic and the motion of the lunar nodes 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 eudoxus s model of planetary motion each of his homocentric spheres is represented 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 the planet along a figure eight curve or hippopede to explain retrograde motion animation depicting eudoxus s model of retrograde planetary motion the two innermost spheres turn with the same period but in opposite directions moving the planet along a figure eight curve or hippopede the five visible planets mercury venus mars jupiter and saturn are assigned four spheres each the outermost explains the daily motion the second explains the planet s motion through the zodiac the third and fourth together explain retrogradation 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 hippopede importance of eudoxan system edit callippus 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 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 autolycus of pitane astronomers responded by introducing the deferent and epicycle which caused a planet to vary its distance however eudoxus s importance to astronomy and in particular to greek astronomy is considerable ethics edit aristotle in the nicomachean ethics 14 attributes to eudoxus an argument in favor of hedonism that is that pleasure is the ultimate good that activity strives for according to aristotle eudoxus put forward the following arguments for this position 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 similarly pleasure s opposite pain is universally avoided which provides additional support for the idea that pleasure is universally considered good people do not seek pleasure as a means to something else but as an end in its own right 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 of all of the things that are good happiness is peculiar for not being praised which may show that it is the crowning good 15 legacy edit craters on mars and the moon are named in his honor an algebraic curve the kampyle of eudoxus is also named after him see also edit eudoxus reals a fairly recently discovered construction of the real numbers named in his honor delian problem incommensurable magnitudes speusippus references edit diogenes laertius viii 86 lasserre françois 1966 die fragmente des eudoxos von knidos de gruyter berlin o connor john j robertson edmund f eudoxus of cnidus mactutor history of mathematics archive university of st andrews hultsch 1907 de santillana george 1940 eudoxus and plato a study in chronology isis 32 2 248 262 doi 10 1086 347693 jstor 226242 diogenes laertius viii 87 sumira sylvia 2014 a brief history of globes globes 400 years of exploration navigation and power chicago chicago university press p 13 doi 10 7208 9780226139142 005 inactive 12 july 2025 isbn 9780226139142 cite book cs1 maint doi inactive as of july 2025 link 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 for dirk struik scientific historical and political essays in honor of dirk j struik boston studies in the philosophy of science vol 15 dordrecht springer pp 225 243 doi 10 1007 978 94 010 2115 9_19 isbn 978 90 277 0379 8 calinger ronald 1982 classics of mathematics oak park illinois moore p 75 isbn 0 935610 13 8 ball 1908 p 54 1 2 kline morris 1972 mathematical thought from ancient to modern times oxford university press pp 48 50 knopp konrad 1951 theory and application of infinite series english 2nd ed london and glasgow blackie son p 7 lloyd ger 1970 early greek science thales to aristotle w w norton p 84 isbn 9780393005837 largely in book ten this particular argument is referenced in book one bibliography edit ball walter william rouse 1908 a short account of the history of mathematics 4th ed dover publications isbn 9780486206301 cite book isbn date incompatibility help evans james 1998 the history and practice of ancient astronomy oxford university press isbn 0 19 509539 1 oclc 185509676 hultsch friedrich 1907 eudoxos von knidos in pauly august wissowa georg eds realencyclopädie der classischen altertumswissenschaft in german vol 6 1 pp 930 950 via wikisource huxley gl 1980 eudoxus of cnidus p 465 7 in the dictionary of scientific biography volume 4 huxley g l 1963 eudoxian topics greek roman and byzantine studies 4 83 96 knorr wilbur richard 1978 archimedes and the pre euclidean proportion theory archives internationales d histoire des sciences 28 183 244 knorr wilbur r 1986 the ancient tradition of geometric problems boston birkhäuser isbn 0 8176 3148 8 lasserre françois 1966 die fragmente des eudoxos von knidos de gruyter berlin laërtius diogenes pythagoreans eudoxus lives of the eminent philosophers vol 2 8 translated by hicks robert drew two volume ed loeb classical library manitius c 1894 hipparchi in arati et eudoxi phaenomena commentariorum libri tres teubner neugebauer o 1975 a history of ancient mathematical astronomy berlin springer verlag isbn 0 387 06995 x van der waerden b l 1988 science awakening 5th ed leiden noordhoff external links edit wikimedia commons has media related to eudoxus of cnidus working model and complete explanation of the eudoxus s spheres video on youtube eudoxus and plato archived 2018 08 16 at the wayback machine a documentary on eudoxus including a description of his planetary model dennis duke statistical dating of the phaenomena of eudoxus dio volume 15 see pages 7 to 23 eudoxus of cnidus britannica com eudoxus of cnidus archived 1997 07 23 at the wayback machine donald allen professor texas a m university eudoxos of knidos eudoxus of cnidus astronomy and homocentric spheres henry mendell cal state u la archived 16 may 2011 herodotus project extensive b w photo essay of cnidus models of planetary motion eudoxus craig mcconnell ph d cal state fullerton archived 19 july 2011 the universe according to eudoxus java applet archived 21 november 2007 v t e platonists ancient academics old plato aristotle eudoxus philip of opus aristonymus coriscus and erastus of scepsis demetrius of amphipolis euaeon of lampsacus heraclides and python of aenus hestiaeus of perinthus lastheneia of mantinea timolaus of cyzicus speusippus axiothea of phlius heraclides ponticus menedemus of pyrrha xenocrates crantor polemon crates of athens skeptics middle arcesilaus diocles of cnidus lacydes telecles and evander hegesinus new carneades hagnon of tarsus metrodorus of stratonicea clitomachus charmadas aeschines of neapolis philo of larissa cicero dio of alexandria middle platonists antiochus eudorus of alexandria anonymous theaetetus commentator philo of alexandria plutarch justin martyr gaius albinus alcinous alexander peloplaton apuleius atticus maximus of tyre numenius of apamea ammonius saccas longinus clement of alexandria origen origen the pagan calcidius neoplatonists plotinus students amelius porphyry of tyre iamblichus sopater eustathius of cappadocia sosipatra aedesius dexippus chrysanthius theodorus of asine julian salutius maximus of ephesus eusebius of myndus priscus of epirus antoninus hypatia gaius marius victorinus augustine macrobius boethius academy plutarch of athens asclepigenia hierocles syrianus hermias aedesia proclus marinus isidore ammonius hermiae asclepiodotus hegias zenodotus agapius damascius simplicius priscian john philoponus olympiodorus david the invincible pseudo dionysius the areopagite medieval john scotus eriugena al farabi anselm peter abelard bernard gilbert thierry henry of ghent bonaventure theodoric of freiberg meister eckhart berthold of moosburg paul of venice modern renaissance florentine academy plethon marsilio ficino cristoforo landino giovanni pico della mirandola giordano bruno blaise pascal robert greville cambridge ralph cudworth henry more anne conway thomas taylor emanuel swedenborg josiah royce aleksei losev contemporary analytic gottlob frege bertrand russell kurt gödel roderick chisholm w v o quine jan łukasiewicz crispin wright edward n zalta penelope maddy continental edmund husserl leo strauss simone weil alain badiou v t e ancient greek astronomy astronomers aglaonice agrippa anaximander andronicus apollonius aratus aristarchus aristyllus attalus autolycus bion callippus cleomedes cleostratus conon eratosthenes euctemon eudoxus geminus heraclides hicetas hipparchus hippocrates of chios hypsicles menelaus meton oenopides philip of opus philolaus posidonius ptolemy pytheas seleucus sosigenes of alexandria sosigenes the peripatetic strabo thales...