Eudoxus of Cnidus

How to use this course

Eudoxus is a mathematician without a surviving book. Not one of his works exists in its own words. His theory of proportion reaches us as Book V of Euclid’s Elements, credited to him by an ancient note in the margin. His proofs about cones and pyramids are known because Archimedes names him as their author. His planetary model survives in a few lines of Aristotle and a commentary by Simplicius written nine centuries later. His star catalogue survives because a poet put it into verse and Hipparchus criticized both.

The course therefore states, for every idea, how we know it is his. The biographical lessons rest mainly on the short life by Diogenes Laertius (3rd century AD), which cites earlier writers by name.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches or derivations, and exercises with folded solutions. Citations such as V.5 refer to Euclid’s Elements, Book V, in Heath’s numbering. The companion course on Euclid covers the rest of the Elements.

Timeline

Two chronologies exist. Nineteenth-century scholars, following a reconstruction published by Friedrich Hultsch, placed his life at 408–355 BC. Most twentieth-century historians, following George de Santillana (1940), prefer about 390–338 BC. The relative order of events below comes from Diogenes Laertius; the absolute dates follow the later chronology and are approximate.

Cnidus, Tarentum and the sources

Cnidus was a Greek city on a long peninsula at the south-west tip of Asia Minor, in Caria, known for its medical school and later for Praxiteles’ statue of Aphrodite. Eudoxus was born there, the son of Aeschines. His name means “of good repute”. Diogenes Laertius reports that his admirers later punned on it, calling him Endoxos, “illustrious”.

The sources and their reliability

• Diogenes Laertius, Lives of the Eminent Philosophers VIII.86–91 (3rd century AD), is the only continuous biography. It is short, but it names its authorities: the Alexandrian scholar Callimachus’ catalogue of the Library, Sotion, Hermippus, Eratosthenes, Apollodorus’ Chronicle and Favorinus. This makes it better than most ancient lives.
• Aristotle, who probably knew him at the Academy, reports his planetary model (Metaphysics Λ.8) and his view of pleasure (Nicomachean Ethics I.12 and X.2).
• Archimedes credits him with the first proofs about pyramids and cones and with the use of the lemma now called the Archimedean axiom.
• Proclus (5th century AD), drawing on Eudemus’ lost history of geometry, summarizes his mathematics in a few lines.
• Strabo, Hipparchus and Simplicius preserve the astronomy.

Teachers

Citing Callimachus, Diogenes says Eudoxus learned geometry from Archytas of Tarentum and medicine from Philistion of Locri, a Sicilian physician. Archytas, a Pythagorean statesman and mathematician, had solved the problem of doubling the cube with a three-dimensional construction and worked on the theory of means and of musical intervals. All three subjects reappear in Eudoxus’ work (Lesson 07). Philistion’s medicine was part of the same world: Diogenes calls Eudoxus astronomer, geometer, physician and legislator.

Athens and Egypt

Two months at the Piraeus

At about 23, Diogenes says, Eudoxus sailed to Athens with the physician Theomedon, who paid his way. He was drawn by the fame of the Socratics. Too poor to live in the city, he lodged at the harbour of the Piraeus and each day walked up to Athens, a trip of some eleven kilometres each way, to hear the lectures, returning at night. After two months he went home. Sotion counts him among Plato’s pupils; another report in Diogenes says that Plato at first took no notice of him.

Sixteen months in Egypt

His friends then raised money for a voyage to Egypt, which he made with a physician named Chrysippus. He carried a letter of introduction from Agesilaus, king of Sparta, to the pharaoh Nectanebo, who commended him to the priests. Diogenes says he stayed a year and four months and shaved his beard and eyebrows in the Egyptian priestly manner. Strabo, writing in the first century BC, was shown an observatory between Heliopolis and Cercesura where Eudoxus was said to have worked, and the rooms at Heliopolis where Plato and Eudoxus were said to have lived. Strabo’s guides were telling a story to visitors three centuries later, so this is tradition, not evidence.

Favorinus, quoted by Diogenes, adds that the sacred Apis bull licked Eudoxus’ cloak, and the priests foretold that he would be famous but short-lived. This is a legend shaped by the outcome.

What Eudoxus learned in Egypt is unknown. Egyptian astronomy had a 365-day civil calendar and long records of star risings, and his later calendar work uses 365-day years (Lesson 10). The mathematics associated with him has no Egyptian parallel.

A school at Cyzicus

From Egypt Eudoxus went north to Cyzicus, a Greek city on the Sea of Marmara (the Propontis), where he taught and attracted many pupils. He also visited Mausolus, the satrap of Caria whose tomb gave its name to every later mausoleum. Cyzicus produced several astronomers in the next generation. Polemarchus of Cyzicus was an associate of Eudoxus, and Callippus of Cyzicus, who later refined the planetary model, studied with Polemarchus according to Simplicius. Hipparchus noted that the star data in Eudoxus’ Phaenomena fit a latitude where the longest day is about fifteen hours, which suits the latitude of Cyzicus better than Cnidus or Athens (Lesson 08).

Pupils and successors

Proclus, following Eudemus, names Menaechmus as a pupil of Eudoxus. Menaechmus is credited with introducing the conic sections, and his brother Dinostratus with using the quadratrix to square the circle. Proclus also lists Amyclas of Heraclea, a friend of Plato, and says these men made geometry more perfect still. Diogenes names Chrysippus of Cnidus, son of Erineus, who heard Eudoxus on the gods, the cosmos and the heavens. Eudoxus’ own family included a son, Aristagoras, and three daughters, Actis, Philtis and Delphis.

By the time he returned to Athens, Eudoxus led his own school. This is the setting for the work of Part II: a theory of ratio for all magnitudes, a rigorous method for curved figures, a catalogue of the sky and a geometrical model of the planets.

The theory of proportion

For the general reader. The Pythagoreans thought every pair of lengths could be measured by a common unit. The side and diagonal of a square showed that they could not. Every theorem about similar figures that had been proved with common units was suddenly in doubt. Eudoxus repaired the damage with a definition of “same ratio” that never needs a common unit. It is the definition in Book V of Euclid’s Elements.

Attribution

An ancient scholium (marginal note) in the manuscripts of the Elements says that Book V is the discovery of Eudoxus. Archimedes, in the preface to Quadrature of the Parabola, says that earlier geometers used the lemma in Definition V.4 to prove the theorems on circles, spheres, pyramids and cones, and elsewhere names Eudoxus as the first to prove those theorems. Proclus says Eudoxus “increased the number of general theorems”. Together these make the attribution secure in substance, though Euclid’s text may not be Eudoxus’ wording.

The definitions

In V.5, \(a\) and \(b\) are of one kind (say lines) and \(c,d\) may be of another (say areas). The definition never multiplies a line by a line, so it is meaningful for any magnitudes that can be added and compared. Reading \(ma>nb\) as \(a/b>n/m\), it says that two ratios are equal when they exceed, equal and fall short of exactly the same fractions.

The main theorems

Sketch of V.16. By V.15, \(ma:mb=a:b\) and \(nc:nd=c:d\), so \(ma:mb=nc:nd\). By V.14, if four magnitudes are proportional and the first is greater than, equal to, or less than the third, the second is correspondingly related to the fourth. Hence \(ma\gtreqless nc\iff mb\gtreqless nd\) for all \(m,n\), which is \(a:c=b:d\) by V.5. Alternation needs all four magnitudes to be of one kind, since \(a\) is compared with \(c\).

Ratios and real numbers

For the general reader. To say whether two ratios are equal, Eudoxus compares each with every fraction. Twenty-two centuries later Richard Dedekind defined a real number as a way of splitting the fractions into those below and those above. The two ideas look very alike. They are not the same, and seeing exactly how they differ teaches a good deal about both.

From ratios to cuts

Given magnitudes \(a,b\) of one kind that have a ratio, set \[ L(a{:}b)=\{\,n/m\in\mathbb Q_{>0} : nb<ma\,\},\qquad U(a{:}b)=\{\,n/m\in\mathbb Q_{>0} : nb>ma\,\}. \] The condition depends only on \(n/m\), since multiplying \(m\) and \(n\) by the same integer preserves it. Definition V.5 says exactly that \(a:b=c:d\) if and only if \(L(a{:}b)=L(c{:}d)\) and \(U(a{:}b)=U(c{:}d)\). Definition V.7 says that \(a:b>c:d\) if and only if \(L(a{:}b)\supsetneq L(c{:}d)\).

V.4 guarantees that \(L\) and \(U\) are non-empty, so the supremum is a positive real. The difference from Dedekind is one of direction. Eudoxus takes the magnitudes as given, by geometry, and uses fractions to compare their ratios. Dedekind has no magnitudes to start from; he constructs the reals from cuts so that every cut has a number. Eudoxus never asserts that every cut is realized by some ratio, and Greek geometry did not need it to be. Thomas Heath stressed the closeness of the two theories; others, such as the historian Jean-Louis Gardies, stress the difference in purpose. When Rudolf Lipschitz suggested in 1876 that Euclid already had the cut idea, Dedekind replied that Euclid’s theory lacks the completeness principle, which was the point of his own.

The Archimedean axiom

Definition V.4 rules out magnitudes that are infinitely small compared with others. Euclid knew such things: III.16 shows that the “horn angle” between a circle and its tangent is less than every rectilinear angle. No multiple of the horn angle exceeds a rectilinear angle, so by V.4 they have no ratio, and Book V does not apply to them.

Eudoxus reals

A modern construction of the real numbers, named in his honour, uses no fractions at all. A function \(f:\mathbb Z\to\mathbb Z\) is an almost homomorphism if \(f(m+n)-f(m)-f(n)\) is bounded. Two such functions are equivalent if their difference is bounded. The equivalence classes, with addition of functions and composition as multiplication, form a field isomorphic to \(\mathbb R\); the class of \(f\) corresponds to its “slope” \(\lim f(n)/n\). Norbert A’Campo and others have studied this construction; the idea is to measure a slope by comparing \(f(n)\) with \(n\) for all multiples at once, in the spirit of Definition V.5.

The method of exhaustion

For the general reader. How can you prove that a cone is exactly one third of a cylinder? Democritus had stated it. Eudoxus proved it, by fitting into the solid a sequence of simpler solids that leave less and less over, and showing by contradiction that any other answer would lead to absurdity. The method uses no infinitesimals and no limits, yet it does the work of integral calculus.

Attribution

In the preface to On the Sphere and Cylinder I, Archimedes says that Eudoxus first proved that a pyramid is one third of the prism, and a cone one third of the cylinder, with the same base and height. In the preface to The Method he adds that Democritus stated these facts first but without proof. Euclid’s Book XII, which proves them, is therefore taken to follow Eudoxus. The name “method of exhaustion” is modern, from the seventeenth century.

By V.4, some multiple \(N\varepsilon\) exceeds \(A\). Halving at least \(N-1\) times reduces \(A\) below \(\varepsilon\), since after \(k\) steps the remainder is less than \(A/2^{k}\le A/(k+1)\), and \(A/(k+1)<\varepsilon\) once \(k+1\ge N\). In modern terms the remainders \(r_k\) satisfy \(r_k<A/2^k\to0\).

The pyramid

Step 1 (XII.3). A triangular pyramid \(ABCD\) can be cut, by joining midpoints of its edges, into two pyramids similar to the whole, each with half the linear size, and two prisms of equal volume. The two prisms together exceed half the pyramid.

Step 2 (XII.4–XII.5). Repeating the cut in both small pyramids, the prisms “exhaust” the pyramid by X.1. Comparing two pyramids of equal height with bases \(B_1,B_2\), the prisms at each stage are in ratio \(B_1:B_2\). A double reductio, as in XII.2, gives: pyramids of equal height are to one another as their bases.

Step 3 (XII.7). A triangular prism splits into three triangular pyramids, which are equal in pairs by Step 2. Hence each is one third of the prism. Any pyramid with a polygonal base is a sum of triangular ones.

The cone

Let \(V\) be the cylinder and \(K\) the cone. Suppose \(3K<V\). Inscribe in the base circle a square, then an octagon, and so on. The prism on each polygon takes more than half of what the previous prism left of the cylinder, so by X.1 some prism \(P\) satisfies \(V-P<V-3K\), so \(P>3K\). The pyramid on the same polygon, inside the cone, is \(P/3>K\), which is absurd. Suppose \(3K>V\): the symmetric argument with pyramids inside the cone gives a contradiction. So \(3K=V\).

Means, the section and the cube

For the general reader. Proclus credits Eudoxus with three things besides general theorems: new kinds of “means” between numbers, many theorems on “the section”, and the use of analysis. Eratosthenes adds that he solved the famous problem of doubling the cube using curved lines. The details are lost, so this lesson presents what the sources say and the mathematics around it, clearly labelled where it is reconstruction.

Means

The Pythagoreans, Archytas among them, studied three means of \(a>c\): \(b\) is the arithmetic mean if \(a-b=b-c\), the geometric mean if \(a:b=b:c\), and the harmonic mean if \((a-b):(b-c)=a:c\). Proclus says Eudoxus added three more “proportions” to these three, without naming them. Later writers, Nicomachus and Pappus, list ten means; the next three on their lists are the “subcontrary” means, for example the fourth: \((a-b):(b-c)=c:a\). Historians often take these to be Eudoxus’ additions, but no ancient text says so.

The duplication of the cube

To double a cube of side \(a\), one needs \(x\) with \(x^3=2a^3\). Hippocrates of Chios showed that it suffices to insert two mean proportionals between \(a\) and \(2a\):

Archytas found the means by intersecting a cylinder, a cone and a torus. Eutocius (6th century AD) preserves a letter ascribed to Eratosthenes that lists Eudoxus among the solvers, saying he used “curved lines”; Eutocius says the version of Eudoxus’ solution he saw contained an error and did not copy it. Paul Tannery proposed that Eudoxus used the curve now called the kampyle of Eudoxus, \(a^2x^4=b^4(x^2+y^2)\), but Heath doubted it, and the attribution is conjecture. Menaechmus, Eudoxus’ pupil, solved the problem with two conic sections, as in the exercise below.

“The section”

Proclus says Eudoxus multiplied the theorems about “the section” that had begun with Plato, applying the method of analysis to them. Many historians take “the section” to be the division of a line in extreme and mean ratio, the golden section, and suggest that Elements XIII.1–5, a group of propositions on that division, go back to Eudoxus. Others read “the section” differently. The identification is plausible and unproven.

The sphere of the sky

For the general reader. Eudoxus wrote the first systematic description of the constellations, the Phaenomena, with a second version called the Mirror. It placed the stars on a single rotating sphere and described which constellations lie on the important circles of that sphere and which rise as others set. The work is lost, but Hipparchus quotes it, and the poet Aratus turned it into verse.

The two-sphere model

The Earth is a small sphere at the centre of a large sphere of stars that turns once a day about the celestial poles. The Sun moves slowly along the ecliptic, a great circle tilted to the celestial equator by the obliquity \(\varepsilon\approx24^\circ\) in Eudoxus’ time. For an observer at latitude \(\varphi\), the celestial pole stands at altitude \(\varphi\) and the horizon cuts each daily circle into a visible and an invisible arc. Hipparchus reports that Eudoxus placed the solstices and equinoxes at the middle of their zodiacal signs, and that he spoke of a star that always stays in the same place at the pole; Hipparchus objected that no star lay at the pole.

Derivation. Use axes with \(z\) toward the celestial pole. A star at hour angle \(H\) has unit vector \((\cos\delta\cos H,\ \cos\delta\sin H,\ \sin\delta)\). The local zenith is \((\cos\varphi,0,\sin\varphi)\). The star is on the horizon when the dot product vanishes: \(\cos\varphi\cos\delta\cos H+\sin\varphi\sin\delta=0\).

Hipparchus’ test

Hipparchus noticed that Eudoxus’ risings and settings fit a place where the longest day lasts about 15 hours, and that this was not Athens. Greek geographers used the length of the longest day to label climates, so the observation is a statement of latitude. Spherical geometry of this kind was later systematized by Autolycus of Pitane and in Euclid’s own Phaenomena; Theodosius’ Spherics may also draw on Eudoxus.

Homocentric spheres and the hippopede

For the general reader. Planets do not move steadily against the stars. They slow down, stop, go backwards for a while, and then resume. Eudoxus produced the first mathematical model of this behaviour. Each planet rides on a nest of spheres, all centred on the Earth, each turning uniformly about its own axis. Two of these spheres, turning in opposite senses, make the planet trace a figure eight, and when this figure eight is carried along the zodiac, the planet can appear to go backwards.

Sources and structure

Eudoxus’ book On Speeds is lost. Aristotle (Metaphysics Λ.8) gives the number of spheres; Simplicius, commenting on Aristotle’s On the Heavens in the sixth century AD and drawing on Eudemus and Sosigenes, describes how they move. The Moon and the Sun have three spheres each and each of the five planets four; with one sphere for the fixed stars, that makes 27 in all, the count Aristotle gives for Eudoxus. For a planet:

1. The outermost sphere turns once a day about the celestial poles.
2. The second turns about the poles of the ecliptic in the planet’s zodiacal period \(Z\), carrying it around the zodiac.
3. The third has its poles on the ecliptic and turns in the synodic period \(S\).
4. The fourth, whose axis is inclined to the third’s by an angle \(i\), turns in the same period \(S\) in the opposite sense; the planet sits on its equator.

Simplicius calls the curve produced by the third and fourth spheres the hippopede, “horse fetter”, the figure-eight track used in training horses. Giovanni Schiaparelli’s reconstruction of 1877 made this geometry precise.

The curve

Work on the unit sphere in a frame where the third sphere’s axis is \(e_3\) and the mean planet sits at \(e_2\). Let \(R_u(\theta)\) be rotation by \(\theta\) about axis \(u\), let \(u=(\sin i,0,\cos i)\) be the fourth sphere’s axis at time 0, and \(t=2\pi T/S\) the phase at time \(T\). The planet is at \(P(t)=R_{e_3}(t)\,R_u(-t)\,e_2\), which works out to

Derivation. (2): longitude is \(\lambda=\arctan\big(z/y\big)\) along the great circle through \(e_2,e_3\); at \(t=\pi/2\), \(P=(0,\cos i,-\sin i)\), so \(\lambda=-i\). Latitude is the \(x\)-component, \(-\tfrac{1-\cos i}{2}\sin2t\), with maximum modulus \(\tfrac{1-\cos i}{2}=\sin^2\tfrac i2\). (3): put \(\rho=\sin^2\tfrac i2\); then \(x^2+(y-1+\rho)^2=\rho^2\) is a direct check using \(1-\cos i=2\rho\). (4): differentiate at \(t=0\): \(\dot z=-\sin i\), \(y=1\).

Successes and failures

For Saturn and Jupiter, \(S/Z\) is small and a modest inclination gives retrograde loops. For Mars, the synodic period (about 780 days) exceeds the zodiacal period (about 687 days), so \(S/Z>1\) and no inclination satisfies the condition. For Venus, \(Z\) is one year and \(S\approx584\) days, with the same result. Simplicius’ figures for the periods are close to the true ones, and for Mars the transmitted text is usually emended. Some modern reconstructions, such as Ido Yavetz’s, relax Schiaparelli’s assumptions and obtain better behaviour, so the extent of the failure is debated.

A further objection was decisive. Simplicius records that the brightness of Mars and Venus and the apparent size of the Moon vary, so their distances from the Earth must change. Polemarchus knew of the variation and dismissed it as imperceptible; Autolycus of Pitane tried and failed to account for it. Homocentric spheres keep every planet at a fixed distance. Callippus added seven spheres to improve the fit; Aristotle added further “unrolling” spheres to connect the systems mechanically, reaching 55. By the second century BC astronomers had turned to eccentrics and epicycles, which Ptolemy brought to completion.

Calendar, sizes and the Earth

For the general reader. Eudoxus also worked on practical astronomy: a calendar cycle that keeps months and years in step, an estimate of how much bigger the Sun is than the Moon, and possibly one of the earliest figures for the size of the Earth. These were rough by later standards, but they were among the first quantitative claims of Greek astronomy.

The Octaeteris

Diogenes says some placed the composition of the Octaeteris (eight-year cycle) in Egypt. Its reconstructed form, based on later Greek sources, fits eight Egyptian years of 365 days with 99 lunar months, 49 of 30 days and 50 of 29 days.

The sizes of Sun and Moon

In The Sand Reckoner, Archimedes records that Eudoxus made the Sun’s diameter about nine times the Moon’s, that Phidias (Archimedes’ father) made it twelve, and that Aristarchus of Samos tried to prove it between 18 and 20. Eudoxus seems to have been the first to make the Sun much larger than the Moon.

The Earth

In On the Heavens II.14, Aristotle cites mathematicians who make the circumference of the Earth 400,000 stades. He does not name them. The figure is often attributed to Eudoxus, which is plausible given his geography in seven books, the Circuit of the Earth, known from about a hundred quotations, but it is not attested.

Athens, Cnidus and the good life

Return to Athens

Eudoxus came back to Athens with many pupils. Diogenes reports that some said he did so to vex Plato, who had once passed him over, and that at a banquet he introduced the custom of reclining in a semicircle because of the numbers present. Late sources say he took charge of the Academy while Plato was in Sicily around 367 BC; this is not attested in early texts and should be treated with caution. What is clear is that he was a presence in the Academy in Aristotle’s time there, and that Aristotle discusses his views on mathematics, astronomy and ethics with respect.

Philosophy

In the Nicomachean Ethics (X.2), Aristotle reports that Eudoxus held pleasure to be the good, arguing that all creatures, rational and not, seek it. Aristotle adds that his arguments were believed more for the excellence of his character than for themselves, since he was thought to be unusually self-controlled. In I.12 Aristotle records another argument of his: pleasure is not praised, which shows it is above things that are praised, as the gods and the good are. In the Metaphysics (A.9), Aristotle mentions Eudoxus’ view that the Platonic Forms are present in things by being mixed into them, as white is mixed into a white object, and rejects it.

Cnidus

Eudoxus ended his life at Cnidus. Diogenes cites a decree in his honour and, from Hermippus, his fame as a lawgiver for his fellow citizens. Strabo mentions his observatory there. Vitruvius credits him with a sundial called the arachnē (spider), though some sources give it to others. He wrote on theology, astronomy and meteorology, and his geography covered peoples and constitutions as well as places. He died in his fifty-third year.

Legacy: from Aristotle to Dedekind

Astronomy

The homocentric spheres became the physical cosmos of Aristotle, refined by Callippus, and through Aristotle they shaped medieval cosmology for two thousand years, long after astronomers had replaced them with epicycles for computation. Eudoxus’ principle that the heavens should be explained by combinations of uniform circular motions governed astronomy until Johannes Kepler. His Phaenomena, versified by Aratus around 275 BC, became one of the most read poems of antiquity; Hipparchus’ Commentary on Aratus and Eudoxus, his only surviving work, preserves direct quotations.

Mathematics

Book V of the Elements made it possible to state theorems about similar figures and incommensurable magnitudes rigorously, and Book XII gave the template that Archimedes used for his areas and volumes. The method of exhaustion was revived in the seventeenth century by Luca Valerio, Gregory of Saint-Vincent and others, and its logic of approximation underlies the epsilon arguments of nineteenth-century analysis. Richard Dedekind’s cuts (1872) and Karl Weierstrass’s rigorous limits completed a programme that Eudoxus began.

Honours

• Craters on the Moon and on Mars are named Eudoxus.
• The kampyle of Eudoxus carries his name, by conjectural attribution.
• The “Eudoxus reals” (Lesson 05) are a modern construction named after him.

Connections

People linked to Eudoxus of Cnidus across this collection. A name without a link has no page here yet. See Eudoxus of Cnidus in the network of all 337 lives →

Teachers and advisors

• Archytas of Tarentum
• Philistion of Locri

Students

• Chrysippus of Cnidus
• Menaechmus
• Polemarchus of Cyzicus

Influenced

• Archimedes
• Aristarchus of Samos
• Aristotle
• Euclid
• Hipparchus
• Richard Dedekind
• Aratus of Soli
• Callippus of Cyzicus

Colleagues and collaborators

• Plato

Also mentioned in this life

• Eratosthenes
• Johannes Kepler
• Pappus of Alexandria
• Proclus
• Claudius Ptolemy
• Karl Weierstrass

Fields

• geometry 32 lives
• astronomy 50 lives
• geography 5 lives

On CoursesHub

• Astrophysics for Beginners

Sources and further reading

Ancient sources

1. Diogenes Laertius, Lives of the Eminent Philosophers VIII.86–91, transl. C. D. Yonge (Wikisource).
2. Aristotle, Metaphysics A.9 and Λ.8; Nicomachean Ethics I.12 and X.2; On the Heavens II.14.
3. Archimedes, prefaces to On the Sphere and Cylinder I, Quadrature of the Parabola and The Method; The Sand Reckoner.
4. Proclus, A Commentary on the First Book of Euclid’s Elements, transl. G. R. Morrow, Princeton University Press, 1970.
5. Strabo, Geography 2.5.14 and 17.1.29–30.
6. F. Lasserre, Die Fragmente des Eudoxos von Knidos, De Gruyter, 1966.

Biography and history

1. J. J. O’Connor and E. F. Robertson, Eudoxus of Cnidus, MacTutor History of Mathematics.
2. Eudoxus of Cnidus, Wikipedia.
3. Eudoxus of Cnidus, Biographical Encyclopedia of Astronomers (via MacTutor).
4. G. de Santillana, Eudoxus and Plato: a study in chronology, Isis 32 (1940).
5. G. L. Huxley, Eudoxus of Cnidus, Dictionary of Scientific Biography, vol. 4, 1971.

Mathematics and astronomy

1. T. L. Heath, The Thirteen Books of Euclid’s Elements, vols. 2–3, Cambridge, 1908; and A History of Greek Mathematics, vol. 1, Oxford, 1921.
2. D. E. Joyce, Euclid’s Elements, online edition (Books V and XII).
3. G. Schiaparelli, Le sfere omocentriche di Eudosso, di Callippo e di Aristotele, Milan, 1877.
4. I. Yavetz, On the homocentric spheres of Eudoxus, Archive for History of Exact Sciences 52 (1998).
5. O. Neugebauer, A History of Ancient Mathematical Astronomy, Springer, 1975.
6. N. A’Campo, A natural construction for the real numbers, arXiv:math/0301015.
7. Homocentric spheres and Hippopede, Wikipedia.