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Eudoxus of Cnidus (Εύδοξος ο Κνίδιος), (408-355 BC) son of Aeschines, Greek mathematician, astronomer, geographer and philosopher, whose genius was apparent from a very young age. He studied first with the famous Pythagorean **Archytas of Tarentum****;** later a wealthy patron, impressed by the ability of this poor student, paid his way to Athens so that he could study at Plato's Academy. Here he was trained in the Platonic method, but continued to work out his own philosophy, which in physics was quite similar to that of Aristippus. He spent 16 months in Egypt during the reign of Nectanebo I (380-363) studying astronomy (Heliopolis, a Cairo suburb now). The learned men of Egypt so admired his genius that they built him an observatory! He studied medicine with Philistius (Φιλίστιος) on Sicily.

At an age of 23 he went to Athens to study philosophy and rhetorics in **Plato's** Academy. When he returned to Greece he founded a school at Cyzicus in the Propontis, which attracted students from all over Greece. He died at an age of 53 in Cnidus.

**Work**

**MATHEMATICS**

"*Axiom of continuity*" (Eudoxus-Archimedes): Discussed by Archimedes and one of the foundations of modern mathematics. The basis of both integral and differential calculus, it was first applied by Eudoxus and later expanded by Archimedes. Centuries later, Newton and Leibniz based their work on this theorem.

"*Method of exhaustion*": For the calculation of the volume of the pyramid and cone. Archimedes notes in a letter to Eratosthenes that Eudoxus was the first to prove that the cone and the pyramid are one-third respectively of the cylinder and prism with the same base and height and that the formula was known to Democritus was he could not provide a proof.

"*Analysis and synthesis in geometry*": Perfected by Eudoxus.

"*Delian problem*": According to Eutocius, Eudoxus solved it by means of a "curved line". His proof has been lost.

"*Incommensurables*": Eudoxus developed a general theory of proportion applicable to incommensurable as well as to commensurable magnitudes, as Euclid explains in his Elements (books V and VI). Correlation of straight segments without the use of numbers. His theory of proportion enabled people to compared the irrational to the rationals (before Eudoxus, these are treated as different objects):

Length of a irrational number λ is determined by rational lengths less than and greater than λ. Then λ_{1} = λ_{2} if for any rational r < λ_{1} we have r < λ_{2 }and vice versa (similarly λ_{1} < λ_{2} if there is rational r < λ_{2 }but r > λ_{1 }). Dedekind in 1872 (*Dedekind cut*) defined √2 as the pair of two sets (lower and upper) of positive rationals *L*_{√2} = {r: r^{2}< 2} and *U*_{√2} = {r: r^{2}>2}.

"*Theory of irrational numbers*": Developed in the 19th century by Dedekind on the basis of Eudoxus' approach. Although ideas similar to the Method of Exhaustion was known before him, he is the one responsible for formulating this early method of Integration. This method is later extensively used by Archimedes to derived areas and volumes of other objects. He used it to prove the following formulae: The volume V of a pyramid or cone is V = 1/3 (base area)* (height).

Eudoxus is considered to have found the Hippopede (Horse Foot) curve which in today terms can be written as a special form of the polar equation r^{2} = 4b(a-bsin^{2}(theta)). It is mentioned also by Proclus and called also Proclus Hippopede. It is connected with sections of the spiric surface generated by a circle revolving about a straight line (axis of revolution) and always remaining in the same plane as this axis. For some parameters it looks like the mathematical infinite symbol (looks like the number 8). It is connected also to the Figure 8 that can be obtained by the location of the Sun in the sky(**Analemma**)

**ASTRONOMY**

*...the observatory of Eudoxus at Cnidus is not much higher than the dwelling-houses, and from there, it is said, Eudoxus saw the star Canopus,* Strabo, Geography, Book 2

Eudoxus was the founder of the first known observatory. Eudoxus made observations of the star Canopus (the lowest visible bright star in Cnidus). He described the constellations, and was the first to construct and use a planisphere and to explain the apparent movements of the heavenly bodies by a geometric model: in other words, he introduced mathematics into astronomy. He is therefore considered the founder of mathematical astronomy.

Papyrus with Astronomy from Eudoxus , another Papyrus (more images)

"*Theory of concentric spheres*": Interpretation of the apparent movement of the planets, using a spherical lemniscate, which he devised. This theory became the foundation of the science of astronomy. Eudoxus also wrote a related treatise entitled "On speeds", which studied the movements of the seven celestial bodies: Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn. His system was universally admired and was accepted by Aristotle, who described it in his "Metaphysics". It was later more fully worked out by his pupil, Callippus.

"*On making spheres*": Eudoxus constructed a mechanical representation of Autolycus' theory on the movement of the planets.

He was the first to calculate the distance of the Sun and the Moon from the Earth.

"*Phenomena and Enoptron*": Treatise on astronomy, discussed by Aratus. It describes the position of the constellations in the heavenly sphere, and their risings and settings. Classical Greek constellations.

"*Mathematical explanation of the observed motions of the stars*"

The founder of celestial mechanics, Eudoxus invented a method to calculate the distance of the sun and the moon. Aristarchus of Samos based his work on Eudoxus' method.

He proposed a calendar of the solar year, with 3 three years of 365 days followed by a fourth with 366. This calendar was adopted 300 years later by Julius Caesar (the Julian Calendar).

"*Octaeteris*": Calendaric treatise, based on an eight-year cycle, on the adaptation of the lunar to the solar year.

In astronomy devised an ingenious planetary system based on spheres.

The spherical earth is at rest at the center.

Around this center, 27 concentric spheres rotate.

The exterior one caries the fixed stars,

The others account for the sun, moon, and five planets.

Each planet requires four spheres, the sun and moon, three each.

The motion of the 2 external spheres causes the planet to move on a figure eight loop, described by the Hippopede necessary to explain the retrograde motion. Of course there are problems with this model as it is based on spheres. Additional to that it implies that all seasons must have the same length it could not predict the motion of the planet Mars.

**GEOGRAPHY - METEOROLOGY
**

Strabo considered him the fourth great Greek geographer, placing him immediately after Democritus, and noting that Eudoxus was the first to apply mathematical axioms to geography. Eudoxus calculated that the ratio of the length to the width of the world was 2:1.

"

*Signs and portents - Observations*": Observations on the weather and a study of the winds.

"

*Ges periodos*": Treatise containing a wealth of geographical information.

Eudoxus was particularly interested in the climate in various parts of the world, and in the zones of the terrestrial globe with similar astronomical data (appearance of the night sky, length of longest day, etc.).

He constructed a planisphere.

**INVENTIONS**

The major contribution to the development of mechanics and technology made by this great mathematician and astronomer was the introduction, in about 360 BC, of two particularly important instruments: the astrolabe and the

*polos*. The astrolabe was a genuine Greek invention - not of course in the form in which we know it today - that has been attributed to Eudoxus on the basis of work done by F. Nau in 1899; the first Hellenistic accounts of this device, e.g. in Philoponos of Alexandria, came centuries later (500-550 AD). Vitruvius tells us that Eudoxus used an instrument that he called

*arachne*(spider). Others have argued that the initial form of the instrument, the plane astrolabe, was discovered by Hipparchus in about 150 BC. The

*polos*was a more complicated arrangement of interlinked rings that was used to tell the time. Both were derived from the older

*gnomon*and sundial, which had developed into two different types of instruments: the

*uranosphere*, supported by a system of rings, and the plane sundial, which finally supplanted it. The astrolabe began its career as an instrument that could tell the time by night as well as by day; the Arabs developed it into an astronomical instrument in about 810 AD, while in Western Europe it ended up as an instrument for navigation.

The *polos* was a more complex device. In essence it was a portable time-piece, that followed the path of the shadow of the sun across a circle marked off into segments corresponding to the constellations in the zodiac. Like a modern watch, it indicated both the hour and the month. Indeed, in the measurement of time in particular the ancient engineers produced truly magnificent instruments, that are admired even today.

His works include *Mirror* and *Phaenomena*, as well as others which are lost.

**Comments of Aristotle**

*Eudoxus supposed that the motion of the sun or of the moon involves, in either case, three spheres, of which the first is the sphere of the fixed stars, and the second moves in the circle which runs along the middle of the zodiac, and the third in the circle which is inclined across the breadth of the zodiac; but the circle in which the moon moves is inclined at a greater angle than that in which the sun moves. And the motion of the planets involves, in each case, four spheres, and of these also the first and second are the same as the first two mentioned above (for the sphere of the fixed stars is that which moves all the other spheres, and that which is placed beneath this and has its movement in the circle which bisects the zodiac is common to all), but the poles of the third sphere of each planet are in the circle which bisects the zodiac, and the motion of the fourth sphere is in the circle which is inclined at an angle to the equator of the third sphere; and the poles of the third sphere are different for each of the other planets, but those of Venus and Mercury are the same.
Callippus made the position of the spheres the same as Eudoxus did, but while he assigned the same number as Eudoxus did to Jupiter and to Saturn, he thought two more spheres should be added to the sun and two to the moon, if one is to explain the observed facts; and one more to each of the other planets.
But it is necessary, if all the spheres combined are to explain the observed facts, that for each of the planets there should be other spheres (one fewer than those hitherto assigned) which counteract those already mentioned and bring back to the same position the outermost sphere of the star which in each case is situated below the star in question; for only thus can all the forces at work produce the observed motion of the planets. Since, then, the spheres involved in the movement of the planets themselves are--eight for Saturn and Jupiter and twenty-five for the others, and of these only those involved in the movement of the lowest-situated planet need not be counteracted the spheres which counteract those of the outermost two planets will be six in number, and the spheres which counteract those of the next four planets will be sixteen; therefore the number of all the spheres--both those which move the planets and those which counteract these--will be fifty-five. And if one were not to add to the moon and to the sun the movements we mentioned, the whole set of spheres will be forty-seven in number. *Aristotle, Metaphysics, Book XII

**LINKS**

The Homocentric Spheres of Eudoxus.

**The crystalline celestial spheres **

**Models of Planetary Motion from Antiquity to the Renaissance**

**Eudoxus and an Introduction to the method of exhaustion**

**Eudoxus Meets Cayley** (PDF File)

**Reports in Greek**

Το ξεπέρασμα της πρώτης κρίσης

Eudoxus Lunar Crater

Heath, Greek Astronomy , Dover Publications, Incorporated, 1991 reprint (from 1932 original)

C.M. Linton. From Eudoxus to Einstein : A History of Mathematical Astronomy C.M. Linton. Cambridge, UK ; New York : Cambridge University Press, 2004

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