For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy .< ref name =" Stillwell Infinite Series Early Results "> Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.

These errors are partly to be attributed to Eudoxus himself, and partly to the way in which Aratus has used the materials supplied by him.

It may have been inspired by the non-Ptolemaic system of Girolamo Fracastoro, who used either 77 or 79 orbs in his system inspired by Eudoxus of Cnidus.

He was the first to realize that the concentric spheres of Eudoxus of Cnidus and Callippus, unlike those used by many astronomers of later times, were not to be taken as material objects, but only as part of an algorithm similar to the modern Fourier series.

It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Gauss.

For the proof of the results Archimedes used the Method of exhaustion of Eudoxus.

Eudoxus of Cnidus used the idea of a sphere to explain how the sun created differing climatic zones based on latitude.

Callippus, a Greek astronomer of the 4th century, added seven spheres to Eudoxus ' original 27 ( in addition to the planetary spheres, Eudoxus included a sphere for the fixed stars ).

The cosmological model of concentric or homocentric spheres, developed by Eudoxus, Callippus, and Aristotle, employed celestial spheres all of which had the same center, the Earth.

Instead of bands, Plato's student Eudoxus developed a planetary model using concentric spheres for all the planets, with three spheres each for his models of the Moon and the Sun and four each for the models of the other five planets, thus making 26 spheres in all. Callippus modified this system, using five spheres for his models of the Sun, Moon, Mercury, Venus, and Mars and retaining four spheres for the models of Jupiter and Saturn, thus making 33 spheres in all.

* Phaenomena ( Φαινόμενα ) and Entropon ( Ἔντροπον ), on spherical astronomy, probably based on observations made by Eudoxus in Egypt and Cnidus

In Aristotle's fully developed celestial model, the spherical Earth is at the centre of the universe and the planets are moved by either 47 or 55 interconnected spheres that form a unified planetary system, whereas in the models of Eudoxus and Callippus each planet's individual set of spheres were not connected to those of the next planet.

Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes.

* Eudoxus of Cnidus, Greek philosopher and astronomer who has expanded on Plato's ideas ( or 355 BC ) ( b. 410 BC or 408 BC )

Eudoxus of Cnidus, who worked with Plato, developed a less mythical, more mathematical explanation of the planets ' motion based on Plato's dictum stating that all phenomena in the heavens can be explained with uniform circular motion.

" It is possible that the dispute between Speusippus and Eudoxus influenced Plato's Philebus ( esp.

" ( According to Simplicius, Plato's colleague Eudoxus was the first to have worked on this problem.

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.

By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Eudoxus.

Induced by the need to understand and operate with incommensurable quantities, Eudoxus established what may have been the first deductive organization of mathematics on the basis of explicit axioms.

Although the models of Eudoxus and Callippus qualitatively describe the major features of the motion of the planets, they fail to account exactly for these motions and therefore cannot provide quantitative predictions.

Astronomers such as Eudoxus ( contemporary with Plato ) observed planetary motions and cycles, and created a geocentric cosmological model that would be accepted by Aristotle – this model generally lasted until Ptolemy, who added epicycles to explain the retrograde motion of Mars.

The first geometrical, three-dimensional models to explain the apparent motion of the planets were developed in the 4th century BC by Eudoxus of Cnidus and Callippus of Cyzicus.

* The Ptolemaic model of planetary motion: Based on the geometrical model of Eudoxus of Cnidus, Ptolemy's Almagest, demonstrated that calculations could compute the exact positions of the Sun, Moon, stars, and planets in the future and in the past, and showed how these computational models were derived from astronomical observations.

However, Eudoxus ' importance to Greek astronomy is considerable, as he was the first to attempt a mathematical explanation of the planets.

He observed the movements of the planets and attempted to use Eudoxus ' scheme of connected spheres to account for their movements.

Theodosius of Bithynia's important work, Sphaerics, may be based on a work of Eudoxus.

Eudoxus is considered by some to be the greatest of classical Greek mathematicians, and in all antiquity, second only to Archimedes.

The Phaenomena appears to be based on two prose works — Phaenomena and Enoptron ( Ἔνοπτρον " Mirror ", presumably a descriptive image of the heavens )— by Eudoxus of Cnidus, written about a century earlier.

Aristotle says the exact number of spheres, and hence the number of movers, is to be determined by astronomical investigation, but he added additional spheres to those proposed by Eudoxus and Callippus, to counteract the motion of the outer spheres.

Eudoxus also accepts that the good will be that at which all people aim, but identifies this as pleasure, as opposed to Speusippus ' exclusive focus on moral goods.

Fra Mauro also comments that the account of this expedition, together with the relation by Strabo of the travels of Eudoxus of Cyzicus from Arabia to Gibraltar through the southern Ocean in Antiquity, led him to believe that the Indian Ocean was not a closed sea and that Africa could be circumnavigated by her southern end ( Text from Fra Mauro map, 11, G2 ).

Becker also showed that all the theorems of Euclidean proportion theory could be proved using an earlier alternative to the Eudoxus technique which Becker found stated in Aristotle's Topics, and which Becker attributes to Theatetus.

He criticized both Aristotle and Eudoxus for their imperfect theory of celestial spheres and also the use of epicycles, which he felt to be inconsistent with Aristotle's philosophical postulates.

Some Pythagoreans, such as Eudoxus ' teacher Archytas, had believed that only arithmetic could provide a basis for proofs.

Eudoxus was able to restore confidence in the use of proportionalities by providing an astounding definition for the meaning of the equality between two ratios.

We are fairly well informed about the contents of Phaenomena, for Eudoxus ' prose text was the basis for a poem of the same name by Aratus.

Aristotle, in The Nicomachean Ethics 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:

* Eudoxus of Cnidus develops the method of exhaustion for mathematically determining the area under a curve.

According to Plutarch, Plato gave the problem to Eudoxus and Archytas and Menaechmus, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry ( Plut., Quaestiones convivales VIII. ii, 718ef ).

Purbach is also noted for his great attempt to reconcile the opposing theories of the universe, the so-called homocentric spheres of Eudoxus of Cnidus and Aristotle, with Ptolemy's epicyclic trains.

When Eudoxus was returning from his second voyage to India the wind forced him south of the Gulf of Aden and down the coast of Africa for some distance.

To the south of Aristoteles lies the slightly smaller crater Eudoxus and these two form a distinctive pair for a telescope observer.

Euclid is known for his Elements, much of which was drawn from his predecessor Eudoxus of Cnidus.

Hippopedes were also investigated by Proclus ( for whom they are sometimes called Hippopedes of Proclus ) and Eudoxus.