Claudius Ptolemy (90-168) created a wealth of astronomical knowledge from his home in Alexandria, Egypt. Ptolemy's success at synthesizing and refining ideas and improvements in astronomy helped make his Almagest so popular that earlier works fell out of circulation. In this method, a planet orbits a circle whose center is itself on a second circle. For the Greeks these were the wandering stars. (ii) Empirical observation: But the heavenly bodies (with the exception of the stars, which appear always to move in simple circular motion about the earth) do not demonstrate simple circular motions - inlcuding the phenomena of retrograde motion, varying brightnesses, and varying velocities. In Aristotle's words, "In the whole range of time past, so far as our inherited records reach, no change appears to have taken place either in the whole scheme of the outermost heaven or in any of its proper parts." In Homer’s Iliad and Odyssey, stars and constellations are mentioned, including Orion, the Great Bear (Ursa Major), Boötes, Sirius, and the Pleiades. Eudoxus (408-355 BC) was a student of Plato's who attempted a model made up of concentric spheres. This view, associated with a materialist perspective on life and death, was most clearly expressed in the epic poem of Lucretius (d. 55BC), The Nature of the Universe, and represented a radical departure from the traditional views of Plato and Aristotle, which latter enjoyed great popularity and near universal assent. When the Earth is in-between the sun and the moon it causes a lunar eclipse and measuring the size of the Earth's shadow on the moon provided part of the information he needed to calculate its size. (It can be proved mathematically that the epicycle and eccentric methods give equivalent results). Aristotle accepted the four Platonic elements of Earth, Water, Air and Fire as the basis for phenomena on both the Earth (the planet) and in the atmosphere, but he added a fifth element -- known as "aether" -- as the matter of the heavenly bodies (moon, planets, sun, and stars). (iii) The solar seasons were not quite equal, as should be expected for perfectly circular motion. (ii) The cosmos consists of many different worlds, randomnly formed by the collision of atoms, not just the one earth-centered world of Aristotle. Babylonian and Egyptian astronomers developed systems that became the basis for Greek astronomy, while societies in the Americas, China and India developed their own. In this method, a planet orbits a circle whose center is itself on a second circle. (ii) The cosmos consists of many different worlds, randomnly formed by the collision of atoms, not just the one earth-centered world of Aristotle. In contrast to the terrestrial, the celestial region of the heavens had a fundamentally different nature. The number of epicycles, eccentrics, and equants used was "ad hoc" -- just enough were used to obtain the desired result, but there was no way of knowing in advance how many were required. For decades and indeed centuries the theory, contained in his book "Almagest" (Arabic for "The Greatest") enabled accurate predictions to be made. Eudoxus of Cnidus constructed the first Greek theory of planetary motion of which any details are known. But there were problems with the theory as expressed above, particularly as concerns the orbits of the sun and the planets, which did not seem to describe a perfect circle (by naked eye observation, using at most a sighting post). 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