Chris Peterson wrote:neufer wrote:To explain both Mars's retrograde motion and variable distance
in a geocentric model always required
epicycles which,
if small enough,
resembled off center circles NOT ellipses.
Yes, but have you seen epicycles used to explain the varying distance of the Moon? I don't recall seeing that explicitly. While epicycles cause a planet's distance from Earth to change, I don't think that effect was observed, only the effect on position- retrograde and prograde motion.
The moon is not treated explicitly
(except for the "unusual" fact that it shines by reflected light from the sun)
because
ALL planets are on epicycles.
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http://en.wikipedia.org/wiki/Deferent_and_epicycle wrote:
<<
In the Ptolemaic system of astronomy, the epicycle (literally: on the circle in Greek) was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets. It was designed by Apollonius of Perga at the end of the 3rd century BC. In particular it explained the retrograde motion of the five planets known at the time. Secondarily, it also explained changes in the apparent distances of the planets from Earth.
http://en.wikipedia.org/wiki/Apollonius_of_Perga wrote:
<<Apollonius of Perga [Pergaeus] (Ancient Greek: Ἀπολλώνιος) (ca. 262 BC–ca. 190 BC) was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes. It was Apollonius who gave the ellipse, the parabola, and the hyperbola the names by which we know them. The hypothesis of eccentric orbits, or equivalently, deferent and epicycles, to explain the apparent motion of the planets
and the varying speed of the Moon, are also attributed to him. Apollonius' theorem demonstrates that the two models are equivalent given the right parameters. Ptolemy describes this theorem in the Almagest XII.1. Apollonius also researched the lunar theory, for which he is said to have been called Epsilon (ε). The crater Apollonius on the Moon is named in his honor.>>
In the Ptolemaic system, the planets are assumed to move in a small circle, called an epicycle, which in turn moves along a larger circle called a deferent. Both circles rotate eastward and are roughly parallel to the plane of the Sun's orbit (ecliptic). The orbits of planets in this system are epitrochoids.
The deferent was a circle centered around a point halfway between the equant and the earth. The epicycle rotated on the deferent with uniform motion, not with respect to the center, but with respect to the off-center point called the equant. The rate at which the planet moved on the epicycle was fixed such that the angle between the center of the epicycle and the planet was the same as the angle between the earth and the sun.
Ptolemy did not predict the relative sizes of the planetary deferents in the Almagest. All of his calculations were done with respect to a normalized deferent. This is not to say that he believed the planets were all equidistant. He did guess at an ordering of the planets. Later he calculated their distances in the Planetary Hypotheses.
For superior planets the planet would typically rotate in the night sky slower than the stars. Each night the planet would "lag" a little behind the star. This is prograde motion. Occasionally, near opposition, the planet would appear to rotate in the night sky faster than the stars. This is retrograde motion. Ptolemy's model, in part, sought to explain this behavior.
The inferior planets were always observed to be near the sun, appearing only shortly before sunrise or shortly after sunset. To accommodate this, Ptolemy's model fixed the motion of Mercury and Venus so that the line from the equant point to the center of the epicycle was always parallel to the earth-sun line.>>
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http://en.wikipedia.org/wiki/Geocentric_model wrote:
<<In the 4th century BC, two influential Greek philosophers wrote works based on the geocentric model. These were Plato and his student Aristotle. According to Plato, the Earth was a sphere, stationary at the center of the universe. The stars and planets were carried around the Earth on spheres or circles, arranged in the order (outwards from the center): Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn, fixed stars. In the "Myth of Er," a section of the Republic, Plato describes the cosmos as the Spindle of Necessity, attended by the Sirens and turned by the three Fates. 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. Aristotle elaborated on Eudoxus' system. In the fully developed Aristotelian system, the spherical Earth is at the center of the universe. All heavenly bodies are attached to 56 concentric spheres which rotate around the Earth . (The number is so high because several transparent spheres are needed for each planet.) The Moon is on the innermost sphere. Thus it touches the realm of Earth, which contaminates it, causing the dark spots (macula) and the ability to go through lunar phases. It is not perfect like the other heavenly bodies, which shine by their own light.>>
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[quote="Chris Peterson"][quote="neufer"]To explain both Mars's retrograde motion and variable distance
in a geocentric model always required [b]epicycles[/b] which,
if small enough, [b][url=http://en.wikipedia.org/wiki/File:1550_SACROBOSCO_Tractatus_de_Sphaera_-_(16)_Ex_Libris_rare_-_Mario_Taddei_.JPG]resembled off center circles[/url] NOT ellipses[/b].[/quote]
Yes, but have you seen epicycles used to explain the varying distance of the Moon? I don't recall seeing that explicitly. While epicycles cause a planet's distance from Earth to change, I don't think that effect was observed, only the effect on position- retrograde and prograde motion.[/quote]
The moon is not treated explicitly
(except for the "unusual" fact that it shines by reflected light from the sun)
because [b]ALL planets are on epicycles[/b].
----------------------------------------------------------------
[quote=" http://en.wikipedia.org/wiki/Deferent_and_epicycle"]
<<[b]In the Ptolemaic system of astronomy, the epicycle (literally: on the circle in Greek) was a geometric model used to explain the variations in speed and direction of the apparent motion of the [color=#0000FF]Moon[/color], Sun, and planets.[/b] It was designed by Apollonius of Perga at the end of the 3rd century BC. In particular it explained the retrograde motion of the five planets known at the time. Secondarily, it also explained changes in the apparent distances of the planets from Earth.
[quote=" http://en.wikipedia.org/wiki/Apollonius_of_Perga"]
<<Apollonius of Perga [Pergaeus] (Ancient Greek: Ἀπολλώνιος) (ca. 262 BC–ca. 190 BC) was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes. It was Apollonius who gave the ellipse, the parabola, and the hyperbola the names by which we know them. The hypothesis of eccentric orbits, or equivalently, deferent and epicycles, to explain the apparent motion of the planets [b]and the varying speed of the [color=#0000FF]Moon[/color][/b], are also attributed to him. Apollonius' theorem demonstrates that the two models are equivalent given the right parameters. Ptolemy describes this theorem in the Almagest XII.1. Apollonius also researched the lunar theory, for which he is said to have been called Epsilon (ε). The crater Apollonius on the Moon is named in his honor.>>[/quote]
In the Ptolemaic system, the planets are assumed to move in a small circle, called an epicycle, which in turn moves along a larger circle called a deferent. Both circles rotate eastward and are roughly parallel to the plane of the Sun's orbit (ecliptic). The orbits of planets in this system are epitrochoids.
The deferent was a circle centered around a point halfway between the equant and the earth. The epicycle rotated on the deferent with uniform motion, not with respect to the center, but with respect to the off-center point called the equant. The rate at which the planet moved on the epicycle was fixed such that the angle between the center of the epicycle and the planet was the same as the angle between the earth and the sun.
Ptolemy did not predict the relative sizes of the planetary deferents in the Almagest. All of his calculations were done with respect to a normalized deferent. This is not to say that he believed the planets were all equidistant. He did guess at an ordering of the planets. Later he calculated their distances in the Planetary Hypotheses.
For superior planets the planet would typically rotate in the night sky slower than the stars. Each night the planet would "lag" a little behind the star. This is prograde motion. Occasionally, near opposition, the planet would appear to rotate in the night sky faster than the stars. This is retrograde motion. Ptolemy's model, in part, sought to explain this behavior.
The inferior planets were always observed to be near the sun, appearing only shortly before sunrise or shortly after sunset. To accommodate this, Ptolemy's model fixed the motion of Mercury and Venus so that the line from the equant point to the center of the epicycle was always parallel to the earth-sun line.>>[/quote]----------------------------------------------------------------
[quote=" http://en.wikipedia.org/wiki/Geocentric_model"]
<<In the 4th century BC, two influential Greek philosophers wrote works based on the geocentric model. These were Plato and his student Aristotle. According to Plato, the Earth was a sphere, stationary at the center of the universe. The stars and planets were carried around the Earth on spheres or circles, arranged in the order (outwards from the center): Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn, fixed stars. In the "Myth of Er," a section of the Republic, Plato describes the cosmos as the Spindle of Necessity, attended by the Sirens and turned by the three Fates. 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. Aristotle elaborated on Eudoxus' system. In the fully developed Aristotelian system, the spherical Earth is at the center of the universe. All heavenly bodies are attached to 56 concentric spheres which rotate around the Earth . (The number is so high because several transparent spheres are needed for each planet.) The Moon is on the innermost sphere. Thus it touches the realm of Earth, which contaminates it, causing the dark spots (macula) and the ability to go through lunar phases. It is not perfect like the other heavenly bodies, which shine by their own light.>>[/quote]--------------------------------