# Lecture 13: The Harmony of the Spheres: Greek Astronomy

## Key Ideas:

Early Geocentric Systems:
• Anaximander
• Pythagoras, Eudoxus, & Aristotle

Early Heliocentric System:

• Aristarchus of Samos

Epicyclic Geocentric Systems:

• Hipparchus of Nicaea
• Claudius Ptolemy

## Summary of Celestial Motions

Fixed Stars:
• Uniform daily motion about the celestial poles.

The Sun:

• Daily motion around the celestial poles (rising and setting).
• Eastward motion along the Ecliptic over a year
• The seasons are of unequal length - the Sun moves a little faster in winter, slower in summer.

The Moon:

• Daily motion around the celestial poles.
• Eastward motion near the Ecliptic over a month.

The Planets:

• Daily motions about the celestial poles.
• Generally eastward motion near the Ecliptic at different speeds for each planet.
• Superior planets move westward (retrograde) around opposition
• Inferior planets move retrograde at inferior conjunction
• Superior Planets are brightest at opposition.

Any successful description of the Solar System must explain all these facts.

## The Geocentric System

Geocentric = Earth-Centered

Anaximander of Miletus (611-546 BC)

Among the first Greek philosophers to suggest a geocentric system:

• Earth was a flat disk (cylinder) fixed and unmoving at the center.
• Sun, Moon & Stars were affixed to rotating crystalline spheres centered on the Earth.
• Sun, Moon & Stars were physical objects.
This was an interesting mix of old (flat earth) and new ideas that set the basic themes for much of what was to follow.

Pythagoras of Samos (569-475 BC)

Philosopher & Mathematician, founded the Pythagorean school. Taught that spheres are the perfect geometric shapes.

Pythagorean Model:

• Spherical Earth fixed at the center
• Planets & Stars on concentric crystalline spheres
• Sizes were ratios of small numbers (e.g., 2:1, 3:2)

Vibrations from their rubbing together created a harmonious "Music of the Spheres."

## Eudoxus of Cnidos (b. 408 BC)

A pupil of Plato, Eudoxus elaborated a geocentric model composed of crystalline spheres, incorporating the Platonic ideal of uniform circular motion.

System of 27 Spheres:

• 1 for the fixed stars
• 3 each for the Sun and Moon
• 4 each for the 5 planets
Spheres within spheres in perfect circular motion combine to give retrograde motions.

Spheres within Spheres

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Four Spheres for each planet:

• One was aligned with the celestial poles, turning once a day to give rising & setting.
• Second was tilted 23.5°, rotated slowly in the opposite direction to give the usual west-to-east drift of the planets relative to the fixed stars.
• Third & Fourth were introduced to produce the periodic retrograde motions of the planets.
All were in uniform circular motion about their axes.

## Aristotle (384-322 BC)

Another pupil of Plato, the tutor of Alexander the Great, and considered the greatest general authority in antiquity. Aristotle wrote about virtual everything known at his time.

His On the Heavens refined Eudoxus' system:

• 55 crystalline spheres within spheres

Incorporated physical reasoning:

• Earth fixed and unmoving at the center as it was too big to move, including rotation.
• All spheres were in uniform circular motion.
• Combination of perfect motions produced net retrograde and non-uniform motions.

Earth, Air, Fire, & Water

The Aristotelian World View made certain basic assumptions:

• The Earth was a sphere, fixed and unmoving at the center of the Universe.
• The natural state of motion on the Earth was rest.
• The natural state of the Heavens was unceasing uniform circular motion (or combinations of such motions).
A rotating or revolving Earth would thus be "unnatural" in this world view.

Preserving Appearances

The models proposed sought primarily to "preserve appearances":

• Started from philosophical ideals.
• Built systems to conform to these ideals.
• Adjusted to provide accurate predictions of planetary motions.
• No need to explain the physical "causes" of the motions.
The Aristotelian view of the world was that planetary motions belonged to the perfect celestial realm, and were thus to be understood without needing to inquire as to their physical "causes". In this sense, Aristotle linked Astronomy with Mathematics, where one explored, say, the geometric properties of triangles without any need to ask why they were that way. Mathematics and Astronomy were seen as expressions of perfection in and of themselves. This is unlike the study of Physics, which to Aristotle was concerned with understanding the causes of phenomena in the ever-changing realm of the Earth. This conceptual separation of Astronomy from Physics was to have a profound impact upon the Medieval and Renaissance mind centuries later.

## The Heliocentric System:

Heliocentric = Sun-Centered

Alternative viewpoint to the Geocentric System:

• Puts the Sun, not the Earth, at the center.
• The Earth rotates and revolves around the Sun.
• Stars are on an outermost celestial sphere.
The complex non-uniform and retrograde motions are now a consequence of viewing the moving planets from a moving Earth.

## Aristarchus of Samos (310-230BC)

Proposed a Heliocentric system. It seems that his reasoning was the large size he found for the Sun.
• Showed geometrically that the Sun was at least 20x further than the Moon.
• Really 400x further: sound method, but inadequate data.
• Meant Sun was 5x bigger than the Earth (more like 109x, again, sound method but inadequate data).
• Makes it even more absurd that the Sun should move, if the Earth was too large to move according to Aristotle.

(Click on the image to view at full scale [Size: 9Kb])

We know none of the details of Aristarchus' Heliocentric model. We only have his treatise on the distances and sizes of the Moon and the Sun, from which we gather what his motivations might have been. We know of his Heliocentric model only from mention of it (usually dismissive) by others who came after him. Our primary surviving source is the Sand Reckoner of Archimedes.

The heliocentric picutre never caught on, perhaps because it was considered too radical given deeply ingrained notions about uniform circular motion and the immobility of the Earth.

## Hipparchus of Nicaea (165-127 BC)

Greatest astronomer of the classical period:
• Discovered the Precession of the Equinoxes
• Developed the system of stellar magnitudes
• Introduced the Babylonian angular notation of 360° in a circle.

Developed a New and Improved Geocentric System:

• Introduced epicycles, building on ideas of Apollonius of Perga.
• Located the Earth slightly off-center (eccentric)

(Click on the image to view at full scale [Size: 7Kb])

The main circle is the Deferent, to which is afixed a second, smaller circle called the Epicycle ("on the circle" in Greek), to which the planet is affixed. The two circles rotate counterclockwise at different rates fine-tuned to make the apparent motions as seen from the Earth come out right. Additional epicycles can be added to further fine-tune the system. Notice that the Earth is not exactly at the center of the Deferent, but slightly offset at the "eccentric" point.

## Successes of Epicycles & Eccentrics

Epicyclic models have a number of successes:
• Combined motion of deferent and epicycle produces retrograde motion.
• Superior planets are brighter at opposition, when moving retrograde.
• Placing the Earth at an eccentric away from the deferent center helps explain the observed non-uniform motions of the Sun, Moon, and Planets.
• Can fine-tune the model by adding epicycles

Note:

What distinguishes Hipparchus' geocentric model from all previous models is that it was firmly grounded in observational data, many observations of which he made himself (by all accounts Hipparchus was the supreme observational astronomer of the classical period). In many ways, this work marks a turning point between models motivated primarily by philosophical aesthetics and models based at least in part upon observational data. Hipparchus still sought to preserve appearances, and chose a reasonable (to him) geometric model to represent planetary motions mathematically.

## Claudius Ptolemais (Ptolemy - c. 150 AD)

Astronomer and Geographer of the late classical age based at Alexandria, then a colonial outpost of the Roman Empire.

Wrote the Mathematical Syntaxis:

• Compilation of all Mathematical & Astronomical knowledge of his time.
• Known to us in Arabic translations that hailed it "Al Magest" ("The Greatest").
• Vastly influential work in medieval Europe after the 11th century.

Elaborated Hipparchus' geocentric system, adding extra features to better preserve appearances.

The Equant

Ptolemy introduced the Equant, a geometric device to account for observed changes in a planet's speed as it moved around the Earth.

(Click on the image to view at full scale [Size: 6Kb])

• The Epicycle still moves about the center of the Deferent, but
• Uniform circular motion about the center of the deferent is replaced by uniform angular motion about an off-center equant point.
This is a very complex construction that, again, was introduced to preserve appearances, this time by replacing the pure Aristotelian and Platonic ideal of Uniform Circular Motion with a slightly modified notion of uniform angular motion about the equant.

## The Ultimate Geocentric System

Ptolemy's final geocentric system was quite complex:
• 40 epicycles and deferents.
• Equants and eccentrics for all planets, the Moon, and Sun
• With only minor modifications, it provided accurate predictions of the motions of the planets, Sun, and Moon.

It was to prevail virtually unchallenged for nearly 1500 years.