Download Ptolemy and the Puzzle of the Planets

Ptolemy and the Puzzle of the
Planets
Puzzle of “wandering stars”
Irregular speeds through sky
 Move W to E, roughly along ecliptic
 Retrograde from E to W, varying loops
(not the major luminaries)
 Changing brightness
 Maximal elongations for Mercury (28°)
and Venus (47°)

Babylonians on the planets
Earliest known planetary observations
 Venus tablet (-1760)

 Dates

of appearances/disappearances
Predictive planet astrology (-300)
 Lists
of dates for oppositions, entry into
zodiacal signs
 Based on linear zig-zag functions
 NO geometrical model or explanatory
(structural) theory
Plato’s legacy
‘Save the phenomena’ quantitatively
 Only uniform, circular motion
 Crystalline spheres, concentric to the
Earth at the center of the cosmos
 Spheres may have tilted axes

 Eudoxus’s
hippopede (all retrograde loops
have fixed shape and size)
Aristotle’s legacy
Celestial/terrestrial realms
 Aether and circular motion in heavens
 Heavy earth at center of cosmos
 Plenum cosmos of 56 spheres
 Physical rather than quantitative or
predictive model

Task of lecture
Greek measurements of the cosmos
 Apollonius’s invention of non-Platonic
mathematical models for planetary
motion (-200)
 Ptolemy’s mathematical models,
influential for 1400 years (+150)

Measuring the cosmos

Eratosthenes (c. -270, Alexandria)
 Circumference

of the Earth
Aristarchus (c. -290)
 Relative
Sun - Moon distances
 Absolute Sun- Moon sizes

Hipparchus (c. -130)
 850
stellar positions (long. and lat.)
 Precession of equinoxes
 Constructed lunar & solar models
Eratosthenes on the
circumference of the Earth
Assumes:
--Spherical earth
--Incoming solar rays
are parallel
--Euclidean geometry
Sunlight at
noon
Alexandria
α
α
Earth
Syrene
Alexandria to Syrene =
5000 stades (measured)
α= 1/50 circle (measured)
Thus, circumference =
250,000 stades!
Aristarchus on bisected Moon
(relative distances)
Moon
Measure α when Moon is
exactly at quarter
If α = 87°, ES/EM = 19
Sun
α
Earth
Aristarchus on lunar eclipses
(relative sizes)
Moon
Earth
Sun
Measure length of time Moon remains in shadow
Finds Dias = 6 3/4 Diae, Diam = 1/3 Diae
Hipparchus’s armillary sphere
Latitude (β)
Longitude (λ)
Apollonius’ models (c. -200)
Planet
slower
α
Center
Earth
Earth
faster
Eccentric (off-center
Earth) model
Epicycle model
(both equivalent, but
anti-Aristotlian!)
Ptolemy (ca. +100 - 170)
Alexandria in Egypt
 Works on Geography, Optics, Harmonica,
and in astronomy/astrology:

Tetrabiblos
 Handy Tables
 Planetary Hypotheses
 Mathematical syntaxis (compilation) = “The
Greatest” = Almagest

Models of the Almagest

Seven predictive, quantitative, independent
models for 5 planets, Sun & Moon
Eccentric (saves unequal speeds) from Apollonius
 Epicycle (saves retrograde motion, brightness)
 Equant (saves varying retro loops) - NEW
 Central cranks for Mercury and Moon - NEW
 Methods for determining parameters from
selected observations (Ptolemy’s most original
contribution)
 Result: Add ca. 20 numbers and get λ, β

Superior planetary model
Observed planetary position
Epicycle
Center of
Uniform
motion
Planet
α
Equant
Earth
Deferent (roughly in plane of ecliptic)
Evaluation of Ptolemy models



Violates uniform motion (equant)
Violates concentric spheres (epicycle)
Saves all known phenomena




but creates problem for lunar sizes!
Predicts positions to ± 1° accuracy
No absolute distances, physical status of the
circles remains ambiguous
Unexplained dependency of models on solar
position
Unexplained links to Sun!
Mercury
Earth
Sun
Venus
Mars
Jupiter
Conclusions re Ptolemy

Defines mathematical astronomy until 1600



Contrasts with physical cosmology of Aristotle &
Eudoxus (concentric spheres)
Successfully “saves the phenomena” with
circles
Cheating on Plato (uniform circular motion)
creates major research problem for medieval
Islamic astronomers