Claudius Ptolemy
Saving the Heavens
SC/STS 3760, V 1
Logic at its Best
• Where Plato and Aristotle agreed was
over the role of reason and precise logical
thinking.
• Plato: From abstraction to new
abstraction.
• Aristotle: From empirical generalizations to
unknown truths.
SC/STS 3760, V 2
Mathematical Reasoning
• Plato’s Academy excelled in training
mathematicians.
• Aristotle’s Lyceum excelled in working out
logical systems.
• They came together in a great
mathematical system.
SC/STS 3760, V 3
The Structure of Ancient Greek
Civilization
• Ancient Greek
civilization is divided
into two major
periods, marked by
the death of
Alexander the Great.
SC/STS 3760, V 4
Hellenic Period
• From the time of Homer to the death of
Alexander is the Hellenic Period, 800-323
BCE.
– When the written Greek language evolved.
– When the major literary and philosophical
works were written.
– When the Greek colonies grew strong and
were eventually pulled together into an empire
by Alexander the Great.
SC/STS 3760, V 5
Hellenistic Period
• From the death of Alexander to the
annexation of the Greek peninsula into the
Roman Empire, and then on with
diminishing influence until the fall of Rome.
• 323 BCE to 27 BCE, but really continuing
its influence until the 5th century CE.
SC/STS 3760, V 6
Science in the Hellenistic Age
• The great philosophical works were written
in the Hellenic Age.
• The most important scientific works from
Ancient Greece came from the Hellenistic
Age.
SC/STS 3760, V 7
Alexandria, Egypt
• Alexander the Great conquered Egypt,
where a city near the mouth of the Nile
was founded in his honour.
• Ptolemy Soter, Alexander’s general in
Egypt, established a great center of
learning and research in Alexandria: The
Museum.
SC/STS 3760, V 8
The Museum
• The Museum – temple to the Muses –
became the greatest research centre of
ancient times, attracting scholars from all
over the ancient world.
• Its centerpiece was the Library, the
greatest collection of written works in
antiquity, about 600,000 papyrus rolls.
SC/STS 3760, V 9
Euclid
• Euclid headed up
mathematical studies at
the Museum.
• Little else is known about
his life. He may have
studied at Plato’s
Academy.
SC/STS 3760, V 10
Euclid’s Elements
• Euclid is now remembered for only one
work, called The Elements.
• 13 ―books‖ or volumes.
• Contains almost every known
mathematical theorem, with logical proofs.
SC/STS 3760, V 11
The Influence of the Elements
• Euclid’s Elements is the second most
widely published book in the world, after
the Bible.
• It was the definitive and basic textbook of
mathematics used in schools up to the
early 20th century.
SC/STS 3760, V 12
Axioms
• What makes Euclid’s Elements distinctive
is that it starts with stated assumptions
and derives all results from them,
systematically.
• The style of argument is Aristotelian logic.
• The subject matter is Platonic forms.
SC/STS 3760, V 13
Axioms, 2
• The axioms, or assumptions, are divided
into three types:
– Definitions
– Postulates
– Common notions
• All are assumed true.
SC/STS 3760, V 14
Definitions
• The definitions simply clarify what is meant by technical
terms. E.g.,
– 1. A point is that which has no part.
– 2. A line is breadthless length.
– 10. When a straight line set up on a straight line makes the
adjacent angles equal to one another, each of the equal angles
is right, and the straight line standing on the other is called a
perpendicular to that on which it stands. …
– 15. A circle is a plane figure contained by one line such that all
the straight lines falling upon it from one point among those lying
within the figure are equal to one another.
SC/STS 3760, V 15
Postulates
• There are 5 postulates.
• The first 3 are ―construction‖ postulates, saying
that he will assume that he can produce
(Platonic) figures that meet his ideal definitions:
– 1. To draw a straight line from any point to any point.
– 2. To produce a finite straight line continuously in a
straight line.
– 3. To describe a circle with any centre and distance.
SC/STS 3760, V 16
Postulate 4
• 4. That all right angles are equal to one another.
• Note that the equality of right angles was not
rigorously implied by the definition.
– 10. When a straight line set up on a straight line
makes the adjacent angles equal to one another,
each of the equal angles is right….
– There could be other right angles not equal to these.
The postulate rules that out.
SC/STS 3760, V 17
The Controversial Postulate 5
d a e
c
b
f g
• 5. That, if a straight line falling on two straight lines
make the interior angles on the same side less than
two right angles, the two straight lines, if produced
indefinitely, meet on that side on which are the
angles less than the two right angles.
SC/STS 3760, V 18
The ―Parallel‖ Postulate
d a e
c
b
f g
• One of Euclid’s definitions was that lines
are parallel if they never meet.
• Postulate 5, usually called the parallel
postulate, gives a criterion for lines not
being parallel.
SC/STS 3760, V 19
The Common Notions
• Finally, Euclid adds 5 ―common notions‖ for
completeness. These are really essentially logical
principles rather than specifically mathematical ideas:
– 1. Things which are equal to the same thing are also equal to
one another.
– 2. If equals be added to equals, the wholes are equal.
– 3. If equals be subtracted from equals, the remainders are equal.
– 4. Things which coincide with one another are equal to one
another.
– 5. The whole is greater than the part.
SC/STS 3760, V 20
An Axiomatic System
• After all this preamble, Euclid is finally
ready to prove some mathematical
propositions.
• The virtue of this approach is that the
assumptions are all laid out ahead.
Nothing that follows makes further
assumptions.
SC/STS 3760, V 21
Axiomatic Systems
• The assumptions are clear and can be referred to.
• The deductive arguments are also clear and can be
examined for logical flaws.
• The truth of any proposition then depends entirely on the
assumptions and on the logical steps.
• And, the system builds. Once some propositions are
established, they can be used to establish others.
– Aristotle’s methodology applied to mathematics.
SC/STS 3760, V 22
Building Knowledge with an
Axiomatic System
• Generally agreed upon premises ("obviously"
true)
• Tight logical implication
• Proofs by:
– 1. Construction
– 2. Exhaustion
– 3. Reductio ad absurdum (reduction to absurdity)
• -- assume a premise to be true
• -- deduce an absurd result
SC/STS 3760, V 23
Example: Proposition IX.20
• There is no limit to the number of prime
numbers
• Proved by
– 1. Constructing a new number.
– 2. Considering the consequences whether it is
prime or not (method of exhaustion).
– 3. Showing that there is a contraction if there
is not another prime number. (reduction ad
absurdum).
SC/STS 3760, V 24
Proof of Proposition IX.20
• Example 1: {2,3,5}
• Given a set of prime numbers, • Q=2x3x5+1 =31
{P1,P2,P3,...Pk} • Q is prime, so the original set
• 1. Let Q = P1P2P3...Pk + 1 was not complete.
(Multiply them all together and 31 is not 2, 3, or 5
add 1)
• 2. Q is either a new prime or a
composite • Example 2: {3,5,7}
• Q=3x5x7+1 =106
• 3. If a new prime, the given set
• Q is composite.
of primes is not complete.
SC/STS 3760, V 25
Proof of Proposition IX.20
• 4. If a composite, Q must be • Q=106=2x53.
divisible by a prime number. • Let G=2.
• -- Due to Proposition VII.31, • G is a new prime
previously proven. (not 3, 5, or 7).
• -- Let that prime number be G. • If G was one of 3,
• 5. G is either a new prime or one of 5, or 7, then it
the original set, {P1,P2,P3,...Pk} would be divisible
• 6. If G is one of the original set, it is into 3x5x7=105.
divisible into P1P2P3...Pk If so, G is • But it is divisible
also divisible into 1, (since G is into 106.
divisible into Q) • Therefore it would
• 7. This is an absurdity. be divisible into 1.
• This is absurd.
SC/STS 3760, V 26
Proof of Proposition IX.20
• Follow the absurdity backwards.
• Trace back to assumption (line 6), that G was
one of the original set. That must be false.
• The only remaining possibilities are that Q is a
new prime, or G is a new prime.
• In any case, there is a prime other than the
original set.
– Since the original set was of arbitrary size, there is
always another prime, no matter how many are
already accounted for.
SC/STS 3760, V 27
Euclid’s Elements at work
• Euclid’s Elements quickly became the
standard text for teaching mathematics at
the Museum at Alexandria.
• Philosophical questions about the world
could now be attacked with exact
mathematical reasoning.
SC/STS 3760, V 28
Eratosthenes of Cyrene
• 276 - 194 BCE
• Born in Cyrene, in North
Africa (now in Lybia).
• Studied at Plato’s Academy.
• Appointed Librarian at the
Museum in Alexandria.
SC/STS 3760, V 29
―Beta‖
• Eratosthenes was prolific. He worked in
many fields. He was a:
– Poet
– Historian
– Mathematician
– Astronomer
– Geographer
• He was nicknamed ―Beta.‖
– Not the best at anything, but the second best
at many things.
SC/STS 3760, V 30
Eratosthenes’ Map
• He coined the word ―geography‖ and drew one
of the first maps of the world (above).
SC/STS 3760, V 31
Using Euclid
• Eratosthenes made very clever use of a
few scant observations, plus a theorem
from Euclid to decide one of the great
unanswered questions about the world.
SC/STS 3760, V 32
His data
• Eratosthenes had heard
that in the town of Syene
(now Aswan) in the south
of Egypt, at noon on the
summer solstice (June 21
for us) the sun was
directly overhead.
– I.e. A perfectly upright pole
(a gnomon) cast no
shadow.
– Or, one could look directly
down in a well and see
one’s reflection.
SC/STS 3760, V 33
His data, 2
• Based on reports
from on a heavily
travelled trade
route,
Eratosthenes
calculated that
Alexandria was
5000 stadia
north of Syene.
SC/STS 3760, V 34
His data, 3
• Eratosthenes then
measured the angle
formed by the sun’s
rays and the upright
pole (gnomon) at
noon at the solstice in
Alexandria. (Noon
marked by when the
shadow is shortest.)
• The angle was 7°12’.
SC/STS 3760, V 35
Proposition I.29 from Euclid
b a
a b
b a
a b
A straight line falling on parallel straight lines makes the
alternate angles equal to one another, the exterior angle
equal to the interior and opposite angle, and the interior
angles on the same side equal to two right angles.
SC/STS 3760, V 36
• Eratosthenes reasoned that by I.29, the angle produced
by the sun’s rays falling on the gnomon at Alexandria is
equal to the angle between Syene and Alexandria at the
centre of the Earth.
SC/STS 3760, V 37
Calculating the size of the Earth
• The angle at the gnomon, α,
was 7°12’, therefore the
angle at the centre of the
Earth, β, was is also 7°12’
which is 1/50 of a complete
circle.
• Therefore the circumference
7°12’ x 50 = 360°
of the Earth had to be stadia
50 x 5000 = 250,000
= 250,000 stadia.
SC/STS 3760, V 38
Eratosthenes’ working assumptions
• 1. The Sun is very far away, so any light
coming from it can be regarded as
traveling in parallel lines.
• 2. The Earth is a perfect sphere.
• 3. A vertical shaft or a gnomon extended
downwards will pass directly through the
center of the Earth.
• 4. Alexandria is directly north of Syene, or
close enough for these purposes.
SC/STS 3760, V 39
A slight correction
• Later Eratosthenes made a somewhat
finer observation and calculation and
concluded that the circumference was
252,000 stadia.
• So, how good was his estimate.
– It depends….
SC/STS 3760, V 40
What, exactly, are stadia?
• Stadia are long
measures of length
in ancient times.
• A stade (singular of
stadia) is the length
of a stadium.
– And that was…?
SC/STS 3760, V 41
Stadium lengths
• In Greece the typical stadium was 185
metres.
• In Egypt, where Eratosthenes was, the
stade unit was 157.5 metres.
SC/STS 3760, V 42
Comparative figures
Circumference
Stade Length In Stadia In km
157.5 m 250,000 39,375
157.5 m 252,000 39,690
185 m 250,000 46,250
185 m 252,000 46,620
Compared to the modern figure for polar
circumference of 39,942 km, Eratosthenes was off
by at worst 17% and at best by under 1%.
SC/STS 3760, V 43
An astounding achievement
• Eratosthenes showed that relatively simple
mathematics was sufficient to determine
answers to many of the perplexing
questions about nature.
SC/STS 3760, V 44
Hipparchus of Rhodes
• Hipparchus of Rhodes
• Became a famous
astronomer in Alexandria.
• Around 150 BCE developed
a new tool for measuring
relative distances of the
stars from each other by the
visual angle between them.
SC/STS 3760, V 45
The Table of Chords
• Hipparchus invented the table of chords, a list of the ratio
of the size of the chord of a circle to its radius associated
with the angle from the centre of the circle that spans the
chord.
• The equivalent of the sine function in trigonometry.
SC/STS 3760, V 46
Precession of the equinoxes
• Hipparchus also calculated that there is a very
slow shift in the heavens that makes the solar
year not quite match the siderial (―star‖) year.
– This is called precession of the equinoxes. He noted
that the equinoxes come slightly earlier every year.
– The entire cycle takes about 26,000 years to
complete.
• Hipparchus was able to discover this shift and to
calculate its duration accurately, but the ancients
had no understanding what might be its cause.
SC/STS 3760, V 47
The Problem of the Planets, again
• 300 years after Hipparchus, another
astronomer uses his calculating devices to
create a complete system of the heavens,
accounting for the weird motions of the
planets.
• Finally a system of geometric motions is
devised to account for the positions of the
planets in the sky mathematically.
SC/STS 3760, V 48
Claudius Ptolemy
• Lived about 150 CE,
and worked in
Alexandria at the
Museum.
SC/STS 3760, V 49
Ptolemy’s Geography
• Like
Eratosthenes,
Ptolemy studied
the Earth as well
as the heavens.
• One of his major
works was his
Geography, one
of the first
realistic atlases
of the known
world.
SC/STS 3760, V 50
The Almagest
• Ptolemy’s major work was his Mathematical
Composition.
• In later years it was referred as The Greatest
(Composition), in Greek, Megiste.
• When translated into Arabic it was called al
Megiste.
• When the work was translated into Latin and
later English, it was called The Almagest.
SC/STS 3760, V 51
The Almagest, 2
• The Almagest attempts to do for
astronomy what Euclid did for
mathematics:
– Start with stated assumptions.
– Use logic and established mathematical
theorems to demonstrate further results.
– Make one coherent system
• It even had 13 books, like Euclid.
SC/STS 3760, V 52
Euclid-like assumptions
1. The heavens move spherically.
2. The Earth is spherical.
3. Earth is in the middle of the heavens.
4. The Earth has the ratio of a point to the
heavens.
5. The Earth is immobile.
SC/STS 3760, V 53
Plato versus Aristotle
• Euclid’s assumptions were about
mathematical objects.
– Matters of definition.
– Platonic forms, idealized.
• Ptolemy’s assumptions were about the
physical world.
– Matters of judgement and decision.
– Empirical assessments and common sense.
SC/STS 3760, V 54
Ptolemy’s Universe
• The basic framework of Ptolemy’s view of the
cosmos is the Empedocles’ two-sphere model:
– Earth in the center, with the four elements.
– The celestial sphere at the outside, holding the fixed
stars and making a complete revolution once a day.
• The seven wandering stars—planets—were
deemed to be somewhere between the Earth
and the celestial sphere.
SC/STS 3760, V 55
The Eudoxus-Aristotle system for
the Planets
• In the system of
Eudoxus, extended
by Aristotle, the
planets were the
visible dots
embedded on nested
rotating spherical
shells, centered on
the Earth.
SC/STS 3760, V 56
The Eudoxus-Aristotle system for
the Planets, 2
• The motions of the
visible planet were the
result of combinations of
circular motions of the
spherical shells.
– For Eudoxus, these may
have just been geometric,
i.e. abstract, paths.
– For Aristotle the spherical
shells were real physical
objects, made of the fifth
element.
SC/STS 3760, V 57
The Ptolemaic system
• Ptolemy’s system was purely geometric,
like Eudoxus, with combinations of circular
motions.
– But they did not involve spheres centered on
the Earth.
– Instead they used a device that had been
invented by Hipparchus 300 years before:
Epicycles and Deferents.
SC/STS 3760, V 58
Epicycles and Deferents
• Ptolemy’s system for each
planet involves a large
(imaginary) circle around
the Earth, called the
deferent, on which revolves
a smaller circle, the
epicycle.
• The visible planet sits on
the edge of the epicycle.
• Both deferent and epicycle
revolve in the same
direction.
SC/STS 3760, V 59
Accounting for Retrograde Motion
• The combined motions of the deferent and epicycle
make the planet appear to turn and go backwards
against the fixed stars.
SC/STS 3760, V 60
Saving the Appearances
• An explanation for the strange apparent
motion of the planets as ―acceptable‖
motions for perfect heavenly bodies.
– The planets do not start and stop and change
their minds. They just go round in circles,
eternally.
SC/STS 3760, V 61
How did it fit the facts?
• The main problem with Eudoxus’ and
Aristotle’s models was that they did not
track that observed motions of the planets
very well.
• Ptolemy’s was much better at putting the
planet in the place where it is actually
seen.
SC/STS 3760, V 62
But only up to a point….
• Ptolemy’s basic model was better than
anything before, but still planets deviated
a lot from where his model said they
should be.
• First solution:
– Vary the relative sizes of epicycle, deferent,
and rates of motion.
SC/STS 3760, V 63
Second solution: The Eccentric
• Another tack:
• Move the centre of
the deferent away
from the Earth.
• The planet still
goes around the
epicycle and the
epicycle goes
around the
deferent.
SC/STS 3760, V 64
Third Solution: The Equant Point
• The most complex
solution was to define
another ―centre‖ for
the deferent.
• The equant point was
the same distance
from the centre of the
deferent as the Earth,
but on the other side.
SC/STS 3760, V 65
Third Solution: The Equant Point, 2
• The epicycle
maintained a
constant distance
from the physical
centre of the
deferent, while
maintaining a
constant angular
motion around the
equant point.
SC/STS 3760, V 66
Ptolemy’s system worked
• Unlike other astronomers, Ptolemy
actually could specify where in the sky a
star or planet would appear throughout its
cycle – within acceptable limits.
• He ―saved the appearances.‖
– He produced an abstract, mathematical
account that explained the sensible
phenomena by reference to Platonic forms.
SC/STS 3760, V 67
But did it make any sense?
• Ptolemy gave no reasons why the planets
should turn about circles attached to
circles in arbitrary positions in the sky.
• Despite its bizarre account, Ptolemy’s
model remained the standard
cosmological view for 1400 years.
SC/STS 3760, V 68