1

11SC/STS 3760, VSC/STS 3760, V

Claudius PtolemyClaudius Ptolemy

Saving the HeavensSaving the Heavens

SC/STS 3760, VSC/STS 3760, V 22

Logic at its BestLogic 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, VSC/STS 3760, V 33

Mathematical ReasoningMathematical 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.

2

SC/STS 3760, VSC/STS 3760, V 44

The Structure of Ancient Greek The Structure of Ancient Greek CivilizationCivilization

• Ancient Greek civilization is divided into two major periods, marked by the death of Alexander the Great.

SC/STS 3760, VSC/STS 3760, V 55

Hellenic PeriodHellenic 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, VSC/STS 3760, V 66

Hellenistic PeriodHellenistic 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.

3

SC/STS 3760, VSC/STS 3760, V 77

Science in the Hellenistic AgeScience 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, VSC/STS 3760, V 88

Alexandria, EgyptAlexandria, 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, VSC/STS 3760, V 99

The MuseumThe 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.

4

SC/STS 3760, VSC/STS 3760, V 1010

EuclidEuclid

• 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, VSC/STS 3760, V 1111

EuclidEuclid’’s s ElementsElements

• 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, VSC/STS 3760, V 1212

The Influence of the The Influence of the ElementsElements

• 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.

5

SC/STS 3760, VSC/STS 3760, V 1313

AxiomsAxioms

• 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, VSC/STS 3760, V 1414

Axioms, 2Axioms, 2

• The axioms, or assumptions, are divided into three types:– Definitions– Postulates– Common notions

• All are assumed true.

SC/STS 3760, VSC/STS 3760, V 1515

DefinitionsDefinitions• 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.

6

SC/STS 3760, VSC/STS 3760, V 1616

PostulatesPostulates• 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, VSC/STS 3760, V 1717

Postulate 4Postulate 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, VSC/STS 3760, V 1818

The Controversial Postulate 5The Controversial Postulate 5

• 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.

a

b

c

d

f g

e

7

SC/STS 3760, VSC/STS 3760, V 1919

The The ““ParallelParallel”” PostulatePostulate

• 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 notbeing parallel.

a

b

c

d

f g

e

SC/STS 3760, VSC/STS 3760, V 2020

The Common NotionsThe 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, VSC/STS 3760, V 2121

An Axiomatic SystemAn 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.

8

SC/STS 3760, VSC/STS 3760, V 2222

Axiomatic SystemsAxiomatic 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, VSC/STS 3760, V 2323

Building Knowledge with an Building Knowledge with an Axiomatic SystemAxiomatic 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, VSC/STS 3760, V 2424

Example: Proposition IX.20Example: 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).

9

SC/STS 3760, VSC/STS 3760, V 2525

Proof of Proposition IX.20Proof of Proposition IX.20

• Given a set of prime numbers, {P1,P2,P3,...Pk}

• 1. Let Q = P1P2P3...Pk + 1 (Multiply them all together and add 1)

• 2. Q is either a new prime or a composite

• 3. If a new prime, the given set of primes is not complete.

• Example 1: {2,3,5}• Q=2x3x5+1 =31• Q is prime, so the original set

was not complete.31 is not 2, 3, or 5

• Example 2: {3,5,7}• Q=3x5x7+1 =106• Q is composite.

SC/STS 3760, VSC/STS 3760, V 2626

Proof of Proposition IX.20Proof of Proposition IX.20

• 4. If a composite, Q must be divisible by a prime number.

• -- Due to Proposition VII.31, previously proven.

• -- Let that prime number be G.• 5. G is either a new prime or one of

the original set, {P1,P2,P3,...Pk}• 6. If G is one of the original set, it is

divisible into P1P2P3...Pk If so, G is also divisible into 1, (since G is divisible into Q)

• 7. This is an absurdity.

• Q=106=2x53.• Let G=2.• G is a new prime

(not 3, 5, or 7).• If G was one of 3,

5, or 7, then it would be divisible into 3x5x7=105.

• But it is divisible into 106.

• Therefore it would be divisible into 1.

• This is absurd.

SC/STS 3760, VSC/STS 3760, V 2727

Proof of Proposition IX.20Proof 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.

10

SC/STS 3760, VSC/STS 3760, V 2828

EuclidEuclid’’s s ElementsElements at workat 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, VSC/STS 3760, V 2929

Eratosthenes of CyreneEratosthenes 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, VSC/STS 3760, V 3030

““BetaBeta””

• 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.

11

SC/STS 3760, VSC/STS 3760, V 3131

EratosthenesEratosthenes’’ MapMap

• He coined the word “geography” and drew one of the first maps of the world (above).

SC/STS 3760, VSC/STS 3760, V 3232

Using EuclidUsing 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, VSC/STS 3760, V 3333

His dataHis 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.

12

SC/STS 3760, VSC/STS 3760, V 3434

His data, 2His 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, VSC/STS 3760, V 3535

His data, 3His 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, VSC/STS 3760, V 3636

Proposition I.29 from EuclidProposition I.29 from Euclid

a

b

b

b

ba

aa

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.

13

SC/STS 3760, VSC/STS 3760, V 3737

• 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, VSC/STS 3760, V 3838

Calculating the size of the EarthCalculating 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 of the Earth had to be stadia = 250,000 stadia.

7°12’ x 50 = 360°

50 x 5000 = 250,000

SC/STS 3760, VSC/STS 3760, V 3939

EratosthenesEratosthenes’’ working assumptionsworking 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.

14

SC/STS 3760, VSC/STS 3760, V 4040

A slight correctionA 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, VSC/STS 3760, V 4141

What, exactly, are What, exactly, are stadiastadia??

• 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, VSC/STS 3760, V 4242

Stadium lengthsStadium lengths

• In Greece the typical stadium was 185 metres.

• In Egypt, where Eratosthenes was, the stade unit was 157.5 metres.

15

SC/STS 3760, VSC/STS 3760, V 4343

Comparative figuresComparative figuresCircumference

46,620252,000185 m

46,250250,000185 m

39,690252,000157.5 m

39,375250,000157.5 mIn kmIn StadiaStade Length

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, VSC/STS 3760, V 4444

An astounding achievementAn astounding achievement

• Eratosthenes showed that relatively simple mathematics was sufficient to determine answers to many of the perplexing questions about nature.

SC/STS 3760, VSC/STS 3760, V 4545

Hipparchus of RhodesHipparchus 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.

16

SC/STS 3760, VSC/STS 3760, V 4646

The Table of ChordsThe 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, VSC/STS 3760, V 4747

Precession of the equinoxesPrecession 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, VSC/STS 3760, V 4848

The Problem of the Planets, againThe 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.

17

SC/STS 3760, VSC/STS 3760, V 4949

Claudius PtolemyClaudius Ptolemy

• Lived about 150 CE, and worked in Alexandria at the Museum.

SC/STS 3760, VSC/STS 3760, V 5050

PtolemyPtolemy’’s Geographys 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, VSC/STS 3760, V 5151

The AlmagestThe 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.

18

SC/STS 3760, VSC/STS 3760, V 5252

The Almagest, 2The 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, VSC/STS 3760, V 5353

EuclidEuclid--like assumptionslike 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, VSC/STS 3760, V 5454

Plato versus AristotlePlato 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.

19

SC/STS 3760, VSC/STS 3760, V 5555

PtolemyPtolemy’’s Universes 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, VSC/STS 3760, V 5656

The The EudoxusEudoxus--Aristotle system for Aristotle system for the Planetsthe 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, VSC/STS 3760, V 5757

The The EudoxusEudoxus--Aristotle system for Aristotle system for the Planets, 2the 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.

20

SC/STS 3760, VSC/STS 3760, V 5858

The Ptolemaic systemThe 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, VSC/STS 3760, V 5959

Epicycles and Epicycles and DeferentsDeferents• 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, VSC/STS 3760, V 6060

Accounting for Retrograde MotionAccounting for Retrograde Motion

• The combined motions of the deferent and epicycle make the planet appear to turn and go backwards against the fixed stars.

21

SC/STS 3760, VSC/STS 3760, V 6161

Saving the AppearancesSaving 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, VSC/STS 3760, V 6262

How did it fit the facts?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, VSC/STS 3760, V 6363

But only up to a pointBut 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.

22

SC/STS 3760, VSC/STS 3760, V 6464

Second solution: The EccentricSecond 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, VSC/STS 3760, V 6565

Third Solution: The Equant PointThird 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, VSC/STS 3760, V 6666

Third Solution: The Equant Point, 2Third Solution: The Equant Point, 2

• The epicycle maintained a constant distancefrom the physical centre of the deferent, while maintaining a constant angular motion around the equant point.

23

SC/STS 3760, VSC/STS 3760, V 6767

PtolemyPtolemy’’s system workeds 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, VSC/STS 3760, V 6868

But did it make any sense?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.