AMA1D01C { Ancient Greecemajlee/AMA1D01C/lec07_5.pdfIntroduction I Some of the major players:...

$: AMA1D01C { Ancient Greecemajlee/AMA1D01C/lec07_5.pdfIntroduction I Some of the major players: Pythagoras (569-475 BC), Plato (429-347 BC), Aristotle (384-322 BC), Euclid (325-265 BC),$
AMA1D01C – Ancient Greece

Dr Joseph Lee, Dr Louis Leung

Hong Kong Polytechnic University

2017

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Ancient Greece

References

These notes mainly follow material from the following book:

I Katz, V. A History of Mathematics: an Introduction.Addison-Wesley, 1998.

and also use material from the following sources:

I Burton, D. The History of Mathematics: an Introduction.McGraw-Hill, 2011.

I Cajori, F. A History of Mathematics. MacMillan, 1893.

I MacTutor History of Mathematics Archive, University of StAndrews. http://www-history.mcs.st-and.ac.uk/


http://www-history.mcs.st-and.ac.uk/

Introduction

I Some of the major players: Pythagoras (569-475 BC), Plato(429-347 BC), Aristotle (384-322 BC), Euclid (325-265 BC),Archimedes (287-212 BC), Appolonius (262-190 BC),Ptolemy (AD 85-165), Diophantus (AD 200-284)


Early Greek Mathematics

I No complete text dating earlier than 300 BC

I Fragments exist and later works refer to earlier works

I Most complete reference can be found in the commentary toBook I of Euclid’s Elements written by Proclus in the 5thcentury AD

I Thought to be a summary of a history written by Eudemus ofRhodes in around 320 BC

I Original of Eudemus’ work was lost


Thales

Thales

I Earliest Greek mathematician mentioned was Thales

I From Miletus in Asia Minor (Asian part of modern dayTurkey)

I Many stories recorded about him: prediction of a solar eclipsein 585 BC, application of the angle-side-angle criterion fortriangle congruence, proving that the base angles of anisosceles triangle are equal, proving the diameter of a circledivide the circle into 2 equal parts

I The proofs themselves are lost, but it looks like there’s astrong logical flavour to his mathematics


Pythagoras

I From the island of Samos, off the coast of Asia Minor

I No surviving works

I All we know about the Pythagoreans was learned throughlater writers

I He was friends with Polycrates, the ruler of Samos

I Polycrates had an alliance with Egypt, and Pythagoras visitedthe country in 535 BC with a letter of introduction fromPolycrates


Pythagoras

I When Persia invaded Egypt in 525 BC (Pythagoras waspossibly on a different visit in the country), Pythagoras wastaken as a prisoner of war and sent to Babylon

I He was influenced by thoughts in both countries

I Left Samos in 518 BC, in order to escape his diplomaticresponsibilities and have more time to think

I Settled in Crotona, a Greek town in southern Italy

I Note: The area of Greek influence was much bigger thanmodern day Greece

I Gathered a group of disciples later known as the“Pythagoreans”


Greek Influence

Figure: Extent of Greek Influence. Source:https://commons.wikimedia.org/wiki/File:

Greek_Colonization_Archaic_Period.png


https://commons.wikimedia.org/wiki/File:Greek_Colonization_Archaic_Period.png

https://commons.wikimedia.org/wiki/File:Greek_Colonization_Archaic_Period.png

Pythagoras

I They believed numbers (positive integers) form the basis of allphysical phenomena

I Motions of the planets can be given in terms of ratios ofnumbers

I Same for the musical harmonies

I Today, “Pythagorean tuning” refers to a tuning system basedon products of the simple ratios 2

3 and 12 .

I For comparison, “equal temperament” is a system based onthe number 12

√2.


Pythagoras

I Using pictures, they managed to prove 1 + 3 = 22,1 + 3 + 5 = 32, 1 + 3 + 5 + 7 = 42, and so on

I Construction of Pythagorean triples: there is evidence to showthat they know if n is odd, then (n, n

2−12 , n

2+12 ) form a

Pythagorean triple

I Also, if m is even (m, m2

2 − 1, m2

2 + 1) form a Pythagoreantriple

I Not a table, but a rule


Sum of consecutive odds

Figure: 1 + 3 + 5 + 7 = 42


Incommensurability of the Diagonal

I Two lengths are said to be commensurable if they are bothmultiples of some shorter length

I In modern language, it means that the ratio of the twolengths is a rational number

I It was discovered around 430 BC that the diagonal and side ofa square are not commensurable

I Note if the year was correct, it was after Pythagoras’ death

I A big shock - as it went against their belief that integers formthe foundation of the universe

I But it was not enough to give something as simple as thelength of the diagonal of a unit square



I How was it discovered? Hint was in Aristotle’s work

I He noted that if the side and the diagonal are commensurable,then one may get an odd number which is equal to an evennumber

I A1 = Area of AGFE, A2 = Area of DBHI

I A1 = 2A2, so A1 is even, and side AG is even, so A1 is amultiple of 4

I Therefore A2 is even, which implies side DB is even

I Looks like the Greeks had the notion of a proof

I Very different from Egyptian or Babylonian mathematics,which emphasized on calculations



Figure: Incommensurability of the diagonal



A similar proof in more algebraic language:

I Let ( ab )2 = 2, where a and b are integers with no common

factor other than 1

I We have a2 = 2b2, so a2 is an even number

I Since 2 is a prime number, if a2 is an even number, a is aneven number.

I Let a = 2k , we have 4k2 = 2b2, so b2 = 2k2. Therefore b2 isan even number and b is an even number.

I Contradiction.


Plato

I Major legacy was his philosophy on mathematicsI Founded the Academy in 385 BCI An unverifiable story states that the line AΓEΩMETPHTOΣ

MH∆EIΣ EIΣITΩ (“Let no one ignorant of geometryenter”) was inscribed over the door to the Academy

I Plato distinguished between ideal, non-physical mathematicalobjects (e.g., “The Circle”) and daily approximations (e.g.,any circle we draw on paper)

I Platonism is the school of philosophy, inspired by Plato, thatbelieves in existence of abstract objects independent of thehuman mind

I Allegory of the Cave: We are all chained prisoners in a cavewho are not allowed to turn our heads. There are objectsoutside of the cave of which we can see only the shadow (acircle we draw on paper), but not the object itself (“TheCircle”).


Plato

“Those who are to take part in the highest functions of state mustbe induced to approach it, not in an amateur spirit, butperseveringly, until, by the aid of pure thought, they come to seethe real nature of number. They are to practise calculation, notlike merchants or shopkeepers for purposes of buying and selling,but with a view of war and to help in the conversion of the soulitself from the world of becoming to truth and reality.” (Note thepractical purpose of war.)“As we were saying, it has a great power of leading the mindupwards and forcing it to reason about pure numbers, refusing todiscuss collections of material things which can be seen andtouched.” (Plato, The Republic. Translated by F. Cornford.)


Zeno

Zeno of Elea

I Born in the 5th century BC

I Most famous for the paradoxes he proposed

I Version 1: To move from point A to point B, a person mustfirst reach the midpoint, but he must also reach the midpointbetween A and that midpoint, and so on. The person mustcover an infinity of midpoints, so it is impossible for him tomove from point A to point B.


Zeno

Zeno of Elea

I Version 2: For a quicker runner to overtake a slower runner,he must first reach the point where the slow runner started,by which time the slower runner will have run a distance, sothe quicker runner must reach the point where the slowerrunner is now, by which time the slower runner will have runanother distance, and so on. Therefore the quicker runner willnever be able to catch up to the slower runner.

I His paradoxes were refuted by Aristotle


Aristotle

Aristotle (384-322 BC)

I Studied at Plato’s Academy from the age of 18 until Plato’sdeath in 347 BC

I Later invited to the court of Philip II of Macedon to teach hisson Alexander (later Alexander the Great)

I Then returned to Athens to found his own school, the Lyceum(root of the French word “lycee”, secondary school)

I List of works include: Physics, Metaphysics, Prior Analytics,Posterior Analytics, Poetics, Rhetoric, Politics, The AthenianConstitution, On Dreams, Virtues and Vices, On Youth andOld Age, On Life and Death, On Breathing


Aristotle

Figure: Aristotle tutoring Alexander, by J L G Ferris 1895. Source:http://www.alexanderstomb.com/main/imageslibrary/

alexander/index.htm


http://www.alexanderstomb.com/main/imageslibrary/alexander/index.htm

http://www.alexanderstomb.com/main/imageslibrary/alexander/index.htm

Aristotle

Logic

I Aristotle believed that arguments should be built out ofsyllogisms

I Syllogism: “Discourse in which, certain things being stated,something other than what is stated follows of necessity fromtheir being so” – Aristotle, Prior Analytics I, 1, 24, 19Translation: Great Books Edition, Chicago: EncyclopediaBritannica 1952

I A syllogism therefore contains certain statements that aretaken as true and some other statements which must be trueby consequence


Aristotle

Syllogism example

I All men are mortal

I Socrates is a man

I Therefore, Socrates is mortal


Aristotle

Logic

I Allows one to use “old knowledge” to produce “newknowledge”

I However one cannot obtain all knowledge as results ofsyllogisms

I We must start somewhere with truths which we acceptwithout argument

I Postulate: Basic truth peculiar to a particular science

I Example: Euclid’s “(One is always able) To draw a straightline from any point to any point.”

I Axiom: Basic truth common to all sciences

I Example: Euclid’s “Things which are equal to the same thingare also equal to one another.”


Aristotle

How did Aristotle refute Zeno’s paradoxes?

I While space can be infinitely divided according to Zeno, timecan also be divided likewise.

I He said “while a thing in a finite time cannot come in contactwith things quantitatively infinite, it can come in contact withthings infinite in respect to divisibility, for in this sense timeitself is also infinite.”

I He distinguished between “infinite” and “infinitely divisible”.

I However he did not have the tool of “convergent infiniteseries”, where a sum of infinitely many terms may be finite.


Chrysippus

Later on, Chrysippus (280-206 BC) analyzed more forms ofargument

I Modus ponens (“the method that affirms”)

I Modus tollens (“the method that denies”)

I Hypothetical syllogism

I Disjunctive syllogism


Modus ponens

Modus ponens

I If P, then Q.

I P.

I Therefore, Q.


Modus ponens

Modus ponens example:

I If this drink contains sugar, then this drink is sweet.

I This drink contains sugar.

I Therefore, this drink is sweet.


Modus tollens

Modus tollens

I If P, then Q.

I Not Q.

I Therefore, not P.


Modus tollens

Modus tollens example:


I This drink is not sweet.

I Therefore, this drink does not contain sugar.


Hypothetical syllogism


I If P, then Q.

I If Q, then R.

I Therefore, if P, then R.



Hypothetical syllogism example:


I If this drink is sweet, then Emma will not drink it.

I Therefore, if this drink contains sugar, then Emma will notdrink it.


Disjunctive syllogism


I P or Q.

I Not P.

I Therefore, Q.



Disjunctive syllogism example:

I Emma’s car is red, or Emma’s car is blue.

I Emma’s car is not red.

I Therefore, Emma’s car is blue.


Museum of Alexandria

I A research institute

I Built around 280 BC by Ptolemy I Soter (not to be confusedwith Ptolemy the astronomer)

I Buildings were destroyed in 272 AD in a civil war under theRoman emperor Aurelian

I Fellows of the museum received stipends, free meals, and wereexempt from taxes

I The famous Library of Alexandria is part of it

I Museum – “Temple of the Muses”

I Muses – nine goddesses inspiring learning and the arts;daughters of Zeus


Muses

The nine muses:

I Calliope (epic poetry), Clio (history), Euterpe (lyric poetry),Thalia (comedy), Malpomene (tragedy), Terpsichore (dance),Erato (love poetry), Polyhymnia (sacred poetry; hymns),Urania (astronomy)


Muses

Figure: Nine Muses, by Samuel Griswold Goodrich. Source:https://commons.wikimedia.org/wiki/File:

Nine_Muses_-_Samuel_Griswold_Goodrich_(1832).jpg


https://commons.wikimedia.org/wiki/File:Nine_Muses_-_Samuel_Griswold_Goodrich_(1832).jpg

https://commons.wikimedia.org/wiki/File:Nine_Muses_-_Samuel_Griswold_Goodrich_(1832).jpg

Alexandria

Figure: Alexandria on a modern map. Source: Google Map


Euclid

I Not much is known about his life

I It is believed that he taught and wrote at the Museum ofAlexandria

I Died in Alexandria in 265 BC


The Elements

I Thirteen books

I Definitions, axioms, theorems, proofs

I His way of thinking influenced modern mathematics, whichfollow an axiomatic approach.

I To the extreme, you forget the “meaning” of a mathematicalobject and just consider how they interact with othermathematical objects.

I For example, you don’t care what “lines” and “points” mean,all you care about is that two points give a line (the one thatjoins them) and two (non-parallel) lines give a point (wherethey intersect)


The Elements

Some of the definitions from Book I:

I 1. A point is that which has no part.

I 2. A line is breadthless length

I 4. A straight line is a line which lies evenly with the points onitself.

I 15. A circle is a plane figure contained by one line such thatall the straight lines meeting it from one point among thoselying within the figure are equal to one another;

I 16. and the point is called the centre of the circle.

I 23. Parallel straight lines are straight lines which, being in thesame plane and being produced indefinitely in both directions,do not meet one another in either direction.


The Elements

I Note by modern standard the definitions are not gooddefinitions

I It was not clear what the terms “part”, “breadthless”,“evenly” mean.

I In modern mathematics, the word “set” remains undefined,but we have a list of well-defined operations which we areallowed to perform on sets.


The Elements

Postulates (truths peculiar to the science of geometry):

I 1. (One is always able) To draw a straight line from any pointto any point.

I 2. (One is always able) To produce a finite straight linecontinuously in a straight line.

I 3. (One is always able) To describe (construct) a circle withany centre and distance.

I 4. That all right angles are equal to one another.

I 5. That, if a straight line intersecting two straight lines makethe interior angles on the same side less than two right angles,the two straight lines, if produced indefinitely, meet on thatside on which the angles are less than two right angles.


The Elements

Common notions (axioms, truths common to all sciences, logicaltruths):

I 1. Things which are equal to the same thing are also equal toone another.

I 2. If equals are added to equals, the wholes are equal.

I 3. If equals are subtracted from equals, the remainders areequal.

I 4. Things which coincide with one another are equal to oneanother.

I 5. The whole is greater than the part.


The Elements

Book I, Proposition I: To construct an equilateral triangle on agiven finite straight line. (A possibility kind of proposition)

I This is the very first proposition, so Euclid could only use thedefinitions, postulates and axioms

I By Postulate 3, he could construct one circle with centre Aand radius AB and another with centre B and radius BA

I The two circles intersect at a point C

I By Postulate 1, he could draw the lines AC and BC

I By Definition 15, AC equals AB and BC equals BA

I By Common Notion 1, AC, AB and BC are equal

I Gap: How did Euclid know the two circles intersect?

I Some postulate of continuity (if a line crosses from one side ofa line to the other side, the two lines must intersect) isnecessary

I Such problems will be dealt with in 19th-century mathematics


The Elements

Figure:


The Elements

Some of the definitions from Book VII:

I 1. A unit is that by virtue of which each of the things thatexist is called one.

I 2. A number is a multitude composed of units.

I 3. A number is a part of a number, the less of the greater,when it measures the greater;

I 4. but parts when it does not measure it.

I 11. A prime number is that which is measured by the unitalone.

I 12. Numbers prime to one another are those which aremeasured by the unit alone as a common measure.

I 15. A number is said to multiply a number when that which ismultiplied is added to itself as many times as there are unitsin the other, and thus some number is produced.


The Elements

Book IX, Proposition XX: Prime numbers are more than anyassigned multitude of primes. (In more modern mathematicallanguage: Given any finite set of prime numbers, there is at leastone prime number which does not belong to the set.)

I Given any fixed number of prime numbers, you can alwaysfind one more, i.e., there are infinitely many prime numbers.

I Let A, B, C be three prime numbers

I Consider ABC + 1

I If ABC + 1 is prime, we have a new prime

I If not, then ABC + 1 has some prime factor G . If G is eitherA, B or C , then G is a factor of 1, a contradiction

I Therefore G is a prime distinct from A, B or C

I Note: Euclid gave his proof with three primes, but the sameproof may be given for any finite number of primesp1, p2, . . . , pn. Consider p1p2 . . . pn + 1.


The Elements

Book XIII

I Devoted to the study of regular polyhedra (also known as“Platonic solids”)

I Each face is a regular polygon

I An equal number of faces meet at each vertex

I There are five: tetrahedron (four triangles, three meeting ateach vertex), cube (six squares, three meeting at each vertex),octahedron (eight triangles, four meeting at each vertex),dodecahedron (twelve pentagons, three meeting at eachvertex), icosahedron (twenty triangles, five meeting at eachvertex)

I Book XIII contained a proof that those are the only ones


The Elements

Figure: Platonic solids. Source:http://www.maths.gla.ac.uk/~ajb/3H-WP/platonic_solids.gif


http://www.maths.gla.ac.uk/~ajb/3H-WP/platonic_solids.gif

Archimedes

Archimedes (287-212 BC)

I Born in Syracuse

I Highly probable that he studied in Alexandria

I Familiar with all work previously done in mathematics

I Later returned to Syracuse where he helped King Hieron byapplying his knowledge to construct war-engines

I Finally the Romans took the city and Archimedes was killedby a Roman soldier

I Last words were said to be “Don’t disturb my circles”,referring to a picture he was contemplating when the Romansoldier approached him

I The Roman general Marcellus admired him and constructed atomb in his honour, with a sphere inscribed in a cylinder


Archimedes

Figure: Death of Archimedes, by Thomas Degeorge 1815. Source:https://www.math.nyu.edu/~crorres/Archimedes/Death/

Degeorge/degeorge.png


https://www.math.nyu.edu/~crorres/Archimedes/Death/Degeorge/degeorge.png

https://www.math.nyu.edu/~crorres/Archimedes/Death/Degeorge/degeorge.png

Archimedes

On the Measurement of the Circle

I Proposition 1: The area of any circle is equal to the area of aright triangle in which one of the legs is equal to the radiusand the other to the circumference.

I Exhaustion argument: Let K be the area of the given triangleand A be the area of the circle.

I Suppose A > K . By inscribing in the circle polygons ofincreasing numbers of sides, eventually gets a polygon witharea P with A− P < A− K . Therefore P > K

I The perpendicular from the centre of the circle to themidpoint of a side of the polygon is shorter than the radius,and the perimeter of the polygon is less than thecircumference. Therefore P < K . CONTRADICTION.

I Therefore A must be less than equal to KI Similarly assuming A < K will lead to another contradictionI Therefore A = K


Archimedes

On the Measurement of the Circle

I Proposition 3: The ratio of the circumference of any circle toits diameter is less than 31

7 but greater than 31071

I Proved by finding the ratios of the perimeters of the inscribedand circumscribed 96-sided polygons to the diameter


On the Equilibrium of Planes

On the Equilibrium of Planes:

I Mathematical theory of the lever



Some Postulates:

I 1. Equal weights at equal distances are in equilibrium, andequal weights at unequal distances are not in equilibrium butincline toward the weight which is at the greater distance.

I 2. If, when weights at certain distances are in equilibrium,something is added to one of the weights, they are not inequilibrium but incline toward the weight to which theaddition was made

I 3. Similarly, if anything is taken away from one of theweights, they are not in equilibrium but incline toward theweight from which nothing was taken

I 6. If magnitudes at certain distances are in equilibrium, othermagnitudes equal to them will also be in equilibrium at thesame distances



Some Propositions:

I 3. Suppose A and B are unequal weights with A > B whichbalance at point C . Let AC = a, BC = b. Then a < b.Conversely, if the weights balance at a < b, then A > B

I 6, 7. Two magnitudes, whether commensurable (Prop 6) orincommensurable (Prop 7), balance at distances inverselyproportional to the magnitudes.


Archimedes

Figure: On the Equilibrium of Planes. Proposition 3


Appolonius

Appolonius

I Born in Perga, studied at Alexandria under successors toEuclid, and composed the first draft of The Conic Sectionsthere

I Later moved to Pergamum, which had a new university andlibrary modeled after those in Alexandria


Appolonius

Conic Sections

I Eight books

I First four books have been passed down to us in the originalGreek, and the next three books were unknown in Europeuntil Arabic translations were found. The eighth book is lost.

I Intersection of a plane and cones gives three types of curves:ellipses, parabolas and hyperbolas


Appolonius

Figure: Conic sections. Source:http://mathworld.wolfram.com/ConicSection.html


http://mathworld.wolfram.com/ConicSection.html

Appolonius

Conic Sections

I Appolonius discovered what were equivalent to modernequations of the parabolas, ellipses and hyperbolas

I Studied asymptotes to the hyperbolas (in Greek, asymptotosmeans “not capable of meeting”)

I Showed how to construct a hyperbola given a point on thehyperbola and its asymptotes

I Also studied tangent lines (a line which touches the curve butdoes not cut the curve)


Ptolemy

Ptolemy

I Native of Egypt

I Major works include Geography and Mathematicki Syntaxis(“Mathematical Collection”)

I Later Mathematicki Syntaxis became known as MegistiSyntaxis (“The Greatest Collection”), and the Arabs called ital-magisti. Now people refer to it as the Almagest.

I First recorded observation was made in 125 AD, last one wasin 151 AD


Ptolemy

Almagest

I Composed of 13 books and is considered the culmination(peak) of Greek astronomy

I Contains a table of chords from 12 degree to 180 degrees in

intervals of 12 degree

I Ptolemy did all his computations in a base-60 system

I Square roots were involved but Ptolemy did not describe howhe calculated them

I A commentary by Theon in the fourth century explained amethod Ptolemy could have used

I Also contains work on plane and spherical trigonometry (withobvious astronomical implications)


Diophantus

I Lived in Alexandria

I Major work is called Arithmetica, which has 13 books, butonly 6 survived in Greek

I Four others (4 to 7) were recently discovered in an Arabic(translated) version


Diophantus

Arithmetica

I Like the Rhind Papyrus, it is a collection of problems

I Only positive rational answers were allowed

I For example, 4x + 20 = 4 has no solution

I We look at two examples (given in modern notation)


Diophantus

Arithmetica Example 1. Book I, Problem 17: Find four numberssuch that when any three of them are added together, their sum isone of four given numbers. Say the given sums are 20, 22, 24, and27.

I Solution: Let x be the sum of the four numbers. The fournumbers are, respectively, x − 20, x − 22, x − 24 and x − 27

I We have x = (x − 20) + (x − 22) + (x − 24) + (x − 27).

I Therefore x = 31 and the numbers are 11, 9, 7 and 4.


Diophantus

Arithmetica Example 2. Book II, Problem 8: Divide a given squarenumber, say 16, into the sum of two squares.

I Let x2 be one of the squares

I 16− x2 = (2x − 4)2

I The 4 is meant to cancel the 16, the choice of 2 was arbitrary

I The equation becomes 5x2 = 16x . The positive solution isx = 16

5

I Therefore one square is 25625 , and the other is 16− 256

25 = 14425


Diophantus

I In modern mathematics, a Diophantine equation is anequation for which only integer solutions are allowed.


Decline of Greek Mathematics

I The Romans held a utilitarian view towards mathematics

I Focus was on application of arithmetic and geometry toengineering and architecture

I “The Greeks held the geometer in the highest honour;accordingly nothing made more brilliant progress among themthan mathematics. But we have established as the limits ofthis art its usefulness in measuring and counting.” –Cicero,Roman politician


AMA1D01C { Ancient Greecemajlee/AMA1D01C/lec07_5.pdfIntroduction I Some of the major players:...

Documents

Transcript of AMA1D01C { Ancient Greecemajlee/AMA1D01C/lec07_5.pdfIntroduction I Some of the major players:...