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 follow material from the following books:
I
Burton, D. The History of Mathematics: an Introduction.
McGraw-Hill, 2011.
I
Cajori, F. A History of Mathematics. MacMillan, 1893.
I
Katz, V. A History of Mathematics: an Introduction.
Addison-Wesley, 1998.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
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)
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Early Greek Mathematics
I
No complete text dating earlier than 300 BC
I
However fragments exist and there are references in later
works to earlier works
I
Most complete reference can be found in the commentary to
Book I of Euclid’s Elements written by Proclus in the 5th
century AD
I
Thought to be a summary of a history written by Eudemus of
Rhodes in around 320 BC
I
Original was lost
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Thales
Thales
I
Earliest Greek mathematician mentioned was Thales
I
From Miletus in Asia Minor (Asian part of modern day
Turkey)
I
Many stories recorded about him: prediction of a solar eclipse
in 585 BC, application of the ASA criterion for triangle
congruence, proving that the base angles of an isosceles
triangle are equal, proving the diameter of a circle divide the
circle into 2 equal parts
I
The proofs themselves are lost, but it looks like there’s a
strong logical flavour to his mathematics
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Pythagoras
I
No surviving works
I
All we know about the Pythagoreans must be learned through
later writers
I
Forced to leave his native island of Samos, off the coast of
Asia Minor
I
Settled in Crotona, a Greek town in southern Italy
I
Note: The area of Greek influence was much bigger than
modern day Greece
I
Gathered a group of disciples later known as the
“Pythagoreans”
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Greek Influence
Figure: Extent of Greek Influence. Source:
https://commons.wikimedia.org/wiki/File:
Greek_Colonization_Archaic_Period.png
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Pythagoras
I
They believed numbers (positive integers) form the basis of all
physical phenomena
I
Motions of the planets can be given in terms of ratios of
numbers
I
Same for the musical harmonies
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 show
2
2
that they know if n is odd, then (n, n 2−1 , n 2+1 ) form a
Pythagorean triple
I
Also, if m is even (m, m2 − 1, m2 + 1) form a Pythagorean
triple
2
Dr Joseph Lee, Dr Louis Leung
2
AMA1D01C – Ancient Greece
Sum of consecutive odds
Figure: 1 + 3 + 5 + 7 = 42
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Incommensurability of the Diagonal
I
Two lengths are said to be commensurable if they are both
multiples of some shorter length
I
In modern language, it means that the ratio of the two
lengths is a rational number
I
It was discovered around 430 BC that the diagonal and side of
a square are not commensurable
I
Note if the year was correct, it was after Pythagoras’ death
I
A big shock
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Incommensurability of the Diagonal
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 even
number
I
A1 = Area of AGFE, A2 = Area of DBHI
I
A1 = 2A2 , so A1 is even, and side AG is even, so A1 is a
multiple 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
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Incommensurability of the Diagonal
Figure: Incommensurability of the diagonal
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Plato
I
Major legacy was his philosophy on mathematics
I
Founded the Academy in 385 BC
I
An unverifiable story states that the line AΓEΩMETPHTOΣ
MH∆EIΣ EIΣITΩ (“Let no one ignorant of geometry
enter”) was inscribed over the door to the Academy
I
Plato distinguished between ideal, non-physical mathematical
objects (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, that
believes in existence of abstract objects independent of the
human mind
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Plato
“Those who are to take part in the highest functions of state must
be induced to approach it, not in an amateur spirit, but
perseveringly, until, by the aid of pure thought, they come to see
the real nature of number. They are to practise calculation, not
like merchants or shopkeepers for purposes of buying and selling,
but with a view of war and to help in the conversion of the soul
itself from the world of becoming to truth and reality.”
“As we were saying, it has a great power of leading the mind
upwards and forcing it to reason about pure numbers, refusing to
discuss collections of material things which can be seen and
touched.” (Plato, The Republic. Translated by F. Cornford.)
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Aristotle
Aristotle (384-322 BC)
I
Studied at Plato’s Academy from the age of 18 until Plato’s
death in 347 BC
I
Later invited to the court of Philip II of Macedon to teach his
son Alexander (later Alexander the Great)
I
Then returned to Athens to found his own school, the Lyceum
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Aristotle
Figure: Aristotle tutoring Alexander, by J L G Ferris 1895. Source:
http://www.alexanderstomb.com/main/imageslibrary/
alexander/index.htm
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Aristotle
Logic
I
Aristotle believed that arguments should be built out of
syllogisms
I
Syllogism: “Discourse in which, certain things being stated,
something other than what is stated follows of necessity from
their being so”
I
A syllogism therefore contains certain statements that are
taken as true and some other statements which must be true
by consequence
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Aristotle
Syllogism example
I
All men are mortal
I
Socrates is a man
I
Therefore, Socrates is mortal
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Aristotle
Logic
I
Allows one to use “old knowledge” to impart “new
knowledge”
I
However one cannot obtain all knowledge as results of
syllogisms
I
We must start somewhere with truths which we accept
without argument
I
Postulate: Basic truth peculiar to a particular science
I
Axiom: Basic truth common to all sciences
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Chrysippus
Later on, Chrysippus (280-206 BC) analyzed more forms of
argument
I
Modus ponens
I
Modus tollens
I
Hypothetical syllogism
I
Disjunctive syllogism
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Modus ponens
Modus ponens
I
If P, then Q.
I
P.
I
Therefore, Q.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
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.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Modus tollens
Modus tollens
I
If P, then Q.
I
Not Q.
I
Therefore, not P.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Modus tollens
Modus tollens example:
I
If this drink contains sugar, then this drink is sweet.
I
This drink is not sweet.
I
Therefore, this drink does not contain sugar.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Hypothetical syllogism
Hypothetical syllogism
I
If P, then Q.
I
If Q, then R.
I
Therefore, if P, then R.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Hypothetical syllogism
Hypothetical syllogism example:
I
If this drink contains sugar, then this drink is sweet.
I
If this drink is sweet, then Emma will not drink it.
I
Therefore, if this drink contains sugar, then Emma will not
drink it.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Disjunctive syllogism
Disjunctive syllogism
I
P or Q.
I
Not P.
I
Therefore, Q.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Disjunctive syllogism
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.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Museum of Alexandria
I
A research institute
I
Built around 280 BC by Ptolemy I Soter (not to be confused
with Ptolemy the astornomer)
I
Buildings were destroyed in 272 AD in a civil war under the
Roman emperor Aurelian
I
Fellows of the museum received stipends, free board, and were
exempt 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
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
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)
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Muses
Figure: Nine Muses, by Samuel Griswold Goodrich. Source:
https://commons.wikimedia.org/wiki/File:
Nine_Muses_-_Samuel_Griswold_Goodrich_(1832).jpg
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Alexandria
Figure: Alexandria on a modern map. Source: Google Map
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Euclid
I
Not much is known about his life
I
It is believed that he taught and wrote at the Museum of
Alexandria
I
Died in Alexandria in 265 BC
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
The Elements
I
Thirteen books
I
Definitions, axioms, theorems, proofs
I
His way of thinking influenced modern mathematics, which
follow an axiomatic approach.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
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 on
itself.
I
15. A circle is a plane figure contained by one line such that
all the straight lines meeting it from one point among those
lying 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 the
same plane and being produced indefinitely in both directions,
do not meet one another in either direction.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
The Elements
Postulates (truths peculiar to the science of geometry):
I
1. To draw a straight line from any point to any point.
I
2. To produce a finite straight line continuously in a straight
line.
I
3. To describe a circle with any 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 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 the angles are less than two right angles.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
The Elements
Common notions (axioms, truths common to all sciences):
I
1. Things which are equal to the same thing are also equal to
one another.
I
2. If equals are added to equals, the wholes are equal.
I
3. If equals are subtracted from equals, the remainders are
equal.
I
4. Things which coincide with one another are equal to one
another.
I
5. The whole is greater than the part.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
The Elements
Book I, Proposition I: To construct an equilateral triangle on a
given finite straight line. (A possibility kind of proposition)
I This is the very first proposition, so Euclid could only use the
definitions, postulates and axioms
I By Postulate 3, he could construct one circle with centre A
and 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 of
a line to the other side, the two lines must intersect) is
necessary
I Such problems will be dealt with in 19th-century mathematics
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
The Elements
Figure:
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
The Elements
Some of the definitions from Book VII:
I
1. A unit is that by virtue of which each of the things that
exist 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 unit
alone.
I
12. Numbers prime to one another are those which are
measured by the unit alone as a common measure.
I
15. A number is said to multiply a number when that which is
multiplied is added to itself as many times as there are units
in the other, and thus some number is produced.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
The Elements
Book IX, Proposition XX: Prime numbers are more than any
assigned multitude of primes
I
Given any fixed number of prime numbers, you can always
find 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 either
A, 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 same
proof may be given for any finite number of primes
p1 , p2 , . . . , pn . Consider p1 p2 . . . pn + 1.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
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 at
each vertex), cube (six squares, three meeting at each vertex),
octahedron (eight triangles, four meeting at each vertex),
dodecahedron (twelve pentagons, three meeting at each
vertex), icosahedron (twenty triangles, five meeting at each
vertex)
I
Book XIII contained a proof that those are the only ones
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
The Elements
Figure: Platonic solids. Source:
http://www.maths.gla.ac.uk/~ajb/3H-WP/platonic_solids.gif
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
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 mathmetaics
I
Later returned to Syracuse where he helped King Hieron by
applying his knowledge to construct war-engines
I
Finally the Romans took the city and Archimedes was killed
by a Roman soldier
I
Last words were said to be “Don’t disturb my circles”,
referring to a picture he was contemplating when the Roman
soldier approached him
I
The Roman general Marcellus admired him and constructed a
tomb in his honour, with a sphere inscribed in a cylinder
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Archimedes
Figure: Death of Archimedes, by Thomas Degeorge 1815. Source:
https://www.math.nyu.edu/~crorres/Archimedes/Death/
Degeorge/degeorge.png
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Archimedes
On the Measurement of the Circle
I Proposition 1: The area of any circle is equal to the area of a
right triangle in which one of the legs is equal to the radius
and the other to the circumference.
I Exhaustion argument: Let K be the area of the given triangle
and A be the area of the circle.
I Suppose A > K . By inscribing in the circle polygons of
increasing numbers of sides, eventually gets a polygon with
area P with A − P < A − K . Therefore P > K
I The perpendicular from the centre of the circle to the
midpoint of a side of the polygon is shorter than the radius,
and the perimeter of the polygon is less than the
circumference. Therefore P < K . CONTRADICTION.
I Therefore A must be less than equal to K
I Similarly assuming A < K will lead to another contradiction
I Therefore A = K
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Archimedes
On the Measurement of the Circle
I
Proposition 3: The ratio of the circumference of any circle to
its diameter is less than 3 17 but greater than 3 10
71
I
Proved by finding the ratios of the perimeters of the inscribed
and circumscribed 96-sided polygons to the diameter
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
On the Equilibrium of Planes
On the Equilibrium of Planes:
I
Mathematical theory of the lever
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
On the Equilibrium of Planes
Some Postulates:
I
1. Equal weights at equal distances are in equilibrium, and
equal weights at unequal distances are not in equilibrium but
incline 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 in
equilibrium but incline toward the weight to which the
addition was made
I
3. Similarly, if anything is taken away from one of the
weights, they are not in equilibrium but incline toward the
weight from which nothing was taken
I
6. If magnitudes at certain distances are in equilibrium, other
magnitudes equal to them will also be in equilibrium at the
same distances
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
On the Equilibrium of Planes
Some Propositions:
I
3. Suppose A and B are unequal weights with A > B which
balance 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) or
incommensurable (Prop 7), balance at distances inversely
proportional to the magnitudes.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Archimedes
Figure: On the Equilibrium of Planes. Proposition 3
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Appolonius
Appolonius
I
Born in Perga, studied at Alexandria under successors to
Euclid, and composed the first draft of The Conic Sections
there
I
Later moved to Pergamum, which had a new university and
library modeled after those in Alexandria
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Appolonius
Conic Sections
I
Eight books
I
First four books have been passed down to us in the original
Greek, and the next three books were unknown in Europe
until 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
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Appolonius
Figure: Conic sections. Source:
http://mathworld.wolfram.com/ConicSection.html
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Appolonius
Conic Sections
I
Appolonius discovered what were equivalent to modern
equations of the parabolas, ellipses and hyperbolas
I
Studied asymptotes to the hyperbolas (in Greek, asymptotos
means “not capable of meeting”)
I
Showed how to construct a hyperbola given a point on the
hyperbola and its asymptotes
I
Also studied tangent lines (a line which touches the curve but
does not cut the curve)
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Ptolemy
Ptolemy
I
Native of Egypt
I
Major works include Geography and Mathematicki Syntaxis
(“Mathematical Collection”)
I
Later Mathematicki Syntaxis became known as Megisti
Syntaxis (“The Greatest Collection”), and the Arabs called it
al-magisti. Now people refer to it as the Almagest.
I
First recorded observation was made in 125 AD, last one was
in 151 AD
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Ptolemy
Almagest
I
Composed of 13 books and is consdered the culmination of
Greek astronomy
I
Contains a table of chords from
intervals of 21 degree
I
Ptolemy did all his computations in a base-60 system
I
Square roots were involved but Ptolemy did not describe how
he calculated them
I
A commentary by Theon in the fourth century explained a
method Ptolemy could have used
I
Also contains work on plane and spherical trigonometry (with
obvious astronomical implications)
Dr Joseph Lee, Dr Louis Leung
1
2
degree to 180 degrees in
AMA1D01C – Ancient Greece
Diophantus
I
Lived in Alexandria
I
Major work is called Arithmetica, which has 13 books, but
only 6 survived in Greek
I
Four others (4 to 7) were recently discovered in an Arabic
(translated) version
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
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)
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Diophantus
Arithmetica Example 1. Book I, Problem 17: Find four numbers
such that when any three of them are added together, their sum is
one of four given numbers. Say the given sums are 20, 22, 24, and
27.
I
Solution: Let x be the sum of the four numbers. The four
numbers 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.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Diophantus
Arithmetica Example 2. Book II, Problem 8: Divide a given square
number, say 16, into the sum of two squares.
I
Let x 2 be one of the squares
I
16 − x 2 = (2x − 4)2
I
The 4 is meant to cancel the 16, the choice of 2 was arbitrary
I
The equation becomes 5x 2 = 16x. The positive solution is
x = 16
5
I
Therefore one square if
256
25 ,
Dr Joseph Lee, Dr Louis Leung
and the other is 16 −
AMA1D01C – Ancient Greece
256
25
=
144
25
Diophantus
I
In modern mathematics, a Diophantine equation is an
equation for which only integer solutions are allowed.
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece
Decline of Greek Mathematics
I
The Romans held a utilitarian view towards mathematics
I
Focus was on application of arithmetic and geometry to
engineering and architecture
I
“The Greeks held the geometer in the highest honour;
accordingly nothing made more brilliant progress among them
than mathematics. But we have established as the limits of
this art its usefulness in measuring and counting.” –Cicero,
Roman politician
Dr Joseph Lee, Dr Louis Leung
AMA1D01C – Ancient Greece