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Lecture 4, MATH 210G.02, Fall 2016
Greek Mathematcs and
Philosophy
Goals:
Learn a few of the theorems proved by Greek
mathematcians
Understand some of the Greek philosophy in
which mathematcs was developed
Lecture 4, MATH 210G.02, Fall 2015
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Greek Mathematcs and
Philosophy
Period 1: 650 BC-400 BC (pre-Plato)
Period 2: 400 BC – 300 BC (Plato, Euclid)
Period 3: 300 BC – 200 BC (Archimedes,
Appolonius, Eratosthenes
q
Thales (624-547 BC): father of mathematcal
proof
In the diagram, the rato of
the segments AD and DB is
the same as the rato of the
segments AE and EC
A) True
B) False
Pythagoras
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(c. 580-500 BC)
In the windmill diagram, the
area of the square with side
a plus the area of the square
with side b equals the area
of the square with side c
A)
True
B)
False
Pythagorean philosophy
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Transmigraton of souls,
purifcaton rites; developed rules of living
believed would enable their soul to achieve a
higher rank among the gods.
Theory that numbers consttute the true
nature of things, including music
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C
D
E
F
G
A
B
C
1
9/8
81/64
4/3
3/2
27/16
243/128
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The diatonic: rato of highest to lowest pitch is 2:1,
produces the interval of an octave.
Octave in turn divided into ffth and fourth, with ratos 3:2 and 4:3 …
up a ffth + up a fourth = up an octave.
ffth … divided into three whole tones, each corresponding to the rato of
9:8 and a remainder with a rato of 256:243
fourth into two whole tones with same remainder.
harmony… combinaton… of … ratos of numbers
… whole cosmos … and individual do not arise by a chance combinatons …
must be fted together in a "pleasing" (harmonic) way in accordance with
number for an order to arise.
htps://en.wikipedia.org/wiki/Pythagorean_tuning
π
π
π
π
π
π
Believed the number system … and universe… based on their
sum (10)
… swore by the “Tetractys” rather than by the gods.
Odd numbers were masculine and even were feminine.
Hippasos …discovered irratonal numbers…was executed.
Hints of “heliocentric theory”
discovery that music was based on proportonal intervals of
numbers 1—4
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"Bless us, divine number, thou
who generated gods and men!
O holy, holy Tetractys, thou
that containest the root and
source of the eternally fowing
creaton! For the divine number
begins with the profound, pure
unity untl it comes to the holy
four; then it begets the mother
of all, the all-comprising, allbounding, the frst-born, the
never-swerving, the nevertring holy ten, the keyholder of
all"
Clicker queston
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The number 10 is a perfect number, that is, it
is equal to the sum of all of the smaller whole
numbers that divide into it.
A) True
B) False
…it was beter to learn none of the truth about mathematcs, God, and the
universe at all than to learn a litle without learning all
Pythagoreans … believed… when someone was "in doubt as to what he
should say, he should always remain silent”
Pythagoreans’ inner circle,“mathematkoi” ("mathematcians”); outer
circle, “akousmatkoi” ("listeners”)
… the akousmatkoi were the exoteric disciples who… listened to lectures
that Pythagoras gave out loud from behind a veil.
Pythagorean theory of numbers stll debated among scholars.
Pythagoras believed in "harmony of the spheres”… that the planets and
stars moved according to mathematcal equatons, which corresponded to
musical notes and thus produced a symphony
Music of the Spheres
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A)
B)
The square root of two is a ratonal number
(the rato of two whole numbers)
True
False
The Pythagorean Theorem
The Pythagorean Theorem
Which of the two diagrams provide “visual
proof” of the Pythagoran theorem?
A)
Left diagram only
B) Right diagram only
C) Both diagrams together
Plato (428 BC – 348 BC),
Plato’s Cave Analogy
Note ratos: AB:BC :: CD:DE:: AC:CE
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A)
B)
C)
D)
In Plato’s Divided Line, Mathematcs falls
under the following category:
Highest form of true knowledge
Second highest form of true knowledge
A form of belief, but not true knowledge
A form of percepton
Plato (left) and Aristotle (right)
Aristotle (384 BC – 322 BC)
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Aristotle’s logic: the syllogism
Major premise: All humans are
mortal.
Minor premise: Socrates is a
human.
Conclusion: Socrates is mortal.
Epictetus and The Stoics (c 300 BC)
Stoics believed … knowledge atained through use of reason… Truth distnguishable
from fallacy; *even if, in practce, only an approximaton can be made.
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Modality (potentality vs actuality).
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Conditonal statements. (if…then)
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Meaning and truth
Euclid’s “Elements”
arranged in order many of Eudoxus's theorems,
perfected many of Theaetetus's, and brought to
irrefutable demonstraton theorems only loosely
proved by his predecessors
Ptolemy once asked him if there were a shorter
way to study geometry than the Elements, …
In his aim he was a Platonist, being in sympathy
with this philosophy, whence he made the end of
the whole "Elements" the constructon of the socalled Platonic fgures.
The axiomatc method
The Elements begins with defnitons and five postulates.
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There are also axioms which Euclid calls 'common notons'.
These are not specifc geometrical propertes but rather
general assumptons which allow mathematcs to proceed as
a deductve science. For example:
“Things which are equal to the same thing are equal to each
other.””
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Euclid's Postulates
A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefnitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one
endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles
on one side is less than two right angles, then the two lines inevitably must intersect each other
on that side if extended far enough. This postulate is equivalent to what is known as the parallel
postulate.
Euclid's ffth postulate cannot be proven from others, though atempted by many people.
Euclid used only 1—4 for the frst 28 propositons of the Elements, but was forced to invoke the
parallel postulate on the 29th.
In 1823,Bolyai and Lobachevsky independently realized that entrely self-consistent "nonEuclidean geometries" could be created in which the parallel postulate did not hold.
Euclid's Postulates
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefnitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment
as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of
the inner angles on one side is less than two right angles, then the two lines
inevitably must intersect each other on that side if extended far enough. This
postulate is equivalent to what is known as the parallel postulate.
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Euclid's ffth postulate cannot be proven from others, though
atempted by many people.
Euclid used only 1—4 for the frst 28 propositons of the
Elements, but was forced to invoke the parallel postulate on
the 29th.
In 1823,Bolyai and Lobachevsky independently realized that
entrely self-consistent "non-Euclidean geometries" could be
created in which the parallel postulate did not hold.
Non-Euclidean geometries 1
Non-Euclidean geometries2
Non-Euclidean geometries 3
Clicker queston
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Euclid’s ffth postulate, the “parallel
postulate” can be proven to be a consequence
of the other four postulates
A) True
B) False
Archimedes
Possibly the greatest
mathematcian ever;
Theoretcal and
practcal
Other cultures
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Avicenna (980-1037): propositonal logic
~ risk analysis
Parallels in India, China,
Medieval (1200-1600)
Occam (1288-1347)
Exercises (solutons on subsequent slide)
Explain how the Pythagorean theorem follows from the picture
using the formula for the area of a trapezoid
Soluton: the Area of a right quadrilateral is the length of the
base tmes the average height of the sides, that is, ½(a+b)(a+b).
The quadrilateral is also the union of three right triangles, two
with area ab/2, the third with area c^2/2. Setng the two areas
equal and multplying both by two gives (a+b)^2=2ab+c^2.
Multplying out the left hand side and cancelling 2ab on both
sides gives a^2+b^2=c^2
Explain how the Pythagorean theorem follows from the picture
Soluton: The big
square has area
c^2. It is made up
of 4 triangles of
area ab/2 and a
small square of
area (a-b)^2.
Altogether,
c^2=2ab+(ab)^2=a^2+b^2 after
multplying out and
cancelling 2ab.
Advanced: Explain how the
Pythagorean theorem
follows from the picture
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Soluton can be found here:
Euclid's proof
Prove that the area of
the big hexagon is the
sum of the areas of the
smaller ones
Soluton: the trick is that
the area of a (regular)
hexagon is a fxed
multple of that of the
square of its side. This
area is 3sqrt(3)/2 s^2
since a hexagon is the
union of six equilateral
triangles which have area
Assuming:
the area of a semicircle of
diameter d is
Prove that the area of the
big semicircle is the sum of
the areas of the smaller
ones
Soluton: Given the hint, if
the triangle has sides a,b,c
then the areas of the
semicircles are pi a^2/8, pi
b^2/8 and pi c^2/8.
Multplying all terms by
8/pi and applying the
Pythagorean theorem gives
the result.
Some practce problems
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If a=3 and b=4, what is the length c of the
hypotenuse of the triangle?
c
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4
Some practce problems
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Soluton: by the Pythagorean theorem,
c^2=9+16=25 so c=5
c
3
4
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e
Soluton: If a=5, b=4, c=3, d=3, and e=√5,
Then the frst hypotenuse is sqrt(25+16)
The second hypotenuse is sqrt(41+9)
The third is sqrt(50+9)
f
The fourth is sqrt(59+5)
=sqrt(64)=8
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a
d
c
3
b
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e
Soluton:
If a=5, b=4, c=3, d=3, and e=√5,
fnd f.
d
c
f
3
b
4
a
Explain the lengths of the sides of
the Pythagorean spiral
Soluton: Just apply the Pythagorean theorem
successively to each successive triangle. The square of
side of next hypotenuse is 1+square of side of previous
hypotenuse.
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A ladder is 10 feet long. When the top of the
ladder just touches the top a wall, the
botom of the ladder is 6 feet from the wall.
How high is the wall?
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Soluton: The ladder’s length of 10 feet is the
length of the hypotenuse of a right triangle
whose height is the height along the wall
and whose base is the base along the
ground. So 10^2 = 6^2+h^2 where h is the
height of the wall, or h^2=64, h=9 How high
is the wall?
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TV screen size is measured diagonally across the
screen. A widescreen TV has an aspect rato of
16:9, meaning the rato of its width to its height is
16/9. If Joe has a cabinet that is 34 inches wide,
what is the largest size wide screen TV that he can
ft in the cabinet?
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Soluton: W=(16/9)H where W is the width and H
is the height of the TV. The square of the diagonal
is W^2+H^2=W^2(1+(9/16)^2). If W^2\leq 34^2
then D\leq 34x sqrt(1+(9/16)^2)\approx 39
Advanced: Extra Credit
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The spherical law of cosines states that, on a
spherical triangle. Cos (c/R) = (cos a/R) (cos
b/R) + (sin a/R) (sin b/R) cos γ where R is the
radius of the sphere. If the Earth’s radius is
6,371 km, fnd the distance from:
from Seatle (48°N, 2°E) to Paris (48°N,
122°W) if traveling due east?
from Lincoln, NE (40°N, 96°W) to Sydney,
Australia (34°S, 151°E).