#### Transcript Lecture 4: Greek Mathematics - Department of Mathematical Sciences

Lecture 4, MATH 210G.02, Fall 2016
Greek Mathematics and
Philosophy
Goals:
Learn a few of the theorems proved by Greek
mathematicians
Understand some of the Greek philosophy in
which mathematics was developed
Lecture 4, MATH 210G.02, Fall 2015
Greek Mathematics 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
Thales (624-547 BC): father of mathematical
proof
In the diagram, the ratio of
is the same as the ratio
of the segments AE and
EC
A) True
B) False
Pythagoras
• (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
☺Transmigration of souls,
☺purification rites; developed rules of living
believed would enable their soul to achieve a
higher rank among the gods.
☺Theory that numbers constitute the true
nature of things, including music
•
•
•
•
•
•
•
•
•
C
D
E
F
G
A
B
C
1
9/8
81/64
4/3
3/2
27/16
243/128
2
The diatonic: ratio of highest to lowest pitch is 2:1,
produces the interval of an octave.
Octave in turn divided into fifth and fourth, with ratios 3:2 and 4:3 …
up a fifth + up a fourth = up an octave.
fifth … divided into three whole tones, each corresponding to the ratio of
9:8 and a remainder with a ratio of 256:243
fourth into two whole tones with same remainder.
harmony… combination… of … ratios of numbers
… whole cosmos … and individual do not arise by a chance combinations …
must be fitted together in a "pleasing" (harmonic) way in accordance with
number for an order to arise.
https://en.wikipedia.org/wiki/Pythagorean_tuning
π Believed the number system … and universe… based on their
sum (10)
π … swore by the “Tetractys” rather than by the gods.
π Odd numbers were masculine and even were feminine.
π Hippasos …discovered irrational numbers…was executed.
π Hints of “heliocentric theory”
π discovery that music was based on proportional intervals of
numbers 1—4
• "Bless us, divine number, thou
who generated gods and men!
O holy, holy Tetractys, thou that
containest the root and source
of the eternally flowing
creation! For the divine number
begins with the profound, pure
unity until it comes to the holy
four; then it begets the mother
of all, the all-comprising, allbounding, the first-born, the
never-swerving, the never-tiring
holy ten, the keyholder of all"
Clicker question
• 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 better to learn none of the truth about mathematics, God, and the
universe at all than to learn a little without learning all
Pythagoreans … believed… when someone was "in doubt as to what he
should say, he should always remain silent”
Pythagoreans’ inner circle,“mathematikoi” ("mathematicians”); outer
circle, “akousmatikoi” ("listeners”)
… the akousmatikoi were the exoteric disciples who… listened to lectures
that Pythagoras gave out loud from behind a veil.
Pythagorean theory of numbers still debated among scholars.
Pythagoras believed in "harmony of the spheres”… that the planets and
stars moved according to mathematical equations, which corresponded to
musical notes and thus produced a symphony
Music of the Spheres
• The square root of two is a rational number
(the ratio of two whole numbers)
A) True
B) 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 ratios: AB:BC :: CD:DE:: AC:CE
• In Plato’s Divided Line, Mathematics falls
under the following category:
A) Highest form of true knowledge
B) Second highest form of true knowledge
C) A form of belief, but not true knowledge
D) A form of perception
Plato (left) and Aristotle (right)
Aristotle (384 BC – 322 BC)
• 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 attained through use of reason… Truth distinguishable
from fallacy; *even if, in practice, only an approximation can be made.
• Modality (potentiality vs actuality).
• Conditional statements. (if…then)
• Meaning and truth
Euclid’s “Elements”
arranged in order many of Eudoxus's theorems,
perfected many of Theaetetus's, and brought to
irrefutable demonstration 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 construction of the socalled Platonic figures.
The axiomatic method
• The Elements begins with definitions and five postulates.
• There are also axioms which Euclid calls 'common notions'.
These are not specific geometrical properties but rather
general assumptions which allow mathematics to proceed as
a deductive science. For example:
“Things which are equal to the same thing are equal to each
other.””
Euclid's Postulates
A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely 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 fifth postulate cannot be proven from others, though attempted by many people.
Euclid used only 1—4 for the first 28 propositions of the Elements, but was forced to invoke the
parallel postulate on the 29th.
In 1823,Bolyai and Lobachevsky independently realized that entirely 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 indefinitely 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 fifth postulate cannot be proven from others, though
attempted by many people.
• Euclid used only 1—4 for the first 28 propositions of the
Elements, but was forced to invoke the parallel postulate on
the 29th.
• In 1823,Bolyai and Lobachevsky independently realized that
entirely 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 question
• Euclid’s fifth postulate, the “parallel
postulate” can be proven to be a consequence
of the other four postulates
• A) True
• B) False
Archimedes
Possibly the greatest
mathematician ever;
Theoretical and
practical
Other cultures
• Avicenna (980-1037): propositional logic
~ risk analysis
• Parallels in India, China,
• Medieval (1200-1600)
• Occam (1288-1347)
Exercises (solutions on subsequent slide)
Explain how the Pythagorean theorem follows from the picture
using the formula for the area of a trapezoid
Solution: the Area of a right quadrilateral is the length of the
base times 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. Setting the two areas
equal and multiplying both by two gives (a+b)^2=2ab+c^2.
Multiplying 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
Solution: The big
square has area
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
multiplying out and
cancelling 2ab.
Pythagorean theorem
follows from the picture
• Solution 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
Solution: the trick is that
the area of a (regular)
hexagon is a fixed
multiple 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
sqrt(3)/4 s^2
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
Solution: 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.
Multiplying all terms by
8/pi and applying the
Pythagorean theorem gives
the result.
Some practice problems
• If a=3 and b=4, what is the length c of the
hypotenuse of the triangle?
c
3
4
Some practice problems
• Solution: by the Pythagorean theorem,
c^2=9+16=25 so c=5
c
3
4
•
•
•
•
•
•
e
Solution: If a=5, b=4, c=3, d=3, and e=√5,
Then the first 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
4
a
d
c
3
b
e
• Solution:
• If a=5, b=4, c=3, d=3, and e=√5,
• find f.
d
c
f
3
b
4
a
Explain the lengths of the sides of
the Pythagorean spiral
Solution: 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.
• A ladder is 10 feet long. When the top of the
ladder just touches the top a wall, the
bottom of the ladder is 6 feet from the wall.
• How high is the wall?
• Solution: 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?
• TV screen size is measured diagonally across the
screen. A widescreen TV has an aspect ratio of
16:9, meaning the ratio 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
fit in the cabinet?
• Solution: 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